{"version":2,"work":"paper-viii","title":"Paper VIII: Packet Operator Algorithms, Exact Compilation, and GPU Realization","source_revision":"cdaf825cc0547fe58973b1770b7cba2fff9bcfa4","sources":[{"repository_path":"papers/paper-viii/current/README.md","file_url":"documents/files/3c3839ca148c49b03826f808a354e6ace6682a74/README.md","url":"read/paper-viii/readme-a3af416b0da1.html","sha256":"4f697b23a4bc211f660ed4aa549bc2470a6165c9f29ca7bc9befa5ebd2c377f1","git_blob_sha1":"3c3839ca148c49b03826f808a354e6ace6682a74","size":5769,"source_url":"read/paper-viii/readme-a3af416b0da1-source.html","passages":0,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/README.txt","file_url":"documents/files/6a8eff1c6bc2030b9919eefa96309ae46b259579/README.txt","url":"read/paper-viii/readme-db957c890210.html","sha256":"8f217e654bdbb36223390d1209a5fab1666122b22fe3544c23ce32698b24ae9e","git_blob_sha1":"6a8eff1c6bc2030b9919eefa96309ae46b259579","size":3985,"passages":0,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/traditional/Paper_VIII_Traditional_Final.tex","file_url":"documents/files/535543f360e32416693655046337c894ffbede77/Paper_VIII_Traditional_Final.tex","url":"read/paper-viii/paper-viii-traditional-final-83368012da3f.html","sha256":"9f767602dfcc1145ba7d593d17dc2c95676bed88bd78a6a24a4fe553455bbaba","git_blob_sha1":"535543f360e32416693655046337c894ffbede77","size":13868,"passages":0,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/traditional/atlas_definitions.tex","file_url":"documents/files/92fc6baad30c7c65a8a10ef5812a589fdd3ce930/atlas_definitions.tex","url":"read/paper-viii/atlas-definitions-964ee3be53df.html","sha256":"681fcf4146bc80f6b4eb5b7baecd4f474acf2ab5c144037b451241ae58ef72b4","git_blob_sha1":"92fc6baad30c7c65a8a10ef5812a589fdd3ce930","size":3150,"passages":0,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\operatorname{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":"","OO":"\\mathcal O","FF":"\\mathbb F","QQ":"\\mathbb Q","ZZ":"\\mathbb Z","Supp":"\\operatorname{Supp}","Fitt":"\\operatorname{Fitt}","Ann":"\\operatorname{Ann}","coker":"\\operatorname{coker}","im":"\\operatorname{im}","whot":"\\mathbin{\\widehat\\otimes}","m":"\\mathfrak m","p":"\\mathfrak p","kk":"\\mathbb k","AA":"\\mathcal A","BB":"\\mathcal B","Hull":"\\operatorname{Hull}","srank":"\\operatorname{srank}","Span":"\\operatorname{span}","Ob":"\\operatorname{Ob}","src":"\\operatorname{src}","tgt":"\\operatorname{tgt}","Res":"\\operatorname{Res}","sech":"\\operatorname{sech}","Dfun":"\\mathsf D","Efun":"\\mathsf E","Tfun":"\\mathsf T","Sfun":"\\mathsf S","Cfun":"\\mathsf C","Thetafun":"\\symup{\\Theta}","ones":"\\mathbf{1}","EE":"\\mathbb{E}","Var":"\\operatorname{Var}","HH":"\\mathcal H","dd":"\\mathrm d"}},{"repository_path":"papers/paper-viii/current/editable/traditional/atlas_sources/extended_pages.tex","file_url":"documents/files/70cce2e17b0b4a5bf2bc2f03d990c8b40ef299ed/extended_pages.tex","url":"read/paper-viii/extended-pages-b2a3010f21f9.html","sha256":"22fddc4df4b2c028d38eabbb38e3a2c7ebaf255f2d9a86df11b4da0a04a9b78a","git_blob_sha1":"70cce2e17b0b4a5bf2bc2f03d990c8b40ef299ed","size":7242,"passages":45,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/traditional/atlas_sources/pages.tex","file_url":"documents/files/84aff5b1eeeb6ea61cf3cdfea009273e702fc6cc/pages.tex","url":"read/paper-viii/pages-228115711d19.html","sha256":"68c4b4587d21504016421d275bc2c24d5d70965e0633ab92e7e673a609370fa8","git_blob_sha1":"84aff5b1eeeb6ea61cf3cdfea009273e702fc6cc","size":44470,"passages":284,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/traditional/monograph_atlas.tex","file_url":"documents/files/ee9193c73386e854c2b141f110706c24f270d7a4/monograph_atlas.tex","url":"read/paper-viii/monograph-atlas-475d52fc4acb.html","sha256":"b53abd8a2280271c3830ee8a90af0ee76bec662aeb6dfd2e7e0ccae03121e5a9","git_blob_sha1":"ee9193c73386e854c2b141f110706c24f270d7a4","size":271,"passages":0,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/traditional/monograph_examples.tex","file_url":"documents/files/9263af89bca0d34d6946451af0c4804e8f4f8ca7/monograph_examples.tex","url":"read/paper-viii/monograph-examples-a1f6849bc832.html","sha256":"0fb4c0314e1bb6ee2b52b970019462160830ee5fcb5a7b1c0915514999ccf2d0","git_blob_sha1":"9263af89bca0d34d6946451af0c4804e8f4f8ca7","size":6790,"passages":50,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/traditional/parts/01_manuscript.tex","file_url":"documents/files/785019f7ea6935ce0cc7c51dec33a45b6e7b8072/01_manuscript.tex","url":"read/paper-viii/01-manuscript-c2c518e9fb78.html","sha256":"ef616bca9291ac766d492b88f3166be4bc2b39f9ba6034dea3a4bcf2d2bf5d7f","git_blob_sha1":"785019f7ea6935ce0cc7c51dec33a45b6e7b8072","size":83873,"passages":573,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/traditional/parts/02_supplement_01.tex","file_url":"documents/files/873c570809e4fc6f4dcfe74a08a3bf71204c013e/02_supplement_01.tex","url":"read/paper-viii/02-supplement-01-35292b02451f.html","sha256":"ce9ff2d9068d57ddf748f2928a20e672a526ba7acbb92c1873ca2d3669b54346","git_blob_sha1":"873c570809e4fc6f4dcfe74a08a3bf71204c013e","size":11451,"passages":65,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/traditional/parts/02_supplement_02.tex","file_url":"documents/files/64b2d93779c3c7f74abc21b3279a85a3d4a38d13/02_supplement_02.tex","url":"read/paper-viii/02-supplement-02-27c9cf0f3613.html","sha256":"b719070578f63fa2d91ebea181d52140d046fdb68f3b52f59c95b57147992714","git_blob_sha1":"64b2d93779c3c7f74abc21b3279a85a3d4a38d13","size":17432,"passages":93,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/traditional/parts/02_supplement_03.tex","file_url":"documents/files/df326e0a7bffb6580456365fe867499cc9b671ca/02_supplement_03.tex","url":"read/paper-viii/02-supplement-03-9057e635351f.html","sha256":"84e71bf194e9d9204c669e2d98bd3c4e13d751029ea96ee86e6965827204933e","git_blob_sha1":"df326e0a7bffb6580456365fe867499cc9b671ca","size":16797,"passages":87,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":"","arraystretch":"1.18"}},{"repository_path":"papers/paper-viii/current/editable/traditional/parts/02_supplement_04.tex","file_url":"documents/files/9dd7dde49ae0755062384f958683150efd2122bd/02_supplement_04.tex","url":"read/paper-viii/02-supplement-04-ca8ce15ed4ed.html","sha256":"910b9f04343326b5dfd1e525b8200772ab824cda389222e7a07e951536de00be","git_blob_sha1":"9dd7dde49ae0755062384f958683150efd2122bd","size":14375,"passages":109,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/traditional/parts/02_supplement_05.tex","file_url":"documents/files/b9f6a471363347a00d8c5b05e0e757ea5f88bfc6/02_supplement_05.tex","url":"read/paper-viii/02-supplement-05-31b4f84fdb83.html","sha256":"8dfd9b4a8caf0ce225a2d5d137c5da47d3895a7cd2ea8185ab948e3f4a8ac353","git_blob_sha1":"b9f6a471363347a00d8c5b05e0e757ea5f88bfc6","size":16846,"passages":125,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/traditional/parts/02_supplement_06.tex","file_url":"documents/files/17f734d181170bc93e750a82331f1956d0dc00cf/02_supplement_06.tex","url":"read/paper-viii/02-supplement-06-90e54f4ed68d.html","sha256":"7179ee3328fb901301208b53884220c69e8c87cb54218919c6fb71cacc4c646a","git_blob_sha1":"17f734d181170bc93e750a82331f1956d0dc00cf","size":15483,"passages":90,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":"","arraystretch":"1.18"}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/amortization.tex","file_url":"documents/files/f8b87f29ef4332e29f74f0e113064abb6a5e09e7/amortization.tex","url":"read/paper-viii/amortization-144c1cbb6b8d.html","sha256":"a4a63d154ef1df3904bb7ca67d21dfbb29265ffd266c89a5e09fbd6ae1cb7b11","git_blob_sha1":"f8b87f29ef4332e29f74f0e113064abb6a5e09e7","size":2496,"passages":5,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/batch_shapes.tex","file_url":"documents/files/63dda1ec2ccb93ce2a0750ff7728b6dc660329a8/batch_shapes.tex","url":"read/paper-viii/batch-shapes-1a5da60d5710.html","sha256":"1b555147f9ecebcb86d1342759c00b9a4786a742dbce846477f53a6bd8870edd","git_blob_sha1":"63dda1ec2ccb93ce2a0750ff7728b6dc660329a8","size":5585,"passages":38,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/bdlr_product.tex","file_url":"documents/files/0fda6318998346a0ba498d532776d68f2cd36e3e/bdlr_product.tex","url":"read/paper-viii/bdlr-product-bdb1773a8f7f.html","sha256":"e547cde0957094852e0e7b3cfd8dabf7ea488baa0ff3d0c2909921076d7cba74","git_blob_sha1":"0fda6318998346a0ba498d532776d68f2cd36e3e","size":3839,"passages":28,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/block_error.tex","file_url":"documents/files/dfae253d021f8390b063c0f69510d7e597897036/block_error.tex","url":"read/paper-viii/block-error-dbde341a4cad.html","sha256":"6714c087ca906102792ea14bd87c13db3c7510e8403431f6a3a4530c283f2b82","git_blob_sha1":"dfae253d021f8390b063c0f69510d7e597897036","size":3391,"passages":23,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/cancellation.tex","file_url":"documents/files/ea9c99959eb9d1df6cf792a81675b0335267d598/cancellation.tex","url":"read/paper-viii/cancellation-139fa9ef4a2c.html","sha256":"82b20c8bc29ff100fe30a0da49ac378546b24433028ba60782b692b30389472a","git_blob_sha1":"ea9c99959eb9d1df6cf792a81675b0335267d598","size":4580,"passages":35,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/circulant.tex","file_url":"documents/files/030019dca0e4b4df43cb9ff6804ba38b9974da7d/circulant.tex","url":"read/paper-viii/circulant-e172e832b4a8.html","sha256":"0c97d523ad649edff0a8ee65174722d4eaed003cc1b4fdc405a0692a97ef46cf","git_blob_sha1":"030019dca0e4b4df43cb9ff6804ba38b9974da7d","size":3396,"passages":25,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/compact_core.tex","file_url":"documents/files/cd60a28e7e38da65d5da0edd09181925be2c1d47/compact_core.tex","url":"read/paper-viii/compact-core-4b8548b7a221.html","sha256":"ef0de17556685cfa02b56ac92593e4468855f3d4310053458801507ef79ef842","git_blob_sha1":"cd60a28e7e38da65d5da0edd09181925be2c1d47","size":4260,"passages":34,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/complex_core.tex","file_url":"documents/files/02f017e0e09d78785dbed83c8f75308079b89348/complex_core.tex","url":"read/paper-viii/complex-core-c76e1ced4b4f.html","sha256":"c450ffe53b9cd22c690656bf7a734522b166250447c543e4d8c8ef9db0b176e3","git_blob_sha1":"02f017e0e09d78785dbed83c8f75308079b89348","size":536,"passages":4,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/contract_match.tex","file_url":"documents/files/75dafede8ea109193402ae041690af8b21ba9842/contract_match.tex","url":"read/paper-viii/contract-match-7bccb215a730.html","sha256":"48ce14e39e95ce8ab1678be786e81518966085dffd98b9fcd347a36abddcbf81","git_blob_sha1":"75dafede8ea109193402ae041690af8b21ba9842","size":2933,"passages":0,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/discrete.tex","file_url":"documents/files/8dc99694c6a250f44eaf50473983b16c5e864beb/discrete.tex","url":"read/paper-viii/discrete-687818158670.html","sha256":"271916c099efe4745b625ef237519cb84b16da8c9dd7d41d2669f478e4a983c2","git_blob_sha1":"8dc99694c6a250f44eaf50473983b16c5e864beb","size":6188,"passages":16,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/end_to_end.tex","file_url":"documents/files/7ac16b385576eafd36233effd536430df51d28ba/end_to_end.tex","url":"read/paper-viii/end-to-end-a37ff6c3270f.html","sha256":"12fa24c590b3a61503f1283e22f3a319a5dfd468db42c27a987fb435ace57358","git_blob_sha1":"7ac16b385576eafd36233effd536430df51d28ba","size":3342,"passages":24,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/error_chain.tex","file_url":"documents/files/112c712610016084b0e769394526065c6fdcc4c6/error_chain.tex","url":"read/paper-viii/error-chain-dd41cf2be449.html","sha256":"57db254aa8769f0945d2c90ab356bfd8ba4b67b2e6df19877b75a8d763dcf30d","git_blob_sha1":"112c712610016084b0e769394526065c6fdcc4c6","size":1974,"passages":8,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/factored_application.tex","file_url":"documents/files/a8c79d83924bc470d06bb306bcc651519cc48910/factored_application.tex","url":"read/paper-viii/factored-application-01b21b02bb33.html","sha256":"0b45a0d35b3fd9028bedbbba77e4bb88ebeb74086f29f1a1a6f92792968080a0","git_blob_sha1":"a8c79d83924bc470d06bb306bcc651519cc48910","size":3882,"passages":31,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/inverse_closure.tex","file_url":"documents/files/c05180a8dd75c8a63a3ea05a9a40df95fc1e5099/inverse_closure.tex","url":"read/paper-viii/inverse-closure-c38aa7db2ea1.html","sha256":"f461bedc7a5d0028880b148623363a187081a08d60cbe2b95476c97125de0cd4","git_blob_sha1":"c05180a8dd75c8a63a3ea05a9a40df95fc1e5099","size":651,"passages":9,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/kronecker.tex","file_url":"documents/files/2ea518b736d37dee6e99c7f5cea3bfaeb05c9cca/kronecker.tex","url":"read/paper-viii/kronecker-84e9c5005ace.html","sha256":"3094db5576178b529ffef32aa8ddfe81db8b48dc6f05aa6712fe98cb1518f39b","git_blob_sha1":"2ea518b736d37dee6e99c7f5cea3bfaeb05c9cca","size":2168,"passages":18,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/krylov.tex","file_url":"documents/files/c8db9b6f4dadda0995bc8e8d0c36a7b457b25b07/krylov.tex","url":"read/paper-viii/krylov-79c3a7e0df4d.html","sha256":"58f6cf193436a8a78d6d854cd5e9bc8ebfac3ff87b608e4259f1b23a81a2e112","git_blob_sha1":"c8db9b6f4dadda0995bc8e8d0c36a7b457b25b07","size":1758,"passages":4,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/local_update.tex","file_url":"documents/files/df771e1737bbed751db81fffe3aac8daa6e469ee/local_update.tex","url":"read/paper-viii/local-update-ae5d249b84ae.html","sha256":"86f0825faa42d1f5c306239fb3308aea46846d93f8a0f93c2b2f3735340e5026","git_blob_sha1":"df771e1737bbed751db81fffe3aac8daa6e469ee","size":2129,"passages":7,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/optimality.tex","file_url":"documents/files/cd12edd66fbae3900f7f2eb66b30e32901922f42/optimality.tex","url":"read/paper-viii/optimality-3907ca9844cf.html","sha256":"093a5fc6b2e638179fc88a9a5a367bf400bcd887886215b25479cd7717475e92","git_blob_sha1":"cd12edd66fbae3900f7f2eb66b30e32901922f42","size":4453,"passages":29,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/precision.tex","file_url":"documents/files/9ce68d7d22d5eca757cd01679af171ed78fc4518/precision.tex","url":"read/paper-viii/precision-471b055b8b01.html","sha256":"25decf724f188e69a151c7ded0ead2f757c06f6169c6672b06282b61e103605a","git_blob_sha1":"9ce68d7d22d5eca757cd01679af171ed78fc4518","size":1953,"passages":7,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/preconditioning.tex","file_url":"documents/files/8d3002bd0f5ad074ec714e41fabad364e0f91355/preconditioning.tex","url":"read/paper-viii/preconditioning-4e7ec0aeb53f.html","sha256":"286bedf11de1d96755c07813a150f2f73a4383472e61995963fab7b9105074a2","git_blob_sha1":"8d3002bd0f5ad074ec714e41fabad364e0f91355","size":9119,"passages":4,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/quantization.tex","file_url":"documents/files/1a2ab9ce41c8205bb3d423aa31c52c9ec536ab3b/quantization.tex","url":"read/paper-viii/quantization-434e47cd58a2.html","sha256":"064730e54c926564642da3f558b5d17cc994c35ccfbebd7aa8254ce1766ee577","git_blob_sha1":"1a2ab9ce41c8205bb3d423aa31c52c9ec536ab3b","size":711,"passages":4,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/schur_fill.tex","file_url":"documents/files/7c3faf68a8d5faa72b77aecf6cf45e612eafb5f3/schur_fill.tex","url":"read/paper-viii/schur-fill-f1fbdc916960.html","sha256":"15b0e6736a12ae1f391b02e6288f6b5997dfd071c99570ddf3ea96088194ea38","git_blob_sha1":"7c3faf68a8d5faa72b77aecf6cf45e612eafb5f3","size":2905,"passages":21,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/shared_family.tex","file_url":"documents/files/b66957ede1f06624e51a645666ae1bd169178ae7/shared_family.tex","url":"read/paper-viii/shared-family-4c5fa0e606a3.html","sha256":"410487aef8b94ad23e06d2734d38ed2115bc1bb4a2d5b106c6b4d886858d2f47","git_blob_sha1":"b66957ede1f06624e51a645666ae1bd169178ae7","size":628,"passages":7,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/three_views.tex","file_url":"documents/files/e85c43f0e868a2b1afd6f73931cef8e8b4437c1b/three_views.tex","url":"read/paper-viii/three-views-b25acb4412eb.html","sha256":"72fc54eb5ac155f70b827885c3ac90f1ca3ff68c49efbc61eaf1bf39a4834eba","git_blob_sha1":"e85c43f0e868a2b1afd6f73931cef8e8b4437c1b","size":7266,"passages":39,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/toeplitz.tex","file_url":"documents/files/023bb06c25f1dc52548a04ace21414664ae36399/toeplitz.tex","url":"read/paper-viii/toeplitz-3c29a1502a2e.html","sha256":"02d783e40d82e068c97bea073a4129b5c7ded2219150439e82e499c31d4449b0","git_blob_sha1":"023bb06c25f1dc52548a04ace21414664ae36399","size":4393,"passages":29,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/tree_frontier.tex","file_url":"documents/files/2caadd098791a738fec6561379b6ea2ebea30e5a/tree_frontier.tex","url":"read/paper-viii/tree-frontier-fed68ce72427.html","sha256":"290527707fec32e4f996dabb1826318d12bc20b6b68327bb9cdf7ff0395bfabd","git_blob_sha1":"2caadd098791a738fec6561379b6ea2ebea30e5a","size":2214,"passages":4,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/triangular.tex","file_url":"documents/files/0dd350e9256f87100c69eb3a0cc5230fec7dbcaa/triangular.tex","url":"read/paper-viii/triangular-7bb6cf6867b8.html","sha256":"37e1637777e795551228007c4112b18afcb143af274853c256a674670e26ac6b","git_blob_sha1":"0dd350e9256f87100c69eb3a0cc5230fec7dbcaa","size":2550,"passages":8,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/walsh.tex","file_url":"documents/files/9cfa27ffd897f9c01f056297fb816a54b10d5f96/walsh.tex","url":"read/paper-viii/walsh-2213ece28222.html","sha256":"cc1a8cf903a608eb50790fb9004f6683c506da5e86929743dafb8f059f4eed1b","git_blob_sha1":"9cfa27ffd897f9c01f056297fb816a54b10d5f96","size":4553,"passages":2,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/woodbury.tex","file_url":"documents/files/6e27ad7ff43393af3fe479ecc4476a80905f3e73/woodbury.tex","url":"read/paper-viii/woodbury-bb3ee4eb9932.html","sha256":"39c4638969eef009ed945348856c1985660130a764ce63dbc6d6fc6fd79bc10a","git_blob_sha1":"6e27ad7ff43393af3fe479ecc4476a80905f3e73","size":1762,"passages":14,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/graphics/work_storage.tex","file_url":"documents/files/cff7b80a01b3044028ad1bb7b6de27357bc649d2/work_storage.tex","url":"read/paper-viii/work-storage-56cc5c878418.html","sha256":"c63e417e2f2bbf49c91bddaf9e094e5d8319dbc54ee77cbcea0d61f8cc01910f","git_blob_sha1":"cff7b80a01b3044028ad1bb7b6de27357bc649d2","size":1621,"passages":12,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/source/extended_pages.tex","file_url":"documents/files/a8942ac74d7817b91870ebd4bfa6c13e61a11ac4/extended_pages.tex","url":"read/paper-viii/extended-pages-fda5329bef4c.html","sha256":"64381afcbd64a29b42fa92b41626b926638ecb68993db63771386ac2216a4512","git_blob_sha1":"a8942ac74d7817b91870ebd4bfa6c13e61a11ac4","size":7486,"passages":45,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/source/pages.tex","file_url":"documents/files/f01ed64af9d15ef610514ecf630b7abdfdfe96ba/pages.tex","url":"read/paper-viii/pages-26d73f15f2e3.html","sha256":"823b8c7a08a1e9e204ff7fdf8e66ee424b553c3ed1dd313c58c258e62304121d","git_blob_sha1":"f01ed64af9d15ef610514ecf630b7abdfdfe96ba","size":46129,"passages":284,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\mathrm{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":""}},{"repository_path":"papers/paper-viii/current/editable/visual_guide/source/visual_macros.tex","file_url":"documents/files/c9784b316cdc510b6151dbf75d9c207c8067200a/visual_macros.tex","url":"read/paper-viii/visual-macros-adb28861efbf.html","sha256":"50931bacf6b4257c2d44566a8e0c2495316e85a5992fde2a05df57c0ec15faec","git_blob_sha1":"c9784b316cdc510b6151dbf75d9c207c8067200a","size":2950,"passages":0,"macros":{"DoNotLoadEpstopdf":"","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","C":"\\mathbb C","R":"\\mathbb R","Q":"\\mathbb Q","Acal":"\\mathcal A","Bcal":"\\mathcal B","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Gcal":"\\mathcal G","Hcal":"\\mathcal H","Ical":"\\mathcal I","Kcal":"\\mathcal K","Lcal":"\\mathcal L","Mcal":"\\mathcal M","Pcal":"\\mathcal P","Qcal":"\\mathcal Q","Rcal":"\\mathcal R","Scal":"\\mathcal S","Tcal":"\\mathcal T","Ucal":"\\mathcal U","Vcal":"\\mathcal V","End":"\\operatorname{End}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{ker}","Spec":"\\operatorname{Spec}","Tr":"\\operatorname{Tr}","tr":"\\operatorname{tr}","rank":"\\operatorname{rank}","diag":"\\operatorname{diag}","cond":"\\operatorname{cond}","supp":"\\operatorname{supp}","flop":"\\operatorname{flop}","bytes":"\\operatorname{bytes}","cost":"\\operatorname{cost}","err":"\\operatorname{err}","dist":"\\operatorname{dist}","e":"\\mathrm e","ii":"\\mathrm i","id":"\\operatorname{id}","one":"\\bm 1","eps":"\\varepsilon","IR":"\\mathsf{IR}","DP":"\\mathsf{DP}","KomorebiTitle":"Packet Operator Algorithms","KomorebiSubtitle":"Exact compilation and GPU realization","KomorebiSeries":"Core series VIII","KomorebiKind":"Modern Monograph","KomorebiEdition":"9 October 2026","KomorebiShortTitle":"Paper VIII","KomorebiTitleSize":"38","KomorebiTitleLeading":"43","KomorebiCoverTitle":"Packet Operator\\\\Algorithms","KomorebiCoverBlurb":"Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.","refname":"","OO":"\\mathcal O","FF":"\\mathbb F","QQ":"\\mathbb Q","ZZ":"\\mathbb Z","Supp":"\\operatorname{Supp}","Fitt":"\\operatorname{Fitt}","Ann":"\\operatorname{Ann}","coker":"\\operatorname{coker}","im":"\\operatorname{im}","whot":"\\mathbin{\\widehat\\otimes}","m":"\\mathfrak m","p":"\\mathfrak p","kk":"\\mathbb k","AA":"\\mathcal A","BB":"\\mathcal B","Hull":"\\operatorname{Hull}","srank":"\\operatorname{srank}","Span":"\\operatorname{span}","Ob":"\\operatorname{Ob}","src":"\\operatorname{src}","tgt":"\\operatorname{tgt}","Res":"\\operatorname{Res}","sech":"\\operatorname{sech}","Dfun":"\\mathsf D","Efun":"\\mathsf E","Tfun":"\\mathsf T","Sfun":"\\mathsf S","Cfun":"\\mathsf C","Thetafun":"\\symup{\\Theta}","ones":"\\mathbf{1}","EE":"\\mathbb{E}","Var":"\\operatorname{Var}","HH":"\\mathcal H","dd":"\\mathrm d"}}],"pdfs":[{"repository_path":"papers/paper-viii/current/papers/Paper_VIII_Traditional_Final.pdf","url":"read/paper-viii/paper-viii-traditional-final-f91411a6fdfe.html","sha256":"0b25167cdddd5c07cc36252ba13abb7db423392d6387ccef0dbd0ddb813b5042"},{"repository_path":"papers/paper-viii/current/papers/Paper_VIII_Visual_Guide_Final.pdf","url":"read/paper-viii/paper-viii-visual-guide-final-dc59d234db65.html","sha256":"a78f1c18c7c36cecbaf9484bc6dc643300c3b5054df338f48be6390af5e520c7"},{"repository_path":"papers/paper-viii/current/papers/Paper_VIII_Lean_Guide_Final.pdf","url":"read/paper-viii/paper-viii-lean-guide-final-1d0c201ebd4d.html","sha256":"a818b2179a581ee3f9c0191f9998e911250dbc25305eaa6b1782232a01b762b2"},{"repository_path":"papers/paper-viii/current/editable/traditional/monograph_figures/01_exact_bdlr_action.pdf","url":"read/paper-viii/01-exact-bdlr-action-5c050a3c0d36.html","sha256":"5d0a7c5cdd80b283655586314c604564bb9d314f735b5edd3da41e692762559f"},{"repository_path":"papers/paper-viii/current/editable/traditional/monograph_figures/02_exact_compiler_frontier.pdf","url":"read/paper-viii/02-exact-compiler-frontier-5b362a05f1c0.html","sha256":"57acb62287d84f8529232cdf9b30ba783f04d72e2ec516697c234496b69ad800"},{"repository_path":"papers/paper-viii/current/editable/traditional/monograph_figures/03_exact_precision_budget.pdf","url":"read/paper-viii/03-exact-precision-budget-d9866fa40f42.html","sha256":"34a46a1e1109e844051f9644424c7f0a34ef381659213de08d1b9ca56d61ae10"},{"repository_path":"papers/paper-viii/current/editable/traditional/figures/01_packet_compilation_architecture.pdf","url":"read/paper-viii/01-packet-compilation-architecture-e3f82b8abcff.html","sha256":"21460704818338f173b51bdd411233bacb2cb82dfc15eedee7d233182255d2cb"},{"repository_path":"papers/paper-viii/current/editable/traditional/figures/02_packet_ir_graph.pdf","url":"read/paper-viii/02-packet-ir-graph-53212adab30c.html","sha256":"7d9e822f26f07fbf4e10b3c0fc977de176ea3bacfcc9ea866e5965be60f55afb"},{"repository_path":"papers/paper-viii/current/editable/traditional/figures/03_ideal_arithmetic_reduction.pdf","url":"read/paper-viii/03-ideal-arithmetic-reduction-944ea26e8bff.html","sha256":"501970f6010597f443ca3dbcbd834c8642412847c89bda7d3bbae8faa7edfa3c"},{"repository_path":"papers/paper-viii/current/editable/traditional/figures/04_roofline_schematic.pdf","url":"read/paper-viii/04-roofline-schematic-251a49885590.html","sha256":"a43d1514ac135de42b8dc883b3eb2567e387c96e82b87e3f15bdf33954af34b3"},{"repository_path":"papers/paper-viii/current/editable/traditional/figures/05_bdlr_structure.pdf","url":"read/paper-viii/05-bdlr-structure-14bd354ce32e.html","sha256":"c261a0bc4bc2075898fdbd807fd5687bd2fe89833f38a4b83beef5aae2e04323"},{"repository_path":"papers/paper-viii/current/editable/traditional/figures/06_recompression_spectrum.pdf","url":"read/paper-viii/06-recompression-spectrum-0b82d42809c9.html","sha256":"3e76f82812b7826de5b982857346acf56e2936da1f3cd079a63803d2c0007f1f"},{"repository_path":"papers/paper-viii/current/editable/traditional/figures/07_path_support.pdf","url":"read/paper-viii/07-path-support-a3ef3e1b2ca6.html","sha256":"94837b788fc0ab7ac48e921993acfae18b302a5cd8328d1adcb8e1f6863fa25f"},{"repository_path":"papers/paper-viii/current/editable/traditional/figures/08_topological_schedule.pdf","url":"read/paper-viii/08-topological-schedule-849c6f06daaa.html","sha256":"e114b62ab2238a4126f4551f2afd2020dbacf27cc19d20ab491a33b1bbbdf8a5"},{"repository_path":"papers/paper-viii/current/editable/traditional/figures/09_packet_preconditioner.pdf","url":"read/paper-viii/09-packet-preconditioner-220831826d75.html","sha256":"fdcbfa9a828c9b106096d704c342185ae88aa5124a8db940a885f1697b512d80"},{"repository_path":"papers/paper-viii/current/editable/traditional/figures/10_precision_allocation.pdf","url":"read/paper-viii/10-precision-allocation-859dc98e31e2.html","sha256":"92a930ba3cecb3ad0dc094c1d1f781f04cfae3ac11cc2aa896aefb342a6cd0f9"},{"repository_path":"papers/paper-viii/current/editable/traditional/figures/11_tree_compiler.pdf","url":"read/paper-viii/11-tree-compiler-c47dc228f7c5.html","sha256":"c684cabc9b73fa597bed2b66fe1ea3851814bd6852288c72691be4d4610fb9df"},{"repository_path":"papers/paper-viii/current/editable/traditional/figures/12_budget_compiler_frontier.pdf","url":"read/paper-viii/12-budget-compiler-frontier-be820a130488.html","sha256":"455fa89fa31b34979075b3592dcc1799a6c6f917c02ece1f1c178ea25ddeb861"},{"repository_path":"papers/paper-viii/current/editable/traditional/figures/22_benchmark_contracts.pdf","url":"read/paper-viii/22-benchmark-contracts-4b9f2e54ac30.html","sha256":"fffd7727821714d84f1fcd45db8e7681ebccee22140f7cb95d70ecb118a88c04"},{"repository_path":"papers/paper-viii/current/editable/traditional/figures/13_dense_gpu_throughput.pdf","url":"read/paper-viii/13-dense-gpu-throughput-808555a22728.html","sha256":"1fd6e9df2880021f372ddffcfa20dc827ee3d8b90cc923de528145efbdf48952"},{"repository_path":"papers/paper-viii/current/editable/traditional/figures/14_accuracy_frontier.pdf","url":"read/paper-viii/14-accuracy-frontier-7f734bd38210.html","sha256":"50a750c445934ead397a07f08a77d13d939b92ad2448258bef7e7f602e10b79b"},{"repository_path":"papers/paper-viii/current/editable/traditional/figures/15_block_diagonal_speedup.pdf","url":"read/paper-viii/15-block-diagonal-speedup-6a4018b163c7.html","sha256":"24deff50980ea3d7c333b39fc83317eb299a8deea59349a2c236d46a28718743"},{"repository_path":"papers/paper-viii/current/editable/traditional/figures/16_bdlr_timings.pdf","url":"read/paper-viii/16-bdlr-timings-311f05dc4a47.html","sha256":"de509a7b09ff8699b39d96a6bc657e59560d247a3f5eef3e72e934f2f50ed567"},{"repository_path":"papers/paper-viii/current/editable/traditional/figures/17_output_contract_timings.pdf","url":"read/paper-viii/17-output-contract-timings-2e1d28a82f7c.html","sha256":"9042916ce8e1ebff5cff320b22f7b1c01ae1268eb869e8ca78df1017d9368597"},{"repository_path":"papers/paper-viii/current/editable/traditional/figures/18_discovery_amortization.pdf","url":"read/paper-viii/18-discovery-amortization-3e874058a876.html","sha256":"b8d5699dd3393f6b2b96f5372a6bb9e7bed9abdb9828642399acb87b2869a17f"},{"repository_path":"papers/paper-viii/current/editable/traditional/figures/19_transfer_pipeline.pdf","url":"read/paper-viii/19-transfer-pipeline-093920ae6c28.html","sha256":"313c6da40e51830f7c687982120e9c056b6f478e8d5b2a2b77d4720575716888"},{"repository_path":"papers/paper-viii/current/editable/traditional/figures/20_negative_controls.pdf","url":"read/paper-viii/20-negative-controls-1119b28a58a3.html","sha256":"38e0aff5adecf0b193e76bc9640af5f52173cba825994cad5473df69d24b728b"},{"repository_path":"papers/paper-viii/current/editable/traditional/figures/21_best_gpu_speedups.pdf","url":"read/paper-viii/21-best-gpu-speedups-907ea4750f51.html","sha256":"06354f76a2c5333420f88f0da369a58dbdcf25ad74caf4f22d21a1124c997070"}],"passages":[{"line":2,"start":133,"end":136,"expression":"\\CC","context":0,"kind":"use","source":4},{"line":2,"start":158,"end":164,"expression":"Q^*Q=I","context":0,"kind":"use","source":4},{"line":3,"start":228,"end":321,"expression":"\nQ_L=\\mat{3/5&0\\\\4\\mathrm i/5&0\\\\0&1},\\qquad\nQ_R=\\mat{1&0\\\\0&\\mathrm i},\\qquad G=\\diag(5,2).\n","context":1,"kind":"use","source":4},{"line":8,"start":385,"end":463,"expression":"\nA=Q_LGQ_R^*=\\mat{3&0\\\\4\\mathrm i&0\\\\0&-2\\mathrm i},\n\\qquad A^*A=\\diag(25,4).\n","context":2,"kind":"use","source":4},{"line":14,"start":563,"end":566,"expression":"Q_L","context":3,"kind":"use","source":4},{"line":14,"start":598,"end":610,"expression":"9/25+16/25=1","context":3,"kind":"use","source":4},{"line":14,"start":657,"end":673,"expression":"9/25-16/25=-7/25","context":3,"kind":"use","source":4},{"line":14,"start":682,"end":690,"expression":"Q_L^TQ_L","context":3,"kind":"use","source":4},{"line":16,"start":857,"end":858,"expression":"G","context":4,"kind":"use","source":4},{"line":16,"start":867,"end":870,"expression":"A_1","context":5,"kind":"use","source":4},{"line":16,"start":963,"end":973,"expression":"\\diag(0,4)","context":6,"kind":"use","source":4},{"line":16,"start":1107,"end":1121,"expression":"\\QQ(\\mathrm i)","context":7,"kind":"use","source":4},{"line":23,"start":1863,"end":1913,"expression":"\nA=I+K=\\mat{1&0&0&0\\\\1&1&0&0\\\\1&0&1&0\\\\0&1&-1&1}.\n","context":8,"kind":"use","source":4},{"line":26,"start":1984,"end":1989,"expression":"K^2=0","context":9,"kind":"use","source":4},{"line":27,"start":2007,"end":2064,"expression":"\nA^{-1}=I-K=\\mat{1&0&0&0\\\\-1&1&0&0\\\\-1&0&1&0\\\\0&-1&1&1}.\n","context":10,"kind":"use","source":4},{"line":32,"start":2169,"end":2174,"expression":"0\\to3","context":11,"kind":"use","source":4},{"line":32,"start":2279,"end":2294,"expression":"(A^{-1})_{30}=0","context":11,"kind":"use","source":4},{"line":34,"start":2501,"end":2504,"expression":"I-K","context":12,"kind":"use","source":4},{"line":36,"start":2943,"end":2944,"expression":"A","context":13,"kind":"use","source":4},{"line":36,"start":2951,"end":2952,"expression":"A","context":14,"kind":"use","source":4},{"line":52,"start":4109,"end":4124,"expression":"(r_1,r_2)=(1,3)","context":15,"kind":"use","source":4},{"line":53,"start":4159,"end":4225,"expression":"\nJ_{\\rm keep}=4+2+3\\cdot7=27,\\qquad\nJ_{\\rm split}=2+8+3\\cdot2=16.\n","context":16,"kind":"use","source":4},{"line":59,"start":4507,"end":4512,"expression":"r_1=1","context":17,"kind":"introduction-passage","source":4},{"line":59,"start":4524,"end":4533,"expression":"r_2=r\\ge0","context":18,"kind":"introduction-passage","source":4},{"line":59,"start":4551,"end":4555,"expression":"6+7r","context":19,"kind":"introduction-passage","source":4},{"line":59,"start":4578,"end":4583,"expression":"10+2r","context":20,"kind":"introduction-passage","source":4},{"line":59,"start":4622,"end":4627,"expression":"r>4/5","context":21,"kind":"introduction-passage","source":4},{"line":59,"start":4662,"end":4665,"expression":"r=1","context":22,"kind":"introduction-passage","source":4},{"line":65,"start":5542,"end":5548,"expression":"h=1/10","context":23,"kind":"use","source":4},{"line":65,"start":5607,"end":5608,"expression":"e","context":23,"kind":"use","source":4},{"line":66,"start":5622,"end":5677,"expression":"\nw=\\lceil e/h\\rceil\\qquad\\text{ensures}\\qquad e\\le hw.\n","context":23,"kind":"use","source":4},{"line":69,"start":5706,"end":5717,"expression":"\\sum w\\le B","context":23,"kind":"use","source":4},{"line":69,"start":5743,"end":5755,"expression":"\\sum e\\le hB","context":23,"kind":"use","source":4},{"line":72,"start":5927,"end":5938,"expression":"(3/50,2/25)","context":24,"kind":"use","source":4},{"line":72,"start":5972,"end":5977,"expression":"(0,0)","context":25,"kind":"use","source":4},{"line":72,"start":6007,"end":6016,"expression":"7/50>1/10","context":26,"kind":"use","source":4},{"line":72,"start":6038,"end":6043,"expression":"(1,1)","context":27,"kind":"use","source":4},{"line":74,"start":6127,"end":6138,"expression":"(3/50,1/50)","context":28,"kind":"use","source":4},{"line":74,"start":6155,"end":6166,"expression":"2/25\\le1/10","context":28,"kind":"use","source":4},{"line":74,"start":6203,"end":6208,"expression":"(1,1)","context":28,"kind":"use","source":4},{"line":76,"start":6458,"end":6459,"expression":"B","context":28,"kind":"use","source":4},{"line":76,"start":6475,"end":6478,"expression":"B+1","context":28,"kind":"use","source":4},{"line":76,"start":6541,"end":6548,"expression":"(B+1)^2","context":29,"kind":"use","source":4},{"line":76,"start":6567,"end":6572,"expression":"B=255","context":30,"kind":"use","source":4},{"line":78,"start":6868,"end":6869,"expression":"B","context":31,"kind":"use","source":4},{"line":5,"start":497,"end":505,"expression":"A=D+uv^T","context":32,"kind":"introduction-passage","source":5},{"line":5,"start":524,"end":525,"expression":"A","context":33,"kind":"introduction-passage","source":5},{"line":5,"start":550,"end":558,"expression":"2\\times2","context":34,"kind":"introduction-passage","source":5},{"line":7,"start":772,"end":773,"expression":"L","context":35,"kind":"use","source":5},{"line":7,"start":780,"end":781,"expression":"R","context":35,"kind":"use","source":5},{"line":7,"start":819,"end":827,"expression":"4\\times3","context":35,"kind":"use","source":5},{"line":7,"start":843,"end":851,"expression":"2\\times3","context":35,"kind":"use","source":5},{"line":19,"start":2347,"end":2355,"expression":"P_\\alpha","context":36,"kind":"use","source":5},{"line":20,"start":2435,"end":2450,"expression":"A_{\\beta\\alpha}","context":37,"kind":"use","source":5},{"line":21,"start":2514,"end":2515,"expression":"E","context":38,"kind":"use","source":5},{"line":22,"start":2604,"end":2610,"expression":"D+UV^*","context":39,"kind":"use","source":5},{"line":23,"start":2696,"end":2705,"expression":"Q_L,G,Q_R","context":40,"kind":"use","source":5},{"line":24,"start":2799,"end":2802,"expression":"E_j","context":41,"kind":"use","source":5},{"line":25,"start":2895,"end":2909,"expression":"s_e,\\epsilon_e","context":42,"kind":"use","source":5},{"line":26,"start":3003,"end":3033,"expression":"(\\kappa_v,\\epsilon_v,\\sigma_v)","context":43,"kind":"use","source":5},{"line":27,"start":3112,"end":3118,"expression":"F_v(b)","context":44,"kind":"use","source":5},{"line":27,"start":3145,"end":3146,"expression":"b","context":44,"kind":"use","source":5},{"line":28,"start":3189,"end":3190,"expression":"R","context":45,"kind":"use","source":5},{"line":31,"start":3358,"end":3373,"expression":"A_{\\beta\\alpha}","context":46,"kind":"use","source":5},{"line":31,"start":3393,"end":3399,"expression":"\\alpha","context":46,"kind":"use","source":5},{"line":31,"start":3412,"end":3417,"expression":"\\beta","context":46,"kind":"use","source":5},{"line":39,"start":4527,"end":4528,"expression":"D","context":47,"kind":"use","source":5},{"line":39,"start":4559,"end":4573,"expression":"\\mat{3&1\\\\1&3}","context":48,"kind":"use","source":5},{"line":39,"start":4584,"end":4601,"expression":"u=v=\\ones\\in\\QQ^6","context":49,"kind":"use","source":5},{"line":39,"start":4610,"end":4618,"expression":"A=D+uv^T","context":49,"kind":"use","source":5},{"line":40,"start":4749,"end":4801,"expression":"\nx=(1,-1,2,0,-2,3)^T,\\qquad v^Tx=3,\\qquad Ax=Dx+3u.\n","context":49,"kind":"use","source":5},{"line":45,"start":4964,"end":4968,"expression":"v^Tx","context":49,"kind":"use","source":5},{"line":45,"start":4999,"end":5000,"expression":"u","context":50,"kind":"use","source":5},{"line":47,"start":5149,"end":5155,"expression":"7+3=10","context":51,"kind":"use","source":5},{"line":47,"start":5178,"end":5184,"expression":"-3+3=0","context":51,"kind":"use","source":5},{"line":51,"start":5867,"end":5868,"expression":"x","context":52,"kind":"use","source":5},{"line":55,"start":6250,"end":6251,"expression":"q","context":53,"kind":"use","source":5},{"line":55,"start":6310,"end":6315,"expression":"2n^2q","context":54,"kind":"use","source":5},{"line":55,"start":6357,"end":6358,"expression":"r","context":55,"kind":"use","source":5},{"line":55,"start":6399,"end":6400,"expression":"b","context":56,"kind":"use","source":5},{"line":56,"start":6409,"end":6462,"expression":"\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n","context":57,"kind":"use","source":5},{"line":61,"start":6683,"end":6694,"expression":"n=6,b=2,r=1","context":58,"kind":"use","source":5},{"line":61,"start":6772,"end":6775,"expression":"72q","context":59,"kind":"use","source":5},{"line":61,"start":6781,"end":6784,"expression":"48q","context":60,"kind":"use","source":5},{"line":61,"start":6798,"end":6801,"expression":"b,r","context":61,"kind":"use","source":5},{"line":61,"start":6815,"end":6823,"expression":"n/(b+2r)","context":62,"kind":"use","source":5},{"line":61,"start":6837,"end":6838,"expression":"n","context":63,"kind":"use","source":5},{"line":63,"start":6991,"end":6992,"expression":"s","context":64,"kind":"use","source":5},{"line":63,"start":7030,"end":7032,"expression":"sM","context":65,"kind":"use","source":5},{"line":65,"start":7222,"end":7223,"expression":"p","context":66,"kind":"use","source":5},{"line":65,"start":7253,"end":7254,"expression":"w","context":66,"kind":"use","source":5},{"line":66,"start":7291,"end":7308,"expression":"\np(2w+1)-w(w+1).\n","context":66,"kind":"use","source":5},{"line":69,"start":7405,"end":7415,"expression":"1\\le p\\le6","context":66,"kind":"use","source":5},{"line":69,"start":7439,"end":7440,"expression":"w","context":66,"kind":"use","source":5},{"line":69,"start":7454,"end":7457,"expression":"w=0","context":66,"kind":"use","source":5},{"line":69,"start":7464,"end":7469,"expression":"w=p-1","context":66,"kind":"use","source":5},{"line":73,"start":7755,"end":7769,"expression":"c=(2,1,0,-1)^T","context":67,"kind":"use","source":5},{"line":73,"start":7783,"end":7806,"expression":"C(c)_{jk}=c_{j-k\\bmod4}","context":67,"kind":"use","source":5},{"line":73,"start":7840,"end":7864,"expression":"F_{jk}=\\mathrm i^{-jk}/2","context":67,"kind":"use","source":5},{"line":74,"start":7928,"end":7995,"expression":"\nC(c)=F^*\\diag(2Fc)F,\\qquad 2Fc=(2,2-2\\mathrm i,2,2+2\\mathrm i)^T.\n","context":67,"kind":"use","source":5},{"line":79,"start":8050,"end":8064,"expression":"x=(1,2,-1,0)^T","context":67,"kind":"use","source":5},{"line":79,"start":8095,"end":8107,"expression":"(0,6,0,-2)^T","context":68,"kind":"use","source":5},{"line":79,"start":8124,"end":8125,"expression":"x","context":69,"kind":"use","source":5},{"line":79,"start":8189,"end":8192,"expression":"F^*","context":70,"kind":"use","source":5},{"line":79,"start":8249,"end":8263,"expression":"\\QQ(\\mathrm i)","context":71,"kind":"use","source":5},{"line":81,"start":8513,"end":8524,"expression":"\\sqrt n\\,Fc","context":72,"kind":"use","source":5},{"line":81,"start":8532,"end":8534,"expression":"Fc","context":73,"kind":"use","source":5},{"line":81,"start":8575,"end":8578,"expression":"C/2","context":74,"kind":"use","source":5},{"line":81,"start":8593,"end":8594,"expression":"C","context":75,"kind":"use","source":5},{"line":88,"start":9392,"end":9444,"expression":"\nT=\\mat{2&-1&5\\\\3&2&-1\\\\4&3&2},\\qquad x=(1,2,-1)^T.\n","context":76,"kind":"use","source":5},{"line":91,"start":9574,"end":9592,"expression":"g=(2,3,4,0,5,-1)^T","context":77,"kind":"use","source":5},{"line":94,"start":9696,"end":9699,"expression":"j<3","context":78,"kind":"use","source":5},{"line":94,"start":9718,"end":9724,"expression":"k\\le j","context":78,"kind":"use","source":5},{"line":94,"start":9761,"end":9768,"expression":"c_{j-k}","context":78,"kind":"use","source":5},{"line":94,"start":9787,"end":9790,"expression":"k>j","context":78,"kind":"use","source":5},{"line":94,"start":9835,"end":9842,"expression":"r_{k-j}","context":79,"kind":"use","source":5},{"line":95,"start":9942,"end":9958,"expression":"\nTx=(-5,8,8)^T.\n","context":80,"kind":"use","source":5},{"line":98,"start":10067,"end":10069,"expression":"Tx","context":81,"kind":"use","source":5},{"line":100,"start":10132,"end":10139,"expression":"(2,3,4)","context":82,"kind":"use","source":5},{"line":100,"start":10283,"end":10285,"expression":"Tx","context":83,"kind":"use","source":5},{"line":102,"start":10343,"end":10344,"expression":"g","context":84,"kind":"use","source":5},{"line":102,"start":10454,"end":10456,"expression":"2n","context":84,"kind":"use","source":5},{"line":107,"start":10911,"end":10973,"expression":"\n(A\\otimes B)\\operatorname{vec}(X)=\\operatorname{vec}(BXA^T).\n","context":85,"kind":"use","source":5},{"line":110,"start":11082,"end":11107,"expression":"A=\\mat{1&\\mathrm i\\\\2&-1}","context":86,"kind":"use","source":5},{"line":110,"start":11125,"end":11141,"expression":"B=\\mat{2&1\\\\0&3}","context":87,"kind":"use","source":5},{"line":110,"start":11148,"end":11164,"expression":"X=\\mat{1&2\\\\3&4}","context":88,"kind":"use","source":5},{"line":113,"start":11235,"end":11246,"expression":"(1,3,2,4)^T","context":89,"kind":"use","source":5},{"line":114,"start":11316,"end":11362,"expression":"\nBXA^T=\\mat{5+8\\mathrm i&2\\\\9+12\\mathrm i&6}.\n","context":90,"kind":"use","source":5},{"line":117,"start":11376,"end":11379,"expression":"A^T","context":91,"kind":"use","source":5},{"line":117,"start":11387,"end":11390,"expression":"A^*","context":91,"kind":"use","source":5},{"line":119,"start":11583,"end":11584,"expression":"A","context":91,"kind":"use","source":5},{"line":119,"start":11596,"end":11597,"expression":"m","context":91,"kind":"use","source":5},{"line":119,"start":11604,"end":11605,"expression":"B","context":91,"kind":"use","source":5},{"line":119,"start":11617,"end":11618,"expression":"n","context":91,"kind":"use","source":5},{"line":119,"start":11656,"end":11662,"expression":"m^2n^2","context":91,"kind":"use","source":5},{"line":119,"start":11692,"end":11699,"expression":"m^2+n^2","context":91,"kind":"use","source":5},{"line":119,"start":11720,"end":11725,"expression":"BXA^T","context":91,"kind":"use","source":5},{"line":119,"start":11734,"end":11745,"expression":"2n^2m+2nm^2","context":91,"kind":"use","source":5},{"line":125,"start":12371,"end":12378,"expression":"H_1=[1]","context":92,"kind":"use","source":5},{"line":126,"start":12386,"end":12438,"expression":"\nH_{2n}=\\mat{H_n&H_n\\\\H_n&-H_n},\\qquad H_nH_n^T=nI.\n","context":93,"kind":"use","source":5},{"line":129,"start":12481,"end":12500,"expression":"(1,2,3,4,5,6,7,8)^T","context":94,"kind":"use","source":5},{"line":132,"start":12786,"end":12794,"expression":"8\\times8","context":94,"kind":"use","source":5},{"line":134,"start":12830,"end":12854,"expression":"(36,-4,-8,0,-16,0,0,0)^T","context":95,"kind":"use","source":5},{"line":134,"start":12902,"end":12909,"expression":"H_8^T/8","context":95,"kind":"use","source":5},{"line":134,"start":13050,"end":13060,"expression":"H_8/\\sqrt8","context":95,"kind":"use","source":5},{"line":142,"start":13894,"end":13908,"expression":"A=D_A+u_av_a^T","context":96,"kind":"use","source":5},{"line":142,"start":13941,"end":13963,"expression":"D_B=\\diag(1,2,3,4,5,6)","context":96,"kind":"use","source":5},{"line":142,"start":13967,"end":13986,"expression":"u_b=(1,0,1,0,1,0)^T","context":96,"kind":"use","source":5},{"line":142,"start":13994,"end":14013,"expression":"v_b=(0,1,0,1,0,1)^T","context":96,"kind":"use","source":5},{"line":142,"start":14024,"end":14038,"expression":"B=D_B+u_bv_b^T","context":96,"kind":"use","source":5},{"line":143,"start":14043,"end":14108,"expression":"\nU_C=[D_Au_b+u_a(v_a^Tu_b)\\ \\ u_a],\\qquad\nV_C=[v_b\\ \\ D_B^Tv_a].\n","context":96,"kind":"use","source":5},{"line":149,"start":14180,"end":14182,"expression":"AB","context":96,"kind":"use","source":5},{"line":149,"start":14246,"end":14251,"expression":"v_b^T","context":97,"kind":"use","source":5},{"line":150,"start":14256,"end":14328,"expression":"\nD_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T\n=\\bigl(D_Au_b+u_a(v_a^Tu_b)\\bigr)v_b^T.\n","context":98,"kind":"use","source":5},{"line":154,"start":14354,"end":14365,"expression":"u_av_a^TD_B","context":99,"kind":"use","source":5},{"line":154,"start":14374,"end":14392,"expression":"AB=D_AD_B+U_CV_C^T","context":100,"kind":"use","source":5},{"line":156,"start":14545,"end":14553,"expression":"v_a^Tu_b","context":101,"kind":"use","source":5},{"line":156,"start":14617,"end":14626,"expression":"3u_av_b^T","context":102,"kind":"use","source":5},{"line":162,"start":15407,"end":15430,"expression":"\\frac18\\mat{3&-1\\\\-1&3}","context":103,"kind":"use","source":5},{"line":162,"start":15440,"end":15451,"expression":"D^{-1}u=u/4","context":104,"kind":"use","source":5},{"line":163,"start":15459,"end":15484,"expression":"\nS=1+u^TD^{-1}u=\\frac52.\n","context":104,"kind":"use","source":5},{"line":167,"start":15541,"end":15603,"expression":"\nA^{-1}=D^{-1}-D^{-1}uS^{-1}u^TD^{-1}\n=D^{-1}-\\frac1{40}uu^T.\n","context":104,"kind":"use","source":5},{"line":173,"start":15671,"end":15672,"expression":"D","context":104,"kind":"use","source":5},{"line":173,"start":15703,"end":15708,"expression":"D=I_2","context":104,"kind":"use","source":5},{"line":173,"start":15712,"end":15717,"expression":"u=e_1","context":104,"kind":"use","source":5},{"line":173,"start":15725,"end":15731,"expression":"v=-e_1","context":104,"kind":"use","source":5},{"line":173,"start":15749,"end":15759,"expression":"\\diag(0,1)","context":104,"kind":"use","source":5},{"line":173,"start":15780,"end":15781,"expression":"S","context":104,"kind":"use","source":5},{"line":175,"start":15936,"end":15937,"expression":"A","context":105,"kind":"use","source":5},{"line":175,"start":16021,"end":16040,"expression":"\\det A=\\det D\\det S","context":105,"kind":"use","source":5},{"line":177,"start":16198,"end":16199,"expression":"S","context":105,"kind":"use","source":5},{"line":177,"start":16270,"end":16280,"expression":"\\|S^{-1}\\|","context":106,"kind":"use","source":5},{"line":177,"start":16368,"end":16375,"expression":"D^{-1}u","context":107,"kind":"use","source":5},{"line":177,"start":16382,"end":16391,"expression":"v^TD^{-1}","context":108,"kind":"use","source":5},{"line":182,"start":16672,"end":16773,"expression":"\nQ_L=\\mat{3/5&0\\\\4/5&0\\\\0&1\\\\0&0},\\quad\nR_L=\\mat{1&0&1\\\\0&1&0},\\quad\nR_R=\\mat{3&0&2\\\\0&2&0\\\\1&0&-1}.\n","context":109,"kind":"use","source":5},{"line":187,"start":16781,"end":16789,"expression":"L=Q_LR_L","context":110,"kind":"use","source":5},{"line":187,"start":16796,"end":16801,"expression":"R=R_R","context":110,"kind":"use","source":5},{"line":187,"start":16830,"end":16831,"expression":"L","context":110,"kind":"use","source":5},{"line":187,"start":16886,"end":16887,"expression":"R","context":110,"kind":"use","source":5},{"line":190,"start":16946,"end":16995,"expression":"\nG=R_LR_R^T=\\mat{5&0&0\\\\0&2&0},\\qquad LR^T=Q_LG.\n","context":111,"kind":"use","source":5},{"line":193,"start":17148,"end":17151,"expression":"Q_L","context":112,"kind":"use","source":5},{"line":193,"start":17195,"end":17196,"expression":"L","context":112,"kind":"use","source":5},{"line":202,"start":18084,"end":18128,"expression":"\nA_0=LR^T=\\mat{3&0&0\\\\4&0&0\\\\0&2&0\\\\0&0&0}.\n","context":113,"kind":"use","source":5},{"line":205,"start":18287,"end":18296,"expression":"\\sqrt{29}","context":114,"kind":"use","source":5},{"line":208,"start":18434,"end":18448,"expression":"C(B)=Q_L^*BQ_R","context":114,"kind":"use","source":5},{"line":208,"start":18514,"end":18515,"expression":"k","context":114,"kind":"use","source":5},{"line":210,"start":18785,"end":18786,"expression":"d","context":115,"kind":"use","source":5},{"line":210,"start":18819,"end":18827,"expression":"0\\le k<d","context":115,"kind":"use","source":5},{"line":210,"start":18895,"end":18896,"expression":"k","context":115,"kind":"use","source":5},{"line":210,"start":18951,"end":18954,"expression":"k+1","context":115,"kind":"use","source":5},{"line":210,"start":19029,"end":19041,"expression":"\\sigma_{k+1}","context":115,"kind":"use","source":5},{"line":212,"start":19210,"end":19215,"expression":"[0,1]","context":116,"kind":"use","source":5},{"line":216,"start":19705,"end":19712,"expression":"(2,3,4)","context":117,"kind":"use","source":5},{"line":216,"start":19730,"end":19735,"expression":"p_0=1","context":117,"kind":"use","source":5},{"line":216,"start":19758,"end":19773,"expression":"(1/10,1/5,1/20)","context":117,"kind":"use","source":5},{"line":217,"start":19830,"end":19922,"expression":"\n\\widehat p_j=a_j\\widehat p_{j-1}+e_j,\\qquad\n\\widehat p_3-p_3=12e_1+4e_2+e_3=\\frac{41}{20}.\n","context":117,"kind":"use","source":5},{"line":223,"start":20067,"end":20088,"expression":"(4+e_3)(3+e_2)(2+e_1)","context":118,"kind":"use","source":5},{"line":223,"start":20105,"end":20112,"expression":"402/125","context":119,"kind":"use","source":5},{"line":223,"start":20120,"end":20125,"expression":"41/20","context":120,"kind":"use","source":5},{"line":225,"start":20478,"end":20482,"expression":"7/20","context":121,"kind":"use","source":5},{"line":231,"start":21155,"end":21156,"expression":"K","context":122,"kind":"use","source":5},{"line":231,"start":21174,"end":21179,"expression":"0\\to1","context":123,"kind":"use","source":5},{"line":231,"start":21183,"end":21188,"expression":"0\\to2","context":124,"kind":"use","source":5},{"line":231,"start":21192,"end":21197,"expression":"1\\to3","context":125,"kind":"use","source":5},{"line":231,"start":21205,"end":21210,"expression":"2\\to3","context":126,"kind":"use","source":5},{"line":231,"start":21232,"end":21240,"expression":"1,1,1,-1","context":127,"kind":"use","source":5},{"line":233,"start":21368,"end":21427,"expression":"\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n","context":128,"kind":"use","source":5},{"line":237,"start":21594,"end":21633,"expression":"\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)","context":129,"kind":"use","source":5},{"line":239,"start":21784,"end":21785,"expression":"h","context":130,"kind":"use","source":5},{"line":239,"start":21811,"end":21818,"expression":"K^{h+1}","context":131,"kind":"use","source":5},{"line":239,"start":21839,"end":21848,"expression":"K^{h+1}=0","context":132,"kind":"use","source":5},{"line":239,"start":21922,"end":21929,"expression":"K^h\\ne0","context":133,"kind":"use","source":5},{"line":239,"start":22018,"end":22023,"expression":"K^2=0","context":134,"kind":"use","source":5},{"line":245,"start":22609,"end":22611,"expression":"-1","context":135,"kind":"use","source":5},{"line":245,"start":22660,"end":22665,"expression":"0\\to2","context":136,"kind":"use","source":5},{"line":245,"start":22669,"end":22674,"expression":"1\\to2","context":137,"kind":"use","source":5},{"line":245,"start":22678,"end":22683,"expression":"1\\to3","context":138,"kind":"use","source":5},{"line":245,"start":22687,"end":22692,"expression":"2\\to4","context":139,"kind":"use","source":5},{"line":245,"start":22696,"end":22701,"expression":"3\\to4","context":140,"kind":"use","source":5},{"line":245,"start":22705,"end":22710,"expression":"3\\to5","context":141,"kind":"use","source":5},{"line":245,"start":22714,"end":22719,"expression":"4\\to6","context":142,"kind":"use","source":5},{"line":245,"start":22727,"end":22732,"expression":"5\\to6","context":143,"kind":"use","source":5},{"line":245,"start":22742,"end":22755,"expression":"(2I+K)x=\\ones","context":144,"kind":"use","source":5},{"line":249,"start":22978,"end":23061,"expression":"\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n","context":145,"kind":"use","source":5},{"line":262,"start":24072,"end":24102,"expression":"\nA=\\mat{2&0&1\\\\0&2&1\\\\1&1&2}.\n","context":146,"kind":"use","source":5},{"line":265,"start":24162,"end":24165,"expression":"[2]","context":147,"kind":"use","source":5},{"line":266,"start":24197,"end":24258,"expression":"\nS=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR}\n=\\mat{3/2&-1/2\\\\-1/2&3/2}.\n","context":148,"kind":"use","source":5},{"line":272,"start":24443,"end":24447,"expression":"-1/2","context":149,"kind":"use","source":5},{"line":274,"start":24585,"end":24589,"expression":"+1/2","context":150,"kind":"use","source":5},{"line":274,"start":24662,"end":24671,"expression":"\\frac32 I","context":150,"kind":"use","source":5},{"line":278,"start":25276,"end":25282,"expression":"S^{-1}","context":151,"kind":"use","source":5},{"line":278,"start":25311,"end":25317,"expression":"A^{-1}","context":151,"kind":"use","source":5},{"line":282,"start":25585,"end":25590,"expression":"A=I_2","context":152,"kind":"use","source":5},{"line":282,"start":25648,"end":25657,"expression":"v=(1,1)^T","context":153,"kind":"use","source":5},{"line":283,"start":25771,"end":25834,"expression":"\np(A)v=p(1)(1,1)^T,\\qquad \\dim\\mathcal K_m(A,v)=1\\quad(m\\ge1).\n","context":154,"kind":"use","source":5},{"line":287,"start":25907,"end":25912,"expression":"\\QQ^2","context":155,"kind":"use","source":5},{"line":289,"start":26122,"end":26134,"expression":"A=\\diag(1,2)","context":156,"kind":"use","source":5},{"line":289,"start":26150,"end":26151,"expression":"v","context":157,"kind":"use","source":5},{"line":289,"start":26158,"end":26168,"expression":"Av=(1,2)^T","context":158,"kind":"use","source":5},{"line":289,"start":26355,"end":26360,"expression":"A=I_2","context":159,"kind":"use","source":5},{"line":291,"start":26476,"end":26481,"expression":"A=D+K","context":160,"kind":"use","source":5},{"line":291,"start":26489,"end":26490,"expression":"D","context":161,"kind":"use","source":5},{"line":292,"start":26516,"end":26577,"expression":"\n\\|A^j-D^j\\|\\le jM^{j-1}\\|K\\|,\n\\qquad M=\\max\\{\\|A\\|,\\|D\\|\\}.\n","context":162,"kind":"use","source":5},{"line":301,"start":27071,"end":27139,"expression":"\nA=\\mat{4&1\\\\1&1},\\quad D=\\diag(4,1),\\quad\nB=D^{-1/2}=\\diag(1/2,1).\n","context":163,"kind":"use","source":5},{"line":305,"start":27148,"end":27174,"expression":"BAB=I+E=\\mat{1&1/2\\\\1/2&1}","context":164,"kind":"use","source":5},{"line":305,"start":27204,"end":27207,"expression":"1/2","context":165,"kind":"use","source":5},{"line":305,"start":27214,"end":27217,"expression":"3/2","context":165,"kind":"use","source":5},{"line":308,"start":27339,"end":27346,"expression":"x^TAx=1","context":165,"kind":"use","source":5},{"line":308,"start":27358,"end":27362,"expression":"x=By","context":165,"kind":"use","source":5},{"line":308,"start":27389,"end":27400,"expression":"y^T(BAB)y=1","context":165,"kind":"use","source":5},{"line":310,"start":27524,"end":27535,"expression":"\\|E\\|_2=1/2","context":166,"kind":"use","source":5},{"line":311,"start":27576,"end":27655,"expression":"\n\\kappa_2(D^{-1/2}AD^{-1/2})\n\\le\\frac{1+\\eta}{1-\\eta},\\qquad \\|E\\|_2\\le\\eta<1,\n","context":167,"kind":"use","source":5},{"line":321,"start":28439,"end":28444,"expression":"\\QQ^4","context":168,"kind":"use","source":5},{"line":322,"start":28476,"end":28521,"expression":"\nA=I+E,\\qquad E=\\frac1{100}(\\ones\\ones^T-I).\n","context":168,"kind":"use","source":5},{"line":325,"start":28566,"end":28567,"expression":"I","context":168,"kind":"use","source":5},{"line":325,"start":28600,"end":28605,"expression":"1/100","context":168,"kind":"use","source":5},{"line":328,"start":28771,"end":28788,"expression":"\\|E\\|_F^2=12/10^4","context":169,"kind":"use","source":5},{"line":328,"start":28845,"end":28850,"expression":"3/100","context":170,"kind":"use","source":5},{"line":328,"start":28908,"end":28914,"expression":"-1/100","context":171,"kind":"use","source":5},{"line":328,"start":28924,"end":28937,"expression":"\\|E\\|_2=3/100","context":172,"kind":"use","source":5},{"line":333,"start":29244,"end":29302,"expression":"\n\\|(\\widehat A-A)x\\|\\le\\sum_e\\|E_e\\|\\,\\|P_{\\alpha(e)}x\\|.\n","context":173,"kind":"use","source":5},{"line":342,"start":30013,"end":30022,"expression":"s=(1,2,4)","context":174,"kind":"use","source":5},{"line":342,"start":30044,"end":30053,"expression":"c=(1,1,2)","context":174,"kind":"use","source":5},{"line":342,"start":30068,"end":30074,"expression":"\\tau=1","context":174,"kind":"use","source":5},{"line":342,"start":30089,"end":30116,"expression":"\\sum_ec_e\\log(1/\\epsilon_e)","context":174,"kind":"use","source":5},{"line":342,"start":30130,"end":30151,"expression":"\\sum_es_e\\epsilon_e=1","context":174,"kind":"use","source":5},{"line":343,"start":30161,"end":30221,"expression":"\n\\epsilon_e=\\frac{c_e}{4s_e},\\qquad\n\\epsilon=(1/4,1/8,1/8).\n","context":174,"kind":"use","source":5},{"line":349,"start":30323,"end":30336,"expression":"(1/4,1/4,1/2)","context":175,"kind":"use","source":5},{"line":351,"start":30566,"end":30582,"expression":"m=(1/10,1/2,1/2)","context":176,"kind":"use","source":5},{"line":351,"start":30647,"end":30659,"expression":"\\lambda=10/3","context":177,"kind":"use","source":5},{"line":352,"start":30688,"end":30764,"expression":"\n\\epsilon_e=\\min\\{m_e,c_e/(\\lambda s_e)\\},\\qquad\n\\epsilon=(1/10,3/20,3/20).\n","context":178,"kind":"use","source":5},{"line":356,"start":30799,"end":30814,"expression":"(1/10,3/10,3/5)","context":178,"kind":"use","source":5},{"line":362,"start":31557,"end":31592,"expression":"(\\text{integer weight},\\text{cost})","context":179,"kind":"use","source":5},{"line":363,"start":31605,"end":31712,"expression":"\n\\begin{aligned}\n1 &: (0,7),(2,3),(4,1),\\\\\n2 &: (0,6),(1,4),(3,1),\\\\\n3 &: (0,9),(2,5),(5,1).\n\\end{aligned}\n","context":180,"kind":"use","source":5},{"line":370,"start":31734,"end":31740,"expression":"3^3=27","context":181,"kind":"use","source":5},{"line":373,"start":32014,"end":32021,"expression":"(2,0,2)","context":181,"kind":"use","source":5},{"line":373,"start":32035,"end":32042,"expression":"(3,6,5)","context":181,"kind":"use","source":5},{"line":375,"start":32315,"end":32322,"expression":"+\\infty","context":182,"kind":"use","source":5},{"line":377,"start":32422,"end":32426,"expression":"3/50","context":182,"kind":"use","source":5},{"line":377,"start":32433,"end":32437,"expression":"2/25","context":182,"kind":"use","source":5},{"line":377,"start":32473,"end":32477,"expression":"1/10","context":182,"kind":"use","source":5},{"line":377,"start":32547,"end":32556,"expression":"7/50>1/10","context":182,"kind":"use","source":5},{"line":381,"start":32977,"end":32978,"expression":"r","context":183,"kind":"use","source":5},{"line":381,"start":32994,"end":32997,"expression":"a,b","context":184,"kind":"use","source":5},{"line":381,"start":33042,"end":33043,"expression":"r","context":185,"kind":"use","source":5},{"line":381,"start":33119,"end":33120,"expression":"a","context":186,"kind":"use","source":5},{"line":381,"start":33126,"end":33127,"expression":"b","context":186,"kind":"use","source":5},{"line":381,"start":33168,"end":33169,"expression":"a","context":186,"kind":"use","source":5},{"line":381,"start":33214,"end":33215,"expression":"b","context":186,"kind":"use","source":5},{"line":383,"start":33299,"end":33325,"expression":"(\\text{cost},\\text{error})","context":187,"kind":"use","source":5},{"line":384,"start":33335,"end":33376,"expression":"\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n","context":188,"kind":"use","source":5},{"line":387,"start":33425,"end":33438,"expression":"\\{a_0,a_1,b\\}","context":189,"kind":"use","source":5},{"line":387,"start":33499,"end":33516,"expression":"\\cost+\\lambda\\err","context":190,"kind":"use","source":5},{"line":387,"start":33527,"end":33538,"expression":"\\lambda\\ge0","context":191,"kind":"use","source":5},{"line":389,"start":33628,"end":33651,"expression":"8+2\\lambda\\le1+6\\lambda","context":192,"kind":"use","source":5},{"line":389,"start":33661,"end":33674,"expression":"\\lambda\\ge7/4","context":193,"kind":"use","source":5},{"line":389,"start":33719,"end":33734,"expression":"8+2\\lambda\\le10","context":194,"kind":"use","source":5},{"line":389,"start":33744,"end":33755,"expression":"\\lambda\\le1","context":195,"kind":"use","source":5},{"line":395,"start":34341,"end":34350,"expression":"\\lambda=2","context":196,"kind":"use","source":5},{"line":395,"start":34385,"end":34388,"expression":"a_0","context":197,"kind":"use","source":5},{"line":395,"start":34445,"end":34448,"expression":"a_0","context":198,"kind":"use","source":5},{"line":395,"start":34463,"end":34464,"expression":"a","context":199,"kind":"use","source":5},{"line":395,"start":34481,"end":34482,"expression":"r","context":200,"kind":"use","source":5},{"line":395,"start":34505,"end":34506,"expression":"b","context":201,"kind":"use","source":5},{"line":400,"start":35082,"end":35085,"expression":"3/2","context":202,"kind":"use","source":5},{"line":400,"start":35141,"end":35144,"expression":"3/2","context":203,"kind":"use","source":5},{"line":411,"start":36583,"end":36591,"expression":"2\\times2","context":204,"kind":"use","source":5},{"line":411,"start":36626,"end":36627,"expression":"q","context":205,"kind":"use","source":5},{"line":412,"start":36681,"end":36736,"expression":"\nB_{\\rm stream}=8\\cdot3(4+2q+4q)=96+144q\\text{ bytes}.\n","context":206,"kind":"use","source":5},{"line":415,"start":36762,"end":36765,"expression":"q=1","context":207,"kind":"use","source":5},{"line":415,"start":36785,"end":36788,"expression":"q=4","context":208,"kind":"use","source":5},{"line":417,"start":36949,"end":36950,"expression":"P","context":209,"kind":"use","source":5},{"line":417,"start":36967,"end":36968,"expression":"W","context":209,"kind":"use","source":5},{"line":417,"start":36972,"end":36989,"expression":"T\\ge\\max(F/P,B/W)","context":209,"kind":"use","source":5},{"line":422,"start":37564,"end":37596,"expression":"\nT_b(R)=7R,\\qquad T_c(R)=30+2R.\n","context":210,"kind":"use","source":5},{"line":426,"start":37623,"end":37626,"expression":"R=6","context":211,"kind":"use","source":5},{"line":426,"start":37685,"end":37688,"expression":"R=7","context":211,"kind":"use","source":5},{"line":427,"start":37772,"end":37818,"expression":"\n\\left\\lfloor\\frac{30}{7-2}\\right\\rfloor+1=7.\n","context":212,"kind":"use","source":5},{"line":438,"start":38922,"end":38928,"expression":"12/2=6","context":213,"kind":"use","source":5},{"line":451,"start":40645,"end":40651,"expression":"D_AD_B","context":214,"kind":"use","source":5},{"line":451,"start":40688,"end":40695,"expression":"U_C,V_C","context":215,"kind":"use","source":5},{"line":451,"start":40724,"end":40743,"expression":"x=(1,-1,2,0,-2,3)^T","context":216,"kind":"use","source":5},{"line":452,"start":40748,"end":40799,"expression":"\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n","context":217,"kind":"use","source":5},{"line":458,"start":40884,"end":40904,"expression":"(1,-5,18,6,-12,44)^T","context":218,"kind":"use","source":5},{"line":458,"start":40926,"end":40947,"expression":"(25,21,25,21,25,21)^T","context":219,"kind":"use","source":5},{"line":459,"start":40965,"end":40989,"expression":"\n(26,16,43,27,13,65)^T.\n","context":220,"kind":"use","source":5},{"line":462,"start":41030,"end":41031,"expression":"A","context":221,"kind":"use","source":5},{"line":462,"start":41038,"end":41039,"expression":"B","context":222,"kind":"use","source":5},{"line":462,"start":41097,"end":41098,"expression":"x","context":223,"kind":"use","source":5},{"line":8,"start":511,"end":572,"expression":"\nD=\\diag(2,3,4,5),\\qquad\nu=(1,0,1,0)^T,\\qquad v=(1,1,0,0)^T,\n","context":224,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L7"},{"line":12,"start":587,"end":595,"expression":"A=D+uv^T","context":224,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L7"},{"line":13,"start":744,"end":757,"expression":"x=(1,2,3,4)^T","context":224,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L7"},{"line":13,"start":772,"end":778,"expression":"v^Tx=3","context":224,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L7"},{"line":14,"start":788,"end":834,"expression":"\nDx=(2,6,12,20)^T,\\qquad u(v^Tx)=(3,0,3,0)^T,\n","context":224,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L7"},{"line":17,"start":841,"end":857,"expression":"Ax=(5,6,15,20)^T","context":224,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L7"},{"line":25,"start":1279,"end":1283,"expression":"y=Ax","context":225,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L25"},{"line":25,"start":1293,"end":1294,"expression":"D","context":225,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L25"},{"line":26,"start":1313,"end":1367,"expression":"\nD^{-1}u=(1/2,0,1/4,0)^T,\\qquad\n1+v^TD^{-1}u=3/2\\ne0.\n","context":225,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L25"},{"line":31,"start":1452,"end":1505,"expression":"\n\\frac{v^TD^{-1}y}{1+v^TD^{-1}u}\n=\\frac{9/2}{3/2}=3.\n","context":225,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L25"},{"line":36,"start":1567,"end":1622,"expression":"\nD^{-1}y-3D^{-1}u\n=(5/2,2,15/4,4)^T-(3/2,0,3/4,0)^T=x.\n","context":225,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L25"},{"line":42,"start":1925,"end":1927,"expression":"Ax","context":226,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L41"},{"line":42,"start":1960,"end":1961,"expression":"A","context":226,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L41"},{"line":47,"start":2231,"end":2232,"expression":"r","context":227,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L47"},{"line":47,"start":2248,"end":2251,"expression":"a,b","context":227,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L47"},{"line":47,"start":2259,"end":2260,"expression":"a","context":227,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L47"},{"line":47,"start":2276,"end":2283,"expression":"a_1,a_2","context":227,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L47"},{"line":50,"start":2437,"end":2438,"expression":"r","context":227,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L49"},{"line":50,"start":2443,"end":2444,"expression":"a","context":227,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L49"},{"line":50,"start":2449,"end":2450,"expression":"b","context":227,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L49"},{"line":50,"start":2455,"end":2458,"expression":"a_1","context":227,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L49"},{"line":50,"start":2463,"end":2466,"expression":"a_2","context":227,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L49"},{"line":59,"start":2705,"end":2706,"expression":"r","context":228,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L57"},{"line":60,"start":2719,"end":2722,"expression":"a,b","context":229,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L57"},{"line":61,"start":2735,"end":2744,"expression":"a_1,a_2,b","context":230,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L57"},{"line":64,"start":2823,"end":2834,"expression":"\\lambda\\ge0","context":231,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L64"},{"line":64,"start":2859,"end":2869,"expression":"1+6\\lambda","context":231,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L64"},{"line":64,"start":2873,"end":2883,"expression":"3+4\\lambda","context":231,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L64"},{"line":64,"start":2891,"end":2900,"expression":"5+\\lambda","context":231,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L64"},{"line":65,"start":2947,"end":3016,"expression":"\n3+4\\lambda\\le1+6\\lambda\\quad\\text{and}\\quad\n3+4\\lambda\\le5+\\lambda.\n","context":231,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L64"},{"line":69,"start":3047,"end":3058,"expression":"\\lambda\\ge1","context":231,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L64"},{"line":69,"start":3065,"end":3078,"expression":"\\lambda\\le2/3","context":231,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L64"},{"line":69,"start":3173,"end":3184,"expression":"\\lambda=4/5","context":232,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L64"},{"line":69,"start":3214,"end":3218,"expression":"29/5","context":233,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L64"},{"line":77,"start":3635,"end":3643,"expression":"\\err\\le4","context":234,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L77"},{"line":77,"start":3883,"end":3890,"expression":"\\lambda","context":234,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L77"},{"line":79,"start":3996,"end":3997,"expression":"a","context":235,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L77"},{"line":79,"start":4013,"end":4023,"expression":"2+4\\lambda","context":235,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L77"},{"line":79,"start":4031,"end":4040,"expression":"4+\\lambda","context":235,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L77"},{"line":79,"start":4047,"end":4048,"expression":"r","context":235,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L77"},{"line":79,"start":4064,"end":4074,"expression":"1+6\\lambda","context":235,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L77"},{"line":79,"start":4082,"end":4096,"expression":"1+F_\\lambda(a)","context":235,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L77"},{"line":93,"start":4932,"end":4939,"expression":"4+1+1=6","context":236,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L93"},{"line":93,"start":4952,"end":4959,"expression":"0+2+1=3","context":236,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L93"},{"line":112,"start":5831,"end":5849,"expression":"(\\err,\\cost)=(5,1)","context":237,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L112"},{"line":112,"start":5856,"end":5861,"expression":"(0,4)","context":237,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L112"},{"line":112,"start":5920,"end":5925,"expression":"(7,2)","context":237,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L112"},{"line":112,"start":5929,"end":5934,"expression":"(5,4)","context":237,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L112"},{"line":112,"start":5938,"end":5943,"expression":"(2,5)","context":237,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L112"},{"line":112,"start":5951,"end":5956,"expression":"(0,7)","context":237,"kind":"use","source":7,"reading_url":"read/paper-viii/three-exact-models.html#L112"},{"line":6,"start":458,"end":493,"expression":"\nX=\\bigoplus_{\\alpha\\in I}X_\\alpha\n","context":238,"kind":"use","source":8,"reading_url":"read/paper-viii/introduction.html#L3"},{"line":10,"start":545,"end":630,"expression":"\nP_\\alpha P_\\beta=\\delta_{\\alpha\\beta}P_\\alpha,\n\\qquad\n\\sum_{\\alpha\\in I}P_\\alpha=I.\n","context":238,"kind":"use","source":8,"reading_url":"read/paper-viii/introduction.html#L3"},{"line":16,"start":665,"end":701,"expression":"\nA_{\\beta\\alpha}=P_\\beta AP_\\alpha.\n","context":238,"kind":"use","source":8,"reading_url":"read/paper-viii/introduction.html#L3"},{"line":36,"start":4449,"end":4545,"expression":"\n\\text{representation},\n\\qquad\n\\text{device},\n\\qquad\n\\text{kernel},\n\\qquad\n\\text{amortization}.\n","context":239,"kind":"use","source":8,"reading_url":"read/paper-viii/introduction.html#L3"},{"line":62,"start":6029,"end":6061,"expression":"\n\\Pcal=(X_\\alpha)_{\\alpha\\in I}\n","context":240,"kind":"definition-passage","definition_title":"Packet layout","source":8,"reading_url":"read/paper-viii/packet-operators.html#L60"},{"line":66,"start":6082,"end":6118,"expression":"\nX=\\bigoplus_{\\alpha\\in I}X_\\alpha.\n","context":240,"kind":"definition-passage","definition_title":"Packet layout","source":8,"reading_url":"read/paper-viii/packet-operators.html#L60"},{"line":70,"start":6129,"end":6191,"expression":"\nn_\\alpha=\\dim X_\\alpha,\n\\qquad\nn=\\sum_{\\alpha\\in I}n_\\alpha.\n","context":240,"kind":"definition-passage","definition_title":"Packet layout","source":8,"reading_url":"read/paper-viii/packet-operators.html#L60"},{"line":75,"start":6239,"end":6265,"expression":"\\iota_\\alpha:X_\\alpha\\to X","context":240,"kind":"definition-passage","definition_title":"Packet layout","source":8,"reading_url":"read/paper-viii/packet-operators.html#L60"},{"line":75,"start":6272,"end":6296,"expression":"\\pi_\\alpha:X\\to X_\\alpha","context":241,"kind":"definition-passage","definition_title":"Packet layout","source":8,"reading_url":"read/paper-viii/packet-operators.html#L60"},{"line":75,"start":6336,"end":6367,"expression":"P_\\alpha=\\iota_\\alpha\\pi_\\alpha","context":242,"kind":"definition-passage","definition_title":"Packet layout","source":8,"reading_url":"read/paper-viii/packet-operators.html#L60"},{"line":80,"start":6541,"end":6597,"expression":"\n\\IR=(\\Pcal,E,(B_{\\beta\\alpha})_{(\\beta,\\alpha)\\in E}),\n","context":243,"kind":"definition-passage","definition_title":"Packet operator representation","source":8,"reading_url":"read/paper-viii/packet-operators.html#L78"},{"line":84,"start":6608,"end":6630,"expression":"\nE\\subseteq I\\times I\n","context":243,"kind":"definition-passage","definition_title":"Packet operator representation","source":8,"reading_url":"read/paper-viii/packet-operators.html#L78"},{"line":88,"start":6639,"end":6689,"expression":"\nB_{\\beta\\alpha}:X_\\alpha\\longrightarrow X_\\beta.\n","context":243,"kind":"definition-passage","definition_title":"Packet operator representation","source":8,"reading_url":"read/paper-viii/packet-operators.html#L78"},{"line":91,"start":6808,"end":6825,"expression":"P_\\beta AP_\\alpha","context":244,"kind":"definition-passage","definition_title":"Packet operator representation","source":8,"reading_url":"read/paper-viii/packet-operators.html#L78"},{"line":91,"start":6880,"end":6899,"expression":"X_\\alpha\\to X_\\beta","context":245,"kind":"definition-passage","definition_title":"Packet operator representation","source":8,"reading_url":"read/paper-viii/packet-operators.html#L78"},{"line":91,"start":6937,"end":6974,"expression":"\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha","context":246,"kind":"definition-passage","definition_title":"Packet operator representation","source":8,"reading_url":"read/paper-viii/packet-operators.html#L78"},{"line":92,"start":7036,"end":7130,"expression":"\n\\operatorname{eval}(\\IR)\n=\n\\sum_{(\\beta,\\alpha)\\in E}\n\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha,\n","context":247,"kind":"definition-passage","definition_title":"Packet operator representation","source":8,"reading_url":"read/paper-viii/packet-operators.html#L78"},{"line":98,"start":7141,"end":7165,"expression":"\\iota_\\beta:X_\\beta\\to X","context":248,"kind":"definition-passage","definition_title":"Packet operator representation","source":8,"reading_url":"read/paper-viii/packet-operators.html#L78"},{"line":102,"start":7307,"end":7318,"expression":"A\\in\\End(X)","context":249,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L101"},{"line":103,"start":7328,"end":7369,"expression":"\nB_{\\beta\\alpha}=\\pi_\\beta A\\iota_\\alpha\n","context":249,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L101"},{"line":106,"start":7382,"end":7383,"expression":"E","context":249,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L101"},{"line":107,"start":7429,"end":7489,"expression":"\n\\boxed{\nA=\n\\sum_{(\\beta,\\alpha)\\in E}\nP_\\beta AP_\\alpha.\n}\n","context":249,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L101"},{"line":114,"start":7531,"end":7532,"expression":"Z","context":249,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L101"},{"line":115,"start":7538,"end":7612,"expression":"\nAZ\n=\n\\sum_{(\\beta,\\alpha)\\in E}\n\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha Z.\n","context":249,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L101"},{"line":124,"start":7720,"end":7721,"expression":"A","context":250,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L123"},{"line":125,"start":7727,"end":7840,"expression":"\nA\n=\n\\left(\\sum_\\beta P_\\beta\\right)\nA\n\\left(\\sum_\\alpha P_\\alpha\\right)\n=\n\\sum_{\\alpha,\\beta}P_\\beta AP_\\alpha.\n","context":250,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L123"},{"line":134,"start":7902,"end":7903,"expression":"Z","context":250,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L123"},{"line":138,"start":8092,"end":8095,"expression":"G_A","context":251,"kind":"definition-passage","definition_title":"Support graph","source":8,"reading_url":"read/paper-viii/packet-operators.html#L137"},{"line":138,"start":8115,"end":8116,"expression":"I","context":251,"kind":"definition-passage","definition_title":"Support graph","source":8,"reading_url":"read/paper-viii/packet-operators.html#L137"},{"line":139,"start":8145,"end":8173,"expression":"\n\\alpha\\longrightarrow\\beta\n","context":251,"kind":"definition-passage","definition_title":"Support graph","source":8,"reading_url":"read/paper-viii/packet-operators.html#L137"},{"line":142,"start":8183,"end":8205,"expression":"P_\\beta AP_\\alpha\\neq0","context":251,"kind":"definition-passage","definition_title":"Support graph","source":8,"reading_url":"read/paper-viii/packet-operators.html#L137"},{"line":153,"start":8601,"end":8614,"expression":"A,B\\in\\End(X)","context":252,"kind":"introduction-passage","source":8,"reading_url":"read/paper-viii/packet-operators.html#L152"},{"line":154,"start":8625,"end":8708,"expression":"\n\\boxed{\n(AB)_{\\gamma\\alpha}\n=\n\\sum_{\\beta\\in I}\nA_{\\gamma\\beta}B_{\\beta\\alpha}.\n}\n","context":252,"kind":"introduction-passage","source":8,"reading_url":"read/paper-viii/packet-operators.html#L152"},{"line":163,"start":8727,"end":8771,"expression":"\n\\supp(AB)\n\\subseteq\n\\supp(A)\\circ\\supp(B),\n","context":252,"kind":"introduction-passage","source":8,"reading_url":"read/paper-viii/packet-operators.html#L152"},{"line":169,"start":8819,"end":8848,"expression":"\n(\\gamma,\\alpha)\\in E\\circ F\n","context":252,"kind":"introduction-passage","source":8,"reading_url":"read/paper-viii/packet-operators.html#L152"},{"line":172,"start":8863,"end":8868,"expression":"\\beta","context":252,"kind":"introduction-passage","source":8,"reading_url":"read/paper-viii/packet-operators.html#L152"},{"line":173,"start":8883,"end":8933,"expression":"\n(\\gamma,\\beta)\\in E,\n\\qquad\n(\\beta,\\alpha)\\in F.\n","context":252,"kind":"introduction-passage","source":8,"reading_url":"read/paper-viii/packet-operators.html#L152"},{"line":181,"start":9039,"end":9040,"expression":"A","context":253,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L180"},{"line":181,"start":9049,"end":9050,"expression":"B","context":253,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L180"},{"line":182,"start":9056,"end":9119,"expression":"\nP_\\gamma ABP_\\alpha\n=\n\\sum_\\beta\nP_\\gamma AP_\\beta BP_\\alpha.\n","context":253,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L180"},{"line":193,"start":9384,"end":9424,"expression":"\n(A^*)_{\\alpha\\beta}=A_{\\beta\\alpha}^*.\n","context":254,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L191"},{"line":196,"start":9432,"end":9433,"expression":"A","context":254,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L191"},{"line":196,"start":9442,"end":9443,"expression":"B","context":254,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L191"},{"line":196,"start":9471,"end":9472,"expression":"X","context":254,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L191"},{"line":196,"start":9481,"end":9482,"expression":"Y","context":254,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L191"},{"line":197,"start":9493,"end":9583,"expression":"\n(A\\otimes B)_{(\\beta,\\delta),(\\alpha,\\gamma)}\n=\nA_{\\beta\\alpha}\\otimes B_{\\delta\\gamma}.\n","context":254,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L191"},{"line":206,"start":9742,"end":9786,"expression":"\n(P_\\beta AP_\\alpha)^*=P_\\alpha A^*P_\\beta.\n","context":255,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L204"},{"line":210,"start":9824,"end":9934,"expression":"\n(P_\\beta\\otimes Q_\\delta)\n(A\\otimes B)\n(P_\\alpha\\otimes Q_\\gamma)\n=\nA_{\\beta\\alpha}\\otimes B_{\\delta\\gamma}.\n","context":255,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L204"},{"line":230,"start":10910,"end":10911,"expression":"R","context":256,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L229"},{"line":230,"start":10955,"end":10956,"expression":"Y","context":256,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L229"},{"line":231,"start":11006,"end":11034,"expression":"\n\\operatorname{decode}(R)=Y\n","context":256,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L229"},{"line":234,"start":11064,"end":11085,"expression":"\\operatorname{decode}","context":256,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L229"},{"line":234,"start":11138,"end":11139,"expression":"R","context":256,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L229"},{"line":238,"start":11493,"end":11494,"expression":"R","context":257,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L237"},{"line":242,"start":11840,"end":11843,"expression":"n^2","context":258,"kind":"use","source":8,"reading_url":"read/paper-viii/packet-operators.html#L241"},{"line":250,"start":12595,"end":12596,"expression":"n","context":259,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L249"},{"line":250,"start":12604,"end":12605,"expression":"n","context":259,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L249"},{"line":250,"start":12628,"end":12629,"expression":"q","context":259,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L249"},{"line":251,"start":12668,"end":12704,"expression":"\n\\boxed{\n\\flop_{\\rm dense}=2n^2q.\n}\n","context":259,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L249"},{"line":256,"start":12754,"end":12755,"expression":"E","context":259,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L249"},{"line":257,"start":12777,"end":12857,"expression":"\n\\boxed{\n\\flop_{\\rm packet}\n=\n2q\n\\sum_{(\\beta,\\alpha)\\in E}\nn_\\beta n_\\alpha.\n}\n","context":259,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L249"},{"line":267,"start":12872,"end":12944,"expression":"\n\\boxed{\nM_{\\rm packet}\n=\n\\sum_{(\\beta,\\alpha)\\in E}\nn_\\beta n_\\alpha\n}\n","context":259,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L249"},{"line":279,"start":13048,"end":13063,"expression":"B_{\\beta\\alpha}","context":260,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L278"},{"line":279,"start":13074,"end":13081,"expression":"n_\\beta","context":260,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L278"},{"line":279,"start":13089,"end":13097,"expression":"n_\\alpha","context":260,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L278"},{"line":279,"start":13128,"end":13136,"expression":"n_\\alpha","context":260,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L278"},{"line":279,"start":13144,"end":13145,"expression":"q","context":260,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L278"},{"line":279,"start":13162,"end":13181,"expression":"2n_\\beta n_\\alpha q","context":260,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L278"},{"line":279,"start":13216,"end":13232,"expression":"n_\\beta n_\\alpha","context":260,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L278"},{"line":283,"start":13452,"end":13457,"expression":"p\\ge1","context":261,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L282"},{"line":283,"start":13464,"end":13469,"expression":"b\\ge1","context":261,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L282"},{"line":283,"start":13488,"end":13492,"expression":"n=pb","context":261,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L282"},{"line":283,"start":13504,"end":13505,"expression":"n","context":261,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L282"},{"line":283,"start":13524,"end":13525,"expression":"p","context":261,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L282"},{"line":283,"start":13560,"end":13561,"expression":"b","context":261,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L282"},{"line":284,"start":13571,"end":13621,"expression":"\n\\flop_{\\rm block}=2qnb,\n\\qquad\nM_{\\rm block}=nb.\n","context":261,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L282"},{"line":289,"start":13743,"end":13748,"expression":"q\\ge1","context":261,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L282"},{"line":290,"start":13753,"end":13777,"expression":"\n\\boxed{\n\\frac{n}{b}.\n}\n","context":261,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L282"},{"line":298,"start":13860,"end":13863,"expression":"n/b","context":262,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L297"},{"line":298,"start":13906,"end":13909,"expression":"b^2","context":262,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L297"},{"line":303,"start":14131,"end":14142,"expression":"\nA=D+UV^*,\n","context":263,"kind":"definition-passage","definition_title":"Block diagonal plus low rank operator","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L301"},{"line":307,"start":14153,"end":14219,"expression":"\nD=\\bigoplus_{\\alpha\\in I}D_\\alpha,\n\\qquad\nU,V\\in\\CC^{n\\times r}.\n","context":263,"kind":"definition-passage","definition_title":"Block diagonal plus low rank operator","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L301"},{"line":316,"start":14381,"end":14382,"expression":"q","context":264,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L315"},{"line":317,"start":14405,"end":14420,"expression":"\nAZ=DZ+U(V^*Z)\n","context":264,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L315"},{"line":321,"start":14442,"end":14499,"expression":"\n\\boxed{\n2q\n\\left(\n\\sum_\\alpha n_\\alpha^2+2nr\n\\right).\n}\n","context":264,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L315"},{"line":330,"start":14530,"end":14568,"expression":"\n\\boxed{\n\\sum_\\alpha n_\\alpha^2+2nr\n}\n","context":264,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L315"},{"line":335,"start":14585,"end":14589,"expression":"n=pb","context":264,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L315"},{"line":335,"start":14607,"end":14614,"expression":"p,b\\ge1","context":264,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L315"},{"line":335,"start":14733,"end":14738,"expression":"q\\ge1","context":265,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L315"},{"line":336,"start":14743,"end":14770,"expression":"\n\\boxed{\n\\frac{n}{b+2r}.\n}\n","context":266,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L315"},{"line":345,"start":14874,"end":14901,"expression":"\n2q\\sum_\\alpha n_\\alpha^2.\n","context":267,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L343"},{"line":348,"start":14919,"end":14923,"expression":"V^*Z","context":267,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L343"},{"line":348,"start":14932,"end":14939,"expression":"U(V^*Z)","context":267,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L343"},{"line":348,"start":14949,"end":14953,"expression":"2nrq","context":267,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L343"},{"line":349,"start":15053,"end":15081,"expression":"\n\\sum_\\alpha n_\\alpha^2=nb.\n","context":267,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L343"},{"line":362,"start":15512,"end":15513,"expression":"s","context":268,"kind":"definition-passage","definition_title":"One pass streaming model","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L361"},{"line":366,"start":15900,"end":15901,"expression":"q","context":269,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L365"},{"line":367,"start":15929,"end":16045,"expression":"\n\\boxed{\nB_{\\rm stream}\n=\ns\n\\sum_{(\\beta,\\alpha)\\in E}\n\\left(\n n_\\beta n_\\alpha\n +n_\\alpha q\n +2n_\\beta q\n\\right)\n}\n","context":269,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L365"},{"line":389,"start":16549,"end":16580,"expression":"\n\\Ical\n=\n\\frac{\\flop}{\\bytes}.\n","context":270,"kind":"definition-passage","definition_title":"Arithmetic intensity","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L387"},{"line":397,"start":16718,"end":16721,"expression":"P>0","context":271,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L396"},{"line":397,"start":16766,"end":16769,"expression":"W>0","context":271,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L396"},{"line":397,"start":16819,"end":16820,"expression":"F","context":271,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L396"},{"line":397,"start":16840,"end":16841,"expression":"B","context":271,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L396"},{"line":398,"start":16862,"end":16926,"expression":"\n\\boxed{\nT\\geq\n\\max\\left\\{\n\\frac{F}{P},\n\\frac{B}{W}\n\\right\\}.\n}\n","context":271,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L396"},{"line":408,"start":17009,"end":17028,"expression":"\n\\min\\{P,W\\Ical\\}.\n","context":272,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L396"},{"line":414,"start":17128,"end":17137,"expression":"T\\geq F/P","context":273,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L413"},{"line":414,"start":17172,"end":17181,"expression":"T\\geq B/W","context":273,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L413"},{"line":417,"start":17406,"end":17407,"expression":"F","context":274,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L417"},{"line":417,"start":17416,"end":17417,"expression":"B","context":274,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L417"},{"line":427,"start":17910,"end":17915,"expression":"M\\ge0","context":275,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L426"},{"line":427,"start":17940,"end":17943,"expression":"s>0","context":275,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L426"},{"line":427,"start":17955,"end":17960,"expression":"n\\ge0","context":275,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L426"},{"line":428,"start":18015,"end":18033,"expression":"\nn\\leq\\sqrt{M/s}.\n","context":275,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L426"},{"line":432,"start":18091,"end":18108,"expression":"\nn(b+2r)s\\leq M.\n","context":275,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L426"},{"line":442,"start":18427,"end":18433,"expression":"D+UV^*","context":276,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L441"},{"line":442,"start":18481,"end":18484,"expression":"n^2","context":276,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L441"},{"line":443,"start":18520,"end":18534,"expression":"\n2n^2r+O(n^2)\n","context":276,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L441"},{"line":450,"start":18685,"end":18689,"expression":"UV^*","context":277,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L449"},{"line":450,"start":18708,"end":18709,"expression":"n","context":277,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L449"},{"line":450,"start":18717,"end":18718,"expression":"r","context":277,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L449"},{"line":450,"start":18736,"end":18737,"expression":"r","context":277,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L449"},{"line":450,"start":18745,"end":18746,"expression":"n","context":277,"kind":"use","source":8,"reading_url":"read/paper-viii/arithmetic-storage-and-movement.html#L449"},{"line":458,"start":19336,"end":19343,"expression":"p,b\\ge1","context":278,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L457"},{"line":458,"start":19359,"end":19363,"expression":"n=pb","context":278,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L457"},{"line":458,"start":19375,"end":19380,"expression":"p=n/b","context":278,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L457"},{"line":458,"start":19408,"end":19409,"expression":"b","context":278,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L457"},{"line":459,"start":19455,"end":19497,"expression":"\n\\abs{i-j}\\leq w,\n\\qquad\n0\\leq w\\leq p-1.\n","context":278,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L457"},{"line":465,"start":19533,"end":19570,"expression":"\n\\boxed{\nN_{p,w}\n=\np(2w+1)-w(w+1).\n}\n","context":278,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L457"},{"line":472,"start":19579,"end":19580,"expression":"q","context":278,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L457"},{"line":473,"start":19655,"end":19712,"expression":"\n\\boxed{\n2qb^2N_{p,w}\n\\quad\\text{and}\\quad\nb^2N_{p,w}.\n}\n","context":279,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L457"},{"line":483,"start":19812,"end":19813,"expression":"p","context":280,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L482"},{"line":483,"start":19836,"end":19837,"expression":"k","context":280,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L482"},{"line":483,"start":19849,"end":19862,"expression":"1\\leq k\\leq w","context":280,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L482"},{"line":483,"start":19915,"end":19921,"expression":"2(p-k)","context":280,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L482"},{"line":484,"start":19944,"end":19986,"expression":"\np+2\\sum_{k=1}^{w}(p-k)\n=\np(2w+1)-w(w+1).\n","context":280,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L482"},{"line":494,"start":20218,"end":20229,"expression":"\nA=S+UV^*,\n","context":281,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L492"},{"line":497,"start":20240,"end":20241,"expression":"S","context":281,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L492"},{"line":497,"start":20250,"end":20251,"expression":"z","context":281,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L492"},{"line":497,"start":20283,"end":20304,"expression":"U,V\\in\\CC^{n\\times r}","context":281,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L492"},{"line":497,"start":20405,"end":20406,"expression":"A","context":281,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L492"},{"line":497,"start":20414,"end":20415,"expression":"q","context":281,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L492"},{"line":498,"start":20443,"end":20464,"expression":"\n\\boxed{\n2q(z+2nr)\n}\n","context":281,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L492"},{"line":504,"start":20499,"end":20516,"expression":"\n\\boxed{\nz+2nr\n}\n","context":281,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L492"},{"line":513,"start":20665,"end":20668,"expression":"2zq","context":282,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L512"},{"line":513,"start":20707,"end":20711,"expression":"2nrq","context":282,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L512"},{"line":513,"start":20760,"end":20761,"expression":"S","context":282,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L512"},{"line":517,"start":20955,"end":20960,"expression":"n\\ge1","context":283,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L516"},{"line":517,"start":20981,"end":20990,"expression":"c\\in\\CC^n","context":283,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L516"},{"line":517,"start":21020,"end":21021,"expression":"n","context":283,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L516"},{"line":518,"start":21037,"end":21057,"expression":"\nC(c)_{jk}=c_{j-k}.\n","context":283,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L516"},{"line":521,"start":21066,"end":21067,"expression":"F","context":283,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L516"},{"line":522,"start":21111,"end":21179,"expression":"\nF_{jk}\n=\n\\frac{1}{\\sqrt n}\n\\exp\\left(-\\frac{2\\pi\\ii jk}{n}\\right).\n","context":283,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L516"},{"line":529,"start":21189,"end":21231,"expression":"\n\\boxed{\nC(c)\n=\nF^*\\diag(\\sqrt n\\,Fc)F.\n}\n","context":283,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L516"},{"line":536,"start":21266,"end":21267,"expression":"n","context":284,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L516"},{"line":540,"start":21442,"end":21443,"expression":"k","context":285,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L539"},{"line":541,"start":21453,"end":21523,"expression":"\nv^{(k)}_j\n=\n\\frac{1}{\\sqrt n}\n\\exp\\left(\\frac{2\\pi\\ii jk}{n}\\right).\n","context":285,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L539"},{"line":548,"start":21533,"end":21671,"expression":"\n(C(c)v^{(k)})_j\n=\n\\sum_{r=0}^{n-1}c_rv^{(k)}_{j-r}\n=\n\\left(\n\\sum_{r=0}^{n-1}\nc_r\\exp\\left(-\\frac{2\\pi\\ii rk}{n}\\right)\n\\right)v^{(k)}_j.\n","context":285,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L539"},{"line":558,"start":21694,"end":21707,"expression":"\\sqrt n(Fc)_k","context":285,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L539"},{"line":558,"start":21725,"end":21732,"expression":"v^{(k)}","context":285,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L539"},{"line":558,"start":21756,"end":21759,"expression":"F^*","context":285,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L539"},{"line":561,"start":21903,"end":21913,"expression":"O(n\\log n)","context":286,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L561"},{"line":564,"start":22177,"end":22182,"expression":"n\\ge1","context":287,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L563"},{"line":564,"start":22203,"end":22222,"expression":"T\\in\\CC^{n\\times n}","context":287,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L563"},{"line":565,"start":22257,"end":22285,"expression":"\n(c_0,c_1,\\ldots,c_{n-1})^T\n","context":287,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L563"},{"line":569,"start":22304,"end":22331,"expression":"\n(c_0,r_1,\\ldots,r_{n-1}).\n","context":287,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L563"},{"line":573,"start":22340,"end":22405,"expression":"\ng\n=\n(c_0,c_1,\\ldots,c_{n-1},0,r_{n-1},\\ldots,r_1)^T\n\\in\\CC^{2n}\n","context":287,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L563"},{"line":579,"start":22421,"end":22422,"expression":"n","context":287,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L563"},{"line":579,"start":22436,"end":22445,"expression":"x\\in\\CC^n","context":287,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L563"},{"line":579,"start":22461,"end":22462,"expression":"n","context":287,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L563"},{"line":579,"start":22500,"end":22511,"expression":"C(g)(x,0)^T","context":288,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L563"},{"line":579,"start":22522,"end":22524,"expression":"Tx","context":289,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L563"},{"line":583,"start":22602,"end":22611,"expression":"0\\leq j<n","context":290,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L582"},{"line":583,"start":22669,"end":22676,"expression":"k\\leq j","context":290,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L582"},{"line":583,"start":22707,"end":22714,"expression":"c_{j-k}","context":290,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L582"},{"line":583,"start":22732,"end":22735,"expression":"k>j","context":290,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L582"},{"line":583,"start":22774,"end":22781,"expression":"r_{k-j}","context":290,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L582"},{"line":583,"start":22862,"end":22863,"expression":"j","context":290,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L582"},{"line":583,"start":22894,"end":22895,"expression":"x","context":290,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L582"},{"line":587,"start":23058,"end":23077,"expression":"A\\in\\CC^{m\\times m}","context":291,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L586"},{"line":587,"start":23083,"end":23102,"expression":"B\\in\\CC^{n\\times n}","context":291,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L586"},{"line":587,"start":23112,"end":23131,"expression":"X\\in\\CC^{n\\times m}","context":291,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L586"},{"line":588,"start":23170,"end":23244,"expression":"\n\\boxed{\n(A\\otimes B)\\operatorname{vec}(X)\n=\n\\operatorname{vec}(BXA^T).\n}\n","context":291,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L586"},{"line":596,"start":23281,"end":23304,"expression":"\n\\boxed{\n2n^2m+2nm^2\n}\n","context":291,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L586"},{"line":601,"start":23320,"end":23327,"expression":"n^2+m^2","context":291,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L586"},{"line":601,"start":23358,"end":23364,"expression":"n^2m^2","context":292,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L586"},{"line":605,"start":23448,"end":23453,"expression":"(i,j)","context":293,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L604"},{"line":605,"start":23467,"end":23472,"expression":"BXA^T","context":293,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L604"},{"line":606,"start":23480,"end":23539,"expression":"\n\\sum_{k=1}^{n}\\sum_{\\ell=1}^{m}\nB_{ik}X_{k\\ell}A_{j\\ell}.\n","context":293,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L604"},{"line":610,"start":23579,"end":23612,"expression":"(A\\otimes B)\\operatorname{vec}(X)","context":293,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L604"},{"line":610,"start":23639,"end":23640,"expression":"B","context":293,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L604"},{"line":610,"start":23657,"end":23660,"expression":"A^T","context":293,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L604"},{"line":614,"start":23848,"end":23855,"expression":"k,d\\ge0","context":294,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L613"},{"line":614,"start":23872,"end":23873,"expression":"k","context":294,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L613"},{"line":614,"start":23900,"end":23901,"expression":"d","context":294,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L613"},{"line":614,"start":23929,"end":23932,"expression":"d^k","context":294,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L613"},{"line":614,"start":23962,"end":23963,"expression":"k","context":294,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L613"},{"line":615,"start":24037,"end":24059,"expression":"\n\\boxed{\n2k d^{k+1}\n}\n","context":294,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L613"},{"line":620,"start":24083,"end":24087,"expression":"kd^2","context":294,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L613"},{"line":624,"start":24164,"end":24167,"expression":"k=0","context":295,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L623"},{"line":624,"start":24251,"end":24256,"expression":"k\\ge1","context":295,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L623"},{"line":624,"start":24273,"end":24280,"expression":"d^{k-1}","context":295,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L623"},{"line":624,"start":24300,"end":24301,"expression":"d","context":295,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L623"},{"line":624,"start":24334,"end":24335,"expression":"d","context":295,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L623"},{"line":624,"start":24343,"end":24344,"expression":"d","context":295,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L623"},{"line":624,"start":24362,"end":24366,"expression":"2d^2","context":295,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L623"},{"line":624,"start":24390,"end":24398,"expression":"2d^{k+1}","context":295,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L623"},{"line":624,"start":24417,"end":24418,"expression":"k","context":295,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L623"},{"line":628,"start":24580,"end":24585,"expression":"N=2^m","context":296,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L627"},{"line":629,"start":24602,"end":24678,"expression":"\nH_1=[1],\n\\qquad\nH_{2N}\n=\n\\begin{bmatrix}\nH_N&H_N\\\\\nH_N&-H_N\n\\end{bmatrix}.\n","context":297,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L627"},{"line":640,"start":24688,"end":24702,"expression":"\nH_NH_N^T=NI.\n","context":298,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L627"},{"line":643,"start":24719,"end":24723,"expression":"H_Nx","context":298,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L627"},{"line":643,"start":24756,"end":24764,"expression":"N\\log_2N","context":298,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L627"},{"line":647,"start":24985,"end":24992,"expression":"x_0+x_1","context":299,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L646"},{"line":647,"start":25001,"end":25008,"expression":"x_0-x_1","context":299,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L646"},{"line":647,"start":25025,"end":25026,"expression":"N","context":299,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L646"},{"line":647,"start":25098,"end":25101,"expression":"N/2","context":299,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L646"},{"line":648,"start":25112,"end":25144,"expression":"\nT(N)=2T(N/2)+N,\n\\qquad\nT(1)=0,\n","context":299,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L646"},{"line":653,"start":25167,"end":25180,"expression":"T(N)=N\\log_2N","context":299,"kind":"use","source":8,"reading_url":"read/paper-viii/structured-primitives.html#L646"},{"line":669,"start":25699,"end":25738,"expression":"\nA=D_A+U_AV_A^*,\n\\qquad\nB=D_B+U_BV_B^*\n","context":300,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L667"},{"line":675,"start":25785,"end":25806,"expression":"\nAB=D_AD_B+U_CV_C^*,\n","context":300,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L667"},{"line":679,"start":25817,"end":25879,"expression":"\nU_C\n=\n\\begin{bmatrix}\nD_AU_B+U_A(V_A^*U_B)&U_A\n\\end{bmatrix}\n","context":300,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L667"},{"line":687,"start":25888,"end":25939,"expression":"\nV_C\n=\n\\begin{bmatrix}\nV_B&D_B^*V_A\n\\end{bmatrix}.\n","context":300,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L667"},{"line":694,"start":25990,"end":25997,"expression":"r_A+r_B","context":300,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L667"},{"line":699,"start":26089,"end":26149,"expression":"\nAB\n=\nD_AD_B\n+D_AU_BV_B^*\n+U_AV_A^*D_B\n+U_A(V_A^*U_B)V_B^*.\n","context":301,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L697"},{"line":707,"start":26205,"end":26210,"expression":"V_B^*","context":301,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L697"},{"line":708,"start":26238,"end":26256,"expression":"\nU_A(D_B^*V_A)^*.\n","context":301,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L697"},{"line":716,"start":26458,"end":26498,"expression":"\nA_j=D_j+U_jV_j^*,\n\\qquad\nj=1,\\ldots,m,\n","context":302,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L714"},{"line":722,"start":26571,"end":26586,"expression":"\nA_m\\cdots A_1\n","context":302,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L714"},{"line":726,"start":26650,"end":26679,"expression":"\n\\boxed{\n\\sum_{j=1}^m r_j.\n}\n","context":302,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L714"},{"line":738,"start":26990,"end":26991,"expression":"D","context":303,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L737"},{"line":739,"start":27000,"end":27016,"expression":"\nS=I+V^*D^{-1}U\n","context":303,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L737"},{"line":743,"start":27042,"end":27101,"expression":"\n\\boxed{\n(D+UV^*)^{-1}\n=\nD^{-1}\n-D^{-1}US^{-1}V^*D^{-1}.\n}\n","context":303,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L737"},{"line":754,"start":27205,"end":27211,"expression":"D+UV^*","context":304,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L753"},{"line":754,"start":27236,"end":27245,"expression":"V^*D^{-1}","context":304,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L753"},{"line":755,"start":27253,"end":27281,"expression":"\nU-US^{-1}(I+V^*D^{-1}U)=0.\n","context":304,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L753"},{"line":761,"start":27531,"end":27532,"expression":"D","context":305,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L761"},{"line":761,"start":27579,"end":27580,"expression":"S","context":305,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L761"},{"line":765,"start":27720,"end":27810,"expression":"\n\\norm{(D+UV^*)^{-1}}\n\\leq\n\\norm{D^{-1}}\n+\n\\norm{D^{-1}U}\n\\norm{S^{-1}}\n\\norm{V^*D^{-1}}.\n","context":306,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L763"},{"line":783,"start":28184,"end":28185,"expression":"M","context":307,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L782"},{"line":784,"start":28210,"end":28243,"expression":"\n\\sigma_1\\geq\\sigma_2\\geq\\cdots,\n","context":307,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L782"},{"line":787,"start":28301,"end":28306,"expression":"k\\ge0","context":307,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L782"},{"line":787,"start":28353,"end":28356,"expression":"M_k","context":307,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L782"},{"line":787,"start":28382,"end":28383,"expression":"k","context":307,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L782"},{"line":788,"start":28415,"end":28444,"expression":"\n\\norm{M-M_k}_2=\\sigma_{k+1}\n","context":307,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L782"},{"line":792,"start":28453,"end":28520,"expression":"\n\\norm{M-M_k}_{\\rm F}\n=\n\\left(\n\\sum_{j>k}\\sigma_j^2\n\\right)^{1/2}.\n","context":307,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L782"},{"line":801,"start":28629,"end":28630,"expression":"n","context":308,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L801"},{"line":805,"start":28837,"end":28887,"expression":"\nL\\in\\CC^{m\\times r},\n\\qquad\nR\\in\\CC^{n\\times r}.\n","context":309,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L803"},{"line":811,"start":28923,"end":28951,"expression":"\nL=Q_LR_L,\n\\qquad\nR=Q_RR_R,\n","context":310,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L803"},{"line":816,"start":28977,"end":28980,"expression":"Q_L","context":311,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L803"},{"line":816,"start":28989,"end":28992,"expression":"Q_R","context":312,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L803"},{"line":817,"start":29034,"end":29046,"expression":"\nG=R_LR_R^*\n","context":313,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L803"},{"line":821,"start":29059,"end":29074,"expression":"\nG=W\\Sigma Z^*\n","context":313,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L803"},{"line":824,"start":29190,"end":29195,"expression":"k\\ge0","context":314,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L803"},{"line":824,"start":29206,"end":29217,"expression":"\\min\\{k,d\\}","context":315,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L803"},{"line":824,"start":29235,"end":29236,"expression":"d","context":316,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L803"},{"line":824,"start":29319,"end":29320,"expression":"d","context":317,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L803"},{"line":825,"start":29341,"end":29401,"expression":"\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n","context":318,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L803"},{"line":831,"start":29411,"end":29421,"expression":"\nL_kR_k^*\n","context":319,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L803"},{"line":834,"start":29449,"end":29450,"expression":"k","context":320,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L803"},{"line":834,"start":29472,"end":29476,"expression":"LR^*","context":321,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L803"},{"line":839,"start":29827,"end":29846,"expression":"\nLR^*\n=\nQ_LGQ_R^*.\n","context":322,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L837"},{"line":844,"start":30031,"end":30032,"expression":"k","context":322,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L837"},{"line":844,"start":30046,"end":30047,"expression":"B","context":322,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L837"},{"line":844,"start":30067,"end":30076,"expression":"Q_L^*BQ_R","context":322,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L837"},{"line":844,"start":30096,"end":30097,"expression":"k","context":322,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L837"},{"line":844,"start":30295,"end":30296,"expression":"G","context":323,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L837"},{"line":859,"start":31188,"end":31202,"expression":"A_1,\\ldots,A_m","context":324,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L858"},{"line":860,"start":31232,"end":31255,"expression":"\n\\widehat A_j=A_j+E_j.\n","context":324,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L858"},{"line":864,"start":31265,"end":31463,"expression":"\n\\boxed{\n\\left\\|\n\\widehat A_m\\cdots\\widehat A_1\n-\nA_m\\cdots A_1\n\\right\\|\n\\leq\n\\sum_{j=1}^m\n\\left(\n\\prod_{i=j+1}^m\\norm{\\widehat A_i}\n\\right)\n\\norm{E_j}\n\\left(\n\\prod_{i=1}^{j-1}\\norm{A_i}\n\\right).\n}\n","context":324,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L858"},{"line":887,"start":31587,"end":31707,"expression":"\n\\widehat A_m\\cdots\\widehat A_1\n-A_m\\cdots A_1\n=\n\\sum_{j=1}^m\n\\widehat A_m\\cdots\\widehat A_{j+1}\nE_j\nA_{j-1}\\cdots A_1.\n","context":325,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L885"},{"line":901,"start":31900,"end":31931,"expression":"\nP_0=I,\n\\qquad\nP_j=A_jP_{j-1},\n","context":326,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L899"},{"line":907,"start":32002,"end":32066,"expression":"\n\\widehat P_0=I,\n\\qquad\n\\widehat P_j\n=\nA_j\\widehat P_{j-1}+E_j,\n","context":326,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L899"},{"line":914,"start":32077,"end":32080,"expression":"E_j","context":326,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L899"},{"line":914,"start":32128,"end":32129,"expression":"j","context":326,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L899"},{"line":915,"start":32194,"end":32304,"expression":"\n\\boxed{\n\\norm{\\widehat P_m-P_m}\n\\leq\n\\sum_{j=1}^{m}\n\\left(\n\\prod_{i=j+1}^{m}\\norm{A_i}\n\\right)\n\\norm{E_j}.\n}\n","context":327,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L899"},{"line":926,"start":32312,"end":32334,"expression":"\\norm{E_j}\\leq\\delta_j","context":328,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L899"},{"line":926,"start":32362,"end":32370,"expression":"\\delta_j","context":329,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L899"},{"line":930,"start":32476,"end":32496,"expression":"D_j=\\widehat P_j-P_j","context":330,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L929"},{"line":931,"start":32507,"end":32542,"expression":"\nD_j=A_jD_{j-1}+E_j,\n\\qquad\nD_0=0.\n","context":330,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L929"},{"line":937,"start":32575,"end":32626,"expression":"\nD_m\n=\n\\sum_{j=1}^{m}\nA_mA_{m-1}\\cdots A_{j+1}E_j.\n","context":330,"kind":"use","source":8,"reading_url":"read/paper-viii/low-rank-algebra.html#L929"},{"line":951,"start":32998,"end":32999,"expression":"m","context":331,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L950"},{"line":952,"start":33005,"end":33027,"expression":"\nP_\\gamma A^mP_\\alpha\n","context":331,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L950"},{"line":956,"start":33085,"end":33180,"expression":"\n\\alpha=\\alpha_0\n\\longrightarrow\n\\alpha_1\n\\longrightarrow\\cdots\\longrightarrow\n\\alpha_m=\\gamma\n","context":331,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L950"},{"line":963,"start":33188,"end":33191,"expression":"G_A","context":331,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L950"},{"line":964,"start":33215,"end":33270,"expression":"\nA_{\\alpha_m\\alpha_{m-1}}\n\\cdots\nA_{\\alpha_1\\alpha_0}.\n","context":331,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L950"},{"line":973,"start":33453,"end":33454,"expression":"A","context":332,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L972"},{"line":973,"start":33509,"end":33510,"expression":"m","context":332,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L972"},{"line":978,"start":33667,"end":33693,"expression":"\np(z)=\\sum_{j=0}^dc_jz^j,\n","context":333,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L976"},{"line":982,"start":33703,"end":33716,"expression":"\n\\supp(p(A))\n","context":333,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L976"},{"line":985,"start":33775,"end":33778,"expression":"A^j","context":333,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L976"},{"line":985,"start":33787,"end":33800,"expression":"0\\leq j\\leq d","context":333,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L976"},{"line":993,"start":34099,"end":34100,"expression":"K","context":334,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L992"},{"line":993,"start":34208,"end":34209,"expression":"h","context":334,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L992"},{"line":994,"start":34226,"end":34238,"expression":"\nK^{h+1}=0.\n","context":334,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L992"},{"line":1000,"start":34347,"end":34350,"expression":"h+1","context":335,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L999"},{"line":1000,"start":34371,"end":34378,"expression":"K^{h+1}","context":335,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L999"},{"line":1013,"start":34941,"end":34942,"expression":"G","context":336,"kind":"definition-passage","definition_title":"Topological level","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1012"},{"line":1017,"start":35250,"end":35255,"expression":"A=D+K","context":337,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1016"},{"line":1017,"start":35344,"end":35352,"expression":"D_\\alpha","context":337,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1016"},{"line":1018,"start":35382,"end":35388,"expression":"\nAx=b\n","context":337,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1016"},{"line":1025,"start":35635,"end":35640,"expression":"\\beta","context":338,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1024"},{"line":1026,"start":35648,"end":35723,"expression":"\nD_\\beta x_\\beta\n=\nb_\\beta\n-\n\\sum_{\\alpha\\to\\beta}K_{\\beta\\alpha}x_\\alpha.\n","context":338,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1024"},{"line":1033,"start":35749,"end":35754,"expression":"\\beta","context":338,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1024"},{"line":1033,"start":35818,"end":35825,"expression":"x_\\beta","context":338,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1024"},{"line":1033,"start":35868,"end":35875,"expression":"D_\\beta","context":338,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1024"},{"line":1054,"start":37078,"end":37079,"expression":"A","context":339,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1053"},{"line":1054,"start":37230,"end":37236,"expression":"\\alpha","context":339,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1053"},{"line":1054,"start":37251,"end":37256,"expression":"\\beta","context":339,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1053"},{"line":1054,"start":37264,"end":37267,"expression":"G_A","context":339,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1053"},{"line":1055,"start":37278,"end":37315,"expression":"\n\\boxed{\nP_\\beta A^{-1}P_\\alpha=0.\n}\n","context":339,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1053"},{"line":1063,"start":37390,"end":37391,"expression":"R","context":340,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1062"},{"line":1063,"start":37433,"end":37439,"expression":"\\alpha","context":340,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1062"},{"line":1063,"start":37455,"end":37461,"expression":"\\alpha","context":340,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1062"},{"line":1064,"start":37475,"end":37513,"expression":"\nX_R=\\bigoplus_{\\gamma\\in R}X_\\gamma.\n","context":340,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1062"},{"line":1067,"start":37540,"end":37541,"expression":"R","context":340,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1062"},{"line":1067,"start":37550,"end":37567,"expression":"AX_R\\subseteq X_R","context":340,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1062"},{"line":1067,"start":37592,"end":37593,"expression":"A","context":340,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1062"},{"line":1067,"start":37630,"end":37633,"expression":"X_R","context":340,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1062"},{"line":1067,"start":37673,"end":37676,"expression":"X_R","context":340,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1062"},{"line":1067,"start":37688,"end":37710,"expression":"A^{-1}X_R\\subseteq X_R","context":340,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1062"},{"line":1067,"start":37722,"end":37735,"expression":"\\beta\\notin R","context":340,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1062"},{"line":1067,"start":37761,"end":37767,"expression":"A^{-1}","context":341,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1062"},{"line":1071,"start":37972,"end":37973,"expression":"R","context":342,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1070"},{"line":1071,"start":37997,"end":37998,"expression":"H","context":342,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1070"},{"line":1071,"start":38029,"end":38035,"expression":"A_{HH}","context":342,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1070"},{"line":1072,"start":38088,"end":38127,"expression":"\nS\n=\nA_{RR}\n-\nA_{RH}A_{HH}^{-1}A_{HR}.\n","context":342,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1070"},{"line":1079,"start":38148,"end":38166,"expression":"({\\beta},{\\alpha})","context":342,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1070"},{"line":1079,"start":38216,"end":38222,"expression":"A_{RR}","context":342,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1070"},{"line":1079,"start":38262,"end":38268,"expression":"\\alpha","context":343,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1070"},{"line":1079,"start":38297,"end":38302,"expression":"\\beta","context":344,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1070"},{"line":1084,"start":38501,"end":38543,"expression":"\nP_\\beta A_{RH}A_{HH}^{-1}A_{HR}P_\\alpha.\n","context":345,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1082"},{"line":1087,"start":38576,"end":38590,"expression":"A_{HR}P_\\alpha","context":345,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1082"},{"line":1087,"start":38676,"end":38690,"expression":"P_\\beta A_{RH}","context":345,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1082"},{"line":1087,"start":38704,"end":38709,"expression":"\\beta","context":345,"kind":"use","source":8,"reading_url":"read/paper-viii/transfer-graphs.html#L1082"},{"line":1095,"start":39265,"end":39266,"expression":"A","context":346,"kind":"definition-passage","definition_title":"Krylov space","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1094"},{"line":1095,"start":39281,"end":39282,"expression":"v","context":346,"kind":"definition-passage","definition_title":"Krylov space","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1094"},{"line":1095,"start":39311,"end":39312,"expression":"m","context":346,"kind":"definition-passage","definition_title":"Krylov space","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1094"},{"line":1096,"start":39325,"end":39387,"expression":"\n\\Kcal_m(A,v)\n=\n\\operatorname{span}\n\\{v,Av,\\ldots,A^{m-1}v\\}.\n","context":346,"kind":"definition-passage","definition_title":"Krylov space","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1094"},{"line":1106,"start":39533,"end":39549,"expression":"\n[A,P_\\alpha]=0\n","context":347,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1104"},{"line":1110,"start":39575,"end":39628,"expression":"\nv=\\sum_\\alpha v_\\alpha,\n\\qquad\nv_\\alpha=P_\\alpha v,\n","context":347,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1104"},{"line":1116,"start":39638,"end":39677,"expression":"\nA^jv\n=\n\\sum_\\alpha A_\\alpha^jv_\\alpha\n","context":347,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1104"},{"line":1121,"start":39692,"end":39698,"expression":"j\\geq0","context":347,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1104"},{"line":1122,"start":39710,"end":39740,"expression":"\nA_\\alpha=P_\\alpha AP_\\alpha.\n","context":347,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1104"},{"line":1126,"start":39751,"end":39820,"expression":"\n\\Kcal_m(A,v)\n\\subseteq\n\\bigoplus_\\alpha\n\\Kcal_m(A_\\alpha,v_\\alpha).\n","context":347,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1104"},{"line":1136,"start":39913,"end":39940,"expression":"\nA^jP_\\alpha=P_\\alpha A^j.\n","context":348,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1134"},{"line":1139,"start":39993,"end":39994,"expression":"v","context":348,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1134"},{"line":1144,"start":40158,"end":40166,"expression":"\nA=D+K,\n","context":349,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1142"},{"line":1147,"start":40177,"end":40178,"expression":"D","context":349,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1142"},{"line":1147,"start":40230,"end":40231,"expression":"j","context":349,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1142"},{"line":1148,"start":40237,"end":40286,"expression":"\n\\boxed{\n\\norm{A^j-D^j}\n\\leq\njM^{j-1}\\norm{K},\n}\n","context":349,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1142"},{"line":1156,"start":40297,"end":40327,"expression":"\nM=\\max\\{\\norm{A},\\norm{D}\\}.\n","context":349,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1142"},{"line":1160,"start":40346,"end":40437,"expression":"\n\\dist(A^jv,\\bigoplus_\\alpha\\Kcal_{j+1}(D_\\alpha,v_\\alpha))\n\\leq\njM^{j-1}\\norm{K}\\norm{v}.\n","context":349,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1142"},{"line":1169,"start":40537,"end":40580,"expression":"\nA^j-D^j\n=\n\\sum_{k=0}^{j-1}\nA^{j-1-k}KD^k.\n","context":350,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1167"},{"line":1175,"start":40612,"end":40627,"expression":"M^{j-1}\\norm{K}","context":350,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1167"},{"line":1175,"start":40644,"end":40648,"expression":"D^jv","context":350,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1167"},{"line":1180,"start":40852,"end":40859,"expression":"\nA=D+K\n","context":351,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1178"},{"line":1183,"start":40983,"end":40984,"expression":"D","context":351,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1178"},{"line":1183,"start":40993,"end":41001,"expression":"D^{-1/2}","context":351,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1178"},{"line":1183,"start":41057,"end":41058,"expression":"D","context":351,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1178"},{"line":1184,"start":41071,"end":41092,"expression":"\nE=D^{-1/2}KD^{-1/2}\n","context":351,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1178"},{"line":1188,"start":41114,"end":41137,"expression":"\n\\norm{E}_2\\leq\\eta<1.\n","context":351,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1178"},{"line":1192,"start":41147,"end":41213,"expression":"\n\\boxed{\n\\cond_2(D^{-1/2}AD^{-1/2})\n\\leq\n\\frac{1+\\eta}{1-\\eta}.\n}\n","context":351,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1178"},{"line":1202,"start":41306,"end":41307,"expression":"x","context":352,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1201"},{"line":1203,"start":41313,"end":41335,"expression":"\n\\abs{x^*Ex}\\leq\\eta.\n","context":352,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1201"},{"line":1206,"start":41365,"end":41368,"expression":"I+E","context":352,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1201"},{"line":1206,"start":41381,"end":41396,"expression":"[1-\\eta,1+\\eta]","context":352,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1201"},{"line":1210,"start":41635,"end":41636,"expression":"A","context":353,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1209"},{"line":1210,"start":41643,"end":41644,"expression":"M","context":353,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1209"},{"line":1211,"start":41726,"end":41764,"expression":"\n\\kappa\n=\n\\cond_2(M^{-1/2}AM^{-1/2}).\n","context":353,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1209"},{"line":1217,"start":41859,"end":41955,"expression":"\n\\norm{x_m-x_*}_A\n\\leq\n2\n\\left(\n\\frac{\\sqrt\\kappa-1}{\\sqrt\\kappa+1}\n\\right)^m\n\\norm{x_0-x_*}_A.\n","context":353,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1209"},{"line":1239,"start":42665,"end":42698,"expression":"\n\\kappa\n=\n\\frac{1+\\eta}{1-\\eta}.\n","context":354,"kind":"use","source":8,"reading_url":"read/paper-viii/krylov-and-preconditioning.html#L1237"},{"line":1256,"start":43175,"end":43241,"expression":"\nA=\n\\sum_{e\\in E}A_e,\n\\qquad\n\\widehat A\n=\n\\sum_{e\\in E}(A_e+E_e),\n","context":355,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1254"},{"line":1264,"start":43257,"end":43258,"expression":"e","context":355,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1254"},{"line":1264,"start":43280,"end":43289,"expression":"\\alpha(e)","context":355,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1254"},{"line":1265,"start":43306,"end":43358,"expression":"\nA_e=A_eP_{\\alpha(e)},\n\\qquad\nE_e=E_eP_{\\alpha(e)}.\n","context":355,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1254"},{"line":1271,"start":43368,"end":43455,"expression":"\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\leq\n\\sum_{e\\in E}\n\\norm{E_e}\n\\norm{P_{\\alpha(e)}x}.\n}\n","context":355,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1254"},{"line":1281,"start":43463,"end":43498,"expression":"\n\\norm{E_e}\n\\leq\n\\eps_e\\norm{A_e},\n","context":355,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1254"},{"line":1287,"start":43508,"end":43601,"expression":"\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\leq\n\\sum_{e\\in E}\n\\eps_e\\norm{A_e}\n\\norm{P_{\\alpha(e)}x}.\n}\n","context":356,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1254"},{"line":1300,"start":43702,"end":43746,"expression":"\n(\\widehat A-A)x\n=\n\\sum_eE_eP_{\\alpha(e)}x.\n","context":357,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1298"},{"line":1309,"start":43993,"end":44012,"expression":"A\\in\\RR^{m\\times p}","context":358,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1308"},{"line":1309,"start":44019,"end":44038,"expression":"B\\in\\RR^{p\\times n}","context":358,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1308"},{"line":1309,"start":44134,"end":44135,"expression":"u","context":358,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1308"},{"line":1309,"start":44169,"end":44173,"expression":"pu<1","context":358,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1308"},{"line":1309,"start":44197,"end":44207,"expression":"\\widehat C","context":358,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1308"},{"line":1310,"start":44259,"end":44326,"expression":"\n|\\widehat C-AB|\\le\\gamma_p|A||B|,\n\\qquad\\gamma_p=\\frac{pu}{1-pu}.\n","context":358,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1308"},{"line":1322,"start":45179,"end":45182,"expression":"A_j","context":359,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1321"},{"line":1322,"start":45240,"end":45252,"expression":"\\widehat A_j","context":359,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1321"},{"line":1322,"start":45262,"end":45316,"expression":"E_j=\\widehat A_j-A_j=\\sum_e\\Delta_{j,e}P_{\\alpha(j,e)}","context":359,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1321"},{"line":1323,"start":45346,"end":45412,"expression":"\n\\|E_j\\|\\le\\delta_j:=\\sum_e\\|\\Delta_{j,e}\\|\\,\\|P_{\\alpha(j,e)}\\|.\n","context":359,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1321"},{"line":1326,"start":45450,"end":45483,"expression":"\\delta_j\\le\\sum_e\\|\\Delta_{j,e}\\|","context":359,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1321"},{"line":1326,"start":45576,"end":45584,"expression":"\\delta_j","context":360,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1321"},{"line":1330,"start":45809,"end":45846,"expression":"\\|P_{\\alpha}x\\|\\le\\|P_{\\alpha}\\|\\|x\\|","context":361,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1329"},{"line":1336,"start":46222,"end":46223,"expression":"E","context":362,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1335"},{"line":1336,"start":46261,"end":46266,"expression":"s_e>0","context":362,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1335"},{"line":1336,"start":46290,"end":46295,"expression":"c_e>0","context":362,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1335"},{"line":1336,"start":46316,"end":46322,"expression":"\\tau>0","context":362,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1335"},{"line":1337,"start":46336,"end":46382,"expression":"\n\\min_{\\eps_e>0}\n\\sum_e c_e\\log\\frac1{\\eps_e}\n","context":362,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1335"},{"line":1342,"start":46398,"end":46421,"expression":"\n\\sum_es_e\\eps_e=\\tau.\n","context":362,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1335"},{"line":1346,"start":46450,"end":46501,"expression":"\n\\boxed{\n\\eps_e\n=\n\\frac{\\tau c_e}{s_e\\sum_fc_f}.\n}\n","context":362,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1335"},{"line":1357,"start":46604,"end":46640,"expression":"\n-\\frac{c_e}{\\eps_e}+\\lambda s_e=0.\n","context":363,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1355"},{"line":1361,"start":46650,"end":46683,"expression":"\n\\eps_e=\\frac{c_e}{\\lambda s_e}.\n","context":363,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1355"},{"line":1365,"start":46709,"end":46742,"expression":"\n\\lambda=\\frac{\\sum_fc_f}{\\tau}.\n","context":363,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1355"},{"line":1372,"start":47117,"end":47122,"expression":"m_e>0","context":364,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1371"},{"line":1373,"start":47144,"end":47163,"expression":"\n0<\\eps_e\\leq m_e.\n","context":364,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1371"},{"line":1377,"start":47175,"end":47200,"expression":"\n0<\\tau\\leq\\sum_es_em_e.\n","context":364,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1371"},{"line":1380,"start":47223,"end":47232,"expression":"\\lambda>0","context":364,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1371"},{"line":1381,"start":47290,"end":47360,"expression":"\n\\boxed{\n\\eps_e\n=\n\\min\\left\\{\nm_e,\n\\frac{c_e}{\\lambda s_e}\n\\right\\}\n}\n","context":364,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1371"},{"line":1392,"start":47369,"end":47392,"expression":"\n\\sum_es_e\\eps_e=\\tau.\n","context":365,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1371"},{"line":1399,"start":47645,"end":47648,"expression":"m_e","context":366,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1398"},{"line":1400,"start":47699,"end":47776,"expression":"\n\\phi(\\lambda)\n=\n\\sum_es_e\n\\min\\left\\{\nm_e,\n\\frac{c_e}{\\lambda s_e}\n\\right\\}\n","context":366,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1398"},{"line":1409,"start":47937,"end":47941,"expression":"\\phi","context":367,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1398"},{"line":1409,"start":48047,"end":48054,"expression":"(m_e)_e","context":368,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1398"},{"line":1409,"start":48067,"end":48098,"expression":"0<\\lambda\\le\\min_e c_e/(s_em_e)","context":369,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1398"},{"line":1422,"start":48699,"end":48700,"expression":"e","context":370,"kind":"definition-passage","definition_title":"Discrete precision option","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1421"},{"line":1422,"start":48734,"end":48735,"expression":"k","context":370,"kind":"definition-passage","definition_title":"Discrete precision option","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1421"},{"line":1422,"start":48771,"end":48778,"expression":"c_{e,k}","context":370,"kind":"definition-passage","definition_title":"Discrete precision option","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1421"},{"line":1422,"start":48822,"end":48829,"expression":"w_{e,k}","context":370,"kind":"definition-passage","definition_title":"Discrete precision option","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1421"},{"line":1426,"start":49046,"end":49047,"expression":"B","context":371,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1425"},{"line":1427,"start":49060,"end":49068,"expression":"\nF_j(b)\n","context":371,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1425"},{"line":1430,"start":49131,"end":49132,"expression":"j","context":371,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1425"},{"line":1430,"start":49169,"end":49170,"expression":"b","context":371,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1425"},{"line":1431,"start":49181,"end":49238,"expression":"\nF_0(0)=0,\n\\qquad\nF_0(b)=+\\infty\n\\quad\\text{for }b\\neq0.\n","context":371,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1425"},{"line":1437,"start":49247,"end":49265,"expression":"F_{j-1}(u)=+\\infty","context":371,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1425"},{"line":1437,"start":49279,"end":49282,"expression":"u<0","context":371,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1425"},{"line":1438,"start":49321,"end":49393,"expression":"\n\\boxed{\nF_j(b)\n=\n\\min_k\n\\left\\{\nF_{j-1}(b-w_{j,k})+c_{j,k}\n\\right\\}.\n}\n","context":372,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1425"},{"line":1448,"start":49413,"end":49419,"expression":"F_m(b)","context":373,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1425"},{"line":1448,"start":49429,"end":49442,"expression":"0\\leq b\\leq B","context":374,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1425"},{"line":1452,"start":49604,"end":49605,"expression":"j","context":375,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1451"},{"line":1452,"start":49637,"end":49638,"expression":"k","context":375,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1451"},{"line":1452,"start":49651,"end":49652,"expression":"j","context":375,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1451"},{"line":1452,"start":49727,"end":49730,"expression":"j-1","context":375,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1451"},{"line":1452,"start":49762,"end":49771,"expression":"b-w_{j,k}","context":375,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1451"},{"line":1452,"start":49910,"end":49911,"expression":"j","context":375,"kind":"use","source":8,"reading_url":"read/paper-viii/mixed-precision.html#L1451"},{"line":1468,"start":51103,"end":51113,"expression":"(P_\\alpha)","context":376,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1467"},{"line":1469,"start":51157,"end":51207,"expression":"\n\\Pi_{\\Pcal}(A)\n=\n\\sum_\\alpha P_\\alpha AP_\\alpha.\n","context":376,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1467"},{"line":1475,"start":51217,"end":51323,"expression":"\n\\boxed{\n\\norm{A-\\Pi_{\\Pcal}(A)}_{\\rm F}^2\n=\n\\sum_{\\alpha\\neq\\beta}\n\\norm{P_\\beta AP_\\alpha}_{\\rm F}^2.\n}\n","context":376,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1467"},{"line":1484,"start":51346,"end":51439,"expression":"\n\\boxed{\n\\norm{A-\\Pi_{\\Pcal}(A)}_2\n\\leq\n\\sum_{\\alpha\\neq\\beta}\n\\norm{P_\\beta AP_\\alpha}_2.\n}\n","context":376,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1467"},{"line":1504,"start":52550,"end":52551,"expression":"v","context":377,"kind":"definition-passage","definition_title":"Candidate representation tree","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1503"},{"line":1505,"start":52560,"end":52600,"expression":"\n\\kappa_v\\geq0,\n\\qquad\n\\epsilon_v\\geq0,\n","context":377,"kind":"definition-passage","definition_title":"Candidate representation tree","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1503"},{"line":1510,"start":52648,"end":52649,"expression":"v","context":377,"kind":"definition-passage","definition_title":"Candidate representation tree","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1503"},{"line":1510,"start":52713,"end":52726,"expression":"\\sigma_v\\geq0","context":377,"kind":"definition-passage","definition_title":"Candidate representation tree","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1503"},{"line":1513,"start":52871,"end":52876,"expression":"\\Lcal","context":378,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1513"},{"line":1514,"start":52885,"end":52958,"expression":"\n\\cost(\\Lcal)\n=\n\\sum_{v\\in\\Lcal}\\kappa_v\n+\n\\sum_{u\\text{ split}}\\sigma_u\n","context":378,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1513"},{"line":1522,"start":52967,"end":53008,"expression":"\n\\err(\\Lcal)=\\sum_{v\\in\\Lcal}\\epsilon_v.\n","context":378,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1513"},{"line":1525,"start":53101,"end":53109,"expression":"\\kappa_v","context":378,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1513"},{"line":1528,"start":53280,"end":53292,"expression":"\\lambda\\geq0","context":379,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1527"},{"line":1529,"start":53308,"end":53363,"expression":"\nJ_\\lambda(\\Lcal)\n=\n\\cost(\\Lcal)\n+\n\\lambda\\err(\\Lcal).\n","context":379,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1527"},{"line":1536,"start":53383,"end":53384,"expression":"v","context":379,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1527"},{"line":1537,"start":53394,"end":53437,"expression":"\nK_\\lambda(v)\n=\n\\kappa_v+\\lambda\\epsilon_v\n","context":379,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1527"},{"line":1543,"start":53446,"end":53521,"expression":"\nS_\\lambda(v)\n=\n\\sigma_v\n+\n\\sum_{w\\in\\operatorname{child}(v)}F_\\lambda(w).\n","context":379,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1527"},{"line":1551,"start":53551,"end":53579,"expression":"\nF_\\lambda(v)=K_\\lambda(v).\n","context":379,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1527"},{"line":1555,"start":53621,"end":53681,"expression":"\n\\boxed{\nF_\\lambda(v)\n=\n\\min\\{K_\\lambda(v),S_\\lambda(v)\\}\n}\n","context":380,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1527"},{"line":1562,"start":53789,"end":53790,"expression":"v","context":381,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1527"},{"line":1566,"start":53897,"end":53898,"expression":"v","context":382,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1565"},{"line":1566,"start":53936,"end":53937,"expression":"v","context":382,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1565"},{"line":1566,"start":53974,"end":53986,"expression":"K_\\lambda(v)","context":382,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1565"},{"line":1566,"start":54012,"end":54013,"expression":"v","context":382,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1565"},{"line":1566,"start":54133,"end":54145,"expression":"F_\\lambda(w)","context":382,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1565"},{"line":1566,"start":54184,"end":54196,"expression":"S_\\lambda(v)","context":382,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1565"},{"line":1579,"start":54899,"end":54921,"expression":"A^{(1)},\\ldots,A^{(m)}","context":383,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1578"},{"line":1579,"start":54969,"end":54970,"expression":"j","context":383,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1578"},{"line":1579,"start":55002,"end":55005,"expression":"r_j","context":383,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1578"},{"line":1580,"start":55052,"end":55118,"expression":"\n\\kappa_v\n=\n\\kappa_v^{\\rm setup}\n+\n\\sum_{j=1}^m r_j\\kappa_v^{(j)}\n","context":383,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1578"},{"line":1588,"start":55127,"end":55175,"expression":"\n\\epsilon_v\n=\n\\sum_{j=1}^m r_j\\epsilon_v^{(j)}.\n","context":383,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1578"},{"line":1601,"start":55661,"end":55685,"expression":"0\\leq\\lambda_1<\\lambda_2","context":384,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1600"},{"line":1601,"start":55715,"end":55722,"expression":"\\Lcal_1","context":384,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1600"},{"line":1601,"start":55731,"end":55738,"expression":"\\Lcal_2","context":384,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1600"},{"line":1602,"start":55783,"end":55827,"expression":"\n\\boxed{\n\\err(\\Lcal_2)\n\\leq\n\\err(\\Lcal_1)\n}\n","context":384,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1600"},{"line":1610,"start":55836,"end":55883,"expression":"\n\\boxed{\n\\cost(\\Lcal_2)\n\\geq\n\\cost(\\Lcal_1).\n}\n","context":384,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1600"},{"line":1621,"start":55975,"end":56057,"expression":"\n\\cost(\\Lcal_1)+\\lambda_1\\err(\\Lcal_1)\n\\leq\n\\cost(\\Lcal_2)+\\lambda_1\\err(\\Lcal_2)\n","context":385,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1619"},{"line":1627,"start":56066,"end":56149,"expression":"\n\\cost(\\Lcal_2)+\\lambda_2\\err(\\Lcal_2)\n\\leq\n\\cost(\\Lcal_1)+\\lambda_2\\err(\\Lcal_1).\n","context":385,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1619"},{"line":1633,"start":56185,"end":56257,"expression":"\n(\\lambda_2-\\lambda_1)\n\\bigl(\n\\err(\\Lcal_2)-\\err(\\Lcal_1)\n\\bigr)\n\\leq0.\n","context":385,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1619"},{"line":1644,"start":56509,"end":56518,"expression":"\\lambda>0","context":386,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1643"},{"line":1645,"start":56580,"end":56595,"expression":"\n(\\cost,\\err).\n","context":386,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1643"},{"line":1652,"start":57023,"end":57031,"expression":"-\\lambda","context":387,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1651"},{"line":1658,"start":57364,"end":57367,"expression":"w_v","context":388,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1657"},{"line":1658,"start":57377,"end":57384,"expression":"B\\in\\NN","context":388,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1657"},{"line":1658,"start":57404,"end":57405,"expression":"v","context":388,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1657"},{"line":1658,"start":57418,"end":57424,"expression":"F_v(b)","context":388,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1657"},{"line":1658,"start":57480,"end":57481,"expression":"v","context":388,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1657"},{"line":1658,"start":57511,"end":57512,"expression":"b","context":388,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1657"},{"line":1659,"start":57546,"end":57569,"expression":"\nF_v(w_v)\\leq\\kappa_v.\n","context":388,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1657"},{"line":1662,"start":57672,"end":57680,"expression":"\\sigma_v","context":389,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1657"},{"line":1663,"start":57711,"end":57751,"expression":"\nF_{\\rm root}(b),\n\\qquad\n0\\leq b\\leq B,\n","context":390,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1657"},{"line":1668,"start":57830,"end":57831,"expression":"B","context":391,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1657"},{"line":1672,"start":58113,"end":58121,"expression":"\\sigma_v","context":392,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1671"},{"line":1676,"start":58505,"end":58506,"expression":"B","context":393,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1675"},{"line":1676,"start":58600,"end":58601,"expression":"B","context":393,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1675"},{"line":1676,"start":58680,"end":58681,"expression":"B","context":393,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1675"},{"line":1680,"start":58793,"end":58796,"expression":"B+1","context":394,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1679"},{"line":1680,"start":58868,"end":58875,"expression":"(B+1)^2","context":394,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1679"},{"line":1693,"start":59419,"end":59420,"expression":"v","context":395,"kind":"definition-passage","definition_title":"Decision gap","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1692"},{"line":1694,"start":59430,"end":59470,"expression":"\ng_v\n=\n\\abs{K_\\lambda(v)-S_\\lambda(v)}.\n","context":395,"kind":"definition-passage","definition_title":"Decision gap","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1692"},{"line":1703,"start":59785,"end":59790,"expression":"g_v/2","context":396,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1702"},{"line":1707,"start":60105,"end":60108,"expression":"g_v","context":397,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1706"},{"line":1707,"start":60167,"end":60172,"expression":"g_v/2","context":397,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1706"},{"line":1707,"start":60226,"end":60231,"expression":"g_v/2","context":397,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1706"},{"line":1711,"start":60540,"end":60541,"expression":"u","context":398,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1710"},{"line":1711,"start":60749,"end":60750,"expression":"u","context":398,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1710"},{"line":1711,"start":60813,"end":60814,"expression":"u","context":399,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1710"},{"line":1715,"start":61258,"end":61259,"expression":"u","context":400,"kind":"use","source":8,"reading_url":"read/paper-viii/exact-compilation.html#L1714"},{"line":1731,"start":62285,"end":62286,"expression":"B","context":401,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1730"},{"line":1731,"start":62327,"end":62328,"expression":"g","context":401,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1730"},{"line":1731,"start":62470,"end":62471,"expression":"g","context":401,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1730"},{"line":1731,"start":62496,"end":62497,"expression":"B","context":401,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1730"},{"line":1753,"start":64025,"end":64031,"expression":"s\\geq0","context":402,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1752"},{"line":1753,"start":64065,"end":64076,"expression":"t_b,t_c\\ge0","context":402,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1752"},{"line":1753,"start":64131,"end":64138,"expression":"R\\in\\NN","context":402,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1752"},{"line":1753,"start":64163,"end":64170,"expression":"t_b>t_c","context":402,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1752"},{"line":1753,"start":64206,"end":64207,"expression":"R","context":402,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1752"},{"line":1754,"start":64230,"end":64253,"expression":"\nR>\n\\frac{s}{t_b-t_c}.\n","context":402,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1752"},{"line":1759,"start":64294,"end":64364,"expression":"\n\\boxed{\nR_{\\min}\n=\n\\left\\lfloor\n\\frac{s}{t_b-t_c}\n\\right\\rfloor+1.\n}\n","context":402,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1752"},{"line":1768,"start":64372,"end":64383,"expression":"t_b\\leq t_c","context":403,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1752"},{"line":1773,"start":64520,"end":64528,"expression":"\ns+Rt_c\n","context":404,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1771"},{"line":1777,"start":64538,"end":64545,"expression":"\nRt_b.\n","context":404,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1771"},{"line":1784,"start":64762,"end":64767,"expression":"c\\ge0","context":405,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1783"},{"line":1784,"start":64796,"end":64803,"expression":"t_o\\ge0","context":405,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1783"},{"line":1784,"start":64824,"end":64836,"expression":"0\\le t_r<t_o","context":405,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1783"},{"line":1785,"start":64873,"end":64896,"expression":"\nR>\n\\frac{c}{t_o-t_r}.