{"collection_revision":"2f443f44c190a64c713d6ea71c3d99ce48f78252","source":"papers/paper-vi/current/editable/traditional/atlas-pages.tex","source_sha256":"07d06ea377c9e5a295002c9eb1d27a213795209467e7272b8a0aea73809ea9e3","definitions":"papers/paper-vi/current/editable/traditional/atlas-definitions.tex","definitions_sha256":"87f31eedf5fb33ac00a0d47eea521f41713371efe6a68d1d7679629ca7b9f36d","macro_source":"papers/paper-vi/current/editable/traditional/Paper_VI_Traditional_Final.tex","macros":{"cA":"\\mathcal A","cB":"\\mathcal B","cC":"\\mathcal C","cE":"\\mathcal E","cF":"\\mathcal F","cG":"\\mathcal G","cH":"\\mathcal H","cK":"\\mathcal K","cL":"\\mathcal L","cP":"\\mathcal P","cR":"\\mathcal R","cS":"\\mathcal S","cT":"\\mathcal T","bbC":"\\mathbb C","bbR":"\\mathbb R","bbZ":"\\mathbb Z","bbN":"\\mathbb N","bbP":"\\mathbb P","Exp":"\\mathbb E","diag":"\\operatorname{diag}","rank":"\\operatorname{rank}","Ran":"\\operatorname{Ran}","Ker":"\\operatorname{Ker}","Var":"\\operatorname{Var}","Cov":"\\operatorname{Cov}","Spec":"\\operatorname{Spec}","Id":"\\operatorname{Id}","Tr":"\\operatorname{Tr}","Dfun":"\\mathsf D","Sfun":"\\mathsf S","CC":"\\mathbb C","RR":"\\mathbb R","NN":"\\mathbb N","Pcal":"\\mathcal P","Tcal":"\\mathcal T","Kcal":"\\mathcal K","Bcal":"\\mathcal B","Ecal":"\\mathcal E","Fcal":"\\mathcal F","Hcal":"\\mathcal H","Lcal":"\\mathcal L","Rcal":"\\mathcal R","Scal":"\\mathcal S","Ocal":"\\mathcal O","Ob":"\\operatorname{Ob}","tr":"\\operatorname{tr}","supp":"\\operatorname{supp}","osc":"\\operatorname{osc}","id":"\\mathrm{id}","e":"\\mathrm e","ii":"\\mathrm i","one":"\\bm 1","dd":"\\,\\mathrm d","simplex":"\\mathop{\\rm int}\\Delta","EE":"\\mathbb E","PP":"\\mathbb P","Acal":"\\mathcal A","Ccal":"\\mathcal C","Dcal":"\\mathcal D","Mcal":"\\mathcal M","Qcal":"\\mathcal Q","Vcal":"\\mathcal V","TV":"\\operatorname{TV}","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}","End":"\\operatorname{End}","src":"\\operatorname{src}","tgt":"\\operatorname{tgt}","Res":"\\operatorname{Res}","sech":"\\operatorname{sech}","Efun":"\\mathsf E","Tfun":"\\mathsf T","Cfun":"\\mathsf C","Thetafun":"\\symup{\\Theta}","ones":"\\mathbf{1}","norm":{"arguments":1,"body":"\\left\\lVert#1\\right\\rVert"},"mat":{"arguments":1,"body":"\\begin{pmatrix}#1\\end{pmatrix}"},"abs":{"arguments":1,"body":"\\left\\lvert#1\\right\\rvert"},"ip":{"arguments":2,"body":"\\left\\langle#1,#2\\right\\rangle"},"bm":{"arguments":1,"body":"\\boldsymbol{#1}","unbraced":true},"diagmat":{"arguments":1,"body":"\\operatorname{diag}(#1)"}},"source_pdf":"papers/paper-vi/current/papers/Paper_VI_Visual_Guide_Final.pdf","source_pdf_sha256":"1c13f03f789d0c6d6ea465fb734b9305ddaaf7f575225932bfaccd1b25cd3a0b","index":"read/paper-vi/atlas/index.html","total_plates":42,"html_plates":[{"number":1,"url":"read/paper-vi/atlas/v01.html","start_line":5,"end_line":22,"math_expressions":10,"math_sequence_sha256":"1bdb5f126cca451765efb8a016023398a1184bf731618a8d9c3cd2e0f9ccd5e1","nodes":[{"kind":"text","id":"L6","line":6,"end_line":7},{"kind":"figure","id":"L7","line":7,"end_line":7},{"kind":"readerroadmap","id":"L8","line":8,"end_line":8},{"kind":"section","id":"L9","line":9,"end_line":9},{"kind":"text","id":"L10","line":10,"end_line":14},{"kind":"tabularx","id":"L14","line