\n","context":405,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1783"},{"line":1792,"start":65015,"end":65016,"expression":"c","context":406,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1791"},{"line":1796,"start":65225,"end":65232,"expression":"t_c\\ge0","context":407,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1795"},{"line":1796,"start":65288,"end":65295,"expression":"t_e\\ge0","context":407,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1795"},{"line":1796,"start":65342,"end":65347,"expression":"m\\ge0","context":407,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1795"},{"line":1796,"start":65368,"end":65375,"expression":"R\\in\\NN","context":407,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1795"},{"line":1797,"start":65423,"end":65445,"expression":"\nR>\n\\frac{m}{t_c-t_e}\n","context":407,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1795"},{"line":1801,"start":65454,"end":65461,"expression":"t_c>t_e","context":407,"kind":"use","source":8,"reading_url":"read/paper-viii/device-scheduling.html#L1795"},{"line":1886,"start":70822,"end":70826,"expression":"4096","context":408,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1886"},{"line":1895,"start":71086,"end":71104,"expression":"2.18\\times10^{-15}","context":409,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1888"},{"line":1896,"start":71139,"end":71156,"expression":"5.76\\times10^{-7}","context":410,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1888"},{"line":1897,"start":71191,"end":71208,"expression":"2.94\\times10^{-4}","context":411,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1888"},{"line":1898,"start":71243,"end":71260,"expression":"3.60\\times10^{-4}","context":412,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1888"},{"line":1899,"start":71295,"end":71312,"expression":"2.88\\times10^{-3}","context":413,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1888"},{"line":1924,"start":72175,"end":72179,"expression":"8192","context":414,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1924"},{"line":1924,"start":72187,"end":72191,"expression":"8192","context":414,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1924"},{"line":1924,"start":72234,"end":72236,"expression":"64","context":414,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1924"},{"line":1924,"start":72245,"end":72247,"expression":"32","context":414,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1924"},{"line":1925,"start":72306,"end":72327,"expression":"\n0.046032\\text{ ms}.\n","context":414,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1924"},{"line":1929,"start":72339,"end":72348,"expression":"\n14.8919\n","context":414,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1924"},{"line":1933,"start":72472,"end":72482,"expression":"\n235.7404\n","context":414,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1924"},{"line":1937,"start":72610,"end":72630,"expression":"\n8.17\\times10^{-8}.\n","context":415,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1924"},{"line":1948,"start":72949,"end":72951,"expression":"64","context":416,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1948"},{"line":1948,"start":72966,"end":72967,"expression":"8","context":416,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1948"},{"line":1949,"start":72996,"end":73017,"expression":"\n0.079872\\text{ ms}.\n","context":416,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1948"},{"line":1953,"start":73029,"end":73038,"expression":"\n10.9355\n","context":416,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1948"},{"line":1957,"start":73142,"end":73152,"expression":"\n153.6527\n","context":416,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1948"},{"line":1984,"start":74590,"end":74593,"expression":"512","context":417,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1984"},{"line":1984,"start":74624,"end":74630,"expression":"2.1102","context":417,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1984"},{"line":1984,"start":74662,"end":74668,"expression":"0.0969","context":417,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1984"},{"line":1984,"start":74705,"end":74711,"expression":"0.0151","context":417,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1984"},{"line":1984,"start":74770,"end":74772,"expression":"26","context":417,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1984"},{"line":1984,"start":74796,"end":74800,"expression":"1024","context":417,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1984"},{"line":1984,"start":74836,"end":74842,"expression":"5.2726","context":417,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1984"},{"line":1984,"start":74848,"end":74854,"expression":"0.1896","context":417,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1984"},{"line":1984,"start":74864,"end":74870,"expression":"0.0241","context":417,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1984"},{"line":1984,"start":74886,"end":74888,"expression":"32","context":417,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1984"},{"line":1993,"start":75126,"end":75130,"expression":"2048","context":418,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1993"},{"line":1993,"start":75189,"end":75195,"expression":"1.9993","context":418,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1993"},{"line":1993,"start":75238,"end":75244,"expression":"7.3036","context":418,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1993"},{"line":1993,"start":75274,"end":75280,"expression":"6.3084","context":418,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1993"},{"line":1993,"start":75330,"end":75336,"expression":"3.6557","context":418,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1993"},{"line":1993,"start":75383,"end":75389,"expression":"4.2368","context":418,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L1993"},{"line":2007,"start":75895,"end":75898,"expression":"512","context":419,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L2006"},{"line":2007,"start":75953,"end":75956,"expression":"128","context":419,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L2006"},{"line":2007,"start":75966,"end":75972,"expression":"7.0950","context":419,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L2006"},{"line":2007,"start":76035,"end":76041,"expression":"1.9597","context":419,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L2006"},{"line":2008,"start":76069,"end":76073,"expression":"2048","context":419,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L2006"},{"line":2008,"start":76120,"end":76126,"expression":"5.6079","context":419,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L2006"},{"line":2008,"start":76171,"end":76177,"expression":"0.0860","context":419,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L2006"},{"line":2009,"start":76240,"end":76247,"expression":"22.0848","context":419,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L2006"},{"line":2009,"start":76293,"end":76300,"expression":"12.6286","context":420,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L2006"},{"line":2011,"start":76504,"end":76509,"expression":"65536","context":421,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L2006"},{"line":2011,"start":76544,"end":76550,"expression":"262144","context":422,"kind":"use","source":8,"reading_url":"read/paper-viii/recorded-measurements.html#L2006"},{"line":2044,"start":78741,"end":78746,"expression":"37392","context":423,"kind":"use","source":8,"reading_url":"read/paper-viii/implementation.html#L2044"},{"line":5,"start":640,"end":641,"expression":"R","context":424,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L4-2"},{"line":5,"start":655,"end":656,"expression":"I","context":424,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L4-2"},{"line":5,"start":684,"end":698,"expression":"(P_a)_{a\\in I}","context":424,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L4-2"},{"line":6,"start":722,"end":794,"expression":"\nP_bP_a=\\begin{cases}P_a&b=a,\\\\0&b\\ne a,\\end{cases}\n\\qquad \\sum_aP_a=1.\n","context":425,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L4-2"},{"line":10,"start":804,"end":818,"expression":"A_{ba}=P_bAP_a","context":426,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L4-2"},{"line":10,"start":829,"end":853,"expression":"E_A=\\{(b,a):A_{ba}\\ne0\\}","context":426,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L4-2"},{"line":13,"start":1006,"end":1045,"expression":"\n\\boxed{A=\\sum_{(b,a)\\in E_A} A_{ba}.}\n","context":427,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L4-2"},{"line":17,"start":1252,"end":1283,"expression":"\nAZ=\\sum_{(b,a)\\in E_A}A_{ba}Z\n","context":427,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L4-2"},{"line":20,"start":1335,"end":1336,"expression":"Z","context":427,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L4-2"},{"line":23,"start":1420,"end":1460,"expression":"\n\\boxed{(AB)_{ba}=\\sum_c A_{bc}B_{ca}.}\n","context":428,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L4-2"},{"line":26,"start":1519,"end":1520,"expression":"c","context":428,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L4-2"},{"line":26,"start":1532,"end":1542,"expression":"A_{bc}\\ne0","context":428,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L4-2"},{"line":26,"start":1549,"end":1559,"expression":"B_{ca}\\ne0","context":428,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L4-2"},{"line":26,"start":1576,"end":1615,"expression":"\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)","context":428,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L4-2"},{"line":29,"start":1769,"end":1793,"expression":"\n(A_{ba})^*=(A^*)_{ab}.\n","context":429,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L4-2"},{"line":33,"start":1866,"end":1935,"expression":"\n(P_b\\otimes Q_d)(A\\otimes B)(P_a\\otimes Q_c)\n=A_{ba}\\otimes B_{dc}.\n","context":429,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L4-2"},{"line":46,"start":2793,"end":2794,"expression":"n","context":430,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L45-2"},{"line":46,"start":2975,"end":2978,"expression":"n+1","context":430,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L45-2"},{"line":46,"start":3027,"end":3028,"expression":"n","context":430,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L45-2"},{"line":49,"start":3101,"end":3209,"expression":"\nF_0(b,a)&=\\begin{cases}P_a&b=a,\\\\0&b\\ne a,\\end{cases}\\\\\nF_{n+1}(b,a)&=\\sum_{c:\\,A_{bc}\\ne0}A_{bc}F_n(c,a).\n","context":431,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L45-2"},{"line":54,"start":3248,"end":3290,"expression":"\n\\boxed{P_bA^nP_a=F_n(b,a)\\qquad(n\\ge0).}\n","context":432,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L45-2"},{"line":57,"start":3491,"end":3494,"expression":"A^n","context":432,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L45-2"},{"line":59,"start":3562,"end":3565,"expression":"h+1","context":433,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L45-2"},{"line":60,"start":3610,"end":3630,"expression":"\n\\boxed{A^{h+1}=0.}\n","context":433,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L45-2"},{"line":64,"start":3794,"end":3884,"expression":"\n\\ell:I\\longrightarrow\\{0,\\ldots,h\\},\\qquad\nA_{bc}\\ne0\\ \\Longrightarrow\\ \\ell(c)<\\ell(b).\n","context":433,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L45-2"},{"line":68,"start":3912,"end":3913,"expression":"n","context":434,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L45-2"},{"line":68,"start":3953,"end":3954,"expression":"n","context":435,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L45-2"},{"line":71,"start":4192,"end":4220,"expression":"\np(A)=\\sum_{j=0}^{d}c_jA^j,\n","context":436,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L45-2"},{"line":74,"start":4331,"end":4334,"expression":"j=0","context":436,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L45-2"},{"line":77,"start":4480,"end":4536,"expression":"\nA=\\begin{bmatrix}1&1\\\\-1&-1\\end{bmatrix},\\qquad A^2=0.\n","context":437,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L45-2"},{"line":80,"start":4561,"end":4567,"expression":"A_{00}","context":438,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L45-2"},{"line":80,"start":4677,"end":4680,"expression":"A^2","context":438,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L45-2"},{"line":85,"start":4890,"end":4904,"expression":"A=D_A+U_AV_A^*","context":439,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L84-2"},{"line":85,"start":4911,"end":4925,"expression":"B=D_B+U_BV_B^*","context":439,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L84-2"},{"line":85,"start":4959,"end":4962,"expression":"r_A","context":439,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L84-2"},{"line":85,"start":4969,"end":4972,"expression":"r_B","context":439,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L84-2"},{"line":86,"start":5018,"end":5126,"expression":"\nU_C&=\\begin{bmatrix}D_AU_B+U_A(V_A^*U_B)&U_A\\end{bmatrix},\\\\\nV_C&=\\begin{bmatrix}V_B&D_B^*V_A\\end{bmatrix}\n","context":440,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L84-2"},{"line":91,"start":5152,"end":5181,"expression":"\n\\boxed{AB=D_AD_B+U_CV_C^*.}\n","context":441,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L84-2"},{"line":94,"start":5229,"end":5236,"expression":"r_B+r_A","context":441,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L84-2"},{"line":95,"start":5379,"end":5409,"expression":"\n\\rank(AB-D_AD_B)\\le r_A+r_B.\n","context":441,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L84-2"},{"line":98,"start":5504,"end":5510,"expression":"D_AD_B","context":442,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L84-2"},{"line":100,"start":5621,"end":5622,"expression":"D","context":443,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L84-2"},{"line":101,"start":5630,"end":5646,"expression":"\nS=I+V^*D^{-1}U\n","context":443,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L84-2"},{"line":105,"start":5716,"end":5771,"expression":"\n\\boxed{(D+UV^*)^{-1}\n=D^{-1}-D^{-1}US^{-1}V^*D^{-1}.}\n","context":443,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L84-2"},{"line":109,"start":5880,"end":5886,"expression":"D+UV^*","context":443,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L84-2"},{"line":109,"start":6000,"end":6001,"expression":"Z","context":444,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L84-2"},{"line":112,"start":6264,"end":6363,"expression":"\n\\boxed{\n\\norm{(D+UV^*)^{-1}}\n\\le\\norm{D^{-1}}+\n\\norm{D^{-1}U}\\,\\norm{S^{-1}}\\,\\norm{V^*D^{-1}}.\n}\n","context":445,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L84-2"},{"line":126,"start":6970,"end":6971,"expression":"X","context":446,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L125-2"},{"line":126,"start":7079,"end":7085,"expression":"X\\to X","context":447,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L125-2"},{"line":126,"start":7119,"end":7123,"expression":"a(e)","context":448,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L125-2"},{"line":126,"start":7161,"end":7162,"expression":"e","context":449,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L125-2"},{"line":127,"start":7179,"end":7208,"expression":"\n\\Delta_e=\\Delta_e P_{a(e)}.\n","context":450,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L125-2"},{"line":131,"start":7217,"end":7271,"expression":"\nA=\\sum_e A_e,\\qquad \\widehat A=\\sum_e(A_e+\\Delta_e).\n","context":451,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L125-2"},{"line":135,"start":7335,"end":7423,"expression":"\n\\widehat A-A=\\sum_e\\Delta_e P_{a(e)},\\qquad\n(\\widehat A-A)x=\\sum_e\\Delta_e(P_{a(e)}x).\n","context":452,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L125-2"},{"line":140,"start":7514,"end":7592,"expression":"\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}x}.\n}\n","context":453,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L125-2"},{"line":146,"start":7599,"end":7640,"expression":"\\norm{\\Delta_e}\\le\\varepsilon_e\\norm{A_e}","context":454,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L125-2"},{"line":147,"start":7685,"end":7771,"expression":"\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\varepsilon_e\\norm{A_e}\\,\\norm{P_{a(e)}x}.\n}\n","context":455,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L125-2"},{"line":156,"start":8084,"end":8147,"expression":"\n\\norm{\\widehat A-A}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}}\n","context":456,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L125-2"},{"line":160,"start":8161,"end":8175,"expression":"\\norm{P_a}\\le1","context":456,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L125-2"},{"line":161,"start":8197,"end":8251,"expression":"\n\\boxed{\\norm{\\widehat A-A}\\le\\sum_e\\norm{\\Delta_e}.}\n","context":456,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L125-2"},{"line":166,"start":8481,"end":8489,"expression":"\\Delta_e","context":457,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L125-2"},{"line":172,"start":8993,"end":9090,"expression":"\n\\sum_e c_e\\log\\frac1{\\varepsilon_e},\\qquad\n\\varepsilon_e>0,\\qquad \\sum_e s_e\\varepsilon_e=\\tau,\n","context":458,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L170-2"},{"line":176,"start":9167,"end":9173,"expression":"\\tau>0","context":458,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L170-2"},{"line":177,"start":9215,"end":9322,"expression":"\ns_e>0\\ \\text{and}\\ \\varepsilon_e>0\\ \\text{for all }e\n\\quad\\Longrightarrow\\quad\n\\sum_e s_e\\varepsilon_e>0.\n","context":459,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L170-2"},{"line":184,"start":9527,"end":9533,"expression":"\\tau>0","context":460,"kind":"use","source":9,"reading_url":"read/paper-viii/appendix-exact-semantics.html#L170-2"},{"line":5,"start":842,"end":850,"expression":"\\kappa_v","context":461,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L4-2"},{"line":5,"start":865,"end":875,"expression":"\\epsilon_v","context":461,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L4-2"},{"line":5,"start":898,"end":906,"expression":"\\sigma_v","context":461,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L4-2"},{"line":9,"start":1262,"end":1263,"expression":"L","context":462,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L4-2"},{"line":9,"start":1308,"end":1315,"expression":"\\lambda","context":462,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L4-2"},{"line":10,"start":1337,"end":1376,"expression":"\nJ_\\lambda(L)=\\cost(L)+\\lambda\\err(L).\n","context":462,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L4-2"},{"line":13,"start":1464,"end":1490,"expression":"\\kappa_v+\\lambda\\epsilon_v","context":462,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L4-2"},{"line":13,"start":1499,"end":1507,"expression":"\\sigma_v","context":462,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L4-2"},{"line":16,"start":1577,"end":1619,"expression":"\nF_\\lambda(v)=\\kappa_v+\\lambda\\epsilon_v.\n","context":463,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L4-2"},{"line":20,"start":1644,"end":1766,"expression":"\n\\boxed{F_\\lambda(v)=\\min\\left\\{\\kappa_v+\\lambda\\epsilon_v,\n\\;\\sigma_v+\\sum_{w\\in\\mathrm{child}(v)}F_\\lambda(w)\\right\\}.}\n","context":463,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L4-2"},{"line":24,"start":1811,"end":1840,"expression":"F_\\lambda(v)\\leq J_\\lambda(L)","context":463,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L4-2"},{"line":24,"start":1970,"end":1982,"expression":"F_\\lambda(v)","context":464,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L4-2"},{"line":28,"start":2385,"end":2388,"expression":"r_j","context":465,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L4-2"},{"line":29,"start":2458,"end":2564,"expression":"\n\\kappa_v=\\kappa_v^{\\mathrm{setup}}+\\sum_jr_j\\kappa_v^{(j)},\n\\qquad \\epsilon_v=\\sum_jr_j\\epsilon_v^{(j)}.\n","context":465,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L4-2"},{"line":33,"start":2739,"end":2741,"expression":"\\Q","context":466,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L4-2"},{"line":33,"start":2748,"end":2750,"expression":"\\R","context":467,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L4-2"},{"line":40,"start":3179,"end":3186,"expression":"L_1,L_2","context":468,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L39-2"},{"line":40,"start":3209,"end":3233,"expression":"0\\leq\\lambda_1<\\lambda_2","context":468,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L39-2"},{"line":41,"start":3284,"end":3417,"expression":"\n\\cost(L_1)+\\lambda_1\\err(L_1)&\\leq\\cost(L_2)+\\lambda_1\\err(L_2),\\\\\n\\cost(L_2)+\\lambda_2\\err(L_2)&\\leq\\cost(L_1)+\\lambda_2\\err(L_1).\n","context":469,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L39-2"},{"line":46,"start":3491,"end":3556,"expression":"\n\\boxed{\\err(L_2)\\leq\\err(L_1),\\qquad \\cost(L_2)\\geq\\cost(L_1).}\n","context":470,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L39-2"},{"line":51,"start":3783,"end":3792,"expression":"\\lambda>0","context":471,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L39-2"},{"line":51,"start":3959,"end":3966,"expression":"\\lambda","context":471,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L39-2"},{"line":54,"start":4034,"end":4070,"expression":"\n(0,2),\\qquad (3/2,1),\\qquad (2,0),\n","context":472,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L39-2"},{"line":57,"start":4195,"end":4207,"expression":"\\lambda\\in\\Q","context":472,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L39-2"},{"line":58,"start":4212,"end":4245,"expression":"\n\\min\\{2\\lambda,2\\}<3/2+\\lambda.\n","context":472,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L39-2"},{"line":64,"start":4442,"end":4525,"expression":"\nK=\\kappa_v+\\lambda\\epsilon_v,\\qquad\nS=\\sigma_v+\\sum_wF_\\lambda(w),\\qquad g=|K-S|.\n","context":473,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L39-2"},{"line":69,"start":4560,"end":4591,"expression":"\n|K'-K|<g/2,\\qquad |S'-S|<g/2,\n","context":473,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L39-2"},{"line":73,"start":4608,"end":4657,"expression":"\nK'\\leq S'\\quad\\Longleftrightarrow\\quad K\\leq S.\n","context":473,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L39-2"},{"line":78,"start":5026,"end":5029,"expression":"g=0","context":474,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L39-2"},{"line":85,"start":5467,"end":5468,"expression":"v","context":475,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L82-2"},{"line":85,"start":5521,"end":5522,"expression":"j","context":475,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L82-2"},{"line":86,"start":5603,"end":5734,"expression":"\n\\boxed{\n\\operatorname{step}(v)=\n\\min\\left\\{K,\\;\\sigma+\n\\sum_i\\begin{cases}\nv,&i=j,\\\\\n\\widehat F_i,&i\\ne j.\n\\end{cases}\\right\\}.\n}\n","context":476,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L82-2"},{"line":96,"start":5879,"end":5891,"expression":"\\widehat F_i","context":477,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L82-2"},{"line":100,"start":6323,"end":6325,"expression":"T'","context":478,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L82-2"},{"line":100,"start":6346,"end":6347,"expression":"C","context":478,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L82-2"},{"line":101,"start":6364,"end":6451,"expression":"\n\\operatorname{updateValue}\\bigl(C,F(T')\\bigr)=F\\bigl(\\operatorname{fill}(C,T')\\bigr).\n","context":478,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L82-2"},{"line":115,"start":7704,"end":7705,"expression":"E","context":479,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L114-2"},{"line":116,"start":7747,"end":7782,"expression":"\ns_e>0,\\qquad c_e>0,\\qquad \\tau>0,\n","context":479,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L114-2"},{"line":119,"start":7797,"end":7808,"expression":"C=\\sum_ec_e","context":479,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L114-2"},{"line":120,"start":7841,"end":7958,"expression":"\n\\min_{\\varepsilon_e>0}\\sum_ec_e\\log\\frac1{\\varepsilon_e}\n\\qquad\\text{subject to}\\qquad\n\\sum_es_e\\varepsilon_e=\\tau.\n","context":479,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L114-2"},{"line":126,"start":8003,"end":8050,"expression":"\n\\boxed{\\varepsilon_e^*=\\frac{\\tau c_e}{s_eC}}\n","context":480,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L114-2"},{"line":131,"start":8252,"end":8263,"expression":"\\varepsilon","context":481,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L114-2"},{"line":132,"start":8271,"end":8306,"expression":"r_e=\\varepsilon_e/\\varepsilon_e^*>0","context":481,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L114-2"},{"line":133,"start":8338,"end":8355,"expression":"\n\\sum_ec_er_e=C.\n","context":481,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L114-2"},{"line":136,"start":8398,"end":8412,"expression":"\\log r\\leq r-1","context":481,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L114-2"},{"line":137,"start":8423,"end":8465,"expression":"\n\\sum_ec_e\\log r_e\\leq\\sum_ec_e(r_e-1)=0.\n","context":481,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L114-2"},{"line":141,"start":8533,"end":8586,"expression":"\nJ(\\varepsilon)-J(\\varepsilon^*)=-\\sum_ec_e\\log r_e.\n","context":481,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L114-2"},{"line":144,"start":8600,"end":8635,"expression":"J(\\varepsilon)\\geq J(\\varepsilon^*)","context":481,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L114-2"},{"line":146,"start":8685,"end":8692,"expression":"r_e\\ne1","context":482,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L114-2"},{"line":146,"start":8757,"end":8762,"expression":"c_e>0","context":482,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L114-2"},{"line":147,"start":8809,"end":8896,"expression":"\n\\varepsilon\\ne\\varepsilon^*\\quad\\Longrightarrow\\quad\nJ(\\varepsilon)>J(\\varepsilon^*).\n","context":482,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L114-2"},{"line":161,"start":9899,"end":9982,"expression":"\n\\coprod_{e\\in E}\\bigl(\\operatorname{Fin}(m_e)\\times\\operatorname{Fin}(n_e)\\bigr).\n","context":483,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L159-2"},{"line":170,"start":10323,"end":10324,"expression":"q","context":484,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L168"},{"line":172,"start":10358,"end":10361,"expression":"n^2","context":485,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L168"},{"line":172,"start":10366,"end":10371,"expression":"2qn^2","context":485,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L168"},{"line":173,"start":10392,"end":10404,"expression":"\\sum_em_en_e","context":486,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L168"},{"line":173,"start":10409,"end":10423,"expression":"2q\\sum_em_en_e","context":486,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L168"},{"line":174,"start":10452,"end":10454,"expression":"nb","context":487,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L168"},{"line":174,"start":10459,"end":10463,"expression":"2qnb","context":487,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L168"},{"line":175,"start":10475,"end":10491,"expression":"\\sum_i n_i^2+2nr","context":488,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L168"},{"line":175,"start":10496,"end":10516,"expression":"2q(\\sum_i n_i^2+2nr)","context":488,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L168"},{"line":176,"start":10540,"end":10547,"expression":"n(b+2r)","context":489,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L168"},{"line":176,"start":10552,"end":10561,"expression":"2qn(b+2r)","context":489,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L168"},{"line":177,"start":10589,"end":10594,"expression":"z+2nr","context":490,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L168"},{"line":177,"start":10599,"end":10608,"expression":"2q(z+2nr)","context":490,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L168"},{"line":181,"start":10694,"end":10700,"expression":"n=|I|b","context":491,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L181"},{"line":181,"start":10774,"end":10777,"expression":"n/b","context":491,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L181"},{"line":181,"start":10806,"end":10814,"expression":"n/(b+2r)","context":491,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L181"},{"line":183,"start":11088,"end":11089,"expression":"s","context":492,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L181"},{"line":184,"start":11112,"end":11161,"expression":"\nB_{\\mathrm{stream}}=s\\sum_e(m_en_e+n_eq+2m_eq).\n","context":492,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L181"},{"line":189,"start":11349,"end":11357,"expression":"F\\leq PT","context":493,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L181"},{"line":189,"start":11361,"end":11369,"expression":"B\\leq WT","context":493,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L181"},{"line":190,"start":11378,"end":11401,"expression":"\nT\\geq\\max\\{F/P,B/W\\}.\n","context":493,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L181"},{"line":193,"start":11459,"end":11480,"expression":"F/T\\leq\\min\\{P,WF/B\\}","context":493,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L181"},{"line":193,"start":11532,"end":11562,"expression":"n^2s\\leq M\\iff n\\leq\\sqrt{M/s}","context":493,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L181"},{"line":193,"start":11581,"end":11584,"expression":"n,M","context":493,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L181"},{"line":193,"start":11600,"end":11601,"expression":"s","context":493,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L181"},{"line":195,"start":11717,"end":11726,"expression":"2n^2r+n^2","context":494,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L181"},{"line":195,"start":11748,"end":11751,"expression":"n^2","context":494,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L181"},{"line":195,"start":11798,"end":11801,"expression":"n^2","context":494,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L181"},{"line":195,"start":11814,"end":11817,"expression":"n,r","context":494,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L181"},{"line":195,"start":11882,"end":11888,"expression":"O(n^2)","context":494,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L181"},{"line":203,"start":12263,"end":12402,"expression":"\n\\bigl(\\operatorname{decode}(\\operatorname{compressed}(x)),\n\\;t_{\\mathrm{run}}(x)+t_{\\mathrm{decode}}(\\operatorname{compressed}(x))\\bigr).\n","context":495,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L201-2"},{"line":211,"start":13243,"end":13244,"expression":"s","context":496,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L201-2"},{"line":211,"start":13262,"end":13265,"expression":"t_b","context":496,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L201-2"},{"line":211,"start":13284,"end":13287,"expression":"t_c","context":496,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L201-2"},{"line":211,"start":13295,"end":13302,"expression":"t_c<t_b","context":496,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L201-2"},{"line":212,"start":13319,"end":13390,"expression":"\n\\boxed{s+Rt_c<Rt_b\\quad\\Longleftrightarrow\\quad\nR>\\frac{s}{t_b-t_c}.}\n","context":496,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L201-2"},{"line":216,"start":13398,"end":13404,"expression":"s\\geq0","context":496,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L201-2"},{"line":217,"start":13474,"end":13538,"expression":"\n\\boxed{R_{\\min}=\\left\\lfloor\\frac{s}{t_b-t_c}\\right\\rfloor+1.}\n","context":496,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L201-2"},{"line":220,"start":13652,"end":13653,"expression":"6","context":497,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L201-2"},{"line":220,"start":13666,"end":13667,"expression":"5","context":498,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L201-2"},{"line":220,"start":13681,"end":13682,"expression":"3","context":499,"kind":"use","source":10,"reading_url":"read/paper-viii/appendix-compilation-and-costs.html#L201-2"},{"line":6,"start":645,"end":705,"expression":"\nA_j=D_j+U_jV_j^*,\\qquad U_j,V_j\\in\\mathbb C^{n\\times r_j},\n","context":500,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L4-2"},{"line":9,"start":718,"end":721,"expression":"D_j","context":500,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L4-2"},{"line":10,"start":760,"end":809,"expression":"\nP_m=A_m\\cdots A_1,\\qquad D^{(m)}=D_m\\cdots D_1.\n","context":500,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L4-2"},{"line":13,"start":856,"end":863,"expression":"L_m,R_m","context":500,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L4-2"},{"line":14,"start":877,"end":908,"expression":"\n\\boxed{P_m=D^{(m)}+L_mR_m^*.}\n","context":500,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L4-2"},{"line":17,"start":1014,"end":1030,"expression":"\\sum_{j=1}^m r_j","context":501,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L4-2"},{"line":19,"start":1066,"end":1100,"expression":"P_{m-1}=D^{(m-1)}+L_{m-1}R_{m-1}^*","context":502,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L4-2"},{"line":20,"start":1137,"end":1244,"expression":"\nL_m&=\\bigl[D_mL_{m-1}+U_m(V_m^*L_{m-1})\\quad U_m\\bigr],\\\\\nR_m&=\\bigl[R_{m-1}\\quad (D^{(m-1)})^*V_m\\bigr].\n","context":503,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L4-2"},{"line":27,"start":1495,"end":1558,"expression":"\n\\boxed{\\operatorname{rank}(P_m-D^{(m)})\\leq\\sum_{j=1}^m r_j.}\n","context":504,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L4-2"},{"line":30,"start":1679,"end":1686,"expression":"D^{(m)}","context":504,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L4-2"},{"line":30,"start":1798,"end":1803,"expression":"r_j=0","context":504,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L4-2"},{"line":35,"start":2192,"end":2197,"expression":"P_0=I","context":505,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L35"},{"line":35,"start":2204,"end":2222,"expression":"P_{j+1}=A_{j+1}P_j","context":505,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L35"},{"line":35,"start":2251,"end":2263,"expression":"\\widehat A_j","context":505,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L35"},{"line":36,"start":2305,"end":2400,"expression":"\n\\widehat P_{j+1}-P_{j+1}\n=\\widehat A_{j+1}(\\widehat P_j-P_j)\n +(\\widehat A_{j+1}-A_{j+1})P_j.\n","context":505,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L35"},{"line":41,"start":2596,"end":2597,"expression":"j","context":506,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L35"},{"line":41,"start":2648,"end":2651,"expression":"j+1","context":507,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L35"},{"line":46,"start":2782,"end":2802,"expression":"\\widehat A_j=A_j+E_j","context":508,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L45-2"},{"line":47,"start":2842,"end":3025,"expression":"\n\\boxed{\n\\norm{\\widehat A_m\\cdots\\widehat A_1-A_m\\cdots A_1}\n\\leq\\sum_{j=1}^m\n \\left(\\prod_{i=j+1}^m\\norm{\\widehat A_i}\\right)\n \\norm{E_j}\n \\left(\\prod_{i=1}^{j-1}\\norm{A_i}\\right).}\n","context":508,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L45-2"},{"line":55,"start":3208,"end":3230,"expression":"\\delta_j\\geq\\norm{E_j}","context":509,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L45-2"},{"line":58,"start":3428,"end":3495,"expression":"\n\\widetilde P_0=I,\\qquad\n\\widetilde P_j=A_j\\widetilde P_{j-1}+E_j.\n","context":510,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L45-2"},{"line":62,"start":3504,"end":3507,"expression":"E_j","context":510,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L45-2"},{"line":63,"start":3571,"end":3627,"expression":"\n\\widetilde P_j-P_j=A_j(\\widetilde P_{j-1}-P_{j-1})+E_j\n","context":510,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L45-2"},{"line":67,"start":3643,"end":3772,"expression":"\n\\boxed{\\norm{\\widetilde P_m-P_m}\n\\leq\\sum_{j=1}^m\\left(\\prod_{i=j+1}^m\\norm{A_i}\\right)\\delta_j,\n\\quad \\norm{E_j}\\leq\\delta_j.}\n","context":510,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L45-2"},{"line":75,"start":4069,"end":4182,"expression":"\nA_j=\\sum_e A_{j,e},\\qquad\n\\widehat A_j=\\sum_e(A_{j,e}+\\Delta_{j,e}),\\qquad\n\\Delta_{j,e}=\\Delta_{j,e}P_{j,a(e)}.\n","context":511,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L45-2"},{"line":81,"start":4255,"end":4338,"expression":"\n\\norm{\\widehat A_j-A_j}\\leq\n\\delta_j:=\\sum_e\\norm{\\Delta_{j,e}}\\norm{P_{j,a(e)}}.\n","context":511,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L45-2"},{"line":85,"start":4376,"end":4384,"expression":"\\delta_j","context":511,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L45-2"},{"line":85,"start":4496,"end":4504,"expression":"\\norm{x}","context":512,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L45-2"},{"line":93,"start":4949,"end":5013,"expression":"\n\\mathcal K_m(A,v)=\\operatorname{span}\\{v,Av,\\ldots,A^{m-1}v\\}.\n","context":513,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L91-2"},{"line":96,"start":5056,"end":5059,"expression":"P_i","context":513,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L91-2"},{"line":96,"start":5071,"end":5072,"expression":"A","context":513,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L91-2"},{"line":96,"start":5095,"end":5098,"expression":"P_i","context":513,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L91-2"},{"line":96,"start":5106,"end":5117,"expression":"A_i=P_iAP_i","context":513,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L91-2"},{"line":97,"start":5160,"end":5183,"expression":"\nA^jv=\\sum_i A_i^jP_iv\n","context":513,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L91-2"},{"line":101,"start":5202,"end":5273,"expression":"\n\\boxed{\\mathcal K_m(A,v)\\subseteq\n\\bigoplus_i\\mathcal K_m(A_i,P_iv).}\n","context":513,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L91-2"},{"line":105,"start":5286,"end":5289,"expression":"j=0","context":514,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L91-2"},{"line":105,"start":5297,"end":5308,"expression":"\\sum_iP_i=I","context":515,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L91-2"},{"line":105,"start":5371,"end":5375,"expression":"P_iv","context":516,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L91-2"},{"line":107,"start":5537,"end":5540,"expression":"P_i","context":517,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L91-2"},{"line":107,"start":5597,"end":5598,"expression":"i","context":517,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L91-2"},{"line":109,"start":5817,"end":5822,"expression":"A=D+K","context":518,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L91-2"},{"line":109,"start":5832,"end":5833,"expression":"D","context":518,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L91-2"},{"line":110,"start":5865,"end":5895,"expression":"\nM=\\max\\{\\norm{A},\\norm{D}\\}.\n","context":518,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L91-2"},{"line":113,"start":5926,"end":5927,"expression":"j","context":518,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L91-2"},{"line":114,"start":5957,"end":6003,"expression":"\n\\boxed{\\norm{A^j-D^j}\\leq jM^{j-1}\\norm{K}.}\n","context":518,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L91-2"},{"line":117,"start":6138,"end":6139,"expression":"M","context":518,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L91-2"},{"line":117,"start":6154,"end":6157,"expression":"M=0","context":518,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L91-2"},{"line":120,"start":6211,"end":6272,"expression":"\n\\mathcal L_{j+1}=\\bigoplus_i\\mathcal K_{j+1}(P_iDP_i,P_iv).\n","context":519,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L91-2"},{"line":123,"start":6362,"end":6385,"expression":"D^jv\\in\\mathcal L_{j+1}","context":519,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L91-2"},{"line":124,"start":6451,"end":6534,"expression":"\n\\boxed{\\operatorname{dist}(A^jv,\\mathcal L_{j+1})\n\\leq jM^{j-1}\\norm{K}\\norm{v}.}\n","context":519,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L91-2"},{"line":135,"start":7213,"end":7214,"expression":"S","context":520,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L132-2"},{"line":136,"start":7268,"end":7408,"expression":"\n\\begin{array}{ll}\n\\code{none}: & \\text{no cost satisfies }S,\\\\\n\\code{some}(v): & S(v)\\text{ and }v\\leq z\\text{ whenever }S(z).\n\\end{array}\n","context":521,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L132-2"},{"line":144,"start":7589,"end":7592,"expression":"F,G","context":522,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L132-2"},{"line":145,"start":7621,"end":7673,"expression":"\n(F\\star G)(b)=\\min_{0\\leq k\\leq b}\\{F(k)+G(b-k)\\},\n","context":522,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L132-2"},{"line":150,"start":8026,"end":8031,"expression":"(w,c)","context":523,"kind":"introduction-passage","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L132-2"},{"line":152,"start":8243,"end":8244,"expression":"D","context":524,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L132-2"},{"line":153,"start":8324,"end":8381,"expression":"\nF(b)=\\min_{(w,c)\\text{ allowed},\\;w\\leq b}\\{D(b-w)+c\\}.\n","context":524,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L132-2"},{"line":156,"start":8405,"end":8429,"expression":"\\min_{0\\leq b\\leq B}F(b)","context":524,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L132-2"},{"line":156,"start":8524,"end":8525,"expression":"B","context":524,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L132-2"},{"line":158,"start":8686,"end":8693,"expression":"w\\leq b","context":525,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L132-2"},{"line":167,"start":9601,"end":9602,"expression":"v","context":526,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L164-2"},{"line":167,"start":9611,"end":9619,"expression":"\\kappa_v","context":526,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L164-2"},{"line":167,"start":9647,"end":9650,"expression":"w_v","context":526,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L164-2"},{"line":167,"start":9679,"end":9687,"expression":"\\sigma_v","context":526,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L164-2"},{"line":168,"start":9762,"end":9952,"expression":"\nF_v(b)=\\min\\left\\{\n\\begin{array}{l}\n\\kappa_v\\quad\\text{if }b=w_v,\\\\[2pt]\n\\displaystyle\\sigma_v+\n\\min_{\\sum_i b_i=b}\\sum_iF_{v_i}(b_i)\\quad\\text{if }v\\text{ is internal}.\n\\end{array}\\right.\n","context":527,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L164-2"},{"line":178,"start":10175,"end":10176,"expression":"b","context":528,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L164-2"},{"line":178,"start":10197,"end":10203,"expression":"F_v(b)","context":528,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L164-2"},{"line":178,"start":10270,"end":10271,"expression":"b","context":528,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L164-2"},{"line":181,"start":10457,"end":10507,"expression":"\n\\boxed{\\min_{0\\leq b\\leq B}F_{\\mathrm{root}}(b)}\n","context":529,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L164-2"},{"line":187,"start":10833,"end":10865,"expression":"\n(0,4),\\qquad(3,2),\\qquad(4,0).\n","context":530,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L164-2"},{"line":190,"start":10951,"end":10958,"expression":"\\lambda","context":530,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L164-2"},{"line":190,"start":11019,"end":11033,"expression":"\\lambda\\geq3/2","context":530,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L164-2"},{"line":190,"start":11083,"end":11097,"expression":"\\lambda\\leq1/2","context":530,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L164-2"},{"line":197,"start":11980,"end":11983,"expression":"B+1","context":531,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L196-2"},{"line":201,"start":12361,"end":12368,"expression":"b\\leq B","context":532,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L196-2"},{"line":202,"start":12385,"end":12450,"expression":"\n\\operatorname{lookup}(\\operatorname{storedTree}(B,t),b)=F_t(b).\n","context":532,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L196-2"},{"line":205,"start":12565,"end":12566,"expression":"b","context":532,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L196-2"},{"line":205,"start":12607,"end":12608,"expression":"b","context":532,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L196-2"},{"line":208,"start":12862,"end":12910,"expression":"\n(b,k),\\qquad 0\\leq b\\leq B,\\quad0\\leq k\\leq b.\n","context":533,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L196-2"},{"line":212,"start":12996,"end":13027,"expression":"\n\\sum_{b=0}^B(b+1)\\leq(B+1)^2.\n","context":533,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L196-2"},{"line":215,"start":13117,"end":13118,"expression":"N","context":533,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L196-2"},{"line":217,"start":13183,"end":13189,"expression":"W_B(t)","context":534,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L196-2"},{"line":218,"start":13399,"end":13449,"expression":"\nW_B(t)\\leq2(B+1)N+\\bigl((B+1)+(B+1)^2\\bigr)(N-1)\n","context":534,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L196-2"},{"line":222,"start":13464,"end":13513,"expression":"\n\\boxed{W_B(t)\\leq N\\bigl(3(B+1)+(B+1)^2\\bigr).}\n","context":534,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L196-2"},{"line":225,"start":13618,"end":13619,"expression":"N","context":535,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L196-2"},{"line":225,"start":13645,"end":13646,"expression":"B","context":536,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L196-2"},{"line":227,"start":13982,"end":13983,"expression":"B","context":537,"kind":"use","source":11,"reading_url":"read/paper-viii/appendix-product-chains-and-budgets.html#L196-2"},{"line":5,"start":655,"end":658,"expression":"n>0","context":538,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L4-2"},{"line":5,"start":690,"end":710,"expression":"\\mathbb Z/n\\mathbb Z","context":538,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L4-2"},{"line":6,"start":746,"end":766,"expression":"\nC(c)_{jk}=c_{j-k}.\n","context":538,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L4-2"},{"line":10,"start":800,"end":892,"expression":"\nF_{jk}=\\frac{1}{\\sqrt n}\\,\\chi(-jk),\n\\qquad\n\\chi(a)=\\exp\\!\\left(\\frac{2\\pi i a}{n}\\right).\n","context":538,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L4-2"},{"line":16,"start":990,"end":1088,"expression":"\n\\frac{1}{\\sqrt n}\\exp\\!\\left(-\\frac{2\\pi i\\,\\operatorname{val}(j)\\operatorname{val}(k)}n\\right).\n","context":539,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L4-2"},{"line":22,"start":1364,"end":1399,"expression":"\nFx=n^{-1/2}\\operatorname{DFT}(x).\n","context":540,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L4-2"},{"line":25,"start":1428,"end":1432,"expression":"F^*x","context":540,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L4-2"},{"line":26,"start":1587,"end":1618,"expression":"\n\\boxed{F^*F=I,\\qquad FF^*=I.}\n","context":540,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L4-2"},{"line":32,"start":1747,"end":1825,"expression":"\n\\operatorname{DFT}(C(c)x)_k\n=\\operatorname{DFT}(c)_k\\operatorname{DFT}(x)_k.\n","context":541,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L4-2"},{"line":36,"start":1873,"end":1920,"expression":"FC(c)=\\operatorname{diag}(\\operatorname{DFT}c)F","context":541,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L4-2"},{"line":37,"start":1990,"end":2042,"expression":"\n\\boxed{C(c)=F^*\\operatorname{diag}(\\sqrt n\\,Fc)F.}\n","context":541,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L4-2"},{"line":40,"start":2144,"end":2147,"expression":"n=1","context":542,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L4-2"},{"line":42,"start":2296,"end":2306,"expression":"O(n\\log n)","context":543,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L4-2"},{"line":47,"start":2582,"end":2585,"expression":"n>0","context":544,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":48,"start":2597,"end":2665,"expression":"\nT_{ij}=\n\\begin{cases}\nc_{i-j},&j\\le i,\\\\\nr_{j-i},&i<j.\n\\end{cases}\n","context":545,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":55,"start":2697,"end":2700,"expression":"c_0","context":546,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":55,"start":2719,"end":2722,"expression":"r_0","context":546,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":55,"start":2772,"end":2774,"expression":"2n","context":546,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":56,"start":2781,"end":2827,"expression":"\ng=(c_0,\\ldots,c_{n-1},0,r_{n-1},\\ldots,r_1),\n","context":546,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":59,"start":2906,"end":2907,"expression":"n","context":546,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":62,"start":2967,"end":3004,"expression":"\nC(g)_{ij}=T_{ij}\\qquad(0\\le i,j<n).\n","context":547,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":65,"start":3012,"end":3018,"expression":"j\\le i","context":547,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":65,"start":3072,"end":3075,"expression":"i-j","context":547,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":65,"start":3083,"end":3086,"expression":"i<j","context":547,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":65,"start":3111,"end":3117,"expression":"2n+i-j","context":547,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":65,"start":3160,"end":3163,"expression":"j-i","context":547,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":68,"start":3309,"end":3352,"expression":"\n\\boxed{\\bigl(C(g)(x,0)^T\\bigr)_{0:n}=Tx.}\n","context":548,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":73,"start":3510,"end":3528,"expression":"A\\in\\C^{m\\times m}","context":549,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":73,"start":3532,"end":3550,"expression":"B\\in\\C^{n\\times n}","context":549,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":73,"start":3557,"end":3575,"expression":"X\\in\\C^{n\\times m}","context":549,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":74,"start":3592,"end":3662,"expression":"\n\\boxed{(A\\otimes B)\\operatorname{vec}(X)=\\operatorname{vec}(BXA^T).}\n","context":549,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":77,"start":3739,"end":3744,"expression":"(j,i)","context":549,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":77,"start":3754,"end":3760,"expression":"X_{ij}","context":549,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":77,"start":3879,"end":3881,"expression":"\\C","context":550,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":80,"start":3971,"end":4058,"expression":"\n\\boxed{2n^2m+2nm^2\\ \\text{modeled operations},\\qquad n^2+m^2\\ \\text{factor entries}.}\n","context":551,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":83,"start":4096,"end":4102,"expression":"n^2m^2","context":551,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L46-2"},{"line":88,"start":4346,"end":4364,"expression":"A_0,\\ldots,A_{k-1}","context":552,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L87-2"},{"line":88,"start":4394,"end":4395,"expression":"d","context":552,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L87-2"},{"line":88,"start":4457,"end":4458,"expression":"k","context":552,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L87-2"},{"line":88,"start":4471,"end":4487,"expression":"\\{0,\\ldots,d-1\\}","context":552,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L87-2"},{"line":88,"start":4578,"end":4581,"expression":"d^k","context":552,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L87-2"},{"line":91,"start":4669,"end":4725,"expression":"\nM_0=[1],\\qquad M_{k+1}=A_0\\otimes M_k(A_1,\\ldots,A_k).\n","context":553,"kind":"introduction-passage","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L87-2"},{"line":94,"start":4873,"end":4876,"expression":"A_0","context":553,"kind":"introduction-passage","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L87-2"},{"line":95,"start":4922,"end":5016,"expression":"\ny_{i,u}=\\sum_{j=0}^{d-1}(A_0)_{ij}\\,\n\\operatorname{Apply}_{k}(A_1,\\ldots,A_k;x_{j,\\cdot})_u.\n","context":553,"kind":"introduction-passage","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L87-2"},{"line":100,"start":5097,"end":5143,"expression":"\n\\boxed{\\operatorname{Apply}_k(A;x)=M_k(A)x.}\n","context":554,"kind":"introduction-passage","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L87-2"},{"line":105,"start":5279,"end":5282,"expression":"k+1","context":555,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L87-2"},{"line":105,"start":5314,"end":5315,"expression":"d","context":555,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L87-2"},{"line":106,"start":5423,"end":5460,"expression":"\nW_0=0,\\qquad W_{k+1}=dW_k+2d^{k+2}.\n","context":555,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L87-2"},{"line":110,"start":5493,"end":5517,"expression":"\n\\boxed{W_k=2kd^{k+1}.}\n","context":555,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L87-2"},{"line":114,"start":5562,"end":5578,"expression":"\n\\boxed{k d^2.}\n","context":555,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L87-2"},{"line":117,"start":5607,"end":5610,"expression":"k=0","context":555,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L87-2"},{"line":117,"start":5617,"end":5620,"expression":"d=0","context":555,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L87-2"},{"line":117,"start":5742,"end":5745,"expression":"d=0","context":556,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L87-2"},{"line":125,"start":6473,"end":6517,"expression":"\nH_2=\\begin{pmatrix}1&1\\\\1&-1\\end{pmatrix},\n","context":557,"kind":"introduction-passage","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L123-2"},{"line":128,"start":6555,"end":6562,"expression":"H_1=[1]","context":558,"kind":"introduction-passage","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L123-2"},{"line":129,"start":6570,"end":6641,"expression":"\nH_{2N}=H_2\\otimes H_N\n=\\begin{pmatrix}H_N&H_N\\\\H_N&-H_N\\end{pmatrix}.\n","context":559,"kind":"introduction-passage","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L123-2"},{"line":134,"start":6756,"end":6791,"expression":"\n\\boxed{H_NH_N^T=NI,\\qquad N=2^k.}\n","context":560,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L123-2"},{"line":137,"start":6906,"end":6908,"expression":"\\C","context":560,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L123-2"},{"line":139,"start":6959,"end":6966,"expression":"x_0,x_1","context":561,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L123-2"},{"line":140,"start":7028,"end":7150,"expression":"\n\\operatorname{Walsh}_{2N}(x_0,x_1)\n=\\bigl(\\operatorname{Walsh}_N(x_0+x_1),\n       \\operatorname{Walsh}_N(x_0-x_1)\\bigr).\n","context":561,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L123-2"},{"line":145,"start":7190,"end":7197,"expression":"H_{2N}x","context":561,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L123-2"},{"line":148,"start":7520,"end":7556,"expression":"\nA_0=0,\\qquad A_{k+1}=2A_k+2^{k+1}.\n","context":562,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L123-2"},{"line":152,"start":7629,"end":7657,"expression":"\n\\boxed{A_k=k2^k=N\\log_2N.}\n","context":562,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L123-2"},{"line":155,"start":7794,"end":7797,"expression":"N=1","context":563,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L123-2"},{"line":158,"start":7861,"end":7878,"expression":"\nH_N(H_N^Tx)=Nx,\n","context":564,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L123-2"},{"line":161,"start":7948,"end":7949,"expression":"N","context":564,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L123-2"},{"line":166,"start":8222,"end":8223,"expression":"E","context":565,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":167,"start":8264,"end":8329,"expression":"\ns_e>0,\\qquad c_e>0,\\qquad m_e>0,\n\\qquad 0<\\tau\\le\\sum_e s_em_e.\n","context":565,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":172,"start":8354,"end":8415,"expression":"\n0<\\varepsilon_e\\le m_e,\\qquad \\sum_e s_e\\varepsilon_e=\\tau,\n","context":565,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":176,"start":8490,"end":8539,"expression":"\nJ(\\varepsilon)=\\sum_e c_e\\log(1/\\varepsilon_e).\n","context":566,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":181,"start":8713,"end":8724,"expression":"t=1/\\lambda","context":567,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":182,"start":8736,"end":8798,"expression":"\nu_e(t)=\\min\\{m_e,tc_e/s_e\\},\\qquad\n\\Phi(t)=\\sum_e s_eu_e(t).\n","context":567,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":186,"start":8819,"end":8823,"expression":"\\Phi","context":567,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":186,"start":8864,"end":8873,"expression":"\\Phi(0)=0","context":567,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":187,"start":8907,"end":8936,"expression":"\nT=\\sum_e\\frac{m_es_e}{c_e},\n","context":567,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":190,"start":8974,"end":8980,"expression":"u(T)=m","context":567,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":190,"start":8987,"end":9007,"expression":"\\Phi(T)=\\sum_es_em_e","context":567,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":190,"start":9051,"end":9054,"expression":"t>0","context":567,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":190,"start":9062,"end":9074,"expression":"\\Phi(t)=\\tau","context":567,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":192,"start":9160,"end":9163,"expression":"t<u","context":568,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":192,"start":9170,"end":9177,"expression":"\\Phi(t)","context":568,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":192,"start":9244,"end":9245,"expression":"t","context":568,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":192,"start":9332,"end":9347,"expression":"\\Phi(t)<\\Phi(u)","context":568,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":193,"start":9442,"end":9539,"expression":"\n\\boxed{\\exists!\\lambda>0:\\quad\n\\sum_e s_e\\min\\!\\left\\{m_e,\\frac{c_e}{\\lambda s_e}\\right\\}=\\tau}\n","context":568,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":197,"start":9548,"end":9567,"expression":"0<\\tau<\\sum_es_em_e","context":569,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":199,"start":9688,"end":9695,"expression":"s=c=m=1","context":570,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":199,"start":9715,"end":9724,"expression":"\\lambda=1","context":570,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":199,"start":9731,"end":9742,"expression":"\\lambda=1/2","context":570,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":199,"start":9772,"end":9773,"expression":"1","context":570,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L165-2"},{"line":206,"start":10111,"end":10117,"expression":"u=u(t)","context":571,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L203-2"},{"line":206,"start":10163,"end":10174,"expression":"\\varepsilon","context":571,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L203-2"},{"line":207,"start":10212,"end":10307,"expression":"\n\\boxed{\n c_e\\left(\\frac{\\varepsilon_e}{u_e}-1\\right)\n \\le \\frac{s_e}{t}(\\varepsilon_e-u_e).\n}\n","context":571,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L203-2"},{"line":213,"start":10316,"end":10328,"expression":"u_e=tc_e/s_e","context":571,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L203-2"},{"line":213,"start":10355,"end":10362,"expression":"u_e=m_e","context":571,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L203-2"},{"line":213,"start":10395,"end":10411,"expression":"s_e/t\\le c_e/m_e","context":571,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L203-2"},{"line":213,"start":10421,"end":10442,"expression":"\\varepsilon_e-m_e\\le0","context":571,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L203-2"},{"line":216,"start":10587,"end":10682,"expression":"\n\\sum_e c_e\\left(\\frac{\\varepsilon_e}{u_e}-1\\right)\n\\le\\frac1t\\sum_e s_e(\\varepsilon_e-u_e)=0.\n","context":572,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L203-2"},{"line":220,"start":10718,"end":10731,"expression":"\\log z\\le z-1","context":572,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L203-2"},{"line":220,"start":10738,"end":10741,"expression":"z>0","context":572,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L203-2"},{"line":221,"start":10751,"end":10791,"expression":"\n\\sum_e c_e\\log(\\varepsilon_e/u_e)\\le0.\n","context":572,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L203-2"},{"line":225,"start":10802,"end":10860,"expression":"\nJ(\\varepsilon)-J(u)\n=-\\sum_e c_e\\log(\\varepsilon_e/u_e),\n","context":572,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L203-2"},{"line":229,"start":10874,"end":10896,"expression":"J(u)\\le J(\\varepsilon)","context":572,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L203-2"},{"line":231,"start":10934,"end":10950,"expression":"\\varepsilon\\ne u","context":573,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L203-2"},{"line":231,"start":10994,"end":10995,"expression":"1","context":573,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L203-2"},{"line":232,"start":11115,"end":11187,"expression":"\n\\boxed{\\varepsilon\\ne u\\quad\\Longrightarrow\\quad J(u)<J(\\varepsilon).}\n","context":573,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L203-2"},{"line":250,"start":13410,"end":13428,"expression":"3.34\\times10^{-14}","context":574,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L241-2"},{"line":250,"start":13452,"end":13470,"expression":"2.98\\times10^{-15}","context":574,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L241-2"},{"line":250,"start":13492,"end":13510,"expression":"2.04\\times10^{-13}","context":574,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L241-2"},{"line":250,"start":13553,"end":13561,"expression":"10^{-10}","context":574,"kind":"use","source":12,"reading_url":"read/paper-viii/appendix-transforms-and-precision.html#L241-2"},{"line":5,"start":683,"end":684,"expression":"V","context":575,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":5,"start":705,"end":711,"expression":"u\\to v","context":575,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":5,"start":729,"end":730,"expression":"v","context":575,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":5,"start":749,"end":750,"expression":"u","context":575,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":8,"start":944,"end":1025,"expression":"\n\\operatorname{Anc}(v)=\\{u:\\text{there is a positive-length path }u\\leadsto v\\}.\n","context":576,"kind":"introduction-passage","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":11,"start":1044,"end":1050,"expression":"u\\to v","context":576,"kind":"introduction-passage","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":12,"start":1077,"end":1132,"expression":"\n\\operatorname{Anc}(u)\\subsetneq\\operatorname{Anc}(v).\n","context":576,"kind":"introduction-passage","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":15,"start":1202,"end":1203,"expression":"u","context":576,"kind":"introduction-passage","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":15,"start":1224,"end":1225,"expression":"v","context":576,"kind":"introduction-passage","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":16,"start":1430,"end":1496,"expression":"\n\\boxed{\\ell(v)=1+\\max_{u\\to v}\\ell(u),\\qquad \\max\\varnothing=0.}\n","context":577,"kind":"introduction-passage","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":20,"start":1547,"end":1556,"expression":"\\ell(v)>0","context":578,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":20,"start":1563,"end":1578,"expression":"\\ell(u)<\\ell(v)","context":578,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":20,"start":1606,"end":1607,"expression":"n","context":578,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":20,"start":1621,"end":1622,"expression":"a","context":578,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":20,"start":1628,"end":1629,"expression":"b","context":578,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":21,"start":1643,"end":1666,"expression":"\n\\ell(a)+n\\leq\\ell(b).\n","context":578,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":24,"start":1751,"end":1760,"expression":"\\ell(v)-1","context":578,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":24,"start":1791,"end":1792,"expression":"v","context":578,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":26,"start":1888,"end":1903,"expression":"s:V\\to\\mathbb N","context":579,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":27,"start":2018,"end":2060,"expression":"\n\\ell(v)\\leq s(v)\\quad\\text{for every }v.\n","context":579,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":31,"start":2141,"end":2174,"expression":"\n\\boxed{d=\\max_{v\\in V}\\ell(v).}\n","context":579,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":34,"start":2200,"end":2201,"expression":"d","context":579,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":36,"start":2382,"end":2395,"expression":"V=\\varnothing","context":580,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L4-2"},{"line":41,"start":2964,"end":2965,"expression":"v","context":581,"kind":"introduction-passage","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L40-2"},{"line":41,"start":3002,"end":3005,"expression":"X_v","context":581,"kind":"introduction-passage","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L40-2"},{"line":42,"start":3069,"end":3150,"expression":"\nD_v:X_v\\overset{\\sim}{\\longrightarrow}X_v,\n\\qquad T_{vu}:X_u\\longrightarrow X_v\n","context":581,"kind":"introduction-passage","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L40-2"},{"line":47,"start":3276,"end":3315,"expression":"\n(Ax)_v=D_vx_v+\\sum_{u\\to v}T_{vu}x_u.\n","context":582,"kind":"introduction-passage","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L40-2"},{"line":53,"start":3462,"end":3526,"expression":"\n\\boxed{x_v=D_v^{-1}\\!\\left(b_v-\\sum_{u\\to v}T_{vu}x_u\\right).}\n","context":583,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L40-2"},{"line":56,"start":3539,"end":3542,"expression":"D_v","context":583,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L40-2"},{"line":56,"start":3577,"end":3578,"expression":"v","context":583,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L40-2"},{"line":56,"start":3723,"end":3726,"expression":"D_v","context":583,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L40-2"},{"line":56,"start":3749,"end":3750,"expression":"v","context":583,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L40-2"},{"line":58,"start":3942,"end":3951,"expression":"x^{(0)}=0","context":584,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L40-2"},{"line":59,"start":3959,"end":4119,"expression":"\nx^{(k+1)}_v=\n\\begin{cases}\nD_v^{-1}\\!\\left(b_v-\\displaystyle\\sum_{u\\to v}T_{vu}x^{(k)}_u\\right),\n  &\\ell(v)=k+1,\\\\[5pt]\nx^{(k)}_v,&\\ell(v)\\ne k+1.\n\\end{cases}\n","context":585,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L40-2"},{"line":68,"start":4212,"end":4267,"expression":"\n\\ell(v)\\leq k\\quad\\Longrightarrow\\quad x^{(k)}_v=x_v.\n","context":586,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L40-2"},{"line":71,"start":4300,"end":4303,"expression":"k+1","context":586,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L40-2"},{"line":71,"start":4331,"end":4332,"expression":"k","context":586,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L40-2"},{"line":71,"start":4423,"end":4424,"expression":"d","context":586,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L40-2"},{"line":72,"start":4440,"end":4461,"expression":"\n\\boxed{Ax^{(d)}=b.}\n","context":586,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L40-2"},{"line":82,"start":5137,"end":5142,"expression":"(P_a)","context":587,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":82,"start":5224,"end":5235,"expression":"a\\leadsto b","context":587,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":82,"start":5286,"end":5293,"expression":"P_bAP_a","context":587,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":84,"start":5393,"end":5394,"expression":"a","context":588,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":85,"start":5451,"end":5503,"expression":"\nW_a=\\{x:P_bx=0\\ \\text{whenever }a\\not\\leadsto b\\}.\n","context":588,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":88,"start":5519,"end":5522,"expression":"P_a","context":588,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":88,"start":5583,"end":5585,"expression":"Ax","context":588,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":88,"start":5736,"end":5753,"expression":"AW_a\\subseteq W_a","context":588,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":90,"start":5766,"end":5770,"expression":"BA=I","context":589,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":90,"start":5797,"end":5798,"expression":"A","context":589,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":90,"start":5828,"end":5831,"expression":"W_a","context":589,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":90,"start":5901,"end":5909,"expression":"x\\in W_a","context":589,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":90,"start":5920,"end":5928,"expression":"y\\in W_a","context":589,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":90,"start":5936,"end":5940,"expression":"Ay=x","context":589,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":90,"start":5949,"end":5964,"expression":"Bx=BAy=y\\in W_a","context":589,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":90,"start":5985,"end":5988,"expression":"P_a","context":589,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":91,"start":5999,"end":6060,"expression":"\n\\boxed{a\\not\\leadsto b\\quad\\Longrightarrow\\quad P_bBP_a=0.}\n","context":589,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":94,"start":6096,"end":6097,"expression":"B","context":589,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":97,"start":6287,"end":6322,"expression":"\nS=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR},\n","context":590,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":100,"start":6441,"end":6444,"expression":"j,k","context":590,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":100,"start":6503,"end":6514,"expression":"j\\leadsto k","context":590,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":100,"start":6544,"end":6550,"expression":"S_{ba}","context":590,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":101,"start":6598,"end":6653,"expression":"\na\\longrightarrow j\\ \\leadsto_{H}\\ k\\longrightarrow b.\n","context":590,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":107,"start":7022,"end":7110,"expression":"\nA=\\begin{pmatrix}1&1&1\\\\0&1&1\\\\0&0&1\\end{pmatrix}\n\\quad\\text{has}\\quad (A^{-1})_{13}=0\n","context":591,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":111,"start":7139,"end":7144,"expression":"(1,3)","context":592,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L81-2"},{"line":116,"start":7399,"end":7400,"expression":"p","context":593,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L115-2"},{"line":116,"start":7430,"end":7431,"expression":"w","context":593,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L115-2"},{"line":117,"start":7462,"end":7548,"expression":"\n\\mathcal B_{p,w}\n=\\{(i,j)\\in\\{0,\\ldots,p-1\\}^2:\n i\\leq j+w\\ \\text{and}\\ j\\leq i+w\\}.\n","context":593,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L115-2"},{"line":122,"start":7592,"end":7603,"expression":"|i-j|\\leq w","context":593,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L115-2"},{"line":124,"start":7771,"end":7780,"expression":"0\\leq w<p","context":594,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L115-2"},{"line":125,"start":7814,"end":7866,"expression":"\n\\boxed{N_{p,w}=|\\mathcal B_{p,w}|=p(2w+1)-w(w+1).}\n","context":594,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L115-2"},{"line":128,"start":7956,"end":7957,"expression":"p","context":594,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L115-2"},{"line":128,"start":8141,"end":8144,"expression":"w=0","context":594,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L115-2"},{"line":128,"start":8161,"end":8162,"expression":"p","context":594,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L115-2"},{"line":128,"start":8196,"end":8201,"expression":"w=p-1","context":595,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L115-2"},{"line":128,"start":8210,"end":8213,"expression":"p^2","context":596,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L115-2"},{"line":130,"start":8266,"end":8267,"expression":"b","context":597,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L115-2"},{"line":131,"start":8300,"end":8354,"expression":"\n\\mathcal S=\\mathcal B_{p,w}\\times\\{0,\\ldots,b-1\\}^2.\n","context":597,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L115-2"},{"line":135,"start":8427,"end":8461,"expression":"\n\\boxed{|\\mathcal S|=b^2N_{p,w}.}\n","context":597,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L115-2"},{"line":139,"start":8498,"end":8499,"expression":"q","context":598,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L115-2"},{"line":140,"start":8638,"end":8697,"expression":"\n\\mathcal O=\\{0,1\\}\\times\\{0,\\ldots,q-1\\}\\times\\mathcal S.\n","context":598,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L115-2"},{"line":144,"start":8708,"end":8744,"expression":"\n\\boxed{|\\mathcal O|=2qb^2N_{p,w}.}\n","context":598,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L115-2"},{"line":147,"start":8897,"end":8902,"expression":"p=n/b","context":599,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L115-2"},{"line":147,"start":8934,"end":8937,"expression":"b>0","context":600,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L115-2"},{"line":147,"start":8944,"end":8948,"expression":"n=pb","context":601,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L115-2"},{"line":154,"start":9497,"end":9502,"expression":"(P_a)","context":602,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L153-2"},{"line":154,"start":9651,"end":9652,"expression":"A","context":602,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L153-2"},{"line":155,"start":9664,"end":9706,"expression":"\n\\Pi(A)=\\sum_aP_aAP_a,\n\\qquad E=A-\\Pi(A).\n","context":602,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L153-2"},{"line":160,"start":9764,"end":9789,"expression":"\nE=\\sum_{b\\ne a}P_bAP_a.\n","context":603,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L153-2"},{"line":164,"start":9869,"end":9882,"expression":"(b,a)\\ne(d,c)","context":604,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L153-2"},{"line":165,"start":9943,"end":9995,"expression":"\n\\operatorname{tr}\\bigl((P_bAP_a)^*P_dBP_c\\bigr)=0.\n","context":604,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L153-2"},{"line":168,"start":10002,"end":10008,"expression":"b\\ne d","context":604,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L153-2"},{"line":168,"start":10062,"end":10068,"expression":"a\\ne c","context":604,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L153-2"},{"line":168,"start":10102,"end":10108,"expression":"P_cP_a","context":604,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L153-2"},{"line":168,"start":10165,"end":10168,"expression":"A,B","context":604,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L153-2"},{"line":171,"start":10243,"end":10305,"expression":"\n\\norm{M}_{\\rm F}^2=\\operatorname{Re}\\operatorname{tr}(M^*M),\n","context":605,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L153-2"},{"line":175,"start":10381,"end":10456,"expression":"\n\\boxed{\\norm{A-\\Pi(A)}_{\\rm F}^2\n=\\sum_{b\\ne a}\\norm{P_bAP_a}_{\\rm F}^2.}\n","context":605,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L153-2"},{"line":182,"start":10610,"end":10672,"expression":"\n\\boxed{\\norm{A-\\Pi(A)}_2\n\\leq\\sum_{b\\ne a}\\norm{P_bAP_a}_2.}\n","context":606,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L153-2"},{"line":186,"start":10681,"end":10695,"expression":"\\norm{\\cdot}_2","context":606,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L153-2"},{"line":193,"start":11592,"end":11593,"expression":"D","context":607,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":193,"start":11600,"end":11605,"expression":"A=D+K","context":607,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":194,"start":11639,"end":11684,"expression":"\nS=(\\sqrt D)^{-1},\\qquad B=SAS,\\qquad E=SKS.\n","context":607,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":197,"start":11860,"end":11867,"expression":"\\sqrt D","context":608,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":198,"start":11928,"end":11949,"expression":"\nSDS=I,\\qquad B=I+E.\n","context":609,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":202,"start":11990,"end":12002,"expression":"C=(I+E)^{-1}","context":610,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":202,"start":12009,"end":12028,"expression":"\\norm E_2\\leq\\eta<1","context":610,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":202,"start":12037,"end":12043,"expression":"C=I-EC","context":610,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":203,"start":12061,"end":12119,"expression":"\n(1-\\eta)\\norm C_2\\leq 1,\n\\qquad \\norm{I+E}_2\\leq 1+\\eta.\n","context":610,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":207,"start":12175,"end":12176,"expression":"S","context":610,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":207,"start":12183,"end":12184,"expression":"A","context":610,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":208,"start":12257,"end":12331,"expression":"\n\\boxed{\\kappa_2(B):=\\norm B_2\\norm{B^{-1}}_2\n\\leq\\frac{1+\\eta}{1-\\eta}.}\n","context":610,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":212,"start":12428,"end":12432,"expression":"\\eta","context":611,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":215,"start":12632,"end":12680,"expression":"\nq(\\kappa)=\\frac{\\sqrt\\kappa-1}{\\sqrt\\kappa+1}.\n","context":612,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":218,"start":12749,"end":12750,"expression":"q","context":612,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":218,"start":12783,"end":12799,"expression":"\\kappa_2(B)\\geq1","context":612,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":218,"start":12803,"end":12817,"expression":"q(\\kappa)\\geq0","context":612,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":218,"start":12824,"end":12835,"expression":"\\kappa\\geq1","context":612,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":218,"start":12843,"end":12854,"expression":"q(\\kappa)<1","context":612,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":219,"start":12864,"end":12931,"expression":"\n\\boxed{q(\\kappa_2(B))\\leq q\\!\\left(\\frac{1+\\eta}{1-\\eta}\\right).}\n","context":612,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":222,"start":12939,"end":12950,"expression":"1\\leq\\kappa","context":612,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":222,"start":12957,"end":12965,"expression":"e_0\\geq0","context":612,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":222,"start":12999,"end":13022,"expression":"e_m\\leq2q(\\kappa)^m e_0","context":613,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":222,"start":13053,"end":13073,"expression":"\\kappa\\leq\\bar\\kappa","context":614,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":222,"start":13082,"end":13109,"expression":"e_m\\leq2q(\\bar\\kappa)^m e_0","context":615,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L192-2"},{"line":239,"start":15792,"end":15810,"expression":"1.42\\times10^{-13}","context":616,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L228-2"},{"line":239,"start":15865,"end":15883,"expression":"6.66\\times10^{-15}","context":616,"kind":"use","source":13,"reading_url":"read/paper-viii/appendix-graph-schedules.html#L228-2"},{"line":6,"start":688,"end":749,"expression":"\nL\\in\\C^{m\\times t},\\qquad R\\in\\C^{n\\times t},\\qquad A=LR^*.\n","context":617,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L4-2"},{"line":10,"start":829,"end":933,"expression":"\nS_L=\\operatorname{range}(L),\\quad r_L=\\dim S_L,\n\\qquad S_R=\\operatorname{range}(R),\\quad r_R=\\dim S_R.\n","context":617,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L4-2"},{"line":14,"start":1023,"end":1026,"expression":"Q_L","context":618,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L4-2"},{"line":14,"start":1033,"end":1036,"expression":"Q_R","context":619,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L4-2"},{"line":14,"start":1163,"end":1170,"expression":"R_L,R_R","context":620,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L4-2"},{"line":17,"start":1203,"end":1256,"expression":"\n\\boxed{Q_L^*Q_L=I,\\quad L=Q_LR_L,\\quad R_L=Q_L^*L,}\n","context":621,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L4-2"},{"line":21,"start":1344,"end":1385,"expression":"\n\\boxed{r_L=\\rank L,\\qquad r_R=\\rank R.}\n","context":621,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L4-2"},{"line":27,"start":1594,"end":1628,"expression":"\nG=R_LR_R^*\\in\\C^{r_L\\times r_R}.\n","context":622,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L4-2"},{"line":31,"start":1689,"end":1734,"expression":"\n\\boxed{A=Q_LGQ_R^*,\\qquad \\rank G=\\rank A.}\n","context":622,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L4-2"},{"line":43,"start":2934,"end":2937,"expression":"Q,V","context":623,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L42-2"},{"line":44,"start":2948,"end":3025,"expression":"\n\\mathcal L(G)=QGV^*,\\qquad \\mathcal C(B)=Q^*BV,\n\\qquad Q^*Q=I,\\quad V^*V=I.\n","context":623,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L42-2"},{"line":49,"start":3118,"end":3221,"expression":"\n\\mathcal C(\\mathcal L(G))=G,\n\\qquad \\rank\\mathcal L(G)=\\rank G,\n\\qquad \\rank\\mathcal C(B)\\leq\\rank B,\n","context":623,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L42-2"},{"line":54,"start":3226,"end":3351,"expression":"\n\\boxed{\\norm{\\mathcal L(G)}_\\nu=\\norm{G}_\\nu,\n\\qquad \\norm{\\mathcal C(B)}_\\nu\\leq\\norm B_\\nu,\n\\quad \\nu\\in\\{2,\\mathrm F\\}.}\n","context":624,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L42-2"},{"line":60,"start":3419,"end":3436,"expression":"\n(QG)^*(QG)=G^*G\n","context":625,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L42-2"},{"line":63,"start":3448,"end":3451,"expression":"C^*","context":626,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L42-2"},{"line":66,"start":3754,"end":3840,"expression":"\n\\norm{B-QQ^*B}_{\\mathrm F}^{2}\n=\\norm B_{\\mathrm F}^{2}-\\norm{Q^*B}_{\\mathrm F}^{2}.\n","context":627,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L42-2"},{"line":72,"start":4023,"end":4026,"expression":"G_k","context":628,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L42-2"},{"line":72,"start":4052,"end":4053,"expression":"k","context":628,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L42-2"},{"line":72,"start":4073,"end":4074,"expression":"G","context":628,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L42-2"},{"line":72,"start":4133,"end":4134,"expression":"B","context":628,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L42-2"},{"line":72,"start":4142,"end":4155,"expression":"\\rank B\\leq k","context":628,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L42-2"},{"line":73,"start":4215,"end":4418,"expression":"\n\\begin{aligned}\n\\norm{\\mathcal L(G)-\\mathcal L(G_k)}_\\nu\n&=\\norm{G-G_k}_\\nu\\\\\n&\\leq\\norm{G-\\mathcal C(B)}_\\nu\\\\\n&=\\norm{\\mathcal C(\\mathcal L(G)-B)}_\\nu\\\\\n&\\leq\\norm{\\mathcal L(G)-B}_\\nu.\n\\end{aligned}\n","context":629,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L42-2"},{"line":82,"start":4453,"end":4454,"expression":"B","context":630,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L42-2"},{"line":87,"start":4812,"end":4847,"expression":"\\sigma_1\\geq\\cdots\\geq\\sigma_d\\geq0","context":631,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":88,"start":4855,"end":4955,"expression":"\nD=\\diag(\\sigma_1,\\ldots,\\sigma_d),\\qquad\nD_k=\\diag(\\sigma_1,\\ldots,\\sigma_{\\min(k,d)},0,\\ldots,0).\n","context":631,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":92,"start":5075,"end":5078,"expression":"D_k","context":632,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":92,"start":5105,"end":5106,"expression":"k","context":633,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":94,"start":5160,"end":5163,"expression":"k<d","context":634,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":94,"start":5176,"end":5185,"expression":"d\\times d","context":634,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":94,"start":5195,"end":5196,"expression":"B","context":634,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":94,"start":5215,"end":5216,"expression":"k","context":634,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":94,"start":5253,"end":5256,"expression":"k+1","context":634,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":94,"start":5309,"end":5310,"expression":"J","context":634,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":95,"start":5321,"end":5354,"expression":"\n\\rank(BJ)\\leq\\rank B\\leq k<k+1,\n","context":634,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":98,"start":5397,"end":5398,"expression":"z","context":634,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":98,"start":5406,"end":5411,"expression":"BJz=0","context":634,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":98,"start":5426,"end":5430,"expression":"y=Jz","context":634,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":98,"start":5455,"end":5459,"expression":"By=0","context":634,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":98,"start":5492,"end":5495,"expression":"k+1","context":634,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":101,"start":5619,"end":5706,"expression":"\n\\norm{Dy}^{2}\n=\\sum_{i=1}^{k+1}\\sigma_i^{2}|y_i|^{2}\n\\geq\\sigma_{k+1}^{2}\\norm y^{2}.\n","context":635,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":106,"start":5718,"end":5727,"expression":"(D-B)y=Dy","context":635,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":107,"start":5769,"end":5833,"expression":"\n\\sigma_{k+1}\\norm y\n\\leq\\norm{(D-B)y}\n\\leq\\norm{D-B}_2\\norm y.\n","context":635,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":112,"start":5883,"end":5884,"expression":"y","context":635,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":113,"start":5895,"end":5934,"expression":"\n\\boxed{\\sigma_{k+1}\\leq\\norm{D-B}_2.}\n","context":635,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":116,"start":5979,"end":5984,"expression":"D-D_k","context":636,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":116,"start":5994,"end":6006,"expression":"\\sigma_{k+1}","context":637,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":116,"start":6013,"end":6016,"expression":"D_k","context":638,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":116,"start":6047,"end":6054,"expression":"k\\geq d","context":639,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L86-2"},{"line":123,"start":6605,"end":6606,"expression":"B","context":640,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":123,"start":6614,"end":6627,"expression":"\\rank B\\leq k","context":640,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":123,"start":6663,"end":6664,"expression":"Q","context":640,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":124,"start":6701,"end":6752,"expression":"\nBQ=0,\\qquad Q^*Q=I,\\qquad q=\\dim\\ker B=d-\\rank B.\n","context":640,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":128,"start":6780,"end":6811,"expression":"\nw_i=\\sum_{j=1}^{q}|Q_{ij}|^2.\n","context":640,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":132,"start":6834,"end":6893,"expression":"\n\\boxed{0\\leq w_i\\leq1,\\qquad \\sum_iw_i=q\\geq\\max(d-k,0).}\n","context":640,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":135,"start":6933,"end":6936,"expression":"Q^*","context":641,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":135,"start":7004,"end":7008,"expression":"Q^*Q","context":642,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":137,"start":7070,"end":7073,"expression":"a_i","context":643,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":138,"start":7116,"end":7149,"expression":"\n\\sum_i a_iw_i\\geq\\sum_{i>k}a_i.\n","context":643,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":141,"start":7207,"end":7210,"expression":"k<d","context":643,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":141,"start":7218,"end":7229,"expression":"\\mu=a_{k+1}","context":643,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":141,"start":7273,"end":7278,"expression":"v_i=0","context":643,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":141,"start":7285,"end":7292,"expression":"i\\leq k","context":643,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":141,"start":7299,"end":7304,"expression":"v_i=1","context":643,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":142,"start":7319,"end":7352,"expression":"\na_iv_i+\\mu(w_i-v_i)\\leq a_iw_i.\n","context":643,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":145,"start":7402,"end":7412,"expression":"a_i\\geq\\mu","context":643,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":145,"start":7419,"end":7427,"expression":"w_i\\geq0","context":643,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":145,"start":7461,"end":7471,"expression":"a_i\\leq\\mu","context":644,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":145,"start":7478,"end":7486,"expression":"w_i\\leq1","context":645,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":145,"start":7504,"end":7524,"expression":"\\sum w_i\\geq\\sum v_i","context":646,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":147,"start":7549,"end":7563,"expression":"a_i=\\sigma_i^2","context":647,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":147,"start":7625,"end":7629,"expression":"BQ=0","context":647,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":148,"start":7640,"end":7810,"expression":"\n\\begin{aligned}\n\\norm{D-B}_{\\mathrm F}^{2}\n&\\geq\\norm{(D-B)Q}_{\\mathrm F}^{2}\\\\\n&=\\norm{DQ}_{\\mathrm F}^{2}\n=\\sum_i\\sigma_i^2w_i\n\\geq\\sum_{i>k}\\sigma_i^2.\n\\end{aligned}\n","context":648,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":157,"start":7836,"end":7841,"expression":"D-D_k","context":649,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L122-2"},{"line":162,"start":8232,"end":8242,"expression":"G=R_LR_R^*","context":650,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L161-2"},{"line":163,"start":8276,"end":8370,"expression":"\nG=W\\diag(\\sigma)Z^*,\\qquad W^*W=I,\n\\qquad Z^*Z=I,\\qquad \\sigma_1\\geq\\cdots\\geq\\sigma_d\\geq0.\n","context":650,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L161-2"},{"line":168,"start":8542,"end":8587,"expression":"\nG^*GZ=Z\\diag(\\sigma_1^2,\\ldots,\\sigma_d^2).\n","context":651,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L161-2"},{"line":171,"start":8595,"end":8606,"expression":"h=\\min(d,k)","context":652,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L161-2"},{"line":171,"start":8638,"end":8639,"expression":"h","context":653,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L161-2"},{"line":172,"start":8704,"end":8812,"expression":"\nM_{ij}=\\begin{cases}\\sqrt{\\sigma_j},&i=j,\\\\0,&i\\ne j,\\end{cases}\n\\qquad 1\\leq i\\leq d,\\quad 1\\leq j\\leq h.\n","context":654,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L161-2"},{"line":176,"start":8854,"end":8862,"expression":"MM^*=D_k","context":655,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L161-2"},{"line":177,"start":8874,"end":8911,"expression":"\n\\boxed{L_k=Q_LWM,\\qquad R_k=Q_RZM.}\n","context":655,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L161-2"},{"line":180,"start":8934,"end":8949,"expression":"Q_LWD_kZ^*Q_R^*","context":655,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L161-2"},{"line":180,"start":8973,"end":8974,"expression":"k","context":655,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L161-2"},{"line":180,"start":9026,"end":9027,"expression":"B","context":655,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L161-2"},{"line":180,"start":9035,"end":9048,"expression":"\\rank B\\leq k","context":655,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L161-2"},{"line":181,"start":9053,"end":9137,"expression":"\n\\boxed{\\norm{LR^*-L_kR_k^*}_\\nu\\leq\\norm{LR^*-B}_\\nu,\n\\quad\\nu\\in\\{2,\\mathrm F\\}.}\n","context":655,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L161-2"},{"line":186,"start":9173,"end":9336,"expression":"\n\\norm{LR^*-L_kR_k^*}_2=\n\\begin{cases}\\sigma_{k+1},&k<d,\\\\0,&k\\geq d,\\end{cases}\n\\qquad\n\\norm{LR^*-L_kR_k^*}_{\\mathrm F}\n=\\left(\\sum_{i>k}\\sigma_i^2\\right)^{1/2}.\n","context":656,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L161-2"},{"line":196,"start":9572,"end":9579,"expression":"k\\geq d","context":657,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L161-2"},{"line":238,"start":13242,"end":13259,"expression":"3.7\\times10^{-15}","context":658,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L235-2"},{"line":238,"start":13282,"end":13296,"expression":"2\\times10^{-9}","context":658,"kind":"use","source":14,"reading_url":"read/paper-viii/appendix-small-core-recompression.html#L235-2"},{"line":28,"start":1666,"end":1667,"expression":"R","context":659,"kind":"use","source":15},{"line":33,"start":2329,"end":2331,"expression":"7R","context":660,"kind":"use","source":15},{"line":34,"start":2391,"end":2396,"expression":"30+2R","context":661,"kind":"use","source":15},{"line":35,"start":2455,"end":2456,"expression":"6","context":662,"kind":"use","source":15},{"line":35,"start":2473,"end":2474,"expression":"7","context":663,"kind":"use","source":15},{"line":3,"start":190,"end":196,"expression":"\\times","context":664,"kind":"use","source":16},{"line":5,"start":330,"end":331,"expression":"1","context":665,"kind":"use","source":16},{"line":7,"start":464,"end":465,"expression":"1","context":666,"kind":"use","source":16},{"line":9,"start":599,"end":600,"expression":"1","context":667,"kind":"use","source":16},{"line":11,"start":734,"end":735,"expression":"1","context":668,"kind":"use","source":16},{"line":13,"start":872,"end":878,"expression":"\\times","context":669,"kind":"use","source":16},{"line":15,"start":1012,"end":1013,"expression":"1","context":670,"kind":"use","source":16},{"line":17,"start":1146,"end":1147,"expression":"1","context":671,"kind":"use","source":16},{"line":19,"start":1277,"end":1278,"expression":"1","context":672,"kind":"use","source":16},{"line":21,"start":1412,"end":1413,"expression":"1","context":673,"kind":"use","source":16},{"line":23,"start":1547,"end":1548,"expression":"1","context":674,"kind":"use","source":16},{"line":25,"start":1679,"end":1680,"expression":"1","context":675,"kind":"use","source":16},{"line":27,"start":1817,"end":1823,"expression":"\\times","context":676,"kind":"use","source":16},{"line":29,"start":1957,"end":1958,"expression":"1","context":677,"kind":"use","source":16},{"line":31,"start":2091,"end":2092,"expression":"1","context":678,"kind":"use","source":16},{"line":33,"start":2226,"end":2227,"expression":"1","context":679,"kind":"use","source":16},{"line":35,"start":2361,"end":2362,"expression":"1","context":680,"kind":"use","source":16},{"line":37,"start":2496,"end":2497,"expression":"1","context":681,"kind":"use","source":16},{"line":39,"start":2631,"end":2632,"expression":"1","context":682,"kind":"use","source":16},{"line":41,"start":2769,"end":2775,"expression":"\\times","context":683,"kind":"use","source":16},{"line":43,"start":2909,"end":2910,"expression":"1","context":684,"kind":"use","source":16},{"line":45,"start":3043,"end":3044,"expression":"1","context":685,"kind":"use","source":16},{"line":47,"start":3178,"end":3179,"expression":"1","context":686,"kind":"use","source":16},{"line":49,"start":3313,"end":3314,"expression":"1","context":687,"kind":"use","source":16},{"line":51,"start":3451,"end":3457,"expression":"\\times","context":688,"kind":"use","source":16},{"line":53,"start":3591,"end":3592,"expression":"1","context":689,"kind":"use","source":16},{"line":55,"start":3727,"end":3728,"expression":"1","context":690,"kind":"use","source":16},{"line":57,"start":3862,"end":3863,"expression":"1","context":691,"kind":"use","source":16},{"line":59,"start":3999,"end":4000,"expression":"1","context":692,"kind":"use","source":16},{"line":61,"start":4134,"end":4135,"expression":"1","context":693,"kind":"use","source":16},{"line":63,"start":4271,"end":4272,"expression":"1","context":694,"kind":"use","source":16},{"line":65,"start":4411,"end":4417,"expression":"\\times","context":695,"kind":"use","source":16},{"line":67,"start":4554,"end":4555,"expression":"1","context":696,"kind":"use","source":16},{"line":69,"start":4691,"end":4692,"expression":"1","context":697,"kind":"use","source":16},{"line":71,"start":4827,"end":4828,"expression":"1","context":698,"kind":"use","source":16},{"line":73,"start":4965,"end":4966,"expression":"1","context":699,"kind":"use","source":16},{"line":75,"start":5103,"end":5104,"expression":"1","context":700,"kind":"use","source":16},{"line":77,"start":5240,"end":5241,"expression":"1","context":701,"kind":"use","source":16},{"line":3,"start":189,"end":192,"expression":"U_C","context":702,"kind":"use","source":17},{"line":5,"start":324,"end":325,"expression":"6","context":703,"kind":"use","source":17},{"line":7,"start":458,"end":459,"expression":"1","context":704,"kind":"use","source":17},{"line":9,"start":592,"end":593,"expression":"4","context":705,"kind":"use","source":17},{"line":11,"start":727,"end":728,"expression":"1","context":706,"kind":"use","source":17},{"line":13,"start":860,"end":861,"expression":"6","context":707,"kind":"use","source":17},{"line":15,"start":994,"end":995,"expression":"1","context":708,"kind":"use","source":17},{"line":17,"start":1128,"end":1129,"expression":"4","context":709,"kind":"use","source":17},{"line":19,"start":1263,"end":1264,"expression":"1","context":710,"kind":"use","source":17},{"line":21,"start":1396,"end":1397,"expression":"6","context":711,"kind":"use","source":17},{"line":23,"start":1530,"end":1531,"expression":"1","context":712,"kind":"use","source":17},{"line":25,"start":1663,"end":1664,"expression":"4","context":713,"kind":"use","source":17},{"line":27,"start":1797,"end":1798,"expression":"1","context":714,"kind":"use","source":17},{"line":28,"start":1862,"end":1865,"expression":"V_C","context":715,"kind":"use","source":17},{"line":30,"start":1995,"end":1996,"expression":"0","context":716,"kind":"use","source":17},{"line":32,"start":2129,"end":2130,"expression":"1","context":717,"kind":"use","source":17},{"line":34,"start":2263,"end":2264,"expression":"1","context":718,"kind":"use","source":17},{"line":36,"start":2398,"end":2399,"expression":"2","context":719,"kind":"use","source":17},{"line":38,"start":2529,"end":2530,"expression":"0","context":720,"kind":"use","source":17},{"line":40,"start":2663,"end":2664,"expression":"3","context":721,"kind":"use","source":17},{"line":42,"start":2797,"end":2798,"expression":"1","context":722,"kind":"use","source":17},{"line":44,"start":2932,"end":2933,"expression":"4","context":723,"kind":"use","source":17},{"line":46,"start":3063,"end":3064,"expression":"0","context":724,"kind":"use","source":17},{"line":48,"start":3197,"end":3198,"expression":"5","context":725,"kind":"use","source":17},{"line":50,"start":3330,"end":3331,"expression":"1","context":726,"kind":"use","source":17},{"line":52,"start":3464,"end":3465,"expression":"6","context":727,"kind":"use","source":17},{"line":53,"start":3518,"end":3536,"expression":"AB=D_AD_B+U_CV_C^T","context":728,"kind":"use","source":17},{"line":54,"start":3589,"end":3607,"expression":"U_A(V_A^TU_B)V_B^T","context":729,"kind":"use","source":17},{"line":3,"start":189,"end":193,"expression":"100E","context":730,"kind":"use","source":18},{"line":5,"start":347,"end":348,"expression":"0","context":731,"kind":"use","source":18},{"line":7,"start":481,"end":482,"expression":"1","context":732,"kind":"use","source":18},{"line":9,"start":615,"end":616,"expression":"1","context":733,"kind":"use","source":18},{"line":11,"start":749,"end":750,"expression":"1","context":734,"kind":"use","source":18},{"line":13,"start":883,"end":884,"expression":"1","context":735,"kind":"use","source":18},{"line":15,"start":1016,"end":1017,"expression":"0","context":736,"kind":"use","source":18},{"line":17,"start":1151,"end":1152,"expression":"1","context":737,"kind":"use","source":18},{"line":19,"start":1286,"end":1287,"expression":"1","context":738,"kind":"use","source":18},{"line":21,"start":1420,"end":1421,"expression":"1","context":739,"kind":"use","source":18},{"line":23,"start":1555,"end":1556,"expression":"1","context":740,"kind":"use","source":18},{"line":25,"start":1688,"end":1689,"expression":"0","context":741,"kind":"use","source":18},{"line":27,"start":1823,"end":1824,"expression":"1","context":742,"kind":"use","source":18},{"line":29,"start":1957,"end":1958,"expression":"1","context":743,"kind":"use","source":18},{"line":31,"start":2092,"end":2093,"expression":"1","context":744,"kind":"use","source":18},{"line":33,"start":2227,"end":2228,"expression":"1","context":745,"kind":"use","source":18},{"line":35,"start":2360,"end":2361,"expression":"0","context":746,"kind":"use","source":18},{"line":37,"start":2491,"end":2496,"expression":"1/100","context":747,"kind":"use","source":18},{"line":40,"start":2742,"end":2743,"expression":"1","context":748,"kind":"use","source":18},{"line":43,"start":2985,"end":2986,"expression":"3","context":749,"kind":"use","source":18},{"line":46,"start":3240,"end":3250,"expression":"2 \\sqrt{3}","context":750,"kind":"use","source":18},{"line":47,"start":3316,"end":3333,"expression":"\\|E\\|_F^2=12/10^4","context":751,"kind":"use","source":18},{"line":47,"start":3343,"end":3368,"expression":"E\\mathbf1=(3/100)\\mathbf1","context":752,"kind":"use","source":18},{"line":3,"start":188,"end":189,"expression":"K","context":753,"kind":"use","source":19},{"line":5,"start":315,"end":316,"expression":"0","context":754,"kind":"use","source":19},{"line":7,"start":442,"end":443,"expression":"0","context":755,"kind":"use","source":19},{"line":9,"start":569,"end":570,"expression":"0","context":756,"kind":"use","source":19},{"line":11,"start":696,"end":697,"expression":"0","context":757,"kind":"use","source":19},{"line":13,"start":823,"end":824,"expression":"1","context":758,"kind":"use","source":19},{"line":15,"start":948,"end":949,"expression":"0","context":759,"kind":"use","source":19},{"line":17,"start":1073,"end":1074,"expression":"0","context":760,"kind":"use","source":19},{"line":19,"start":1198,"end":1199,"expression":"0","context":761,"kind":"use","source":19},{"line":21,"start":1325,"end":1326,"expression":"1","context":762,"kind":"use","source":19},{"line":23,"start":1450,"end":1451,"expression":"0","context":763,"kind":"use","source":19},{"line":25,"start":1575,"end":1576,"expression":"0","context":764,"kind":"use","source":19},{"line":27,"start":1700,"end":1701,"expression":"0","context":765,"kind":"use","source":19},{"line":29,"start":1827,"end":1828,"expression":"0","context":766,"kind":"use","source":19},{"line":31,"start":1956,"end":1957,"expression":"1","context":767,"kind":"use","source":19},{"line":33,"start":2085,"end":2087,"expression":"-1","context":768,"kind":"use","source":19},{"line":35,"start":2213,"end":2214,"expression":"0","context":769,"kind":"use","source":19},{"line":38,"start":2424,"end":2425,"expression":"0","context":770,"kind":"use","source":19},{"line":40,"start":2551,"end":2552,"expression":"0","context":771,"kind":"use","source":19},{"line":42,"start":2676,"end":2677,"expression":"0","context":772,"kind":"use","source":19},{"line":44,"start":2801,"end":2802,"expression":"0","context":773,"kind":"use","source":19},{"line":46,"start":2924,"end":2925,"expression":"0","context":774,"kind":"use","source":19},{"line":48,"start":3049,"end":3050,"expression":"0","context":775,"kind":"use","source":19},{"line":50,"start":3172,"end":3173,"expression":"0","context":776,"kind":"use","source":19},{"line":52,"start":3295,"end":3296,"expression":"0","context":777,"kind":"use","source":19},{"line":54,"start":3418,"end":3419,"expression":"0","context":778,"kind":"use","source":19},{"line":56,"start":3543,"end":3544,"expression":"0","context":779,"kind":"use","source":19},{"line":58,"start":3666,"end":3667,"expression":"0","context":780,"kind":"use","source":19},{"line":60,"start":3789,"end":3790,"expression":"0","context":781,"kind":"use","source":19},{"line":62,"start":3916,"end":3917,"expression":"2","context":782,"kind":"use","source":19},{"line":64,"start":4043,"end":4044,"expression":"0","context":783,"kind":"use","source":19},{"line":66,"start":4168,"end":4169,"expression":"0","context":784,"kind":"use","source":19},{"line":68,"start":4293,"end":4294,"expression":"0","context":785,"kind":"use","source":19},{"line":69,"start":4345,"end":4353,"expression":"1+(-1)=0","context":786,"kind":"use","source":19},{"line":70,"start":4407,"end":4412,"expression":"K^2=0","context":787,"kind":"use","source":19},{"line":3,"start":189,"end":193,"expression":"C(c)","context":788,"kind":"use","source":20},{"line":5,"start":327,"end":328,"expression":"2","context":789,"kind":"use","source":20},{"line":7,"start":463,"end":465,"expression":"-1","context":790,"kind":"use","source":20},{"line":9,"start":598,"end":599,"expression":"0","context":791,"kind":"use","source":20},{"line":11,"start":734,"end":735,"expression":"1","context":792,"kind":"use","source":20},{"line":13,"start":870,"end":871,"expression":"1","context":793,"kind":"use","source":20},{"line":15,"start":1007,"end":1008,"expression":"2","context":794,"kind":"use","source":20},{"line":17,"start":1144,"end":1146,"expression":"-1","context":795,"kind":"use","source":20},{"line":19,"start":1280,"end":1281,"expression":"0","context":796,"kind":"use","source":20},{"line":21,"start":1414,"end":1415,"expression":"0","context":797,"kind":"use","source":20},{"line":23,"start":1551,"end":1552,"expression":"1","context":798,"kind":"use","source":20},{"line":25,"start":1688,"end":1689,"expression":"2","context":799,"kind":"use","source":20},{"line":27,"start":1825,"end":1827,"expression":"-1","context":800,"kind":"use","source":20},{"line":29,"start":1962,"end":1964,"expression":"-1","context":801,"kind":"use","source":20},{"line":31,"start":2098,"end":2099,"expression":"0","context":802,"kind":"use","source":20},{"line":33,"start":2235,"end":2236,"expression":"1","context":803,"kind":"use","source":20},{"line":35,"start":2372,"end":2373,"expression":"2","context":804,"kind":"use","source":20},{"line":36,"start":2438,"end":2448,"expression":"\\sqrt4\\,Fc","context":805,"kind":"use","source":20},{"line":38,"start":2582,"end":2583,"expression":"2","context":806,"kind":"use","source":20},{"line":40,"start":2718,"end":2725,"expression":"2 - 2 i","context":807,"kind":"use","source":20},{"line":42,"start":2860,"end":2861,"expression":"2","context":808,"kind":"use","source":20},{"line":44,"start":2996,"end":3003,"expression":"2 + 2 i","context":809,"kind":"use","source":20},{"line":45,"start":3055,"end":3086,"expression":"F_{jk}=\\frac12\\,\\mathrm i^{-jk}","context":810,"kind":"use","source":20},{"line":46,"start":3138,"end":3160,"expression":"C=F^*\\diag(\\sqrt4 Fc)F","context":811,"kind":"use","source":20},{"line":47,"start":3210,"end":3218,"expression":"\\sqrt4=2","context":812,"kind":"use","source":20},{"line":3,"start":190,"end":191,"expression":"L","context":813,"kind":"use","source":21},{"line":3,"start":200,"end":201,"expression":"2","context":814,"kind":"use","source":21},{"line":5,"start":335,"end":346,"expression":"\\frac{3}{5}","context":815,"kind":"use","source":21},{"line":7,"start":478,"end":479,"expression":"0","context":816,"kind":"use","source":21},{"line":9,"start":613,"end":624,"expression":"\\frac{3}{5}","context":817,"kind":"use","source":21},{"line":11,"start":758,"end":769,"expression":"\\frac{4}{5}","context":818,"kind":"use","source":21},{"line":13,"start":901,"end":902,"expression":"0","context":819,"kind":"use","source":21},{"line":15,"start":1036,"end":1047,"expression":"\\frac{4}{5}","context":820,"kind":"use","source":21},{"line":17,"start":1179,"end":1180,"expression":"0","context":821,"kind":"use","source":21},{"line":19,"start":1314,"end":1315,"expression":"1","context":822,"kind":"use","source":21},{"line":21,"start":1447,"end":1448,"expression":"0","context":823,"kind":"use","source":21},{"line":23,"start":1580,"end":1581,"expression":"0","context":824,"kind":"use","source":21},{"line":25,"start":1713,"end":1714,"expression":"0","context":825,"kind":"use","source":21},{"line":27,"start":1846,"end":1847,"expression":"0","context":826,"kind":"use","source":21},{"line":28,"start":1912,"end":1913,"expression":"R","context":827,"kind":"use","source":21},{"line":28,"start":1922,"end":1923,"expression":"3","context":828,"kind":"use","source":21},{"line":30,"start":2057,"end":2058,"expression":"3","context":829,"kind":"use","source":21},{"line":32,"start":2190,"end":2191,"expression":"0","context":830,"kind":"use","source":21},{"line":34,"start":2325,"end":2326,"expression":"2","context":831,"kind":"use","source":21},{"line":36,"start":2458,"end":2459,"expression":"0","context":832,"kind":"use","source":21},{"line":38,"start":2593,"end":2594,"expression":"2","context":833,"kind":"use","source":21},{"line":40,"start":2726,"end":2727,"expression":"0","context":834,"kind":"use","source":21},{"line":42,"start":2861,"end":2862,"expression":"1","context":835,"kind":"use","source":21},{"line":44,"start":2994,"end":2995,"expression":"0","context":836,"kind":"use","source":21},{"line":46,"start":3129,"end":3131,"expression":"-1","context":837,"kind":"use","source":21},{"line":47,"start":3197,"end":3198,"expression":"G","context":838,"kind":"use","source":21},{"line":47,"start":3202,"end":3210,"expression":"2\\times3","context":839,"kind":"use","source":21},{"line":49,"start":3345,"end":3346,"expression":"5","context":840,"kind":"use","source":21},{"line":51,"start":3481,"end":3482,"expression":"0","context":841,"kind":"use","source":21},{"line":53,"start":3618,"end":3619,"expression":"0","context":842,"kind":"use","source":21},{"line":55,"start":3753,"end":3754,"expression":"0","context":843,"kind":"use","source":21},{"line":57,"start":3892,"end":3893,"expression":"2","context":844,"kind":"use","source":21},{"line":59,"start":4030,"end":4031,"expression":"0","context":845,"kind":"use","source":21},{"line":60,"start":4084,"end":4093,"expression":"LR^T=Q_LG","context":846,"kind":"use","source":21},{"line":2,"start":74,"end":104,"expression":"A^*A=\\operatorname{diag}(25,4)","context":847,"kind":"use","source":22},{"line":7,"start":390,"end":392,"expression":"\\n","context":847,"kind":"use","source":22},{"line":7,"start":421,"end":423,"expression":"\\v","context":848,"kind":"use","source":22},{"line":8,"start":460,"end":514,"expression":"Q_L^*Q_L=I\\qquad Q_L^TQ_L=\\operatorname{diag}(-7/25,1)","context":849,"kind":"use","source":22},{"line":7,"start":473,"end":479,"expression":"\\infty","context":850,"kind":"use","source":24},{"line":9,"start":610,"end":616,"expression":"\\infty","context":851,"kind":"use","source":24},{"line":11,"start":747,"end":753,"expression":"\\infty","context":852,"kind":"use","source":24},{"line":13,"start":882,"end":888,"expression":"\\infty","context":853,"kind":"use","source":24},{"line":15,"start":1019,"end":1025,"expression":"\\infty","context":854,"kind":"use","source":24},{"line":17,"start":1156,"end":1162,"expression":"\\infty","context":855,"kind":"use","source":24},{"line":19,"start":1296,"end":1302,"expression":"\\infty","context":856,"kind":"use","source":24},{"line":21,"start":1436,"end":1442,"expression":"\\infty","context":857,"kind":"use","source":24},{"line":26,"start":1797,"end":1803,"expression":"\\infty","context":858,"kind":"use","source":24},{"line":30,"start":2072,"end":2078,"expression":"\\infty","context":859,"kind":"use","source":24},{"line":34,"start":2345,"end":2351,"expression":"\\infty","context":860,"kind":"use","source":24},{"line":36,"start":2485,"end":2491,"expression":"\\infty","context":861,"kind":"use","source":24},{"line":38,"start":2628,"end":2634,"expression":"\\infty","context":862,"kind":"use","source":24},{"line":40,"start":2771,"end":2777,"expression":"\\infty","context":863,"kind":"use","source":24},{"line":55,"start":3807,"end":3813,"expression":"\\infty","context":864,"kind":"use","source":24},{"line":59,"start":4088,"end":4094,"expression":"\\infty","context":865,"kind":"use","source":24},{"line":3,"start":191,"end":197,"expression":"V_C^Tx","context":866,"kind":"use","source":25},{"line":5,"start":331,"end":332,"expression":"2","context":867,"kind":"use","source":25},{"line":7,"start":466,"end":468,"expression":"13","context":868,"kind":"use","source":25},{"line":8,"start":534,"end":541,"expression":"D_AD_Bx","context":869,"kind":"use","source":25},{"line":10,"start":676,"end":677,"expression":"1","context":870,"kind":"use","source":25},{"line":12,"start":812,"end":814,"expression":"-5","context":871,"kind":"use","source":25},{"line":14,"start":949,"end":951,"expression":"18","context":872,"kind":"use","source":25},{"line":16,"start":1086,"end":1087,"expression":"6","context":873,"kind":"use","source":25},{"line":18,"start":1221,"end":1224,"expression":"-12","context":874,"kind":"use","source":25},{"line":20,"start":1358,"end":1360,"expression":"44","context":875,"kind":"use","source":25},{"line":21,"start":1426,"end":1437,"expression":"U_C(V_C^Tx)","context":876,"kind":"use","source":25},{"line":23,"start":1572,"end":1574,"expression":"25","context":877,"kind":"use","source":25},{"line":25,"start":1709,"end":1711,"expression":"21","context":878,"kind":"use","source":25},{"line":27,"start":1846,"end":1848,"expression":"25","context":879,"kind":"use","source":25},{"line":29,"start":1983,"end":1985,"expression":"21","context":880,"kind":"use","source":25},{"line":31,"start":2119,"end":2121,"expression":"25","context":881,"kind":"use","source":25},{"line":33,"start":2255,"end":2257,"expression":"21","context":882,"kind":"use","source":25},{"line":34,"start":2324,"end":2329,"expression":"(AB)x","context":883,"kind":"use","source":25},{"line":36,"start":2468,"end":2470,"expression":"26","context":884,"kind":"use","source":25},{"line":38,"start":2609,"end":2611,"expression":"16","context":885,"kind":"use","source":25},{"line":40,"start":2750,"end":2752,"expression":"43","context":886,"kind":"use","source":25},{"line":42,"start":2891,"end":2893,"expression":"27","context":887,"kind":"use","source":25},{"line":44,"start":3031,"end":3033,"expression":"13","context":888,"kind":"use","source":25},{"line":46,"start":3171,"end":3173,"expression":"65","context":889,"kind":"use","source":25},{"line":7,"start":476,"end":488,"expression":"\\frac{1}{10}","context":890,"kind":"use","source":26},{"line":10,"start":714,"end":725,"expression":"\\frac{1}{5}","context":891,"kind":"use","source":26},{"line":13,"start":960,"end":972,"expression":"\\frac{1}{20}","context":892,"kind":"use","source":26},{"line":18,"start":1316,"end":1327,"expression":"\\frac{6}{5}","context":893,"kind":"use","source":26},{"line":21,"start":1556,"end":1567,"expression":"\\frac{4}{5}","context":894,"kind":"use","source":26},{"line":24,"start":1806,"end":1818,"expression":"\\frac{1}{20}","context":895,"kind":"use","source":26},{"line":25,"start":1885,"end":1921,"expression":"\\widehat p_j=a_j\\widehat p_{j-1}+e_j","context":896,"kind":"use","source":26},{"line":25,"start":1930,"end":1951,"expression":"(a_1,a_2,a_3)=(2,3,4)","context":897,"kind":"use","source":26},{"line":3,"start":189,"end":190,"expression":"x","context":898,"kind":"use","source":27},{"line":5,"start":322,"end":323,"expression":"1","context":899,"kind":"use","source":27},{"line":7,"start":456,"end":458,"expression":"-1","context":900,"kind":"use","source":27},{"line":9,"start":590,"end":591,"expression":"2","context":901,"kind":"use","source":27},{"line":11,"start":722,"end":723,"expression":"0","context":902,"kind":"use","source":27},{"line":13,"start":855,"end":857,"expression":"-2","context":903,"kind":"use","source":27},{"line":15,"start":989,"end":990,"expression":"3","context":904,"kind":"use","source":27},{"line":16,"start":1054,"end":1056,"expression":"Dx","context":905,"kind":"use","source":27},{"line":18,"start":1189,"end":1190,"expression":"2","context":906,"kind":"use","source":27},{"line":20,"start":1324,"end":1326,"expression":"-2","context":907,"kind":"use","source":27},{"line":22,"start":1459,"end":1460,"expression":"6","context":908,"kind":"use","source":27},{"line":24,"start":1594,"end":1595,"expression":"2","context":909,"kind":"use","source":27},{"line":26,"start":1728,"end":1730,"expression":"-3","context":910,"kind":"use","source":27},{"line":28,"start":1863,"end":1864,"expression":"7","context":911,"kind":"use","source":27},{"line":29,"start":1928,"end":1935,"expression":"u(v^Tx)","context":912,"kind":"use","source":27},{"line":31,"start":2067,"end":2068,"expression":"3","context":913,"kind":"use","source":27},{"line":33,"start":2201,"end":2202,"expression":"3","context":914,"kind":"use","source":27},{"line":35,"start":2334,"end":2335,"expression":"3","context":915,"kind":"use","source":27},{"line":37,"start":2468,"end":2469,"expression":"3","context":916,"kind":"use","source":27},{"line":39,"start":2601,"end":2602,"expression":"3","context":917,"kind":"use","source":27},{"line":41,"start":2734,"end":2735,"expression":"3","context":918,"kind":"use","source":27},{"line":42,"start":2800,"end":2802,"expression":"Ax","context":919,"kind":"use","source":27},{"line":44,"start":2938,"end":2939,"expression":"5","context":920,"kind":"use","source":27},{"line":46,"start":3076,"end":3077,"expression":"1","context":921,"kind":"use","source":27},{"line":48,"start":3213,"end":3214,"expression":"9","context":922,"kind":"use","source":27},{"line":50,"start":3351,"end":3352,"expression":"5","context":923,"kind":"use","source":27},{"line":52,"start":3486,"end":3487,"expression":"0","context":924,"kind":"use","source":27},{"line":54,"start":3623,"end":3625,"expression":"10","context":925,"kind":"use","source":27},{"line":55,"start":3689,"end":3690,"expression":"+","context":926,"kind":"use","source":27},{"line":56,"start":3751,"end":3752,"expression":"=","context":927,"kind":"use","source":27},{"line":57,"start":3813,"end":3819,"expression":"v^Tx=3","context":928,"kind":"use","source":27},{"line":3,"start":152,"end":154,"expression":"\\n","context":929,"kind":"use","source":28},{"line":4,"start":199,"end":200,"expression":"1","context":929,"kind":"use","source":28},{"line":5,"start":249,"end":250,"expression":"1","context":929,"kind":"use","source":28},{"line":6,"start":300,"end":301,"expression":"1","context":929,"kind":"use","source":28},{"line":7,"start":351,"end":353,"expression":"-1","context":929,"kind":"use","source":28},{"line":9,"start":489,"end":494,"expression":"0\\to3","context":930,"kind":"use","source":28},{"line":10,"start":530,"end":535,"expression":"K^2=0","context":931,"kind":"use","source":28},{"line":11,"start":569,"end":579,"expression":"A^{-1}=I-K","context":932,"kind":"use","source":28},{"line":12,"start":614,"end":629,"expression":"(A^{-1})_{30}=0","context":933,"kind":"use","source":28},{"line":3,"start":188,"end":189,"expression":"X","context":934,"kind":"use","source":29},{"line":5,"start":323,"end":324,"expression":"1","context":935,"kind":"use","source":29},{"line":7,"start":458,"end":459,"expression":"2","context":936,"kind":"use","source":29},{"line":9,"start":593,"end":594,"expression":"3","context":937,"kind":"use","source":29},{"line":11,"start":728,"end":729,"expression":"4","context":938,"kind":"use","source":29},{"line":12,"start":794,"end":815,"expression":"\\operatorname{vec}(X)","context":939,"kind":"use","source":29},{"line":14,"start":948,"end":949,"expression":"1","context":940,"kind":"use","source":29},{"line":16,"start":1082,"end":1083,"expression":"3","context":941,"kind":"use","source":29},{"line":18,"start":1216,"end":1217,"expression":"2","context":942,"kind":"use","source":29},{"line":20,"start":1350,"end":1351,"expression":"4","context":943,"kind":"use","source":29},{"line":21,"start":1414,"end":1419,"expression":"BXA^T","context":944,"kind":"use","source":29},{"line":23,"start":1549,"end":1556,"expression":"5 + 8 i","context":945,"kind":"use","source":29},{"line":25,"start":1687,"end":1688,"expression":"2","context":946,"kind":"use","source":29},{"line":27,"start":1818,"end":1826,"expression":"9 + 12 i","context":947,"kind":"use","source":29},{"line":29,"start":1957,"end":1958,"expression":"6","context":948,"kind":"use","source":29},{"line":30,"start":2023,"end":2026,"expression":"A^T","context":949,"kind":"use","source":29},{"line":30,"start":2034,"end":2037,"expression":"A^*","context":950,"kind":"use","source":29},{"line":31,"start":2087,"end":2146,"expression":"(A\\otimes B)\\operatorname{vec}(X)=\\operatorname{vec}(BXA^T)","context":950,"kind":"use","source":29},{"line":23,"start":1485,"end":1502,"expression":"A=I_2,\\ v=(1,1)^T","context":951,"kind":"use","source":30},{"line":24,"start":1553,"end":1570,"expression":"p(A)v=p(1)(1,1)^T","context":952,"kind":"use","source":30},{"line":25,"start":1621,"end":1644,"expression":"\\dim\\mathcal K_m(A,v)=1","context":953,"kind":"use","source":30},{"line":26,"start":1734,"end":1735,"expression":"2","context":954,"kind":"use","source":30},{"line":3,"start":174,"end":176,"expression":"a0","context":955,"kind":"use","source":31},{"line":8,"start":442,"end":444,"expression":"a1","context":956,"kind":"use","source":31},{"line":13,"start":704,"end":705,"expression":"a","context":957,"kind":"use","source":31},{"line":18,"start":967,"end":969,"expression":"b0","context":958,"kind":"use","source":31},{"line":23,"start":1229,"end":1231,"expression":"b1","context":959,"kind":"use","source":31},{"line":28,"start":1490,"end":1491,"expression":"b","context":960,"kind":"use","source":31},{"line":33,"start":1754,"end":1755,"expression":"r","context":961,"kind":"use","source":31},{"line":3,"start":189,"end":193,"expression":"LR^T","context":962,"kind":"use","source":32},{"line":5,"start":326,"end":327,"expression":"3","context":963,"kind":"use","source":32},{"line":7,"start":458,"end":459,"expression":"0","context":964,"kind":"use","source":32},{"line":9,"start":590,"end":591,"expression":"0","context":965,"kind":"use","source":32},{"line":11,"start":725,"end":726,"expression":"4","context":966,"kind":"use","source":32},{"line":13,"start":858,"end":859,"expression":"0","context":967,"kind":"use","source":32},{"line":15,"start":991,"end":992,"expression":"0","context":968,"kind":"use","source":32},{"line":17,"start":1121,"end":1122,"expression":"0","context":969,"kind":"use","source":32},{"line":19,"start":1253,"end":1254,"expression":"2","context":970,"kind":"use","source":32},{"line":21,"start":1383,"end":1384,"expression":"0","context":971,"kind":"use","source":32},{"line":23,"start":1516,"end":1517,"expression":"0","context":972,"kind":"use","source":32},{"line":25,"start":1649,"end":1650,"expression":"0","context":973,"kind":"use","source":32},{"line":27,"start":1782,"end":1783,"expression":"0","context":974,"kind":"use","source":32},{"line":30,"start":1993,"end":1994,"expression":"3","context":975,"kind":"use","source":32},{"line":32,"start":2125,"end":2126,"expression":"0","context":976,"kind":"use","source":32},{"line":34,"start":2257,"end":2258,"expression":"0","context":977,"kind":"use","source":32},{"line":36,"start":2392,"end":2393,"expression":"4","context":978,"kind":"use","source":32},{"line":38,"start":2525,"end":2526,"expression":"0","context":979,"kind":"use","source":32},{"line":40,"start":2658,"end":2659,"expression":"0","context":980,"kind":"use","source":32},{"line":42,"start":2788,"end":2789,"expression":"0","context":981,"kind":"use","source":32},{"line":44,"start":2918,"end":2919,"expression":"0","context":982,"kind":"use","source":32},{"line":46,"start":3048,"end":3049,"expression":"0","context":983,"kind":"use","source":32},{"line":48,"start":3181,"end":3182,"expression":"0","context":984,"kind":"use","source":32},{"line":50,"start":3314,"end":3315,"expression":"0","context":985,"kind":"use","source":32},{"line":52,"start":3447,"end":3448,"expression":"0","context":986,"kind":"use","source":32},{"line":57,"start":3802,"end":3803,"expression":"5","context":987,"kind":"use","source":32},{"line":60,"start":4038,"end":4039,"expression":"2","context":988,"kind":"use","source":32},{"line":63,"start":4280,"end":4281,"expression":"0","context":989,"kind":"use","source":32},{"line":64,"start":4365,"end":4382,"expression":"\\sqrt{29},\\ 2,\\ 0","context":990,"kind":"use","source":32},{"line":4,"start":248,"end":261,"expression":"s_e\\epsilon_e","context":991,"kind":"use","source":33},{"line":7,"start":488,"end":499,"expression":"\\frac{1}{4}","context":992,"kind":"use","source":33},{"line":10,"start":726,"end":737,"expression":"\\frac{1}{4}","context":993,"kind":"use","source":33},{"line":13,"start":962,"end":973,"expression":"\\frac{1}{2}","context":994,"kind":"use","source":33},{"line":18,"start":1318,"end":1330,"expression":"\\frac{1}{10}","context":995,"kind":"use","source":33},{"line":21,"start":1559,"end":1571,"expression":"\\frac{3}{10}","context":996,"kind":"use","source":33},{"line":24,"start":1799,"end":1810,"expression":"\\frac{3}{5}","context":997,"kind":"use","source":33},{"line":21,"start":4493,"end":4500,"expression":"x^TAx=1","context":998,"kind":"use","source":34},{"line":40,"start":8961,"end":8972,"expression":"y^T(I+E)y=1","context":999,"kind":"use","source":34},{"line":41,"start":9069,"end":9076,"expression":"1/2,3/2","context":1000,"kind":"use","source":34},{"line":41,"start":9086,"end":9096,"expression":"\\kappa_2=3","context":1001,"kind":"use","source":34},{"line":4,"start":261,"end":263,"expression":"\\v","context":1002,"kind":"use","source":35},{"line":7,"start":426,"end":435,"expression":"7/50>1/10","context":1003,"kind":"use","source":35},{"line":9,"start":530,"end":541,"expression":"2/25\\le1/10","context":1004,"kind":"use","source":35},{"line":11,"start":678,"end":688,"expression":"h(1+1)=1/5","context":1005,"kind":"use","source":35},{"line":3,"start":189,"end":190,"expression":"A","context":1006,"kind":"use","source":36},{"line":5,"start":342,"end":343,"expression":"2","context":1007,"kind":"use","source":36},{"line":7,"start":474,"end":475,"expression":"0","context":1008,"kind":"use","source":36},{"line":9,"start":607,"end":608,"expression":"1","context":1009,"kind":"use","source":36},{"line":11,"start":740,"end":741,"expression":"0","context":1010,"kind":"use","source":36},{"line":13,"start":875,"end":876,"expression":"2","context":1011,"kind":"use","source":36},{"line":15,"start":1009,"end":1010,"expression":"1","context":1012,"kind":"use","source":36},{"line":17,"start":1144,"end":1145,"expression":"1","context":1013,"kind":"use","source":36},{"line":19,"start":1279,"end":1280,"expression":"1","context":1014,"kind":"use","source":36},{"line":21,"start":1413,"end":1414,"expression":"2","context":1015,"kind":"use","source":36},{"line":22,"start":1477,"end":1509,"expression":"S=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR}","context":1016,"kind":"use","source":36},{"line":24,"start":1643,"end":1654,"expression":"\\frac{3}{2}","context":1017,"kind":"use","source":36},{"line":26,"start":1788,"end":1801,"expression":"- \\frac{1}{2}","context":1018,"kind":"use","source":36},{"line":28,"start":1935,"end":1948,"expression":"- \\frac{1}{2}","context":1019,"kind":"use","source":36},{"line":30,"start":2082,"end":2093,"expression":"\\frac{3}{2}","context":1020,"kind":"use","source":36},{"line":33,"start":2310,"end":2321,"expression":"\\frac{3}{2}","context":1021,"kind":"use","source":36},{"line":35,"start":2456,"end":2457,"expression":"0","context":1022,"kind":"use","source":36},{"line":37,"start":2592,"end":2593,"expression":"0","context":1023,"kind":"use","source":36},{"line":39,"start":2730,"end":2741,"expression":"\\frac{3}{2}","context":1024,"kind":"use","source":36},{"line":40,"start":2830,"end":2834,"expression":"-1/2","context":1025,"kind":"use","source":36},{"line":40,"start":2845,"end":2849,"expression":"+1/2","context":1026,"kind":"use","source":36},{"line":2,"start":87,"end":90,"expression":"r_2","context":1027,"kind":"use","source":37},{"line":4,"start":190,"end":196,"expression":"6+7r_2","context":1028,"kind":"use","source":37},{"line":5,"start":256,"end":263,"expression":"10+2r_2","context":1028,"kind":"use","source":37},{"line":7,"start":362,"end":365,"expression":"4/5","context":1028,"kind":"use","source":37},{"line":8,"start":445,"end":447,"expression":"\\x","context":1029,"kind":"use","source":37},{"line":10,"start":567,"end":569,"expression":"27","context":1030,"kind":"use","source":37},{"line":10,"start":604,"end":606,"expression":"16","context":1031,"kind":"use","source":37},{"line":3,"start":189,"end":190,"expression":"A","context":1032,"kind":"use","source":38},{"line":5,"start":342,"end":343,"expression":"4","context":1033,"kind":"use","source":38},{"line":7,"start":476,"end":477,"expression":"2","context":1034,"kind":"use","source":38},{"line":9,"start":609,"end":610,"expression":"1","context":1035,"kind":"use","source":38},{"line":11,"start":743,"end":744,"expression":"1","context":1036,"kind":"use","source":38},{"line":13,"start":876,"end":877,"expression":"1","context":1037,"kind":"use","source":38},{"line":15,"start":1009,"end":1010,"expression":"1","context":1038,"kind":"use","source":38},{"line":17,"start":1143,"end":1144,"expression":"2","context":1039,"kind":"use","source":38},{"line":19,"start":1278,"end":1279,"expression":"4","context":1040,"kind":"use","source":38},{"line":21,"start":1412,"end":1413,"expression":"1","context":1041,"kind":"use","source":38},{"line":23,"start":1547,"end":1548,"expression":"1","context":1042,"kind":"use","source":38},{"line":25,"start":1681,"end":1682,"expression":"1","context":1043,"kind":"use","source":38},{"line":27,"start":1815,"end":1816,"expression":"1","context":1044,"kind":"use","source":38},{"line":29,"start":1948,"end":1949,"expression":"1","context":1045,"kind":"use","source":38},{"line":31,"start":2082,"end":2083,"expression":"1","context":1046,"kind":"use","source":38},{"line":33,"start":2215,"end":2216,"expression":"4","context":1047,"kind":"use","source":38},{"line":35,"start":2349,"end":2350,"expression":"2","context":1048,"kind":"use","source":38},{"line":37,"start":2482,"end":2483,"expression":"1","context":1049,"kind":"use","source":38},{"line":39,"start":2615,"end":2616,"expression":"1","context":1050,"kind":"use","source":38},{"line":41,"start":2749,"end":2750,"expression":"1","context":1051,"kind":"use","source":38},{"line":43,"start":2884,"end":2885,"expression":"1","context":1052,"kind":"use","source":38},{"line":45,"start":3018,"end":3019,"expression":"2","context":1053,"kind":"use","source":38},{"line":47,"start":3153,"end":3154,"expression":"4","context":1054,"kind":"use","source":38},{"line":49,"start":3287,"end":3288,"expression":"1","context":1055,"kind":"use","source":38},{"line":51,"start":3421,"end":3422,"expression":"1","context":1056,"kind":"use","source":38},{"line":53,"start":3554,"end":3555,"expression":"1","context":1057,"kind":"use","source":38},{"line":55,"start":3688,"end":3689,"expression":"1","context":1058,"kind":"use","source":38},{"line":57,"start":3821,"end":3822,"expression":"1","context":1059,"kind":"use","source":38},{"line":59,"start":3955,"end":3956,"expression":"1","context":1060,"kind":"use","source":38},{"line":61,"start":4088,"end":4089,"expression":"4","context":1061,"kind":"use","source":38},{"line":63,"start":4221,"end":4222,"expression":"2","context":1062,"kind":"use","source":38},{"line":65,"start":4354,"end":4355,"expression":"1","context":1063,"kind":"use","source":38},{"line":67,"start":4488,"end":4489,"expression":"1","context":1064,"kind":"use","source":38},{"line":69,"start":4621,"end":4622,"expression":"1","context":1065,"kind":"use","source":38},{"line":71,"start":4755,"end":4756,"expression":"1","context":1066,"kind":"use","source":38},{"line":73,"start":4888,"end":4889,"expression":"2","context":1067,"kind":"use","source":38},{"line":75,"start":5021,"end":5022,"expression":"4","context":1068,"kind":"use","source":38},{"line":81,"start":5398,"end":5400,"expression":"36","context":1069,"kind":"use","source":38},{"line":84,"start":5638,"end":5640,"expression":"24","context":1070,"kind":"use","source":38},{"line":3,"start":189,"end":190,"expression":"T","context":1071,"kind":"use","source":39},{"line":5,"start":323,"end":324,"expression":"2","context":1072,"kind":"use","source":39},{"line":7,"start":457,"end":459,"expression":"-1","context":1073,"kind":"use","source":39},{"line":9,"start":591,"end":592,"expression":"5","context":1074,"kind":"use","source":39},{"line":11,"start":726,"end":727,"expression":"3","context":1075,"kind":"use","source":39},{"line":13,"start":861,"end":862,"expression":"2","context":1076,"kind":"use","source":39},{"line":15,"start":995,"end":997,"expression":"-1","context":1077,"kind":"use","source":39},{"line":17,"start":1131,"end":1132,"expression":"4","context":1078,"kind":"use","source":39},{"line":19,"start":1266,"end":1267,"expression":"3","context":1079,"kind":"use","source":39},{"line":21,"start":1400,"end":1401,"expression":"2","context":1080,"kind":"use","source":39},{"line":22,"start":1483,"end":1484,"expression":"g","context":1081,"kind":"use","source":39},{"line":24,"start":1618,"end":1619,"expression":"2","context":1082,"kind":"use","source":39},{"line":26,"start":1754,"end":1755,"expression":"3","context":1083,"kind":"use","source":39},{"line":28,"start":1890,"end":1891,"expression":"4","context":1084,"kind":"use","source":39},{"line":30,"start":2024,"end":2025,"expression":"0","context":1085,"kind":"use","source":39},{"line":32,"start":2160,"end":2161,"expression":"5","context":1086,"kind":"use","source":39},{"line":34,"start":2296,"end":2298,"expression":"-1","context":1087,"kind":"use","source":39},{"line":37,"start":2507,"end":2508,"expression":"1","context":1088,"kind":"use","source":39},{"line":39,"start":2643,"end":2644,"expression":"2","context":1089,"kind":"use","source":39},{"line":41,"start":2779,"end":2781,"expression":"-1","context":1090,"kind":"use","source":39},{"line":43,"start":2914,"end":2915,"expression":"0","context":1091,"kind":"use","source":39},{"line":45,"start":3048,"end":3049,"expression":"0","context":1092,"kind":"use","source":39},{"line":47,"start":3182,"end":3183,"expression":"0","context":1093,"kind":"use","source":39},{"line":50,"start":3395,"end":3397,"expression":"-5","context":1094,"kind":"use","source":39},{"line":52,"start":3535,"end":3536,"expression":"8","context":1095,"kind":"use","source":39},{"line":54,"start":3674,"end":3675,"expression":"8","context":1096,"kind":"use","source":39},{"line":56,"start":3813,"end":3814,"expression":"5","context":1097,"kind":"use","source":39},{"line":58,"start":3952,"end":3953,"expression":"1","context":1098,"kind":"use","source":39},{"line":60,"start":4091,"end":4092,"expression":"9","context":1099,"kind":"use","source":39},{"line":31,"start":1966,"end":1969,"expression":"B=2","context":1100,"kind":"use","source":40},{"line":31,"start":1978,"end":1979,"expression":"8","context":1101,"kind":"use","source":40},{"line":33,"start":2127,"end":2150,"expression":"8+2\\lambda\\le1+6\\lambda","context":1102,"kind":"use","source":40},{"line":33,"start":2158,"end":2173,"expression":"8+2\\lambda\\le10","context":1103,"kind":"use","source":40},{"line":21,"start":1399,"end":1420,"expression":"x=(2I+K)^{-1}\\mathbf1","context":1104,"kind":"use","source":41},{"line":23,"start":1555,"end":1566,"expression":"\\frac{1}{2}","context":1105,"kind":"use","source":41},{"line":25,"start":1702,"end":1713,"expression":"\\frac{1}{2}","context":1106,"kind":"use","source":41},{"line":27,"start":1848,"end":1849,"expression":"1","context":1107,"kind":"use","source":41},{"line":29,"start":1985,"end":1996,"expression":"\\frac{3}{4}","context":1108,"kind":"use","source":41},{"line":31,"start":2131,"end":2143,"expression":"\\frac{11}{8}","context":1109,"kind":"use","source":41},{"line":33,"start":2278,"end":2289,"expression":"\\frac{7}{8}","context":1110,"kind":"use","source":41},{"line":35,"start":2425,"end":2437,"expression":"\\frac{13}{8}","context":1111,"kind":"use","source":41},{"line":71,"start":4480,"end":4492,"expression":"8\\log_2 8=24","context":1112,"kind":"use","source":42},{"line":71,"start":4519,"end":4530,"expression":"H_8H_8^T=8I","context":1113,"kind":"use","source":42},{"line":3,"start":202,"end":208,"expression":"D^{-1}","context":1114,"kind":"use","source":43},{"line":5,"start":342,"end":353,"expression":"\\frac{3}{8}","context":1115,"kind":"use","source":43},{"line":7,"start":487,"end":500,"expression":"- \\frac{1}{8}","context":1116,"kind":"use","source":43},{"line":9,"start":634,"end":647,"expression":"- \\frac{1}{8}","context":1117,"kind":"use","source":43},{"line":11,"start":781,"end":792,"expression":"\\frac{3}{8}","context":1118,"kind":"use","source":43},{"line":12,"start":844,"end":862,"expression":"D^{-1}u=\\tfrac14 u","context":1119,"kind":"use","source":43},{"line":13,"start":915,"end":938,"expression":"S=1+u^TD^{-1}u=\\tfrac52","context":1120,"kind":"use","source":43},{"line":14,"start":990,"end":1019,"expression":"A^{-1}=D^{-1}-\\tfrac1{40}uu^T","context":1121,"kind":"use","source":43},{"line":15,"start":1085,"end":1095,"expression":"I-e_1e_1^T","context":1122,"kind":"use","source":43},{"line":17,"start":1230,"end":1231,"expression":"0","context":1123,"kind":"use","source":43},{"line":19,"start":1366,"end":1367,"expression":"0","context":1124,"kind":"use","source":43},{"line":21,"start":1502,"end":1503,"expression":"0","context":1125,"kind":"use","source":43},{"line":23,"start":1640,"end":1641,"expression":"1","context":1126,"kind":"use","source":43},{"line":24,"start":1722,"end":1723,"expression":"D","context":1127,"kind":"use","source":43},{"line":4,"start":226,"end":234,"expression":"n/(b+2r)","context":1128,"kind":"use","source":44},{"line":4,"start":242,"end":249,"expression":"b=2,r=1","context":1129,"kind":"use","source":44},{"line":7,"start":473,"end":484,"expression":"\\frac{3}{2}","context":1130,"kind":"use","source":44},{"line":10,"start":703,"end":704,"expression":"3","context":1131,"kind":"use","source":44},{"line":13,"start":925,"end":926,"expression":"6","context":1132,"kind":"use","source":44},{"line":16,"start":1141,"end":1143,"expression":"12","context":1133,"kind":"use","source":44},{"line":17,"start":1230,"end":1231,"expression":"n","context":1134,"kind":"use","source":44},{"line":19,"start":1387,"end":1389,"expression":"36","context":1135,"kind":"use","source":44},{"line":19,"start":1401,"end":1404,"expression":"72q","context":1136,"kind":"use","source":44},{"line":20,"start":1480,"end":1482,"expression":"24","context":1137,"kind":"use","source":44},{"line":20,"start":1494,"end":1497,"expression":"48q","context":1138,"kind":"use","source":44},{"line":21,"start":1575,"end":1578,"expression":"3/2","context":1139,"kind":"use","source":44},{"line":2,"start":200,"end":203,"expression":"\\CC","context":1140,"kind":"use","source":45},{"line":2,"start":225,"end":231,"expression":"Q^*Q=I","context":1140,"kind":"use","source":45},{"line":3,"start":295,"end":388,"expression":"\nQ_L=\\mat{3/5&0\\\\4\\mathrm i/5&0\\\\0&1},\\qquad\nQ_R=\\mat{1&0\\\\0&\\mathrm i},\\qquad G=\\diag(5,2).\n","context":1141,"kind":"use","source":45},{"line":8,"start":452,"end":530,"expression":"\nA=Q_LGQ_R^*=\\mat{3&0\\\\4\\mathrm i&0\\\\0&-2\\mathrm i},\n\\qquad A^*A=\\diag(25,4).\n","context":2,"kind":"use","source":45},{"line":14,"start":630,"end":633,"expression":"Q_L","context":3,"kind":"use","source":45},{"line":14,"start":665,"end":677,"expression":"9/25+16/25=1","context":3,"kind":"use","source":45},{"line":14,"start":724,"end":740,"expression":"9/25-16/25=-7/25","context":3,"kind":"use","source":45},{"line":14,"start":749,"end":757,"expression":"Q_L^TQ_L","context":3,"kind":"use","source":45},{"line":16,"start":924,"end":925,"expression":"G","context":4,"kind":"use","source":45},{"line":16,"start":934,"end":937,"expression":"A_1","context":5,"kind":"use","source":45},{"line":16,"start":1030,"end":1040,"expression":"\\diag(0,4)","context":6,"kind":"use","source":45},{"line":16,"start":1174,"end":1188,"expression":"\\QQ(\\mathrm i)","context":7,"kind":"use","source":45},{"line":23,"start":1995,"end":2045,"expression":"\nA=I+K=\\mat{1&0&0&0\\\\1&1&0&0\\\\1&0&1&0\\\\0&1&-1&1}.\n","context":1142,"kind":"use","source":45},{"line":26,"start":2116,"end":2121,"expression":"K^2=0","context":9,"kind":"use","source":45},{"line":27,"start":2139,"end":2196,"expression":"\nA^{-1}=I-K=\\mat{1&0&0&0\\\\-1&1&0&0\\\\-1&0&1&0\\\\0&-1&1&1}.\n","context":10,"kind":"use","source":45},{"line":32,"start":2301,"end":2306,"expression":"0\\to3","context":11,"kind":"use","source":45},{"line":32,"start":2411,"end":2426,"expression":"(A^{-1})_{30}=0","context":11,"kind":"use","source":45},{"line":34,"start":2633,"end":2636,"expression":"I-K","context":12,"kind":"use","source":45},{"line":36,"start":3075,"end":3076,"expression":"A","context":13,"kind":"use","source":45},{"line":36,"start":3083,"end":3084,"expression":"A","context":14,"kind":"use","source":45},{"line":52,"start":4288,"end":4303,"expression":"(r_1,r_2)=(1,3)","context":15,"kind":"use","source":45},{"line":53,"start":4338,"end":4404,"expression":"\nJ_{\\rm keep}=4+2+3\\cdot7=27,\\qquad\nJ_{\\rm split}=2+8+3\\cdot2=16.\n","context":16,"kind":"use","source":45},{"line":59,"start":4686,"end":4691,"expression":"r_1=1","context":17,"kind":"introduction-passage","source":45},{"line":59,"start":4703,"end":4712,"expression":"r_2=r\\ge0","context":18,"kind":"introduction-passage","source":45},{"line":59,"start":4730,"end":4734,"expression":"6+7r","context":19,"kind":"introduction-passage","source":45},{"line":59,"start":4757,"end":4762,"expression":"10+2r","context":20,"kind":"introduction-passage","source":45},{"line":59,"start":4801,"end":4806,"expression":"r>4/5","context":21,"kind":"introduction-passage","source":45},{"line":59,"start":4841,"end":4844,"expression":"r=1","context":22,"kind":"introduction-passage","source":45},{"line":65,"start":5786,"end":5792,"expression":"h=1/10","context":1143,"kind":"use","source":45},{"line":65,"start":5851,"end":5852,"expression":"e","context":1143,"kind":"use","source":45},{"line":66,"start":5866,"end":5921,"expression":"\nw=\\lceil e/h\\rceil\\qquad\\text{ensures}\\qquad e\\le hw.\n","context":1143,"kind":"use","source":45},{"line":69,"start":5950,"end":5961,"expression":"\\sum w\\le B","context":1143,"kind":"use","source":45},{"line":69,"start":5987,"end":5999,"expression":"\\sum e\\le hB","context":1143,"kind":"use","source":45},{"line":72,"start":6171,"end":6182,"expression":"(3/50,2/25)","context":24,"kind":"use","source":45},{"line":72,"start":6216,"end":6221,"expression":"(0,0)","context":25,"kind":"use","source":45},{"line":72,"start":6251,"end":6260,"expression":"7/50>1/10","context":26,"kind":"use","source":45},{"line":72,"start":6282,"end":6287,"expression":"(1,1)","context":27,"kind":"use","source":45},{"line":74,"start":6371,"end":6382,"expression":"(3/50,1/50)","context":28,"kind":"use","source":45},{"line":74,"start":6399,"end":6410,"expression":"2/25\\le1/10","context":28,"kind":"use","source":45},{"line":74,"start":6447,"end":6452,"expression":"(1,1)","context":28,"kind":"use","source":45},{"line":76,"start":6702,"end":6703,"expression":"B","context":28,"kind":"use","source":45},{"line":76,"start":6719,"end":6722,"expression":"B+1","context":28,"kind":"use","source":45},{"line":76,"start":6785,"end":6792,"expression":"(B+1)^2","context":29,"kind":"use","source":45},{"line":76,"start":6811,"end":6816,"expression":"B=255","context":30,"kind":"use","source":45},{"line":78,"start":7112,"end":7113,"expression":"B","context":31,"kind":"use","source":45},{"line":5,"start":558,"end":566,"expression":"A=D+uv^T","context":32,"kind":"introduction-passage","source":46},{"line":5,"start":585,"end":586,"expression":"A","context":33,"kind":"introduction-passage","source":46},{"line":5,"start":611,"end":619,"expression":"2\\times2","context":34,"kind":"introduction-passage","source":46},{"line":7,"start":833,"end":834,"expression":"L","context":35,"kind":"use","source":46},{"line":7,"start":841,"end":842,"expression":"R","context":35,"kind":"use","source":46},{"line":7,"start":880,"end":888,"expression":"4\\times3","context":35,"kind":"use","source":46},{"line":7,"start":904,"end":912,"expression":"2\\times3","context":35,"kind":"use","source":46},{"line":19,"start":2501,"end":2509,"expression":"P_\\alpha","context":36,"kind":"use","source":46},{"line":20,"start":2589,"end":2604,"expression":"A_{\\beta\\alpha}","context":37,"kind":"use","source":46},{"line":21,"start":2668,"end":2669,"expression":"E","context":38,"kind":"use","source":46},{"line":22,"start":2758,"end":2764,"expression":"D+UV^*","context":39,"kind":"use","source":46},{"line":23,"start":2850,"end":2859,"expression":"Q_L,G,Q_R","context":40,"kind":"use","source":46},{"line":24,"start":2953,"end":2956,"expression":"E_j","context":41,"kind":"use","source":46},{"line":25,"start":3049,"end":3063,"expression":"s_e,\\epsilon_e","context":42,"kind":"use","source":46},{"line":26,"start":3157,"end":3187,"expression":"(\\kappa_v,\\epsilon_v,\\sigma_v)","context":43,"kind":"use","source":46},{"line":27,"start":3266,"end":3272,"expression":"F_v(b)","context":44,"kind":"use","source":46},{"line":27,"start":3299,"end":3300,"expression":"b","context":44,"kind":"use","source":46},{"line":28,"start":3343,"end":3344,"expression":"R","context":45,"kind":"use","source":46},{"line":31,"start":3512,"end":3527,"expression":"A_{\\beta\\alpha}","context":46,"kind":"use","source":46},{"line":31,"start":3547,"end":3553,"expression":"\\alpha","context":46,"kind":"use","source":46},{"line":31,"start":3566,"end":3571,"expression":"\\beta","context":46,"kind":"use","source":46},{"line":39,"start":4730,"end":4731,"expression":"D","context":1144,"kind":"use","source":46},{"line":39,"start":4762,"end":4776,"expression":"\\mat{3&1\\\\1&3}","context":1145,"kind":"use","source":46},{"line":39,"start":4787,"end":4804,"expression":"u=v=\\ones\\in\\QQ^6","context":49,"kind":"use","source":46},{"line":39,"start":4813,"end":4821,"expression":"A=D+uv^T","context":49,"kind":"use","source":46},{"line":40,"start":4952,"end":5004,"expression":"\nx=(1,-1,2,0,-2,3)^T,\\qquad v^Tx=3,\\qquad Ax=Dx+3u.\n","context":49,"kind":"use","source":46},{"line":45,"start":5167,"end":5171,"expression":"v^Tx","context":49,"kind":"use","source":46},{"line":45,"start":5202,"end":5203,"expression":"u","context":50,"kind":"use","source":46},{"line":47,"start":5352,"end":5358,"expression":"7+3=10","context":51,"kind":"use","source":46},{"line":47,"start":5381,"end":5387,"expression":"-3+3=0","context":51,"kind":"use","source":46},{"line":51,"start":6070,"end":6071,"expression":"x","context":52,"kind":"use","source":46},{"line":55,"start":6558,"end":6559,"expression":"q","context":1146,"kind":"use","source":46},{"line":55,"start":6618,"end":6623,"expression":"2n^2q","context":1147,"kind":"use","source":46},{"line":55,"start":6665,"end":6666,"expression":"r","context":1148,"kind":"use","source":46},{"line":55,"start":6707,"end":6708,"expression":"b","context":1149,"kind":"use","source":46},{"line":56,"start":6717,"end":6770,"expression":"\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n","context":1150,"kind":"use","source":46},{"line":61,"start":6991,"end":7002,"expression":"n=6,b=2,r=1","context":58,"kind":"use","source":46},{"line":61,"start":7080,"end":7083,"expression":"72q","context":59,"kind":"use","source":46},{"line":61,"start":7089,"end":7092,"expression":"48q","context":60,"kind":"use","source":46},{"line":61,"start":7106,"end":7109,"expression":"b,r","context":61,"kind":"use","source":46},{"line":61,"start":7123,"end":7131,"expression":"n/(b+2r)","context":62,"kind":"use","source":46},{"line":61,"start":7145,"end":7146,"expression":"n","context":63,"kind":"use","source":46},{"line":63,"start":7299,"end":7300,"expression":"s","context":64,"kind":"use","source":46},{"line":63,"start":7338,"end":7340,"expression":"sM","context":65,"kind":"use","source":46},{"line":65,"start":7530,"end":7531,"expression":"p","context":66,"kind":"use","source":46},{"line":65,"start":7561,"end":7562,"expression":"w","context":66,"kind":"use","source":46},{"line":66,"start":7599,"end":7616,"expression":"\np(2w+1)-w(w+1).\n","context":66,"kind":"use","source":46},{"line":69,"start":7713,"end":7723,"expression":"1\\le p\\le6","context":66,"kind":"use","source":46},{"line":69,"start":7747,"end":7748,"expression":"w","context":66,"kind":"use","source":46},{"line":69,"start":7762,"end":7765,"expression":"w=0","context":66,"kind":"use","source":46},{"line":69,"start":7772,"end":7777,"expression":"w=p-1","context":66,"kind":"use","source":46},{"line":73,"start":8095,"end":8109,"expression":"c=(2,1,0,-1)^T","context":1151,"kind":"use","source":46},{"line":73,"start":8123,"end":8146,"expression":"C(c)_{jk}=c_{j-k\\bmod4}","context":1151,"kind":"use","source":46},{"line":73,"start":8180,"end":8204,"expression":"F_{jk}=\\mathrm i^{-jk}/2","context":1151,"kind":"use","source":46},{"line":74,"start":8268,"end":8335,"expression":"\nC(c)=F^*\\diag(2Fc)F,\\qquad 2Fc=(2,2-2\\mathrm i,2,2+2\\mathrm i)^T.\n","context":1151,"kind":"use","source":46},{"line":79,"start":8390,"end":8404,"expression":"x=(1,2,-1,0)^T","context":1152,"kind":"use","source":46},{"line":79,"start":8435,"end":8447,"expression":"(0,6,0,-2)^T","context":1153,"kind":"use","source":46},{"line":79,"start":8464,"end":8465,"expression":"x","context":1154,"kind":"use","source":46},{"line":79,"start":8529,"end":8532,"expression":"F^*","context":70,"kind":"use","source":46},{"line":79,"start":8589,"end":8603,"expression":"\\QQ(\\mathrm i)","context":71,"kind":"use","source":46},{"line":81,"start":8853,"end":8864,"expression":"\\sqrt n\\,Fc","context":72,"kind":"use","source":46},{"line":81,"start":8872,"end":8874,"expression":"Fc","context":73,"kind":"use","source":46},{"line":81,"start":8915,"end":8918,"expression":"C/2","context":74,"kind":"use","source":46},{"line":81,"start":8933,"end":8934,"expression":"C","context":75,"kind":"use","source":46},{"line":88,"start":9766,"end":9818,"expression":"\nT=\\mat{2&-1&5\\\\3&2&-1\\\\4&3&2},\\qquad x=(1,2,-1)^T.\n","context":1155,"kind":"use","source":46},{"line":91,"start":9948,"end":9966,"expression":"g=(2,3,4,0,5,-1)^T","context":77,"kind":"use","source":46},{"line":94,"start":10070,"end":10073,"expression":"j<3","context":78,"kind":"use","source":46},{"line":94,"start":10092,"end":10098,"expression":"k\\le j","context":78,"kind":"use","source":46},{"line":94,"start":10135,"end":10142,"expression":"c_{j-k}","context":78,"kind":"use","source":46},{"line":94,"start":10161,"end":10164,"expression":"k>j","context":78,"kind":"use","source":46},{"line":94,"start":10209,"end":10216,"expression":"r_{k-j}","context":79,"kind":"use","source":46},{"line":95,"start":10316,"end":10332,"expression":"\nTx=(-5,8,8)^T.