":14,"end_line":20},{"kind":"readerroadmap","id":"L21","line":21,"end_line":21}]},{"number":2,"url":"read/paper-vi/atlas/v02.html","start_line":23,"end_line":52,"math_expressions":28,"math_sequence_sha256":"84bb1367806c6ddf7871e4a5d26980011b70e7f1ce57fc89b29785ab04152d15","nodes":[{"kind":"text","id":"L24","line":24,"end_line":26},{"kind":"tabularx","id":"L26","line":26,"end_line":39},{"kind":"section","id":"L40","line":40,"end_line":40},{"kind":"text","id":"L41","line":41,"end_line":51},{"kind":"readerroadmap","id":"L51","line":51,"end_line":51}]},{"number":3,"url":"read/paper-vi/atlas/v03.html","start_line":53,"end_line":73,"math_expressions":23,"math_sequence_sha256":"7b38312a5190b88520226ab30441a9b6d8320913171bf96e6260dbd465e4abb3","nodes":[{"kind":"text","id":"L54","line":54,"end_line":62},{"kind":"figure","id":"L62","line":62,"end_line":62},{"kind":"readerroadmap","id":"L63","line":63,"end_line":63},{"kind":"section","id":"L64","line":64,"end_line":64},{"kind":"text","id":"L65","line":65,"end_line":72},{"kind":"readerroadmap","id":"L72","line":72,"end_line":72}]},{"number":4,"url":"read/paper-vi/atlas/v04.html","start_line":74,"end_line":97,"math_expressions":36,"math_sequence_sha256":"d6710ce934f90355dd9b4cebb5f777d9ea6c006ed29c1ea2c1a6b2c6b5e51859","nodes":[{"kind":"text","id":"L75","line":75,"end_line":80},{"kind":"figure","id":"L80","line":80,"end_line":80},{"kind":"readerroadmap","id":"L81","line":81,"end_line":81},{"kind":"section","id":"L82","line":82,"end_line":82},{"kind":"text","id":"L83","line":83,"end_line":96},{"kind":"readerroadmap","id":"L96","line":96,"end_line":96}]},{"number":5,"url":"read/paper-vi/atlas/v05.html","start_line":98,"end_line":123,"math_expressions":40,"math_sequence_sha256":"8cf751474227b51d7842c1a20f4eb5c2e707d86350cc346c52a488804b690ae0","nodes":[{"kind":"text","id":"L99","line":99,"end_line":105},{"kind":"figure","id":"L105","line":105,"end_line":105},{"kind":"readerroadmap","id":"L106","line":106,"end_line":106},{"kind":"section","id":"L107","line":107,"end_line":107},{"kind":"text","id":"L108","line":108,"end_line":122},{"kind":"readerroadmap","id":"L122","line":122,"end_line":122}]},{"number":6,"url":"read/paper-vi/atlas/v06.html","start_line":124,"end_line":147,"math_expressions":20,"math_sequence_sha256":"7e599ed3e8885c99636318736e0af7bf35bf8e510cb500b06df46ba39f2decd4","nodes":[{"kind":"text","id":"L125","line":125,"end_line":131},{"kind":"figure","id":"L131","line":131,"end_line":131},{"kind":"readerroadmap","id":"L132","line":132,"end_line":132},{"kind":"section","id":"L133","line":133,"end_line":133},{"kind":"text","id":"L134","line":134,"end_line":146},{"kind":"readerroadmap","id":"L146","line":146,"end_line":146}]},{"number":7,"url":"read/paper-vi/atlas/v07.html","start_line":148,"end_line":169,"math_expressions":27,"math_sequence_sha256":"7dd9308db2530f2a9f6233e4a4b3fd0ee040549c6a8248f1d5ff755aab7cc273","nodes":[{"kind":"text","id":"L149","line":149,"end_line":156},{"kind":"figure","id":"L156","line":156,"end_line":156},{"kind":"readerroadmap","id":"L157","line":157,"end_line":157},{"kind":"section","id":"L158","line":158,"end_line":158},{"kind":"text","id":"L159","line":159,"end_line":168},{"kind":"readerroadmap","id":"L168","line":168,"end_line":168}]},{"number":8,