\n","context":80,"kind":"use","source":46},{"line":98,"start":10441,"end":10443,"expression":"Tx","context":81,"kind":"use","source":46},{"line":100,"start":10506,"end":10513,"expression":"(2,3,4)","context":82,"kind":"use","source":46},{"line":100,"start":10657,"end":10659,"expression":"Tx","context":83,"kind":"use","source":46},{"line":102,"start":10717,"end":10718,"expression":"g","context":84,"kind":"use","source":46},{"line":102,"start":10828,"end":10830,"expression":"2n","context":84,"kind":"use","source":46},{"line":107,"start":11332,"end":11394,"expression":"\n(A\\otimes B)\\operatorname{vec}(X)=\\operatorname{vec}(BXA^T).\n","context":1156,"kind":"use","source":46},{"line":110,"start":11503,"end":11528,"expression":"A=\\mat{1&\\mathrm i\\\\2&-1}","context":1157,"kind":"use","source":46},{"line":110,"start":11546,"end":11562,"expression":"B=\\mat{2&1\\\\0&3}","context":87,"kind":"use","source":46},{"line":110,"start":11569,"end":11585,"expression":"X=\\mat{1&2\\\\3&4}","context":88,"kind":"use","source":46},{"line":113,"start":11656,"end":11667,"expression":"(1,3,2,4)^T","context":89,"kind":"use","source":46},{"line":114,"start":11737,"end":11783,"expression":"\nBXA^T=\\mat{5+8\\mathrm i&2\\\\9+12\\mathrm i&6}.\n","context":90,"kind":"use","source":46},{"line":117,"start":11797,"end":11800,"expression":"A^T","context":91,"kind":"use","source":46},{"line":117,"start":11808,"end":11811,"expression":"A^*","context":91,"kind":"use","source":46},{"line":119,"start":12004,"end":12005,"expression":"A","context":91,"kind":"use","source":46},{"line":119,"start":12017,"end":12018,"expression":"m","context":91,"kind":"use","source":46},{"line":119,"start":12025,"end":12026,"expression":"B","context":91,"kind":"use","source":46},{"line":119,"start":12038,"end":12039,"expression":"n","context":91,"kind":"use","source":46},{"line":119,"start":12077,"end":12083,"expression":"m^2n^2","context":91,"kind":"use","source":46},{"line":119,"start":12113,"end":12120,"expression":"m^2+n^2","context":91,"kind":"use","source":46},{"line":119,"start":12141,"end":12146,"expression":"BXA^T","context":91,"kind":"use","source":46},{"line":119,"start":12155,"end":12166,"expression":"2n^2m+2nm^2","context":91,"kind":"use","source":46},{"line":125,"start":12824,"end":12831,"expression":"H_1=[1]","context":1158,"kind":"use","source":46},{"line":126,"start":12839,"end":12891,"expression":"\nH_{2n}=\\mat{H_n&H_n\\\\H_n&-H_n},\\qquad H_nH_n^T=nI.\n","context":1159,"kind":"use","source":46},{"line":129,"start":12934,"end":12953,"expression":"(1,2,3,4,5,6,7,8)^T","context":94,"kind":"use","source":46},{"line":132,"start":13239,"end":13247,"expression":"8\\times8","context":94,"kind":"use","source":46},{"line":134,"start":13283,"end":13307,"expression":"(36,-4,-8,0,-16,0,0,0)^T","context":95,"kind":"use","source":46},{"line":134,"start":13355,"end":13362,"expression":"H_8^T/8","context":95,"kind":"use","source":46},{"line":134,"start":13503,"end":13513,"expression":"H_8/\\sqrt8","context":95,"kind":"use","source":46},{"line":142,"start":14399,"end":14413,"expression":"A=D_A+u_av_a^T","context":1160,"kind":"use","source":46},{"line":142,"start":14446,"end":14468,"expression":"D_B=\\diag(1,2,3,4,5,6)","context":1160,"kind":"use","source":46},{"line":142,"start":14472,"end":14491,"expression":"u_b=(1,0,1,0,1,0)^T","context":1160,"kind":"use","source":46},{"line":142,"start":14499,"end":14518,"expression":"v_b=(0,1,0,1,0,1)^T","context":1160,"kind":"use","source":46},{"line":142,"start":14529,"end":14543,"expression":"B=D_B+u_bv_b^T","context":1160,"kind":"use","source":46},{"line":143,"start":14548,"end":14613,"expression":"\nU_C=[D_Au_b+u_a(v_a^Tu_b)\\ \\ u_a],\\qquad\nV_C=[v_b\\ \\ D_B^Tv_a].\n","context":1160,"kind":"use","source":46},{"line":149,"start":14685,"end":14687,"expression":"AB","context":1161,"kind":"use","source":46},{"line":149,"start":14751,"end":14756,"expression":"v_b^T","context":1162,"kind":"use","source":46},{"line":150,"start":14761,"end":14833,"expression":"\nD_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T\n=\\bigl(D_Au_b+u_a(v_a^Tu_b)\\bigr)v_b^T.\n","context":1163,"kind":"use","source":46},{"line":154,"start":14859,"end":14870,"expression":"u_av_a^TD_B","context":99,"kind":"use","source":46},{"line":154,"start":14879,"end":14897,"expression":"AB=D_AD_B+U_CV_C^T","context":100,"kind":"use","source":46},{"line":156,"start":15050,"end":15058,"expression":"v_a^Tu_b","context":101,"kind":"use","source":46},{"line":156,"start":15122,"end":15131,"expression":"3u_av_b^T","context":102,"kind":"use","source":46},{"line":162,"start":15959,"end":15982,"expression":"\\frac18\\mat{3&-1\\\\-1&3}","context":1164,"kind":"use","source":46},{"line":162,"start":15992,"end":16003,"expression":"D^{-1}u=u/4","context":104,"kind":"use","source":46},{"line":163,"start":16011,"end":16036,"expression":"\nS=1+u^TD^{-1}u=\\frac52.\n","context":104,"kind":"use","source":46},{"line":167,"start":16093,"end":16155,"expression":"\nA^{-1}=D^{-1}-D^{-1}uS^{-1}u^TD^{-1}\n=D^{-1}-\\frac1{40}uu^T.\n","context":104,"kind":"use","source":46},{"line":173,"start":16223,"end":16224,"expression":"D","context":104,"kind":"use","source":46},{"line":173,"start":16255,"end":16260,"expression":"D=I_2","context":104,"kind":"use","source":46},{"line":173,"start":16264,"end":16269,"expression":"u=e_1","context":104,"kind":"use","source":46},{"line":173,"start":16277,"end":16283,"expression":"v=-e_1","context":104,"kind":"use","source":46},{"line":173,"start":16301,"end":16311,"expression":"\\diag(0,1)","context":104,"kind":"use","source":46},{"line":173,"start":16332,"end":16333,"expression":"S","context":104,"kind":"use","source":46},{"line":175,"start":16488,"end":16489,"expression":"A","context":105,"kind":"use","source":46},{"line":175,"start":16573,"end":16592,"expression":"\\det A=\\det D\\det S","context":105,"kind":"use","source":46},{"line":177,"start":16750,"end":16751,"expression":"S","context":105,"kind":"use","source":46},{"line":177,"start":16822,"end":16832,"expression":"\\|S^{-1}\\|","context":106,"kind":"use","source":46},{"line":177,"start":16920,"end":16927,"expression":"D^{-1}u","context":107,"kind":"use","source":46},{"line":177,"start":16934,"end":16943,"expression":"v^TD^{-1}","context":108,"kind":"use","source":46},{"line":182,"start":17274,"end":17375,"expression":"\nQ_L=\\mat{3/5&0\\\\4/5&0\\\\0&1\\\\0&0},\\quad\nR_L=\\mat{1&0&1\\\\0&1&0},\\quad\nR_R=\\mat{3&0&2\\\\0&2&0\\\\1&0&-1}.\n","context":1165,"kind":"use","source":46},{"line":187,"start":17383,"end":17391,"expression":"L=Q_LR_L","context":110,"kind":"use","source":46},{"line":187,"start":17398,"end":17403,"expression":"R=R_R","context":110,"kind":"use","source":46},{"line":187,"start":17432,"end":17433,"expression":"L","context":110,"kind":"use","source":46},{"line":187,"start":17488,"end":17489,"expression":"R","context":110,"kind":"use","source":46},{"line":190,"start":17548,"end":17597,"expression":"\nG=R_LR_R^T=\\mat{5&0&0\\\\0&2&0},\\qquad LR^T=Q_LG.\n","context":111,"kind":"use","source":46},{"line":193,"start":17750,"end":17753,"expression":"Q_L","context":112,"kind":"use","source":46},{"line":193,"start":17797,"end":17798,"expression":"L","context":112,"kind":"use","source":46},{"line":202,"start":18766,"end":18810,"expression":"\nA_0=LR^T=\\mat{3&0&0\\\\4&0&0\\\\0&2&0\\\\0&0&0}.\n","context":1166,"kind":"use","source":46},{"line":205,"start":18969,"end":18978,"expression":"\\sqrt{29}","context":114,"kind":"use","source":46},{"line":208,"start":19116,"end":19130,"expression":"C(B)=Q_L^*BQ_R","context":114,"kind":"use","source":46},{"line":208,"start":19196,"end":19197,"expression":"k","context":114,"kind":"use","source":46},{"line":210,"start":19467,"end":19468,"expression":"d","context":115,"kind":"use","source":46},{"line":210,"start":19501,"end":19509,"expression":"0\\le k<d","context":115,"kind":"use","source":46},{"line":210,"start":19577,"end":19578,"expression":"k","context":115,"kind":"use","source":46},{"line":210,"start":19633,"end":19636,"expression":"k+1","context":115,"kind":"use","source":46},{"line":210,"start":19711,"end":19723,"expression":"\\sigma_{k+1}","context":115,"kind":"use","source":46},{"line":212,"start":19892,"end":19897,"expression":"[0,1]","context":116,"kind":"use","source":46},{"line":216,"start":20437,"end":20444,"expression":"(2,3,4)","context":1167,"kind":"use","source":46},{"line":216,"start":20462,"end":20467,"expression":"p_0=1","context":1167,"kind":"use","source":46},{"line":216,"start":20490,"end":20505,"expression":"(1/10,1/5,1/20)","context":1167,"kind":"use","source":46},{"line":217,"start":20562,"end":20654,"expression":"\n\\widehat p_j=a_j\\widehat p_{j-1}+e_j,\\qquad\n\\widehat p_3-p_3=12e_1+4e_2+e_3=\\frac{41}{20}.\n","context":1167,"kind":"use","source":46},{"line":223,"start":20799,"end":20820,"expression":"(4+e_3)(3+e_2)(2+e_1)","context":1168,"kind":"use","source":46},{"line":223,"start":20837,"end":20844,"expression":"402/125","context":119,"kind":"use","source":46},{"line":223,"start":20852,"end":20857,"expression":"41/20","context":120,"kind":"use","source":46},{"line":225,"start":21210,"end":21214,"expression":"7/20","context":121,"kind":"use","source":46},{"line":231,"start":21951,"end":21952,"expression":"K","context":1169,"kind":"use","source":46},{"line":231,"start":21970,"end":21975,"expression":"0\\to1","context":1170,"kind":"use","source":46},{"line":231,"start":21979,"end":21984,"expression":"0\\to2","context":1171,"kind":"use","source":46},{"line":231,"start":21988,"end":21993,"expression":"1\\to3","context":1172,"kind":"use","source":46},{"line":231,"start":22001,"end":22006,"expression":"2\\to3","context":1173,"kind":"use","source":46},{"line":231,"start":22028,"end":22036,"expression":"1,1,1,-1","context":1174,"kind":"use","source":46},{"line":233,"start":22164,"end":22223,"expression":"\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n","context":1175,"kind":"use","source":46},{"line":237,"start":22390,"end":22429,"expression":"\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)","context":129,"kind":"use","source":46},{"line":239,"start":22580,"end":22581,"expression":"h","context":130,"kind":"use","source":46},{"line":239,"start":22607,"end":22614,"expression":"K^{h+1}","context":131,"kind":"use","source":46},{"line":239,"start":22635,"end":22644,"expression":"K^{h+1}=0","context":132,"kind":"use","source":46},{"line":239,"start":22718,"end":22725,"expression":"K^h\\ne0","context":133,"kind":"use","source":46},{"line":239,"start":22814,"end":22819,"expression":"K^2=0","context":134,"kind":"use","source":46},{"line":245,"start":23471,"end":23473,"expression":"-1","context":1176,"kind":"use","source":46},{"line":245,"start":23522,"end":23527,"expression":"0\\to2","context":1177,"kind":"use","source":46},{"line":245,"start":23531,"end":23536,"expression":"1\\to2","context":1178,"kind":"use","source":46},{"line":245,"start":23540,"end":23545,"expression":"1\\to3","context":1179,"kind":"use","source":46},{"line":245,"start":23549,"end":23554,"expression":"2\\to4","context":1180,"kind":"use","source":46},{"line":245,"start":23558,"end":23563,"expression":"3\\to4","context":1181,"kind":"use","source":46},{"line":245,"start":23567,"end":23572,"expression":"3\\to5","context":1182,"kind":"use","source":46},{"line":245,"start":23576,"end":23581,"expression":"4\\to6","context":1183,"kind":"use","source":46},{"line":245,"start":23589,"end":23594,"expression":"5\\to6","context":1184,"kind":"use","source":46},{"line":245,"start":23604,"end":23617,"expression":"(2I+K)x=\\ones","context":1185,"kind":"use","source":46},{"line":249,"start":23840,"end":23923,"expression":"\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n","context":145,"kind":"use","source":46},{"line":262,"start":24983,"end":25013,"expression":"\nA=\\mat{2&0&1\\\\0&2&1\\\\1&1&2}.\n","context":1186,"kind":"use","source":46},{"line":265,"start":25073,"end":25076,"expression":"[2]","context":147,"kind":"use","source":46},{"line":266,"start":25108,"end":25169,"expression":"\nS=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR}\n=\\mat{3/2&-1/2\\\\-1/2&3/2}.\n","context":148,"kind":"use","source":46},{"line":272,"start":25354,"end":25358,"expression":"-1/2","context":149,"kind":"use","source":46},{"line":274,"start":25496,"end":25500,"expression":"+1/2","context":150,"kind":"use","source":46},{"line":274,"start":25573,"end":25582,"expression":"\\frac32 I","context":150,"kind":"use","source":46},{"line":278,"start":26187,"end":26193,"expression":"S^{-1}","context":151,"kind":"use","source":46},{"line":278,"start":26222,"end":26228,"expression":"A^{-1}","context":151,"kind":"use","source":46},{"line":282,"start":26545,"end":26550,"expression":"A=I_2","context":1187,"kind":"use","source":46},{"line":282,"start":26608,"end":26617,"expression":"v=(1,1)^T","context":1188,"kind":"use","source":46},{"line":283,"start":26731,"end":26794,"expression":"\np(A)v=p(1)(1,1)^T,\\qquad \\dim\\mathcal K_m(A,v)=1\\quad(m\\ge1).\n","context":1189,"kind":"use","source":46},{"line":287,"start":26867,"end":26872,"expression":"\\QQ^2","context":1190,"kind":"use","source":46},{"line":289,"start":27082,"end":27094,"expression":"A=\\diag(1,2)","context":156,"kind":"use","source":46},{"line":289,"start":27110,"end":27111,"expression":"v","context":157,"kind":"use","source":46},{"line":289,"start":27118,"end":27128,"expression":"Av=(1,2)^T","context":158,"kind":"use","source":46},{"line":289,"start":27315,"end":27320,"expression":"A=I_2","context":159,"kind":"use","source":46},{"line":291,"start":27436,"end":27441,"expression":"A=D+K","context":160,"kind":"use","source":46},{"line":291,"start":27449,"end":27450,"expression":"D","context":161,"kind":"use","source":46},{"line":292,"start":27476,"end":27537,"expression":"\n\\|A^j-D^j\\|\\le jM^{j-1}\\|K\\|,\n\\qquad M=\\max\\{\\|A\\|,\\|D\\|\\}.\n","context":162,"kind":"use","source":46},{"line":301,"start":28099,"end":28167,"expression":"\nA=\\mat{4&1\\\\1&1},\\quad D=\\diag(4,1),\\quad\nB=D^{-1/2}=\\diag(1/2,1).\n","context":1191,"kind":"use","source":46},{"line":305,"start":28176,"end":28202,"expression":"BAB=I+E=\\mat{1&1/2\\\\1/2&1}","context":164,"kind":"use","source":46},{"line":305,"start":28232,"end":28235,"expression":"1/2","context":165,"kind":"use","source":46},{"line":305,"start":28242,"end":28245,"expression":"3/2","context":165,"kind":"use","source":46},{"line":308,"start":28367,"end":28374,"expression":"x^TAx=1","context":165,"kind":"use","source":46},{"line":308,"start":28386,"end":28390,"expression":"x=By","context":165,"kind":"use","source":46},{"line":308,"start":28417,"end":28428,"expression":"y^T(BAB)y=1","context":165,"kind":"use","source":46},{"line":310,"start":28552,"end":28563,"expression":"\\|E\\|_2=1/2","context":166,"kind":"use","source":46},{"line":311,"start":28604,"end":28683,"expression":"\n\\kappa_2(D^{-1/2}AD^{-1/2})\n\\le\\frac{1+\\eta}{1-\\eta},\\qquad \\|E\\|_2\\le\\eta<1,\n","context":167,"kind":"use","source":46},{"line":321,"start":29521,"end":29526,"expression":"\\QQ^4","context":1192,"kind":"use","source":46},{"line":322,"start":29558,"end":29603,"expression":"\nA=I+E,\\qquad E=\\frac1{100}(\\ones\\ones^T-I).\n","context":1192,"kind":"use","source":46},{"line":325,"start":29648,"end":29649,"expression":"I","context":1192,"kind":"use","source":46},{"line":325,"start":29682,"end":29687,"expression":"1/100","context":1192,"kind":"use","source":46},{"line":328,"start":29853,"end":29870,"expression":"\\|E\\|_F^2=12/10^4","context":1193,"kind":"use","source":46},{"line":328,"start":29927,"end":29932,"expression":"3/100","context":170,"kind":"use","source":46},{"line":328,"start":29990,"end":29996,"expression":"-1/100","context":171,"kind":"use","source":46},{"line":328,"start":30006,"end":30019,"expression":"\\|E\\|_2=3/100","context":172,"kind":"use","source":46},{"line":333,"start":30326,"end":30384,"expression":"\n\\|(\\widehat A-A)x\\|\\le\\sum_e\\|E_e\\|\\,\\|P_{\\alpha(e)}x\\|.\n","context":173,"kind":"use","source":46},{"line":342,"start":31145,"end":31154,"expression":"s=(1,2,4)","context":1194,"kind":"use","source":46},{"line":342,"start":31176,"end":31185,"expression":"c=(1,1,2)","context":1194,"kind":"use","source":46},{"line":342,"start":31200,"end":31206,"expression":"\\tau=1","context":1194,"kind":"use","source":46},{"line":342,"start":31221,"end":31248,"expression":"\\sum_ec_e\\log(1/\\epsilon_e)","context":1194,"kind":"use","source":46},{"line":342,"start":31262,"end":31283,"expression":"\\sum_es_e\\epsilon_e=1","context":1194,"kind":"use","source":46},{"line":343,"start":31293,"end":31353,"expression":"\n\\epsilon_e=\\frac{c_e}{4s_e},\\qquad\n\\epsilon=(1/4,1/8,1/8).\n","context":1194,"kind":"use","source":46},{"line":349,"start":31455,"end":31468,"expression":"(1/4,1/4,1/2)","context":1195,"kind":"use","source":46},{"line":351,"start":31698,"end":31714,"expression":"m=(1/10,1/2,1/2)","context":176,"kind":"use","source":46},{"line":351,"start":31779,"end":31791,"expression":"\\lambda=10/3","context":177,"kind":"use","source":46},{"line":352,"start":31820,"end":31896,"expression":"\n\\epsilon_e=\\min\\{m_e,c_e/(\\lambda s_e)\\},\\qquad\n\\epsilon=(1/10,3/20,3/20).\n","context":178,"kind":"use","source":46},{"line":356,"start":31931,"end":31946,"expression":"(1/10,3/10,3/5)","context":178,"kind":"use","source":46},{"line":362,"start":32733,"end":32768,"expression":"(\\text{integer weight},\\text{cost})","context":1196,"kind":"use","source":46},{"line":363,"start":32781,"end":32888,"expression":"\n\\begin{aligned}\n1 &: (0,7),(2,3),(4,1),\\\\\n2 &: (0,6),(1,4),(3,1),\\\\\n3 &: (0,9),(2,5),(5,1).\n\\end{aligned}\n","context":1197,"kind":"use","source":46},{"line":370,"start":32910,"end":32916,"expression":"3^3=27","context":181,"kind":"use","source":46},{"line":373,"start":33190,"end":33197,"expression":"(2,0,2)","context":181,"kind":"use","source":46},{"line":373,"start":33211,"end":33218,"expression":"(3,6,5)","context":181,"kind":"use","source":46},{"line":375,"start":33491,"end":33498,"expression":"+\\infty","context":182,"kind":"use","source":46},{"line":377,"start":33598,"end":33602,"expression":"3/50","context":182,"kind":"use","source":46},{"line":377,"start":33609,"end":33613,"expression":"2/25","context":182,"kind":"use","source":46},{"line":377,"start":33649,"end":33653,"expression":"1/10","context":182,"kind":"use","source":46},{"line":377,"start":33723,"end":33732,"expression":"7/50>1/10","context":182,"kind":"use","source":46},{"line":381,"start":34222,"end":34223,"expression":"r","context":1198,"kind":"use","source":46},{"line":381,"start":34239,"end":34242,"expression":"a,b","context":1199,"kind":"use","source":46},{"line":381,"start":34287,"end":34288,"expression":"r","context":1200,"kind":"use","source":46},{"line":381,"start":34364,"end":34365,"expression":"a","context":1201,"kind":"use","source":46},{"line":381,"start":34371,"end":34372,"expression":"b","context":1201,"kind":"use","source":46},{"line":381,"start":34413,"end":34414,"expression":"a","context":1201,"kind":"use","source":46},{"line":381,"start":34459,"end":34460,"expression":"b","context":1202,"kind":"use","source":46},{"line":383,"start":34544,"end":34570,"expression":"(\\text{cost},\\text{error})","context":1203,"kind":"use","source":46},{"line":384,"start":34580,"end":34621,"expression":"\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n","context":1204,"kind":"use","source":46},{"line":387,"start":34670,"end":34683,"expression":"\\{a_0,a_1,b\\}","context":189,"kind":"use","source":46},{"line":387,"start":34744,"end":34761,"expression":"\\cost+\\lambda\\err","context":190,"kind":"use","source":46},{"line":387,"start":34772,"end":34783,"expression":"\\lambda\\ge0","context":191,"kind":"use","source":46},{"line":389,"start":34873,"end":34896,"expression":"8+2\\lambda\\le1+6\\lambda","context":192,"kind":"use","source":46},{"line":389,"start":34906,"end":34919,"expression":"\\lambda\\ge7/4","context":193,"kind":"use","source":46},{"line":389,"start":34964,"end":34979,"expression":"8+2\\lambda\\le10","context":194,"kind":"use","source":46},{"line":389,"start":34989,"end":35000,"expression":"\\lambda\\le1","context":195,"kind":"use","source":46},{"line":395,"start":35638,"end":35647,"expression":"\\lambda=2","context":1205,"kind":"use","source":46},{"line":395,"start":35682,"end":35685,"expression":"a_0","context":1206,"kind":"use","source":46},{"line":395,"start":35742,"end":35745,"expression":"a_0","context":1207,"kind":"use","source":46},{"line":395,"start":35760,"end":35761,"expression":"a","context":1208,"kind":"use","source":46},{"line":395,"start":35778,"end":35779,"expression":"r","context":1209,"kind":"use","source":46},{"line":395,"start":35802,"end":35803,"expression":"b","context":1210,"kind":"use","source":46},{"line":400,"start":36379,"end":36382,"expression":"3/2","context":202,"kind":"use","source":46},{"line":400,"start":36438,"end":36441,"expression":"3/2","context":203,"kind":"use","source":46},{"line":411,"start":37956,"end":37964,"expression":"2\\times2","context":204,"kind":"use","source":46},{"line":411,"start":37999,"end":38000,"expression":"q","context":205,"kind":"use","source":46},{"line":412,"start":38054,"end":38109,"expression":"\nB_{\\rm stream}=8\\cdot3(4+2q+4q)=96+144q\\text{ bytes}.\n","context":206,"kind":"use","source":46},{"line":415,"start":38135,"end":38138,"expression":"q=1","context":207,"kind":"use","source":46},{"line":415,"start":38158,"end":38161,"expression":"q=4","context":208,"kind":"use","source":46},{"line":417,"start":38322,"end":38323,"expression":"P","context":209,"kind":"use","source":46},{"line":417,"start":38340,"end":38341,"expression":"W","context":209,"kind":"use","source":46},{"line":417,"start":38345,"end":38362,"expression":"T\\ge\\max(F/P,B/W)","context":209,"kind":"use","source":46},{"line":422,"start":39007,"end":39039,"expression":"\nT_b(R)=7R,\\qquad T_c(R)=30+2R.\n","context":1211,"kind":"use","source":46},{"line":426,"start":39066,"end":39069,"expression":"R=6","context":1212,"kind":"use","source":46},{"line":426,"start":39128,"end":39131,"expression":"R=7","context":1213,"kind":"use","source":46},{"line":427,"start":39215,"end":39261,"expression":"\n\\left\\lfloor\\frac{30}{7-2}\\right\\rfloor+1=7.\n","context":1214,"kind":"use","source":46},{"line":438,"start":40433,"end":40439,"expression":"12/2=6","context":1215,"kind":"use","source":46},{"line":451,"start":42222,"end":42228,"expression":"D_AD_B","context":1216,"kind":"use","source":46},{"line":451,"start":42265,"end":42272,"expression":"U_C,V_C","context":1217,"kind":"use","source":46},{"line":451,"start":42301,"end":42320,"expression":"x=(1,-1,2,0,-2,3)^T","context":1218,"kind":"use","source":46},{"line":452,"start":42325,"end":42376,"expression":"\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n","context":1219,"kind":"use","source":46},{"line":458,"start":42461,"end":42481,"expression":"(1,-5,18,6,-12,44)^T","context":1220,"kind":"use","source":46},{"line":458,"start":42503,"end":42524,"expression":"(25,21,25,21,25,21)^T","context":1221,"kind":"use","source":46},{"line":459,"start":42542,"end":42566,"expression":"\n(26,16,43,27,13,65)^T.\n","context":220,"kind":"use","source":46},{"line":462,"start":42607,"end":42608,"expression":"A","context":221,"kind":"use","source":46},{"line":462,"start":42615,"end":42616,"expression":"B","context":222,"kind":"use","source":46},{"line":462,"start":42674,"end":42675,"expression":"x","context":223,"kind":"use","source":46}],"entries":[{"id":"symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"definition-passage","occurrences":[459,1,3,55,99,103,129,184,419,425,462,470,471,493,496,514,531,538,597,598,599,600,602,603,604,605,609,610,611,612,613,616,629,630,636,638,639,642,742,748,966,984,985,988,989,994,995,996,998,1057,1059,1063,1064,1067,1110,1114,1116,1117,1203,1204,1207,1395,1400,1431,1436,1438,1439,1441,1444,1445,1446,1482,1485,1486,1504,1505,1506,1510,1512,1516,1517,1673,1709,1712,1817,2011,2013,2065,2109,2113,2139,2194]},{"id":"symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"definition-passage","occurrences":[459,24,25,45,1238,1315,1332,2034,2035,2055,5,12,21,41,42,55,69,70,73,74,80,85,91,99,127,135,137,145,149,150,153,155,160,189,190,195,196,206,209,210,222,223,228,240,244,251,253,277,292,294,296,308,314,317,322,330,337,338,355,356,357,358,366,367,369,370,371,372,462,463,464,468,476,492,493,495,506,513,514,520,521,561,562,575,584,591,595,597,598,599,601,602,605,609,610,611,613,616,622,646,647,648,650,653,657,658,672,673,674,676,725,730,734,740,743,744,748,749,750,778,800,801,803,821,825,827,828,841,848,849,868,869,890,892,969,971,974,975,984,985,988,989,990,991,994,995,996,998,1004,1022,1023,1025,1026,1030,1035,1041,1042,1048,1076,1077,1078,1079,1080,1081,1084,1086,1094,1099,1104,1106,1110,1114,1116,1117,1122,1124,1126,1127,1128,1130,1132,1133,1134,1150,1157,1158,1159,1163,1166,1174,1183,1189,1192,1193,1221,1231,1242,1244,1245,1256,1258,1321,1339,1341,1372,1375,1405,1408,1409,1412,1415,1417,1423,1462,1464,1469,1470,1472,1474,1476,1490,1494,1517,1523,1590,1648,1671,1759,1789,1812,1814,1867,1873,1894,1896,1897,1972,1973,1974,1990,1998,2015,2022,2031,2051,2052,2065,2079,2080,2083,2084,2090,2095,2101,2109,2137,2145,2147,2155,2159,2160,2163,2165,2170,2199,2200,2205,2206,2216,2219,2220,2232,2233,2238,2250,2254,2261,2263,2287,2302,2304,2306,2318,2324,2327,2332]},{"id":"symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","occurrences":[383,384,385,388,389,395,459,460,477,716,871,22,23,27,45,411,1238,1240,1241,1250,1251,1252,1310,1315,1332,2032,2033,2037,2055,1,2,3,5,6,12,13,14,16,20,21,28,30,43,68,69,70,73,74,80,81,94,95,96,97,98,99,100,109,110,116,122,123,124,125,127,138,139,145,146,147,148,149,150,153,155,159,160,161,163,164,165,169,174,175,176,179,182,184,193,195,206,210,212,222,223,224,226,232,233,234,236,238,239,240,242,243,244,247,248,249,250,253,256,259,261,262,263,265,266,268,269,270,274,298,308,309,310,314,315,316,317,318,321,322,329,330,331,332,333,334,335,337,338,339,361,371,372,373,379,380,381,398,400,402,404,418,419,424,425,426,429,437,439,440,450,454,462,465,473,476,502,503,506,513,514,528,530,531,534,535,544,560,566,577,586,587,588,595,596,597,598,599,600,602,605,607,609,610,613,622,623,626,629,630,635,639,646,647,648,649,650,653,656,657,658,661,666,672,676,678,680,688,692,704,717,718,719,721,723,725,729,731,734,738,747,749,750,753,757,763,766,767,776,777,778,779,780,785,786,788,800,801,803,813,814,821,822,825,827,828,829,830,841,842,890,955,956,957,958,959,961,966,967,971,972,975,979,980,981,984,985,988,989,994,995,1003,1004,1005,1013,1022,1025,1026,1030,1042,1045,1048,1052,1055,1056,1057,1059,1060,1062,1063,1082,1086,1106,1110,1112,1114,1115,1116,1117,1118,1120,1121,1122,1124,1127,1128,1130,1132,1133,1134,1135,1138,1142,1143,1145,1146,1150,1152,1156,1157,1163,1166,1174,1185,1189,1199,1200,1202,1204,1205,1206,1207,1208,1211,1215,1217,1223,1227,1244,1245,1247,1248,1249,1253,1256,1258,1259,1260,1261,1265,1266,1267,1268,1270,1271,1272,1273,1275,1280,1282,1283,1284,1286,1289,1290,1293,1297,1326,1328,1333,1338,1339,1340,1344,1349,1353,1358,1359,1361,1363,1368,1372,1375,1377,1379,1382,1383,1385,1386,1387,1389,1392,1393,1395,1400,1401,1405,1406,1408,1409,1411,1415,1417,1431,1432,1436,1437,1438,1439,1441,1442,1443,1444,1446,1452,1455,1466,1467,1468,1470,1471,1482,1483,1484,1490,1491,1493,1500,1501,1502,1504,1505,1506,1507,1509,1510,1511,1517,1590,1613,1614,1648,1649,1672,1673,1674,1708,1709,1712,1759,1760,1790,1791,1798,1799,1800,1818,1819,1820,1821,1866,1867,1869,1873,1884,1970,1974,1982,1983,1989,1990,1991,1999,2011,2012,2013,2015,2016,2022,2023,2024,2026,2030,2031,2038,2040,2053,2078,2079,2080,2083,2084,2090,2091,2104,2105,2106,2107,2108,2109,2110,2119,2120,2126,2132,2133,2134,2135,2137,2148,2149,2155,2156,2157,2158,2159,2160,2163,2165,2169,2170,2171,2173,2174,2175,2179,2184,2185,2186,2189,2192,2194,2203,2205,2216,2220,2222,2232,2233,2234,2236,2242,2243,2244,2246,2248,2249,2250,2252,2253,2254,2257,2258,2259,2260,2263,2266,2269,2271,2272,2273,2275,2276,2278,2279,2280,2284,2308,2318,2319,2320,2324,2325,2326,2327,2328,2331,2332]},{"id":"symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"definition-passage","occurrences":[392,406,409,459,713,716,45,46,410,411,412,1238,1239,1240,1241,1332,2055,2056,3,9,12,14,16,18,19,53,63,69,70,122,123,127,128,129,130,136,145,150,151,153,155,161,168,169,182,207,224,226,231,232,234,236,238,239,240,242,243,244,247,249,251,253,260,319,322,326,330,334,335,340,341,381,397,398,400,402,403,404,416,418,419,420,424,425,426,462,514,518,557,560,565,566,567,569,597,598,599,600,601,602,604,605,606,645,646,647,648,649,650,653,657,658,660,662,663,664,667,668,676,678,684,687,688,694,695,698,700,703,704,706,709,710,711,717,719,721,722,723,725,728,729,730,731,734,740,745,747,748,750,753,754,755,756,757,758,763,764,766,813,814,815,838,956,957,958,959,961,963,965,966,967,971,972,973,975,976,979,981,982,983,984,986,988,989,990,991,992,1004,1005,1006,1007,1008,1009,1011,1110,1112,1122,1124,1127,1128,1130,1132,1133,1134,1135,1138,1140,1142,1143,1144,1150,1152,1154,1159,1224,1227,1233,1258,1259,1344,1347,1351,1352,1353,1354,1358,1359,1363,1368,1391,1392,1393,1395,1399,1401,1402,1405,1406,1414,1431,1439,1590,1591,1709,1732,1746,1782,1799,1800,1811,1816,1817,1818,1819,1820,1821,1866,1874,1884,1902,1991,2013,2019,2022,2024,2026,2028,2029,2063,2073,2079,2080,2132,2133,2137,2138,2139,2140,2146,2155,2160,2161,2163,2165,2171,2178,2179,2192,2217,2234,2236,2241,2242,2244,2246,2248,2249,2250,2252,2253,2254,2257,2259,2261,2263,2270,2329,2332,2336]},{"id":"symbol-6153a08bafbf354e","tex":"A^{m-1}","aliases":"A^m-1","display":"A^{m-1}","kind":"definition-passage","occurrences":[716,1138]},{"id":"symbol-4faecfcca4a3cf4c","tex":"\\abs","aliases":"abs","display":null,"kind":"definition-passage","occurrences":[871,505,742]},{"id":"symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"definition-passage","occurrences":[389,391,394,395,410,411,412,31,32,40,41,42,43,44,122,124,127,132,136,146,149,150,151,154,155,184,207,243,244,248,249,308,313,319,322,327,398,402,417,418,421,424,426,441,476,481,482,485,487,558,560,565,566,567,568,597,598,599,600,601,602,603,604,641,642,759,763,796,805,855,862,863,865,866,867,868,869,880,883,961,964,965,967,985,987,988,989,990,991,992,1086,1088,1089,1090,1167,1168,1178,1183,1184,1185,1188,1189,1191,1192,1193,1195,1196,1225,1227,1353,1359,1361,1362,1395,1399,1406,1408,1415,1419,1423,1441,1442,1443,1446,1450,1451,1452,1453,1460,1464,1466,1468,1471,1472,1474,1479,1480,1482,1501,1502,1514,1515,1516,1590,1591,1732,1746,1811,1818,1970,2041,2042,2050,2051,2052,2053,2054,2132,2134,2137,2142,2146,2156,2159,2160,2161,2164,2165,2194,2217,2253,2254,2258,2259,2318,2323,2329,2332,2337]},{"id":"symbol-6e4af5afb524dbe9","tex":"B_{\\beta\\alpha}","aliases":"B_βα beta alpha","display":"B_{\\beta\\alpha}","kind":"definition-passage","occurrences":[389,391,394,395,411,398,402,441]},{"id":"symbol-7b62fdaa08c020af","tex":"\\bytes","aliases":"bytes operatorname","display":"{\\operatorname{bytes}}","kind":"definition-passage","occurrences":[477]},{"id":"symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"definition-passage","occurrences":[794,1164,96,97,99,105,106,113,262,264,266,270,314,439,440,454,526,528,531,535,536,542,543,544,552,666,773,775,777,778,779,780,785,788,791,803,885,887,889,890,891,892,894,897,898,899,903,904,961,962,963,964,967,971,976,979,1013,1054,1055,1056,1057,1060,1062,1063,1066,1102,1103,1104,1106,1166,1199,1205,1206,1207,1211,1212,1215,1264,1266,1268,1271,1280,1282,1288,1289,1291,1293,1296,1297,1394,1395,1397,1398,1650,1667,1673,2106,2107,2109,2115,2116,2123,2272,2274,2276,2280,2324]},{"id":"symbol-98cd1622d78cdfec","tex":"c_{e,k}","aliases":"c_e,k","display":"c_{e,k}","kind":"definition-passage","occurrences":[794]},{"id":"symbol-167caac527a0def7","tex":"\\CC","aliases":"CC C mathbb","display":"{\\mathbb C}","kind":"definition-passage","occurrences":[460,0,517,526,541,544,546,557,558,559,625,2010]},{"id":"symbol-ff4ec9351d7bdf0c","tex":"\\CC^{n\\times r}","aliases":"CC^n× r C^n× r CC times mathbb times","display":"{\\mathbb C}^{n\\times r}","kind":"definition-passage","occurrences":[460,517,625]},{"id":"symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"definition-passage","occurrences":[459,460,45,1331,1332,2055,55,66,69,70,145,146,149,150,153,154,155,159,160,161,162,163,169,172,173,240,241,242,243,251,319,322,329,330,333,336,337,338,339,462,493,597,598,599,600,602,604,605,608,609,610,611,612,613,614,616,656,657,658,676,677,680,683,725,726,728,729,730,731,733,734,735,736,737,738,740,984,985,988,989,991,992,993,994,995,996,998,1110,1111,1112,1114,1116,1117,1118,1119,1150,1151,1152,1154,1157,1158,1165,1166,1333,1334,1336,1339,1404,1405,1406,1407,1408,1455,1456,1470,1471,1472,1474,1475,1477,1502,1503,1510,1512,1590,1732,1768,1984,1989,1990,1991,1997,2065,2076,2079,2080,2155,2156,2159,2160,2163,2164,2165,2169,2170,2171,2172,2173,2179,2182,2183,2250,2251,2252,2253,2261,2329,2332]},{"id":"symbol-2569dc5eb62aae9e","tex":"D_\\alpha","aliases":"D_α alpha","display":"D_\\alpha","kind":"definition-passage","occurrences":[460,677,730]},{"id":"symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"definition-passage","occurrences":[385,234,1432,1482,1821,2244]},{"id":"symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"definition-passage","occurrences":[389,390,395,413,415,54,57,244,250,251,253,256,259,260,399,400,402,438,439,440,476,646,647,648,650,651,653,654,657,658,738,739,742,743,750,753,754,755,756,757,766,767,771,957,958,959,1053,1068,1127,1128,1129,1130,1131,1132,1133,1263,1392,1393,1406,1408,1409,1410,1411,1412,1592,1613,1614,1867,2064,2067,2254,2260,2261,2263,2266,2269,2270]},{"id":"symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"definition-passage","occurrences":[792,794,795,21,29,30,32,58,164,165,195,196,260,264,265,266,270,308,437,439,440,476,750,751,752,753,754,755,756,757,766,767,768,772,773,775,776,777,778,779,781,782,783,785,786,787,788,790,791,841,900,903,904,1001,1002,1003,1004,1005,1006,1007,1008,1009,1011,1012,1013,1015,1054,1055,1056,1057,1059,1060,1062,1063,1065,1066,1068,1072,1073,1086,1134,1135,1161,1264,1265,1266,1268,1271,1273,1280,1281,1288,1289,1290,1291,1292,1293,1296,1297,1425,1426,1428,1759,1859,1992,2031,2039,2040,2042,2068,2174,2175,2205,2206,2270,2274,2275,2276,2280,2318]},{"id":"symbol-7c808485c3e440e2","tex":"\\operatorname{eval}","aliases":"eval operatorname","display":"\\operatorname{eval}","kind":"definition-passage","occurrences":[395]},{"id":"symbol-4b3cf9d696dc4fcb","tex":"\\flop","aliases":"flop operatorname","display":"{\\operatorname{flop}}","kind":"definition-passage","occurrences":[477,437,439,454]},{"id":"symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"definition-passage","occurrences":[406,675,2,3,8,56,179,629,630,639,644,662,687,1162,1163,1438,1439,1441,1442,1443,1444,1447,1449,1452,1504,1505,1506,1700,1708,2012,2013,2018,2066,2189]},{"id":"symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"definition-passage","occurrences":[871,110,120,544,548,872,873,874,875,881,882,1042,1043,1045,1215,1217,1223,1951,2120,2130]},{"id":"symbol-28a0f54c10c0e63f","tex":"G_A","aliases":"G_A","display":"G_A","kind":"definition-passage","occurrences":[406,662,687]},{"id":"symbol-99b7a1e1268a1b18","tex":"g_v","aliases":"g_v","display":"g_v","kind":"definition-passage","occurrences":[871,872,873,874,875]},{"id":"symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"definition-passage","occurrences":[383,384,385,390,407,460,411,1,12,14,17,139,163,222,229,232,239,244,253,254,379,380,588,609,613,649,650,743,953,954,976,994,1082,1121,1130,1146,1204,1253,1353,1408,1409,1411,1412,1436,1441,1482,1505,1712,1799,1819,1867,1974,1983,1992,2011,2022,2024,2027,2149,2173,2232,2239,2242,2249,2254,2263,2264]},{"id":"symbol-41bd5b0100b2ea43","tex":"\\Ical","aliases":"Ical I mathcal","display":"{\\mathcal I}","kind":"definition-passage","occurrences":[477,483]},{"id":"symbol-ea0ad3db0c0f9c47","tex":"\\IR","aliases":"IR mathsf","display":"{\\mathsf{IR}}","kind":"definition-passage","occurrences":[389,395]},{"id":"symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"definition-passage","occurrences":[871,12,13,14,17,200,206,209,210,211,212,222,240,242,670,672,674,676,680,725,728,730,731,732,734,738,827,829,830,834,1042,1043,1044,1048,1150,1154,1159,1405,1406,1615,1649,1798,1799,1974,2022,2023,2024,2027,2210,2216,2219,2220,2221,2222,2232,2250,2252]},{"id":"symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"definition-passage","occurrences":[793,794,795,1238,1240,1241,21,97,98,112,113,114,115,185,187,188,189,190,439,440,454,510,511,512,513,528,530,533,534,535,536,537,551,552,553,554,566,570,571,573,574,575,576,577,578,579,584,585,619,620,621,622,623,631,632,635,636,637,640,643,731,803,807,810,1163,1188,1199,1200,1201,1205,1233,1235,1237,1242,1244,1245,1246,1247,1253,1258,1259,1339,1340,1341,1342,1364,1365,1367,1447,1448,1451,1452,1455,1456,1457,1458,1461,1462,1464,1469,1470,1472,1474,1475,1476,1477,1478,1480,1484,1488,1489,1490,1492,1502,1503,1507,1510,1511,1512,1513,1515,1516,1517,1518,1672,2031,2107,2108,2122,2123,2124,2125,2195,2197,2198,2199,2200]},{"id":"symbol-fc97a5c4f9fcff61","tex":"K_\\lambda","aliases":"K_λ lambda","display":"K_\\lambda","kind":"definition-passage","occurrences":[871,827,829,830,834]},{"id":"symbol-9feea0dd1279e0f4","tex":"\\Kcal","aliases":"Kcal K mathcal","display":"{\\mathcal K}","kind":"definition-passage","occurrences":[716,722,730]},{"id":"symbol-2def419be484fcdf","tex":"\\Kcal_m","aliases":"Kcal_m K_m Kcal mathcal","display":"{\\mathcal K}_m","kind":"definition-passage","occurrences":[716,722]},{"id":"symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"definition-passage","occurrences":[715,716,131,134,135,137,234,268,270,308,476,557,559,561,562,563,566,586,605,606,607,625,645,647,648,653,658,659,660,661,663,665,722,748,758,781,782,783,785,787,788,790,791,804,838,841,842,901,903,1068,1072,1073,1086,1112,1113,1114,1115,1116,1117,1118,1119,1128,1133,1138,1144,1161,1224,1226,1231,1232,1264,1265,1268,1271,1272,1273,1280,1281,1282,1290,1291,1292,1426,1428,1431,1821,2141,2144,2145,2147,2244,2278,2280,2318]},{"id":"symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"definition-passage","occurrences":[385,460,77,80,81,85,86,105,121,133,134,135,137,139,433,434,435,437,439,440,442,443,444,446,447,450,451,454,456,457,463,464,465,468,469,472,473,476,490,491,492,494,495,497,500,502,503,517,520,521,523,525,526,527,530,531,532,534,535,536,539,540,541,542,543,544,545,546,547,550,558,559,561,562,563,566,624,625,759,968,969,970,971,972,973,977,978,1068,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1086,1091,1092,1094,1095,1096,1097,1098,1110,1161,1197,1198,1200,1201,1202,1207,1208,1209,1210,1214,1215,1216,1217,1221,1223,1225,1226,1231,1232,1318,1321,1387,1389,1431,1998,2004,2087,2090,2091,2095,2096,2115,2131,2143,2144,2145,2147,2149]},{"id":"symbol-9727f5125c78ee28","tex":"n_\\alpha","aliases":"n_α alpha","display":"n_\\alpha","kind":"definition-passage","occurrences":[385,439,440,443,444,446,447,476]},{"id":"symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"definition-passage","occurrences":[388,392,409,52,260,311,313,380,381,400,404,418,425,426,478,482,483,484,649,653,656,660,688,709,710,711,717,718,721,723,753,754,756,757,766,767,770,812,813,814,815,954,955,956,967,971,972,1003,1005,1006,1008,1009,1010,1087,1089,1090,1112,1114,1116,1118,1121,1122,1124,1132,1133,1134,1135,1139,1141,1142,1143,1144,1146,1147,1148,1157,1345,1347,1349,1350,1360,1361,1390,1392,1393,1395,1398,1401,1402,2062,2270,2321,2323]},{"id":"symbol-33a5f2682723cba6","tex":"P_\\alpha","aliases":"P_α alpha","display":"P_\\alpha","kind":"definition-passage","occurrences":[388,392,409,52,380,381,400,404,418,425,426,660,688,709,710,717,718,721,723,812,813,814,815,2062]},{"id":"symbol-aad7607e608d0052","tex":"P_\\beta","aliases":"P_β beta","display":"P_\\beta","kind":"definition-passage","occurrences":[392,409,380,381,400,404,418,425,426,688,709,711,814,815]},{"id":"symbol-813431acda8054c4","tex":"\\Pcal","aliases":"Pcal P mathcal","display":"{\\mathcal P}","kind":"definition-passage","occurrences":[383,389,813,814,815]},{"id":"symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"definition-passage","occurrences":[460,22,23,24,25,26,27,2032,2033,2034,2035,2036,2037,20,78,80,81,84,85,115,282,284,302,308,342,346,351,368,463,464,468,472,476,492,495,498,499,517,520,521,523,535,543,544,554,601,607,625,840,841,842,862,896,897,986,987,990,991,1029,1030,1059,1060,1061,1062,1063,1065,1076,1077,1078,1079,1080,1081,1084,1094,1097,1110,1115,1118,1120,1211,1213,1215,1432,1437,1438,1829,1895,1896,1897,1998,1999,2030,2088,2090,2091,2094,2095,2125,2292,2294,2312,2318]},{"id":"symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"definition-passage","occurrences":[871,160,161,167,169,170,171,226,230,514,515,524,609,610,613,615,616,704,828,830,837,994,995,998,1042,1043,1044,1160,1161,1363,1366,1406,1408,1413,1432,1884,1990,2170,2171,2177,2179,2180,2181,2236,2240]},{"id":"symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"definition-passage","occurrences":[474,21,58,87,88,261,265,266,270,308,437,476,489,491,492,772,776,777,778,779,783,785,786,788,791,841,884,889,890,892,1013,1015,1054,1056,1057,1085,1086,1091,1093,1100,1104,1105,1106,1161,1264,1265,1268,1271,1273,1280,1281,1282,1288,1289,1291,1293,1324,1325,1859,2031,2068,2097,2098,2271,2275,2276,2280,2318]},{"id":"symbol-152abbc9214c8600","tex":"S_\\lambda","aliases":"S_λ lambda","display":"S_\\lambda","kind":"definition-passage","occurrences":[871,828,830,837]},{"id":"symbol-452a3350693f9f7b","tex":"\\operatorname{span}","aliases":"span operatorname","display":"\\operatorname{span}","kind":"definition-passage","occurrences":[716,1138]},{"id":"symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"definition-passage","occurrences":[459,460,55,150,155,320,322,462,471,493,496,514,517,597,598,599,602,604,605,609,610,611,613,616,984,985,988,989,994,995,996,998,1110,1117,1564,1590,1591,1739,2065,2160,2165,2330,2332]},{"id":"symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"definition-passage","occurrences":[459,460,55,150,155,320,322,462,470,471,493,496,514,517,597,598,599,600,602,603,604,605,609,610,611,612,613,616,984,985,988,989,994,995,996,998,1110,1117,1306,1324,1326,1328,1440,1441,1577,1590,1591,1729,1739,2065,2160,2165,2330,2332]},{"id":"symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"definition-passage","occurrences":[714,716,816,817,818,819,870,871,45,1310,1311,1312,1314,1315,1329,1330,1331,1332,2055,59,60,68,69,70,71,145,148,149,150,152,153,154,156,157,165,173,233,234,237,238,329,330,332,333,337,338,534,535,537,718,719,722,724,730,733,821,822,823,826,827,828,829,830,831,832,833,834,835,837,841,842,854,856,857,858,860,861,864,872,873,874,875,1017,1018,1019,1023,1024,1025,1026,1027,1028,1030,1042,1046,1048,1138,1143,1144,1147,1157,1158,1159,1161,1170,1171,1172,1173,1174,1176,1307,1308,1316,1317,1322,1323,1325,1326,1333,1334,1335,1336,1337,1339,1340,1491,1493,1494,1499,1775,1791,1819,1820,1821,2069,2070,2078,2079,2080,2081,2155,2158,2159,2160,2162,2163,2164,2166,2167,2175,2183,2243,2244,2247,2248]},{"id":"symbol-4b270ddb72d4b503","tex":"V^*","aliases":"V^*","display":"V^*","kind":"definition-passage","occurrences":[459,55,462,470,471,493,496,514,609,610,611,612,613,616,994,995,996,998,1441,2065]},{"id":"symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"definition-passage","occurrences":[795,1164,30,31,90,91,93,94,95,505,506,508,511,513,803,810,828,836,854,860,1026,1042,1166,1169,1172,1174,1371,1372,1373,1374,1375,1377,1379,1382,1383,1386,1483,1484,1488,1494,1496,1498,1499,1502,2040,2041,2100,2101,2103,2104,2105]},{"id":"symbol-19ce956d274d340e","tex":"w_{e,k}","aliases":"w_e,k","display":"w_{e,k}","kind":"definition-passage","occurrences":[795]},{"id":"symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"definition-passage","occurrences":[383,384,385,386,387,391,393,396,410,1330,1331,122,125,127,136,379,397,422,559,560,565,566,567,692,694,696,697,698,999,1000,1226,1227,1229,1801,1806,1811,1818,2132,2135,2137,2146]},{"id":"symbol-1e9458c4c7da9d3b","tex":"X_\\alpha","aliases":"X_α alpha","display":"X_\\alpha","kind":"definition-passage","occurrences":[383,384,385,386,387,391,393,379]},{"id":"symbol-e5a69bbb6bda64f3","tex":"X_\\beta","aliases":"X_β beta","display":"X_\\beta","kind":"definition-passage","occurrences":[391,393,396]},{"id":"symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"definition-passage","occurrences":[390,460,47,2057,50,51,141,306,517,541,557,558,559,625,758,759,906,907,908,909,910,918,1068,1110,1224,1225,1226,1302,1303,1304,1382,1385,1429,1430,1431,1438,1459,1519,1520,1526,1531,1538,1545,1550,1557,1701,2060,2061,2151,2316]},{"id":"symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"definition-passage","occurrences":[383,384,385,386,387,388,389,391,392,393,394,395,408,409,460,411,413,415,52,53,63,64,260,379,380,381,398,400,402,404,418,419,424,425,426,439,440,441,443,444,446,447,463,464,469,473,476,660,661,663,677,680,685,688,690,691,705,707,709,710,717,718,719,721,722,723,730,752,753,754,756,757,766,767,770,812,813,814,815,2062,2063,2073,2074,2270]},{"id":"symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"definition-passage","occurrences":[389,391,392,393,394,395,396,408,409,411,414,415,53,63,65,380,381,398,400,402,404,418,419,424,425,426,439,440,441,442,446,447,476,679,680,681,682,683,686,688,699,705,708,709,711,712,814,815,2063,2073,2075]},{"id":"symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"definition-passage","occurrences":[817,58,59,264,265,266,270,822,827,842,1018,1023,1025,1026,1030,1042,1859,2068,2069,2274,2275,2276,2280]},{"id":"symbol-f6991c3ad9fdd14c","tex":"\\epsilon_v","aliases":"ϵ_v epsilon","display":"\\epsilon_v","kind":"definition-passage","occurrences":[817,59,822,827,842,1018,1023,1025,1026,1030,1042,2069]},{"id":"symbol-95d58d27f67c1fbd","tex":"\\iota","aliases":"ι iota","display":"\\iota","kind":"definition-passage","occurrences":[386,388,394,395,396,398,402]},{"id":"symbol-300f133d4213a5fd","tex":"\\iota_\\alpha","aliases":"ι_α iota alpha","display":"\\iota_\\alpha","kind":"definition-passage","occurrences":[386,388,398]},{"id":"symbol-c039735a9807331b","tex":"\\iota_\\beta","aliases":"ι_β iota beta","display":"\\iota_\\beta","kind":"definition-passage","occurrences":[394,395,396,402]},{"id":"symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"definition-passage","occurrences":[817,59,251,747,748,749,821,823,827,841,860,1017,1023,1025,1026,1030,1042,1171,1174,1415,1417,1419,1420,1421,1422,1423,1424,1426,1427,1869,2069,2261]},{"id":"symbol-fd0a204d0362614f","tex":"\\kappa_v","aliases":"κ_v kappa","display":"\\kappa_v","kind":"definition-passage","occurrences":[817,59,821,823,827,841,860,1017,1023,1025,1026,1030,1042,1171,1174,2069]},{"id":"symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"definition-passage","occurrences":[871,269,270,292,293,294,295,296,297,298,354,355,356,357,358,359,360,361,364,366,367,369,370,778,779,780,784,785,788,791,824,825,827,828,829,830,834,836,837,843,848,849,850,851,853,1021,1022,1023,1025,1026,1027,1028,1034,1035,1037,1038,1040,1041,1042,1180,1181,1182,1267,1280,1283,1284,1972,1973,2279,2280,2302,2303,2304,2305,2306,2307,2308]},{"id":"symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"definition-passage","occurrences":[387,388,394,395,398,402,530,534,535,1200,1201]},{"id":"symbol-bd82674cd48e7508","tex":"\\pi_\\alpha","aliases":"π_α pi alpha","display":"\\pi_\\alpha","kind":"definition-passage","occurrences":[387,388,394,395,402]},{"id":"symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"definition-passage","occurrences":[819,59,190,618,622,623,821,828,861,864,1019,1024,1026,1042,1048,1173,1174,1454,1455,1470,1472,1474,1476,1500,1502,1505,1506,1509,1517,2069,2200]},{"id":"symbol-7702fce301ebe3bf","tex":"\\sigma_v","aliases":"σ_v sigma","display":"\\sigma_v","kind":"definition-passage","occurrences":[819,59,828,861,864,1019,1024,1026,1042,1173,1174,2069]},{"id":"symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"definition-passage","occurrences":[716,1238,1240,542,543,544,605,645,838,976,1138,1215,1233,1236,1372,1382,1385,1455,1506]},{"id":"symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"definition-passage","occurrences":[386,387,393,396,1311,1315,1332,15,201,202,203,204,214,215,216,217,218,219,220,221,680,1000,1307,1324,1333,1339,1797,2025,2211,2212,2213,2214,2224,2225,2226,2227,2228,2229,2230,2231]},{"id":"symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","occurrences":[383,384,385,389,395,460,410,411,413,415,68,379,380,397,400,402,439,440,476,517,526,541,544,546,557,558,559,625,692,750,754,756,758,759,821,822,828,855,886,902,954,958,959,1026,1040,1068,1110,1158,1224,1225,1226,1326,1356,1357,1359,1372,1431,1438,1443,1516,2078]},{"id":"symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"definition-passage","occurrences":[385,395,411,1240,1332,31,32,260,264,265,380,400,402,404,418,439,440,463,464,469,473,476,513,535,566,607,623,647,648,653,658,666,680,718,719,731,750,754,756,757,766,767,768,775,776,777,780,783,786,788,813,814,815,821,822,828,841,842,955,958,959,961,971,979,1004,1005,1006,1008,1009,1011,1013,1015,1026,1030,1042,1048,1055,1056,1060,1062,1063,1072,1073,1076,1077,1086,1115,1118,1128,1133,1134,1135,1143,1146,1174,1189,1264,1265,1266,1268,1271,1273,1280,1281,1293,1296,1297,1333,1339,1392,1393,1401,1402,1470,1483,1484,1488,1499,1502,1517,2041,2042,2270,2274,2275]},{"id":"symbol-7354371b4dfcd811","tex":"\\sum_{(\\beta,\\alpha)\\in E}","aliases":"∑_(β,α)∈ E sum beta alpha in","display":"\\sum_{(\\beta,\\alpha)\\in E}","kind":"definition-passage","occurrences":[395,400,402,439,440,476]},{"id":"symbol-cabedb544f677980","tex":"\\sum_{\\alpha\\in I}","aliases":"∑_α∈ I sum alpha in","display":"\\sum_{\\alpha\\in I}","kind":"definition-passage","occurrences":[385,380]},{"id":"symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"definition-passage","occurrences":[409,800,814,815]},{"id":"symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"definition-passage","occurrences":[817,819,482,484,485,618,720,824,847,884,1036,1064,1089,1105,1129,1181,1419,1420,1421,1425,1454,1470,1478,1484,1488,1495,1496,1499,1502,1505,1517,1518]},{"id":"symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"definition-passage","occurrences":[390,412,207,694,698,722,965,1144,1352,2217]},{"id":"symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"definition-passage","occurrences":[384,460,379,692,722,730,1144,1157]},{"id":"symbol-90b882d77bb3725c","tex":"\\bigoplus_{\\alpha\\in I}","aliases":"⊕_α∈ I bigoplus alpha in","display":"\\bigoplus_{\\alpha\\in I}","kind":"definition-passage","occurrences":[384,460,379]},{"id":"symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"definition-passage","occurrences":[391,408,1331,661,976,1367]},{"id":"symbol-993220b6cfa76019","tex":"A_0","aliases":"A_0","display":"A_0","kind":"introduction-passage","occurrences":[1238,1239,1240,182,1233,1258,2192]},{"id":"symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"introduction-passage","occurrences":[1238,1240,9,606,645,647,648,1112,1128,2019]},{"id":"symbol-7d8e3b8668ae2ddd","tex":"A_k","aliases":"A_k","display":"A_k","kind":"introduction-passage","occurrences":[1238,1240,1258,1259]},{"id":"symbol-f70174408316e1b5","tex":"A_{\\gamma\\beta}","aliases":"A_γβ gamma beta","display":"A_{\\gamma\\beta}","kind":"introduction-passage","occurrences":[411]},{"id":"symbol-b8c748f09e8f469d","tex":"\\operatorname{Anc}","aliases":"Anc operatorname","display":"\\operatorname{Anc}","kind":"introduction-passage","occurrences":[1310,1312]},{"id":"symbol-50c181940765c5f1","tex":"\\operatorname{Apply}","aliases":"Apply operatorname","display":"\\operatorname{Apply}","kind":"introduction-passage","occurrences":[1240,1241]},{"id":"symbol-a6d440b3f2f61b84","tex":"\\operatorname{Apply}_k","aliases":"Apply_k operatorname","display":"\\operatorname{Apply}_k","kind":"introduction-passage","occurrences":[1241]},{"id":"symbol-e9c15d19ce3d5bef","tex":"\\operatorname{Apply}_{k}","aliases":"Apply_k operatorname","display":"\\operatorname{Apply}_{k}","kind":"introduction-passage","occurrences":[1240]},{"id":"symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","occurrences":[1240,186,187,437,570,572,573,575,576,579,580,581,582,583,584,632,633,634,666,669,967,979,1234,1236,1237,1243,1244,1245,1246,1248,1249,1326,1327,1343,1344,1394,1395,1396,1454,1455,1458,1459,1478,1482,1484,1489,1505,1506,1507,1509,1517,1518,2196,2197]},{"id":"symbol-ef97823489e8f12d","tex":"D_v","aliases":"D_v","display":"D_v","kind":"introduction-passage","occurrences":[1331,1332,1334,1336]},{"id":"symbol-a2b16f8c547b88cb","tex":"\\End","aliases":"End operatorname","display":"{\\operatorname{End}}","kind":"introduction-passage","occurrences":[410,397]},{"id":"symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"introduction-passage","occurrences":[413,415,60,80,98,99,103,105,106,256,313,370,480,482,484,486,529,530,531,536,538,623,797,800,801,803,804,814,828,829,830,836,857,860,862,971,972,1025,1026,1027,1028,1042,1052,1087,1089,1090,1162,1163,1166,1167,1174,1176,1178,1185,1200,1202,1203,1204,1206,1207,1400,1401,1613,1667,1672,1673,2070,2090,2108,2109,2113,2115,2116,2266,2323]},{"id":"symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"introduction-passage","occurrences":[1250,1251,1252,138,139,143,144,226,587,588,589,702,703,704,709,710,711,1253,1257,1261,1363,1367,1884,1983,2148,2149,2153,2154,2236]},{"id":"symbol-63e9c3e80a66add1","tex":"H_1","aliases":"H_1","display":"H_1","kind":"introduction-passage","occurrences":[1251,138,587,2148]},{"id":"symbol-66fc929643bbe3da","tex":"H_2","aliases":"H_2","display":"H_2","kind":"introduction-passage","occurrences":[1250,1252]},{"id":"symbol-c9b3c767be0a1967","tex":"H_{2N}","aliases":"H_2N","display":"H_{2N}","kind":"introduction-passage","occurrences":[1252,587,1257]},{"id":"symbol-ac7885f6694a994f","tex":"H_N","aliases":"H_N","display":"H_N","kind":"introduction-passage","occurrences":[1252,587,588,589,1253,1261]},{"id":"symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"introduction-passage","occurrences":[1240,21,505,564,566,647,653,1048,1049,1076,1077,1128,1133,1139,1141,1142,1143,1144,1146,1147,1148,1149,1157,1174,1200,1201,1211,1217,1218,1219,1220,1221,1222,1228,1229,1372,1373,1470,1483,1484,1487,1488,1491,1492,1493,1494,1495,1496,1497,1498,1499,1500,1502,1509,1517,1669,1671,1812,1814,2031]},{"id":"symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"introduction-passage","occurrences":[1240,57,97,98,111,112,113,114,115,195,242,505,528,530,534,535,550,551,552,553,554,555,564,566,605,607,623,646,647,648,649,650,651,652,653,654,655,656,657,658,666,668,669,719,720,723,727,728,730,731,732,733,764,765,766,767,768,769,797,798,801,803,806,808,809,810,811,839,840,841,842,979,980,1029,1030,1047,1048,1110,1111,1115,1118,1120,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1143,1145,1153,1154,1157,1158,1159,1199,1200,1201,1211,1217,1218,1219,1220,1221,1222,1228,1229,1364,1365,1367,1372,1373,1483,1509,1672,1759,2067,2107,2108,2121,2122,2123,2124,2125,2205,2252]},{"id":"symbol-345427490e88d80b","tex":"\\leadsto","aliases":"leadsto","display":null,"kind":"introduction-passage","occurrences":[1310,1346,1349,1361,1365,1367]},{"id":"symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"introduction-passage","occurrences":[1238,1241,80,88,242,440,454,488,491,492,617,620,622,623,728,729,730,732,746,747,1091,1092,1152,1154,1155,1156,1159,1400,1509,1510,1511,2090,2098,2252]},{"id":"symbol-a66db8ec4f89a9e7","tex":"M_0","aliases":"M_0","display":"M_0","kind":"introduction-passage","occurrences":[1238]},{"id":"symbol-c9f29e7d9e7a175b","tex":"M_k","aliases":"M_k","display":"M_k","kind":"introduction-passage","occurrences":[1238,1241,620,622,623]},{"id":"symbol-3db6ddc80894285f","tex":"M_{k+1}","aliases":"M_k+1","display":"M_{k+1}","kind":"introduction-passage","occurrences":[1238]},{"id":"symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"introduction-passage","occurrences":[1315,242,313,482,729,1089,1152,1326,1484,2252,2323]},{"id":"symbol-b61b8cfdaf3918dd","tex":"\\max_{u\\to v}","aliases":"max_u→ v max to","display":"\\max_{u\\to v}","kind":"introduction-passage","occurrences":[1315]},{"id":"symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"introduction-passage","occurrences":[1252,506,508,586,587,588,589,590,593,594,595,596,1190,1192,1193,1194,1253,1256,1257,1259,1260,1261,1262,1375,1383,1386]},{"id":"symbol-0e517349dec3b6ba","tex":"\\overset","aliases":"overset","display":null,"kind":"introduction-passage","occurrences":[1331]},{"id":"symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"introduction-passage","occurrences":[22,2032,20,543,544,1215,2030]},{"id":"symbol-6d6362ab00b61fc6","tex":"r_2","aliases":"r_2","display":"r_2","kind":"introduction-passage","occurrences":[23,2033,20,1895,1896,1897,2030]},{"id":"symbol-4e4ee8a109009a53","tex":"\\subsetneq","aliases":"subsetneq","display":null,"kind":"introduction-passage","occurrences":[1312]},{"id":"symbol-1084f45e6b3e947e","tex":"\\supp","aliases":"supp operatorname","display":"{\\operatorname{supp}}","kind":"introduction-passage","occurrences":[412,207,667,965,2217]},{"id":"symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"introduction-passage","occurrences":[45,1331,1332,2055,7,69,70,71,96,99,100,101,109,110,116,117,119,122,126,127,128,136,139,140,142,143,145,147,148,149,150,152,153,154,155,156,157,160,161,173,179,182,233,234,238,247,249,253,313,314,321,322,323,324,325,329,330,331,332,333,334,337,338,339,482,484,485,541,542,544,548,549,560,565,569,588,595,596,1050,1052,1087,1088,1089,1090,1211,1217,1223,1227,1253,1261,1271,1272,1273,1333,1339,1590,1591,1708,1712,1729,1739,1775,1791,1811,1816,1818,1819,1820,1830,1866,1867,1941,1983,1990,1991,1992,2017,2079,2080,2081,2106,2109,2110,2111,2119,2120,2126,2127,2129,2132,2136,2137,2138,2146,2149,2150,2152,2153,2155,2157,2158,2159,2160,2162,2163,2164,2165,2166,2167,2170,2171,2183,2189,2192,2243,2244,2248,2257,2259,2263,2323,2324,2331,2332,2333,2334,2335]},{"id":"symbol-014a85fddb5fe786","tex":"T_{vu}","aliases":"T_vu","display":"T_{vu}","kind":"introduction-passage","occurrences":[1331,1332,1333,1339]},{"id":"symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"introduction-passage","occurrences":[45,1240,1310,1311,1312,1313,1315,1331,1332,2055,68,69,70,72,145,147,149,150,153,154,156,157,159,160,161,164,172,329,330,333,337,338,339,760,761,763,801,802,821,841,876,877,878,879,1268,1272,1276,1279,1286,1288,1289,1290,1293,1296,1297,1298,1299,1301,1307,1309,1317,1333,1339,1775,1989,1990,1991,2078,2079,2080,2082,2155,2157,2159,2160,2163,2164,2166,2167,2169,2170,2171,2174,2182]},{"id":"symbol-36077cb07719c8a1","tex":"v^T","aliases":"v^T","display":"v^T","kind":"introduction-passage","occurrences":[45,2055,69,70,71,173,330,332,333,337,338,1775,1791,2079,2080,2081,2183]},{"id":"symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"introduction-passage","occurrences":[1240,1241,1332,70,71,75,100,102,109,116,117,119,222,223,247,248,260,321,322,328,331,332,333,334,335,339,340,546,548,549,556,589,591,592,678,680,682,741,742,748,754,756,757,770,1005,1006,1008,1099,1137,1202,1203,1205,1223,1255,1256,1257,1261,1333,1338,1339,1340,1344,1349,1351,1356,1358,1359,1729,1732,1739,1746,1761,1768,1775,1782,1791,1866,1974,2080,2081,2085,2110,2112,2119,2126,2127,2129,2232,2233,2257,2258,2270,2331,2332,2338]},{"id":"symbol-c070a883f58aab2d","tex":"x_{j,\\cdot}","aliases":"x_j,· cdot","display":"x_{j,\\cdot}","kind":"introduction-passage","occurrences":[1240]},{"id":"symbol-708bfd9721bf49c2","tex":"X_u","aliases":"X_u","display":"X_u","kind":"introduction-passage","occurrences":[1331]},{"id":"symbol-6590090cc9a70e3b","tex":"x_u","aliases":"x_u","display":"x_u","kind":"introduction-passage","occurrences":[1332,1333]},{"id":"symbol-08a6f9f108660d1a","tex":"X_v","aliases":"X_v","display":"X_v","kind":"introduction-passage","occurrences":[1330,1331]},{"id":"symbol-67383509ca3cbfa6","tex":"x_v","aliases":"x_v","display":"x_v","kind":"introduction-passage","occurrences":[1332,1333,1340]},{"id":"symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"introduction-passage","occurrences":[1240,248,249,335,338,339,1357,1358,1359,1467,1468,1470,1471,1472,1473,1867,2258,2259]},{"id":"symbol-46014f7f0635c4a0","tex":"y_{i,u}","aliases":"y_i,u","display":"y_{i,u}","kind":"introduction-passage","occurrences":[1240]},{"id":"symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"introduction-passage","occurrences":[1240,21,308,1403,2031,2318]},{"id":"symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"introduction-passage","occurrences":[411,413,415,418,424,426,660,661,692,763]},{"id":"symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"introduction-passage","occurrences":[1315,566,976,1316,1317,1321,1322,1325,1326,1339,1340]},{"id":"symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"introduction-passage","occurrences":[1315,1328]},{"id":"symbol-d57a4808dbd2b57e","tex":"\\sum_{j=0}^{d-1}","aliases":"∑_j=0^d-1 sum","display":"\\sum_{j=0}^{d-1}","kind":"introduction-passage","occurrences":[1240]},{"id":"symbol-1ea15b9f71040065","tex":"\\sum_{u\\to v}","aliases":"∑_u→ v sum to","display":"\\sum_{u\\to v}","kind":"introduction-passage","occurrences":[1332,1333,1339]},{"id":"symbol-1b2a0f40b316674f","tex":"\\sum_{\\beta\\in I}","aliases":"∑_β∈ I sum beta in","display":"\\sum_{\\beta\\in I}","kind":"introduction-passage","occurrences":[411]},{"id":"symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"introduction-passage","occurrences":[412,413,207,965,2217]},{"id":"symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"introduction-passage","occurrences":[1331]},{"id":"symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"introduction-passage","occurrences":[23,2033,234,293,295,313,354,359,448,449,455,466,467,488,490,501,525,540,570,578,619,631,885,894,895,899,900,901,972,2244,2303,2305,2323]},{"id":"symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"introduction-passage","occurrences":[1238,1252,122,424,426,560,567,967,1227,1818,2132]},{"id":"symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","occurrences":[1614,1974]},{"id":"symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","occurrences":[145,150,153,154,156,157,195,283,285,287,291,299,300,301,308,343,344,345,347,349,350,352,353,365,370,439,440,476,954,955,956,957,958,959,961,964,966,967,971,972,1001,1003,1005,1006,1008,1009,1010,1134,1135,1200,1319,1321,1345,1346,1347,1348,1349,1350,1352,1355,1356,1357,1359,1360,1361,1366,1367,1390,1392,1393,1394,1395,1397,1398,1401,1402,1487,1488,1490,1494,1495,1497,1500,1759,1760,1823,1824,1825,2155,2160,2163,2164,2166,2167,2205,2293,2295,2297,2301,2309,2310,2311,2318]},{"id":"symbol-0aa786b643f5da71","tex":"A^{(1)}","aliases":"A^(1)","display":"A^{(1)}","kind":"use","occurrences":[838]},{"id":"symbol-6973a6c793fe345f","tex":"A^{(m)}","aliases":"A^(m)","display":"A^{(m)}","kind":"use","occurrences":[838]},{"id":"symbol-e0c9947b1daf0522","tex":"A^*","aliases":"A^*","display":"A^*","kind":"use","occurrences":[3,129,419,425,966,1709,1817,2013,2139]},{"id":"symbol-677eab50351e7675","tex":"A^{-1}","aliases":"A^-1","display":"A^{-1}","kind":"use","occurrences":[14,16,161,231,688,698,700,1368,1799,1800,1991,2024,2026,2171,2241]},{"id":"symbol-8838c457f7756577","tex":"A^2","aliases":"A^2","display":"A^2","kind":"use","occurrences":[981,983]},{"id":"symbol-06d6907fddca4380","tex":"A^{h+1}","aliases":"A^h+1","display":"A^{h+1}","kind":"use","occurrences":[975]},{"id":"symbol-afc7e5c12f805c36","tex":"A^j","aliases":"A^j","display":"A^j","kind":"use","occurrences":[242,668,719,723,728,730,731,979,1143,1154,1159,2252]},{"id":"symbol-fab2b96c59335d56","tex":"A^{j-1-k}","aliases":"A^j-1-k","display":"A^{j-1-k}","kind":"use","occurrences":[731]},{"id":"symbol-339ddf229ead4fcf","tex":"A^m","aliases":"A^m","display":"A^m","kind":"use","occurrences":[660]},{"id":"symbol-9a52b1bc77ae2b6d","tex":"A^n","aliases":"A^n","display":"A^n","kind":"use","occurrences":[972,973]},{"id":"symbol-8f914025617b7ebc","tex":"A^T","aliases":"A^T","display":"A^T","kind":"use","occurrences":[122,127,128,136,560,565,569,1227,1811,1816,1818,2132,2137,2138,2146]},{"id":"symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","occurrences":[291,299,300,2301,2309,2310]},{"id":"symbol-0f75896d5ca1818d","tex":"A_{00}","aliases":"A_00","display":"A_{00}","kind":"use","occurrences":[982]},{"id":"symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","occurrences":[291,345,349,353,1760,2301]},{"id":"symbol-e5cda8318149bdb8","tex":"a_2","aliases":"a_2","display":"a_2","kind":"use","occurrences":[345,350,353,1760]},{"id":"symbol-f2cfa673011ea148","tex":"a_3","aliases":"a_3","display":"a_3","kind":"use","occurrences":[1760]},{"id":"symbol-94cffe8dd0af28a5","tex":"A_{ba}","aliases":"A_ba","display":"A_{ba}","kind":"use","occurrences":[956,957,958,959,966,967]},{"id":"symbol-7000b8c39e7a4cdb","tex":"A_{bc}","aliases":"A_bc","display":"A_{bc}","kind":"use","occurrences":[961,963,971,976]},{"id":"symbol-7764e407096f3048","tex":"A_e","aliases":"A_e","display":"A_e","kind":"use","occurrences":[750,753,755,756,1004,1007,1008]},{"id":"symbol-74c2307a2295b9fc","tex":"A_{HH}","aliases":"A_HH","display":"A_{HH}","kind":"use","occurrences":[703]},{"id":"symbol-4a8722c909d9ace9","tex":"A_{HH}^{-1}","aliases":"A_HH^-1","display":"A_{HH}^{-1}","kind":"use","occurrences":[226,704,709,1363,1884,2236]},{"id":"symbol-574944342ee92d11","tex":"A_{HR}","aliases":"A_HR","display":"A_{HR}","kind":"use","occurrences":[226,704,709,710,1363,1884,2236]},{"id":"symbol-574cc7908da88025","tex":"A_i","aliases":"A_i","display":"A_i","kind":"use","occurrences":[647,653,1128,1133,1142,1144]},{"id":"symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"use","occurrences":[1487,1488,1494,1495,1497,1500]},{"id":"symbol-3c32e4b0e0900b11","tex":"A_i^j","aliases":"A_i^j","display":"A_i^j","kind":"use","occurrences":[1143]},{"id":"symbol-c0307cd43e859d23","tex":"A_j","aliases":"A_j","display":"A_j","kind":"use","occurrences":[605,646,649,650,657,764,766,1110,1127,1130,1132,1134,1135]},{"id":"symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","occurrences":[195,1759,2205]},{"id":"symbol-2167913f89021b35","tex":"A_{j+1}","aliases":"A_j+1","display":"A_{j+1}","kind":"use","occurrences":[658,1122,1124]},{"id":"symbol-68fee95ac83fd47a","tex":"A_{j,e}","aliases":"A_j,e","display":"A_{j,e}","kind":"use","occurrences":[1134]},{"id":"symbol-0841717db411db13","tex":"A_{j-1}","aliases":"A_j-1","display":"A_{j-1}","kind":"use","occurrences":[648]},{"id":"symbol-91f49c5985231a83","tex":"A_{j\\ell}","aliases":"A_jℓ ell","display":"A_{j\\ell}","kind":"use","occurrences":[566]},{"id":"symbol-d8f016d5b63ca3cf","tex":"A_{k+1}","aliases":"A_k+1","display":"A_{k+1}","kind":"use","occurrences":[1258]},{"id":"symbol-e383dbc0f9ccfa0b","tex":"a_{k+1}","aliases":"a_k+1","display":"a_{k+1}","kind":"use","occurrences":[1490]},{"id":"symbol-da2a817868989569","tex":"A_{k-1}","aliases":"A_k-1","display":"A_{k-1}","kind":"use","occurrences":[1233]},{"id":"symbol-e0115225a292e38f","tex":"A_m","aliases":"A_m","display":"A_m","kind":"use","occurrences":[606,645,647,648,658,1112,1128]},{"id":"symbol-605293bc8cd6a683","tex":"A_{m-1}","aliases":"A_m-1","display":"A_{m-1}","kind":"use","occurrences":[658]},{"id":"symbol-8b601bf3f58468ef","tex":"A_{RH}","aliases":"A_RH","display":"A_{RH}","kind":"use","occurrences":[226,704,709,711,1363,1884,2236]},{"id":"symbol-56009cd5bba2cd88","tex":"A_{RR}","aliases":"A_RR","display":"A_{RR}","kind":"use","occurrences":[226,704,706,1363,1884,2236]},{"id":"symbol-edac8f4090c64b8b","tex":"A_\\alpha","aliases":"A_α alpha","display":"A_\\alpha","kind":"use","occurrences":[721,722]},{"id":"symbol-84b88103b8092dfe","tex":"A_\\alpha^j","aliases":"A_α^j alpha","display":"A_\\alpha^j","kind":"use","occurrences":[719]},{"id":"symbol-903a870c12766098","tex":"A_{\\alpha_1\\alpha_0}","aliases":"A_α_1α_0 alpha alpha","display":"A_{\\alpha_1\\alpha_0}","kind":"use","occurrences":[663]},{"id":"symbol-bb2c326e063bfa7f","tex":"A_{\\alpha_m\\alpha_{m-1}}","aliases":"A_α_mα_m-1 alpha alpha","display":"A_{\\alpha_m\\alpha_{m-1}}","kind":"use","occurrences":[663]},{"id":"symbol-5498b3bc9fb8719d","tex":"A_{\\beta\\alpha}","aliases":"A_βα beta alpha","display":"A_{\\beta\\alpha}","kind":"use","occurrences":[53,63,381,424,426,2063,2073]},{"id":"symbol-935d18e87c9add19","tex":"A_{\\beta\\alpha}^*","aliases":"A_βα^* beta alpha","display":"A_{\\beta\\alpha}^*","kind":"use","occurrences":[419]},{"id":"symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"use","occurrences":[1372,1375,1382]},{"id":"symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","occurrences":[60,61,79,80,81,84,85,147,148,149,150,152,153,156,157,283,286,288,291,303,314,343,348,352,353,449,450,453,454,456,457,458,465,466,468,473,492,501,502,503,504,508,678,680,797,799,800,803,804,805,810,857,859,862,885,887,889,890,891,893,955,956,957,958,959,961,963,966,967,971,972,976,1074,1075,1078,1079,1082,1083,1084,1101,1103,1104,1106,1163,1166,1167,1169,1174,1175,1176,1177,1178,1184,1185,1186,1187,1188,1189,1320,1321,1333,1339,1344,1346,1347,1349,1361,1366,1367,1381,1382,1383,1386,1387,1388,1389,1393,1394,1395,1396,1401,1402,1826,1827,1828,1998,1999,2070,2071,2089,2090,2091,2094,2095,2157,2158,2159,2160,2162,2163,2166,2167,2293,2296,2298,2301,2313,2324]},{"id":"symbol-73ec8038e3175b44","tex":"B^{-1}","aliases":"B^-1","display":"B^{-1}","kind":"use","occurrences":[1415]},{"id":"symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","occurrences":[458,508,1383,1386]},{"id":"symbol-5b1e637f78e0d2bb","tex":"B_{\\mathrm F}^{2}","aliases":"B_ F^2 mathrm","display":"B_{\\mathrm F}^{2}","kind":"use","occurrences":[1446]},{"id":"symbol-8efd426e3c3875e3","tex":"B_2","aliases":"B_2","display":"B_2","kind":"use","occurrences":[1415]},{"id":"symbol-b5384093c6e0cfdb","tex":"B_{ca}","aliases":"B_ca","display":"B_{ca}","kind":"use","occurrences":[961,964]},{"id":"symbol-7bb9adbb22b5e88e","tex":"B_{dc}","aliases":"B_dc","display":"B_{dc}","kind":"use","occurrences":[967]},{"id":"symbol-a34b13535dfa2adc","tex":"b_i","aliases":"b_i","display":"b_i","kind":"use","occurrences":[1174]},{"id":"symbol-643bdb7bc30d61f1","tex":"B_{ik}","aliases":"B_ik","display":"B_{ik}","kind":"use","occurrences":[566]},{"id":"symbol-4c82c8901bdffe44","tex":"\\mathcal B_{p,w}","aliases":"B_p,w mathcal","display":"\\mathcal B_{p,w}","kind":"use","occurrences":[1372,1375,1382]},{"id":"symbol-336970a90c727278","tex":"B_{\\rm stream}","aliases":"B_rm stream rm","display":"B_{\\rm stream}","kind":"use","occurrences":[308,476,2318]},{"id":"symbol-b9846458a14f54b6","tex":"B_{\\mathrm{stream}}","aliases":"B_stream mathrm","display":"B_{\\mathrm{stream}}","kind":"use","occurrences":[1086]},{"id":"symbol-69b376b6bbe4d3ac","tex":"b_v","aliases":"b_v","display":"b_v","kind":"use","occurrences":[1333,1339]},{"id":"symbol-5982aa6063dafe04","tex":"b_\\beta","aliases":"b_β beta","display":"b_\\beta","kind":"use","occurrences":[680]},{"id":"symbol-b0a7f57cb21faf55","tex":"B_{\\delta\\gamma}","aliases":"B_δγ delta gamma","display":"B_{\\delta\\gamma}","kind":"use","occurrences":[424,426]},{"id":"symbol-0a092bc56762bbd9","tex":"B_\\nu","aliases":"B_ν nu","display":"B_\\nu","kind":"use","occurrences":[1443]},{"id":"symbol-c6822eaf3f6b3d51","tex":"\\bar\\kappa","aliases":"barκ bar kappa","display":"\\bar\\kappa","kind":"use","occurrences":[1427,1428]},{"id":"symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","occurrences":[97,2107]},{"id":"symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","occurrences":[97,99,107,108,150,155,184,320,322,528,531,535,548,598,599,600,988,989,1051,1052,1055,1057,1060,1199,1205,1206,1207,1217,1223,1409,1411,1412,1445,1564,1577,1590,1650,1673,1729,1739,2107,2109,2117,2118,2160,2165,2194,2330,2332]},{"id":"symbol-2f40d444413ea162","tex":"\\C","aliases":"C mathbb","display":"{\\mathbb C}","kind":"use","occurrences":[1224,1225,1226,1230,1254,1431,1438]},{"id":"symbol-2e12e550b72be021","tex":"\\mathbb C","aliases":"C mathbb","display":"\\mathbb C","kind":"use","occurrences":[1110]},{"id":"symbol-db3c7ac2786221e9","tex":"\\mathcal C","aliases":"C mathcal","display":"\\mathcal C","kind":"use","occurrences":[1441,1442,1443,1452]},{"id":"symbol-6023b95954c36f1d","tex":"C^*","aliases":"C^*","display":"C^*","kind":"use","occurrences":[1445]},{"id":"symbol-512937534243ab85","tex":"\\C^{m\\times m}","aliases":"C^m× m C times mathbb times","display":"{\\mathbb C}^{m\\times m}","kind":"use","occurrences":[1224]},{"id":"symbol-eee5bcabc685dee4","tex":"\\C^{m\\times t}","aliases":"C^m× t C times mathbb times","display":"{\\mathbb C}^{m\\times t}","kind":"use","occurrences":[1431]},{"id":"symbol-8898d320fd158845","tex":"\\C^{n\\times m}","aliases":"C^n× m C times mathbb times","display":"{\\mathbb C}^{n\\times m}","kind":"use","occurrences":[1226]},{"id":"symbol-1dc82af4d957fdd4","tex":"\\C^{n\\times n}","aliases":"C^n× n C times mathbb times","display":"{\\mathbb C}^{n\\times n}","kind":"use","occurrences":[1225]},{"id":"symbol-e06675c775fa7c9d","tex":"\\mathbb C^{n\\times r_j}","aliases":"C^n× r_j mathbb times","display":"\\mathbb C^{n\\times r_j}","kind":"use","occurrences":[1110]},{"id":"symbol-c2e1f9b36ab7f058","tex":"\\C^{n\\times t}","aliases":"C^n× t C times mathbb times","display":"{\\mathbb C}^{n\\times t}","kind":"use","occurrences":[1431]},{"id":"symbol-2ccfe315d05aabc8","tex":"\\C^{r_L\\times r_R}","aliases":"C^r_L× r_R C times mathbb times","display":"{\\mathbb C}^{r_L\\times r_R}","kind":"use","occurrences":[1438]},{"id":"symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"use","occurrences":[542,543,544,1212,1215]},{"id":"symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","occurrences":[542,544]},{"id":"symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"use","occurrences":[1412]},{"id":"symbol-2a62ce5b7390bdd7","tex":"c_e","aliases":"c_e","display":"c_e","kind":"use","occurrences":[264,266,270,773,775,777,778,779,785,788,791,1013,1054,1055,1056,1057,1060,1062,1063,1066,1264,1266,1268,1271,1280,1288,1289,1291,1293,1296,1297,2274,2276,2280]},{"id":"symbol-cdae72f972852da1","tex":"c_f","aliases":"c_f","display":"c_f","kind":"use","occurrences":[777,780]},{"id":"symbol-8eb98997bbe300af","tex":"c_{i-j}","aliases":"c_i-j","display":"c_{i-j}","kind":"use","occurrences":[1211]},{"id":"symbol-60b3c954bda772f4","tex":"c_j","aliases":"c_j","display":"c_j","kind":"use","occurrences":[666,979]},{"id":"symbol-00ed39c395ad1950","tex":"c_{j,k}","aliases":"c_j,k","display":"c_{j,k}","kind":"use","occurrences":[803]},{"id":"symbol-40c460a40f8ac2ac","tex":"c_{j-k}","aliases":"c_j-k","display":"c_{j-k}","kind":"use","occurrences":[113,528,552,1199,2123]},{"id":"symbol-b26dec79c0d88ba5","tex":"c_{j-k\\bmod4}","aliases":"c_j-kbmod4 bmod","display":"c_{j-k\\bmod4}","kind":"use","occurrences":[97,2107]},{"id":"symbol-b7e8c8a0a1fa875e","tex":"c_{n-1}","aliases":"c_n-1","display":"c_{n-1}","kind":"use","occurrences":[542,544,1215]},{"id":"symbol-5f900eab61babb52","tex":"c_r","aliases":"c_r","display":"c_r","kind":"use","occurrences":[535]},{"id":"symbol-0c1fa1d6163c6fb8","tex":"\\CC^{2n}","aliases":"CC^2n C^2n CC mathbb","display":"{\\mathbb C}^{2n}","kind":"use","occurrences":[544]},{"id":"symbol-b9f0fa654a6d5449","tex":"\\CC^{m\\times m}","aliases":"CC^m× m C^m× m CC times mathbb times","display":"{\\mathbb C}^{m\\times m}","kind":"use","occurrences":[557]},{"id":"symbol-ac276f231fb12d59","tex":"\\CC^{m\\times r}","aliases":"CC^m× r C^m× r CC times mathbb times","display":"{\\mathbb C}^{m\\times r}","kind":"use","occurrences":[625]},{"id":"symbol-cf090df8943be9f2","tex":"\\CC^n","aliases":"CC^n C^n CC mathbb","display":"{\\mathbb C}^n","kind":"use","occurrences":[526,546]},{"id":"symbol-8e82c63e4ddfb564","tex":"\\CC^{n\\times m}","aliases":"CC^n× m C^n× m CC times mathbb times","display":"{\\mathbb C}^{n\\times m}","kind":"use","occurrences":[559]},{"id":"symbol-3e1e698420bbbca9","tex":"\\CC^{n\\times n}","aliases":"CC^n× n C^n× n CC times mathbb times","display":"{\\mathbb C}^{n\\times n}","kind":"use","occurrences":[541,558]},{"id":"symbol-f9a7361238345df3","tex":"\\mathrm{child}","aliases":"child mathrm","display":"\\mathrm{child}","kind":"use","occurrences":[1026]},{"id":"symbol-860a844f873baee1","tex":"\\operatorname{child}","aliases":"child operatorname","display":"\\operatorname{child}","kind":"use","occurrences":[828]},{"id":"symbol-269cd99c3ea8db81","tex":"\\code","aliases":"code","display":null,"kind":"use","occurrences":[1161]},{"id":"symbol-52ad9950a5d7bf10","tex":"\\operatorname{compressed}","aliases":"compressed operatorname","display":"\\operatorname{compressed}","kind":"use","occurrences":[1099]},{"id":"symbol-3f4e7b64e9db023c","tex":"\\cond","aliases":"cond operatorname","display":"{\\operatorname{cond}}","kind":"use","occurrences":[740,747]},{"id":"symbol-aa303ae817004b47","tex":"\\cond_2","aliases":"cond_2 cond operatorname","display":"{\\operatorname{cond}}_2","kind":"use","occurrences":[740,747]},{"id":"symbol-cef2f13a7966bbb0","tex":"\\coprod","aliases":"coprod","display":null,"kind":"use","occurrences":[1068]},{"id":"symbol-a0eafdcac63578ce","tex":"\\coprod_{e\\in E}","aliases":"coprod_e∈ E coprod in","display":null,"kind":"use","occurrences":[1068]},{"id":"symbol-e1a4d4510c259fa0","tex":"\\cost","aliases":"cost operatorname","display":"{\\operatorname{cost}}","kind":"use","occurrences":[292,373,821,825,847,848,849,852,1022,1035,1036,2302]},{"id":"symbol-cb7ca0be63af352d","tex":"D^{(m)}","aliases":"D^(m)","display":"D^{(m)}","kind":"use","occurrences":[1112,1114,1118,1119]},{"id":"symbol-fd12667298c0030a","tex":"D^{(m-1)}","aliases":"D^(m-1)","display":"D^{(m-1)}","kind":"use","occurrences":[1116,1117]},{"id":"symbol-10a6a0ef946e6792","tex":"D^{-1}","aliases":"D^-1","display":"D^{-1}","kind":"use","occurrences":[159,160,161,172,173,337,338,339,609,610,612,613,616,994,995,998,1984,1989,1990,1991,2169,2170,2171,2182,2183]},{"id":"symbol-6311982fa9099aea","tex":"D^{-1/2}","aliases":"D^-1/2","display":"D^{-1/2}","kind":"use","occurrences":[243,251,736,738,740,2253,2261]},{"id":"symbol-9329fd1deca2e237","tex":"d^2","aliases":"d^2","display":"d^2","kind":"use","occurrences":[576,583,1246]},{"id":"symbol-87a4659e9ddda329","tex":"D^j","aliases":"D^j","display":"D^j","kind":"use","occurrences":[242,728,731,733,1154,1158,2252]},{"id":"symbol-2f80fcde2a727a6d","tex":"D^k","aliases":"D^k","display":"D^k","kind":"use","occurrences":[731]},{"id":"symbol-7d80d76ad0ed63a4","tex":"d^k","aliases":"d^k","display":"d^k","kind":"use","occurrences":[573,1237]},{"id":"symbol-9d209e4a42d264b0","tex":"d^{k+1}","aliases":"d^k+1","display":"d^{k+1}","kind":"use","occurrences":[575,584,1245]},{"id":"symbol-c64f3c404f15cb37","tex":"d^{k+2}","aliases":"d^k+2","display":"d^{k+2}","kind":"use","occurrences":[1244]},{"id":"symbol-b43f40b2f922a574","tex":"d^{k-1}","aliases":"d^k-1","display":"d^{k-1}","kind":"use","occurrences":[579]},{"id":"symbol-04edc0a682114fdd","tex":"D_0","aliases":"D_0","display":"D_0","kind":"use","occurrences":[657]},{"id":"symbol-99e5f8e078f9fe31","tex":"D_1","aliases":"D_1","display":"D_1","kind":"use","occurrences":[1112]},{"id":"symbol-63ed82a00b685c9b","tex":"D_A","aliases":"D_A","display":"D_A","kind":"use","occurrences":[145,150,153,155,319,322,597,598,599,602,984,988,989,991,992,1590,1732,2155,2160,2163,2165,2329,2332]},{"id":"symbol-ee5bb7d36741efec","tex":"D_B","aliases":"D_B","display":"D_B","kind":"use","occurrences":[146,149,154,155,319,322,597,598,602,985,989,991,992,1590,1732,2156,2159,2164,2165,2329,2332]},{"id":"symbol-7f62021dadd3fc99","tex":"D_B^*","aliases":"D_B^*","display":"D_B^*","kind":"use","occurrences":[600,604,988]},{"id":"symbol-60f01eac58353ae3","tex":"D_B^T","aliases":"D_B^T","display":"D_B^T","kind":"use","occurrences":[150,2160]},{"id":"symbol-b1670fb432754538","tex":"D_j","aliases":"D_j","display":"D_j","kind":"use","occurrences":[605,656,657,1110,1111]},{"id":"symbol-0b05c681f47bcdf3","tex":"D_{j-1}","aliases":"D_j-1","display":"D_{j-1}","kind":"use","occurrences":[657]},{"id":"symbol-a572acccc9b7cf05","tex":"D_k","aliases":"D_k","display":"D_k","kind":"use","occurrences":[1455,1456,1475,1477,1503,1510,1512]},{"id":"symbol-71efea616c614868","tex":"D_m","aliases":"D_m","display":"D_m","kind":"use","occurrences":[658,1112,1117]},{"id":"symbol-1b008a80da8b3621","tex":"D_v^{-1}","aliases":"D_v^-1","display":"D_v^{-1}","kind":"use","occurrences":[1333,1339]},{"id":"symbol-de786ac59a3f75f3","tex":"D_\\beta","aliases":"D_β beta","display":"D_\\beta","kind":"use","occurrences":[680,683]},{"id":"symbol-f5ae8519ff6d0dff","tex":"\\mathrm{decode}","aliases":"decode mathrm","display":"\\mathrm{decode}","kind":"use","occurrences":[1099]},{"id":"symbol-1e31f1e54430399f","tex":"\\operatorname{decode}","aliases":"decode operatorname","display":"\\operatorname{decode}","kind":"use","occurrences":[429,430,1099]},{"id":"symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","occurrences":[169,2179]},{"id":"symbol-0658d980d8d1bc6a","tex":"\\operatorname{DFT}","aliases":"DFT operatorname","display":"\\operatorname{DFT}","kind":"use","occurrences":[1202,1205,1206]},{"id":"symbol-3b6b37c3ee83ea6f","tex":"\\diag","aliases":"diag operatorname","display":"{\\operatorname{diag}}","kind":"use","occurrences":[2,3,10,99,146,166,236,243,329,531,1455,1505,1506,1673,2012,2013,2020,2109,2156,2176,2246,2253]},{"id":"symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","occurrences":[1206,1207,1709,1712]},{"id":"symbol-558c1dc971656de6","tex":"\\dist","aliases":"dist operatorname","display":"{\\operatorname{dist}}","kind":"use","occurrences":[730]},{"id":"symbol-6bdeb30989e12919","tex":"\\operatorname{dist}","aliases":"dist operatorname","display":"\\operatorname{dist}","kind":"use","occurrences":[1159]},{"id":"symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","occurrences":[1425,1426,1428]},{"id":"symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","occurrences":[164,165,195,196,1992,2174,2175,2205,2206]},{"id":"symbol-115653a560af9b18","tex":"e_1^T","aliases":"e_1^T","display":"e_1^T","kind":"use","occurrences":[1992]},{"id":"symbol-98af5c47feb2b63c","tex":"E_2","aliases":"E_2","display":"E_2","kind":"use","occurrences":[1410]},{"id":"symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","occurrences":[195,196,2205,2206]},{"id":"symbol-2cf646d1eeab02b1","tex":"e_3","aliases":"e_3","display":"e_3","kind":"use","occurrences":[195,196,2205,2206]},{"id":"symbol-8db0cea46fdd60f3","tex":"E_A","aliases":"E_A","display":"E_A","kind":"use","occurrences":[957,958,959]},{"id":"symbol-a20a0c8f1552bcce","tex":"E_e","aliases":"E_e","display":"E_e","kind":"use","occurrences":[260,750,753,754,755,757,2270]},{"id":"symbol-ba526470e4ff9e1e","tex":"E_j","aliases":"E_j","display":"E_j","kind":"use","occurrences":[57,646,647,648,650,651,653,654,657,658,766,767,1127,1128,1129,1130,1131,1132,1133,2067]},{"id":"symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"use","occurrences":[195,1759,2205]},{"id":"symbol-0e681b3e1a1af15b","tex":"e_m","aliases":"e_m","display":"e_m","kind":"use","occurrences":[1426,1428]},{"id":"symbol-8b6f5031e4dc5e6e","tex":"\\eps","aliases":"eps ε varepsilon","display":"{\\varepsilon}","kind":"use","occurrences":[755,756,775,776,777,778,779,782,785,786]},{"id":"symbol-bc04c9966dd2873a","tex":"\\eps_e","aliases":"eps_e ε_e eps varepsilon","display":"{\\varepsilon}_e","kind":"use","occurrences":[755,756,775,776,777,778,779,782,785,786]},{"id":"symbol-7340ca44d667f098","tex":"\\err","aliases":"err operatorname","display":"{\\operatorname{err}}","kind":"use","occurrences":[292,363,373,822,825,846,848,849,850,852,1022,1035,1036,2302]},{"id":"symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"use","occurrences":[530,534,535,1200,1201]},{"id":"symbol-484de1c254ac2bb8","tex":"\\mathrm F","aliases":"F mathrm","display":"\\mathrm F","kind":"use","occurrences":[1443,1446,1502,1516,1517]},{"id":"symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","occurrences":[777,780]},{"id":"symbol-a3382034c436fa8d","tex":"F^*","aliases":"F^*","display":"F^*","kind":"use","occurrences":[99,103,531,538,1203,1204,1207,1673,2109,2113]},{"id":"symbol-62f1baf4056016b8","tex":"F^2","aliases":"F^2","display":"F^2","kind":"use","occurrences":[256,1613,2266]},{"id":"symbol-575255af81659400","tex":"F_0","aliases":"F_0","display":"F_0","kind":"use","occurrences":[800,971]},{"id":"symbol-74cb89ef83acdc31","tex":"F_j","aliases":"F_j","display":"F_j","kind":"use","occurrences":[797,803]},{"id":"symbol-7ac4acd8af2778de","tex":"F_{j-1}","aliases":"F_j-1","display":"F_{j-1}","kind":"use","occurrences":[801,803]},{"id":"symbol-6bf87b4a8f3226a8","tex":"F_{jk}","aliases":"F_jk","display":"F_{jk}","kind":"use","occurrences":[98,530,1200,1672,2108]},{"id":"symbol-8e49021145d6ab04","tex":"F_m","aliases":"F_m","display":"F_m","kind":"use","occurrences":[804]},{"id":"symbol-c57cafb013c6775f","tex":"F_n","aliases":"F_n","display":"F_n","kind":"use","occurrences":[971,972]},{"id":"symbol-7817222f46df5e65","tex":"F_{n+1}","aliases":"F_n+1","display":"F_{n+1}","kind":"use","occurrences":[971]},{"id":"symbol-faf064f796e64232","tex":"F_{\\rm root}","aliases":"F_rm root rm","display":"F_{\\rm root}","kind":"use","occurrences":[862]},{"id":"symbol-21c255185d10999e","tex":"F_{\\mathrm{root}}","aliases":"F_root mathrm","display":"F_{\\mathrm{root}}","kind":"use","occurrences":[1178]},{"id":"symbol-3dd40d0b9ff1c8ed","tex":"F_t","aliases":"F_t","display":"F_t","kind":"use","occurrences":[1185]},{"id":"symbol-78b29d109052da32","tex":"F_v","aliases":"F_v","display":"F_v","kind":"use","occurrences":[60,857,860,1174,1176,2070]},{"id":"symbol-e68207ff3f414911","tex":"F_{v_i}","aliases":"F_v_i","display":"F_{v_i}","kind":"use","occurrences":[1174]},{"id":"symbol-832e56d8bd1b7868","tex":"F_\\lambda","aliases":"F_λ lambda","display":"F_\\lambda","kind":"use","occurrences":[370,828,829,830,836,1025,1026,1027,1028,1042]},{"id":"symbol-642d1aafbe0d4290","tex":"\\operatorname{fill}","aliases":"fill operatorname","display":"\\operatorname{fill}","kind":"use","occurrences":[1052]},{"id":"symbol-280dad33f94a39d1","tex":"\\operatorname{Fin}","aliases":"Fin operatorname","display":"\\operatorname{Fin}","kind":"use","occurrences":[1068]},{"id":"symbol-2b0a62a84fa62d7d","tex":"\\flop_{\\rm block}","aliases":"flop_rm block flop rm operatorname rm","display":"{\\operatorname{flop}}_{\\rm block}","kind":"use","occurrences":[454]},{"id":"symbol-cde9aac4c00078d2","tex":"\\flop_{\\rm dense}","aliases":"flop_rm dense flop rm operatorname rm","display":"{\\operatorname{flop}}_{\\rm dense}","kind":"use","occurrences":[437]},{"id":"symbol-82a45cf5e0bc4cc8","tex":"\\flop_{\\rm packet}","aliases":"flop_rm packet flop rm operatorname rm","display":"{\\operatorname{flop}}_{\\rm packet}","kind":"use","occurrences":[439]},{"id":"symbol-25f0be3ddda5cf8c","tex":"G^*","aliases":"G^*","display":"G^*","kind":"use","occurrences":[1444,1506]},{"id":"symbol-103f7de3482048f3","tex":"G_k","aliases":"G_k","display":"G_k","kind":"use","occurrences":[1447,1452]},{"id":"symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","occurrences":[28,30,32,208,209,210,211,671,672,673,674,974,975,976,1507,1508,1509,1873,2038,2040,2042,2218,2219,2220,2221]},{"id":"symbol-0ea241c4c1df1f56","tex":"H_{2n}","aliases":"H_2n","display":"H_{2n}","kind":"use","occurrences":[139,2149]},{"id":"symbol-902dedcbd6a5754e","tex":"H_8","aliases":"H_8","display":"H_8","kind":"use","occurrences":[144,1983,2154]},{"id":"symbol-7a7409aaabc384b9","tex":"H_8^T","aliases":"H_8^T","display":"H_8^T","kind":"use","occurrences":[143,1983,2153]},{"id":"symbol-348cda0e12172b60","tex":"H_n","aliases":"H_n","display":"H_n","kind":"use","occurrences":[139,2149]},{"id":"symbol-b557fbd91b225d1c","tex":"H_N^T","aliases":"H_N^T","display":"H_N^T","kind":"use","occurrences":[588,1253,1261]},{"id":"symbol-e71e9ced609ec71d","tex":"H_n^T","aliases":"H_n^T","display":"H_n^T","kind":"use","occurrences":[139,2149]},{"id":"symbol-ab39294284c03d46","tex":"\\mathrm i","aliases":"i mathrm","display":"\\mathrm i","kind":"use","occurrences":[2,3,11,98,99,104,123,127,1672,2012,2013,2021,2108,2109,2114,2133,2137]},{"id":"symbol-b966b3d3d2ca3fe9","tex":"\\mathrm i^{-jk}","aliases":"i^-jk mathrm","display":"\\mathrm i^{-jk}","kind":"use","occurrences":[98,1672,2108]},{"id":"symbol-e2ed4b2197fa2598","tex":"I_2","aliases":"I_2","display":"I_2","kind":"use","occurrences":[163,232,239,1819,2173,2242,2249]},{"id":"symbol-4a54158041a35b80","tex":"\\iff","aliases":"iff","display":"\\iff","kind":"use","occurrences":[1091]},{"id":"symbol-88b326798e7011cf","tex":"\\ii","aliases":"ii i mathrm","display":"{\\mathrm i}","kind":"use","occurrences":[530,534,535]},{"id":"symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","occurrences":[21,825,1022,1027,1063,1064,1067,1266,1297,1298,1301,1463,1464,1466,1467,2031]},{"id":"symbol-98e38db59701cf95","tex":"J_{\\rm keep}","aliases":"J_rm keep rm","display":"J_{\\rm keep}","kind":"use","occurrences":[21,2031]},{"id":"symbol-b1a429be9f79e459","tex":"J_{\\rm split}","aliases":"J_rm split rm","display":"J_{\\rm split}","kind":"use","occurrences":[21,2031]},{"id":"symbol-fc399b7a1160964d","tex":"J_\\lambda","aliases":"J_λ lambda","display":"J_\\lambda","kind":"use","occurrences":[825,1022,1027]},{"id":"symbol-9da7de32ab512a2d","tex":"\\mathcal K","aliases":"K mathcal","display":"\\mathcal K","kind":"use","occurrences":[234,1138,1144,1157,1821,2244]},{"id":"symbol-fb9dab5747004561","tex":"K'","aliases":"K'","display":"K'","kind":"use","occurrences":[1043,1044]},{"id":"symbol-4b779dcac03cb4e5","tex":"K^2","aliases":"K^2","display":"K^2","kind":"use","occurrences":[13,206,212,1649,1798,2023,2216,2222]},{"id":"symbol-a7c895cee857ce71","tex":"K^h","aliases":"K^h","display":"K^h","kind":"use","occurrences":[211,2221]},{"id":"symbol-ba73dafd81ea4478","tex":"K^{h+1}","aliases":"K^h+1","display":"K^{h+1}","kind":"use","occurrences":[209,210,672,674,2219,2220]},{"id":"symbol-0af29e9b97d6dd25","tex":"K_{10}","aliases":"K_10","display":"K_{10}","kind":"use","occurrences":[206,2216]},{"id":"symbol-d9f725106db7a34c","tex":"K_{20}","aliases":"K_20","display":"K_{20}","kind":"use","occurrences":[206,2216]},{"id":"symbol-a4a6203c6556b0cb","tex":"K_{31}","aliases":"K_31","display":"K_{31}","kind":"use","occurrences":[206,2216]},{"id":"symbol-21a51e221f2f6f08","tex":"K_{32}","aliases":"K_32","display":"K_{32}","kind":"use","occurrences":[206,2216]},{"id":"symbol-b1fd09588ac918c1","tex":"\\mathcal K_{j+1}","aliases":"K_j+1 mathcal","display":"\\mathcal K_{j+1}","kind":"use","occurrences":[1157]},{"id":"symbol-4b35ad5d2218e1cf","tex":"\\mathcal K_m","aliases":"K_m mathcal","display":"\\mathcal K_m","kind":"use","occurrences":[234,1138,1144,1821,2244]},{"id":"symbol-44cc17134afaa418","tex":"K_{\\beta\\alpha}","aliases":"K_βα beta alpha","display":"K_{\\beta\\alpha}","kind":"use","occurrences":[680]},{"id":"symbol-ccedf2f845a654d7","tex":"\\Kcal_{j+1}","aliases":"Kcal_j+1 K_j+1 Kcal mathcal","display":"{\\mathcal K}_{j+1}","kind":"use","occurrences":[730]},{"id":"symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","occurrences":[1482]},{"id":"symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","occurrences":[2,3,4,7,48,56,174,175,177,179,180,181,182,184,625,626,627,629,635,636,638,639,642,1020,1022,1027,1033,1035,1036,1113,1114,1116,1117,1431,1432,1433,1435,1436,1437,1438,1439,1504,1511,1512,1516,1517,1675,1708,1712,1830,2012,2013,2014,2017,2058,2066,2184,2185,2187,2189,2190,2191,2192,2194]},{"id":"symbol-264918da85881789","tex":"\\mathcal L","aliases":"L mathcal","display":"\\mathcal L","kind":"use","occurrences":[1157,1158,1159,1441,1442,1443,1452]},{"id":"symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","occurrences":[21,454,1161,1174,2031]},{"id":"symbol-aacfb5e5cd28e2b2","tex":"L_1","aliases":"L_1","display":"L_1","kind":"use","occurrences":[1033,1035,1036]},{"id":"symbol-b4a121d33c2f6cd4","tex":"L_2","aliases":"L_2","display":"L_2","kind":"use","occurrences":[1033,1035,1036]},{"id":"symbol-0945233af0503d7f","tex":"\\mathcal L_{j+1}","aliases":"L_j+1 mathcal","display":"\\mathcal L_{j+1}","kind":"use","occurrences":[1157,1158,1159]},{"id":"symbol-dfc293305a0744ec","tex":"L_k","aliases":"L_k","display":"L_k","kind":"use","occurrences":[635,636,1511,1516,1517]},{"id":"symbol-ce43d1e9219f5a36","tex":"L_m","aliases":"L_m","display":"L_m","kind":"use","occurrences":[1113,1114,1117]},{"id":"symbol-66053b0fba2aa69a","tex":"L_{m-1}","aliases":"L_m-1","display":"L_{m-1}","kind":"use","occurrences":[1116,1117]},{"id":"symbol-3755b02b61222ecc","tex":"\\Lcal","aliases":"Lcal L mathcal","display":"{\\mathcal L}","kind":"use","occurrences":[820,821,822,825,844,845,846,847,848,849,850]},{"id":"symbol-271ba6ea9d5e2027","tex":"\\Lcal_1","aliases":"Lcal_1 L_1 Lcal mathcal","display":"{\\mathcal L}_1","kind":"use","occurrences":[844,846,847,848,849,850]},{"id":"symbol-cc64d46d6f1e674d","tex":"\\Lcal_2","aliases":"Lcal_2 L_2 Lcal mathcal","display":"{\\mathcal L}_2","kind":"use","occurrences":[845,846,847,848,849,850]},{"id":"symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","occurrences":[30,2040]},{"id":"symbol-5f10b967e4b2f95d","tex":"\\leadsto_{H}","aliases":"leadsto_H leadsto","display":null,"kind":"use","occurrences":[1367]},{"id":"symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"use","occurrences":[317,890,1106,2327]},{"id":"symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","occurrences":[264,539,590,596,775,1013,1056,1061,1062,1063,1209,1259,1266,1294,1296,1297,1982,2274]},{"id":"symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"use","occurrences":[590,596,1259,1982]},{"id":"symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","occurrences":[1044,1104]},{"id":"symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","occurrences":[976,1015,1067,1301,1340,1361]},{"id":"symbol-e0639ea84119f55a","tex":"\\operatorname{lookup}","aliases":"lookup operatorname","display":"\\operatorname{lookup}","kind":"use","occurrences":[1185]},{"id":"symbol-fc33b4e3162aeb75","tex":"M^*","aliases":"M^*","display":"M^*","kind":"use","occurrences":[1400,1510]},{"id":"symbol-ca52a161c084be9c","tex":"M^{-1/2}","aliases":"M^-1/2","display":"M^{-1/2}","kind":"use","occurrences":[747]},{"id":"symbol-068897646ca3bf5a","tex":"m^2","aliases":"m^2","display":"m^2","kind":"use","occurrences":[134,135,137,561,562,563,1231,1232,2144,2145,2147]},{"id":"symbol-0a98f829b8404676","tex":"M^{j-1}","aliases":"M^j-1","display":"M^{j-1}","kind":"use","occurrences":[242,728,730,732,1154,1159,2252]},{"id":"symbol-2eb18a26b2462763","tex":"m_e","aliases":"m_e","display":"m_e","kind":"use","occurrences":[270,781,782,783,785,787,788,790,791,1068,1072,1073,1086,1264,1265,1268,1271,1273,1280,1281,1290,1291,1292,2280]},{"id":"symbol-83d3f2e0b215ee50","tex":"M_{ij}","aliases":"M_ij","display":"M_{ij}","kind":"use","occurrences":[1509]},{"id":"symbol-12fcf6e1ba955880","tex":"M_{\\rm block}","aliases":"M_rm block rm","display":"M_{\\rm block}","kind":"use","occurrences":[454]},{"id":"symbol-27220d8a886e4747","tex":"M_{\\rm packet}","aliases":"M_rm packet rm","display":"M_{\\rm packet}","kind":"use","occurrences":[440]},{"id":"symbol-956f614ae0645ab5","tex":"\\mat","aliases":"mat","display":null,"kind":"use","occurrences":[2,3,12,14,67,109,123,124,125,127,139,158,174,179,182,224,226,243,244,2012,2013,2022,2024,2077,2119,2133,2134,2135,2137,2149,2168,2184,2189,2192,2234,2236,2253,2254]},{"id":"symbol-b29f89d48389ba9c","tex":"\\max_{v\\in V}","aliases":"max_v∈ V max in","display":"\\max_{v\\in V}","kind":"use","occurrences":[1326]},{"id":"symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","occurrences":[270,483,632,775,785,788,791,803,830,890,1026,1041,1048,1056,1090,1106,1163,1166,1167,1174,1178,1268,1280,1455,1507,2280]},{"id":"symbol-83370d7c90a0d024","tex":"\\min_{(w,c)\\text{ allowed},\\;w\\leq b}","aliases":"min_(w,c) allowed, w≤ b min text leq","display":"\\min_{(w,c)\\text{ allowed},\\;w\\leq b}","kind":"use","occurrences":[1166]},{"id":"symbol-08929fcbd469af5d","tex":"\\min_{0\\leq b\\leq B}","aliases":"min_0≤ b≤ B min leq leq","display":"\\min_{0\\leq b\\leq B}","kind":"use","occurrences":[1167,1178]},{"id":"symbol-c9ccf46f5921c4a5","tex":"\\min_{0\\leq k\\leq b}","aliases":"min_0≤ k≤ b min leq leq","display":"\\min_{0\\leq k\\leq b}","kind":"use","occurrences":[1163]},{"id":"symbol-1ace88525e5ee17f","tex":"\\min_e","aliases":"min_e min","display":"\\min_e","kind":"use","occurrences":[791]},{"id":"symbol-b765b9bd7781a783","tex":"\\min_{\\eps_e>0}","aliases":"min_eps_e>0 min_ε_e>0 min eps min varepsilon","display":"\\min_{{\\varepsilon}_e>0}","kind":"use","occurrences":[775]},{"id":"symbol-567e9df79d4789c7","tex":"\\min_k","aliases":"min_k min","display":"\\min_k","kind":"use","occurrences":[803]},{"id":"symbol-ab2c688fc16fea97","tex":"\\min_{\\varepsilon_e>0}","aliases":"min_ε_e>0 min varepsilon","display":"\\min_{\\varepsilon_e>0}","kind":"use","occurrences":[1056]},{"id":"symbol-cd9b1bf9b23006d6","tex":"\\min_{\\sum_i b_i=b}","aliases":"min_∑_i b_i=b min sum","display":"\\min_{\\sum_i b_i=b}","kind":"use","occurrences":[1174]},{"id":"symbol-3f620354ed5db565","tex":"\\mathbb N","aliases":"N mathbb","display":"\\mathbb N","kind":"use","occurrences":[1324]},{"id":"symbol-a80e516bfb196e1c","tex":"\\n","aliases":"n","display":null,"kind":"use","occurrences":[1710,1792]},{"id":"symbol-79616ac402fc2170","tex":"n^{-1/2}","aliases":"n^-1/2","display":"n^{-1/2}","kind":"use","occurrences":[1202]},{"id":"symbol-12dcb271d25c745a","tex":"n^2","aliases":"n^2","display":"n^2","kind":"use","occurrences":[77,134,135,137,433,437,494,495,561,562,563,1070,1071,1091,1094,1095,1096,1098,1231,1232,2087,2144,2145,2147]},{"id":"symbol-b9588331116f694e","tex":"n_e","aliases":"n_e","display":"n_e","kind":"use","occurrences":[1068,1072,1073,1086]},{"id":"symbol-b6801e9881994d7d","tex":"n_i^2","aliases":"n_i^2","display":"n_i^2","kind":"use","occurrences":[1076,1077]},{"id":"symbol-af2a18fd20397da4","tex":"N_{p,w}","aliases":"N_p,w","display":"N_{p,w}","kind":"use","occurrences":[506,508,1375,1383,1386]},{"id":"symbol-3b600c930ff971ba","tex":"n_\\alpha^2","aliases":"n_α^2 alpha","display":"n_\\alpha^2","kind":"use","occurrences":[463,464,469,473]},{"id":"symbol-8205d7811a8a9f9c","tex":"n_\\beta","aliases":"n_β beta","display":"n_\\beta","kind":"use","occurrences":[439,440,442,446,447,476]},{"id":"symbol-90f600277c50100a","tex":"\\NN","aliases":"NN N mathbb","display":"{\\mathbb N}","kind":"use","occurrences":[855,886,902]},{"id":"symbol-7bc2fe684c29431d","tex":"\\norm","aliases":"norm","display":null,"kind":"use","occurrences":[616,622,623,647,653,654,728,729,730,732,739,748,754,755,756,814,815,998,1006,1007,1008,1009,1010,1011,1128,1129,1133,1135,1137,1152,1154,1159,1400,1401,1402,1403,1410,1412,1415,1443,1446,1452,1470,1472,1474,1502,1516,1517]},{"id":"symbol-e16b60493b1a650b","tex":"\\not","aliases":"not","display":null,"kind":"use","occurrences":[1349,1361]},{"id":"symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","occurrences":[495,539,1098,1209]},{"id":"symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"use","occurrences":[1385,1386]},{"id":"symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","occurrences":[454,862,895,896,897,1161]},{"id":"symbol-c1a14ba7424a4a8c","tex":"\\ones","aliases":"ones","display":null,"kind":"use","occurrences":[68,222,253,2078,2232,2263]},{"id":"symbol-3d48a1e60ecf7c1c","tex":"\\ones^T","aliases":"ones^T ones","display":null,"kind":"use","occurrences":[253,2263]},{"id":"symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"use","occurrences":[21,89,91,92,95,193,195,234,439,440,448,450,452,465,466,501,502,503,505,506,508,509,512,513,666,667,758,759,761,763,841,979,1174,1339,1370,1372,1374,1375,1376,1378,1379,1380,1382,1383,1386,1387,1389,1820,2031,2099,2101,2102,2105,2203,2205,2244]},{"id":"symbol-79867b3b5190754f","tex":"p^2","aliases":"p^2","display":"p^2","kind":"use","occurrences":[1380]},{"id":"symbol-e50b5dbfe7add137","tex":"P_0","aliases":"P_0","display":"P_0","kind":"use","occurrences":[649,1121]},{"id":"symbol-db6b2ac0be28aba6","tex":"p_0","aliases":"p_0","display":"p_0","kind":"use","occurrences":[193,2203]},{"id":"symbol-7c2443102c181d52","tex":"p_3","aliases":"p_3","display":"p_3","kind":"use","occurrences":[195,2205]},{"id":"symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"use","occurrences":[954,955,956,967,971,972,1010,1345,1347,1350,1360,1361,1390,1392,1393,1395,1398,1401,1402]},{"id":"symbol-07a802f35729938f","tex":"P_{a(e)}","aliases":"P_a(e)","display":"P_{a(e)}","kind":"use","occurrences":[1003,1005,1006,1008,1009]},{"id":"symbol-5e3a3e3d08311a9a","tex":"P_b","aliases":"P_b","display":"P_b","kind":"use","occurrences":[955,956,967,972,1347,1349,1361,1393,1395,1401,1402]},{"id":"symbol-3270e71bf1ae54c5","tex":"P_c","aliases":"P_c","display":"P_c","kind":"use","occurrences":[1395,1398]},{"id":"symbol-14f8900152d6716a","tex":"P_d","aliases":"P_d","display":"P_d","kind":"use","occurrences":[1395]},{"id":"symbol-429e7f1087bafa6e","tex":"P_i","aliases":"P_i","display":"P_i","kind":"use","occurrences":[1139,1141,1142,1143,1144,1146,1147,1148,1157]},{"id":"symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","occurrences":[649,656,1122,1124,1132]},{"id":"symbol-7a5d03f7fbf7ca50","tex":"P_{j+1}","aliases":"P_j+1","display":"P_{j+1}","kind":"use","occurrences":[1122,1124]},{"id":"symbol-5e6b6a6054c3c5d9","tex":"P_{j,a(e)}","aliases":"P_j,a(e)","display":"P_{j,a(e)}","kind":"use","occurrences":[1134,1135]},{"id":"symbol-2646be694d017280","tex":"P_{j-1}","aliases":"P_j-1","display":"P_{j-1}","kind":"use","occurrences":[649,1132]},{"id":"symbol-49f3b852810c6d4e","tex":"P_m","aliases":"P_m","display":"P_m","kind":"use","occurrences":[653,1112,1114,1118,1133]},{"id":"symbol-5266a4839b2b51a8","tex":"P_{m-1}","aliases":"P_m-1","display":"P_{m-1}","kind":"use","occurrences":[1116]},{"id":"symbol-5b1d3ec03a20b6b9","tex":"P_{\\alpha}","aliases":"P_α alpha","display":"P_{\\alpha}","kind":"use","occurrences":[770]},{"id":"symbol-056b9963f80df264","tex":"P_{\\alpha(e)}","aliases":"P_α(e) alpha","display":"P_{\\alpha(e)}","kind":"use","occurrences":[260,753,754,756,757,2270]},{"id":"symbol-e2bbf7f75432b536","tex":"P_{\\alpha(j,e)}","aliases":"P_α(j,e) alpha","display":"P_{\\alpha(j,e)}","kind":"use","occurrences":[766,767]},{"id":"symbol-9813ba63ac7cffd1","tex":"P_\\gamma","aliases":"P_γ gamma","display":"P_\\gamma","kind":"use","occurrences":[418,660]},{"id":"symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","occurrences":[1,2,3,4,7,56,174,175,179,180,184,426,626,627,628,635,639,642,967,1433,1434,1436,1439,1440,1441,1444,1446,1481,1482,1483,1485,1486,1501,1502,1511,1512,1708,1712,2011,2012,2013,2014,2017,2066,2184,2185,2189,2190,2194]},{"id":"symbol-3f115f6cb4125cf9","tex":"\\Q","aliases":"Q mathbb","display":"{\\mathbb Q}","kind":"use","occurrences":[1031,1040]},{"id":"symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","occurrences":[76,77,80,82,83,307,308,309,310,436,437,439,445,446,454,455,461,463,467,469,472,475,476,507,508,519,520,522,523,1069,1071,1073,1075,1077,1079,1081,1086,1384,1385,1386,1417,1418,1420,1422,1423,1426,1428,1482,1483,1484,2006,2008,2086,2087,2090,2092,2093,2317,2318,2319,2320]},{"id":"symbol-d76d885e9e3c5ba5","tex":"Q^*","aliases":"Q^*","display":"Q^*","kind":"use","occurrences":[1,1441,1446,1482,1485,1486,2011]},{"id":"symbol-a4591bb144ad9c3f","tex":"Q_c","aliases":"Q_c","display":"Q_c","kind":"use","occurrences":[967]},{"id":"symbol-83dc7e104ed7732b","tex":"Q_d","aliases":"Q_d","display":"Q_d","kind":"use","occurrences":[967]},{"id":"symbol-dfe49e85066a6b84","tex":"Q_{ij}","aliases":"Q_ij","display":"Q_{ij}","kind":"use","occurrences":[1483]},{"id":"symbol-9bd193e7de59f86f","tex":"Q_L","aliases":"Q_L","display":"Q_L","kind":"use","occurrences":[2,3,4,7,56,174,175,179,180,626,627,635,639,1433,1436,1439,1511,1512,1708,1712,2012,2013,2014,2017,2066,2184,2185,2189,2190]},{"id":"symbol-9bcb84d1006a5379","tex":"Q_L^*","aliases":"Q_L^*","display":"Q_L^*","kind":"use","occurrences":[184,642,1436,1712,2194]},{"id":"symbol-5c06fc40d19e1dce","tex":"Q_L^T","aliases":"Q_L^T","display":"Q_L^T","kind":"use","occurrences":[7,1712,2017]},{"id":"symbol-808125305b3a9f1c","tex":"Q_R","aliases":"Q_R","display":"Q_R","kind":"use","occurrences":[2,56,184,626,628,635,642,1434,1511,2012,2066,2194]},{"id":"symbol-b2e800860e3d6ef1","tex":"Q_R^*","aliases":"Q_R^*","display":"Q_R^*","kind":"use","occurrences":[3,639,1439,1512,2013]},{"id":"symbol-5a423e1823849ed7","tex":"Q_\\gamma","aliases":"Q_γ gamma","display":"Q_\\gamma","kind":"use","occurrences":[426]},{"id":"symbol-aa7fbe64d6b9e7ee","tex":"Q_\\delta","aliases":"Q_δ delta","display":"Q_\\delta","kind":"use","occurrences":[426]},{"id":"symbol-221aa401f4b73008","tex":"\\QQ","aliases":"QQ","display":null,"kind":"use","occurrences":[11,68,104,235,252,2021,2078,2114,2245,2262]},{"id":"symbol-b9a6d997b7294e64","tex":"\\QQ^2","aliases":"QQ^2 QQ","display":null,"kind":"use","occurrences":[235,2245]},{"id":"symbol-4dafa9764bf19601","tex":"\\QQ^4","aliases":"QQ^4 QQ","display":null,"kind":"use","occurrences":[252,2262]},{"id":"symbol-0989b37fff5d66a9","tex":"\\QQ^6","aliases":"QQ^6 QQ","display":null,"kind":"use","occurrences":[68,2078]},{"id":"symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","occurrences":[2,3,49,56,62,174,175,176,178,179,182,184,226,314,315,316,427,429,431,432,625,626,628,629,635,636,638,639,642,689,692,693,694,696,697,698,699,701,704,706,709,710,711,886,888,889,890,892,893,897,902,903,952,1104,1106,1113,1114,1116,1117,1363,1431,1432,1434,1435,1436,1437,1438,1439,1504,1511,1512,1516,1517,1521,1522,1523,1689,1708,1830,1884,2012,2013,2059,2066,2072,2184,2185,2186,2188,2189,2192,2194,2236,2324,2325,2326]},{"id":"symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R mathbb","display":"{\\mathbb R}","kind":"use","occurrences":[1032]},{"id":"symbol-055d5ee422435b80","tex":"R^*","aliases":"R^*","display":"R^*","kind":"use","occurrences":[638,639,1431,1516,1517]},{"id":"symbol-42f97ca6df89f2f9","tex":"R^T","aliases":"R^T","display":"R^T","kind":"use","occurrences":[179,182,1708,1830,2189,2192]},{"id":"symbol-a6be10a383c0ff11","tex":"r_0","aliases":"r_0","display":"r_0","kind":"use","occurrences":[1213]},{"id":"symbol-6e93838ea3bf8d2c","tex":"r_A","aliases":"r_A","display":"r_A","kind":"use","occurrences":[601,986,990,991]},{"id":"symbol-c6016f5cf675a3ae","tex":"r_B","aliases":"r_B","display":"r_B","kind":"use","occurrences":[601,987,990,991]},{"id":"symbol-b3975fa3c319f1d3","tex":"r_e","aliases":"r_e","display":"r_e","kind":"use","occurrences":[1059,1060,1062,1063,1065]},{"id":"symbol-35e4b1420672d392","tex":"r_j","aliases":"r_j","display":"r_j","kind":"use","occurrences":[607,840,841,842,1029,1030,1110,1115,1118,1120]},{"id":"symbol-f5a0eed759a6f31c","tex":"r_{j-i}","aliases":"r_j-i","display":"r_{j-i}","kind":"use","occurrences":[1211]},{"id":"symbol-a64e897db39c6a7f","tex":"R_k","aliases":"R_k","display":"R_k","kind":"use","occurrences":[635,1511]},{"id":"symbol-61d7b9e7617499dd","tex":"r_{k-j}","aliases":"r_k-j","display":"r_{k-j}","kind":"use","occurrences":[115,554,2125]},{"id":"symbol-eae30801c8c7f23d","tex":"R_k^*","aliases":"R_k^*","display":"R_k^*","kind":"use","occurrences":[636,1516,1517]},{"id":"symbol-61bdbf136c2dac84","tex":"R_L","aliases":"R_L","display":"R_L","kind":"use","occurrences":[174,175,179,626,629,1435,1436,1438,1504,2184,2185,2189]},{"id":"symbol-d96a67e19d75bf1d","tex":"r_L","aliases":"r_L","display":"r_L","kind":"use","occurrences":[1432,1437,1438]},{"id":"symbol-25ee15775b880083","tex":"R_m","aliases":"R_m","display":"R_m","kind":"use","occurrences":[1113,1117]},{"id":"symbol-d151cfb191c2c72b","tex":"R_{m-1}","aliases":"R_m-1","display":"R_{m-1}","kind":"use","occurrences":[1117]},{"id":"symbol-147aec0806fe40f4","tex":"R_{m-1}^*","aliases":"R_m-1^*","display":"R_{m-1}^*","kind":"use","occurrences":[1116]},{"id":"symbol-f02b5ce794f70074","tex":"R_m^*","aliases":"R_m^*","display":"R_m^*","kind":"use","occurrences":[1114]},{"id":"symbol-0f7c3db2a7afd485","tex":"R_{\\min}","aliases":"R_min min","display":"R_{\\min}","kind":"use","occurrences":[890,1106]},{"id":"symbol-9d4ab28ea1385868","tex":"r_{n-1}","aliases":"r_n-1","display":"r_{n-1}","kind":"use","occurrences":[543,544,1215]},{"id":"symbol-ed27d154616b4c67","tex":"R_R","aliases":"R_R","display":"R_R","kind":"use","occurrences":[174,176,626,1435,2184,2186]},{"id":"symbol-ace9c120d24521df","tex":"r_R","aliases":"r_R","display":"r_R","kind":"use","occurrences":[1432,1437,1438]},{"id":"symbol-dbe6bc0f4ac42150","tex":"R_R^*","aliases":"R_R^*","display":"R_R^*","kind":"use","occurrences":[629,1438,1504]},{"id":"symbol-868ac8e8bd426dc4","tex":"R_R^T","aliases":"R_R^T","display":"R_R^T","kind":"use","occurrences":[179,2189]},{"id":"symbol-9bf28df79e66226e","tex":"\\operatorname{range}","aliases":"range operatorname","display":"\\operatorname{range}","kind":"use","occurrences":[1432]},{"id":"symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","occurrences":[1118]},{"id":"symbol-6621036c9088e172","tex":"\\rank","aliases":"rank operatorname","display":"{\\operatorname{rank}}","kind":"use","occurrences":[991,1437,1439,1442,1451,1464,1480,1482,1515]},{"id":"symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","occurrences":[30,2040]},{"id":"symbol-1002ed943b3a0c5e","tex":"\\operatorname{Re}","aliases":"Re operatorname","display":"\\operatorname{Re}","kind":"use","occurrences":[1400]},{"id":"symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"use","occurrences":[317,890,1106,2327]},{"id":"symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"use","occurrences":[21,308,437,439,440,454,476,623,814,841,862,1400,1401,2031,2318]},{"id":"symbol-a58e4c2e82e7cd1b","tex":"\\mathrm{root}","aliases":"root mathrm","display":"\\mathrm{root}","kind":"use","occurrences":[1178]},{"id":"symbol-a62a596621079394","tex":"\\RR","aliases":"RR R mathbb","display":"{\\mathbb R}","kind":"use","occurrences":[758,759]},{"id":"symbol-162e5e4a459cac13","tex":"\\RR^{m\\times p}","aliases":"RR^m× p R^m× p RR times mathbb times","display":"{\\mathbb R}^{m\\times p}","kind":"use","occurrences":[758]},{"id":"symbol-2831aff814dfd9d4","tex":"\\RR^{p\\times n}","aliases":"RR^p× n R^p× n RR times mathbb times","display":"{\\mathbb R}^{p\\times n}","kind":"use","occurrences":[759]},{"id":"symbol-c2b4530a730b06f1","tex":"\\mathrm{run}","aliases":"run mathrm","display":"\\mathrm{run}","kind":"use","occurrences":[1099]},{"id":"symbol-dc5c5f4f378f97aa","tex":"\\mathcal S","aliases":"S mathcal","display":"\\mathcal S","kind":"use","occurrences":[1382,1383,1385]},{"id":"symbol-a43e0d8c06476226","tex":"S'","aliases":"S'","display":"S'","kind":"use","occurrences":[1043,1044]},{"id":"symbol-cf8c7f65df694c86","tex":"S^{-1}","aliases":"S^-1","display":"S^{-1}","kind":"use","occurrences":[161,171,230,610,613,616,995,998,2171,2181,2240]},{"id":"symbol-cc4fbf61a6a9f156","tex":"S_{ba}","aliases":"S_ba","display":"S_{ba}","kind":"use","occurrences":[1366]},{"id":"symbol-7aa9c815c9ae3803","tex":"s_e","aliases":"s_e","display":"s_e","kind":"use","occurrences":[58,265,266,270,772,776,777,778,779,783,785,786,788,791,1013,1015,1054,1056,1057,1264,1265,1268,1271,1273,1280,1281,1288,1289,1291,1293,1859,2068,2275,2276,2280]},{"id":"symbol-85a648879a8456a4","tex":"S_L","aliases":"S_L","display":"S_L","kind":"use","occurrences":[1432]},{"id":"symbol-f7af78049dcddf01","tex":"S_R","aliases":"S_R","display":"S_R","kind":"use","occurrences":[1432]},{"id":"symbol-91ad15719e4542fe","tex":"\\mathrm{setup}","aliases":"setup mathrm","display":"\\mathrm{setup}","kind":"use","occurrences":[1030]},{"id":"symbol-89d7944b7e472825","tex":"\\operatorname{step}","aliases":"step operatorname","display":"\\operatorname{step}","kind":"use","occurrences":[1048]},{"id":"symbol-327fd569cd80a61b","tex":"\\operatorname{storedTree}","aliases":"storedTree operatorname","display":"\\operatorname{storedTree}","kind":"use","occurrences":[1185]},{"id":"symbol-d6008072c780cd95","tex":"\\mathrm{stream}","aliases":"stream mathrm","display":"\\mathrm{stream}","kind":"use","occurrences":[1086]},{"id":"symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","occurrences":[21,308,439,440,476,841,862,885,887,889,890,891,892,893,895,896,897,899,900,903,904,1099,1101,1102,1103,1104,1106,1174,1185,1191,1192,1193,1267,1268,1274,1275,1276,1277,1278,1279,1286,1288,1289,1291,1293,1339,1431,2031,2318]},{"id":"symbol-6da3583735a2a7a0","tex":"T'","aliases":"T'","display":"T'","kind":"use","occurrences":[1050,1052]},{"id":"symbol-cefa01be96058c3e","tex":"T_b","aliases":"T_b","display":"T_b","kind":"use","occurrences":[314,2324]},{"id":"symbol-d1367c12dada8011","tex":"t_b","aliases":"t_b","display":"t_b","kind":"use","occurrences":[885,887,889,890,891,893,1101,1103,1104,1106]},{"id":"symbol-0692c76d8494573e","tex":"T_c","aliases":"T_c","display":"T_c","kind":"use","occurrences":[314,2324]},{"id":"symbol-04db525251126e68","tex":"t_c","aliases":"t_c","display":"t_c","kind":"use","occurrences":[885,887,889,890,891,892,899,903,904,1102,1103,1104,1106]},{"id":"symbol-aa6a46659a84add3","tex":"t_{\\mathrm{decode}}","aliases":"t_decode mathrm","display":"t_{\\mathrm{decode}}","kind":"use","occurrences":[1099]},{"id":"symbol-c78b7188b7b592c0","tex":"t_e","aliases":"t_e","display":"t_e","kind":"use","occurrences":[900,903,904]},{"id":"symbol-e9a7aef78fcda23e","tex":"T_{ij}","aliases":"T_ij","display":"T_{ij}","kind":"use","occurrences":[1211,1217]},{"id":"symbol-dcee229e3aaf9526","tex":"t_o","aliases":"t_o","display":"t_o","kind":"use","occurrences":[895,896,897]},{"id":"symbol-aaffcbbdb2c59acf","tex":"t_r","aliases":"t_r","display":"t_r","kind":"use","occurrences":[896,897]},{"id":"symbol-4775952797c5d985","tex":"t_{\\mathrm{run}}","aliases":"t_run mathrm","display":"t_{\\mathrm{run}}","kind":"use","occurrences":[1099]},{"id":"symbol-6e51d2fa53a6aa7c","tex":"\\operatorname{tr}","aliases":"tr operatorname","display":"\\operatorname{tr}","kind":"use","occurrences":[1395,1400]},{"id":"symbol-7c05ecd0a14b319f","tex":"u^T","aliases":"u^T","display":"u^T","kind":"use","occurrences":[160,161,1990,1991,2170,2171]},{"id":"symbol-e105a73706668fb5","tex":"U_A","aliases":"U_A","display":"U_A","kind":"use","occurrences":[597,599,602,604,984,988,1591]},{"id":"symbol-91c677571574a0b9","tex":"u_a","aliases":"u_a","display":"u_a","kind":"use","occurrences":[145,150,153,154,157,2155,2160,2163,2164,2167]},{"id":"symbol-91de088f77bc89c5","tex":"U_B","aliases":"U_B","display":"U_B","kind":"use","occurrences":[597,599,602,985,988,1591]},{"id":"symbol-e21dd43c611f6c4e","tex":"u_b","aliases":"u_b","display":"u_b","kind":"use","occurrences":[147,149,150,153,156,2157,2159,2160,2163,2166]},{"id":"symbol-cfbfd121cc430e57","tex":"U_C","aliases":"U_C","display":"U_C","kind":"use","occurrences":[150,155,320,322,598,599,988,989,1564,1590,1739,2160,2165,2330,2332]},{"id":"symbol-b1bfe02ef752d496","tex":"u_e","aliases":"u_e","display":"u_e","kind":"use","occurrences":[1268,1288,1289,1290,1293,1296,1297]},{"id":"symbol-8e0ecb4b8f25eabc","tex":"U_j","aliases":"U_j","display":"U_j","kind":"use","occurrences":[605,1110]},{"id":"symbol-25121c20156ac918","tex":"U_m","aliases":"U_m","display":"U_m","kind":"use","occurrences":[1117]},{"id":"symbol-fb939028bb05fbbc","tex":"\\operatorname{updateValue}","aliases":"updateValue operatorname","display":"\\operatorname{updateValue}","kind":"use","occurrences":[1052]},{"id":"symbol-1b5cec8c46451b53","tex":"\\v","aliases":"v","display":null,"kind":"use","occurrences":[1711,1870]},{"id":"symbol-b684dfa3e6bcfe1e","tex":"v^{(k)}","aliases":"v^(k)","display":"v^{(k)}","kind":"use","occurrences":[535,537]},{"id":"symbol-2fe163fa16dae77c","tex":"v^{(k)}_j","aliases":"v^(k)_j","display":"v^{(k)}_j","kind":"use","occurrences":[534,535]},{"id":"symbol-b5eff5be3aa59d19","tex":"v^{(k)}_{j-r}","aliases":"v^(k)_j-r","display":"v^{(k)}_{j-r}","kind":"use","occurrences":[535]},{"id":"symbol-8c41a1fdf1eb317c","tex":"V_A","aliases":"V_A","display":"V_A","kind":"use","occurrences":[600,604,988]},{"id":"symbol-93a28f0d222e68e7","tex":"v_a","aliases":"v_a","display":"v_a","kind":"use","occurrences":[150,2160]},{"id":"symbol-0367417d53502da1","tex":"V_A^*","aliases":"V_A^*","display":"V_A^*","kind":"use","occurrences":[597,599,602,984,988]},{"id":"symbol-db1a7ed5fdbbc416","tex":"V_A^T","aliases":"V_A^T","display":"V_A^T","kind":"use","occurrences":[1591]},{"id":"symbol-612c002c1c1e9a93","tex":"v_a^T","aliases":"v_a^T","display":"v_a^T","kind":"use","occurrences":[145,150,153,154,156,2155,2160,2163,2164,2166]},{"id":"symbol-9f5cc10c11d8d9e7","tex":"V_B","aliases":"V_B","display":"V_B","kind":"use","occurrences":[600,988]},{"id":"symbol-3234e8a9dca18cb9","tex":"v_b","aliases":"v_b","display":"v_b","kind":"use","occurrences":[148,150,2158,2160]},{"id":"symbol-d343ba286305f1ad","tex":"V_B^*","aliases":"V_B^*","display":"V_B^*","kind":"use","occurrences":[597,602,603,985]},{"id":"symbol-12d0f7ef25761981","tex":"V_B^T","aliases":"V_B^T","display":"V_B^T","kind":"use","occurrences":[1591]},{"id":"symbol-fe377465e680a401","tex":"v_b^T","aliases":"v_b^T","display":"v_b^T","kind":"use","occurrences":[149,152,153,157,2159,2162,2163,2167]},{"id":"symbol-9336b2ab1b4f322f","tex":"V_C","aliases":"V_C","display":"V_C","kind":"use","occurrences":[150,320,600,988,1577,2160,2330]},{"id":"symbol-b0f0625a5bf696ae","tex":"V_C^*","aliases":"V_C^*","display":"V_C^*","kind":"use","occurrences":[598,989]},{"id":"symbol-1c63837778b9b8ed","tex":"V_C^T","aliases":"V_C^T","display":"V_C^T","kind":"use","occurrences":[155,322,1590,1729,1739,2165,2332]},{"id":"symbol-23dfe8b37c1f5d9f","tex":"v_i","aliases":"v_i","display":"v_i","kind":"use","occurrences":[1174,1491,1493,1494,1499]},{"id":"symbol-3de92b22860491e7","tex":"V_j","aliases":"V_j","display":"V_j","kind":"use","occurrences":[1110]},{"id":"symbol-eb483cd80b51511d","tex":"V_j^*","aliases":"V_j^*","display":"V_j^*","kind":"use","occurrences":[605,1110]},{"id":"symbol-8b86c046f5cd21ee","tex":"V_m","aliases":"V_m","display":"V_m","kind":"use","occurrences":[1117]},{"id":"symbol-83eb9ad5dd648cb4","tex":"V_m^*","aliases":"V_m^*","display":"V_m^*","kind":"use","occurrences":[1117]},{"id":"symbol-c3a76197560b59e1","tex":"v_\\alpha","aliases":"v_α alpha","display":"v_\\alpha","kind":"use","occurrences":[718,719,722,730]},{"id":"symbol-022a12d8dd418b72","tex":"\\operatorname{val}","aliases":"val operatorname","display":"\\operatorname{val}","kind":"use","occurrences":[1201]},{"id":"symbol-f7cc81212c08d083","tex":"\\operatorname{vec}","aliases":"vec operatorname","display":"\\operatorname{vec}","kind":"use","occurrences":[122,560,567,1227,1806,1818,2132]},{"id":"symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","occurrences":[312,313,479,482,483,485,630,635,1088,1089,1090,1191,1192,1193,1244,1245,1349,1352,1355,1356,1357,1359,1505,1511,1512,2322,2323]},{"id":"symbol-73ffecb5618fb3cc","tex":"W^*","aliases":"W^*","display":"W^*","kind":"use","occurrences":[1505]},{"id":"symbol-b9d7c18b6334d33d","tex":"W_0","aliases":"W_0","display":"W_0","kind":"use","occurrences":[1244]},{"id":"symbol-0289b7ef692ee27e","tex":"W_a","aliases":"W_a","display":"W_a","kind":"use","occurrences":[1349,1352,1355,1356,1357,1359]},{"id":"symbol-f39670f030db3b1a","tex":"W_B","aliases":"W_B","display":"W_B","kind":"use","occurrences":[1191,1192,1193]},{"id":"symbol-cc7e0d5f7e350ab2","tex":"w_i","aliases":"w_i","display":"w_i","kind":"use","occurrences":[1483,1484,1488,1494,1496,1498,1499,1502]},{"id":"symbol-a8da97db1cfb2f53","tex":"w_{j,k}","aliases":"w_j,k","display":"w_{j,k}","kind":"use","occurrences":[803,810]},{"id":"symbol-cf76327e749de1b7","tex":"W_k","aliases":"W_k","display":"W_k","kind":"use","occurrences":[635,1244,1245]},{"id":"symbol-b5731a3740a4e086","tex":"W_{k+1}","aliases":"W_k+1","display":"W_{k+1}","kind":"use","occurrences":[1244]},{"id":"symbol-e6e9e4d147adf393","tex":"w_v","aliases":"w_v","display":"w_v","kind":"use","occurrences":[854,860,1172,1174]},{"id":"symbol-f33ee59cc7d16b42","tex":"\\operatorname{Walsh}","aliases":"Walsh operatorname","display":"\\operatorname{Walsh}","kind":"use","occurrences":[1256]},{"id":"symbol-986d47d11f30d4d9","tex":"\\operatorname{Walsh}_{2N}","aliases":"Walsh_2N operatorname","display":"\\operatorname{Walsh}_{2N}","kind":"use","occurrences":[1256]},{"id":"symbol-b72fa7a71d0d77ae","tex":"\\operatorname{Walsh}_N","aliases":"Walsh_N operatorname","display":"\\operatorname{Walsh}_N","kind":"use","occurrences":[1256]},{"id":"symbol-8d3781ce495edcf4","tex":"\\widehat A","aliases":"widehat A widehat","display":"\\widehat A","kind":"use","occurrences":[260,646,647,648,750,754,756,757,765,766,1004,1005,1006,1008,1009,1011,1123,1124,1127,1128,1134,1135,2270]},{"id":"symbol-891994962d96636a","tex":"\\widehat A_1","aliases":"widehat A_1 widehat","display":"\\widehat A_1","kind":"use","occurrences":[647,648,1128]},{"id":"symbol-4895bc4181eddb45","tex":"\\widehat A_i","aliases":"widehat A_i widehat","display":"\\widehat A_i","kind":"use","occurrences":[647,1128]},{"id":"symbol-9708fbc1470a12ef","tex":"\\widehat A_j","aliases":"widehat A_j widehat","display":"\\widehat A_j","kind":"use","occurrences":[646,765,766,1123,1127,1134,1135]},{"id":"symbol-ed3d5df42b309969","tex":"\\widehat A_{j+1}","aliases":"widehat A_j+1 widehat","display":"\\widehat A_{j+1}","kind":"use","occurrences":[648,1124]},{"id":"symbol-7a01d17f6b25e6b1","tex":"\\widehat A_m","aliases":"widehat A_m widehat","display":"\\widehat A_m","kind":"use","occurrences":[647,648,1128]},{"id":"symbol-7bc1f2bd77a0ea58","tex":"\\widehat C","aliases":"widehat C widehat","display":"\\widehat C","kind":"use","occurrences":[762,763]},{"id":"symbol-9a999477715e268b","tex":"\\widehat F","aliases":"widehat F widehat","display":"\\widehat F","kind":"use","occurrences":[1048,1049]},{"id":"symbol-d2f6188722a2e8ff","tex":"\\widehat F_i","aliases":"widehat F_i widehat","display":"\\widehat F_i","kind":"use","occurrences":[1048,1049]},{"id":"symbol-08422a7c74004217","tex":"\\widehat P","aliases":"widehat P widehat","display":"\\widehat P","kind":"use","occurrences":[650,653,656,1124]},{"id":"symbol-2a584b32cb183cf3","tex":"\\widehat p","aliases":"widehat p widehat","display":"\\widehat p","kind":"use","occurrences":[195,1759,2205]},{"id":"symbol-62308a18a459ffc6","tex":"\\widehat P_0","aliases":"widehat P_0 widehat","display":"\\widehat P_0","kind":"use","occurrences":[650]},{"id":"symbol-add64233d3bae23a","tex":"\\widehat p_3","aliases":"widehat p_3 widehat","display":"\\widehat p_3","kind":"use","occurrences":[195,2205]},{"id":"symbol-22cdc1586433efd2","tex":"\\widehat P_j","aliases":"widehat P_j widehat","display":"\\widehat P_j","kind":"use","occurrences":[650,656,1124]},{"id":"symbol-f3a9183aa4d4607b","tex":"\\widehat p_j","aliases":"widehat p_j widehat","display":"\\widehat p_j","kind":"use","occurrences":[195,1759,2205]},{"id":"symbol-bd0609390bade526","tex":"\\widehat P_{j+1}","aliases":"widehat P_j+1 widehat","display":"\\widehat P_{j+1}","kind":"use","occurrences":[1124]},{"id":"symbol-5a26fa77c1b9a117","tex":"\\widehat P_{j-1}","aliases":"widehat P_j-1 widehat","display":"\\widehat P_{j-1}","kind":"use","occurrences":[650]},{"id":"symbol-6c1d9a673bbb0544","tex":"\\widehat p_{j-1}","aliases":"widehat p_j-1 widehat","display":"\\widehat p_{j-1}","kind":"use","occurrences":[195,1759,2205]},{"id":"symbol-266ff731dd16faeb","tex":"\\widehat P_m","aliases":"widehat P_m widehat","display":"\\widehat P_m","kind":"use","occurrences":[653]},{"id":"symbol-352b3ac4a4067415","tex":"\\widetilde P","aliases":"widetilde P widetilde","display":"\\widetilde P","kind":"use","occurrences":[1130,1132,1133]},{"id":"symbol-7efd54fa5d68cf48","tex":"\\widetilde P_0","aliases":"widetilde P_0 widetilde","display":"\\widetilde P_0","kind":"use","occurrences":[1130]},{"id":"symbol-064ad848deb9dca0","tex":"\\widetilde P_j","aliases":"widetilde P_j widetilde","display":"\\widetilde P_j","kind":"use","occurrences":[1130,1132]},{"id":"symbol-7bf0caffa4ff570d","tex":"\\widetilde P_{j-1}","aliases":"widetilde P_j-1 widetilde","display":"\\widetilde P_{j-1}","kind":"use","occurrences":[1130,1132]},{"id":"symbol-7217c1cd533b14e6","tex":"\\widetilde P_m","aliases":"widetilde P_m widetilde","display":"\\widetilde P_m","kind":"use","occurrences":[1133]},{"id":"symbol-a86d6dedf5bd962e","tex":"\\x","aliases":"x","display":null,"kind":"use","occurrences":[1899]},{"id":"symbol-dab4e8a89da28b3b","tex":"x^{(0)}","aliases":"x^(0)","display":"x^{(0)}","kind":"use","occurrences":[1338]},{"id":"symbol-7696a5e0b656722b","tex":"x^{(d)}","aliases":"x^(d)","display":"x^{(d)}","kind":"use","occurrences":[1344]},{"id":"symbol-0190f670081497c9","tex":"x^{(k)}_u","aliases":"x^(k)_u","display":"x^{(k)}_u","kind":"use","occurrences":[1339]},{"id":"symbol-55b833de72597be9","tex":"x^{(k)}_v","aliases":"x^(k)_v","display":"x^{(k)}_v","kind":"use","occurrences":[1339,1340]},{"id":"symbol-c44fc36b58974583","tex":"x^{(k+1)}_v","aliases":"x^(k+1)_v","display":"x^{(k+1)}_v","kind":"use","occurrences":[1339]},{"id":"symbol-6e7e9fdf289546b6","tex":"x^*","aliases":"x^*","display":"x^*","kind":"use","occurrences":[742]},{"id":"symbol-0514a60dd6e60420","tex":"x^T","aliases":"x^T","display":"x^T","kind":"use","occurrences":[247,1866,2257]},{"id":"symbol-14a3332b34948231","tex":"x_*","aliases":"x_*","display":"x_*","kind":"use","occurrences":[748]},{"id":"symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","occurrences":[591,592,748,1255,1256]},{"id":"symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","occurrences":[591,592,1255,1256]},{"id":"symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","occurrences":[223,2233]},{"id":"symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","occurrences":[223,2233]},{"id":"symbol-ca7008ae447ee9dc","tex":"x_4","aliases":"x_4","display":"x_4","kind":"use","occurrences":[223,2233]},{"id":"symbol-98391915cbb7346b","tex":"x_5","aliases":"x_5","display":"x_5","kind":"use","occurrences":[223,2233]},{"id":"symbol-3864fe3cacc7c90d","tex":"x_6","aliases":"x_6","display":"x_6","kind":"use","occurrences":[223,2233]},{"id":"symbol-0298cc60bfc2d7e5","tex":"X_{ij}","aliases":"X_ij","display":"X_{ij}","kind":"use","occurrences":[1229]},{"id":"symbol-2357b435bdbbf889","tex":"X_{k\\ell}","aliases":"X_kℓ ell","display":"X_{k\\ell}","kind":"use","occurrences":[566]},{"id":"symbol-5efdca41184961ce","tex":"x_m","aliases":"x_m","display":"x_m","kind":"use","occurrences":[748]},{"id":"symbol-e798a8a7ea91ac3e","tex":"X_R","aliases":"X_R","display":"X_R","kind":"use","occurrences":[692,694,696,697,698]},{"id":"symbol-54ac43616fbfd36c","tex":"x_\\alpha","aliases":"x_α alpha","display":"x_\\alpha","kind":"use","occurrences":[680]},{"id":"symbol-bcf7586068a6a313","tex":"x_\\beta","aliases":"x_β beta","display":"x_\\beta","kind":"use","occurrences":[680,682]},{"id":"symbol-c5c7030cb4e0b4b6","tex":"X_\\gamma","aliases":"X_γ gamma","display":"X_\\gamma","kind":"use","occurrences":[692]},{"id":"symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","occurrences":[423,428,429]},{"id":"symbol-3590b2065493ac41","tex":"y^{2}","aliases":"y^2","display":"y^{2}","kind":"use","occurrences":[1470]},{"id":"symbol-87e8ed00b8e4c9d2","tex":"y^T","aliases":"y^T","display":"y^T","kind":"use","occurrences":[249,1867,2259]},{"id":"symbol-274ec82875bc15b2","tex":"y_i","aliases":"y_i","display":"y_i","kind":"use","occurrences":[1470]},{"id":"symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","occurrences":[401,402,405,462,470,471,630,635,959,960,997,1505,1506,1511,1512]},{"id":"symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"use","occurrences":[1198]},{"id":"symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","occurrences":[516,520,521,522,666,1080,1081,1161,1294,1295,1465,1466,1467]},{"id":"symbol-5e30e315795c2b5b","tex":"Z^*","aliases":"Z^*","display":"Z^*","kind":"use","occurrences":[630,1505,1512]},{"id":"symbol-38e30a14620f14a0","tex":"z^j","aliases":"z^j","display":"z^j","kind":"use","occurrences":[666]},{"id":"symbol-d533223c5565c514","tex":"Z_k","aliases":"Z_k","display":"Z_k","kind":"use","occurrences":[635]},{"id":"symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","occurrences":[763,1042,1043,1082,1373,1375,1383,1386,1470,1483]},{"id":"symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"use","occurrences":[1483]},{"id":"symbol-3e679c26bc7fd384","tex":"|^{2}","aliases":"|^2","display":"|^{2}","kind":"use","occurrences":[1470]},{"id":"symbol-59ce46150d0cfe77","tex":"\\alpha_0","aliases":"α_0 alpha","display":"\\alpha_0","kind":"use","occurrences":[661,663]},{"id":"symbol-d3ec1c46c6686070","tex":"\\alpha_1","aliases":"α_1 alpha","display":"\\alpha_1","kind":"use","occurrences":[661,663]},{"id":"symbol-da4f2068e2750ae4","tex":"\\alpha_m","aliases":"α_m alpha","display":"\\alpha_m","kind":"use","occurrences":[661,663]},{"id":"symbol-20abd9c0a72b9797","tex":"\\alpha_{m-1}","aliases":"α_m-1 alpha","display":"\\alpha_{m-1}","kind":"use","occurrences":[663]},{"id":"symbol-3e52bdbf6bbe4050","tex":"\\gamma_p","aliases":"γ_p gamma","display":"\\gamma_p","kind":"use","occurrences":[763]},{"id":"symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","occurrences":[766,767,768,1003,1004,1005,1006,1007,1009,1011,1012,1134,1135]},{"id":"symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","occurrences":[380,424,426,654,655,767,768,769,1129,1133,1135,1136]},{"id":"symbol-c5b085f18c150ac1","tex":"\\Delta_e","aliases":"Δ_e Delta","display":"\\Delta_e","kind":"use","occurrences":[1003,1004,1005,1006,1007,1009,1011,1012]},{"id":"symbol-e76cde478bccee9e","tex":"\\delta_j","aliases":"δ_j delta","display":"\\delta_j","kind":"use","occurrences":[654,655,767,768,769,1129,1133,1135,1136]},{"id":"symbol-c5a1bc4b1c0866ff","tex":"\\Delta_{j,e}","aliases":"Δ_j,e Delta","display":"\\Delta_{j,e}","kind":"use","occurrences":[766,767,768,1134,1135]},{"id":"symbol-0fcb1c3cf5e3bf16","tex":"\\delta_{\\alpha\\beta}","aliases":"δ_αβ delta alpha beta","display":"\\delta_{\\alpha\\beta}","kind":"use","occurrences":[380]},{"id":"symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","occurrences":[1007,1008,1013,1015,1056,1057,1058,1059,1063,1064,1067,1265,1266,1287,1288,1292,1293,1296,1297,1298,1299,1301]},{"id":"symbol-be8d5809df598ef7","tex":"\\varepsilon^*","aliases":"ε^* varepsilon","display":"\\varepsilon^*","kind":"use","occurrences":[1063,1064,1067]},{"id":"symbol-56db928a7ce0a173","tex":"\\epsilon_e","aliases":"ϵ_e epsilon","display":"\\epsilon_e","kind":"use","occurrences":[58,264,265,266,270,1859,2068,2274,2275,2276,2280]},{"id":"symbol-2fc46a82cd7318e7","tex":"\\varepsilon_e","aliases":"ε_e varepsilon","display":"\\varepsilon_e","kind":"use","occurrences":[1007,1008,1013,1015,1056,1059,1265,1266,1288,1292,1293,1296,1297]},{"id":"symbol-cb0e8c76ef72f0c6","tex":"\\varepsilon_e^*","aliases":"ε_e^* varepsilon","display":"\\varepsilon_e^*","kind":"use","occurrences":[1057,1059]},{"id":"symbol-e2a74941d662acb4","tex":"\\epsilon_v^{(j)}","aliases":"ϵ_v^(j) epsilon","display":"\\epsilon_v^{(j)}","kind":"use","occurrences":[842,1030]},{"id":"symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","occurrences":[251,739,740,742,744,749,1410,1412,1415,1416,1423,2261]},{"id":"symbol-93055c50da4a404a","tex":"\\kappa_2","aliases":"κ_2 kappa","display":"\\kappa_2","kind":"use","occurrences":[251,1415,1419,1423,1869,2261]},{"id":"symbol-39c423c6294d6173","tex":"\\kappa_v^{(j)}","aliases":"κ_v^(j) kappa","display":"\\kappa_v^{(j)}","kind":"use","occurrences":[841,1030]},{"id":"symbol-a9680dd9eec29c6c","tex":"\\kappa_v^{\\rm setup}","aliases":"κ_v^rm setup kappa rm","display":"\\kappa_v^{\\rm setup}","kind":"use","occurrences":[841]},{"id":"symbol-8e6997cad3f90314","tex":"\\kappa_v^{\\mathrm{setup}}","aliases":"κ_v^setup kappa mathrm","display":"\\kappa_v^{\\mathrm{setup}}","kind":"use","occurrences":[1030]},{"id":"symbol-326f3d290f6131a7","tex":"\\lambda_1","aliases":"λ_1 lambda","display":"\\lambda_1","kind":"use","occurrences":[843,848,850,1034,1035]},{"id":"symbol-975e1a0c6d000741","tex":"\\lambda_2","aliases":"λ_2 lambda","display":"\\lambda_2","kind":"use","occurrences":[843,849,850,1034,1035]},{"id":"symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","occurrences":[1490,1494,1495,1497]},{"id":"symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"use","occurrences":[1443,1452,1516]},{"id":"symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"use","occurrences":[813,814,815,1392,1401,1402]},{"id":"symbol-3315617fe9f782aa","tex":"\\Pi_{\\Pcal}","aliases":"Π_Pcal Π_ P Pi Pcal Pi mathcal","display":"\\Pi_{{\\mathcal P}}","kind":"use","occurrences":[813,814,815]},{"id":"symbol-9eac9da427d5101f","tex":"\\pi_\\beta","aliases":"π_β pi beta","display":"\\pi_\\beta","kind":"use","occurrences":[398]},{"id":"symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"use","occurrences":[630,635]},{"id":"symbol-5048c4b9123eb530","tex":"\\sigma_1","aliases":"σ_1 sigma","display":"\\sigma_1","kind":"use","occurrences":[618,1454,1455,1505]},{"id":"symbol-2cce6a80c926063f","tex":"\\sigma_1^2","aliases":"σ_1^2 sigma","display":"\\sigma_1^2","kind":"use","occurrences":[1506]},{"id":"symbol-15fde97ab40a2e94","tex":"\\sigma_2","aliases":"σ_2 sigma","display":"\\sigma_2","kind":"use","occurrences":[618]},{"id":"symbol-58c1a4cd175eaace","tex":"\\sigma_d","aliases":"σ_d sigma","display":"\\sigma_d","kind":"use","occurrences":[1454,1455,1505]},{"id":"symbol-3443a030a9ee99f2","tex":"\\sigma_d^2","aliases":"σ_d^2 sigma","display":"\\sigma_d^2","kind":"use","occurrences":[1506]},{"id":"symbol-f9ab434fbb0d31ef","tex":"\\sigma_i^2","aliases":"σ_i^2 sigma","display":"\\sigma_i^2","kind":"use","occurrences":[1500,1502,1517]},{"id":"symbol-652e4283f0c8bec9","tex":"\\sigma_i^{2}","aliases":"σ_i^2 sigma","display":"\\sigma_i^{2}","kind":"use","occurrences":[1470]},{"id":"symbol-8de4fa9e7f9c5804","tex":"\\sigma_j","aliases":"σ_j sigma","display":"\\sigma_j","kind":"use","occurrences":[1509]},{"id":"symbol-76a89e17dd1077d6","tex":"\\sigma_j^2","aliases":"σ_j^2 sigma","display":"\\sigma_j^2","kind":"use","occurrences":[623]},{"id":"symbol-2b44b17ec5f59013","tex":"\\sigma_{k+1}","aliases":"σ_k+1 sigma","display":"\\sigma_{k+1}","kind":"use","occurrences":[190,622,1472,1474,1476,1517,2200]},{"id":"symbol-d1737ab7785be77f","tex":"\\sigma_{k+1}^{2}","aliases":"σ_k+1^2 sigma","display":"\\sigma_{k+1}^{2}","kind":"use","occurrences":[1470]},{"id":"symbol-4fd51bd4c6f1161b","tex":"\\Sigma_k^{1/2}","aliases":"Σ_k^1/2 Sigma","display":"\\Sigma_k^{1/2}","kind":"use","occurrences":[635]},{"id":"symbol-0931a7c425476227","tex":"\\sigma_{\\min(k,d)}","aliases":"σ_min(k,d) sigma min","display":"\\sigma_{\\min(k,d)}","kind":"use","occurrences":[1455]},{"id":"symbol-b17d423cde5b3e30","tex":"\\sigma_u","aliases":"σ_u sigma","display":"\\sigma_u","kind":"use","occurrences":[821]},{"id":"symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","occurrences":[263,774,776,777,780,783,786,1013,1014,1016,1054,1056,1057,1264,1265,1275,1280,1281,2273]},{"id":"symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"use","occurrences":[1268,1269,1270,1273,1275,1277,1279]},{"id":"symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","occurrences":[788,789]},{"id":"symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"use","occurrences":[1200]},{"id":"symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","occurrences":[1280]},{"id":"symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","occurrences":[699]},{"id":"symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","occurrences":[647,653,1128,1133]},{"id":"symbol-75c8e2436af69090","tex":"\\prod_{i=1}^{j-1}","aliases":"∏_i=1^j-1 prod","display":"\\prod_{i=1}^{j-1}","kind":"use","occurrences":[647,1128]},{"id":"symbol-3c3bdaa5a997770c","tex":"\\prod_{i=j+1}^m","aliases":"∏_i=j+1^m prod","display":"\\prod_{i=j+1}^m","kind":"use","occurrences":[647,1128,1133]},{"id":"symbol-2e717462c0866d39","tex":"\\prod_{i=j+1}^{m}","aliases":"∏_i=j+1^m prod","display":"\\prod_{i=j+1}^{m}","kind":"use","occurrences":[653]},{"id":"symbol-0ace3cb3f12efd36","tex":"\\sum_{(b,a)\\in E_A}","aliases":"∑_(b,a)∈ E_A sum in","display":"\\sum_{(b,a)\\in E_A}","kind":"use","occurrences":[958,959]},{"id":"symbol-9314650eb3dee5bd","tex":"\\sum_a","aliases":"∑_a sum","display":"\\sum_a","kind":"use","occurrences":[955,1392]},{"id":"symbol-7bfc5a7087fd58a2","tex":"\\sum_{b=0}^B","aliases":"∑_b=0^B sum","display":"\\sum_{b=0}^B","kind":"use","occurrences":[1189]},{"id":"symbol-2046ba257a075646","tex":"\\sum_{b\\ne a}","aliases":"∑_b≠ a sum ne","display":"\\sum_{b\\ne a}","kind":"use","occurrences":[1393,1401,1402]},{"id":"symbol-3004d1c3c6d1b821","tex":"\\sum_c","aliases":"∑_c sum","display":"\\sum_c","kind":"use","occurrences":[961]},{"id":"symbol-3974edc19cef25f4","tex":"\\sum_{c:\\,A_{bc}\\ne0}","aliases":"∑_c: A_bc≠0 sum ne","display":"\\sum_{c:\\,A_{bc}\\ne0}","kind":"use","occurrences":[971]},{"id":"symbol-ec1616068a6ee172","tex":"\\sum_e","aliases":"∑_e sum","display":"\\sum_e","kind":"use","occurrences":[260,264,265,757,766,767,768,775,776,783,786,788,1004,1005,1006,1008,1009,1011,1013,1015,1055,1056,1060,1062,1063,1072,1073,1086,1134,1135,1264,1265,1266,1268,1271,1273,1280,1281,1293,1296,1297,2270,2274,2275]},{"id":"symbol-6d1546afc3ef11e5","tex":"\\sum_{e\\in E}","aliases":"∑_e∈ E sum in","display":"\\sum_{e\\in E}","kind":"use","occurrences":[750,754,756]},{"id":"symbol-f7155be421cc3965","tex":"\\sum_f","aliases":"∑_f sum","display":"\\sum_f","kind":"use","occurrences":[777,780]},{"id":"symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","occurrences":[1048,1076,1077,1143,1146,1174,1484,1488,1502]},{"id":"symbol-5bed34448f0dc86f","tex":"\\sum_{i=1}^{k+1}","aliases":"∑_i=1^k+1 sum","display":"\\sum_{i=1}^{k+1}","kind":"use","occurrences":[1470]},{"id":"symbol-995c1e5022b02f2d","tex":"\\sum_{i>k}","aliases":"∑_i>k sum","display":"\\sum_{i>k}","kind":"use","occurrences":[1488,1502,1517]},{"id":"symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","occurrences":[1030]},{"id":"symbol-b845f4dcb81f67c5","tex":"\\sum_{j=0}^d","aliases":"∑_j=0^d sum","display":"\\sum_{j=0}^d","kind":"use","occurrences":[666]},{"id":"symbol-cfe7b3387ef9124c","tex":"\\sum_{j=0}^{d}","aliases":"∑_j=0^d sum","display":"\\sum_{j=0}^{d}","kind":"use","occurrences":[979]},{"id":"symbol-24dfb51124d8ae91","tex":"\\sum_{j=1}^m","aliases":"∑_j=1^m sum","display":"\\sum_{j=1}^m","kind":"use","occurrences":[607,647,648,841,842,1115,1118,1128,1133]},{"id":"symbol-55bfd4be1daf726b","tex":"\\sum_{j=1}^{m}","aliases":"∑_j=1^m sum","display":"\\sum_{j=1}^{m}","kind":"use","occurrences":[653,658]},{"id":"symbol-93f65e678e57f837","tex":"\\sum_{j=1}^{q}","aliases":"∑_j=1^q sum","display":"\\sum_{j=1}^{q}","kind":"use","occurrences":[1483]},{"id":"symbol-2f357ff52e6c2d2b","tex":"\\sum_{j>k}","aliases":"∑_j>k sum","display":"\\sum_{j>k}","kind":"use","occurrences":[623]},{"id":"symbol-c98e9b3ccc8ca2c6","tex":"\\sum_{k=0}^{j-1}","aliases":"∑_k=0^j-1 sum","display":"\\sum_{k=0}^{j-1}","kind":"use","occurrences":[731]},{"id":"symbol-1f3736c162765e82","tex":"\\sum_{k=1}^{n}","aliases":"∑_k=1^n sum","display":"\\sum_{k=1}^{n}","kind":"use","occurrences":[566]},{"id":"symbol-3d357445fed9cde3","tex":"\\sum_{k=1}^{w}","aliases":"∑_k=1^w sum","display":"\\sum_{k=1}^{w}","kind":"use","occurrences":[513]},{"id":"symbol-2753495994955ca2","tex":"\\sum_{r=0}^{n-1}","aliases":"∑_r=0^n-1 sum","display":"\\sum_{r=0}^{n-1}","kind":"use","occurrences":[535]},{"id":"symbol-7ecb3223d556bc7c","tex":"\\sum_{u\\text{ split}}","aliases":"∑_u split sum text","display":"\\sum_{u\\text{ split}}","kind":"use","occurrences":[821]},{"id":"symbol-582da013421a516b","tex":"\\sum_{v\\in\\Lcal}","aliases":"∑_v∈Lcal ∑_v∈ L sum in Lcal sum in mathcal","display":"\\sum_{v\\in{\\mathcal L}}","kind":"use","occurrences":[821,822]},{"id":"symbol-2b95868d24f2bd48","tex":"\\sum_w","aliases":"∑_w sum","display":"\\sum_w","kind":"use","occurrences":[1042]},{"id":"symbol-1c79b8a11020f120","tex":"\\sum_{w\\in\\mathrm{child}(v)}","aliases":"∑_w∈child(v) sum in mathrm","display":"\\sum_{w\\in\\mathrm{child}(v)}","kind":"use","occurrences":[1026]},{"id":"symbol-a38ac3f06fcc5327","tex":"\\sum_{w\\in\\operatorname{child}(v)}","aliases":"∑_w∈child(v) sum in operatorname","display":"\\sum_{w\\in\\operatorname{child}(v)}","kind":"use","occurrences":[828]},{"id":"symbol-6d8e30d9f2646a56","tex":"\\sum_\\alpha","aliases":"∑_α sum alpha","display":"\\sum_\\alpha","kind":"use","occurrences":[404,463,464,469,473,718,719,813]},{"id":"symbol-722d121a5352a8a2","tex":"\\sum_{\\alpha,\\beta}","aliases":"∑_α,β sum alpha beta","display":"\\sum_{\\alpha,\\beta}","kind":"use","occurrences":[404]},{"id":"symbol-97bc4b69d2c44d19","tex":"\\sum_{\\alpha\\to\\beta}","aliases":"∑_α→β sum alpha to beta","display":"\\sum_{\\alpha\\to\\beta}","kind":"use","occurrences":[680]},{"id":"symbol-1fa499eb28c28965","tex":"\\sum_{\\alpha\\neq\\beta}","aliases":"∑_α≠β sum alpha neq beta","display":"\\sum_{\\alpha\\neq\\beta}","kind":"use","occurrences":[814,815]},{"id":"symbol-dd5a8bdc2b931502","tex":"\\sum_\\beta","aliases":"∑_β sum beta","display":"\\sum_\\beta","kind":"use","occurrences":[404,418]},{"id":"symbol-22bd389693d3f179","tex":"\\sum_{\\ell=1}^{m}","aliases":"∑_ℓ=1^m sum ell","display":"\\sum_{\\ell=1}^{m}","kind":"use","occurrences":[566]},{"id":"symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","occurrences":[277,800,801,1713,1714,1715,1716,1717,1718,1719,1720,1721,1722,1723,1724,1725,1726,1727,1728,2287]},{"id":"symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","occurrences":[211,337,955,957,963,964,971,976,1048,1065,1067,1299,1301,1339,1393,1394,1396,1397,1401,1402,1509,2221]},{"id":"symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","occurrences":[30,31,32,38,92,112,187,242,251,260,294,296,297,358,360,363,763,767,768,770,791,896,991,998,1006,1007,1008,1009,1010,1011,1211,1217,1218,1264,1265,1288,1291,1292,1293,1294,1296,1298,1872,1972,1973,2040,2041,2042,2048,2102,2122,2197,2252,2261,2270,2304,2306,2307]},{"id":"symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"use","occurrences":[491,492,505,511,550,551,616,647,653,654,669,728,730,739,740,742,748,754,755,756,782,783,805,815,843,846,848,849,850,860,862,891,1027,1034,1035,1036,1044,1061,1062,1087,1088,1090,1091,1118,1128,1133,1135,1154,1159,1161,1163,1166,1167,1169,1178,1182,1184,1188,1189,1192,1193,1321,1325,1340,1372,1373,1374,1402,1410,1412,1415,1423,1424,1426,1427,1428,1442,1443,1451,1452,1464,1472,1474,1480,1484,1492,1494,1497,1498,1509,1515,1516]},{"id":"symbol-3c62641491514988","tex":"\\bigoplus_i","aliases":"⊕_i bigoplus","display":"\\bigoplus_i","kind":"use","occurrences":[1144,1157]},{"id":"symbol-9b591bcc37a61369","tex":"\\bigoplus_\\alpha","aliases":"⊕_α bigoplus alpha","display":"\\bigoplus_\\alpha","kind":"use","occurrences":[722,730]},{"id":"symbol-7f618b24b380de7c","tex":"\\bigoplus_{\\gamma\\in R}","aliases":"⊕_γ∈ R bigoplus gamma in","display":"\\bigoplus_{\\gamma\\in R}","kind":"use","occurrences":[692]},{"id":"symbol-362138992adfe804","tex":"\\star","aliases":"⋆ star","display":"\\star","kind":"use","occurrences":[1163]},{"id":"symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","occurrences":[606,618,647,648,658,661,663,1112,1128,1454,1505]}],"contexts":["\\vpage{29}{A complex range needs an adjoint}{THE INNER PRODUCT IS PART OF THE CONTRACT}{}\nThe earlier compact example is real. Over $\\CC$, an isometry means $Q^*Q=I$, with conjugation. Take the exact Gaussian-rational factors\n\\[\nQ_L=\\mat{3/5&0","\\vpage{29}{A complex range needs an adjoint}{THE INNER PRODUCT IS PART OF THE CONTRACT}{}\nThe earlier compact example is real. Over $\\CC$, an isometry means $Q^*Q=I$, with conjugation. Take the exact Gaussian-rational factors\n\\[\nQ_L=\\mat{3/5&0\\\\4\\mathrm i/5&0\\\\0&1},\\qquad\nQ_R=\\mat{1&0\\\\0&\\mathrm i},\\qquad G=\\diag(5,2).\n\\]\nTheir product is a genuinely complex rectangular operator:\n\\[\nA=Q_LGQ_R^*=\\mat{3&0","0&\\mathrm i},\\qquad G=\\diag(5,2).\n\\]\nTheir product is a genuinely complex rectangular operator:\n\\[\nA=Q_LGQ_R^*=\\mat{3&0\\\\4\\mathrm i&0\\\\0&-2\\mathrm i},\n\\qquad A^*A=\\diag(25,4).\n\\]\n\\vfig{complex_core}\n\\head{The small calculation exposes a wrong convention}\nThe first column of $Q_L$ has Hermitian squared length $9/25+16/25=1$. Its bilinear square without conjugation is $9/25-16/25=-7/25$. Thus $Q_L^TQ_L$ is not the identity. The portable negative control replaces the adjoint by transpose and rejects the resulting isometry claim.","0&-2\\mathrm i},\n\\qquad A^*A=\\diag(25,4).\n\\]\n\\vfig{complex_core}\n\\head{The small calculation exposes a wrong convention}\nThe first column of $Q_L$ has Hermitian squared length $9/25+16/25=1$. Its bilinear square without conjugation is $9/25-16/25=-7/25$. Thus $Q_L^TQ_L$ is not the identity. The portable negative control replaces the adjoint by transpose and rejects the resulting isometry claim.","Keeping the first diagonal entry of $G$ gives $A_1$ with its second column zero. The residual has squared Frobenius norm four and Gram matrix $\\diag(0,4)$, hence spectral norm two. Both errors agree with the discarded core singular value. Every displayed arithmetic operation occurs in $\\QQ(\\mathrm i)$.\n\\head{What the finite check establishes}\nIt checks both isometries, the reconstructed ambient matrix, its Gram matrix, and the explicit rank-one residual. Global best-rank optimality still comes from the manuscript's compression and lower-bound proof; this two-column calculation is a worked instance of that argument. No existence theorem or numerical SVD routine is executed.\n\\takeaway{Conjugation preserves the complex Euclidean no","Keeping the first diagonal entry of $G$ gives $A_1$ with its second column zero. The residual has squared Frobenius norm four and Gram matrix $\\diag(0,4)$, hence spectral norm two. Both errors agree with the discarded core singular value. Every displayed arithmetic operation occurs in $\\QQ(\\mathrm i)$.\n\\head{What the finite check establishes}\nIt checks both isometries, the reconstructed ambient matrix, its Gram matrix, and the explicit rank-one residual. Global best-rank optimality still comes from the manuscript's compression and lower-bound proof; this two-column calculation is a worked instance of that argument. No existence theorem or numerical SVD routine is executed.\n\\takeaway{Conjugation preserves the complex Euclidean norm. A transp","Keeping the first diagonal entry of $G$ gives $A_1$ with its second column zero. The residual has squared Frobenius norm four and Gram matrix $\\diag(0,4)$, hence spectral norm two. Both errors agree with the discarded core singular value. Every displayed arithmetic operation occurs in $\\QQ(\\mathrm i)$.\n\\head{What the finite check establishes}\nIt checks both isometries, the reconstructed ambient matrix, its Gram matrix, and the explicit rank-one residual. Global best-rank optimality still comes from the manuscript's compression and lower-bound proof; this two-column calculation is a worked instance of that argument. No existence theorem or numerical SVD routine is executed.\n\\takeaway{Conjugation preserves the complex Euclidean norm. A transpose is correct in the vectorization identity on \\pagerefV{7}, but an adjoint is required for an isometr","Keeping the first diagonal entry of $G$ gives $A_1$ with its second column zero. The residual has squared Frobenius norm four and Gram matrix $\\diag(0,4)$, hence spectral norm two. Both errors agree with the discarded core singular value. Every displayed arithmetic operation occurs in $\\QQ(\\mathrm i)$.\n\\head{What the finite check establishes}\nIt checks both isometries, the reconstructed ambient matrix, its Gram matrix, and the explicit rank-one residual. Global best-rank optimality still comes from the manuscript's compression and lower-bound proof; this two-column calculation is a worked instance of that argument. No existence theorem or numerical SVD routine is executed.\n\\takeaway{Conjugation preserves the complex Euclidean norm. A transpose is correct in the vectorization identity on \\pagerefV{7}, but an adjoint is required for an isometric range lift.}","\\vpage{30}{Inverse reachability is still an inclusion}{A FINITE INVERSE WITH A CANCELLED ROUTE}{}\nReuse the signed diamond from \\pagerefV{14}, now as a complete operator:\n\\[\nA=I+K=\\mat{1&0&0&0\\\\1&1&0&0\\\\1&0&1&0\\\\0&1&-1&1}.\n\\]\nThe two routes from coordinate zero to coordinate three cancel, so $K^2=0$. Consequently\n\\[\nA^{-1}=I-K=\\mat{1&0&0&0","0&1&-1&1}.\n\\]\nThe two routes from coordinate zero to coordinate three cancel, so $K^2=0$. Consequently\n\\[\nA^{-1}=I-K=\\mat{1&0&0&0","0&1&-1&1}.\n\\]\nThe two routes from coordinate zero to coordinate three cancel, so $K^2=0$. Consequently\n\\[\nA^{-1}=I-K=\\mat{1&0&0&0\\\\-1&1&0&0\\\\-1&0&1&0\\\\0&-1&1&1}.\n\\]\n\\vfig[.52]{inverse_closure}\n\\head{What the closure permits}\nThe directed transitive closure includes $0\\to3$, because it records existence of a route rather than the signed sum of its coefficients. Nevertheless $(A^{-1})_{30}=0$. Reachability is therefore an upper envelope for inverse support, just as it is for product support.","0&-1&1&1}.\n\\]\n\\vfig[.52]{inverse_closure}\n\\head{What the closure permits}\nThe directed transitive closure includes $0\\to3$, because it records existence of a route rather than the signed sum of its coefficients. Nevertheless $(A^{-1})_{30}=0$. Reachability is therefore an upper envelope for inverse support, just as it is for product support.","The portable checker computes the inverse independently by rational row elimination, compares it with $I-K$, constructs Boolean reachability, and checks every inverse entry outside that closure. It separately rejects the assertion that every reachable transfer is nonzero.\n\\head{Why the general proof is broader}\nThis example happens to have a terminating Neumann polynomial. The general finite-dimensional theorem does not require nilpotence or a small norm. For a fixed source coordinate, its reachable-coordinate subspace is invariant under $A$. If $A$ is invertible, the restricted map is injective. Equal finite domain and codomain dimensions make it surjective, so the inverse preserves the same subspace.","checks every inverse entry outside that closure. It separately rejects the assertion that every reachable transfer is nonzero.\n\\head{Why the general proof is broader}\nThis example happens to have a terminating Neumann polynomial. The general finite-dimensional theorem does not require nilpotence or a small norm. For a fixed source coordinate, its reachable-coordinate subspace is invariant under $A$. If $A$ is invertible, the restricted map is injective. Equal finite domain and codomain dimensions make it surjective, so the inverse preserves the same subspace.","every inverse entry outside that closure. It separately rejects the assertion that every reachable transfer is nonzero.\n\\head{Why the general proof is broader}\nThis example happens to have a terminating Neumann polynomial. The general finite-dimensional theorem does not require nilpotence or a small norm. For a fixed source coordinate, its reachable-coordinate subspace is invariant under $A$. If $A$ is invertible, the restricted map is injective. Equal finite domain and codomain dimensions make it surjective, so the inverse preserves the same subspace.","bottomrule\n\\end{tabular}\n\\end{center}\n\\vfig[.45]{shared_family}\nWith reuse weights $(r_1,r_2)=(1,3)$, the aggregated objectives are\n\\[\nJ_{\\rm keep}=4+2+3\\cdot7=27,\\qquad\nJ_{\\rm split}=2+8+3\\cdot2=16.\n\\]\nThe exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem.\n\\head{A hand-computable decision threshold}\nFix $r_1=1$ and vary $r_2=r\\ge0$. Keeping costs $6+7r$ and splitting costs $10+2r$. Splitting is strictly cheaper when $r>4/5$; its first integer reuse win is $r=1$. The checker evaluates both admissible choices for integers zero through four and checks the displayed totals at three.\n\\head{Where the common-tree proof app","bottomrule\n\\end{tabular}\n\\end{center}\n\\vfig[.45]{shared_family}\nWith reuse weights $(r_1,r_2)=(1,3)$, the aggregated objectives are\n\\[\nJ_{\\rm keep}=4+2+3\\cdot7=27,\\qquad\nJ_{\\rm split}=2+8+3\\cdot2=16.\n\\]\nThe exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem.\n\\head{A hand-computable decision threshold}\nFix $r_1=1$ and vary $r_2=r\\ge0$. Keeping costs $6+7r$ and splitting costs $10+2r$. Splitting is strictly cheaper when $r>4/5$; its first integer reuse win is $r=1$. The checker evaluates both admissible choices for integers zero through four and checks the displayed totals at three.\n\\head{Where the common-tree proof applies}\nThe family corollary aggregates local keep costs and errors, then uses the same exact subtree r","$(r_1,r_2)=(1,3)$, the aggregated objectives are\n\\[\nJ_{\\rm keep}=4+2+3\\cdot7=27,\\qquad\nJ_{\\rm split}=2+8+3\\cdot2=16.