"url":"read/paper-vi/atlas/v08.html","start_line":170,"end_line":194,"math_expressions":21,"math_sequence_sha256":"ec4829d10d6ba92d5243406f4081a8247579f2599e35476f6c8204957627eff3","nodes":[{"kind":"text","id":"L171","line":171,"end_line":176},{"kind":"figure","id":"L176","line":176,"end_line":176},{"kind":"readerroadmap","id":"L177","line":177,"end_line":177},{"kind":"section","id":"L178","line":178,"end_line":178},{"kind":"text","id":"L179","line":179,"end_line":193},{"kind":"readerroadmap","id":"L193","line":193,"end_line":193}]},{"number":9,"url":"read/paper-vi/atlas/v09.html","start_line":195,"end_line":213,"math_expressions":22,"math_sequence_sha256":"9955ed56b23f62413447d33a482bb7095df06df66050824bfdea645edf087dec","nodes":[{"kind":"text","id":"L196","line":196,"end_line":201},{"kind":"figure","id":"L201","line":201,"end_line":201},{"kind":"readerroadmap","id":"L202","line":202,"end_line":202},{"kind":"section","id":"L203","line":203,"end_line":203},{"kind":"text","id":"L204","line":204,"end_line":212},{"kind":"readerroadmap","id":"L212","line":212,"end_line":212}]},{"number":10,"url":"read/paper-vi/atlas/v10.html","start_line":214,"end_line":236,"math_expressions":25,"math_sequence_sha256":"8e3e8b4e958b740561e1b48ffe7565e9bc1e4b6255f560ec094a8f12e7f6b7a4","nodes":[{"kind":"text","id":"L215","line":215,"end_line":221},{"kind":"figure","id":"L221","line":221,"end_line":221},{"kind":"readerroadmap","id":"L222","line":222,"end_line":222},{"kind":"section","id":"L223","line":223,"end_line":223},{"kind":"text","id":"L224","line":224,"end_line":235},{"kind":"readerroadmap","id":"L235","line":235,"end_line":235}]},{"number":11,"url":"read/paper-vi/atlas/v11.html","start_line":237,"end_line":257,"math_expressions":22,"math_sequence_sha256":"e34dddbb84a5e0cdf0893ab7ad2653d7c20177627613f34914939a590d887fe5","nodes":[{"kind":"text","id":"L238","line":238,"end_line":243},{"kind":"figure","id":"L243","line":243,"end_line":243},{"kind":"readerroadmap","id":"L244","line":244,"end_line":244},{"kind":"section","id":"L245","line":245,"end_line":245},{"kind":"text","id":"L246","line":246,"end_line":256},{"kind":"readerroadmap","id":"L256","line":256,"end_line":256}]},{"number":12,"url":"read/paper-vi/atlas/v12.html","start_line":258,"end_line":281,"math_expressions":22,"math_sequence_sha256":"274f5d964477ac645dbb292397d54b9cfb87989f71452459c1215552dac44d84","nodes":[{"kind":"text","id":"L259","line":259,"end_line":264},{"kind":"figure","id":"L264","line":264,"end_line":264},{"kind":"readerroadmap","id":"L265","line":265,"end_line":265},{"kind":"section","id":"L266","line":266,"end_line":266},{"kind":"text","id":"L267","line":267,"end_line":280},{"kind":"readerroadmap","id":"L280","line":280,"end_line":280}]},{"number":13,"url":"read/paper-vi/atlas/v13.html","start_line":282,"end_line":303,"math_expressions":21,"math_sequence_sha256":"1d268ba2fc0741745c93ae54c896ca7ff473d14bb4dd5a253903ace53d98c9bb","nodes":[{"kind":"text","id":"L283","line":283,"end_line":286},{"kind":"figure","id":"L286","line":286,"end_line":286},{"kind":"readerroadmap","id":"L287","line":287,"end_line":287},{"kind":"section","id":"L288","line":288,"end_line":288},{"kind":"text","id":"L289","line":289,"end_line":302},{"kind":"readerroadmap","id":"L302","line":302