\n\\]\nThe exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem.\n\\head{A hand-computable decision threshold}\nFix $r_1=1$ and vary $r_2=r\\ge0$. Keeping costs $6+7r$ and splitting costs $10+2r$. Splitting is strictly cheaper when $r>4/5$; its first integer reuse win is $r=1$. The checker evaluates both admissible choices for integers zero through four and checks the displayed totals at three.\n\\head{Where the common-tree proof applies}\nThe family corollary aggregates local keep costs and errors, then uses the same exact subtree recurrence. In this example, each row denotes the sum of the selected local leaf terms; it does not conceal a nonlocal discount. A setup shared by distant unrelated nodes, overlapping candidates, or a global resource constraint would need an enlarged state or a different model.\n\\take…","$, the aggregated objectives are\n\\[\nJ_{\\rm keep}=4+2+3\\cdot7=27,\\qquad\nJ_{\\rm split}=2+8+3\\cdot2=16.\n\\]\nThe exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem.\n\\head{A hand-computable decision threshold}\nFix $r_1=1$ and vary $r_2=r\\ge0$. Keeping costs $6+7r$ and splitting costs $10+2r$. Splitting is strictly cheaper when $r>4/5$; its first integer reuse win is $r=1$. The checker evaluates both admissible choices for integers zero through four and checks the displayed totals at three.\n\\head{Where the common-tree proof applies}\nThe family corollary aggregates local keep costs and errors, then uses the same exact subtree recurrence. In this example, each row denotes the sum of the selected local leaf terms; it does not conceal a nonlocal discount. A setup shared by distant unrelated nodes, overlapping candidates, or a global resource constraint would need an enlarged state or a different model.\n\\takeaway{Reuse weigh…","s are\n\\[\nJ_{\\rm keep}=4+2+3\\cdot7=27,\\qquad\nJ_{\\rm split}=2+8+3\\cdot2=16.\n\\]\nThe exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem.\n\\head{A hand-computable decision threshold}\nFix $r_1=1$ and vary $r_2=r\\ge0$. Keeping costs $6+7r$ and splitting costs $10+2r$. Splitting is strictly cheaper when $r>4/5$; its first integer reuse win is $r=1$. The checker evaluates both admissible choices for integers zero through four and checks the displayed totals at three.\n\\head{Where the common-tree proof applies}\nThe family corollary aggregates local keep costs and errors, then uses the same exact subtree recurrence. In this example, each row denotes the sum of the selected local leaf terms; it does not conceal a nonlocal discount. A setup shared by distant unrelated nodes, overlapping candidates, or a global resource constraint would need an enlarged state or a different model.\n\\takeaway{Reuse weights change the objective bef…","\\cdot7=27,\\qquad\nJ_{\\rm split}=2+8+3\\cdot2=16.\n\\]\nThe exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem.\n\\head{A hand-computable decision threshold}\nFix $r_1=1$ and vary $r_2=r\\ge0$. Keeping costs $6+7r$ and splitting costs $10+2r$. Splitting is strictly cheaper when $r>4/5$; its first integer reuse win is $r=1$. The checker evaluates both admissible choices for integers zero through four and checks the displayed totals at three.\n\\head{Where the common-tree proof applies}\nThe family corollary aggregates local keep costs and errors, then uses the same exact subtree recurrence. In this example, each row denotes the sum of the selected local leaf terms; it does not conceal a nonlocal discount. A setup shared by distant unrelated nodes, overlapping candidates, or a global resource constraint would need an enlarged state or a different model.\n\\takeaway{Reuse weights change the objective before compilation. Shared opt…","6.\n\\]\nThe exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem.\n\\head{A hand-computable decision threshold}\nFix $r_1=1$ and vary $r_2=r\\ge0$. Keeping costs $6+7r$ and splitting costs $10+2r$. Splitting is strictly cheaper when $r>4/5$; its first integer reuse win is $r=1$. The checker evaluates both admissible choices for integers zero through four and checks the displayed totals at three.\n\\head{Where the common-tree proof applies}\nThe family corollary aggregates local keep costs and errors, then uses the same exact subtree recurrence. In this example, each row denotes the sum of the selected local leaf terms; it does not conceal a nonlocal discount. A setup shared by distant unrelated nodes, overlapping candidates, or a global resource constraint would need an enlarged state or a different model.\n\\takeaway{Reuse weights change the objective before compilation. Shared optimality is exact for the supplied common tre…","Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem.\n\\head{A hand-computable decision threshold}\nFix $r_1=1$ and vary $r_2=r\\ge0$. Keeping costs $6+7r$ and splitting costs $10+2r$. Splitting is strictly cheaper when $r>4/5$; its first integer reuse win is $r=1$. The checker evaluates both admissible choices for integers zero through four and checks the displayed totals at three.\n\\head{Where the common-tree proof applies}\nThe family corollary aggregates local keep costs and errors, then uses the same exact subtree recurrence. In this example, each row denotes the sum of the selected local leaf terms; it does not conceal a nonlocal discount. A setup shared by distant unrelated nodes, overlapping candidates, or a global resource constraint would need an enlarged state or a different model.\n\\takeaway{Reuse weights change the objective before compilation. Shared optimality is exact for the supplied common tree and additive family costs, rather than …","\\vpage{32}{Safe integers can still be conservative}{QUANTIZATION AND THE SIZE OF THE BUDGET TABLE}{}\nLet one integer unit represent $h=1/10$ of certified error. For an individual nonnegative error $e$, choosing\n\\[\nw=\\lceil e/h\\rceil\\qquad\\text{ensures}\\qquad e\\le hw.\n\\]\nThus integer feasibility $\\sum w\\le B$ implies the real bound $\\sum e\\le hB$. This implication is checked before applying the exact option or tree recurrence.\n\\vfig{quantization}\n\\head{A false certificate and a conservative rejection}\nFor errors $(3/50,2/25)$, rounding down assigns weights $(0,0)$ although their real sum is $7/50>1/10$. Rounding up gives $(1,1)$ and correctly excludes the assignment from budget one.","nit represent $h=1/10$ of certified error. For an individual nonnegative error $e$, choosing\n\\[\nw=\\lceil e/h\\rceil\\qquad\\text{ensures}\\qquad e\\le hw.\n\\]\nThus integer feasibility $\\sum w\\le B$ implies the real bound $\\sum e\\le hB$. This implication is checked before applying the exact option or tree recurrence.\n\\vfig{quantization}\n\\head{A false certificate and a conservative rejection}\nFor errors $(3/50,2/25)$, rounding down assigns weights $(0,0)$ although their real sum is $7/50>1/10$. Rounding up gives $(1,1)$ and correctly excludes the assignment from budget one.","r an individual nonnegative error $e$, choosing\n\\[\nw=\\lceil e/h\\rceil\\qquad\\text{ensures}\\qquad e\\le hw.\n\\]\nThus integer feasibility $\\sum w\\le B$ implies the real bound $\\sum e\\le hB$. This implication is checked before applying the exact option or tree recurrence.\n\\vfig{quantization}\n\\head{A false certificate and a conservative rejection}\nFor errors $(3/50,2/25)$, rounding down assigns weights $(0,0)$ although their real sum is $7/50>1/10$. Rounding up gives $(1,1)$ and correctly excludes the assignment from budget one.","e$, choosing\n\\[\nw=\\lceil e/h\\rceil\\qquad\\text{ensures}\\qquad e\\le hw.\n\\]\nThus integer feasibility $\\sum w\\le B$ implies the real bound $\\sum e\\le hB$. This implication is checked before applying the exact option or tree recurrence.\n\\vfig{quantization}\n\\head{A false certificate and a conservative rejection}\nFor errors $(3/50,2/25)$, rounding down assigns weights $(0,0)$ although their real sum is $7/50>1/10$. Rounding up gives $(1,1)$ and correctly excludes the assignment from budget one.","eil\\qquad\\text{ensures}\\qquad e\\le hw.\n\\]\nThus integer feasibility $\\sum w\\le B$ implies the real bound $\\sum e\\le hB$. This implication is checked before applying the exact option or tree recurrence.\n\\vfig{quantization}\n\\head{A false certificate and a conservative rejection}\nFor errors $(3/50,2/25)$, rounding down assigns weights $(0,0)$ although their real sum is $7/50>1/10$. Rounding up gives $(1,1)$ and correctly excludes the assignment from budget one.","Now change the errors to $(3/50,1/50)$. Their sum is $2/25\\le1/10$, but separate ceilings still give $(1,1)$. The integer model rejects a genuinely real-feasible assignment. Conservative rounding protects soundness; it need not preserve every optimum of the unquantized problem.\n\\head{Why the complexity is pseudo polynomial}\nFor a declared integer budget $B$, a table has $B+1$ possible exact-weight states. A direct merge checks at most $(B+1)^2$ state pairs. At $B=255$, this means 256 states and 65,536 candidate pairs at a merge, even though the budget fits in eight binary digits.","Their sum is $2/25\\le1/10$, but separate ceilings still give $(1,1)$. The integer model rejects a genuinely real-feasible assignment. Conservative rounding protects soundness; it need not preserve every optimum of the unquantized problem.\n\\head{Why the complexity is pseudo polynomial}\nFor a declared integer budget $B$, a table has $B+1$ possible exact-weight states. A direct merge checks at most $(B+1)^2$ state pairs. At $B=255$, this means 256 states and 65,536 candidate pairs at a merge, even though the budget fits in eight binary digits.",", but separate ceilings still give $(1,1)$. The integer model rejects a genuinely real-feasible assignment. Conservative rounding protects soundness; it need not preserve every optimum of the unquantized problem.\n\\head{Why the complexity is pseudo polynomial}\nFor a declared integer budget $B$, a table has $B+1$ possible exact-weight states. A direct merge checks at most $(B+1)^2$ state pairs. At $B=255$, this means 256 states and 65,536 candidate pairs at a merge, even though the budget fits in eight binary digits.","With a bounded number of children, the paper's direct tree implementation therefore has quadratic budget dependence at each node. Polynomial dependence on the numerical value of $B$ does not mean polynomial dependence on the binary input length. These are counts of declared states and pairs, not measured runtimes or claims that every state is reachable.\n\\takeaway{The dynamic program is exact for its integer data. Conservative conversion and the numerical size of the budget are separate obligations that determine its accuracy and practical scale.}","ctored operator, and a compiled execution plan are different objects. They may denote the same linear map, but they need not store the same data, return the same output, or incur the same costs. This guide makes those distinctions visible through calculations rather than a pipeline of boxes.\n\\vfig{three_views}\n\\head{Three models to carry through the paper}\nThe first is a six-dimensional operator $A=D+uv^T$. Every entry of $A$ is nonzero, yet three $2\\times2$ diagonal blocks and two thin vectors suffice to apply it exactly. Pages \\pagerefV{3}, \\pagerefV{9}, \\pagerefV{10}, and \\pagerefV{27} reuse this operator or its factors.","iled execution plan are different objects. They may denote the same linear map, but they need not store the same data, return the same output, or incur the same costs. This guide makes those distinctions visible through calculations rather than a pipeline of boxes.\n\\vfig{three_views}\n\\head{Three models to carry through the paper}\nThe first is a six-dimensional operator $A=D+uv^T$. Every entry of $A$ is nonzero, yet three $2\\times2$ diagonal blocks and two thin vectors suffice to apply it exactly. Pages \\pagerefV{3}, \\pagerefV{9}, \\pagerefV{10}, and \\pagerefV{27} reuse this operator or its factors.","fferent objects. They may denote the same linear map, but they need not store the same data, return the same output, or incur the same costs. This guide makes those distinctions visible through calculations rather than a pipeline of boxes.\n\\vfig{three_views}\n\\head{Three models to carry through the paper}\nThe first is a six-dimensional operator $A=D+uv^T$. Every entry of $A$ is nonzero, yet three $2\\times2$ diagonal blocks and two thin vectors suffice to apply it exactly. Pages \\pagerefV{3}, \\pagerefV{9}, \\pagerefV{10}, and \\pagerefV{27} reuse this operator or its factors.","The second starts with redundant factors $L$ and $R$. Actual range coordinates reduce a $4\\times3$ product to a $2\\times3$ core. Pages \\pagerefV{11}--\\pagerefV{12} connect rank, singular tails, and the geometric reason a small-core optimum is also an ambient optimum.","midrule\n$P_\\alpha$ & Coordinate projector & Which input or output coordinates does it select?","$A_{\\beta\\alpha}$ & A block map & What are its source and target dimensions?","$E$ & Active block addresses & Is this exact support, or only a structural upper bound?","$D+UV^*$ & Factored operator & Is the desired output a vector, factors, or a dense array?","$Q_L,G,Q_R$ & Range factors and core & Are the outer columns orthonormal, with exact reconstruction?","$E_j$ & An error operator & Is it a factor perturbation or an additive error after a product?","$s_e,\\epsilon_e$ & Sensitivity and tolerance & Is the budget input-dependent or uniform over unit inputs?","$(\\kappa_v,\\epsilon_v,\\sigma_v)$ & Candidate-tree data & Do costs, errors, and split overheads really add?","$F_v(b)$ & A budget table & Does $b$ mean exact weight, or an upper bound?","$R$ & Reuse count & Which one-time costs have been included?","bottomrule\n\\end{tabularx}\n\\head{Conventions used in the examples}\nMatrices act on column vectors. A block $A_{\\beta\\alpha}$ goes from source $\\alpha$ to target $\\beta$. Products act from right to left. Real examples use transpose; complex examples use adjoint unless a vectorization identity explicitly requires transpose. Column-major vectorization stacks the columns of a matrix.","\\vpage{3}{A dense-looking matrix, one scalar coupling}{THE RUNNING SIX-DIMENSIONAL OPERATOR}{}\nLet $D$ have three identical blocks $\\mat{3&1","\\vpage{3}{A dense-looking matrix, one scalar coupling}{THE RUNNING SIX-DIMENSIONAL OPERATOR}{}\nLet $D$ have three identical blocks $\\mat{3&1\\\\1&3}$ and put $u=v=\\ones\\in\\QQ^6$. Then $A=D+uv^T$ has diagonal entries four, within-block off-diagonal entries two, and every cross-block entry one. All 36 entries are nonzero.\n\\[\nx=(1,-1,2,0,-2,3)^T,\\qquad v^Tx=3,\\qquad Ax=Dx+3u.\n\\]\n\\vfig{factored_application}\n\\head{What was avoided?}\nThe global part does not require 36 independent coefficients. It depends on the input through one scalar, $v^Tx$, which is broadcast through $u$. The diagonal part is computed by three separate local matrix products. Nothing is approximated and no channel is discarded.","1&3}$ and put $u=v=\\ones\\in\\QQ^6$. Then $A=D+uv^T$ has diagonal entries four, within-block off-diagonal entries two, and every cross-block entry one. All 36 entries are nonzero.\n\\[\nx=(1,-1,2,0,-2,3)^T,\\qquad v^Tx=3,\\qquad Ax=Dx+3u.\n\\]\n\\vfig{factored_application}\n\\head{What was avoided?}\nThe global part does not require 36 independent coefficients. It depends on the input through one scalar, $v^Tx$, which is broadcast through $u$. The diagonal part is computed by three separate local matrix products. Nothing is approximated and no channel is discarded.","^6$. Then $A=D+uv^T$ has diagonal entries four, within-block off-diagonal entries two, and every cross-block entry one. All 36 entries are nonzero.\n\\[\nx=(1,-1,2,0,-2,3)^T,\\qquad v^Tx=3,\\qquad Ax=Dx+3u.\n\\]\n\\vfig{factored_application}\n\\head{What was avoided?}\nThe global part does not require 36 independent coefficients. It depends on the input through one scalar, $v^Tx$, which is broadcast through $u$. The diagonal part is computed by three separate local matrix products. Nothing is approximated and no channel is discarded.","The sixth output is $7+3=10$, while the fifth is $-3+3=0$. A zero result can arise through cancellation even when the corresponding row contains no zero coefficient. The output vector is therefore not a support map of the input matrix.\n\\head{What has not been established?}\nThis computation proves neither that the supplied factors are the cheapest representation nor that six small tasks outperform one dense kernel. For example, the repeated diagonal blocks and all-one vectors admit further special handling. The stated BDLR model deliberately treats the blocks as dense and the factors as stored vectors so that the paper's general formulas apply literally.","The checker constructs the complete matrix independently, multiplies it by $x$, and compares all six entries with the factored route. The same input returns in the product calculation on \\pagerefV{27}.\n\\takeaway{A matrix can have full entrywise support and still admit a small exact representation. Entry sparsity and algebraic compressibility are different properties.}","\\vpage{4}{Work removed is not time measured}{ARITHMETIC, STORAGE, AND CAPACITY}{}\nFor $q$ right-hand sides, the paper models dense application by $2n^2q$ operations. A block-diagonal-plus-rank-$r$ representation with equal block width $b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory i","\\vpage{4}{Work removed is not time measured}{ARITHMETIC, STORAGE, AND CAPACITY}{}\nFor $q$ right-hand sides, the paper models dense application by $2n^2q$ operations. A block-diagonal-plus-rank-$r$ representation with equal block width $b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation a","\\vpage{4}{Work removed is not time measured}{ARITHMETIC, STORAGE, AND CAPACITY}{}\nFor $q$ right-hand sides, the paper models dense application by $2n^2q$ operations. A block-diagonal-plus-rank-$r$ representation with equal block width $b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs,","\\vpage{4}{Work removed is not time measured}{ARITHMETIC, STORAGE, AND CAPACITY}{}\nFor $q$ right-hand sides, the paper models dense application by $2n^2q$ operations. A block-diagonal-plus-rank-$r$ representation with equal block width $b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs, workspace, allocator overhead, and stored","\\vpage{4}{Work removed is not time measured}{ARITHMETIC, STORAGE, AND CAPACITY}{}\nFor $q$ right-hand sides, the paper models dense application by $2n^2q$ operations. A block-diagonal-plus-rank-$r$ representation with equal block width $b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs, workspace, allocator overhead, and stored indices are additional. A matrix-only capacity inequality is n","dels dense application by $2n^2q$ operations. A block-diagonal-plus-rank-$r$ representation with equal block width $b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs, workspace, allocator overhead, and stored indices are additional. A matrix-only capacity inequality is not a complete device-memory fit test.","on with equal block width $b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs, workspace, allocator overhead, and stored indices are additional. A matrix-only capacity inequality is not a complete device-memory fit test.","qual block width $b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs, workspace, allocator overhead, and stored indices are additional. A matrix-only capacity inequality is not a complete device-memory fit test.","$b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs, workspace, allocator overhead, and stored indices are additional. A matrix-only capacity inequality is not a complete device-memory fit test.","(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs, workspace, allocator overhead, and stored indices are additional. A matrix-only capacity inequality is not a complete device-memory fit test.",")\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs, workspace, allocator overhead, and stored indices are additional. A matrix-only capacity inequality is not a complete device-memory fit test.","every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs, workspace, allocator overhead, and stored indices are additional. A matrix-only capacity inequality is not a complete device-memory fit test.","ation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs, workspace, allocator overhead, and stored indices are additional. A matrix-only capacity inequality is not a complete device-memory fit test.","For a block band of $p$ packets and half-bandwidth $w$, the exact active-block count is\n\\[\np(2w+1)-w(w+1).\n\\]\nThe subtraction removes overcounted edge diagonals. The checker enumerates all addresses for $1\\le p\\le6$ and every admissible $w$, including $w=0$ and $w=p-1$.\n\\takeaway{An arithmetic or storage ratio explains a representation advantage. Kernel launch cost, traffic, utilization, and conversion determine whether that advantage becomes a time advantage.}","\\vpage{5}{Four coefficients, four exact frequencies}{CIRCULANT MULTIPLICATION}{}\nTake $c=(2,1,0,-1)^T$ and define $C(c)_{jk}=c_{j-k\\bmod4}$. The unitary Fourier matrix is $F_{jk}=\\mathrm i^{-jk}/2$. Its normalization is visible in the exact diagonalization:\n\\[\nC(c)=F^*\\diag(2Fc)F,\\qquad 2Fc=(2,2-2\\mathrm i,2,2+2\\mathrm i)^T.\n\\]\n\\vfig{circulant}\n\\head{Follow an actual input}\nFor $x=(1,2,-1,0)^T$, dense multiplication gives $(0,6,0,-2)^T$. Transforming $x$, multiplying by the four displayed eigenvalues, and applying $F^*$ gives exactly the same vector. Every entry belongs to $\\QQ(\\mathrm i)$, so the check needs no approximate trigonometric values.",", four exact frequencies}{CIRCULANT MULTIPLICATION}{}\nTake $c=(2,1,0,-1)^T$ and define $C(c)_{jk}=c_{j-k\\bmod4}$. The unitary Fourier matrix is $F_{jk}=\\mathrm i^{-jk}/2$. Its normalization is visible in the exact diagonalization:\n\\[\nC(c)=F^*\\diag(2Fc)F,\\qquad 2Fc=(2,2-2\\mathrm i,2,2+2\\mathrm i)^T.\n\\]\n\\vfig{circulant}\n\\head{Follow an actual input}\nFor $x=(1,2,-1,0)^T$, dense multiplication gives $(0,6,0,-2)^T$. Transforming $x$, multiplying by the four displayed eigenvalues, and applying $F^*$ gives exactly the same vector. Every entry belongs to $\\QQ(\\mathrm i)$, so the check needs no approximate trigonometric values.","CULANT MULTIPLICATION}{}\nTake $c=(2,1,0,-1)^T$ and define $C(c)_{jk}=c_{j-k\\bmod4}$. The unitary Fourier matrix is $F_{jk}=\\mathrm i^{-jk}/2$. Its normalization is visible in the exact diagonalization:\n\\[\nC(c)=F^*\\diag(2Fc)F,\\qquad 2Fc=(2,2-2\\mathrm i,2,2+2\\mathrm i)^T.\n\\]\n\\vfig{circulant}\n\\head{Follow an actual input}\nFor $x=(1,2,-1,0)^T$, dense multiplication gives $(0,6,0,-2)^T$. Transforming $x$, multiplying by the four displayed eigenvalues, and applying $F^*$ gives exactly the same vector. Every entry belongs to $\\QQ(\\mathrm i)$, so the check needs no approximate trigonometric values.","jk}=c_{j-k\\bmod4}$. The unitary Fourier matrix is $F_{jk}=\\mathrm i^{-jk}/2$. Its normalization is visible in the exact diagonalization:\n\\[\nC(c)=F^*\\diag(2Fc)F,\\qquad 2Fc=(2,2-2\\mathrm i,2,2+2\\mathrm i)^T.\n\\]\n\\vfig{circulant}\n\\head{Follow an actual input}\nFor $x=(1,2,-1,0)^T$, dense multiplication gives $(0,6,0,-2)^T$. Transforming $x$, multiplying by the four displayed eigenvalues, and applying $F^*$ gives exactly the same vector. Every entry belongs to $\\QQ(\\mathrm i)$, so the check needs no approximate trigonometric values.","athrm i^{-jk}/2$. Its normalization is visible in the exact diagonalization:\n\\[\nC(c)=F^*\\diag(2Fc)F,\\qquad 2Fc=(2,2-2\\mathrm i,2,2+2\\mathrm i)^T.\n\\]\n\\vfig{circulant}\n\\head{Follow an actual input}\nFor $x=(1,2,-1,0)^T$, dense multiplication gives $(0,6,0,-2)^T$. Transforming $x$, multiplying by the four displayed eigenvalues, and applying $F^*$ gives exactly the same vector. Every entry belongs to $\\QQ(\\mathrm i)$, so the check needs no approximate trigonometric values.","There is a common normalization trap. With an unnormalized discrete transform, the convolution multiplier has one convention. With the unitary transform used here, the eigenvalue vector is $\\sqrt n\\,Fc$, not $Fc$. Dropping the factor two reconstructs $C/2$ rather than $C$ in this example. The invalid control deliberately tries that change and rejects it.\n\\head{Where the fast algorithm enters}\nThe identity specifies the represented operator. An FFT is an additional factorization that evaluates the transforms efficiently. The algebraic diagonalization does not certify a particular floating-point FFT, data layout, or launch policy. For a tiny four-point example, direct multiplication may be the simpler execution path even though the frequency representation is exact.\n\\takeaway{Transform coordinates change both the storage pattern and the normalization conventions. Verify the inverse transfor","There is a common normalization trap. With an unnormalized discrete transform, the convolution multiplier has one convention. With the unitary transform used here, the eigenvalue vector is $\\sqrt n\\,Fc$, not $Fc$. Dropping the factor two reconstructs $C/2$ rather than $C$ in this example. The invalid control deliberately tries that change and rejects it.\n\\head{Where the fast algorithm enters}\nThe identity specifies the represented operator. An FFT is an additional factorization that evaluates the transforms efficiently. The algebraic diagonalization does not certify a particular floating-point FFT, data layout, or launch policy. For a tiny four-point example, direct multiplication may be the simpler execution path even though the frequency representation is exact.\n\\takeaway{Transform coordinates change both the storage pattern and the normalization conventions. Verify the inverse transform and conv","There is a common normalization trap. With an unnormalized discrete transform, the convolution multiplier has one convention. With the unitary transform used here, the eigenvalue vector is $\\sqrt n\\,Fc$, not $Fc$. Dropping the factor two reconstructs $C/2$ rather than $C$ in this example. The invalid control deliberately tries that change and rejects it.\n\\head{Where the fast algorithm enters}\nThe identity specifies the represented operator. An FFT is an additional factorization that evaluates the transforms efficiently. The algebraic diagonalization does not certify a particular floating-point FFT, data layout, or launch policy. For a tiny four-point example, direct multiplication may be the simpler execution path even though the frequency representation is exact.\n\\takeaway{Transform coordinates change both the storage pattern and the normalization conventions. Verify the inverse transform and convolution scale before interpreting a frequenc","There is a common normalization trap. With an unnormalized discrete transform, the convolution multiplier has one convention. With the unitary transform used here, the eigenvalue vector is $\\sqrt n\\,Fc$, not $Fc$. Dropping the factor two reconstructs $C/2$ rather than $C$ in this example. The invalid control deliberately tries that change and rejects it.\n\\head{Where the fast algorithm enters}\nThe identity specifies the represented operator. An FFT is an additional factorization that evaluates the transforms efficiently. The algebraic diagonalization does not certify a particular floating-point FFT, data layout, or launch policy. For a tiny four-point example, direct multiplication may be the simpler execution path even though the frequency representation is exact.\n\\takeaway{Transform coordinates change both the storage pattern and the normalization conventions. Verify the inverse transform and convolution scale before interpreting a frequency-domain product","\\vpage{6}{Padding is part of the theorem}{TOEPLITZ EMBEDDING WITHOUT WRAPAROUND}{}\nConsider\n\\[\nT=\\mat{2&-1&5\\\\3&2&-1\\\\4&3&2},\\qquad x=(1,2,-1)^T.\n\\]\nThe diagonal pattern is Toeplitz, not circulant. To use a circular convolution, embed it into dimension six with first column $g=(2,3,4,0,5,-1)^T$ and pad the input with three zeros.\n\\vfig{toeplitz}\n\\head{Read the wrapped indices}\nFor output index $j<3$, an input index $k\\le j$ uses the first-column coefficient $c_{j-k}$. An input index $k>j$ wraps to the stored first-row coefficient $r_{k-j}$. The padded input positions contribute nothing. Keeping the first three outputs therefore gives\n\\[\nTx=(-5,8,8)^T.\n\\]\nThe remaining circular outputs are workspace consequences of the embedding, not additional components of $Tx$.\n\\head{A concrete failure mode}\nUsing only the first column $(2,3,4)$ as","4&3&2},\\qquad x=(1,2,-1)^T.\n\\]\nThe diagonal pattern is Toeplitz, not circulant. To use a circular convolution, embed it into dimension six with first column $g=(2,3,4,0,5,-1)^T$ and pad the input with three zeros.\n\\vfig{toeplitz}\n\\head{Read the wrapped indices}\nFor output index $j<3$, an input index $k\\le j$ uses the first-column coefficient $c_{j-k}$. An input index $k>j$ wraps to the stored first-row coefficient $r_{k-j}$. The padded input positions contribute nothing. Keeping the first three outputs therefore gives\n\\[\nTx=(-5,8,8)^T.\n\\]\nThe remaining circular outputs are workspace consequences of the embedding, not additional components of $Tx$.\n\\head{A concrete failure mode}\nUsing only the first column $(2,3,4)$ as a three-dimensional circulant changes the upper-triangular coefficients. That is a different matrix. The checker compares its output with $Tx$ and r","4&3&2},\\qquad x=(1,2,-1)^T.\n\\]\nThe diagonal pattern is Toeplitz, not circulant. To use a circular convolution, embed it into dimension six with first column $g=(2,3,4,0,5,-1)^T$ and pad the input with three zeros.\n\\vfig{toeplitz}\n\\head{Read the wrapped indices}\nFor output index $j<3$, an input index $k\\le j$ uses the first-column coefficient $c_{j-k}$. An input index $k>j$ wraps to the stored first-row coefficient $r_{k-j}$. The padded input positions contribute nothing. Keeping the first three outputs therefore gives\n\\[\nTx=(-5,8,8)^T.\n\\]\nThe remaining circular outputs are workspace consequences of the embedding, not additional components of $Tx$.\n\\head{A concrete failure mode}\nUsing only the first column $(2,3,4)$ as a three-dimensional circulant changes the upper-triangular coefficients. That is a different matrix. The checker compares its output with $Tx$ and rejects the unpadded shortcut.","2,-1)^T.\n\\]\nThe diagonal pattern is Toeplitz, not circulant. To use a circular convolution, embed it into dimension six with first column $g=(2,3,4,0,5,-1)^T$ and pad the input with three zeros.\n\\vfig{toeplitz}\n\\head{Read the wrapped indices}\nFor output index $j<3$, an input index $k\\le j$ uses the first-column coefficient $c_{j-k}$. An input index $k>j$ wraps to the stored first-row coefficient $r_{k-j}$. The padded input positions contribute nothing. Keeping the first three outputs therefore gives\n\\[\nTx=(-5,8,8)^T.\n\\]\nThe remaining circular outputs are workspace consequences of the embedding, not additional components of $Tx$.\n\\head{A concrete failure mode}\nUsing only the first column $(2,3,4)$ as a three-dimensional circulant changes the upper-triangular coefficients. That is a different matrix. The checker compares its output with $Tx$ and rejects the unpadded shortcut.","imension six with first column $g=(2,3,4,0,5,-1)^T$ and pad the input with three zeros.\n\\vfig{toeplitz}\n\\head{Read the wrapped indices}\nFor output index $j<3$, an input index $k\\le j$ uses the first-column coefficient $c_{j-k}$. An input index $k>j$ wraps to the stored first-row coefficient $r_{k-j}$. The padded input positions contribute nothing. Keeping the first three outputs therefore gives\n\\[\nTx=(-5,8,8)^T.\n\\]\nThe remaining circular outputs are workspace consequences of the embedding, not additional components of $Tx$.\n\\head{A concrete failure mode}\nUsing only the first column $(2,3,4)$ as a three-dimensional circulant changes the upper-triangular coefficients. That is a different matrix. The checker compares its output with $Tx$ and rejects the unpadded shortcut.","d indices}\nFor output index $j<3$, an input index $k\\le j$ uses the first-column coefficient $c_{j-k}$. An input index $k>j$ wraps to the stored first-row coefficient $r_{k-j}$. The padded input positions contribute nothing. Keeping the first three outputs therefore gives\n\\[\nTx=(-5,8,8)^T.\n\\]\nThe remaining circular outputs are workspace consequences of the embedding, not additional components of $Tx$.\n\\head{A concrete failure mode}\nUsing only the first column $(2,3,4)$ as a three-dimensional circulant changes the upper-triangular coefficients. That is a different matrix. The checker compares its output with $Tx$ and rejects the unpadded shortcut.","he first-column coefficient $c_{j-k}$. An input index $k>j$ wraps to the stored first-row coefficient $r_{k-j}$. The padded input positions contribute nothing. Keeping the first three outputs therefore gives\n\\[\nTx=(-5,8,8)^T.\n\\]\nThe remaining circular outputs are workspace consequences of the embedding, not additional components of $Tx$.\n\\head{A concrete failure mode}\nUsing only the first column $(2,3,4)$ as a three-dimensional circulant changes the upper-triangular coefficients. That is a different matrix. The checker compares its output with $Tx$ and rejects the unpadded shortcut.","nothing. Keeping the first three outputs therefore gives\n\\[\nTx=(-5,8,8)^T.\n\\]\nThe remaining circular outputs are workspace consequences of the embedding, not additional components of $Tx$.\n\\head{A concrete failure mode}\nUsing only the first column $(2,3,4)$ as a three-dimensional circulant changes the upper-triangular coefficients. That is a different matrix. The checker compares its output with $Tx$ and rejects the unpadded shortcut.","The middle zero in $g$ is not a discarded Toeplitz coefficient: it separates the lower and reversed upper diagonals in the chosen $2n$ embedding. Other embeddings require their own index identities. It is not sufficient for an array merely to have the right length or to produce a plausible plot.\n\\takeaway{Zero padding prevents the wrong circular interactions. It is an algebraic part of the operator representation, not an optional performance tweak.}","\\vpage{7}{A transpose, not an adjoint}{KRONECKER ACTION AND ARRAY ORDER}{}\nWith column-major vectorization, the tensor identity is\n\\[\n(A\\otimes B)\\operatorname{vec}(X)=\\operatorname{vec}(BXA^T).\n\\]\nHere the transpose records index order, not a Hermitian inner product. Use the explicitly complex matrix $A=\\mat{1&\\mathrm i","\\vpage{7}{A transpose, not an adjoint}{KRONECKER ACTION AND ARRAY ORDER}{}\nWith column-major vectorization, the tensor identity is\n\\[\n(A\\otimes B)\\operatorname{vec}(X)=\\operatorname{vec}(BXA^T).\n\\]\nHere the transpose records index order, not a Hermitian inner product. Use the explicitly complex matrix $A=\\mat{1&\\mathrm i\\\\2&-1}$, together with $B=\\mat{2&1","2&-1}$, together with $B=\\mat{2&1\\\\0&3}$ and $X=\\mat{1&2","0&3}$ and $X=\\mat{1&2\\\\3&4}$.\n\\vfig{kronecker}\n\\head{Check the coordinates}\nThe stacked input is $(1,3,2,4)^T$. Direct tensor multiplication agrees with stacking the columns of\n\\[\nBXA^T=\\mat{5+8\\mathrm i&2","3&4}$.\n\\vfig{kronecker}\n\\head{Check the coordinates}\nThe stacked input is $(1,3,2,4)^T$. Direct tensor multiplication agrees with stacking the columns of\n\\[\nBXA^T=\\mat{5+8\\mathrm i&2","3&4}$.\n\\vfig{kronecker}\n\\head{Check the coordinates}\nThe stacked input is $(1,3,2,4)^T$. Direct tensor multiplication agrees with stacking the columns of\n\\[\nBXA^T=\\mat{5+8\\mathrm i&2\\\\9+12\\mathrm i&6}.\n\\]\nReplacing $A^T$ with $A^*$ changes the signs of the imaginary terms. A real-only test would conceal that error, which is why the worked model uses an imaginary coefficient.\n\\head{Why modewise contraction matters}\nFor $A$ of order $m$ and $B$ of order $n$, the explicit tensor matrix stores $m^2n^2$ entries. Its factors store $m^2+n^2$. Applying through $BXA^T$ costs $2n^2m+2nm^2$ modeled operations, rather than forming the full tensor matrix first.","9+12\\mathrm i&6}.\n\\]\nReplacing $A^T$ with $A^*$ changes the signs of the imaginary terms. A real-only test would conceal that error, which is why the worked model uses an imaginary coefficient.\n\\head{Why modewise contraction matters}\nFor $A$ of order $m$ and $B$ of order $n$, the explicit tensor matrix stores $m^2n^2$ entries. Its factors store $m^2+n^2$. Applying through $BXA^T$ costs $2n^2m+2nm^2$ modeled operations, rather than forming the full tensor matrix first.","\\vpage{8}{See every butterfly value}{WALSH RECURSION IN EIGHT COORDINATES}{}\nThe unnormalized Walsh matrix satisfies $H_1=[1]$ and\n\\[\nH_{2n}=\\mat{H_n&H_n","\\vpage{8}{See every butterfly value}{WALSH RECURSION IN EIGHT COORDINATES}{}\nThe unnormalized Walsh matrix satisfies $H_1=[1]$ and\n\\[\nH_{2n}=\\mat{H_n&H_n\\\\H_n&-H_n},\\qquad H_nH_n^T=nI.\n\\]\nApply the three add/subtract stages to $(1,2,3,4,5,6,7,8)^T$. The full intermediate ledger is displayed below.\n\\vfig{walsh}\n\\head{Arithmetic can be counted exactly}\nEach stage visits eight coordinates, producing one sum or difference per coordinate. Three stages therefore use exactly 24 additions or subtractions. No multiplication by a dense $8\\times8$ matrix is necessary.","H_n&-H_n},\\qquad H_nH_n^T=nI.\n\\]\nApply the three add/subtract stages to $(1,2,3,4,5,6,7,8)^T$. The full intermediate ledger is displayed below.\n\\vfig{walsh}\n\\head{Arithmetic can be counted exactly}\nEach stage visits eight coordinates, producing one sum or difference per coordinate. Three stages therefore use exactly 24 additions or subtractions. No multiplication by a dense $8\\times8$ matrix is necessary.","The result $(36,-4,-8,0,-16,0,0,0)^T$ still reconstructs the input: the inverse is $H_8^T/8$. Because the displayed transform is unnormalized, its squared output norm is eight times the squared input norm. The normalized transform $H_8/\\sqrt8$ would instead be orthogonal.\n\\head{What the zeros mean}\nThe zero coefficients arise from exact cancellation among the input values. They do not imply that the transform's defining matrix is sparse. The butterfly structure, not the final coefficient support, explains the arithmetic reduction.","\\vpage{9}{The cross term has somewhere to go}{CLOSED LOW-RANK PRODUCT ALGEBRA}{}\nLet $A=D_A+u_av_a^T$ be the running operator. Take $D_B=\\diag(1,2,3,4,5,6)$, $u_b=(1,0,1,0,1,0)^T$, and $v_b=(0,1,0,1,0,1)^T$. Define $B=D_B+u_bv_b^T$.\n\\[\nU_C=[D_Au_b+u_a(v_a^Tu_b)\\ \\ u_a],\\qquad\nV_C=[v_b\\ \\ D_B^Tv_a].\n\\]\n\\vfig{bdlr_product}\n\\head{Why there are only two columns}\nExpanding $AB$ gives three correction terms, but two share the right factor $v_b^T$:\n\\[\nD_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T\n=\\bigl(D_Au_b+u_a(v_a^Tu_b)\\bigr)v_b^T.\n\\]\nThe remaining term is $u_av_a^TD_B$. Thus $AB=D_AD_B+U_CV_C^T$ with at most two correction directions. Counting the three expanded summands as independent ranks would overestimate this representation.","to go}{CLOSED LOW-RANK PRODUCT ALGEBRA}{}\nLet $A=D_A+u_av_a^T$ be the running operator. Take $D_B=\\diag(1,2,3,4,5,6)$, $u_b=(1,0,1,0,1,0)^T$, and $v_b=(0,1,0,1,0,1)^T$. Define $B=D_B+u_bv_b^T$.\n\\[\nU_C=[D_Au_b+u_a(v_a^Tu_b)\\ \\ u_a],\\qquad\nV_C=[v_b\\ \\ D_B^Tv_a].\n\\]\n\\vfig{bdlr_product}\n\\head{Why there are only two columns}\nExpanding $AB$ gives three correction terms, but two share the right factor $v_b^T$:\n\\[\nD_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T\n=\\bigl(D_Au_b+u_a(v_a^Tu_b)\\bigr)v_b^T.\n\\]\nThe remaining term is $u_av_a^TD_B$. Thus $AB=D_AD_B+U_CV_C^T$ with at most two correction directions. Counting the three expanded summands as independent ranks would overestimate this representation.","OSED LOW-RANK PRODUCT ALGEBRA}{}\nLet $A=D_A+u_av_a^T$ be the running operator. Take $D_B=\\diag(1,2,3,4,5,6)$, $u_b=(1,0,1,0,1,0)^T$, and $v_b=(0,1,0,1,0,1)^T$. Define $B=D_B+u_bv_b^T$.\n\\[\nU_C=[D_Au_b+u_a(v_a^Tu_b)\\ \\ u_a],\\qquad\nV_C=[v_b\\ \\ D_B^Tv_a].\n\\]\n\\vfig{bdlr_product}\n\\head{Why there are only two columns}\nExpanding $AB$ gives three correction terms, but two share the right factor $v_b^T$:\n\\[\nD_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T\n=\\bigl(D_Au_b+u_a(v_a^Tu_b)\\bigr)v_b^T.\n\\]\nThe remaining term is $u_av_a^TD_B$. Thus $AB=D_AD_B+U_CV_C^T$ with at most two correction directions. Counting the three expanded summands as independent ranks would overestimate this representation.",",3,4,5,6)$, $u_b=(1,0,1,0,1,0)^T$, and $v_b=(0,1,0,1,0,1)^T$. Define $B=D_B+u_bv_b^T$.\n\\[\nU_C=[D_Au_b+u_a(v_a^Tu_b)\\ \\ u_a],\\qquad\nV_C=[v_b\\ \\ D_B^Tv_a].\n\\]\n\\vfig{bdlr_product}\n\\head{Why there are only two columns}\nExpanding $AB$ gives three correction terms, but two share the right factor $v_b^T$:\n\\[\nD_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T\n=\\bigl(D_Au_b+u_a(v_a^Tu_b)\\bigr)v_b^T.\n\\]\nThe remaining term is $u_av_a^TD_B$. Thus $AB=D_AD_B+U_CV_C^T$ with at most two correction directions. Counting the three expanded summands as independent ranks would overestimate this representation.","0,1,0,1,0)^T$, and $v_b=(0,1,0,1,0,1)^T$. Define $B=D_B+u_bv_b^T$.\n\\[\nU_C=[D_Au_b+u_a(v_a^Tu_b)\\ \\ u_a],\\qquad\nV_C=[v_b\\ \\ D_B^Tv_a].\n\\]\n\\vfig{bdlr_product}\n\\head{Why there are only two columns}\nExpanding $AB$ gives three correction terms, but two share the right factor $v_b^T$:\n\\[\nD_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T\n=\\bigl(D_Au_b+u_a(v_a^Tu_b)\\bigr)v_b^T.\n\\]\nThe remaining term is $u_av_a^TD_B$. Thus $AB=D_AD_B+U_CV_C^T$ with at most two correction directions. Counting the three expanded summands as independent ranks would overestimate this representation.","The scalar $v_a^Tu_b$ is three. Removing the quadratic term would therefore remove $3u_av_b^T$, which is nonzero. The checker compares the complete product to the factor construction and rejects a calculation that omits this contribution.\n\\head{Before and after recompression}\nA rank bound on stored factors is not always the actual matrix rank. Repeated products may introduce dependent columns. Recompression can remove those dependencies exactly, or remove small singular directions approximately under an explicit error budget.\n\\takeaway{Closure of a representation class is an algebraic statement about all product terms. It is not permission to drop an interaction merely because it is second order in the factors.","The scalar $v_a^Tu_b$ is three. Removing the quadratic term would therefore remove $3u_av_b^T$, which is nonzero. The checker compares the complete product to the factor construction and rejects a calculation that omits this contribution.\n\\head{Before and after recompression}\nA rank bound on stored factors is not always the actual matrix rank. Repeated products may introduce dependent columns. Recompression can remove those dependencies exactly, or remove small singular directions approximately under an explicit error budget.\n\\takeaway{Closure of a representation class is an algebraic statement about all product terms. It is not permission to drop an interaction merely because it is second order in the factors.}","\\vpage{10}{A large inverse, a small denominator}{WOODBURY WITH BOTH INVERTIBILITY CHECKS}{}\nFor the running operator, each diagonal block has inverse $\\frac18\\mat{3&-1\\\\-1&3}$. Hence $D^{-1}u=u/4$ and\n\\[\nS=1+u^TD^{-1}u=\\frac52.\n\\]\nThe Woodbury identity reduces the entire inverse to\n\\[\nA^{-1}=D^{-1}-D^{-1}uS^{-1}u^TD^{-1}\n=D^{-1}-\\frac1{40}uu^T.\n\\]\n\\vfig{woodbury}\n\\head{Two hypotheses, not one}\nInvertibility of $D$ alone does not suffice. For $D=I_2$, $u=e_1$, and $v=-e_1$, the update is $\\diag(0,1)$. The small matrix $S$ is zero, exactly where the correction formula would require an invalid division.","-1&3}$. Hence $D^{-1}u=u/4$ and\n\\[\nS=1+u^TD^{-1}u=\\frac52.\n\\]\nThe Woodbury identity reduces the entire inverse to\n\\[\nA^{-1}=D^{-1}-D^{-1}uS^{-1}u^TD^{-1}\n=D^{-1}-\\frac1{40}uu^T.\n\\]\n\\vfig{woodbury}\n\\head{Two hypotheses, not one}\nInvertibility of $D$ alone does not suffice. For $D=I_2$, $u=e_1$, and $v=-e_1$, the update is $\\diag(0,1)$. The small matrix $S$ is zero, exactly where the correction formula would require an invalid division.","For the running model, the checker multiplies the proposed inverse by $A$ in both orders and obtains the identity. It also checks the determinant relation $\\det A=\\det D\\det S$. These tests verify the actual factors and sign of the correction, not just a numerical solve residual.\n\\head{Conditioning survives the reduction}\nA small $S$ can still be ill-conditioned. Reducing its dimension does not bound $\\|S^{-1}\\|$. The manuscript's inverse estimate keeps that term explicit, along with the norms of $D^{-1}u$ and $v^TD^{-1}$.\n\\takeaway{The hidden coupling has not vanished. Its solvability and sensitivity are concentrated in a smaller operator that still has to be invertible and numerically controlled.}","he running model, the checker multiplies the proposed inverse by $A$ in both orders and obtains the identity. It also checks the determinant relation $\\det A=\\det D\\det S$. These tests verify the actual factors and sign of the correction, not just a numerical solve residual.\n\\head{Conditioning survives the reduction}\nA small $S$ can still be ill-conditioned. Reducing its dimension does not bound $\\|S^{-1}\\|$. The manuscript's inverse estimate keeps that term explicit, along with the norms of $D^{-1}u$ and $v^TD^{-1}$.\n\\takeaway{The hidden coupling has not vanished. Its solvability and sensitivity are concentrated in a smaller operator that still has to be invertible and numerically controlled.}","e identity. It also checks the determinant relation $\\det A=\\det D\\det S$. These tests verify the actual factors and sign of the correction, not just a numerical solve residual.\n\\head{Conditioning survives the reduction}\nA small $S$ can still be ill-conditioned. Reducing its dimension does not bound $\\|S^{-1}\\|$. The manuscript's inverse estimate keeps that term explicit, along with the norms of $D^{-1}u$ and $v^TD^{-1}$.\n\\takeaway{The hidden coupling has not vanished. Its solvability and sensitivity are concentrated in a smaller operator that still has to be invertible and numerically controlled.}","also checks the determinant relation $\\det A=\\det D\\det S$. These tests verify the actual factors and sign of the correction, not just a numerical solve residual.\n\\head{Conditioning survives the reduction}\nA small $S$ can still be ill-conditioned. Reducing its dimension does not bound $\\|S^{-1}\\|$. The manuscript's inverse estimate keeps that term explicit, along with the norms of $D^{-1}u$ and $v^TD^{-1}$.\n\\takeaway{The hidden coupling has not vanished. Its solvability and sensitivity are concentrated in a smaller operator that still has to be invertible and numerically controlled.}","\\vpage{11}{Stored width is not intrinsic rank}{ACTUAL RANGE COORDINATES}{}\nChoose the isometry\n\\[\nQ_L=\\mat{3/5&0\\\\4/5&0\\\\0&1\\\\0&0},\\quad\nR_L=\\mat{1&0&1\\\\0&1&0},\\quad\nR_R=\\mat{3&0&2\\\\0&2&0\\\\1&0&-1}.\n\\]\nSet $L=Q_LR_L$ and $R=R_R$. The two equal columns of $L$ do not create a third range direction. In contrast, $R$ has rank three.\n\\vfig{compact_core}\nThe actual core is\n\\[\nG=R_LR_R^T=\\mat{5&0&0","1&0&-1}.\n\\]\nSet $L=Q_LR_L$ and $R=R_R$. The two equal columns of $L$ do not create a third range direction. In contrast, $R$ has rank three.\n\\vfig{compact_core}\nThe actual core is\n\\[\nG=R_LR_R^T=\\mat{5&0&0","1&0&-1}.\n\\]\nSet $L=Q_LR_L$ and $R=R_R$. The two equal columns of $L$ do not create a third range direction. In contrast, $R$ has rank three.\n\\vfig{compact_core}\nThe actual core is\n\\[\nG=R_LR_R^T=\\mat{5&0&0\\\\0&2&0},\\qquad LR^T=Q_LG.\n\\]\nIts dimensions are the two actual input ranks, two and three. This is not inferred by merely discarding a column with a small norm. The range factor $Q_L$ has orthonormal columns and reconstructs $L$ exactly.\n\\head{The preserved formal boundary}\nThe sixth release constructs orthonormal bases of the actual column spaces. It then assumes exact ordered thin-SVD data for the core, as the manuscript instructs the reader to take an SVD. It does not assume an optimality or tail-error conclusion. Nor does it certify a floating-point QR or SVD implementation simply because those routines use similar notation.","0&2&0},\\qquad LR^T=Q_LG.\n\\]\nIts dimensions are the two actual input ranks, two and three. This is not inferred by merely discarding a column with a small norm. The range factor $Q_L$ has orthonormal columns and reconstructs $L$ exactly.\n\\head{The preserved formal boundary}\nThe sixth release constructs orthonormal bases of the actual column spaces. It then assumes exact ordered thin-SVD data for the core, as the manuscript instructs the reader to take an SVD. It does not assume an optimality or tail-error conclusion. Nor does it certify a floating-point QR or SVD implementation simply because those routines use similar notation.","\\vpage{12}{Why a core optimum is global}{COMPRESSION, KERNEL WITNESSES, AND TAILS}{}\nThe product from the previous page is\n\\[\nA_0=LR^T=\\mat{3&0&0\\\\4&0&0\\\\0&2&0\\\\0&0&0}.\n\\]\nKeeping only its first singular direction gives a rank-one approximation with spectral and Frobenius error exactly two. Keeping none gives errors five and $\\sqrt{29}$; keeping both gives zero.\n\\vfig{optimality}\n\\head{Why an outside competitor cannot do better}\nFor isometric outer factors, compression $C(B)=Q_L^*BQ_R$ cannot increase rank or either matrix norm. Every ambient rank-$k$ competitor therefore supplies an admissible core competitor. The lifted core residual has exactly the core residual norm. These two facts transfer a core optimum to the entire ambient matrix space, not just to competitors already lying in the supplied ranges.","0&0&0}.\n\\]\nKeeping only its first singular direction gives a rank-one approximation with spectral and Frobenius error exactly two. Keeping none gives errors five and $\\sqrt{29}$; keeping both gives zero.\n\\vfig{optimality}\n\\head{Why an outside competitor cannot do better}\nFor isometric outer factors, compression $C(B)=Q_L^*BQ_R$ cannot increase rank or either matrix norm. Every ambient rank-$k$ competitor therefore supplies an admissible core competitor. The lifted core residual has exactly the core residual norm. These two facts transfer a core optimum to the entire ambient matrix space, not just to competitors already lying in the supplied ranges.","For a $d$-dimensional diagonal core and $0\\le k<d$, the spectral lower bound has a concrete witness. A rank-at-most-$k$ competitor has a nonzero kernel vector in the first $k+1$ coordinates. On that vector the diagonal matrix acts with norm at least $\\sigma_{k+1}$ times the vector norm. The competitor cannot cancel that direction.","The Frobenius proof uses an orthonormal basis of the competitor's kernel. Its row weights lie in $[0,1]$ and sum to the kernel dimension. Weighting the decreasing squared singular values forces at least the discarded tail. The two norm arguments are distinct; one is not a graph of the other.\n\\takeaway{Optimality is not inferred from the appearance of the singular-value plot. It follows from compression and a lower bound valid against every rank-bounded competitor.}","\\vpage{13}{Errors enter at different places}{FACTOR PERTURBATION VERSUS RECOMPRESSION}{}\nLet the exact scalar factors be $(2,3,4)$ and start from $p_0=1$. If additive errors $(1/10,1/5,1/20)$ enter \\emph{after} each partial multiplication, then\n\\[\n\\widehat p_j=a_j\\widehat p_{j-1}+e_j,\\qquad\n\\widehat p_3-p_3=12e_1+4e_2+e_3=\\frac{41}{20}.\n\\]\n\\vfig{error_chain}\n\\head{The same local numbers, a different model}\nIf instead the factors themselves are perturbed, the computed product is $(4+e_3)(3+e_2)(2+e_1)$. Its error is $402/125$, not $41/20$. The factor-perturbation telescoping formula contains already-perturbed later factors; the additive recompression formula contains the original later factors.","N}{}\nLet the exact scalar factors be $(2,3,4)$ and start from $p_0=1$. If additive errors $(1/10,1/5,1/20)$ enter \\emph{after} each partial multiplication, then\n\\[\n\\widehat p_j=a_j\\widehat p_{j-1}+e_j,\\qquad\n\\widehat p_3-p_3=12e_1+4e_2+e_3=\\frac{41}{20}.\n\\]\n\\vfig{error_chain}\n\\head{The same local numbers, a different model}\nIf instead the factors themselves are perturbed, the computed product is $(4+e_3)(3+e_2)(2+e_1)$. Its error is $402/125$, not $41/20$. The factor-perturbation telescoping formula contains already-perturbed later factors; the additive recompression formula contains the original later factors.","(2,3,4)$ and start from $p_0=1$. If additive errors $(1/10,1/5,1/20)$ enter \\emph{after} each partial multiplication, then\n\\[\n\\widehat p_j=a_j\\widehat p_{j-1}+e_j,\\qquad\n\\widehat p_3-p_3=12e_1+4e_2+e_3=\\frac{41}{20}.\n\\]\n\\vfig{error_chain}\n\\head{The same local numbers, a different model}\nIf instead the factors themselves are perturbed, the computed product is $(4+e_3)(3+e_2)(2+e_1)$. Its error is $402/125$, not $41/20$. The factor-perturbation telescoping formula contains already-perturbed later factors; the additive recompression formula contains the original later factors.","art from $p_0=1$. If additive errors $(1/10,1/5,1/20)$ enter \\emph{after} each partial multiplication, then\n\\[\n\\widehat p_j=a_j\\widehat p_{j-1}+e_j,\\qquad\n\\widehat p_3-p_3=12e_1+4e_2+e_3=\\frac{41}{20}.\n\\]\n\\vfig{error_chain}\n\\head{The same local numbers, a different model}\nIf instead the factors themselves are perturbed, the computed product is $(4+e_3)(3+e_2)(2+e_1)$. Its error is $402/125$, not $41/20$. The factor-perturbation telescoping formula contains already-perturbed later factors; the additive recompression formula contains the original later factors.","The visual comparison uses the additive model. The first error is multiplied by both remaining factors. The last error is not amplified at all. Summing only the local error magnitudes gives $7/20$, which is smaller than the actual final error and therefore not a valid bound here.\n\\head{In a matrix chain}\nThe corresponding formulas replace scalar magnitudes with operator norms and respect multiplication order. A bound can be conservative when contributions cancel, but it must not omit possible amplification. Singular-tail errors from a recompression step become useful only after they are inserted into the correct global error model.\n\\takeaway{An error tolerance needs an entry point. A bound for perturbing factors and a bound for compressing partial products are not interchangeable.}","\\vpage{14}{Two routes can cancel exactly}{SUPPORT IS AN INCLUSION}{}\nLet $K$ have transfers $0\\to1$, $0\\to2$, $1\\to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This","\\vpage{14}{Two routes can cancel exactly}{SUPPORT IS AN INCLUSION}{}\nLet $K$ have transfers $0\\to1$, $0\\to2$, $1\\to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on th","\\vpage{14}{Two routes can cancel exactly}{SUPPORT IS AN INCLUSION}{}\nLet $K$ have transfers $0\\to1$, $0\\to2$, $1\\to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpote","\\vpage{14}{Two routes can cancel exactly}{SUPPORT IS AN INCLUSION}{}\nLet $K$ have transfers $0\\to1$, $0\\to2$, $1\\to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index","\\vpage{14}{Two routes can cancel exactly}{SUPPORT IS AN INCLUSION}{}\nLet $K$ have transfers $0\\to1$, $0\\to2$, $1\\to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index, not an asse","\\vpage{14}{Two routes can cancel exactly}{SUPPORT IS AN INCLUSION}{}\nLet $K$ have transfers $0\\to1$, $0\\to2$, $1\\to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index, not an assertion that $K^h\\ne0$. Here the","\\vpage{14}{Two routes can cancel exactly}{SUPPORT IS AN INCLUSION}{}\nLet $K$ have transfers $0\\to1$, $0\\to2$, $1\\to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index, not an assertion that $K^h\\ne0$. Here the graph has paths of length two, while coefficient cancellation already gives $K^2=0$.","to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index, not an assertion that $K^h\\ne0$. Here the graph has paths of length two, while coefficient cancellation already gives $K^2=0$.","1}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index, not an assertion that $K^h\\ne0$. Here the graph has paths of length two, while coefficient cancellation already gives $K^2=0$.",",\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index, not an assertion that $K^h\\ne0$. Here the graph has paths of length two, while coefficient cancellation already gives $K^2=0$.","x of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index, not an assertion that $K^h\\ne0$. Here the graph has paths of length two, while coefficient cancellation already gives $K^2=0$.","dicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index, not an assertion that $K^h\\ne0$. Here the graph has paths of length two, while coefficient cancellation already gives $K^2=0$.","A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index, not an assertion that $K^h\\ne0$. Here the graph has paths of length two, while coefficient cancellation already gives $K^2=0$.","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimite","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary an","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attaina","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If o","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If only one e","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If only one execution","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If only one execution slot is a","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If only one execution slot is available,","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If only one execution slot is available, seven tasks","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If only one execution slot is available, seven tasks require seven slots of",", put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If only one execution slot is available, seven tasks require seven slots of work. If task durations differ, the relevant critical path is weighted.","\\vpage{16}{Elimination creates new interactions}{SCHUR FILL AND ITS CANCELLATION}{}\nRetain the first two coordinates of\n\\[\nA=\\mat{2&0&1\\\\0&2&1\\\\1&1&2}.\n\\]\nThe third coordinate is hidden. Its invertible block is $[2]$, so eliminating it produces\n\\[\nS=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR}\n=\\mat{3/2&-1/2","1&1&2}.\n\\]\nThe third coordinate is hidden. Its invertible block is $[2]$, so eliminating it produces\n\\[\nS=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR}\n=\\mat{3/2&-1/2","1&1&2}.\n\\]\nThe third coordinate is hidden. Its invertible block is $[2]$, so eliminating it produces\n\\[\nS=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR}\n=\\mat{3/2&-1/2\\\\-1/2&3/2}.\n\\]\n\\vfig{schur_fill}\n\\head{The hidden route has a sign}\nThe retained coordinates had no direct coupling, yet both touch the hidden coordinate. The resulting correction couples them by $-1/2$. This is fill: a new active retained block produced by an eliminated route.","-1/2&3/2}.\n\\]\n\\vfig{schur_fill}\n\\head{The hidden route has a sign}\nThe retained coordinates had no direct coupling, yet both touch the hidden coordinate. The resulting correction couples them by $-1/2$. This is fill: a new active retained block produced by an eliminated route.","Now change both original retained off-diagonal entries to $+1/2$. The same hidden correction cancels them, leaving the diagonal matrix $\\frac32 I$. Thus a structural route can create or remove visible coupling depending on the direct coefficient already present.\n\\head{A stronger inverse statement}\nFor an invertible finite packet matrix, inverse support lies in the directed transitive closure. A reachable-coordinate subspace is invariant under the matrix. Injectivity of its restriction implies surjectivity in finite dimension, so the inverse preserves it as well. This proof uses finite-dimensional invertibility, not a convergent geometric series.","The checker verifies both Schur complements and, in each invertible example, equality between $S^{-1}$ and the retained block of $A^{-1}$.\n\\takeaway{Removing coordinates does not remove their influence. It changes the retained operator, with possible fill, cancellation, and new conditioning constraints.}","\\vpage{17}{Invariant does not mean independently addressable}{A STRICT KRYLOV INCLUSION}{}\nTake $A=I_2$, split into the two coordinate packets, and start with $v=(1,1)^T$. Every packet is invariant. Nevertheless every polynomial applies the \\emph{same} scalar to both coordinates:\n\\[\np(A)v=p(1)(1,1)^T,\\qquad \\dim\\mathcal K_m(A,v)=1\\quad(m\\ge1).\n\\]\n\\vfig{krylov}\nThe direct sum of the two local Krylov spans is all of $\\QQ^2$. It permits independent coefficients in the two packets. The global Krylov space does not. This is why the manuscript states an inclusion, not an unconditional equality.\n\\head{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors","\\vpage{17}{Invariant does not mean independently addressable}{A STRICT KRYLOV INCLUSION}{}\nTake $A=I_2$, split into the two coordinate packets, and start with $v=(1,1)^T$. Every packet is invariant. Nevertheless every polynomial applies the \\emph{same} scalar to both coordinates:\n\\[\np(A)v=p(1)(1,1)^T,\\qquad \\dim\\mathcal K_m(A,v)=1\\quad(m\\ge1).\n\\]\n\\vfig{krylov}\nThe direct sum of the two local Krylov spans is all of $\\QQ^2$. It permits independent coefficients in the two packets. The global Krylov space does not. This is why the manuscript states an inclusion, not an unconditional equality.\n\\head{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=","\\vpage{17}{Invariant does not mean independently addressable}{A STRICT KRYLOV INCLUSION}{}\nTake $A=I_2$, split into the two coordinate packets, and start with $v=(1,1)^T$. Every packet is invariant. Nevertheless every polynomial applies the \\emph{same} scalar to both coordinates:\n\\[\np(A)v=p(1)(1,1)^T,\\qquad \\dim\\mathcal K_m(A,v)=1\\quad(m\\ge1).\n\\]\n\\vfig{krylov}\nThe direct sum of the two local Krylov spans is all of $\\QQ^2$. It permits independent coefficients in the two packets. The global Krylov space does not. This is why the manuscript states an inclusion, not an unconditional equality.\n\\head{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=I_2$. Both changes enlarge the addressing power, but in different ways.\n\\head{When invariance is only approximate}\nIf $A=D+K$ with $D$ packet diagonal, then\n\\[\n\\|A^j-D^j\\|\\le jM","t does not mean independently addressable}{A STRICT KRYLOV INCLUSION}{}\nTake $A=I_2$, split into the two coordinate packets, and start with $v=(1,1)^T$. Every packet is invariant. Nevertheless every polynomial applies the \\emph{same} scalar to both coordinates:\n\\[\np(A)v=p(1)(1,1)^T,\\qquad \\dim\\mathcal K_m(A,v)=1\\quad(m\\ge1).\n\\]\n\\vfig{krylov}\nThe direct sum of the two local Krylov spans is all of $\\QQ^2$. It permits independent coefficients in the two packets. The global Krylov space does not. This is why the manuscript states an inclusion, not an unconditional equality.\n\\head{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=I_2$. Both changes enlarge the addressing power, but in different ways.\n\\head{When invariance is only approximate}\nIf $A=D+K$ with $D$ packet diagonal, then\n\\[\n\\|A^j-D^j\\|\\le jM^{j-1}\\|K\\|,\n\\qquad M=\\max\\{\\|A\\|,\\|D\\|\\}.\n\\]\nThis controls departure fro…","es the \\emph{same} scalar to both coordinates:\n\\[\np(A)v=p(1)(1,1)^T,\\qquad \\dim\\mathcal K_m(A,v)=1\\quad(m\\ge1).\n\\]\n\\vfig{krylov}\nThe direct sum of the two local Krylov spans is all of $\\QQ^2$. It permits independent coefficients in the two packets. The global Krylov space does not. This is why the manuscript states an inclusion, not an unconditional equality.\n\\head{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=I_2$. Both changes enlarge the addressing power, but in different ways.\n\\head{When invariance is only approximate}\nIf $A=D+K$ with $D$ packet diagonal, then\n\\[\n\\|A^j-D^j\\|\\le jM^{j-1}\\|K\\|,\n\\qquad M=\\max\\{\\|A\\|,\\|D\\|\\}.\n\\]\nThis controls departure from the decoupled iterate. It is not a claim that the dimension or numerical conditioning of a Krylov basis varies continuously. Near dependence can remain difficult even when the operator error is small.\n\\takeaway{A …","both coordinates:\n\\[\np(A)v=p(1)(1,1)^T,\\qquad \\dim\\mathcal K_m(A,v)=1\\quad(m\\ge1).\n\\]\n\\vfig{krylov}\nThe direct sum of the two local Krylov spans is all of $\\QQ^2$. It permits independent coefficients in the two packets. The global Krylov space does not. This is why the manuscript states an inclusion, not an unconditional equality.\n\\head{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=I_2$. Both changes enlarge the addressing power, but in different ways.\n\\head{When invariance is only approximate}\nIf $A=D+K$ with $D$ packet diagonal, then\n\\[\n\\|A^j-D^j\\|\\le jM^{j-1}\\|K\\|,\n\\qquad M=\\max\\{\\|A\\|,\\|D\\|\\}.\n\\]\nThis controls departure from the decoupled iterate. It is not a claim that the dimension or numerical conditioning of a Krylov basis varies continuously. Near dependence can remain difficult even when the operator error is small.\n\\takeaway{A single global polynomial can","ordinates:\n\\[\np(A)v=p(1)(1,1)^T,\\qquad \\dim\\mathcal K_m(A,v)=1\\quad(m\\ge1).\n\\]\n\\vfig{krylov}\nThe direct sum of the two local Krylov spans is all of $\\QQ^2$. It permits independent coefficients in the two packets. The global Krylov space does not. This is why the manuscript states an inclusion, not an unconditional equality.\n\\head{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=I_2$. Both changes enlarge the addressing power, but in different ways.\n\\head{When invariance is only approximate}\nIf $A=D+K$ with $D$ packet diagonal, then\n\\[\n\\|A^j-D^j\\|\\le jM^{j-1}\\|K\\|,\n\\qquad M=\\max\\{\\|A\\|,\\|D\\|\\}.\n\\]\nThis controls departure from the decoupled iterate. It is not a claim that the dimension or numerical conditioning of a Krylov basis varies continuously. Near dependence can remain difficult even when the operator error is small.\n\\takeaway{A single global polynomial can lock ot…","he two packets. The global Krylov space does not. This is why the manuscript states an inclusion, not an unconditional equality.\n\\head{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=I_2$. Both changes enlarge the addressing power, but in different ways.\n\\head{When invariance is only approximate}\nIf $A=D+K$ with $D$ packet diagonal, then\n\\[\n\\|A^j-D^j\\|\\le jM^{j-1}\\|K\\|,\n\\qquad M=\\max\\{\\|A\\|,\\|D\\|\\}.\n\\]\nThis controls departure from the decoupled iterate. It is not a claim that the dimension or numerical conditioning of a Krylov basis varies continuously. Near dependence can remain difficult even when the operator error is small.\n\\takeaway{A single global polynomial can lock otherwise invariant packets together. Independence of subspaces is not the same as independence of the allowed actions.}","uality.\n\\head{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=I_2$. Both changes enlarge the addressing power, but in different ways.\n\\head{When invariance is only approximate}\nIf $A=D+K$ with $D$ packet diagonal, then\n\\[\n\\|A^j-D^j\\|\\le jM^{j-1}\\|K\\|,\n\\qquad M=\\max\\{\\|A\\|,\\|D\\|\\}.\n\\]\nThis controls departure from the decoupled iterate. It is not a claim that the dimension or numerical conditioning of a Krylov basis varies continuously. Near dependence can remain difficult even when the operator error is small.\n\\takeaway{A single global polynomial can lock otherwise invariant packets together. Independence of subspaces is not the same as independence of the allowed actions.}","{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=I_2$. Both changes enlarge the addressing power, but in different ways.\n\\head{When invariance is only approximate}\nIf $A=D+K$ with $D$ packet diagonal, then\n\\[\n\\|A^j-D^j\\|\\le jM^{j-1}\\|K\\|,\n\\qquad M=\\max\\{\\|A\\|,\\|D\\|\\}.\n\\]\nThis controls departure from the decoupled iterate. It is not a claim that the dimension or numerical conditioning of a Krylov basis varies continuously. Near dependence can remain difficult even when the operator error is small.\n\\takeaway{A single global polynomial can lock otherwise invariant packets together. Independence of subspaces is not the same as independence of the allowed actions.}","With $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=I_2$. Both changes enlarge the addressing power, but in different ways.\n\\head{When invariance is only approximate}\nIf $A=D+K$ with $D$ packet diagonal, then\n\\[\n\\|A^j-D^j\\|\\le jM^{j-1}\\|K\\|,\n\\qquad M=\\max\\{\\|A\\|,\\|D\\|\\}.\n\\]\nThis controls departure from the decoupled iterate. It is not a claim that the dimension or numerical conditioning of a Krylov basis varies continuously. Near dependence can remain difficult even when the operator error is small.\n\\takeaway{A single global polynomial can lock otherwise invariant packets together. Independence of subspaces is not the same as independence of the allowed actions.}","\\vpage{18}{Measure coupling after normalization}{BLOCK JACOBI AS A GEOMETRIC CHANGE}{}\nLet\n\\[\nA=\\mat{4&1\\\\1&1},\\quad D=\\diag(4,1),\\quad\nB=D^{-1/2}=\\diag(1/2,1).\n\\]\nThen $BAB=I+E=\\mat{1&1/2","1&1},\\quad D=\\diag(4,1),\\quad\nB=D^{-1/2}=\\diag(1/2,1).\n\\]\nThen $BAB=I+E=\\mat{1&1/2\\\\1/2&1}$. Its exact eigenvalues are $1/2$ and $3/2$, giving condition number three.\n\\vfig{preconditioning}\n\\head{The unit circles of the energy norms}\nThe left ellipse is $x^TAx=1$. Writing $x=By$ gives the right ellipse $y^T(BAB)y=1$. The drawing samples these explicitly defined curves; the norm and eigenvalue equalities are checked symbolically.","1/2&1}$. Its exact eigenvalues are $1/2$ and $3/2$, giving condition number three.\n\\vfig{preconditioning}\n\\head{The unit circles of the energy norms}\nThe left ellipse is $x^TAx=1$. Writing $x=By$ gives the right ellipse $y^T(BAB)y=1$. The drawing samples these explicitly defined curves; the norm and eigenvalue equalities are checked symbolically.","Here $\\|E\\|_2=1/2$. The manuscript's general inequality\n\\[\n\\kappa_2(D^{-1/2}AD^{-1/2})\n\\le\\frac{1+\\eta}{1-\\eta},\\qquad \\|E\\|_2\\le\\eta<1,\n\\]\nis attained. The raw off-diagonal entry is one; its meaningful size is one-half only after normalization by the diagonal energy scales.\n\\head{Keep the convergence input separate}\nThe condition bound can be inserted into a conjugate-gradient convergence estimate under that method's positive-definite assumptions. The CG estimate is one of the separately tracked classical inputs. The algebraic normalization theorem does not certify every stopping rule, floating-point iteration, or arbitrary preconditioner.\n\\takeaway{A small raw coupling is not scale-invariant. Normalize aga","Here $\\|E\\|_2=1/2$. The manuscript's general inequality\n\\[\n\\kappa_2(D^{-1/2}AD^{-1/2})\n\\le\\frac{1+\\eta}{1-\\eta},\\qquad \\|E\\|_2\\le\\eta<1,\n\\]\nis attained. The raw off-diagonal entry is one; its meaningful size is one-half only after normalization by the diagonal energy scales.\n\\head{Keep the convergence input separate}\nThe condition bound can be inserted into a conjugate-gradient convergence estimate under that method's positive-definite assumptions. The CG estimate is one of the separately tracked classical inputs. The algebraic normalization theorem does not certify every stopping rule, floating-point iteration, or arbitrary preconditioner.\n\\takeaway{A small raw coupling is not scale-invariant. Normalize against the diagonal blocks before deciding whether the coupling is weak enough for a condition bound.}","\\vpage{19}{Twelve small blocks, one coherent error}{DISCOVERY AND PERTURBATION CERTIFICATES}{}\nSplit $\\QQ^4$ into scalar packets and let\n\\[\nA=I+E,\\qquad E=\\frac1{100}(\\ones\\ones^T-I).\n\\]\nDropping every off-diagonal block leaves $I$. Each discarded entry is only $1/100$, but the total error is not bounded by that single-entry size.\n\\vfig{block_error}\n\\head{Two norms answer different questions}\nThere are twelve omitted entries, so $\\|E\\|_F^2=12/10^4$. The all-one vector is an eigenvector with eigenvalue $3/100$, while the three orthogonal directions have eigenvalue $-1/100$. Hence $\\|E\\|_2=3/100$.","e coherent error}{DISCOVERY AND PERTURBATION CERTIFICATES}{}\nSplit $\\QQ^4$ into scalar packets and let\n\\[\nA=I+E,\\qquad E=\\frac1{100}(\\ones\\ones^T-I).\n\\]\nDropping every off-diagonal block leaves $I$. Each discarded entry is only $1/100$, but the total error is not bounded by that single-entry size.\n\\vfig{block_error}\n\\head{Two norms answer different questions}\nThere are twelve omitted entries, so $\\|E\\|_F^2=12/10^4$. The all-one vector is an eigenvector with eigenvalue $3/100$, while the three orthogonal directions have eigenvalue $-1/100$. Hence $\\|E\\|_2=3/100$.","into scalar packets and let\n\\[\nA=I+E,\\qquad E=\\frac1{100}(\\ones\\ones^T-I).\n\\]\nDropping every off-diagonal block leaves $I$. Each discarded entry is only $1/100$, but the total error is not bounded by that single-entry size.\n\\vfig{block_error}\n\\head{Two norms answer different questions}\nThere are twelve omitted entries, so $\\|E\\|_F^2=12/10^4$. The all-one vector is an eigenvector with eigenvalue $3/100$, while the three orthogonal directions have eigenvalue $-1/100$. Hence $\\|E\\|_2=3/100$.","s\\ones^T-I).\n\\]\nDropping every off-diagonal block leaves $I$. Each discarded entry is only $1/100$, but the total error is not bounded by that single-entry size.\n\\vfig{block_error}\n\\head{Two norms answer different questions}\nThere are twelve omitted entries, so $\\|E\\|_F^2=12/10^4$. The all-one vector is an eigenvector with eigenvalue $3/100$, while the three orthogonal directions have eigenvalue $-1/100$. Hence $\\|E\\|_2=3/100$.","Dropping every off-diagonal block leaves $I$. Each discarded entry is only $1/100$, but the total error is not bounded by that single-entry size.\n\\vfig{block_error}\n\\head{Two norms answer different questions}\nThere are twelve omitted entries, so $\\|E\\|_F^2=12/10^4$. The all-one vector is an eigenvector with eigenvalue $3/100$, while the three orthogonal directions have eigenvalue $-1/100$. Hence $\\|E\\|_2=3/100$.","For orthogonal coordinate blocks, the squared Frobenius norms add exactly. Operator norms instead require a suitable bound or further spectral information. The maximum omitted block norm is generally insufficient.\n\\head{Input sensitivity is additional information}\nThe packet perturbation estimate is\n\\[\n\\|(\\widehat A-A)x\\|\\le\\sum_e\\|E_e\\|\\,\\|P_{\\alpha(e)}x\\|.\n\\]\nIt can be much tighter for an input that scarcely occupies an affected source packet. To obtain a uniform operator bound, the input dependence must be bounded uniformly over unit vectors. A certificate for one input is not automatically a certificate for every right-hand side.","\\vpage{20}{Spend error where it is cheapest}{CONTINUOUS AND CLIPPED PRECISION}{}\nTake sensitivities $s=(1,2,4)$, cost coefficients $c=(1,1,2)$, and budget $\\tau=1$. Minimizing $\\sum_ec_e\\log(1/\\epsilon_e)$ subject to $\\sum_es_e\\epsilon_e=1$ gives\n\\[\n\\epsilon_e=\\frac{c_e}{4s_e},\\qquad\n\\epsilon=(1/4,1/8,1/8).\n\\]\n\\vfig{precision}\n\\head{The budget is weighted}\nThe three contributions to the error allowance are $(1/4,1/4,1/2)$. Equal cost coefficients give equal weighted contributions, not equal tolerances. The third edge has twice the cost coefficient of the second, which offsets its doubled sensitivity in this example.","Spend error where it is cheapest}{CONTINUOUS AND CLIPPED PRECISION}{}\nTake sensitivities $s=(1,2,4)$, cost coefficients $c=(1,1,2)$, and budget $\\tau=1$. Minimizing $\\sum_ec_e\\log(1/\\epsilon_e)$ subject to $\\sum_es_e\\epsilon_e=1$ gives\n\\[\n\\epsilon_e=\\frac{c_e}{4s_e},\\qquad\n\\epsilon=(1/4,1/8,1/8).\n\\]\n\\vfig{precision}\n\\head{The budget is weighted}\nThe three contributions to the error allowance are $(1/4,1/4,1/2)$. Equal cost coefficients give equal weighted contributions, not equal tolerances. The third edge has twice the cost coefficient of the second, which offsets its doubled sensitivity in this example.","Now impose upper tolerances $m=(1/10,1/2,1/2)$. The first edge must stop at its tighter cap. With multiplier $\\lambda=10/3$, the clipped solution is\n\\[\n\\epsilon_e=\\min\\{m_e,c_e/(\\lambda s_e)\\},\\qquad\n\\epsilon=(1/10,3/20,3/20).\n\\]\nIts weighted contributions are $(1/10,3/10,3/5)$ and still sum to one. The checker verifies the active cap, the multiplier equations, and the budget exactly.\n\\head{From tolerance to a real data type}\nThe continuous objective is a model. Actual kernels offer discrete precisions and may use different accumulation types. A tolerance below one does not identify a particular hardware format. The finite option program on the next page is the correct next step once admissible costs and error bounds have been supplied","Now impose upper tolerances $m=(1/10,1/2,1/2)$. The first edge must stop at its tighter cap. With multiplier $\\lambda=10/3$, the clipped solution is\n\\[\n\\epsilon_e=\\min\\{m_e,c_e/(\\lambda s_e)\\},\\qquad\n\\epsilon=(1/10,3/20,3/20).\n\\]\nIts weighted contributions are $(1/10,3/10,3/5)$ and still sum to one. The checker verifies the active cap, the multiplier equations, and the budget exactly.\n\\head{From tolerance to a real data type}\nThe continuous objective is a model. Actual kernels offer discrete precisions and may use different accumulation types. A tolerance below one does not identify a particular hardware format. The finite option program on the next page is the correct next step once admissible costs and error bounds have been supplied.\n\\takeaway{Precision allocation is an optimization over certified sensitivit","Now impose upper tolerances $m=(1/10,1/2,1/2)$. The first edge must stop at its tighter cap. With multiplier $\\lambda=10/3$, the clipped solution is\n\\[\n\\epsilon_e=\\min\\{m_e,c_e/(\\lambda s_e)\\},\\qquad\n\\epsilon=(1/10,3/20,3/20).\n\\]\nIts weighted contributions are $(1/10,3/10,3/5)$ and still sum to one. The checker verifies the active cap, the multiplier equations, and the budget exactly.\n\\head{From tolerance to a real data type}\nThe continuous objective is a model. Actual kernels offer discrete precisions and may use different accumulation types. A tolerance below one does not identify a particular hardware format. The finite option program on the next page is the correct next step once admissible costs and error bounds have been supplied.\n\\takeaway{Precision allocation is an optimization over certified sensitivities and costs. It does not infer error bounds from a data-type name alone.}","\\vpage{21}{An exact budget is not a greedy choice}{THREE EDGES, TWENTY-SEVEN ASSIGNMENTS}{}\nEach edge has three allowed $(\\text{integer weight},\\text{cost})$ options:\n\\[\n\\begin{aligned}\n1 &: (0,7),(2,3),(4,1),","\\vpage{21}{An exact budget is not a greedy choice}{THREE EDGES, TWENTY-SEVEN ASSIGNMENTS}{}\nEach edge has three allowed $(\\text{integer weight},\\text{cost})$ options:\n\\[\n\\begin{aligned}\n1 &: (0,7),(2,3),(4,1),\\\\\n2 &: (0,6),(1,4),(3,1),\\\\\n3 &: (0,9),(2,5),(5,1).\n\\end{aligned}\n\\]\nThere are exactly $3^3=27$ full assignments. The table stores the minimum cost at each \\emph{exact} weight; columns zero through eight are displayed.\n\\vfig{discrete}\n\\head{Read the final row correctly}\nAt budget four, minimize across columns zero through four. The optimum costs 14: choose weights $(2,0,2)$ with costs $(3,6,5)$. The zero-weight choice on edge two is its first, more expensive option. A locally cheap choice can consume too much of the shared budget.","3 &: (0,9),(2,5),(5,1).\n\\end{aligned}\n\\]\nThere are exactly $3^3=27$ full assignments. The table stores the minimum cost at each \\emph{exact} weight; columns zero through eight are displayed.\n\\vfig{discrete}\n\\head{Read the final row correctly}\nAt budget four, minimize across columns zero through four. The optimum costs 14: choose weights $(2,0,2)$ with costs $(3,6,5)$. The zero-weight choice on edge two is its first, more expensive option. A locally cheap choice can consume too much of the shared budget.","The checker compares the recurrence with complete enumeration for every budget from zero through twelve. Infeasible states remain $+\\infty$; they are not zero-cost states.\n\\head{Quantization must be conservative}\nSuppose real errors are $3/50$ and $2/25$, and one integer unit represents $1/10$. Rounding down gives two zero weights, even though the true sum is $7/50>1/10$. Rounding up gives a certified upper envelope instead. The budget method is exact for its integers; it cannot repair unsafe integers supplied by a preprocessing step.\n\\takeaway{The table optimizes the declared option set exactly. Accuracy depends separately on the validity and conservativeness of the option weights.}","\\vpage{22}{A penalty can miss the budget optimum}{FIVE PRUNINGS, ONE NONCONVEX FRONTIER}{}\nA root $r$ has children $a,b$, each with two terminal children. Keeping $r$ costs one with error six; splitting it adds overhead one. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. T","\\vpage{22}{A penalty can miss the budget optimum}{FIVE PRUNINGS, ONE NONCONVEX FRONTIER}{}\nA root $r$ has children $a,b$, each with two terminal children. Keeping $r$ costs one with error six; splitting it adds overhead one. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full spl","\\vpage{22}{A penalty can miss the budget optimum}{FIVE PRUNINGS, ONE NONCONVEX FRONTIER}{}\nA root $r$ has children $a,b$, each with two terminal children. Keeping $r$ costs one with error six; splitting it adds overhead one. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence","\\vpage{22}{A penalty can miss the budget optimum}{FIVE PRUNINGS, ONE NONCONVEX FRONTIER}{}\nA root $r$ has children $a,b$, each with two terminal children. Keeping $r$ costs one with error six; splitting it adds overhead one. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","can miss the budget optimum}{FIVE PRUNINGS, ONE NONCONVEX FRONTIER}{}\nA root $r$ has children $a,b$, each with two terminal children. Keeping $r$ costs one with error six; splitting it adds overhead one. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","UNINGS, ONE NONCONVEX FRONTIER}{}\nA root $r$ has children $a,b$, each with two terminal children. Keeping $r$ costs one with error six; splitting it adds overhead one. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","ildren. Keeping $r$ costs one with error six; splitting it adds overhead one. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","ne. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","ier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","rror})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","(7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","\\vpage{23}{Recompile the changed part, not everything}{LOCAL DEPENDENCIES AND STRICT MARGINS}{}\nUse the same tree with $\\lambda=2$. Increase only the leaf cost at $a_0$ from two to three. The exact subtree values change at $a_0$, its parent $a$, and the root $r$. Every value in the $b$ subtree remains untouched.\n\\vfig{local_update}\n\\head{Why the update is local}\nA dynamic-programming value depends only on the local keep/split data and its children's values. A changed leaf can therefore affect its ancestors but not a disjoint sibling subtree. Recomputing the dependency closure yields the same global optimum as a fresh traversal. In this example, the root value increases from ten to eleven.\n\\head{Strict gaps protect decisions}\nBefore the update, the root keep score is 13 and its split score is 10, giving gap three. If e","\\vpage{23}{Recompile the changed part, not everything}{LOCAL DEPENDENCIES AND STRICT MARGINS}{}\nUse the same tree with $\\lambda=2$. Increase only the leaf cost at $a_0$ from two to three. The exact subtree values change at $a_0$, its parent $a$, and the root $r$. Every value in the $b$ subtree remains untouched.\n\\vfig{local_update}\n\\head{Why the update is local}\nA dynamic-programming value depends only on the local keep/split data and its children's values. A changed leaf can therefore affect its ancestors but not a disjoint sibling subtree. Recomputing the dependency closure yields the same global optimum as a fresh traversal. In this example, the root value increases from ten to eleven.\n\\head{Strict gaps protect decisions}\nBefore the update, the root keep score is 13 and its split score is 10, giving gap three. If each score changes by less than $3/2$,","\\vpage{23}{Recompile the changed part, not everything}{LOCAL DEPENDENCIES AND STRICT MARGINS}{}\nUse the same tree with $\\lambda=2$. Increase only the leaf cost at $a_0$ from two to three. The exact subtree values change at $a_0$, its parent $a$, and the root $r$. Every value in the $b$ subtree remains untouched.\n\\vfig{local_update}\n\\head{Why the update is local}\nA dynamic-programming value depends only on the local keep/split data and its children's values. A changed leaf can therefore affect its ancestors but not a disjoint sibling subtree. Recomputing the dependency closure yields the same global optimum as a fresh traversal. In this example, the root value increases from ten to eleven.\n\\head{Strict gaps protect decisions}\nBefore the update, the root keep score is 13 and its split score is 10, giving gap three. If each score changes by less than $3/2$, their strict ordering is retained. Allowing exactly $3/2$ in","\\vpage{23}{Recompile the changed part, not everything}{LOCAL DEPENDENCIES AND STRICT MARGINS}{}\nUse the same tree with $\\lambda=2$. Increase only the leaf cost at $a_0$ from two to three. The exact subtree values change at $a_0$, its parent $a$, and the root $r$. Every value in the $b$ subtree remains untouched.\n\\vfig{local_update}\n\\head{Why the update is local}\nA dynamic-programming value depends only on the local keep/split data and its children's values. A changed leaf can therefore affect its ancestors but not a disjoint sibling subtree. Recomputing the dependency closure yields the same global optimum as a fresh traversal. In this example, the root value increases from ten to eleven.\n\\head{Strict gaps protect decisions}\nBefore the update, the root keep score is 13 and its split score is 10, giving gap three. If each score changes by less than $3/2$, their strict ordering is retained. Allowing exactly $3/2$ in both directions","\\vpage{23}{Recompile the changed part, not everything}{LOCAL DEPENDENCIES AND STRICT MARGINS}{}\nUse the same tree with $\\lambda=2$. Increase only the leaf cost at $a_0$ from two to three. The exact subtree values change at $a_0$, its parent $a$, and the root $r$. Every value in the $b$ subtree remains untouched.\n\\vfig{local_update}\n\\head{Why the update is local}\nA dynamic-programming value depends only on the local keep/split data and its children's values. A changed leaf can therefore affect its ancestors but not a disjoint sibling subtree. Recomputing the dependency closure yields the same global optimum as a fresh traversal. In this example, the root value increases from ten to eleven.\n\\head{Strict gaps protect decisions}\nBefore the update, the root keep score is 13 and its split score is 10, giving gap three. If each score changes by less than $3/2$, their strict ordering is retained. Allowing exactly $3/2$ in both directions can produce a tie","\\vpage{23}{Recompile the changed part, not everything}{LOCAL DEPENDENCIES AND STRICT MARGINS}{}\nUse the same tree with $\\lambda=2$. Increase only the leaf cost at $a_0$ from two to three. The exact subtree values change at $a_0$, its parent $a$, and the root $r$. Every value in the $b$ subtree remains untouched.\n\\vfig{local_update}\n\\head{Why the update is local}\nA dynamic-programming value depends only on the local keep/split data and its children's values. A changed leaf can therefore affect its ancestors but not a disjoint sibling subtree. Recomputing the dependency closure yields the same global optimum as a fresh traversal. In this example, the root value increases from ten to eleven.\n\\head{Strict gaps protect decisions}\nBefore the update, the root keep score is 13 and its split score is 10, giving gap three. If each score changes by less than $3/2$, their strict ordering is retained. Allowing exactly $3/2$ in both directions can produce a tie, so the strict inequali","A changed leaf can therefore affect its ancestors but not a disjoint sibling subtree. Recomputing the dependency closure yields the same global optimum as a fresh traversal. In this example, the root value increases from ten to eleven.\n\\head{Strict gaps protect decisions}\nBefore the update, the root keep score is 13 and its split score is 10, giving gap three. If each score changes by less than $3/2$, their strict ordering is retained. Allowing exactly $3/2$ in both directions can produce a tie, so the strict inequality in the stability statement matters.","a disjoint sibling subtree. Recomputing the dependency closure yields the same global optimum as a fresh traversal. In this example, the root value increases from ten to eleven.\n\\head{Strict gaps protect decisions}\nBefore the update, the root keep score is 13 and its split score is 10, giving gap three. If each score changes by less than $3/2$, their strict ordering is retained. Allowing exactly $3/2$ in both directions can produce a tie, so the strict inequality in the stability statement matters.","The batching statement assumes compatible backend support and independent block tasks. If several blocks accumulate into one target, the reduction or accumulation policy is additional work. Equal matrix dimensions alone do not remove that dependency. Different accumulation types also define different shape classes under the paper's convention.\n\\head{Count the data separately}\nFor three diagonal $2\\times2$ blocks, eight-byte scalars, and $q$ right-hand sides, the stated one-pass model moves\n\\[\nB_{\\rm stream}=8\\cdot3(4+2q+4q)=96+144q\\text{ bytes}.\n\\]\nIt gives 240 bytes at $q=1$ and 672 bytes at $q=4$. It deliberately assumes no reuse across edges. A cache-aware implementation may have another traffic model, but it must state one.","backend support and independent block tasks. If several blocks accumulate into one target, the reduction or accumulation policy is additional work. Equal matrix dimensions alone do not remove that dependency. Different accumulation types also define different shape classes under the paper's convention.\n\\head{Count the data separately}\nFor three diagonal $2\\times2$ blocks, eight-byte scalars, and $q$ right-hand sides, the stated one-pass model moves\n\\[\nB_{\\rm stream}=8\\cdot3(4+2q+4q)=96+144q\\text{ bytes}.\n\\]\nIt gives 240 bytes at $q=1$ and 672 bytes at $q=4$. It deliberately assumes no reuse across edges. A cache-aware implementation may have another traffic model, but it must state one.","blocks accumulate into one target, the reduction or accumulation policy is additional work. Equal matrix dimensions alone do not remove that dependency. Different accumulation types also define different shape classes under the paper's convention.\n\\head{Count the data separately}\nFor three diagonal $2\\times2$ blocks, eight-byte scalars, and $q$ right-hand sides, the stated one-pass model moves\n\\[\nB_{\\rm stream}=8\\cdot3(4+2q+4q)=96+144q\\text{ bytes}.\n\\]\nIt gives 240 bytes at $q=1$ and 672 bytes at $q=4$. It deliberately assumes no reuse across edges. A cache-aware implementation may have another traffic model, but it must state one.","ional work. Equal matrix dimensions alone do not remove that dependency. Different accumulation types also define different shape classes under the paper's convention.\n\\head{Count the data separately}\nFor three diagonal $2\\times2$ blocks, eight-byte scalars, and $q$ right-hand sides, the stated one-pass model moves\n\\[\nB_{\\rm stream}=8\\cdot3(4+2q+4q)=96+144q\\text{ bytes}.\n\\]\nIt gives 240 bytes at $q=1$ and 672 bytes at $q=4$. It deliberately assumes no reuse across edges. A cache-aware implementation may have another traffic model, but it must state one.","x dimensions alone do not remove that dependency. Different accumulation types also define different shape classes under the paper's convention.\n\\head{Count the data separately}\nFor three diagonal $2\\times2$ blocks, eight-byte scalars, and $q$ right-hand sides, the stated one-pass model moves\n\\[\nB_{\\rm stream}=8\\cdot3(4+2q+4q)=96+144q\\text{ bytes}.\n\\]\nIt gives 240 bytes at $q=1$ and 672 bytes at $q=4$. It deliberately assumes no reuse across edges. A cache-aware implementation may have another traffic model, but it must state one.","Given operation capacity $P$ and bandwidth $W$, $T\\ge\\max(F/P,B/W)$ is a lower bound, not an achieved runtime. Launch overhead, imperfect overlap, and utilization can make the actual time much larger.\n\\takeaway{Grouping can remove launches while leaving the same products and data movement. Arithmetic, traffic, and scheduling are related constraints, not one interchangeable cost.}","\\vpage{25}{The first win comes after the tie}{SETUP AND REUSE IN ONE LINEAR MODEL}{}\nSuppose the baseline costs seven units per use. A candidate costs 30 units to set up and two per use. These are synthetic constants chosen to expose the exact threshold:\n\\[\nT_b(R)=7R,\\qquad T_c(R)=30+2R.\n\\]\n\\vfig{amortization}\nAt $R=6$, both cost 42. The candidate becomes strictly faster at $R=7$, where it costs 44 instead of 49. The minimum profitable integer reuse count is\n\\[\n\\left\\lfloor\\frac{30}{7-2}\\right\\rfloor+1=7.\n\\]\nReplacing the floor-plus-one rule with a ceiling would be wrong when the ratio is already an integer and strict improvement is required.\n\\head{Which costs belong to setup?}\nDiscovery, compilation, host/device transfer, graph capture, and factor preparation can have different reuse lifetimes. If a matrix changes but its shape does not, a captured schedule might remain reusable while its numerical factors do not. A single setup number is meaningful only after those lifetim","\\vpage{25}{The first win comes after the tie}{SETUP AND REUSE IN ONE LINEAR MODEL}{}\nSuppose the baseline costs seven units per use. A candidate costs 30 units to set up and two per use. These are synthetic constants chosen to expose the exact threshold:\n\\[\nT_b(R)=7R,\\qquad T_c(R)=30+2R.\n\\]\n\\vfig{amortization}\nAt $R=6$, both cost 42. The candidate becomes strictly faster at $R=7$, where it costs 44 instead of 49. The minimum profitable integer reuse count is\n\\[\n\\left\\lfloor\\frac{30}{7-2}\\right\\rfloor+1=7.\n\\]\nReplacing the floor-plus-one rule with a ceiling would be wrong when the ratio is already an integer and strict improvement is required.\n\\head{Which costs belong to setup?}\nDiscovery, compilation, host/device transfer, graph capture, and factor preparation can have different reuse lifetimes. If a matrix changes but its shape does not, a captured schedule might remain reusable while its numerical factors do not. A single setup number is meaningful only after those lifetimes are fixed.","ONE LINEAR MODEL}{}\nSuppose the baseline costs seven units per use. A candidate costs 30 units to set up and two per use. These are synthetic constants chosen to expose the exact threshold:\n\\[\nT_b(R)=7R,\\qquad T_c(R)=30+2R.\n\\]\n\\vfig{amortization}\nAt $R=6$, both cost 42. The candidate becomes strictly faster at $R=7$, where it costs 44 instead of 49. The minimum profitable integer reuse count is\n\\[\n\\left\\lfloor\\frac{30}{7-2}\\right\\rfloor+1=7.\n\\]\nReplacing the floor-plus-one rule with a ceiling would be wrong when the ratio is already an integer and strict improvement is required.\n\\head{Which costs belong to setup?}\nDiscovery, compilation, host/device transfer, graph capture, and factor preparation can have different reuse lifetimes. If a matrix changes but its shape does not, a captured schedule might remain reusable while its numerical factors do not. A single setup number is meaningful only after those lifetimes are fixed.","\\vpage{26}{Same output before comparing times}{A BENCHMARK ROW IS A CONTRACT}{}\nConsider four synthetic timing rows for a product. Row B returns factors, while row A returns an explicit matrix. A ratio $12/2=6$ does not compare two implementations of the same returned object. Adding the decoding cost changes the candidate to row C.\n\\vfig{contract_match}\n\\head{What a fair match retains}\nThe input values, dimensions, layout, source precision, correctness reference, returned output, and timing scope must agree. Row D changes the input precision; that can be a useful accuracy tradeoff, but it is not the same case merely because the output has the same shape.","\\vpage{27}{Close the loop on the running example}{FACTORS, APPLICATION, AND AN INDEPENDENT REFERENCE}{}\nReturn to the two operators from \\pagerefV{9}. The exact product representation has diagonal part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization","\\vpage{27}{Close the loop on the running example}{FACTORS, APPLICATION, AND AN INDEPENDENT REFERENCE}{}\nReturn to the two operators from \\pagerefV{9}. The exact product representation has diagonal part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growt","\\vpage{27}{Close the loop on the running example}{FACTORS, APPLICATION, AND AN INDEPENDENT REFERENCE}{}\nReturn to the two operators from \\pagerefV{9}. The exact product representation has diagonal part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growth and decide whether recompression is useful. If","\\vpage{27}{Close the loop on the running example}{FACTORS, APPLICATION, AND AN INDEPENDENT REFERENCE}{}\nReturn to the two operators from \\pagerefV{9}. The exact product representation has diagonal part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growth and decide whether recompression is useful. If approximation is introduced, transport its error throug","xample}{FACTORS, APPLICATION, AND AN INDEPENDENT REFERENCE}{}\nReturn to the two operators from \\pagerefV{9}. The exact product representation has diagonal part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growth and decide whether recompression is useful. If approximation is introduced, transport its error through the later operators rather than reporting only a local singular tail.","ENDENT REFERENCE}{}\nReturn to the two operators from \\pagerefV{9}. The exact product representation has diagonal part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growth and decide whether recompression is useful. If approximation is introduced, transport its error through the later operators rather than reporting only a local singular tail.","perators from \\pagerefV{9}. The exact product representation has diagonal part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growth and decide whether recompression is useful. If approximation is introduced, transport its error through the later operators rather than reporting only a local singular tail.","diagonal part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growth and decide whether recompression is useful. If approximation is introduced, transport its error through the later operators rather than reporting only a local singular tail.","part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growth and decide whether recompression is useful. If approximation is introduced, transport its error through the later operators rather than reporting only a local singular tail.","Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growth and decide whether recompression is useful. If approximation is introduced, transport its error through the later operators rather than reporting only a local singular tail.","\\subsection*{A low rank action and its inverse}\nSet\n\\[\nD=\\diag(2,3,4,5),\\qquad\nu=(1,0,1,0)^T,\\qquad v=(1,1,0,0)^T,\n\\]\nand define $A=D+uv^T$. Its entries are shown in the figure. The same operator can be stored by its diagonal and two thin vectors rather than by every matrix entry.\nFor $x=(1,2,3,4)^T$, first form $v^Tx=3$. Then\n\\[\nDx=(2,6,12,20)^T,\\qquad u(v^Tx)=(3,0,3,0)^T,\n\\]\nso $Ax=(5,6,15,20)^T$. Only the first and third output coordinates receive the correction.","The inverse starts with the same representation. Write $y=Ax$. Since $D$ is invertible,\n\\[\nD^{-1}u=(1/2,0,1/4,0)^T,\\qquad\n1+v^TD^{-1}u=3/2\\ne0.\n\\]\nThe scalar Woodbury core is therefore invertible. Its correction coefficient is\n\\[\n\\frac{v^TD^{-1}y}{1+v^TD^{-1}u}\n=\\frac{9/2}{3/2}=3.\n\\]\nSubtracting that correction recovers the original input:\n\\[\nD^{-1}y-3D^{-1}u\n=(5/2,2,15/4,4)^T-(3/2,0,3/4,0)^T=x.\n\\]","\\begin{interpretation}[title={What the output contract preserves}]\nThe diagonal and thin vectors use twelve scalar slots, compared with sixteen for the explicit matrix. This small storage count assumes the full thin vectors are stored. It does not predict elapsed time. A compressed action returns $Ax$. Materializing all entries of $A$ produces another output and incurs its own work. The general formulas appear in \\cref{thm:exact_evaluation,thm:woodbury}.\n\\end{interpretation}","\\clearpage\n\\subsection*{A budget can select a point that no Lagrangian selects}\nConsider a rooted candidate tree. The root $r$ has children $a,b$, and $a$ has children $a_1,a_2$. Split overheads are zero. Each retained node has the following declared cost and integer error:\n\\begin{center}\n\\begin{tabular}{lrrrrr}\\toprule\nNode & $r$ & $a$ & $b$ & $a_1$ & $a_2$","midrule\n$r$ & 1 & 6","$a,b$ & 3 & 4","$a_1,a_2,b$ & 5 & 1","bottomrule\n\\end{tabular}\n\\end{center}\nFor a Lagrangian multiplier $\\lambda\\ge0$, their objectives are $1+6\\lambda$, $3+4\\lambda$, and $5+\\lambda$. The middle pruning can be optimal only if\n\\[\n3+4\\lambda\\le1+6\\lambda\\quad\\text{and}\\quad\n3+4\\lambda\\le5+\\lambda.\n\\]\nThese inequalities require $\\lambda\\ge1$ and $\\lambda\\le2/3$ simultaneously. There is no such multiplier. The root and full split exchange optimality at $\\lambda=4/5$, where both have objective $29/5$.","abular}\n\\end{center}\nFor a Lagrangian multiplier $\\lambda\\ge0$, their objectives are $1+6\\lambda$, $3+4\\lambda$, and $5+\\lambda$. The middle pruning can be optimal only if\n\\[\n3+4\\lambda\\le1+6\\lambda\\quad\\text{and}\\quad\n3+4\\lambda\\le5+\\lambda.\n\\]\nThese inequalities require $\\lambda\\ge1$ and $\\lambda\\le2/3$ simultaneously. There is no such multiplier. The root and full split exchange optimality at $\\lambda=4/5$, where both have objective $29/5$.","tiplier $\\lambda\\ge0$, their objectives are $1+6\\lambda$, $3+4\\lambda$, and $5+\\lambda$. The middle pruning can be optimal only if\n\\[\n3+4\\lambda\\le1+6\\lambda\\quad\\text{and}\\quad\n3+4\\lambda\\le5+\\lambda.\n\\]\nThese inequalities require $\\lambda\\ge1$ and $\\lambda\\le2/3$ simultaneously. There is no such multiplier. The root and full split exchange optimality at $\\lambda=4/5$, where both have objective $29/5$.","The budget problem still selects that middle pruning. Under the constraint $\\err\\le4$, the root is infeasible. The two feasible costs are three and five, so the minimum is three. Under budget one, only the full split is feasible. Under budget six, the root is cheapest. Thus the budget program adds something that changing $\\lambda$ cannot recover.","The dynamic recurrence gives the same Lagrangian choices without listing prunings. At $a$, it compares $2+4\\lambda$ with $4+\\lambda$. At $r$, it compares $1+6\\lambda$ with $1+F_\\lambda(a)$. Its exactness follows from the finite tree and additive model in \\cref{thm:tree_compiler,thm:budget_compiler}. Overlapping decompositions or nonlocal execution costs need another argument.","bottomrule\n\\end{tabular}\n\\end{center}\nThe all-fast assignment FFF has cost three and error eight. It exceeds budget three. Changing only edge one to A gives AFF, with cost $4+1+1=6$ and error $0+2+1=3$. Changing only edges two and three gives FAA, with cost seven and error five. That choice still exceeds the budget.","The exact recurrence reaches the same result. After edge one, only $(\\err,\\cost)=(5,1)$ and $(0,4)$ are reachable. After edge two, the reachable states are $(7,2)$, $(5,4)$, $(2,5)$, and $(0,7)$. Extending these with either option on edge three gives the final eight assignments. Among states of error at most three, the lowest cost is six.","This article begins with a finite direct sum\n\\[\nX=\\bigoplus_{\\alpha\\in I}X_\\alpha\n\\]\nand a complete family of coordinate projectors\n\\[\nP_\\alpha P_\\beta=\\delta_{\\alpha\\beta}P_\\alpha,\n\\qquad\n\\sum_{\\alpha\\in I}P_\\alpha=I.\n\\]\nAn operator has packet blocks\n\\[\nA_{\\beta\\alpha}=P_\\beta AP_\\alpha.\n\\]\nThe active addresses form a directed graph. Block dimensions, block types, approximation tolerances, and output requirements turn this graph into a computational intermediate representation.","The article separates four sources of gain:\n\\[\n\\text{representation},\n\\qquad\n\\text{device},\n\\qquad\n\\text{kernel},\n\\qquad\n\\text{amortization}.\n\\]\nA single speedup number can mix all four. The benchmark protocol therefore requires case identity, output contract, timing scope, source precision, correctness reference, discovery time, compilation time, and reuse count. The archived rows do not satisfy every field of this ideal contract: source-value hashes and explicit correctness-reference fields are absent; setup, discovery, and reuse metadata are sparse. Recorded ratios are consequently descriptive comparisons under their stated scopes, rather than full-provenance fair matches.","\\begin{definition}[Packet layout]\\label{main:definition:001}\\label{def:packet_layout}\nA packet layout is a finite family of finite dimensional Hilbert spaces\n\\[\n\\Pcal=(X_\\alpha)_{\\alpha\\in I}\n\\]\nwith direct sum\n\\[\nX=\\bigoplus_{\\alpha\\in I}X_\\alpha.\n\\]\nWrite\n\\[\nn_\\alpha=\\dim X_\\alpha,\n\\qquad\nn=\\sum_{\\alpha\\in I}n_\\alpha.\n\\]\nThe coordinate inclusion and projection are $\\iota_\\alpha:X_\\alpha\\to X$ and $\\pi_\\alpha:X\\to X_\\alpha$. The ambient coordinate projector is $P_\\alpha=\\iota_\\alpha\\pi_\\alpha$.\n\\end{definition}","gin{definition}[Packet layout]\\label{main:definition:001}\\label{def:packet_layout}\nA packet layout is a finite family of finite dimensional Hilbert spaces\n\\[\n\\Pcal=(X_\\alpha)_{\\alpha\\in I}\n\\]\nwith direct sum\n\\[\nX=\\bigoplus_{\\alpha\\in I}X_\\alpha.\n\\]\nWrite\n\\[\nn_\\alpha=\\dim X_\\alpha,\n\\qquad\nn=\\sum_{\\alpha\\in I}n_\\alpha.\n\\]\nThe coordinate inclusion and projection are $\\iota_\\alpha:X_\\alpha\\to X$ and $\\pi_\\alpha:X\\to X_\\alpha$. The ambient coordinate projector is $P_\\alpha=\\iota_\\alpha\\pi_\\alpha$.\n\\end{definition}","def:packet_layout}\nA packet layout is a finite family of finite dimensional Hilbert spaces\n\\[\n\\Pcal=(X_\\alpha)_{\\alpha\\in I}\n\\]\nwith direct sum\n\\[\nX=\\bigoplus_{\\alpha\\in I}X_\\alpha.\n\\]\nWrite\n\\[\nn_\\alpha=\\dim X_\\alpha,\n\\qquad\nn=\\sum_{\\alpha\\in I}n_\\alpha.\n\\]\nThe coordinate inclusion and projection are $\\iota_\\alpha:X_\\alpha\\to X$ and $\\pi_\\alpha:X\\to X_\\alpha$. The ambient coordinate projector is $P_\\alpha=\\iota_\\alpha\\pi_\\alpha$.\n\\end{definition}","\\begin{definition}[Packet operator representation]\\label{main:definition:002}\\label{def:packet_ir}\nA finite packet operator representation consists of\n\\[\n\\IR=(\\Pcal,E,(B_{\\beta\\alpha})_{(\\beta,\\alpha)\\in E}),\n\\]\nwhere\n\\[\nE\\subseteq I\\times I\n\\]\nand\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\longrightarrow X_\\beta.\n\\]\nLocal blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation $P_\\beta AP_\\alpha$ denotes either the ambient block or its restriction $X_\\alpha\\to X_\\beta$ according to the displayed domain; $\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha$ is the corresponding ambient extension. Its evaluation is\n\\[\n\\operatorname{eval}(\\IR)\n=\n\\sum_{(\\beta,\\alpha)\\in E}\n\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha,\n\\]\nwhere \\(\\iota_\\beta:X_\\beta\\to X\\) is the coordinate inclusion.\n\\end{definition}","acket operator representation]\\label{main:definition:002}\\label{def:packet_ir}\nA finite packet operator representation consists of\n\\[\n\\IR=(\\Pcal,E,(B_{\\beta\\alpha})_{(\\beta,\\alpha)\\in E}),\n\\]\nwhere\n\\[\nE\\subseteq I\\times I\n\\]\nand\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\longrightarrow X_\\beta.\n\\]\nLocal blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation $P_\\beta AP_\\alpha$ denotes either the ambient block or its restriction $X_\\alpha\\to X_\\beta$ according to the displayed domain; $\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha$ is the corresponding ambient extension. Its evaluation is\n\\[\n\\operatorname{eval}(\\IR)\n=\n\\sum_{(\\beta,\\alpha)\\in E}\n\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha,\n\\]\nwhere \\(\\iota_\\beta:X_\\beta\\to X\\) is the coordinate inclusion.\n\\end{definition}","et_ir}\nA finite packet operator representation consists of\n\\[\n\\IR=(\\Pcal,E,(B_{\\beta\\alpha})_{(\\beta,\\alpha)\\in E}),\n\\]\nwhere\n\\[\nE\\subseteq I\\times I\n\\]\nand\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\longrightarrow X_\\beta.\n\\]\nLocal blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation $P_\\beta AP_\\alpha$ denotes either the ambient block or its restriction $X_\\alpha\\to X_\\beta$ according to the displayed domain; $\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha$ is the corresponding ambient extension. Its evaluation is\n\\[\n\\operatorname{eval}(\\IR)\n=\n\\sum_{(\\beta,\\alpha)\\in E}\n\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha,\n\\]\nwhere \\(\\iota_\\beta:X_\\beta\\to X\\) is the coordinate inclusion.\n\\end{definition}","f\n\\[\n\\IR=(\\Pcal,E,(B_{\\beta\\alpha})_{(\\beta,\\alpha)\\in E}),\n\\]\nwhere\n\\[\nE\\subseteq I\\times I\n\\]\nand\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\longrightarrow X_\\beta.\n\\]\nLocal blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation $P_\\beta AP_\\alpha$ denotes either the ambient block or its restriction $X_\\alpha\\to X_\\beta$ according to the displayed domain; $\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha$ is the corresponding ambient extension. Its evaluation is\n\\[\n\\operatorname{eval}(\\IR)\n=\n\\sum_{(\\beta,\\alpha)\\in E}\n\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha,\n\\]\nwhere \\(\\iota_\\beta:X_\\beta\\to X\\) is the coordinate inclusion.\n\\end{definition}","\\[\nB_{\\beta\\alpha}:X_\\alpha\\longrightarrow X_\\beta.\n\\]\nLocal blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation $P_\\beta AP_\\alpha$ denotes either the ambient block or its restriction $X_\\alpha\\to X_\\beta$ according to the displayed domain; $\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha$ is the corresponding ambient extension. Its evaluation is\n\\[\n\\operatorname{eval}(\\IR)\n=\n\\sum_{(\\beta,\\alpha)\\in E}\n\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha,\n\\]\nwhere \\(\\iota_\\beta:X_\\beta\\to X\\) is the coordinate inclusion.\n\\end{definition}","tensions by inclusion and projection. In particular, the notation $P_\\beta AP_\\alpha$ denotes either the ambient block or its restriction $X_\\alpha\\to X_\\beta$ according to the displayed domain; $\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha$ is the corresponding ambient extension. Its evaluation is\n\\[\n\\operatorname{eval}(\\IR)\n=\n\\sum_{(\\beta,\\alpha)\\in E}\n\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha,\n\\]\nwhere \\(\\iota_\\beta:X_\\beta\\to X\\) is the coordinate inclusion.\n\\end{definition}","\\begin{theorem}[Exact evaluation]\\label{main:theorem:001}\\label{thm:exact_evaluation}\nLet \\(A\\in\\End(X)\\). Set\n\\[\nB_{\\beta\\alpha}=\\pi_\\beta A\\iota_\\alpha\n\\]\nand let \\(E\\) contain exactly the nonzero blocks. Then\n\\[\n\\boxed{\nA=\n\\sum_{(\\beta,\\alpha)\\in E}\nP_\\beta AP_\\alpha.\n}\n\\]\nFor every matrix of right hand sides \\(Z\\),\n\\[\nAZ\n=\n\\sum_{(\\beta,\\alpha)\\in E}\n\\iota_\\beta B_{\\beta\\alpha}\\pi_\\alpha Z.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:001}\nInsert the identity on both sides of \\(A\\):\n\\[\nA\n=\n\\left(\\sum_\\beta P_\\beta\\right)\nA\n\\left(\\sum_\\alpha P_\\alpha\\right)\n=\n\\sum_{\\alpha,\\beta}P_\\beta AP_\\alpha.\n\\]\nThe zero blocks can be removed. Applying the identity to \\(Z\\) gives the second formula.\n\\phantomsection\\label{main:proof:001:end}\\end{proof}","\\begin{definition}[Support graph]\\label{main:definition:003}\\label{def:support_graph}\nThe support graph \\(G_A\\) has vertex set \\(I\\). It has a directed edge\n\\[\n\\alpha\\longrightarrow\\beta\n\\]\nwhen \\(P_\\beta AP_\\alpha\\neq0\\).\n\\end{definition}","\\begin{theorem}[Block composition]\\label{main:theorem:002}\\label{thm:block_composition}\nLet \\(A,B\\in\\End(X)\\). Then\n\\[\n\\boxed{\n(AB)_{\\gamma\\alpha}\n=\n\\sum_{\\beta\\in I}\nA_{\\gamma\\beta}B_{\\beta\\alpha}.\n}\n\\]\nConsequently,\n\\[\n\\supp(AB)\n\\subseteq\n\\supp(A)\\circ\\supp(B),\n\\]\nwhere relational composition is defined by\n\\[\n(\\gamma,\\alpha)\\in E\\circ F\n\\]\nwhen some \\(\\beta\\) satisfies\n\\[\n(\\gamma,\\beta)\\in E,\n\\qquad\n(\\beta,\\alpha)\\in F.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:002}\nInsert the packet identity between \\(A\\) and \\(B\\):\n\\[\nP_\\gamma ABP_\\alpha\n=\n\\sum_\\beta\nP_\\gamma AP_\\beta BP_\\alpha.\n\\]\nA summand can be nonzero only when both block addresses are active.\n\\phantomsection\\label{main:proof:002:end}\\end{proof}","\\begin{corollary}[Adjoint and tensor semantics]\\label{main:corollary:001}\\label{cor:adjoint_tensor}\nThe adjoint representation has blocks\n\\[\n(A^*)_{\\alpha\\beta}=A_{\\beta\\alpha}^*.\n\\]\nIf \\(A\\) and \\(B\\) are packet resolved on \\(X\\) and \\(Y\\), then\n\\[\n(A\\otimes B)_{(\\beta,\\delta),(\\alpha,\\gamma)}\n=\nA_{\\beta\\alpha}\\otimes B_{\\delta\\gamma}.\n\\]\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{main:proof:003}\nBoth identities follow by inserting the corresponding coordinate projectors and using\n\\[\n(P_\\beta AP_\\alpha)^*=P_\\alpha A^*P_\\beta.\n\\]\nThe tensor identity follows from\n\\[\n(P_\\beta\\otimes Q_\\delta)\n(A\\otimes B)\n(P_\\alpha\\otimes Q_\\gamma)\n=\nA_{\\beta\\alpha}\\otimes B_{\\delta\\gamma}.\n\\]\n\\phantomsection\\label{main:proof:003:end}\\end{proof}","\\begin{proposition}[Contract conversion]\\label{main:proposition:001}\\label{prop:contract_conversion}\nSuppose one algorithm returns a compressed representation \\(R\\) and another returns an explicit object \\(Y\\). A direct timing comparison is valid only if\n\\[\n\\operatorname{decode}(R)=Y\n\\]\nand the time required by \\(\\operatorname{decode}\\) is included, or if the downstream task consumes \\(R\\) directly.\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{main:proof:004}\nWithout decoding, the two procedures return different mathematical objects. A timing ratio between different outputs does not compare implementations of the same map. Including the decoding cost gives two procedures with the same output. If the next operation consumes \\(R\\), decoding is not part of the declared computation.\n\\phantomsection\\label{main:proof:004:end}\\end{proof}","\\begin{remark}\\label{main:remark:001}\\label{rem:semantic_boundary}\nA compressed product and an explicit matrix product may both be exact. They still have different contracts. This distinction becomes decisive when materialization costs \\(n^2\\) entries.\n\\end{remark}","\\begin{theorem}[Dense and packet application counts]\\label{main:theorem:003}\\label{thm:packet_counts}\nA dense \\(n\\) by \\(n\\) matrix applied to \\(q\\) right hand sides has modeled cost\n\\[\n\\boxed{\n\\flop_{\\rm dense}=2n^2q.\n}\n\\]\nA packet representation with active edge set \\(E\\) has modeled cost\n\\[\n\\boxed{\n\\flop_{\\rm packet}\n=\n2q\n\\sum_{(\\beta,\\alpha)\\in E}\nn_\\beta n_\\alpha.\n}\n\\]\nIt stores\n\\[\n\\boxed{\nM_{\\rm packet}\n=\n\\sum_{(\\beta,\\alpha)\\in E}\nn_\\beta n_\\alpha\n}\n\\]\nscalar entries.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:005}\nThe active block \\(B_{\\beta\\alpha}\\) is an \\(n_\\beta\\) by \\(n_\\alpha\\) matrix. Applying it to an \\(n_\\alpha\\) by \\(q\\) input costs \\(2n_\\beta n_\\alpha q\\) modeled operations and stores \\(n_\\beta n_\\alpha\\) scalars. Summation over the active addresses gives both formulas.\n\\phantomsection\\label{main:proof:005:end}\\end{proof}","\\begin{corollary}[Equal diagonal packets]\\label{main:corollary:002}\\label{cor:equal_blocks}\nLet $p\\ge1$ and $b\\ge1$ be integers and $n=pb$. Suppose $n$ is divided into $p$ equal diagonal packets of width $b$. Then\n\\[\n\\flop_{\\rm block}=2qnb,\n\\qquad\nM_{\\rm block}=nb.\n\\]\nThe dense-to-block storage reduction factor is the following; the arithmetic reduction factor has the same value when $q\\ge1$:\n\\[\n\\boxed{\n\\frac{n}{b}.\n}\n\\]\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{main:proof:006}\nThere are \\(n/b\\) active diagonal blocks. Each contains \\(b^2\\) entries.\n\\phantomsection\\label{main:proof:006:end}\\end{proof}","\\begin{definition}[Block diagonal plus low rank operator]\\label{main:definition:006}\\label{def:bdlr}\nA block diagonal plus low rank operator has the form\n\\[\nA=D+UV^*,\n\\]\nwhere\n\\[\nD=\\bigoplus_{\\alpha\\in I}D_\\alpha,\n\\qquad\nU,V\\in\\CC^{n\\times r}.\n\\]\nThe abbreviation BDLR will be used below.\n\\end{definition}","\\begin{theorem}[BDLR application and storage]\\label{main:theorem:004}\\label{thm:bdlr_counts}\nFor \\(q\\) right hand sides,\n\\[\nAZ=DZ+U(V^*Z)\n\\]\nhas modeled cost\n\\[\n\\boxed{\n2q\n\\left(\n\\sum_\\alpha n_\\alpha^2+2nr\n\\right).\n}\n\\]\nThe representation stores\n\\[\n\\boxed{\n\\sum_\\alpha n_\\alpha^2+2nr\n}\n\\]\nscalars. For $n=pb$, with integers $p,b\\ge1$, the ideal dense storage reduction factor is the following; the arithmetic reduction factor has the same value when $q\\ge1$:\n\\[\n\\boxed{\n\\frac{n}{b+2r}.\n}\n\\]\n\\end{theorem}","{main:theorem:004}\\label{thm:bdlr_counts}\nFor \\(q\\) right hand sides,\n\\[\nAZ=DZ+U(V^*Z)\n\\]\nhas modeled cost\n\\[\n\\boxed{\n2q\n\\left(\n\\sum_\\alpha n_\\alpha^2+2nr\n\\right).\n}\n\\]\nThe representation stores\n\\[\n\\boxed{\n\\sum_\\alpha n_\\alpha^2+2nr\n}\n\\]\nscalars. For $n=pb$, with integers $p,b\\ge1$, the ideal dense storage reduction factor is the following; the arithmetic reduction factor has the same value when $q\\ge1$:\n\\[\n\\boxed{\n\\frac{n}{b+2r}.\n}\n\\]\n\\end{theorem}","rem:004}\\label{thm:bdlr_counts}\nFor \\(q\\) right hand sides,\n\\[\nAZ=DZ+U(V^*Z)\n\\]\nhas modeled cost\n\\[\n\\boxed{\n2q\n\\left(\n\\sum_\\alpha n_\\alpha^2+2nr\n\\right).\n}\n\\]\nThe representation stores\n\\[\n\\boxed{\n\\sum_\\alpha n_\\alpha^2+2nr\n}\n\\]\nscalars. For $n=pb$, with integers $p,b\\ge1$, the ideal dense storage reduction factor is the following; the arithmetic reduction factor has the same value when $q\\ge1$:\n\\[\n\\boxed{\n\\frac{n}{b+2r}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:007}\nThe block diagonal product costs\n\\[\n2q\\sum_\\alpha n_\\alpha^2.\n\\]\nThe products \\(V^*Z\\) and \\(U(V^*Z)\\) cost \\(2nrq\\) each. Storage consists of the diagonal blocks and the two low rank factors. For equal packets,\n\\[\n\\sum_\\alpha n_\\alpha^2=nb.\n\\]\n\\phantomsection\\label{main:proof:007:end}\\end{proof}","\\begin{definition}[One pass streaming model]\\label{main:definition:007}\\label{def:streaming_model}\nFix a scalar size \\(s\\) bytes. For every active block, the one pass streaming model reads the block once, reads its source packet once, and reads and writes its target packet once. It does not assume cache reuse across edges.\n\\end{definition}","\\begin{proposition}[Streaming byte count]\\label{main:proposition:002}\\label{prop:streaming_bytes}\nUnder \\cref{def:streaming_model}, applying a packet operator to \\(q\\) right hand sides moves\n\\[\n\\boxed{\nB_{\\rm stream}\n=\ns\n\\sum_{(\\beta,\\alpha)\\in E}\n\\left(\n n_\\beta n_\\alpha\n +n_\\alpha q\n +2n_\\beta q\n\\right)\n}\n\\]\nbytes.\n\\end{proposition}","\\begin{definition}[Arithmetic intensity]\\label{main:definition:008}\\label{def:intensity}\nWhen the modeled byte traffic is strictly positive, the modeled arithmetic intensity is\n\\[\n\\Ical\n=\n\\frac{\\flop}{\\bytes}.\n\\]\n\\end{definition}","\\begin{theorem}[Roofline lower bound]\\label{main:theorem:005}\\label{thm:roofline}\nSuppose a device sustains at most $P>0$ modeled operations per second and at most $W>0$ bytes per second. A computation that requires \\(F\\) operations and \\(B\\) bytes satisfies\n\\[\n\\boxed{\nT\\geq\n\\max\\left\\{\n\\frac{F}{P},\n\\frac{B}{W}\n\\right\\}.\n}\n\\]\nFor positive elapsed time and positive byte traffic, its work rate is at most\n\\[\n\\min\\{P,W\\Ical\\}.\n\\]\n\\end{theorem}","heorem}[Roofline lower bound]\\label{main:theorem:005}\\label{thm:roofline}\nSuppose a device sustains at most $P>0$ modeled operations per second and at most $W>0$ bytes per second. A computation that requires \\(F\\) operations and \\(B\\) bytes satisfies\n\\[\n\\boxed{\nT\\geq\n\\max\\left\\{\n\\frac{F}{P},\n\\frac{B}{W}\n\\right\\}.\n}\n\\]\nFor positive elapsed time and positive byte traffic, its work rate is at most\n\\[\n\\min\\{P,W\\Ical\\}.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:009}\nThe operation capacity gives \\(T\\geq F/P\\). The bandwidth capacity gives \\(T\\geq B/W\\). Both restrictions hold at once.\n\\phantomsection\\label{main:proof:009:end}\\end{proof}","The roofline model is a guide rather than an exact execution law \\cite{WilliamsWatermanPatterson2009}. Packet structure changes both \\(F\\) and \\(B\\). An irregular representation may remove arithmetic while lowering intensity enough to become bandwidth or launch limited.","\\begin{corollary}[Capacity threshold]\\label{main:corollary:003}\\label{cor:capacity}\nLet a device have available memory $M\\ge0$ bytes and scalar size $s>0$, and let $n\\ge0$. The matrix representation alone can fit only when\n\\[\nn\\leq\\sqrt{M/s}.\n\\]\nThe equal packet BDLR representation alone fits when\n\\[\nn(b+2r)s\\leq M.\n\\]\n\\end{corollary}","\\begin{proposition}[Materialization cost]\\label{main:proposition:003}\\label{prop:materialization_cost}\nMaterializing \\(D+UV^*\\) as an explicit matrix requires storage for \\(n^2\\) scalars and modeled arithmetic\n\\[\n2n^2r+O(n^2)\n\\]\nwhen the low rank correction is formed by dense multiplication.\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{main:proof:011}\nThe product \\(UV^*\\) multiplies an \\(n\\) by \\(r\\) matrix by an \\(r\\) by \\(n\\) matrix. Adding the diagonal part changes only the lower order term.\n\\phantomsection\\label{main:proof:011:end}\\end{proof}","\\begin{proposition}[Equal block banded count]\\label{main:proposition:004}\\label{prop:block_banded_count}\nLet $p,b\\ge1$ be integers, $n=pb$, and let $p=n/b$ equal packets have width $b$. Suppose a block is active precisely when\n\\[\n\\abs{i-j}\\leq w,\n\\qquad\n0\\leq w\\leq p-1.\n\\]\nThe number of active blocks is\n\\[\n\\boxed{\nN_{p,w}\n=\np(2w+1)-w(w+1).\n}\n\\]\nFor \\(q\\) right hand sides, the modeled application cost and scalar storage are\n\\[\n\\boxed{\n2qb^2N_{p,w}\n\\quad\\text{and}\\quad\nb^2N_{p,w}.\n}\n\\]\n\\end{proposition}","ck banded count]\\label{main:proposition:004}\\label{prop:block_banded_count}\nLet $p,b\\ge1$ be integers, $n=pb$, and let $p=n/b$ equal packets have width $b$. Suppose a block is active precisely when\n\\[\n\\abs{i-j}\\leq w,\n\\qquad\n0\\leq w\\leq p-1.\n\\]\nThe number of active blocks is\n\\[\n\\boxed{\nN_{p,w}\n=\np(2w+1)-w(w+1).\n}\n\\]\nFor \\(q\\) right hand sides, the modeled application cost and scalar storage are\n\\[\n\\boxed{\n2qb^2N_{p,w}\n\\quad\\text{and}\\quad\nb^2N_{p,w}.\n}\n\\]\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{main:proof:012}\nThe diagonal contributes \\(p\\) blocks. At offset \\(k\\), where \\(1\\leq k\\leq w\\), the upper and lower block diagonals contribute \\(2(p-k)\\) blocks. Therefore\n\\[\np+2\\sum_{k=1}^{w}(p-k)\n=\np(2w+1)-w(w+1).\n\\]\nThe cost and storage follow from \\cref{thm:packet_counts}.\n\\phantomsection\\label{main:proof:012:end}\\end{proof}","\\begin{proposition}[Sparse plus low rank application]\\label{main:proposition:005}\\label{prop:sparse_low_rank}\nLet\n\\[\nA=S+UV^*,\n\\]\nwhere \\(S\\) has \\(z\\) stored scalar nonzeros and \\(U,V\\in\\CC^{n\\times r}\\). In the model that charges two operations per stored sparse entry and right hand side, applying \\(A\\) to \\(q\\) right hand sides costs\n\\[\n\\boxed{\n2q(z+2nr)\n}\n\\]\nmodeled operations and stores\n\\[\n\\boxed{\nz+2nr\n}\n\\]\nscalar values, apart from sparse index metadata.\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{main:proof:013}\nThe sparse product costs \\(2zq\\). The two dense thin products cost \\(2nrq\\) each. The stored values are the nonzeros of \\(S\\) and the entries of the two factors.\n\\phantomsection\\label{main:proof:013:end}\\end{proof}","\\begin{theorem}[Circulant spectral representation]\\label{main:theorem:006}\\label{thm:circulant}\nLet $n\\ge1$ be an integer and $c\\in\\CC^n$, with indices taken modulo $n$, and define\n\\[\nC(c)_{jk}=c_{j-k}.\n\\]\nLet \\(F\\) be the unitary discrete Fourier matrix\n\\[\nF_{jk}\n=\n\\frac{1}{\\sqrt n}\n\\exp\\left(-\\frac{2\\pi\\ii jk}{n}\\right).\n\\]\nThen\n\\[\n\\boxed{\nC(c)\n=\nF^*\\diag(\\sqrt n\\,Fc)F.\n}\n\\]\nThe operator is determined by \\(n\\) scalars and can be applied through two Fourier transforms and one pointwise product.\n\\end{theorem}","em}[Circulant spectral representation]\\label{main:theorem:006}\\label{thm:circulant}\nLet $n\\ge1$ be an integer and $c\\in\\CC^n$, with indices taken modulo $n$, and define\n\\[\nC(c)_{jk}=c_{j-k}.\n\\]\nLet \\(F\\) be the unitary discrete Fourier matrix\n\\[\nF_{jk}\n=\n\\frac{1}{\\sqrt n}\n\\exp\\left(-\\frac{2\\pi\\ii jk}{n}\\right).\n\\]\nThen\n\\[\n\\boxed{\nC(c)\n=\nF^*\\diag(\\sqrt n\\,Fc)F.\n}\n\\]\nThe operator is determined by \\(n\\) scalars and can be applied through two Fourier transforms and one pointwise product.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:014}\nFor each frequency \\(k\\), let\n\\[\nv^{(k)}_j\n=\n\\frac{1}{\\sqrt n}\n\\exp\\left(\\frac{2\\pi\\ii jk}{n}\\right).\n\\]\nThen\n\\[\n(C(c)v^{(k)})_j\n=\n\\sum_{r=0}^{n-1}c_rv^{(k)}_{j-r}\n=\n\\left(\n\\sum_{r=0}^{n-1}\nc_r\\exp\\left(-\\frac{2\\pi\\ii rk}{n}\\right)\n\\right)v^{(k)}_j.\n\\]\nThe eigenvalue is \\(\\sqrt n(Fc)_k\\). The vectors \\(v^{(k)}\\) are the columns of \\(F^*\\), which gives the formula.\n\\phantomsection\\label{main:proof:014:end}\\end{proof}","The fast Fourier transform evaluates the two transforms in \\(O(n\\log n)\\) arithmetic for the usual composite lengths \\cite{CooleyTukey1965}. This is an algorithmic factorization of the exact identity in \\cref{thm:circulant}.","\\begin{corollary}[Toeplitz circulant embedding]\\label{main:corollary:004}\\label{cor:toeplitz_embedding}\nLet $n\\ge1$ be an integer and $T\\in\\CC^{n\\times n}$ be Toeplitz, with first column\n\\[\n(c_0,c_1,\\ldots,c_{n-1})^T\n\\]\nand first row\n\\[\n(c_0,r_1,\\ldots,r_{n-1}).\n\\]\nSet\n\\[\ng\n=\n(c_0,c_1,\\ldots,c_{n-1},0,r_{n-1},\\ldots,r_1)^T\n\\in\\CC^{2n}\n\\]\nand append \\(n\\) zeros to \\(x\\in\\CC^n\\). The first \\(n\\) entries of the circulant product \\(C(g)(x,0)^T\\) equal \\(Tx\\).\n\\end{corollary}","lant embedding]\\label{main:corollary:004}\\label{cor:toeplitz_embedding}\nLet $n\\ge1$ be an integer and $T\\in\\CC^{n\\times n}$ be Toeplitz, with first column\n\\[\n(c_0,c_1,\\ldots,c_{n-1})^T\n\\]\nand first row\n\\[\n(c_0,r_1,\\ldots,r_{n-1}).\n\\]\nSet\n\\[\ng\n=\n(c_0,c_1,\\ldots,c_{n-1},0,r_{n-1},\\ldots,r_1)^T\n\\in\\CC^{2n}\n\\]\nand append \\(n\\) zeros to \\(x\\in\\CC^n\\). The first \\(n\\) entries of the circulant product \\(C(g)(x,0)^T\\) equal \\(Tx\\).\n\\end{corollary}","main:corollary:004}\\label{cor:toeplitz_embedding}\nLet $n\\ge1$ be an integer and $T\\in\\CC^{n\\times n}$ be Toeplitz, with first column\n\\[\n(c_0,c_1,\\ldots,c_{n-1})^T\n\\]\nand first row\n\\[\n(c_0,r_1,\\ldots,r_{n-1}).\n\\]\nSet\n\\[\ng\n=\n(c_0,c_1,\\ldots,c_{n-1},0,r_{n-1},\\ldots,r_1)^T\n\\in\\CC^{2n}\n\\]\nand append \\(n\\) zeros to \\(x\\in\\CC^n\\). The first \\(n\\) entries of the circulant product \\(C(g)(x,0)^T\\) equal \\(Tx\\).\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{main:proof:015}\nFor \\(0\\leq j<n\\), the circular convolution sum separates into indices \\(k\\leq j\\), where the coefficient is \\(c_{j-k}\\), and indices \\(k>j\\), where the wrapped coefficient is \\(r_{k-j}\\). The padded coordinates do not contribute. The resulting sum is exactly the \\(j\\)th Toeplitz row applied to \\(x\\).\n\\phantomsection\\label{main:proof:015:end}\\end{proof}","\\begin{theorem}[Kronecker modewise application]\\label{main:theorem:007}\\label{thm:kronecker_apply}\nLet \\(A\\in\\CC^{m\\times m}\\), \\(B\\in\\CC^{n\\times n}\\), and \\(X\\in\\CC^{n\\times m}\\). With column major vectorization,\n\\[\n\\boxed{\n(A\\otimes B)\\operatorname{vec}(X)\n=\n\\operatorname{vec}(BXA^T).\n}\n\\]\nThe right side has modeled cost\n\\[\n\\boxed{\n2n^2m+2nm^2\n}\n\\]\nand stores \\(n^2+m^2\\) factor entries instead of \\(n^2m^2\\) entries.\n\\end{theorem}","n{theorem}[Kronecker modewise application]\\label{main:theorem:007}\\label{thm:kronecker_apply}\nLet \\(A\\in\\CC^{m\\times m}\\), \\(B\\in\\CC^{n\\times n}\\), and \\(X\\in\\CC^{n\\times m}\\). With column major vectorization,\n\\[\n\\boxed{\n(A\\otimes B)\\operatorname{vec}(X)\n=\n\\operatorname{vec}(BXA^T).\n}\n\\]\nThe right side has modeled cost\n\\[\n\\boxed{\n2n^2m+2nm^2\n}\n\\]\nand stores \\(n^2+m^2\\) factor entries instead of \\(n^2m^2\\) entries.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:016}\nThe \\((i,j)\\) entry of \\(BXA^T\\) is\n\\[\n\\sum_{k=1}^{n}\\sum_{\\ell=1}^{m}\nB_{ik}X_{k\\ell}A_{j\\ell}.\n\\]\nThis is the coordinate formula for \\((A\\otimes B)\\operatorname{vec}(X)\\). Multiplying first by \\(B\\) and then by \\(A^T\\) gives the stated count.\n\\phantomsection\\label{main:proof:016:end}\\end{proof}","\\begin{corollary}[Equal tensor factors]\\label{main:corollary:005}\\label{cor:tensor_modewise}\nFor integers $k,d\\ge0$, a product of $k$ square factors of order $d$, acting on a tensor with $d^k$ entries, can be applied by $k$ modewise contractions, reusing intermediate slices, with modeled cost\n\\[\n\\boxed{\n2k d^{k+1}\n}\n\\]\nand factor storage \\(kd^2\\).\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{main:proof:017}\nFor $k=0$ there are no contractions and both modeled work and factor storage are zero. For $k\\ge1$, one mode has $d^{k-1}$ fibers of length $d$. Multiplying each fiber by a \\(d\\) by \\(d\\) matrix costs \\(2d^2\\), so one mode costs \\(2d^{k+1}\\). Sum over the \\(k\\) modes.\n\\phantomsection\\label{main:proof:017:end}\\end{proof}","\\begin{theorem}[Walsh Hadamard recursion]\\label{main:theorem:008}\\label{thm:walsh_hadamard}\nLet \\(N=2^m\\), and define\n\\[\nH_1=[1],\n\\qquad\nH_{2N}\n=\n\\begin{bmatrix}\nH_N&H_N","\\begin{theorem}[Walsh Hadamard recursion]\\label{main:theorem:008}\\label{thm:walsh_hadamard}\nLet \\(N=2^m\\), and define\n\\[\nH_1=[1],\n\\qquad\nH_{2N}\n=\n\\begin{bmatrix}\nH_N&H_N\\\\\nH_N&-H_N\n\\end{bmatrix}.\n\\]\nThen\n\\[\nH_NH_N^T=NI.\n\\]\nThe product \\(H_Nx\\) can be evaluated in exactly \\(N\\log_2N\\) additions or subtractions through the recursion.\n\\end{theorem}","H_N&-H_N\n\\end{bmatrix}.\n\\]\nThen\n\\[\nH_NH_N^T=NI.\n\\]\nThe product \\(H_Nx\\) can be evaluated in exactly \\(N\\log_2N\\) additions or subtractions through the recursion.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:018}\nThe orthogonality identity follows by induction from block multiplication. One recursive stage forms \\(x_0+x_1\\) and \\(x_0-x_1\\), which uses \\(N\\) additions or subtractions, and then applies two transforms of size \\(N/2\\). Thus\n\\[\nT(N)=2T(N/2)+N,\n\\qquad\nT(1)=0,\n\\]\nwhose solution is \\(T(N)=N\\log_2N\\).\n\\phantomsection\\label{main:proof:018:end}\\end{proof}","\\begin{theorem}[Exact BDLR product]\\label{main:theorem:009}\\label{thm:bdlr_product}\nLet\n\\[\nA=D_A+U_AV_A^*,\n\\qquad\nB=D_B+U_BV_B^*\n\\]\nuse the same diagonal packet layout. Then\n\\[\nAB=D_AD_B+U_CV_C^*,\n\\]\nwhere\n\\[\nU_C\n=\n\\begin{bmatrix}\nD_AU_B+U_A(V_A^*U_B)&U_A\n\\end{bmatrix}\n\\]\nand\n\\[\nV_C\n=\n\\begin{bmatrix}\nV_B&D_B^*V_A\n\\end{bmatrix}.\n\\]\nThe unrecompressed correction rank is at most \\(r_A+r_B\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:019}\nExpand the product:\n\\[\nAB\n=\nD_AD_B\n+D_AU_BV_B^*\n+U_AV_A^*D_B\n+U_A(V_A^*U_B)V_B^*.\n\\]\nThe second and fourth terms share the right factor \\(V_B^*\\). The third term equals\n\\[\nU_A(D_B^*V_A)^*.\n\\]\nConcatenation gives the stated factors.\n\\phantomsection\\label{main:proof:019:end}\\end{proof}","\\begin{corollary}[Rank growth in a product chain]\\label{main:corollary:006}\\label{cor:rank_growth}\nLet\n\\[\nA_j=D_j+U_jV_j^*,\n\\qquad\nj=1,\\ldots,m,\n\\]\nshare one diagonal packet layout. Before recompression, the product\n\\[\nA_m\\cdots A_1\n\\]\nhas a BDLR representation whose correction rank is at most\n\\[\n\\boxed{\n\\sum_{j=1}^m r_j.\n}\n\\]\n\\end{corollary}","\\begin{theorem}[Woodbury inversion]\\label{main:theorem:010}\\label{thm:woodbury}\nAssume \\(D\\) and\n\\[\nS=I+V^*D^{-1}U\n\\]\nare invertible. Then\n\\[\n\\boxed{\n(D+UV^*)^{-1}\n=\nD^{-1}\n-D^{-1}US^{-1}V^*D^{-1}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:021}\nMultiply the proposed inverse by \\(D+UV^*\\). The coefficient of \\(V^*D^{-1}\\) is\n\\[\nU-US^{-1}(I+V^*D^{-1}U)=0.\n\\]\nThe remaining term is the identity. The right inverse calculation is the same.\n\\phantomsection\\label{main:proof:021:end}\\end{proof}","The formula and its history are discussed in \\cite{Hager1989}. It reduces one large solve to packet solves with \\(D\\) and one dense solve with the small matrix \\(S\\).","\\begin{corollary}[Inverse norm bound]\\label{main:corollary:007}\\label{cor:inverse_norm}\nUnder the hypotheses of \\cref{thm:woodbury},\n\\[\n\\norm{(D+UV^*)^{-1}}\n\\leq\n\\norm{D^{-1}}\n+\n\\norm{D^{-1}U}\n\\norm{S^{-1}}\n\\norm{V^*D^{-1}}.\n\\]\n\\end{corollary}","\\begin{standardfact}[Eckart Young Mirsky theorem]\\label{main:standardfact:001}\\label{fact:eym}\nIf \\(M\\) has singular values\n\\[\n\\sigma_1\\geq\\sigma_2\\geq\\cdots,\n\\]\npad this finite sequence with zeros. For each integer $k\\ge0$, its truncated singular value decomposition $M_k$ is a best rank-at-most-$k$ approximation. It satisfies\n\\[\n\\norm{M-M_k}_2=\\sigma_{k+1}\n\\]\nand\n\\[\n\\norm{M-M_k}_{\\rm F}\n=\n\\left(\n\\sum_{j>k}\\sigma_j^2\n\\right)^{1/2}.\n\\]\n\\end{standardfact}","This is classical \\cite{EckartYoung1936,Mirsky1960}. The next theorem shows that an \\(n\\) dimensional singular value decomposition is unnecessary when a correction is already factored.","\\begin{theorem}[Small core recompression]\\label{main:theorem:011}\\label{thm:small_core_recompression}\nLet\n\\[\nL\\in\\CC^{m\\times r},\n\\qquad\nR\\in\\CC^{n\\times r}.\n\\]\nTake compact QR factorizations\n\\[\nL=Q_LR_L,\n\\qquad\nR=Q_RR_R,\n\\]\nwhere the columns of \\(Q_L\\) and \\(Q_R\\) are orthonormal. Form the small core\n\\[\nG=R_LR_R^*\n\\]\nand let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions; tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equa","\\begin{theorem}[Small core recompression]\\label{main:theorem:011}\\label{thm:small_core_recompression}\nLet\n\\[\nL\\in\\CC^{m\\times r},\n\\qquad\nR\\in\\CC^{n\\times r}.\n\\]\nTake compact QR factorizations\n\\[\nL=Q_LR_L,\n\\qquad\nR=Q_RR_R,\n\\]\nwhere the columns of \\(Q_L\\) and \\(Q_R\\) are orthonormal. Form the small core\n\\[\nG=R_LR_R^*\n\\]\nand let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions; tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduce","\\begin{theorem}[Small core recompression]\\label{main:theorem:011}\\label{thm:small_core_recompression}\nLet\n\\[\nL\\in\\CC^{m\\times r},\n\\qquad\nR\\in\\CC^{n\\times r}.\n\\]\nTake compact QR factorizations\n\\[\nL=Q_LR_L,\n\\qquad\nR=Q_RR_R,\n\\]\nwhere the columns of \\(Q_L\\) and \\(Q_R\\) are orthonormal. Form the small core\n\\[\nG=R_LR_R^*\n\\]\nand let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions; tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduced QR may retain extra zero si","\\begin{theorem}[Small core recompression]\\label{main:theorem:011}\\label{thm:small_core_recompression}\nLet\n\\[\nL\\in\\CC^{m\\times r},\n\\qquad\nR\\in\\CC^{n\\times r}.\n\\]\nTake compact QR factorizations\n\\[\nL=Q_LR_L,\n\\qquad\nR=Q_RR_R,\n\\]\nwhere the columns of \\(Q_L\\) and \\(Q_R\\) are orthonormal. Form the small core\n\\[\nG=R_LR_R^*\n\\]\nand let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions; tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduced QR may retain extra zero singular direc","\\begin{theorem}[Small core recompression]\\label{main:theorem:011}\\label{thm:small_core_recompression}\nLet\n\\[\nL\\in\\CC^{m\\times r},\n\\qquad\nR\\in\\CC^{n\\times r}.\n\\]\nTake compact QR factorizations\n\\[\nL=Q_LR_L,\n\\qquad\nR=Q_RR_R,\n\\]\nwhere the columns of \\(Q_L\\) and \\(Q_R\\) are orthonormal. Form the small core\n\\[\nG=R_LR_R^*\n\\]\nand let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions; tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduced QR may retain extra zero singular directions without changing the conclusion.\n\\end{theorem}","011}\\label{thm:small_core_recompression}\nLet\n\\[\nL\\in\\CC^{m\\times r},\n\\qquad\nR\\in\\CC^{n\\times r}.\n\\]\nTake compact QR factorizations\n\\[\nL=Q_LR_L,\n\\qquad\nR=Q_RR_R,\n\\]\nwhere the columns of \\(Q_L\\) and \\(Q_R\\) are orthonormal. Form the small core\n\\[\nG=R_LR_R^*\n\\]\nand let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions; tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduced QR may retain extra zero singular directions without changing the conclusion.\n\\end{theorem}","mall_core_recompression}\nLet\n\\[\nL\\in\\CC^{m\\times r},\n\\qquad\nR\\in\\CC^{n\\times r}.\n\\]\nTake compact QR factorizations\n\\[\nL=Q_LR_L,\n\\qquad\nR=Q_RR_R,\n\\]\nwhere the columns of \\(Q_L\\) and \\(Q_R\\) are orthonormal. Form the small core\n\\[\nG=R_LR_R^*\n\\]\nand let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions; tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduced QR may retain extra zero singular directions without changing the conclusion.\n\\end{theorem}","\\[\nL\\in\\CC^{m\\times r},\n\\qquad\nR\\in\\CC^{n\\times r}.\n\\]\nTake compact QR factorizations\n\\[\nL=Q_LR_L,\n\\qquad\nR=Q_RR_R,\n\\]\nwhere the columns of \\(Q_L\\) and \\(Q_R\\) are orthonormal. Form the small core\n\\[\nG=R_LR_R^*\n\\]\nand let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions; tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduced QR may retain extra zero singular directions without changing the conclusion.\n\\end{theorem}","s\n\\[\nL=Q_LR_L,\n\\qquad\nR=Q_RR_R,\n\\]\nwhere the columns of \\(Q_L\\) and \\(Q_R\\) are orthonormal. Form the small core\n\\[\nG=R_LR_R^*\n\\]\nand let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions; tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduced QR may retain extra zero singular directions without changing the conclusion.\n\\end{theorem}","R=Q_RR_R,\n\\]\nwhere the columns of \\(Q_L\\) and \\(Q_R\\) are orthonormal. Form the small core\n\\[\nG=R_LR_R^*\n\\]\nand let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions; tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduced QR may retain extra zero singular directions without changing the conclusion.\n\\end{theorem}","Form the small core\n\\[\nG=R_LR_R^*\n\\]\nand let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions; tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduced QR may retain extra zero singular directions without changing the conclusion.\n\\end{theorem}","and let\n\\[\nG=W\\Sigma Z^*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions; tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduced QR may retain extra zero singular directions without changing the conclusion.\n\\end{theorem}","*\n\\]\nbe an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer $k\\ge0$, retain $\\min\\{k,d\\}$ columns, where $d$ is the number of supplied thin singular directions; tail singular values beyond $d$ are zero. Define\n\\[\nL_k=Q_LW_k\\Sigma_k^{1/2},\n\\qquad\nR_k=Q_RZ_k\\Sigma_k^{1/2}.\n\\]\nThen\n\\[\nL_kR_k^*\n\\]\nis a best rank at most \\(k\\) approximation to \\(LR^*\\). Its spectral and Frobenius errors are the singular value tails in \\cref{fact:eym}. The core dimensions equal the column ranks of the two input factors. A noncompact reduced QR may retain extra zero singular directions without changing the conclusion.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:023}\nThe exact factor product is\n\\[\nLR^*\n=\nQ_LGQ_R^*.\n\\]\nThe two outer maps are isometries on their column spaces. They preserve the nonzero singular values and the spectral and Frobenius norms of the core. For every ambient rank-at-most-$k$ competitor $B$, its compression $Q_L^*BQ_R$ has rank at most $k$, and compression cannot increase either norm. Thus the core best-approximation inequality also lower-bounds the error of every ambient competitor. Truncating the singular value decomposition of \\(G\\) and applying \\cref{fact:eym} gives the result. The compact QR dimensions equal the ranks of the factor column spaces.\n\\phantomsection\\label{main:proof:023:end}\\end{proof}","umn spaces. They preserve the nonzero singular values and the spectral and Frobenius norms of the core. For every ambient rank-at-most-$k$ competitor $B$, its compression $Q_L^*BQ_R$ has rank at most $k$, and compression cannot increase either norm. Thus the core best-approximation inequality also lower-bounds the error of every ambient competitor. Truncating the singular value decomposition of \\(G\\) and applying \\cref{fact:eym} gives the result. The compact QR dimensions equal the ranks of the factor column spaces.\n\\phantomsection\\label{main:proof:023:end}\\end{proof}","\\begin{theorem}[Product chain perturbation]\\label{main:theorem:012}\\label{thm:product_chain_error}\nLet \\(A_1,\\ldots,A_m\\) be exact factors and let\n\\[\n\\widehat A_j=A_j+E_j.\n\\]\nThen\n\\[\n\\boxed{\n\\left\\|\n\\widehat A_m\\cdots\\widehat A_1\n-\nA_m\\cdots A_1\n\\right\\|\n\\leq\n\\sum_{j=1}^m\n\\left(\n\\prod_{i=j+1}^m\\norm{\\widehat A_i}\n\\right)\n\\norm{E_j}\n\\left(\n\\prod_{i=1}^{j-1}\\norm{A_i}\n\\right).\n}\n\\]\nEmpty products are one.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:024}\nUse the telescoping identity\n\\[\n\\widehat A_m\\cdots\\widehat A_1\n-A_m\\cdots A_1\n=\n\\sum_{j=1}^m\n\\widehat A_m\\cdots\\widehat A_{j+1}\nE_j\nA_{j-1}\\cdots A_1.\n\\]\nTake norms term by term.\n\\phantomsection\\label{main:proof:024:end}\\end{proof}","\\begin{theorem}[Certified repeated recompression]\\label{main:theorem:013}\\label{cor:recompression_chain}\nLet\n\\[\nP_0=I,\n\\qquad\nP_j=A_jP_{j-1},\n\\]\nbe exact partial products. Suppose an implemented chain satisfies\n\\[\n\\widehat P_0=I,\n\\qquad\n\\widehat P_j\n=\nA_j\\widehat P_{j-1}+E_j,\n\\]\nwhere \\(E_j\\) is the additive error introduced after the \\(j\\)th multiplication, for example by factor recompression. Then\n\\[\n\\boxed{\n\\norm{\\widehat P_m-P_m}\n\\leq\n\\sum_{j=1}^{m}\n\\left(\n\\prod_{i=j+1}^{m}\\norm{A_i}\n\\right)\n\\norm{E_j}.\n}\n\\]\nIf \\(\\norm{E_j}\\leq\\delta_j\\), the same formula with \\(\\delta_j\\) gives a certified final error.\n\\end{theorem}","n{theorem}[Certified repeated recompression]\\label{main:theorem:013}\\label{cor:recompression_chain}\nLet\n\\[\nP_0=I,\n\\qquad\nP_j=A_jP_{j-1},\n\\]\nbe exact partial products. Suppose an implemented chain satisfies\n\\[\n\\widehat P_0=I,\n\\qquad\n\\widehat P_j\n=\nA_j\\widehat P_{j-1}+E_j,\n\\]\nwhere \\(E_j\\) is the additive error introduced after the \\(j\\)th multiplication, for example by factor recompression. Then\n\\[\n\\boxed{\n\\norm{\\widehat P_m-P_m}\n\\leq\n\\sum_{j=1}^{m}\n\\left(\n\\prod_{i=j+1}^{m}\\norm{A_i}\n\\right)\n\\norm{E_j}.\n}\n\\]\nIf \\(\\norm{E_j}\\leq\\delta_j\\), the same formula with \\(\\delta_j\\) gives a certified final error.\n\\end{theorem}","ad\nP_j=A_jP_{j-1},\n\\]\nbe exact partial products. Suppose an implemented chain satisfies\n\\[\n\\widehat P_0=I,\n\\qquad\n\\widehat P_j\n=\nA_j\\widehat P_{j-1}+E_j,\n\\]\nwhere \\(E_j\\) is the additive error introduced after the \\(j\\)th multiplication, for example by factor recompression. Then\n\\[\n\\boxed{\n\\norm{\\widehat P_m-P_m}\n\\leq\n\\sum_{j=1}^{m}\n\\left(\n\\prod_{i=j+1}^{m}\\norm{A_i}\n\\right)\n\\norm{E_j}.\n}\n\\]\nIf \\(\\norm{E_j}\\leq\\delta_j\\), the same formula with \\(\\delta_j\\) gives a certified final error.\n\\end{theorem}","uppose an implemented chain satisfies\n\\[\n\\widehat P_0=I,\n\\qquad\n\\widehat P_j\n=\nA_j\\widehat P_{j-1}+E_j,\n\\]\nwhere \\(E_j\\) is the additive error introduced after the \\(j\\)th multiplication, for example by factor recompression. Then\n\\[\n\\boxed{\n\\norm{\\widehat P_m-P_m}\n\\leq\n\\sum_{j=1}^{m}\n\\left(\n\\prod_{i=j+1}^{m}\\norm{A_i}\n\\right)\n\\norm{E_j}.\n}\n\\]\nIf \\(\\norm{E_j}\\leq\\delta_j\\), the same formula with \\(\\delta_j\\) gives a certified final error.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:025}\nSet \\(D_j=\\widehat P_j-P_j\\). Then\n\\[\nD_j=A_jD_{j-1}+E_j,\n\\qquad\nD_0=0.\n\\]\nRepeated substitution gives\n\\[\nD_m\n=\n\\sum_{j=1}^{m}\nA_mA_{m-1}\\cdots A_{j+1}E_j.\n\\]\nTaking norms yields the result.\n\\phantomsection\\label{main:proof:025:end}\\end{proof}","\\begin{theorem}[Path support of powers]\\label{main:theorem:014}\\label{thm:path_support}\nFor every positive integer \\(m\\),\n\\[\nP_\\gamma A^mP_\\alpha\n\\]\nis the sum over directed walks (vertices may repeat)\n\\[\n\\alpha=\\alpha_0\n\\longrightarrow\n\\alpha_1\n\\longrightarrow\\cdots\\longrightarrow\n\\alpha_m=\\gamma\n\\]\nin \\(G_A\\), with contribution\n\\[\nA_{\\alpha_m\\alpha_{m-1}}\n\\cdots\nA_{\\alpha_1\\alpha_0}.\n\\]\nIf no such path exists, the block vanishes.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:026}\nInsert the packet identity between every two consecutive factors of \\(A\\). Every nonzero term determines one path of length \\(m\\).\n\\phantomsection\\label{main:proof:026:end}\\end{proof}","\\begin{corollary}[Polynomial support]\\label{main:corollary:008}\\label{cor:polynomial_support}\nIf\n\\[\np(z)=\\sum_{j=0}^dc_jz^j,\n\\]\nthen\n\\[\n\\supp(p(A))\n\\]\nis contained in the union of the support relations of \\(A^j\\) for \\(0\\leq j\\leq d\\).\n\\end{corollary}","\\begin{corollary}[Acyclic transfer nilpotence]\\label{main:corollary:009}\\label{cor:acyclic_nilpotence}\nLet \\(K\\) contain only strict transfer blocks. If its support graph is acyclic and its longest directed path has \\(h\\) edges, then\n\\[\nK^{h+1}=0.\n\\]\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{main:proof:028}\nThere is no directed path of length \\(h+1\\). Every block of \\(K^{h+1}\\) therefore vanishes by \\cref{thm:path_support}.\n\\phantomsection\\label{main:proof:028:end}\\end{proof}","\\begin{definition}[Topological level]\\label{main:definition:009}\\label{def:topological_level}\nLet \\(G\\) be a directed acyclic graph. The first level contains its source vertices. Remove them. The next level contains the new sources. Continue until every vertex has been assigned.\n\\end{definition}","\\begin{theorem}[Level schedule for a triangular solve]\\label{main:theorem:015}\\label{thm:level_schedule}\nLet \\(A=D+K\\) be block lower triangular after a topological ordering. Assume every diagonal block \\(D_\\alpha\\) is invertible. The solve\n\\[\nAx=b\n\\]\ncan be completed in one sequential stage per topological level. Blocks in the same level may be solved independently after all earlier levels are known.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:029}\nThe equation in packet \\(\\beta\\) is\n\\[\nD_\\beta x_\\beta\n=\nb_\\beta\n-\n\\sum_{\\alpha\\to\\beta}K_{\\beta\\alpha}x_\\alpha.\n\\]\nEvery predecessor of \\(\\beta\\) lies in an earlier level. Once those values are available, \\(x_\\beta\\) is obtained from the local solve with \\(D_\\beta\\). Vertices in the same level have no directed edge between them, so their solves are independent.\n\\phantomsection\\label{main:proof:029:end}\\end{proof}","\\begin{theorem}[Inverse support lies in the transitive closure]\\label{main:theorem:016}\\label{thm:inverse_support}\nLet \\(A\\) be an invertible packet operator. Reachability here includes a zero-length path from a packet to itself. If there is no directed path from packet \\(\\alpha\\) to packet \\(\\beta\\) in \\(G_A\\), then\n\\[\n\\boxed{\nP_\\beta A^{-1}P_\\alpha=0.\n}\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:031}\nLet \\(R\\) be the set of packets reachable from \\(\\alpha\\), including \\(\\alpha\\), and set\n\\[\nX_R=\\bigoplus_{\\gamma\\in R}X_\\gamma.\n\\]\nNo active edge leaves \\(R\\), so \\(AX_R\\subseteq X_R\\). The restriction of \\(A\\) to the finite dimensional space \\(X_R\\) is injective. It is therefore onto \\(X_R\\). Hence \\(A^{-1}X_R\\subseteq X_R\\). Since \\(\\beta\\notin R\\), the stated block of \\(A^{-1}\\) vanishes.\n\\phantomsection\\label{main:proof:031:end}\\end{proof}","\\label{main:proof:031}\nLet \\(R\\) be the set of packets reachable from \\(\\alpha\\), including \\(\\alpha\\), and set\n\\[\nX_R=\\bigoplus_{\\gamma\\in R}X_\\gamma.\n\\]\nNo active edge leaves \\(R\\), so \\(AX_R\\subseteq X_R\\). The restriction of \\(A\\) to the finite dimensional space \\(X_R\\) is injective. It is therefore onto \\(X_R\\). Hence \\(A^{-1}X_R\\subseteq X_R\\). Since \\(\\beta\\notin R\\), the stated block of \\(A^{-1}\\) vanishes.\n\\phantomsection\\label{main:proof:031:end}\\end{proof}","\\begin{proposition}[Schur support inclusion]\\label{main:proposition:006}\\label{prop:schur_fill}\nSplit the packets into retained indices \\(R\\) and hidden indices \\(H\\). Suppose the hidden block \\(A_{HH}\\) is invertible. The retained Schur complement is\n\\[\nS\n=\nA_{RR}\n-\nA_{RH}A_{HH}^{-1}A_{HR}.\n\\]\nA retained block $({\\beta},{\\alpha})$ can be active only if it was already active in $A_{RR}$, or if there is a directed walk from $\\alpha$ through hidden packets to $\\beta$. The internal hidden path may have length zero, so entrance and exit may use the same hidden packet.\n\\end{proposition}","pport inclusion]\\label{main:proposition:006}\\label{prop:schur_fill}\nSplit the packets into retained indices \\(R\\) and hidden indices \\(H\\). Suppose the hidden block \\(A_{HH}\\) is invertible. The retained Schur complement is\n\\[\nS\n=\nA_{RR}\n-\nA_{RH}A_{HH}^{-1}A_{HR}.\n\\]\nA retained block $({\\beta},{\\alpha})$ can be active only if it was already active in $A_{RR}$, or if there is a directed walk from $\\alpha$ through hidden packets to $\\beta$. The internal hidden path may have length zero, so entrance and exit may use the same hidden packet.\n\\end{proposition}","tion:006}\\label{prop:schur_fill}\nSplit the packets into retained indices \\(R\\) and hidden indices \\(H\\). Suppose the hidden block \\(A_{HH}\\) is invertible. The retained Schur complement is\n\\[\nS\n=\nA_{RR}\n-\nA_{RH}A_{HH}^{-1}A_{HR}.\n\\]\nA retained block $({\\beta},{\\alpha})$ can be active only if it was already active in $A_{RR}$, or if there is a directed walk from $\\alpha$ through hidden packets to $\\beta$. The internal hidden path may have length zero, so entrance and exit may use the same hidden packet.\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{main:proof:032}\nThe correction block is\n\\[\nP_\\beta A_{RH}A_{HH}^{-1}A_{HR}P_\\alpha.\n\\]\nIt can be nonzero only when \\(A_{HR}P_\\alpha\\) enters a hidden packet, the inverse hidden operator connects hidden packets, and \\(P_\\beta A_{RH}\\) exits to \\(\\beta\\). By \\cref{thm:inverse_support}, the hidden inverse can connect only packets joined by a hidden directed path. Cancellation may still remove an allowed block.\n\\phantomsection\\label{main:proof:032:end}\\end{proof}","\\begin{definition}[Krylov space]\\label{main:definition:010}\\label{def:krylov}\nFor a linear operator \\(A\\), a vector \\(v\\), and a positive integer \\(m\\), define\n\\[\n\\Kcal_m(A,v)\n=\n\\operatorname{span}\n\\{v,Av,\\ldots,A^{m-1}v\\}.\n\\]\n\\end{definition}","\\begin{proposition}[Invariant packet Krylov decomposition]\\label{main:proposition:007}\\label{prop:krylov_invariant}\nAssume\n\\[\n[A,P_\\alpha]=0\n\\]\nfor every packet. If\n\\[\nv=\\sum_\\alpha v_\\alpha,\n\\qquad\nv_\\alpha=P_\\alpha v,\n\\]\nthen\n\\[\nA^jv\n=\n\\sum_\\alpha A_\\alpha^jv_\\alpha\n\\]\nfor every \\(j\\geq0\\), where\n\\[\nA_\\alpha=P_\\alpha AP_\\alpha.\n\\]\nHence\n\\[\n\\Kcal_m(A,v)\n\\subseteq\n\\bigoplus_\\alpha\n\\Kcal_m(A_\\alpha,v_\\alpha).\n\\]\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{main:proof:033}\nCommutation gives\n\\[\nA^jP_\\alpha=P_\\alpha A^j.\n\\]\nApply this identity to each packet component of \\(v\\).\n\\phantomsection\\label{main:proof:033:end}\\end{proof}","\\begin{theorem}[Approximate packet Krylov bound]\\label{main:theorem:017}\\label{thm:approx_krylov}\nWrite\n\\[\nA=D+K,\n\\]\nwhere \\(D\\) is packet diagonal. For every positive integer \\(j\\),\n\\[\n\\boxed{\n\\norm{A^j-D^j}\n\\leq\njM^{j-1}\\norm{K},\n}\n\\]\nwhere\n\\[\nM=\\max\\{\\norm{A},\\norm{D}\\}.\n\\]\nConsequently,\n\\[\n\\dist(A^jv,\\bigoplus_\\alpha\\Kcal_{j+1}(D_\\alpha,v_\\alpha))\n\\leq\njM^{j-1}\\norm{K}\\norm{v}.\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:034}\nUse the telescoping identity\n\\[\nA^j-D^j\n=\n\\sum_{k=0}^{j-1}\nA^{j-1-k}KD^k.\n\\]\nEach term has norm at most \\(M^{j-1}\\norm{K}\\). The vector \\(D^jv\\) lies in the direct sum of local Krylov spaces.\n\\phantomsection\\label{main:proof:034:end}\\end{proof}","\\begin{theorem}[Block Jacobi condition bound]\\label{main:theorem:018}\\label{thm:block_jacobi}\nLet\n\\[\nA=D+K\n\\]\nact on a nonempty finite coordinate space and be Hermitian positive definite with positive definite block diagonal part $D$. Here $D^{-1/2}$ is the actual inverse of the positive square root of $D$. Suppose\n\\[\nE=D^{-1/2}KD^{-1/2}\n\\]\nis Hermitian and\n\\[\n\\norm{E}_2\\leq\\eta<1.\n\\]\nThen\n\\[\n\\boxed{\n\\cond_2(D^{-1/2}AD^{-1/2})\n\\leq\n\\frac{1+\\eta}{1-\\eta}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:035}\nFor every unit vector \\(x\\),\n\\[\n\\abs{x^*Ex}\\leq\\eta.\n\\]\nThe Rayleigh quotient of \\(I+E\\) lies in \\([1-\\eta,1+\\eta]\\). The quotient of the extreme eigenvalue bounds gives the result.\n\\phantomsection\\label{main:proof:035:end}\\end{proof}","\\begin{standardfact}[Preconditioned conjugate gradient convergence]\\label{main:standardfact:002}\\label{fact:cg}\nLet $A$ and $M$ be Hermitian positive definite on a nonempty finite coordinate space, and set\n\\[\n\\kappa\n=\n\\cond_2(M^{-1/2}AM^{-1/2}).\n\\]\nIn exact arithmetic, the energy norm error of preconditioned conjugate gradient satisfies\n\\[\n\\norm{x_m-x_*}_A\n\\leq\n2\n\\left(\n\\frac{\\sqrt\\kappa-1}{\\sqrt\\kappa+1}\n\\right)^m\n\\norm{x_0-x_*}_A.\n\\]\n\\end{standardfact}","\\begin{corollary}[Certified packet preconditioning]\\label{main:corollary:011}\\label{cor:certified_pcg}\nUnder the hypotheses of \\cref{thm:block_jacobi}, the convergence factor in \\cref{fact:cg} is bounded by the value obtained from\n\\[\n\\kappa\n=\n\\frac{1+\\eta}{1-\\eta}.\n\\]\n\\end{corollary}","\\begin{theorem}[Packet perturbation certificate]\\label{main:theorem:019}\\label{thm:packet_error}\nLet\n\\[\nA=\n\\sum_{e\\in E}A_e,\n\\qquad\n\\widehat A\n=\n\\sum_{e\\in E}(A_e+E_e),\n\\]\nwhere edge \\(e\\) starts at packet \\(\\alpha(e)\\), and assume\n\\[\nA_e=A_eP_{\\alpha(e)},\n\\qquad\nE_e=E_eP_{\\alpha(e)}.\n\\]\nThen\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\leq\n\\sum_{e\\in E}\n\\norm{E_e}\n\\norm{P_{\\alpha(e)}x}.\n}\n\\]\nIf\n\\[\n\\norm{E_e}\n\\leq\n\\eps_e\\norm{A_e},\n\\]\nthen\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\leq\n\\sum_{e\\in E}\n\\eps_e\\norm{A_e}\n\\norm{P_{\\alpha(e)}x}.\n}\n\\]\n\\end{theorem}","certificate]\\label{main:theorem:019}\\label{thm:packet_error}\nLet\n\\[\nA=\n\\sum_{e\\in E}A_e,\n\\qquad\n\\widehat A\n=\n\\sum_{e\\in E}(A_e+E_e),\n\\]\nwhere edge \\(e\\) starts at packet \\(\\alpha(e)\\), and assume\n\\[\nA_e=A_eP_{\\alpha(e)},\n\\qquad\nE_e=E_eP_{\\alpha(e)}.\n\\]\nThen\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\leq\n\\sum_{e\\in E}\n\\norm{E_e}\n\\norm{P_{\\alpha(e)}x}.\n}\n\\]\nIf\n\\[\n\\norm{E_e}\n\\leq\n\\eps_e\\norm{A_e},\n\\]\nthen\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\leq\n\\sum_{e\\in E}\n\\eps_e\\norm{A_e}\n\\norm{P_{\\alpha(e)}x}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:037}\nExact packet evaluation gives\n\\[\n(\\widehat A-A)x\n=\n\\sum_eE_eP_{\\alpha(e)}x.\n\\]\nApply the triangle inequality and the operator norm bound to each term.\n\\phantomsection\\label{main:proof:037:end}\\end{proof}","\\begin{standardfact}[Floating point matrix product model]\\label{main:standardfact:003}\\label{fact:rounding}\nFor real $A\\in\\RR^{m\\times p}$ and $B\\in\\RR^{p\\times n}$, consider ordinary dot-product matrix multiplication in uniform precision with unit roundoff $u$, no overflow or underflow, and $pu<1$. Its computed output $\\widehat C$ satisfies the componentwise forward-error bound\n\\[\n|\\widehat C-AB|\\le\\gamma_p|A||B|,\n\\qquad\\gamma_p=\\frac{pu}{1-pu}.\n\\]\nAbsolute values and the inequality are entrywise. Complex arithmetic, fused kernels, mixed accumulation, and tensor kernels require their own stated constants. This fact does not assert one common small input perturbation for every column, and an input-dependent rounded action is not automatically a fixed linear operator.\n\\end{standardfact}","\\begin{corollary}[Mixed precision product chain]\\label{main:corollary:012}\\label{cor:mixed_chain}\nIf each fixed linear factor $A_j$ is replaced by a fixed packetwise linear approximation $\\widehat A_j$, write $E_j=\\widehat A_j-A_j=\\sum_e\\Delta_{j,e}P_{\\alpha(j,e)}$. A uniform certificate is\n\\[\n\\|E_j\\|\\le\\delta_j:=\\sum_e\\|\\Delta_{j,e}\\|\\,\\|P_{\\alpha(j,e)}\\|.\n\\]\nFor orthogonal packet projectors, $\\delta_j\\le\\sum_e\\|\\Delta_{j,e}\\|$. The final operator error is bounded by \\cref{thm:product_chain_error} with these uniform $\\delta_j$. The corresponding vector error is this operator bound multiplied by the input norm.\n\\end{corollary}","r $A_j$ is replaced by a fixed packetwise linear approximation $\\widehat A_j$, write $E_j=\\widehat A_j-A_j=\\sum_e\\Delta_{j,e}P_{\\alpha(j,e)}$. A uniform certificate is\n\\[\n\\|E_j\\|\\le\\delta_j:=\\sum_e\\|\\Delta_{j,e}\\|\\,\\|P_{\\alpha(j,e)}\\|.\n\\]\nFor orthogonal packet projectors, $\\delta_j\\le\\sum_e\\|\\Delta_{j,e}\\|$. The final operator error is bounded by \\cref{thm:product_chain_error} with these uniform $\\delta_j$. The corresponding vector error is this operator bound multiplied by the input norm.\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{main:proof:038}\nTake the supremum over unit inputs in \\cref{thm:packet_error}, using $\\|P_{\\alpha}x\\|\\le\\|P_{\\alpha}\\|\\|x\\|$. Substitute the resulting uniform operator error bounds into \\cref{thm:product_chain_error}; the action estimate follows from submultiplicativity.\n\\phantomsection\\label{main:proof:038:end}\\end{proof}","\\begin{theorem}[Continuous precision allocation]\\label{main:theorem:020}\\label{thm:continuous_precision}\nLet $E$ be a finite nonempty edge set, let $s_e>0$ be sensitivities and $c_e>0$ be costs, and let $\\tau>0$. Consider\n\\[\n\\min_{\\eps_e>0}\n\\sum_e c_e\\log\\frac1{\\eps_e}\n\\]\nsubject to\n\\[\n\\sum_es_e\\eps_e=\\tau.\n\\]\nThe unique minimizer is\n\\[\n\\boxed{\n\\eps_e\n=\n\\frac{\\tau c_e}{s_e\\sum_fc_f}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:039}\nThe Lagrangian derivative gives\n\\[\n-\\frac{c_e}{\\eps_e}+\\lambda s_e=0.\n\\]\nThus\n\\[\n\\eps_e=\\frac{c_e}{\\lambda s_e}.\n\\]\nThe constraint gives\n\\[\n\\lambda=\\frac{\\sum_fc_f}{\\tau}.\n\\]\nThe objective is strictly convex on the positive orthant, so the stationary point is the unique minimizer.\n\\phantomsection\\label{main:proof:039:end}\\end{proof}","\\begin{theorem}[Clipped precision allocation]\\label{main:theorem:021}\\label{thm:clipped_precision}\nRetain the finite nonempty edge set, strictly positive sensitivities and costs, and add strictly positive caps $m_e>0$ with upper bounds\n\\[\n0<\\eps_e\\leq m_e.\n\\]\nAssume\n\\[\n0<\\tau\\leq\\sum_es_em_e.\n\\]\nThere is a unique \\(\\lambda>0\\), except at the maximal tolerance endpoint, such that\n\\[\n\\boxed{\n\\eps_e\n=\n\\min\\left\\{\nm_e,\n\\frac{c_e}{\\lambda s_e}\n\\right\\}\n}\n\\]\nand\n\\[\n\\sum_es_e\\eps_e=\\tau.\n\\]\nThis vector minimizes the same objective over the clipped feasible set.\n\\end{theorem}","m:021}\\label{thm:clipped_precision}\nRetain the finite nonempty edge set, strictly positive sensitivities and costs, and add strictly positive caps $m_e>0$ with upper bounds\n\\[\n0<\\eps_e\\leq m_e.\n\\]\nAssume\n\\[\n0<\\tau\\leq\\sum_es_em_e.\n\\]\nThere is a unique \\(\\lambda>0\\), except at the maximal tolerance endpoint, such that\n\\[\n\\boxed{\n\\eps_e\n=\n\\min\\left\\{\nm_e,\n\\frac{c_e}{\\lambda s_e}\n\\right\\}\n}\n\\]\nand\n\\[\n\\sum_es_e\\eps_e=\\tau.\n\\]\nThis vector minimizes the same objective over the clipped feasible set.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:040}\nThe Karush Kuhn Tucker conditions give the unconstrained form on every nonactive coordinate and equality with \\(m_e\\) on every active upper bound. The weighted sum\n\\[\n\\phi(\\lambda)\n=\n\\sum_es_e\n\\min\\left\\{\nm_e,\n\\frac{c_e}{\\lambda s_e}\n\\right\\}\n\\]\nis continuous and nonincreasing. It decreases from the maximal feasible value to zero. For a strictly submaximal budget, once any coordinate is unsaturated, $\\phi$ is strictly decreasing, which gives a unique multiplier. At maximal budget the only feasible vector is $(m_e)_e$ and every $0<\\lambda\\le\\min_e c_e/(s_em_e)$ produces it. Thus the minimizing vector is unique at this endpoint while its multiplier need not be. Strict convexity gives uniqueness of the minimizer in every case.\n\\phantomsection\\label{main:proof:040:end}\\end{proof}","Karush Kuhn Tucker conditions give the unconstrained form on every nonactive coordinate and equality with \\(m_e\\) on every active upper bound. The weighted sum\n\\[\n\\phi(\\lambda)\n=\n\\sum_es_e\n\\min\\left\\{\nm_e,\n\\frac{c_e}{\\lambda s_e}\n\\right\\}\n\\]\nis continuous and nonincreasing. It decreases from the maximal feasible value to zero. For a strictly submaximal budget, once any coordinate is unsaturated, $\\phi$ is strictly decreasing, which gives a unique multiplier. At maximal budget the only feasible vector is $(m_e)_e$ and every $0<\\lambda\\le\\min_e c_e/(s_em_e)$ produces it. Thus the minimizing vector is unique at this endpoint while its multiplier need not be. Strict convexity gives uniqueness of the minimizer in every case.\n\\phantomsection\\label{main:proof:040:end}\\end{proof}","e\\) on every active upper bound. The weighted sum\n\\[\n\\phi(\\lambda)\n=\n\\sum_es_e\n\\min\\left\\{\nm_e,\n\\frac{c_e}{\\lambda s_e}\n\\right\\}\n\\]\nis continuous and nonincreasing. It decreases from the maximal feasible value to zero. For a strictly submaximal budget, once any coordinate is unsaturated, $\\phi$ is strictly decreasing, which gives a unique multiplier. At maximal budget the only feasible vector is $(m_e)_e$ and every $0<\\lambda\\le\\min_e c_e/(s_em_e)$ produces it. Thus the minimizing vector is unique at this endpoint while its multiplier need not be. Strict convexity gives uniqueness of the minimizer in every case.\n\\phantomsection\\label{main:proof:040:end}\\end{proof}","upper bound. The weighted sum\n\\[\n\\phi(\\lambda)\n=\n\\sum_es_e\n\\min\\left\\{\nm_e,\n\\frac{c_e}{\\lambda s_e}\n\\right\\}\n\\]\nis continuous and nonincreasing. It decreases from the maximal feasible value to zero. For a strictly submaximal budget, once any coordinate is unsaturated, $\\phi$ is strictly decreasing, which gives a unique multiplier. At maximal budget the only feasible vector is $(m_e)_e$ and every $0<\\lambda\\le\\min_e c_e/(s_em_e)$ produces it. Thus the minimizing vector is unique at this endpoint while its multiplier need not be. Strict convexity gives uniqueness of the minimizer in every case.\n\\phantomsection\\label{main:proof:040:end}\\end{proof}","\\begin{definition}[Discrete precision option]\\label{main:definition:011}\\label{def:dtype_option}\nFor edge \\(e\\), a discrete precision option \\(k\\) has a measured or modeled cost \\(c_{e,k}\\) and a nonnegative integer error weight \\(w_{e,k}\\). Exactly one option must be selected on every edge.\n\\end{definition}","\\begin{theorem}[Exact discrete precision program]\\label{main:theorem:022}\\label{thm:discrete_precision}\nLet the total integer error budget be \\(B\\). Define\n\\[\nF_j(b)\n\\]\nas the minimum cost for assigning precisions to the first \\(j\\) edges with total weight exactly \\(b\\). Then\n\\[\nF_0(0)=0,\n\\qquad\nF_0(b)=+\\infty\n\\quad\\text{for }b\\neq0.\n\\]\nSet \\(F_{j-1}(u)=+\\infty\\) whenever \\(u<0\\) or the state is unreachable. Then\n\\[\n\\boxed{\nF_j(b)\n=\n\\min_k\n\\left\\{\nF_{j-1}(b-w_{j,k})+c_{j,k}\n\\right\\}.\n}\n\\]\nThe minimum of \\(F_m(b)\\) over \\(0\\leq b\\leq B\\) is the exact minimum cost for the quantized model.\n\\end{theorem}","ct discrete precision program]\\label{main:theorem:022}\\label{thm:discrete_precision}\nLet the total integer error budget be \\(B\\). Define\n\\[\nF_j(b)\n\\]\nas the minimum cost for assigning precisions to the first \\(j\\) edges with total weight exactly \\(b\\). Then\n\\[\nF_0(0)=0,\n\\qquad\nF_0(b)=+\\infty\n\\quad\\text{for }b\\neq0.\n\\]\nSet \\(F_{j-1}(u)=+\\infty\\) whenever \\(u<0\\) or the state is unreachable. Then\n\\[\n\\boxed{\nF_j(b)\n=\n\\min_k\n\\left\\{\nF_{j-1}(b-w_{j,k})+c_{j,k}\n\\right\\}.\n}\n\\]\nThe minimum of \\(F_m(b)\\) over \\(0\\leq b\\leq B\\) is the exact minimum cost for the quantized model.\n\\end{theorem}","total integer error budget be \\(B\\). Define\n\\[\nF_j(b)\n\\]\nas the minimum cost for assigning precisions to the first \\(j\\) edges with total weight exactly \\(b\\). Then\n\\[\nF_0(0)=0,\n\\qquad\nF_0(b)=+\\infty\n\\quad\\text{for }b\\neq0.\n\\]\nSet \\(F_{j-1}(u)=+\\infty\\) whenever \\(u<0\\) or the state is unreachable. Then\n\\[\n\\boxed{\nF_j(b)\n=\n\\min_k\n\\left\\{\nF_{j-1}(b-w_{j,k})+c_{j,k}\n\\right\\}.\n}\n\\]\nThe minimum of \\(F_m(b)\\) over \\(0\\leq b\\leq B\\) is the exact minimum cost for the quantized model.\n\\end{theorem}","rror budget be \\(B\\). Define\n\\[\nF_j(b)\n\\]\nas the minimum cost for assigning precisions to the first \\(j\\) edges with total weight exactly \\(b\\). Then\n\\[\nF_0(0)=0,\n\\qquad\nF_0(b)=+\\infty\n\\quad\\text{for }b\\neq0.\n\\]\nSet \\(F_{j-1}(u)=+\\infty\\) whenever \\(u<0\\) or the state is unreachable. Then\n\\[\n\\boxed{\nF_j(b)\n=\n\\min_k\n\\left\\{\nF_{j-1}(b-w_{j,k})+c_{j,k}\n\\right\\}.\n}\n\\]\nThe minimum of \\(F_m(b)\\) over \\(0\\leq b\\leq B\\) is the exact minimum cost for the quantized model.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:041}\nEvery feasible assignment for the first \\(j\\) edges ends with one option \\(k\\) on edge \\(j\\). Removing it leaves an optimal or suboptimal assignment for the first \\(j-1\\) edges with residual weight \\(b-w_{j,k}\\). Replacing a suboptimal prefix by an optimal one can only lower cost. The recurrence therefore has optimal substructure. Induction on \\(j\\) proves the result.\n\\phantomsection\\label{main:proof:041:end}\\end{proof}","\\begin{proposition}[Exact block discovery certificate]\\label{main:proposition:008}\\label{prop:block_discovery_certificate}\nFor an orthogonal packet system \\((P_\\alpha)\\), define the block diagonal compression\n\\[\n\\Pi_{\\Pcal}(A)\n=\n\\sum_\\alpha P_\\alpha AP_\\alpha.\n\\]\nThen\n\\[\n\\boxed{\n\\norm{A-\\Pi_{\\Pcal}(A)}_{\\rm F}^2\n=\n\\sum_{\\alpha\\neq\\beta}\n\\norm{P_\\beta AP_\\alpha}_{\\rm F}^2.\n}\n\\]\nIn operator norm,\n\\[\n\\boxed{\n\\norm{A-\\Pi_{\\Pcal}(A)}_2\n\\leq\n\\sum_{\\alpha\\neq\\beta}\n\\norm{P_\\beta AP_\\alpha}_2.\n}\n\\]\nThus any proposed packet discovery has an exact finite residual certificate after its projectors are known.\n\\end{proposition}","\\begin{definition}[Candidate representation tree]\\label{main:definition:012}\\label{def:candidate_tree}\nA candidate representation tree is a finite rooted tree. Every node \\(v\\) has\n\\[\n\\kappa_v\\geq0,\n\\qquad\n\\epsilon_v\\geq0,\n\\]\nrepresenting the cost and error of keeping \\(v\\) as one leaf. Every internal node also has a split overhead \\(\\sigma_v\\geq0\\). An admissible pruning contains exactly one selected node on every root to terminal path.\n\\end{definition}","The cost and error of a pruning \\(\\Lcal\\) are\n\\[\n\\cost(\\Lcal)\n=\n\\sum_{v\\in\\Lcal}\\kappa_v\n+\n\\sum_{u\\text{ split}}\\sigma_u\n\\]\nand\n\\[\n\\err(\\Lcal)=\\sum_{v\\in\\Lcal}\\epsilon_v.\n\\]\nThe model can include arithmetic, storage, launch, transfer, and precision costs inside \\(\\kappa_v\\). It remains exact only when the declared totals are additive.","\\begin{theorem}[Exact Lagrangian packet compiler]\\label{main:theorem:023}\\label{thm:tree_compiler}\nFix \\(\\lambda\\geq0\\) and define\n\\[\nJ_\\lambda(\\Lcal)\n=\n\\cost(\\Lcal)\n+\n\\lambda\\err(\\Lcal).\n\\]\nFor every node \\(v\\), set\n\\[\nK_\\lambda(v)\n=\n\\kappa_v+\\lambda\\epsilon_v\n\\]\nand\n\\[\nS_\\lambda(v)\n=\n\\sigma_v\n+\n\\sum_{w\\in\\operatorname{child}(v)}F_\\lambda(w).\n\\]\nFor a terminal node, set\n\\[\nF_\\lambda(v)=K_\\lambda(v).\n\\]\nFor an internal node, the recurrence\n\\[\n\\boxed{\nF_\\lambda(v)\n=\n\\min\\{K_\\lambda(v),S_\\lambda(v)\\}\n}\n\\]\ncomputes the exact minimum Lagrangian objective over every admissible pruning of the subtree rooted at \\(v\\).\n\\end{theorem}","er]\\label{main:theorem:023}\\label{thm:tree_compiler}\nFix \\(\\lambda\\geq0\\) and define\n\\[\nJ_\\lambda(\\Lcal)\n=\n\\cost(\\Lcal)\n+\n\\lambda\\err(\\Lcal).\n\\]\nFor every node \\(v\\), set\n\\[\nK_\\lambda(v)\n=\n\\kappa_v+\\lambda\\epsilon_v\n\\]\nand\n\\[\nS_\\lambda(v)\n=\n\\sigma_v\n+\n\\sum_{w\\in\\operatorname{child}(v)}F_\\lambda(w).\n\\]\nFor a terminal node, set\n\\[\nF_\\lambda(v)=K_\\lambda(v).\n\\]\nFor an internal node, the recurrence\n\\[\n\\boxed{\nF_\\lambda(v)\n=\n\\min\\{K_\\lambda(v),S_\\lambda(v)\\}\n}\n\\]\ncomputes the exact minimum Lagrangian objective over every admissible pruning of the subtree rooted at \\(v\\).\n\\end{theorem}","et\n\\[\nK_\\lambda(v)\n=\n\\kappa_v+\\lambda\\epsilon_v\n\\]\nand\n\\[\nS_\\lambda(v)\n=\n\\sigma_v\n+\n\\sum_{w\\in\\operatorname{child}(v)}F_\\lambda(w).\n\\]\nFor a terminal node, set\n\\[\nF_\\lambda(v)=K_\\lambda(v).\n\\]\nFor an internal node, the recurrence\n\\[\n\\boxed{\nF_\\lambda(v)\n=\n\\min\\{K_\\lambda(v),S_\\lambda(v)\\}\n}\n\\]\ncomputes the exact minimum Lagrangian objective over every admissible pruning of the subtree rooted at \\(v\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:043}\nA pruning of the subtree rooted at \\(v\\) has two possible forms. It keeps \\(v\\), in which case its objective is \\(K_\\lambda(v)\\). Otherwise it splits \\(v\\) and independently chooses an admissible pruning in every child subtree. By induction, the optimal child values are \\(F_\\lambda(w)\\). The split objective is therefore \\(S_\\lambda(v)\\). Taking the smaller value is exact.\n\\phantomsection\\label{main:proof:043:end}\\end{proof}","\\begin{corollary}[Shared compilation for an operator family]\\label{main:corollary:013}\\label{cor:shared_compiler}\nLet operators \\(A^{(1)},\\ldots,A^{(m)}\\) use the same candidate tree. Give operator \\(j\\) a nonnegative reuse weight \\(r_j\\). Replace the local keep cost and error by\n\\[\n\\kappa_v\n=\n\\kappa_v^{\\rm setup}\n+\n\\sum_{j=1}^m r_j\\kappa_v^{(j)}\n\\]\nand\n\\[\n\\epsilon_v\n=\n\\sum_{j=1}^m r_j\\epsilon_v^{(j)}.\n\\]\nThen \\cref{thm:tree_compiler} gives the exact optimal shared pruning for the additive weighted objective.\n\\end{corollary}","\\begin{proposition}[Monotone Lagrangian frontier]\\label{main:proposition:009}\\label{prop:lagrangian_frontier}\nLet \\(0\\leq\\lambda_1<\\lambda_2\\). Choose optimal prunings \\(\\Lcal_1\\) and \\(\\Lcal_2\\) for the two Lagrangian parameters. Then\n\\[\n\\boxed{\n\\err(\\Lcal_2)\n\\leq\n\\err(\\Lcal_1)\n}\n\\]\nand\n\\[\n\\boxed{\n\\cost(\\Lcal_2)\n\\geq\n\\cost(\\Lcal_1).\n}\n\\]\n\\end{proposition}","\\begin{proof}\\phantomsection\\label{main:proof:045}\nOptimality gives\n\\[\n\\cost(\\Lcal_1)+\\lambda_1\\err(\\Lcal_1)\n\\leq\n\\cost(\\Lcal_2)+\\lambda_1\\err(\\Lcal_2)\n\\]\nand\n\\[\n\\cost(\\Lcal_2)+\\lambda_2\\err(\\Lcal_2)\n\\leq\n\\cost(\\Lcal_1)+\\lambda_2\\err(\\Lcal_1).\n\\]\nAdding the inequalities yields\n\\[\n(\\lambda_2-\\lambda_1)\n\\bigl(\n\\err(\\Lcal_2)-\\err(\\Lcal_1)\n\\bigr)\n\\leq0.\n\\]\nThe error ordering follows. Substitution into the first inequality gives the cost ordering.\n\\phantomsection\\label{main:proof:045:end}\\end{proof}","\\begin{corollary}[Supported Pareto points]\\label{main:corollary:014}\\label{cor:supported_pareto}\nFor \\(\\lambda>0\\), every Lagrangian optimum is Pareto optimal for the pair\n\\[\n(\\cost,\\err).\n\\]\nThe quantized budget compiler may recover Pareto points that are not optimal for any Lagrangian parameter.\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{main:proof:046}\nIf another pruning had no larger cost and no larger error, with one strict inequality, its Lagrangian objective would be smaller. This contradicts optimality. A finite nonconvex frontier can contain points that are not supported by a line of slope \\(-\\lambda\\), so the budget formulation is genuinely additional.\n\\phantomsection\\label{main:proof:046:end}\\end{proof}","\\begin{theorem}[Quantized budget dynamic program]\\label{main:theorem:024}\\label{thm:budget_compiler}\nAssume every keep error is a nonnegative integer \\(w_v\\). Let \\(B\\in\\NN\\). For each node \\(v\\), define \\(F_v(b)\\) as the minimum cost of a pruning of the subtree at \\(v\\) with total error exactly \\(b\\). The keep option contributes\n\\[\nF_v(w_v)\\leq\\kappa_v.\n\\]\nThe split option is obtained by min plus convolution of the child tables, followed by addition of \\(\\sigma_v\\). The smallest value among\n\\[\nF_{\\rm root}(b),\n\\qquad\n0\\leq b\\leq B,\n\\]\nis the exact minimum cost among prunings whose quantized error is at most \\(B\\).\n\\end{theorem}",":theorem:024}\\label{thm:budget_compiler}\nAssume every keep error is a nonnegative integer \\(w_v\\). Let \\(B\\in\\NN\\). For each node \\(v\\), define \\(F_v(b)\\) as the minimum cost of a pruning of the subtree at \\(v\\) with total error exactly \\(b\\). The keep option contributes\n\\[\nF_v(w_v)\\leq\\kappa_v.\n\\]\nThe split option is obtained by min plus convolution of the child tables, followed by addition of \\(\\sigma_v\\). The smallest value among\n\\[\nF_{\\rm root}(b),\n\\qquad\n0\\leq b\\leq B,\n\\]\nis the exact minimum cost among prunings whose quantized error is at most \\(B\\).\n\\end{theorem}","}\nAssume every keep error is a nonnegative integer \\(w_v\\). Let \\(B\\in\\NN\\). For each node \\(v\\), define \\(F_v(b)\\) as the minimum cost of a pruning of the subtree at \\(v\\) with total error exactly \\(b\\). The keep option contributes\n\\[\nF_v(w_v)\\leq\\kappa_v.\n\\]\nThe split option is obtained by min plus convolution of the child tables, followed by addition of \\(\\sigma_v\\). The smallest value among\n\\[\nF_{\\rm root}(b),\n\\qquad\n0\\leq b\\leq B,\n\\]\nis the exact minimum cost among prunings whose quantized error is at most \\(B\\).\n\\end{theorem}","the minimum cost of a pruning of the subtree at \\(v\\) with total error exactly \\(b\\). The keep option contributes\n\\[\nF_v(w_v)\\leq\\kappa_v.\n\\]\nThe split option is obtained by min plus convolution of the child tables, followed by addition of \\(\\sigma_v\\). The smallest value among\n\\[\nF_{\\rm root}(b),\n\\qquad\n0\\leq b\\leq B,\n\\]\nis the exact minimum cost among prunings whose quantized error is at most \\(B\\).\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:047}\nThe keep case is immediate. In the split case, an admissible pruning is the union of one admissible pruning from each child. Its error is the sum of the child errors and its cost is the sum of their costs plus \\(\\sigma_v\\). Repeated min plus convolution enumerates every possible distribution of the integer budget across the child subtrees. Induction on the tree proves exactness.\n\\phantomsection\\label{main:proof:047:end}\\end{proof}","\\begin{corollary}[Pseudo polynomial complexity]\\label{main:corollary:015}\\label{cor:budget_complexity}\nIf the maximum number of children is bounded and the budget is \\(B\\), the quantized compiler has pseudo polynomial complexity in the number of tree nodes and \\(B\\). A direct pairwise convolution implementation has quadratic dependence on \\(B\\) at each merge.\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{main:proof:048}\nEvery table has at most \\(B+1\\) entries. Merging two tables by direct convolution examines at most \\((B+1)^2\\) pairs. A bounded number of merges occurs at every node.\n\\phantomsection\\label{main:proof:048:end}\\end{proof}","\\begin{definition}[Decision gap]\\label{main:definition:013}\\label{def:decision_gap}\nAt an internal node \\(v\\), let\n\\[\ng_v\n=\n\\abs{K_\\lambda(v)-S_\\lambda(v)}.\n\\]\nThis is the gap between the keep and split scores after the child subproblems have been solved.\n\\end{definition}","\\begin{theorem}[Compiler stability]\\label{main:theorem:025}\\label{thm:compiler_stability}\nSuppose the solved keep score and split score at every internal node change in absolute value by less than $g_v/2$. These are scores after child subproblems have been solved, rather than unconstrained changes in raw local costs. Then every strict keep or split decision in the original optimal pruning remains unchanged.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:049}\nIf the original keep score is lower by \\(g_v\\), then after perturbation it can increase by less than \\(g_v/2\\), while the split score can decrease by less than \\(g_v/2\\). The order remains strict. The split case is the same. Apply the argument from the leaves upward.\n\\phantomsection\\label{main:proof:049:end}\\end{proof}","\\begin{theorem}[Local recompilation]\\label{main:theorem:026}\\label{thm:local_recompilation}\nSuppose local costs change only inside the subtree rooted at $u$, while ancestor and sibling local models are unchanged. Use correct cached sibling values from the old solve. Then every dynamic programming value outside that subtree and outside the ancestor chain from \\(u\\) to the root remains unchanged. Recomputing the subtree at \\(u\\) and that ancestor chain gives the exact new global optimum.\n\\end{theorem}","pilation]\\label{main:theorem:026}\\label{thm:local_recompilation}\nSuppose local costs change only inside the subtree rooted at $u$, while ancestor and sibling local models are unchanged. Use correct cached sibling values from the old solve. Then every dynamic programming value outside that subtree and outside the ancestor chain from \\(u\\) to the root remains unchanged. Recomputing the subtree at \\(u\\) and that ancestor chain gives the exact new global optimum.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:050}\nA node value depends only on its own local data and the values of its children. Nodes outside the changed subtree and outside the ancestor chain have the same local data and the same child values. Their values remain unchanged. Every affected dependency lies inside the changed subtree or on the unique path from \\(u\\) to the root.\n\\phantomsection\\label{main:proof:050:end}\\end{proof}","\\begin{proposition}[Launch reduction by shape]\\label{main:proposition:010}\\label{prop:shape_launch}\nSuppose \\(B\\) independent active blocks fall into \\(g\\) shape classes, and suppose the backend accepts one batched array or pointer list for each class. Grouped execution then requires at most \\(g\\) launches instead of \\(B\\), without changing the arithmetic or the output.\n\\end{proposition}","\\begin{theorem}[Amortization threshold]\\label{main:theorem:028}\\label{thm:amortization}\nLet \\(s\\geq0\\) be a one time setup cost. Let $t_b,t_c\\ge0$ be the baseline and candidate times per use and let $R\\in\\NN$ be a reuse count. If \\(t_b>t_c\\), the candidate is faster after \\(R\\) uses exactly when\n\\[\nR>\n\\frac{s}{t_b-t_c}.\n\\]\nThe smallest integer reuse count is\n\\[\n\\boxed{\nR_{\\min}\n=\n\\left\\lfloor\n\\frac{s}{t_b-t_c}\n\\right\\rfloor+1.\n}\n\\]\nIf \\(t_b\\leq t_c\\), no positive reuse count amortizes a positive setup cost.\n\\end{theorem}","abel{main:theorem:028}\\label{thm:amortization}\nLet \\(s\\geq0\\) be a one time setup cost. Let $t_b,t_c\\ge0$ be the baseline and candidate times per use and let $R\\in\\NN$ be a reuse count. If \\(t_b>t_c\\), the candidate is faster after \\(R\\) uses exactly when\n\\[\nR>\n\\frac{s}{t_b-t_c}.\n\\]\nThe smallest integer reuse count is\n\\[\n\\boxed{\nR_{\\min}\n=\n\\left\\lfloor\n\\frac{s}{t_b-t_c}\n\\right\\rfloor+1.\n}\n\\]\nIf \\(t_b\\leq t_c\\), no positive reuse count amortizes a positive setup cost.\n\\end{theorem}","\\begin{proof}\\phantomsection\\label{main:proof:053}\nCompare\n\\[\ns+Rt_c\n\\]\nwith\n\\[\nRt_b.\n\\]\nRearranging gives the stated inequality.\n\\phantomsection\\label{main:proof:053:end}\\end{proof}","\\begin{corollary}[Graph replay threshold]\\label{main:corollary:016}\\label{cor:graph_threshold}\nIf graph capture costs $c\\ge0$, ordinary execution costs $t_o\\ge0$, and replay costs $0\\le t_r<t_o$, then capture is profitable when\n\\[\nR>\n\\frac{c}{t_o-t_r}.\n\\]\n\\end{corollary}","\\begin{proof}\\phantomsection\\label{main:proof:054}\nApply \\cref{thm:amortization} with setup cost \\(c\\).\n\\phantomsection\\label{main:proof:054:end}\\end{proof}","\\begin{corollary}[Materialization threshold]\\label{main:corollary:017}\\label{cor:materialization_threshold}\nSuppose a compressed representation costs $t_c\\ge0$ per downstream use, an explicit representation costs $t_e\\ge0$ per use, and one time materialization costs $m\\ge0$, with reuse count $R\\in\\NN$. Materialization is profitable exactly when\n\\[\nR>\n\\frac{m}{t_c-t_e}\n\\]\nand \\(t_c>t_e\\).\n\\end{corollary}","At dimension \\(4096\\), the device resident dense rows gave the values in \\cref{tab:dense_frontier}.","\\midrule\nFP64 & 541.2803 & 253.91 & \\(2.18\\times10^{-15}\\)","FP32 & 18.1309 & 7580.35 & \\(5.76\\times10^{-7}\\)","TF32 & 8.6999 & 15797.75 & \\(2.94\\times10^{-4}\\)","FP16 & 4.0264 & 34134.72 & \\(3.60\\times10^{-4}\\)","BF16 & 4.1482 & 33132.00 & \\(2.88\\times10^{-3}\\)","For an \\(8192\\) by \\(8192\\) equal block operator with block width \\(64\\) and \\(32\\) right hand sides, the batched packet calculation took\n\\[\n0.046032\\text{ ms}.\n\\]\nIt was\n\\[\n14.8919\n\\]\ntimes faster than the recorded device resident dense GPU row with the same recorded case, output, and timing scope and\n\\[\n235.7404\n\\]\ntimes faster than the recorded CPU dense row (resident CPU scope versus device-resident GPU scope). The relative error was\n\\[\n8.17\\times10^{-8}.\n\\]","ator with block width \\(64\\) and \\(32\\) right hand sides, the batched packet calculation took\n\\[\n0.046032\\text{ ms}.\n\\]\nIt was\n\\[\n14.8919\n\\]\ntimes faster than the recorded device resident dense GPU row with the same recorded case, output, and timing scope and\n\\[\n235.7404\n\\]\ntimes faster than the recorded CPU dense row (resident CPU scope versus device-resident GPU scope). The relative error was\n\\[\n8.17\\times10^{-8}.\n\\]","For BDLR application with the same dimension and right hand side count, block width \\(64\\), and rank \\(8\\), CUDA Graph replay took\n\\[\n0.079872\\text{ ms}.\n\\]\nIt was\n\\[\n10.9355\n\\]\ntimes faster than the recorded dense GPU row (graph-replay scope versus device-resident scope) and\n\\[\n153.6527\n\\]\ntimes faster than the recorded CPU dense row (resident CPU scope versus device-resident GPU scope). Capture time was excluded from the replay scope and must be amortized separately.","For hidden permuted blocks at dimension \\(512\\), structure discovery took \\(2.1102\\) ms, dense application took \\(0.0969\\) ms, and packet application took \\(0.0151\\) ms. The exact threshold in \\cref{thm:amortization} is \\(26\\) uses. At dimension \\(1024\\), the corresponding values were \\(5.2726\\), \\(0.1896\\), and \\(0.0241\\) ms, giving \\(32\\) uses.","For one \\(2048\\) dense FP32 product, device resident graph replay took \\(1.9993\\) ms. The end to end pageable path took \\(7.3036\\) ms. The pinned path took \\(6.3084\\) ms. Two serial device resident products took \\(3.6557\\) ms, while the tested two stream path took \\(4.2368\\) ms.","\\begin{itemize}\n\\item At dimension \\(512\\), recursive Strassen multiplication with leaf size \\(128\\) took \\(7.0950\\) ms, while the recorded OpenBLAS dense multiplication took \\(1.9597\\) ms.\n\\item At dimension \\(2048\\), the tested recursive HODLR GPU path took \\(5.6079\\) ms, while dense GPU multiplication took \\(0.0860\\) ms.\n\\item The tested FP32 iterative refinement solve took \\(22.0848\\) ms, while the direct FP64 GPU solve took \\(12.6286\\) ms with a comparable final error.\n\\item The two stream dense path was slower than the serial two product path.\n\\item The GPU Walsh Hadamard transform was slower than the CPU implementation at length \\(65536\\), then became faster at length \\(262144\\).\n\\end{itemize}","\\(512\\), recursive Strassen multiplication with leaf size \\(128\\) took \\(7.0950\\) ms, while the recorded OpenBLAS dense multiplication took \\(1.9597\\) ms.\n\\item At dimension \\(2048\\), the tested recursive HODLR GPU path took \\(5.6079\\) ms, while dense GPU multiplication took \\(0.0860\\) ms.\n\\item The tested FP32 iterative refinement solve took \\(22.0848\\) ms, while the direct FP64 GPU solve took \\(12.6286\\) ms with a comparable final error.\n\\item The two stream dense path was slower than the serial two product path.\n\\item The GPU Walsh Hadamard transform was slower than the CPU implementation at length \\(65536\\), then became faster at length \\(262144\\).\n\\end{itemize}","GPU path took \\(5.6079\\) ms, while dense GPU multiplication took \\(0.0860\\) ms.\n\\item The tested FP32 iterative refinement solve took \\(22.0848\\) ms, while the direct FP64 GPU solve took \\(12.6286\\) ms with a comparable final error.\n\\item The two stream dense path was slower than the serial two product path.\n\\item The GPU Walsh Hadamard transform was slower than the CPU implementation at length \\(65536\\), then became faster at length \\(262144\\).\n\\end{itemize}","GPU multiplication took \\(0.0860\\) ms.\n\\item The tested FP32 iterative refinement solve took \\(22.0848\\) ms, while the direct FP64 GPU solve took \\(12.6286\\) ms with a comparable final error.\n\\item The two stream dense path was slower than the serial two product path.\n\\item The GPU Walsh Hadamard transform was slower than the CPU implementation at length \\(65536\\), then became faster at length \\(262144\\).\n\\end{itemize}","The strengthened release records a deterministic verifier with $37392$ checks in nine groups. It covers packet evaluation, adjoints, composition, structured primitives, recompression optimality, product chain error, graph support, Schur fill, tree compilation, precision assignment, and the benchmark corpus. The strengthened release records eleven passing unit tests and twenty three passing original-source tests for BDLR operations, Krylov methods, mixed precision, measured data schema, and figure generation.","\\subsection{Exact packet semantics}\\label{supp1:subsection:001}\\textit{Source items 2.3, 2.5, and 2.6}\\par\nLet $R$ be a ring, $I$ a finite index type, and $(P_a)_{a\\in I}$ a family satisfying\n\\[\nP_bP_a=\\begin{cases}P_a&b=a,","\\subsection{Exact packet semantics}\\label{supp1:subsection:001}\\textit{Source items 2.3, 2.5, and 2.6}\\par\nLet $R$ be a ring, $I$ a finite index type, and $(P_a)_{a\\in I}$ a family satisfying\n\\[\nP_bP_a=\\begin{cases}P_a&b=a,\\\\0&b\\ne a,\\end{cases}\n\\qquad \\sum_aP_a=1.\n\\]\nWrite $A_{ba}=P_bAP_a$ and let $E_A=\\{(b,a):A_{ba}\\ne0\\}$. These are exact algebraic zeros, not values discarded by a floating-point threshold.","0&b\\ne a,\\end{cases}\n\\qquad \\sum_aP_a=1.\n\\]\nWrite $A_{ba}=P_bAP_a$ and let $E_A=\\{(b,a):A_{ba}\\ne0\\}$. These are exact algebraic zeros, not values discarded by a floating-point threshold.","\\textbf{Reconstruction and action.} The checked finite sum is\n\\[\n\\boxed{A=\\sum_{(b,a)\\in E_A} A_{ba}.}\n\\]\nThe proof inserts the complete projector sum on both sides and removes only zero summands. Applying this identity to an arbitrary module gives exact action. A separate finite-matrix theorem establishes\n\\[\nAZ=\\sum_{(b,a)\\in E_A}A_{ba}Z\n\\]\nfor every rectangular batch of right-hand sides $Z$.","\\textbf{Composition.} Inserting the projector identity between factors yields\n\\[\n\\boxed{(AB)_{ba}=\\sum_c A_{bc}B_{ca}.}\n\\]\nIf the left side is nonzero, at least one intermediate $c$ has both $A_{bc}\\ne0$ and $B_{ca}\\ne0$. Consequently $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$. No equality of supports is asserted.","\\textbf{Adjoint and tensor products.} With an involution and self-adjoint projectors, the formalization proves\n\\[\n(A_{ba})^*=(A^*)_{ab}.\n\\]\nFor actual finite matrices and actual Kronecker products, it proves\n\\[\n(P_b\\otimes Q_d)(A\\otimes B)(P_a\\otimes Q_c)\n=A_{ba}\\otimes B_{dc}.\n\\]\nThe self-adjointness hypothesis is explicit in the adjoint theorem. The tensor theorem is not a statement about an uninterpreted tensor symbol.","\\subsection{Support propagation and nilpotence}\\label{supp1:subsection:002}\\textit{Source items 6.1, 6.2, and 6.3}\\par\nA path of length $n$ in the Lean predicate \\code{TransferPath} is a directed walk: vertices may repeat. This is the notion required by matrix powers. At length zero the endpoints coincide. At length $n+1$, a nonzero last block is appended to a length-$n$ walk.","Define the active contribution sum recursively by\n\\begin{align*}\nF_0(b,a)&=\\begin{cases}P_a&b=a,\\\\0&b\\ne a,\\end{cases}\\\\\nF_{n+1}(b,a)&=\\sum_{c:\\,A_{bc}\\ne0}A_{bc}F_n(c,a).\n\\end{align*}\nThe checked identity is\n\\[\n\\boxed{P_bA^nP_a=F_n(b,a)\\qquad(n\\ge0).}\n\\]\nIts proof is induction using the exact composition theorem, not an assumed support recurrence. When no active walk exists, the corresponding contribution sum is zero, and hence the actual block of $A^n$ vanishes.","F_{n+1}(b,a)&=\\sum_{c:\\,A_{bc}\\ne0}A_{bc}F_n(c,a).\n\\end{align*}\nThe checked identity is\n\\[\n\\boxed{P_bA^nP_a=F_n(b,a)\\qquad(n\\ge0).}\n\\]\nIts proof is induction using the exact composition theorem, not an assumed support recurrence. When no active walk exists, the corresponding contribution sum is zero, and hence the actual block of $A^n$ vanishes.","\\textbf{Nilpotence certificate.} If no walk of length $h+1$ exists between any pair of packets, then\n\\[\n\\boxed{A^{h+1}=0.}\n\\]\nEvery block vanishes, and reconstruction forces the whole operator to vanish. A second theorem derives the required absence of walks from a supplied level map\n\\[\n\\ell:I\\longrightarrow\\{0,\\ldots,h\\},\\qquad\nA_{bc}\\ne0\\ \\Longrightarrow\\ \\ell(c)<\\ell(b).\n\\]\nIt proves that a length-$n$ walk increases the level by at least $n$. This certifies the acyclic case with a bounded height. This block does not implement or verify an algorithm that constructs a topological ordering or computes the longest path.","bf{Nilpotence certificate.} If no walk of length $h+1$ exists between any pair of packets, then\n\\[\n\\boxed{A^{h+1}=0.}\n\\]\nEvery block vanishes, and reconstruction forces the whole operator to vanish. A second theorem derives the required absence of walks from a supplied level map\n\\[\n\\ell:I\\longrightarrow\\{0,\\ldots,h\\},\\qquad\nA_{bc}\\ne0\\ \\Longrightarrow\\ \\ell(c)<\\ell(b).\n\\]\nIt proves that a length-$n$ walk increases the level by at least $n$. This certifies the acyclic case with a bounded height. This block does not implement or verify an algorithm that constructs a topological ordering or computes the longest path.","length $h+1$ exists between any pair of packets, then\n\\[\n\\boxed{A^{h+1}=0.}\n\\]\nEvery block vanishes, and reconstruction forces the whole operator to vanish. A second theorem derives the required absence of walks from a supplied level map\n\\[\n\\ell:I\\longrightarrow\\{0,\\ldots,h\\},\\qquad\nA_{bc}\\ne0\\ \\Longrightarrow\\ \\ell(c)<\\ell(b).\n\\]\nIt proves that a length-$n$ walk increases the level by at least $n$. This certifies the acyclic case with a bounded height. This block does not implement or verify an algorithm that constructs a topological ordering or computes the longest path.","\\textbf{Polynomial support.} For the actual finite sum\n\\[\np(A)=\\sum_{j=0}^{d}c_jA^j,\n\\]\nthe checked support lies in the union of the support relations of the powers. The formalization treats the $j=0$ contribution explicitly, so the constant term is not accidentally omitted.","\\textbf{Cancellation control.} Lean verifies the rational example\n\\[\nA=\\begin{bmatrix}1&1\\\\-1&-1\\end{bmatrix},\\qquad A^2=0.\n\\]\nThe coordinate block $A_{00}$ is nonzero, and a length-two active walk from zero to zero exists. Nevertheless the corresponding block of $A^2$ is zero. This is an exact counterexample to replacing support inclusion by equality.","-1&-1\\end{bmatrix},\\qquad A^2=0.\n\\]\nThe coordinate block $A_{00}$ is nonzero, and a length-two active walk from zero to zero exists. Nevertheless the corresponding block of $A^2$ is zero. This is an exact counterexample to replacing support inclusion by equality.","\\subsection{Low-rank algebra and inversion}\\label{supp1:subsection:003}\\textit{Source items 5.1, 5.3, and 5.4}\\par\nLet $A=D_A+U_AV_A^*$ and $B=D_B+U_BV_B^*$ have rectangular factor widths $r_A$ and $r_B$. The formalization constructs\n\\begin{align*}\nU_C&=\\begin{bmatrix}D_AU_B+U_A(V_A^*U_B)&U_A\\end{bmatrix},","\\subsection{Low-rank algebra and inversion}\\label{supp1:subsection:003}\\textit{Source items 5.1, 5.3, and 5.4}\\par\nLet $A=D_A+U_AV_A^*$ and $B=D_B+U_BV_B^*$ have rectangular factor widths $r_A$ and $r_B$. The formalization constructs\n\\begin{align*}\nU_C&=\\begin{bmatrix}D_AU_B+U_A(V_A^*U_B)&U_A\\end{bmatrix},\\\\\nV_C&=\\begin{bmatrix}V_B&D_B^*V_A\\end{bmatrix}\n\\end{align*}\nand proves\n\\[\n\\boxed{AB=D_AD_B+U_CV_C^*.}\n\\]\nThese are actual concatenated matrices with $r_B+r_A$ columns. Expanding the product and grouping the terms with common right factors proves the identity. The separate rank theorem establishes\n\\[\n\\rank(AB-D_AD_B)\\le r_A+r_B.\n\\]\nIt bounds actual matrix rank, not just the number of stored columns. Packet diagonality of $D_AD_B$ follows from diagonality of both factors and the checked block-composition formula.","V_C&=\\begin{bmatrix}V_B&D_B^*V_A\\end{bmatrix}\n\\end{align*}\nand proves\n\\[\n\\boxed{AB=D_AD_B+U_CV_C^*.}\n\\]\nThese are actual concatenated matrices with $r_B+r_A$ columns. Expanding the product and grouping the terms with common right factors proves the identity. The separate rank theorem establishes\n\\[\n\\rank(AB-D_AD_B)\\le r_A+r_B.\n\\]\nIt bounds actual matrix rank, not just the number of stored columns. Packet diagonality of $D_AD_B$ follows from diagonality of both factors and the checked block-composition formula.","D_B^*V_A\\end{bmatrix}\n\\end{align*}\nand proves\n\\[\n\\boxed{AB=D_AD_B+U_CV_C^*.}\n\\]\nThese are actual concatenated matrices with $r_B+r_A$ columns. Expanding the product and grouping the terms with common right factors proves the identity. The separate rank theorem establishes\n\\[\n\\rank(AB-D_AD_B)\\le r_A+r_B.\n\\]\nIt bounds actual matrix rank, not just the number of stored columns. Packet diagonality of $D_AD_B$ follows from diagonality of both factors and the checked block-composition formula.","\\textbf{Woodbury.} For $D$ and\n\\[\nS=I+V^*D^{-1}U\n\\]\nboth invertible, Lean checks the ordinary matrix inverse formula\n\\[\n\\boxed{(D+UV^*)^{-1}\n=D^{-1}-D^{-1}US^{-1}V^*D^{-1}.}\n\\]\nThe proof specializes Mathlib's classical Woodbury identity. A separate theorem derives invertibility of $D+UV^*$ from the two stated assumptions. A further theorem verifies that multiplying the proposed inverse by any batch $Z$ solves the updated system. Invertibility of the update is not inserted as an extra premise.","xtbf{Woodbury.} For $D$ and\n\\[\nS=I+V^*D^{-1}U\n\\]\nboth invertible, Lean checks the ordinary matrix inverse formula\n\\[\n\\boxed{(D+UV^*)^{-1}\n=D^{-1}-D^{-1}US^{-1}V^*D^{-1}.}\n\\]\nThe proof specializes Mathlib's classical Woodbury identity. A separate theorem derives invertibility of $D+UV^*$ from the two stated assumptions. A further theorem verifies that multiplying the proposed inverse by any batch $Z$ solves the updated system. Invertibility of the update is not inserted as an extra premise.","\\textbf{Norm bound.} Over the real or complex scalars, all matrix norms in the following theorem are Hilbert-space operator norms, including the rectangular factors:\n\\[\n\\boxed{\n\\norm{(D+UV^*)^{-1}}\n\\le\\norm{D^{-1}}+\n\\norm{D^{-1}U}\\,\\norm{S^{-1}}\\,\\norm{V^*D^{-1}}.\n}\n\\]\nAfter the checked inverse identity, the proof uses the triangle inequality and two rectangular operator-norm product bounds.","\\subsection{Packet perturbation certificates}\\label{supp1:subsection:004}\\textit{Source item 8.1 and two uniform consequences}\\par\nLet $X$ be a normed space over a nontrivially normed field. The formal objects are actual continuous linear maps $X\\to X$, with their operator norm. Let $a(e)$ identify the source packet of edge $e$, and suppose\n\\[\n\\Delta_e=\\Delta_e P_{a(e)}.\n\\]\nSet\n\\[\nA=\\sum_e A_e,\\qquad \\widehat A=\\sum_e(A_e+\\Delta_e).\n\\]\nThe exact operator and vector identities are proved first:\n\\[\n\\widehat A-A=\\sum_e\\Delta_e P_{a(e)},\\qquad\n(\\widehat A-A)x=\\sum_e\\Delta_e(P_{a(e)}x).\n\\]\nApplying the finite triangle inequality and the defining operator-norm estimate gives\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nIf $\\norm{\\Delta_e}\\le\\varepsilon_e\\norm{A_e}$, then the formal relative cer","\\subsection{Packet perturbation certificates}\\label{supp1:subsection:004}\\textit{Source item 8.1 and two uniform consequences}\\par\nLet $X$ be a normed space over a nontrivially normed field. The formal objects are actual continuous linear maps $X\\to X$, with their operator norm. Let $a(e)$ identify the source packet of edge $e$, and suppose\n\\[\n\\Delta_e=\\Delta_e P_{a(e)}.\n\\]\nSet\n\\[\nA=\\sum_e A_e,\\qquad \\widehat A=\\sum_e(A_e+\\Delta_e).\n\\]\nThe exact operator and vector identities are proved first:\n\\[\n\\widehat A-A=\\sum_e\\Delta_e P_{a(e)},\\qquad\n(\\widehat A-A)x=\\sum_e\\Delta_e(P_{a(e)}x).\n\\]\nApplying the finite triangle inequality and the defining operator-norm estimate gives\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nIf $\\norm{\\Delta_e}\\le\\varepsilon_e\\norm{A_e}$, then the formal relative certificate is\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\varepsilon_e\\norm{A_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nThe source","\\subsection{Packet perturbation certificates}\\label{supp1:subsection:004}\\textit{Source item 8.1 and two uniform consequences}\\par\nLet $X$ be a normed space over a nontrivially normed field. The formal objects are actual continuous linear maps $X\\to X$, with their operator norm. Let $a(e)$ identify the source packet of edge $e$, and suppose\n\\[\n\\Delta_e=\\Delta_e P_{a(e)}.\n\\]\nSet\n\\[\nA=\\sum_e A_e,\\qquad \\widehat A=\\sum_e(A_e+\\Delta_e).\n\\]\nThe exact operator and vector identities are proved first:\n\\[\n\\widehat A-A=\\sum_e\\Delta_e P_{a(e)},\\qquad\n(\\widehat A-A)x=\\sum_e\\Delta_e(P_{a(e)}x).\n\\]\nApplying the finite triangle inequality and the defining operator-norm estimate gives\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nIf $\\norm{\\Delta_e}\\le\\varepsilon_e\\norm{A_e}$, then the formal relative certificate is\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\varepsilon_e\\norm{A_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nThe source additionally assumes that each exact b","\\subsection{Packet perturbation certificates}\\label{supp1:subsection:004}\\textit{Source item 8.1 and two uniform consequences}\\par\nLet $X$ be a normed space over a nontrivially normed field. The formal objects are actual continuous linear maps $X\\to X$, with their operator norm. Let $a(e)$ identify the source packet of edge $e$, and suppose\n\\[\n\\Delta_e=\\Delta_e P_{a(e)}.\n\\]\nSet\n\\[\nA=\\sum_e A_e,\\qquad \\widehat A=\\sum_e(A_e+\\Delta_e).\n\\]\nThe exact operator and vector identities are proved first:\n\\[\n\\widehat A-A=\\sum_e\\Delta_e P_{a(e)},\\qquad\n(\\widehat A-A)x=\\sum_e\\Delta_e(P_{a(e)}x).\n\\]\nApplying the finite triangle inequality and the defining operator-norm estimate gives\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nIf $\\norm{\\Delta_e}\\le\\varepsilon_e\\norm{A_e}$, then the formal relative certificate is\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\varepsilon_e\\norm{A_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nThe source additionally assumes that each exact block is source supported. The proof of","\\subsection{Packet perturbation certificates}\\label{supp1:subsection:004}\\textit{Source item 8.1 and two uniform consequences}\\par\nLet $X$ be a normed space over a nontrivially normed field. The formal objects are actual continuous linear maps $X\\to X$, with their operator norm. Let $a(e)$ identify the source packet of edge $e$, and suppose\n\\[\n\\Delta_e=\\Delta_e P_{a(e)}.\n\\]\nSet\n\\[\nA=\\sum_e A_e,\\qquad \\widehat A=\\sum_e(A_e+\\Delta_e).\n\\]\nThe exact operator and vector identities are proved first:\n\\[\n\\widehat A-A=\\sum_e\\Delta_e P_{a(e)},\\qquad\n(\\widehat A-A)x=\\sum_e\\Delta_e(P_{a(e)}x).\n\\]\nApplying the finite triangle inequality and the defining operator-norm estimate gives\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nIf $\\norm{\\Delta_e}\\le\\varepsilon_e\\norm{A_e}$, then the formal relative certificate is\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\varepsilon_e\\norm{A_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nThe source additionally assumes that each exact block is source supported. The proof of this bound does not need that assumption. The","\\subsection{Packet perturbation certificates}\\label{supp1:subsection:004}\\textit{Source item 8.1 and two uniform consequences}\\par\nLet $X$ be a normed space over a nontrivially normed field. The formal objects are actual continuous linear maps $X\\to X$, with their operator norm. Let $a(e)$ identify the source packet of edge $e$, and suppose\n\\[\n\\Delta_e=\\Delta_e P_{a(e)}.\n\\]\nSet\n\\[\nA=\\sum_e A_e,\\qquad \\widehat A=\\sum_e(A_e+\\Delta_e).\n\\]\nThe exact operator and vector identities are proved first:\n\\[\n\\widehat A-A=\\sum_e\\Delta_e P_{a(e)},\\qquad\n(\\widehat A-A)x=\\sum_e\\Delta_e(P_{a(e)}x).\n\\]\nApplying the finite triangle inequality and the defining operator-norm estimate gives\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nIf $\\norm{\\Delta_e}\\le\\varepsilon_e\\norm{A_e}$, then the formal relative certificate is\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\varepsilon_e\\norm{A_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nThe source additionally assumes that each exact block is source supported. The proof of this bound does not need that assumption. The checked statement therefor…","two uniform consequences}\\par\nLet $X$ be a normed space over a nontrivially normed field. The formal objects are actual continuous linear maps $X\\to X$, with their operator norm. Let $a(e)$ identify the source packet of edge $e$, and suppose\n\\[\n\\Delta_e=\\Delta_e P_{a(e)}.\n\\]\nSet\n\\[\nA=\\sum_e A_e,\\qquad \\widehat A=\\sum_e(A_e+\\Delta_e).\n\\]\nThe exact operator and vector identities are proved first:\n\\[\n\\widehat A-A=\\sum_e\\Delta_e P_{a(e)},\\qquad\n(\\widehat A-A)x=\\sum_e\\Delta_e(P_{a(e)}x).\n\\]\nApplying the finite triangle inequality and the defining operator-norm estimate gives\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nIf $\\norm{\\Delta_e}\\le\\varepsilon_e\\norm{A_e}$, then the formal relative certificate is\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\varepsilon_e\\norm{A_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nThe source additionally assumes that each exact block is source supported. The proof of this bound does not need that assumption. The checked statement therefore uses weaker sufficient hypotheses without weakening the conclusion.","Let $a(e)$ identify the source packet of edge $e$, and suppose\n\\[\n\\Delta_e=\\Delta_e P_{a(e)}.\n\\]\nSet\n\\[\nA=\\sum_e A_e,\\qquad \\widehat A=\\sum_e(A_e+\\Delta_e).\n\\]\nThe exact operator and vector identities are proved first:\n\\[\n\\widehat A-A=\\sum_e\\Delta_e P_{a(e)},\\qquad\n(\\widehat A-A)x=\\sum_e\\Delta_e(P_{a(e)}x).\n\\]\nApplying the finite triangle inequality and the defining operator-norm estimate gives\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nIf $\\norm{\\Delta_e}\\le\\varepsilon_e\\norm{A_e}$, then the formal relative certificate is\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\varepsilon_e\\norm{A_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nThe source additionally assumes that each exact block is source supported. The proof of this bound does not need that assumption. The checked statement therefore uses weaker sufficient hypotheses without weakening the conclusion.","_{a(e)}.\n\\]\nSet\n\\[\nA=\\sum_e A_e,\\qquad \\widehat A=\\sum_e(A_e+\\Delta_e).\n\\]\nThe exact operator and vector identities are proved first:\n\\[\n\\widehat A-A=\\sum_e\\Delta_e P_{a(e)},\\qquad\n(\\widehat A-A)x=\\sum_e\\Delta_e(P_{a(e)}x).\n\\]\nApplying the finite triangle inequality and the defining operator-norm estimate gives\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nIf $\\norm{\\Delta_e}\\le\\varepsilon_e\\norm{A_e}$, then the formal relative certificate is\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\varepsilon_e\\norm{A_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nThe source additionally assumes that each exact block is source supported. The proof of this bound does not need that assumption. The checked statement therefore uses weaker sufficient hypotheses without weakening the conclusion.","perator and vector identities are proved first:\n\\[\n\\widehat A-A=\\sum_e\\Delta_e P_{a(e)},\\qquad\n(\\widehat A-A)x=\\sum_e\\Delta_e(P_{a(e)}x).\n\\]\nApplying the finite triangle inequality and the defining operator-norm estimate gives\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nIf $\\norm{\\Delta_e}\\le\\varepsilon_e\\norm{A_e}$, then the formal relative certificate is\n\\[\n\\boxed{\n\\norm{(\\widehat A-A)x}\n\\le\\sum_e\\varepsilon_e\\norm{A_e}\\,\\norm{P_{a(e)}x}.\n}\n\\]\nThe source additionally assumes that each exact block is source supported. The proof of this bound does not need that assumption. The checked statement therefore uses weaker sufficient hypotheses without weakening the conclusion.","\\textbf{Uniform operator estimates.} Two additional checked consequences are\n\\[\n\\norm{\\widehat A-A}\n\\le\\sum_e\\norm{\\Delta_e}\\,\\norm{P_{a(e)}}\n\\]\nand, when $\\norm{P_a}\\le1$ for every packet,\n\\[\n\\boxed{\\norm{\\widehat A-A}\\le\\sum_e\\norm{\\Delta_e}.}\n\\]\nThese estimates are uniform in the input vector and can be used as inputs to later chain analyses. The chain results themselves are not included in this block's completed inventory.","\\textbf{Implementation boundary.} The maps $\\Delta_e$ and their bounds must be supplied or justified for a numerical implementation. This formalization does not establish IEEE rounding behavior, prevent overflow or underflow, model a CUDA kernel, or certify the recorded device speedups. Those are distinct claims and require separate evidence.","\\subsection{Statement correction and verification}\\label{supp1:subsection:005}\\textit{Boundary audit for item 8.4; evidence for this release}\\par\n\\textbf{Positive budget is necessary.} The source minimizes\n\\[\n\\sum_e c_e\\log\\frac1{\\varepsilon_e},\\qquad\n\\varepsilon_e>0,\\qquad \\sum_e s_e\\varepsilon_e=\\tau,\n\\]\nwith positive sensitivities and costs, but it does not explicitly impose $\\tau>0$. For a nonempty edge set, Lean proves\n\\[\ns_e>0\\ \\text{and}\\ \\varepsilon_e>0\\ \\text{for all }e\n\\quad\\Longrightarrow\\quad\n\\sum_e s_e\\varepsilon_e>0.\n\\]\nIt follows that every nonpositive budget is infeasible. A one-edge, unit-sensitivity, zero-budget instance is checked as an exact boundary example.","on and verification}\\label{supp1:subsection:005}\\textit{Boundary audit for item 8.4; evidence for this release}\\par\n\\textbf{Positive budget is necessary.} The source minimizes\n\\[\n\\sum_e c_e\\log\\frac1{\\varepsilon_e},\\qquad\n\\varepsilon_e>0,\\qquad \\sum_e s_e\\varepsilon_e=\\tau,\n\\]\nwith positive sensitivities and costs, but it does not explicitly impose $\\tau>0$. For a nonempty edge set, Lean proves\n\\[\ns_e>0\\ \\text{and}\\ \\varepsilon_e>0\\ \\text{for all }e\n\\quad\\Longrightarrow\\quad\n\\sum_e s_e\\varepsilon_e>0.\n\\]\nIt follows that every nonpositive budget is infeasible. A one-edge, unit-sensitivity, zero-budget instance is checked as an exact boundary example.","The amended nonempty-index statement should require $\\tau>0$. Nonemptiness also makes the sum of positive costs in the displayed optimizer's denominator positive. The empty edge set should be treated separately. \\textbf{Existence, optimality, and uniqueness of the corrected optimizer are not proved in this release.} Item 8.4 remains boundary-audit-only.","\\subsection{Exact finite-tree compilation}\\label{supp2:subsection:001}\\textit{Source items 9.4 and 9.5}\\par\nA candidate tree is either a terminal node or an internal node with a nonempty finite family of children. Arbitrary finite branching is supported. Every node carries a label from which keep cost $\\kappa_v$, keep error $\\epsilon_v$, and split overhead $\\sigma_v$ are read.","The cost and error of a cut $L$ are computed recursively. With multiplier $\\lambda$, its objective is\n\\[\nJ_\\lambda(L)=\\cost(L)+\\lambda\\err(L).\n\\]\nLean proves that this objective equals the recursively accumulated local keep score $\\kappa_v+\\lambda\\epsilon_v$, with $\\sigma_v$ charged at every split.","\\textbf{Recurrence.} For terminal nodes,\n\\[\nF_\\lambda(v)=\\kappa_v+\\lambda\\epsilon_v.\n\\]\nFor internal nodes,\n\\[\n\\boxed{F_\\lambda(v)=\\min\\left\\{\\kappa_v+\\lambda\\epsilon_v,\n\\;\\sigma_v+\\sum_{w\\in\\mathrm{child}(v)}F_\\lambda(w)\\right\\}.}\n\\]\nThe checked result has two parts. First, $F_\\lambda(v)\\leq J_\\lambda(L)$ for every legal cut of the subtree. Second, the constructed function \\code{compile} returns a legal cut with objective exactly $F_\\lambda(v)$. Thus the minimum is attained, not merely bounded below.","des,\n\\[\nF_\\lambda(v)=\\kappa_v+\\lambda\\epsilon_v.\n\\]\nFor internal nodes,\n\\[\n\\boxed{F_\\lambda(v)=\\min\\left\\{\\kappa_v+\\lambda\\epsilon_v,\n\\;\\sigma_v+\\sum_{w\\in\\mathrm{child}(v)}F_\\lambda(w)\\right\\}.}\n\\]\nThe checked result has two parts. First, $F_\\lambda(v)\\leq J_\\lambda(L)$ for every legal cut of the subtree. Second, the constructed function \\code{compile} returns a legal cut with objective exactly $F_\\lambda(v)$. Thus the minimum is attained, not merely bounded below.","\\textbf{Shared operator families.} For reuse weights $r_j$, the local keep model can aggregate setup and per-operator costs:\n\\[\n\\kappa_v=\\kappa_v^{\\mathrm{setup}}+\\sum_jr_j\\kappa_v^{(j)},\n\\qquad \\epsilon_v=\\sum_jr_j\\epsilon_v^{(j)}.\n\\]\nLean proves the aggregation identity at the cut level, including split overhead, before applying the compiler theorem. The proof is valid over an ordered field, including $\\Q$ and $\\R$. The intended nonnegative model is a specialization.","Shared operator families.} For reuse weights $r_j$, the local keep model can aggregate setup and per-operator costs:\n\\[\n\\kappa_v=\\kappa_v^{\\mathrm{setup}}+\\sum_jr_j\\kappa_v^{(j)},\n\\qquad \\epsilon_v=\\sum_jr_j\\epsilon_v^{(j)}.\n\\]\nLean proves the aggregation identity at the cut level, including split overhead, before applying the compiler theorem. The proof is valid over an ordered field, including $\\Q$ and $\\R$. The intended nonnegative model is a specialization.","erator families.} For reuse weights $r_j$, the local keep model can aggregate setup and per-operator costs:\n\\[\n\\kappa_v=\\kappa_v^{\\mathrm{setup}}+\\sum_jr_j\\kappa_v^{(j)},\n\\qquad \\epsilon_v=\\sum_jr_j\\epsilon_v^{(j)}.\n\\]\nLean proves the aggregation identity at the cut level, including split overhead, before applying the compiler theorem. The proof is valid over an ordered field, including $\\Q$ and $\\R$. The intended nonnegative model is a specialization.","\\subsection{Frontiers and decision stability}\\label{supp2:subsection:002}\\textit{Source items 9.6, 9.7, and 9.11}\\par\nLet $L_1,L_2$ be actual optima at $0\\leq\\lambda_1<\\lambda_2$. Their optimality inequalities are\n\\begin{align*}\n\\cost(L_1)+\\lambda_1\\err(L_1)&\\leq\\cost(L_2)+\\lambda_1\\err(L_2),","\\subsection{Frontiers and decision stability}\\label{supp2:subsection:002}\\textit{Source items 9.6, 9.7, and 9.11}\\par\nLet $L_1,L_2$ be actual optima at $0\\leq\\lambda_1<\\lambda_2$. Their optimality inequalities are\n\\begin{align*}\n\\cost(L_1)+\\lambda_1\\err(L_1)&\\leq\\cost(L_2)+\\lambda_1\\err(L_2),\\\\\n\\cost(L_2)+\\lambda_2\\err(L_2)&\\leq\\cost(L_1)+\\lambda_2\\err(L_1).\n\\end{align*}\nAdding them and using the strict multiplier ordering gives\n\\[\n\\boxed{\\err(L_2)\\leq\\err(L_1),\\qquad \\cost(L_2)\\geq\\cost(L_1).}\n\\]\nThese conclusions hold for any choices among tied optima. For the cuts returned by the compiler, the required optimality premises are supplied by its constructive correctness theorem.","\\cost(L_2)+\\lambda_2\\err(L_2)&\\leq\\cost(L_1)+\\lambda_2\\err(L_1).\n\\end{align*}\nAdding them and using the strict multiplier ordering gives\n\\[\n\\boxed{\\err(L_2)\\leq\\err(L_1),\\qquad \\cost(L_2)\\geq\\cost(L_1).}\n\\]\nThese conclusions hold for any choices among tied optima. For the cuts returned by the compiler, the required optimality premises are supplied by its constructive correctness theorem.","\\textbf{Supported Pareto points.} For $\\lambda>0$, a scalarized optimum cannot be dominated in both cost and error with one strict improvement. Lean proves this directly from the two inequalities and positivity of $\\lambda$.","The converse is not asserted. For the three cost/error points\n\\[\n(0,2),\\qquad (3/2,1),\\qquad (2,0),\n\\]\nthe middle point is undominated, but it is not selected by scalarization. A checked rational inequality gives, for every $\\lambda\\in\\Q$,\n\\[\n\\min\\{2\\lambda,2\\}<3/2+\\lambda.\n\\]\nThe quantized-budget compiler, which can recover unsupported points, remains a separate unformalized item in this release.","\\textbf{Strict half-gap stability.} At a solved internal node write\n\\[\nK=\\kappa_v+\\lambda\\epsilon_v,\\qquad\nS=\\sigma_v+\\sum_wF_\\lambda(w),\\qquad g=|K-S|.\n\\]\nIf the updated scores satisfy\n\\[\n|K'-K|<g/2,\\qquad |S'-S|<g/2,\n\\]\nLean proves\n\\[\nK'\\leq S'\\quad\\Longleftrightarrow\\quad K\\leq S.\n\\]\nApplying this condition recursively at every internal node proves equality of the entire returned cuts before and after perturbation, not only equality of one numerical minimum.","The hypotheses concern \\emph{solved} split scores, including changes propagated from descendants. A bound on raw local costs alone is not substituted for this condition. At an exact tie $g=0$, the strict inequalities cannot hold. No robustness at ties is claimed.","Let $v$ be the new optimum of the edited subtree, and let $j$ be the edited child slot at one frame. The update performed at that frame is\n\\[\n\\boxed{\n\\operatorname{step}(v)=\n\\min\\left\\{K,\\;\\sigma+\n\\sum_i\\begin{cases}\nv,&i=j,","Let $v$ be the new optimum of the edited subtree, and let $j$ be the edited child slot at one frame. The update performed at that frame is\n\\[\n\\boxed{\n\\operatorname{step}(v)=\n\\min\\left\\{K,\\;\\sigma+\n\\sum_i\\begin{cases}\nv,&i=j,\\\\\n\\widehat F_i,&i\\ne j.\n\\end{cases}\\right\\}.\n}\n\\]\nThis result is passed to the next ancestor. The update function does not recursively evaluate sibling subtrees. It reads their stored values $\\widehat F_i$.","\\widehat F_i,&i\\ne j.\n\\end{cases}\\right\\}.\n}\n\\]\nThis result is passed to the next ancestor. The update function does not recursively evaluate sibling subtrees. It reads their stored values $\\widehat F_i$.","\\textbf{Correctness of the actual update.} For any edited subtree $T'$ and valid context $C$, Lean proves\n\\[\n\\operatorname{updateValue}\\bigl(C,F(T')\\bigr)=F\\bigl(\\operatorname{fill}(C,T')\\bigr).\n\\]\nThe proof inducts through the context. At each frame, the updated child contributes the new value, and each other child contributes its unchanged cached optimum. The frame recurrence is therefore exactly the full-tree recurrence.","\\subsection{The continuous allocation theorem is complete}\\label{supp2:subsection:004}\\textit{Source item 8.4, with an explicit positive budget}\\par\nLet $E$ be a finite nonempty edge set. Assume\n\\[\ns_e>0,\\qquad c_e>0,\\qquad \\tau>0,\n\\]\nand define $C=\\sum_ec_e$. The optimization problem is\n\\[\n\\min_{\\varepsilon_e>0}\\sum_ec_e\\log\\frac1{\\varepsilon_e}\n\\qquad\\text{subject to}\\qquad\n\\sum_es_e\\varepsilon_e=\\tau.\n\\]\nLean constructs the displayed candidate\n\\[\n\\boxed{\\varepsilon_e^*=\\frac{\\tau c_e}{s_eC}}\n\\]\nand proves strict positivity and the exact budget equality. It then proves global optimality and uniqueness for the actual logarithmic objective.","lete}\\label{supp2:subsection:004}\\textit{Source item 8.4, with an explicit positive budget}\\par\nLet $E$ be a finite nonempty edge set. Assume\n\\[\ns_e>0,\\qquad c_e>0,\\qquad \\tau>0,\n\\]\nand define $C=\\sum_ec_e$. The optimization problem is\n\\[\n\\min_{\\varepsilon_e>0}\\sum_ec_e\\log\\frac1{\\varepsilon_e}\n\\qquad\\text{subject to}\\qquad\n\\sum_es_e\\varepsilon_e=\\tau.\n\\]\nLean constructs the displayed candidate\n\\[\n\\boxed{\\varepsilon_e^*=\\frac{\\tau c_e}{s_eC}}\n\\]\nand proves strict positivity and the exact budget equality. It then proves global optimality and uniqueness for the actual logarithmic objective.","\\textbf{Global comparison.} For any other feasible $\\varepsilon$, set\n$r_e=\\varepsilon_e/\\varepsilon_e^*>0$. The budget identities give\n\\[\n\\sum_ec_er_e=C.\n\\]\nUsing the standard analytic inequality $\\log r\\leq r-1$ yields\n\\[\n\\sum_ec_e\\log r_e\\leq\\sum_ec_e(r_e-1)=0.\n\\]\nLean separately proves the exact objective-difference identity\n\\[\nJ(\\varepsilon)-J(\\varepsilon^*)=-\\sum_ec_e\\log r_e.\n\\]\nTherefore $J(\\varepsilon)\\geq J(\\varepsilon^*)$.","If the feasible vectors differ, at least one $r_e\\ne1$. At that coordinate the logarithm inequality is strict. Since $c_e>0$, the finite weighted sum is strict, giving\n\\[\n\\varepsilon\\ne\\varepsilon^*\\quad\\Longrightarrow\\quad\nJ(\\varepsilon)>J(\\varepsilon^*).\n\\]\nThe final theorem characterizes every feasible global minimizer as exactly the constructed formula. No unproved stationary-point, convexity, or KKT optimality assertion is supplied as a hypothesis.","\\subsection{Explicit storage and operation models}\\label{supp2:subsection:005}\\textit{Source items 3.1, 3.2, 3.4, 3.6, 3.8--3.10, and 4.2}\\par\nFor a finite collection of rectangular blocks, the scalar addresses are actual elements of\n\\[\n\\coprod_{e\\in E}\\bigl(\\operatorname{Fin}(m_e)\\times\\operatorname{Fin}(n_e)\\bigr).\n\\]\nThe corresponding operation addresses include a right-hand-side index and one of two scalar-operation tags. Lean proves their cardinalities rather than merely defining a formula named ``cost.''","\\begin{center}\\small\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{tabular}{@{}lll@{}}\n\\toprule\nRepresentation & Scalar storage & Modeled work for $q$ right sides","\\midrule\nDense & $n^2$ & $2qn^2$","Packet blocks & $\\sum_em_en_e$ & $2q\\sum_em_en_e$","Equal diagonal blocks & $nb$ & $2qnb$","BDLR & $\\sum_i n_i^2+2nr$ & $2q(\\sum_i n_i^2+2nr)$","Equal-block BDLR & $n(b+2r)$ & $2qn(b+2r)$","Sparse plus low rank & $z+2nr$ & $2q(z+2nr)$","\\bottomrule\n\\end{tabular}\n\\end{center}\nFor equal blocks, the layout condition is $n=|I|b$. Under nonzero denominator conditions, the dense-to-block reduction is $n/b$ and the BDLR reduction is $n/(b+2r)$. The BDLR output thin product accumulates into the diagonal result under the declared two-operations-per-coefficient model.","\\textbf{Streaming.} The explicit event type counts one block read, one source read, and a target read plus write for every edge. With scalar size $s$, its byte count is\n\\[\nB_{\\mathrm{stream}}=s\\sum_e(m_en_e+n_eq+2m_eq).\n\\]\nNo cross-edge cache reuse is assumed. Sparse index metadata and temporary workspace remain separate.","\\textbf{Capacity and roofline.} Positive physical capacities and the inequalities $F\\leq PT$, $B\\leq WT$ give\n\\[\nT\\geq\\max\\{F/P,B/W\\}.\n\\]\nFor positive bytes and elapsed time, the rate form is $F/T\\leq\\min\\{P,WF/B\\}$. The dense memory bound is the exact equivalence $n^2s\\leq M\\iff n\\leq\\sqrt{M/s}$ for nonnegative $n,M$ and positive $s$.","\\textbf{Materialization.} The declared dense product followed by one addition at each output entry has exactly $2n^2r+n^2$ modeled events and $n^2$ output entries. The remainder is bounded by $n^2$ for every $n,r$, giving a concrete quadratic certificate for the manuscript's $O(n^2)$ term.","\\subsection{Contracts, grouping, and amortization}\\label{supp2:subsection:006}\\textit{Source items 2.9, 10.2, 10.4--10.6, and 11.3}\\par\n\\textbf{Output conversion.} A decoded run returns the actual pair\n\\[\n\\bigl(\\operatorname{decode}(\\operatorname{compressed}(x)),\n\\;t_{\\mathrm{run}}(x)+t_{\\mathrm{decode}}(\\operatorname{compressed}(x))\\bigr).\n\\]\nUnder the declared output equality, Lean proves that this returns the same explicit result and charges the conversion time. It also proves the downstream composition identity when compressed output is consumed directly. Omitting a positive decode cost strictly understates the modeled explicit-output time. This formalizes the protocol; it cannot certify the truth of measured times.","\\textbf{Exact profitability.} With setup $s$, baseline time $t_b$, candidate time $t_c$, and $t_c<t_b$, Lean proves\n\\[\n\\boxed{s+Rt_c<Rt_b\\quad\\Longleftrightarrow\\quad\nR>\\frac{s}{t_b-t_c}.}\n\\]\nFor $s\\geq0$ and natural-number reuse counts, the smallest profitable count is\n\\[\n\\boxed{R_{\\min}=\\left\\lfloor\\frac{s}{t_b-t_c}\\right\\rfloor+1.}\n\\]\nIt is proved both profitable and minimal. At equality there is only break-even. A checked example gives setup $6$, baseline $5$, candidate $3$: three uses break even, and four are the first profitable count. Zero setup gives a first profitable count of one when there is a per-use advantage.","ine time $t_b$, candidate time $t_c$, and $t_c<t_b$, Lean proves\n\\[\n\\boxed{s+Rt_c<Rt_b\\quad\\Longleftrightarrow\\quad\nR>\\frac{s}{t_b-t_c}.}\n\\]\nFor $s\\geq0$ and natural-number reuse counts, the smallest profitable count is\n\\[\n\\boxed{R_{\\min}=\\left\\lfloor\\frac{s}{t_b-t_c}\\right\\rfloor+1.}\n\\]\nIt is proved both profitable and minimal. At equality there is only break-even. A checked example gives setup $6$, baseline $5$, candidate $3$: three uses break even, and four are the first profitable count. Zero setup gives a first profitable count of one when there is a per-use advantage.",", candidate time $t_c$, and $t_c<t_b$, Lean proves\n\\[\n\\boxed{s+Rt_c<Rt_b\\quad\\Longleftrightarrow\\quad\nR>\\frac{s}{t_b-t_c}.}\n\\]\nFor $s\\geq0$ and natural-number reuse counts, the smallest profitable count is\n\\[\n\\boxed{R_{\\min}=\\left\\lfloor\\frac{s}{t_b-t_c}\\right\\rfloor+1.}\n\\]\nIt is proved both profitable and minimal. At equality there is only break-even. A checked example gives setup $6$, baseline $5$, candidate $3$: three uses break even, and four are the first profitable count. Zero setup gives a first profitable count of one when there is a per-use advantage.","e $t_c$, and $t_c<t_b$, Lean proves\n\\[\n\\boxed{s+Rt_c<Rt_b\\quad\\Longleftrightarrow\\quad\nR>\\frac{s}{t_b-t_c}.}\n\\]\nFor $s\\geq0$ and natural-number reuse counts, the smallest profitable count is\n\\[\n\\boxed{R_{\\min}=\\left\\lfloor\\frac{s}{t_b-t_c}\\right\\rfloor+1.}\n\\]\nIt is proved both profitable and minimal. At equality there is only break-even. A checked example gives setup $6$, baseline $5$, candidate $3$: three uses break even, and four are the first profitable count. Zero setup gives a first profitable count of one when there is a per-use advantage.","\\subsection{Low-rank factors through a product chain}\\label{supp3:subsection:001}\\textit{Source item 5.2}\\par\nLet\n\\[\nA_j=D_j+U_jV_j^*,\\qquad U_j,V_j\\in\\mathbb C^{n\\times r_j},\n\\]\nwith all $D_j$ diagonal in one packet layout. Set\n\\[\nP_m=A_m\\cdots A_1,\\qquad D^{(m)}=D_m\\cdots D_1.\n\\]\nThe formalization constructs thin matrices $L_m,R_m$ such that\n\\[\n\\boxed{P_m=D^{(m)}+L_mR_m^*.}\n\\]\nTheir column index type is a recursively constructed disjoint sum. Its cardinality is proved to equal $\\sum_{j=1}^m r_j$.","tit{Source item 5.2}\\par\nLet\n\\[\nA_j=D_j+U_jV_j^*,\\qquad U_j,V_j\\in\\mathbb C^{n\\times r_j},\n\\]\nwith all $D_j$ diagonal in one packet layout. Set\n\\[\nP_m=A_m\\cdots A_1,\\qquad D^{(m)}=D_m\\cdots D_1.\n\\]\nThe formalization constructs thin matrices $L_m,R_m$ such that\n\\[\n\\boxed{P_m=D^{(m)}+L_mR_m^*.}\n\\]\nTheir column index type is a recursively constructed disjoint sum. Its cardinality is proved to equal $\\sum_{j=1}^m r_j$.","\\textbf{Construction.} Suppose $P_{m-1}=D^{(m-1)}+L_{m-1}R_{m-1}^*$. The new factors are\n\\begin{align*}\nL_m&=\\bigl[D_mL_{m-1}+U_m(V_m^*L_{m-1})\\quad U_m\\bigr],","\\textbf{Construction.} Suppose $P_{m-1}=D^{(m-1)}+L_{m-1}R_{m-1}^*$. The new factors are\n\\begin{align*}\nL_m&=\\bigl[D_mL_{m-1}+U_m(V_m^*L_{m-1})\\quad U_m\\bigr],\\\\\nR_m&=\\bigl[R_{m-1}\\quad (D^{(m-1)})^*V_m\\bigr].\n\\end{align*}\nSubstitution into the already checked two-factor product identity proves exactness. This is not a rank-only existence argument. The recursively defined factors are themselves audited.","Applying the ordinary matrix-rank inequality gives\n\\[\n\\boxed{\\operatorname{rank}(P_m-D^{(m)})\\leq\\sum_{j=1}^m r_j.}\n\\]\nThe bound concerns the actual correction matrix, not just the number of allocated columns. The proof also shows that $D^{(m)}$ stays packet diagonal. The empty chain has identity base, empty column type, and zero correction. Individual $r_j=0$ are permitted.","\\subsubsection{Actual noncommutative products}\nThe error module defines the left-accumulated product recursively, with $P_0=I$ and $P_{j+1}=A_{j+1}P_j$. For a perturbed sequence $\\widehat A_j$, it derives the discrepancy recursion\n\\[\n\\widehat P_{j+1}-P_{j+1}\n=\\widehat A_{j+1}(\\widehat P_j-P_j)\n +(\\widehat A_{j+1}-A_{j+1})P_j.\n\\]\nThe identity follows from the actual products. It is not supplied as an assumption to a norm estimate, and it does not commute the operators. Lean uses zero-based indices internally, so index $j$ in source code corresponds to manuscript factor $j+1$.","I$ and $P_{j+1}=A_{j+1}P_j$. For a perturbed sequence $\\widehat A_j$, it derives the discrepancy recursion\n\\[\n\\widehat P_{j+1}-P_{j+1}\n=\\widehat A_{j+1}(\\widehat P_j-P_j)\n +(\\widehat A_{j+1}-A_{j+1})P_j.\n\\]\nThe identity follows from the actual products. It is not supplied as an assumption to a norm estimate, and it does not commute the operators. Lean uses zero-based indices internally, so index $j$ in source code corresponds to manuscript factor $j+1$.","e $\\widehat A_j$, it derives the discrepancy recursion\n\\[\n\\widehat P_{j+1}-P_{j+1}\n=\\widehat A_{j+1}(\\widehat P_j-P_j)\n +(\\widehat A_{j+1}-A_{j+1})P_j.\n\\]\nThe identity follows from the actual products. It is not supplied as an assumption to a norm estimate, and it does not commute the operators. Lean uses zero-based indices internally, so index $j$ in source code corresponds to manuscript factor $j+1$.","\\subsection{Error propagation and recompression}\\label{supp3:subsection:002}\\textit{Source items 5.8, 5.9, and 8.3}\\par\nFor $\\widehat A_j=A_j+E_j$, the checked product-chain bound is\n\\[\n\\boxed{\n\\norm{\\widehat A_m\\cdots\\widehat A_1-A_m\\cdots A_1}\n\\leq\\sum_{j=1}^m\n \\left(\\prod_{i=j+1}^m\\norm{\\widehat A_i}\\right)\n \\norm{E_j}\n \\left(\\prod_{i=1}^{j-1}\\norm{A_i}\\right).}\n\\]\nEmpty products are one. The proof first solves a scalar gain recurrence as a finite sum, then bounds the actual discrepancy recursion by it. A version with certified upper bounds $\\delta_j\\geq\\norm{E_j}$ is included. The product-norm argument uses a norm-one identity, as in operators on a nonzero normed space.","checked product-chain bound is\n\\[\n\\boxed{\n\\norm{\\widehat A_m\\cdots\\widehat A_1-A_m\\cdots A_1}\n\\leq\\sum_{j=1}^m\n \\left(\\prod_{i=j+1}^m\\norm{\\widehat A_i}\\right)\n \\norm{E_j}\n \\left(\\prod_{i=1}^{j-1}\\norm{A_i}\\right).}\n\\]\nEmpty products are one. The proof first solves a scalar gain recurrence as a finite sum, then bounds the actual discrepancy recursion by it. A version with certified upper bounds $\\delta_j\\geq\\norm{E_j}$ is included. The product-norm argument uses a norm-one identity, as in operators on a nonzero normed space.","\\textbf{Post-product recompression is different.} Define the implemented iterates by\n\\[\n\\widetilde P_0=I,\\qquad\n\\widetilde P_j=A_j\\widetilde P_{j-1}+E_j.\n\\]\nHere $E_j$ is a residual introduced after multiplication. Lean derives\n\\[\n\\widetilde P_j-P_j=A_j(\\widetilde P_{j-1}-P_{j-1})+E_j\n\\]\nand proves\n\\[\n\\boxed{\\norm{\\widetilde P_m-P_m}\n\\leq\\sum_{j=1}^m\\left(\\prod_{i=j+1}^m\\norm{A_i}\\right)\\delta_j,\n\\quad \\norm{E_j}\\leq\\delta_j.}\n\\]\nThis theorem works in a normed ring without the extra norm-one condition. It includes the empty iteration. There is no exact-prefix factor in the local residual term, because the residual is already measured at the post-product output.","\\textbf{Packetwise mixed precision.} At each stage let\n\\[\nA_j=\\sum_e A_{j,e},\\qquad\n\\widehat A_j=\\sum_e(A_{j,e}+\\Delta_{j,e}),\\qquad\n\\Delta_{j,e}=\\Delta_{j,e}P_{j,a(e)}.\n\\]\nThe previous packet certificate supplies the uniform operator bound\n\\[\n\\norm{\\widehat A_j-A_j}\\leq\n\\delta_j:=\\sum_e\\norm{\\Delta_{j,e}}\\norm{P_{j,a(e)}}.\n\\]\nThe new theorem substitutes these $\\delta_j$ into the actual product-chain estimate. The resulting vector error is at most the operator certificate times $\\norm{x}$. A vector-specific packet bound is not silently reused as an operator norm.","\\widehat A_j=\\sum_e(A_{j,e}+\\Delta_{j,e}),\\qquad\n\\Delta_{j,e}=\\Delta_{j,e}P_{j,a(e)}.\n\\]\nThe previous packet certificate supplies the uniform operator bound\n\\[\n\\norm{\\widehat A_j-A_j}\\leq\n\\delta_j:=\\sum_e\\norm{\\Delta_{j,e}}\\norm{P_{j,a(e)}}.\n\\]\nThe new theorem substitutes these $\\delta_j$ into the actual product-chain estimate. The resulting vector error is at most the operator certificate times $\\norm{x}$. A vector-specific packet bound is not silently reused as an operator norm.","\\subsection{Exact and approximate packet Krylov spaces}\\label{supp3:subsection:003}\\textit{Source items 7.2 and 7.3}\\par\nWrite\n\\[\n\\mathcal K_m(A,v)=\\operatorname{span}\\{v,Av,\\ldots,A^{m-1}v\\}.\n\\]\nFor a complete finite projector family $P_i$, suppose $A$ commutes with every $P_i$. Let $A_i=P_iAP_i$. Lean proves the actual orbit identity\n\\[\nA^jv=\\sum_i A_i^jP_iv\n\\]\nand therefore\n\\[\n\\boxed{\\mathcal K_m(A,v)\\subseteq\n\\bigoplus_i\\mathcal K_m(A_i,P_iv).}\n\\]\nThe case $j=0$ uses $\\sum_iP_i=I$. The zeroth local power is the ambient identity, applied to $P_iv$, rather than an incorrectly substituted local projector power.","3:subsection:003}\\textit{Source items 7.2 and 7.3}\\par\nWrite\n\\[\n\\mathcal K_m(A,v)=\\operatorname{span}\\{v,Av,\\ldots,A^{m-1}v\\}.\n\\]\nFor a complete finite projector family $P_i$, suppose $A$ commutes with every $P_i$. Let $A_i=P_iAP_i$. Lean proves the actual orbit identity\n\\[\nA^jv=\\sum_i A_i^jP_iv\n\\]\nand therefore\n\\[\n\\boxed{\\mathcal K_m(A,v)\\subseteq\n\\bigoplus_i\\mathcal K_m(A_i,P_iv).}\n\\]\nThe case $j=0$ uses $\\sum_iP_i=I$. The zeroth local power is the ambient identity, applied to $P_iv$, rather than an incorrectly substituted local projector power.","n:003}\\textit{Source items 7.2 and 7.3}\\par\nWrite\n\\[\n\\mathcal K_m(A,v)=\\operatorname{span}\\{v,Av,\\ldots,A^{m-1}v\\}.\n\\]\nFor a complete finite projector family $P_i$, suppose $A$ commutes with every $P_i$. Let $A_i=P_iAP_i$. Lean proves the actual orbit identity\n\\[\nA^jv=\\sum_i A_i^jP_iv\n\\]\nand therefore\n\\[\n\\boxed{\\mathcal K_m(A,v)\\subseteq\n\\bigoplus_i\\mathcal K_m(A_i,P_iv).}\n\\]\nThe case $j=0$ uses $\\sum_iP_i=I$. The zeroth local power is the ambient identity, applied to $P_iv$, rather than an incorrectly substituted local projector power.","eratorname{span}\\{v,Av,\\ldots,A^{m-1}v\\}.\n\\]\nFor a complete finite projector family $P_i$, suppose $A$ commutes with every $P_i$. Let $A_i=P_iAP_i$. Lean proves the actual orbit identity\n\\[\nA^jv=\\sum_i A_i^jP_iv\n\\]\nand therefore\n\\[\n\\boxed{\\mathcal K_m(A,v)\\subseteq\n\\bigoplus_i\\mathcal K_m(A_i,P_iv).}\n\\]\nThe case $j=0$ uses $\\sum_iP_i=I$. The zeroth local power is the ambient identity, applied to $P_iv$, rather than an incorrectly substituted local projector power.","\\textbf{The direct sum is justified.} Each local space is proved to lie in the actual range of $P_i$. Projecting a finite sum of local vectors selects its $i$th summand. Consequently a zero sum of local vectors has every summand zero. The proof does not assert equality between the global Krylov space and the full sum of local Krylov spaces.","For approximate invariance let $A=D+K$, where $D$ is packet diagonal, and set\n\\[\nM=\\max\\{\\norm{A},\\norm{D}\\}.\n\\]\nFor every positive integer $j$, the operator estimate is\n\\[\n\\boxed{\\norm{A^j-D^j}\\leq jM^{j-1}\\norm{K}.}\n\\]\nThe proof is noncommutative and uses actual powers. Its positive-index version works even in a trivial normed ring. No division by $M$ is used, so $M=0$ causes no exception.","Define the comparison space\n\\[\n\\mathcal L_{j+1}=\\bigoplus_i\\mathcal K_{j+1}(P_iDP_i,P_iv).\n\\]\nPacket diagonality is first shown to imply commutation. The exact theorem then proves $D^jv\\in\\mathcal L_{j+1}$. Using that vector as a witness in the infimum distance gives\n\\[\n\\boxed{\\operatorname{dist}(A^jv,\\mathcal L_{j+1})\n\\leq jM^{j-1}\\norm{K}\\norm{v}.}\n\\]\nThis statement uses Lean's actual metric infimum distance to the subspace. It does not assume that the comparison vector lies there. The block-Jacobi condition bound and the preconditioned conjugate-gradient corollary are not completed by these Krylov results.","For a predicate $S$ on costs, the common minimum specification means:\n\\[\n\\begin{array}{ll}\n\\code{none}: & \\text{no cost satisfies }S,","For a predicate $S$ on costs, the common minimum specification means:\n\\[\n\\begin{array}{ll}\n\\code{none}: & \\text{no cost satisfies }S,\\\\\n\\code{some}(v): & S(v)\\text{ and }v\\leq z\\text{ whenever }S(z).\n\\end{array}\n\\]\nBoth attainment and minimality are included. Optional minimum and optional addition are proved to implement union and independent sum of feasible sets.","For exact-weight tables $F,G$, min-plus convolution is\n\\[\n(F\\star G)(b)=\\min_{0\\leq k\\leq b}\\{F(k)+G(b-k)\\},\n\\]\nwith an infeasible sum whenever either operand is infeasible. Its correctness theorem identifies every finite answer with an actual feasible pair of component weights and costs. Repeated convolution handles arbitrary finite lists of components.","\\textbf{Independent precision assignments.} For every edge, a finite list contains its permitted pairs $(w,c)$ of quantized error and cost. The legal assignment type chooses exactly one listed pair per edge. It is defined independently of the dynamic program. Costs and weights are then summed over that assignment.","If $D$ is the already-solved tail table, the next edge has exact-weight recurrence\n\\[\nF(b)=\\min_{(w,c)\\text{ allowed},\\;w\\leq b}\\{D(b-w)+c\\}.\n\\]\nThe final answer is $\\min_{0\\leq b\\leq B}F(b)$. Lean proves that this is the exact minimum over legal assignments with total error at most $B$, or \\code{none} exactly when no such assignment exists.","\\textbf{The subtraction guard matters.} Natural subtraction truncates at zero. Without the condition $w\\leq b$, an option of weight one could be accepted at budget zero. The implemented program checks the condition before subtraction. Dedicated regressions also cover a zero-weight option, an empty option set, and the empty collection of edges. The last case has cost and error zero.","Keeping node $v$ costs $\\kappa_v$ and incurs integer error $w_v$. Splitting costs overhead $\\sigma_v$ in addition to the selected child cuts. The exact-weight recurrence is\n\\[\nF_v(b)=\\min\\left\\{\n\\begin{array}{l}\n\\kappa_v\\quad\\text{if }b=w_v,","Keeping node $v$ costs $\\kappa_v$ and incurs integer error $w_v$. Splitting costs overhead $\\sigma_v$ in addition to the selected child cuts. The exact-weight recurrence is\n\\[\nF_v(b)=\\min\\left\\{\n\\begin{array}{l}\n\\kappa_v\\quad\\text{if }b=w_v,\\\\[2pt]\n\\displaystyle\\sigma_v+\n\\min_{\\sum_i b_i=b}\\sum_iF_{v_i}(b_i)\\quad\\text{if }v\\text{ is internal}.\n\\end{array}\\right.\n\\]\nNonexistent alternatives are infeasible. The child sum is implemented by the proved finite min-plus convolution, not by assuming an optimal allocation.","\\textbf{Exact semantics.} For every tree and every integer weight $b$, Lean proves that $F_v(b)$ is a minimum of costs of independent legal cuts of exact weight $b$. Thus each finite answer has an attaining cut and is a lower bound for every other legal cut at that weight. Infeasibility is equivalent to the absence of such a cut.","The root scan\n\\[\n\\boxed{\\min_{0\\leq b\\leq B}F_{\\mathrm{root}}(b)}\n\\]\nis therefore the exact budget-constrained optimum. Increasing the budget cannot increase a finite optimum. The theorem retains feasibility hypotheses where necessary rather than treating \\code{none} as an ordinary real cost.","\\textbf{Why the budget program adds something.} Consider the three admissible cost/error pairs\n\\[\n(0,4),\\qquad(3,2),\\qquad(4,0).\n\\]\nAt budget two the answer is cost three. The middle pair cannot minimize cost plus $\\lambda$ times error: comparison with the first pair would require $\\lambda\\geq3/2$, while comparison with the third would require $\\lambda\\leq1/2$. The executable suite checks the budget answer on an actual unary candidate tree with these cuts. It is not obtained merely by varying the previous Lagrangian compiler's multiplier.","\\subsection{Cached arrays and a work certificate}\\label{supp3:subsection:006}\\textit{Source item 9.9 and implemented forms of 8.7 and 9.8}\\par\nA recursive value equation alone need not be an efficient program. Repeated calls can recompute the same subtree table. The new implementation therefore stores each table as an array-backed \\code{Vector} with exactly $B+1$ optional entries.","\\textbf{Entrywise implementation theorem.} For every $b\\leq B$, Lean proves\n\\[\n\\operatorname{lookup}(\\operatorname{storedTree}(B,t),b)=F_t(b).\n\\]\nThere is a parallel theorem for stored discrete precision. The proof establishes first that an output entry at $b$ depends only on input entries at most $b$, then proceeds through cached convolution and the tree recursion. Consequently the optimality, attainment, and infeasibility specification applies to the actual stored result.","\\textbf{Counted candidates.} One full convolution table examines indices\n\\[\n(b,k),\\qquad 0\\leq b\\leq B,\\quad0\\leq k\\leq b.\n\\]\nThese form a concrete finite dependent-sum type. Its cardinality is proved to be\n\\[\n\\sum_{b=0}^B(b+1)\\leq(B+1)^2.\n\\]\nFor a nonempty finite tree, the number of child merges plus one equals the node count $N$. This identity is proved for arbitrary finite branching.","Let $W_B(t)$ be the declared unit-cost count of table-cell constructions and tested min-plus candidates. The model charges the empty child-combination table as well as the output cells and merge candidates. Lean proves\n\\[\nW_B(t)\\leq2(B+1)N+\\bigl((B+1)+(B+1)^2\\bigr)(N-1)\n\\]\nand hence\n\\[\n\\boxed{W_B(t)\\leq N\\bigl(3(B+1)+(B+1)^2\\bigr).}\n\\]\nThe root minimum scan adds a linear pass. Thus the declared tabulation model is pseudo-polynomial in $N$ and the numeric budget $B$.","ount of table-cell constructions and tested min-plus candidates. The model charges the empty child-combination table as well as the output cells and merge candidates. Lean proves\n\\[\nW_B(t)\\leq2(B+1)N+\\bigl((B+1)+(B+1)^2\\bigr)(N-1)\n\\]\nand hence\n\\[\n\\boxed{W_B(t)\\leq N\\bigl(3(B+1)+(B+1)^2\\bigr).}\n\\]\nThe root minimum scan adds a linear pass. Thus the declared tabulation model is pseudo-polynomial in $N$ and the numeric budget $B$.","tions and tested min-plus candidates. The model charges the empty child-combination table as well as the output cells and merge candidates. Lean proves\n\\[\nW_B(t)\\leq2(B+1)N+\\bigl((B+1)+(B+1)^2\\bigr)(N-1)\n\\]\nand hence\n\\[\n\\boxed{W_B(t)\\leq N\\bigl(3(B+1)+(B+1)^2\\bigr).}\n\\]\nThe root minimum scan adds a linear pass. Thus the declared tabulation model is pseudo-polynomial in $N$ and the numeric budget $B$.","\\textbf{Scope of this certificate.} These are table and candidate counts, not a measured execution trace or a theorem about every instruction emitted by the compiler. Rational bit complexity, allocation overhead, memory traffic, and hardware timing are not included. The certificate is not polynomial in the binary input length of $B$.","\\subsection{Circulant Fourier diagonalization}\\label{supp4:subsection:001}\\textit{Source item 4.3; module CirculantFourier}\\par\nLet $n>0$, with coordinates indexed by $\\mathbb Z/n\\mathbb Z$. Define the actual circulant by\n\\[\nC(c)_{jk}=c_{j-k}.\n\\]\nThe formal Fourier matrix is\n\\[\nF_{jk}=\\frac{1}{\\sqrt n}\\,\\chi(-jk),\n\\qquad\n\\chi(a)=\\exp\\!\\left(\\frac{2\\pi i a}{n}\\right).\n\\]\nThe standard character is not left abstract: a checked theorem identifies these entries with\n\\[\n\\frac{1}{\\sqrt n}\\exp\\!\\left(-\\frac{2\\pi i\\,\\operatorname{val}(j)\\operatorname{val}(k)}n\\right).\n\\]\nThe value representatives enter only this explicit exponential formula. All modular algebra uses the residue classes themselves.","n:001}\\textit{Source item 4.3; module CirculantFourier}\\par\nLet $n>0$, with coordinates indexed by $\\mathbb Z/n\\mathbb Z$. Define the actual circulant by\n\\[\nC(c)_{jk}=c_{j-k}.\n\\]\nThe formal Fourier matrix is\n\\[\nF_{jk}=\\frac{1}{\\sqrt n}\\,\\chi(-jk),\n\\qquad\n\\chi(a)=\\exp\\!\\left(\\frac{2\\pi i a}{n}\\right).\n\\]\nThe standard character is not left abstract: a checked theorem identifies these entries with\n\\[\n\\frac{1}{\\sqrt n}\\exp\\!\\left(-\\frac{2\\pi i\\,\\operatorname{val}(j)\\operatorname{val}(k)}n\\right).\n\\]\nThe value representatives enter only this explicit exponential formula. All modular algebra uses the residue classes themselves.","\\textbf{Normalization and inversion.} The proof identifies matrix application with the ordinary counting-measure discrete Fourier transform:\n\\[\nFx=n^{-1/2}\\operatorname{DFT}(x).\n\\]\nIt separately identifies $F^*x$ with the appropriately scaled inverse transform. Mathlib's checked Fourier inversion theorem and the proved scalar normalization yield both identities\n\\[\n\\boxed{F^*F=I,\\qquad FF^*=I.}\n\\]\nNo unitarity assumption is inserted.","\\textbf{Convolution and reconstruction.} An explicit bijective change of index proves\n\\[\n\\operatorname{DFT}(C(c)x)_k\n=\\operatorname{DFT}(c)_k\\operatorname{DFT}(x)_k.\n\\]\nThis gives the matrix intertwining relation $FC(c)=\\operatorname{diag}(\\operatorname{DFT}c)F$. Multiplication by the proved inverse reconstructs the circulant:\n\\[\n\\boxed{C(c)=F^*\\operatorname{diag}(\\sqrt n\\,Fc)F.}\n\\]\nThe two-transform application formula is also checked for every input vector. The result includes $n=1$ and arbitrary complex coefficients. Hermiticity is not required.","\\[\n\\operatorname{DFT}(C(c)x)_k\n=\\operatorname{DFT}(c)_k\\operatorname{DFT}(x)_k.\n\\]\nThis gives the matrix intertwining relation $FC(c)=\\operatorname{diag}(\\operatorname{DFT}c)F$. Multiplication by the proved inverse reconstructs the circulant:\n\\[\n\\boxed{C(c)=F^*\\operatorname{diag}(\\sqrt n\\,Fc)F.}\n\\]\nThe two-transform application formula is also checked for every input vector. The result includes $n=1$ and arbitrary complex coefficients. Hermiticity is not required.","\\textbf{Claim boundary.} This verifies the exact Fourier representation, not an $O(n\\log n)$ implementation of the transform. The latter is an additional algorithmic factorization of the same linear map.","\\subsection{Toeplitz padding and column-major vectorization}\\label{supp4:subsection:002}\\textit{Source items 4.4 and 4.5}\\par\n\\textbf{Toeplitz embedding.} For $n>0$, define\n\\[\nT_{ij}=\n\\begin{cases}\nc_{i-j},&j\\le i,","\\subsection{Toeplitz padding and column-major vectorization}\\label{supp4:subsection:002}\\textit{Source items 4.4 and 4.5}\\par\n\\textbf{Toeplitz embedding.} For $n>0$, define\n\\[\nT_{ij}=\n\\begin{cases}\nc_{i-j},&j\\le i,\\\\\nr_{j-i},&i<j.\n\\end{cases}\n\\]\nThe diagonal is supplied by $c_0$; the coordinate $r_0$ is unused. The constructed generator of length $2n$ is\n\\[\ng=(c_0,\\ldots,c_{n-1},0,r_{n-1},\\ldots,r_1),\n\\]\nand the padding map is the actual finite-index sum eliminator that appends $n$ zeros.","r_{j-i},&i<j.\n\\end{cases}\n\\]\nThe diagonal is supplied by $c_0$; the coordinate $r_0$ is unused. The constructed generator of length $2n$ is\n\\[\ng=(c_0,\\ldots,c_{n-1},0,r_{n-1},\\ldots,r_1),\n\\]\nand the padding map is the actual finite-index sum eliminator that appends $n$ zeros.","Lean first proves the upper-left block identity\n\\[\nC(g)_{ij}=T_{ij}\\qquad(0\\le i,j<n).\n\\]\nFor $j\\le i$, finite modular subtraction has its ordinary value $i-j$. For $i<j$, it has wrapped value $2n+i-j$, whose reversed generator coordinate is $j-i$. Both branches are checked with the actual finite-index arithmetic.","Splitting the product sum into the original and padded halves then gives\n\\[\n\\boxed{\\bigl(C(g)(x,0)^T\\bigr)_{0:n}=Tx.}\n\\]\nThe generator's first half and its middle zero are checked separately. Neither symmetry nor conjugation is involved.","\\textbf{Kronecker application.} For $A\\in\\C^{m\\times m}$, $B\\in\\C^{n\\times n}$ and $X\\in\\C^{n\\times m}$, Lean proves\n\\[\n\\boxed{(A\\otimes B)\\operatorname{vec}(X)=\\operatorname{vec}(BXA^T).}\n\\]\nThe formal vectorization is explicitly column-major: its pair coordinate $(j,i)$ stores $X_{ij}$. The right factor is the \\emph{ordinary transpose}, not the conjugate transpose. This distinction is essential over $\\C$.","f{Kronecker application.} For $A\\in\\C^{m\\times m}$, $B\\in\\C^{n\\times n}$ and $X\\in\\C^{n\\times m}$, Lean proves\n\\[\n\\boxed{(A\\otimes B)\\operatorname{vec}(X)=\\operatorname{vec}(BXA^T).}\n\\]\nThe formal vectorization is explicitly column-major: its pair coordinate $(j,i)$ stores $X_{ij}$. The right factor is the \\emph{ordinary transpose}, not the conjugate transpose. This distinction is essential over $\\C$.","Finite types of multiplication/accumulation events and stored scalar addresses give\n\\[\n\\boxed{2n^2m+2nm^2\\ \\text{modeled operations},\\qquad n^2+m^2\\ \\text{factor entries}.}\n\\]\nThe explicit Kronecker matrix has $n^2m^2$ entries. As in the manuscript, the operation model assigns two scalar operations to each multiply-accumulate term.","\\subsection{Arbitrary tensor factors}\\label{supp4:subsection:003}\\textit{Source item 4.6; module StructuredTensor}\\par\nLet $A_0,\\ldots,A_{k-1}$ be square factors of order $d$. Tensor coordinates are represented by a nested product of $k$ copies of $\\{0,\\ldots,d-1\\}$. The empty product is a singleton. Lean proves that the coordinate type has cardinality $d^k$.","The represented matrix is defined recursively using the actual Kronecker product:\n\\[\nM_0=[1],\\qquad M_{k+1}=A_0\\otimes M_k(A_1,\\ldots,A_k).\n\\]\nThe algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies $A_0$ to the resulting first-coordinate fibers:\n\\[\ny_{i,u}=\\sum_{j=0}^{d-1}(A_0)_{ij}\\,\n\\operatorname{Apply}_{k}(A_1,\\ldots,A_k;x_{j,\\cdot})_u.\n\\]\nInduction, the Kronecker entry formula, and finite-sum distributivity prove\n\\[\n\\boxed{\\operatorname{Apply}_k(A;x)=M_k(A)x.}\n\\]\nThus correctness is not inferred merely from agreeing dimension or storage formulas.","otimes M_k(A_1,\\ldots,A_k).\n\\]\nThe algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies $A_0$ to the resulting first-coordinate fibers:\n\\[\ny_{i,u}=\\sum_{j=0}^{d-1}(A_0)_{ij}\\,\n\\operatorname{Apply}_{k}(A_1,\\ldots,A_k;x_{j,\\cdot})_u.\n\\]\nInduction, the Kronecker entry formula, and finite-sum distributivity prove\n\\[\n\\boxed{\\operatorname{Apply}_k(A;x)=M_k(A)x.}\n\\]\nThus correctness is not inferred merely from agreeing dimension or storage formulas.","\\textbf{Operation events.} The event type for $k+1$ modes is a disjoint union of $d$ copies of the trailing event type and the events for the final fiber contraction. Its cardinality obeys\n\\[\nW_0=0,\\qquad W_{k+1}=dW_k+2d^{k+2}.\n\\]\nLean proves the closed form\n\\[\n\\boxed{W_k=2kd^{k+1}.}\n\\]\nThe factor-address type has cardinality\n\\[\n\\boxed{k d^2.}\n\\]\nThese statements include $k=0$ and $d=0$. With no factors, the action is the identity on the singleton coordinate type and incurs zero modeled arithmetic. With $d=0$ and at least one factor, the coordinate type is empty.","ent type and the events for the final fiber contraction. Its cardinality obeys\n\\[\nW_0=0,\\qquad W_{k+1}=dW_k+2d^{k+2}.\n\\]\nLean proves the closed form\n\\[\n\\boxed{W_k=2kd^{k+1}.}\n\\]\nThe factor-address type has cardinality\n\\[\n\\boxed{k d^2.}\n\\]\nThese statements include $k=0$ and $d=0$. With no factors, the action is the identity on the singleton coordinate type and incurs zero modeled arithmetic. With $d=0$ and at least one factor, the coordinate type is empty.","\\subsection{Walsh butterfly and exact addition count}\\label{supp4:subsection:004}\\textit{Source item 4.7; module WalshTransform}\\par\nThe two-point matrix is constructed as\n\\[\nH_2=\\begin{pmatrix}1&1\\\\1&-1\\end{pmatrix},\n\\]\nand the full family is defined by $H_1=[1]$ and\n\\[\nH_{2N}=H_2\\otimes H_N\n=\\begin{pmatrix}H_N&H_N","1&-1\\end{pmatrix},\n\\]\nand the full family is defined by $H_1=[1]$ and\n\\[\nH_{2N}=H_2\\otimes H_N\n=\\begin{pmatrix}H_N&H_N","1&-1\\end{pmatrix},\n\\]\nand the full family is defined by $H_1=[1]$ and\n\\[\nH_{2N}=H_2\\otimes H_N\n=\\begin{pmatrix}H_N&H_N\\\\H_N&-H_N\\end{pmatrix}.\n\\]\nThe two-point orthogonality identity is checked explicitly. Kronecker multiplication and induction then prove\n\\[\n\\boxed{H_NH_N^T=NI,\\qquad N=2^k.}\n\\]\nThe ordinary transpose is sufficient because all constructed entries are real, even though the scalar field is $\\C$.","H_N&-H_N\\end{pmatrix}.\n\\]\nThe two-point orthogonality identity is checked explicitly. Kronecker multiplication and induction then prove\n\\[\n\\boxed{H_NH_N^T=NI,\\qquad N=2^k.}\n\\]\nThe ordinary transpose is sufficient because all constructed entries are real, even though the scalar field is $\\C$.","\\textbf{Actual butterfly.} For the two halves $x_0,x_1$ of the input, the independently defined procedure returns\n\\[\n\\operatorname{Walsh}_{2N}(x_0,x_1)\n=\\bigl(\\operatorname{Walsh}_N(x_0+x_1),\n       \\operatorname{Walsh}_N(x_0-x_1)\\bigr).\n\\]\nLean proves that this vector equals $H_{2N}x$. The argument expands the two first-coordinate cases and applies finite-sum distributivity. It does not assume that the recursive algorithm already represents the matrix.","\\textbf{Exact arithmetic count.} Each pair of slices forms one sum and one difference at every trailing coordinate. The event type therefore gives\n\\[\nA_0=0,\\qquad A_{k+1}=2A_k+2^{k+1}.\n\\]\nThe closed form and its logarithmic re-expression are both checked:\n\\[\n\\boxed{A_k=k2^k=N\\log_2N.}\n\\]\nThe logarithm in the formal count is the natural-number binary logarithm, with its power-of-two identity proved by Mathlib. The case $N=1$ has zero arithmetic events.","c count.} Each pair of slices forms one sum and one difference at every trailing coordinate. The event type therefore gives\n\\[\nA_0=0,\\qquad A_{k+1}=2A_k+2^{k+1}.\n\\]\nThe closed form and its logarithmic re-expression are both checked:\n\\[\n\\boxed{A_k=k2^k=N\\log_2N.}\n\\]\nThe logarithm in the formal count is the natural-number binary logarithm, with its power-of-two identity proved by Mathlib. The case $N=1$ has zero arithmetic events.","A further action theorem gives\n\\[\nH_N(H_N^Tx)=Nx,\n\\]\nwhich identifies the inverse after division by the nonzero scalar $N$. This result concerns exact arithmetic, not floating-point error accumulation through a butterfly implementation.","\\subsection{Clipped precision: existence and multiplier uniqueness}\\label{supp4:subsection:005}\\textit{Source item 8.5; module ClippedPrecision}\\par\nLet $E$ be a finite nonempty set and suppose\n\\[\ns_e>0,\\qquad c_e>0,\\qquad m_e>0,\n\\qquad 0<\\tau\\le\\sum_e s_em_e.\n\\]\nThe feasible set is\n\\[\n0<\\varepsilon_e\\le m_e,\\qquad \\sum_e s_e\\varepsilon_e=\\tau,\n\\]\nand the objective is the same logarithmic objective used in item 8.4:\n\\[\nJ(\\varepsilon)=\\sum_e c_e\\log(1/\\varepsilon_e).\n\\]\nStrictly positive caps are stated explicitly. A zero or negative cap cannot admit a positive tolerance at that coordinate.","ecision: existence and multiplier uniqueness}\\label{supp4:subsection:005}\\textit{Source item 8.5; module ClippedPrecision}\\par\nLet $E$ be a finite nonempty set and suppose\n\\[\ns_e>0,\\qquad c_e>0,\\qquad m_e>0,\n\\qquad 0<\\tau\\le\\sum_e s_em_e.\n\\]\nThe feasible set is\n\\[\n0<\\varepsilon_e\\le m_e,\\qquad \\sum_e s_e\\varepsilon_e=\\tau,\n\\]\nand the objective is the same logarithmic objective used in item 8.4:\n\\[\nJ(\\varepsilon)=\\sum_e c_e\\log(1/\\varepsilon_e).\n\\]\nStrictly positive caps are stated explicitly. A zero or negative cap cannot admit a positive tolerance at that coordinate.","\\textbf{Constructive parameterization.} Write $t=1/\\lambda$ and set\n\\[\nu_e(t)=\\min\\{m_e,tc_e/s_e\\},\\qquad\n\\Phi(t)=\\sum_e s_eu_e(t).\n\\]\nLean proves that $\\Phi$ is continuous and nondecreasing, with $\\Phi(0)=0$. At the finite positive scale\n\\[\nT=\\sum_e\\frac{m_es_e}{c_e},\n\\]\nevery coordinate is saturated, so $u(T)=m$ and $\\Phi(T)=\\sum_es_em_e$. The intermediate value theorem produces $t>0$ with $\\Phi(t)=\\tau$. The multiplier is not assumed to exist.","\\textbf{Strictness below saturation.} If $t<u$ and $\\Phi(t)$ is below the maximum, at least one coordinate is unsaturated at $t$. That coordinate strictly increases; every other coordinate is nondecreasing. Hence $\\Phi(t)<\\Phi(u)$. This proves uniqueness of the scale and, by reciprocation, the exact manuscript statement\n\\[\n\\boxed{\\exists!\\lambda>0:\\quad\n\\sum_e s_e\\min\\!\\left\\{m_e,\\frac{c_e}{\\lambda s_e}\\right\\}=\\tau}\n\\]\nwhen $0<\\tau<\\sum_es_em_e$.","ation.} If $t<u$ and $\\Phi(t)$ is below the maximum, at least one coordinate is unsaturated at $t$. That coordinate strictly increases; every other coordinate is nondecreasing. Hence $\\Phi(t)<\\Phi(u)$. This proves uniqueness of the scale and, by reciprocation, the exact manuscript statement\n\\[\n\\boxed{\\exists!\\lambda>0:\\quad\n\\sum_e s_e\\min\\!\\left\\{m_e,\\frac{c_e}{\\lambda s_e}\\right\\}=\\tau}\n\\]\nwhen $0<\\tau<\\sum_es_em_e$.","At the maximal budget, the minimizing vector remains unique but the multiplier need not be. The one-coordinate case $s=c=m=1$ is checked: both $\\lambda=1$ and $\\lambda=1/2$ produce the same tolerance $1$.","Fix the feasible vector $u=u(t)$ obtained above and any feasible competitor $\\varepsilon$. For each coordinate, Lean proves\n\\[\n\\boxed{\n c_e\\left(\\frac{\\varepsilon_e}{u_e}-1\\right)\n \\le \\frac{s_e}{t}(\\varepsilon_e-u_e).\n}\n\\]\nWhen $u_e=tc_e/s_e$, this is equality. When $u_e=m_e$, the capped condition implies $s_e/t\\le c_e/m_e$, while $\\varepsilon_e-m_e\\le0$. Multiplying by this nonpositive difference gives the required inequality in the correct direction.","The equal-budget constraint then yields\n\\[\n\\sum_e c_e\\left(\\frac{\\varepsilon_e}{u_e}-1\\right)\n\\le\\frac1t\\sum_e s_e(\\varepsilon_e-u_e)=0.\n\\]\nApplying the checked inequality $\\log z\\le z-1$ for $z>0$ gives\n\\[\n\\sum_e c_e\\log(\\varepsilon_e/u_e)\\le0.\n\\]\nSince\n\\[\nJ(\\varepsilon)-J(u)\n=-\\sum_e c_e\\log(\\varepsilon_e/u_e),\n\\]\nwe obtain $J(u)\\le J(\\varepsilon)$ for every feasible competitor.","If $\\varepsilon\\ne u$, one coordinate has ratio different from $1$. The logarithmic inequality is strict there, and its positive weight preserves strictness in the sum. Consequently,\n\\[\n\\boxed{\\varepsilon\\ne u\\quad\\Longrightarrow\\quad J(u)<J(\\varepsilon).}\n\\]\nThis proves the unique global minimizing vector for every positive feasible budget, including the maximal endpoint.","A separate complex128 Fourier regression covered 96 cases at lengths 1 through 24. Maximum absolute errors were approximately $3.34\\times10^{-14}$ for diagonalization, $2.98\\times10^{-15}$ for unitarity, and $2.04\\times10^{-13}$ for application, all below the declared $10^{-10}$ threshold. These numerical checks are not substituted for the symbolic Lean proof.","\\subsection{Constructing the optimal level schedule}\\label{supp5:subsection:001}\\textit{Source items 6.6 and 10.3; module GraphSchedules}\\par\nLet $V$ be finite and let $u\\to v$ mean that task $v$ depends on task $u$. Assume there is no nonempty directed cycle. A topological order or valid level function is \\emph{not} supplied as an additional hypothesis.","For each vertex, define its strict ancestor set\n\\[\n\\operatorname{Anc}(v)=\\{u:\\text{there is a positive-length path }u\\leadsto v\\}.\n\\]\nFor every edge $u\\to v$, the proof establishes\n\\[\n\\operatorname{Anc}(u)\\subsetneq\\operatorname{Anc}(v).\n\\]\nThe inclusion follows by appending the edge. It is strict because $u$ is an ancestor of $v$ but cannot be its own strict ancestor in an acyclic graph. Cardinality therefore decreases when following predecessors. This constructs a well-founded relation on which the level recursion is defined:\n\\[\n\\boxed{\\ell(v)=1+\\max_{u\\to v}\\ell(u),\\qquad \\max\\varnothing=0.}\n\\]","r every edge $u\\to v$, the proof establishes\n\\[\n\\operatorname{Anc}(u)\\subsetneq\\operatorname{Anc}(v).\n\\]\nThe inclusion follows by appending the edge. It is strict because $u$ is an ancestor of $v$ but cannot be its own strict ancestor in an acyclic graph. Cardinality therefore decreases when following predecessors. This constructs a well-founded relation on which the level recursion is defined:\n\\[\n\\boxed{\\ell(v)=1+\\max_{u\\to v}\\ell(u),\\qquad \\max\\varnothing=0.}\n\\]","\\textbf{Validity and attainment.} Lean proves $\\ell(v)>0$ and $\\ell(u)<\\ell(v)$ on every edge. A path of $n$ edges from $a$ to $b$ satisfies\n\\[\n\\ell(a)+n\\leq\\ell(b).\n\\]\nConversely, predecessor-max attainment and induction produce an actual path with $\\ell(v)-1$ edges ending at each vertex $v$. The result is an equality with a longest path, not merely an upper bound.","A valid schedule $s:V\\to\\mathbb N$ assigns positive stages and strictly increases on each edge. Induction proves the stronger pointwise statement\n\\[\n\\ell(v)\\leq s(v)\\quad\\text{for every }v.\n\\]\nConsequently the constructed schedule achieves the least possible makespan,\n\\[\n\\boxed{d=\\max_{v\\in V}\\ell(v).}\n\\]\nFor a nonempty graph, $d$ equals the number of vertices on an attained longest path. Vertices of the same level have no dependency edge in either direction.","\\textbf{Empty graph and interpretation.} When $V=\\varnothing$, the checked depth is zero. The phrase ``one plus the longest edge-path length'' applies to nonempty graphs. These are ideal discrete scheduling statements. Weighted task costs, a finite processor budget, launch latency and communication are not part of this model. The definitions are mathematical constructions, not a certified hardware scheduler.","\\subsection{Forward substitution with an actual stage evaluator}\\label{supp5:subsection:002}\\textit{Source item 6.5; module TriangularSolve}\\par\nUse the graph and levels constructed on the previous page. A vertex $v$ carries an arbitrary vector space $X_v$ over a field. The spaces may have different dimensions. Let\n\\[\nD_v:X_v\\overset{\\sim}{\\longrightarrow}X_v,\n\\qquad T_{vu}:X_u\\longrightarrow X_v\n\\]\nbe an invertible diagonal map and linear transfer maps. The operator being solved is defined by its actual block action:\n\\[\n(Ax)_v=D_vx_v+\\sum_{u\\to v}T_{vu}x_u.\n\\]\nNo solution, solve equation or precomputed topological certificate is assumed.","iangularSolve}\\par\nUse the graph and levels constructed on the previous page. A vertex $v$ carries an arbitrary vector space $X_v$ over a field. The spaces may have different dimensions. Let\n\\[\nD_v:X_v\\overset{\\sim}{\\longrightarrow}X_v,\n\\qquad T_{vu}:X_u\\longrightarrow X_v\n\\]\nbe an invertible diagonal map and linear transfer maps. The operator being solved is defined by its actual block action:\n\\[\n(Ax)_v=D_vx_v+\\sum_{u\\to v}T_{vu}x_u.\n\\]\nNo solution, solve equation or precomputed topological certificate is assumed.","\\textbf{Constructed solution.} Well-founded recursion defines\n\\[\n\\boxed{x_v=D_v^{-1}\\!\\left(b_v-\\sum_{u\\to v}T_{vu}x_u\\right).}\n\\]\nApplying $D_v$ proves the original equation at $v$. If another vector satisfies all equations, induction over predecessors makes each previously used coordinate identical, and invertibility of $D_v$ forces agreement at $v$. Thus the construction gives existence and uniqueness, and the block operator is bijective.","\\textbf{Parallel stages.} A second, independent definition is an explicit staged evaluator. Set $x^{(0)}=0$ and\n\\[\nx^{(k+1)}_v=\n\\begin{cases}\nD_v^{-1}\\!\\left(b_v-\\displaystyle\\sum_{u\\to v}T_{vu}x^{(k)}_u\\right),\n  &\\ell(v)=k+1,","\\textbf{Parallel stages.} A second, independent definition is an explicit staged evaluator. Set $x^{(0)}=0$ and\n\\[\nx^{(k+1)}_v=\n\\begin{cases}\nD_v^{-1}\\!\\left(b_v-\\displaystyle\\sum_{u\\to v}T_{vu}x^{(k)}_u\\right),\n  &\\ell(v)=k+1,\\\\[5pt]\nx^{(k)}_v,&\\ell(v)\\ne k+1.\n\\end{cases}\n\\]\nIt reads the previous-stage vector, not partially updated coordinates. The invariant is\n\\[\n\\ell(v)\\leq k\\quad\\Longrightarrow\\quad x^{(k)}_v=x_v.\n\\]\nEvery predecessor of a level-$k+1$ vertex has level at most $k$, so the recurrence uses only coordinates already proved correct. At the canonical depth $d$ this yields\n\\[\n\\boxed{Ax^{(d)}=b.}\n\\]\nThe local independence theorem separately proves that changing coordinates outside the strict earlier levels cannot change a vertex's local solve.","[5pt]\nx^{(k)}_v,&\\ell(v)\\ne k+1.\n\\end{cases}\n\\]\nIt reads the previous-stage vector, not partially updated coordinates. The invariant is\n\\[\n\\ell(v)\\leq k\\quad\\Longrightarrow\\quad x^{(k)}_v=x_v.\n\\]\nEvery predecessor of a level-$k+1$ vertex has level at most $k$, so the recurrence uses only coordinates already proved correct. At the canonical depth $d$ this yields\n\\[\n\\boxed{Ax^{(d)}=b.}\n\\]\nThe local independence theorem separately proves that changing coordinates outside the strict earlier levels cannot change a vertex's local solve.","\\subsection{Support of inverses and Schur elimination}\\label{supp5:subsection:003}\\textit{Source items 6.7 and 6.8; modules InverseSupport and SchurSupport}\\par\nLet $(P_a)$ be a complete idempotent packet family in a finite-dimensional algebra. Define $a\\leadsto b$ by an actual finite path through nonzero blocks $P_bAP_a$. \\textbf{Zero-length paths are included}, so every packet reaches itself.","Fix the source packet $a$. In the regular representation, define the submodule\n\\[\nW_a=\\{x:P_bx=0\\ \\text{whenever }a\\not\\leadsto b\\}.\n\\]\nIt contains $P_a$. Completeness of the projectors expands each component of $Ax$ into packet transfers. If a transfer into an unreachable packet were nonzero from a reachable packet, it would extend a path, a contradiction. Thus $AW_a\\subseteq W_a$.","Suppose $BA=I$. Left multiplication by $A$ is injective, including on $W_a$. Finite-dimensionality makes that restricted map surjective. Given $x\\in W_a$, choose $y\\in W_a$ with $Ay=x$. Then $Bx=BAy=y\\in W_a$. Applying this to $P_a$ proves\n\\[\n\\boxed{a\\not\\leadsto b\\quad\\Longrightarrow\\quad P_bBP_a=0.}\n\\]\nThe matrix specialization takes $B$ to Mathlib's actual nonsingular matrix inverse, with an invertible determinant. No inverse-sparsity hypothesis is inserted.","\\textbf{Hidden elimination.} For the actual Schur complement\n\\[\nS=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR},\n\\]\nthe hidden inverse is expanded through the hidden projectors. A nonzero retained correction block requires indices $j,k$ with nonzero entrance and exit blocks and a hidden path $j\\leadsto k$. Therefore a nonzero block $S_{ba}$ requires either a direct retained block, or\n\\[\na\\longrightarrow j\\ \\leadsto_{H}\\ k\\longrightarrow b.\n\\]\nThe entrance and exit matrices may be rectangular; retained and hidden coordinate sizes and packet systems are independently typed. The hidden matrix must be invertible. A zero-length hidden path permits entrance and exit through the same hidden packet.","\\textbf{Inclusion is not equality.} Cancellation can remove a block permitted by a graph route. For example,\n\\[\nA=\\begin{pmatrix}1&1&1\\\\0&1&1\\\\0&0&1\\end{pmatrix}\n\\quad\\text{has}\\quad (A^{-1})_{13}=0\n\\]\ndespite a direct nonzero $(1,3)$ block. The implementation proves support inclusion and the corresponding no-route zero certificate, not a converse.","0&0&1\\end{pmatrix}\n\\quad\\text{has}\\quad (A^{-1})_{13}=0\n\\]\ndespite a direct nonzero $(1,3)$ block. The implementation proves support inclusion and the corresponding no-route zero certificate, not a converse.","\\subsection{Exact banded addresses and modeled work}\\label{supp5:subsection:004}\\textit{Source item 4.1; module BandedCount}\\par\nFor $p$ packets and half-bandwidth $w$, the actual address set is\n\\[\n\\mathcal B_{p,w}\n=\\{(i,j)\\in\\{0,\\ldots,p-1\\}^2:\n i\\leq j+w\\ \\text{and}\\ j\\leq i+w\\}.\n\\]\nThe predicate is the integer version of $|i-j|\\leq w$. The count is first defined as a double sum of indicators, then identified with the cardinality of this subtype. It is not defined by the target closed form.","Under $0\\leq w<p$, the formal proof establishes\n\\[\n\\boxed{N_{p,w}=|\\mathcal B_{p,w}|=p(2w+1)-w(w+1).}\n\\]\nThe proof uses finite prefix counting, the newly exposed boundary row and column when $p$ increases, and the saturated initial band. An intermediate subtraction-free identity avoids relying on natural-number subtraction without the required inequality. The diagonal case $w=0$ gives exactly $p$ blocks, and the saturated case $w=p-1$ gives $p^2$.","oof establishes\n\\[\n\\boxed{N_{p,w}=|\\mathcal B_{p,w}|=p(2w+1)-w(w+1).}\n\\]\nThe proof uses finite prefix counting, the newly exposed boundary row and column when $p$ increases, and the saturated initial band. An intermediate subtraction-free identity avoids relying on natural-number subtraction without the required inequality. The diagonal case $w=0$ gives exactly $p$ blocks, and the saturated case $w=p-1$ gives $p^2$.","s\n\\[\n\\boxed{N_{p,w}=|\\mathcal B_{p,w}|=p(2w+1)-w(w+1).}\n\\]\nThe proof uses finite prefix counting, the newly exposed boundary row and column when $p$ increases, and the saturated initial band. An intermediate subtraction-free identity avoids relying on natural-number subtraction without the required inequality. The diagonal case $w=0$ gives exactly $p$ blocks, and the saturated case $w=p-1$ gives $p^2$.","\\textbf{Stored scalars.} With equal block width $b$, the storage-address type is\n\\[\n\\mathcal S=\\mathcal B_{p,w}\\times\\{0,\\ldots,b-1\\}^2.\n\\]\nIts cardinality is the actual number of represented scalar entries:\n\\[\n\\boxed{|\\mathcal S|=b^2N_{p,w}.}\n\\]","\\textbf{Arithmetic events.} For $q$ right-hand sides, the declared event type adds a right-hand-side index and one of two operation roles, multiplication or accumulation:\n\\[\n\\mathcal O=\\{0,1\\}\\times\\{0,\\ldots,q-1\\}\\times\\mathcal S.\n\\]\nHence\n\\[\n\\boxed{|\\mathcal O|=2qb^2N_{p,w}.}\n\\]\nThese identities include zero right-hand-side count. The abstract storage formula also admits zero block width. To use the manuscript's substitution $p=n/b$, one must separately require $b>0$ and $n=pb$.","$q$ right-hand sides, the declared event type adds a right-hand-side index and one of two operation roles, multiplication or accumulation:\n\\[\n\\mathcal O=\\{0,1\\}\\times\\{0,\\ldots,q-1\\}\\times\\mathcal S.\n\\]\nHence\n\\[\n\\boxed{|\\mathcal O|=2qb^2N_{p,w}.}\n\\]\nThese identities include zero right-hand-side count. The abstract storage formula also admits zero block width. To use the manuscript's substitution $p=n/b$, one must separately require $b>0$ and $n=pb$.","ent type adds a right-hand-side index and one of two operation roles, multiplication or accumulation:\n\\[\n\\mathcal O=\\{0,1\\}\\times\\{0,\\ldots,q-1\\}\\times\\mathcal S.\n\\]\nHence\n\\[\n\\boxed{|\\mathcal O|=2qb^2N_{p,w}.}\n\\]\nThese identities include zero right-hand-side count. The abstract storage formula also admits zero block width. To use the manuscript's substitution $p=n/b$, one must separately require $b>0$ and $n=pb$.","dds a right-hand-side index and one of two operation roles, multiplication or accumulation:\n\\[\n\\mathcal O=\\{0,1\\}\\times\\{0,\\ldots,q-1\\}\\times\\mathcal S.\n\\]\nHence\n\\[\n\\boxed{|\\mathcal O|=2qb^2N_{p,w}.}\n\\]\nThese identities include zero right-hand-side count. The abstract storage formula also admits zero block width. To use the manuscript's substitution $p=n/b$, one must separately require $b>0$ and $n=pb$.","\\subsection{Exact residual certificates for supplied packets}\\label{supp5:subsection:005}\\textit{Source item 9.2; module DiscoveryCertificate}\\par\nLet $(P_a)$ be a complete orthogonal projector system on a finite complex coordinate space. The projectors need not be coordinate masks. For an actual matrix $A$, define\n\\[\n\\Pi(A)=\\sum_aP_aAP_a,\n\\qquad E=A-\\Pi(A).\n\\]\nReconstruction gives the exact off-diagonal residual\n\\[\nE=\\sum_{b\\ne a}P_bAP_a.\n\\]","esidual certificates for supplied packets}\\label{supp5:subsection:005}\\textit{Source item 9.2; module DiscoveryCertificate}\\par\nLet $(P_a)$ be a complete orthogonal projector system on a finite complex coordinate space. The projectors need not be coordinate masks. For an actual matrix $A$, define\n\\[\n\\Pi(A)=\\sum_aP_aAP_a,\n\\qquad E=A-\\Pi(A).\n\\]\nReconstruction gives the exact off-diagonal residual\n\\[\nE=\\sum_{b\\ne a}P_bAP_a.\n\\]","\\textbf{Hilbert--Schmidt orthogonality.} For distinct ordered packet pairs $(b,a)\\ne(d,c)$, adjoint multiplication and projector orthogonality give\n\\[\n\\operatorname{tr}\\bigl((P_bAP_a)^*P_dBP_c\\bigr)=0.\n\\]\nIf $b\\ne d$, the central projector product vanishes. Otherwise $a\\ne c$, and cyclicity of trace brings $P_cP_a$ together. This is proved for arbitrary input matrices $A,B$.","Lean identifies the actual Frobenius norm with its trace expression,\n\\[\n\\norm{M}_{\\rm F}^2=\\operatorname{Re}\\operatorname{tr}(M^*M),\n\\]\nthen expands the finite sum and removes each cross term. The result is\n\\[\n\\boxed{\\norm{A-\\Pi(A)}_{\\rm F}^2\n=\\sum_{b\\ne a}\\norm{P_bAP_a}_{\\rm F}^2.}\n\\]\nNo Pythagorean equality for the target residual is assumed.","\\textbf{Operator norm.} In a separate scoped matrix norm, the triangle inequality gives\n\\[\n\\boxed{\\norm{A-\\Pi(A)}_2\n\\leq\\sum_{b\\ne a}\\norm{P_bAP_a}_2.}\n\\]\nHere $\\norm{\\cdot}_2$ is the Euclidean induced operator norm, not the Frobenius norm. The formal declarations use distinct norm instances, preventing the two certificates from being conflated. Self-adjointness is used for Frobenius orthogonality; the sum estimate itself requires only the complete packet decomposition.","\\subsection{Block-Jacobi normalization and the CG factor}\\label{supp5:subsection:006}\\textit{Source items 7.4 and 7.6; module BlockJacobi}\\par\nWork on a nonempty finite complex coordinate space with Euclidean operator norms. Assume $D$ and $A=D+K$ are positive definite. Define\n\\[\nS=(\\sqrt D)^{-1},\\qquad B=SAS,\\qquad E=SKS.\n\\]\nThe square root is the actual positive matrix square root supplied by continuous functional calculus; the inverse is the actual matrix inverse. Positive definiteness makes $\\sqrt D$ a unit. Its square identity and inverse identities prove\n\\[\nSDS=I,\\qquad B=I+E.\n\\]","tems 7.4 and 7.6; module BlockJacobi}\\par\nWork on a nonempty finite complex coordinate space with Euclidean operator norms. Assume $D$ and $A=D+K$ are positive definite. Define\n\\[\nS=(\\sqrt D)^{-1},\\qquad B=SAS,\\qquad E=SKS.\n\\]\nThe square root is the actual positive matrix square root supplied by continuous functional calculus; the inverse is the actual matrix inverse. Positive definiteness makes $\\sqrt D$ a unit. Its square identity and inverse identities prove\n\\[\nSDS=I,\\qquad B=I+E.\n\\]","complex coordinate space with Euclidean operator norms. Assume $D$ and $A=D+K$ are positive definite. Define\n\\[\nS=(\\sqrt D)^{-1},\\qquad B=SAS,\\qquad E=SKS.\n\\]\nThe square root is the actual positive matrix square root supplied by continuous functional calculus; the inverse is the actual matrix inverse. Positive definiteness makes $\\sqrt D$ a unit. Its square identity and inverse identities prove\n\\[\nSDS=I,\\qquad B=I+E.\n\\]","\\textbf{A direct inverse bound.} If $C=(I+E)^{-1}$ and $\\norm E_2\\leq\\eta<1$, then $C=I-EC$. Consequently\n\\[\n(1-\\eta)\\norm C_2\\leq 1,\n\\qquad \\norm{I+E}_2\\leq 1+\\eta.\n\\]\nThe constructed normalization is invertible because $S$ and $A$ are. Combining the estimates gives the actual condition number bound\n\\[\n\\boxed{\\kappa_2(B):=\\norm B_2\\norm{B^{-1}}_2\n\\leq\\frac{1+\\eta}{1-\\eta}.}\n\\]\nThis norm argument does not require a separately assumed Hermitian perturbation. Nonnegative $\\eta$ follows from the norm bound. Nonemptiness is explicit because the condition-number and norm-of-identity convention uses a nonzero coordinate space.","$, then $C=I-EC$. Consequently\n\\[\n(1-\\eta)\\norm C_2\\leq 1,\n\\qquad \\norm{I+E}_2\\leq 1+\\eta.\n\\]\nThe constructed normalization is invertible because $S$ and $A$ are. Combining the estimates gives the actual condition number bound\n\\[\n\\boxed{\\kappa_2(B):=\\norm B_2\\norm{B^{-1}}_2\n\\leq\\frac{1+\\eta}{1-\\eta}.}\n\\]\nThis norm argument does not require a separately assumed Hermitian perturbation. Nonnegative $\\eta$ follows from the norm bound. Nonemptiness is explicit because the condition-number and norm-of-identity convention uses a nonzero coordinate space.","\\textbf{The preconditioning corollary.} Define\n\\[\nq(\\kappa)=\\frac{\\sqrt\\kappa-1}{\\sqrt\\kappa+1}.\n\\]\nMonotonicity of square root and positive denominators prove that $q$ is monotone. Lean also proves $\\kappa_2(B)\\geq1$, $q(\\kappa)\\geq0$ for $\\kappa\\geq1$, and $q(\\kappa)<1$. Thus\n\\[\n\\boxed{q(\\kappa_2(B))\\leq q\\!\\left(\\frac{1+\\eta}{1-\\eta}\\right).}\n\\]\nFor $1\\leq\\kappa$ and $e_0\\geq0$, an explicit classical premise $e_m\\leq2q(\\kappa)^m e_0$ and a certified upper bound $\\kappa\\leq\\bar\\kappa$ yield $e_m\\leq2q(\\bar\\kappa)^m e_0$.","onditioning corollary.} Define\n\\[\nq(\\kappa)=\\frac{\\sqrt\\kappa-1}{\\sqrt\\kappa+1}.\n\\]\nMonotonicity of square root and positive denominators prove that $q$ is monotone. Lean also proves $\\kappa_2(B)\\geq1$, $q(\\kappa)\\geq0$ for $\\kappa\\geq1$, and $q(\\kappa)<1$. Thus\n\\[\n\\boxed{q(\\kappa_2(B))\\leq q\\!\\left(\\frac{1+\\eta}{1-\\eta}\\right).}\n\\]\nFor $1\\leq\\kappa$ and $e_0\\geq0$, an explicit classical premise $e_m\\leq2q(\\kappa)^m e_0$ and a certified upper bound $\\kappa\\leq\\bar\\kappa$ yield $e_m\\leq2q(\\bar\\kappa)^m e_0$.","t\\kappa-1}{\\sqrt\\kappa+1}.\n\\]\nMonotonicity of square root and positive denominators prove that $q$ is monotone. Lean also proves $\\kappa_2(B)\\geq1$, $q(\\kappa)\\geq0$ for $\\kappa\\geq1$, and $q(\\kappa)<1$. Thus\n\\[\n\\boxed{q(\\kappa_2(B))\\leq q\\!\\left(\\frac{1+\\eta}{1-\\eta}\\right).}\n\\]\nFor $1\\leq\\kappa$ and $e_0\\geq0$, an explicit classical premise $e_m\\leq2q(\\kappa)^m e_0$ and a certified upper bound $\\kappa\\leq\\bar\\kappa$ yield $e_m\\leq2q(\\bar\\kappa)^m e_0$.","Monotonicity of square root and positive denominators prove that $q$ is monotone. Lean also proves $\\kappa_2(B)\\geq1$, $q(\\kappa)\\geq0$ for $\\kappa\\geq1$, and $q(\\kappa)<1$. Thus\n\\[\n\\boxed{q(\\kappa_2(B))\\leq q\\!\\left(\\frac{1+\\eta}{1-\\eta}\\right).}\n\\]\nFor $1\\leq\\kappa$ and $e_0\\geq0$, an explicit classical premise $e_m\\leq2q(\\kappa)^m e_0$ and a certified upper bound $\\kappa\\leq\\bar\\kappa$ yield $e_m\\leq2q(\\bar\\kappa)^m e_0$.","Separate numerical tests passed 64 complex projector cases and 120 positive-matrix normalization/CG-factor cases. The largest squared-Frobenius identity residual was $1.42\\times10^{-13}$; the largest positive condition-bound violation was $6.66\\times10^{-15}$, within the declared tolerance. These are independent Python mirrors, not Lean execution, new universal proofs or GPU measurements.","\\subsection{Compact factors from actual column spaces}\\label{supp6:subsection:001}\\textit{Modules CompactRangeFactors and SmallCoreTransfer}\\par\nLet\n\\[\nL\\in\\C^{m\\times t},\\qquad R\\in\\C^{n\\times t},\\qquad A=LR^*.\n\\]\nUse the ordinary Euclidean inner products on the coordinate spaces. Define\n\\[\nS_L=\\operatorname{range}(L),\\quad r_L=\\dim S_L,\n\\qquad S_R=\\operatorname{range}(R),\\quad r_R=\\dim S_R.\n\\]\nThe implementation constructs an orthonormal basis of each actual range subspace. Let $Q_L$ and $Q_R$ be the matrices of the corresponding isometric inclusions. Express the range-restricted input maps in these bases to obtain $R_L,R_R$.","tit{Modules CompactRangeFactors and SmallCoreTransfer}\\par\nLet\n\\[\nL\\in\\C^{m\\times t},\\qquad R\\in\\C^{n\\times t},\\qquad A=LR^*.\n\\]\nUse the ordinary Euclidean inner products on the coordinate spaces. Define\n\\[\nS_L=\\operatorname{range}(L),\\quad r_L=\\dim S_L,\n\\qquad S_R=\\operatorname{range}(R),\\quad r_R=\\dim S_R.\n\\]\nThe implementation constructs an orthonormal basis of each actual range subspace. Let $Q_L$ and $Q_R$ be the matrices of the corresponding isometric inclusions. Express the range-restricted input maps in these bases to obtain $R_L,R_R$.","s CompactRangeFactors and SmallCoreTransfer}\\par\nLet\n\\[\nL\\in\\C^{m\\times t},\\qquad R\\in\\C^{n\\times t},\\qquad A=LR^*.\n\\]\nUse the ordinary Euclidean inner products on the coordinate spaces. Define\n\\[\nS_L=\\operatorname{range}(L),\\quad r_L=\\dim S_L,\n\\qquad S_R=\\operatorname{range}(R),\\quad r_R=\\dim S_R.\n\\]\nThe implementation constructs an orthonormal basis of each actual range subspace. Let $Q_L$ and $Q_R$ be the matrices of the corresponding isometric inclusions. Express the range-restricted input maps in these bases to obtain $R_L,R_R$.","inary Euclidean inner products on the coordinate spaces. Define\n\\[\nS_L=\\operatorname{range}(L),\\quad r_L=\\dim S_L,\n\\qquad S_R=\\operatorname{range}(R),\\quad r_R=\\dim S_R.\n\\]\nThe implementation constructs an orthonormal basis of each actual range subspace. Let $Q_L$ and $Q_R$ be the matrices of the corresponding isometric inclusions. Express the range-restricted input maps in these bases to obtain $R_L,R_R$.","The checked identities are\n\\[\n\\boxed{Q_L^*Q_L=I,\\quad L=Q_LR_L,\\quad R_L=Q_L^*L,}\n\\]\nwith the analogous identities on the right. The dimension theorem is not an input:\n\\[\n\\boxed{r_L=\\rank L,\\qquad r_R=\\rank R.}\n\\]\nThe coordinate factors have full row rank. The construction therefore handles rank-deficient input factors without retaining artificial column directions.","Define the small core by actual multiplication,\n\\[\nG=R_LR_R^*\\in\\C^{r_L\\times r_R}.\n\\]\nFinite matrix associativity and adjoint identities give\n\\[\n\\boxed{A=Q_LGQ_R^*,\\qquad \\rank G=\\rank A.}\n\\]\nThe rank identity follows from both multiplication inequalities and the left inverse obtained by compression. It is not inferred from agreeing dimensions.","\\subsection{Why a small-core optimum is a global optimum}\\label{supp6:subsection:002}\\textit{Module SmallCoreNorms; arbitrary ambient competitors}\\par\nFor rectangular isometries $Q,V$ define\n\\[\n\\mathcal L(G)=QGV^*,\\qquad \\mathcal C(B)=Q^*BV,\n\\qquad Q^*Q=I,\\quad V^*V=I.\n\\]\nThe following statements are proved for the actual matrix rank and actual matrix norms:\n\\[\n\\mathcal C(\\mathcal L(G))=G,\n\\qquad \\rank\\mathcal L(G)=\\rank G,\n\\qquad \\rank\\mathcal C(B)\\leq\\rank B,\n\\]\n\\[\n\\boxed{\\norm{\\mathcal L(G)}_\\nu=\\norm{G}_\\nu,\n\\qquad \\norm{\\mathcal C(B)}_\\nu\\leq\\norm B_\\nu,\n\\quad \\nu\\in\\{2,\\mathrm F\\}.}\n\\]\nFor the spectral norm, the proof uses the Gram matrix identity\n\\[\n(QG)^*(QG)=G^*G\n\\]\nand the $C^*$-norm square identity, together with right-side adjoint symmetry. Compression follows from submultiplicativity and the isometry norm bound, including empty spaces.","ubsection:002}\\textit{Module SmallCoreNorms; arbitrary ambient competitors}\\par\nFor rectangular isometries $Q,V$ define\n\\[\n\\mathcal L(G)=QGV^*,\\qquad \\mathcal C(B)=Q^*BV,\n\\qquad Q^*Q=I,\\quad V^*V=I.\n\\]\nThe following statements are proved for the actual matrix rank and actual matrix norms:\n\\[\n\\mathcal C(\\mathcal L(G))=G,\n\\qquad \\rank\\mathcal L(G)=\\rank G,\n\\qquad \\rank\\mathcal C(B)\\leq\\rank B,\n\\]\n\\[\n\\boxed{\\norm{\\mathcal L(G)}_\\nu=\\norm{G}_\\nu,\n\\qquad \\norm{\\mathcal C(B)}_\\nu\\leq\\norm B_\\nu,\n\\quad \\nu\\in\\{2,\\mathrm F\\}.}\n\\]\nFor the spectral norm, the proof uses the Gram matrix identity\n\\[\n(QG)^*(QG)=G^*G\n\\]\nand the $C^*$-norm square identity, together with right-side adjoint symmetry. Compression follows from submultiplicativity and the isometry norm bound, including empty spaces.","*V=I.\n\\]\nThe following statements are proved for the actual matrix rank and actual matrix norms:\n\\[\n\\mathcal C(\\mathcal L(G))=G,\n\\qquad \\rank\\mathcal L(G)=\\rank G,\n\\qquad \\rank\\mathcal C(B)\\leq\\rank B,\n\\]\n\\[\n\\boxed{\\norm{\\mathcal L(G)}_\\nu=\\norm{G}_\\nu,\n\\qquad \\norm{\\mathcal C(B)}_\\nu\\leq\\norm B_\\nu,\n\\quad \\nu\\in\\{2,\\mathrm F\\}.}\n\\]\nFor the spectral norm, the proof uses the Gram matrix identity\n\\[\n(QG)^*(QG)=G^*G\n\\]\nand the $C^*$-norm square identity, together with right-side adjoint symmetry. Compression follows from submultiplicativity and the isometry norm bound, including empty spaces.","ents are proved for the actual matrix rank and actual matrix norms:\n\\[\n\\mathcal C(\\mathcal L(G))=G,\n\\qquad \\rank\\mathcal L(G)=\\rank G,\n\\qquad \\rank\\mathcal C(B)\\leq\\rank B,\n\\]\n\\[\n\\boxed{\\norm{\\mathcal L(G)}_\\nu=\\norm{G}_\\nu,\n\\qquad \\norm{\\mathcal C(B)}_\\nu\\leq\\norm B_\\nu,\n\\quad \\nu\\in\\{2,\\mathrm F\\}.}\n\\]\nFor the spectral norm, the proof uses the Gram matrix identity\n\\[\n(QG)^*(QG)=G^*G\n\\]\nand the $C^*$-norm square identity, together with right-side adjoint symmetry. Compression follows from submultiplicativity and the isometry norm bound, including empty spaces.","For the Frobenius norm, trace cyclicity proves left and right isometric invariance. The proof also establishes the projection identity\n\\[\n\\norm{B-QQ^*B}_{\\mathrm F}^{2}\n=\\norm B_{\\mathrm F}^{2}-\\norm{Q^*B}_{\\mathrm F}^{2}.\n\\]\nNonnegativity of the left side gives left compression contractivity; the right version follows by adjoints. No norm-preservation or contractivity hypothesis is inserted.","Suppose $G_k$ is a best rank-at-most-$k$ approximation to $G$ in either norm. Take an \\emph{arbitrary} ambient matrix $B$ with $\\rank B\\leq k$. Compression is an admissible small-core competitor and\n\\[\n\\begin{aligned}\n\\norm{\\mathcal L(G)-\\mathcal L(G_k)}_\\nu\n&=\\norm{G-G_k}_\\nu","Suppose $G_k$ is a best rank-at-most-$k$ approximation to $G$ in either norm. Take an \\emph{arbitrary} ambient matrix $B$ with $\\rank B\\leq k$. Compression is an admissible small-core competitor and\n\\[\n\\begin{aligned}\n\\norm{\\mathcal L(G)-\\mathcal L(G_k)}_\\nu\n&=\\norm{G-G_k}_\\nu\\\\\n&\\leq\\norm{G-\\mathcal C(B)}_\\nu\\\\\n&=\\norm{\\mathcal C(\\mathcal L(G)-B)}_\\nu\\\\\n&\\leq\\norm{\\mathcal L(G)-B}_\\nu.\n\\end{aligned}\n\\]\nThis is the key global bridge: $B$ is not required to lie in the supplied row and column spaces. The next two pages discharge the intermediate small-core optimality obligation, rather than leaving it as a hypothesis in the final theorem.","&\\leq\\norm{\\mathcal L(G)-B}_\\nu.\n\\end{aligned}\n\\]\nThis is the key global bridge: $B$ is not required to lie in the supplied row and column spaces. The next two pages discharge the intermediate small-core optimality obligation, rather than leaving it as a hypothesis in the final theorem.","\\subsection{Spectral optimality from a kernel witness}\\label{supp6:subsection:003}\\textit{Modules DiagonalTruncation and SpectralOptimality}\\par\nLet $\\sigma_1\\geq\\cdots\\geq\\sigma_d\\geq0$ and\n\\[\nD=\\diag(\\sigma_1,\\ldots,\\sigma_d),\\qquad\nD_k=\\diag(\\sigma_1,\\ldots,\\sigma_{\\min(k,d)},0,\\ldots,0).\n\\]\nLean uses indices beginning at zero; the formulas on this page use the conventional one-based notation. The rank of $D_k$ is proved to be at most $k$, even when retained singular values are zero.","pectral optimality from a kernel witness}\\label{supp6:subsection:003}\\textit{Modules DiagonalTruncation and SpectralOptimality}\\par\nLet $\\sigma_1\\geq\\cdots\\geq\\sigma_d\\geq0$ and\n\\[\nD=\\diag(\\sigma_1,\\ldots,\\sigma_d),\\qquad\nD_k=\\diag(\\sigma_1,\\ldots,\\sigma_{\\min(k,d)},0,\\ldots,0).\n\\]\nLean uses indices beginning at zero; the formulas on this page use the conventional one-based notation. The rank of $D_k$ is proved to be at most $k$, even when retained singular values are zero.","el witness}\\label{supp6:subsection:003}\\textit{Modules DiagonalTruncation and SpectralOptimality}\\par\nLet $\\sigma_1\\geq\\cdots\\geq\\sigma_d\\geq0$ and\n\\[\nD=\\diag(\\sigma_1,\\ldots,\\sigma_d),\\qquad\nD_k=\\diag(\\sigma_1,\\ldots,\\sigma_{\\min(k,d)},0,\\ldots,0).\n\\]\nLean uses indices beginning at zero; the formulas on this page use the conventional one-based notation. The rank of $D_k$ is proved to be at most $k$, even when retained singular values are zero.","For $k<d$, take any $d\\times d$ matrix $B$ of rank at most $k$. Restrict its domain to the first $k+1$ coordinates using the actual coordinate inclusion $J$. Since\n\\[\n\\rank(BJ)\\leq\\rank B\\leq k<k+1,\n\\]\nrank-nullity supplies a nonzero vector $z$ with $BJz=0$. The vector $y=Jz$ is nonzero, satisfies $By=0$ and is supported in the first $k+1$ coordinates. The injection and the support property are checked separately.","Monotonicity of the diagonal entries gives\n\\[\n\\norm{Dy}^{2}\n=\\sum_{i=1}^{k+1}\\sigma_i^{2}|y_i|^{2}\n\\geq\\sigma_{k+1}^{2}\\norm y^{2}.\n\\]\nBecause $(D-B)y=Dy$, the operator-norm inequality implies\n\\[\n\\sigma_{k+1}\\norm y\n\\leq\\norm{(D-B)y}\n\\leq\\norm{D-B}_2\\norm y.\n\\]\nCancellation of the strictly positive norm of $y$ proves\n\\[\n\\boxed{\\sigma_{k+1}\\leq\\norm{D-B}_2.}\n\\]\nThe norm of the actual diagonal residual $D-D_k$ equals $\\sigma_{k+1}$, so $D_k$ attains the lower bound. If $k\\geq d$, the residual is zero and optimality follows from norm nonnegativity.","onicity of the diagonal entries gives\n\\[\n\\norm{Dy}^{2}\n=\\sum_{i=1}^{k+1}\\sigma_i^{2}|y_i|^{2}\n\\geq\\sigma_{k+1}^{2}\\norm y^{2}.\n\\]\nBecause $(D-B)y=Dy$, the operator-norm inequality implies\n\\[\n\\sigma_{k+1}\\norm y\n\\leq\\norm{(D-B)y}\n\\leq\\norm{D-B}_2\\norm y.\n\\]\nCancellation of the strictly positive norm of $y$ proves\n\\[\n\\boxed{\\sigma_{k+1}\\leq\\norm{D-B}_2.}\n\\]\nThe norm of the actual diagonal residual $D-D_k$ equals $\\sigma_{k+1}$, so $D_k$ attains the lower bound. If $k\\geq d$, the residual is zero and optimality follows from norm nonnegativity.","diagonal entries gives\n\\[\n\\norm{Dy}^{2}\n=\\sum_{i=1}^{k+1}\\sigma_i^{2}|y_i|^{2}\n\\geq\\sigma_{k+1}^{2}\\norm y^{2}.\n\\]\nBecause $(D-B)y=Dy$, the operator-norm inequality implies\n\\[\n\\sigma_{k+1}\\norm y\n\\leq\\norm{(D-B)y}\n\\leq\\norm{D-B}_2\\norm y.\n\\]\nCancellation of the strictly positive norm of $y$ proves\n\\[\n\\boxed{\\sigma_{k+1}\\leq\\norm{D-B}_2.}\n\\]\nThe norm of the actual diagonal residual $D-D_k$ equals $\\sigma_{k+1}$, so $D_k$ attains the lower bound. If $k\\geq d$, the residual is zero and optimality follows from norm nonnegativity.","ves\n\\[\n\\norm{Dy}^{2}\n=\\sum_{i=1}^{k+1}\\sigma_i^{2}|y_i|^{2}\n\\geq\\sigma_{k+1}^{2}\\norm y^{2}.\n\\]\nBecause $(D-B)y=Dy$, the operator-norm inequality implies\n\\[\n\\sigma_{k+1}\\norm y\n\\leq\\norm{(D-B)y}\n\\leq\\norm{D-B}_2\\norm y.\n\\]\nCancellation of the strictly positive norm of $y$ proves\n\\[\n\\boxed{\\sigma_{k+1}\\leq\\norm{D-B}_2.}\n\\]\nThe norm of the actual diagonal residual $D-D_k$ equals $\\sigma_{k+1}$, so $D_k$ attains the lower bound. If $k\\geq d$, the residual is zero and optimality follows from norm nonnegativity.","k+1}\\sigma_i^{2}|y_i|^{2}\n\\geq\\sigma_{k+1}^{2}\\norm y^{2}.\n\\]\nBecause $(D-B)y=Dy$, the operator-norm inequality implies\n\\[\n\\sigma_{k+1}\\norm y\n\\leq\\norm{(D-B)y}\n\\leq\\norm{D-B}_2\\norm y.\n\\]\nCancellation of the strictly positive norm of $y$ proves\n\\[\n\\boxed{\\sigma_{k+1}\\leq\\norm{D-B}_2.}\n\\]\nThe norm of the actual diagonal residual $D-D_k$ equals $\\sigma_{k+1}$, so $D_k$ attains the lower bound. If $k\\geq d$, the residual is zero and optimality follows from norm nonnegativity.","\\subsection{Frobenius optimality from kernel weights}\\label{supp6:subsection:004}\\textit{Modules KernelFrame and FrobeniusOptimality}\\par\nFor an arbitrary competitor $B$ with $\\rank B\\leq k$, construct an orthonormal matrix $Q$ spanning its actual kernel. Then\n\\[\nBQ=0,\\qquad Q^*Q=I,\\qquad q=\\dim\\ker B=d-\\rank B.\n\\]\nDefine the row weights\n\\[\nw_i=\\sum_{j=1}^{q}|Q_{ij}|^2.\n\\]\nThe proof derives\n\\[\n\\boxed{0\\leq w_i\\leq1,\\qquad \\sum_iw_i=q\\geq\\max(d-k,0).}\n\\]\nThe upper bound uses contraction of $Q^*$ on a coordinate vector. The weight sum follows from the trace of $Q^*Q$ and rank-nullity.","s KernelFrame and FrobeniusOptimality}\\par\nFor an arbitrary competitor $B$ with $\\rank B\\leq k$, construct an orthonormal matrix $Q$ spanning its actual kernel. Then\n\\[\nBQ=0,\\qquad Q^*Q=I,\\qquad q=\\dim\\ker B=d-\\rank B.\n\\]\nDefine the row weights\n\\[\nw_i=\\sum_{j=1}^{q}|Q_{ij}|^2.\n\\]\nThe proof derives\n\\[\n\\boxed{0\\leq w_i\\leq1,\\qquad \\sum_iw_i=q\\geq\\max(d-k,0).}\n\\]\nThe upper bound uses contraction of $Q^*$ on a coordinate vector. The weight sum follows from the trace of $Q^*Q$ and rank-nullity.","$B$ with $\\rank B\\leq k$, construct an orthonormal matrix $Q$ spanning its actual kernel. Then\n\\[\nBQ=0,\\qquad Q^*Q=I,\\qquad q=\\dim\\ker B=d-\\rank B.\n\\]\nDefine the row weights\n\\[\nw_i=\\sum_{j=1}^{q}|Q_{ij}|^2.\n\\]\nThe proof derives\n\\[\n\\boxed{0\\leq w_i\\leq1,\\qquad \\sum_iw_i=q\\geq\\max(d-k,0).}\n\\]\nThe upper bound uses contraction of $Q^*$ on a coordinate vector. The weight sum follows from the trace of $Q^*Q$ and rank-nullity.","For any nonnegative decreasing sequence $a_i$, weights with these properties satisfy\n\\[\n\\sum_i a_iw_i\\geq\\sum_{i>k}a_i.\n\\]\nThis finite weighted inequality is itself proved. For $k<d$, use $\\mu=a_{k+1}$ and compare the summands to tail weights $v_i=0$ for $i\\leq k$ and $v_i=1$ otherwise:\n\\[\na_iv_i+\\mu(w_i-v_i)\\leq a_iw_i.\n\\]\nFor a retained coordinate the inequality uses $a_i\\geq\\mu$ and $w_i\\geq0$. For a tail coordinate it uses $a_i\\leq\\mu$ and $w_i\\leq1$. Summation and $\\sum w_i\\geq\\sum v_i$ give the result.","equence $a_i$, weights with these properties satisfy\n\\[\n\\sum_i a_iw_i\\geq\\sum_{i>k}a_i.\n\\]\nThis finite weighted inequality is itself proved. For $k<d$, use $\\mu=a_{k+1}$ and compare the summands to tail weights $v_i=0$ for $i\\leq k$ and $v_i=1$ otherwise:\n\\[\na_iv_i+\\mu(w_i-v_i)\\leq a_iw_i.\n\\]\nFor a retained coordinate the inequality uses $a_i\\geq\\mu$ and $w_i\\geq0$. For a tail coordinate it uses $a_i\\leq\\mu$ and $w_i\\leq1$. Summation and $\\sum w_i\\geq\\sum v_i$ give the result.","ights with these properties satisfy\n\\[\n\\sum_i a_iw_i\\geq\\sum_{i>k}a_i.\n\\]\nThis finite weighted inequality is itself proved. For $k<d$, use $\\mu=a_{k+1}$ and compare the summands to tail weights $v_i=0$ for $i\\leq k$ and $v_i=1$ otherwise:\n\\[\na_iv_i+\\mu(w_i-v_i)\\leq a_iw_i.\n\\]\nFor a retained coordinate the inequality uses $a_i\\geq\\mu$ and $w_i\\geq0$. For a tail coordinate it uses $a_i\\leq\\mu$ and $w_i\\leq1$. Summation and $\\sum w_i\\geq\\sum v_i$ give the result.","s satisfy\n\\[\n\\sum_i a_iw_i\\geq\\sum_{i>k}a_i.\n\\]\nThis finite weighted inequality is itself proved. For $k<d$, use $\\mu=a_{k+1}$ and compare the summands to tail weights $v_i=0$ for $i\\leq k$ and $v_i=1$ otherwise:\n\\[\na_iv_i+\\mu(w_i-v_i)\\leq a_iw_i.\n\\]\nFor a retained coordinate the inequality uses $a_i\\geq\\mu$ and $w_i\\geq0$. For a tail coordinate it uses $a_i\\leq\\mu$ and $w_i\\leq1$. Summation and $\\sum w_i\\geq\\sum v_i$ give the result.","Set $a_i=\\sigma_i^2$. Right compression cannot increase the Frobenius norm, and $BQ=0$, hence\n\\[\n\\begin{aligned}\n\\norm{D-B}_{\\mathrm F}^{2}\n&\\geq\\norm{(D-B)Q}_{\\mathrm F}^{2}","Set $a_i=\\sigma_i^2$. Right compression cannot increase the Frobenius norm, and $BQ=0$, hence\n\\[\n\\begin{aligned}\n\\norm{D-B}_{\\mathrm F}^{2}\n&\\geq\\norm{(D-B)Q}_{\\mathrm F}^{2}\\\\\n&=\\norm{DQ}_{\\mathrm F}^{2}\n=\\sum_i\\sigma_i^2w_i\n\\geq\\sum_{i>k}\\sigma_i^2.\n\\end{aligned}\n\\]\nDirect calculation of $D-D_k$ attains equality. Thus the best-rank Frobenius assertion is proved from kernel geometry and finite sums, independently of the spectral lower-bound proof. Repeated and zero singular values require no uniqueness claim.","&=\\norm{DQ}_{\\mathrm F}^{2}\n=\\sum_i\\sigma_i^2w_i\n\\geq\\sum_{i>k}\\sigma_i^2.\n\\end{aligned}\n\\]\nDirect calculation of $D-D_k$ attains equality. Thus the best-rank Frobenius assertion is proved from kernel geometry and finite sums, independently of the spectral lower-bound proof. Repeated and zero singular values require no uniqueness claim.","\\subsection{Thin output factors and the final theorem}\\label{supp6:subsection:005}\\textit{Modules CompactTruncation and SmallCoreRecompression}\\par\nFor the actual core $G=R_LR_R^*$, supply ordered thin SVD data\n\\[\nG=W\\diag(\\sigma)Z^*,\\qquad W^*W=I,\n\\qquad Z^*Z=I,\\qquad \\sigma_1\\geq\\cdots\\geq\\sigma_d\\geq0.\n\\]\nThese are precisely the fields of the formal SVD structure. It contains no rank-optimality or tail-error field. A derived theorem checks the Gram eigenvector equation\n\\[\nG^*GZ=Z\\diag(\\sigma_1^2,\\ldots,\\sigma_d^2).\n\\]\nLet $h=\\min(d,k)$ and retain exactly the first $h$ column indices. The rectangular square-root mask has entries\n\\[\nM_{ij}=\\begin{cases}\\sqrt{\\sigma_j},&i=j,","05}\\textit{Modules CompactTruncation and SmallCoreRecompression}\\par\nFor the actual core $G=R_LR_R^*$, supply ordered thin SVD data\n\\[\nG=W\\diag(\\sigma)Z^*,\\qquad W^*W=I,\n\\qquad Z^*Z=I,\\qquad \\sigma_1\\geq\\cdots\\geq\\sigma_d\\geq0.\n\\]\nThese are precisely the fields of the formal SVD structure. It contains no rank-optimality or tail-error field. A derived theorem checks the Gram eigenvector equation\n\\[\nG^*GZ=Z\\diag(\\sigma_1^2,\\ldots,\\sigma_d^2).\n\\]\nLet $h=\\min(d,k)$ and retain exactly the first $h$ column indices. The rectangular square-root mask has entries\n\\[\nM_{ij}=\\begin{cases}\\sqrt{\\sigma_j},&i=j,","ompression}\\par\nFor the actual core $G=R_LR_R^*$, supply ordered thin SVD data\n\\[\nG=W\\diag(\\sigma)Z^*,\\qquad W^*W=I,\n\\qquad Z^*Z=I,\\qquad \\sigma_1\\geq\\cdots\\geq\\sigma_d\\geq0.\n\\]\nThese are precisely the fields of the formal SVD structure. It contains no rank-optimality or tail-error field. A derived theorem checks the Gram eigenvector equation\n\\[\nG^*GZ=Z\\diag(\\sigma_1^2,\\ldots,\\sigma_d^2).\n\\]\nLet $h=\\min(d,k)$ and retain exactly the first $h$ column indices. The rectangular square-root mask has entries\n\\[\nM_{ij}=\\begin{cases}\\sqrt{\\sigma_j},&i=j,","_R^*$, supply ordered thin SVD data\n\\[\nG=W\\diag(\\sigma)Z^*,\\qquad W^*W=I,\n\\qquad Z^*Z=I,\\qquad \\sigma_1\\geq\\cdots\\geq\\sigma_d\\geq0.\n\\]\nThese are precisely the fields of the formal SVD structure. It contains no rank-optimality or tail-error field. A derived theorem checks the Gram eigenvector equation\n\\[\nG^*GZ=Z\\diag(\\sigma_1^2,\\ldots,\\sigma_d^2).\n\\]\nLet $h=\\min(d,k)$ and retain exactly the first $h$ column indices. The rectangular square-root mask has entries\n\\[\nM_{ij}=\\begin{cases}\\sqrt{\\sigma_j},&i=j,","W^*W=I,\n\\qquad Z^*Z=I,\\qquad \\sigma_1\\geq\\cdots\\geq\\sigma_d\\geq0.\n\\]\nThese are precisely the fields of the formal SVD structure. It contains no rank-optimality or tail-error field. A derived theorem checks the Gram eigenvector equation\n\\[\nG^*GZ=Z\\diag(\\sigma_1^2,\\ldots,\\sigma_d^2).\n\\]\nLet $h=\\min(d,k)$ and retain exactly the first $h$ column indices. The rectangular square-root mask has entries\n\\[\nM_{ij}=\\begin{cases}\\sqrt{\\sigma_j},&i=j,\\\\0,&i\\ne j,\\end{cases}\n\\qquad 1\\leq i\\leq d,\\quad 1\\leq j\\leq h.\n\\]\nLean proves its index cardinality and $MM^*=D_k$. Define\n\\[\n\\boxed{L_k=Q_LWM,\\qquad R_k=Q_RZM.}\n\\]\nThe product equals $Q_LWD_kZ^*Q_R^*$ and has rank at most $k$. The principal theorem proves, for every ambient $B$ with $\\rank B\\leq k$,\n\\[\n\\boxed{\\norm{LR^*-L_kR_k^*}_\\nu\\leq\\norm{LR^*-B}_\\nu,\n\\quad\\nu\\in\\{2,\\mathrm F\\}.}\n\\]\nIt also gives the exact errors\n\\[\n\\norm{LR^*-L_kR_k^*}_2=\n\\begin{cases}\\sigma_{k+1},&k<d,","0,&i\\ne j,\\end{cases}\n\\qquad 1\\leq i\\leq d,\\quad 1\\leq j\\leq h.\n\\]\nLean proves its index cardinality and $MM^*=D_k$. Define\n\\[\n\\boxed{L_k=Q_LWM,\\qquad R_k=Q_RZM.}\n\\]\nThe product equals $Q_LWD_kZ^*Q_R^*$ and has rank at most $k$. The principal theorem proves, for every ambient $B$ with $\\rank B\\leq k$,\n\\[\n\\boxed{\\norm{LR^*-L_kR_k^*}_\\nu\\leq\\norm{LR^*-B}_\\nu,\n\\quad\\nu\\in\\{2,\\mathrm F\\}.}\n\\]\nIt also gives the exact errors\n\\[\n\\norm{LR^*-L_kR_k^*}_2=\n\\begin{cases}\\sigma_{k+1},&k<d,","uad 1\\leq i\\leq d,\\quad 1\\leq j\\leq h.\n\\]\nLean proves its index cardinality and $MM^*=D_k$. Define\n\\[\n\\boxed{L_k=Q_LWM,\\qquad R_k=Q_RZM.}\n\\]\nThe product equals $Q_LWD_kZ^*Q_R^*$ and has rank at most $k$. The principal theorem proves, for every ambient $B$ with $\\rank B\\leq k$,\n\\[\n\\boxed{\\norm{LR^*-L_kR_k^*}_\\nu\\leq\\norm{LR^*-B}_\\nu,\n\\quad\\nu\\in\\{2,\\mathrm F\\}.}\n\\]\nIt also gives the exact errors\n\\[\n\\norm{LR^*-L_kR_k^*}_2=\n\\begin{cases}\\sigma_{k+1},&k<d,\\\\0,&k\\geq d,\\end{cases}\n\\qquad\n\\norm{LR^*-L_kR_k^*}_{\\mathrm F}\n=\\left(\\sum_{i>k}\\sigma_i^2\\right)^{1/2}.\n\\]\nBoth the squared and square-root Frobenius formulas are checked. All these statements are combined in\\par\n\\path{KomorebiPaperVIII.SmallCore.small_core_optimal_recompression}.","\\textbf{Endpoints.} Rank zero produces the zero output; $k\\geq d$ reconstructs the original product. An empty SVD certificate is constructed for a zero core, and diagonal SVD certificates are constructed explicitly. Zero singular values are never inverted. At repeated singular values the result asserts optimality, not uniqueness of the minimizer.","The largest relative tail/reconstruction discrepancy was below $3.7\\times10^{-15}$, against a declared $2\\times10^{-9}$ tolerance. The mirror uses numerical rank thresholds and a numerical SVD to construct column bases. It is not compiled or extracted Lean code, a proof for all inputs, or a GPU benchmark. The formal result uses exact ranks and exact factorization equations.","node[font=\\small,align=center,anchor=east,font=\\scriptsize] at (0.74,2.9) {60};\n\\draw[black!13,line width=.25pt] (0.9,3.95) -- (8.9,3.95);\n\\node[font=\\small,align=center,anchor=east,font=\\scriptsize] at (0.74,3.95) {90};\n\\draw[->,line width=.6pt] (0.9,0.8) -- (9.02,0.8);\n\\draw[->,line width=.6pt] (0.9,0.8) -- (0.9,4.07);\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.9,0.25) {reuse count $R$};\n\\node[font=\\small,align=center,anchor=west,font=\\scriptsize] at (0.9,4.35) {toy time units};\n\\draw[line width=1.1pt] plot coordinates {(0.9,0.8) (1.566667,1.045) (2.233333,1.29) (2.9,1.535) (3.566667,1.78) (4.233333,2.025) (4.9,2.27) (5.566667,2.515) (6.233333,2.76) (6.9,3.005) (7.566667,3.25) (8.233333,3.495) (8.9,3.74)};\n\\draw[dashed,line width=1.1pt] plot coordinates {(0.9,1.85) (1.566667,1.92) (2.233333,1.99) (2.9,2.06) (3.566667,2.13) (4.233333,2.2) (4.9,2.27) (5.566667,2.34) (6.233333,2.41) (6.9,2.48) (7.566667,2.55) (8.233333,2.62) (8.9,2.69)};\n\\path[fill=black] (4.9,2.27) circle (0.075);\n\\node[font=\\small,align=center] at (11.6,3.4) {solid: $7R$};\n\\node[font=\\small,align=center]","33,2.76) (6.9,3.005) (7.566667,3.25) (8.233333,3.495) (8.9,3.74)};\n\\draw[dashed,line width=1.1pt] plot coordinates {(0.9,1.85) (1.566667,1.92) (2.233333,1.99) (2.9,2.06) (3.566667,2.13) (4.233333,2.2) (4.9,2.27) (5.566667,2.34) (6.233333,2.41) (6.9,2.48) (7.566667,2.55) (8.233333,2.62) (8.9,2.69)};\n\\path[fill=black] (4.9,2.27) circle (0.075);\n\\node[font=\\small,align=center] at (11.6,3.4) {solid: $7R$};\n\\node[font=\\small,align=center] at (11.6,2.3) {dashed: $30+2R$};\n\\node[font=\\small,align=center] at (11.6,1.2) {tie at $6$","4)};\n\\draw[dashed,line width=1.1pt] plot coordinates {(0.9,1.85) (1.566667,1.92) (2.233333,1.99) (2.9,2.06) (3.566667,2.13) (4.233333,2.2) (4.9,2.27) (5.566667,2.34) (6.233333,2.41) (6.9,2.48) (7.566667,2.55) (8.233333,2.62) (8.9,2.69)};\n\\path[fill=black] (4.9,2.27) circle (0.075);\n\\node[font=\\small,align=center] at (11.6,3.4) {solid: $7R$};\n\\node[font=\\small,align=center] at (11.6,2.3) {dashed: $30+2R$};\n\\node[font=\\small,align=center] at (11.6,1.2) {tie at $6$","(1.566667,1.92) (2.233333,1.99) (2.9,2.06) (3.566667,2.13) (4.233333,2.2) (4.9,2.27) (5.566667,2.34) (6.233333,2.41) (6.9,2.48) (7.566667,2.55) (8.233333,2.62) (8.9,2.69)};\n\\path[fill=black] (4.9,2.27) circle (0.075);\n\\node[font=\\small,align=center] at (11.6,3.4) {solid: $7R$};\n\\node[font=\\small,align=center] at (11.6,2.3) {dashed: $30+2R$};\n\\node[font=\\small,align=center] at (11.6,1.2) {tie at $6$","first win at $7$};\n\\end{tikzpicture}","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.17,4.6) {2$\\times$2};\n\\path[draw=black!30,fill=black!8] (0.75,3.48) rectangle (1.17,3.9);\n\\node[font=\\small,align=center,font=\\small] at (0.96,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (1.17,3.48) rectangle (1.59,3.9);\n\\node[font=\\small,align=center,font=\\small] at (1.38,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (0.75,3.06) rectangle (1.17,3.48);\n\\node[font=\\small,align=center,font=\\small] at (0.96,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (1.17,3.06) rectangle (1.59,3.48);\n\\node[font=\\small,align","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.17,4.6) {2$\\times$2};\n\\path[draw=black!30,fill=black!8] (0.75,3.48) rectangle (1.17,3.9);\n\\node[font=\\small,align=center,font=\\small] at (0.96,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (1.17,3.48) rectangle (1.59,3.9);\n\\node[font=\\small,align=center,font=\\small] at (1.38,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (0.75,3.06) rectangle (1.17,3.48);\n\\node[font=\\small,align=center,font=\\small] at (0.96,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (1.17,3.06) rectangle (1.59,3.48);\n\\node[font=\\small,align=center,font=\\small] at (1.38,3.27) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.3,1.8) {task 1};\n\\node[font=\\small,al","in=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.17,4.6) {2$\\times$2};\n\\path[draw=black!30,fill=black!8] (0.75,3.48) rectangle (1.17,3.9);\n\\node[font=\\small,align=center,font=\\small] at (0.96,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (1.17,3.48) rectangle (1.59,3.9);\n\\node[font=\\small,align=center,font=\\small] at (1.38,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (0.75,3.06) rectangle (1.17,3.48);\n\\node[font=\\small,align=center,font=\\small] at (0.96,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (1.17,3.06) rectangle (1.59,3.48);\n\\node[font=\\small,align=center,font=\\small] at (1.38,3.27) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.3,1.8) {task 1};\n\\node[font=\\small,align=center,font=\\small] at (3.58,4.6) {2$\\times$3};\n\\path[draw=black!30,fill=black!8] (2.95,3.48) rectangle (3.37,3.9);\n\\node[font=\\sm",";\n\\path[draw=black!30,fill=black!8] (0.75,3.48) rectangle (1.17,3.9);\n\\node[font=\\small,align=center,font=\\small] at (0.96,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (1.17,3.48) rectangle (1.59,3.9);\n\\node[font=\\small,align=center,font=\\small] at (1.38,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (0.75,3.06) rectangle (1.17,3.48);\n\\node[font=\\small,align=center,font=\\small] at (0.96,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (1.17,3.06) rectangle (1.59,3.48);\n\\node[font=\\small,align=center,font=\\small] at (1.38,3.27) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.3,1.8) {task 1};\n\\node[font=\\small,align=center,font=\\small] at (3.58,4.6) {2$\\times$3};\n\\path[draw=black!30,fill=black!8] (2.95,3.48) rectangle (3.37,3.9);\n\\node[font=\\small,align=center,font=\\small] at (3.16,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (3.37,3.48) rectangle (3.79,3.9);\n\\node[font=\\sma","\\path[draw=black!30,fill=black!8] (1.17,3.48) rectangle (1.59,3.9);\n\\node[font=\\small,align=center,font=\\small] at (1.38,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (0.75,3.06) rectangle (1.17,3.48);\n\\node[font=\\small,align=center,font=\\small] at (0.96,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (1.17,3.06) rectangle (1.59,3.48);\n\\node[font=\\small,align=center,font=\\small] at (1.38,3.27) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.3,1.8) {task 1};\n\\node[font=\\small,align=center,font=\\small] at (3.58,4.6) {2$\\times$3};\n\\path[draw=black!30,fill=black!8] (2.95,3.48) rectangle (3.37,3.9);\n\\node[font=\\small,align=center,font=\\small] at (3.16,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (3.37,3.48) rectangle (3.79,3.9);\n\\node[font=\\small,align=center,font=\\small] at (3.58,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (3.79,3.48) rectangle (4.21,3.9);\n\\node[font=\\smal","th[draw=black!30,fill=black!8] (0.75,3.06) rectangle (1.17,3.48);\n\\node[font=\\small,align=center,font=\\small] at (0.96,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (1.17,3.06) rectangle (1.59,3.48);\n\\node[font=\\small,align=center,font=\\small] at (1.38,3.27) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.3,1.8) {task 1};\n\\node[font=\\small,align=center,font=\\small] at (3.58,4.6) {2$\\times$3};\n\\path[draw=black!30,fill=black!8] (2.95,3.48) rectangle (3.37,3.9);\n\\node[font=\\small,align=center,font=\\small] at (3.16,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (3.37,3.48) rectangle (3.79,3.9);\n\\node[font=\\small,align=center,font=\\small] at (3.58,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (3.79,3.48) rectangle (4.21,3.9);\n\\node[font=\\small,align=center,font=\\small] at (4,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (2.95,3.06) rectangle (3.37,3.48);\n\\node[font=\\small,align=cen","aw=black!30,fill=black!8] (1.17,3.06) rectangle (1.59,3.48);\n\\node[font=\\small,align=center,font=\\small] at (1.38,3.27) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.3,1.8) {task 1};\n\\node[font=\\small,align=center,font=\\small] at (3.58,4.6) {2$\\times$3};\n\\path[draw=black!30,fill=black!8] (2.95,3.48) rectangle (3.37,3.9);\n\\node[font=\\small,align=center,font=\\small] at (3.16,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (3.37,3.48) rectangle (3.79,3.9);\n\\node[font=\\small,align=center,font=\\small] at (3.58,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (3.79,3.48) rectangle (4.21,3.9);\n\\node[font=\\small,align=center,font=\\small] at (4,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (2.95,3.06) rectangle (3.37,3.48);\n\\node[font=\\small,align=center,font=\\small] at (3.16,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (3.37,3.06) rectangle (3.79,3.48);\n\\node[font=\\small,align=cen","ont=\\small,align=center,font=\\scriptsize] at (1.3,1.8) {task 1};\n\\node[font=\\small,align=center,font=\\small] at (3.58,4.6) {2$\\times$3};\n\\path[draw=black!30,fill=black!8] (2.95,3.48) rectangle (3.37,3.9);\n\\node[font=\\small,align=center,font=\\small] at (3.16,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (3.37,3.48) rectangle (3.79,3.9);\n\\node[font=\\small,align=center,font=\\small] at (3.58,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (3.79,3.48) rectangle (4.21,3.9);\n\\node[font=\\small,align=center,font=\\small] at (4,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (2.95,3.06) rectangle (3.37,3.48);\n\\node[font=\\small,align=center,font=\\small] at (3.16,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (3.37,3.06) rectangle (3.79,3.48);\n\\node[font=\\small,align=center,font=\\small] at (3.58,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (3.79,3.06) rectangle (4.21,3.48);\n\\node[font=\\small,align=ce","s$3};\n\\path[draw=black!30,fill=black!8] (2.95,3.48) rectangle (3.37,3.9);\n\\node[font=\\small,align=center,font=\\small] at (3.16,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (3.37,3.48) rectangle (3.79,3.9);\n\\node[font=\\small,align=center,font=\\small] at (3.58,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (3.79,3.48) rectangle (4.21,3.9);\n\\node[font=\\small,align=center,font=\\small] at (4,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (2.95,3.06) rectangle (3.37,3.48);\n\\node[font=\\small,align=center,font=\\small] at (3.16,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (3.37,3.06) rectangle (3.79,3.48);\n\\node[font=\\small,align=center,font=\\small] at (3.58,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (3.79,3.06) rectangle (4.21,3.48);\n\\node[font=\\small,align=center,font=\\small] at (4,3.27) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.5,1.8) {task 2};\n\\node[font=\\small,alig","1$};\n\\path[draw=black!30,fill=black!8] (3.37,3.48) rectangle (3.79,3.9);\n\\node[font=\\small,align=center,font=\\small] at (3.58,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (3.79,3.48) rectangle (4.21,3.9);\n\\node[font=\\small,align=center,font=\\small] at (4,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (2.95,3.06) rectangle (3.37,3.48);\n\\node[font=\\small,align=center,font=\\small] at (3.16,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (3.37,3.06) rectangle (3.79,3.48);\n\\node[font=\\small,align=center,font=\\small] at (3.58,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (3.79,3.06) rectangle (4.21,3.48);\n\\node[font=\\small,align=center,font=\\small] at (4,3.27) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.5,1.8) {task 2};\n\\node[font=\\small,align=center,font=\\small] at (5.57,4.6) {3$\\times$2};\n\\path[draw=black!30,fill=black!8] (5.15,3.48) rectangle (5.57,3.9);\n\\node[font=\\small","$};\n\\path[draw=black!30,fill=black!8] (3.79,3.48) rectangle (4.21,3.9);\n\\node[font=\\small,align=center,font=\\small] at (4,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (2.95,3.06) rectangle (3.37,3.48);\n\\node[font=\\small,align=center,font=\\small] at (3.16,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (3.37,3.06) rectangle (3.79,3.48);\n\\node[font=\\small,align=center,font=\\small] at (3.58,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (3.79,3.06) rectangle (4.21,3.48);\n\\node[font=\\small,align=center,font=\\small] at (4,3.27) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.5,1.8) {task 2};\n\\node[font=\\small,align=center,font=\\small] at (5.57,4.6) {3$\\times$2};\n\\path[draw=black!30,fill=black!8] (5.15,3.48) rectangle (5.57,3.9);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (5.57,3.48) rectangle (5.99,3.9);\n\\node[font=\\small,","};\n\\path[draw=black!30,fill=black!8] (2.95,3.06) rectangle (3.37,3.48);\n\\node[font=\\small,align=center,font=\\small] at (3.16,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (3.37,3.06) rectangle (3.79,3.48);\n\\node[font=\\small,align=center,font=\\small] at (3.58,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (3.79,3.06) rectangle (4.21,3.48);\n\\node[font=\\small,align=center,font=\\small] at (4,3.27) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.5,1.8) {task 2};\n\\node[font=\\small,align=center,font=\\small] at (5.57,4.6) {3$\\times$2};\n\\path[draw=black!30,fill=black!8] (5.15,3.48) rectangle (5.57,3.9);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (5.57,3.48) rectangle (5.99,3.9);\n\\node[font=\\small,align=center,font=\\small] at (5.78,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (5.15,3.06) rectangle (5.57,3.48);\n\\node[font=\\sma","\\path[draw=black!30,fill=black!8] (3.37,3.06) rectangle (3.79,3.48);\n\\node[font=\\small,align=center,font=\\small] at (3.58,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (3.79,3.06) rectangle (4.21,3.48);\n\\node[font=\\small,align=center,font=\\small] at (4,3.27) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.5,1.8) {task 2};\n\\node[font=\\small,align=center,font=\\small] at (5.57,4.6) {3$\\times$2};\n\\path[draw=black!30,fill=black!8] (5.15,3.48) rectangle (5.57,3.9);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (5.57,3.48) rectangle (5.99,3.9);\n\\node[font=\\small,align=center,font=\\small] at (5.78,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (5.15,3.06) rectangle (5.57,3.48);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (5.57,3.06) rectangle (5.99,3.48);\n\\node[font=\\small,align","[draw=black!30,fill=black!8] (3.79,3.06) rectangle (4.21,3.48);\n\\node[font=\\small,align=center,font=\\small] at (4,3.27) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.5,1.8) {task 2};\n\\node[font=\\small,align=center,font=\\small] at (5.57,4.6) {3$\\times$2};\n\\path[draw=black!30,fill=black!8] (5.15,3.48) rectangle (5.57,3.9);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (5.57,3.48) rectangle (5.99,3.9);\n\\node[font=\\small,align=center,font=\\small] at (5.78,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (5.15,3.06) rectangle (5.57,3.48);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (5.57,3.06) rectangle (5.99,3.48);\n\\node[font=\\small,align=center,font=\\small] at (5.78,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (5.15,2.64) rectangle (5.57,3.06);\n\\node[font=\\small,align","ont=\\small,align=center,font=\\scriptsize] at (3.5,1.8) {task 2};\n\\node[font=\\small,align=center,font=\\small] at (5.57,4.6) {3$\\times$2};\n\\path[draw=black!30,fill=black!8] (5.15,3.48) rectangle (5.57,3.9);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (5.57,3.48) rectangle (5.99,3.9);\n\\node[font=\\small,align=center,font=\\small] at (5.78,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (5.15,3.06) rectangle (5.57,3.48);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (5.57,3.06) rectangle (5.99,3.48);\n\\node[font=\\small,align=center,font=\\small] at (5.78,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (5.15,2.64) rectangle (5.57,3.06);\n\\node[font=\\small,align=center,font=\\small] at (5.36,2.85) {$1$};\n\\path[draw=black!30,fill=black!8] (5.57,2.64) rectangle (5.99,3.06);\n\\node[font=\\small,alig",";\n\\path[draw=black!30,fill=black!8] (5.15,3.48) rectangle (5.57,3.9);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (5.57,3.48) rectangle (5.99,3.9);\n\\node[font=\\small,align=center,font=\\small] at (5.78,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (5.15,3.06) rectangle (5.57,3.48);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (5.57,3.06) rectangle (5.99,3.48);\n\\node[font=\\small,align=center,font=\\small] at (5.78,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (5.15,2.64) rectangle (5.57,3.06);\n\\node[font=\\small,align=center,font=\\small] at (5.36,2.85) {$1$};\n\\path[draw=black!30,fill=black!8] (5.57,2.64) rectangle (5.99,3.06);\n\\node[font=\\small,align=center,font=\\small] at (5.78,2.85) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.7,1.8) {task 3};\n\\node[font=\\small,a","\\path[draw=black!30,fill=black!8] (5.57,3.48) rectangle (5.99,3.9);\n\\node[font=\\small,align=center,font=\\small] at (5.78,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (5.15,3.06) rectangle (5.57,3.48);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (5.57,3.06) rectangle (5.99,3.48);\n\\node[font=\\small,align=center,font=\\small] at (5.78,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (5.15,2.64) rectangle (5.57,3.06);\n\\node[font=\\small,align=center,font=\\small] at (5.36,2.85) {$1$};\n\\path[draw=black!30,fill=black!8] (5.57,2.64) rectangle (5.99,3.06);\n\\node[font=\\small,align=center,font=\\small] at (5.78,2.85) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.7,1.8) {task 3};\n\\node[font=\\small,align=center,font=\\small] at (7.77,4.6) {2$\\times$2};\n\\path[draw=black!30,fill=black!8] (7.35,3.48) rectangle (7.77,3.9);\n\\node[font=\\sm","\\path[draw=black!30,fill=black!8] (5.15,3.06) rectangle (5.57,3.48);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (5.57,3.06) rectangle (5.99,3.48);\n\\node[font=\\small,align=center,font=\\small] at (5.78,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (5.15,2.64) rectangle (5.57,3.06);\n\\node[font=\\small,align=center,font=\\small] at (5.36,2.85) {$1$};\n\\path[draw=black!30,fill=black!8] (5.57,2.64) rectangle (5.99,3.06);\n\\node[font=\\small,align=center,font=\\small] at (5.78,2.85) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.7,1.8) {task 3};\n\\node[font=\\small,align=center,font=\\small] at (7.77,4.6) {2$\\times$2};\n\\path[draw=black!30,fill=black!8] (7.35,3.48) rectangle (7.77,3.9);\n\\node[font=\\small,align=center,font=\\small] at (7.56,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (7.77,3.48) rectangle (8.19,3.9);\n\\node[font=\\sma","\\path[draw=black!30,fill=black!8] (5.57,3.06) rectangle (5.99,3.48);\n\\node[font=\\small,align=center,font=\\small] at (5.78,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (5.15,2.64) rectangle (5.57,3.06);\n\\node[font=\\small,align=center,font=\\small] at (5.36,2.85) {$1$};\n\\path[draw=black!30,fill=black!8] (5.57,2.64) rectangle (5.99,3.06);\n\\node[font=\\small,align=center,font=\\small] at (5.78,2.85) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.7,1.8) {task 3};\n\\node[font=\\small,align=center,font=\\small] at (7.77,4.6) {2$\\times$2};\n\\path[draw=black!30,fill=black!8] (7.35,3.48) rectangle (7.77,3.9);\n\\node[font=\\small,align=center,font=\\small] at (7.56,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (7.77,3.48) rectangle (8.19,3.9);\n\\node[font=\\small,align=center,font=\\small] at (7.98,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (7.35,3.06) rectangle (7.77,3.48);\n\\node[font=\\sma","th[draw=black!30,fill=black!8] (5.15,2.64) rectangle (5.57,3.06);\n\\node[font=\\small,align=center,font=\\small] at (5.36,2.85) {$1$};\n\\path[draw=black!30,fill=black!8] (5.57,2.64) rectangle (5.99,3.06);\n\\node[font=\\small,align=center,font=\\small] at (5.78,2.85) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.7,1.8) {task 3};\n\\node[font=\\small,align=center,font=\\small] at (7.77,4.6) {2$\\times$2};\n\\path[draw=black!30,fill=black!8] (7.35,3.48) rectangle (7.77,3.9);\n\\node[font=\\small,align=center,font=\\small] at (7.56,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (7.77,3.48) rectangle (8.19,3.9);\n\\node[font=\\small,align=center,font=\\small] at (7.98,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (7.35,3.06) rectangle (7.77,3.48);\n\\node[font=\\small,align=center,font=\\small] at (7.56,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (7.77,3.06) rectangle (8.19,3.48);\n\\node[font=\\small,align","aw=black!30,fill=black!8] (5.57,2.64) rectangle (5.99,3.06);\n\\node[font=\\small,align=center,font=\\small] at (5.78,2.85) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.7,1.8) {task 3};\n\\node[font=\\small,align=center,font=\\small] at (7.77,4.6) {2$\\times$2};\n\\path[draw=black!30,fill=black!8] (7.35,3.48) rectangle (7.77,3.9);\n\\node[font=\\small,align=center,font=\\small] at (7.56,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (7.77,3.48) rectangle (8.19,3.9);\n\\node[font=\\small,align=center,font=\\small] at (7.98,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (7.35,3.06) rectangle (7.77,3.48);\n\\node[font=\\small,align=center,font=\\small] at (7.56,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (7.77,3.06) rectangle (8.19,3.48);\n\\node[font=\\small,align=center,font=\\small] at (7.98,3.27) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7.9,1.8) {task 4};\n\\node[font=\\small,al","ont=\\small,align=center,font=\\scriptsize] at (5.7,1.8) {task 3};\n\\node[font=\\small,align=center,font=\\small] at (7.77,4.6) {2$\\times$2};\n\\path[draw=black!30,fill=black!8] (7.35,3.48) rectangle (7.77,3.9);\n\\node[font=\\small,align=center,font=\\small] at (7.56,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (7.77,3.48) rectangle (8.19,3.9);\n\\node[font=\\small,align=center,font=\\small] at (7.98,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (7.35,3.06) rectangle (7.77,3.48);\n\\node[font=\\small,align=center,font=\\small] at (7.56,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (7.77,3.06) rectangle (8.19,3.48);\n\\node[font=\\small,align=center,font=\\small] at (7.98,3.27) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7.9,1.8) {task 4};\n\\node[font=\\small,align=center,font=\\small] at (9.97,4.6) {3$\\times$2};\n\\path[draw=black!30,fill=black!8] (9.55,3.48) rectangle (9.97,3.9);\n\\node[font=\\sm",";\n\\path[draw=black!30,fill=black!8] (7.35,3.48) rectangle (7.77,3.9);\n\\node[font=\\small,align=center,font=\\small] at (7.56,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (7.77,3.48) rectangle (8.19,3.9);\n\\node[font=\\small,align=center,font=\\small] at (7.98,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (7.35,3.06) rectangle (7.77,3.48);\n\\node[font=\\small,align=center,font=\\small] at (7.56,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (7.77,3.06) rectangle (8.19,3.48);\n\\node[font=\\small,align=center,font=\\small] at (7.98,3.27) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7.9,1.8) {task 4};\n\\node[font=\\small,align=center,font=\\small] at (9.97,4.6) {3$\\times$2};\n\\path[draw=black!30,fill=black!8] (9.55,3.48) rectangle (9.97,3.9);\n\\node[font=\\small,align=center,font=\\small] at (9.76,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (9.97,3.48) rectangle (10.39,3.9);\n\\node[font=\\sm","\\path[draw=black!30,fill=black!8] (7.77,3.48) rectangle (8.19,3.9);\n\\node[font=\\small,align=center,font=\\small] at (7.98,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (7.35,3.06) rectangle (7.77,3.48);\n\\node[font=\\small,align=center,font=\\small] at (7.56,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (7.77,3.06) rectangle (8.19,3.48);\n\\node[font=\\small,align=center,font=\\small] at (7.98,3.27) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7.9,1.8) {task 4};\n\\node[font=\\small,align=center,font=\\small] at (9.97,4.6) {3$\\times$2};\n\\path[draw=black!30,fill=black!8] (9.55,3.48) rectangle (9.97,3.9);\n\\node[font=\\small,align=center,font=\\small] at (9.76,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (9.97,3.48) rectangle (10.39,3.9);\n\\node[font=\\small,align=center,font=\\small] at (10.18,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (9.55,3.06) rectangle (9.97,3.48);\n\\node[font=\\s","th[draw=black!30,fill=black!8] (7.35,3.06) rectangle (7.77,3.48);\n\\node[font=\\small,align=center,font=\\small] at (7.56,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (7.77,3.06) rectangle (8.19,3.48);\n\\node[font=\\small,align=center,font=\\small] at (7.98,3.27) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7.9,1.8) {task 4};\n\\node[font=\\small,align=center,font=\\small] at (9.97,4.6) {3$\\times$2};\n\\path[draw=black!30,fill=black!8] (9.55,3.48) rectangle (9.97,3.9);\n\\node[font=\\small,align=center,font=\\small] at (9.76,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (9.97,3.48) rectangle (10.39,3.9);\n\\node[font=\\small,align=center,font=\\small] at (10.18,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (9.55,3.06) rectangle (9.97,3.48);\n\\node[font=\\small,align=center,font=\\small] at (9.76,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (9.97,3.06) rectangle (10.39,3.48);\n\\node[font=\\small,al","aw=black!30,fill=black!8] (7.77,3.06) rectangle (8.19,3.48);\n\\node[font=\\small,align=center,font=\\small] at (7.98,3.27) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7.9,1.8) {task 4};\n\\node[font=\\small,align=center,font=\\small] at (9.97,4.6) {3$\\times$2};\n\\path[draw=black!30,fill=black!8] (9.55,3.48) rectangle (9.97,3.9);\n\\node[font=\\small,align=center,font=\\small] at (9.76,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (9.97,3.48) rectangle (10.39,3.9);\n\\node[font=\\small,align=center,font=\\small] at (10.18,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (9.55,3.06) rectangle (9.97,3.48);\n\\node[font=\\small,align=center,font=\\small] at (9.76,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (9.97,3.06) rectangle (10.39,3.48);\n\\node[font=\\small,align=center,font=\\small] at (10.18,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (9.55,2.64) rectangle (9.97,3.06);\n\\node[font=\\small,a","t=\\small,align=center,font=\\scriptsize] at (7.9,1.8) {task 4};\n\\node[font=\\small,align=center,font=\\small] at (9.97,4.6) {3$\\times$2};\n\\path[draw=black!30,fill=black!8] (9.55,3.48) rectangle (9.97,3.9);\n\\node[font=\\small,align=center,font=\\small] at (9.76,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (9.97,3.48) rectangle (10.39,3.9);\n\\node[font=\\small,align=center,font=\\small] at (10.18,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (9.55,3.06) rectangle (9.97,3.48);\n\\node[font=\\small,align=center,font=\\small] at (9.76,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (9.97,3.06) rectangle (10.39,3.48);\n\\node[font=\\small,align=center,font=\\small] at (10.18,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (9.55,2.64) rectangle (9.97,3.06);\n\\node[font=\\small,align=center,font=\\small] at (9.76,2.85) {$1$};\n\\path[draw=black!30,fill=black!8] (9.97,2.64) rectangle (10.39,3.06);\n\\node[font=\\small,a","\\path[draw=black!30,fill=black!8] (9.55,3.48) rectangle (9.97,3.9);\n\\node[font=\\small,align=center,font=\\small] at (9.76,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (9.97,3.48) rectangle (10.39,3.9);\n\\node[font=\\small,align=center,font=\\small] at (10.18,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (9.55,3.06) rectangle (9.97,3.48);\n\\node[font=\\small,align=center,font=\\small] at (9.76,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (9.97,3.06) rectangle (10.39,3.48);\n\\node[font=\\small,align=center,font=\\small] at (10.18,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (9.55,2.64) rectangle (9.97,3.06);\n\\node[font=\\small,align=center,font=\\small] at (9.76,2.85) {$1$};\n\\path[draw=black!30,fill=black!8] (9.97,2.64) rectangle (10.39,3.06);\n\\node[font=\\small,align=center,font=\\small] at (10.18,2.85) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.1,1.8) {task 5};\n\\node[font=\\sm","th[draw=black!30,fill=black!8] (9.97,3.48) rectangle (10.39,3.9);\n\\node[font=\\small,align=center,font=\\small] at (10.18,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (9.55,3.06) rectangle (9.97,3.48);\n\\node[font=\\small,align=center,font=\\small] at (9.76,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (9.97,3.06) rectangle (10.39,3.48);\n\\node[font=\\small,align=center,font=\\small] at (10.18,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (9.55,2.64) rectangle (9.97,3.06);\n\\node[font=\\small,align=center,font=\\small] at (9.76,2.85) {$1$};\n\\path[draw=black!30,fill=black!8] (9.97,2.64) rectangle (10.39,3.06);\n\\node[font=\\small,align=center,font=\\small] at (10.18,2.85) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.1,1.8) {task 5};\n\\node[font=\\small,align=center,font=\\small] at (12.38,4.6) {2$\\times$3};\n\\path[draw=black!30,fill=black!8] (11.75,3.48) rectangle (12.17,3.9);\n\\node[fo","ath[draw=black!30,fill=black!8] (9.55,3.06) rectangle (9.97,3.48);\n\\node[font=\\small,align=center,font=\\small] at (9.76,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (9.97,3.06) rectangle (10.39,3.48);\n\\node[font=\\small,align=center,font=\\small] at (10.18,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (9.55,2.64) rectangle (9.97,3.06);\n\\node[font=\\small,align=center,font=\\small] at (9.76,2.85) {$1$};\n\\path[draw=black!30,fill=black!8] (9.97,2.64) rectangle (10.39,3.06);\n\\node[font=\\small,align=center,font=\\small] at (10.18,2.85) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.1,1.8) {task 5};\n\\node[font=\\small,align=center,font=\\small] at (12.38,4.6) {2$\\times$3};\n\\path[draw=black!30,fill=black!8] (11.75,3.48) rectangle (12.17,3.9);\n\\node[font=\\small,align=center,font=\\small] at (11.96,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (12.17,3.48) rectangle (12.59,3.9);\n\\node[","h[draw=black!30,fill=black!8] (9.97,3.06) rectangle (10.39,3.48);\n\\node[font=\\small,align=center,font=\\small] at (10.18,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (9.55,2.64) rectangle (9.97,3.06);\n\\node[font=\\small,align=center,font=\\small] at (9.76,2.85) {$1$};\n\\path[draw=black!30,fill=black!8] (9.97,2.64) rectangle (10.39,3.06);\n\\node[font=\\small,align=center,font=\\small] at (10.18,2.85) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.1,1.8) {task 5};\n\\node[font=\\small,align=center,font=\\small] at (12.38,4.6) {2$\\times$3};\n\\path[draw=black!30,fill=black!8] (11.75,3.48) rectangle (12.17,3.9);\n\\node[font=\\small,align=center,font=\\small] at (11.96,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (12.17,3.48) rectangle (12.59,3.9);\n\\node[font=\\small,align=center,font=\\small] at (12.38,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (12.59,3.48) rectangle (13.01,3.9);\n\\node[","raw=black!30,fill=black!8] (9.55,2.64) rectangle (9.97,3.06);\n\\node[font=\\small,align=center,font=\\small] at (9.76,2.85) {$1$};\n\\path[draw=black!30,fill=black!8] (9.97,2.64) rectangle (10.39,3.06);\n\\node[font=\\small,align=center,font=\\small] at (10.18,2.85) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.1,1.8) {task 5};\n\\node[font=\\small,align=center,font=\\small] at (12.38,4.6) {2$\\times$3};\n\\path[draw=black!30,fill=black!8] (11.75,3.48) rectangle (12.17,3.9);\n\\node[font=\\small,align=center,font=\\small] at (11.96,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (12.17,3.48) rectangle (12.59,3.9);\n\\node[font=\\small,align=center,font=\\small] at (12.38,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (12.59,3.48) rectangle (13.01,3.9);\n\\node[font=\\small,align=center,font=\\small] at (12.8,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (11.75,3.06) rectangle (12.17,3.48);\n\\node[font=\\sm","k!30,fill=black!8] (9.97,2.64) rectangle (10.39,3.06);\n\\node[font=\\small,align=center,font=\\small] at (10.18,2.85) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.1,1.8) {task 5};\n\\node[font=\\small,align=center,font=\\small] at (12.38,4.6) {2$\\times$3};\n\\path[draw=black!30,fill=black!8] (11.75,3.48) rectangle (12.17,3.9);\n\\node[font=\\small,align=center,font=\\small] at (11.96,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (12.17,3.48) rectangle (12.59,3.9);\n\\node[font=\\small,align=center,font=\\small] at (12.38,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (12.59,3.48) rectangle (13.01,3.9);\n\\node[font=\\small,align=center,font=\\small] at (12.8,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (11.75,3.06) rectangle (12.17,3.48);\n\\node[font=\\small,align=center,font=\\small] at (11.96,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (12.17,3.06) rectangle (12.59,3.48);\n\\node[font=\\sm","ll,align=center,font=\\scriptsize] at (10.1,1.8) {task 5};\n\\node[font=\\small,align=center,font=\\small] at (12.38,4.6) {2$\\times$3};\n\\path[draw=black!30,fill=black!8] (11.75,3.48) rectangle (12.17,3.9);\n\\node[font=\\small,align=center,font=\\small] at (11.96,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (12.17,3.48) rectangle (12.59,3.9);\n\\node[font=\\small,align=center,font=\\small] at (12.38,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (12.59,3.48) rectangle (13.01,3.9);\n\\node[font=\\small,align=center,font=\\small] at (12.8,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (11.75,3.06) rectangle (12.17,3.48);\n\\node[font=\\small,align=center,font=\\small] at (11.96,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (12.17,3.06) rectangle (12.59,3.48);\n\\node[font=\\small,align=center,font=\\small] at (12.38,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (12.59,3.06) rectangle (13.01,3.48);\n\\node[font=\\s","[draw=black!30,fill=black!8] (11.75,3.48) rectangle (12.17,3.9);\n\\node[font=\\small,align=center,font=\\small] at (11.96,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (12.17,3.48) rectangle (12.59,3.9);\n\\node[font=\\small,align=center,font=\\small] at (12.38,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (12.59,3.48) rectangle (13.01,3.9);\n\\node[font=\\small,align=center,font=\\small] at (12.8,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (11.75,3.06) rectangle (12.17,3.48);\n\\node[font=\\small,align=center,font=\\small] at (11.96,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (12.17,3.06) rectangle (12.59,3.48);\n\\node[font=\\small,align=center,font=\\small] at (12.38,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (12.59,3.06) rectangle (13.01,3.48);\n\\node[font=\\small,align=center,font=\\small] at (12.8,3.27) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.3,1.8) {task 6};\n\\node[font","draw=black!30,fill=black!8] (12.17,3.48) rectangle (12.59,3.9);\n\\node[font=\\small,align=center,font=\\small] at (12.38,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (12.59,3.48) rectangle (13.01,3.9);\n\\node[font=\\small,align=center,font=\\small] at (12.8,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (11.75,3.06) rectangle (12.17,3.48);\n\\node[font=\\small,align=center,font=\\small] at (11.96,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (12.17,3.06) rectangle (12.59,3.48);\n\\node[font=\\small,align=center,font=\\small] at (12.38,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (12.59,3.06) rectangle (13.01,3.48);\n\\node[font=\\small,align=center,font=\\small] at (12.8,3.27) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.3,1.8) {task 6};\n\\node[font=\\small,align=center,font=\\small] at (7,0.95) {Three shape classes, six independent tasks.};\n\\node[font=\\small,align=center,font=\\scriptsi","raw=black!30,fill=black!8] (12.59,3.48) rectangle (13.01,3.9);\n\\node[font=\\small,align=center,font=\\small] at (12.8,3.69) {$1$};\n\\path[draw=black!30,fill=black!8] (11.75,3.06) rectangle (12.17,3.48);\n\\node[font=\\small,align=center,font=\\small] at (11.96,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (12.17,3.06) rectangle (12.59,3.48);\n\\node[font=\\small,align=center,font=\\small] at (12.38,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (12.59,3.06) rectangle (13.01,3.48);\n\\node[font=\\small,align=center,font=\\small] at (12.8,3.27) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.3,1.8) {task 6};\n\\node[font=\\small,align=center,font=\\small] at (7,0.95) {Three shape classes, six independent tasks.};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.1) {Grouping does not remove arithmetic, data transfer, or shared-output dependencies.};\n\\end{tikzpicture}","aw=black!30,fill=black!8] (11.75,3.06) rectangle (12.17,3.48);\n\\node[font=\\small,align=center,font=\\small] at (11.96,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (12.17,3.06) rectangle (12.59,3.48);\n\\node[font=\\small,align=center,font=\\small] at (12.38,3.27) {$1$};\n\\path[draw=black!30,fill=black!8] (12.59,3.06) rectangle (13.01,3.48);\n\\node[font=\\small,align=center,font=\\small] at (12.8,3.27) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.3,1.8) {task 6};\n\\node[font=\\small,align=center,font=\\small] at (7,0.95) {Three shape classes, six independent tasks.};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.1) {Grouping does not remove arithmetic, data transfer, or shared-output dependencies.};\n\\end{tikzpicture}","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.32,4.5) {$U_C$};\n\\path[draw=black!30,fill=black!8] (0.8,3.28) rectangle (1.32,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.06,3.54) {$6$};\n\\path[draw=black!30,fill=black!8] (1.32,3.28) rectangle (1.84,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.58,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,2.76) rectangle (1.32,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.06,3.02) {$4$};\n\\path[draw=black!30,fill=black!8] (1.32,2.76) rectangle (1.84,3.28);\n\\node[font=\\small,align=ce","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.32,4.5) {$U_C$};\n\\path[draw=black!30,fill=black!8] (0.8,3.28) rectangle (1.32,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.06,3.54) {$6$};\n\\path[draw=black!30,fill=black!8] (1.32,3.28) rectangle (1.84,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.58,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,2.76) rectangle (1.32,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.06,3.02) {$4$};\n\\path[draw=black!30,fill=black!8] (1.32,2.76) rectangle (1.84,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.58,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,2.24) rectangle (1.32,2.76);\n\\node[font=\\small,align=c","ine join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.32,4.5) {$U_C$};\n\\path[draw=black!30,fill=black!8] (0.8,3.28) rectangle (1.32,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.06,3.54) {$6$};\n\\path[draw=black!30,fill=black!8] (1.32,3.28) rectangle (1.84,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.58,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,2.76) rectangle (1.32,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.06,3.02) {$4$};\n\\path[draw=black!30,fill=black!8] (1.32,2.76) rectangle (1.84,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.58,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,2.24) rectangle (1.32,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.06,2.5) {$6$};\n\\path[draw=black!30,fill=black!8] (1.32,2.24) rectangle (1.84,2.76);\n\\node[font=\\small,align=c","$};\n\\path[draw=black!30,fill=black!8] (0.8,3.28) rectangle (1.32,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.06,3.54) {$6$};\n\\path[draw=black!30,fill=black!8] (1.32,3.28) rectangle (1.84,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.58,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,2.76) rectangle (1.32,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.06,3.02) {$4$};\n\\path[draw=black!30,fill=black!8] (1.32,2.76) rectangle (1.84,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.58,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,2.24) rectangle (1.32,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.06,2.5) {$6$};\n\\path[draw=black!30,fill=black!8] (1.32,2.24) rectangle (1.84,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.58,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,1.72) rectangle (1.32,2.24);\n\\node[font=\\small,align=ce",";\n\\path[draw=black!30,fill=black!8] (1.32,3.28) rectangle (1.84,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.58,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,2.76) rectangle (1.32,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.06,3.02) {$4$};\n\\path[draw=black!30,fill=black!8] (1.32,2.76) rectangle (1.84,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.58,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,2.24) rectangle (1.32,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.06,2.5) {$6$};\n\\path[draw=black!30,fill=black!8] (1.32,2.24) rectangle (1.84,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.58,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,1.72) rectangle (1.32,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.98) {$4$};\n\\path[draw=black!30,fill=black!8] (1.32,1.72) rectangle (1.84,2.24);\n\\node[font=\\small,align=ce","};\n\\path[draw=black!30,fill=black!8] (0.8,2.76) rectangle (1.32,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.06,3.02) {$4$};\n\\path[draw=black!30,fill=black!8] (1.32,2.76) rectangle (1.84,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.58,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,2.24) rectangle (1.32,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.06,2.5) {$6$};\n\\path[draw=black!30,fill=black!8] (1.32,2.24) rectangle (1.84,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.58,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,1.72) rectangle (1.32,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.98) {$4$};\n\\path[draw=black!30,fill=black!8] (1.32,1.72) rectangle (1.84,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.58,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,1.2) rectangle (1.32,1.72);\n\\node[font=\\small,align=ce","};\n\\path[draw=black!30,fill=black!8] (1.32,2.76) rectangle (1.84,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.58,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,2.24) rectangle (1.32,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.06,2.5) {$6$};\n\\path[draw=black!30,fill=black!8] (1.32,2.24) rectangle (1.84,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.58,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,1.72) rectangle (1.32,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.98) {$4$};\n\\path[draw=black!30,fill=black!8] (1.32,1.72) rectangle (1.84,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.58,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,1.2) rectangle (1.32,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.46) {$6$};\n\\path[draw=black!30,fill=black!8] (1.32,1.2) rectangle (1.84,1.72);\n\\node[font=\\small,align=ce","$};\n\\path[draw=black!30,fill=black!8] (0.8,2.24) rectangle (1.32,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.06,2.5) {$6$};\n\\path[draw=black!30,fill=black!8] (1.32,2.24) rectangle (1.84,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.58,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,1.72) rectangle (1.32,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.98) {$4$};\n\\path[draw=black!30,fill=black!8] (1.32,1.72) rectangle (1.84,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.58,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,1.2) rectangle (1.32,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.46) {$6$};\n\\path[draw=black!30,fill=black!8] (1.32,1.2) rectangle (1.84,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.58,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,0.68) rectangle (1.32,1.2);\n\\node[font=\\small,align=cen",";\n\\path[draw=black!30,fill=black!8] (1.32,2.24) rectangle (1.84,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.58,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,1.72) rectangle (1.32,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.98) {$4$};\n\\path[draw=black!30,fill=black!8] (1.32,1.72) rectangle (1.84,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.58,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,1.2) rectangle (1.32,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.46) {$6$};\n\\path[draw=black!30,fill=black!8] (1.32,1.2) rectangle (1.84,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.58,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,0.68) rectangle (1.32,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.06,0.94) {$4$};\n\\path[draw=black!30,fill=black!8] (1.32,0.68) rectangle (1.84,1.2);\n\\node[font=\\small,align=cent","};\n\\path[draw=black!30,fill=black!8] (0.8,1.72) rectangle (1.32,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.98) {$4$};\n\\path[draw=black!30,fill=black!8] (1.32,1.72) rectangle (1.84,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.58,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,1.2) rectangle (1.32,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.46) {$6$};\n\\path[draw=black!30,fill=black!8] (1.32,1.2) rectangle (1.84,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.58,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,0.68) rectangle (1.32,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.06,0.94) {$4$};\n\\path[draw=black!30,fill=black!8] (1.32,0.68) rectangle (1.84,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.58,0.94) {$1$};\n\\node[font=\\small,align=center,font=\\small] at (5.62,4.5) {$V_C$};\n\\path[draw=black!30,fill=whi","};\n\\path[draw=black!30,fill=black!8] (1.32,1.72) rectangle (1.84,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.58,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,1.2) rectangle (1.32,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.46) {$6$};\n\\path[draw=black!30,fill=black!8] (1.32,1.2) rectangle (1.84,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.58,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,0.68) rectangle (1.32,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.06,0.94) {$4$};\n\\path[draw=black!30,fill=black!8] (1.32,0.68) rectangle (1.84,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.58,0.94) {$1$};\n\\node[font=\\small,align=center,font=\\small] at (5.62,4.5) {$V_C$};\n\\path[draw=black!30,fill=white] (5.1,3.28) rectangle (5.62,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.54) {$0$};\n\\path[draw=black!30,fill=black!","1$};\n\\path[draw=black!30,fill=black!8] (0.8,1.2) rectangle (1.32,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.46) {$6$};\n\\path[draw=black!30,fill=black!8] (1.32,1.2) rectangle (1.84,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.58,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,0.68) rectangle (1.32,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.06,0.94) {$4$};\n\\path[draw=black!30,fill=black!8] (1.32,0.68) rectangle (1.84,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.58,0.94) {$1$};\n\\node[font=\\small,align=center,font=\\small] at (5.62,4.5) {$V_C$};\n\\path[draw=black!30,fill=white] (5.1,3.28) rectangle (5.62,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.54) {$0$};\n\\path[draw=black!30,fill=black!8] (5.62,3.28) rectangle (6.14,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.88,3.54) {$1$};\n\\path[draw=black!30,fill=black","$};\n\\path[draw=black!30,fill=black!8] (1.32,1.2) rectangle (1.84,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.58,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,0.68) rectangle (1.32,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.06,0.94) {$4$};\n\\path[draw=black!30,fill=black!8] (1.32,0.68) rectangle (1.84,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.58,0.94) {$1$};\n\\node[font=\\small,align=center,font=\\small] at (5.62,4.5) {$V_C$};\n\\path[draw=black!30,fill=white] (5.1,3.28) rectangle (5.62,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.54) {$0$};\n\\path[draw=black!30,fill=black!8] (5.62,3.28) rectangle (6.14,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.88,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (5.1,2.76) rectangle (5.62,3.28);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.02) {$1$};\n\\path[draw=black!30,fill=black","1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.58,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,0.68) rectangle (1.32,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.06,0.94) {$4$};\n\\path[draw=black!30,fill=black!8] (1.32,0.68) rectangle (1.84,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.58,0.94) {$1$};\n\\node[font=\\small,align=center,font=\\small] at (5.62,4.5) {$V_C$};\n\\path[draw=black!30,fill=white] (5.1,3.28) rectangle (5.62,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.54) {$0$};\n\\path[draw=black!30,fill=black!8] (5.62,3.28) rectangle (6.14,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.88,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (5.1,2.76) rectangle (5.62,3.28);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (5.62,2.76) rectangle (6.14,3.28);\n\\node[font=\\small,align=cent",",1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.06,0.94) {$4$};\n\\path[draw=black!30,fill=black!8] (1.32,0.68) rectangle (1.84,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.58,0.94) {$1$};\n\\node[font=\\small,align=center,font=\\small] at (5.62,4.5) {$V_C$};\n\\path[draw=black!30,fill=white] (5.1,3.28) rectangle (5.62,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.54) {$0$};\n\\path[draw=black!30,fill=black!8] (5.62,3.28) rectangle (6.14,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.88,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (5.1,2.76) rectangle (5.62,3.28);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (5.62,2.76) rectangle (6.14,3.28);\n\\node[font=\\small,align=center,font=\\small] at (5.88,3.02) {$2$};\n\\path[draw=black!30,fill=white] (5.1,2.24) rectangle (5.62,2.76);\n\\node[font=\\small,align=cen",",1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.58,0.94) {$1$};\n\\node[font=\\small,align=center,font=\\small] at (5.62,4.5) {$V_C$};\n\\path[draw=black!30,fill=white] (5.1,3.28) rectangle (5.62,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.54) {$0$};\n\\path[draw=black!30,fill=black!8] (5.62,3.28) rectangle (6.14,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.88,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (5.1,2.76) rectangle (5.62,3.28);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (5.62,2.76) rectangle (6.14,3.28);\n\\node[font=\\small,align=center,font=\\small] at (5.88,3.02) {$2$};\n\\path[draw=black!30,fill=white] (5.1,2.24) rectangle (5.62,2.76);\n\\node[font=\\small,align=center,font=\\small] at (5.36,2.5) {$0$};\n\\path[draw=black!30,fill=black!8] (5.62,2.24) rectangle (6.14,2.76);\n\\node[font=\\small,align=cen","_C$};\n\\path[draw=black!30,fill=white] (5.1,3.28) rectangle (5.62,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.54) {$0$};\n\\path[draw=black!30,fill=black!8] (5.62,3.28) rectangle (6.14,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.88,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (5.1,2.76) rectangle (5.62,3.28);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (5.62,2.76) rectangle (6.14,3.28);\n\\node[font=\\small,align=center,font=\\small] at (5.88,3.02) {$2$};\n\\path[draw=black!30,fill=white] (5.1,2.24) rectangle (5.62,2.76);\n\\node[font=\\small,align=center,font=\\small] at (5.36,2.5) {$0$};\n\\path[draw=black!30,fill=black!8] (5.62,2.24) rectangle (6.14,2.76);\n\\node[font=\\small,align=center,font=\\small] at (5.88,2.5) {$3$};\n\\path[draw=black!30,fill=black!8] (5.1,1.72) rectangle (5.62,2.24);\n\\node[font=\\small,align=cent",";\n\\path[draw=black!30,fill=black!8] (5.62,3.28) rectangle (6.14,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.88,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (5.1,2.76) rectangle (5.62,3.28);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (5.62,2.76) rectangle (6.14,3.28);\n\\node[font=\\small,align=center,font=\\small] at (5.88,3.02) {$2$};\n\\path[draw=black!30,fill=white] (5.1,2.24) rectangle (5.62,2.76);\n\\node[font=\\small,align=center,font=\\small] at (5.36,2.5) {$0$};\n\\path[draw=black!30,fill=black!8] (5.62,2.24) rectangle (6.14,2.76);\n\\node[font=\\small,align=center,font=\\small] at (5.88,2.5) {$3$};\n\\path[draw=black!30,fill=black!8] (5.1,1.72) rectangle (5.62,2.24);\n\\node[font=\\small,align=center,font=\\small] at (5.36,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (5.62,1.72) rectangle (6.14,2.24);\n\\node[font=\\small,align=cent","1$};\n\\path[draw=black!30,fill=black!8] (5.1,2.76) rectangle (5.62,3.28);\n\\node[font=\\small,align=center,font=\\small] at (5.36,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (5.62,2.76) rectangle (6.14,3.28);\n\\node[font=\\small,align=center,font=\\small] at (5.88,3.02) {$2$};\n\\path[draw=black!30,fill=white] (5.1,2.24) rectangle (5.62,2.76);\n\\node[font=\\small,align=center,font=\\small] at (5.36,2.5) {$0$};\n\\path[draw=black!30,fill=black!8] (5.62,2.24) rectangle (6.14,2.76);\n\\node[font=\\small,align=center,font=\\small] at (5.88,2.5) {$3$};\n\\path[draw=black!30,fill=black!8] (5.1,1.72) rectangle (5.62,2.24);\n\\node[font=\\small,align=center,font=\\small] at (5.36,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (5.62,1.72) rectangle (6.14,2.24);\n\\node[font=\\small,align=center,font=\\small] at (5.88,1.98) {$4$};\n\\path[draw=black!30,fill=white] (5.1,1.2) rectangle (5.62,1.72);\n\\node[font=\\small,align=cent","1$};\n\\path[draw=black!30,fill=black!8] (5.62,2.76) rectangle (6.14,3.28);\n\\node[font=\\small,align=center,font=\\small] at (5.88,3.02) {$2$};\n\\path[draw=black!30,fill=white] (5.1,2.24) rectangle (5.62,2.76);\n\\node[font=\\small,align=center,font=\\small] at (5.36,2.5) {$0$};\n\\path[draw=black!30,fill=black!8] (5.62,2.24) rectangle (6.14,2.76);\n\\node[font=\\small,align=center,font=\\small] at (5.88,2.5) {$3$};\n\\path[draw=black!30,fill=black!8] (5.1,1.72) rectangle (5.62,2.24);\n\\node[font=\\small,align=center,font=\\small] at (5.36,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (5.62,1.72) rectangle (6.14,2.24);\n\\node[font=\\small,align=center,font=\\small] at (5.88,1.98) {$4$};\n\\path[draw=black!30,fill=white] (5.1,1.2) rectangle (5.62,1.72);\n\\node[font=\\small,align=center,font=\\small] at (5.36,1.46) {$0$};\n\\path[draw=black!30,fill=black!8] (5.62,1.2) rectangle (6.14,1.72);\n\\node[font=\\small,align=cent","$2$};\n\\path[draw=black!30,fill=white] (5.1,2.24) rectangle (5.62,2.76);\n\\node[font=\\small,align=center,font=\\small] at (5.36,2.5) {$0$};\n\\path[draw=black!30,fill=black!8] (5.62,2.24) rectangle (6.14,2.76);\n\\node[font=\\small,align=center,font=\\small] at (5.88,2.5) {$3$};\n\\path[draw=black!30,fill=black!8] (5.1,1.72) rectangle (5.62,2.24);\n\\node[font=\\small,align=center,font=\\small] at (5.36,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (5.62,1.72) rectangle (6.14,2.24);\n\\node[font=\\small,align=center,font=\\small] at (5.88,1.98) {$4$};\n\\path[draw=black!30,fill=white] (5.1,1.2) rectangle (5.62,1.72);\n\\node[font=\\small,align=center,font=\\small] at (5.36,1.46) {$0$};\n\\path[draw=black!30,fill=black!8] (5.62,1.2) rectangle (6.14,1.72);\n\\node[font=\\small,align=center,font=\\small] at (5.88,1.46) {$5$};\n\\path[draw=black!30,fill=black!8] (5.1,0.68) rectangle (5.62,1.2);\n\\node[font=\\small,align=cente",";\n\\path[draw=black!30,fill=black!8] (5.62,2.24) rectangle (6.14,2.76);\n\\node[font=\\small,align=center,font=\\small] at (5.88,2.5) {$3$};\n\\path[draw=black!30,fill=black!8] (5.1,1.72) rectangle (5.62,2.24);\n\\node[font=\\small,align=center,font=\\small] at (5.36,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (5.62,1.72) rectangle (6.14,2.24);\n\\node[font=\\small,align=center,font=\\small] at (5.88,1.98) {$4$};\n\\path[draw=black!30,fill=white] (5.1,1.2) rectangle (5.62,1.72);\n\\node[font=\\small,align=center,font=\\small] at (5.36,1.46) {$0$};\n\\path[draw=black!30,fill=black!8] (5.62,1.2) rectangle (6.14,1.72);\n\\node[font=\\small,align=center,font=\\small] at (5.88,1.46) {$5$};\n\\path[draw=black!30,fill=black!8] (5.1,0.68) rectangle (5.62,1.2);\n\\node[font=\\small,align=center,font=\\small] at (5.36,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (5.62,0.68) rectangle (6.14,1.2);\n\\node[font=\\small,align=center","3$};\n\\path[draw=black!30,fill=black!8] (5.1,1.72) rectangle (5.62,2.24);\n\\node[font=\\small,align=center,font=\\small] at (5.36,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (5.62,1.72) rectangle (6.14,2.24);\n\\node[font=\\small,align=center,font=\\small] at (5.88,1.98) {$4$};\n\\path[draw=black!30,fill=white] (5.1,1.2) rectangle (5.62,1.72);\n\\node[font=\\small,align=center,font=\\small] at (5.36,1.46) {$0$};\n\\path[draw=black!30,fill=black!8] (5.62,1.2) rectangle (6.14,1.72);\n\\node[font=\\small,align=center,font=\\small] at (5.88,1.46) {$5$};\n\\path[draw=black!30,fill=black!8] (5.1,0.68) rectangle (5.62,1.2);\n\\node[font=\\small,align=center,font=\\small] at (5.36,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (5.62,0.68) rectangle (6.14,1.2);\n\\node[font=\\small,align=center,font=\\small] at (5.88,0.94) {$6$};\n\\node[font=\\small,align=center] at (10.3,3.45) {$AB=D_AD_B+U_CV_C^T$};\n\\node[font=\\small,align=","1$};\n\\path[draw=black!30,fill=black!8] (5.62,1.72) rectangle (6.14,2.24);\n\\node[font=\\small,align=center,font=\\small] at (5.88,1.98) {$4$};\n\\path[draw=black!30,fill=white] (5.1,1.2) rectangle (5.62,1.72);\n\\node[font=\\small,align=center,font=\\small] at (5.36,1.46) {$0$};\n\\path[draw=black!30,fill=black!8] (5.62,1.2) rectangle (6.14,1.72);\n\\node[font=\\small,align=center,font=\\small] at (5.88,1.46) {$5$};\n\\path[draw=black!30,fill=black!8] (5.1,0.68) rectangle (5.62,1.2);\n\\node[font=\\small,align=center,font=\\small] at (5.36,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (5.62,0.68) rectangle (6.14,1.2);\n\\node[font=\\small,align=center,font=\\small] at (5.88,0.94) {$6$};\n\\node[font=\\small,align=center] at (10.3,3.45) {$AB=D_AD_B+U_CV_C^T$};\n\\node[font=\\small,align=center] at (10.3,2.15) {$U_A(V_A^TU_B)V_B^T$};\n\\node[font=\\small,align=center,font=\\small] at (10.3,1.35) {must be retained};\n\\node[fo","{$4$};\n\\path[draw=black!30,fill=white] (5.1,1.2) rectangle (5.62,1.72);\n\\node[font=\\small,align=center,font=\\small] at (5.36,1.46) {$0$};\n\\path[draw=black!30,fill=black!8] (5.62,1.2) rectangle (6.14,1.72);\n\\node[font=\\small,align=center,font=\\small] at (5.88,1.46) {$5$};\n\\path[draw=black!30,fill=black!8] (5.1,0.68) rectangle (5.62,1.2);\n\\node[font=\\small,align=center,font=\\small] at (5.36,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (5.62,0.68) rectangle (6.14,1.2);\n\\node[font=\\small,align=center,font=\\small] at (5.88,0.94) {$6$};\n\\node[font=\\small,align=center] at (10.3,3.45) {$AB=D_AD_B+U_CV_C^T$};\n\\node[font=\\small,align=center] at (10.3,2.15) {$U_A(V_A^TU_B)V_B^T$};\n\\node[font=\\small,align=center,font=\\small] at (10.3,1.35) {must be retained};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.1) {Two input rank-one corrections produce at most rank two, not three.};\n\\end{tikzp","$};\n\\path[draw=black!30,fill=black!8] (5.62,1.2) rectangle (6.14,1.72);\n\\node[font=\\small,align=center,font=\\small] at (5.88,1.46) {$5$};\n\\path[draw=black!30,fill=black!8] (5.1,0.68) rectangle (5.62,1.2);\n\\node[font=\\small,align=center,font=\\small] at (5.36,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (5.62,0.68) rectangle (6.14,1.2);\n\\node[font=\\small,align=center,font=\\small] at (5.88,0.94) {$6$};\n\\node[font=\\small,align=center] at (10.3,3.45) {$AB=D_AD_B+U_CV_C^T$};\n\\node[font=\\small,align=center] at (10.3,2.15) {$U_A(V_A^TU_B)V_B^T$};\n\\node[font=\\small,align=center,font=\\small] at (10.3,1.35) {must be retained};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.1) {Two input rank-one corrections produce at most rank two, not three.};\n\\end{tikzpicture}","ngle (6.14,1.72);\n\\node[font=\\small,align=center,font=\\small] at (5.88,1.46) {$5$};\n\\path[draw=black!30,fill=black!8] (5.1,0.68) rectangle (5.62,1.2);\n\\node[font=\\small,align=center,font=\\small] at (5.36,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (5.62,0.68) rectangle (6.14,1.2);\n\\node[font=\\small,align=center,font=\\small] at (5.88,0.94) {$6$};\n\\node[font=\\small,align=center] at (10.3,3.45) {$AB=D_AD_B+U_CV_C^T$};\n\\node[font=\\small,align=center] at (10.3,2.15) {$U_A(V_A^TU_B)V_B^T$};\n\\node[font=\\small,align=center,font=\\small] at (10.3,1.35) {must be retained};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.1) {Two input rank-one corrections produce at most rank two, not three.};\n\\end{tikzpicture}","1.46) {$5$};\n\\path[draw=black!30,fill=black!8] (5.1,0.68) rectangle (5.62,1.2);\n\\node[font=\\small,align=center,font=\\small] at (5.36,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (5.62,0.68) rectangle (6.14,1.2);\n\\node[font=\\small,align=center,font=\\small] at (5.88,0.94) {$6$};\n\\node[font=\\small,align=center] at (10.3,3.45) {$AB=D_AD_B+U_CV_C^T$};\n\\node[font=\\small,align=center] at (10.3,2.15) {$U_A(V_A^TU_B)V_B^T$};\n\\node[font=\\small,align=center,font=\\small] at (10.3,1.35) {must be retained};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.1) {Two input rank-one corrections produce at most rank two, not three.};\n\\end{tikzpicture}","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (2.26,4.5) {$100E$: twelve omitted entries};\n\\path[draw=black!30,fill=white] (0.7,3.02) rectangle (1.48,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.09,3.41) {$0$};\n\\path[draw=black!30,fill=black!8] (1.48,3.02) rectangle (2.26,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.87,3.41) {$1$};\n\\path[draw=black!30,fill=black!8] (2.26,3.02) rectangle (3.04,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.65,3.41) {$1$};\n\\path[draw=black!30,fill=black!8] (3.04,3.02) rectangle (3.82,3.8);\n\\node","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (2.26,4.5) {$100E$: twelve omitted entries};\n\\path[draw=black!30,fill=white] (0.7,3.02) rectangle (1.48,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.09,3.41) {$0$};\n\\path[draw=black!30,fill=black!8] (1.48,3.02) rectangle (2.26,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.87,3.41) {$1$};\n\\path[draw=black!30,fill=black!8] (2.26,3.02) rectangle (3.04,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.65,3.41) {$1$};\n\\path[draw=black!30,fill=black!8] (3.04,3.02) rectangle (3.82,3.8);\n\\node[font=\\small,align=center,font=\\small] at (3.43,3.41) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,2.24) rectangle (1.48,3.02);\n\\node[font=\\small,align=ce","se as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (2.26,4.5) {$100E$: twelve omitted entries};\n\\path[draw=black!30,fill=white] (0.7,3.02) rectangle (1.48,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.09,3.41) {$0$};\n\\path[draw=black!30,fill=black!8] (1.48,3.02) rectangle (2.26,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.87,3.41) {$1$};\n\\path[draw=black!30,fill=black!8] (2.26,3.02) rectangle (3.04,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.65,3.41) {$1$};\n\\path[draw=black!30,fill=black!8] (3.04,3.02) rectangle (3.82,3.8);\n\\node[font=\\small,align=center,font=\\small] at (3.43,3.41) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,2.24) rectangle (1.48,3.02);\n\\node[font=\\small,align=center,font=\\small] at (1.09,2.63) {$1$};\n\\path[draw=black!30,fill=white] (1.48,2.24) rectangle (2.26,3.02);\n\\node[font=\\small,align=cen","ies};\n\\path[draw=black!30,fill=white] (0.7,3.02) rectangle (1.48,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.09,3.41) {$0$};\n\\path[draw=black!30,fill=black!8] (1.48,3.02) rectangle (2.26,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.87,3.41) {$1$};\n\\path[draw=black!30,fill=black!8] (2.26,3.02) rectangle (3.04,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.65,3.41) {$1$};\n\\path[draw=black!30,fill=black!8] (3.04,3.02) rectangle (3.82,3.8);\n\\node[font=\\small,align=center,font=\\small] at (3.43,3.41) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,2.24) rectangle (1.48,3.02);\n\\node[font=\\small,align=center,font=\\small] at (1.09,2.63) {$1$};\n\\path[draw=black!30,fill=white] (1.48,2.24) rectangle (2.26,3.02);\n\\node[font=\\small,align=center,font=\\small] at (1.87,2.63) {$0$};\n\\path[draw=black!30,fill=black!8] (2.26,2.24) rectangle (3.04,3.02);\n\\node[font=\\small,align=ce","};\n\\path[draw=black!30,fill=black!8] (1.48,3.02) rectangle (2.26,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.87,3.41) {$1$};\n\\path[draw=black!30,fill=black!8] (2.26,3.02) rectangle (3.04,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.65,3.41) {$1$};\n\\path[draw=black!30,fill=black!8] (3.04,3.02) rectangle (3.82,3.8);\n\\node[font=\\small,align=center,font=\\small] at (3.43,3.41) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,2.24) rectangle (1.48,3.02);\n\\node[font=\\small,align=center,font=\\small] at (1.09,2.63) {$1$};\n\\path[draw=black!30,fill=white] (1.48,2.24) rectangle (2.26,3.02);\n\\node[font=\\small,align=center,font=\\small] at (1.87,2.63) {$0$};\n\\path[draw=black!30,fill=black!8] (2.26,2.24) rectangle (3.04,3.02);\n\\node[font=\\small,align=center,font=\\small] at (2.65,2.63) {$1$};\n\\path[draw=black!30,fill=black!8] (3.04,2.24) rectangle (3.82,3.02);\n\\node[font=\\small,align=c","};\n\\path[draw=black!30,fill=black!8] (2.26,3.02) rectangle (3.04,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.65,3.41) {$1$};\n\\path[draw=black!30,fill=black!8] (3.04,3.02) rectangle (3.82,3.8);\n\\node[font=\\small,align=center,font=\\small] at (3.43,3.41) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,2.24) rectangle (1.48,3.02);\n\\node[font=\\small,align=center,font=\\small] at (1.09,2.63) {$1$};\n\\path[draw=black!30,fill=white] (1.48,2.24) rectangle (2.26,3.02);\n\\node[font=\\small,align=center,font=\\small] at (1.87,2.63) {$0$};\n\\path[draw=black!30,fill=black!8] (2.26,2.24) rectangle (3.04,3.02);\n\\node[font=\\small,align=center,font=\\small] at (2.65,2.63) {$1$};\n\\path[draw=black!30,fill=black!8] (3.04,2.24) rectangle (3.82,3.02);\n\\node[font=\\small,align=center,font=\\small] at (3.43,2.63) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,1.46) rectangle (1.48,2.24);\n\\node[font=\\small,align=c","$};\n\\path[draw=black!30,fill=black!8] (3.04,3.02) rectangle (3.82,3.8);\n\\node[font=\\small,align=center,font=\\small] at (3.43,3.41) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,2.24) rectangle (1.48,3.02);\n\\node[font=\\small,align=center,font=\\small] at (1.09,2.63) {$1$};\n\\path[draw=black!30,fill=white] (1.48,2.24) rectangle (2.26,3.02);\n\\node[font=\\small,align=center,font=\\small] at (1.87,2.63) {$0$};\n\\path[draw=black!30,fill=black!8] (2.26,2.24) rectangle (3.04,3.02);\n\\node[font=\\small,align=center,font=\\small] at (2.65,2.63) {$1$};\n\\path[draw=black!30,fill=black!8] (3.04,2.24) rectangle (3.82,3.02);\n\\node[font=\\small,align=center,font=\\small] at (3.43,2.63) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,1.46) rectangle (1.48,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.09,1.85) {$1$};\n\\path[draw=black!30,fill=black!8] (1.48,1.46) rectangle (2.26,2.24);\n\\node[font=\\small,align","};\n\\path[draw=black!30,fill=black!8] (0.7,2.24) rectangle (1.48,3.02);\n\\node[font=\\small,align=center,font=\\small] at (1.09,2.63) {$1$};\n\\path[draw=black!30,fill=white] (1.48,2.24) rectangle (2.26,3.02);\n\\node[font=\\small,align=center,font=\\small] at (1.87,2.63) {$0$};\n\\path[draw=black!30,fill=black!8] (2.26,2.24) rectangle (3.04,3.02);\n\\node[font=\\small,align=center,font=\\small] at (2.65,2.63) {$1$};\n\\path[draw=black!30,fill=black!8] (3.04,2.24) rectangle (3.82,3.02);\n\\node[font=\\small,align=center,font=\\small] at (3.43,2.63) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,1.46) rectangle (1.48,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.09,1.85) {$1$};\n\\path[draw=black!30,fill=black!8] (1.48,1.46) rectangle (2.26,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.87,1.85) {$1$};\n\\path[draw=black!30,fill=white] (2.26,1.46) rectangle (3.04,2.24);\n\\node[font=\\small,align=c",";\n\\path[draw=black!30,fill=white] (1.48,2.24) rectangle (2.26,3.02);\n\\node[font=\\small,align=center,font=\\small] at (1.87,2.63) {$0$};\n\\path[draw=black!30,fill=black!8] (2.26,2.24) rectangle (3.04,3.02);\n\\node[font=\\small,align=center,font=\\small] at (2.65,2.63) {$1$};\n\\path[draw=black!30,fill=black!8] (3.04,2.24) rectangle (3.82,3.02);\n\\node[font=\\small,align=center,font=\\small] at (3.43,2.63) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,1.46) rectangle (1.48,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.09,1.85) {$1$};\n\\path[draw=black!30,fill=black!8] (1.48,1.46) rectangle (2.26,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.87,1.85) {$1$};\n\\path[draw=black!30,fill=white] (2.26,1.46) rectangle (3.04,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.65,1.85) {$0$};\n\\path[draw=black!30,fill=black!8] (3.04,1.46) rectangle (3.82,2.24);\n\\node[font=\\small,align=c","\\path[draw=black!30,fill=black!8] (2.26,2.24) rectangle (3.04,3.02);\n\\node[font=\\small,align=center,font=\\small] at (2.65,2.63) {$1$};\n\\path[draw=black!30,fill=black!8] (3.04,2.24) rectangle (3.82,3.02);\n\\node[font=\\small,align=center,font=\\small] at (3.43,2.63) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,1.46) rectangle (1.48,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.09,1.85) {$1$};\n\\path[draw=black!30,fill=black!8] (1.48,1.46) rectangle (2.26,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.87,1.85) {$1$};\n\\path[draw=black!30,fill=white] (2.26,1.46) rectangle (3.04,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.65,1.85) {$0$};\n\\path[draw=black!30,fill=black!8] (3.04,1.46) rectangle (3.82,2.24);\n\\node[font=\\small,align=center,font=\\small] at (3.43,1.85) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,0.68) rectangle (1.48,1.46);\n\\node[font=\\small,align=c","\\path[draw=black!30,fill=black!8] (3.04,2.24) rectangle (3.82,3.02);\n\\node[font=\\small,align=center,font=\\small] at (3.43,2.63) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,1.46) rectangle (1.48,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.09,1.85) {$1$};\n\\path[draw=black!30,fill=black!8] (1.48,1.46) rectangle (2.26,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.87,1.85) {$1$};\n\\path[draw=black!30,fill=white] (2.26,1.46) rectangle (3.04,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.65,1.85) {$0$};\n\\path[draw=black!30,fill=black!8] (3.04,1.46) rectangle (3.82,2.24);\n\\node[font=\\small,align=center,font=\\small] at (3.43,1.85) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,0.68) rectangle (1.48,1.46);\n\\node[font=\\small,align=center,font=\\small] at (1.09,1.07) {$1$};\n\\path[draw=black!30,fill=black!8] (1.48,0.68) rectangle (2.26,1.46);\n\\node[font=\\small,align=c","};\n\\path[draw=black!30,fill=black!8] (0.7,1.46) rectangle (1.48,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.09,1.85) {$1$};\n\\path[draw=black!30,fill=black!8] (1.48,1.46) rectangle (2.26,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.87,1.85) {$1$};\n\\path[draw=black!30,fill=white] (2.26,1.46) rectangle (3.04,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.65,1.85) {$0$};\n\\path[draw=black!30,fill=black!8] (3.04,1.46) rectangle (3.82,2.24);\n\\node[font=\\small,align=center,font=\\small] at (3.43,1.85) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,0.68) rectangle (1.48,1.46);\n\\node[font=\\small,align=center,font=\\small] at (1.09,1.07) {$1$};\n\\path[draw=black!30,fill=black!8] (1.48,0.68) rectangle (2.26,1.46);\n\\node[font=\\small,align=center,font=\\small] at (1.87,1.07) {$1$};\n\\path[draw=black!30,fill=black!8] (2.26,0.68) rectangle (3.04,1.46);\n\\node[font=\\small,align",";\n\\path[draw=black!30,fill=black!8] (1.48,1.46) rectangle (2.26,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.87,1.85) {$1$};\n\\path[draw=black!30,fill=white] (2.26,1.46) rectangle (3.04,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.65,1.85) {$0$};\n\\path[draw=black!30,fill=black!8] (3.04,1.46) rectangle (3.82,2.24);\n\\node[font=\\small,align=center,font=\\small] at (3.43,1.85) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,0.68) rectangle (1.48,1.46);\n\\node[font=\\small,align=center,font=\\small] at (1.09,1.07) {$1$};\n\\path[draw=black!30,fill=black!8] (1.48,0.68) rectangle (2.26,1.46);\n\\node[font=\\small,align=center,font=\\small] at (1.87,1.07) {$1$};\n\\path[draw=black!30,fill=black!8] (2.26,0.68) rectangle (3.04,1.46);\n\\node[font=\\small,align=center,font=\\small] at (2.65,1.07) {$1$};\n\\path[draw=black!30,fill=white] (3.04,0.68) rectangle (3.82,1.46);\n\\node[font=\\small,align=c","};\n\\path[draw=black!30,fill=white] (2.26,1.46) rectangle (3.04,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.65,1.85) {$0$};\n\\path[draw=black!30,fill=black!8] (3.04,1.46) rectangle (3.82,2.24);\n\\node[font=\\small,align=center,font=\\small] at (3.43,1.85) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,0.68) rectangle (1.48,1.46);\n\\node[font=\\small,align=center,font=\\small] at (1.09,1.07) {$1$};\n\\path[draw=black!30,fill=black!8] (1.48,0.68) rectangle (2.26,1.46);\n\\node[font=\\small,align=center,font=\\small] at (1.87,1.07) {$1$};\n\\path[draw=black!30,fill=black!8] (2.26,0.68) rectangle (3.04,1.46);\n\\node[font=\\small,align=center,font=\\small] at (2.65,1.07) {$1$};\n\\path[draw=black!30,fill=white] (3.04,0.68) rectangle (3.82,1.46);\n\\node[font=\\small,align=center,font=\\small] at (3.43,1.07) {$0$};\n\\draw[line width=.6pt] (7.35,0.85) -- (12.85,0.85);\n\\node[font=\\small,align=center] at (10.1,","\\path[draw=black!30,fill=black!8] (3.04,1.46) rectangle (3.82,2.24);\n\\node[font=\\small,align=center,font=\\small] at (3.43,1.85) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,0.68) rectangle (1.48,1.46);\n\\node[font=\\small,align=center,font=\\small] at (1.09,1.07) {$1$};\n\\path[draw=black!30,fill=black!8] (1.48,0.68) rectangle (2.26,1.46);\n\\node[font=\\small,align=center,font=\\small] at (1.87,1.07) {$1$};\n\\path[draw=black!30,fill=black!8] (2.26,0.68) rectangle (3.04,1.46);\n\\node[font=\\small,align=center,font=\\small] at (2.65,1.07) {$1$};\n\\path[draw=black!30,fill=white] (3.04,0.68) rectangle (3.82,1.46);\n\\node[font=\\small,align=center,font=\\small] at (3.43,1.07) {$0$};\n\\draw[line width=.6pt] (7.35,0.85) -- (12.85,0.85);\n\\node[font=\\small,align=center] at (10.1,4.25) {error norms, in units of $1/100$};\n\\path[draw=black,fill=black!12,line width=.6pt] (7.79,0.85) rectangle (8.743333,1.5125);\n\\nod","\\path[draw=black!30,fill=black!8] (0.7,0.68) rectangle (1.48,1.46);\n\\node[font=\\small,align=center,font=\\small] at (1.09,1.07) {$1$};\n\\path[draw=black!30,fill=black!8] (1.48,0.68) rectangle (2.26,1.46);\n\\node[font=\\small,align=center,font=\\small] at (1.87,1.07) {$1$};\n\\path[draw=black!30,fill=black!8] (2.26,0.68) rectangle (3.04,1.46);\n\\node[font=\\small,align=center,font=\\small] at (2.65,1.07) {$1$};\n\\path[draw=black!30,fill=white] (3.04,0.68) rectangle (3.82,1.46);\n\\node[font=\\small,align=center,font=\\small] at (3.43,1.07) {$0$};\n\\draw[line width=.6pt] (7.35,0.85) -- (12.85,0.85);\n\\node[font=\\small,align=center] at (10.1,4.25) {error norms, in units of $1/100$};\n\\path[draw=black,fill=black!12,line width=.6pt] (7.79,0.85) rectangle (8.743333,1.5125);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.266667,0.55) {one block};\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.266",";\n\\path[draw=black!30,fill=black!8] (1.48,0.68) rectangle (2.26,1.46);\n\\node[font=\\small,align=center,font=\\small] at (1.87,1.07) {$1$};\n\\path[draw=black!30,fill=black!8] (2.26,0.68) rectangle (3.04,1.46);\n\\node[font=\\small,align=center,font=\\small] at (2.65,1.07) {$1$};\n\\path[draw=black!30,fill=white] (3.04,0.68) rectangle (3.82,1.46);\n\\node[font=\\small,align=center,font=\\small] at (3.43,1.07) {$0$};\n\\draw[line width=.6pt] (7.35,0.85) -- (12.85,0.85);\n\\node[font=\\small,align=center] at (10.1,4.25) {error norms, in units of $1/100$};\n\\path[draw=black,fill=black!12,line width=.6pt] (7.79,0.85) rectangle (8.743333,1.5125);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.266667,0.55) {one block};\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.266667,1.7325) {$1$};\n\\path[draw=black,fill=black!12,line width=.6pt] (9.623333,0.85) rectangle (10.576667,2.8375);\n\\node[font=\\small,al","$1$};\n\\path[draw=black!30,fill=black!8] (2.26,0.68) rectangle (3.04,1.46);\n\\node[font=\\small,align=center,font=\\small] at (2.65,1.07) {$1$};\n\\path[draw=black!30,fill=white] (3.04,0.68) rectangle (3.82,1.46);\n\\node[font=\\small,align=center,font=\\small] at (3.43,1.07) {$0$};\n\\draw[line width=.6pt] (7.35,0.85) -- (12.85,0.85);\n\\node[font=\\small,align=center] at (10.1,4.25) {error norms, in units of $1/100$};\n\\path[draw=black,fill=black!12,line width=.6pt] (7.79,0.85) rectangle (8.743333,1.5125);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.266667,0.55) {one block};\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.266667,1.7325) {$1$};\n\\path[draw=black,fill=black!12,line width=.6pt] (9.623333,0.85) rectangle (10.576667,2.8375);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.1,0.55) {spectral};\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.1,3.0575) {$3$};\n\\path[","at (3.43,1.07) {$0$};\n\\draw[line width=.6pt] (7.35,0.85) -- (12.85,0.85);\n\\node[font=\\small,align=center] at (10.1,4.25) {error norms, in units of $1/100$};\n\\path[draw=black,fill=black!12,line width=.6pt] (7.79,0.85) rectangle (8.743333,1.5125);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.266667,0.55) {one block};\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.266667,1.7325) {$1$};\n\\path[draw=black,fill=black!12,line width=.6pt] (9.623333,0.85) rectangle (10.576667,2.8375);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.1,0.55) {spectral};\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.1,3.0575) {$3$};\n\\path[draw=black,fill=black!12,line width=.6pt] (11.456667,0.85) rectangle (12.41,3.144967);\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.933333,0.55) {Frobenius};\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.933333,3.364967) {$2 \\","5);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.266667,0.55) {one block};\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.266667,1.7325) {$1$};\n\\path[draw=black,fill=black!12,line width=.6pt] (9.623333,0.85) rectangle (10.576667,2.8375);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.1,0.55) {spectral};\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.1,3.0575) {$3$};\n\\path[draw=black,fill=black!12,line width=.6pt] (11.456667,0.85) rectangle (12.41,3.144967);\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.933333,0.55) {Frobenius};\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.933333,3.364967) {$2 \\sqrt{3}$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.1) {$\\|E\\|_F^2=12/10^4$, while $E\\mathbf1=(3/100)\\mathbf1$.};\n\\end{tikzpicture}","\\node[font=\\small,align=center,font=\\scriptsize] at (10.1,0.55) {spectral};\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.1,3.0575) {$3$};\n\\path[draw=black,fill=black!12,line width=.6pt] (11.456667,0.85) rectangle (12.41,3.144967);\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.933333,0.55) {Frobenius};\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.933333,3.364967) {$2 \\sqrt{3}$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.1) {$\\|E\\|_F^2=12/10^4$, while $E\\mathbf1=(3/100)\\mathbf1$.};\n\\end{tikzpicture}","\\node[font=\\small,align=center,font=\\scriptsize] at (10.1,3.0575) {$3$};\n\\path[draw=black,fill=black!12,line width=.6pt] (11.456667,0.85) rectangle (12.41,3.144967);\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.933333,0.55) {Frobenius};\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.933333,3.364967) {$2 \\sqrt{3}$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.1) {$\\|E\\|_F^2=12/10^4$, while $E\\mathbf1=(3/100)\\mathbf1$.};\n\\end{tikzpicture}","nter,font=\\scriptsize] at (10.1,3.0575) {$3$};\n\\path[draw=black,fill=black!12,line width=.6pt] (11.456667,0.85) rectangle (12.41,3.144967);\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.933333,0.55) {Frobenius};\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.933333,3.364967) {$2 \\sqrt{3}$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.1) {$\\|E\\|_F^2=12/10^4$, while $E\\mathbf1=(3/100)\\mathbf1$.};\n\\end{tikzpicture}","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (2.3,4.3) {$K$};\n\\path[draw=black!30,fill=white] (0.7,2.8) rectangle (1.5,3.6);\n\\node[font=\\small,align=center,font=\\small] at (1.1,3.2) {$0$};\n\\path[draw=black!30,fill=white] (1.5,2.8) rectangle (2.3,3.6);\n\\node[font=\\small,align=center,font=\\small] at (1.9,3.2) {$0$};\n\\path[draw=black!30,fill=white] (2.3,2.8) rectangle (3.1,3.6);\n\\node[font=\\small,align=center,font=\\small] at (2.7,3.2) {$0$};\n\\path[draw=black!30,fill=white] (3.1,2.8) rectangle (3.9,3.6);\n\\node[font=\\small,align=center,font=\\small] at (3.5,","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (2.3,4.3) {$K$};\n\\path[draw=black!30,fill=white] (0.7,2.8) rectangle (1.5,3.6);\n\\node[font=\\small,align=center,font=\\small] at (1.1,3.2) {$0$};\n\\path[draw=black!30,fill=white] (1.5,2.8) rectangle (2.3,3.6);\n\\node[font=\\small,align=center,font=\\small] at (1.9,3.2) {$0$};\n\\path[draw=black!30,fill=white] (2.3,2.8) rectangle (3.1,3.6);\n\\node[font=\\small,align=center,font=\\small] at (2.7,3.2) {$0$};\n\\path[draw=black!30,fill=white] (3.1,2.8) rectangle (3.9,3.6);\n\\node[font=\\small,align=center,font=\\small] at (3.5,3.2) {$0$};\n\\path[draw=black!30,fill=black!8] (0.7,2) rectangle (1.5,2.8);\n\\node[font=\\small,align=center,font=\\small] at (1.1,","line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (2.3,4.3) {$K$};\n\\path[draw=black!30,fill=white] (0.7,2.8) rectangle (1.5,3.6);\n\\node[font=\\small,align=center,font=\\small] at (1.1,3.2) {$0$};\n\\path[draw=black!30,fill=white] (1.5,2.8) rectangle (2.3,3.6);\n\\node[font=\\small,align=center,font=\\small] at (1.9,3.2) {$0$};\n\\path[draw=black!30,fill=white] (2.3,2.8) rectangle (3.1,3.6);\n\\node[font=\\small,align=center,font=\\small] at (2.7,3.2) {$0$};\n\\path[draw=black!30,fill=white] (3.1,2.8) rectangle (3.9,3.6);\n\\node[font=\\small,align=center,font=\\small] at (3.5,3.2) {$0$};\n\\path[draw=black!30,fill=black!8] (0.7,2) rectangle (1.5,2.8);\n\\node[font=\\small,align=center,font=\\small] at (1.1,2.4) {$1$};\n\\path[draw=black!30,fill=white] (1.5,2) rectangle (2.3,2.8);\n\\node[font=\\small,align=center,font=\\small] at (1.9,2.","ll] at (2.3,4.3) {$K$};\n\\path[draw=black!30,fill=white] (0.7,2.8) rectangle (1.5,3.6);\n\\node[font=\\small,align=center,font=\\small] at (1.1,3.2) {$0$};\n\\path[draw=black!30,fill=white] (1.5,2.8) rectangle (2.3,3.6);\n\\node[font=\\small,align=center,font=\\small] at (1.9,3.2) {$0$};\n\\path[draw=black!30,fill=white] (2.3,2.8) rectangle (3.1,3.6);\n\\node[font=\\small,align=center,font=\\small] at (2.7,3.2) {$0$};\n\\path[draw=black!30,fill=white] (3.1,2.8) rectangle (3.9,3.6);\n\\node[font=\\small,align=center,font=\\small] at (3.5,3.2) {$0$};\n\\path[draw=black!30,fill=black!8] (0.7,2) rectangle (1.5,2.8);\n\\node[font=\\small,align=center,font=\\small] at (1.1,2.4) {$1$};\n\\path[draw=black!30,fill=white] (1.5,2) rectangle (2.3,2.8);\n\\node[font=\\small,align=center,font=\\small] at (1.9,2.4) {$0$};\n\\path[draw=black!30,fill=white] (2.3,2) rectangle (3.1,2.8);\n\\node[font=\\small,align=center,font=\\small] at (2.7,2.4)","ll] at (1.1,3.2) {$0$};\n\\path[draw=black!30,fill=white] (1.5,2.8) rectangle (2.3,3.6);\n\\node[font=\\small,align=center,font=\\small] at (1.9,3.2) {$0$};\n\\path[draw=black!30,fill=white] (2.3,2.8) rectangle (3.1,3.6);\n\\node[font=\\small,align=center,font=\\small] at (2.7,3.2) {$0$};\n\\path[draw=black!30,fill=white] (3.1,2.8) rectangle (3.9,3.6);\n\\node[font=\\small,align=center,font=\\small] at (3.5,3.2) {$0$};\n\\path[draw=black!30,fill=black!8] (0.7,2) rectangle (1.5,2.8);\n\\node[font=\\small,align=center,font=\\small] at (1.1,2.4) {$1$};\n\\path[draw=black!30,fill=white] (1.5,2) rectangle (2.3,2.8);\n\\node[font=\\small,align=center,font=\\small] at (1.9,2.4) {$0$};\n\\path[draw=black!30,fill=white] (2.3,2) rectangle (3.1,2.8);\n\\node[font=\\small,align=center,font=\\small] at (2.7,2.4) {$0$};\n\\path[draw=black!30,fill=white] (3.1,2) rectangle (3.9,2.8);\n\\node[font=\\small,align=center,font=\\small] at (3.5,2.4) {","ll] at (1.9,3.2) {$0$};\n\\path[draw=black!30,fill=white] (2.3,2.8) rectangle (3.1,3.6);\n\\node[font=\\small,align=center,font=\\small] at (2.7,3.2) {$0$};\n\\path[draw=black!30,fill=white] (3.1,2.8) rectangle (3.9,3.6);\n\\node[font=\\small,align=center,font=\\small] at (3.5,3.2) {$0$};\n\\path[draw=black!30,fill=black!8] (0.7,2) rectangle (1.5,2.8);\n\\node[font=\\small,align=center,font=\\small] at (1.1,2.4) {$1$};\n\\path[draw=black!30,fill=white] (1.5,2) rectangle (2.3,2.8);\n\\node[font=\\small,align=center,font=\\small] at (1.9,2.4) {$0$};\n\\path[draw=black!30,fill=white] (2.3,2) rectangle (3.1,2.8);\n\\node[font=\\small,align=center,font=\\small] at (2.7,2.4) {$0$};\n\\path[draw=black!30,fill=white] (3.1,2) rectangle (3.9,2.8);\n\\node[font=\\small,align=center,font=\\small] at (3.5,2.4) {$0$};\n\\path[draw=black!30,fill=black!8] (0.7,1.2) rectangle (1.5,2);\n\\node[font=\\small,align=center,font=\\small] at (1.1,1.6) {","mall] at (2.7,3.2) {$0$};\n\\path[draw=black!30,fill=white] (3.1,2.8) rectangle (3.9,3.6);\n\\node[font=\\small,align=center,font=\\small] at (3.5,3.2) {$0$};\n\\path[draw=black!30,fill=black!8] (0.7,2) rectangle (1.5,2.8);\n\\node[font=\\small,align=center,font=\\small] at (1.1,2.4) {$1$};\n\\path[draw=black!30,fill=white] (1.5,2) rectangle (2.3,2.8);\n\\node[font=\\small,align=center,font=\\small] at (1.9,2.4) {$0$};\n\\path[draw=black!30,fill=white] (2.3,2) rectangle (3.1,2.8);\n\\node[font=\\small,align=center,font=\\small] at (2.7,2.4) {$0$};\n\\path[draw=black!30,fill=white] (3.1,2) rectangle (3.9,2.8);\n\\node[font=\\small,align=center,font=\\small] at (3.5,2.4) {$0$};\n\\path[draw=black!30,fill=black!8] (0.7,1.2) rectangle (1.5,2);\n\\node[font=\\small,align=center,font=\\small] at (1.1,1.6) {$1$};\n\\path[draw=black!30,fill=white] (1.5,1.2) rectangle (2.3,2);\n\\node[font=\\small,align=center,font=\\small] at (1.9,1.6) {","\\small] at (3.5,3.2) {$0$};\n\\path[draw=black!30,fill=black!8] (0.7,2) rectangle (1.5,2.8);\n\\node[font=\\small,align=center,font=\\small] at (1.1,2.4) {$1$};\n\\path[draw=black!30,fill=white] (1.5,2) rectangle (2.3,2.8);\n\\node[font=\\small,align=center,font=\\small] at (1.9,2.4) {$0$};\n\\path[draw=black!30,fill=white] (2.3,2) rectangle (3.1,2.8);\n\\node[font=\\small,align=center,font=\\small] at (2.7,2.4) {$0$};\n\\path[draw=black!30,fill=white] (3.1,2) rectangle (3.9,2.8);\n\\node[font=\\small,align=center,font=\\small] at (3.5,2.4) {$0$};\n\\path[draw=black!30,fill=black!8] (0.7,1.2) rectangle (1.5,2);\n\\node[font=\\small,align=center,font=\\small] at (1.1,1.6) {$1$};\n\\path[draw=black!30,fill=white] (1.5,1.2) rectangle (2.3,2);\n\\node[font=\\small,align=center,font=\\small] at (1.9,1.6) {$0$};\n\\path[draw=black!30,fill=white] (2.3,1.2) rectangle (3.1,2);\n\\node[font=\\small,align=center,font=\\small] at (2.7,1.6) {","t=\\small] at (1.1,2.4) {$1$};\n\\path[draw=black!30,fill=white] (1.5,2) rectangle (2.3,2.8);\n\\node[font=\\small,align=center,font=\\small] at (1.9,2.4) {$0$};\n\\path[draw=black!30,fill=white] (2.3,2) rectangle (3.1,2.8);\n\\node[font=\\small,align=center,font=\\small] at (2.7,2.4) {$0$};\n\\path[draw=black!30,fill=white] (3.1,2) rectangle (3.9,2.8);\n\\node[font=\\small,align=center,font=\\small] at (3.5,2.4) {$0$};\n\\path[draw=black!30,fill=black!8] (0.7,1.2) rectangle (1.5,2);\n\\node[font=\\small,align=center,font=\\small] at (1.1,1.6) {$1$};\n\\path[draw=black!30,fill=white] (1.5,1.2) rectangle (2.3,2);\n\\node[font=\\small,align=center,font=\\small] at (1.9,1.6) {$0$};\n\\path[draw=black!30,fill=white] (2.3,1.2) rectangle (3.1,2);\n\\node[font=\\small,align=center,font=\\small] at (2.7,1.6) {$0$};\n\\path[draw=black!30,fill=white] (3.1,1.2) rectangle (3.9,2);\n\\node[font=\\small,align=center,font=\\small] at (3.5,1.6) {","\\small] at (1.9,2.4) {$0$};\n\\path[draw=black!30,fill=white] (2.3,2) rectangle (3.1,2.8);\n\\node[font=\\small,align=center,font=\\small] at (2.7,2.4) {$0$};\n\\path[draw=black!30,fill=white] (3.1,2) rectangle (3.9,2.8);\n\\node[font=\\small,align=center,font=\\small] at (3.5,2.4) {$0$};\n\\path[draw=black!30,fill=black!8] (0.7,1.2) rectangle (1.5,2);\n\\node[font=\\small,align=center,font=\\small] at (1.1,1.6) {$1$};\n\\path[draw=black!30,fill=white] (1.5,1.2) rectangle (2.3,2);\n\\node[font=\\small,align=center,font=\\small] at (1.9,1.6) {$0$};\n\\path[draw=black!30,fill=white] (2.3,1.2) rectangle (3.1,2);\n\\node[font=\\small,align=center,font=\\small] at (2.7,1.6) {$0$};\n\\path[draw=black!30,fill=white] (3.1,1.2) rectangle (3.9,2);\n\\node[font=\\small,align=center,font=\\small] at (3.5,1.6) {$0$};\n\\path[draw=black!30,fill=white] (0.7,0.4) rectangle (1.5,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.1,0.8) {","\\small] at (2.7,2.4) {$0$};\n\\path[draw=black!30,fill=white] (3.1,2) rectangle (3.9,2.8);\n\\node[font=\\small,align=center,font=\\small] at (3.5,2.4) {$0$};\n\\path[draw=black!30,fill=black!8] (0.7,1.2) rectangle (1.5,2);\n\\node[font=\\small,align=center,font=\\small] at (1.1,1.6) {$1$};\n\\path[draw=black!30,fill=white] (1.5,1.2) rectangle (2.3,2);\n\\node[font=\\small,align=center,font=\\small] at (1.9,1.6) {$0$};\n\\path[draw=black!30,fill=white] (2.3,1.2) rectangle (3.1,2);\n\\node[font=\\small,align=center,font=\\small] at (2.7,1.6) {$0$};\n\\path[draw=black!30,fill=white] (3.1,1.2) rectangle (3.9,2);\n\\node[font=\\small,align=center,font=\\small] at (3.5,1.6) {$0$};\n\\path[draw=black!30,fill=white] (0.7,0.4) rectangle (1.5,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.1,0.8) {$0$};\n\\path[draw=black!30,fill=black!8] (1.5,0.4) rectangle (2.3,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.9,0.","\\small] at (3.5,2.4) {$0$};\n\\path[draw=black!30,fill=black!8] (0.7,1.2) rectangle (1.5,2);\n\\node[font=\\small,align=center,font=\\small] at (1.1,1.6) {$1$};\n\\path[draw=black!30,fill=white] (1.5,1.2) rectangle (2.3,2);\n\\node[font=\\small,align=center,font=\\small] at (1.9,1.6) {$0$};\n\\path[draw=black!30,fill=white] (2.3,1.2) rectangle (3.1,2);\n\\node[font=\\small,align=center,font=\\small] at (2.7,1.6) {$0$};\n\\path[draw=black!30,fill=white] (3.1,1.2) rectangle (3.9,2);\n\\node[font=\\small,align=center,font=\\small] at (3.5,1.6) {$0$};\n\\path[draw=black!30,fill=white] (0.7,0.4) rectangle (1.5,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.1,0.8) {$0$};\n\\path[draw=black!30,fill=black!8] (1.5,0.4) rectangle (2.3,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.9,0.8) {$1$};\n\\path[draw=black!30,fill=black!8] (2.3,0.4) rectangle (3.1,1.2);\n\\node[font=\\small,align=center,font=\\small] at (2.","t=\\small] at (1.1,1.6) {$1$};\n\\path[draw=black!30,fill=white] (1.5,1.2) rectangle (2.3,2);\n\\node[font=\\small,align=center,font=\\small] at (1.9,1.6) {$0$};\n\\path[draw=black!30,fill=white] (2.3,1.2) rectangle (3.1,2);\n\\node[font=\\small,align=center,font=\\small] at (2.7,1.6) {$0$};\n\\path[draw=black!30,fill=white] (3.1,1.2) rectangle (3.9,2);\n\\node[font=\\small,align=center,font=\\small] at (3.5,1.6) {$0$};\n\\path[draw=black!30,fill=white] (0.7,0.4) rectangle (1.5,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.1,0.8) {$0$};\n\\path[draw=black!30,fill=black!8] (1.5,0.4) rectangle (2.3,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.9,0.8) {$1$};\n\\path[draw=black!30,fill=black!8] (2.3,0.4) rectangle (3.1,1.2);\n\\node[font=\\small,align=center,font=\\small] at (2.7,0.8) {$-1$};\n\\path[draw=black!30,fill=white] (3.1,0.4) rectangle (3.9,1.2);\n\\node[font=\\small,align=center,font=\\small] at","\\small] at (1.9,1.6) {$0$};\n\\path[draw=black!30,fill=white] (2.3,1.2) rectangle (3.1,2);\n\\node[font=\\small,align=center,font=\\small] at (2.7,1.6) {$0$};\n\\path[draw=black!30,fill=white] (3.1,1.2) rectangle (3.9,2);\n\\node[font=\\small,align=center,font=\\small] at (3.5,1.6) {$0$};\n\\path[draw=black!30,fill=white] (0.7,0.4) rectangle (1.5,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.1,0.8) {$0$};\n\\path[draw=black!30,fill=black!8] (1.5,0.4) rectangle (2.3,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.9,0.8) {$1$};\n\\path[draw=black!30,fill=black!8] (2.3,0.4) rectangle (3.1,1.2);\n\\node[font=\\small,align=center,font=\\small] at (2.7,0.8) {$-1$};\n\\path[draw=black!30,fill=white] (3.1,0.4) rectangle (3.9,1.2);\n\\node[font=\\small,align=center,font=\\small] at (3.5,0.8) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (7.2,4.3) {path counts at length two};\n\\path[draw=black!30,fill","ll] at (2.7,1.6) {$0$};\n\\path[draw=black!30,fill=white] (3.1,1.2) rectangle (3.9,2);\n\\node[font=\\small,align=center,font=\\small] at (3.5,1.6) {$0$};\n\\path[draw=black!30,fill=white] (0.7,0.4) rectangle (1.5,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.1,0.8) {$0$};\n\\path[draw=black!30,fill=black!8] (1.5,0.4) rectangle (2.3,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.9,0.8) {$1$};\n\\path[draw=black!30,fill=black!8] (2.3,0.4) rectangle (3.1,1.2);\n\\node[font=\\small,align=center,font=\\small] at (2.7,0.8) {$-1$};\n\\path[draw=black!30,fill=white] (3.1,0.4) rectangle (3.9,1.2);\n\\node[font=\\small,align=center,font=\\small] at (3.5,0.8) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (7.2,4.3) {path counts at length two};\n\\path[draw=black!30,fill=white] (5.6,2.8) rectangle (6.4,3.6);\n\\node[font=\\small,align=center,font=\\small] at (6,3.2) {$0$};\n\\path[draw=black!30,fill=whi","at (3.5,1.6) {$0$};\n\\path[draw=black!30,fill=white] (0.7,0.4) rectangle (1.5,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.1,0.8) {$0$};\n\\path[draw=black!30,fill=black!8] (1.5,0.4) rectangle (2.3,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.9,0.8) {$1$};\n\\path[draw=black!30,fill=black!8] (2.3,0.4) rectangle (3.1,1.2);\n\\node[font=\\small,align=center,font=\\small] at (2.7,0.8) {$-1$};\n\\path[draw=black!30,fill=white] (3.1,0.4) rectangle (3.9,1.2);\n\\node[font=\\small,align=center,font=\\small] at (3.5,0.8) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (7.2,4.3) {path counts at length two};\n\\path[draw=black!30,fill=white] (5.6,2.8) rectangle (6.4,3.6);\n\\node[font=\\small,align=center,font=\\small] at (6,3.2) {$0$};\n\\path[draw=black!30,fill=white] (6.4,2.8) rectangle (7.2,3.6);\n\\node[font=\\small,align=center,font=\\small] at (6.8,3.2) {$0$};\n\\path[draw=black!30,fill=white]","t (1.1,0.8) {$0$};\n\\path[draw=black!30,fill=black!8] (1.5,0.4) rectangle (2.3,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.9,0.8) {$1$};\n\\path[draw=black!30,fill=black!8] (2.3,0.4) rectangle (3.1,1.2);\n\\node[font=\\small,align=center,font=\\small] at (2.7,0.8) {$-1$};\n\\path[draw=black!30,fill=white] (3.1,0.4) rectangle (3.9,1.2);\n\\node[font=\\small,align=center,font=\\small] at (3.5,0.8) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (7.2,4.3) {path counts at length two};\n\\path[draw=black!30,fill=white] (5.6,2.8) rectangle (6.4,3.6);\n\\node[font=\\small,align=center,font=\\small] at (6,3.2) {$0$};\n\\path[draw=black!30,fill=white] (6.4,2.8) rectangle (7.2,3.6);\n\\node[font=\\small,align=center,font=\\small] at (6.8,3.2) {$0$};\n\\path[draw=black!30,fill=white] (7.2,2.8) rectangle (8,3.6);\n\\node[font=\\small,align=center,font=\\small] at (7.6,3.2) {$0$};\n\\path[draw=black!30,fill=white] (",";\n\\node[font=\\small,align=center,font=\\small] at (2.7,0.8) {$-1$};\n\\path[draw=black!30,fill=white] (3.1,0.4) rectangle (3.9,1.2);\n\\node[font=\\small,align=center,font=\\small] at (3.5,0.8) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (7.2,4.3) {path counts at length two};\n\\path[draw=black!30,fill=white] (5.6,2.8) rectangle (6.4,3.6);\n\\node[font=\\small,align=center,font=\\small] at (6,3.2) {$0$};\n\\path[draw=black!30,fill=white] (6.4,2.8) rectangle (7.2,3.6);\n\\node[font=\\small,align=center,font=\\small] at (6.8,3.2) {$0$};\n\\path[draw=black!30,fill=white] (7.2,2.8) rectangle (8,3.6);\n\\node[font=\\small,align=center,font=\\small] at (7.6,3.2) {$0$};\n\\path[draw=black!30,fill=white] (8,2.8) rectangle (8.8,3.6);\n\\node[font=\\small,align=center,font=\\small] at (8.4,3.2) {$0$};\n\\path[draw=black!30,fill=white] (5.6,2) rectangle (6.4,2.8);\n\\node[font=\\small,align=center,font=\\small] at (6,2.4) {$0",");\n\\node[font=\\small,align=center,font=\\small] at (3.5,0.8) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (7.2,4.3) {path counts at length two};\n\\path[draw=black!30,fill=white] (5.6,2.8) rectangle (6.4,3.6);\n\\node[font=\\small,align=center,font=\\small] at (6,3.2) {$0$};\n\\path[draw=black!30,fill=white] (6.4,2.8) rectangle (7.2,3.6);\n\\node[font=\\small,align=center,font=\\small] at (6.8,3.2) {$0$};\n\\path[draw=black!30,fill=white] (7.2,2.8) rectangle (8,3.6);\n\\node[font=\\small,align=center,font=\\small] at (7.6,3.2) {$0$};\n\\path[draw=black!30,fill=white] (8,2.8) rectangle (8.8,3.6);\n\\node[font=\\small,align=center,font=\\small] at (8.4,3.2) {$0$};\n\\path[draw=black!30,fill=white] (5.6,2) rectangle (6.4,2.8);\n\\node[font=\\small,align=center,font=\\small] at (6,2.4) {$0$};\n\\path[draw=black!30,fill=white] (6.4,2) rectangle (7.2,2.8);\n\\node[font=\\small,align=center,font=\\small] at (6.8,2.4) {$0$}","path counts at length two};\n\\path[draw=black!30,fill=white] (5.6,2.8) rectangle (6.4,3.6);\n\\node[font=\\small,align=center,font=\\small] at (6,3.2) {$0$};\n\\path[draw=black!30,fill=white] (6.4,2.8) rectangle (7.2,3.6);\n\\node[font=\\small,align=center,font=\\small] at (6.8,3.2) {$0$};\n\\path[draw=black!30,fill=white] (7.2,2.8) rectangle (8,3.6);\n\\node[font=\\small,align=center,font=\\small] at (7.6,3.2) {$0$};\n\\path[draw=black!30,fill=white] (8,2.8) rectangle (8.8,3.6);\n\\node[font=\\small,align=center,font=\\small] at (8.4,3.2) {$0$};\n\\path[draw=black!30,fill=white] (5.6,2) rectangle (6.4,2.8);\n\\node[font=\\small,align=center,font=\\small] at (6,2.4) {$0$};\n\\path[draw=black!30,fill=white] (6.4,2) rectangle (7.2,2.8);\n\\node[font=\\small,align=center,font=\\small] at (6.8,2.4) {$0$};\n\\path[draw=black!30,fill=white] (7.2,2) rectangle (8,2.8);\n\\node[font=\\small,align=center,font=\\small] at (7.6,2.4) {$0$};","t=\\small] at (6,3.2) {$0$};\n\\path[draw=black!30,fill=white] (6.4,2.8) rectangle (7.2,3.6);\n\\node[font=\\small,align=center,font=\\small] at (6.8,3.2) {$0$};\n\\path[draw=black!30,fill=white] (7.2,2.8) rectangle (8,3.6);\n\\node[font=\\small,align=center,font=\\small] at (7.6,3.2) {$0$};\n\\path[draw=black!30,fill=white] (8,2.8) rectangle (8.8,3.6);\n\\node[font=\\small,align=center,font=\\small] at (8.4,3.2) {$0$};\n\\path[draw=black!30,fill=white] (5.6,2) rectangle (6.4,2.8);\n\\node[font=\\small,align=center,font=\\small] at (6,2.4) {$0$};\n\\path[draw=black!30,fill=white] (6.4,2) rectangle (7.2,2.8);\n\\node[font=\\small,align=center,font=\\small] at (6.8,2.4) {$0$};\n\\path[draw=black!30,fill=white] (7.2,2) rectangle (8,2.8);\n\\node[font=\\small,align=center,font=\\small] at (7.6,2.4) {$0$};\n\\path[draw=black!30,fill=white] (8,2) rectangle (8.8,2.8);\n\\node[font=\\small,align=center,font=\\small] at (8.4,2.4) {$0$};\n\\p","ont=\\small] at (6.8,3.2) {$0$};\n\\path[draw=black!30,fill=white] (7.2,2.8) rectangle (8,3.6);\n\\node[font=\\small,align=center,font=\\small] at (7.6,3.2) {$0$};\n\\path[draw=black!30,fill=white] (8,2.8) rectangle (8.8,3.6);\n\\node[font=\\small,align=center,font=\\small] at (8.4,3.2) {$0$};\n\\path[draw=black!30,fill=white] (5.6,2) rectangle (6.4,2.8);\n\\node[font=\\small,align=center,font=\\small] at (6,2.4) {$0$};\n\\path[draw=black!30,fill=white] (6.4,2) rectangle (7.2,2.8);\n\\node[font=\\small,align=center,font=\\small] at (6.8,2.4) {$0$};\n\\path[draw=black!30,fill=white] (7.2,2) rectangle (8,2.8);\n\\node[font=\\small,align=center,font=\\small] at (7.6,2.4) {$0$};\n\\path[draw=black!30,fill=white] (8,2) rectangle (8.8,2.8);\n\\node[font=\\small,align=center,font=\\small] at (8.4,2.4) {$0$};\n\\path[draw=black!30,fill=white] (5.6,1.2) rectangle (6.4,2);\n\\node[font=\\small,align=center,font=\\small] at (6,1.6) {$0$};\n\\p","ont=\\small] at (7.6,3.2) {$0$};\n\\path[draw=black!30,fill=white] (8,2.8) rectangle (8.8,3.6);\n\\node[font=\\small,align=center,font=\\small] at (8.4,3.2) {$0$};\n\\path[draw=black!30,fill=white] (5.6,2) rectangle (6.4,2.8);\n\\node[font=\\small,align=center,font=\\small] at (6,2.4) {$0$};\n\\path[draw=black!30,fill=white] (6.4,2) rectangle (7.2,2.8);\n\\node[font=\\small,align=center,font=\\small] at (6.8,2.4) {$0$};\n\\path[draw=black!30,fill=white] (7.2,2) rectangle (8,2.8);\n\\node[font=\\small,align=center,font=\\small] at (7.6,2.4) {$0$};\n\\path[draw=black!30,fill=white] (8,2) rectangle (8.8,2.8);\n\\node[font=\\small,align=center,font=\\small] at (8.4,2.4) {$0$};\n\\path[draw=black!30,fill=white] (5.6,1.2) rectangle (6.4,2);\n\\node[font=\\small,align=center,font=\\small] at (6,1.6) {$0$};\n\\path[draw=black!30,fill=white] (6.4,1.2) rectangle (7.2,2);\n\\node[font=\\small,align=center,font=\\small] at (6.8,1.6) {$0$};\n\\p",",font=\\small] at (8.4,3.2) {$0$};\n\\path[draw=black!30,fill=white] (5.6,2) rectangle (6.4,2.8);\n\\node[font=\\small,align=center,font=\\small] at (6,2.4) {$0$};\n\\path[draw=black!30,fill=white] (6.4,2) rectangle (7.2,2.8);\n\\node[font=\\small,align=center,font=\\small] at (6.8,2.4) {$0$};\n\\path[draw=black!30,fill=white] (7.2,2) rectangle (8,2.8);\n\\node[font=\\small,align=center,font=\\small] at (7.6,2.4) {$0$};\n\\path[draw=black!30,fill=white] (8,2) rectangle (8.8,2.8);\n\\node[font=\\small,align=center,font=\\small] at (8.4,2.4) {$0$};\n\\path[draw=black!30,fill=white] (5.6,1.2) rectangle (6.4,2);\n\\node[font=\\small,align=center,font=\\small] at (6,1.6) {$0$};\n\\path[draw=black!30,fill=white] (6.4,1.2) rectangle (7.2,2);\n\\node[font=\\small,align=center,font=\\small] at (6.8,1.6) {$0$};\n\\path[draw=black!30,fill=white] (7.2,1.2) rectangle (8,2);\n\\node[font=\\small,align=center,font=\\small] at (7.6,1.6) {$0$};\n\\p","er,font=\\small] at (6,2.4) {$0$};\n\\path[draw=black!30,fill=white] (6.4,2) rectangle (7.2,2.8);\n\\node[font=\\small,align=center,font=\\small] at (6.8,2.4) {$0$};\n\\path[draw=black!30,fill=white] (7.2,2) rectangle (8,2.8);\n\\node[font=\\small,align=center,font=\\small] at (7.6,2.4) {$0$};\n\\path[draw=black!30,fill=white] (8,2) rectangle (8.8,2.8);\n\\node[font=\\small,align=center,font=\\small] at (8.4,2.4) {$0$};\n\\path[draw=black!30,fill=white] (5.6,1.2) rectangle (6.4,2);\n\\node[font=\\small,align=center,font=\\small] at (6,1.6) {$0$};\n\\path[draw=black!30,fill=white] (6.4,1.2) rectangle (7.2,2);\n\\node[font=\\small,align=center,font=\\small] at (6.8,1.6) {$0$};\n\\path[draw=black!30,fill=white] (7.2,1.2) rectangle (8,2);\n\\node[font=\\small,align=center,font=\\small] at (7.6,1.6) {$0$};\n\\path[draw=black!30,fill=white] (8,1.2) rectangle (8.8,2);\n\\node[font=\\small,align=center,font=\\small] at (8.4,1.6) {$0$};\n\\p","er,font=\\small] at (6.8,2.4) {$0$};\n\\path[draw=black!30,fill=white] (7.2,2) rectangle (8,2.8);\n\\node[font=\\small,align=center,font=\\small] at (7.6,2.4) {$0$};\n\\path[draw=black!30,fill=white] (8,2) rectangle (8.8,2.8);\n\\node[font=\\small,align=center,font=\\small] at (8.4,2.4) {$0$};\n\\path[draw=black!30,fill=white] (5.6,1.2) rectangle (6.4,2);\n\\node[font=\\small,align=center,font=\\small] at (6,1.6) {$0$};\n\\path[draw=black!30,fill=white] (6.4,1.2) rectangle (7.2,2);\n\\node[font=\\small,align=center,font=\\small] at (6.8,1.6) {$0$};\n\\path[draw=black!30,fill=white] (7.2,1.2) rectangle (8,2);\n\\node[font=\\small,align=center,font=\\small] at (7.6,1.6) {$0$};\n\\path[draw=black!30,fill=white] (8,1.2) rectangle (8.8,2);\n\\node[font=\\small,align=center,font=\\small] at (8.4,1.6) {$0$};\n\\path[draw=black!30,fill=black!8] (5.6,0.4) rectangle (6.4,1.2);\n\\node[font=\\small,align=center,font=\\small] at (6,0.8) {$2$}",",font=\\small] at (7.6,2.4) {$0$};\n\\path[draw=black!30,fill=white] (8,2) rectangle (8.8,2.8);\n\\node[font=\\small,align=center,font=\\small] at (8.4,2.4) {$0$};\n\\path[draw=black!30,fill=white] (5.6,1.2) rectangle (6.4,2);\n\\node[font=\\small,align=center,font=\\small] at (6,1.6) {$0$};\n\\path[draw=black!30,fill=white] (6.4,1.2) rectangle (7.2,2);\n\\node[font=\\small,align=center,font=\\small] at (6.8,1.6) {$0$};\n\\path[draw=black!30,fill=white] (7.2,1.2) rectangle (8,2);\n\\node[font=\\small,align=center,font=\\small] at (7.6,1.6) {$0$};\n\\path[draw=black!30,fill=white] (8,1.2) rectangle (8.8,2);\n\\node[font=\\small,align=center,font=\\small] at (8.4,1.6) {$0$};\n\\path[draw=black!30,fill=black!8] (5.6,0.4) rectangle (6.4,1.2);\n\\node[font=\\small,align=center,font=\\small] at (6,0.8) {$2$};\n\\path[draw=black!30,fill=white] (6.4,0.4) rectangle (7.2,1.2);\n\\node[font=\\small,align=center,font=\\small] at (6.8,0.8) {$0",",font=\\small] at (8.4,2.4) {$0$};\n\\path[draw=black!30,fill=white] (5.6,1.2) rectangle (6.4,2);\n\\node[font=\\small,align=center,font=\\small] at (6,1.6) {$0$};\n\\path[draw=black!30,fill=white] (6.4,1.2) rectangle (7.2,2);\n\\node[font=\\small,align=center,font=\\small] at (6.8,1.6) {$0$};\n\\path[draw=black!30,fill=white] (7.2,1.2) rectangle (8,2);\n\\node[font=\\small,align=center,font=\\small] at (7.6,1.6) {$0$};\n\\path[draw=black!30,fill=white] (8,1.2) rectangle (8.8,2);\n\\node[font=\\small,align=center,font=\\small] at (8.4,1.6) {$0$};\n\\path[draw=black!30,fill=black!8] (5.6,0.4) rectangle (6.4,1.2);\n\\node[font=\\small,align=center,font=\\small] at (6,0.8) {$2$};\n\\path[draw=black!30,fill=white] (6.4,0.4) rectangle (7.2,1.2);\n\\node[font=\\small,align=center,font=\\small] at (6.8,0.8) {$0$};\n\\path[draw=black!30,fill=white] (7.2,0.4) rectangle (8,1.2);\n\\node[font=\\small,align=center,font=\\small] at (7.6,0.8) {","er,font=\\small] at (6,1.6) {$0$};\n\\path[draw=black!30,fill=white] (6.4,1.2) rectangle (7.2,2);\n\\node[font=\\small,align=center,font=\\small] at (6.8,1.6) {$0$};\n\\path[draw=black!30,fill=white] (7.2,1.2) rectangle (8,2);\n\\node[font=\\small,align=center,font=\\small] at (7.6,1.6) {$0$};\n\\path[draw=black!30,fill=white] (8,1.2) rectangle (8.8,2);\n\\node[font=\\small,align=center,font=\\small] at (8.4,1.6) {$0$};\n\\path[draw=black!30,fill=black!8] (5.6,0.4) rectangle (6.4,1.2);\n\\node[font=\\small,align=center,font=\\small] at (6,0.8) {$2$};\n\\path[draw=black!30,fill=white] (6.4,0.4) rectangle (7.2,1.2);\n\\node[font=\\small,align=center,font=\\small] at (6.8,0.8) {$0$};\n\\path[draw=black!30,fill=white] (7.2,0.4) rectangle (8,1.2);\n\\node[font=\\small,align=center,font=\\small] at (7.6,0.8) {$0$};\n\\path[draw=black!30,fill=white] (8,0.4) rectangle (8.8,1.2);\n\\node[font=\\small,align=center,font=\\small] at (8.4,0.8)","ont=\\small] at (6.8,1.6) {$0$};\n\\path[draw=black!30,fill=white] (7.2,1.2) rectangle (8,2);\n\\node[font=\\small,align=center,font=\\small] at (7.6,1.6) {$0$};\n\\path[draw=black!30,fill=white] (8,1.2) rectangle (8.8,2);\n\\node[font=\\small,align=center,font=\\small] at (8.4,1.6) {$0$};\n\\path[draw=black!30,fill=black!8] (5.6,0.4) rectangle (6.4,1.2);\n\\node[font=\\small,align=center,font=\\small] at (6,0.8) {$2$};\n\\path[draw=black!30,fill=white] (6.4,0.4) rectangle (7.2,1.2);\n\\node[font=\\small,align=center,font=\\small] at (6.8,0.8) {$0$};\n\\path[draw=black!30,fill=white] (7.2,0.4) rectangle (8,1.2);\n\\node[font=\\small,align=center,font=\\small] at (7.6,0.8) {$0$};\n\\path[draw=black!30,fill=white] (8,0.4) rectangle (8.8,1.2);\n\\node[font=\\small,align=center,font=\\small] at (8.4,0.8) {$0$};\n\\node[font=\\small,align=center] at (11.85,3) {$1+(-1)=0$};\n\\node[font=\\small,align=center] at (11.85,1.65) {$K^2=0$};\n\\","\\small] at (7.6,1.6) {$0$};\n\\path[draw=black!30,fill=white] (8,1.2) rectangle (8.8,2);\n\\node[font=\\small,align=center,font=\\small] at (8.4,1.6) {$0$};\n\\path[draw=black!30,fill=black!8] (5.6,0.4) rectangle (6.4,1.2);\n\\node[font=\\small,align=center,font=\\small] at (6,0.8) {$2$};\n\\path[draw=black!30,fill=white] (6.4,0.4) rectangle (7.2,1.2);\n\\node[font=\\small,align=center,font=\\small] at (6.8,0.8) {$0$};\n\\path[draw=black!30,fill=white] (7.2,0.4) rectangle (8,1.2);\n\\node[font=\\small,align=center,font=\\small] at (7.6,0.8) {$0$};\n\\path[draw=black!30,fill=white] (8,0.4) rectangle (8.8,1.2);\n\\node[font=\\small,align=center,font=\\small] at (8.4,0.8) {$0$};\n\\node[font=\\small,align=center] at (11.85,3) {$1+(-1)=0$};\n\\node[font=\\small,align=center] at (11.85,1.65) {$K^2=0$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.1) {Structural support is an upper bound. It does not encode destructiv","mall] at (8.4,1.6) {$0$};\n\\path[draw=black!30,fill=black!8] (5.6,0.4) rectangle (6.4,1.2);\n\\node[font=\\small,align=center,font=\\small] at (6,0.8) {$2$};\n\\path[draw=black!30,fill=white] (6.4,0.4) rectangle (7.2,1.2);\n\\node[font=\\small,align=center,font=\\small] at (6.8,0.8) {$0$};\n\\path[draw=black!30,fill=white] (7.2,0.4) rectangle (8,1.2);\n\\node[font=\\small,align=center,font=\\small] at (7.6,0.8) {$0$};\n\\path[draw=black!30,fill=white] (8,0.4) rectangle (8.8,1.2);\n\\node[font=\\small,align=center,font=\\small] at (8.4,0.8) {$0$};\n\\node[font=\\small,align=center] at (11.85,3) {$1+(-1)=0$};\n\\node[font=\\small,align=center] at (11.85,1.65) {$K^2=0$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.1) {Structural support is an upper bound. It does not encode destructive cancellation.};\n\\end{tikzpicture}","t=\\small] at (6,0.8) {$2$};\n\\path[draw=black!30,fill=white] (6.4,0.4) rectangle (7.2,1.2);\n\\node[font=\\small,align=center,font=\\small] at (6.8,0.8) {$0$};\n\\path[draw=black!30,fill=white] (7.2,0.4) rectangle (8,1.2);\n\\node[font=\\small,align=center,font=\\small] at (7.6,0.8) {$0$};\n\\path[draw=black!30,fill=white] (8,0.4) rectangle (8.8,1.2);\n\\node[font=\\small,align=center,font=\\small] at (8.4,0.8) {$0$};\n\\node[font=\\small,align=center] at (11.85,3) {$1+(-1)=0$};\n\\node[font=\\small,align=center] at (11.85,1.65) {$K^2=0$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.1) {Structural support is an upper bound. It does not encode destructive cancellation.};\n\\end{tikzpicture}","=white] (6.4,0.4) rectangle (7.2,1.2);\n\\node[font=\\small,align=center,font=\\small] at (6.8,0.8) {$0$};\n\\path[draw=black!30,fill=white] (7.2,0.4) rectangle (8,1.2);\n\\node[font=\\small,align=center,font=\\small] at (7.6,0.8) {$0$};\n\\path[draw=black!30,fill=white] (8,0.4) rectangle (8.8,1.2);\n\\node[font=\\small,align=center,font=\\small] at (8.4,0.8) {$0$};\n\\node[font=\\small,align=center] at (11.85,3) {$1+(-1)=0$};\n\\node[font=\\small,align=center] at (11.85,1.65) {$K^2=0$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.1) {Structural support is an upper bound. It does not encode destructive cancellation.};\n\\end{tikzpicture}","=center,font=\\small] at (6.8,0.8) {$0$};\n\\path[draw=black!30,fill=white] (7.2,0.4) rectangle (8,1.2);\n\\node[font=\\small,align=center,font=\\small] at (7.6,0.8) {$0$};\n\\path[draw=black!30,fill=white] (8,0.4) rectangle (8.8,1.2);\n\\node[font=\\small,align=center,font=\\small] at (8.4,0.8) {$0$};\n\\node[font=\\small,align=center] at (11.85,3) {$1+(-1)=0$};\n\\node[font=\\small,align=center] at (11.85,1.65) {$K^2=0$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.1) {Structural support is an upper bound. It does not encode destructive cancellation.};\n\\end{tikzpicture}","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (2.16,4.4) {$C(c)$};\n\\path[draw=black!30,fill=black!8] (0.7,2.97) rectangle (1.43,3.7);\n\\node[font=\\small,align=center,font=\\small] at (1.065,3.335) {$2$};\n\\path[draw=black!30,fill=black!8] (1.43,2.97) rectangle (2.16,3.7);\n\\node[font=\\small,align=center,font=\\small] at (1.795,3.335) {$-1$};\n\\path[draw=black!30,fill=white] (2.16,2.97) rectangle (2.89,3.7);\n\\node[font=\\small,align=center,font=\\small] at (2.525,3.335) {$0$};\n\\path[draw=black!30,fill=black!8] (2.89,2.97) rectangle (3.62,3.7);\n\\node[font=\\small,alig","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (2.16,4.4) {$C(c)$};\n\\path[draw=black!30,fill=black!8] (0.7,2.97) rectangle (1.43,3.7);\n\\node[font=\\small,align=center,font=\\small] at (1.065,3.335) {$2$};\n\\path[draw=black!30,fill=black!8] (1.43,2.97) rectangle (2.16,3.7);\n\\node[font=\\small,align=center,font=\\small] at (1.795,3.335) {$-1$};\n\\path[draw=black!30,fill=white] (2.16,2.97) rectangle (2.89,3.7);\n\\node[font=\\small,align=center,font=\\small] at (2.525,3.335) {$0$};\n\\path[draw=black!30,fill=black!8] (2.89,2.97) rectangle (3.62,3.7);\n\\node[font=\\small,align=center,font=\\small] at (3.255,3.335) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,2.24) rectangle (1.43,2.97);\n\\node[font=\\small,ali","oin=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (2.16,4.4) {$C(c)$};\n\\path[draw=black!30,fill=black!8] (0.7,2.97) rectangle (1.43,3.7);\n\\node[font=\\small,align=center,font=\\small] at (1.065,3.335) {$2$};\n\\path[draw=black!30,fill=black!8] (1.43,2.97) rectangle (2.16,3.7);\n\\node[font=\\small,align=center,font=\\small] at (1.795,3.335) {$-1$};\n\\path[draw=black!30,fill=white] (2.16,2.97) rectangle (2.89,3.7);\n\\node[font=\\small,align=center,font=\\small] at (2.525,3.335) {$0$};\n\\path[draw=black!30,fill=black!8] (2.89,2.97) rectangle (3.62,3.7);\n\\node[font=\\small,align=center,font=\\small] at (3.255,3.335) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,2.24) rectangle (1.43,2.97);\n\\node[font=\\small,align=center,font=\\small] at (1.065,2.605) {$1$};\n\\path[draw=black!30,fill=black!8] (1.43,2.24) rectangle (2.16,2.97);\n\\node[font=\\small,ali","path[draw=black!30,fill=black!8] (0.7,2.97) rectangle (1.43,3.7);\n\\node[font=\\small,align=center,font=\\small] at (1.065,3.335) {$2$};\n\\path[draw=black!30,fill=black!8] (1.43,2.97) rectangle (2.16,3.7);\n\\node[font=\\small,align=center,font=\\small] at (1.795,3.335) {$-1$};\n\\path[draw=black!30,fill=white] (2.16,2.97) rectangle (2.89,3.7);\n\\node[font=\\small,align=center,font=\\small] at (2.525,3.335) {$0$};\n\\path[draw=black!30,fill=black!8] (2.89,2.97) rectangle (3.62,3.7);\n\\node[font=\\small,align=center,font=\\small] at (3.255,3.335) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,2.24) rectangle (1.43,2.97);\n\\node[font=\\small,align=center,font=\\small] at (1.065,2.605) {$1$};\n\\path[draw=black!30,fill=black!8] (1.43,2.24) rectangle (2.16,2.97);\n\\node[font=\\small,align=center,font=\\small] at (1.795,2.605) {$2$};\n\\path[draw=black!30,fill=black!8] (2.16,2.24) rectangle (2.89,2.97);\n\\node[font=\\small,","ath[draw=black!30,fill=black!8] (1.43,2.97) rectangle (2.16,3.7);\n\\node[font=\\small,align=center,font=\\small] at (1.795,3.335) {$-1$};\n\\path[draw=black!30,fill=white] (2.16,2.97) rectangle (2.89,3.7);\n\\node[font=\\small,align=center,font=\\small] at (2.525,3.335) {$0$};\n\\path[draw=black!30,fill=black!8] (2.89,2.97) rectangle (3.62,3.7);\n\\node[font=\\small,align=center,font=\\small] at (3.255,3.335) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,2.24) rectangle (1.43,2.97);\n\\node[font=\\small,align=center,font=\\small] at (1.065,2.605) {$1$};\n\\path[draw=black!30,fill=black!8] (1.43,2.24) rectangle (2.16,2.97);\n\\node[font=\\small,align=center,font=\\small] at (1.795,2.605) {$2$};\n\\path[draw=black!30,fill=black!8] (2.16,2.24) rectangle (2.89,2.97);\n\\node[font=\\small,align=center,font=\\small] at (2.525,2.605) {$-1$};\n\\path[draw=black!30,fill=white] (2.89,2.24) rectangle (3.62,2.97);\n\\node[font=\\small,","path[draw=black!30,fill=white] (2.16,2.97) rectangle (2.89,3.7);\n\\node[font=\\small,align=center,font=\\small] at (2.525,3.335) {$0$};\n\\path[draw=black!30,fill=black!8] (2.89,2.97) rectangle (3.62,3.7);\n\\node[font=\\small,align=center,font=\\small] at (3.255,3.335) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,2.24) rectangle (1.43,2.97);\n\\node[font=\\small,align=center,font=\\small] at (1.065,2.605) {$1$};\n\\path[draw=black!30,fill=black!8] (1.43,2.24) rectangle (2.16,2.97);\n\\node[font=\\small,align=center,font=\\small] at (1.795,2.605) {$2$};\n\\path[draw=black!30,fill=black!8] (2.16,2.24) rectangle (2.89,2.97);\n\\node[font=\\small,align=center,font=\\small] at (2.525,2.605) {$-1$};\n\\path[draw=black!30,fill=white] (2.89,2.24) rectangle (3.62,2.97);\n\\node[font=\\small,align=center,font=\\small] at (3.255,2.605) {$0$};\n\\path[draw=black!30,fill=white] (0.7,1.51) rectangle (1.43,2.24);\n\\node[font=\\small,al","h[draw=black!30,fill=black!8] (2.89,2.97) rectangle (3.62,3.7);\n\\node[font=\\small,align=center,font=\\small] at (3.255,3.335) {$1$};\n\\path[draw=black!30,fill=black!8] (0.7,2.24) rectangle (1.43,2.97);\n\\node[font=\\small,align=center,font=\\small] at (1.065,2.605) {$1$};\n\\path[draw=black!30,fill=black!8] (1.43,2.24) rectangle (2.16,2.97);\n\\node[font=\\small,align=center,font=\\small] at (1.795,2.605) {$2$};\n\\path[draw=black!30,fill=black!8] (2.16,2.24) rectangle (2.89,2.97);\n\\node[font=\\small,align=center,font=\\small] at (2.525,2.605) {$-1$};\n\\path[draw=black!30,fill=white] (2.89,2.24) rectangle (3.62,2.97);\n\\node[font=\\small,align=center,font=\\small] at (3.255,2.605) {$0$};\n\\path[draw=black!30,fill=white] (0.7,1.51) rectangle (1.43,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.065,1.875) {$0$};\n\\path[draw=black!30,fill=black!8] (1.43,1.51) rectangle (2.16,2.24);\n\\node[font=\\small,al","[draw=black!30,fill=black!8] (0.7,2.24) rectangle (1.43,2.97);\n\\node[font=\\small,align=center,font=\\small] at (1.065,2.605) {$1$};\n\\path[draw=black!30,fill=black!8] (1.43,2.24) rectangle (2.16,2.97);\n\\node[font=\\small,align=center,font=\\small] at (1.795,2.605) {$2$};\n\\path[draw=black!30,fill=black!8] (2.16,2.24) rectangle (2.89,2.97);\n\\node[font=\\small,align=center,font=\\small] at (2.525,2.605) {$-1$};\n\\path[draw=black!30,fill=white] (2.89,2.24) rectangle (3.62,2.97);\n\\node[font=\\small,align=center,font=\\small] at (3.255,2.605) {$0$};\n\\path[draw=black!30,fill=white] (0.7,1.51) rectangle (1.43,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.065,1.875) {$0$};\n\\path[draw=black!30,fill=black!8] (1.43,1.51) rectangle (2.16,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.795,1.875) {$1$};\n\\path[draw=black!30,fill=black!8] (2.16,1.51) rectangle (2.89,2.24);\n\\node[font=\\small,ali","[draw=black!30,fill=black!8] (1.43,2.24) rectangle (2.16,2.97);\n\\node[font=\\small,align=center,font=\\small] at (1.795,2.605) {$2$};\n\\path[draw=black!30,fill=black!8] (2.16,2.24) rectangle (2.89,2.97);\n\\node[font=\\small,align=center,font=\\small] at (2.525,2.605) {$-1$};\n\\path[draw=black!30,fill=white] (2.89,2.24) rectangle (3.62,2.97);\n\\node[font=\\small,align=center,font=\\small] at (3.255,2.605) {$0$};\n\\path[draw=black!30,fill=white] (0.7,1.51) rectangle (1.43,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.065,1.875) {$0$};\n\\path[draw=black!30,fill=black!8] (1.43,1.51) rectangle (2.16,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.795,1.875) {$1$};\n\\path[draw=black!30,fill=black!8] (2.16,1.51) rectangle (2.89,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.525,1.875) {$2$};\n\\path[draw=black!30,fill=black!8] (2.89,1.51) rectangle (3.62,2.24);\n\\node[font=\\small,a","ath[draw=black!30,fill=black!8] (2.16,2.24) rectangle (2.89,2.97);\n\\node[font=\\small,align=center,font=\\small] at (2.525,2.605) {$-1$};\n\\path[draw=black!30,fill=white] (2.89,2.24) rectangle (3.62,2.97);\n\\node[font=\\small,align=center,font=\\small] at (3.255,2.605) {$0$};\n\\path[draw=black!30,fill=white] (0.7,1.51) rectangle (1.43,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.065,1.875) {$0$};\n\\path[draw=black!30,fill=black!8] (1.43,1.51) rectangle (2.16,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.795,1.875) {$1$};\n\\path[draw=black!30,fill=black!8] (2.16,1.51) rectangle (2.89,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.525,1.875) {$2$};\n\\path[draw=black!30,fill=black!8] (2.89,1.51) rectangle (3.62,2.24);\n\\node[font=\\small,align=center,font=\\small] at (3.255,1.875) {$-1$};\n\\path[draw=black!30,fill=black!8] (0.7,0.78) rectangle (1.43,1.51);\n\\node[font=\\smal","path[draw=black!30,fill=white] (2.89,2.24) rectangle (3.62,2.97);\n\\node[font=\\small,align=center,font=\\small] at (3.255,2.605) {$0$};\n\\path[draw=black!30,fill=white] (0.7,1.51) rectangle (1.43,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.065,1.875) {$0$};\n\\path[draw=black!30,fill=black!8] (1.43,1.51) rectangle (2.16,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.795,1.875) {$1$};\n\\path[draw=black!30,fill=black!8] (2.16,1.51) rectangle (2.89,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.525,1.875) {$2$};\n\\path[draw=black!30,fill=black!8] (2.89,1.51) rectangle (3.62,2.24);\n\\node[font=\\small,align=center,font=\\small] at (3.255,1.875) {$-1$};\n\\path[draw=black!30,fill=black!8] (0.7,0.78) rectangle (1.43,1.51);\n\\node[font=\\small,align=center,font=\\small] at (1.065,1.145) {$-1$};\n\\path[draw=black!30,fill=white] (1.43,0.78) rectangle (2.16,1.51);\n\\node[font=\\small","th[draw=black!30,fill=white] (0.7,1.51) rectangle (1.43,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.065,1.875) {$0$};\n\\path[draw=black!30,fill=black!8] (1.43,1.51) rectangle (2.16,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.795,1.875) {$1$};\n\\path[draw=black!30,fill=black!8] (2.16,1.51) rectangle (2.89,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.525,1.875) {$2$};\n\\path[draw=black!30,fill=black!8] (2.89,1.51) rectangle (3.62,2.24);\n\\node[font=\\small,align=center,font=\\small] at (3.255,1.875) {$-1$};\n\\path[draw=black!30,fill=black!8] (0.7,0.78) rectangle (1.43,1.51);\n\\node[font=\\small,align=center,font=\\small] at (1.065,1.145) {$-1$};\n\\path[draw=black!30,fill=white] (1.43,0.78) rectangle (2.16,1.51);\n\\node[font=\\small,align=center,font=\\small] at (1.795,1.145) {$0$};\n\\path[draw=black!30,fill=black!8] (2.16,0.78) rectangle (2.89,1.51);\n\\node[font=\\small","draw=black!30,fill=black!8] (1.43,1.51) rectangle (2.16,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.795,1.875) {$1$};\n\\path[draw=black!30,fill=black!8] (2.16,1.51) rectangle (2.89,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.525,1.875) {$2$};\n\\path[draw=black!30,fill=black!8] (2.89,1.51) rectangle (3.62,2.24);\n\\node[font=\\small,align=center,font=\\small] at (3.255,1.875) {$-1$};\n\\path[draw=black!30,fill=black!8] (0.7,0.78) rectangle (1.43,1.51);\n\\node[font=\\small,align=center,font=\\small] at (1.065,1.145) {$-1$};\n\\path[draw=black!30,fill=white] (1.43,0.78) rectangle (2.16,1.51);\n\\node[font=\\small,align=center,font=\\small] at (1.795,1.145) {$0$};\n\\path[draw=black!30,fill=black!8] (2.16,0.78) rectangle (2.89,1.51);\n\\node[font=\\small,align=center,font=\\small] at (2.525,1.145) {$1$};\n\\path[draw=black!30,fill=black!8] (2.89,0.78) rectangle (3.62,1.51);\n\\node[font=\\small,","draw=black!30,fill=black!8] (2.16,1.51) rectangle (2.89,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.525,1.875) {$2$};\n\\path[draw=black!30,fill=black!8] (2.89,1.51) rectangle (3.62,2.24);\n\\node[font=\\small,align=center,font=\\small] at (3.255,1.875) {$-1$};\n\\path[draw=black!30,fill=black!8] (0.7,0.78) rectangle (1.43,1.51);\n\\node[font=\\small,align=center,font=\\small] at (1.065,1.145) {$-1$};\n\\path[draw=black!30,fill=white] (1.43,0.78) rectangle (2.16,1.51);\n\\node[font=\\small,align=center,font=\\small] at (1.795,1.145) {$0$};\n\\path[draw=black!30,fill=black!8] (2.16,0.78) rectangle (2.89,1.51);\n\\node[font=\\small,align=center,font=\\small] at (2.525,1.145) {$1$};\n\\path[draw=black!30,fill=black!8] (2.89,0.78) rectangle (3.62,1.51);\n\\node[font=\\small,align=center,font=\\small] at (3.255,1.145) {$2$};\n\\node[font=\\small,align=center,font=\\small] at (5.675,4.4) {$\\sqrt4\\,Fc$};\n\\path[draw=b","[draw=black!30,fill=black!8] (2.89,1.51) rectangle (3.62,2.24);\n\\node[font=\\small,align=center,font=\\small] at (3.255,1.875) {$-1$};\n\\path[draw=black!30,fill=black!8] (0.7,0.78) rectangle (1.43,1.51);\n\\node[font=\\small,align=center,font=\\small] at (1.065,1.145) {$-1$};\n\\path[draw=black!30,fill=white] (1.43,0.78) rectangle (2.16,1.51);\n\\node[font=\\small,align=center,font=\\small] at (1.795,1.145) {$0$};\n\\path[draw=black!30,fill=black!8] (2.16,0.78) rectangle (2.89,1.51);\n\\node[font=\\small,align=center,font=\\small] at (2.525,1.145) {$1$};\n\\path[draw=black!30,fill=black!8] (2.89,0.78) rectangle (3.62,1.51);\n\\node[font=\\small,align=center,font=\\small] at (3.255,1.145) {$2$};\n\\node[font=\\small,align=center,font=\\small] at (5.675,4.4) {$\\sqrt4\\,Fc$};\n\\path[draw=black!30,fill=black!8] (4.95,2.97) rectangle (6.4,3.7);\n\\node[font=\\small,align=center,font=\\small] at (5.675,3.335) {$2$};\n\\path[draw=b","h[draw=black!30,fill=black!8] (0.7,0.78) rectangle (1.43,1.51);\n\\node[font=\\small,align=center,font=\\small] at (1.065,1.145) {$-1$};\n\\path[draw=black!30,fill=white] (1.43,0.78) rectangle (2.16,1.51);\n\\node[font=\\small,align=center,font=\\small] at (1.795,1.145) {$0$};\n\\path[draw=black!30,fill=black!8] (2.16,0.78) rectangle (2.89,1.51);\n\\node[font=\\small,align=center,font=\\small] at (2.525,1.145) {$1$};\n\\path[draw=black!30,fill=black!8] (2.89,0.78) rectangle (3.62,1.51);\n\\node[font=\\small,align=center,font=\\small] at (3.255,1.145) {$2$};\n\\node[font=\\small,align=center,font=\\small] at (5.675,4.4) {$\\sqrt4\\,Fc$};\n\\path[draw=black!30,fill=black!8] (4.95,2.97) rectangle (6.4,3.7);\n\\node[font=\\small,align=center,font=\\small] at (5.675,3.335) {$2$};\n\\path[draw=black!30,fill=black!8] (4.95,2.24) rectangle (6.4,2.97);\n\\node[font=\\small,align=center,font=\\small] at (5.675,2.605) {$2 - 2 i$};\n\\path[d","h[draw=black!30,fill=white] (1.43,0.78) rectangle (2.16,1.51);\n\\node[font=\\small,align=center,font=\\small] at (1.795,1.145) {$0$};\n\\path[draw=black!30,fill=black!8] (2.16,0.78) rectangle (2.89,1.51);\n\\node[font=\\small,align=center,font=\\small] at (2.525,1.145) {$1$};\n\\path[draw=black!30,fill=black!8] (2.89,0.78) rectangle (3.62,1.51);\n\\node[font=\\small,align=center,font=\\small] at (3.255,1.145) {$2$};\n\\node[font=\\small,align=center,font=\\small] at (5.675,4.4) {$\\sqrt4\\,Fc$};\n\\path[draw=black!30,fill=black!8] (4.95,2.97) rectangle (6.4,3.7);\n\\node[font=\\small,align=center,font=\\small] at (5.675,3.335) {$2$};\n\\path[draw=black!30,fill=black!8] (4.95,2.24) rectangle (6.4,2.97);\n\\node[font=\\small,align=center,font=\\small] at (5.675,2.605) {$2 - 2 i$};\n\\path[draw=black!30,fill=black!8] (4.95,1.51) rectangle (6.4,2.24);\n\\node[font=\\small,align=center,font=\\small] at (5.675,1.875) {$2$};\n\\path[dr","de[font=\\small,align=center,font=\\small] at (1.795,1.145) {$0$};\n\\path[draw=black!30,fill=black!8] (2.16,0.78) rectangle (2.89,1.51);\n\\node[font=\\small,align=center,font=\\small] at (2.525,1.145) {$1$};\n\\path[draw=black!30,fill=black!8] (2.89,0.78) rectangle (3.62,1.51);\n\\node[font=\\small,align=center,font=\\small] at (3.255,1.145) {$2$};\n\\node[font=\\small,align=center,font=\\small] at (5.675,4.4) {$\\sqrt4\\,Fc$};\n\\path[draw=black!30,fill=black!8] (4.95,2.97) rectangle (6.4,3.7);\n\\node[font=\\small,align=center,font=\\small] at (5.675,3.335) {$2$};\n\\path[draw=black!30,fill=black!8] (4.95,2.24) rectangle (6.4,2.97);\n\\node[font=\\small,align=center,font=\\small] at (5.675,2.605) {$2 - 2 i$};\n\\path[draw=black!30,fill=black!8] (4.95,1.51) rectangle (6.4,2.24);\n\\node[font=\\small,align=center,font=\\small] at (5.675,1.875) {$2$};\n\\path[draw=black!30,fill=black!8] (4.95,0.78) rectangle (6.4,1.51);\n\\node[font=\\sma","=\\small,align=center,font=\\small] at (2.525,1.145) {$1$};\n\\path[draw=black!30,fill=black!8] (2.89,0.78) rectangle (3.62,1.51);\n\\node[font=\\small,align=center,font=\\small] at (3.255,1.145) {$2$};\n\\node[font=\\small,align=center,font=\\small] at (5.675,4.4) {$\\sqrt4\\,Fc$};\n\\path[draw=black!30,fill=black!8] (4.95,2.97) rectangle (6.4,3.7);\n\\node[font=\\small,align=center,font=\\small] at (5.675,3.335) {$2$};\n\\path[draw=black!30,fill=black!8] (4.95,2.24) rectangle (6.4,2.97);\n\\node[font=\\small,align=center,font=\\small] at (5.675,2.605) {$2 - 2 i$};\n\\path[draw=black!30,fill=black!8] (4.95,1.51) rectangle (6.4,2.24);\n\\node[font=\\small,align=center,font=\\small] at (5.675,1.875) {$2$};\n\\path[draw=black!30,fill=black!8] (4.95,0.78) rectangle (6.4,1.51);\n\\node[font=\\small,align=center,font=\\small] at (5.675,1.145) {$2 + 2 i$};\n\\node[font=\\small,align=center] at (10.7,3.4) {$F_{jk}=\\frac12\\,\\mathrm i^{-","t=\\small,align=center,font=\\small] at (3.255,1.145) {$2$};\n\\node[font=\\small,align=center,font=\\small] at (5.675,4.4) {$\\sqrt4\\,Fc$};\n\\path[draw=black!30,fill=black!8] (4.95,2.97) rectangle (6.4,3.7);\n\\node[font=\\small,align=center,font=\\small] at (5.675,3.335) {$2$};\n\\path[draw=black!30,fill=black!8] (4.95,2.24) rectangle (6.4,2.97);\n\\node[font=\\small,align=center,font=\\small] at (5.675,2.605) {$2 - 2 i$};\n\\path[draw=black!30,fill=black!8] (4.95,1.51) rectangle (6.4,2.24);\n\\node[font=\\small,align=center,font=\\small] at (5.675,1.875) {$2$};\n\\path[draw=black!30,fill=black!8] (4.95,0.78) rectangle (6.4,1.51);\n\\node[font=\\small,align=center,font=\\small] at (5.675,1.145) {$2 + 2 i$};\n\\node[font=\\small,align=center] at (10.7,3.4) {$F_{jk}=\\frac12\\,\\mathrm i^{-jk}$};\n\\node[font=\\small,align=center] at (10.7,2.2) {$C=F^*\\diag(\\sqrt4 Fc)F$};\n\\node[font=\\small,align=center] at (10.7,1) {$\\sqrt4=2$ is es","aw=black!30,fill=black!8] (4.95,2.97) rectangle (6.4,3.7);\n\\node[font=\\small,align=center,font=\\small] at (5.675,3.335) {$2$};\n\\path[draw=black!30,fill=black!8] (4.95,2.24) rectangle (6.4,2.97);\n\\node[font=\\small,align=center,font=\\small] at (5.675,2.605) {$2 - 2 i$};\n\\path[draw=black!30,fill=black!8] (4.95,1.51) rectangle (6.4,2.24);\n\\node[font=\\small,align=center,font=\\small] at (5.675,1.875) {$2$};\n\\path[draw=black!30,fill=black!8] (4.95,0.78) rectangle (6.4,1.51);\n\\node[font=\\small,align=center,font=\\small] at (5.675,1.145) {$2 + 2 i$};\n\\node[font=\\small,align=center] at (10.7,3.4) {$F_{jk}=\\frac12\\,\\mathrm i^{-jk}$};\n\\node[font=\\small,align=center] at (10.7,2.2) {$C=F^*\\diag(\\sqrt4 Fc)F$};\n\\node[font=\\small,align=center] at (10.7,1) {$\\sqrt4=2$ is essential.};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.05) {A unitary transform changes the normalization of the convolutio","w=black!30,fill=black!8] (4.95,2.24) rectangle (6.4,2.97);\n\\node[font=\\small,align=center,font=\\small] at (5.675,2.605) {$2 - 2 i$};\n\\path[draw=black!30,fill=black!8] (4.95,1.51) rectangle (6.4,2.24);\n\\node[font=\\small,align=center,font=\\small] at (5.675,1.875) {$2$};\n\\path[draw=black!30,fill=black!8] (4.95,0.78) rectangle (6.4,1.51);\n\\node[font=\\small,align=center,font=\\small] at (5.675,1.145) {$2 + 2 i$};\n\\node[font=\\small,align=center] at (10.7,3.4) {$F_{jk}=\\frac12\\,\\mathrm i^{-jk}$};\n\\node[font=\\small,align=center] at (10.7,2.2) {$C=F^*\\diag(\\sqrt4 Fc)F$};\n\\node[font=\\small,align=center] at (10.7,1) {$\\sqrt4=2$ is essential.};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.05) {A unitary transform changes the normalization of the convolution eigenvalues.};\n\\end{tikzpicture}","\\node[font=\\small,align=center,font=\\small] at (5.675,2.605) {$2 - 2 i$};\n\\path[draw=black!30,fill=black!8] (4.95,1.51) rectangle (6.4,2.24);\n\\node[font=\\small,align=center,font=\\small] at (5.675,1.875) {$2$};\n\\path[draw=black!30,fill=black!8] (4.95,0.78) rectangle (6.4,1.51);\n\\node[font=\\small,align=center,font=\\small] at (5.675,1.145) {$2 + 2 i$};\n\\node[font=\\small,align=center] at (10.7,3.4) {$F_{jk}=\\frac12\\,\\mathrm i^{-jk}$};\n\\node[font=\\small,align=center] at (10.7,2.2) {$C=F^*\\diag(\\sqrt4 Fc)F$};\n\\node[font=\\small,align=center] at (10.7,1) {$\\sqrt4=2$ is essential.};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.05) {A unitary transform changes the normalization of the convolution eigenvalues.};\n\\end{tikzpicture}","w=black!30,fill=black!8] (4.95,1.51) rectangle (6.4,2.24);\n\\node[font=\\small,align=center,font=\\small] at (5.675,1.875) {$2$};\n\\path[draw=black!30,fill=black!8] (4.95,0.78) rectangle (6.4,1.51);\n\\node[font=\\small,align=center,font=\\small] at (5.675,1.145) {$2 + 2 i$};\n\\node[font=\\small,align=center] at (10.7,3.4) {$F_{jk}=\\frac12\\,\\mathrm i^{-jk}$};\n\\node[font=\\small,align=center] at (10.7,2.2) {$C=F^*\\diag(\\sqrt4 Fc)F$};\n\\node[font=\\small,align=center] at (10.7,1) {$\\sqrt4=2$ is essential.};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.05) {A unitary transform changes the normalization of the convolution eigenvalues.};\n\\end{tikzpicture}","mall,align=center,font=\\small] at (5.675,1.875) {$2$};\n\\path[draw=black!30,fill=black!8] (4.95,0.78) rectangle (6.4,1.51);\n\\node[font=\\small,align=center,font=\\small] at (5.675,1.145) {$2 + 2 i$};\n\\node[font=\\small,align=center] at (10.7,3.4) {$F_{jk}=\\frac12\\,\\mathrm i^{-jk}$};\n\\node[font=\\small,align=center] at (10.7,2.2) {$C=F^*\\diag(\\sqrt4 Fc)F$};\n\\node[font=\\small,align=center] at (10.7,1) {$\\sqrt4=2$ is essential.};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.05) {A unitary transform changes the normalization of the convolution eigenvalues.};\n\\end{tikzpicture}","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.425,4.5) {$L$: rank $2$};\n\\path[draw=black!30,fill=black!8] (0.45,3.15) rectangle (1.1,3.8);\n\\node[font=\\small,align=center,font=\\small] at (0.775,3.475) {$\\frac{3}{5}$};\n\\path[draw=black!30,fill=white] (1.1,3.15) rectangle (1.75,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.425,3.475) {$0$};\n\\path[draw=black!30,fill=black!8] (1.75,3.15) rectangle (2.4,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.075,3.475) {$\\frac{3}{5}$};\n\\path[draw=black!30,fill=black!8] (0.45,2.5) rectangle (1.1,3.1","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.425,4.5) {$L$: rank $2$};\n\\path[draw=black!30,fill=black!8] (0.45,3.15) rectangle (1.1,3.8);\n\\node[font=\\small,align=center,font=\\small] at (0.775,3.475) {$\\frac{3}{5}$};\n\\path[draw=black!30,fill=white] (1.1,3.15) rectangle (1.75,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.425,3.475) {$0$};\n\\path[draw=black!30,fill=black!8] (1.75,3.15) rectangle (2.4,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.075,3.475) {$\\frac{3}{5}$};\n\\path[draw=black!30,fill=black!8] (0.45,2.5) rectangle (1.1,3.15);\n\\node[","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.425,4.5) {$L$: rank $2$};\n\\path[draw=black!30,fill=black!8] (0.45,3.15) rectangle (1.1,3.8);\n\\node[font=\\small,align=center,font=\\small] at (0.775,3.475) {$\\frac{3}{5}$};\n\\path[draw=black!30,fill=white] (1.1,3.15) rectangle (1.75,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.425,3.475) {$0$};\n\\path[draw=black!30,fill=black!8] (1.75,3.15) rectangle (2.4,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.075,3.475) {$\\frac{3}{5}$};\n\\path[draw=black!30,fill=black!8] (0.45,2.5) rectangle (1.1,3.15);\n\\node[font=\\small,align=center,font=\\small] at (0.775,2.825) {$\\frac{4}{5}$};\n\\path[draw=black!30,fill=white] (1.1,2.5) rectangle (1.75,3.15);\n\\node[fo","h[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.425,4.5) {$L$: rank $2$};\n\\path[draw=black!30,fill=black!8] (0.45,3.15) rectangle (1.1,3.8);\n\\node[font=\\small,align=center,font=\\small] at (0.775,3.475) {$\\frac{3}{5}$};\n\\path[draw=black!30,fill=white] (1.1,3.15) rectangle (1.75,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.425,3.475) {$0$};\n\\path[draw=black!30,fill=black!8] (1.75,3.15) rectangle (2.4,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.075,3.475) {$\\frac{3}{5}$};\n\\path[draw=black!30,fill=black!8] (0.45,2.5) rectangle (1.1,3.15);\n\\node[font=\\small,align=center,font=\\small] at (0.775,2.825) {$\\frac{4}{5}$};\n\\path[draw=black!30,fill=white] (1.1,2.5) rectangle (1.75,3.15);\n\\node[font=\\small,align=center,font=\\small] at (1.425,2.825) {$0$};\n\\path[draw=black!30,fill=black!8] (1.75,2.5) rectangle (2.4,3.15);\n\\node[","aw=black!30,fill=black!8] (0.45,3.15) rectangle (1.1,3.8);\n\\node[font=\\small,align=center,font=\\small] at (0.775,3.475) {$\\frac{3}{5}$};\n\\path[draw=black!30,fill=white] (1.1,3.15) rectangle (1.75,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.425,3.475) {$0$};\n\\path[draw=black!30,fill=black!8] (1.75,3.15) rectangle (2.4,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.075,3.475) {$\\frac{3}{5}$};\n\\path[draw=black!30,fill=black!8] (0.45,2.5) rectangle (1.1,3.15);\n\\node[font=\\small,align=center,font=\\small] at (0.775,2.825) {$\\frac{4}{5}$};\n\\path[draw=black!30,fill=white] (1.1,2.5) rectangle (1.75,3.15);\n\\node[font=\\small,align=center,font=\\small] at (1.425,2.825) {$0$};\n\\path[draw=black!30,fill=black!8] (1.75,2.5) rectangle (2.4,3.15);\n\\node[font=\\small,align=center,font=\\small] at (2.075,2.825) {$\\frac{4}{5}$};\n\\path[draw=black!30,fill=white] (0.45,1.85) rectangle (1.1,2.5);\n\\node[fo","aw=black!30,fill=white] (1.1,3.15) rectangle (1.75,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.425,3.475) {$0$};\n\\path[draw=black!30,fill=black!8] (1.75,3.15) rectangle (2.4,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.075,3.475) {$\\frac{3}{5}$};\n\\path[draw=black!30,fill=black!8] (0.45,2.5) rectangle (1.1,3.15);\n\\node[font=\\small,align=center,font=\\small] at (0.775,2.825) {$\\frac{4}{5}$};\n\\path[draw=black!30,fill=white] (1.1,2.5) rectangle (1.75,3.15);\n\\node[font=\\small,align=center,font=\\small] at (1.425,2.825) {$0$};\n\\path[draw=black!30,fill=black!8] (1.75,2.5) rectangle (2.4,3.15);\n\\node[font=\\small,align=center,font=\\small] at (2.075,2.825) {$\\frac{4}{5}$};\n\\path[draw=black!30,fill=white] (0.45,1.85) rectangle (1.1,2.5);\n\\node[font=\\small,align=center,font=\\small] at (0.775,2.175) {$0$};\n\\path[draw=black!30,fill=black!8] (1.1,1.85) rectangle (1.75,2.5);\n\\node[font=\\small,","0,fill=black!8] (1.75,3.15) rectangle (2.4,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.075,3.475) {$\\frac{3}{5}$};\n\\path[draw=black!30,fill=black!8] (0.45,2.5) rectangle (1.1,3.15);\n\\node[font=\\small,align=center,font=\\small] at (0.775,2.825) {$\\frac{4}{5}$};\n\\path[draw=black!30,fill=white] (1.1,2.5) rectangle (1.75,3.15);\n\\node[font=\\small,align=center,font=\\small] at (1.425,2.825) {$0$};\n\\path[draw=black!30,fill=black!8] (1.75,2.5) rectangle (2.4,3.15);\n\\node[font=\\small,align=center,font=\\small] at (2.075,2.825) {$\\frac{4}{5}$};\n\\path[draw=black!30,fill=white] (0.45,1.85) rectangle (1.1,2.5);\n\\node[font=\\small,align=center,font=\\small] at (0.775,2.175) {$0$};\n\\path[draw=black!30,fill=black!8] (1.1,1.85) rectangle (1.75,2.5);\n\\node[font=\\small,align=center,font=\\small] at (1.425,2.175) {$1$};\n\\path[draw=black!30,fill=white] (1.75,1.85) rectangle (2.4,2.5);\n\\node[font=\\small,","aw=black!30,fill=black!8] (0.45,2.5) rectangle (1.1,3.15);\n\\node[font=\\small,align=center,font=\\small] at (0.775,2.825) {$\\frac{4}{5}$};\n\\path[draw=black!30,fill=white] (1.1,2.5) rectangle (1.75,3.15);\n\\node[font=\\small,align=center,font=\\small] at (1.425,2.825) {$0$};\n\\path[draw=black!30,fill=black!8] (1.75,2.5) rectangle (2.4,3.15);\n\\node[font=\\small,align=center,font=\\small] at (2.075,2.825) {$\\frac{4}{5}$};\n\\path[draw=black!30,fill=white] (0.45,1.85) rectangle (1.1,2.5);\n\\node[font=\\small,align=center,font=\\small] at (0.775,2.175) {$0$};\n\\path[draw=black!30,fill=black!8] (1.1,1.85) rectangle (1.75,2.5);\n\\node[font=\\small,align=center,font=\\small] at (1.425,2.175) {$1$};\n\\path[draw=black!30,fill=white] (1.75,1.85) rectangle (2.4,2.5);\n\\node[font=\\small,align=center,font=\\small] at (2.075,2.175) {$0$};\n\\path[draw=black!30,fill=white] (0.45,1.2) rectangle (1.1,1.85);\n\\node[font=\\small,align=center","draw=black!30,fill=white] (1.1,2.5) rectangle (1.75,3.15);\n\\node[font=\\small,align=center,font=\\small] at (1.425,2.825) {$0$};\n\\path[draw=black!30,fill=black!8] (1.75,2.5) rectangle (2.4,3.15);\n\\node[font=\\small,align=center,font=\\small] at (2.075,2.825) {$\\frac{4}{5}$};\n\\path[draw=black!30,fill=white] (0.45,1.85) rectangle (1.1,2.5);\n\\node[font=\\small,align=center,font=\\small] at (0.775,2.175) {$0$};\n\\path[draw=black!30,fill=black!8] (1.1,1.85) rectangle (1.75,2.5);\n\\node[font=\\small,align=center,font=\\small] at (1.425,2.175) {$1$};\n\\path[draw=black!30,fill=white] (1.75,1.85) rectangle (2.4,2.5);\n\\node[font=\\small,align=center,font=\\small] at (2.075,2.175) {$0$};\n\\path[draw=black!30,fill=white] (0.45,1.2) rectangle (1.1,1.85);\n\\node[font=\\small,align=center,font=\\small] at (0.775,1.525) {$0$};\n\\path[draw=black!30,fill=white] (1.1,1.2) rectangle (1.75,1.85);\n\\node[font=\\small,align=center","aw=black!30,fill=black!8] (1.75,2.5) rectangle (2.4,3.15);\n\\node[font=\\small,align=center,font=\\small] at (2.075,2.825) {$\\frac{4}{5}$};\n\\path[draw=black!30,fill=white] (0.45,1.85) rectangle (1.1,2.5);\n\\node[font=\\small,align=center,font=\\small] at (0.775,2.175) {$0$};\n\\path[draw=black!30,fill=black!8] (1.1,1.85) rectangle (1.75,2.5);\n\\node[font=\\small,align=center,font=\\small] at (1.425,2.175) {$1$};\n\\path[draw=black!30,fill=white] (1.75,1.85) rectangle (2.4,2.5);\n\\node[font=\\small,align=center,font=\\small] at (2.075,2.175) {$0$};\n\\path[draw=black!30,fill=white] (0.45,1.2) rectangle (1.1,1.85);\n\\node[font=\\small,align=center,font=\\small] at (0.775,1.525) {$0$};\n\\path[draw=black!30,fill=white] (1.1,1.2) rectangle (1.75,1.85);\n\\node[font=\\small,align=center,font=\\small] at (1.425,1.525) {$0$};\n\\path[draw=black!30,fill=white] (1.75,1.2) rectangle (2.4,1.85);\n\\node[font=\\small,align=center,f","$};\n\\path[draw=black!30,fill=white] (0.45,1.85) rectangle (1.1,2.5);\n\\node[font=\\small,align=center,font=\\small] at (0.775,2.175) {$0$};\n\\path[draw=black!30,fill=black!8] (1.1,1.85) rectangle (1.75,2.5);\n\\node[font=\\small,align=center,font=\\small] at (1.425,2.175) {$1$};\n\\path[draw=black!30,fill=white] (1.75,1.85) rectangle (2.4,2.5);\n\\node[font=\\small,align=center,font=\\small] at (2.075,2.175) {$0$};\n\\path[draw=black!30,fill=white] (0.45,1.2) rectangle (1.1,1.85);\n\\node[font=\\small,align=center,font=\\small] at (0.775,1.525) {$0$};\n\\path[draw=black!30,fill=white] (1.1,1.2) rectangle (1.75,1.85);\n\\node[font=\\small,align=center,font=\\small] at (1.425,1.525) {$0$};\n\\path[draw=black!30,fill=white] (1.75,1.2) rectangle (2.4,1.85);\n\\node[font=\\small,align=center,font=\\small] at (2.075,1.525) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (5.525,4.5) {$R$: rank $3$};\n\\path[draw=black!30,f","$};\n\\path[draw=black!30,fill=black!8] (1.1,1.85) rectangle (1.75,2.5);\n\\node[font=\\small,align=center,font=\\small] at (1.425,2.175) {$1$};\n\\path[draw=black!30,fill=white] (1.75,1.85) rectangle (2.4,2.5);\n\\node[font=\\small,align=center,font=\\small] at (2.075,2.175) {$0$};\n\\path[draw=black!30,fill=white] (0.45,1.2) rectangle (1.1,1.85);\n\\node[font=\\small,align=center,font=\\small] at (0.775,1.525) {$0$};\n\\path[draw=black!30,fill=white] (1.1,1.2) rectangle (1.75,1.85);\n\\node[font=\\small,align=center,font=\\small] at (1.425,1.525) {$0$};\n\\path[draw=black!30,fill=white] (1.75,1.2) rectangle (2.4,1.85);\n\\node[font=\\small,align=center,font=\\small] at (2.075,1.525) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (5.525,4.5) {$R$: rank $3$};\n\\path[draw=black!30,fill=black!8] (4.55,3.15) rectangle (5.2,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.875,3.475) {$3$};\n\\path[draw=black!30","$1$};\n\\path[draw=black!30,fill=white] (1.75,1.85) rectangle (2.4,2.5);\n\\node[font=\\small,align=center,font=\\small] at (2.075,2.175) {$0$};\n\\path[draw=black!30,fill=white] (0.45,1.2) rectangle (1.1,1.85);\n\\node[font=\\small,align=center,font=\\small] at (0.775,1.525) {$0$};\n\\path[draw=black!30,fill=white] (1.1,1.2) rectangle (1.75,1.85);\n\\node[font=\\small,align=center,font=\\small] at (1.425,1.525) {$0$};\n\\path[draw=black!30,fill=white] (1.75,1.2) rectangle (2.4,1.85);\n\\node[font=\\small,align=center,font=\\small] at (2.075,1.525) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (5.525,4.5) {$R$: rank $3$};\n\\path[draw=black!30,fill=black!8] (4.55,3.15) rectangle (5.2,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.875,3.475) {$3$};\n\\path[draw=black!30,fill=white] (5.2,3.15) rectangle (5.85,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.525,3.475) {$0$};\n\\path[draw=black!30","$0$};\n\\path[draw=black!30,fill=white] (0.45,1.2) rectangle (1.1,1.85);\n\\node[font=\\small,align=center,font=\\small] at (0.775,1.525) {$0$};\n\\path[draw=black!30,fill=white] (1.1,1.2) rectangle (1.75,1.85);\n\\node[font=\\small,align=center,font=\\small] at (1.425,1.525) {$0$};\n\\path[draw=black!30,fill=white] (1.75,1.2) rectangle (2.4,1.85);\n\\node[font=\\small,align=center,font=\\small] at (2.075,1.525) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (5.525,4.5) {$R$: rank $3$};\n\\path[draw=black!30,fill=black!8] (4.55,3.15) rectangle (5.2,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.875,3.475) {$3$};\n\\path[draw=black!30,fill=white] (5.2,3.15) rectangle (5.85,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.525,3.475) {$0$};\n\\path[draw=black!30,fill=black!8] (5.85,3.15) rectangle (6.5,3.8);\n\\node[font=\\small,align=center,font=\\small] at (6.175,3.475) {$2$};\n\\path[draw=black!","85);\n\\node[font=\\small,align=center,font=\\small] at (0.775,1.525) {$0$};\n\\path[draw=black!30,fill=white] (1.1,1.2) rectangle (1.75,1.85);\n\\node[font=\\small,align=center,font=\\small] at (1.425,1.525) {$0$};\n\\path[draw=black!30,fill=white] (1.75,1.2) rectangle (2.4,1.85);\n\\node[font=\\small,align=center,font=\\small] at (2.075,1.525) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (5.525,4.5) {$R$: rank $3$};\n\\path[draw=black!30,fill=black!8] (4.55,3.15) rectangle (5.2,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.875,3.475) {$3$};\n\\path[draw=black!30,fill=white] (5.2,3.15) rectangle (5.85,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.525,3.475) {$0$};\n\\path[draw=black!30,fill=black!8] (5.85,3.15) rectangle (6.5,3.8);\n\\node[font=\\small,align=center,font=\\small] at (6.175,3.475) {$2$};\n\\path[draw=black!30,fill=white] (4.55,2.5) rectangle (5.2,3.15);\n\\node[font=\\small,","[font=\\small,align=center,font=\\small] at (0.775,1.525) {$0$};\n\\path[draw=black!30,fill=white] (1.1,1.2) rectangle (1.75,1.85);\n\\node[font=\\small,align=center,font=\\small] at (1.425,1.525) {$0$};\n\\path[draw=black!30,fill=white] (1.75,1.2) rectangle (2.4,1.85);\n\\node[font=\\small,align=center,font=\\small] at (2.075,1.525) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (5.525,4.5) {$R$: rank $3$};\n\\path[draw=black!30,fill=black!8] (4.55,3.15) rectangle (5.2,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.875,3.475) {$3$};\n\\path[draw=black!30,fill=white] (5.2,3.15) rectangle (5.85,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.525,3.475) {$0$};\n\\path[draw=black!30,fill=black!8] (5.85,3.15) rectangle (6.5,3.8);\n\\node[font=\\small,align=center,font=\\small] at (6.175,3.475) {$2$};\n\\path[draw=black!30,fill=white] (4.55,2.5) rectangle (5.2,3.15);\n\\node[font=\\small,align=cent","ont=\\small,align=center,font=\\small] at (1.425,1.525) {$0$};\n\\path[draw=black!30,fill=white] (1.75,1.2) rectangle (2.4,1.85);\n\\node[font=\\small,align=center,font=\\small] at (2.075,1.525) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (5.525,4.5) {$R$: rank $3$};\n\\path[draw=black!30,fill=black!8] (4.55,3.15) rectangle (5.2,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.875,3.475) {$3$};\n\\path[draw=black!30,fill=white] (5.2,3.15) rectangle (5.85,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.525,3.475) {$0$};\n\\path[draw=black!30,fill=black!8] (5.85,3.15) rectangle (6.5,3.8);\n\\node[font=\\small,align=center,font=\\small] at (6.175,3.475) {$2$};\n\\path[draw=black!30,fill=white] (4.55,2.5) rectangle (5.2,3.15);\n\\node[font=\\small,align=center,font=\\small] at (4.875,2.825) {$0$};\n\\path[draw=black!30,fill=black!8] (5.2,2.5) rectangle (5.85,3.15);\n\\node[font=\\small,align=cent","ont=\\small,align=center,font=\\small] at (2.075,1.525) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (5.525,4.5) {$R$: rank $3$};\n\\path[draw=black!30,fill=black!8] (4.55,3.15) rectangle (5.2,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.875,3.475) {$3$};\n\\path[draw=black!30,fill=white] (5.2,3.15) rectangle (5.85,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.525,3.475) {$0$};\n\\path[draw=black!30,fill=black!8] (5.85,3.15) rectangle (6.5,3.8);\n\\node[font=\\small,align=center,font=\\small] at (6.175,3.475) {$2$};\n\\path[draw=black!30,fill=white] (4.55,2.5) rectangle (5.2,3.15);\n\\node[font=\\small,align=center,font=\\small] at (4.875,2.825) {$0$};\n\\path[draw=black!30,fill=black!8] (5.2,2.5) rectangle (5.85,3.15);\n\\node[font=\\small,align=center,font=\\small] at (5.525,2.825) {$2$};\n\\path[draw=black!30,fill=white] (5.85,2.5) rectangle (6.5,3.15);\n\\node[font=\\small,align=cent",";\n\\path[draw=black!30,fill=black!8] (4.55,3.15) rectangle (5.2,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.875,3.475) {$3$};\n\\path[draw=black!30,fill=white] (5.2,3.15) rectangle (5.85,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.525,3.475) {$0$};\n\\path[draw=black!30,fill=black!8] (5.85,3.15) rectangle (6.5,3.8);\n\\node[font=\\small,align=center,font=\\small] at (6.175,3.475) {$2$};\n\\path[draw=black!30,fill=white] (4.55,2.5) rectangle (5.2,3.15);\n\\node[font=\\small,align=center,font=\\small] at (4.875,2.825) {$0$};\n\\path[draw=black!30,fill=black!8] (5.2,2.5) rectangle (5.85,3.15);\n\\node[font=\\small,align=center,font=\\small] at (5.525,2.825) {$2$};\n\\path[draw=black!30,fill=white] (5.85,2.5) rectangle (6.5,3.15);\n\\node[font=\\small,align=center,font=\\small] at (6.175,2.825) {$0$};\n\\path[draw=black!30,fill=black!8] (4.55,1.85) rectangle (5.2,2.5);\n\\node[font=\\small,align=cent","$};\n\\path[draw=black!30,fill=white] (5.2,3.15) rectangle (5.85,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.525,3.475) {$0$};\n\\path[draw=black!30,fill=black!8] (5.85,3.15) rectangle (6.5,3.8);\n\\node[font=\\small,align=center,font=\\small] at (6.175,3.475) {$2$};\n\\path[draw=black!30,fill=white] (4.55,2.5) rectangle (5.2,3.15);\n\\node[font=\\small,align=center,font=\\small] at (4.875,2.825) {$0$};\n\\path[draw=black!30,fill=black!8] (5.2,2.5) rectangle (5.85,3.15);\n\\node[font=\\small,align=center,font=\\small] at (5.525,2.825) {$2$};\n\\path[draw=black!30,fill=white] (5.85,2.5) rectangle (6.5,3.15);\n\\node[font=\\small,align=center,font=\\small] at (6.175,2.825) {$0$};\n\\path[draw=black!30,fill=black!8] (4.55,1.85) rectangle (5.2,2.5);\n\\node[font=\\small,align=center,font=\\small] at (4.875,2.175) {$1$};\n\\path[draw=black!30,fill=white] (5.2,1.85) rectangle (5.85,2.5);\n\\node[font=\\small,align=cent",";\n\\path[draw=black!30,fill=black!8] (5.85,3.15) rectangle (6.5,3.8);\n\\node[font=\\small,align=center,font=\\small] at (6.175,3.475) {$2$};\n\\path[draw=black!30,fill=white] (4.55,2.5) rectangle (5.2,3.15);\n\\node[font=\\small,align=center,font=\\small] at (4.875,2.825) {$0$};\n\\path[draw=black!30,fill=black!8] (5.2,2.5) rectangle (5.85,3.15);\n\\node[font=\\small,align=center,font=\\small] at (5.525,2.825) {$2$};\n\\path[draw=black!30,fill=white] (5.85,2.5) rectangle (6.5,3.15);\n\\node[font=\\small,align=center,font=\\small] at (6.175,2.825) {$0$};\n\\path[draw=black!30,fill=black!8] (4.55,1.85) rectangle (5.2,2.5);\n\\node[font=\\small,align=center,font=\\small] at (4.875,2.175) {$1$};\n\\path[draw=black!30,fill=white] (5.2,1.85) rectangle (5.85,2.5);\n\\node[font=\\small,align=center,font=\\small] at (5.525,2.175) {$0$};\n\\path[draw=black!30,fill=black!8] (5.85,1.85) rectangle (6.5,2.5);\n\\node[font=\\small,align=cent","$};\n\\path[draw=black!30,fill=white] (4.55,2.5) rectangle (5.2,3.15);\n\\node[font=\\small,align=center,font=\\small] at (4.875,2.825) {$0$};\n\\path[draw=black!30,fill=black!8] (5.2,2.5) rectangle (5.85,3.15);\n\\node[font=\\small,align=center,font=\\small] at (5.525,2.825) {$2$};\n\\path[draw=black!30,fill=white] (5.85,2.5) rectangle (6.5,3.15);\n\\node[font=\\small,align=center,font=\\small] at (6.175,2.825) {$0$};\n\\path[draw=black!30,fill=black!8] (4.55,1.85) rectangle (5.2,2.5);\n\\node[font=\\small,align=center,font=\\small] at (4.875,2.175) {$1$};\n\\path[draw=black!30,fill=white] (5.2,1.85) rectangle (5.85,2.5);\n\\node[font=\\small,align=center,font=\\small] at (5.525,2.175) {$0$};\n\\path[draw=black!30,fill=black!8] (5.85,1.85) rectangle (6.5,2.5);\n\\node[font=\\small,align=center,font=\\small] at (6.175,2.175) {$-1$};\n\\node[font=\\small,align=center,font=\\small] at (10.055,4.5) {$G$: $2\\times3$};\n\\path[draw=bl",";\n\\path[draw=black!30,fill=black!8] (5.2,2.5) rectangle (5.85,3.15);\n\\node[font=\\small,align=center,font=\\small] at (5.525,2.825) {$2$};\n\\path[draw=black!30,fill=white] (5.85,2.5) rectangle (6.5,3.15);\n\\node[font=\\small,align=center,font=\\small] at (6.175,2.825) {$0$};\n\\path[draw=black!30,fill=black!8] (4.55,1.85) rectangle (5.2,2.5);\n\\node[font=\\small,align=center,font=\\small] at (4.875,2.175) {$1$};\n\\path[draw=black!30,fill=white] (5.2,1.85) rectangle (5.85,2.5);\n\\node[font=\\small,align=center,font=\\small] at (5.525,2.175) {$0$};\n\\path[draw=black!30,fill=black!8] (5.85,1.85) rectangle (6.5,2.5);\n\\node[font=\\small,align=center,font=\\small] at (6.175,2.175) {$-1$};\n\\node[font=\\small,align=center,font=\\small] at (10.055,4.5) {$G$: $2\\times3$};\n\\path[draw=black!30,fill=black!8] (8.75,2.93) rectangle (9.62,3.8);\n\\node[font=\\small,align=center,font=\\small] at (9.185,3.365) {$5$};\n\\path[draw=b","$};\n\\path[draw=black!30,fill=white] (5.85,2.5) rectangle (6.5,3.15);\n\\node[font=\\small,align=center,font=\\small] at (6.175,2.825) {$0$};\n\\path[draw=black!30,fill=black!8] (4.55,1.85) rectangle (5.2,2.5);\n\\node[font=\\small,align=center,font=\\small] at (4.875,2.175) {$1$};\n\\path[draw=black!30,fill=white] (5.2,1.85) rectangle (5.85,2.5);\n\\node[font=\\small,align=center,font=\\small] at (5.525,2.175) {$0$};\n\\path[draw=black!30,fill=black!8] (5.85,1.85) rectangle (6.5,2.5);\n\\node[font=\\small,align=center,font=\\small] at (6.175,2.175) {$-1$};\n\\node[font=\\small,align=center,font=\\small] at (10.055,4.5) {$G$: $2\\times3$};\n\\path[draw=black!30,fill=black!8] (8.75,2.93) rectangle (9.62,3.8);\n\\node[font=\\small,align=center,font=\\small] at (9.185,3.365) {$5$};\n\\path[draw=black!30,fill=white] (9.62,2.93) rectangle (10.49,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.055,3.365) {$0$};\n\\path[dra",";\n\\path[draw=black!30,fill=black!8] (4.55,1.85) rectangle (5.2,2.5);\n\\node[font=\\small,align=center,font=\\small] at (4.875,2.175) {$1$};\n\\path[draw=black!30,fill=white] (5.2,1.85) rectangle (5.85,2.5);\n\\node[font=\\small,align=center,font=\\small] at (5.525,2.175) {$0$};\n\\path[draw=black!30,fill=black!8] (5.85,1.85) rectangle (6.5,2.5);\n\\node[font=\\small,align=center,font=\\small] at (6.175,2.175) {$-1$};\n\\node[font=\\small,align=center,font=\\small] at (10.055,4.5) {$G$: $2\\times3$};\n\\path[draw=black!30,fill=black!8] (8.75,2.93) rectangle (9.62,3.8);\n\\node[font=\\small,align=center,font=\\small] at (9.185,3.365) {$5$};\n\\path[draw=black!30,fill=white] (9.62,2.93) rectangle (10.49,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.055,3.365) {$0$};\n\\path[draw=black!30,fill=white] (10.49,2.93) rectangle (11.36,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.925,3.365) {$0$};\n\\path[dr","\\node[font=\\small,align=center,font=\\small] at (4.875,2.175) {$1$};\n\\path[draw=black!30,fill=white] (5.2,1.85) rectangle (5.85,2.5);\n\\node[font=\\small,align=center,font=\\small] at (5.525,2.175) {$0$};\n\\path[draw=black!30,fill=black!8] (5.85,1.85) rectangle (6.5,2.5);\n\\node[font=\\small,align=center,font=\\small] at (6.175,2.175) {$-1$};\n\\node[font=\\small,align=center,font=\\small] at (10.055,4.5) {$G$: $2\\times3$};\n\\path[draw=black!30,fill=black!8] (8.75,2.93) rectangle (9.62,3.8);\n\\node[font=\\small,align=center,font=\\small] at (9.185,3.365) {$5$};\n\\path[draw=black!30,fill=white] (9.62,2.93) rectangle (10.49,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.055,3.365) {$0$};\n\\path[draw=black!30,fill=white] (10.49,2.93) rectangle (11.36,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.925,3.365) {$0$};\n\\path[draw=black!30,fill=white] (8.75,2.06) rectangle (9.62,2.93);\n\\node[fo","e[font=\\small,align=center,font=\\small] at (4.875,2.175) {$1$};\n\\path[draw=black!30,fill=white] (5.2,1.85) rectangle (5.85,2.5);\n\\node[font=\\small,align=center,font=\\small] at (5.525,2.175) {$0$};\n\\path[draw=black!30,fill=black!8] (5.85,1.85) rectangle (6.5,2.5);\n\\node[font=\\small,align=center,font=\\small] at (6.175,2.175) {$-1$};\n\\node[font=\\small,align=center,font=\\small] at (10.055,4.5) {$G$: $2\\times3$};\n\\path[draw=black!30,fill=black!8] (8.75,2.93) rectangle (9.62,3.8);\n\\node[font=\\small,align=center,font=\\small] at (9.185,3.365) {$5$};\n\\path[draw=black!30,fill=white] (9.62,2.93) rectangle (10.49,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.055,3.365) {$0$};\n\\path[draw=black!30,fill=white] (10.49,2.93) rectangle (11.36,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.925,3.365) {$0$};\n\\path[draw=black!30,fill=white] (8.75,2.06) rectangle (9.62,2.93);\n\\node[font=\\small,al","all,align=center,font=\\small] at (5.525,2.175) {$0$};\n\\path[draw=black!30,fill=black!8] (5.85,1.85) rectangle (6.5,2.5);\n\\node[font=\\small,align=center,font=\\small] at (6.175,2.175) {$-1$};\n\\node[font=\\small,align=center,font=\\small] at (10.055,4.5) {$G$: $2\\times3$};\n\\path[draw=black!30,fill=black!8] (8.75,2.93) rectangle (9.62,3.8);\n\\node[font=\\small,align=center,font=\\small] at (9.185,3.365) {$5$};\n\\path[draw=black!30,fill=white] (9.62,2.93) rectangle (10.49,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.055,3.365) {$0$};\n\\path[draw=black!30,fill=white] (10.49,2.93) rectangle (11.36,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.925,3.365) {$0$};\n\\path[draw=black!30,fill=white] (8.75,2.06) rectangle (9.62,2.93);\n\\node[font=\\small,align=center,font=\\small] at (9.185,2.495) {$0$};\n\\path[draw=black!30,fill=black!8] (9.62,2.06) rectangle (10.49,2.93);\n\\node[font=\\small,","ll,align=center,font=\\small] at (6.175,2.175) {$-1$};\n\\node[font=\\small,align=center,font=\\small] at (10.055,4.5) {$G$: $2\\times3$};\n\\path[draw=black!30,fill=black!8] (8.75,2.93) rectangle (9.62,3.8);\n\\node[font=\\small,align=center,font=\\small] at (9.185,3.365) {$5$};\n\\path[draw=black!30,fill=white] (9.62,2.93) rectangle (10.49,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.055,3.365) {$0$};\n\\path[draw=black!30,fill=white] (10.49,2.93) rectangle (11.36,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.925,3.365) {$0$};\n\\path[draw=black!30,fill=white] (8.75,2.06) rectangle (9.62,2.93);\n\\node[font=\\small,align=center,font=\\small] at (9.185,2.495) {$0$};\n\\path[draw=black!30,fill=black!8] (9.62,2.06) rectangle (10.49,2.93);\n\\node[font=\\small,align=center,font=\\small] at (10.055,2.495) {$2$};\n\\path[draw=black!30,fill=white] (10.49,2.06) rectangle (11.36,2.93);\n\\node[font=\\smal","h[draw=black!30,fill=black!8] (8.75,2.93) rectangle (9.62,3.8);\n\\node[font=\\small,align=center,font=\\small] at (9.185,3.365) {$5$};\n\\path[draw=black!30,fill=white] (9.62,2.93) rectangle (10.49,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.055,3.365) {$0$};\n\\path[draw=black!30,fill=white] (10.49,2.93) rectangle (11.36,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.925,3.365) {$0$};\n\\path[draw=black!30,fill=white] (8.75,2.06) rectangle (9.62,2.93);\n\\node[font=\\small,align=center,font=\\small] at (9.185,2.495) {$0$};\n\\path[draw=black!30,fill=black!8] (9.62,2.06) rectangle (10.49,2.93);\n\\node[font=\\small,align=center,font=\\small] at (10.055,2.495) {$2$};\n\\path[draw=black!30,fill=white] (10.49,2.06) rectangle (11.36,2.93);\n\\node[font=\\small,align=center,font=\\small] at (10.925,2.495) {$0$};\n\\node[font=\\small,align=center] at (10.1,0.85) {$LR^T=Q_LG$};\n\\node[font=\\small,alig","th[draw=black!30,fill=white] (9.62,2.93) rectangle (10.49,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.055,3.365) {$0$};\n\\path[draw=black!30,fill=white] (10.49,2.93) rectangle (11.36,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.925,3.365) {$0$};\n\\path[draw=black!30,fill=white] (8.75,2.06) rectangle (9.62,2.93);\n\\node[font=\\small,align=center,font=\\small] at (9.185,2.495) {$0$};\n\\path[draw=black!30,fill=black!8] (9.62,2.06) rectangle (10.49,2.93);\n\\node[font=\\small,align=center,font=\\small] at (10.055,2.495) {$2$};\n\\path[draw=black!30,fill=white] (10.49,2.06) rectangle (11.36,2.93);\n\\node[font=\\small,align=center,font=\\small] at (10.925,2.495) {$0$};\n\\node[font=\\small,align=center] at (10.1,0.85) {$LR^T=Q_LG$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (6.8,0.15) {The stored width is three; the left range has only two independent directions.};\n\\end{tikzpic","draw=black!30,fill=white] (10.49,2.93) rectangle (11.36,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.925,3.365) {$0$};\n\\path[draw=black!30,fill=white] (8.75,2.06) rectangle (9.62,2.93);\n\\node[font=\\small,align=center,font=\\small] at (9.185,2.495) {$0$};\n\\path[draw=black!30,fill=black!8] (9.62,2.06) rectangle (10.49,2.93);\n\\node[font=\\small,align=center,font=\\small] at (10.055,2.495) {$2$};\n\\path[draw=black!30,fill=white] (10.49,2.06) rectangle (11.36,2.93);\n\\node[font=\\small,align=center,font=\\small] at (10.925,2.495) {$0$};\n\\node[font=\\small,align=center] at (10.1,0.85) {$LR^T=Q_LG$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (6.8,0.15) {The stored width is three; the left range has only two independent directions.};\n\\end{tikzpicture}","raw=black!30,fill=white] (8.75,2.06) rectangle (9.62,2.93);\n\\node[font=\\small,align=center,font=\\small] at (9.185,2.495) {$0$};\n\\path[draw=black!30,fill=black!8] (9.62,2.06) rectangle (10.49,2.93);\n\\node[font=\\small,align=center,font=\\small] at (10.055,2.495) {$2$};\n\\path[draw=black!30,fill=white] (10.49,2.06) rectangle (11.36,2.93);\n\\node[font=\\small,align=center,font=\\small] at (10.925,2.495) {$0$};\n\\node[font=\\small,align=center] at (10.1,0.85) {$LR^T=Q_LG$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (6.8,0.15) {The stored width is three; the left range has only two independent directions.};\n\\end{tikzpicture}",".93);\n\\node[font=\\small,align=center,font=\\small] at (9.185,2.495) {$0$};\n\\path[draw=black!30,fill=black!8] (9.62,2.06) rectangle (10.49,2.93);\n\\node[font=\\small,align=center,font=\\small] at (10.055,2.495) {$2$};\n\\path[draw=black!30,fill=white] (10.49,2.06) rectangle (11.36,2.93);\n\\node[font=\\small,align=center,font=\\small] at (10.925,2.495) {$0$};\n\\node[font=\\small,align=center] at (10.1,0.85) {$LR^T=Q_LG$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (6.8,0.15) {The stored width is three; the left range has only two independent directions.};\n\\end{tikzpicture}","\\begin{tikzpicture}[font=\\sffamily\\small]\n\\node[anchor=west] at (0,3.4) {$A^*A=\\operatorname{diag}(25,4)$};\n\\draw[->] (0,0)--(8,0) node[right] {retained singular directions};\n\\draw[->] (0,0)--(0,2.5) node[above] {squared error};\n\\draw[thick,black!65] (1,2.03)--(3.5,.28)--(6,0);\n\\foreach \\x/\\y/\\n/\\v in {1/2.03/0/29,3.5/.28/1/4,6/0/2/0}{\n\\fill (\\x,\\y) circle (2pt);\\node[below] at (\\x,0) {$\\n$};\\node[above] at (\\x,\\y) {$\\v$};}\n\\node[anchor=west] at (0,-.9) {$Q_L^*Q_L=I\\qquad Q_L^TQ_L=\\operatorname{diag}(-7/25,1)$};\n\\end{tikzpicture}","ont=\\sffamily\\small]\n\\node[anchor=west] at (0,3.4) {$A^*A=\\operatorname{diag}(25,4)$};\n\\draw[->] (0,0)--(8,0) node[right] {retained singular directions};\n\\draw[->] (0,0)--(0,2.5) node[above] {squared error};\n\\draw[thick,black!65] (1,2.03)--(3.5,.28)--(6,0);\n\\foreach \\x/\\y/\\n/\\v in {1/2.03/0/29,3.5/.28/1/4,6/0/2/0}{\n\\fill (\\x,\\y) circle (2pt);\\node[below] at (\\x,0) {$\\n$};\\node[above] at (\\x,\\y) {$\\v$};}\n\\node[anchor=west] at (0,-.9) {$Q_L^*Q_L=I\\qquad Q_L^TQ_L=\\operatorname{diag}(-7/25,1)$};\n\\end{tikzpicture}","at (0,3.4) {$A^*A=\\operatorname{diag}(25,4)$};\n\\draw[->] (0,0)--(8,0) node[right] {retained singular directions};\n\\draw[->] (0,0)--(0,2.5) node[above] {squared error};\n\\draw[thick,black!65] (1,2.03)--(3.5,.28)--(6,0);\n\\foreach \\x/\\y/\\n/\\v in {1/2.03/0/29,3.5/.28/1/4,6/0/2/0}{\n\\fill (\\x,\\y) circle (2pt);\\node[below] at (\\x,0) {$\\n$};\\node[above] at (\\x,\\y) {$\\v$};}\n\\node[anchor=west] at (0,-.9) {$Q_L^*Q_L=I\\qquad Q_L^TQ_L=\\operatorname{diag}(-7/25,1)$};\n\\end{tikzpicture}","\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,anchor=west,font=\\scriptsize] at (0.5,3.5) {edges 0};\n\\path[draw=black!25,fill=black!8] (1.7,3.2) rectangle (2.7,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (2.2,3.5) {0};\n\\path[draw=black!25,fill=white] (2.9,3.2) rectangle (3.9,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.4,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (4.1,3.2) rectangle (5.1,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.6,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (5.3,3.2) rectangle (6.3,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.8,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (6.5,3.2) rectangle (7.5,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (7.7,3.2) rectangle (8.7,3.8);\n\\node[font=\\small,align=","0};\n\\path[draw=black!25,fill=black!8] (1.7,3.2) rectangle (2.7,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (2.2,3.5) {0};\n\\path[draw=black!25,fill=white] (2.9,3.2) rectangle (3.9,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.4,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (4.1,3.2) rectangle (5.1,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.6,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (5.3,3.2) rectangle (6.3,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.8,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (6.5,3.2) rectangle (7.5,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (7.7,3.2) rectangle (8.7,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.2,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (8.9,3.2) rectangle (9.9,3.8);\n\\node[font=\\small,align=","path[draw=black!25,fill=white] (2.9,3.2) rectangle (3.9,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.4,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (4.1,3.2) rectangle (5.1,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.6,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (5.3,3.2) rectangle (6.3,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.8,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (6.5,3.2) rectangle (7.5,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (7.7,3.2) rectangle (8.7,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.2,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (8.9,3.2) rectangle (9.9,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (9.4,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (10.1,3.2) rectangle (11.1,3.8);\n\\node[font=\\small,alig","\\path[draw=black!25,fill=white] (4.1,3.2) rectangle (5.1,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.6,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (5.3,3.2) rectangle (6.3,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.8,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (6.5,3.2) rectangle (7.5,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (7.7,3.2) rectangle (8.7,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.2,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (8.9,3.2) rectangle (9.9,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (9.4,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (10.1,3.2) rectangle (11.1,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.6,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (11.3,3.2) rectangle (12.3,3.8);\n\\node[font=\\small","\\path[draw=black!25,fill=white] (5.3,3.2) rectangle (6.3,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.8,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (6.5,3.2) rectangle (7.5,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (7.7,3.2) rectangle (8.7,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.2,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (8.9,3.2) rectangle (9.9,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (9.4,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (10.1,3.2) rectangle (11.1,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.6,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (11.3,3.2) rectangle (12.3,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.8,3.5) {$\\infty$};\n\\node[font=\\small,align=center,anchor=west,font=\\scriptsize] at (0.5,2.66) {edg","\\path[draw=black!25,fill=white] (6.5,3.2) rectangle (7.5,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (7.7,3.2) rectangle (8.7,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.2,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (8.9,3.2) rectangle (9.9,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (9.4,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (10.1,3.2) rectangle (11.1,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.6,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (11.3,3.2) rectangle (12.3,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.8,3.5) {$\\infty$};\n\\node[font=\\small,align=center,anchor=west,font=\\scriptsize] at (0.5,2.66) {edges 1};\n\\path[draw=black!25,fill=black!8] (1.7,2.36) rectangle (2.7,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (2.2,2.66)","h[draw=black!25,fill=white] (7.7,3.2) rectangle (8.7,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.2,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (8.9,3.2) rectangle (9.9,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (9.4,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (10.1,3.2) rectangle (11.1,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.6,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (11.3,3.2) rectangle (12.3,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.8,3.5) {$\\infty$};\n\\node[font=\\small,align=center,anchor=west,font=\\scriptsize] at (0.5,2.66) {edges 1};\n\\path[draw=black!25,fill=black!8] (1.7,2.36) rectangle (2.7,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (2.2,2.66) {7};\n\\path[draw=black!25,fill=white] (2.9,2.36) rectangle (3.9,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.4,2.66) {$\\inft","raw=black!25,fill=white] (8.9,3.2) rectangle (9.9,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (9.4,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (10.1,3.2) rectangle (11.1,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.6,3.5) {$\\infty$};\n\\path[draw=black!25,fill=white] (11.3,3.2) rectangle (12.3,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.8,3.5) {$\\infty$};\n\\node[font=\\small,align=center,anchor=west,font=\\scriptsize] at (0.5,2.66) {edges 1};\n\\path[draw=black!25,fill=black!8] (1.7,2.36) rectangle (2.7,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (2.2,2.66) {7};\n\\path[draw=black!25,fill=white] (2.9,2.36) rectangle (3.9,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.4,2.66) {$\\infty$};\n\\path[draw=black!25,fill=black!8] (4.1,2.36) rectangle (5.1,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.6,2.66) {3};","nter,font=\\scriptsize] at (11.8,3.5) {$\\infty$};\n\\node[font=\\small,align=center,anchor=west,font=\\scriptsize] at (0.5,2.66) {edges 1};\n\\path[draw=black!25,fill=black!8] (1.7,2.36) rectangle (2.7,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (2.2,2.66) {7};\n\\path[draw=black!25,fill=white] (2.9,2.36) rectangle (3.9,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.4,2.66) {$\\infty$};\n\\path[draw=black!25,fill=black!8] (4.1,2.36) rectangle (5.1,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.6,2.66) {3};\n\\path[draw=black!25,fill=white] (5.3,2.36) rectangle (6.3,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.8,2.66) {$\\infty$};\n\\path[draw=black!25,fill=black!8] (6.5,2.36) rectangle (7.5,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,2.66) {1};\n\\path[draw=black!25,fill=white] (7.7,2.36) rectangle (8.7,2.96);\n\\node[font=\\small,align","[draw=black!25,fill=white] (2.9,2.36) rectangle (3.9,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.4,2.66) {$\\infty$};\n\\path[draw=black!25,fill=black!8] (4.1,2.36) rectangle (5.1,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.6,2.66) {3};\n\\path[draw=black!25,fill=white] (5.3,2.36) rectangle (6.3,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.8,2.66) {$\\infty$};\n\\path[draw=black!25,fill=black!8] (6.5,2.36) rectangle (7.5,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,2.66) {1};\n\\path[draw=black!25,fill=white] (7.7,2.36) rectangle (8.7,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.2,2.66) {$\\infty$};\n\\path[draw=black!25,fill=white] (8.9,2.36) rectangle (9.9,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (9.4,2.66) {$\\infty$};\n\\path[draw=black!25,fill=white] (10.1,2.36) rectangle (11.1,2.96);\n\\node[font=\\smal","th[draw=black!25,fill=white] (5.3,2.36) rectangle (6.3,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.8,2.66) {$\\infty$};\n\\path[draw=black!25,fill=black!8] (6.5,2.36) rectangle (7.5,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,2.66) {1};\n\\path[draw=black!25,fill=white] (7.7,2.36) rectangle (8.7,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.2,2.66) {$\\infty$};\n\\path[draw=black!25,fill=white] (8.9,2.36) rectangle (9.9,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (9.4,2.66) {$\\infty$};\n\\path[draw=black!25,fill=white] (10.1,2.36) rectangle (11.1,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.6,2.66) {$\\infty$};\n\\path[draw=black!25,fill=white] (11.3,2.36) rectangle (12.3,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.8,2.66) {$\\infty$};\n\\node[font=\\small,align=center,anchor=west,font=\\scriptsize] at (0.5,1","th[draw=black!25,fill=black!8] (6.5,2.36) rectangle (7.5,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,2.66) {1};\n\\path[draw=black!25,fill=white] (7.7,2.36) rectangle (8.7,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.2,2.66) {$\\infty$};\n\\path[draw=black!25,fill=white] (8.9,2.36) rectangle (9.9,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (9.4,2.66) {$\\infty$};\n\\path[draw=black!25,fill=white] (10.1,2.36) rectangle (11.1,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.6,2.66) {$\\infty$};\n\\path[draw=black!25,fill=white] (11.3,2.36) rectangle (12.3,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.8,2.66) {$\\infty$};\n\\node[font=\\small,align=center,anchor=west,font=\\scriptsize] at (0.5,1.82) {edges 2};\n\\path[draw=black!25,fill=black!8] (1.7,1.52) rectangle (2.7,2.12);\n\\node[font=\\small,align=center,font=\\scriptsize] at (2.2,","ack!25,fill=white] (7.7,2.36) rectangle (8.7,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.2,2.66) {$\\infty$};\n\\path[draw=black!25,fill=white] (8.9,2.36) rectangle (9.9,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (9.4,2.66) {$\\infty$};\n\\path[draw=black!25,fill=white] (10.1,2.36) rectangle (11.1,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.6,2.66) {$\\infty$};\n\\path[draw=black!25,fill=white] (11.3,2.36) rectangle (12.3,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.8,2.66) {$\\infty$};\n\\node[font=\\small,align=center,anchor=west,font=\\scriptsize] at (0.5,1.82) {edges 2};\n\\path[draw=black!25,fill=black!8] (1.7,1.52) rectangle (2.7,2.12);\n\\node[font=\\small,align=center,font=\\scriptsize] at (2.2,1.82) {13};\n\\path[draw=black!25,fill=black!8] (2.9,1.52) rectangle (3.9,2.12);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.4,1.82) {","!25,fill=white] (8.9,2.36) rectangle (9.9,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (9.4,2.66) {$\\infty$};\n\\path[draw=black!25,fill=white] (10.1,2.36) rectangle (11.1,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.6,2.66) {$\\infty$};\n\\path[draw=black!25,fill=white] (11.3,2.36) rectangle (12.3,2.96);\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.8,2.66) {$\\infty$};\n\\node[font=\\small,align=center,anchor=west,font=\\scriptsize] at (0.5,1.82) {edges 2};\n\\path[draw=black!25,fill=black!8] (1.7,1.52) rectangle (2.7,2.12);\n\\node[font=\\small,align=center,font=\\scriptsize] at (2.2,1.82) {13};\n\\path[draw=black!25,fill=black!8] (2.9,1.52) rectangle (3.9,2.12);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.4,1.82) {11};\n\\path[draw=black!25,fill=black!8] (4.1,1.52) rectangle (5.1,2.12);\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.6,1.82) {9};\n\\pa",";\n\\path[draw=black!25,fill=black!8] (6.5,1.52) rectangle (7.5,2.12);\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,1.82) {7};\n\\path[draw=black!25,fill=black!8] (7.7,1.52) rectangle (8.7,2.12);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.2,1.82) {4};\n\\path[draw=black!25,fill=white] (8.9,1.52) rectangle (9.9,2.12);\n\\node[font=\\small,align=center,font=\\scriptsize] at (9.4,1.82) {$\\infty$};\n\\path[draw=black!25,fill=black!8] (10.1,1.52) rectangle (11.1,2.12);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.6,1.82) {2};\n\\path[draw=black!25,fill=white] (11.3,1.52) rectangle (12.3,2.12);\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.8,1.82) {$\\infty$};\n\\node[font=\\small,align=center,anchor=west,font=\\scriptsize] at (0.5,0.98) {edges 3};\n\\path[draw=black!25,fill=black!8] (1.7,0.68) rectangle (2.7,1.28);\n\\node[font=\\small,align=center,font=\\scriptsize] at (2.2,0.98)","black!25,fill=white] (8.9,1.52) rectangle (9.9,2.12);\n\\node[font=\\small,align=center,font=\\scriptsize] at (9.4,1.82) {$\\infty$};\n\\path[draw=black!25,fill=black!8] (10.1,1.52) rectangle (11.1,2.12);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.6,1.82) {2};\n\\path[draw=black!25,fill=white] (11.3,1.52) rectangle (12.3,2.12);\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.8,1.82) {$\\infty$};\n\\node[font=\\small,align=center,anchor=west,font=\\scriptsize] at (0.5,0.98) {edges 3};\n\\path[draw=black!25,fill=black!8] (1.7,0.68) rectangle (2.7,1.28);\n\\node[font=\\small,align=center,font=\\scriptsize] at (2.2,0.98) {22};\n\\path[draw=black!25,fill=black!8] (2.9,0.68) rectangle (3.9,1.28);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.4,0.98) {20};\n\\path[draw=black!25,fill=black!8] (4.1,0.68) rectangle (5.1,1.28);\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.6,0.98) {18};\n\\p","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.075,4.35) {$V_C^Tx$};\n\\path[draw=black!30,fill=black!8] (0.7,2.9) rectangle (1.45,3.65);\n\\node[font=\\small,align=center,font=\\small] at (1.075,3.275) {$2$};\n\\path[draw=black!30,fill=black!8] (0.7,2.15) rectangle (1.45,2.9);\n\\node[font=\\small,align=center,font=\\small] at (1.075,2.525) {$13$};\n\\node[font=\\small,align=center,font=\\small] at (4.355,4.55) {$D_AD_Bx$};\n\\path[draw=black!30,fill=black!8] (4.1,3.34) rectangle (4.61,3.85);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.595) {$1$};\n\\path[draw=black","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.075,4.35) {$V_C^Tx$};\n\\path[draw=black!30,fill=black!8] (0.7,2.9) rectangle (1.45,3.65);\n\\node[font=\\small,align=center,font=\\small] at (1.075,3.275) {$2$};\n\\path[draw=black!30,fill=black!8] (0.7,2.15) rectangle (1.45,2.9);\n\\node[font=\\small,align=center,font=\\small] at (1.075,2.525) {$13$};\n\\node[font=\\small,align=center,font=\\small] at (4.355,4.55) {$D_AD_Bx$};\n\\path[draw=black!30,fill=black!8] (4.1,3.34) rectangle (4.61,3.85);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.595) {$1$};\n\\path[draw=black!30,fill=black!8] (4.1,2.83) rectangle (4.61,3.34);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.085) {$-5$};\n\\path[draw=bla","=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.075,4.35) {$V_C^Tx$};\n\\path[draw=black!30,fill=black!8] (0.7,2.9) rectangle (1.45,3.65);\n\\node[font=\\small,align=center,font=\\small] at (1.075,3.275) {$2$};\n\\path[draw=black!30,fill=black!8] (0.7,2.15) rectangle (1.45,2.9);\n\\node[font=\\small,align=center,font=\\small] at (1.075,2.525) {$13$};\n\\node[font=\\small,align=center,font=\\small] at (4.355,4.55) {$D_AD_Bx$};\n\\path[draw=black!30,fill=black!8] (4.1,3.34) rectangle (4.61,3.85);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.595) {$1$};\n\\path[draw=black!30,fill=black!8] (4.1,2.83) rectangle (4.61,3.34);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.085) {$-5$};\n\\path[draw=black!30,fill=black!8] (4.1,2.32) rectangle (4.61,2.83);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.575) {$18$};\n\\path[draw=bl","[font=\\small,align=center,font=\\small] at (1.075,4.35) {$V_C^Tx$};\n\\path[draw=black!30,fill=black!8] (0.7,2.9) rectangle (1.45,3.65);\n\\node[font=\\small,align=center,font=\\small] at (1.075,3.275) {$2$};\n\\path[draw=black!30,fill=black!8] (0.7,2.15) rectangle (1.45,2.9);\n\\node[font=\\small,align=center,font=\\small] at (1.075,2.525) {$13$};\n\\node[font=\\small,align=center,font=\\small] at (4.355,4.55) {$D_AD_Bx$};\n\\path[draw=black!30,fill=black!8] (4.1,3.34) rectangle (4.61,3.85);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.595) {$1$};\n\\path[draw=black!30,fill=black!8] (4.1,2.83) rectangle (4.61,3.34);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.085) {$-5$};\n\\path[draw=black!30,fill=black!8] (4.1,2.32) rectangle (4.61,2.83);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.575) {$18$};\n\\path[draw=black!30,fill=black!8] (4.1,1.81) rectangle (4.61,2.32);\n\\node[font=\\small,","nt=\\small,align=center,font=\\small] at (1.075,3.275) {$2$};\n\\path[draw=black!30,fill=black!8] (0.7,2.15) rectangle (1.45,2.9);\n\\node[font=\\small,align=center,font=\\small] at (1.075,2.525) {$13$};\n\\node[font=\\small,align=center,font=\\small] at (4.355,4.55) {$D_AD_Bx$};\n\\path[draw=black!30,fill=black!8] (4.1,3.34) rectangle (4.61,3.85);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.595) {$1$};\n\\path[draw=black!30,fill=black!8] (4.1,2.83) rectangle (4.61,3.34);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.085) {$-5$};\n\\path[draw=black!30,fill=black!8] (4.1,2.32) rectangle (4.61,2.83);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.575) {$18$};\n\\path[draw=black!30,fill=black!8] (4.1,1.81) rectangle (4.61,2.32);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.065) {$6$};\n\\path[draw=black!30,fill=black!8] (4.1,1.3) rectangle (4.61,1.81);\n\\node[font=\\small,a","t=\\small,align=center,font=\\small] at (1.075,2.525) {$13$};\n\\node[font=\\small,align=center,font=\\small] at (4.355,4.55) {$D_AD_Bx$};\n\\path[draw=black!30,fill=black!8] (4.1,3.34) rectangle (4.61,3.85);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.595) {$1$};\n\\path[draw=black!30,fill=black!8] (4.1,2.83) rectangle (4.61,3.34);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.085) {$-5$};\n\\path[draw=black!30,fill=black!8] (4.1,2.32) rectangle (4.61,2.83);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.575) {$18$};\n\\path[draw=black!30,fill=black!8] (4.1,1.81) rectangle (4.61,2.32);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.065) {$6$};\n\\path[draw=black!30,fill=black!8] (4.1,1.3) rectangle (4.61,1.81);\n\\node[font=\\small,align=center,font=\\small] at (4.355,1.555) {$-12$};\n\\path[draw=black!30,fill=black!8] (4.1,0.79) rectangle (4.61,1.3);\n\\node[font=\\small,a","h[draw=black!30,fill=black!8] (4.1,3.34) rectangle (4.61,3.85);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.595) {$1$};\n\\path[draw=black!30,fill=black!8] (4.1,2.83) rectangle (4.61,3.34);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.085) {$-5$};\n\\path[draw=black!30,fill=black!8] (4.1,2.32) rectangle (4.61,2.83);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.575) {$18$};\n\\path[draw=black!30,fill=black!8] (4.1,1.81) rectangle (4.61,2.32);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.065) {$6$};\n\\path[draw=black!30,fill=black!8] (4.1,1.3) rectangle (4.61,1.81);\n\\node[font=\\small,align=center,font=\\small] at (4.355,1.555) {$-12$};\n\\path[draw=black!30,fill=black!8] (4.1,0.79) rectangle (4.61,1.3);\n\\node[font=\\small,align=center,font=\\small] at (4.355,1.045) {$44$};\n\\node[font=\\small,align=center,font=\\small] at (7.555,4.55) {$U_C(V_C^Tx)$};\n\\path[draw","[draw=black!30,fill=black!8] (4.1,2.83) rectangle (4.61,3.34);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.085) {$-5$};\n\\path[draw=black!30,fill=black!8] (4.1,2.32) rectangle (4.61,2.83);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.575) {$18$};\n\\path[draw=black!30,fill=black!8] (4.1,1.81) rectangle (4.61,2.32);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.065) {$6$};\n\\path[draw=black!30,fill=black!8] (4.1,1.3) rectangle (4.61,1.81);\n\\node[font=\\small,align=center,font=\\small] at (4.355,1.555) {$-12$};\n\\path[draw=black!30,fill=black!8] (4.1,0.79) rectangle (4.61,1.3);\n\\node[font=\\small,align=center,font=\\small] at (4.355,1.045) {$44$};\n\\node[font=\\small,align=center,font=\\small] at (7.555,4.55) {$U_C(V_C^Tx)$};\n\\path[draw=black!30,fill=black!8] (7.3,3.34) rectangle (7.81,3.85);\n\\node[font=\\small,align=center,font=\\small] at (7.555,3.595) {$25$};\n\\path[dra","th[draw=black!30,fill=black!8] (4.1,2.32) rectangle (4.61,2.83);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.575) {$18$};\n\\path[draw=black!30,fill=black!8] (4.1,1.81) rectangle (4.61,2.32);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.065) {$6$};\n\\path[draw=black!30,fill=black!8] (4.1,1.3) rectangle (4.61,1.81);\n\\node[font=\\small,align=center,font=\\small] at (4.355,1.555) {$-12$};\n\\path[draw=black!30,fill=black!8] (4.1,0.79) rectangle (4.61,1.3);\n\\node[font=\\small,align=center,font=\\small] at (4.355,1.045) {$44$};\n\\node[font=\\small,align=center,font=\\small] at (7.555,4.55) {$U_C(V_C^Tx)$};\n\\path[draw=black!30,fill=black!8] (7.3,3.34) rectangle (7.81,3.85);\n\\node[font=\\small,align=center,font=\\small] at (7.555,3.595) {$25$};\n\\path[draw=black!30,fill=black!8] (7.3,2.83) rectangle (7.81,3.34);\n\\node[font=\\small,align=center,font=\\small] at (7.555,3.085) {$21$};\n\\path[dra","th[draw=black!30,fill=black!8] (4.1,1.81) rectangle (4.61,2.32);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.065) {$6$};\n\\path[draw=black!30,fill=black!8] (4.1,1.3) rectangle (4.61,1.81);\n\\node[font=\\small,align=center,font=\\small] at (4.355,1.555) {$-12$};\n\\path[draw=black!30,fill=black!8] (4.1,0.79) rectangle (4.61,1.3);\n\\node[font=\\small,align=center,font=\\small] at (4.355,1.045) {$44$};\n\\node[font=\\small,align=center,font=\\small] at (7.555,4.55) {$U_C(V_C^Tx)$};\n\\path[draw=black!30,fill=black!8] (7.3,3.34) rectangle (7.81,3.85);\n\\node[font=\\small,align=center,font=\\small] at (7.555,3.595) {$25$};\n\\path[draw=black!30,fill=black!8] (7.3,2.83) rectangle (7.81,3.34);\n\\node[font=\\small,align=center,font=\\small] at (7.555,3.085) {$21$};\n\\path[draw=black!30,fill=black!8] (7.3,2.32) rectangle (7.81,2.83);\n\\node[font=\\small,align=center,font=\\small] at (7.555,2.575) {$25$};\n\\path[dr","de[font=\\small,align=center,font=\\small] at (4.355,2.065) {$6$};\n\\path[draw=black!30,fill=black!8] (4.1,1.3) rectangle (4.61,1.81);\n\\node[font=\\small,align=center,font=\\small] at (4.355,1.555) {$-12$};\n\\path[draw=black!30,fill=black!8] (4.1,0.79) rectangle (4.61,1.3);\n\\node[font=\\small,align=center,font=\\small] at (4.355,1.045) {$44$};\n\\node[font=\\small,align=center,font=\\small] at (7.555,4.55) {$U_C(V_C^Tx)$};\n\\path[draw=black!30,fill=black!8] (7.3,3.34) rectangle (7.81,3.85);\n\\node[font=\\small,align=center,font=\\small] at (7.555,3.595) {$25$};\n\\path[draw=black!30,fill=black!8] (7.3,2.83) rectangle (7.81,3.34);\n\\node[font=\\small,align=center,font=\\small] at (7.555,3.085) {$21$};\n\\path[draw=black!30,fill=black!8] (7.3,2.32) rectangle (7.81,2.83);\n\\node[font=\\small,align=center,font=\\small] at (7.555,2.575) {$25$};\n\\path[draw=black!30,fill=black!8] (7.3,1.81) rectangle (7.81,2.32);\n\\node[font=\\small","all,align=center,font=\\small] at (4.355,1.555) {$-12$};\n\\path[draw=black!30,fill=black!8] (4.1,0.79) rectangle (4.61,1.3);\n\\node[font=\\small,align=center,font=\\small] at (4.355,1.045) {$44$};\n\\node[font=\\small,align=center,font=\\small] at (7.555,4.55) {$U_C(V_C^Tx)$};\n\\path[draw=black!30,fill=black!8] (7.3,3.34) rectangle (7.81,3.85);\n\\node[font=\\small,align=center,font=\\small] at (7.555,3.595) {$25$};\n\\path[draw=black!30,fill=black!8] (7.3,2.83) rectangle (7.81,3.34);\n\\node[font=\\small,align=center,font=\\small] at (7.555,3.085) {$21$};\n\\path[draw=black!30,fill=black!8] (7.3,2.32) rectangle (7.81,2.83);\n\\node[font=\\small,align=center,font=\\small] at (7.555,2.575) {$25$};\n\\path[draw=black!30,fill=black!8] (7.3,1.81) rectangle (7.81,2.32);\n\\node[font=\\small,align=center,font=\\small] at (7.555,2.065) {$21$};\n\\path[draw=black!30,fill=black!8] (7.3,1.3) rectangle (7.81,1.81);\n\\node[font=\\small,","all,align=center,font=\\small] at (4.355,1.045) {$44$};\n\\node[font=\\small,align=center,font=\\small] at (7.555,4.55) {$U_C(V_C^Tx)$};\n\\path[draw=black!30,fill=black!8] (7.3,3.34) rectangle (7.81,3.85);\n\\node[font=\\small,align=center,font=\\small] at (7.555,3.595) {$25$};\n\\path[draw=black!30,fill=black!8] (7.3,2.83) rectangle (7.81,3.34);\n\\node[font=\\small,align=center,font=\\small] at (7.555,3.085) {$21$};\n\\path[draw=black!30,fill=black!8] (7.3,2.32) rectangle (7.81,2.83);\n\\node[font=\\small,align=center,font=\\small] at (7.555,2.575) {$25$};\n\\path[draw=black!30,fill=black!8] (7.3,1.81) rectangle (7.81,2.32);\n\\node[font=\\small,align=center,font=\\small] at (7.555,2.065) {$21$};\n\\path[draw=black!30,fill=black!8] (7.3,1.3) rectangle (7.81,1.81);\n\\node[font=\\small,align=center,font=\\small] at (7.555,1.555) {$25$};\n\\path[draw=black!30,fill=black!8] (7.3,0.79) rectangle (7.81,1.3);\n\\node[font=\\small,a","[draw=black!30,fill=black!8] (7.3,3.34) rectangle (7.81,3.85);\n\\node[font=\\small,align=center,font=\\small] at (7.555,3.595) {$25$};\n\\path[draw=black!30,fill=black!8] (7.3,2.83) rectangle (7.81,3.34);\n\\node[font=\\small,align=center,font=\\small] at (7.555,3.085) {$21$};\n\\path[draw=black!30,fill=black!8] (7.3,2.32) rectangle (7.81,2.83);\n\\node[font=\\small,align=center,font=\\small] at (7.555,2.575) {$25$};\n\\path[draw=black!30,fill=black!8] (7.3,1.81) rectangle (7.81,2.32);\n\\node[font=\\small,align=center,font=\\small] at (7.555,2.065) {$21$};\n\\path[draw=black!30,fill=black!8] (7.3,1.3) rectangle (7.81,1.81);\n\\node[font=\\small,align=center,font=\\small] at (7.555,1.555) {$25$};\n\\path[draw=black!30,fill=black!8] (7.3,0.79) rectangle (7.81,1.3);\n\\node[font=\\small,align=center,font=\\small] at (7.555,1.045) {$21$};\n\\node[font=\\small,align=center,font=\\small] at (10.805,4.55) {$(AB)x$};\n\\path[draw=blac","[draw=black!30,fill=black!8] (7.3,2.83) rectangle (7.81,3.34);\n\\node[font=\\small,align=center,font=\\small] at (7.555,3.085) {$21$};\n\\path[draw=black!30,fill=black!8] (7.3,2.32) rectangle (7.81,2.83);\n\\node[font=\\small,align=center,font=\\small] at (7.555,2.575) {$25$};\n\\path[draw=black!30,fill=black!8] (7.3,1.81) rectangle (7.81,2.32);\n\\node[font=\\small,align=center,font=\\small] at (7.555,2.065) {$21$};\n\\path[draw=black!30,fill=black!8] (7.3,1.3) rectangle (7.81,1.81);\n\\node[font=\\small,align=center,font=\\small] at (7.555,1.555) {$25$};\n\\path[draw=black!30,fill=black!8] (7.3,0.79) rectangle (7.81,1.3);\n\\node[font=\\small,align=center,font=\\small] at (7.555,1.045) {$21$};\n\\node[font=\\small,align=center,font=\\small] at (10.805,4.55) {$(AB)x$};\n\\path[draw=black!30,fill=black!8] (10.55,3.34) rectangle (11.06,3.85);\n\\node[font=\\small,align=center,font=\\small] at (10.805,3.595) {$26$};\n\\path[draw=","h[draw=black!30,fill=black!8] (7.3,2.32) rectangle (7.81,2.83);\n\\node[font=\\small,align=center,font=\\small] at (7.555,2.575) {$25$};\n\\path[draw=black!30,fill=black!8] (7.3,1.81) rectangle (7.81,2.32);\n\\node[font=\\small,align=center,font=\\small] at (7.555,2.065) {$21$};\n\\path[draw=black!30,fill=black!8] (7.3,1.3) rectangle (7.81,1.81);\n\\node[font=\\small,align=center,font=\\small] at (7.555,1.555) {$25$};\n\\path[draw=black!30,fill=black!8] (7.3,0.79) rectangle (7.81,1.3);\n\\node[font=\\small,align=center,font=\\small] at (7.555,1.045) {$21$};\n\\node[font=\\small,align=center,font=\\small] at (10.805,4.55) {$(AB)x$};\n\\path[draw=black!30,fill=black!8] (10.55,3.34) rectangle (11.06,3.85);\n\\node[font=\\small,align=center,font=\\small] at (10.805,3.595) {$26$};\n\\path[draw=black!30,fill=black!8] (10.55,2.83) rectangle (11.06,3.34);\n\\node[font=\\small,align=center,font=\\small] at (10.805,3.085) {$16$};\n\\path[","th[draw=black!30,fill=black!8] (7.3,1.81) rectangle (7.81,2.32);\n\\node[font=\\small,align=center,font=\\small] at (7.555,2.065) {$21$};\n\\path[draw=black!30,fill=black!8] (7.3,1.3) rectangle (7.81,1.81);\n\\node[font=\\small,align=center,font=\\small] at (7.555,1.555) {$25$};\n\\path[draw=black!30,fill=black!8] (7.3,0.79) rectangle (7.81,1.3);\n\\node[font=\\small,align=center,font=\\small] at (7.555,1.045) {$21$};\n\\node[font=\\small,align=center,font=\\small] at (10.805,4.55) {$(AB)x$};\n\\path[draw=black!30,fill=black!8] (10.55,3.34) rectangle (11.06,3.85);\n\\node[font=\\small,align=center,font=\\small] at (10.805,3.595) {$26$};\n\\path[draw=black!30,fill=black!8] (10.55,2.83) rectangle (11.06,3.34);\n\\node[font=\\small,align=center,font=\\small] at (10.805,3.085) {$16$};\n\\path[draw=black!30,fill=black!8] (10.55,2.32) rectangle (11.06,2.83);\n\\node[font=\\small,align=center,font=\\small] at (10.805,2.575) {$43$};\n\\","e[font=\\small,align=center,font=\\small] at (7.555,2.065) {$21$};\n\\path[draw=black!30,fill=black!8] (7.3,1.3) rectangle (7.81,1.81);\n\\node[font=\\small,align=center,font=\\small] at (7.555,1.555) {$25$};\n\\path[draw=black!30,fill=black!8] (7.3,0.79) rectangle (7.81,1.3);\n\\node[font=\\small,align=center,font=\\small] at (7.555,1.045) {$21$};\n\\node[font=\\small,align=center,font=\\small] at (10.805,4.55) {$(AB)x$};\n\\path[draw=black!30,fill=black!8] (10.55,3.34) rectangle (11.06,3.85);\n\\node[font=\\small,align=center,font=\\small] at (10.805,3.595) {$26$};\n\\path[draw=black!30,fill=black!8] (10.55,2.83) rectangle (11.06,3.34);\n\\node[font=\\small,align=center,font=\\small] at (10.805,3.085) {$16$};\n\\path[draw=black!30,fill=black!8] (10.55,2.32) rectangle (11.06,2.83);\n\\node[font=\\small,align=center,font=\\small] at (10.805,2.575) {$43$};\n\\path[draw=black!30,fill=black!8] (10.55,1.81) rectangle (11.06,2.32);\n\\n","small,align=center,font=\\small] at (7.555,1.555) {$25$};\n\\path[draw=black!30,fill=black!8] (7.3,0.79) rectangle (7.81,1.3);\n\\node[font=\\small,align=center,font=\\small] at (7.555,1.045) {$21$};\n\\node[font=\\small,align=center,font=\\small] at (10.805,4.55) {$(AB)x$};\n\\path[draw=black!30,fill=black!8] (10.55,3.34) rectangle (11.06,3.85);\n\\node[font=\\small,align=center,font=\\small] at (10.805,3.595) {$26$};\n\\path[draw=black!30,fill=black!8] (10.55,2.83) rectangle (11.06,3.34);\n\\node[font=\\small,align=center,font=\\small] at (10.805,3.085) {$16$};\n\\path[draw=black!30,fill=black!8] (10.55,2.32) rectangle (11.06,2.83);\n\\node[font=\\small,align=center,font=\\small] at (10.805,2.575) {$43$};\n\\path[draw=black!30,fill=black!8] (10.55,1.81) rectangle (11.06,2.32);\n\\node[font=\\small,align=center,font=\\small] at (10.805,2.065) {$27$};\n\\path[draw=black!30,fill=black!8] (10.55,1.3) rectangle (11.06,1.81);\n\\no",",align=center,font=\\small] at (7.555,1.045) {$21$};\n\\node[font=\\small,align=center,font=\\small] at (10.805,4.55) {$(AB)x$};\n\\path[draw=black!30,fill=black!8] (10.55,3.34) rectangle (11.06,3.85);\n\\node[font=\\small,align=center,font=\\small] at (10.805,3.595) {$26$};\n\\path[draw=black!30,fill=black!8] (10.55,2.83) rectangle (11.06,3.34);\n\\node[font=\\small,align=center,font=\\small] at (10.805,3.085) {$16$};\n\\path[draw=black!30,fill=black!8] (10.55,2.32) rectangle (11.06,2.83);\n\\node[font=\\small,align=center,font=\\small] at (10.805,2.575) {$43$};\n\\path[draw=black!30,fill=black!8] (10.55,1.81) rectangle (11.06,2.32);\n\\node[font=\\small,align=center,font=\\small] at (10.805,2.065) {$27$};\n\\path[draw=black!30,fill=black!8] (10.55,1.3) rectangle (11.06,1.81);\n\\node[font=\\small,align=center,font=\\small] at (10.805,1.555) {$13$};\n\\path[draw=black!30,fill=black!8] (10.55,0.79) rectangle (11.06,1.3);\n\\nod","30,fill=black!8] (10.55,3.34) rectangle (11.06,3.85);\n\\node[font=\\small,align=center,font=\\small] at (10.805,3.595) {$26$};\n\\path[draw=black!30,fill=black!8] (10.55,2.83) rectangle (11.06,3.34);\n\\node[font=\\small,align=center,font=\\small] at (10.805,3.085) {$16$};\n\\path[draw=black!30,fill=black!8] (10.55,2.32) rectangle (11.06,2.83);\n\\node[font=\\small,align=center,font=\\small] at (10.805,2.575) {$43$};\n\\path[draw=black!30,fill=black!8] (10.55,1.81) rectangle (11.06,2.32);\n\\node[font=\\small,align=center,font=\\small] at (10.805,2.065) {$27$};\n\\path[draw=black!30,fill=black!8] (10.55,1.3) rectangle (11.06,1.81);\n\\node[font=\\small,align=center,font=\\small] at (10.805,1.555) {$13$};\n\\path[draw=black!30,fill=black!8] (10.55,0.79) rectangle (11.06,1.3);\n\\node[font=\\small,align=center,font=\\small] at (10.805,1.045) {$65$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.1) {Two intermedia","30,fill=black!8] (10.55,2.83) rectangle (11.06,3.34);\n\\node[font=\\small,align=center,font=\\small] at (10.805,3.085) {$16$};\n\\path[draw=black!30,fill=black!8] (10.55,2.32) rectangle (11.06,2.83);\n\\node[font=\\small,align=center,font=\\small] at (10.805,2.575) {$43$};\n\\path[draw=black!30,fill=black!8] (10.55,1.81) rectangle (11.06,2.32);\n\\node[font=\\small,align=center,font=\\small] at (10.805,2.065) {$27$};\n\\path[draw=black!30,fill=black!8] (10.55,1.3) rectangle (11.06,1.81);\n\\node[font=\\small,align=center,font=\\small] at (10.805,1.555) {$13$};\n\\path[draw=black!30,fill=black!8] (10.55,0.79) rectangle (11.06,1.3);\n\\node[font=\\small,align=center,font=\\small] at (10.805,1.045) {$65$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.1) {Two intermediate scalars and six output coordinates, with an exact dense reference.};\n\\end{tikzpicture}","!30,fill=black!8] (10.55,2.32) rectangle (11.06,2.83);\n\\node[font=\\small,align=center,font=\\small] at (10.805,2.575) {$43$};\n\\path[draw=black!30,fill=black!8] (10.55,1.81) rectangle (11.06,2.32);\n\\node[font=\\small,align=center,font=\\small] at (10.805,2.065) {$27$};\n\\path[draw=black!30,fill=black!8] (10.55,1.3) rectangle (11.06,1.81);\n\\node[font=\\small,align=center,font=\\small] at (10.805,1.555) {$13$};\n\\path[draw=black!30,fill=black!8] (10.55,0.79) rectangle (11.06,1.3);\n\\node[font=\\small,align=center,font=\\small] at (10.805,1.045) {$65$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.1) {Two intermediate scalars and six output coordinates, with an exact dense reference.};\n\\end{tikzpicture}","k!30,fill=black!8] (10.55,1.81) rectangle (11.06,2.32);\n\\node[font=\\small,align=center,font=\\small] at (10.805,2.065) {$27$};\n\\path[draw=black!30,fill=black!8] (10.55,1.3) rectangle (11.06,1.81);\n\\node[font=\\small,align=center,font=\\small] at (10.805,1.555) {$13$};\n\\path[draw=black!30,fill=black!8] (10.55,0.79) rectangle (11.06,1.3);\n\\node[font=\\small,align=center,font=\\small] at (10.805,1.045) {$65$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.1) {Two intermediate scalars and six output coordinates, with an exact dense reference.};\n\\end{tikzpicture}","ath[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\draw[line width=.6pt] (0.8,0.9) -- (5.8,0.9);\n\\node[font=\\small,align=center] at (3.3,4.35) {local injected errors};\n\\path[draw=black,fill=black!12,line width=.6pt] (1.2,0.9) rectangle (2.066667,1.125);\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.633333,0.6) {1};\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.633333,1.345) {$\\frac{1}{10}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (2.866667,0.9) rectangle (3.733333,1.35);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.3,0.6) {2};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.3,1.57) {$\\frac{1}{5}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (4.533333,0.9) rectangle (5.4,1.0125);\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.966667,0.6) {3};\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.966667,1.2325) {$\\frac{1}{20}$};\n\\draw[line w","(2.066667,1.125);\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.633333,0.6) {1};\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.633333,1.345) {$\\frac{1}{10}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (2.866667,0.9) rectangle (3.733333,1.35);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.3,0.6) {2};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.3,1.57) {$\\frac{1}{5}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (4.533333,0.9) rectangle (5.4,1.0125);\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.966667,0.6) {3};\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.966667,1.2325) {$\\frac{1}{20}$};\n\\draw[line width=.6pt] (8,0.9) -- (13,0.9);\n\\node[font=\\small,align=center] at (10.5,4.35) {final contributions};\n\\path[draw=black,fill=black!12,line width=.6pt] (8.4,0.9) rectangle (9.266667,3.6);\n\\node[font=\\small,align=center,font=\\scriptsize] at","ngle (3.733333,1.35);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.3,0.6) {2};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.3,1.57) {$\\frac{1}{5}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (4.533333,0.9) rectangle (5.4,1.0125);\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.966667,0.6) {3};\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.966667,1.2325) {$\\frac{1}{20}$};\n\\draw[line width=.6pt] (8,0.9) -- (13,0.9);\n\\node[font=\\small,align=center] at (10.5,4.35) {final contributions};\n\\path[draw=black,fill=black!12,line width=.6pt] (8.4,0.9) rectangle (9.266667,3.6);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.833333,0.6) {1};\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.833333,3.82) {$\\frac{6}{5}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (10.066667,0.9) rectangle (10.933333,2.7);\n\\node[font=\\small,align=center,font=\\scriptsize] a","er,font=\\scriptsize] at (4.966667,1.2325) {$\\frac{1}{20}$};\n\\draw[line width=.6pt] (8,0.9) -- (13,0.9);\n\\node[font=\\small,align=center] at (10.5,4.35) {final contributions};\n\\path[draw=black,fill=black!12,line width=.6pt] (8.4,0.9) rectangle (9.266667,3.6);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.833333,0.6) {1};\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.833333,3.82) {$\\frac{6}{5}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (10.066667,0.9) rectangle (10.933333,2.7);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.5,0.6) {2};\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.5,2.92) {$\\frac{4}{5}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (11.733333,0.9) rectangle (12.6,1.0125);\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.166667,0.6) {3};\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.166667,1.2325) {$\\frac{1}{20}$};\n\\node","e (9.266667,3.6);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.833333,0.6) {1};\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.833333,3.82) {$\\frac{6}{5}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (10.066667,0.9) rectangle (10.933333,2.7);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.5,0.6) {2};\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.5,2.92) {$\\frac{4}{5}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (11.733333,0.9) rectangle (12.6,1.0125);\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.166667,0.6) {3};\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.166667,1.2325) {$\\frac{1}{20}$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.05) {$\\widehat p_j=a_j\\widehat p_{j-1}+e_j$, with $(a_1,a_2,a_3)=(2,3,4)$.};\n\\end{tikzpicture}","10.933333,2.7);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.5,0.6) {2};\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.5,2.92) {$\\frac{4}{5}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (11.733333,0.9) rectangle (12.6,1.0125);\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.166667,0.6) {3};\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.166667,1.2325) {$\\frac{1}{20}$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.05) {$\\widehat p_j=a_j\\widehat p_{j-1}+e_j$, with $(a_1,a_2,a_3)=(2,3,4)$.};\n\\end{tikzpicture}","{2};\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.5,2.92) {$\\frac{4}{5}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (11.733333,0.9) rectangle (12.6,1.0125);\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.166667,0.6) {3};\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.166667,1.2325) {$\\frac{1}{20}$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.05) {$\\widehat p_j=a_j\\widehat p_{j-1}+e_j$, with $(a_1,a_2,a_3)=(2,3,4)$.};\n\\end{tikzpicture}","iptsize] at (10.5,2.92) {$\\frac{4}{5}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (11.733333,0.9) rectangle (12.6,1.0125);\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.166667,0.6) {3};\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.166667,1.2325) {$\\frac{1}{20}$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.05) {$\\widehat p_j=a_j\\widehat p_{j-1}+e_j$, with $(a_1,a_2,a_3)=(2,3,4)$.};\n\\end{tikzpicture}","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.06,4.5) {$x$};\n\\path[draw=black!30,fill=black!8] (0.8,3.28) rectangle (1.32,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.06,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,2.76) rectangle (1.32,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.06,3.02) {$-1$};\n\\path[draw=black!30,fill=black!8] (0.8,2.24) rectangle (1.32,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.06,2.5) {$2$};\n\\path[draw=black!30,fill=white] (0.8,1.72) rectangle (1.32,2.24);\n\\node[font=\\small,align=cente","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.06,4.5) {$x$};\n\\path[draw=black!30,fill=black!8] (0.8,3.28) rectangle (1.32,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.06,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,2.76) rectangle (1.32,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.06,3.02) {$-1$};\n\\path[draw=black!30,fill=black!8] (0.8,2.24) rectangle (1.32,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.06,2.5) {$2$};\n\\path[draw=black!30,fill=white] (0.8,1.72) rectangle (1.32,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.98) {$0$};\n\\path[draw=black!30,fill=black!8] (0.8,1.2) rectangle (1.32,1.72);\n\\node[font=\\small,align=cente",",line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.06,4.5) {$x$};\n\\path[draw=black!30,fill=black!8] (0.8,3.28) rectangle (1.32,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.06,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,2.76) rectangle (1.32,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.06,3.02) {$-1$};\n\\path[draw=black!30,fill=black!8] (0.8,2.24) rectangle (1.32,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.06,2.5) {$2$};\n\\path[draw=black!30,fill=white] (0.8,1.72) rectangle (1.32,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.98) {$0$};\n\\path[draw=black!30,fill=black!8] (0.8,1.2) rectangle (1.32,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.46) {$-2$};\n\\path[draw=black!30,fill=black!8] (0.8,0.68) rectangle (1.32,1.2);\n\\node[font=\\small,align=center","$};\n\\path[draw=black!30,fill=black!8] (0.8,3.28) rectangle (1.32,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.06,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (0.8,2.76) rectangle (1.32,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.06,3.02) {$-1$};\n\\path[draw=black!30,fill=black!8] (0.8,2.24) rectangle (1.32,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.06,2.5) {$2$};\n\\path[draw=black!30,fill=white] (0.8,1.72) rectangle (1.32,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.98) {$0$};\n\\path[draw=black!30,fill=black!8] (0.8,1.2) rectangle (1.32,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.46) {$-2$};\n\\path[draw=black!30,fill=black!8] (0.8,0.68) rectangle (1.32,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.06,0.94) {$3$};\n\\node[font=\\small,align=center,font=\\small] at (4.51,4.5) {$Dx$};\n\\path[draw=black!30,fill=black!","1$};\n\\path[draw=black!30,fill=black!8] (0.8,2.76) rectangle (1.32,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.06,3.02) {$-1$};\n\\path[draw=black!30,fill=black!8] (0.8,2.24) rectangle (1.32,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.06,2.5) {$2$};\n\\path[draw=black!30,fill=white] (0.8,1.72) rectangle (1.32,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.98) {$0$};\n\\path[draw=black!30,fill=black!8] (0.8,1.2) rectangle (1.32,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.46) {$-2$};\n\\path[draw=black!30,fill=black!8] (0.8,0.68) rectangle (1.32,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.06,0.94) {$3$};\n\\node[font=\\small,align=center,font=\\small] at (4.51,4.5) {$Dx$};\n\\path[draw=black!30,fill=black!8] (4.25,3.28) rectangle (4.77,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.51,3.54) {$2$};\n\\path[draw=black!30,fill=blac","$-1$};\n\\path[draw=black!30,fill=black!8] (0.8,2.24) rectangle (1.32,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.06,2.5) {$2$};\n\\path[draw=black!30,fill=white] (0.8,1.72) rectangle (1.32,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.98) {$0$};\n\\path[draw=black!30,fill=black!8] (0.8,1.2) rectangle (1.32,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.46) {$-2$};\n\\path[draw=black!30,fill=black!8] (0.8,0.68) rectangle (1.32,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.06,0.94) {$3$};\n\\node[font=\\small,align=center,font=\\small] at (4.51,4.5) {$Dx$};\n\\path[draw=black!30,fill=black!8] (4.25,3.28) rectangle (4.77,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.51,3.54) {$2$};\n\\path[draw=black!30,fill=black!8] (4.25,2.76) rectangle (4.77,3.28);\n\\node[font=\\small,align=center,font=\\small] at (4.51,3.02) {$-2$};\n\\path[draw=black!30,fill=bl","$2$};\n\\path[draw=black!30,fill=white] (0.8,1.72) rectangle (1.32,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.98) {$0$};\n\\path[draw=black!30,fill=black!8] (0.8,1.2) rectangle (1.32,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.46) {$-2$};\n\\path[draw=black!30,fill=black!8] (0.8,0.68) rectangle (1.32,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.06,0.94) {$3$};\n\\node[font=\\small,align=center,font=\\small] at (4.51,4.5) {$Dx$};\n\\path[draw=black!30,fill=black!8] (4.25,3.28) rectangle (4.77,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.51,3.54) {$2$};\n\\path[draw=black!30,fill=black!8] (4.25,2.76) rectangle (4.77,3.28);\n\\node[font=\\small,align=center,font=\\small] at (4.51,3.02) {$-2$};\n\\path[draw=black!30,fill=black!8] (4.25,2.24) rectangle (4.77,2.76);\n\\node[font=\\small,align=center,font=\\small] at (4.51,2.5) {$6$};\n\\path[draw=black!30,fill=b","2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.98) {$0$};\n\\path[draw=black!30,fill=black!8] (0.8,1.2) rectangle (1.32,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.46) {$-2$};\n\\path[draw=black!30,fill=black!8] (0.8,0.68) rectangle (1.32,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.06,0.94) {$3$};\n\\node[font=\\small,align=center,font=\\small] at (4.51,4.5) {$Dx$};\n\\path[draw=black!30,fill=black!8] (4.25,3.28) rectangle (4.77,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.51,3.54) {$2$};\n\\path[draw=black!30,fill=black!8] (4.25,2.76) rectangle (4.77,3.28);\n\\node[font=\\small,align=center,font=\\small] at (4.51,3.02) {$-2$};\n\\path[draw=black!30,fill=black!8] (4.25,2.24) rectangle (4.77,2.76);\n\\node[font=\\small,align=center,font=\\small] at (4.51,2.5) {$6$};\n\\path[draw=black!30,fill=black!8] (4.25,1.72) rectangle (4.77,2.24);\n\\node[font=\\small,align","72);\n\\node[font=\\small,align=center,font=\\small] at (1.06,1.46) {$-2$};\n\\path[draw=black!30,fill=black!8] (0.8,0.68) rectangle (1.32,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.06,0.94) {$3$};\n\\node[font=\\small,align=center,font=\\small] at (4.51,4.5) {$Dx$};\n\\path[draw=black!30,fill=black!8] (4.25,3.28) rectangle (4.77,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.51,3.54) {$2$};\n\\path[draw=black!30,fill=black!8] (4.25,2.76) rectangle (4.77,3.28);\n\\node[font=\\small,align=center,font=\\small] at (4.51,3.02) {$-2$};\n\\path[draw=black!30,fill=black!8] (4.25,2.24) rectangle (4.77,2.76);\n\\node[font=\\small,align=center,font=\\small] at (4.51,2.5) {$6$};\n\\path[draw=black!30,fill=black!8] (4.25,1.72) rectangle (4.77,2.24);\n\\node[font=\\small,align=center,font=\\small] at (4.51,1.98) {$2$};\n\\path[draw=black!30,fill=black!8] (4.25,1.2) rectangle (4.77,1.72);\n\\node[font=\\small,align","2);\n\\node[font=\\small,align=center,font=\\small] at (1.06,0.94) {$3$};\n\\node[font=\\small,align=center,font=\\small] at (4.51,4.5) {$Dx$};\n\\path[draw=black!30,fill=black!8] (4.25,3.28) rectangle (4.77,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.51,3.54) {$2$};\n\\path[draw=black!30,fill=black!8] (4.25,2.76) rectangle (4.77,3.28);\n\\node[font=\\small,align=center,font=\\small] at (4.51,3.02) {$-2$};\n\\path[draw=black!30,fill=black!8] (4.25,2.24) rectangle (4.77,2.76);\n\\node[font=\\small,align=center,font=\\small] at (4.51,2.5) {$6$};\n\\path[draw=black!30,fill=black!8] (4.25,1.72) rectangle (4.77,2.24);\n\\node[font=\\small,align=center,font=\\small] at (4.51,1.98) {$2$};\n\\path[draw=black!30,fill=black!8] (4.25,1.2) rectangle (4.77,1.72);\n\\node[font=\\small,align=center,font=\\small] at (4.51,1.46) {$-3$};\n\\path[draw=black!30,fill=black!8] (4.25,0.68) rectangle (4.77,1.2);\n\\node[font=\\small,align=","\\path[draw=black!30,fill=black!8] (4.25,3.28) rectangle (4.77,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.51,3.54) {$2$};\n\\path[draw=black!30,fill=black!8] (4.25,2.76) rectangle (4.77,3.28);\n\\node[font=\\small,align=center,font=\\small] at (4.51,3.02) {$-2$};\n\\path[draw=black!30,fill=black!8] (4.25,2.24) rectangle (4.77,2.76);\n\\node[font=\\small,align=center,font=\\small] at (4.51,2.5) {$6$};\n\\path[draw=black!30,fill=black!8] (4.25,1.72) rectangle (4.77,2.24);\n\\node[font=\\small,align=center,font=\\small] at (4.51,1.98) {$2$};\n\\path[draw=black!30,fill=black!8] (4.25,1.2) rectangle (4.77,1.72);\n\\node[font=\\small,align=center,font=\\small] at (4.51,1.46) {$-3$};\n\\path[draw=black!30,fill=black!8] (4.25,0.68) rectangle (4.77,1.2);\n\\node[font=\\small,align=center,font=\\small] at (4.51,0.94) {$7$};\n\\node[font=\\small,align=center,font=\\small] at (7.96,4.5) {$u(v^Tx)$};\n\\path[draw=black!30,f","\\path[draw=black!30,fill=black!8] (4.25,2.76) rectangle (4.77,3.28);\n\\node[font=\\small,align=center,font=\\small] at (4.51,3.02) {$-2$};\n\\path[draw=black!30,fill=black!8] (4.25,2.24) rectangle (4.77,2.76);\n\\node[font=\\small,align=center,font=\\small] at (4.51,2.5) {$6$};\n\\path[draw=black!30,fill=black!8] (4.25,1.72) rectangle (4.77,2.24);\n\\node[font=\\small,align=center,font=\\small] at (4.51,1.98) {$2$};\n\\path[draw=black!30,fill=black!8] (4.25,1.2) rectangle (4.77,1.72);\n\\node[font=\\small,align=center,font=\\small] at (4.51,1.46) {$-3$};\n\\path[draw=black!30,fill=black!8] (4.25,0.68) rectangle (4.77,1.2);\n\\node[font=\\small,align=center,font=\\small] at (4.51,0.94) {$7$};\n\\node[font=\\small,align=center,font=\\small] at (7.96,4.5) {$u(v^Tx)$};\n\\path[draw=black!30,fill=black!8] (7.7,3.28) rectangle (8.22,3.8);\n\\node[font=\\small,align=center,font=\\small] at (7.96,3.54) {$3$};\n\\path[draw=black!30,fil",";\n\\path[draw=black!30,fill=black!8] (4.25,2.24) rectangle (4.77,2.76);\n\\node[font=\\small,align=center,font=\\small] at (4.51,2.5) {$6$};\n\\path[draw=black!30,fill=black!8] (4.25,1.72) rectangle (4.77,2.24);\n\\node[font=\\small,align=center,font=\\small] at (4.51,1.98) {$2$};\n\\path[draw=black!30,fill=black!8] (4.25,1.2) rectangle (4.77,1.72);\n\\node[font=\\small,align=center,font=\\small] at (4.51,1.46) {$-3$};\n\\path[draw=black!30,fill=black!8] (4.25,0.68) rectangle (4.77,1.2);\n\\node[font=\\small,align=center,font=\\small] at (4.51,0.94) {$7$};\n\\node[font=\\small,align=center,font=\\small] at (7.96,4.5) {$u(v^Tx)$};\n\\path[draw=black!30,fill=black!8] (7.7,3.28) rectangle (8.22,3.8);\n\\node[font=\\small,align=center,font=\\small] at (7.96,3.54) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,2.76) rectangle (8.22,3.28);\n\\node[font=\\small,align=center,font=\\small] at (7.96,3.02) {$3$};\n\\path[draw=black!30,fill","\\path[draw=black!30,fill=black!8] (4.25,1.72) rectangle (4.77,2.24);\n\\node[font=\\small,align=center,font=\\small] at (4.51,1.98) {$2$};\n\\path[draw=black!30,fill=black!8] (4.25,1.2) rectangle (4.77,1.72);\n\\node[font=\\small,align=center,font=\\small] at (4.51,1.46) {$-3$};\n\\path[draw=black!30,fill=black!8] (4.25,0.68) rectangle (4.77,1.2);\n\\node[font=\\small,align=center,font=\\small] at (4.51,0.94) {$7$};\n\\node[font=\\small,align=center,font=\\small] at (7.96,4.5) {$u(v^Tx)$};\n\\path[draw=black!30,fill=black!8] (7.7,3.28) rectangle (8.22,3.8);\n\\node[font=\\small,align=center,font=\\small] at (7.96,3.54) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,2.76) rectangle (8.22,3.28);\n\\node[font=\\small,align=center,font=\\small] at (7.96,3.02) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,2.24) rectangle (8.22,2.76);\n\\node[font=\\small,align=center,font=\\small] at (7.96,2.5) {$3$};\n\\path[draw=black!30,fill=","24);\n\\node[font=\\small,align=center,font=\\small] at (4.51,1.98) {$2$};\n\\path[draw=black!30,fill=black!8] (4.25,1.2) rectangle (4.77,1.72);\n\\node[font=\\small,align=center,font=\\small] at (4.51,1.46) {$-3$};\n\\path[draw=black!30,fill=black!8] (4.25,0.68) rectangle (4.77,1.2);\n\\node[font=\\small,align=center,font=\\small] at (4.51,0.94) {$7$};\n\\node[font=\\small,align=center,font=\\small] at (7.96,4.5) {$u(v^Tx)$};\n\\path[draw=black!30,fill=black!8] (7.7,3.28) rectangle (8.22,3.8);\n\\node[font=\\small,align=center,font=\\small] at (7.96,3.54) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,2.76) rectangle (8.22,3.28);\n\\node[font=\\small,align=center,font=\\small] at (7.96,3.02) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,2.24) rectangle (8.22,2.76);\n\\node[font=\\small,align=center,font=\\small] at (7.96,2.5) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,1.72) rectangle (8.22,2.24);\n\\node[font=\\small,align=cent","\\node[font=\\small,align=center,font=\\small] at (4.51,1.46) {$-3$};\n\\path[draw=black!30,fill=black!8] (4.25,0.68) rectangle (4.77,1.2);\n\\node[font=\\small,align=center,font=\\small] at (4.51,0.94) {$7$};\n\\node[font=\\small,align=center,font=\\small] at (7.96,4.5) {$u(v^Tx)$};\n\\path[draw=black!30,fill=black!8] (7.7,3.28) rectangle (8.22,3.8);\n\\node[font=\\small,align=center,font=\\small] at (7.96,3.54) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,2.76) rectangle (8.22,3.28);\n\\node[font=\\small,align=center,font=\\small] at (7.96,3.02) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,2.24) rectangle (8.22,2.76);\n\\node[font=\\small,align=center,font=\\small] at (7.96,2.5) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,1.72) rectangle (8.22,2.24);\n\\node[font=\\small,align=center,font=\\small] at (7.96,1.98) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,1.2) rectangle (8.22,1.72);\n\\node[font=\\small,align=cent","\\node[font=\\small,align=center,font=\\small] at (4.51,0.94) {$7$};\n\\node[font=\\small,align=center,font=\\small] at (7.96,4.5) {$u(v^Tx)$};\n\\path[draw=black!30,fill=black!8] (7.7,3.28) rectangle (8.22,3.8);\n\\node[font=\\small,align=center,font=\\small] at (7.96,3.54) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,2.76) rectangle (8.22,3.28);\n\\node[font=\\small,align=center,font=\\small] at (7.96,3.02) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,2.24) rectangle (8.22,2.76);\n\\node[font=\\small,align=center,font=\\small] at (7.96,2.5) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,1.72) rectangle (8.22,2.24);\n\\node[font=\\small,align=center,font=\\small] at (7.96,1.98) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,1.2) rectangle (8.22,1.72);\n\\node[font=\\small,align=center,font=\\small] at (7.96,1.46) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,0.68) rectangle (8.22,1.2);\n\\node[font=\\small,align=cente",")$};\n\\path[draw=black!30,fill=black!8] (7.7,3.28) rectangle (8.22,3.8);\n\\node[font=\\small,align=center,font=\\small] at (7.96,3.54) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,2.76) rectangle (8.22,3.28);\n\\node[font=\\small,align=center,font=\\small] at (7.96,3.02) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,2.24) rectangle (8.22,2.76);\n\\node[font=\\small,align=center,font=\\small] at (7.96,2.5) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,1.72) rectangle (8.22,2.24);\n\\node[font=\\small,align=center,font=\\small] at (7.96,1.98) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,1.2) rectangle (8.22,1.72);\n\\node[font=\\small,align=center,font=\\small] at (7.96,1.46) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,0.68) rectangle (8.22,1.2);\n\\node[font=\\small,align=center,font=\\small] at (7.96,0.94) {$3$};\n\\node[font=\\small,align=center,font=\\small] at (11.41,4.5) {$Ax$};\n\\path[draw=black!30,fill=blac","$};\n\\path[draw=black!30,fill=black!8] (7.7,2.76) rectangle (8.22,3.28);\n\\node[font=\\small,align=center,font=\\small] at (7.96,3.02) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,2.24) rectangle (8.22,2.76);\n\\node[font=\\small,align=center,font=\\small] at (7.96,2.5) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,1.72) rectangle (8.22,2.24);\n\\node[font=\\small,align=center,font=\\small] at (7.96,1.98) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,1.2) rectangle (8.22,1.72);\n\\node[font=\\small,align=center,font=\\small] at (7.96,1.46) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,0.68) rectangle (8.22,1.2);\n\\node[font=\\small,align=center,font=\\small] at (7.96,0.94) {$3$};\n\\node[font=\\small,align=center,font=\\small] at (11.41,4.5) {$Ax$};\n\\path[draw=black!30,fill=black!8] (11.15,3.28) rectangle (11.67,3.8);\n\\node[font=\\small,align=center,font=\\small] at (11.41,3.54) {$5$};\n\\path[draw=black!30,fill=b","3$};\n\\path[draw=black!30,fill=black!8] (7.7,2.24) rectangle (8.22,2.76);\n\\node[font=\\small,align=center,font=\\small] at (7.96,2.5) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,1.72) rectangle (8.22,2.24);\n\\node[font=\\small,align=center,font=\\small] at (7.96,1.98) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,1.2) rectangle (8.22,1.72);\n\\node[font=\\small,align=center,font=\\small] at (7.96,1.46) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,0.68) rectangle (8.22,1.2);\n\\node[font=\\small,align=center,font=\\small] at (7.96,0.94) {$3$};\n\\node[font=\\small,align=center,font=\\small] at (11.41,4.5) {$Ax$};\n\\path[draw=black!30,fill=black!8] (11.15,3.28) rectangle (11.67,3.8);\n\\node[font=\\small,align=center,font=\\small] at (11.41,3.54) {$5$};\n\\path[draw=black!30,fill=black!8] (11.15,2.76) rectangle (11.67,3.28);\n\\node[font=\\small,align=center,font=\\small] at (11.41,3.02) {$1$};\n\\path[draw=black!30,f","3$};\n\\path[draw=black!30,fill=black!8] (7.7,1.72) rectangle (8.22,2.24);\n\\node[font=\\small,align=center,font=\\small] at (7.96,1.98) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,1.2) rectangle (8.22,1.72);\n\\node[font=\\small,align=center,font=\\small] at (7.96,1.46) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,0.68) rectangle (8.22,1.2);\n\\node[font=\\small,align=center,font=\\small] at (7.96,0.94) {$3$};\n\\node[font=\\small,align=center,font=\\small] at (11.41,4.5) {$Ax$};\n\\path[draw=black!30,fill=black!8] (11.15,3.28) rectangle (11.67,3.8);\n\\node[font=\\small,align=center,font=\\small] at (11.41,3.54) {$5$};\n\\path[draw=black!30,fill=black!8] (11.15,2.76) rectangle (11.67,3.28);\n\\node[font=\\small,align=center,font=\\small] at (11.41,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (11.15,2.24) rectangle (11.67,2.76);\n\\node[font=\\small,align=center,font=\\small] at (11.41,2.5) {$9$};\n\\path[draw=black!","2.24);\n\\node[font=\\small,align=center,font=\\small] at (7.96,1.98) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,1.2) rectangle (8.22,1.72);\n\\node[font=\\small,align=center,font=\\small] at (7.96,1.46) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,0.68) rectangle (8.22,1.2);\n\\node[font=\\small,align=center,font=\\small] at (7.96,0.94) {$3$};\n\\node[font=\\small,align=center,font=\\small] at (11.41,4.5) {$Ax$};\n\\path[draw=black!30,fill=black!8] (11.15,3.28) rectangle (11.67,3.8);\n\\node[font=\\small,align=center,font=\\small] at (11.41,3.54) {$5$};\n\\path[draw=black!30,fill=black!8] (11.15,2.76) rectangle (11.67,3.28);\n\\node[font=\\small,align=center,font=\\small] at (11.41,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (11.15,2.24) rectangle (11.67,2.76);\n\\node[font=\\small,align=center,font=\\small] at (11.41,2.5) {$9$};\n\\path[draw=black!30,fill=black!8] (11.15,1.72) rectangle (11.67,2.24);\n\\node[font=\\s",";\n\\node[font=\\small,align=center,font=\\small] at (7.96,1.46) {$3$};\n\\path[draw=black!30,fill=black!8] (7.7,0.68) rectangle (8.22,1.2);\n\\node[font=\\small,align=center,font=\\small] at (7.96,0.94) {$3$};\n\\node[font=\\small,align=center,font=\\small] at (11.41,4.5) {$Ax$};\n\\path[draw=black!30,fill=black!8] (11.15,3.28) rectangle (11.67,3.8);\n\\node[font=\\small,align=center,font=\\small] at (11.41,3.54) {$5$};\n\\path[draw=black!30,fill=black!8] (11.15,2.76) rectangle (11.67,3.28);\n\\node[font=\\small,align=center,font=\\small] at (11.41,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (11.15,2.24) rectangle (11.67,2.76);\n\\node[font=\\small,align=center,font=\\small] at (11.41,2.5) {$9$};\n\\path[draw=black!30,fill=black!8] (11.15,1.72) rectangle (11.67,2.24);\n\\node[font=\\small,align=center,font=\\small] at (11.41,1.98) {$5$};\n\\path[draw=black!30,fill=white] (11.15,1.2) rectangle (11.67,1.72);\n\\node[font=\\sma","de[font=\\small,align=center,font=\\small] at (7.96,0.94) {$3$};\n\\node[font=\\small,align=center,font=\\small] at (11.41,4.5) {$Ax$};\n\\path[draw=black!30,fill=black!8] (11.15,3.28) rectangle (11.67,3.8);\n\\node[font=\\small,align=center,font=\\small] at (11.41,3.54) {$5$};\n\\path[draw=black!30,fill=black!8] (11.15,2.76) rectangle (11.67,3.28);\n\\node[font=\\small,align=center,font=\\small] at (11.41,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (11.15,2.24) rectangle (11.67,2.76);\n\\node[font=\\small,align=center,font=\\small] at (11.41,2.5) {$9$};\n\\path[draw=black!30,fill=black!8] (11.15,1.72) rectangle (11.67,2.24);\n\\node[font=\\small,align=center,font=\\small] at (11.41,1.98) {$5$};\n\\path[draw=black!30,fill=white] (11.15,1.2) rectangle (11.67,1.72);\n\\node[font=\\small,align=center,font=\\small] at (11.41,1.46) {$0$};\n\\path[draw=black!30,fill=black!8] (11.15,0.68) rectangle (11.67,1.2);\n\\node[font=\\smal","raw=black!30,fill=black!8] (11.15,3.28) rectangle (11.67,3.8);\n\\node[font=\\small,align=center,font=\\small] at (11.41,3.54) {$5$};\n\\path[draw=black!30,fill=black!8] (11.15,2.76) rectangle (11.67,3.28);\n\\node[font=\\small,align=center,font=\\small] at (11.41,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (11.15,2.24) rectangle (11.67,2.76);\n\\node[font=\\small,align=center,font=\\small] at (11.41,2.5) {$9$};\n\\path[draw=black!30,fill=black!8] (11.15,1.72) rectangle (11.67,2.24);\n\\node[font=\\small,align=center,font=\\small] at (11.41,1.98) {$5$};\n\\path[draw=black!30,fill=white] (11.15,1.2) rectangle (11.67,1.72);\n\\node[font=\\small,align=center,font=\\small] at (11.41,1.46) {$0$};\n\\path[draw=black!30,fill=black!8] (11.15,0.68) rectangle (11.67,1.2);\n\\node[font=\\small,align=center,font=\\small] at (11.41,0.94) {$10$};\n\\node[font=\\small,align=center,font=\\Large] at (5.15,2.1) {$+$};\n\\node[font=\\small,al","aw=black!30,fill=black!8] (11.15,2.76) rectangle (11.67,3.28);\n\\node[font=\\small,align=center,font=\\small] at (11.41,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (11.15,2.24) rectangle (11.67,2.76);\n\\node[font=\\small,align=center,font=\\small] at (11.41,2.5) {$9$};\n\\path[draw=black!30,fill=black!8] (11.15,1.72) rectangle (11.67,2.24);\n\\node[font=\\small,align=center,font=\\small] at (11.41,1.98) {$5$};\n\\path[draw=black!30,fill=white] (11.15,1.2) rectangle (11.67,1.72);\n\\node[font=\\small,align=center,font=\\small] at (11.41,1.46) {$0$};\n\\path[draw=black!30,fill=black!8] (11.15,0.68) rectangle (11.67,1.2);\n\\node[font=\\small,align=center,font=\\small] at (11.41,0.94) {$10$};\n\\node[font=\\small,align=center,font=\\Large] at (5.15,2.1) {$+$};\n\\node[font=\\small,align=center,font=\\Large] at (9,2.1) {$=$};\n\\node[font=\\small,align=center,font=\\small] at (7,0.1) {$v^Tx=3$ is shared by all six output coo","[draw=black!30,fill=black!8] (11.15,2.24) rectangle (11.67,2.76);\n\\node[font=\\small,align=center,font=\\small] at (11.41,2.5) {$9$};\n\\path[draw=black!30,fill=black!8] (11.15,1.72) rectangle (11.67,2.24);\n\\node[font=\\small,align=center,font=\\small] at (11.41,1.98) {$5$};\n\\path[draw=black!30,fill=white] (11.15,1.2) rectangle (11.67,1.72);\n\\node[font=\\small,align=center,font=\\small] at (11.41,1.46) {$0$};\n\\path[draw=black!30,fill=black!8] (11.15,0.68) rectangle (11.67,1.2);\n\\node[font=\\small,align=center,font=\\small] at (11.41,0.94) {$10$};\n\\node[font=\\small,align=center,font=\\Large] at (5.15,2.1) {$+$};\n\\node[font=\\small,align=center,font=\\Large] at (9,2.1) {$=$};\n\\node[font=\\small,align=center,font=\\small] at (7,0.1) {$v^Tx=3$ is shared by all six output coordinates.};\n\\end{tikzpicture}","[draw=black!30,fill=black!8] (11.15,1.72) rectangle (11.67,2.24);\n\\node[font=\\small,align=center,font=\\small] at (11.41,1.98) {$5$};\n\\path[draw=black!30,fill=white] (11.15,1.2) rectangle (11.67,1.72);\n\\node[font=\\small,align=center,font=\\small] at (11.41,1.46) {$0$};\n\\path[draw=black!30,fill=black!8] (11.15,0.68) rectangle (11.67,1.2);\n\\node[font=\\small,align=center,font=\\small] at (11.41,0.94) {$10$};\n\\node[font=\\small,align=center,font=\\Large] at (5.15,2.1) {$+$};\n\\node[font=\\small,align=center,font=\\Large] at (9,2.1) {$=$};\n\\node[font=\\small,align=center,font=\\small] at (7,0.1) {$v^Tx=3$ is shared by all six output coordinates.};\n\\end{tikzpicture}","\\node[font=\\small,align=center,font=\\small] at (11.41,1.98) {$5$};\n\\path[draw=black!30,fill=white] (11.15,1.2) rectangle (11.67,1.72);\n\\node[font=\\small,align=center,font=\\small] at (11.41,1.46) {$0$};\n\\path[draw=black!30,fill=black!8] (11.15,0.68) rectangle (11.67,1.2);\n\\node[font=\\small,align=center,font=\\small] at (11.41,0.94) {$10$};\n\\node[font=\\small,align=center,font=\\Large] at (5.15,2.1) {$+$};\n\\node[font=\\small,align=center,font=\\Large] at (9,2.1) {$=$};\n\\node[font=\\small,align=center,font=\\small] at (7,0.1) {$v^Tx=3$ is shared by all six output coordinates.};\n\\end{tikzpicture}","5$};\n\\path[draw=black!30,fill=white] (11.15,1.2) rectangle (11.67,1.72);\n\\node[font=\\small,align=center,font=\\small] at (11.41,1.46) {$0$};\n\\path[draw=black!30,fill=black!8] (11.15,0.68) rectangle (11.67,1.2);\n\\node[font=\\small,align=center,font=\\small] at (11.41,0.94) {$10$};\n\\node[font=\\small,align=center,font=\\Large] at (5.15,2.1) {$+$};\n\\node[font=\\small,align=center,font=\\Large] at (9,2.1) {$=$};\n\\node[font=\\small,align=center,font=\\small] at (7,0.1) {$v^Tx=3$ is shared by all six output coordinates.};\n\\end{tikzpicture}",".67,1.72);\n\\node[font=\\small,align=center,font=\\small] at (11.41,1.46) {$0$};\n\\path[draw=black!30,fill=black!8] (11.15,0.68) rectangle (11.67,1.2);\n\\node[font=\\small,align=center,font=\\small] at (11.41,0.94) {$10$};\n\\node[font=\\small,align=center,font=\\Large] at (5.15,2.1) {$+$};\n\\node[font=\\small,align=center,font=\\Large] at (9,2.1) {$=$};\n\\node[font=\\small,align=center,font=\\small] at (7,0.1) {$v^Tx=3$ is shared by all six output coordinates.};\n\\end{tikzpicture}","\\begin{tikzpicture}[font=\\sffamily\\small,>=Latex]\n\\foreach \\x/\\y/\\n in {0/1/0,2/2/1,2/0/2,4/1/3}\n\\node[draw,circle,minimum size=8mm] (n\\n) at (\\x,\\y) {$\\n$};\n\\draw[->,thick] (n0)--node[above left] {$1$}(n1);\n\\draw[->,thick] (n0)--node[below left] {$1$}(n2);\n\\draw[->,thick] (n1)--node[above right] {$1$}(n3);\n\\draw[->,thick] (n2)--node[below right] {$-1$}(n3);\n\\draw[->,dashed,rounded corners=5pt] (n0.north)--(0,2.6)--(4,2.6)--(n3.north);\n\\node[anchor=south] at (2,2.8) {closure permits $0\\to3$};\n\\node[anchor=west] at (5,1.8) {$K^2=0$};\n\\node[anchor=west] at (5,1) {$A^{-1}=I-K$};\n\\node[anchor=west] at (5,.2) {$(A^{-1})_{30}=0$};\n\\end{tikzpicture}",",4/1/3}\n\\node[draw,circle,minimum size=8mm] (n\\n) at (\\x,\\y) {$\\n$};\n\\draw[->,thick] (n0)--node[above left] {$1$}(n1);\n\\draw[->,thick] (n0)--node[below left] {$1$}(n2);\n\\draw[->,thick] (n1)--node[above right] {$1$}(n3);\n\\draw[->,thick] (n2)--node[below right] {$-1$}(n3);\n\\draw[->,dashed,rounded corners=5pt] (n0.north)--(0,2.6)--(4,2.6)--(n3.north);\n\\node[anchor=south] at (2,2.8) {closure permits $0\\to3$};\n\\node[anchor=west] at (5,1.8) {$K^2=0$};\n\\node[anchor=west] at (5,1) {$A^{-1}=I-K$};\n\\node[anchor=west] at (5,.2) {$(A^{-1})_{30}=0$};\n\\end{tikzpicture}","m] (n\\n) at (\\x,\\y) {$\\n$};\n\\draw[->,thick] (n0)--node[above left] {$1$}(n1);\n\\draw[->,thick] (n0)--node[below left] {$1$}(n2);\n\\draw[->,thick] (n1)--node[above right] {$1$}(n3);\n\\draw[->,thick] (n2)--node[below right] {$-1$}(n3);\n\\draw[->,dashed,rounded corners=5pt] (n0.north)--(0,2.6)--(4,2.6)--(n3.north);\n\\node[anchor=south] at (2,2.8) {closure permits $0\\to3$};\n\\node[anchor=west] at (5,1.8) {$K^2=0$};\n\\node[anchor=west] at (5,1) {$A^{-1}=I-K$};\n\\node[anchor=west] at (5,.2) {$(A^{-1})_{30}=0$};\n\\end{tikzpicture}","ick] (n0)--node[above left] {$1$}(n1);\n\\draw[->,thick] (n0)--node[below left] {$1$}(n2);\n\\draw[->,thick] (n1)--node[above right] {$1$}(n3);\n\\draw[->,thick] (n2)--node[below right] {$-1$}(n3);\n\\draw[->,dashed,rounded corners=5pt] (n0.north)--(0,2.6)--(4,2.6)--(n3.north);\n\\node[anchor=south] at (2,2.8) {closure permits $0\\to3$};\n\\node[anchor=west] at (5,1.8) {$K^2=0$};\n\\node[anchor=west] at (5,1) {$A^{-1}=I-K$};\n\\node[anchor=west] at (5,.2) {$(A^{-1})_{30}=0$};\n\\end{tikzpicture}","->,thick] (n0)--node[below left] {$1$}(n2);\n\\draw[->,thick] (n1)--node[above right] {$1$}(n3);\n\\draw[->,thick] (n2)--node[below right] {$-1$}(n3);\n\\draw[->,dashed,rounded corners=5pt] (n0.north)--(0,2.6)--(4,2.6)--(n3.north);\n\\node[anchor=south] at (2,2.8) {closure permits $0\\to3$};\n\\node[anchor=west] at (5,1.8) {$K^2=0$};\n\\node[anchor=west] at (5,1) {$A^{-1}=I-K$};\n\\node[anchor=west] at (5,.2) {$(A^{-1})_{30}=0$};\n\\end{tikzpicture}","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.5,4.3) {$X$};\n\\path[draw=black!30,fill=black!8] (0.55,2.65) rectangle (1.5,3.6);\n\\node[font=\\small,align=center,font=\\small] at (1.025,3.125) {$1$};\n\\path[draw=black!30,fill=black!8] (1.5,2.65) rectangle (2.45,3.6);\n\\node[font=\\small,align=center,font=\\small] at (1.975,3.125) {$2$};\n\\path[draw=black!30,fill=black!8] (0.55,1.7) rectangle (1.5,2.65);\n\\node[font=\\small,align=center,font=\\small] at (1.025,2.175) {$3$};\n\\path[draw=black!30,fill=black!8] (1.5,1.7) rectangle (2.45,2.65);\n\\node[font=\\small,align=center,font=\\small] at (1.975,2.175) {$4$};\n\\node[font=\\small,align=center,font=\\small] at (4.46,4.65) {$\\operatorname{vec}(X)$};\n\\path[draw=black!30,fill=black!8] (4.1,3.23) rectangle (4.82,3.95);\n\\n","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.5,4.3) {$X$};\n\\path[draw=black!30,fill=black!8] (0.55,2.65) rectangle (1.5,3.6);\n\\node[font=\\small,align=center,font=\\small] at (1.025,3.125) {$1$};\n\\path[draw=black!30,fill=black!8] (1.5,2.65) rectangle (2.45,3.6);\n\\node[font=\\small,align=center,font=\\small] at (1.975,3.125) {$2$};\n\\path[draw=black!30,fill=black!8] (0.55,1.7) rectangle (1.5,2.65);\n\\node[font=\\small,align=center,font=\\small] at (1.025,2.175) {$3$};\n\\path[draw=black!30,fill=black!8] (1.5,1.7) rectangle (2.45,2.65);\n\\node[font=\\small,align=center,font=\\small] at (1.975,2.175) {$4$};\n\\node[font=\\small,align=center,font=\\small] at (4.46,4.65) {$\\operatorname{vec}(X)$};\n\\path[draw=black!30,fill=black!8] (4.1,3.23) rectangle (4.82,3.95);\n\\node[font=\\small,align=center,font=\\small] at (4.46,3.59) {$1$};\n\\path[draw=black!30,fill=black!8] (4.1,2.51) rectangle (4.82,3.23);\n\\no","ine join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.5,4.3) {$X$};\n\\path[draw=black!30,fill=black!8] (0.55,2.65) rectangle (1.5,3.6);\n\\node[font=\\small,align=center,font=\\small] at (1.025,3.125) {$1$};\n\\path[draw=black!30,fill=black!8] (1.5,2.65) rectangle (2.45,3.6);\n\\node[font=\\small,align=center,font=\\small] at (1.975,3.125) {$2$};\n\\path[draw=black!30,fill=black!8] (0.55,1.7) rectangle (1.5,2.65);\n\\node[font=\\small,align=center,font=\\small] at (1.025,2.175) {$3$};\n\\path[draw=black!30,fill=black!8] (1.5,1.7) rectangle (2.45,2.65);\n\\node[font=\\small,align=center,font=\\small] at (1.975,2.175) {$4$};\n\\node[font=\\small,align=center,font=\\small] at (4.46,4.65) {$\\operatorname{vec}(X)$};\n\\path[draw=black!30,fill=black!8] (4.1,3.23) rectangle (4.82,3.95);\n\\node[font=\\small,align=center,font=\\small] at (4.46,3.59) {$1$};\n\\path[draw=black!30,fill=black!8] (4.1,2.51) rectangle (4.82,3.23);\n\\node[font=\\small,align=center,font=\\small] at (4.46,2.87) {$3$};\n\\path[draw=black!30,fill=black!8] (4.1,1.79) rectangle (4.82,2.51);\n\\no…","\\path[draw=black!30,fill=black!8] (0.55,2.65) rectangle (1.5,3.6);\n\\node[font=\\small,align=center,font=\\small] at (1.025,3.125) {$1$};\n\\path[draw=black!30,fill=black!8] (1.5,2.65) rectangle (2.45,3.6);\n\\node[font=\\small,align=center,font=\\small] at (1.975,3.125) {$2$};\n\\path[draw=black!30,fill=black!8] (0.55,1.7) rectangle (1.5,2.65);\n\\node[font=\\small,align=center,font=\\small] at (1.025,2.175) {$3$};\n\\path[draw=black!30,fill=black!8] (1.5,1.7) rectangle (2.45,2.65);\n\\node[font=\\small,align=center,font=\\small] at (1.975,2.175) {$4$};\n\\node[font=\\small,align=center,font=\\small] at (4.46,4.65) {$\\operatorname{vec}(X)$};\n\\path[draw=black!30,fill=black!8] (4.1,3.23) rectangle (4.82,3.95);\n\\node[font=\\small,align=center,font=\\small] at (4.46,3.59) {$1$};\n\\path[draw=black!30,fill=black!8] (4.1,2.51) rectangle (4.82,3.23);\n\\node[font=\\small,align=center,font=\\small] at (4.46,2.87) {$3$};\n\\path[draw=black!30,fill=black!8] (4.1,1.79) rectangle (4.82,2.51);\n\\node[font=\\small,align=center,font=\\small] at (4.46,2.15) {$2$};\n\\path[draw=black!30,fill=black!8] (4.1,1.07) rectangle (4.82,1.79);\n\\nod…","\\path[draw=black!30,fill=black!8] (1.5,2.65) rectangle (2.45,3.6);\n\\node[font=\\small,align=center,font=\\small] at (1.975,3.125) {$2$};\n\\path[draw=black!30,fill=black!8] (0.55,1.7) rectangle (1.5,2.65);\n\\node[font=\\small,align=center,font=\\small] at (1.025,2.175) {$3$};\n\\path[draw=black!30,fill=black!8] (1.5,1.7) rectangle (2.45,2.65);\n\\node[font=\\small,align=center,font=\\small] at (1.975,2.175) {$4$};\n\\node[font=\\small,align=center,font=\\small] at (4.46,4.65) {$\\operatorname{vec}(X)$};\n\\path[draw=black!30,fill=black!8] (4.1,3.23) rectangle (4.82,3.95);\n\\node[font=\\small,align=center,font=\\small] at (4.46,3.59) {$1$};\n\\path[draw=black!30,fill=black!8] (4.1,2.51) rectangle (4.82,3.23);\n\\node[font=\\small,align=center,font=\\small] at (4.46,2.87) {$3$};\n\\path[draw=black!30,fill=black!8] (4.1,1.79) rectangle (4.82,2.51);\n\\node[font=\\small,align=center,font=\\small] at (4.46,2.15) {$2$};\n\\path[draw=black!30,fill=black!8] (4.1,1.07) rectangle (4.82,1.79);\n\\node[font=\\small,align=center,font=\\small] at (4.46,1.43) {$4$};\n\\node[font=\\small,align=center,font=\\small] at (8.7,4.3) {$BXA^T$};\n\\path…","\\node[font=\\small,align=center,font=\\small] at (1.975,3.125) {$2$};\n\\path[draw=black!30,fill=black!8] (0.55,1.7) rectangle (1.5,2.65);\n\\node[font=\\small,align=center,font=\\small] at (1.025,2.175) {$3$};\n\\path[draw=black!30,fill=black!8] (1.5,1.7) rectangle (2.45,2.65);\n\\node[font=\\small,align=center,font=\\small] at (1.975,2.175) {$4$};\n\\node[font=\\small,align=center,font=\\small] at (4.46,4.65) {$\\operatorname{vec}(X)$};\n\\path[draw=black!30,fill=black!8] (4.1,3.23) rectangle (4.82,3.95);\n\\node[font=\\small,align=center,font=\\small] at (4.46,3.59) {$1$};\n\\path[draw=black!30,fill=black!8] (4.1,2.51) rectangle (4.82,3.23);\n\\node[font=\\small,align=center,font=\\small] at (4.46,2.87) {$3$};\n\\path[draw=black!30,fill=black!8] (4.1,1.79) rectangle (4.82,2.51);\n\\node[font=\\small,align=center,font=\\small] at (4.46,2.15) {$2$};\n\\path[draw=black!30,fill=black!8] (4.1,1.07) rectangle (4.82,1.79);\n\\node[font=\\small,align=center,font=\\small] at (4.46,1.43) {$4$};\n\\node[font=\\small,align=center,font=\\small] at (8.7,4.3) {$BXA^T$};\n\\path[draw=black!30,fill=black!8] (7.2,2.7) rectangle (8.7,3.6);\n\\node[f…","align=center,font=\\small] at (1.025,2.175) {$3$};\n\\path[draw=black!30,fill=black!8] (1.5,1.7) rectangle (2.45,2.65);\n\\node[font=\\small,align=center,font=\\small] at (1.975,2.175) {$4$};\n\\node[font=\\small,align=center,font=\\small] at (4.46,4.65) {$\\operatorname{vec}(X)$};\n\\path[draw=black!30,fill=black!8] (4.1,3.23) rectangle (4.82,3.95);\n\\node[font=\\small,align=center,font=\\small] at (4.46,3.59) {$1$};\n\\path[draw=black!30,fill=black!8] (4.1,2.51) rectangle (4.82,3.23);\n\\node[font=\\small,align=center,font=\\small] at (4.46,2.87) {$3$};\n\\path[draw=black!30,fill=black!8] (4.1,1.79) rectangle (4.82,2.51);\n\\node[font=\\small,align=center,font=\\small] at (4.46,2.15) {$2$};\n\\path[draw=black!30,fill=black!8] (4.1,1.07) rectangle (4.82,1.79);\n\\node[font=\\small,align=center,font=\\small] at (4.46,1.43) {$4$};\n\\node[font=\\small,align=center,font=\\small] at (8.7,4.3) {$BXA^T$};\n\\path[draw=black!30,fill=black!8] (7.2,2.7) rectangle (8.7,3.6);\n\\node[font=\\small,align=center,font=\\small] at (7.95,3.15) {$5 + 8 i$};\n\\path[draw=black!30,fill=black!8] (8.7,2.7) rectangle (10.2,3.6);\n\\node[font=\\small,alig…",",align=center,font=\\small] at (1.975,2.175) {$4$};\n\\node[font=\\small,align=center,font=\\small] at (4.46,4.65) {$\\operatorname{vec}(X)$};\n\\path[draw=black!30,fill=black!8] (4.1,3.23) rectangle (4.82,3.95);\n\\node[font=\\small,align=center,font=\\small] at (4.46,3.59) {$1$};\n\\path[draw=black!30,fill=black!8] (4.1,2.51) rectangle (4.82,3.23);\n\\node[font=\\small,align=center,font=\\small] at (4.46,2.87) {$3$};\n\\path[draw=black!30,fill=black!8] (4.1,1.79) rectangle (4.82,2.51);\n\\node[font=\\small,align=center,font=\\small] at (4.46,2.15) {$2$};\n\\path[draw=black!30,fill=black!8] (4.1,1.07) rectangle (4.82,1.79);\n\\node[font=\\small,align=center,font=\\small] at (4.46,1.43) {$4$};\n\\node[font=\\small,align=center,font=\\small] at (8.7,4.3) {$BXA^T$};\n\\path[draw=black!30,fill=black!8] (7.2,2.7) rectangle (8.7,3.6);\n\\node[font=\\small,align=center,font=\\small] at (7.95,3.15) {$5 + 8 i$};\n\\path[draw=black!30,fill=black!8] (8.7,2.7) rectangle (10.2,3.6);\n\\node[font=\\small,align=center,font=\\small] at (9.45,3.15) {$2$};\n\\path[draw=black!30,fill=black!8] (7.2,1.8) rectangle (8.7,2.7);\n\\node[font=\\small,align=c…","};\n\\path[draw=black!30,fill=black!8] (4.1,3.23) rectangle (4.82,3.95);\n\\node[font=\\small,align=center,font=\\small] at (4.46,3.59) {$1$};\n\\path[draw=black!30,fill=black!8] (4.1,2.51) rectangle (4.82,3.23);\n\\node[font=\\small,align=center,font=\\small] at (4.46,2.87) {$3$};\n\\path[draw=black!30,fill=black!8] (4.1,1.79) rectangle (4.82,2.51);\n\\node[font=\\small,align=center,font=\\small] at (4.46,2.15) {$2$};\n\\path[draw=black!30,fill=black!8] (4.1,1.07) rectangle (4.82,1.79);\n\\node[font=\\small,align=center,font=\\small] at (4.46,1.43) {$4$};\n\\node[font=\\small,align=center,font=\\small] at (8.7,4.3) {$BXA^T$};\n\\path[draw=black!30,fill=black!8] (7.2,2.7) rectangle (8.7,3.6);\n\\node[font=\\small,align=center,font=\\small] at (7.95,3.15) {$5 + 8 i$};\n\\path[draw=black!30,fill=black!8] (8.7,2.7) rectangle (10.2,3.6);\n\\node[font=\\small,align=center,font=\\small] at (9.45,3.15) {$2$};\n\\path[draw=black!30,fill=black!8] (7.2,1.8) rectangle (8.7,2.7);\n\\node[font=\\small,align=center,font=\\small] at (7.95,2.25) {$9 + 12 i$};\n\\path[draw=black!30,fill=black!8] (8.7,1.8) rectangle (10.2,2.7);\n\\node[font=\\small,al…","};\n\\path[draw=black!30,fill=black!8] (4.1,2.51) rectangle (4.82,3.23);\n\\node[font=\\small,align=center,font=\\small] at (4.46,2.87) {$3$};\n\\path[draw=black!30,fill=black!8] (4.1,1.79) rectangle (4.82,2.51);\n\\node[font=\\small,align=center,font=\\small] at (4.46,2.15) {$2$};\n\\path[draw=black!30,fill=black!8] (4.1,1.07) rectangle (4.82,1.79);\n\\node[font=\\small,align=center,font=\\small] at (4.46,1.43) {$4$};\n\\node[font=\\small,align=center,font=\\small] at (8.7,4.3) {$BXA^T$};\n\\path[draw=black!30,fill=black!8] (7.2,2.7) rectangle (8.7,3.6);\n\\node[font=\\small,align=center,font=\\small] at (7.95,3.15) {$5 + 8 i$};\n\\path[draw=black!30,fill=black!8] (8.7,2.7) rectangle (10.2,3.6);\n\\node[font=\\small,align=center,font=\\small] at (9.45,3.15) {$2$};\n\\path[draw=black!30,fill=black!8] (7.2,1.8) rectangle (8.7,2.7);\n\\node[font=\\small,align=center,font=\\small] at (7.95,2.25) {$9 + 12 i$};\n\\path[draw=black!30,fill=black!8] (8.7,1.8) rectangle (10.2,2.7);\n\\node[font=\\small,align=center,font=\\small] at (9.45,2.25) {$6$};\n\\node[font=\\small,align=center,font=\\small] at (12.2,3.25) {$A^T$","3.23);\n\\node[font=\\small,align=center,font=\\small] at (4.46,2.87) {$3$};\n\\path[draw=black!30,fill=black!8] (4.1,1.79) rectangle (4.82,2.51);\n\\node[font=\\small,align=center,font=\\small] at (4.46,2.15) {$2$};\n\\path[draw=black!30,fill=black!8] (4.1,1.07) rectangle (4.82,1.79);\n\\node[font=\\small,align=center,font=\\small] at (4.46,1.43) {$4$};\n\\node[font=\\small,align=center,font=\\small] at (8.7,4.3) {$BXA^T$};\n\\path[draw=black!30,fill=black!8] (7.2,2.7) rectangle (8.7,3.6);\n\\node[font=\\small,align=center,font=\\small] at (7.95,3.15) {$5 + 8 i$};\n\\path[draw=black!30,fill=black!8] (8.7,2.7) rectangle (10.2,3.6);\n\\node[font=\\small,align=center,font=\\small] at (9.45,3.15) {$2$};\n\\path[draw=black!30,fill=black!8] (7.2,1.8) rectangle (8.7,2.7);\n\\node[font=\\small,align=center,font=\\small] at (7.95,2.25) {$9 + 12 i$};\n\\path[draw=black!30,fill=black!8] (8.7,1.8) rectangle (10.2,2.7);\n\\node[font=\\small,align=center,font=\\small] at (9.45,2.25) {$6$};\n\\node[font=\\small,align=center,font=\\small] at (12.2,3.25) {$A^T$",".51);\n\\node[font=\\small,align=center,font=\\small] at (4.46,2.15) {$2$};\n\\path[draw=black!30,fill=black!8] (4.1,1.07) rectangle (4.82,1.79);\n\\node[font=\\small,align=center,font=\\small] at (4.46,1.43) {$4$};\n\\node[font=\\small,align=center,font=\\small] at (8.7,4.3) {$BXA^T$};\n\\path[draw=black!30,fill=black!8] (7.2,2.7) rectangle (8.7,3.6);\n\\node[font=\\small,align=center,font=\\small] at (7.95,3.15) {$5 + 8 i$};\n\\path[draw=black!30,fill=black!8] (8.7,2.7) rectangle (10.2,3.6);\n\\node[font=\\small,align=center,font=\\small] at (9.45,3.15) {$2$};\n\\path[draw=black!30,fill=black!8] (7.2,1.8) rectangle (8.7,2.7);\n\\node[font=\\small,align=center,font=\\small] at (7.95,2.25) {$9 + 12 i$};\n\\path[draw=black!30,fill=black!8] (8.7,1.8) rectangle (10.2,2.7);\n\\node[font=\\small,align=center,font=\\small] at (9.45,2.25) {$6$};\n\\node[font=\\small,align=center,font=\\small] at (12.2,3.25) {$A^T$",";\n\\node[font=\\small,align=center,font=\\small] at (4.46,1.43) {$4$};\n\\node[font=\\small,align=center,font=\\small] at (8.7,4.3) {$BXA^T$};\n\\path[draw=black!30,fill=black!8] (7.2,2.7) rectangle (8.7,3.6);\n\\node[font=\\small,align=center,font=\\small] at (7.95,3.15) {$5 + 8 i$};\n\\path[draw=black!30,fill=black!8] (8.7,2.7) rectangle (10.2,3.6);\n\\node[font=\\small,align=center,font=\\small] at (9.45,3.15) {$2$};\n\\path[draw=black!30,fill=black!8] (7.2,1.8) rectangle (8.7,2.7);\n\\node[font=\\small,align=center,font=\\small] at (7.95,2.25) {$9 + 12 i$};\n\\path[draw=black!30,fill=black!8] (8.7,1.8) rectangle (10.2,2.7);\n\\node[font=\\small,align=center,font=\\small] at (9.45,2.25) {$6$};\n\\node[font=\\small,align=center,font=\\small] at (12.2,3.25) {$A^T$","T$};\n\\path[draw=black!30,fill=black!8] (7.2,2.7) rectangle (8.7,3.6);\n\\node[font=\\small,align=center,font=\\small] at (7.95,3.15) {$5 + 8 i$};\n\\path[draw=black!30,fill=black!8] (8.7,2.7) rectangle (10.2,3.6);\n\\node[font=\\small,align=center,font=\\small] at (9.45,3.15) {$2$};\n\\path[draw=black!30,fill=black!8] (7.2,1.8) rectangle (8.7,2.7);\n\\node[font=\\small,align=center,font=\\small] at (7.95,2.25) {$9 + 12 i$};\n\\path[draw=black!30,fill=black!8] (8.7,1.8) rectangle (10.2,2.7);\n\\node[font=\\small,align=center,font=\\small] at (9.45,2.25) {$6$};\n\\node[font=\\small,align=center,font=\\small] at (12.2,3.25) {$A^T$","};\n\\path[draw=black!30,fill=black!8] (8.7,2.7) rectangle (10.2,3.6);\n\\node[font=\\small,align=center,font=\\small] at (9.45,3.15) {$2$};\n\\path[draw=black!30,fill=black!8] (7.2,1.8) rectangle (8.7,2.7);\n\\node[font=\\small,align=center,font=\\small] at (7.95,2.25) {$9 + 12 i$};\n\\path[draw=black!30,fill=black!8] (8.7,1.8) rectangle (10.2,2.7);\n\\node[font=\\small,align=center,font=\\small] at (9.45,2.25) {$6$};\n\\node[font=\\small,align=center,font=\\small] at (12.2,3.25) {$A^T$",");\n\\node[font=\\small,align=center,font=\\small] at (9.45,3.15) {$2$};\n\\path[draw=black!30,fill=black!8] (7.2,1.8) rectangle (8.7,2.7);\n\\node[font=\\small,align=center,font=\\small] at (7.95,2.25) {$9 + 12 i$};\n\\path[draw=black!30,fill=black!8] (8.7,1.8) rectangle (10.2,2.7);\n\\node[font=\\small,align=center,font=\\small] at (9.45,2.25) {$6$};\n\\node[font=\\small,align=center,font=\\small] at (12.2,3.25) {$A^T$","not $A^*$};\n\\node[font=\\small,align=center] at (7,0.55) {$(A\\otimes B)\\operatorname{vec}(X)=\\operatorname{vec}(BXA^T)$};\n\\end{tikzpicture}","};\n\\draw[->,line width=.6pt] (0.95,0.75) -- (5.72,0.75);\n\\draw[->,line width=.6pt] (0.95,0.75) -- (0.95,3.97);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.275,0.2) {packet 1};\n\\node[font=\\small,align=center,anchor=west,font=\\scriptsize] at (0.95,4.25) {packet 2};\n\\draw[line width=1.1pt] plot coordinates {(1.26,0.956667) (5.29,3.643333)};\n\\node[font=\\small,align=center] at (3.25,4.25) {$A=I_2,\\ v=(1,1)^T$};\n\\node[font=\\small,align=center] at (9.5,3.3) {$p(A)v=p(1)(1,1)^T$};\n\\node[font=\\small,align=center] at (9.5,2.1) {$\\dim\\mathcal K_m(A,v)=1$};\n\\node[font=\\small,align=center] at (9.5,0.9) {Independent local spans have dimension $2$.};\n\\end{tikzpicture}","ne width=.6pt] (0.95,0.75) -- (0.95,3.97);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.275,0.2) {packet 1};\n\\node[font=\\small,align=center,anchor=west,font=\\scriptsize] at (0.95,4.25) {packet 2};\n\\draw[line width=1.1pt] plot coordinates {(1.26,0.956667) (5.29,3.643333)};\n\\node[font=\\small,align=center] at (3.25,4.25) {$A=I_2,\\ v=(1,1)^T$};\n\\node[font=\\small,align=center] at (9.5,3.3) {$p(A)v=p(1)(1,1)^T$};\n\\node[font=\\small,align=center] at (9.5,2.1) {$\\dim\\mathcal K_m(A,v)=1$};\n\\node[font=\\small,align=center] at (9.5,0.9) {Independent local spans have dimension $2$.};\n\\end{tikzpicture}","enter,font=\\scriptsize] at (3.275,0.2) {packet 1};\n\\node[font=\\small,align=center,anchor=west,font=\\scriptsize] at (0.95,4.25) {packet 2};\n\\draw[line width=1.1pt] plot coordinates {(1.26,0.956667) (5.29,3.643333)};\n\\node[font=\\small,align=center] at (3.25,4.25) {$A=I_2,\\ v=(1,1)^T$};\n\\node[font=\\small,align=center] at (9.5,3.3) {$p(A)v=p(1)(1,1)^T$};\n\\node[font=\\small,align=center] at (9.5,2.1) {$\\dim\\mathcal K_m(A,v)=1$};\n\\node[font=\\small,align=center] at (9.5,0.9) {Independent local spans have dimension $2$.};\n\\end{tikzpicture}","t (0.95,4.25) {packet 2};\n\\draw[line width=1.1pt] plot coordinates {(1.26,0.956667) (5.29,3.643333)};\n\\node[font=\\small,align=center] at (3.25,4.25) {$A=I_2,\\ v=(1,1)^T$};\n\\node[font=\\small,align=center] at (9.5,3.3) {$p(A)v=p(1)(1,1)^T$};\n\\node[font=\\small,align=center] at (9.5,2.1) {$\\dim\\mathcal K_m(A,v)=1$};\n\\node[font=\\small,align=center] at (9.5,0.9) {Independent local spans have dimension $2$.};\n\\end{tikzpicture}","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center] at (1.5,4) {$a0$};\n\\path[draw,fill=black!14] (0.8,2.4) rectangle (2.2,3.1);\n\\node[font=\\small,align=center] at (1.5,2.75) {2};\n\\path[draw,fill=black!14] (0.8,1.25) rectangle (2.2,1.95);\n\\node[font=\\small,align=center] at (1.5,1.6) {3};\n\\node[font=\\small,align=center] at (3.4,4) {$a1$};\n\\path[draw,fill=white] (2.7,2.4) rectangle (4.1,3.1);\n\\node[font=\\small,align=center] at (3.4,2.75) {2};\n\\path[draw,fill=white] (2.7,1.25) rectangle (4.1,1.95);\n\\node[font=\\small,align=center] at (3.4,1.6) {2};\n\\node[font=\\small","line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center] at (1.5,4) {$a0$};\n\\path[draw,fill=black!14] (0.8,2.4) rectangle (2.2,3.1);\n\\node[font=\\small,align=center] at (1.5,2.75) {2};\n\\path[draw,fill=black!14] (0.8,1.25) rectangle (2.2,1.95);\n\\node[font=\\small,align=center] at (1.5,1.6) {3};\n\\node[font=\\small,align=center] at (3.4,4) {$a1$};\n\\path[draw,fill=white] (2.7,2.4) rectangle (4.1,3.1);\n\\node[font=\\small,align=center] at (3.4,2.75) {2};\n\\path[draw,fill=white] (2.7,1.25) rectangle (4.1,1.95);\n\\node[font=\\small,align=center] at (3.4,1.6) {2};\n\\node[font=\\small,align=center] at (5.3,4) {$a$};\n\\path[draw,fill=black!14] (4.6,2.4) rectangle (6,3.1);\n\\node[font=\\small,align=center] at (5.3,2.75) {4};\n\\path[draw,fill=black!14] (4.6,1.25) rectangle (6,1.95);\n\\node[font=\\small,align=center] at (5.3,1.6) {5};\n\\node[font=\\small,alig","black!14] (0.8,1.25) rectangle (2.2,1.95);\n\\node[font=\\small,align=center] at (1.5,1.6) {3};\n\\node[font=\\small,align=center] at (3.4,4) {$a1$};\n\\path[draw,fill=white] (2.7,2.4) rectangle (4.1,3.1);\n\\node[font=\\small,align=center] at (3.4,2.75) {2};\n\\path[draw,fill=white] (2.7,1.25) rectangle (4.1,1.95);\n\\node[font=\\small,align=center] at (3.4,1.6) {2};\n\\node[font=\\small,align=center] at (5.3,4) {$a$};\n\\path[draw,fill=black!14] (4.6,2.4) rectangle (6,3.1);\n\\node[font=\\small,align=center] at (5.3,2.75) {4};\n\\path[draw,fill=black!14] (4.6,1.25) rectangle (6,1.95);\n\\node[font=\\small,align=center] at (5.3,1.6) {5};\n\\node[font=\\small,align=center] at (7.2,4) {$b0$};\n\\path[draw,fill=white] (6.5,2.4) rectangle (7.9,3.1);\n\\node[font=\\small,align=center] at (7.2,2.75) {2};\n\\path[draw,fill=white] (6.5,1.25) rectangle (7.9,1.95);\n\\node[font=\\small,align=center] at (7.2,1.6) {2};\n\\node[font=\\small,ali","l=white] (2.7,1.25) rectangle (4.1,1.95);\n\\node[font=\\small,align=center] at (3.4,1.6) {2};\n\\node[font=\\small,align=center] at (5.3,4) {$a$};\n\\path[draw,fill=black!14] (4.6,2.4) rectangle (6,3.1);\n\\node[font=\\small,align=center] at (5.3,2.75) {4};\n\\path[draw,fill=black!14] (4.6,1.25) rectangle (6,1.95);\n\\node[font=\\small,align=center] at (5.3,1.6) {5};\n\\node[font=\\small,align=center] at (7.2,4) {$b0$};\n\\path[draw,fill=white] (6.5,2.4) rectangle (7.9,3.1);\n\\node[font=\\small,align=center] at (7.2,2.75) {2};\n\\path[draw,fill=white] (6.5,1.25) rectangle (7.9,1.95);\n\\node[font=\\small,align=center] at (7.2,1.6) {2};\n\\node[font=\\small,align=center] at (9.1,4) {$b1$};\n\\path[draw,fill=white] (8.4,2.4) rectangle (9.8,3.1);\n\\node[font=\\small,align=center] at (9.1,2.75) {3};\n\\path[draw,fill=white] (8.4,1.25) rectangle (9.8,1.95);\n\\node[font=\\small,align=center] at (9.1,1.6) {3};\n\\node[font=\\small,align","l=black!14] (4.6,1.25) rectangle (6,1.95);\n\\node[font=\\small,align=center] at (5.3,1.6) {5};\n\\node[font=\\small,align=center] at (7.2,4) {$b0$};\n\\path[draw,fill=white] (6.5,2.4) rectangle (7.9,3.1);\n\\node[font=\\small,align=center] at (7.2,2.75) {2};\n\\path[draw,fill=white] (6.5,1.25) rectangle (7.9,1.95);\n\\node[font=\\small,align=center] at (7.2,1.6) {2};\n\\node[font=\\small,align=center] at (9.1,4) {$b1$};\n\\path[draw,fill=white] (8.4,2.4) rectangle (9.8,3.1);\n\\node[font=\\small,align=center] at (9.1,2.75) {3};\n\\path[draw,fill=white] (8.4,1.25) rectangle (9.8,1.95);\n\\node[font=\\small,align=center] at (9.1,1.6) {3};\n\\node[font=\\small,align=center] at (11,4) {$b$};\n\\path[draw,fill=white] (10.3,2.4) rectangle (11.7,3.1);\n\\node[font=\\small,align=center] at (11,2.75) {5};\n\\path[draw,fill=white] (10.3,1.25) rectangle (11.7,1.95);\n\\node[font=\\small,align=center] at (11,1.6) {5};\n\\node[font=\\small,align","ill=white] (6.5,1.25) rectangle (7.9,1.95);\n\\node[font=\\small,align=center] at (7.2,1.6) {2};\n\\node[font=\\small,align=center] at (9.1,4) {$b1$};\n\\path[draw,fill=white] (8.4,2.4) rectangle (9.8,3.1);\n\\node[font=\\small,align=center] at (9.1,2.75) {3};\n\\path[draw,fill=white] (8.4,1.25) rectangle (9.8,1.95);\n\\node[font=\\small,align=center] at (9.1,1.6) {3};\n\\node[font=\\small,align=center] at (11,4) {$b$};\n\\path[draw,fill=white] (10.3,2.4) rectangle (11.7,3.1);\n\\node[font=\\small,align=center] at (11,2.75) {5};\n\\path[draw,fill=white] (10.3,1.25) rectangle (11.7,1.95);\n\\node[font=\\small,align=center] at (11,1.6) {5};\n\\node[font=\\small,align=center] at (12.9,4) {$r$};\n\\path[draw,fill=black!14] (12.2,2.4) rectangle (13.6,3.1);\n\\node[font=\\small,align=center] at (12.9,2.75) {10};\n\\path[draw,fill=black!14] (12.2,1.25) rectangle (13.6,1.95);\n\\node[font=\\small,align=center] at (12.9,1.6) {11};\n\\node[f","l=white] (8.4,1.25) rectangle (9.8,1.95);\n\\node[font=\\small,align=center] at (9.1,1.6) {3};\n\\node[font=\\small,align=center] at (11,4) {$b$};\n\\path[draw,fill=white] (10.3,2.4) rectangle (11.7,3.1);\n\\node[font=\\small,align=center] at (11,2.75) {5};\n\\path[draw,fill=white] (10.3,1.25) rectangle (11.7,1.95);\n\\node[font=\\small,align=center] at (11,1.6) {5};\n\\node[font=\\small,align=center] at (12.9,4) {$r$};\n\\path[draw,fill=black!14] (12.2,2.4) rectangle (13.6,3.1);\n\\node[font=\\small,align=center] at (12.9,2.75) {10};\n\\path[draw,fill=black!14] (12.2,1.25) rectangle (13.6,1.95);\n\\node[font=\\small,align=center] at (12.9,1.6) {11};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.45) {Top: old values. Bottom: new values. Only a0, a, and r change.};\n\\end{tikzpicture}","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.73,4.5) {$LR^T$};\n\\path[draw=black!30,fill=black!8] (0.65,3.08) rectangle (1.37,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.01,3.44) {$3$};\n\\path[draw=black!30,fill=white] (1.37,3.08) rectangle (2.09,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.73,3.44) {$0$};\n\\path[draw=black!30,fill=white] (2.09,3.08) rectangle (2.81,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.45,3.44) {$0$};\n\\path[draw=black!30,fill=black!8] (0.65,2.36) rectangle (1.37,3.08);\n\\node[font=\\small,align=cente","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.73,4.5) {$LR^T$};\n\\path[draw=black!30,fill=black!8] (0.65,3.08) rectangle (1.37,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.01,3.44) {$3$};\n\\path[draw=black!30,fill=white] (1.37,3.08) rectangle (2.09,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.73,3.44) {$0$};\n\\path[draw=black!30,fill=white] (2.09,3.08) rectangle (2.81,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.45,3.44) {$0$};\n\\path[draw=black!30,fill=black!8] (0.65,2.36) rectangle (1.37,3.08);\n\\node[font=\\small,align=center,font=\\small] at (1.01,2.72) {$4$};\n\\path[draw=black!30,fill=white] (1.37,2.36) rectangle (2.09,3.08);\n\\node[font=\\small,align=center","ine join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.73,4.5) {$LR^T$};\n\\path[draw=black!30,fill=black!8] (0.65,3.08) rectangle (1.37,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.01,3.44) {$3$};\n\\path[draw=black!30,fill=white] (1.37,3.08) rectangle (2.09,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.73,3.44) {$0$};\n\\path[draw=black!30,fill=white] (2.09,3.08) rectangle (2.81,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.45,3.44) {$0$};\n\\path[draw=black!30,fill=black!8] (0.65,2.36) rectangle (1.37,3.08);\n\\node[font=\\small,align=center,font=\\small] at (1.01,2.72) {$4$};\n\\path[draw=black!30,fill=white] (1.37,2.36) rectangle (2.09,3.08);\n\\node[font=\\small,align=center,font=\\small] at (1.73,2.72) {$0$};\n\\path[draw=black!30,fill=white] (2.09,2.36) rectangle (2.81,3.08);\n\\node[font=\\small,align=cente","R^T$};\n\\path[draw=black!30,fill=black!8] (0.65,3.08) rectangle (1.37,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.01,3.44) {$3$};\n\\path[draw=black!30,fill=white] (1.37,3.08) rectangle (2.09,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.73,3.44) {$0$};\n\\path[draw=black!30,fill=white] (2.09,3.08) rectangle (2.81,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.45,3.44) {$0$};\n\\path[draw=black!30,fill=black!8] (0.65,2.36) rectangle (1.37,3.08);\n\\node[font=\\small,align=center,font=\\small] at (1.01,2.72) {$4$};\n\\path[draw=black!30,fill=white] (1.37,2.36) rectangle (2.09,3.08);\n\\node[font=\\small,align=center,font=\\small] at (1.73,2.72) {$0$};\n\\path[draw=black!30,fill=white] (2.09,2.36) rectangle (2.81,3.08);\n\\node[font=\\small,align=center,font=\\small] at (2.45,2.72) {$0$};\n\\path[draw=black!30,fill=white] (0.65,1.64) rectangle (1.37,2.36);\n\\node[font=\\small,align=cent","$3$};\n\\path[draw=black!30,fill=white] (1.37,3.08) rectangle (2.09,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.73,3.44) {$0$};\n\\path[draw=black!30,fill=white] (2.09,3.08) rectangle (2.81,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.45,3.44) {$0$};\n\\path[draw=black!30,fill=black!8] (0.65,2.36) rectangle (1.37,3.08);\n\\node[font=\\small,align=center,font=\\small] at (1.01,2.72) {$4$};\n\\path[draw=black!30,fill=white] (1.37,2.36) rectangle (2.09,3.08);\n\\node[font=\\small,align=center,font=\\small] at (1.73,2.72) {$0$};\n\\path[draw=black!30,fill=white] (2.09,2.36) rectangle (2.81,3.08);\n\\node[font=\\small,align=center,font=\\small] at (2.45,2.72) {$0$};\n\\path[draw=black!30,fill=white] (0.65,1.64) rectangle (1.37,2.36);\n\\node[font=\\small,align=center,font=\\small] at (1.01,2) {$0$};\n\\path[draw=black!30,fill=black!8] (1.37,1.64) rectangle (2.09,2.36);\n\\node[font=\\small,align=center,","0$};\n\\path[draw=black!30,fill=white] (2.09,3.08) rectangle (2.81,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.45,3.44) {$0$};\n\\path[draw=black!30,fill=black!8] (0.65,2.36) rectangle (1.37,3.08);\n\\node[font=\\small,align=center,font=\\small] at (1.01,2.72) {$4$};\n\\path[draw=black!30,fill=white] (1.37,2.36) rectangle (2.09,3.08);\n\\node[font=\\small,align=center,font=\\small] at (1.73,2.72) {$0$};\n\\path[draw=black!30,fill=white] (2.09,2.36) rectangle (2.81,3.08);\n\\node[font=\\small,align=center,font=\\small] at (2.45,2.72) {$0$};\n\\path[draw=black!30,fill=white] (0.65,1.64) rectangle (1.37,2.36);\n\\node[font=\\small,align=center,font=\\small] at (1.01,2) {$0$};\n\\path[draw=black!30,fill=black!8] (1.37,1.64) rectangle (2.09,2.36);\n\\node[font=\\small,align=center,font=\\small] at (1.73,2) {$2$};\n\\path[draw=black!30,fill=white] (2.09,1.64) rectangle (2.81,2.36);\n\\node[font=\\small,align=center,fon","$};\n\\path[draw=black!30,fill=black!8] (0.65,2.36) rectangle (1.37,3.08);\n\\node[font=\\small,align=center,font=\\small] at (1.01,2.72) {$4$};\n\\path[draw=black!30,fill=white] (1.37,2.36) rectangle (2.09,3.08);\n\\node[font=\\small,align=center,font=\\small] at (1.73,2.72) {$0$};\n\\path[draw=black!30,fill=white] (2.09,2.36) rectangle (2.81,3.08);\n\\node[font=\\small,align=center,font=\\small] at (2.45,2.72) {$0$};\n\\path[draw=black!30,fill=white] (0.65,1.64) rectangle (1.37,2.36);\n\\node[font=\\small,align=center,font=\\small] at (1.01,2) {$0$};\n\\path[draw=black!30,fill=black!8] (1.37,1.64) rectangle (2.09,2.36);\n\\node[font=\\small,align=center,font=\\small] at (1.73,2) {$2$};\n\\path[draw=black!30,fill=white] (2.09,1.64) rectangle (2.81,2.36);\n\\node[font=\\small,align=center,font=\\small] at (2.45,2) {$0$};\n\\path[draw=black!30,fill=white] (0.65,0.92) rectangle (1.37,1.64);\n\\node[font=\\small,align=center,font=\\",") {$4$};\n\\path[draw=black!30,fill=white] (1.37,2.36) rectangle (2.09,3.08);\n\\node[font=\\small,align=center,font=\\small] at (1.73,2.72) {$0$};\n\\path[draw=black!30,fill=white] (2.09,2.36) rectangle (2.81,3.08);\n\\node[font=\\small,align=center,font=\\small] at (2.45,2.72) {$0$};\n\\path[draw=black!30,fill=white] (0.65,1.64) rectangle (1.37,2.36);\n\\node[font=\\small,align=center,font=\\small] at (1.01,2) {$0$};\n\\path[draw=black!30,fill=black!8] (1.37,1.64) rectangle (2.09,2.36);\n\\node[font=\\small,align=center,font=\\small] at (1.73,2) {$2$};\n\\path[draw=black!30,fill=white] (2.09,1.64) rectangle (2.81,2.36);\n\\node[font=\\small,align=center,font=\\small] at (2.45,2) {$0$};\n\\path[draw=black!30,fill=white] (0.65,0.92) rectangle (1.37,1.64);\n\\node[font=\\small,align=center,font=\\small] at (1.01,1.28) {$0$};\n\\path[draw=black!30,fill=white] (1.37,0.92) rectangle (2.09,1.64);\n\\node[font=\\small,align=center,fon","2) {$0$};\n\\path[draw=black!30,fill=white] (2.09,2.36) rectangle (2.81,3.08);\n\\node[font=\\small,align=center,font=\\small] at (2.45,2.72) {$0$};\n\\path[draw=black!30,fill=white] (0.65,1.64) rectangle (1.37,2.36);\n\\node[font=\\small,align=center,font=\\small] at (1.01,2) {$0$};\n\\path[draw=black!30,fill=black!8] (1.37,1.64) rectangle (2.09,2.36);\n\\node[font=\\small,align=center,font=\\small] at (1.73,2) {$2$};\n\\path[draw=black!30,fill=white] (2.09,1.64) rectangle (2.81,2.36);\n\\node[font=\\small,align=center,font=\\small] at (2.45,2) {$0$};\n\\path[draw=black!30,fill=white] (0.65,0.92) rectangle (1.37,1.64);\n\\node[font=\\small,align=center,font=\\small] at (1.01,1.28) {$0$};\n\\path[draw=black!30,fill=white] (1.37,0.92) rectangle (2.09,1.64);\n\\node[font=\\small,align=center,font=\\small] at (1.73,1.28) {$0$};\n\\path[draw=black!30,fill=white] (2.09,0.92) rectangle (2.81,1.64);\n\\node[font=\\small,align=center,fo","2.72) {$0$};\n\\path[draw=black!30,fill=white] (0.65,1.64) rectangle (1.37,2.36);\n\\node[font=\\small,align=center,font=\\small] at (1.01,2) {$0$};\n\\path[draw=black!30,fill=black!8] (1.37,1.64) rectangle (2.09,2.36);\n\\node[font=\\small,align=center,font=\\small] at (1.73,2) {$2$};\n\\path[draw=black!30,fill=white] (2.09,1.64) rectangle (2.81,2.36);\n\\node[font=\\small,align=center,font=\\small] at (2.45,2) {$0$};\n\\path[draw=black!30,fill=white] (0.65,0.92) rectangle (1.37,1.64);\n\\node[font=\\small,align=center,font=\\small] at (1.01,1.28) {$0$};\n\\path[draw=black!30,fill=white] (1.37,0.92) rectangle (2.09,1.64);\n\\node[font=\\small,align=center,font=\\small] at (1.73,1.28) {$0$};\n\\path[draw=black!30,fill=white] (2.09,0.92) rectangle (2.81,1.64);\n\\node[font=\\small,align=center,font=\\small] at (2.45,1.28) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (6.13,4.5) {rank-one output};\n\\path[draw=black!30,","2) {$0$};\n\\path[draw=black!30,fill=black!8] (1.37,1.64) rectangle (2.09,2.36);\n\\node[font=\\small,align=center,font=\\small] at (1.73,2) {$2$};\n\\path[draw=black!30,fill=white] (2.09,1.64) rectangle (2.81,2.36);\n\\node[font=\\small,align=center,font=\\small] at (2.45,2) {$0$};\n\\path[draw=black!30,fill=white] (0.65,0.92) rectangle (1.37,1.64);\n\\node[font=\\small,align=center,font=\\small] at (1.01,1.28) {$0$};\n\\path[draw=black!30,fill=white] (1.37,0.92) rectangle (2.09,1.64);\n\\node[font=\\small,align=center,font=\\small] at (1.73,1.28) {$0$};\n\\path[draw=black!30,fill=white] (2.09,0.92) rectangle (2.81,1.64);\n\\node[font=\\small,align=center,font=\\small] at (2.45,1.28) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (6.13,4.5) {rank-one output};\n\\path[draw=black!30,fill=black!8] (5.05,3.08) rectangle (5.77,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.41,3.44) {$3$};\n\\path[draw=black!30",") {$2$};\n\\path[draw=black!30,fill=white] (2.09,1.64) rectangle (2.81,2.36);\n\\node[font=\\small,align=center,font=\\small] at (2.45,2) {$0$};\n\\path[draw=black!30,fill=white] (0.65,0.92) rectangle (1.37,1.64);\n\\node[font=\\small,align=center,font=\\small] at (1.01,1.28) {$0$};\n\\path[draw=black!30,fill=white] (1.37,0.92) rectangle (2.09,1.64);\n\\node[font=\\small,align=center,font=\\small] at (1.73,1.28) {$0$};\n\\path[draw=black!30,fill=white] (2.09,0.92) rectangle (2.81,1.64);\n\\node[font=\\small,align=center,font=\\small] at (2.45,1.28) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (6.13,4.5) {rank-one output};\n\\path[draw=black!30,fill=black!8] (5.05,3.08) rectangle (5.77,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.41,3.44) {$3$};\n\\path[draw=black!30,fill=white] (5.77,3.08) rectangle (6.49,3.8);\n\\node[font=\\small,align=center,font=\\small] at (6.13,3.44) {$0$};\n\\path[draw=black!30,","$0$};\n\\path[draw=black!30,fill=white] (0.65,0.92) rectangle (1.37,1.64);\n\\node[font=\\small,align=center,font=\\small] at (1.01,1.28) {$0$};\n\\path[draw=black!30,fill=white] (1.37,0.92) rectangle (2.09,1.64);\n\\node[font=\\small,align=center,font=\\small] at (1.73,1.28) {$0$};\n\\path[draw=black!30,fill=white] (2.09,0.92) rectangle (2.81,1.64);\n\\node[font=\\small,align=center,font=\\small] at (2.45,1.28) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (6.13,4.5) {rank-one output};\n\\path[draw=black!30,fill=black!8] (5.05,3.08) rectangle (5.77,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.41,3.44) {$3$};\n\\path[draw=black!30,fill=white] (5.77,3.08) rectangle (6.49,3.8);\n\\node[font=\\small,align=center,font=\\small] at (6.13,3.44) {$0$};\n\\path[draw=black!30,fill=white] (6.49,3.08) rectangle (7.21,3.8);\n\\node[font=\\small,align=center,font=\\small] at (6.85,3.44) {$0$};\n\\path[draw=black!30,f","[font=\\small,align=center,font=\\small] at (1.73,1.28) {$0$};\n\\path[draw=black!30,fill=white] (2.09,0.92) rectangle (2.81,1.64);\n\\node[font=\\small,align=center,font=\\small] at (2.45,1.28) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (6.13,4.5) {rank-one output};\n\\path[draw=black!30,fill=black!8] (5.05,3.08) rectangle (5.77,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.41,3.44) {$3$};\n\\path[draw=black!30,fill=white] (5.77,3.08) rectangle (6.49,3.8);\n\\node[font=\\small,align=center,font=\\small] at (6.13,3.44) {$0$};\n\\path[draw=black!30,fill=white] (6.49,3.08) rectangle (7.21,3.8);\n\\node[font=\\small,align=center,font=\\small] at (6.85,3.44) {$0$};\n\\path[draw=black!30,fill=black!8] (5.05,2.36) rectangle (5.77,3.08);\n\\node[font=\\small,align=center,font=\\small] at (5.41,2.72) {$4$};\n\\path[draw=black!30,fill=white] (5.77,2.36) rectangle (6.49,3.08);\n\\node[font=\\small,align=center","e[font=\\small,align=center,font=\\small] at (2.45,1.28) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (6.13,4.5) {rank-one output};\n\\path[draw=black!30,fill=black!8] (5.05,3.08) rectangle (5.77,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.41,3.44) {$3$};\n\\path[draw=black!30,fill=white] (5.77,3.08) rectangle (6.49,3.8);\n\\node[font=\\small,align=center,font=\\small] at (6.13,3.44) {$0$};\n\\path[draw=black!30,fill=white] (6.49,3.08) rectangle (7.21,3.8);\n\\node[font=\\small,align=center,font=\\small] at (6.85,3.44) {$0$};\n\\path[draw=black!30,fill=black!8] (5.05,2.36) rectangle (5.77,3.08);\n\\node[font=\\small,align=center,font=\\small] at (5.41,2.72) {$4$};\n\\path[draw=black!30,fill=white] (5.77,2.36) rectangle (6.49,3.08);\n\\node[font=\\small,align=center,font=\\small] at (6.13,2.72) {$0$};\n\\path[draw=black!30,fill=white] (6.49,2.36) rectangle (7.21,3.08);\n\\node[font=\\small,align=cente","tput};\n\\path[draw=black!30,fill=black!8] (5.05,3.08) rectangle (5.77,3.8);\n\\node[font=\\small,align=center,font=\\small] at (5.41,3.44) {$3$};\n\\path[draw=black!30,fill=white] (5.77,3.08) rectangle (6.49,3.8);\n\\node[font=\\small,align=center,font=\\small] at (6.13,3.44) {$0$};\n\\path[draw=black!30,fill=white] (6.49,3.08) rectangle (7.21,3.8);\n\\node[font=\\small,align=center,font=\\small] at (6.85,3.44) {$0$};\n\\path[draw=black!30,fill=black!8] (5.05,2.36) rectangle (5.77,3.08);\n\\node[font=\\small,align=center,font=\\small] at (5.41,2.72) {$4$};\n\\path[draw=black!30,fill=white] (5.77,2.36) rectangle (6.49,3.08);\n\\node[font=\\small,align=center,font=\\small] at (6.13,2.72) {$0$};\n\\path[draw=black!30,fill=white] (6.49,2.36) rectangle (7.21,3.08);\n\\node[font=\\small,align=center,font=\\small] at (6.85,2.72) {$0$};\n\\path[draw=black!30,fill=white] (5.05,1.64) rectangle (5.77,2.36);\n\\node[font=\\small,align=cent","$3$};\n\\path[draw=black!30,fill=white] (5.77,3.08) rectangle (6.49,3.8);\n\\node[font=\\small,align=center,font=\\small] at (6.13,3.44) {$0$};\n\\path[draw=black!30,fill=white] (6.49,3.08) rectangle (7.21,3.8);\n\\node[font=\\small,align=center,font=\\small] at (6.85,3.44) {$0$};\n\\path[draw=black!30,fill=black!8] (5.05,2.36) rectangle (5.77,3.08);\n\\node[font=\\small,align=center,font=\\small] at (5.41,2.72) {$4$};\n\\path[draw=black!30,fill=white] (5.77,2.36) rectangle (6.49,3.08);\n\\node[font=\\small,align=center,font=\\small] at (6.13,2.72) {$0$};\n\\path[draw=black!30,fill=white] (6.49,2.36) rectangle (7.21,3.08);\n\\node[font=\\small,align=center,font=\\small] at (6.85,2.72) {$0$};\n\\path[draw=black!30,fill=white] (5.05,1.64) rectangle (5.77,2.36);\n\\node[font=\\small,align=center,font=\\small] at (5.41,2) {$0$};\n\\path[draw=black!30,fill=white] (5.77,1.64) rectangle (6.49,2.36);\n\\node[font=\\small,align=center,fo","0$};\n\\path[draw=black!30,fill=white] (6.49,3.08) rectangle (7.21,3.8);\n\\node[font=\\small,align=center,font=\\small] at (6.85,3.44) {$0$};\n\\path[draw=black!30,fill=black!8] (5.05,2.36) rectangle (5.77,3.08);\n\\node[font=\\small,align=center,font=\\small] at (5.41,2.72) {$4$};\n\\path[draw=black!30,fill=white] (5.77,2.36) rectangle (6.49,3.08);\n\\node[font=\\small,align=center,font=\\small] at (6.13,2.72) {$0$};\n\\path[draw=black!30,fill=white] (6.49,2.36) rectangle (7.21,3.08);\n\\node[font=\\small,align=center,font=\\small] at (6.85,2.72) {$0$};\n\\path[draw=black!30,fill=white] (5.05,1.64) rectangle (5.77,2.36);\n\\node[font=\\small,align=center,font=\\small] at (5.41,2) {$0$};\n\\path[draw=black!30,fill=white] (5.77,1.64) rectangle (6.49,2.36);\n\\node[font=\\small,align=center,font=\\small] at (6.13,2) {$0$};\n\\path[draw=black!30,fill=white] (6.49,1.64) rectangle (7.21,2.36);\n\\node[font=\\small,align=center,font=","$};\n\\path[draw=black!30,fill=black!8] (5.05,2.36) rectangle (5.77,3.08);\n\\node[font=\\small,align=center,font=\\small] at (5.41,2.72) {$4$};\n\\path[draw=black!30,fill=white] (5.77,2.36) rectangle (6.49,3.08);\n\\node[font=\\small,align=center,font=\\small] at (6.13,2.72) {$0$};\n\\path[draw=black!30,fill=white] (6.49,2.36) rectangle (7.21,3.08);\n\\node[font=\\small,align=center,font=\\small] at (6.85,2.72) {$0$};\n\\path[draw=black!30,fill=white] (5.05,1.64) rectangle (5.77,2.36);\n\\node[font=\\small,align=center,font=\\small] at (5.41,2) {$0$};\n\\path[draw=black!30,fill=white] (5.77,1.64) rectangle (6.49,2.36);\n\\node[font=\\small,align=center,font=\\small] at (6.13,2) {$0$};\n\\path[draw=black!30,fill=white] (6.49,1.64) rectangle (7.21,2.36);\n\\node[font=\\small,align=center,font=\\small] at (6.85,2) {$0$};\n\\path[draw=black!30,fill=white] (5.05,0.92) rectangle (5.77,1.64);\n\\node[font=\\small,align=center,font=\\sm",") {$4$};\n\\path[draw=black!30,fill=white] (5.77,2.36) rectangle (6.49,3.08);\n\\node[font=\\small,align=center,font=\\small] at (6.13,2.72) {$0$};\n\\path[draw=black!30,fill=white] (6.49,2.36) rectangle (7.21,3.08);\n\\node[font=\\small,align=center,font=\\small] at (6.85,2.72) {$0$};\n\\path[draw=black!30,fill=white] (5.05,1.64) rectangle (5.77,2.36);\n\\node[font=\\small,align=center,font=\\small] at (5.41,2) {$0$};\n\\path[draw=black!30,fill=white] (5.77,1.64) rectangle (6.49,2.36);\n\\node[font=\\small,align=center,font=\\small] at (6.13,2) {$0$};\n\\path[draw=black!30,fill=white] (6.49,1.64) rectangle (7.21,2.36);\n\\node[font=\\small,align=center,font=\\small] at (6.85,2) {$0$};\n\\path[draw=black!30,fill=white] (5.05,0.92) rectangle (5.77,1.64);\n\\node[font=\\small,align=center,font=\\small] at (5.41,1.28) {$0$};\n\\path[draw=black!30,fill=white] (5.77,0.92) rectangle (6.49,1.64);\n\\node[font=\\small,align=center,font=",".72) {$0$};\n\\path[draw=black!30,fill=white] (6.49,2.36) rectangle (7.21,3.08);\n\\node[font=\\small,align=center,font=\\small] at (6.85,2.72) {$0$};\n\\path[draw=black!30,fill=white] (5.05,1.64) rectangle (5.77,2.36);\n\\node[font=\\small,align=center,font=\\small] at (5.41,2) {$0$};\n\\path[draw=black!30,fill=white] (5.77,1.64) rectangle (6.49,2.36);\n\\node[font=\\small,align=center,font=\\small] at (6.13,2) {$0$};\n\\path[draw=black!30,fill=white] (6.49,1.64) rectangle (7.21,2.36);\n\\node[font=\\small,align=center,font=\\small] at (6.85,2) {$0$};\n\\path[draw=black!30,fill=white] (5.05,0.92) rectangle (5.77,1.64);\n\\node[font=\\small,align=center,font=\\small] at (5.41,1.28) {$0$};\n\\path[draw=black!30,fill=white] (5.77,0.92) rectangle (6.49,1.64);\n\\node[font=\\small,align=center,font=\\small] at (6.13,1.28) {$0$};\n\\path[draw=black!30,fill=white] (6.49,0.92) rectangle (7.21,1.64);\n\\node[font=\\small,align=center,fo","5,2.72) {$0$};\n\\path[draw=black!30,fill=white] (5.05,1.64) rectangle (5.77,2.36);\n\\node[font=\\small,align=center,font=\\small] at (5.41,2) {$0$};\n\\path[draw=black!30,fill=white] (5.77,1.64) rectangle (6.49,2.36);\n\\node[font=\\small,align=center,font=\\small] at (6.13,2) {$0$};\n\\path[draw=black!30,fill=white] (6.49,1.64) rectangle (7.21,2.36);\n\\node[font=\\small,align=center,font=\\small] at (6.85,2) {$0$};\n\\path[draw=black!30,fill=white] (5.05,0.92) rectangle (5.77,1.64);\n\\node[font=\\small,align=center,font=\\small] at (5.41,1.28) {$0$};\n\\path[draw=black!30,fill=white] (5.77,0.92) rectangle (6.49,1.64);\n\\node[font=\\small,align=center,font=\\small] at (6.13,1.28) {$0$};\n\\path[draw=black!30,fill=white] (6.49,0.92) rectangle (7.21,1.64);\n\\node[font=\\small,align=center,font=\\small] at (6.85,1.28) {$0$};\n\\draw[line width=.6pt] (9.7,0.9) -- (13.5,0.9);\n\\node[font=\\small,align=center] at (11.6,4.25) {s","1,2) {$0$};\n\\path[draw=black!30,fill=white] (5.77,1.64) rectangle (6.49,2.36);\n\\node[font=\\small,align=center,font=\\small] at (6.13,2) {$0$};\n\\path[draw=black!30,fill=white] (6.49,1.64) rectangle (7.21,2.36);\n\\node[font=\\small,align=center,font=\\small] at (6.85,2) {$0$};\n\\path[draw=black!30,fill=white] (5.05,0.92) rectangle (5.77,1.64);\n\\node[font=\\small,align=center,font=\\small] at (5.41,1.28) {$0$};\n\\path[draw=black!30,fill=white] (5.77,0.92) rectangle (6.49,1.64);\n\\node[font=\\small,align=center,font=\\small] at (6.13,1.28) {$0$};\n\\path[draw=black!30,fill=white] (6.49,0.92) rectangle (7.21,1.64);\n\\node[font=\\small,align=center,font=\\small] at (6.85,1.28) {$0$};\n\\draw[line width=.6pt] (9.7,0.9) -- (13.5,0.9);\n\\node[font=\\small,align=center] at (11.6,4.25) {spectral error};\n\\path[draw=black,fill=black!12,line width=.6pt] (10.004,0.9) rectangle (10.662667,3.5);\n\\node[font=\\small,align=cente",") {$0$};\n\\path[draw=black!30,fill=white] (6.49,1.64) rectangle (7.21,2.36);\n\\node[font=\\small,align=center,font=\\small] at (6.85,2) {$0$};\n\\path[draw=black!30,fill=white] (5.05,0.92) rectangle (5.77,1.64);\n\\node[font=\\small,align=center,font=\\small] at (5.41,1.28) {$0$};\n\\path[draw=black!30,fill=white] (5.77,0.92) rectangle (6.49,1.64);\n\\node[font=\\small,align=center,font=\\small] at (6.13,1.28) {$0$};\n\\path[draw=black!30,fill=white] (6.49,0.92) rectangle (7.21,1.64);\n\\node[font=\\small,align=center,font=\\small] at (6.85,1.28) {$0$};\n\\draw[line width=.6pt] (9.7,0.9) -- (13.5,0.9);\n\\node[font=\\small,align=center] at (11.6,4.25) {spectral error};\n\\path[draw=black,fill=black!12,line width=.6pt] (10.004,0.9) rectangle (10.662667,3.5);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.333333,0.6) {rank 0};\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.333333,3.72) {$5$};\n\\path[dr","$0$};\n\\path[draw=black!30,fill=white] (5.05,0.92) rectangle (5.77,1.64);\n\\node[font=\\small,align=center,font=\\small] at (5.41,1.28) {$0$};\n\\path[draw=black!30,fill=white] (5.77,0.92) rectangle (6.49,1.64);\n\\node[font=\\small,align=center,font=\\small] at (6.13,1.28) {$0$};\n\\path[draw=black!30,fill=white] (6.49,0.92) rectangle (7.21,1.64);\n\\node[font=\\small,align=center,font=\\small] at (6.85,1.28) {$0$};\n\\draw[line width=.6pt] (9.7,0.9) -- (13.5,0.9);\n\\node[font=\\small,align=center] at (11.6,4.25) {spectral error};\n\\path[draw=black,fill=black!12,line width=.6pt] (10.004,0.9) rectangle (10.662667,3.5);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.333333,0.6) {rank 0};\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.333333,3.72) {$5$};\n\\path[draw=black,fill=black!12,line width=.6pt] (11.270667,0.9) rectangle (11.929333,1.94);\n\\node[font=\\small,align=center,font=\\scriptsize]","l,align=center,font=\\small] at (6.85,1.28) {$0$};\n\\draw[line width=.6pt] (9.7,0.9) -- (13.5,0.9);\n\\node[font=\\small,align=center] at (11.6,4.25) {spectral error};\n\\path[draw=black,fill=black!12,line width=.6pt] (10.004,0.9) rectangle (10.662667,3.5);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.333333,0.6) {rank 0};\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.333333,3.72) {$5$};\n\\path[draw=black,fill=black!12,line width=.6pt] (11.270667,0.9) rectangle (11.929333,1.94);\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.6,0.6) {rank 1};\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.6,2.16) {$2$};\n\\path[draw=black,fill=black!12,line width=.6pt] (12.537333,0.9) rectangle (13.196,0.9);\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.866667,0.6) {rank 2};\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.866667,1.12) {$0$};\n\\node[font=\\small,","0.662667,3.5);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.333333,0.6) {rank 0};\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.333333,3.72) {$5$};\n\\path[draw=black,fill=black!12,line width=.6pt] (11.270667,0.9) rectangle (11.929333,1.94);\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.6,0.6) {rank 1};\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.6,2.16) {$2$};\n\\path[draw=black,fill=black!12,line width=.6pt] (12.537333,0.9) rectangle (13.196,0.9);\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.866667,0.6) {rank 2};\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.866667,1.12) {$0$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.1) {Frobenius errors: $\\sqrt{29},\\ 2,\\ 0$. The rank-one choice loses only the second mode.};\n\\end{tikzpicture}","(11.929333,1.94);\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.6,0.6) {rank 1};\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.6,2.16) {$2$};\n\\path[draw=black,fill=black!12,line width=.6pt] (12.537333,0.9) rectangle (13.196,0.9);\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.866667,0.6) {rank 2};\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.866667,1.12) {$0$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.1) {Frobenius errors: $\\sqrt{29},\\ 2,\\ 0$. The rank-one choice loses only the second mode.};\n\\end{tikzpicture}","nk 1};\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.6,2.16) {$2$};\n\\path[draw=black,fill=black!12,line width=.6pt] (12.537333,0.9) rectangle (13.196,0.9);\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.866667,0.6) {rank 2};\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.866667,1.12) {$0$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.1) {Frobenius errors: $\\sqrt{29},\\ 2,\\ 0$. The rank-one choice loses only the second mode.};\n\\end{tikzpicture}","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\draw[line width=.6pt] (0.7,0.8) -- (6.1,0.8);\n\\node[font=\\small,align=center] at (3.4,4.25) {unclipped contributions $s_e\\epsilon_e$};\n\\path[draw=black,fill=black!12,line width=.6pt] (1.132,0.8) rectangle (2.068,1.925);\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.6,0.5) {edge 1};\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.6,2.145) {$\\frac{1}{4}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (2.932,0.8) rectangle (3.868,1.925);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.4,0.5) {edge 2};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.4,2.145) {$\\frac{1}{4}$};\n\\path[draw=black,fil","ounding box] (0,-.6) rectangle (14,4.8);\n\\draw[line width=.6pt] (0.7,0.8) -- (6.1,0.8);\n\\node[font=\\small,align=center] at (3.4,4.25) {unclipped contributions $s_e\\epsilon_e$};\n\\path[draw=black,fill=black!12,line width=.6pt] (1.132,0.8) rectangle (2.068,1.925);\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.6,0.5) {edge 1};\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.6,2.145) {$\\frac{1}{4}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (2.932,0.8) rectangle (3.868,1.925);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.4,0.5) {edge 2};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.4,2.145) {$\\frac{1}{4}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (4.732,0.8) rectangle (5.668,3.05);\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.2,0.5) {edge 3};\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.2,3.27) {$\\frac{1}{2}$};\n\\draw[line width=.6pt]","ectangle (2.068,1.925);\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.6,0.5) {edge 1};\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.6,2.145) {$\\frac{1}{4}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (2.932,0.8) rectangle (3.868,1.925);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.4,0.5) {edge 2};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.4,2.145) {$\\frac{1}{4}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (4.732,0.8) rectangle (5.668,3.05);\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.2,0.5) {edge 3};\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.2,3.27) {$\\frac{1}{2}$};\n\\draw[line width=.6pt] (7.8,0.8) -- (13.2,0.8);\n\\node[font=\\small,align=center] at (10.5,4.25) {clipped contributions};\n\\path[draw=black,fill=black!12,line width=.6pt] (8.232,0.8) rectangle (9.168,1.25);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.7","rectangle (3.868,1.925);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.4,0.5) {edge 2};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.4,2.145) {$\\frac{1}{4}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (4.732,0.8) rectangle (5.668,3.05);\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.2,0.5) {edge 3};\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.2,3.27) {$\\frac{1}{2}$};\n\\draw[line width=.6pt] (7.8,0.8) -- (13.2,0.8);\n\\node[font=\\small,align=center] at (10.5,4.25) {clipped contributions};\n\\path[draw=black,fill=black!12,line width=.6pt] (8.232,0.8) rectangle (9.168,1.25);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.7,0.5) {edge 1};\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.7,1.47) {$\\frac{1}{10}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (10.032,0.8) rectangle (10.968,2.15);\n\\node[font=\\small,align=center,font=\\scriptsize] at (","gn=center,font=\\scriptsize] at (5.2,3.27) {$\\frac{1}{2}$};\n\\draw[line width=.6pt] (7.8,0.8) -- (13.2,0.8);\n\\node[font=\\small,align=center] at (10.5,4.25) {clipped contributions};\n\\path[draw=black,fill=black!12,line width=.6pt] (8.232,0.8) rectangle (9.168,1.25);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.7,0.5) {edge 1};\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.7,1.47) {$\\frac{1}{10}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (10.032,0.8) rectangle (10.968,2.15);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.5,0.5) {edge 2};\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.5,2.37) {$\\frac{3}{10}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (11.832,0.8) rectangle (12.768,3.5);\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.3,0.5) {edge 3};\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.3,3.72) {$\\frac{3}{5}$};\n\\node[font=\\smal","ctangle (9.168,1.25);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.7,0.5) {edge 1};\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.7,1.47) {$\\frac{1}{10}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (10.032,0.8) rectangle (10.968,2.15);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.5,0.5) {edge 2};\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.5,2.37) {$\\frac{3}{10}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (11.832,0.8) rectangle (12.768,3.5);\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.3,0.5) {edge 3};\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.3,3.72) {$\\frac{3}{5}$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.05) {Both allocations spend exactly one unit of error budget.};\n\\end{tikzpicture}","angle (10.968,2.15);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.5,0.5) {edge 2};\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.5,2.37) {$\\frac{3}{10}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (11.832,0.8) rectangle (12.768,3.5);\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.3,0.5) {edge 3};\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.3,3.72) {$\\frac{3}{5}$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.05) {Both allocations spend exactly one unit of error budget.};\n\\end{tikzpicture}","35) (4.395209,1.52007) (4.384452,1.490862) (4.372392,1.462956) (4.359049,1.436397) (4.344441,1.411224) (4.328594,1.387479) (4.311531,1.365197) (4.293278,1.344414) (4.273865,1.325162) (4.253322,1.307471) (4.231681,1.291369) (4.208975,1.276881) (4.185241,1.264029) (4.160515,1.252834) (4.134835,1.243313) (4.108242,1.235481) (4.080778,1.22935)};\n\\node[font=\\small,align=center] at (3.55,4.3) {before: $x^TAx=1$};\n\\draw[black!13,line width=.25pt] (8.720588,0.7) -- (8.720588,3.95);\n\\draw[] (8.720588,0.64) -- (8.720588,0.76);\n\\node[font=\\small,align=center,font=\\scriptsize] at (8.720588,0.44) {-1};\n\\draw[black!13,line width=.25pt] (10.25,0.7) -- (10.25,3.95);\n\\draw[] (10.25,0.64) -- (10.25,0.76);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.25,0.44) {0};\n\\draw[black!13,line width=.25pt] (11.779412,0.7) -- (11.779412,3.95);\n\\draw[] (11.779412,0.64) -- (11.779412,0.76);\n\\node[font=\\small,align=","698) (8.494405,2.769951) (8.488209,2.809509) (8.484763,2.848311) (8.484074,2.886295) (8.486141,2.923403) (8.490963,2.959577) (8.498532,2.99476) (8.508835,3.028898) (8.521857,3.061936) (8.537577,3.093824) (8.555971,3.124511) (8.57701,3.15395) (8.600661,3.182094) (8.626887,3.208901) (8.655648,3.234327) (8.686898,3.258333) (8.720588,3.280882)};\n\\node[font=\\small,align=center] at (10.25,4.3) {after: $y^T(I+E)y=1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,-0.1) {Exact normalized eigenvalues: $1/2,3/2$. Hence $\\kappa_2=3$.};\n\\end{tikzpicture}","490963,2.959577) (8.498532,2.99476) (8.508835,3.028898) (8.521857,3.061936) (8.537577,3.093824) (8.555971,3.124511) (8.57701,3.15395) (8.600661,3.182094) (8.626887,3.208901) (8.655648,3.234327) (8.686898,3.258333) (8.720588,3.280882)};\n\\node[font=\\small,align=center] at (10.25,4.3) {after: $y^T(I+E)y=1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,-0.1) {Exact normalized eigenvalues: $1/2,3/2$. Hence $\\kappa_2=3$.};\n\\end{tikzpicture}","(8.498532,2.99476) (8.508835,3.028898) (8.521857,3.061936) (8.537577,3.093824) (8.555971,3.124511) (8.57701,3.15395) (8.600661,3.182094) (8.626887,3.208901) (8.655648,3.234327) (8.686898,3.258333) (8.720588,3.280882)};\n\\node[font=\\small,align=center] at (10.25,4.3) {after: $y^T(I+E)y=1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,-0.1) {Exact normalized eigenvalues: $1/2,3/2$. Hence $\\kappa_2=3$.};\n\\end{tikzpicture}","\\begin{tikzpicture}[font=\\sffamily\\small]\n\\node[anchor=west] at (0,3.2) {actual error sum\\quad versus\\quad ceiling envelope};\n\\draw[->] (0,0)--(8,0) node[right] {error};\n\\foreach \\x/\\v in {0/0,2/{1/10},4/{1/5}}{\\draw (\\x,0)--(\\x,-.1);\\node[below] at (\\x,-.1) {$\\v$};}\n\\draw[dotted] (2,0)--(2,2.2);\\node[above] at (2,2.2) {budget};\n\\draw[very thick] (0,1.6)--(2.8,1.6);\\fill (2.8,1.6) circle (2pt);\n\\node[right] at (4.3,1.6) {$7/50>1/10$};\n\\draw[very thick] (0,.8)--(1.6,.8);\\fill (1.6,.8) circle (2pt);\n\\node[right] at (4.3,.8) {$2/25\\le1/10$};\n\\draw[thick,dashed] (0,1.3)--(4,1.3);\\draw[thick,dashed] (0,.5)--(4,.5);\n\\node[anchor=west] at (0,-1.1) {Both ceiling envelopes are $h(1+1)=1/5$.};\n\\end{tikzpicture}","sffamily\\small]\n\\node[anchor=west] at (0,3.2) {actual error sum\\quad versus\\quad ceiling envelope};\n\\draw[->] (0,0)--(8,0) node[right] {error};\n\\foreach \\x/\\v in {0/0,2/{1/10},4/{1/5}}{\\draw (\\x,0)--(\\x,-.1);\\node[below] at (\\x,-.1) {$\\v$};}\n\\draw[dotted] (2,0)--(2,2.2);\\node[above] at (2,2.2) {budget};\n\\draw[very thick] (0,1.6)--(2.8,1.6);\\fill (2.8,1.6) circle (2pt);\n\\node[right] at (4.3,1.6) {$7/50>1/10$};\n\\draw[very thick] (0,.8)--(1.6,.8);\\fill (1.6,.8) circle (2pt);\n\\node[right] at (4.3,.8) {$2/25\\le1/10$};\n\\draw[thick,dashed] (0,1.3)--(4,1.3);\\draw[thick,dashed] (0,.5)--(4,.5);\n\\node[anchor=west] at (0,-1.1) {Both ceiling envelopes are $h(1+1)=1/5$.};\n\\end{tikzpicture}","w[->] (0,0)--(8,0) node[right] {error};\n\\foreach \\x/\\v in {0/0,2/{1/10},4/{1/5}}{\\draw (\\x,0)--(\\x,-.1);\\node[below] at (\\x,-.1) {$\\v$};}\n\\draw[dotted] (2,0)--(2,2.2);\\node[above] at (2,2.2) {budget};\n\\draw[very thick] (0,1.6)--(2.8,1.6);\\fill (2.8,1.6) circle (2pt);\n\\node[right] at (4.3,1.6) {$7/50>1/10$};\n\\draw[very thick] (0,.8)--(1.6,.8);\\fill (1.6,.8) circle (2pt);\n\\node[right] at (4.3,.8) {$2/25\\le1/10$};\n\\draw[thick,dashed] (0,1.3)--(4,1.3);\\draw[thick,dashed] (0,.5)--(4,.5);\n\\node[anchor=west] at (0,-1.1) {Both ceiling envelopes are $h(1+1)=1/5$.};\n\\end{tikzpicture}","ed] (2,0)--(2,2.2);\\node[above] at (2,2.2) {budget};\n\\draw[very thick] (0,1.6)--(2.8,1.6);\\fill (2.8,1.6) circle (2pt);\n\\node[right] at (4.3,1.6) {$7/50>1/10$};\n\\draw[very thick] (0,.8)--(1.6,.8);\\fill (1.6,.8) circle (2pt);\n\\node[right] at (4.3,.8) {$2/25\\le1/10$};\n\\draw[thick,dashed] (0,1.3)--(4,1.3);\\draw[thick,dashed] (0,.5)--(4,.5);\n\\node[anchor=west] at (0,-1.1) {Both ceiling envelopes are $h(1+1)=1/5$.};\n\\end{tikzpicture}","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (2.04,4.3) {$A$ before elimination};\n\\path[draw=black!30,fill=black!8] (0.75,2.74) rectangle (1.61,3.6);\n\\node[font=\\small,align=center,font=\\small] at (1.18,3.17) {$2$};\n\\path[draw=black!30,fill=white] (1.61,2.74) rectangle (2.47,3.6);\n\\node[font=\\small,align=center,font=\\small] at (2.04,3.17) {$0$};\n\\path[draw=black!30,fill=black!8] (2.47,2.74) rectangle (3.33,3.6);\n\\node[font=\\small,align=center,font=\\small] at (2.9,3.17) {$1$};\n\\path[draw=black!30,fill=white] (0.75,1.88) rectangle (1.61,2.74);\n\\node[font=","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (2.04,4.3) {$A$ before elimination};\n\\path[draw=black!30,fill=black!8] (0.75,2.74) rectangle (1.61,3.6);\n\\node[font=\\small,align=center,font=\\small] at (1.18,3.17) {$2$};\n\\path[draw=black!30,fill=white] (1.61,2.74) rectangle (2.47,3.6);\n\\node[font=\\small,align=center,font=\\small] at (2.04,3.17) {$0$};\n\\path[draw=black!30,fill=black!8] (2.47,2.74) rectangle (3.33,3.6);\n\\node[font=\\small,align=center,font=\\small] at (2.9,3.17) {$1$};\n\\path[draw=black!30,fill=white] (0.75,1.88) rectangle (1.61,2.74);\n\\node[font=\\small,align=center,font=\\small] at (1.18,2.31) {$0$};\n\\path[draw=black!30,fill=black!8] (1.61,1.88) rectangle (2.47,2.74);\n\\node[font=\\small,align=cente","\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (2.04,4.3) {$A$ before elimination};\n\\path[draw=black!30,fill=black!8] (0.75,2.74) rectangle (1.61,3.6);\n\\node[font=\\small,align=center,font=\\small] at (1.18,3.17) {$2$};\n\\path[draw=black!30,fill=white] (1.61,2.74) rectangle (2.47,3.6);\n\\node[font=\\small,align=center,font=\\small] at (2.04,3.17) {$0$};\n\\path[draw=black!30,fill=black!8] (2.47,2.74) rectangle (3.33,3.6);\n\\node[font=\\small,align=center,font=\\small] at (2.9,3.17) {$1$};\n\\path[draw=black!30,fill=white] (0.75,1.88) rectangle (1.61,2.74);\n\\node[font=\\small,align=center,font=\\small] at (1.18,2.31) {$0$};\n\\path[draw=black!30,fill=black!8] (1.61,1.88) rectangle (2.47,2.74);\n\\node[font=\\small,align=center,font=\\small] at (2.04,2.31) {$2$};\n\\path[draw=black!30,fill=black!8] (2.47,1.88) rectangle (3.33,2.74);\n\\node[font=\\small,align=ce","ion};\n\\path[draw=black!30,fill=black!8] (0.75,2.74) rectangle (1.61,3.6);\n\\node[font=\\small,align=center,font=\\small] at (1.18,3.17) {$2$};\n\\path[draw=black!30,fill=white] (1.61,2.74) rectangle (2.47,3.6);\n\\node[font=\\small,align=center,font=\\small] at (2.04,3.17) {$0$};\n\\path[draw=black!30,fill=black!8] (2.47,2.74) rectangle (3.33,3.6);\n\\node[font=\\small,align=center,font=\\small] at (2.9,3.17) {$1$};\n\\path[draw=black!30,fill=white] (0.75,1.88) rectangle (1.61,2.74);\n\\node[font=\\small,align=center,font=\\small] at (1.18,2.31) {$0$};\n\\path[draw=black!30,fill=black!8] (1.61,1.88) rectangle (2.47,2.74);\n\\node[font=\\small,align=center,font=\\small] at (2.04,2.31) {$2$};\n\\path[draw=black!30,fill=black!8] (2.47,1.88) rectangle (3.33,2.74);\n\\node[font=\\small,align=center,font=\\small] at (2.9,2.31) {$1$};\n\\path[draw=black!30,fill=black!8] (0.75,1.02) rectangle (1.61,1.88);\n\\node[font=\\small,align=c","{$2$};\n\\path[draw=black!30,fill=white] (1.61,2.74) rectangle (2.47,3.6);\n\\node[font=\\small,align=center,font=\\small] at (2.04,3.17) {$0$};\n\\path[draw=black!30,fill=black!8] (2.47,2.74) rectangle (3.33,3.6);\n\\node[font=\\small,align=center,font=\\small] at (2.9,3.17) {$1$};\n\\path[draw=black!30,fill=white] (0.75,1.88) rectangle (1.61,2.74);\n\\node[font=\\small,align=center,font=\\small] at (1.18,2.31) {$0$};\n\\path[draw=black!30,fill=black!8] (1.61,1.88) rectangle (2.47,2.74);\n\\node[font=\\small,align=center,font=\\small] at (2.04,2.31) {$2$};\n\\path[draw=black!30,fill=black!8] (2.47,1.88) rectangle (3.33,2.74);\n\\node[font=\\small,align=center,font=\\small] at (2.9,2.31) {$1$};\n\\path[draw=black!30,fill=black!8] (0.75,1.02) rectangle (1.61,1.88);\n\\node[font=\\small,align=center,font=\\small] at (1.18,1.45) {$1$};\n\\path[draw=black!30,fill=black!8] (1.61,1.02) rectangle (2.47,1.88);\n\\node[font=\\small,align","$};\n\\path[draw=black!30,fill=black!8] (2.47,2.74) rectangle (3.33,3.6);\n\\node[font=\\small,align=center,font=\\small] at (2.9,3.17) {$1$};\n\\path[draw=black!30,fill=white] (0.75,1.88) rectangle (1.61,2.74);\n\\node[font=\\small,align=center,font=\\small] at (1.18,2.31) {$0$};\n\\path[draw=black!30,fill=black!8] (1.61,1.88) rectangle (2.47,2.74);\n\\node[font=\\small,align=center,font=\\small] at (2.04,2.31) {$2$};\n\\path[draw=black!30,fill=black!8] (2.47,1.88) rectangle (3.33,2.74);\n\\node[font=\\small,align=center,font=\\small] at (2.9,2.31) {$1$};\n\\path[draw=black!30,fill=black!8] (0.75,1.02) rectangle (1.61,1.88);\n\\node[font=\\small,align=center,font=\\small] at (1.18,1.45) {$1$};\n\\path[draw=black!30,fill=black!8] (1.61,1.02) rectangle (2.47,1.88);\n\\node[font=\\small,align=center,font=\\small] at (2.04,1.45) {$1$};\n\\path[draw=black!30,fill=black!8] (2.47,1.02) rectangle (3.33,1.88);\n\\node[font=\\small,align","};\n\\path[draw=black!30,fill=white] (0.75,1.88) rectangle (1.61,2.74);\n\\node[font=\\small,align=center,font=\\small] at (1.18,2.31) {$0$};\n\\path[draw=black!30,fill=black!8] (1.61,1.88) rectangle (2.47,2.74);\n\\node[font=\\small,align=center,font=\\small] at (2.04,2.31) {$2$};\n\\path[draw=black!30,fill=black!8] (2.47,1.88) rectangle (3.33,2.74);\n\\node[font=\\small,align=center,font=\\small] at (2.9,2.31) {$1$};\n\\path[draw=black!30,fill=black!8] (0.75,1.02) rectangle (1.61,1.88);\n\\node[font=\\small,align=center,font=\\small] at (1.18,1.45) {$1$};\n\\path[draw=black!30,fill=black!8] (1.61,1.02) rectangle (2.47,1.88);\n\\node[font=\\small,align=center,font=\\small] at (2.04,1.45) {$1$};\n\\path[draw=black!30,fill=black!8] (2.47,1.02) rectangle (3.33,1.88);\n\\node[font=\\small,align=center,font=\\small] at (2.9,1.45) {$2$};\n\\node[font=\\small,align=center,font=\\small] at (6.6,4.3) {$S=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR}$","\\path[draw=black!30,fill=black!8] (1.61,1.88) rectangle (2.47,2.74);\n\\node[font=\\small,align=center,font=\\small] at (2.04,2.31) {$2$};\n\\path[draw=black!30,fill=black!8] (2.47,1.88) rectangle (3.33,2.74);\n\\node[font=\\small,align=center,font=\\small] at (2.9,2.31) {$1$};\n\\path[draw=black!30,fill=black!8] (0.75,1.02) rectangle (1.61,1.88);\n\\node[font=\\small,align=center,font=\\small] at (1.18,1.45) {$1$};\n\\path[draw=black!30,fill=black!8] (1.61,1.02) rectangle (2.47,1.88);\n\\node[font=\\small,align=center,font=\\small] at (2.04,1.45) {$1$};\n\\path[draw=black!30,fill=black!8] (2.47,1.02) rectangle (3.33,1.88);\n\\node[font=\\small,align=center,font=\\small] at (2.9,1.45) {$2$};\n\\node[font=\\small,align=center,font=\\small] at (6.6,4.3) {$S=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR}$};\n\\path[draw=black!30,fill=black!8] (5.65,2.65) rectangle (6.6,3.6);\n\\node[font=\\small,align=center,font=\\small] at (6.125,3.125) {$\\f","\\path[draw=black!30,fill=black!8] (2.47,1.88) rectangle (3.33,2.74);\n\\node[font=\\small,align=center,font=\\small] at (2.9,2.31) {$1$};\n\\path[draw=black!30,fill=black!8] (0.75,1.02) rectangle (1.61,1.88);\n\\node[font=\\small,align=center,font=\\small] at (1.18,1.45) {$1$};\n\\path[draw=black!30,fill=black!8] (1.61,1.02) rectangle (2.47,1.88);\n\\node[font=\\small,align=center,font=\\small] at (2.04,1.45) {$1$};\n\\path[draw=black!30,fill=black!8] (2.47,1.02) rectangle (3.33,1.88);\n\\node[font=\\small,align=center,font=\\small] at (2.9,1.45) {$2$};\n\\node[font=\\small,align=center,font=\\small] at (6.6,4.3) {$S=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR}$};\n\\path[draw=black!30,fill=black!8] (5.65,2.65) rectangle (6.6,3.6);\n\\node[font=\\small,align=center,font=\\small] at (6.125,3.125) {$\\frac{3}{2}$};\n\\path[draw=black!30,fill=black!8] (6.6,2.65) rectangle (7.55,3.6);\n\\node[font=\\small,align=center,font=\\small] at (7.075,3","\\path[draw=black!30,fill=black!8] (0.75,1.02) rectangle (1.61,1.88);\n\\node[font=\\small,align=center,font=\\small] at (1.18,1.45) {$1$};\n\\path[draw=black!30,fill=black!8] (1.61,1.02) rectangle (2.47,1.88);\n\\node[font=\\small,align=center,font=\\small] at (2.04,1.45) {$1$};\n\\path[draw=black!30,fill=black!8] (2.47,1.02) rectangle (3.33,1.88);\n\\node[font=\\small,align=center,font=\\small] at (2.9,1.45) {$2$};\n\\node[font=\\small,align=center,font=\\small] at (6.6,4.3) {$S=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR}$};\n\\path[draw=black!30,fill=black!8] (5.65,2.65) rectangle (6.6,3.6);\n\\node[font=\\small,align=center,font=\\small] at (6.125,3.125) {$\\frac{3}{2}$};\n\\path[draw=black!30,fill=black!8] (6.6,2.65) rectangle (7.55,3.6);\n\\node[font=\\small,align=center,font=\\small] at (7.075,3.125) {$- \\frac{1}{2}$};\n\\path[draw=black!30,fill=black!8] (5.65,1.7) rectangle (6.6,2.65);\n\\node[font=\\small,align=center,font=\\small",".88);\n\\node[font=\\small,align=center,font=\\small] at (1.18,1.45) {$1$};\n\\path[draw=black!30,fill=black!8] (1.61,1.02) rectangle (2.47,1.88);\n\\node[font=\\small,align=center,font=\\small] at (2.04,1.45) {$1$};\n\\path[draw=black!30,fill=black!8] (2.47,1.02) rectangle (3.33,1.88);\n\\node[font=\\small,align=center,font=\\small] at (2.9,1.45) {$2$};\n\\node[font=\\small,align=center,font=\\small] at (6.6,4.3) {$S=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR}$};\n\\path[draw=black!30,fill=black!8] (5.65,2.65) rectangle (6.6,3.6);\n\\node[font=\\small,align=center,font=\\small] at (6.125,3.125) {$\\frac{3}{2}$};\n\\path[draw=black!30,fill=black!8] (6.6,2.65) rectangle (7.55,3.6);\n\\node[font=\\small,align=center,font=\\small] at (7.075,3.125) {$- \\frac{1}{2}$};\n\\path[draw=black!30,fill=black!8] (5.65,1.7) rectangle (6.6,2.65);\n\\node[font=\\small,align=center,font=\\small] at (6.125,2.175) {$- \\frac{1}{2}$};\n\\path[draw=black!30,fill=black!8] (6.6,1.7) rectangle (7.","enter,font=\\small] at (2.04,1.45) {$1$};\n\\path[draw=black!30,fill=black!8] (2.47,1.02) rectangle (3.33,1.88);\n\\node[font=\\small,align=center,font=\\small] at (2.9,1.45) {$2$};\n\\node[font=\\small,align=center,font=\\small] at (6.6,4.3) {$S=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR}$};\n\\path[draw=black!30,fill=black!8] (5.65,2.65) rectangle (6.6,3.6);\n\\node[font=\\small,align=center,font=\\small] at (6.125,3.125) {$\\frac{3}{2}$};\n\\path[draw=black!30,fill=black!8] (6.6,2.65) rectangle (7.55,3.6);\n\\node[font=\\small,align=center,font=\\small] at (7.075,3.125) {$- \\frac{1}{2}$};\n\\path[draw=black!30,fill=black!8] (5.65,1.7) rectangle (6.6,2.65);\n\\node[font=\\small,align=center,font=\\small] at (6.125,2.175) {$- \\frac{1}{2}$};\n\\path[draw=black!30,fill=black!8] (6.6,1.7) rectangle (7.55,2.65);\n\\node[font=\\small,align=center,font=\\small] at (7.075,2.175) {$\\frac{3}{2}$};\n\\node[font=\\small,align=center,font=\\small] at (11.2,4.3)","=\\small] at (2.9,1.45) {$2$};\n\\node[font=\\small,align=center,font=\\small] at (6.6,4.3) {$S=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR}$};\n\\path[draw=black!30,fill=black!8] (5.65,2.65) rectangle (6.6,3.6);\n\\node[font=\\small,align=center,font=\\small] at (6.125,3.125) {$\\frac{3}{2}$};\n\\path[draw=black!30,fill=black!8] (6.6,2.65) rectangle (7.55,3.6);\n\\node[font=\\small,align=center,font=\\small] at (7.075,3.125) {$- \\frac{1}{2}$};\n\\path[draw=black!30,fill=black!8] (5.65,1.7) rectangle (6.6,2.65);\n\\node[font=\\small,align=center,font=\\small] at (6.125,2.175) {$- \\frac{1}{2}$};\n\\path[draw=black!30,fill=black!8] (6.6,1.7) rectangle (7.55,2.65);\n\\node[font=\\small,align=center,font=\\small] at (7.075,2.175) {$\\frac{3}{2}$};\n\\node[font=\\small,align=center,font=\\small] at (11.2,4.3) {cancelling variant};\n\\path[draw=black!30,fill=black!8] (10.25,2.65) rectangle (11.2,3.6);\n\\node[font=\\small,align=center,font=\\small] at (10.725,","ll=black!8] (5.65,2.65) rectangle (6.6,3.6);\n\\node[font=\\small,align=center,font=\\small] at (6.125,3.125) {$\\frac{3}{2}$};\n\\path[draw=black!30,fill=black!8] (6.6,2.65) rectangle (7.55,3.6);\n\\node[font=\\small,align=center,font=\\small] at (7.075,3.125) {$- \\frac{1}{2}$};\n\\path[draw=black!30,fill=black!8] (5.65,1.7) rectangle (6.6,2.65);\n\\node[font=\\small,align=center,font=\\small] at (6.125,2.175) {$- \\frac{1}{2}$};\n\\path[draw=black!30,fill=black!8] (6.6,1.7) rectangle (7.55,2.65);\n\\node[font=\\small,align=center,font=\\small] at (7.075,2.175) {$\\frac{3}{2}$};\n\\node[font=\\small,align=center,font=\\small] at (11.2,4.3) {cancelling variant};\n\\path[draw=black!30,fill=black!8] (10.25,2.65) rectangle (11.2,3.6);\n\\node[font=\\small,align=center,font=\\small] at (10.725,3.125) {$\\frac{3}{2}$};\n\\path[draw=black!30,fill=white] (11.2,2.65) rectangle (12.15,3.6);\n\\node[font=\\small,align=center,font=\\small] at (11.675,3","=black!8] (6.6,2.65) rectangle (7.55,3.6);\n\\node[font=\\small,align=center,font=\\small] at (7.075,3.125) {$- \\frac{1}{2}$};\n\\path[draw=black!30,fill=black!8] (5.65,1.7) rectangle (6.6,2.65);\n\\node[font=\\small,align=center,font=\\small] at (6.125,2.175) {$- \\frac{1}{2}$};\n\\path[draw=black!30,fill=black!8] (6.6,1.7) rectangle (7.55,2.65);\n\\node[font=\\small,align=center,font=\\small] at (7.075,2.175) {$\\frac{3}{2}$};\n\\node[font=\\small,align=center,font=\\small] at (11.2,4.3) {cancelling variant};\n\\path[draw=black!30,fill=black!8] (10.25,2.65) rectangle (11.2,3.6);\n\\node[font=\\small,align=center,font=\\small] at (10.725,3.125) {$\\frac{3}{2}$};\n\\path[draw=black!30,fill=white] (11.2,2.65) rectangle (12.15,3.6);\n\\node[font=\\small,align=center,font=\\small] at (11.675,3.125) {$0$};\n\\path[draw=black!30,fill=white] (10.25,1.7) rectangle (11.2,2.65);\n\\node[font=\\small,align=center,font=\\small] at (10.725,2.175) {$0","mall] at (6.125,2.175) {$- \\frac{1}{2}$};\n\\path[draw=black!30,fill=black!8] (6.6,1.7) rectangle (7.55,2.65);\n\\node[font=\\small,align=center,font=\\small] at (7.075,2.175) {$\\frac{3}{2}$};\n\\node[font=\\small,align=center,font=\\small] at (11.2,4.3) {cancelling variant};\n\\path[draw=black!30,fill=black!8] (10.25,2.65) rectangle (11.2,3.6);\n\\node[font=\\small,align=center,font=\\small] at (10.725,3.125) {$\\frac{3}{2}$};\n\\path[draw=black!30,fill=white] (11.2,2.65) rectangle (12.15,3.6);\n\\node[font=\\small,align=center,font=\\small] at (11.675,3.125) {$0$};\n\\path[draw=black!30,fill=white] (10.25,1.7) rectangle (11.2,2.65);\n\\node[font=\\small,align=center,font=\\small] at (10.725,2.175) {$0$};\n\\path[draw=black!30,fill=black!8] (11.2,1.7) rectangle (12.15,2.65);\n\\node[font=\\small,align=center,font=\\small] at (11.675,2.175) {$\\frac{3}{2}$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.7) {A hidden route","small] at (7.075,2.175) {$\\frac{3}{2}$};\n\\node[font=\\small,align=center,font=\\small] at (11.2,4.3) {cancelling variant};\n\\path[draw=black!30,fill=black!8] (10.25,2.65) rectangle (11.2,3.6);\n\\node[font=\\small,align=center,font=\\small] at (10.725,3.125) {$\\frac{3}{2}$};\n\\path[draw=black!30,fill=white] (11.2,2.65) rectangle (12.15,3.6);\n\\node[font=\\small,align=center,font=\\small] at (11.675,3.125) {$0$};\n\\path[draw=black!30,fill=white] (10.25,1.7) rectangle (11.2,2.65);\n\\node[font=\\small,align=center,font=\\small] at (10.725,2.175) {$0$};\n\\path[draw=black!30,fill=black!8] (11.2,1.7) rectangle (12.15,2.65);\n\\node[font=\\small,align=center,font=\\small] at (11.675,2.175) {$\\frac{3}{2}$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.7) {A hidden route creates $-1/2$; adding $+1/2$ in the original block cancels it.};\n\\end{tikzpicture}","k!30,fill=black!8] (10.25,2.65) rectangle (11.2,3.6);\n\\node[font=\\small,align=center,font=\\small] at (10.725,3.125) {$\\frac{3}{2}$};\n\\path[draw=black!30,fill=white] (11.2,2.65) rectangle (12.15,3.6);\n\\node[font=\\small,align=center,font=\\small] at (11.675,3.125) {$0$};\n\\path[draw=black!30,fill=white] (10.25,1.7) rectangle (11.2,2.65);\n\\node[font=\\small,align=center,font=\\small] at (10.725,2.175) {$0$};\n\\path[draw=black!30,fill=black!8] (11.2,1.7) rectangle (12.15,2.65);\n\\node[font=\\small,align=center,font=\\small] at (11.675,2.175) {$\\frac{3}{2}$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.7) {A hidden route creates $-1/2$; adding $+1/2$ in the original block cancels it.};\n\\end{tikzpicture}","[draw=black!30,fill=white] (11.2,2.65) rectangle (12.15,3.6);\n\\node[font=\\small,align=center,font=\\small] at (11.675,3.125) {$0$};\n\\path[draw=black!30,fill=white] (10.25,1.7) rectangle (11.2,2.65);\n\\node[font=\\small,align=center,font=\\small] at (10.725,2.175) {$0$};\n\\path[draw=black!30,fill=black!8] (11.2,1.7) rectangle (12.15,2.65);\n\\node[font=\\small,align=center,font=\\small] at (11.675,2.175) {$\\frac{3}{2}$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.7) {A hidden route creates $-1/2$; adding $+1/2$ in the original block cancels it.};\n\\end{tikzpicture}","mall] at (11.675,3.125) {$0$};\n\\path[draw=black!30,fill=white] (10.25,1.7) rectangle (11.2,2.65);\n\\node[font=\\small,align=center,font=\\small] at (10.725,2.175) {$0$};\n\\path[draw=black!30,fill=black!8] (11.2,1.7) rectangle (12.15,2.65);\n\\node[font=\\small,align=center,font=\\small] at (11.675,2.175) {$\\frac{3}{2}$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.7) {A hidden route creates $-1/2$; adding $+1/2$ in the original block cancels it.};\n\\end{tikzpicture}","5,3.125) {$0$};\n\\path[draw=black!30,fill=white] (10.25,1.7) rectangle (11.2,2.65);\n\\node[font=\\small,align=center,font=\\small] at (10.725,2.175) {$0$};\n\\path[draw=black!30,fill=black!8] (11.2,1.7) rectangle (12.15,2.65);\n\\node[font=\\small,align=center,font=\\small] at (11.675,2.175) {$\\frac{3}{2}$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,0.7) {A hidden route creates $-1/2$; adding $+1/2$ in the original block cancels it.};\n\\end{tikzpicture}","\\begin{tikzpicture}[font=\\sffamily\\small,>=Latex]\n\\draw[->] (0,0)--(7,0) node[right] {$r_2$};\n\\draw[->] (0,0)--(0,4.5) node[above] {modeled cost};\n\\draw[thick] (0,.6)--(4,3.4) node[right] {$6+7r_2$ : keep};\n\\draw[thick,dashed] (0,1)--(4,1.8) node[right] {$10+2r_2$ : split};\n\\draw[dotted] (.8,0)--(.8,1.16);\\fill (.8,1.16) circle (2pt);\n\\node[below] at (.8,0) {$4/5$};\n\\foreach \\x in {0,2,3,4}{\\draw (\\x,0)--(\\x,-.08);\\node[below] at (\\x,-.1) {$\\x$};}\n\\draw[dotted] (3,0)--(3,2.7);\\fill (3,2.7) circle (2pt);\\fill (3,1.6) circle (2pt);\n\\node[above left] at (3,2.7) {$27$};\\node[below right]","\\begin{tikzpicture}[font=\\sffamily\\small,>=Latex]\n\\draw[->] (0,0)--(7,0) node[right] {$r_2$};\n\\draw[->] (0,0)--(0,4.5) node[above] {modeled cost};\n\\draw[thick] (0,.6)--(4,3.4) node[right] {$6+7r_2$ : keep};\n\\draw[thick,dashed] (0,1)--(4,1.8) node[right] {$10+2r_2$ : split};\n\\draw[dotted] (.8,0)--(.8,1.16);\\fill (.8,1.16) circle (2pt);\n\\node[below] at (.8,0) {$4/5$};\n\\foreach \\x in {0,2,3,4}{\\draw (\\x,0)--(\\x,-.08);\\node[below] at (\\x,-.1) {$\\x$};}\n\\draw[dotted] (3,0)--(3,2.7);\\fill (3,2.7) circle (2pt);\\fill (3,1.6) circle (2pt);\n\\node[above left] at (3,2.7) {$27$};\\node[below right] at (3,1.6) {$16$};\n\\end{tikzpicture}","tex]\n\\draw[->] (0,0)--(7,0) node[right] {$r_2$};\n\\draw[->] (0,0)--(0,4.5) node[above] {modeled cost};\n\\draw[thick] (0,.6)--(4,3.4) node[right] {$6+7r_2$ : keep};\n\\draw[thick,dashed] (0,1)--(4,1.8) node[right] {$10+2r_2$ : split};\n\\draw[dotted] (.8,0)--(.8,1.16);\\fill (.8,1.16) circle (2pt);\n\\node[below] at (.8,0) {$4/5$};\n\\foreach \\x in {0,2,3,4}{\\draw (\\x,0)--(\\x,-.08);\\node[below] at (\\x,-.1) {$\\x$};}\n\\draw[dotted] (3,0)--(3,2.7);\\fill (3,2.7) circle (2pt);\\fill (3,1.6) circle (2pt);\n\\node[above left] at (3,2.7) {$27$};\\node[below right] at (3,1.6) {$16$};\n\\end{tikzpicture}","-(4,3.4) node[right] {$6+7r_2$ : keep};\n\\draw[thick,dashed] (0,1)--(4,1.8) node[right] {$10+2r_2$ : split};\n\\draw[dotted] (.8,0)--(.8,1.16);\\fill (.8,1.16) circle (2pt);\n\\node[below] at (.8,0) {$4/5$};\n\\foreach \\x in {0,2,3,4}{\\draw (\\x,0)--(\\x,-.08);\\node[below] at (\\x,-.1) {$\\x$};}\n\\draw[dotted] (3,0)--(3,2.7);\\fill (3,2.7) circle (2pt);\\fill (3,1.6) circle (2pt);\n\\node[above left] at (3,2.7) {$27$};\\node[below right] at (3,1.6) {$16$};\n\\end{tikzpicture}","};\n\\draw[thick,dashed] (0,1)--(4,1.8) node[right] {$10+2r_2$ : split};\n\\draw[dotted] (.8,0)--(.8,1.16);\\fill (.8,1.16) circle (2pt);\n\\node[below] at (.8,0) {$4/5$};\n\\foreach \\x in {0,2,3,4}{\\draw (\\x,0)--(\\x,-.08);\\node[below] at (\\x,-.1) {$\\x$};}\n\\draw[dotted] (3,0)--(3,2.7);\\fill (3,2.7) circle (2pt);\\fill (3,1.6) circle (2pt);\n\\node[above left] at (3,2.7) {$27$};\\node[below right] at (3,1.6) {$16$};\n\\end{tikzpicture}","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.76,4.5) {$A$: 36 visible entries};\n\\path[draw=black!30,fill=black!8] (0.2,3.28) rectangle (0.72,3.8);\n\\node[font=\\small,align=center,font=\\small] at (0.46,3.54) {$4$};\n\\path[draw=black!30,fill=black!8] (0.72,3.28) rectangle (1.24,3.8);\n\\node[font=\\small,align=center,font=\\small] at (0.98,3.54) {$2$};\n\\path[draw=black!30,fill=black!8] (1.24,3.28) rectangle (1.76,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.5,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,3.28) rectangle (2.28,3.8);\n\\node[fo","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.76,4.5) {$A$: 36 visible entries};\n\\path[draw=black!30,fill=black!8] (0.2,3.28) rectangle (0.72,3.8);\n\\node[font=\\small,align=center,font=\\small] at (0.46,3.54) {$4$};\n\\path[draw=black!30,fill=black!8] (0.72,3.28) rectangle (1.24,3.8);\n\\node[font=\\small,align=center,font=\\small] at (0.98,3.54) {$2$};\n\\path[draw=black!30,fill=black!8] (1.24,3.28) rectangle (1.76,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.5,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,3.28) rectangle (2.28,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.02,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,3.28) rectangle (2.8,3.8);\n\\node[font=\\small,align=cent","ath[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.76,4.5) {$A$: 36 visible entries};\n\\path[draw=black!30,fill=black!8] (0.2,3.28) rectangle (0.72,3.8);\n\\node[font=\\small,align=center,font=\\small] at (0.46,3.54) {$4$};\n\\path[draw=black!30,fill=black!8] (0.72,3.28) rectangle (1.24,3.8);\n\\node[font=\\small,align=center,font=\\small] at (0.98,3.54) {$2$};\n\\path[draw=black!30,fill=black!8] (1.24,3.28) rectangle (1.76,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.5,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,3.28) rectangle (2.28,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.02,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,3.28) rectangle (2.8,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.54,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,3.28) rectangle (3.32,3.8);\n\\node[font=\\small,align=cente","es};\n\\path[draw=black!30,fill=black!8] (0.2,3.28) rectangle (0.72,3.8);\n\\node[font=\\small,align=center,font=\\small] at (0.46,3.54) {$4$};\n\\path[draw=black!30,fill=black!8] (0.72,3.28) rectangle (1.24,3.8);\n\\node[font=\\small,align=center,font=\\small] at (0.98,3.54) {$2$};\n\\path[draw=black!30,fill=black!8] (1.24,3.28) rectangle (1.76,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.5,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,3.28) rectangle (2.28,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.02,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,3.28) rectangle (2.8,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.54,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,3.28) rectangle (3.32,3.8);\n\\node[font=\\small,align=center,font=\\small] at (3.06,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,2.76) rectangle (0.72,3.28);\n\\node[font=\\small,align=cent","$};\n\\path[draw=black!30,fill=black!8] (0.72,3.28) rectangle (1.24,3.8);\n\\node[font=\\small,align=center,font=\\small] at (0.98,3.54) {$2$};\n\\path[draw=black!30,fill=black!8] (1.24,3.28) rectangle (1.76,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.5,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,3.28) rectangle (2.28,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.02,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,3.28) rectangle (2.8,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.54,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,3.28) rectangle (3.32,3.8);\n\\node[font=\\small,align=center,font=\\small] at (3.06,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,2.76) rectangle (0.72,3.28);\n\\node[font=\\small,align=center,font=\\small] at (0.46,3.02) {$2$};\n\\path[draw=black!30,fill=black!8] (0.72,2.76) rectangle (1.24,3.28);\n\\node[font=\\small,align=cen","2$};\n\\path[draw=black!30,fill=black!8] (1.24,3.28) rectangle (1.76,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.5,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,3.28) rectangle (2.28,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.02,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,3.28) rectangle (2.8,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.54,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,3.28) rectangle (3.32,3.8);\n\\node[font=\\small,align=center,font=\\small] at (3.06,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,2.76) rectangle (0.72,3.28);\n\\node[font=\\small,align=center,font=\\small] at (0.46,3.02) {$2$};\n\\path[draw=black!30,fill=black!8] (0.72,2.76) rectangle (1.24,3.28);\n\\node[font=\\small,align=center,font=\\small] at (0.98,3.02) {$4$};\n\\path[draw=black!30,fill=black!8] (1.24,2.76) rectangle (1.76,3.28);\n\\node[font=\\small,align=c","1$};\n\\path[draw=black!30,fill=black!8] (1.76,3.28) rectangle (2.28,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.02,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,3.28) rectangle (2.8,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.54,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,3.28) rectangle (3.32,3.8);\n\\node[font=\\small,align=center,font=\\small] at (3.06,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,2.76) rectangle (0.72,3.28);\n\\node[font=\\small,align=center,font=\\small] at (0.46,3.02) {$2$};\n\\path[draw=black!30,fill=black!8] (0.72,2.76) rectangle (1.24,3.28);\n\\node[font=\\small,align=center,font=\\small] at (0.98,3.02) {$4$};\n\\path[draw=black!30,fill=black!8] (1.24,2.76) rectangle (1.76,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.5,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,2.76) rectangle (2.28,3.28);\n\\node[font=\\small,align=","1$};\n\\path[draw=black!30,fill=black!8] (2.28,3.28) rectangle (2.8,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.54,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,3.28) rectangle (3.32,3.8);\n\\node[font=\\small,align=center,font=\\small] at (3.06,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,2.76) rectangle (0.72,3.28);\n\\node[font=\\small,align=center,font=\\small] at (0.46,3.02) {$2$};\n\\path[draw=black!30,fill=black!8] (0.72,2.76) rectangle (1.24,3.28);\n\\node[font=\\small,align=center,font=\\small] at (0.98,3.02) {$4$};\n\\path[draw=black!30,fill=black!8] (1.24,2.76) rectangle (1.76,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.5,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,2.76) rectangle (2.28,3.28);\n\\node[font=\\small,align=center,font=\\small] at (2.02,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,2.76) rectangle (2.8,3.28);\n\\node[font=\\small,align=","};\n\\path[draw=black!30,fill=black!8] (2.8,3.28) rectangle (3.32,3.8);\n\\node[font=\\small,align=center,font=\\small] at (3.06,3.54) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,2.76) rectangle (0.72,3.28);\n\\node[font=\\small,align=center,font=\\small] at (0.46,3.02) {$2$};\n\\path[draw=black!30,fill=black!8] (0.72,2.76) rectangle (1.24,3.28);\n\\node[font=\\small,align=center,font=\\small] at (0.98,3.02) {$4$};\n\\path[draw=black!30,fill=black!8] (1.24,2.76) rectangle (1.76,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.5,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,2.76) rectangle (2.28,3.28);\n\\node[font=\\small,align=center,font=\\small] at (2.02,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,2.76) rectangle (2.8,3.28);\n\\node[font=\\small,align=center,font=\\small] at (2.54,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,2.76) rectangle (3.32,3.28);\n\\node[font=\\small,align=c",";\n\\path[draw=black!30,fill=black!8] (0.2,2.76) rectangle (0.72,3.28);\n\\node[font=\\small,align=center,font=\\small] at (0.46,3.02) {$2$};\n\\path[draw=black!30,fill=black!8] (0.72,2.76) rectangle (1.24,3.28);\n\\node[font=\\small,align=center,font=\\small] at (0.98,3.02) {$4$};\n\\path[draw=black!30,fill=black!8] (1.24,2.76) rectangle (1.76,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.5,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,2.76) rectangle (2.28,3.28);\n\\node[font=\\small,align=center,font=\\small] at (2.02,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,2.76) rectangle (2.8,3.28);\n\\node[font=\\small,align=center,font=\\small] at (2.54,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,2.76) rectangle (3.32,3.28);\n\\node[font=\\small,align=center,font=\\small] at (3.06,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,2.24) rectangle (0.72,2.76);\n\\node[font=\\small,align=c","\\path[draw=black!30,fill=black!8] (0.72,2.76) rectangle (1.24,3.28);\n\\node[font=\\small,align=center,font=\\small] at (0.98,3.02) {$4$};\n\\path[draw=black!30,fill=black!8] (1.24,2.76) rectangle (1.76,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.5,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,2.76) rectangle (2.28,3.28);\n\\node[font=\\small,align=center,font=\\small] at (2.02,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,2.76) rectangle (2.8,3.28);\n\\node[font=\\small,align=center,font=\\small] at (2.54,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,2.76) rectangle (3.32,3.28);\n\\node[font=\\small,align=center,font=\\small] at (3.06,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,2.24) rectangle (0.72,2.76);\n\\node[font=\\small,align=center,font=\\small] at (0.46,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,2.24) rectangle (1.24,2.76);\n\\node[font=\\small,align=ce",";\n\\path[draw=black!30,fill=black!8] (1.24,2.76) rectangle (1.76,3.28);\n\\node[font=\\small,align=center,font=\\small] at (1.5,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,2.76) rectangle (2.28,3.28);\n\\node[font=\\small,align=center,font=\\small] at (2.02,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,2.76) rectangle (2.8,3.28);\n\\node[font=\\small,align=center,font=\\small] at (2.54,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,2.76) rectangle (3.32,3.28);\n\\node[font=\\small,align=center,font=\\small] at (3.06,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,2.24) rectangle (0.72,2.76);\n\\node[font=\\small,align=center,font=\\small] at (0.46,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,2.24) rectangle (1.24,2.76);\n\\node[font=\\small,align=center,font=\\small] at (0.98,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,2.24) rectangle (1.76,2.76);\n\\node[font=\\small,align=ce",";\n\\path[draw=black!30,fill=black!8] (1.76,2.76) rectangle (2.28,3.28);\n\\node[font=\\small,align=center,font=\\small] at (2.02,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,2.76) rectangle (2.8,3.28);\n\\node[font=\\small,align=center,font=\\small] at (2.54,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,2.76) rectangle (3.32,3.28);\n\\node[font=\\small,align=center,font=\\small] at (3.06,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,2.24) rectangle (0.72,2.76);\n\\node[font=\\small,align=center,font=\\small] at (0.46,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,2.24) rectangle (1.24,2.76);\n\\node[font=\\small,align=center,font=\\small] at (0.98,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,2.24) rectangle (1.76,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.5,2.5) {$4$};\n\\path[draw=black!30,fill=black!8] (1.76,2.24) rectangle (2.28,2.76);\n\\node[font=\\small,align=cen","$};\n\\path[draw=black!30,fill=black!8] (2.28,2.76) rectangle (2.8,3.28);\n\\node[font=\\small,align=center,font=\\small] at (2.54,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,2.76) rectangle (3.32,3.28);\n\\node[font=\\small,align=center,font=\\small] at (3.06,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,2.24) rectangle (0.72,2.76);\n\\node[font=\\small,align=center,font=\\small] at (0.46,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,2.24) rectangle (1.24,2.76);\n\\node[font=\\small,align=center,font=\\small] at (0.98,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,2.24) rectangle (1.76,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.5,2.5) {$4$};\n\\path[draw=black!30,fill=black!8] (1.76,2.24) rectangle (2.28,2.76);\n\\node[font=\\small,align=center,font=\\small] at (2.02,2.5) {$2$};\n\\path[draw=black!30,fill=black!8] (2.28,2.24) rectangle (2.8,2.76);\n\\node[font=\\small,align=cen","$};\n\\path[draw=black!30,fill=black!8] (2.8,2.76) rectangle (3.32,3.28);\n\\node[font=\\small,align=center,font=\\small] at (3.06,3.02) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,2.24) rectangle (0.72,2.76);\n\\node[font=\\small,align=center,font=\\small] at (0.46,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,2.24) rectangle (1.24,2.76);\n\\node[font=\\small,align=center,font=\\small] at (0.98,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,2.24) rectangle (1.76,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.5,2.5) {$4$};\n\\path[draw=black!30,fill=black!8] (1.76,2.24) rectangle (2.28,2.76);\n\\node[font=\\small,align=center,font=\\small] at (2.02,2.5) {$2$};\n\\path[draw=black!30,fill=black!8] (2.28,2.24) rectangle (2.8,2.76);\n\\node[font=\\small,align=center,font=\\small] at (2.54,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,2.24) rectangle (3.32,2.76);\n\\node[font=\\small,align=cent","1$};\n\\path[draw=black!30,fill=black!8] (0.2,2.24) rectangle (0.72,2.76);\n\\node[font=\\small,align=center,font=\\small] at (0.46,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,2.24) rectangle (1.24,2.76);\n\\node[font=\\small,align=center,font=\\small] at (0.98,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,2.24) rectangle (1.76,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.5,2.5) {$4$};\n\\path[draw=black!30,fill=black!8] (1.76,2.24) rectangle (2.28,2.76);\n\\node[font=\\small,align=center,font=\\small] at (2.02,2.5) {$2$};\n\\path[draw=black!30,fill=black!8] (2.28,2.24) rectangle (2.8,2.76);\n\\node[font=\\small,align=center,font=\\small] at (2.54,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,2.24) rectangle (3.32,2.76);\n\\node[font=\\small,align=center,font=\\small] at (3.06,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,1.72) rectangle (0.72,2.24);\n\\node[font=\\small,align=cent","$};\n\\path[draw=black!30,fill=black!8] (0.72,2.24) rectangle (1.24,2.76);\n\\node[font=\\small,align=center,font=\\small] at (0.98,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,2.24) rectangle (1.76,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.5,2.5) {$4$};\n\\path[draw=black!30,fill=black!8] (1.76,2.24) rectangle (2.28,2.76);\n\\node[font=\\small,align=center,font=\\small] at (2.02,2.5) {$2$};\n\\path[draw=black!30,fill=black!8] (2.28,2.24) rectangle (2.8,2.76);\n\\node[font=\\small,align=center,font=\\small] at (2.54,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,2.24) rectangle (3.32,2.76);\n\\node[font=\\small,align=center,font=\\small] at (3.06,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,1.72) rectangle (0.72,2.24);\n\\node[font=\\small,align=center,font=\\small] at (0.46,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,1.72) rectangle (1.24,2.24);\n\\node[font=\\small,align=cen","1$};\n\\path[draw=black!30,fill=black!8] (1.24,2.24) rectangle (1.76,2.76);\n\\node[font=\\small,align=center,font=\\small] at (1.5,2.5) {$4$};\n\\path[draw=black!30,fill=black!8] (1.76,2.24) rectangle (2.28,2.76);\n\\node[font=\\small,align=center,font=\\small] at (2.02,2.5) {$2$};\n\\path[draw=black!30,fill=black!8] (2.28,2.24) rectangle (2.8,2.76);\n\\node[font=\\small,align=center,font=\\small] at (2.54,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,2.24) rectangle (3.32,2.76);\n\\node[font=\\small,align=center,font=\\small] at (3.06,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,1.72) rectangle (0.72,2.24);\n\\node[font=\\small,align=center,font=\\small] at (0.46,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,1.72) rectangle (1.24,2.24);\n\\node[font=\\small,align=center,font=\\small] at (0.98,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,1.72) rectangle (1.76,2.24);\n\\node[font=\\small,align=c","4$};\n\\path[draw=black!30,fill=black!8] (1.76,2.24) rectangle (2.28,2.76);\n\\node[font=\\small,align=center,font=\\small] at (2.02,2.5) {$2$};\n\\path[draw=black!30,fill=black!8] (2.28,2.24) rectangle (2.8,2.76);\n\\node[font=\\small,align=center,font=\\small] at (2.54,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,2.24) rectangle (3.32,2.76);\n\\node[font=\\small,align=center,font=\\small] at (3.06,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,1.72) rectangle (0.72,2.24);\n\\node[font=\\small,align=center,font=\\small] at (0.46,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,1.72) rectangle (1.24,2.24);\n\\node[font=\\small,align=center,font=\\small] at (0.98,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,1.72) rectangle (1.76,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.5,1.98) {$2$};\n\\path[draw=black!30,fill=black!8] (1.76,1.72) rectangle (2.28,2.24);\n\\node[font=\\small,align=","2$};\n\\path[draw=black!30,fill=black!8] (2.28,2.24) rectangle (2.8,2.76);\n\\node[font=\\small,align=center,font=\\small] at (2.54,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,2.24) rectangle (3.32,2.76);\n\\node[font=\\small,align=center,font=\\small] at (3.06,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,1.72) rectangle (0.72,2.24);\n\\node[font=\\small,align=center,font=\\small] at (0.46,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,1.72) rectangle (1.24,2.24);\n\\node[font=\\small,align=center,font=\\small] at (0.98,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,1.72) rectangle (1.76,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.5,1.98) {$2$};\n\\path[draw=black!30,fill=black!8] (1.76,1.72) rectangle (2.28,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.02,1.98) {$4$};\n\\path[draw=black!30,fill=black!8] (2.28,1.72) rectangle (2.8,2.24);\n\\node[font=\\small,align=","};\n\\path[draw=black!30,fill=black!8] (2.8,2.24) rectangle (3.32,2.76);\n\\node[font=\\small,align=center,font=\\small] at (3.06,2.5) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,1.72) rectangle (0.72,2.24);\n\\node[font=\\small,align=center,font=\\small] at (0.46,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,1.72) rectangle (1.24,2.24);\n\\node[font=\\small,align=center,font=\\small] at (0.98,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,1.72) rectangle (1.76,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.5,1.98) {$2$};\n\\path[draw=black!30,fill=black!8] (1.76,1.72) rectangle (2.28,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.02,1.98) {$4$};\n\\path[draw=black!30,fill=black!8] (2.28,1.72) rectangle (2.8,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.54,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,1.72) rectangle (3.32,2.24);\n\\node[font=\\small,align=c",";\n\\path[draw=black!30,fill=black!8] (0.2,1.72) rectangle (0.72,2.24);\n\\node[font=\\small,align=center,font=\\small] at (0.46,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,1.72) rectangle (1.24,2.24);\n\\node[font=\\small,align=center,font=\\small] at (0.98,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,1.72) rectangle (1.76,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.5,1.98) {$2$};\n\\path[draw=black!30,fill=black!8] (1.76,1.72) rectangle (2.28,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.02,1.98) {$4$};\n\\path[draw=black!30,fill=black!8] (2.28,1.72) rectangle (2.8,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.54,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,1.72) rectangle (3.32,2.24);\n\\node[font=\\small,align=center,font=\\small] at (3.06,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,1.2) rectangle (0.72,1.72);\n\\node[font=\\small,align=ce","\\path[draw=black!30,fill=black!8] (0.72,1.72) rectangle (1.24,2.24);\n\\node[font=\\small,align=center,font=\\small] at (0.98,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,1.72) rectangle (1.76,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.5,1.98) {$2$};\n\\path[draw=black!30,fill=black!8] (1.76,1.72) rectangle (2.28,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.02,1.98) {$4$};\n\\path[draw=black!30,fill=black!8] (2.28,1.72) rectangle (2.8,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.54,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,1.72) rectangle (3.32,2.24);\n\\node[font=\\small,align=center,font=\\small] at (3.06,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,1.2) rectangle (0.72,1.72);\n\\node[font=\\small,align=center,font=\\small] at (0.46,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,1.2) rectangle (1.24,1.72);\n\\node[font=\\small,align=cen",";\n\\path[draw=black!30,fill=black!8] (1.24,1.72) rectangle (1.76,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.5,1.98) {$2$};\n\\path[draw=black!30,fill=black!8] (1.76,1.72) rectangle (2.28,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.02,1.98) {$4$};\n\\path[draw=black!30,fill=black!8] (2.28,1.72) rectangle (2.8,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.54,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,1.72) rectangle (3.32,2.24);\n\\node[font=\\small,align=center,font=\\small] at (3.06,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,1.2) rectangle (0.72,1.72);\n\\node[font=\\small,align=center,font=\\small] at (0.46,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,1.2) rectangle (1.24,1.72);\n\\node[font=\\small,align=center,font=\\small] at (0.98,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,1.2) rectangle (1.76,1.72);\n\\node[font=\\small,align=cen",";\n\\path[draw=black!30,fill=black!8] (1.76,1.72) rectangle (2.28,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.02,1.98) {$4$};\n\\path[draw=black!30,fill=black!8] (2.28,1.72) rectangle (2.8,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.54,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,1.72) rectangle (3.32,2.24);\n\\node[font=\\small,align=center,font=\\small] at (3.06,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,1.2) rectangle (0.72,1.72);\n\\node[font=\\small,align=center,font=\\small] at (0.46,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,1.2) rectangle (1.24,1.72);\n\\node[font=\\small,align=center,font=\\small] at (0.98,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,1.2) rectangle (1.76,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.5,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,1.2) rectangle (2.28,1.72);\n\\node[font=\\small,align=cent","$};\n\\path[draw=black!30,fill=black!8] (2.28,1.72) rectangle (2.8,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.54,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (2.8,1.72) rectangle (3.32,2.24);\n\\node[font=\\small,align=center,font=\\small] at (3.06,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,1.2) rectangle (0.72,1.72);\n\\node[font=\\small,align=center,font=\\small] at (0.46,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,1.2) rectangle (1.24,1.72);\n\\node[font=\\small,align=center,font=\\small] at (0.98,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,1.2) rectangle (1.76,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.5,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,1.2) rectangle (2.28,1.72);\n\\node[font=\\small,align=center,font=\\small] at (2.02,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,1.2) rectangle (2.8,1.72);\n\\node[font=\\small,align=cent","$};\n\\path[draw=black!30,fill=black!8] (2.8,1.72) rectangle (3.32,2.24);\n\\node[font=\\small,align=center,font=\\small] at (3.06,1.98) {$1$};\n\\path[draw=black!30,fill=black!8] (0.2,1.2) rectangle (0.72,1.72);\n\\node[font=\\small,align=center,font=\\small] at (0.46,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,1.2) rectangle (1.24,1.72);\n\\node[font=\\small,align=center,font=\\small] at (0.98,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,1.2) rectangle (1.76,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.5,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,1.2) rectangle (2.28,1.72);\n\\node[font=\\small,align=center,font=\\small] at (2.02,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,1.2) rectangle (2.8,1.72);\n\\node[font=\\small,align=center,font=\\small] at (2.54,1.46) {$4$};\n\\path[draw=black!30,fill=black!8] (2.8,1.2) rectangle (3.32,1.72);\n\\node[font=\\small,align=cente","1$};\n\\path[draw=black!30,fill=black!8] (0.2,1.2) rectangle (0.72,1.72);\n\\node[font=\\small,align=center,font=\\small] at (0.46,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,1.2) rectangle (1.24,1.72);\n\\node[font=\\small,align=center,font=\\small] at (0.98,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,1.2) rectangle (1.76,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.5,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,1.2) rectangle (2.28,1.72);\n\\node[font=\\small,align=center,font=\\small] at (2.02,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,1.2) rectangle (2.8,1.72);\n\\node[font=\\small,align=center,font=\\small] at (2.54,1.46) {$4$};\n\\path[draw=black!30,fill=black!8] (2.8,1.2) rectangle (3.32,1.72);\n\\node[font=\\small,align=center,font=\\small] at (3.06,1.46) {$2$};\n\\path[draw=black!30,fill=black!8] (0.2,0.68) rectangle (0.72,1.2);\n\\node[font=\\small,align=cente","$};\n\\path[draw=black!30,fill=black!8] (0.72,1.2) rectangle (1.24,1.72);\n\\node[font=\\small,align=center,font=\\small] at (0.98,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,1.2) rectangle (1.76,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.5,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,1.2) rectangle (2.28,1.72);\n\\node[font=\\small,align=center,font=\\small] at (2.02,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,1.2) rectangle (2.8,1.72);\n\\node[font=\\small,align=center,font=\\small] at (2.54,1.46) {$4$};\n\\path[draw=black!30,fill=black!8] (2.8,1.2) rectangle (3.32,1.72);\n\\node[font=\\small,align=center,font=\\small] at (3.06,1.46) {$2$};\n\\path[draw=black!30,fill=black!8] (0.2,0.68) rectangle (0.72,1.2);\n\\node[font=\\small,align=center,font=\\small] at (0.46,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,0.68) rectangle (1.24,1.2);\n\\node[font=\\small,align=cente","1$};\n\\path[draw=black!30,fill=black!8] (1.24,1.2) rectangle (1.76,1.72);\n\\node[font=\\small,align=center,font=\\small] at (1.5,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,1.2) rectangle (2.28,1.72);\n\\node[font=\\small,align=center,font=\\small] at (2.02,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,1.2) rectangle (2.8,1.72);\n\\node[font=\\small,align=center,font=\\small] at (2.54,1.46) {$4$};\n\\path[draw=black!30,fill=black!8] (2.8,1.2) rectangle (3.32,1.72);\n\\node[font=\\small,align=center,font=\\small] at (3.06,1.46) {$2$};\n\\path[draw=black!30,fill=black!8] (0.2,0.68) rectangle (0.72,1.2);\n\\node[font=\\small,align=center,font=\\small] at (0.46,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,0.68) rectangle (1.24,1.2);\n\\node[font=\\small,align=center,font=\\small] at (0.98,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,0.68) rectangle (1.76,1.2);\n\\node[font=\\small,align=cent","1$};\n\\path[draw=black!30,fill=black!8] (1.76,1.2) rectangle (2.28,1.72);\n\\node[font=\\small,align=center,font=\\small] at (2.02,1.46) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,1.2) rectangle (2.8,1.72);\n\\node[font=\\small,align=center,font=\\small] at (2.54,1.46) {$4$};\n\\path[draw=black!30,fill=black!8] (2.8,1.2) rectangle (3.32,1.72);\n\\node[font=\\small,align=center,font=\\small] at (3.06,1.46) {$2$};\n\\path[draw=black!30,fill=black!8] (0.2,0.68) rectangle (0.72,1.2);\n\\node[font=\\small,align=center,font=\\small] at (0.46,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,0.68) rectangle (1.24,1.2);\n\\node[font=\\small,align=center,font=\\small] at (0.98,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,0.68) rectangle (1.76,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.5,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,0.68) rectangle (2.28,1.2);\n\\node[font=\\small,align=cent","$1$};\n\\path[draw=black!30,fill=black!8] (2.28,1.2) rectangle (2.8,1.72);\n\\node[font=\\small,align=center,font=\\small] at (2.54,1.46) {$4$};\n\\path[draw=black!30,fill=black!8] (2.8,1.2) rectangle (3.32,1.72);\n\\node[font=\\small,align=center,font=\\small] at (3.06,1.46) {$2$};\n\\path[draw=black!30,fill=black!8] (0.2,0.68) rectangle (0.72,1.2);\n\\node[font=\\small,align=center,font=\\small] at (0.46,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,0.68) rectangle (1.24,1.2);\n\\node[font=\\small,align=center,font=\\small] at (0.98,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,0.68) rectangle (1.76,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.5,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,0.68) rectangle (2.28,1.2);\n\\node[font=\\small,align=center,font=\\small] at (2.02,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,0.68) rectangle (2.8,1.2);\n\\node[font=\\small,align=cent","4$};\n\\path[draw=black!30,fill=black!8] (2.8,1.2) rectangle (3.32,1.72);\n\\node[font=\\small,align=center,font=\\small] at (3.06,1.46) {$2$};\n\\path[draw=black!30,fill=black!8] (0.2,0.68) rectangle (0.72,1.2);\n\\node[font=\\small,align=center,font=\\small] at (0.46,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,0.68) rectangle (1.24,1.2);\n\\node[font=\\small,align=center,font=\\small] at (0.98,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,0.68) rectangle (1.76,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.5,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,0.68) rectangle (2.28,1.2);\n\\node[font=\\small,align=center,font=\\small] at (2.02,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,0.68) rectangle (2.8,1.2);\n\\node[font=\\small,align=center,font=\\small] at (2.54,0.94) {$2$};\n\\path[draw=black!30,fill=black!8] (2.8,0.68) rectangle (3.32,1.2);\n\\node[font=\\small,align=cente","2$};\n\\path[draw=black!30,fill=black!8] (0.2,0.68) rectangle (0.72,1.2);\n\\node[font=\\small,align=center,font=\\small] at (0.46,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (0.72,0.68) rectangle (1.24,1.2);\n\\node[font=\\small,align=center,font=\\small] at (0.98,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,0.68) rectangle (1.76,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.5,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,0.68) rectangle (2.28,1.2);\n\\node[font=\\small,align=center,font=\\small] at (2.02,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,0.68) rectangle (2.8,1.2);\n\\node[font=\\small,align=center,font=\\small] at (2.54,0.94) {$2$};\n\\path[draw=black!30,fill=black!8] (2.8,0.68) rectangle (3.32,1.2);\n\\node[font=\\small,align=center,font=\\small] at (3.06,0.94) {$4$};\n\\draw[black!35] (4,0.2) -- (4,4.35);\n\\draw[line width=.6pt] (4.4,0.9) -- (8.1,0.9);\n\\node[font=\\","$};\n\\path[draw=black!30,fill=black!8] (0.72,0.68) rectangle (1.24,1.2);\n\\node[font=\\small,align=center,font=\\small] at (0.98,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (1.24,0.68) rectangle (1.76,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.5,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,0.68) rectangle (2.28,1.2);\n\\node[font=\\small,align=center,font=\\small] at (2.02,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,0.68) rectangle (2.8,1.2);\n\\node[font=\\small,align=center,font=\\small] at (2.54,0.94) {$2$};\n\\path[draw=black!30,fill=black!8] (2.8,0.68) rectangle (3.32,1.2);\n\\node[font=\\small,align=center,font=\\small] at (3.06,0.94) {$4$};\n\\draw[black!35] (4,0.2) -- (4,4.35);\n\\draw[line width=.6pt] (4.4,0.9) -- (8.1,0.9);\n\\node[font=\\small,align=center] at (6.25,4.15) {stored scalars};\n\\path[draw=black,fill=black!12,line width=.6pt] (4.844,0.9) rectangle (5.806,3.4)","1$};\n\\path[draw=black!30,fill=black!8] (1.24,0.68) rectangle (1.76,1.2);\n\\node[font=\\small,align=center,font=\\small] at (1.5,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (1.76,0.68) rectangle (2.28,1.2);\n\\node[font=\\small,align=center,font=\\small] at (2.02,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,0.68) rectangle (2.8,1.2);\n\\node[font=\\small,align=center,font=\\small] at (2.54,0.94) {$2$};\n\\path[draw=black!30,fill=black!8] (2.8,0.68) rectangle (3.32,1.2);\n\\node[font=\\small,align=center,font=\\small] at (3.06,0.94) {$4$};\n\\draw[black!35] (4,0.2) -- (4,4.35);\n\\draw[line width=.6pt] (4.4,0.9) -- (8.1,0.9);\n\\node[font=\\small,align=center] at (6.25,4.15) {stored scalars};\n\\path[draw=black,fill=black!12,line width=.6pt] (4.844,0.9) rectangle (5.806,3.4);\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.325,0.6) {dense};\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.325","1$};\n\\path[draw=black!30,fill=black!8] (1.76,0.68) rectangle (2.28,1.2);\n\\node[font=\\small,align=center,font=\\small] at (2.02,0.94) {$1$};\n\\path[draw=black!30,fill=black!8] (2.28,0.68) rectangle (2.8,1.2);\n\\node[font=\\small,align=center,font=\\small] at (2.54,0.94) {$2$};\n\\path[draw=black!30,fill=black!8] (2.8,0.68) rectangle (3.32,1.2);\n\\node[font=\\small,align=center,font=\\small] at (3.06,0.94) {$4$};\n\\draw[black!35] (4,0.2) -- (4,4.35);\n\\draw[line width=.6pt] (4.4,0.9) -- (8.1,0.9);\n\\node[font=\\small,align=center] at (6.25,4.15) {stored scalars};\n\\path[draw=black,fill=black!12,line width=.6pt] (4.844,0.9) rectangle (5.806,3.4);\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.325,0.6) {dense};\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.325,3.62) {$36$};\n\\path[draw=black,fill=black!12,line width=.6pt] (6.694,0.9) rectangle (7.656,2.566667);\n\\node[font=\\small,align=center","mall] at (3.06,0.94) {$4$};\n\\draw[black!35] (4,0.2) -- (4,4.35);\n\\draw[line width=.6pt] (4.4,0.9) -- (8.1,0.9);\n\\node[font=\\small,align=center] at (6.25,4.15) {stored scalars};\n\\path[draw=black,fill=black!12,line width=.6pt] (4.844,0.9) rectangle (5.806,3.4);\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.325,0.6) {dense};\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.325,3.62) {$36$};\n\\path[draw=black,fill=black!12,line width=.6pt] (6.694,0.9) rectangle (7.656,2.566667);\n\\node[font=\\small,align=center,font=\\scriptsize] at (7.175,0.6) {factors};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7.175,2.786667) {$24$};\n\\draw[black!13,line width=.25pt] (9,0.8) -- (9,3.6);\n\\draw[] (9,0.74) -- (9,0.86);\n\\node[font=\\small,align=center,font=\\scriptsize] at (9,0.54) {0};\n\\draw[black!13,line width=.25pt] (10.466667,0.8) -- (10.466667,3.6);\n\\draw[] (10.466667,0.74) -- (10.466667","tangle (5.806,3.4);\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.325,0.6) {dense};\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.325,3.62) {$36$};\n\\path[draw=black,fill=black!12,line width=.6pt] (6.694,0.9) rectangle (7.656,2.566667);\n\\node[font=\\small,align=center,font=\\scriptsize] at (7.175,0.6) {factors};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7.175,2.786667) {$24$};\n\\draw[black!13,line width=.25pt] (9,0.8) -- (9,3.6);\n\\draw[] (9,0.74) -- (9,0.86);\n\\node[font=\\small,align=center,font=\\scriptsize] at (9,0.54) {0};\n\\draw[black!13,line width=.25pt] (10.466667,0.8) -- (10.466667,3.6);\n\\draw[] (10.466667,0.74) -- (10.466667,0.86);\n\\node[font=\\small,align=center,font=\\scriptsize] at (10.466667,0.54) {2};\n\\draw[black!13,line width=.25pt] (11.933333,0.8) -- (11.933333,3.6);\n\\draw[] (11.933333,0.74) -- (11.933333,0.86);\n\\node[font=\\small,align=center,font=\\script","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.52,4.5) {$T$};\n\\path[draw=black!30,fill=black!8] (0.35,3.02) rectangle (1.13,3.8);\n\\node[font=\\small,align=center,font=\\small] at (0.74,3.41) {$2$};\n\\path[draw=black!30,fill=black!8] (1.13,3.02) rectangle (1.91,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.52,3.41) {$-1$};\n\\path[draw=black!30,fill=black!8] (1.91,3.02) rectangle (2.69,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.3,3.41) {$5$};\n\\path[draw=black!30,fill=black!8] (0.35,2.24) rectangle (1.13,3.02);\n\\node[font=\\small,align=center,font=\\small] at (0.74,2.63) {$3$};\n\\path[draw=black!30,fill=black!8] (1.13,2.24) rectangle (1.91,3.02);\n\\node[font=\\small,align=center,font=\\small] at (1.52,2.63) {$2$};\n\\path[draw=black!30,fill","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.52,4.5) {$T$};\n\\path[draw=black!30,fill=black!8] (0.35,3.02) rectangle (1.13,3.8);\n\\node[font=\\small,align=center,font=\\small] at (0.74,3.41) {$2$};\n\\path[draw=black!30,fill=black!8] (1.13,3.02) rectangle (1.91,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.52,3.41) {$-1$};\n\\path[draw=black!30,fill=black!8] (1.91,3.02) rectangle (2.69,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.3,3.41) {$5$};\n\\path[draw=black!30,fill=black!8] (0.35,2.24) rectangle (1.13,3.02);\n\\node[font=\\small,align=center,font=\\small] at (0.74,2.63) {$3$};\n\\path[draw=black!30,fill=black!8] (1.13,2.24) rectangle (1.91,3.02);\n\\node[font=\\small,align=center,font=\\small] at (1.52,2.63) {$2$};\n\\path[draw=black!30,fill=black!8] (1.91,2.24) rectangle (2.69,3.02);\n\\node[font=\\small,align=center,font=\\small] at (2.3,2.63) {$-1$};\n\\path[draw=black!30,fil","line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.52,4.5) {$T$};\n\\path[draw=black!30,fill=black!8] (0.35,3.02) rectangle (1.13,3.8);\n\\node[font=\\small,align=center,font=\\small] at (0.74,3.41) {$2$};\n\\path[draw=black!30,fill=black!8] (1.13,3.02) rectangle (1.91,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.52,3.41) {$-1$};\n\\path[draw=black!30,fill=black!8] (1.91,3.02) rectangle (2.69,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.3,3.41) {$5$};\n\\path[draw=black!30,fill=black!8] (0.35,2.24) rectangle (1.13,3.02);\n\\node[font=\\small,align=center,font=\\small] at (0.74,2.63) {$3$};\n\\path[draw=black!30,fill=black!8] (1.13,2.24) rectangle (1.91,3.02);\n\\node[font=\\small,align=center,font=\\small] at (1.52,2.63) {$2$};\n\\path[draw=black!30,fill=black!8] (1.91,2.24) rectangle (2.69,3.02);\n\\node[font=\\small,align=center,font=\\small] at (2.3,2.63) {$-1$};\n\\path[draw=black!30,fill=black!8] (0.35,1.46) rectangle (1.13,2.24);\n\\node[font=\\small,align=center,font=\\small] at (0.74,1.85) {$4$};\n\\path[draw=black!30,f…","};\n\\path[draw=black!30,fill=black!8] (0.35,3.02) rectangle (1.13,3.8);\n\\node[font=\\small,align=center,font=\\small] at (0.74,3.41) {$2$};\n\\path[draw=black!30,fill=black!8] (1.13,3.02) rectangle (1.91,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.52,3.41) {$-1$};\n\\path[draw=black!30,fill=black!8] (1.91,3.02) rectangle (2.69,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.3,3.41) {$5$};\n\\path[draw=black!30,fill=black!8] (0.35,2.24) rectangle (1.13,3.02);\n\\node[font=\\small,align=center,font=\\small] at (0.74,2.63) {$3$};\n\\path[draw=black!30,fill=black!8] (1.13,2.24) rectangle (1.91,3.02);\n\\node[font=\\small,align=center,font=\\small] at (1.52,2.63) {$2$};\n\\path[draw=black!30,fill=black!8] (1.91,2.24) rectangle (2.69,3.02);\n\\node[font=\\small,align=center,font=\\small] at (2.3,2.63) {$-1$};\n\\path[draw=black!30,fill=black!8] (0.35,1.46) rectangle (1.13,2.24);\n\\node[font=\\small,align=center,font=\\small] at (0.74,1.85) {$4$};\n\\path[draw=black!30,fill=black!8] (1.13,1.46) rectangle (1.91,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.52,1.85) {$3$};\n\\path[draw=black!30,…",";\n\\path[draw=black!30,fill=black!8] (1.13,3.02) rectangle (1.91,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.52,3.41) {$-1$};\n\\path[draw=black!30,fill=black!8] (1.91,3.02) rectangle (2.69,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.3,3.41) {$5$};\n\\path[draw=black!30,fill=black!8] (0.35,2.24) rectangle (1.13,3.02);\n\\node[font=\\small,align=center,font=\\small] at (0.74,2.63) {$3$};\n\\path[draw=black!30,fill=black!8] (1.13,2.24) rectangle (1.91,3.02);\n\\node[font=\\small,align=center,font=\\small] at (1.52,2.63) {$2$};\n\\path[draw=black!30,fill=black!8] (1.91,2.24) rectangle (2.69,3.02);\n\\node[font=\\small,align=center,font=\\small] at (2.3,2.63) {$-1$};\n\\path[draw=black!30,fill=black!8] (0.35,1.46) rectangle (1.13,2.24);\n\\node[font=\\small,align=center,font=\\small] at (0.74,1.85) {$4$};\n\\path[draw=black!30,fill=black!8] (1.13,1.46) rectangle (1.91,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.52,1.85) {$3$};\n\\path[draw=black!30,fill=black!8] (1.91,1.46) rectangle (2.69,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.3,1.85) {$2$};\n\\node[font=\\small,ali…",";\n\\path[draw=black!30,fill=black!8] (1.91,3.02) rectangle (2.69,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.3,3.41) {$5$};\n\\path[draw=black!30,fill=black!8] (0.35,2.24) rectangle (1.13,3.02);\n\\node[font=\\small,align=center,font=\\small] at (0.74,2.63) {$3$};\n\\path[draw=black!30,fill=black!8] (1.13,2.24) rectangle (1.91,3.02);\n\\node[font=\\small,align=center,font=\\small] at (1.52,2.63) {$2$};\n\\path[draw=black!30,fill=black!8] (1.91,2.24) rectangle (2.69,3.02);\n\\node[font=\\small,align=center,font=\\small] at (2.3,2.63) {$-1$};\n\\path[draw=black!30,fill=black!8] (0.35,1.46) rectangle (1.13,2.24);\n\\node[font=\\small,align=center,font=\\small] at (0.74,1.85) {$4$};\n\\path[draw=black!30,fill=black!8] (1.13,1.46) rectangle (1.91,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.52,1.85) {$3$};\n\\path[draw=black!30,fill=black!8] (1.91,1.46) rectangle (2.69,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.3,1.85) {$2$};\n\\node[font=\\small,align=center,font=\\small] at (4.355,4.5) {circulant column $g$};\n\\path[draw=black!30,fill=black!8] (4.1,3.29) rectangle (4.61,3.8);\n\\node[…","\\path[draw=black!30,fill=black!8] (0.35,2.24) rectangle (1.13,3.02);\n\\node[font=\\small,align=center,font=\\small] at (0.74,2.63) {$3$};\n\\path[draw=black!30,fill=black!8] (1.13,2.24) rectangle (1.91,3.02);\n\\node[font=\\small,align=center,font=\\small] at (1.52,2.63) {$2$};\n\\path[draw=black!30,fill=black!8] (1.91,2.24) rectangle (2.69,3.02);\n\\node[font=\\small,align=center,font=\\small] at (2.3,2.63) {$-1$};\n\\path[draw=black!30,fill=black!8] (0.35,1.46) rectangle (1.13,2.24);\n\\node[font=\\small,align=center,font=\\small] at (0.74,1.85) {$4$};\n\\path[draw=black!30,fill=black!8] (1.13,1.46) rectangle (1.91,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.52,1.85) {$3$};\n\\path[draw=black!30,fill=black!8] (1.91,1.46) rectangle (2.69,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.3,1.85) {$2$};\n\\node[font=\\small,align=center,font=\\small] at (4.355,4.5) {circulant column $g$};\n\\path[draw=black!30,fill=black!8] (4.1,3.29) rectangle (4.61,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.545) {$2$};\n\\path[draw=black!30,fill=black!8] (4.1,2.78) rectangle (4.61,3.29);\n\\node…","\\path[draw=black!30,fill=black!8] (1.13,2.24) rectangle (1.91,3.02);\n\\node[font=\\small,align=center,font=\\small] at (1.52,2.63) {$2$};\n\\path[draw=black!30,fill=black!8] (1.91,2.24) rectangle (2.69,3.02);\n\\node[font=\\small,align=center,font=\\small] at (2.3,2.63) {$-1$};\n\\path[draw=black!30,fill=black!8] (0.35,1.46) rectangle (1.13,2.24);\n\\node[font=\\small,align=center,font=\\small] at (0.74,1.85) {$4$};\n\\path[draw=black!30,fill=black!8] (1.13,1.46) rectangle (1.91,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.52,1.85) {$3$};\n\\path[draw=black!30,fill=black!8] (1.91,1.46) rectangle (2.69,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.3,1.85) {$2$};\n\\node[font=\\small,align=center,font=\\small] at (4.355,4.5) {circulant column $g$};\n\\path[draw=black!30,fill=black!8] (4.1,3.29) rectangle (4.61,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.545) {$2$};\n\\path[draw=black!30,fill=black!8] (4.1,2.78) rectangle (4.61,3.29);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.035) {$3$};\n\\path[draw=black!30,fill=black!8] (4.1,2.27) rectangle (4.61,2.78);\n\\nod…","\\path[draw=black!30,fill=black!8] (1.91,2.24) rectangle (2.69,3.02);\n\\node[font=\\small,align=center,font=\\small] at (2.3,2.63) {$-1$};\n\\path[draw=black!30,fill=black!8] (0.35,1.46) rectangle (1.13,2.24);\n\\node[font=\\small,align=center,font=\\small] at (0.74,1.85) {$4$};\n\\path[draw=black!30,fill=black!8] (1.13,1.46) rectangle (1.91,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.52,1.85) {$3$};\n\\path[draw=black!30,fill=black!8] (1.91,1.46) rectangle (2.69,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.3,1.85) {$2$};\n\\node[font=\\small,align=center,font=\\small] at (4.355,4.5) {circulant column $g$};\n\\path[draw=black!30,fill=black!8] (4.1,3.29) rectangle (4.61,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.545) {$2$};\n\\path[draw=black!30,fill=black!8] (4.1,2.78) rectangle (4.61,3.29);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.035) {$3$};\n\\path[draw=black!30,fill=black!8] (4.1,2.27) rectangle (4.61,2.78);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.525) {$4$};\n\\path[draw=black!30,fill=white] (4.1,1.76) rectangle (4.61,2.27);\n\\node…","\\path[draw=black!30,fill=black!8] (0.35,1.46) rectangle (1.13,2.24);\n\\node[font=\\small,align=center,font=\\small] at (0.74,1.85) {$4$};\n\\path[draw=black!30,fill=black!8] (1.13,1.46) rectangle (1.91,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.52,1.85) {$3$};\n\\path[draw=black!30,fill=black!8] (1.91,1.46) rectangle (2.69,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.3,1.85) {$2$};\n\\node[font=\\small,align=center,font=\\small] at (4.355,4.5) {circulant column $g$};\n\\path[draw=black!30,fill=black!8] (4.1,3.29) rectangle (4.61,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.545) {$2$};\n\\path[draw=black!30,fill=black!8] (4.1,2.78) rectangle (4.61,3.29);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.035) {$3$};\n\\path[draw=black!30,fill=black!8] (4.1,2.27) rectangle (4.61,2.78);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.525) {$4$};\n\\path[draw=black!30,fill=white] (4.1,1.76) rectangle (4.61,2.27);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.015) {$0$};\n\\path[draw=black!30,fill=black!8] (4.1,1.25) rectangle (4.61,1.76);\n\\nod","mall,align=center,font=\\small] at (0.74,1.85) {$4$};\n\\path[draw=black!30,fill=black!8] (1.13,1.46) rectangle (1.91,2.24);\n\\node[font=\\small,align=center,font=\\small] at (1.52,1.85) {$3$};\n\\path[draw=black!30,fill=black!8] (1.91,1.46) rectangle (2.69,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.3,1.85) {$2$};\n\\node[font=\\small,align=center,font=\\small] at (4.355,4.5) {circulant column $g$};\n\\path[draw=black!30,fill=black!8] (4.1,3.29) rectangle (4.61,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.545) {$2$};\n\\path[draw=black!30,fill=black!8] (4.1,2.78) rectangle (4.61,3.29);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.035) {$3$};\n\\path[draw=black!30,fill=black!8] (4.1,2.27) rectangle (4.61,2.78);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.525) {$4$};\n\\path[draw=black!30,fill=white] (4.1,1.76) rectangle (4.61,2.27);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.015) {$0$};\n\\path[draw=black!30,fill=black!8] (4.1,1.25) rectangle (4.61,1.76);\n\\node[font=\\small,align=center,font=\\small] at (4.355,1.505) {$5$};\n\\path[draw=black!3…","mall,align=center,font=\\small] at (1.52,1.85) {$3$};\n\\path[draw=black!30,fill=black!8] (1.91,1.46) rectangle (2.69,2.24);\n\\node[font=\\small,align=center,font=\\small] at (2.3,1.85) {$2$};\n\\node[font=\\small,align=center,font=\\small] at (4.355,4.5) {circulant column $g$};\n\\path[draw=black!30,fill=black!8] (4.1,3.29) rectangle (4.61,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.545) {$2$};\n\\path[draw=black!30,fill=black!8] (4.1,2.78) rectangle (4.61,3.29);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.035) {$3$};\n\\path[draw=black!30,fill=black!8] (4.1,2.27) rectangle (4.61,2.78);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.525) {$4$};\n\\path[draw=black!30,fill=white] (4.1,1.76) rectangle (4.61,2.27);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.015) {$0$};\n\\path[draw=black!30,fill=black!8] (4.1,1.25) rectangle (4.61,1.76);\n\\node[font=\\small,align=center,font=\\small] at (4.355,1.505) {$5$};\n\\path[draw=black!30,fill=black!8] (4.1,0.74) rectangle (4.61,1.25);\n\\node[font=\\small,align=center,font=\\small] at (4.355,0.995) {$-1$};\n\\node[font=\\smal…","all,align=center,font=\\small] at (2.3,1.85) {$2$};\n\\node[font=\\small,align=center,font=\\small] at (4.355,4.5) {circulant column $g$};\n\\path[draw=black!30,fill=black!8] (4.1,3.29) rectangle (4.61,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.545) {$2$};\n\\path[draw=black!30,fill=black!8] (4.1,2.78) rectangle (4.61,3.29);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.035) {$3$};\n\\path[draw=black!30,fill=black!8] (4.1,2.27) rectangle (4.61,2.78);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.525) {$4$};\n\\path[draw=black!30,fill=white] (4.1,1.76) rectangle (4.61,2.27);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.015) {$0$};\n\\path[draw=black!30,fill=black!8] (4.1,1.25) rectangle (4.61,1.76);\n\\node[font=\\small,align=center,font=\\small] at (4.355,1.505) {$5$};\n\\path[draw=black!30,fill=black!8] (4.1,0.74) rectangle (4.61,1.25);\n\\node[font=\\small,align=center,font=\\small] at (4.355,0.995) {$-1$};\n\\node[font=\\small,align=center,font=\\small] at (7.355,4.5) {padded input};\n\\path[draw=black!30,fill=black!8] (7.1,3.29) rectangle (7.61,3.8);\n\\node[font…","ath[draw=black!30,fill=black!8] (4.1,3.29) rectangle (4.61,3.8);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.545) {$2$};\n\\path[draw=black!30,fill=black!8] (4.1,2.78) rectangle (4.61,3.29);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.035) {$3$};\n\\path[draw=black!30,fill=black!8] (4.1,2.27) rectangle (4.61,2.78);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.525) {$4$};\n\\path[draw=black!30,fill=white] (4.1,1.76) rectangle (4.61,2.27);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.015) {$0$};\n\\path[draw=black!30,fill=black!8] (4.1,1.25) rectangle (4.61,1.76);\n\\node[font=\\small,align=center,font=\\small] at (4.355,1.505) {$5$};\n\\path[draw=black!30,fill=black!8] (4.1,0.74) rectangle (4.61,1.25);\n\\node[font=\\small,align=center,font=\\small] at (4.355,0.995) {$-1$};\n\\node[font=\\small,align=center,font=\\small] at (7.355,4.5) {padded input};\n\\path[draw=black!30,fill=black!8] (7.1,3.29) rectangle (7.61,3.8);\n\\node[font=\\small,align=center,font=\\small] at (7.355,3.545) {$1$};\n\\path[draw=black!30,fill=black!8] (7.1,2.78) rectangle (7.61,3.29);\n\\node[font…","path[draw=black!30,fill=black!8] (4.1,2.78) rectangle (4.61,3.29);\n\\node[font=\\small,align=center,font=\\small] at (4.355,3.035) {$3$};\n\\path[draw=black!30,fill=black!8] (4.1,2.27) rectangle (4.61,2.78);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.525) {$4$};\n\\path[draw=black!30,fill=white] (4.1,1.76) rectangle (4.61,2.27);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.015) {$0$};\n\\path[draw=black!30,fill=black!8] (4.1,1.25) rectangle (4.61,1.76);\n\\node[font=\\small,align=center,font=\\small] at (4.355,1.505) {$5$};\n\\path[draw=black!30,fill=black!8] (4.1,0.74) rectangle (4.61,1.25);\n\\node[font=\\small,align=center,font=\\small] at (4.355,0.995) {$-1$};\n\\node[font=\\small,align=center,font=\\small] at (7.355,4.5) {padded input};\n\\path[draw=black!30,fill=black!8] (7.1,3.29) rectangle (7.61,3.8);\n\\node[font=\\small,align=center,font=\\small] at (7.355,3.545) {$1$};\n\\path[draw=black!30,fill=black!8] (7.1,2.78) rectangle (7.61,3.29);\n\\node[font=\\small,align=center,font=\\small] at (7.355,3.035) {$2$};\n\\path[draw=black!30,fill=black!8] (7.1,2.27) rectangle (7.61,2.78);\n\\node[fo…","path[draw=black!30,fill=black!8] (4.1,2.27) rectangle (4.61,2.78);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.525) {$4$};\n\\path[draw=black!30,fill=white] (4.1,1.76) rectangle (4.61,2.27);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.015) {$0$};\n\\path[draw=black!30,fill=black!8] (4.1,1.25) rectangle (4.61,1.76);\n\\node[font=\\small,align=center,font=\\small] at (4.355,1.505) {$5$};\n\\path[draw=black!30,fill=black!8] (4.1,0.74) rectangle (4.61,1.25);\n\\node[font=\\small,align=center,font=\\small] at (4.355,0.995) {$-1$};\n\\node[font=\\small,align=center,font=\\small] at (7.355,4.5) {padded input};\n\\path[draw=black!30,fill=black!8] (7.1,3.29) rectangle (7.61,3.8);\n\\node[font=\\small,align=center,font=\\small] at (7.355,3.545) {$1$};\n\\path[draw=black!30,fill=black!8] (7.1,2.78) rectangle (7.61,3.29);\n\\node[font=\\small,align=center,font=\\small] at (7.355,3.035) {$2$};\n\\path[draw=black!30,fill=black!8] (7.1,2.27) rectangle (7.61,2.78);\n\\node[font=\\small,align=center,font=\\small] at (7.355,2.525) {$-1$};\n\\path[draw=black!30,fill=white] (7.1,1.76) rectangle (7.61,2.27);\n\\node[fon…","path[draw=black!30,fill=white] (4.1,1.76) rectangle (4.61,2.27);\n\\node[font=\\small,align=center,font=\\small] at (4.355,2.015) {$0$};\n\\path[draw=black!30,fill=black!8] (4.1,1.25) rectangle (4.61,1.76);\n\\node[font=\\small,align=center,font=\\small] at (4.355,1.505) {$5$};\n\\path[draw=black!30,fill=black!8] (4.1,0.74) rectangle (4.61,1.25);\n\\node[font=\\small,align=center,font=\\small] at (4.355,0.995) {$-1$};\n\\node[font=\\small,align=center,font=\\small] at (7.355,4.5) {padded input};\n\\path[draw=black!30,fill=black!8] (7.1,3.29) rectangle (7.61,3.8);\n\\node[font=\\small,align=center,font=\\small] at (7.355,3.545) {$1$};\n\\path[draw=black!30,fill=black!8] (7.1,2.78) rectangle (7.61,3.29);\n\\node[font=\\small,align=center,font=\\small] at (7.355,3.035) {$2$};\n\\path[draw=black!30,fill=black!8] (7.1,2.27) rectangle (7.61,2.78);\n\\node[font=\\small,align=center,font=\\small] at (7.355,2.525) {$-1$};\n\\path[draw=black!30,fill=white] (7.1,1.76) rectangle (7.61,2.27);\n\\node[font=\\small,align=center,font=\\small] at (7.355,2.015) {$0$};\n\\path[draw=black!30,fill=white] (7.1,1.25) rectangle (7.61,1.76);\n\\node[font=…","=\\small,align=center,font=\\small] at (4.355,1.505) {$5$};\n\\path[draw=black!30,fill=black!8] (4.1,0.74) rectangle (4.61,1.25);\n\\node[font=\\small,align=center,font=\\small] at (4.355,0.995) {$-1$};\n\\node[font=\\small,align=center,font=\\small] at (7.355,4.5) {padded input};\n\\path[draw=black!30,fill=black!8] (7.1,3.29) rectangle (7.61,3.8);\n\\node[font=\\small,align=center,font=\\small] at (7.355,3.545) {$1$};\n\\path[draw=black!30,fill=black!8] (7.1,2.78) rectangle (7.61,3.29);\n\\node[font=\\small,align=center,font=\\small] at (7.355,3.035) {$2$};\n\\path[draw=black!30,fill=black!8] (7.1,2.27) rectangle (7.61,2.78);\n\\node[font=\\small,align=center,font=\\small] at (7.355,2.525) {$-1$};\n\\path[draw=black!30,fill=white] (7.1,1.76) rectangle (7.61,2.27);\n\\node[font=\\small,align=center,font=\\small] at (7.355,2.015) {$0$};\n\\path[draw=black!30,fill=white] (7.1,1.25) rectangle (7.61,1.76);\n\\node[font=\\small,align=center,font=\\small] at (7.355,1.505) {$0$};\n\\path[draw=black!30,fill=white] (7.1,0.74) rectangle (7.61,1.25);\n\\node[font=\\small,align=center,font=\\small] at (7.355,0.995) {$0$};\n\\node[font=\\small,al…","=\\small,align=center,font=\\small] at (4.355,0.995) {$-1$};\n\\node[font=\\small,align=center,font=\\small] at (7.355,4.5) {padded input};\n\\path[draw=black!30,fill=black!8] (7.1,3.29) rectangle (7.61,3.8);\n\\node[font=\\small,align=center,font=\\small] at (7.355,3.545) {$1$};\n\\path[draw=black!30,fill=black!8] (7.1,2.78) rectangle (7.61,3.29);\n\\node[font=\\small,align=center,font=\\small] at (7.355,3.035) {$2$};\n\\path[draw=black!30,fill=black!8] (7.1,2.27) rectangle (7.61,2.78);\n\\node[font=\\small,align=center,font=\\small] at (7.355,2.525) {$-1$};\n\\path[draw=black!30,fill=white] (7.1,1.76) rectangle (7.61,2.27);\n\\node[font=\\small,align=center,font=\\small] at (7.355,2.015) {$0$};\n\\path[draw=black!30,fill=white] (7.1,1.25) rectangle (7.61,1.76);\n\\node[font=\\small,align=center,font=\\small] at (7.355,1.505) {$0$};\n\\path[draw=black!30,fill=white] (7.1,0.74) rectangle (7.61,1.25);\n\\node[font=\\small,align=center,font=\\small] at (7.355,0.995) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (10.855,4.5) {six outputs};\n\\path[draw=black!30,fill=black!8] (10.6,3.29) rectangle (11.11,3.8);\n\\node[font=\\…","ath[draw=black!30,fill=black!8] (7.1,3.29) rectangle (7.61,3.8);\n\\node[font=\\small,align=center,font=\\small] at (7.355,3.545) {$1$};\n\\path[draw=black!30,fill=black!8] (7.1,2.78) rectangle (7.61,3.29);\n\\node[font=\\small,align=center,font=\\small] at (7.355,3.035) {$2$};\n\\path[draw=black!30,fill=black!8] (7.1,2.27) rectangle (7.61,2.78);\n\\node[font=\\small,align=center,font=\\small] at (7.355,2.525) {$-1$};\n\\path[draw=black!30,fill=white] (7.1,1.76) rectangle (7.61,2.27);\n\\node[font=\\small,align=center,font=\\small] at (7.355,2.015) {$0$};\n\\path[draw=black!30,fill=white] (7.1,1.25) rectangle (7.61,1.76);\n\\node[font=\\small,align=center,font=\\small] at (7.355,1.505) {$0$};\n\\path[draw=black!30,fill=white] (7.1,0.74) rectangle (7.61,1.25);\n\\node[font=\\small,align=center,font=\\small] at (7.355,0.995) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (10.855,4.5) {six outputs};\n\\path[draw=black!30,fill=black!8] (10.6,3.29) rectangle (11.11,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.855,3.545) {$-5$};\n\\path[draw=black!30,fill=black!8] (10.6,2.78) rectangle (11.11,3.29);\n\\node[fo…","ath[draw=black!30,fill=black!8] (7.1,2.78) rectangle (7.61,3.29);\n\\node[font=\\small,align=center,font=\\small] at (7.355,3.035) {$2$};\n\\path[draw=black!30,fill=black!8] (7.1,2.27) rectangle (7.61,2.78);\n\\node[font=\\small,align=center,font=\\small] at (7.355,2.525) {$-1$};\n\\path[draw=black!30,fill=white] (7.1,1.76) rectangle (7.61,2.27);\n\\node[font=\\small,align=center,font=\\small] at (7.355,2.015) {$0$};\n\\path[draw=black!30,fill=white] (7.1,1.25) rectangle (7.61,1.76);\n\\node[font=\\small,align=center,font=\\small] at (7.355,1.505) {$0$};\n\\path[draw=black!30,fill=white] (7.1,0.74) rectangle (7.61,1.25);\n\\node[font=\\small,align=center,font=\\small] at (7.355,0.995) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (10.855,4.5) {six outputs};\n\\path[draw=black!30,fill=black!8] (10.6,3.29) rectangle (11.11,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.855,3.545) {$-5$};\n\\path[draw=black!30,fill=black!8] (10.6,2.78) rectangle (11.11,3.29);\n\\node[font=\\small,align=center,font=\\small] at (10.855,3.035) {$8$};\n\\path[draw=black!30,fill=black!8] (10.6,2.27) rectangle (11.11,2.78);\n\\nod…","\\path[draw=black!30,fill=black!8] (7.1,2.27) rectangle (7.61,2.78);\n\\node[font=\\small,align=center,font=\\small] at (7.355,2.525) {$-1$};\n\\path[draw=black!30,fill=white] (7.1,1.76) rectangle (7.61,2.27);\n\\node[font=\\small,align=center,font=\\small] at (7.355,2.015) {$0$};\n\\path[draw=black!30,fill=white] (7.1,1.25) rectangle (7.61,1.76);\n\\node[font=\\small,align=center,font=\\small] at (7.355,1.505) {$0$};\n\\path[draw=black!30,fill=white] (7.1,0.74) rectangle (7.61,1.25);\n\\node[font=\\small,align=center,font=\\small] at (7.355,0.995) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (10.855,4.5) {six outputs};\n\\path[draw=black!30,fill=black!8] (10.6,3.29) rectangle (11.11,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.855,3.545) {$-5$};\n\\path[draw=black!30,fill=black!8] (10.6,2.78) rectangle (11.11,3.29);\n\\node[font=\\small,align=center,font=\\small] at (10.855,3.035) {$8$};\n\\path[draw=black!30,fill=black!8] (10.6,2.27) rectangle (11.11,2.78);\n\\node[font=\\small,align=center,font=\\small] at (10.855,2.525) {$8$};\n\\path[draw=black!30,fill=black!8] (10.6,1.76) rectangle (11.11,2.27);","};\n\\path[draw=black!30,fill=white] (7.1,1.76) rectangle (7.61,2.27);\n\\node[font=\\small,align=center,font=\\small] at (7.355,2.015) {$0$};\n\\path[draw=black!30,fill=white] (7.1,1.25) rectangle (7.61,1.76);\n\\node[font=\\small,align=center,font=\\small] at (7.355,1.505) {$0$};\n\\path[draw=black!30,fill=white] (7.1,0.74) rectangle (7.61,1.25);\n\\node[font=\\small,align=center,font=\\small] at (7.355,0.995) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (10.855,4.5) {six outputs};\n\\path[draw=black!30,fill=black!8] (10.6,3.29) rectangle (11.11,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.855,3.545) {$-5$};\n\\path[draw=black!30,fill=black!8] (10.6,2.78) rectangle (11.11,3.29);\n\\node[font=\\small,align=center,font=\\small] at (10.855,3.035) {$8$};\n\\path[draw=black!30,fill=black!8] (10.6,2.27) rectangle (11.11,2.78);\n\\node[font=\\small,align=center,font=\\small] at (10.855,2.525) {$8$};\n\\path[draw=black!30,fill=black!8] (10.6,1.76) rectangle (11.11,2.27);\n\\node[font=\\small,align=center,font=\\small] at (10.855,2.015) {$5$};\n\\path[draw=black!30,fill=black!8] (10.6,1.25) rectangle (11.11,1…","=\\small,align=center,font=\\small] at (7.355,1.505) {$0$};\n\\path[draw=black!30,fill=white] (7.1,0.74) rectangle (7.61,1.25);\n\\node[font=\\small,align=center,font=\\small] at (7.355,0.995) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (10.855,4.5) {six outputs};\n\\path[draw=black!30,fill=black!8] (10.6,3.29) rectangle (11.11,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.855,3.545) {$-5$};\n\\path[draw=black!30,fill=black!8] (10.6,2.78) rectangle (11.11,3.29);\n\\node[font=\\small,align=center,font=\\small] at (10.855,3.035) {$8$};\n\\path[draw=black!30,fill=black!8] (10.6,2.27) rectangle (11.11,2.78);\n\\node[font=\\small,align=center,font=\\small] at (10.855,2.525) {$8$};\n\\path[draw=black!30,fill=black!8] (10.6,1.76) rectangle (11.11,2.27);\n\\node[font=\\small,align=center,font=\\small] at (10.855,2.015) {$5$};\n\\path[draw=black!30,fill=black!8] (10.6,1.25) rectangle (11.11,1.76);\n\\node[font=\\small,align=center,font=\\small] at (10.855,1.505) {$1$};\n\\path[draw=black!30,fill=black!8] (10.6,0.74) rectangle (11.11,1.25);\n\\node[font=\\small,align=center,font=\\small] at (10.855,0.995) {$9$};…","l,align=center,font=\\small] at (7.355,0.995) {$0$};\n\\node[font=\\small,align=center,font=\\small] at (10.855,4.5) {six outputs};\n\\path[draw=black!30,fill=black!8] (10.6,3.29) rectangle (11.11,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.855,3.545) {$-5$};\n\\path[draw=black!30,fill=black!8] (10.6,2.78) rectangle (11.11,3.29);\n\\node[font=\\small,align=center,font=\\small] at (10.855,3.035) {$8$};\n\\path[draw=black!30,fill=black!8] (10.6,2.27) rectangle (11.11,2.78);\n\\node[font=\\small,align=center,font=\\small] at (10.855,2.525) {$8$};\n\\path[draw=black!30,fill=black!8] (10.6,1.76) rectangle (11.11,2.27);\n\\node[font=\\small,align=center,font=\\small] at (10.855,2.015) {$5$};\n\\path[draw=black!30,fill=black!8] (10.6,1.25) rectangle (11.11,1.76);\n\\node[font=\\small,align=center,font=\\small] at (10.855,1.505) {$1$};\n\\path[draw=black!30,fill=black!8] (10.6,0.74) rectangle (11.11,1.25);\n\\node[font=\\small,align=center,font=\\small] at (10.855,0.995) {$9$};\n\\path[draw,line width=1.2pt] (10.6,2.27) rectangle (11.11,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.5,2.95) {keep","lack!30,fill=black!8] (10.6,3.29) rectangle (11.11,3.8);\n\\node[font=\\small,align=center,font=\\small] at (10.855,3.545) {$-5$};\n\\path[draw=black!30,fill=black!8] (10.6,2.78) rectangle (11.11,3.29);\n\\node[font=\\small,align=center,font=\\small] at (10.855,3.035) {$8$};\n\\path[draw=black!30,fill=black!8] (10.6,2.27) rectangle (11.11,2.78);\n\\node[font=\\small,align=center,font=\\small] at (10.855,2.525) {$8$};\n\\path[draw=black!30,fill=black!8] (10.6,1.76) rectangle (11.11,2.27);\n\\node[font=\\small,align=center,font=\\small] at (10.855,2.015) {$5$};\n\\path[draw=black!30,fill=black!8] (10.6,1.25) rectangle (11.11,1.76);\n\\node[font=\\small,align=center,font=\\small] at (10.855,1.505) {$1$};\n\\path[draw=black!30,fill=black!8] (10.6,0.74) rectangle (11.11,1.25);\n\\node[font=\\small,align=center,font=\\small] at (10.855,0.995) {$9$};\n\\path[draw,line width=1.2pt] (10.6,2.27) rectangle (11.11,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.5,2.95) {keep","lack!30,fill=black!8] (10.6,2.78) rectangle (11.11,3.29);\n\\node[font=\\small,align=center,font=\\small] at (10.855,3.035) {$8$};\n\\path[draw=black!30,fill=black!8] (10.6,2.27) rectangle (11.11,2.78);\n\\node[font=\\small,align=center,font=\\small] at (10.855,2.525) {$8$};\n\\path[draw=black!30,fill=black!8] (10.6,1.76) rectangle (11.11,2.27);\n\\node[font=\\small,align=center,font=\\small] at (10.855,2.015) {$5$};\n\\path[draw=black!30,fill=black!8] (10.6,1.25) rectangle (11.11,1.76);\n\\node[font=\\small,align=center,font=\\small] at (10.855,1.505) {$1$};\n\\path[draw=black!30,fill=black!8] (10.6,0.74) rectangle (11.11,1.25);\n\\node[font=\\small,align=center,font=\\small] at (10.855,0.995) {$9$};\n\\path[draw,line width=1.2pt] (10.6,2.27) rectangle (11.11,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.5,2.95) {keep","lack!30,fill=black!8] (10.6,2.27) rectangle (11.11,2.78);\n\\node[font=\\small,align=center,font=\\small] at (10.855,2.525) {$8$};\n\\path[draw=black!30,fill=black!8] (10.6,1.76) rectangle (11.11,2.27);\n\\node[font=\\small,align=center,font=\\small] at (10.855,2.015) {$5$};\n\\path[draw=black!30,fill=black!8] (10.6,1.25) rectangle (11.11,1.76);\n\\node[font=\\small,align=center,font=\\small] at (10.855,1.505) {$1$};\n\\path[draw=black!30,fill=black!8] (10.6,0.74) rectangle (11.11,1.25);\n\\node[font=\\small,align=center,font=\\small] at (10.855,0.995) {$9$};\n\\path[draw,line width=1.2pt] (10.6,2.27) rectangle (11.11,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.5,2.95) {keep","lack!30,fill=black!8] (10.6,1.76) rectangle (11.11,2.27);\n\\node[font=\\small,align=center,font=\\small] at (10.855,2.015) {$5$};\n\\path[draw=black!30,fill=black!8] (10.6,1.25) rectangle (11.11,1.76);\n\\node[font=\\small,align=center,font=\\small] at (10.855,1.505) {$1$};\n\\path[draw=black!30,fill=black!8] (10.6,0.74) rectangle (11.11,1.25);\n\\node[font=\\small,align=center,font=\\small] at (10.855,0.995) {$9$};\n\\path[draw,line width=1.2pt] (10.6,2.27) rectangle (11.11,3.8);\n\\node[font=\\small,align=center,font=\\scriptsize] at (12.5,2.95) {keep","[fill=black] (6.388462,1.090909) circle (0.075);\n\\path[fill=black] (4.542308,2.836364) circle (0.075);\n\\path[draw,fill=white] (2.696154,3.418182) circle (0.075);\n\\path[fill=black] (2.696154,3.127273) circle (0.075);\n\\path[fill=black] (0.85,3.709091) circle (0.075);\n\\draw[dashed,black!60] plot coordinates {(0.85,3.709091) (6.388462,1.090909)};\n\\node[font=\\small,align=center] at (10.1,3.6) {Budget $B=2$: cost $8$};\n\\node[font=\\small,align=center] at (10.1,2.35) {Penalty optimum: root or full split};\n\\node[font=\\small,align=center,font=\\small] at (10.1,1) {$8+2\\lambda\\le1+6\\lambda$","(6.388462,1.090909) circle (0.075);\n\\path[fill=black] (4.542308,2.836364) circle (0.075);\n\\path[draw,fill=white] (2.696154,3.418182) circle (0.075);\n\\path[fill=black] (2.696154,3.127273) circle (0.075);\n\\path[fill=black] (0.85,3.709091) circle (0.075);\n\\draw[dashed,black!60] plot coordinates {(0.85,3.709091) (6.388462,1.090909)};\n\\node[font=\\small,align=center] at (10.1,3.6) {Budget $B=2$: cost $8$};\n\\node[font=\\small,align=center] at (10.1,2.35) {Penalty optimum: root or full split};\n\\node[font=\\small,align=center,font=\\small] at (10.1,1) {$8+2\\lambda\\le1+6\\lambda$","\\path[fill=black] (2.696154,3.127273) circle (0.075);\n\\path[fill=black] (0.85,3.709091) circle (0.075);\n\\draw[dashed,black!60] plot coordinates {(0.85,3.709091) (6.388462,1.090909)};\n\\node[font=\\small,align=center] at (10.1,3.6) {Budget $B=2$: cost $8$};\n\\node[font=\\small,align=center] at (10.1,2.35) {Penalty optimum: root or full split};\n\\node[font=\\small,align=center,font=\\small] at (10.1,1) {$8+2\\lambda\\le1+6\\lambda$","and $8+2\\lambda\\le10$","ont=\\scriptsize] at (7.275,1.195) {6};\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.72,0.25) {stage 1};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.57,0.25) {stage 2};\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.42,0.25) {stage 3};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7.27,0.25) {stage 4};\n\\node[font=\\small,align=center,font=\\small] at (10.84,4.8) {$x=(2I+K)^{-1}\\mathbf1$};\n\\path[draw=black!30,fill=black!8] (10.6,3.62) rectangle (11.08,4.1);\n\\node[font=\\small,align=center,font=\\small] at (10.84,3.86) {$\\frac{1}{2}$};\n\\path[draw=black!30,fill=black!8] (10.6,3.14) rectangle (11.08,3.62);\n\\node[font=\\small,align=center,font=\\small] at (10.84,3.38) {$\\frac{1}{2}$};\n\\path[draw=black!30,fill=black!8] (10.6,2.66) rectangle (11.08,3.14);\n\\node[font=\\small,align=center,font=\\small] at (10.84,2.9) {$1$};\n\\path[draw=black!30,fill=black!8] (10.6,2.18) rectangle (11.08,2.66","tsize] at (3.57,0.25) {stage 2};\n\\node[font=\\small,align=center,font=\\scriptsize] at (5.42,0.25) {stage 3};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7.27,0.25) {stage 4};\n\\node[font=\\small,align=center,font=\\small] at (10.84,4.8) {$x=(2I+K)^{-1}\\mathbf1$};\n\\path[draw=black!30,fill=black!8] (10.6,3.62) rectangle (11.08,4.1);\n\\node[font=\\small,align=center,font=\\small] at (10.84,3.86) {$\\frac{1}{2}$};\n\\path[draw=black!30,fill=black!8] (10.6,3.14) rectangle (11.08,3.62);\n\\node[font=\\small,align=center,font=\\small] at (10.84,3.38) {$\\frac{1}{2}$};\n\\path[draw=black!30,fill=black!8] (10.6,2.66) rectangle (11.08,3.14);\n\\node[font=\\small,align=center,font=\\small] at (10.84,2.9) {$1$};\n\\path[draw=black!30,fill=black!8] (10.6,2.18) rectangle (11.08,2.66);\n\\node[font=\\small,align=center,font=\\small] at (10.84,2.42) {$\\frac{3}{4}$};\n\\path[draw=black!30,fill=black!8] (10.6,1.7) rectangle (11.08,2.18","riptsize] at (7.27,0.25) {stage 4};\n\\node[font=\\small,align=center,font=\\small] at (10.84,4.8) {$x=(2I+K)^{-1}\\mathbf1$};\n\\path[draw=black!30,fill=black!8] (10.6,3.62) rectangle (11.08,4.1);\n\\node[font=\\small,align=center,font=\\small] at (10.84,3.86) {$\\frac{1}{2}$};\n\\path[draw=black!30,fill=black!8] (10.6,3.14) rectangle (11.08,3.62);\n\\node[font=\\small,align=center,font=\\small] at (10.84,3.38) {$\\frac{1}{2}$};\n\\path[draw=black!30,fill=black!8] (10.6,2.66) rectangle (11.08,3.14);\n\\node[font=\\small,align=center,font=\\small] at (10.84,2.9) {$1$};\n\\path[draw=black!30,fill=black!8] (10.6,2.18) rectangle (11.08,2.66);\n\\node[font=\\small,align=center,font=\\small] at (10.84,2.42) {$\\frac{3}{4}$};\n\\path[draw=black!30,fill=black!8] (10.6,1.7) rectangle (11.08,2.18);\n\\node[font=\\small,align=center,font=\\small] at (10.84,1.94) {$\\frac{11}{8}$};\n\\path[draw=black!30,fill=black!8] (10.6,1.22) rectangle (11.08,1.7","=black!8] (10.6,3.62) rectangle (11.08,4.1);\n\\node[font=\\small,align=center,font=\\small] at (10.84,3.86) {$\\frac{1}{2}$};\n\\path[draw=black!30,fill=black!8] (10.6,3.14) rectangle (11.08,3.62);\n\\node[font=\\small,align=center,font=\\small] at (10.84,3.38) {$\\frac{1}{2}$};\n\\path[draw=black!30,fill=black!8] (10.6,2.66) rectangle (11.08,3.14);\n\\node[font=\\small,align=center,font=\\small] at (10.84,2.9) {$1$};\n\\path[draw=black!30,fill=black!8] (10.6,2.18) rectangle (11.08,2.66);\n\\node[font=\\small,align=center,font=\\small] at (10.84,2.42) {$\\frac{3}{4}$};\n\\path[draw=black!30,fill=black!8] (10.6,1.7) rectangle (11.08,2.18);\n\\node[font=\\small,align=center,font=\\small] at (10.84,1.94) {$\\frac{11}{8}$};\n\\path[draw=black!30,fill=black!8] (10.6,1.22) rectangle (11.08,1.7);\n\\node[font=\\small,align=center,font=\\small] at (10.84,1.46) {$\\frac{7}{8}$};\n\\path[draw=black!30,fill=black!8] (10.6,0.74) rectangle","k!30,fill=black!8] (10.6,3.14) rectangle (11.08,3.62);\n\\node[font=\\small,align=center,font=\\small] at (10.84,3.38) {$\\frac{1}{2}$};\n\\path[draw=black!30,fill=black!8] (10.6,2.66) rectangle (11.08,3.14);\n\\node[font=\\small,align=center,font=\\small] at (10.84,2.9) {$1$};\n\\path[draw=black!30,fill=black!8] (10.6,2.18) rectangle (11.08,2.66);\n\\node[font=\\small,align=center,font=\\small] at (10.84,2.42) {$\\frac{3}{4}$};\n\\path[draw=black!30,fill=black!8] (10.6,1.7) rectangle (11.08,2.18);\n\\node[font=\\small,align=center,font=\\small] at (10.84,1.94) {$\\frac{11}{8}$};\n\\path[draw=black!30,fill=black!8] (10.6,1.22) rectangle (11.08,1.7);\n\\node[font=\\small,align=center,font=\\small] at (10.84,1.46) {$\\frac{7}{8}$};\n\\path[draw=black!30,fill=black!8] (10.6,0.74) rectangle (11.08,1.22);\n\\node[font=\\small,align=center,font=\\small] at (10.84,0.98) {$\\frac{13}{8}$};\n\\node[font=\\small,align=center,anchor=west,font=\\script","ck!30,fill=black!8] (10.6,2.66) rectangle (11.08,3.14);\n\\node[font=\\small,align=center,font=\\small] at (10.84,2.9) {$1$};\n\\path[draw=black!30,fill=black!8] (10.6,2.18) rectangle (11.08,2.66);\n\\node[font=\\small,align=center,font=\\small] at (10.84,2.42) {$\\frac{3}{4}$};\n\\path[draw=black!30,fill=black!8] (10.6,1.7) rectangle (11.08,2.18);\n\\node[font=\\small,align=center,font=\\small] at (10.84,1.94) {$\\frac{11}{8}$};\n\\path[draw=black!30,fill=black!8] (10.6,1.22) rectangle (11.08,1.7);\n\\node[font=\\small,align=center,font=\\small] at (10.84,1.46) {$\\frac{7}{8}$};\n\\path[draw=black!30,fill=black!8] (10.6,0.74) rectangle (11.08,1.22);\n\\node[font=\\small,align=center,font=\\small] at (10.84,0.98) {$\\frac{13}{8}$};\n\\node[font=\\small,align=center,anchor=west,font=\\scriptsize] at (0.35,4.42) {packet task};\n\\end{tikzpicture}","black!8] (10.6,2.18) rectangle (11.08,2.66);\n\\node[font=\\small,align=center,font=\\small] at (10.84,2.42) {$\\frac{3}{4}$};\n\\path[draw=black!30,fill=black!8] (10.6,1.7) rectangle (11.08,2.18);\n\\node[font=\\small,align=center,font=\\small] at (10.84,1.94) {$\\frac{11}{8}$};\n\\path[draw=black!30,fill=black!8] (10.6,1.22) rectangle (11.08,1.7);\n\\node[font=\\small,align=center,font=\\small] at (10.84,1.46) {$\\frac{7}{8}$};\n\\path[draw=black!30,fill=black!8] (10.6,0.74) rectangle (11.08,1.22);\n\\node[font=\\small,align=center,font=\\small] at (10.84,0.98) {$\\frac{13}{8}$};\n\\node[font=\\small,align=center,anchor=west,font=\\scriptsize] at (0.35,4.42) {packet task};\n\\end{tikzpicture}","black!8] (10.6,1.7) rectangle (11.08,2.18);\n\\node[font=\\small,align=center,font=\\small] at (10.84,1.94) {$\\frac{11}{8}$};\n\\path[draw=black!30,fill=black!8] (10.6,1.22) rectangle (11.08,1.7);\n\\node[font=\\small,align=center,font=\\small] at (10.84,1.46) {$\\frac{7}{8}$};\n\\path[draw=black!30,fill=black!8] (10.6,0.74) rectangle (11.08,1.22);\n\\node[font=\\small,align=center,font=\\small] at (10.84,0.98) {$\\frac{13}{8}$};\n\\node[font=\\small,align=center,anchor=west,font=\\scriptsize] at (0.35,4.42) {packet task};\n\\end{tikzpicture}","8.9,0.07) rectangle (10.05,0.66);\n\\node[font=\\small,align=center] at (9.475,0.365) {0};\n\\path[draw=black!40,fill=white] (10.28,0.07) rectangle (11.43,0.66);\n\\node[font=\\small,align=center] at (10.855,0.365) {0};\n\\path[draw=black!40,fill=white] (11.66,0.07) rectangle (12.81,0.66);\n\\node[font=\\small,align=center] at (12.235,0.365) {0};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,-0.25) {$8\\log_2 8=24$ additions/subtractions; $H_8H_8^T=8I$.};\n\\end{tikzpicture}","[font=\\small,align=center] at (9.475,0.365) {0};\n\\path[draw=black!40,fill=white] (10.28,0.07) rectangle (11.43,0.66);\n\\node[font=\\small,align=center] at (10.855,0.365) {0};\n\\path[draw=black!40,fill=white] (11.66,0.07) rectangle (12.81,0.66);\n\\node[font=\\small,align=center] at (12.235,0.365) {0};\n\\node[font=\\small,align=center,font=\\scriptsize] at (7,-0.25) {$8\\log_2 8=24$ additions/subtractions; $H_8H_8^T=8I$.};\n\\end{tikzpicture}","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.65,4.5) {one block of $D^{-1}$};\n\\path[draw=black!30,fill=black!8] (0.8,2.95) rectangle (1.65,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.225,3.375) {$\\frac{3}{8}$};\n\\path[draw=black!30,fill=black!8] (1.65,2.95) rectangle (2.5,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.075,3.375) {$- \\frac{1}{8}$};\n\\path[draw=black!30,fill=black!8] (0.8,2.1) rectangle (1.65,2.95);\n\\node[font=\\small,align=center,font=\\small] at (1.225,2.525) {$- \\frac{1}{8}$};\n\\path[draw=black!30,fill=black!8] (1.65,2.1) rectangle (2.5,2.95);\n\\node[font=\\small,align=center,font=\\small] at (2.075,2.525) {$\\frac{3}{8}$};\n\\node[font=\\small,align=center] at (6.25,3.5) {$D^{-1}u=\\tfrac14 u$};\n\\node[font=\\small,align=center] at (6.25,2","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.65,4.5) {one block of $D^{-1}$};\n\\path[draw=black!30,fill=black!8] (0.8,2.95) rectangle (1.65,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.225,3.375) {$\\frac{3}{8}$};\n\\path[draw=black!30,fill=black!8] (1.65,2.95) rectangle (2.5,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.075,3.375) {$- \\frac{1}{8}$};\n\\path[draw=black!30,fill=black!8] (0.8,2.1) rectangle (1.65,2.95);\n\\node[font=\\small,align=center,font=\\small] at (1.225,2.525) {$- \\frac{1}{8}$};\n\\path[draw=black!30,fill=black!8] (1.65,2.1) rectangle (2.5,2.95);\n\\node[font=\\small,align=center,font=\\small] at (2.075,2.525) {$\\frac{3}{8}$};\n\\node[font=\\small,align=center] at (6.25,3.5) {$D^{-1}u=\\tfrac14 u$};\n\\node[font=\\small,align=center] at (6.25,2.25) {$S=1+u^TD^{-1}u=\\tfrac52$};\n\\node[font=\\small,align=center] at (6.25,0.9) {$A^{-1}=D^{-1}-\\tfrac1{40}uu^T$};\n\\node[font=\\small,align=center","bounding box] (0,-.6) rectangle (14,4.8);\n\\node[font=\\small,align=center,font=\\small] at (1.65,4.5) {one block of $D^{-1}$};\n\\path[draw=black!30,fill=black!8] (0.8,2.95) rectangle (1.65,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.225,3.375) {$\\frac{3}{8}$};\n\\path[draw=black!30,fill=black!8] (1.65,2.95) rectangle (2.5,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.075,3.375) {$- \\frac{1}{8}$};\n\\path[draw=black!30,fill=black!8] (0.8,2.1) rectangle (1.65,2.95);\n\\node[font=\\small,align=center,font=\\small] at (1.225,2.525) {$- \\frac{1}{8}$};\n\\path[draw=black!30,fill=black!8] (1.65,2.1) rectangle (2.5,2.95);\n\\node[font=\\small,align=center,font=\\small] at (2.075,2.525) {$\\frac{3}{8}$};\n\\node[font=\\small,align=center] at (6.25,3.5) {$D^{-1}u=\\tfrac14 u$};\n\\node[font=\\small,align=center] at (6.25,2.25) {$S=1+u^TD^{-1}u=\\tfrac52$};\n\\node[font=\\small,align=center] at (6.25,0.9) {$A^{-1}=D^{-1}-\\tfrac1{40}uu^T$};\n\\node[font=\\small,align=center,font=\\small] at (11.45,4.25) {$I-e_1e_1^T$};\n\\path[draw=black!30,fill=white] (10.6,2.7) rectangle (11.45,3.55);\n\\node[font=\\small,ali…","ll=black!8] (0.8,2.95) rectangle (1.65,3.8);\n\\node[font=\\small,align=center,font=\\small] at (1.225,3.375) {$\\frac{3}{8}$};\n\\path[draw=black!30,fill=black!8] (1.65,2.95) rectangle (2.5,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.075,3.375) {$- \\frac{1}{8}$};\n\\path[draw=black!30,fill=black!8] (0.8,2.1) rectangle (1.65,2.95);\n\\node[font=\\small,align=center,font=\\small] at (1.225,2.525) {$- \\frac{1}{8}$};\n\\path[draw=black!30,fill=black!8] (1.65,2.1) rectangle (2.5,2.95);\n\\node[font=\\small,align=center,font=\\small] at (2.075,2.525) {$\\frac{3}{8}$};\n\\node[font=\\small,align=center] at (6.25,3.5) {$D^{-1}u=\\tfrac14 u$};\n\\node[font=\\small,align=center] at (6.25,2.25) {$S=1+u^TD^{-1}u=\\tfrac52$};\n\\node[font=\\small,align=center] at (6.25,0.9) {$A^{-1}=D^{-1}-\\tfrac1{40}uu^T$};\n\\node[font=\\small,align=center,font=\\small] at (11.45,4.25) {$I-e_1e_1^T$};\n\\path[draw=black!30,fill=white] (10.6,2.7) rectangle (11.45,3.55);\n\\node[font=\\small,align=center,font=\\small] at (11.025,3.125) {$0$};\n\\path[draw=black!30,fill=white] (11.45,2.7) rectangle (12.3,3.55);\n\\node[font=\\small,align=center,f…","=black!8] (1.65,2.95) rectangle (2.5,3.8);\n\\node[font=\\small,align=center,font=\\small] at (2.075,3.375) {$- \\frac{1}{8}$};\n\\path[draw=black!30,fill=black!8] (0.8,2.1) rectangle (1.65,2.95);\n\\node[font=\\small,align=center,font=\\small] at (1.225,2.525) {$- \\frac{1}{8}$};\n\\path[draw=black!30,fill=black!8] (1.65,2.1) rectangle (2.5,2.95);\n\\node[font=\\small,align=center,font=\\small] at (2.075,2.525) {$\\frac{3}{8}$};\n\\node[font=\\small,align=center] at (6.25,3.5) {$D^{-1}u=\\tfrac14 u$};\n\\node[font=\\small,align=center] at (6.25,2.25) {$S=1+u^TD^{-1}u=\\tfrac52$};\n\\node[font=\\small,align=center] at (6.25,0.9) {$A^{-1}=D^{-1}-\\tfrac1{40}uu^T$};\n\\node[font=\\small,align=center,font=\\small] at (11.45,4.25) {$I-e_1e_1^T$};\n\\path[draw=black!30,fill=white] (10.6,2.7) rectangle (11.45,3.55);\n\\node[font=\\small,align=center,font=\\small] at (11.025,3.125) {$0$};\n\\path[draw=black!30,fill=white] (11.45,2.7) rectangle (12.3,3.55);\n\\node[font=\\small,align=center,font=\\small] at (11.875,3.125) {$0$};\n\\path[draw=black!30,fill=white] (10.6,1.85) rectangle (11.45,2.7);\n\\node[font=\\small,align=center,font=\\small]…","ign=center,font=\\small] at (2.075,3.375) {$- \\frac{1}{8}$};\n\\path[draw=black!30,fill=black!8] (0.8,2.1) rectangle (1.65,2.95);\n\\node[font=\\small,align=center,font=\\small] at (1.225,2.525) {$- \\frac{1}{8}$};\n\\path[draw=black!30,fill=black!8] (1.65,2.1) rectangle (2.5,2.95);\n\\node[font=\\small,align=center,font=\\small] at (2.075,2.525) {$\\frac{3}{8}$};\n\\node[font=\\small,align=center] at (6.25,3.5) {$D^{-1}u=\\tfrac14 u$};\n\\node[font=\\small,align=center] at (6.25,2.25) {$S=1+u^TD^{-1}u=\\tfrac52$};\n\\node[font=\\small,align=center] at (6.25,0.9) {$A^{-1}=D^{-1}-\\tfrac1{40}uu^T$};\n\\node[font=\\small,align=center,font=\\small] at (11.45,4.25) {$I-e_1e_1^T$};\n\\path[draw=black!30,fill=white] (10.6,2.7) rectangle (11.45,3.55);\n\\node[font=\\small,align=center,font=\\small] at (11.025,3.125) {$0$};\n\\path[draw=black!30,fill=white] (11.45,2.7) rectangle (12.3,3.55);\n\\node[font=\\small,align=center,font=\\small] at (11.875,3.125) {$0$};\n\\path[draw=black!30,fill=white] (10.6,1.85) rectangle (11.45,2.7);\n\\node[font=\\small,align=center,font=\\small] at (11.025,2.275) {$0$};\n\\path[draw=black!30,fill=black!8] (11…","black!30,fill=black!8] (0.8,2.1) rectangle (1.65,2.95);\n\\node[font=\\small,align=center,font=\\small] at (1.225,2.525) {$- \\frac{1}{8}$};\n\\path[draw=black!30,fill=black!8] (1.65,2.1) rectangle (2.5,2.95);\n\\node[font=\\small,align=center,font=\\small] at (2.075,2.525) {$\\frac{3}{8}$};\n\\node[font=\\small,align=center] at (6.25,3.5) {$D^{-1}u=\\tfrac14 u$};\n\\node[font=\\small,align=center] at (6.25,2.25) {$S=1+u^TD^{-1}u=\\tfrac52$};\n\\node[font=\\small,align=center] at (6.25,0.9) {$A^{-1}=D^{-1}-\\tfrac1{40}uu^T$};\n\\node[font=\\small,align=center,font=\\small] at (11.45,4.25) {$I-e_1e_1^T$};\n\\path[draw=black!30,fill=white] (10.6,2.7) rectangle (11.45,3.55);\n\\node[font=\\small,align=center,font=\\small] at (11.025,3.125) {$0$};\n\\path[draw=black!30,fill=white] (11.45,2.7) rectangle (12.3,3.55);\n\\node[font=\\small,align=center,font=\\small] at (11.875,3.125) {$0$};\n\\path[draw=black!30,fill=white] (10.6,1.85) rectangle (11.45,2.7);\n\\node[font=\\small,align=center,font=\\small] at (11.025,2.275) {$0$};\n\\path[draw=black!30,fill=black!8] (11.45,1.85) rectangle (12.3,2.7);\n\\node[font=\\small,align=center,font=\\sm…","lign=center,font=\\small] at (1.225,2.525) {$- \\frac{1}{8}$};\n\\path[draw=black!30,fill=black!8] (1.65,2.1) rectangle (2.5,2.95);\n\\node[font=\\small,align=center,font=\\small] at (2.075,2.525) {$\\frac{3}{8}$};\n\\node[font=\\small,align=center] at (6.25,3.5) {$D^{-1}u=\\tfrac14 u$};\n\\node[font=\\small,align=center] at (6.25,2.25) {$S=1+u^TD^{-1}u=\\tfrac52$};\n\\node[font=\\small,align=center] at (6.25,0.9) {$A^{-1}=D^{-1}-\\tfrac1{40}uu^T$};\n\\node[font=\\small,align=center,font=\\small] at (11.45,4.25) {$I-e_1e_1^T$};\n\\path[draw=black!30,fill=white] (10.6,2.7) rectangle (11.45,3.55);\n\\node[font=\\small,align=center,font=\\small] at (11.025,3.125) {$0$};\n\\path[draw=black!30,fill=white] (11.45,2.7) rectangle (12.3,3.55);\n\\node[font=\\small,align=center,font=\\small] at (11.875,3.125) {$0$};\n\\path[draw=black!30,fill=white] (10.6,1.85) rectangle (11.45,2.7);\n\\node[font=\\small,align=center,font=\\small] at (11.025,2.275) {$0$};\n\\path[draw=black!30,fill=black!8] (11.45,1.85) rectangle (12.3,2.7);\n\\node[font=\\small,align=center,font=\\small] at (11.875,2.275) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsi…","(1.65,2.1) rectangle (2.5,2.95);\n\\node[font=\\small,align=center,font=\\small] at (2.075,2.525) {$\\frac{3}{8}$};\n\\node[font=\\small,align=center] at (6.25,3.5) {$D^{-1}u=\\tfrac14 u$};\n\\node[font=\\small,align=center] at (6.25,2.25) {$S=1+u^TD^{-1}u=\\tfrac52$};\n\\node[font=\\small,align=center] at (6.25,0.9) {$A^{-1}=D^{-1}-\\tfrac1{40}uu^T$};\n\\node[font=\\small,align=center,font=\\small] at (11.45,4.25) {$I-e_1e_1^T$};\n\\path[draw=black!30,fill=white] (10.6,2.7) rectangle (11.45,3.55);\n\\node[font=\\small,align=center,font=\\small] at (11.025,3.125) {$0$};\n\\path[draw=black!30,fill=white] (11.45,2.7) rectangle (12.3,3.55);\n\\node[font=\\small,align=center,font=\\small] at (11.875,3.125) {$0$};\n\\path[draw=black!30,fill=white] (10.6,1.85) rectangle (11.45,2.7);\n\\node[font=\\small,align=center,font=\\small] at (11.025,2.275) {$0$};\n\\path[draw=black!30,fill=black!8] (11.45,1.85) rectangle (12.3,2.7);\n\\node[font=\\small,align=center,font=\\small] at (11.875,2.275) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.45,0.7) {invertible $D$","(6.25,3.5) {$D^{-1}u=\\tfrac14 u$};\n\\node[font=\\small,align=center] at (6.25,2.25) {$S=1+u^TD^{-1}u=\\tfrac52$};\n\\node[font=\\small,align=center] at (6.25,0.9) {$A^{-1}=D^{-1}-\\tfrac1{40}uu^T$};\n\\node[font=\\small,align=center,font=\\small] at (11.45,4.25) {$I-e_1e_1^T$};\n\\path[draw=black!30,fill=white] (10.6,2.7) rectangle (11.45,3.55);\n\\node[font=\\small,align=center,font=\\small] at (11.025,3.125) {$0$};\n\\path[draw=black!30,fill=white] (11.45,2.7) rectangle (12.3,3.55);\n\\node[font=\\small,align=center,font=\\small] at (11.875,3.125) {$0$};\n\\path[draw=black!30,fill=white] (10.6,1.85) rectangle (11.45,2.7);\n\\node[font=\\small,align=center,font=\\small] at (11.025,2.275) {$0$};\n\\path[draw=black!30,fill=black!8] (11.45,1.85) rectangle (12.3,2.7);\n\\node[font=\\small,align=center,font=\\small] at (11.875,2.275) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.45,0.7) {invertible $D$","center] at (6.25,0.9) {$A^{-1}=D^{-1}-\\tfrac1{40}uu^T$};\n\\node[font=\\small,align=center,font=\\small] at (11.45,4.25) {$I-e_1e_1^T$};\n\\path[draw=black!30,fill=white] (10.6,2.7) rectangle (11.45,3.55);\n\\node[font=\\small,align=center,font=\\small] at (11.025,3.125) {$0$};\n\\path[draw=black!30,fill=white] (11.45,2.7) rectangle (12.3,3.55);\n\\node[font=\\small,align=center,font=\\small] at (11.875,3.125) {$0$};\n\\path[draw=black!30,fill=white] (10.6,1.85) rectangle (11.45,2.7);\n\\node[font=\\small,align=center,font=\\small] at (11.025,2.275) {$0$};\n\\path[draw=black!30,fill=black!8] (11.45,1.85) rectangle (12.3,2.7);\n\\node[font=\\small,align=center,font=\\small] at (11.875,2.275) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.45,0.7) {invertible $D$","th[draw=black!30,fill=white] (10.6,2.7) rectangle (11.45,3.55);\n\\node[font=\\small,align=center,font=\\small] at (11.025,3.125) {$0$};\n\\path[draw=black!30,fill=white] (11.45,2.7) rectangle (12.3,3.55);\n\\node[font=\\small,align=center,font=\\small] at (11.875,3.125) {$0$};\n\\path[draw=black!30,fill=white] (10.6,1.85) rectangle (11.45,2.7);\n\\node[font=\\small,align=center,font=\\small] at (11.025,2.275) {$0$};\n\\path[draw=black!30,fill=black!8] (11.45,1.85) rectangle (12.3,2.7);\n\\node[font=\\small,align=center,font=\\small] at (11.875,2.275) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.45,0.7) {invertible $D$","[draw=black!30,fill=white] (11.45,2.7) rectangle (12.3,3.55);\n\\node[font=\\small,align=center,font=\\small] at (11.875,3.125) {$0$};\n\\path[draw=black!30,fill=white] (10.6,1.85) rectangle (11.45,2.7);\n\\node[font=\\small,align=center,font=\\small] at (11.025,2.275) {$0$};\n\\path[draw=black!30,fill=black!8] (11.45,1.85) rectangle (12.3,2.7);\n\\node[font=\\small,align=center,font=\\small] at (11.875,2.275) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.45,0.7) {invertible $D$","ign=center,font=\\small] at (11.875,3.125) {$0$};\n\\path[draw=black!30,fill=white] (10.6,1.85) rectangle (11.45,2.7);\n\\node[font=\\small,align=center,font=\\small] at (11.025,2.275) {$0$};\n\\path[draw=black!30,fill=black!8] (11.45,1.85) rectangle (12.3,2.7);\n\\node[font=\\small,align=center,font=\\small] at (11.875,2.275) {$1$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (11.45,0.7) {invertible $D$","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\draw[line width=.6pt] (0.75,0.8) -- (6.75,0.8);\n\\node[font=\\small,align=center] at (3.75,4.2) {$n/(b+2r)$ with $b=2,r=1$};\n\\path[draw=black,fill=black!12,line width=.6pt] (1.11,0.8) rectangle (1.89,1.13125);\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.5,0.5) {6};\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.5,1.35125) {$\\frac{3}{2}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (2.61,0.8) rectangle (3.39,1.4625);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3,0.5) {12};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3,1.6825) {$3$};\n\\path[draw=black,fill=blac","\\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]\n\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\draw[line width=.6pt] (0.75,0.8) -- (6.75,0.8);\n\\node[font=\\small,align=center] at (3.75,4.2) {$n/(b+2r)$ with $b=2,r=1$};\n\\path[draw=black,fill=black!12,line width=.6pt] (1.11,0.8) rectangle (1.89,1.13125);\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.5,0.5) {6};\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.5,1.35125) {$\\frac{3}{2}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (2.61,0.8) rectangle (3.39,1.4625);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3,0.5) {12};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3,1.6825) {$3$};\n\\path[draw=black,fill=black!12,line width","\\path[use as bounding box] (0,-.6) rectangle (14,4.8);\n\\draw[line width=.6pt] (0.75,0.8) -- (6.75,0.8);\n\\node[font=\\small,align=center] at (3.75,4.2) {$n/(b+2r)$ with $b=2,r=1$};\n\\path[draw=black,fill=black!12,line width=.6pt] (1.11,0.8) rectangle (1.89,1.13125);\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.5,0.5) {6};\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.5,1.35125) {$\\frac{3}{2}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (2.61,0.8) rectangle (3.39,1.4625);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3,0.5) {12};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3,1.6825) {$3$};\n\\path[draw=black,fill=black!12,line width=.6pt] (4.11,0.8) rectangle (4.89,2.125);\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.5,0.5) {24};\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.5,2.345) {$6$};\n\\path[draw=black,fill=black!12,line width=.6pt] (5.61,",".11,0.8) rectangle (1.89,1.13125);\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.5,0.5) {6};\n\\node[font=\\small,align=center,font=\\scriptsize] at (1.5,1.35125) {$\\frac{3}{2}$};\n\\path[draw=black,fill=black!12,line width=.6pt] (2.61,0.8) rectangle (3.39,1.4625);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3,0.5) {12};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3,1.6825) {$3$};\n\\path[draw=black,fill=black!12,line width=.6pt] (4.11,0.8) rectangle (4.89,2.125);\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.5,0.5) {24};\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.5,2.345) {$6$};\n\\path[draw=black,fill=black!12,line width=.6pt] (5.61,0.8) rectangle (6.39,3.45);\n\\node[font=\\small,align=center,font=\\scriptsize] at (6,0.5) {48};\n\\node[font=\\small,align=center,font=\\scriptsize] at (6,3.67) {$12$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.75","idth=.6pt] (2.61,0.8) rectangle (3.39,1.4625);\n\\node[font=\\small,align=center,font=\\scriptsize] at (3,0.5) {12};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3,1.6825) {$3$};\n\\path[draw=black,fill=black!12,line width=.6pt] (4.11,0.8) rectangle (4.89,2.125);\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.5,0.5) {24};\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.5,2.345) {$6$};\n\\path[draw=black,fill=black!12,line width=.6pt] (5.61,0.8) rectangle (6.39,3.45);\n\\node[font=\\small,align=center,font=\\scriptsize] at (6,0.5) {48};\n\\node[font=\\small,align=center,font=\\scriptsize] at (6,3.67) {$12$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.75,0.1) {ambient dimension $n$};\n\\node[font=\\small,align=center] at (10.55,3.8) {Six-dimensional running operator};\n\\node[font=\\small,align=center,font=\\small] at (10.55,2.85) {dense: $36$ scalars, $72q$ work};\n\\node[font=\\s","ne width=.6pt] (4.11,0.8) rectangle (4.89,2.125);\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.5,0.5) {24};\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.5,2.345) {$6$};\n\\path[draw=black,fill=black!12,line width=.6pt] (5.61,0.8) rectangle (6.39,3.45);\n\\node[font=\\small,align=center,font=\\scriptsize] at (6,0.5) {48};\n\\node[font=\\small,align=center,font=\\scriptsize] at (6,3.67) {$12$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.75,0.1) {ambient dimension $n$};\n\\node[font=\\small,align=center] at (10.55,3.8) {Six-dimensional running operator};\n\\node[font=\\small,align=center,font=\\small] at (10.55,2.85) {dense: $36$ scalars, $72q$ work};\n\\node[font=\\small,align=center,font=\\small] at (10.55,1.9) {BDLR: $24$ scalars, $48q$ work};\n\\node[font=\\small,align=center,font=\\small] at (10.55,0.95) {ratio: $3/2$, not a timing result};\n\\end{tikzpicture}","riptsize] at (4.5,0.5) {24};\n\\node[font=\\small,align=center,font=\\scriptsize] at (4.5,2.345) {$6$};\n\\path[draw=black,fill=black!12,line width=.6pt] (5.61,0.8) rectangle (6.39,3.45);\n\\node[font=\\small,align=center,font=\\scriptsize] at (6,0.5) {48};\n\\node[font=\\small,align=center,font=\\scriptsize] at (6,3.67) {$12$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.75,0.1) {ambient dimension $n$};\n\\node[font=\\small,align=center] at (10.55,3.8) {Six-dimensional running operator};\n\\node[font=\\small,align=center,font=\\small] at (10.55,2.85) {dense: $36$ scalars, $72q$ work};\n\\node[font=\\small,align=center,font=\\small] at (10.55,1.9) {BDLR: $24$ scalars, $48q$ work};\n\\node[font=\\small,align=center,font=\\small] at (10.55,0.95) {ratio: $3/2$, not a timing result};\n\\end{tikzpicture}",") rectangle (6.39,3.45);\n\\node[font=\\small,align=center,font=\\scriptsize] at (6,0.5) {48};\n\\node[font=\\small,align=center,font=\\scriptsize] at (6,3.67) {$12$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.75,0.1) {ambient dimension $n$};\n\\node[font=\\small,align=center] at (10.55,3.8) {Six-dimensional running operator};\n\\node[font=\\small,align=center,font=\\small] at (10.55,2.85) {dense: $36$ scalars, $72q$ work};\n\\node[font=\\small,align=center,font=\\small] at (10.55,1.9) {BDLR: $24$ scalars, $48q$ work};\n\\node[font=\\small,align=center,font=\\small] at (10.55,0.95) {ratio: $3/2$, not a timing result};\n\\end{tikzpicture}",".39,3.45);\n\\node[font=\\small,align=center,font=\\scriptsize] at (6,0.5) {48};\n\\node[font=\\small,align=center,font=\\scriptsize] at (6,3.67) {$12$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.75,0.1) {ambient dimension $n$};\n\\node[font=\\small,align=center] at (10.55,3.8) {Six-dimensional running operator};\n\\node[font=\\small,align=center,font=\\small] at (10.55,2.85) {dense: $36$ scalars, $72q$ work};\n\\node[font=\\small,align=center,font=\\small] at (10.55,1.9) {BDLR: $24$ scalars, $48q$ work};\n\\node[font=\\small,align=center,font=\\small] at (10.55,0.95) {ratio: $3/2$, not a timing result};\n\\end{tikzpicture}","ode[font=\\small,align=center,font=\\scriptsize] at (6,3.67) {$12$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.75,0.1) {ambient dimension $n$};\n\\node[font=\\small,align=center] at (10.55,3.8) {Six-dimensional running operator};\n\\node[font=\\small,align=center,font=\\small] at (10.55,2.85) {dense: $36$ scalars, $72q$ work};\n\\node[font=\\small,align=center,font=\\small] at (10.55,1.9) {BDLR: $24$ scalars, $48q$ work};\n\\node[font=\\small,align=center,font=\\small] at (10.55,0.95) {ratio: $3/2$, not a timing result};\n\\end{tikzpicture}","l,align=center,font=\\scriptsize] at (6,3.67) {$12$};\n\\node[font=\\small,align=center,font=\\scriptsize] at (3.75,0.1) {ambient dimension $n$};\n\\node[font=\\small,align=center] at (10.55,3.8) {Six-dimensional running operator};\n\\node[font=\\small,align=center,font=\\small] at (10.55,2.85) {dense: $36$ scalars, $72q$ work};\n\\node[font=\\small,align=center,font=\\small] at (10.55,1.9) {BDLR: $24$ scalars, $48q$ work};\n\\node[font=\\small,align=center,font=\\small] at (10.55,0.95) {ratio: $3/2$, not a timing result};\n\\end{tikzpicture}","er,font=\\scriptsize] at (3.75,0.1) {ambient dimension $n$};\n\\node[font=\\small,align=center] at (10.55,3.8) {Six-dimensional running operator};\n\\node[font=\\small,align=center,font=\\small] at (10.55,2.85) {dense: $36$ scalars, $72q$ work};\n\\node[font=\\small,align=center,font=\\small] at (10.55,1.9) {BDLR: $24$ scalars, $48q$ work};\n\\node[font=\\small,align=center,font=\\small] at (10.55,0.95) {ratio: $3/2$, not a timing result};\n\\end{tikzpicture}","\\vpage{29}{A complex range needs an adjoint}{THE INNER PRODUCT IS PART OF THE CONTRACT}{Traditional pp. 18, 79, 80 / Theorem 5.6, Section F.1, Section F.2.}\nThe earlier compact example is real. Over $\\CC$, an isometry means $Q^*Q=I$, with conjugation. Take the exact Gaussian-rational factors\n\\[\nQ_L=\\mat{3/5&0","\\vpage{29}{A complex range needs an adjoint}{THE INNER PRODUCT IS PART OF THE CONTRACT}{Traditional pp. 18, 79, 80 / Theorem 5.6, Section F.1, Section F.2.}\nThe earlier compact example is real. Over $\\CC$, an isometry means $Q^*Q=I$, with conjugation. Take the exact Gaussian-rational factors\n\\[\nQ_L=\\mat{3/5&0\\\\4\\mathrm i/5&0\\\\0&1},\\qquad\nQ_R=\\mat{1&0\\\\0&\\mathrm i},\\qquad G=\\diag(5,2).\n\\]\nTheir product is a genuinely complex rectangular operator:\n\\[\nA=Q_LGQ_R^*=\\mat{3&0","\\vpage{30}{Inverse reachability is still an inclusion}{A FINITE INVERSE WITH A CANCELLED ROUTE}{Traditional pp. 21, 23 / Theorem 6.7, Theorem 6.1, Corollary 6.3.}\nReuse the signed diamond from \\pagerefV{14}, now as a complete operator:\n\\[\nA=I+K=\\mat{1&0&0&0\\\\1&1&0&0\\\\1&0&1&0\\\\0&1&-1&1}.\n\\]\nThe two routes from coordinate zero to coordinate three cancel, so $K^2=0$. Consequently\n\\[\nA^{-1}=I-K=\\mat{1&0&0&0","\\vpage{32}{Safe integers can still be conservative}{QUANTIZATION AND THE SIZE OF THE BUDGET TABLE}{Traditional pp. 29, 33 / Theorem 8.7, Theorem 9.8, Corollary 9.9.}\nLet one integer unit represent $h=1/10$ of certified error. For an individual nonnegative error $e$, choosing\n\\[\nw=\\lceil e/h\\rceil\\qquad\\text{ensures}\\qquad e\\le hw.\n\\]\nThus integer feasibility $\\sum w\\le B$ implies the real bound $\\sum e\\le hB$. This implication is checked before applying the exact option or tree recurrence.\n\\vfig{quantization}\n\\head{A false certificate and a conservative rejection}\nFor errors $(3/50,2/25)$, rounding down assigns weights $(0,0)$ although their real sum is $7/50>1/10$. Rounding up gives $(1,1)$ and correctly excludes the assignment from budget one.","\\vpage{3}{A dense-looking matrix, one scalar coupling}{THE RUNNING SIX-DIMENSIONAL OPERATOR}{Traditional pp. 7, 11 / Theorem 2.3, Theorem 3.4.}\nLet $D$ have three identical blocks $\\mat{3&1","\\vpage{3}{A dense-looking matrix, one scalar coupling}{THE RUNNING SIX-DIMENSIONAL OPERATOR}{Traditional pp. 7, 11 / Theorem 2.3, Theorem 3.4.}\nLet $D$ have three identical blocks $\\mat{3&1\\\\1&3}$ and put $u=v=\\ones\\in\\QQ^6$. Then $A=D+uv^T$ has diagonal entries four, within-block off-diagonal entries two, and every cross-block entry one. All 36 entries are nonzero.\n\\[\nx=(1,-1,2,0,-2,3)^T,\\qquad v^Tx=3,\\qquad Ax=Dx+3u.\n\\]\n\\vfig{factored_application}\n\\head{What was avoided?}\nThe global part does not require 36 independent coefficients. It depends on the input through one scalar, $v^Tx$, which is broadcast through $u$. The diagonal part is computed by three separate local matrix products. Nothing is approximated and no channel is discarded.","\\vpage{4}{Work removed is not time measured}{ARITHMETIC, STORAGE, AND CAPACITY}{Traditional pp. 10, 11, 12, 13 / Theorem 3.1, Corollary 3.2, Theorem 3.4, Corollary 3.9, Proposition 4.1.}\nFor $q$ right-hand sides, the paper models dense application by $2n^2q$ operations. A block-diagonal-plus-rank-$r$ representation with equal block width $b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory i","\\vpage{4}{Work removed is not time measured}{ARITHMETIC, STORAGE, AND CAPACITY}{Traditional pp. 10, 11, 12, 13 / Theorem 3.1, Corollary 3.2, Theorem 3.4, Corollary 3.9, Proposition 4.1.}\nFor $q$ right-hand sides, the paper models dense application by $2n^2q$ operations. A block-diagonal-plus-rank-$r$ representation with equal block width $b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation a","\\vpage{4}{Work removed is not time measured}{ARITHMETIC, STORAGE, AND CAPACITY}{Traditional pp. 10, 11, 12, 13 / Theorem 3.1, Corollary 3.2, Theorem 3.4, Corollary 3.9, Proposition 4.1.}\nFor $q$ right-hand sides, the paper models dense application by $2n^2q$ operations. A block-diagonal-plus-rank-$r$ representation with equal block width $b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs,","\\vpage{4}{Work removed is not time measured}{ARITHMETIC, STORAGE, AND CAPACITY}{Traditional pp. 10, 11, 12, 13 / Theorem 3.1, Corollary 3.2, Theorem 3.4, Corollary 3.9, Proposition 4.1.}\nFor $q$ right-hand sides, the paper models dense application by $2n^2q$ operations. A block-diagonal-plus-rank-$r$ representation with equal block width $b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs, workspace, allocator overhead, and stored","\\vpage{4}{Work removed is not time measured}{ARITHMETIC, STORAGE, AND CAPACITY}{Traditional pp. 10, 11, 12, 13 / Theorem 3.1, Corollary 3.2, Theorem 3.4, Corollary 3.9, Proposition 4.1.}\nFor $q$ right-hand sides, the paper models dense application by $2n^2q$ operations. A block-diagonal-plus-rank-$r$ representation with equal block width $b$ uses\n\\[\nF=2qn(b+2r),\\qquad M=n(b+2r)\\text{ stored scalars}.\n\\]\nThese formulas count the dense block products and two thin products. They follow the manuscript convention rather than adding every scalar accumulation or index operation.\n\\vfig{work_storage}\nFor the running example, $n=6,b=2,r=1$. The model reduces storage from 36 to 24 scalars and application work from $72q$ to $48q$. For fixed $b,r$, the ratio $n/(b+2r)$ grows with $n$. If the rank instead grows proportionally with dimension, that apparent advantage need not grow.\n\\head{Memory is another constraint}\nWith scalar size $s$, the representation alone requires $sM$ bytes. Inputs, outputs, workspace, allocator overhead, and stored indices are additional. A matrix-only capacity inequality …","\\vpage{5}{Four coefficients, four exact frequencies}{CIRCULANT MULTIPLICATION}{Traditional p. 14 / Theorem 4.3.}\nTake $c=(2,1,0,-1)^T$ and define $C(c)_{jk}=c_{j-k\\bmod4}$. The unitary Fourier matrix is $F_{jk}=\\mathrm i^{-jk}/2$. Its normalization is visible in the exact diagonalization:\n\\[\nC(c)=F^*\\diag(2Fc)F,\\qquad 2Fc=(2,2-2\\mathrm i,2,2+2\\mathrm i)^T.\n\\]\n\\vfig{circulant}\n\\head{Follow an actual input}\nFor $x=(1,2,-1,0)^T$, dense multiplication gives $(0,6,0,-2)^T$. Transforming $x$, multiplying by the four displayed eigenvalues, and applying $F^*$ gives exactly the same vector. Every entry belongs to $\\QQ(\\mathrm i)$, so the check needs no approximate trigonometric values.","coefficients, four exact frequencies}{CIRCULANT MULTIPLICATION}{Traditional p. 14 / Theorem 4.3.}\nTake $c=(2,1,0,-1)^T$ and define $C(c)_{jk}=c_{j-k\\bmod4}$. The unitary Fourier matrix is $F_{jk}=\\mathrm i^{-jk}/2$. Its normalization is visible in the exact diagonalization:\n\\[\nC(c)=F^*\\diag(2Fc)F,\\qquad 2Fc=(2,2-2\\mathrm i,2,2+2\\mathrm i)^T.\n\\]\n\\vfig{circulant}\n\\head{Follow an actual input}\nFor $x=(1,2,-1,0)^T$, dense multiplication gives $(0,6,0,-2)^T$. Transforming $x$, multiplying by the four displayed eigenvalues, and applying $F^*$ gives exactly the same vector. Every entry belongs to $\\QQ(\\mathrm i)$, so the check needs no approximate trigonometric values.","ANT MULTIPLICATION}{Traditional p. 14 / Theorem 4.3.}\nTake $c=(2,1,0,-1)^T$ and define $C(c)_{jk}=c_{j-k\\bmod4}$. The unitary Fourier matrix is $F_{jk}=\\mathrm i^{-jk}/2$. Its normalization is visible in the exact diagonalization:\n\\[\nC(c)=F^*\\diag(2Fc)F,\\qquad 2Fc=(2,2-2\\mathrm i,2,2+2\\mathrm i)^T.\n\\]\n\\vfig{circulant}\n\\head{Follow an actual input}\nFor $x=(1,2,-1,0)^T$, dense multiplication gives $(0,6,0,-2)^T$. Transforming $x$, multiplying by the four displayed eigenvalues, and applying $F^*$ gives exactly the same vector. Every entry belongs to $\\QQ(\\mathrm i)$, so the check needs no approximate trigonometric values.","al p. 14 / Theorem 4.3.}\nTake $c=(2,1,0,-1)^T$ and define $C(c)_{jk}=c_{j-k\\bmod4}$. The unitary Fourier matrix is $F_{jk}=\\mathrm i^{-jk}/2$. Its normalization is visible in the exact diagonalization:\n\\[\nC(c)=F^*\\diag(2Fc)F,\\qquad 2Fc=(2,2-2\\mathrm i,2,2+2\\mathrm i)^T.\n\\]\n\\vfig{circulant}\n\\head{Follow an actual input}\nFor $x=(1,2,-1,0)^T$, dense multiplication gives $(0,6,0,-2)^T$. Transforming $x$, multiplying by the four displayed eigenvalues, and applying $F^*$ gives exactly the same vector. Every entry belongs to $\\QQ(\\mathrm i)$, so the check needs no approximate trigonometric values.","\\vpage{6}{Padding is part of the theorem}{TOEPLITZ EMBEDDING WITHOUT WRAPAROUND}{Traditional p. 15 / Corollary 4.4.}\nConsider\n\\[\nT=\\mat{2&-1&5\\\\3&2&-1\\\\4&3&2},\\qquad x=(1,2,-1)^T.\n\\]\nThe diagonal pattern is Toeplitz, not circulant. To use a circular convolution, embed it into dimension six with first column $g=(2,3,4,0,5,-1)^T$ and pad the input with three zeros.\n\\vfig{toeplitz}\n\\head{Read the wrapped indices}\nFor output index $j<3$, an input index $k\\le j$ uses the first-column coefficient $c_{j-k}$. An input index $k>j$ wraps to the stored first-row coefficient $r_{k-j}$. The padded input positions contribute nothing. Keeping the first three outputs therefore gives\n\\[\nTx=(-5,8,8)^T.\n\\]\nThe remaining circular outputs are workspace consequences of the embedding, not additional components of $Tx$.\n\\head{A concrete failure mode}\nUsing only the first column $(2,3,4)$ as","\\vpage{7}{A transpose, not an adjoint}{KRONECKER ACTION AND ARRAY ORDER}{Traditional p. 15 / Theorem 4.5, Corollary 4.6.}\nWith column-major vectorization, the tensor identity is\n\\[\n(A\\otimes B)\\operatorname{vec}(X)=\\operatorname{vec}(BXA^T).\n\\]\nHere the transpose records index order, not a Hermitian inner product. Use the explicitly complex matrix $A=\\mat{1&\\mathrm i","\\vpage{7}{A transpose, not an adjoint}{KRONECKER ACTION AND ARRAY ORDER}{Traditional p. 15 / Theorem 4.5, Corollary 4.6.}\nWith column-major vectorization, the tensor identity is\n\\[\n(A\\otimes B)\\operatorname{vec}(X)=\\operatorname{vec}(BXA^T).\n\\]\nHere the transpose records index order, not a Hermitian inner product. Use the explicitly complex matrix $A=\\mat{1&\\mathrm i\\\\2&-1}$, together with $B=\\mat{2&1","\\vpage{8}{See every butterfly value}{WALSH RECURSION IN EIGHT COORDINATES}{Traditional p. 16 / Theorem 4.7.}\nThe unnormalized Walsh matrix satisfies $H_1=[1]$ and\n\\[\nH_{2n}=\\mat{H_n&H_n","\\vpage{8}{See every butterfly value}{WALSH RECURSION IN EIGHT COORDINATES}{Traditional p. 16 / Theorem 4.7.}\nThe unnormalized Walsh matrix satisfies $H_1=[1]$ and\n\\[\nH_{2n}=\\mat{H_n&H_n\\\\H_n&-H_n},\\qquad H_nH_n^T=nI.\n\\]\nApply the three add/subtract stages to $(1,2,3,4,5,6,7,8)^T$. The full intermediate ledger is displayed below.\n\\vfig{walsh}\n\\head{Arithmetic can be counted exactly}\nEach stage visits eight coordinates, producing one sum or difference per coordinate. Three stages therefore use exactly 24 additions or subtractions. No multiplication by a dense $8\\times8$ matrix is necessary.","\\vpage{9}{The cross term has somewhere to go}{CLOSED LOW-RANK PRODUCT ALGEBRA}{Traditional pp. 16, 17 / Theorem 5.1, Corollary 5.2.}\nLet $A=D_A+u_av_a^T$ be the running operator. Take $D_B=\\diag(1,2,3,4,5,6)$, $u_b=(1,0,1,0,1,0)^T$, and $v_b=(0,1,0,1,0,1)^T$. Define $B=D_B+u_bv_b^T$.\n\\[\nU_C=[D_Au_b+u_a(v_a^Tu_b)\\ \\ u_a],\\qquad\nV_C=[v_b\\ \\ D_B^Tv_a].\n\\]\n\\vfig{bdlr_product}\n\\head{Why there are only two columns}\nExpanding $AB$ gives three correction terms, but two share the right factor $v_b^T$:\n\\[\nD_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T\n=\\bigl(D_Au_b+u_a(v_a^Tu_b)\\bigr)v_b^T.\n\\]\nThe remaining term is $u_av_a^TD_B$. Thus $AB=D_AD_B+U_CV_C^T$ with at most two correction directions. Counting the three expanded summands as independent ranks would overestimate this representation.","has somewhere to go}{CLOSED LOW-RANK PRODUCT ALGEBRA}{Traditional pp. 16, 17 / Theorem 5.1, Corollary 5.2.}\nLet $A=D_A+u_av_a^T$ be the running operator. Take $D_B=\\diag(1,2,3,4,5,6)$, $u_b=(1,0,1,0,1,0)^T$, and $v_b=(0,1,0,1,0,1)^T$. Define $B=D_B+u_bv_b^T$.\n\\[\nU_C=[D_Au_b+u_a(v_a^Tu_b)\\ \\ u_a],\\qquad\nV_C=[v_b\\ \\ D_B^Tv_a].\n\\]\n\\vfig{bdlr_product}\n\\head{Why there are only two columns}\nExpanding $AB$ gives three correction terms, but two share the right factor $v_b^T$:\n\\[\nD_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T\n=\\bigl(D_Au_b+u_a(v_a^Tu_b)\\bigr)v_b^T.\n\\]\nThe remaining term is $u_av_a^TD_B$. Thus $AB=D_AD_B+U_CV_C^T$ with at most two correction directions. Counting the three expanded summands as independent ranks would overestimate this representation.","pp. 16, 17 / Theorem 5.1, Corollary 5.2.}\nLet $A=D_A+u_av_a^T$ be the running operator. Take $D_B=\\diag(1,2,3,4,5,6)$, $u_b=(1,0,1,0,1,0)^T$, and $v_b=(0,1,0,1,0,1)^T$. Define $B=D_B+u_bv_b^T$.\n\\[\nU_C=[D_Au_b+u_a(v_a^Tu_b)\\ \\ u_a],\\qquad\nV_C=[v_b\\ \\ D_B^Tv_a].\n\\]\n\\vfig{bdlr_product}\n\\head{Why there are only two columns}\nExpanding $AB$ gives three correction terms, but two share the right factor $v_b^T$:\n\\[\nD_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T\n=\\bigl(D_Au_b+u_a(v_a^Tu_b)\\bigr)v_b^T.\n\\]\nThe remaining term is $u_av_a^TD_B$. Thus $AB=D_AD_B+U_CV_C^T$ with at most two correction directions. Counting the three expanded summands as independent ranks would overestimate this representation.","7 / Theorem 5.1, Corollary 5.2.}\nLet $A=D_A+u_av_a^T$ be the running operator. Take $D_B=\\diag(1,2,3,4,5,6)$, $u_b=(1,0,1,0,1,0)^T$, and $v_b=(0,1,0,1,0,1)^T$. Define $B=D_B+u_bv_b^T$.\n\\[\nU_C=[D_Au_b+u_a(v_a^Tu_b)\\ \\ u_a],\\qquad\nV_C=[v_b\\ \\ D_B^Tv_a].\n\\]\n\\vfig{bdlr_product}\n\\head{Why there are only two columns}\nExpanding $AB$ gives three correction terms, but two share the right factor $v_b^T$:\n\\[\nD_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T\n=\\bigl(D_Au_b+u_a(v_a^Tu_b)\\bigr)v_b^T.\n\\]\nThe remaining term is $u_av_a^TD_B$. Thus $AB=D_AD_B+U_CV_C^T$ with at most two correction directions. Counting the three expanded summands as independent ranks would overestimate this representation.","\\vpage{10}{A large inverse, a small denominator}{WOODBURY WITH BOTH INVERTIBILITY CHECKS}{Traditional p. 18 / Theorem 5.3, Corollary 5.4.}\nFor the running operator, each diagonal block has inverse $\\frac18\\mat{3&-1\\\\-1&3}$. Hence $D^{-1}u=u/4$ and\n\\[\nS=1+u^TD^{-1}u=\\frac52.\n\\]\nThe Woodbury identity reduces the entire inverse to\n\\[\nA^{-1}=D^{-1}-D^{-1}uS^{-1}u^TD^{-1}\n=D^{-1}-\\frac1{40}uu^T.\n\\]\n\\vfig{woodbury}\n\\head{Two hypotheses, not one}\nInvertibility of $D$ alone does not suffice. For $D=I_2$, $u=e_1$, and $v=-e_1$, the update is $\\diag(0,1)$. The small matrix $S$ is zero, exactly where the correction formula would require an invalid division.","\\vpage{11}{Stored width is not intrinsic rank}{ACTUAL RANGE COORDINATES}{Traditional pp. 18, 79 / Theorem 5.6, Section F.1.}\nChoose the isometry\n\\[\nQ_L=\\mat{3/5&0\\\\4/5&0\\\\0&1\\\\0&0},\\quad\nR_L=\\mat{1&0&1\\\\0&1&0},\\quad\nR_R=\\mat{3&0&2\\\\0&2&0\\\\1&0&-1}.\n\\]\nSet $L=Q_LR_L$ and $R=R_R$. The two equal columns of $L$ do not create a third range direction. In contrast, $R$ has rank three.\n\\vfig{compact_core}\nThe actual core is\n\\[\nG=R_LR_R^T=\\mat{5&0&0","\\vpage{12}{Why a core optimum is global}{COMPRESSION, KERNEL WITNESSES, AND TAILS}{Traditional pp. 18, 80, 81 / Theorem 5.6, Section F.2, Section F.3, Section F.4.}\nThe product from the previous page is\n\\[\nA_0=LR^T=\\mat{3&0&0\\\\4&0&0\\\\0&2&0\\\\0&0&0}.\n\\]\nKeeping only its first singular direction gives a rank-one approximation with spectral and Frobenius error exactly two. Keeping none gives errors five and $\\sqrt{29}$; keeping both gives zero.\n\\vfig{optimality}\n\\head{Why an outside competitor cannot do better}\nFor isometric outer factors, compression $C(B)=Q_L^*BQ_R$ cannot increase rank or either matrix norm. Every ambient rank-$k$ competitor therefore supplies an admissible core competitor. The lifted core residual has exactly the core residual norm. These two facts transfer a core optimum to the entire ambient matrix space, not just to competitors already lying in the supplied ranges.","\\vpage{13}{Errors enter at different places}{FACTOR PERTURBATION VERSUS RECOMPRESSION}{Traditional pp. 19, 20 / Theorem 5.8, Theorem 5.9.}\nLet the exact scalar factors be $(2,3,4)$ and start from $p_0=1$. If additive errors $(1/10,1/5,1/20)$ enter \\emph{after} each partial multiplication, then\n\\[\n\\widehat p_j=a_j\\widehat p_{j-1}+e_j,\\qquad\n\\widehat p_3-p_3=12e_1+4e_2+e_3=\\frac{41}{20}.\n\\]\n\\vfig{error_chain}\n\\head{The same local numbers, a different model}\nIf instead the factors themselves are perturbed, the computed product is $(4+e_3)(3+e_2)(2+e_1)$. Its error is $402/125$, not $41/20$. The factor-perturbation telescoping formula contains already-perturbed later factors; the additive recompression formula contains the original later factors.",".9.}\nLet the exact scalar factors be $(2,3,4)$ and start from $p_0=1$. If additive errors $(1/10,1/5,1/20)$ enter \\emph{after} each partial multiplication, then\n\\[\n\\widehat p_j=a_j\\widehat p_{j-1}+e_j,\\qquad\n\\widehat p_3-p_3=12e_1+4e_2+e_3=\\frac{41}{20}.\n\\]\n\\vfig{error_chain}\n\\head{The same local numbers, a different model}\nIf instead the factors themselves are perturbed, the computed product is $(4+e_3)(3+e_2)(2+e_1)$. Its error is $402/125$, not $41/20$. The factor-perturbation telescoping formula contains already-perturbed later factors; the additive recompression formula contains the original later factors.","\\vpage{14}{Two routes can cancel exactly}{SUPPORT IS AN INCLUSION}{Traditional pp. 8, 21 / Theorem 2.5, Theorem 6.1, Corollary 6.3.}\nLet $K$ have transfers $0\\to1$, $0\\to2$, $1\\to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This","\\vpage{14}{Two routes can cancel exactly}{SUPPORT IS AN INCLUSION}{Traditional pp. 8, 21 / Theorem 2.5, Theorem 6.1, Corollary 6.3.}\nLet $K$ have transfers $0\\to1$, $0\\to2$, $1\\to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on th","\\vpage{14}{Two routes can cancel exactly}{SUPPORT IS AN INCLUSION}{Traditional pp. 8, 21 / Theorem 2.5, Theorem 6.1, Corollary 6.3.}\nLet $K$ have transfers $0\\to1$, $0\\to2$, $1\\to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpote","\\vpage{14}{Two routes can cancel exactly}{SUPPORT IS AN INCLUSION}{Traditional pp. 8, 21 / Theorem 2.5, Theorem 6.1, Corollary 6.3.}\nLet $K$ have transfers $0\\to1$, $0\\to2$, $1\\to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index","\\vpage{14}{Two routes can cancel exactly}{SUPPORT IS AN INCLUSION}{Traditional pp. 8, 21 / Theorem 2.5, Theorem 6.1, Corollary 6.3.}\nLet $K$ have transfers $0\\to1$, $0\\to2$, $1\\to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index, not an asse","\\vpage{14}{Two routes can cancel exactly}{SUPPORT IS AN INCLUSION}{Traditional pp. 8, 21 / Theorem 2.5, Theorem 6.1, Corollary 6.3.}\nLet $K$ have transfers $0\\to1$, $0\\to2$, $1\\to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index, not an assertion that $K^h\\ne0$. Here the","\\vpage{14}{Two routes can cancel exactly}{SUPPORT IS AN INCLUSION}{Traditional pp. 8, 21 / Theorem 2.5, Theorem 6.1, Corollary 6.3.}\nLet $K$ have transfers $0\\to1$, $0\\to2$, $1\\to3$, and $2\\to3$, with coefficients $1,1,1,-1$. There are two length-two routes from packet zero to packet three, but their contributions sum to zero.\n\\vfig{cancellation}\n\\[\n(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0,\n\\qquad K^2=0.\n\\]\nThe matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion $\\supp(AB)\\subseteq\\supp(A)\\circ\\supp(B)$ by equality would be wrong even in this four-dimensional model.\n\\head{What an acyclic graph does guarantee}\nIf the longest strict directed path has $h$ edges, then no term of $K^{h+1}$ can survive. Thus $K^{h+1}=0$. This is an upper bound on the nilpotence index, not an assertion that $K^h\\ne0$. Here the graph has paths of length two, while coefficient cancellation already gives $K^2=0$.","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{Traditional pp. 22, 35 / Theorem 6.5, Corollary 6.6, Theorem 10.3.}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimite","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{Traditional pp. 22, 35 / Theorem 6.5, Corollary 6.6, Theorem 10.3.}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary an","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{Traditional pp. 22, 35 / Theorem 6.5, Corollary 6.6, Theorem 10.3.}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attaina","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{Traditional pp. 22, 35 / Theorem 6.5, Corollary 6.6, Theorem 10.3.}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If o","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{Traditional pp. 22, 35 / Theorem 6.5, Corollary 6.6, Theorem 10.3.}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If only one e","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{Traditional pp. 22, 35 / Theorem 6.5, Corollary 6.6, Theorem 10.3.}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If only one execution","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{Traditional pp. 22, 35 / Theorem 6.5, Corollary 6.6, Theorem 10.3.}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If only one execution slot is a","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{Traditional pp. 22, 35 / Theorem 6.5, Corollary 6.6, Theorem 10.3.}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If only one execution slot is available,","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{Traditional pp. 22, 35 / Theorem 6.5, Corollary 6.6, Theorem 10.3.}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If only one execution slot is available, seven tasks","\\vpage{15}{A solve has a dependency clock}{TOPOLOGICAL LEVELS WITH EXACT OUTPUTS}{Traditional pp. 22, 35 / Theorem 6.5, Corollary 6.6, Theorem 10.3.}\nFor seven scalar packets, put two on the diagonal and $-1$ on each declared incoming edge. The edges are $0\\to2$, $1\\to2$, $1\\to3$, $2\\to4$, $3\\to4$, $3\\to5$, $4\\to6$, and $5\\to6$. Solve $(2I+K)x=\\ones$.\n\\vfig{triangular}\n\\head{Read the stages as dependencies}\nPackets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance,\n\\[\nx_4=\\frac{1+x_2+x_3}{2}=\\frac{11}{8},\\qquad\nx_6=\\frac{1+x_4+x_5}{2}=\\frac{13}{8}.\n\\]\nAll entries in the displayed solution are checked by multiplying the original triangular matrix by the resulting vector.\n\\head{When is four a minimum?}\nThere are four vertices on a longest dependency path. Under the paper's unit-duration, unlimited-concurrency model, four stages are both necessary and attainable. If only one execution slot is available, seven tasks require seven slots of","\\vpage{16}{Elimination creates new interactions}{SCHUR FILL AND ITS CANCELLATION}{Traditional p. 23 / Theorem 6.7, Proposition 6.8.}\nRetain the first two coordinates of\n\\[\nA=\\mat{2&0&1\\\\0&2&1\\\\1&1&2}.\n\\]\nThe third coordinate is hidden. Its invertible block is $[2]$, so eliminating it produces\n\\[\nS=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR}\n=\\mat{3/2&-1/2","\\vpage{17}{Invariant does not mean independently addressable}{A STRICT KRYLOV INCLUSION}{Traditional p. 24 / Proposition 7.2, Theorem 7.3.}\nTake $A=I_2$, split into the two coordinate packets, and start with $v=(1,1)^T$. Every packet is invariant. Nevertheless every polynomial applies the \\emph{same} scalar to both coordinates:\n\\[\np(A)v=p(1)(1,1)^T,\\qquad \\dim\\mathcal K_m(A,v)=1\\quad(m\\ge1).\n\\]\n\\vfig{krylov}\nThe direct sum of the two local Krylov spans is all of $\\QQ^2$. It permits independent coefficients in the two packets. The global Krylov space does not. This is why the manuscript states an inclusion, not an unconditional equality.\n\\head{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors","\\vpage{17}{Invariant does not mean independently addressable}{A STRICT KRYLOV INCLUSION}{Traditional p. 24 / Proposition 7.2, Theorem 7.3.}\nTake $A=I_2$, split into the two coordinate packets, and start with $v=(1,1)^T$. Every packet is invariant. Nevertheless every polynomial applies the \\emph{same} scalar to both coordinates:\n\\[\np(A)v=p(1)(1,1)^T,\\qquad \\dim\\mathcal K_m(A,v)=1\\quad(m\\ge1).\n\\]\n\\vfig{krylov}\nThe direct sum of the two local Krylov spans is all of $\\QQ^2$. It permits independent coefficients in the two packets. The global Krylov space does not. This is why the manuscript states an inclusion, not an unconditional equality.\n\\head{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=","\\vpage{17}{Invariant does not mean independently addressable}{A STRICT KRYLOV INCLUSION}{Traditional p. 24 / Proposition 7.2, Theorem 7.3.}\nTake $A=I_2$, split into the two coordinate packets, and start with $v=(1,1)^T$. Every packet is invariant. Nevertheless every polynomial applies the \\emph{same} scalar to both coordinates:\n\\[\np(A)v=p(1)(1,1)^T,\\qquad \\dim\\mathcal K_m(A,v)=1\\quad(m\\ge1).\n\\]\n\\vfig{krylov}\nThe direct sum of the two local Krylov spans is all of $\\QQ^2$. It permits independent coefficients in the two packets. The global Krylov space does not. This is why the manuscript states an inclusion, not an unconditional equality.\n\\head{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=I_2$. Both changes enlarge the addressing power, but in different ways.\n\\head{When invariance is only approximate}\nIf $A=D+K$ with $D$ packet diagonal, then\n\\[\n\\|A^j-D^j\\|\\le jM","CT KRYLOV INCLUSION}{Traditional p. 24 / Proposition 7.2, Theorem 7.3.}\nTake $A=I_2$, split into the two coordinate packets, and start with $v=(1,1)^T$. Every packet is invariant. Nevertheless every polynomial applies the \\emph{same} scalar to both coordinates:\n\\[\np(A)v=p(1)(1,1)^T,\\qquad \\dim\\mathcal K_m(A,v)=1\\quad(m\\ge1).\n\\]\n\\vfig{krylov}\nThe direct sum of the two local Krylov spans is all of $\\QQ^2$. It permits independent coefficients in the two packets. The global Krylov space does not. This is why the manuscript states an inclusion, not an unconditional equality.\n\\head{What breaks the linkage?}\nWith $A=\\diag(1,2)$, the vectors $v$ and $Av=(1,2)^T$ are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when $A=I_2$. Both changes enlarge the addressing power, but in different ways.\n\\head{When invariance is only approximate}\nIf $A=D+K$ with $D$ packet diagonal, then\n\\[\n\\|A^j-D^j\\|\\le jM^{j-1}\\|K\\|,\n\\qquad M=\\max\\{\\|A\\|,\\|D\\|\\}.\n\\]\nThis controls departure fro…","\\vpage{18}{Measure coupling after normalization}{BLOCK JACOBI AS A GEOMETRIC CHANGE}{Traditional p. 25 / Theorem 7.4, Corollary 7.6, Classical input 7.5.}\nLet\n\\[\nA=\\mat{4&1\\\\1&1},\\quad D=\\diag(4,1),\\quad\nB=D^{-1/2}=\\diag(1/2,1).\n\\]\nThen $BAB=I+E=\\mat{1&1/2","\\vpage{19}{Twelve small blocks, one coherent error}{DISCOVERY AND PERTURBATION CERTIFICATES}{Traditional pp. 26, 30 / Theorem 8.1, Proposition 9.2.}\nSplit $\\QQ^4$ into scalar packets and let\n\\[\nA=I+E,\\qquad E=\\frac1{100}(\\ones\\ones^T-I).\n\\]\nDropping every off-diagonal block leaves $I$. Each discarded entry is only $1/100$, but the total error is not bounded by that single-entry size.\n\\vfig{block_error}\n\\head{Two norms answer different questions}\nThere are twelve omitted entries, so $\\|E\\|_F^2=12/10^4$. The all-one vector is an eigenvector with eigenvalue $3/100$, while the three orthogonal directions have eigenvalue $-1/100$. Hence $\\|E\\|_2=3/100$.","TES}{Traditional pp. 26, 30 / Theorem 8.1, Proposition 9.2.}\nSplit $\\QQ^4$ into scalar packets and let\n\\[\nA=I+E,\\qquad E=\\frac1{100}(\\ones\\ones^T-I).\n\\]\nDropping every off-diagonal block leaves $I$. Each discarded entry is only $1/100$, but the total error is not bounded by that single-entry size.\n\\vfig{block_error}\n\\head{Two norms answer different questions}\nThere are twelve omitted entries, so $\\|E\\|_F^2=12/10^4$. The all-one vector is an eigenvector with eigenvalue $3/100$, while the three orthogonal directions have eigenvalue $-1/100$. Hence $\\|E\\|_2=3/100$.","\\vpage{20}{Spend error where it is cheapest}{CONTINUOUS AND CLIPPED PRECISION}{Traditional pp. 27, 28 / Theorem 8.4, Theorem 8.5.}\nTake sensitivities $s=(1,2,4)$, cost coefficients $c=(1,1,2)$, and budget $\\tau=1$. Minimizing $\\sum_ec_e\\log(1/\\epsilon_e)$ subject to $\\sum_es_e\\epsilon_e=1$ gives\n\\[\n\\epsilon_e=\\frac{c_e}{4s_e},\\qquad\n\\epsilon=(1/4,1/8,1/8).\n\\]\n\\vfig{precision}\n\\head{The budget is weighted}\nThe three contributions to the error allowance are $(1/4,1/4,1/2)$. Equal cost coefficients give equal weighted contributions, not equal tolerances. The third edge has twice the cost coefficient of the second, which offsets its doubled sensitivity in this example.","LIPPED PRECISION}{Traditional pp. 27, 28 / Theorem 8.4, Theorem 8.5.}\nTake sensitivities $s=(1,2,4)$, cost coefficients $c=(1,1,2)$, and budget $\\tau=1$. Minimizing $\\sum_ec_e\\log(1/\\epsilon_e)$ subject to $\\sum_es_e\\epsilon_e=1$ gives\n\\[\n\\epsilon_e=\\frac{c_e}{4s_e},\\qquad\n\\epsilon=(1/4,1/8,1/8).\n\\]\n\\vfig{precision}\n\\head{The budget is weighted}\nThe three contributions to the error allowance are $(1/4,1/4,1/2)$. Equal cost coefficients give equal weighted contributions, not equal tolerances. The third edge has twice the cost coefficient of the second, which offsets its doubled sensitivity in this example.","\\vpage{21}{An exact budget is not a greedy choice}{THREE EDGES, TWENTY-SEVEN ASSIGNMENTS}{Traditional p. 29 / Theorem 8.7, Remark 8.8.}\nEach edge has three allowed $(\\text{integer weight},\\text{cost})$ options:\n\\[\n\\begin{aligned}\n1 &: (0,7),(2,3),(4,1),","\\vpage{21}{An exact budget is not a greedy choice}{THREE EDGES, TWENTY-SEVEN ASSIGNMENTS}{Traditional p. 29 / Theorem 8.7, Remark 8.8.}\nEach edge has three allowed $(\\text{integer weight},\\text{cost})$ options:\n\\[\n\\begin{aligned}\n1 &: (0,7),(2,3),(4,1),\\\\\n2 &: (0,6),(1,4),(3,1),\\\\\n3 &: (0,9),(2,5),(5,1).\n\\end{aligned}\n\\]\nThere are exactly $3^3=27$ full assignments. The table stores the minimum cost at each \\emph{exact} weight; columns zero through eight are displayed.\n\\vfig{discrete}\n\\head{Read the final row correctly}\nAt budget four, minimize across columns zero through four. The optimum costs 14: choose weights $(2,0,2)$ with costs $(3,6,5)$. The zero-weight choice on edge two is its first, more expensive option. A locally cheap choice can consume too much of the shared budget.","\\vpage{22}{A penalty can miss the budget optimum}{FIVE PRUNINGS, ONE NONCONVEX FRONTIER}{Traditional pp. 31, 32, 33 / Theorem 9.4, Corollary 9.7, Theorem 9.8.}\nA root $r$ has children $a,b$, each with two terminal children. Keeping $r$ costs one with error six; splitting it adds overhead one. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. T","\\vpage{22}{A penalty can miss the budget optimum}{FIVE PRUNINGS, ONE NONCONVEX FRONTIER}{Traditional pp. 31, 32, 33 / Theorem 9.4, Corollary 9.7, Theorem 9.8.}\nA root $r$ has children $a,b$, each with two terminal children. Keeping $r$ costs one with error six; splitting it adds overhead one. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full spl","\\vpage{22}{A penalty can miss the budget optimum}{FIVE PRUNINGS, ONE NONCONVEX FRONTIER}{Traditional pp. 31, 32, 33 / Theorem 9.4, Corollary 9.7, Theorem 9.8.}\nA root $r$ has children $a,b$, each with two terminal children. Keeping $r$ costs one with error six; splitting it adds overhead one. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence","\\vpage{22}{A penalty can miss the budget optimum}{FIVE PRUNINGS, ONE NONCONVEX FRONTIER}{Traditional pp. 31, 32, 33 / Theorem 9.4, Corollary 9.7, Theorem 9.8.}\nA root $r$ has children $a,b$, each with two terminal children. Keeping $r$ costs one with error six; splitting it adds overhead one. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","e{22}{A penalty can miss the budget optimum}{FIVE PRUNINGS, ONE NONCONVEX FRONTIER}{Traditional pp. 31, 32, 33 / Theorem 9.4, Corollary 9.7, Theorem 9.8.}\nA root $r$ has children $a,b$, each with two terminal children. Keeping $r$ costs one with error six; splitting it adds overhead one. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","raditional pp. 31, 32, 33 / Theorem 9.4, Corollary 9.7, Theorem 9.8.}\nA root $r$ has children $a,b$, each with two terminal children. Keeping $r$ costs one with error six; splitting it adds overhead one. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","9.4, Corollary 9.7, Theorem 9.8.}\nA root $r$ has children $a,b$, each with two terminal children. Keeping $r$ costs one with error six; splitting it adds overhead one. Keeping either $a$ or $b$ costs three with error two. Splitting $a$ costs four with zero error, and splitting $b$ costs five with zero error.\n\\vfig{tree_frontier}\nThe five complete prunings have $(\\text{cost},\\text{error})$ pairs\n\\[\n(1,6),\\ (7,4),\\ (9,2),\\ (8,2),\\ (10,0).\n\\]\nAt budget two the unique cheapest pruning is $\\{a_0,a_1,b\\}$, costing eight. It is Pareto optimal, but never minimizes $\\cost+\\lambda\\err$ for any $\\lambda\\ge0$.\n\\head{See the incompatible inequalities}\nTo beat the root, that pruning would require $8+2\\lambda\\le1+6\\lambda$, hence $\\lambda\\ge7/4$. To beat the full split, it would require $8+2\\lambda\\le10$, hence $\\lambda\\le1$. No multiplier meets both conditions.","\\vpage{23}{Recompile the changed part, not everything}{LOCAL DEPENDENCIES AND STRICT MARGINS}{Traditional pp. 33, 34 / Theorem 9.11, Theorem 9.12.}\nUse the same tree with $\\lambda=2$. Increase only the leaf cost at $a_0$ from two to three. The exact subtree values change at $a_0$, its parent $a$, and the root $r$. Every value in the $b$ subtree remains untouched.\n\\vfig{local_update}\n\\head{Why the update is local}\nA dynamic-programming value depends only on the local keep/split data and its children's values. A changed leaf can therefore affect its ancestors but not a disjoint sibling subtree. Recomputing the dependency closure yields the same global optimum as a fresh traversal. In this example, the root value increases from ten to eleven.\n\\head{Strict gaps protect decisions}\nBefore the update, the root keep score is 13 and its split score is 10, giving gap three. If e","\\vpage{23}{Recompile the changed part, not everything}{LOCAL DEPENDENCIES AND STRICT MARGINS}{Traditional pp. 33, 34 / Theorem 9.11, Theorem 9.12.}\nUse the same tree with $\\lambda=2$. Increase only the leaf cost at $a_0$ from two to three. The exact subtree values change at $a_0$, its parent $a$, and the root $r$. Every value in the $b$ subtree remains untouched.\n\\vfig{local_update}\n\\head{Why the update is local}\nA dynamic-programming value depends only on the local keep/split data and its children's values. A changed leaf can therefore affect its ancestors but not a disjoint sibling subtree. Recomputing the dependency closure yields the same global optimum as a fresh traversal. In this example, the root value increases from ten to eleven.\n\\head{Strict gaps protect decisions}\nBefore the update, the root keep score is 13 and its split score is 10, giving gap three. If each score changes by less than $3/2$,","\\vpage{23}{Recompile the changed part, not everything}{LOCAL DEPENDENCIES AND STRICT MARGINS}{Traditional pp. 33, 34 / Theorem 9.11, Theorem 9.12.}\nUse the same tree with $\\lambda=2$. Increase only the leaf cost at $a_0$ from two to three. The exact subtree values change at $a_0$, its parent $a$, and the root $r$. Every value in the $b$ subtree remains untouched.\n\\vfig{local_update}\n\\head{Why the update is local}\nA dynamic-programming value depends only on the local keep/split data and its children's values. A changed leaf can therefore affect its ancestors but not a disjoint sibling subtree. Recomputing the dependency closure yields the same global optimum as a fresh traversal. In this example, the root value increases from ten to eleven.\n\\head{Strict gaps protect decisions}\nBefore the update, the root keep score is 13 and its split score is 10, giving gap three. If each score changes by less than $3/2$, their strict ordering is retained. Allowing exactly $3/2$ in","\\vpage{23}{Recompile the changed part, not everything}{LOCAL DEPENDENCIES AND STRICT MARGINS}{Traditional pp. 33, 34 / Theorem 9.11, Theorem 9.12.}\nUse the same tree with $\\lambda=2$. Increase only the leaf cost at $a_0$ from two to three. The exact subtree values change at $a_0$, its parent $a$, and the root $r$. Every value in the $b$ subtree remains untouched.\n\\vfig{local_update}\n\\head{Why the update is local}\nA dynamic-programming value depends only on the local keep/split data and its children's values. A changed leaf can therefore affect its ancestors but not a disjoint sibling subtree. Recomputing the dependency closure yields the same global optimum as a fresh traversal. In this example, the root value increases from ten to eleven.\n\\head{Strict gaps protect decisions}\nBefore the update, the root keep score is 13 and its split score is 10, giving gap three. If each score changes by less than $3/2$, their strict ordering is retained. Allowing exactly $3/2$ in both directions","\\vpage{23}{Recompile the changed part, not everything}{LOCAL DEPENDENCIES AND STRICT MARGINS}{Traditional pp. 33, 34 / Theorem 9.11, Theorem 9.12.}\nUse the same tree with $\\lambda=2$. Increase only the leaf cost at $a_0$ from two to three. The exact subtree values change at $a_0$, its parent $a$, and the root $r$. Every value in the $b$ subtree remains untouched.\n\\vfig{local_update}\n\\head{Why the update is local}\nA dynamic-programming value depends only on the local keep/split data and its children's values. A changed leaf can therefore affect its ancestors but not a disjoint sibling subtree. Recomputing the dependency closure yields the same global optimum as a fresh traversal. In this example, the root value increases from ten to eleven.\n\\head{Strict gaps protect decisions}\nBefore the update, the root keep score is 13 and its split score is 10, giving gap three. If each score changes by less than $3/2$, their strict ordering is retained. Allowing exactly $3/2$ in both directions can produce a tie","\\vpage{23}{Recompile the changed part, not everything}{LOCAL DEPENDENCIES AND STRICT MARGINS}{Traditional pp. 33, 34 / Theorem 9.11, Theorem 9.12.}\nUse the same tree with $\\lambda=2$. Increase only the leaf cost at $a_0$ from two to three. The exact subtree values change at $a_0$, its parent $a$, and the root $r$. Every value in the $b$ subtree remains untouched.\n\\vfig{local_update}\n\\head{Why the update is local}\nA dynamic-programming value depends only on the local keep/split data and its children's values. A changed leaf can therefore affect its ancestors but not a disjoint sibling subtree. Recomputing the dependency closure yields the same global optimum as a fresh traversal. In this example, the root value increases from ten to eleven.\n\\head{Strict gaps protect decisions}\nBefore the update, the root keep score is 13 and its split score is 10, giving gap three. If each score changes by less than $3/2$, their strict ordering is retained. Allowing exactly $3/2$ in both directions can produce a tie, so the strict inequali","\\vpage{25}{The first win comes after the tie}{SETUP AND REUSE IN ONE LINEAR MODEL}{Traditional pp. 35, 36 / Theorem 10.4, Corollary 10.5, Corollary 10.6.}\nSuppose the baseline costs seven units per use. A candidate costs 30 units to set up and two per use. These are synthetic constants chosen to expose the exact threshold:\n\\[\nT_b(R)=7R,\\qquad T_c(R)=30+2R.\n\\]\n\\vfig{amortization}\nAt $R=6$, both cost 42. The candidate becomes strictly faster at $R=7$, where it costs 44 instead of 49. The minimum profitable integer reuse count is\n\\[\n\\left\\lfloor\\frac{30}{7-2}\\right\\rfloor+1=7.\n\\]\nReplacing the floor-plus-one rule with a ceiling would be wrong when the ratio is already an integer and strict improvement is required.\n\\head{Which costs belong to setup?}\nDiscovery, compilation, host/device transfer, graph capture, and factor preparation can have different reuse lifetimes. If a matrix changes but its shape does not, a captured schedule might remain reusable while its numerical factors do not. A single setup number is meaningful only after those lifetim","\\vpage{25}{The first win comes after the tie}{SETUP AND REUSE IN ONE LINEAR MODEL}{Traditional pp. 35, 36 / Theorem 10.4, Corollary 10.5, Corollary 10.6.}\nSuppose the baseline costs seven units per use. A candidate costs 30 units to set up and two per use. These are synthetic constants chosen to expose the exact threshold:\n\\[\nT_b(R)=7R,\\qquad T_c(R)=30+2R.\n\\]\n\\vfig{amortization}\nAt $R=6$, both cost 42. The candidate becomes strictly faster at $R=7$, where it costs 44 instead of 49. The minimum profitable integer reuse count is\n\\[\n\\left\\lfloor\\frac{30}{7-2}\\right\\rfloor+1=7.\n\\]\nReplacing the floor-plus-one rule with a ceiling would be wrong when the ratio is already an integer and strict improvement is required.\n\\head{Which costs belong to setup?}\nDiscovery, compilation, host/device transfer, graph capture, and factor preparation can have different reuse lifetimes. If a matrix changes but its shape does not, a captured schedule might remain reusable while its numerical factors do not. A single setup number is meaningful only after those lifetimes are fixed.","TUP AND REUSE IN ONE LINEAR MODEL}{Traditional pp. 35, 36 / Theorem 10.4, Corollary 10.5, Corollary 10.6.}\nSuppose the baseline costs seven units per use. A candidate costs 30 units to set up and two per use. These are synthetic constants chosen to expose the exact threshold:\n\\[\nT_b(R)=7R,\\qquad T_c(R)=30+2R.\n\\]\n\\vfig{amortization}\nAt $R=6$, both cost 42. The candidate becomes strictly faster at $R=7$, where it costs 44 instead of 49. The minimum profitable integer reuse count is\n\\[\n\\left\\lfloor\\frac{30}{7-2}\\right\\rfloor+1=7.\n\\]\nReplacing the floor-plus-one rule with a ceiling would be wrong when the ratio is already an integer and strict improvement is required.\n\\head{Which costs belong to setup?}\nDiscovery, compilation, host/device transfer, graph capture, and factor preparation can have different reuse lifetimes. If a matrix changes but its shape does not, a captured schedule might remain reusable while its numerical factors do not. A single setup number is meaningful only after those lifetimes are fixed.","5, Corollary 10.6.}\nSuppose the baseline costs seven units per use. A candidate costs 30 units to set up and two per use. These are synthetic constants chosen to expose the exact threshold:\n\\[\nT_b(R)=7R,\\qquad T_c(R)=30+2R.\n\\]\n\\vfig{amortization}\nAt $R=6$, both cost 42. The candidate becomes strictly faster at $R=7$, where it costs 44 instead of 49. The minimum profitable integer reuse count is\n\\[\n\\left\\lfloor\\frac{30}{7-2}\\right\\rfloor+1=7.\n\\]\nReplacing the floor-plus-one rule with a ceiling would be wrong when the ratio is already an integer and strict improvement is required.\n\\head{Which costs belong to setup?}\nDiscovery, compilation, host/device transfer, graph capture, and factor preparation can have different reuse lifetimes. If a matrix changes but its shape does not, a captured schedule might remain reusable while its numerical factors do not. A single setup number is meaningful only after those lifetimes are fixed.","\\vpage{26}{Same output before comparing times}{A BENCHMARK ROW IS A CONTRACT}{Traditional pp. 9, 36, 37 / Proposition 2.9, Section 11, Section 12.}\nConsider four synthetic timing rows for a product. Row B returns factors, while row A returns an explicit matrix. A ratio $12/2=6$ does not compare two implementations of the same returned object. Adding the decoding cost changes the candidate to row C.\n\\vfig{contract_match}\n\\head{What a fair match retains}\nThe input values, dimensions, layout, source precision, correctness reference, returned output, and timing scope must agree. Row D changes the input precision; that can be a useful accuracy tradeoff, but it is not the same case merely because the output has the same shape.","\\vpage{27}{Close the loop on the running example}{FACTORS, APPLICATION, AND AN INDEPENDENT REFERENCE}{Traditional pp. 7, 11, 16 / Theorem 2.3, Theorem 3.4, Theorem 5.1.}\nReturn to the two operators from \\pagerefV{9}. The exact product representation has diagonal part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization","\\vpage{27}{Close the loop on the running example}{FACTORS, APPLICATION, AND AN INDEPENDENT REFERENCE}{Traditional pp. 7, 11, 16 / Theorem 2.3, Theorem 3.4, Theorem 5.1.}\nReturn to the two operators from \\pagerefV{9}. The exact product representation has diagonal part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growt","\\vpage{27}{Close the loop on the running example}{FACTORS, APPLICATION, AND AN INDEPENDENT REFERENCE}{Traditional pp. 7, 11, 16 / Theorem 2.3, Theorem 3.4, Theorem 5.1.}\nReturn to the two operators from \\pagerefV{9}. The exact product representation has diagonal part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growth and decide whether recompression is useful. If","\\vpage{27}{Close the loop on the running example}{FACTORS, APPLICATION, AND AN INDEPENDENT REFERENCE}{Traditional pp. 7, 11, 16 / Theorem 2.3, Theorem 3.4, Theorem 5.1.}\nReturn to the two operators from \\pagerefV{9}. The exact product representation has diagonal part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growth and decide whether recompression is useful. If approximation is introduced, tra…","ional pp. 7, 11, 16 / Theorem 2.3, Theorem 3.4, Theorem 5.1.}\nReturn to the two operators from \\pagerefV{9}. The exact product representation has diagonal part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growth and decide whether recompression is useful. If approximation is introduced, transport its error through the later operators rather than reporting only a local singular tail.","3.4, Theorem 5.1.}\nReturn to the two operators from \\pagerefV{9}. The exact product representation has diagonal part $D_AD_B$ and two-column correction factors $U_C,V_C$. Apply it to the original $x=(1,-1,2,0,-2,3)^T$.\n\\[\nV_C^Tx=(2,13)^T,\\qquad\n(AB)x=D_AD_Bx+U_C(V_C^Tx).\n\\]\n\\vfig{end_to_end}\n\\head{A complete reader calculation}\nThe local contribution is $(1,-5,18,6,-12,44)^T$. The correction is $(25,21,25,21,25,21)^T$. Their sum is\n\\[\n(26,16,43,27,13,65)^T.\n\\]\nThe checker forms the dense matrices $A$ and $B$ independently, multiplies them, applies the product to $x$, and compares all six components. The dense array is a validation reference, not a requirement of the factored application route.\n\\head{Choose what happens next}\nFor another input, reuse the factors. For a required explicit matrix, pay materialization. For a long product chain, track rank growth and decide whether recompression is useful. If approximation is introduced, transport its error through the later operators rather than reporting only a local singular tail."]}