,"end_line":302}]},{"number":14,"url":"read/paper-vi/atlas/v14.html","start_line":304,"end_line":323,"math_expressions":19,"math_sequence_sha256":"496814d7fb17a497e31ddb03de79bd3d1dbae35c16d719a6d86b705181b917ea","nodes":[{"kind":"text","id":"L305","line":305,"end_line":310},{"kind":"figure","id":"L310","line":310,"end_line":310},{"kind":"readerroadmap","id":"L311","line":311,"end_line":311},{"kind":"section","id":"L312","line":312,"end_line":312},{"kind":"text","id":"L313","line":313,"end_line":322},{"kind":"readerroadmap","id":"L322","line":322,"end_line":322}]},{"number":15,"url":"read/paper-vi/atlas/v15.html","start_line":324,"end_line":348,"math_expressions":23,"math_sequence_sha256":"2d549647ba51f0ed2ed40f2069c006abb494b932a5c3f544933c5b1ed9dcfd75","nodes":[{"kind":"text","id":"L325","line":325,"end_line":331},{"kind":"figure","id":"L331","line":331,"end_line":331},{"kind":"readerroadmap","id":"L332","line":332,"end_line":332},{"kind":"section","id":"L333","line":333,"end_line":333},{"kind":"text","id":"L334","line":334,"end_line":347},{"kind":"readerroadmap","id":"L347","line":347,"end_line":347}]},{"number":16,"url":"read/paper-vi/atlas/v16.html","start_line":349,"end_line":372,"math_expressions":24,"math_sequence_sha256":"c9675a8fbe7f5632e8d46a05e9c7bfbe80cb932601915f79f92c63f68db6e837","nodes":[{"kind":"text","id":"L350","line":350,"end_line":358},{"kind":"figure","id":"L358","line":358,"end_line":358},{"kind":"readerroadmap","id":"L359","line":359,"end_line":359},{"kind":"section","id":"L360","line":360,"end_line":360},{"kind":"text","id":"L361","line":361,"end_line":371},{"kind":"readerroadmap","id":"L371","line":371,"end_line":371}]},{"number":17,"url":"read/paper-vi/atlas/v17.html","start_line":373,"end_line":396,"math_expressions":23,"math_sequence_sha256":"61112e56a68ef4c5b8604d5e65d5d421f845eb68025f09a7e3d3ff2306431323","nodes":[{"kind":"text","id":"L374","line":374,"end_line":381},{"kind":"figure","id":"L381","line":381,"end_line":381},{"kind":"readerroadmap","id":"L382","line":382,"end_line":382},{"kind":"section","id":"L383","line":383,"end_line":383},{"kind":"text","id":"L384","line":384,"end_line":395},{"kind":"readerroadmap","id":"L395","line":395,"end_line":395}]},{"number":18,"url":"read/paper-vi/atlas/v18.html","start_line":397,"end_line":421,"math_expressions":31,"math_sequence_sha256":"3244d75964b0c274986eddee2dd842aa330d89d739e94cbe11cbfd3bb20a571b","nodes":[{"kind":"text","id":"L398","line":398,"end_line":407},{"kind":"figure","id":"L407","line":407,"end_line":407},{"kind":"readerroadmap","id":"L408","line":408,"end_line":408},{"kind":"section","id":"L409","line":409,"end_line":409},{"kind":"text","id":"L410","line":410,"end_line":420},{"kind":"readerroadmap","id":"L420","line":420,"end_line":420}]},{"number":19,"url":"read/paper-vi/atlas/v19.html","start_line":422,"end_line":445,"math_expressions":27,"math_sequence_sha256":"dadc378e9f7079f8ba5a8ff28a6bec7ee30c6ae53620b702791cf1f92daa896c","nodes":[{"kind":"text","id":"L423","line":423,"end_line":432},{"kind":"figure","id":"L432","line":432,"end_line":432},{"kind":"readerroadmap","id":"L433","line":433,"end_line":433},{"kind":"section","id":"L434","line":434,"end_line":434},{"kind":"text","id":"L435","line":435,"end_line":444},{"kind":"readerroadmap","id":"L444","line":444,"end_line":444}]},{"number":20,"url":"read/paper-vi/atlas/v20.html","start_line":446,"end_line":465,"math_expressions":46,"math_sequence_sha256":"c3829db33619d3748eedf31437fa39081e8f5a5774cefe639a117d8a8145f31a","nodes":[{"kind":"text","id":"L447","line":447,"end_line":452},{"kind":"figure","id":"L452","line":452,"end_line":452},{"kind":"readerroadmap","id":"L453","line":453,"end_line":453},{"kind":"section","id":"L454","line":454,"end_line":454},{"kind":"text","id":"L455","line":455,"end_line":464},{"kind":"readerroadmap","id":"L464","line":464,"end_line":464}]},{"number":21,"url":"read/paper-vi/atlas/v21.html","start_line":466,"end_line":486,"math_expressions":19,"math_sequence_sha256":"03b0588f5966694d562bbc3d23684810518c1937e88eaaf613533773510e4dfc","nodes":[{"kind":"text","id":"L467","line":467,"end_line":477},{"kind":"figure","id":"L477","line":477,"end_line":477},{"kind":"readerroadmap","id":"L478","line":478,"end_line":478},{"kind":"section","id":"L479","line":479,"end_line":479},{"kind":"text","id":"L480","line":480,"end_line":485},{"kind":"readerroadmap","id":"L485","line":485,"end_line":485}]},{"number":22,"url":"read/paper-vi/atlas/v22.html","start_line":487,"end_line":513,"math_expressions":15,"math_sequence_sha256":"0df9018c68cd262586ebb419e8cca52c5a0289f2ac3944360ec643fc586b999c","nodes":[{"kind":"text","id":"L488","line":488,"end_line":501},{"kind":"figure","id":"L501","line":501,"end_line":501},{"kind":"readerroadmap","id":"L502","line":502,"end_line":502},{"kind":"section","id":"L503","line":503,"end_line":503},{"kind":"text","id":"L504","line":504,"end_line":512},{"kind":"readerroadmap","id":"L512","line":512,"end_line":512}]},{"number":23,"url":"read/paper-vi/atlas/v23.html","start_line":514,"end_line":540,"math_expressions":16,"math_sequence_sha256":"45a908b0d562fecd23fbae76ef8a251c7fe57081bfd8c2df4e3c23690b24dc61","nodes":[{"kind":"text","id":"L515","line":515,"end_line":525},{"kind":"figure","id":"L525","line":525,"end_line":525},{"kind":"readerroadmap","id":"L526","line":526,"end_line":526},{"kind":"section","id":"L527","line":527,"end_line":527},{"kind":"text","id":"L528","line":528,"end_line":539},{"kind":"readerroadmap","id":"L539","line":539,"end_line":539}]},{"number":24,"url":"read/paper-vi/atlas/v24.html","start_line":541,"end_line":567,"math_expressions":18,"math_sequence_sha256":"1fff9feda704c7082f9451f4a85467c905f7e705dd4c92bbc7644e1429a88546","nodes":[{"kind":"text","id":"L542","line":542,"end_line":552},{"kind":"figure","id":"L552","line":552,"end_line":552},{"kind":"readerroadmap","id":"L553","line":553,"end_line":553},{"kind":"section","id":"L554","line":554,"end_line":554},{"kind":"text","id":"L555","line":555,"end_line":566},{"kind":"readerroadmap","id":"L566","line":566,"end_line":566}]},{"number":25,"url":"read/paper-vi/atlas/v25.html","start_line":568,"end_line":592,"math_expressions":39,"math_sequence_sha256":"4f8614d5bb2f2a0417110affcb0f406d7a8261213415cdef7cd4df49d96c9587","nodes":[{"kind":"text","id":"L569","line":569,"end_line":579},{"kind":"figure","id":"L579","line":579,"end_line":579},{"kind":"readerroadmap","id":"L580","line":580,"end_line":580},{"kind":"section","id":"L581","line":581,"end_line":581},{"kind":"text","id":"L582","line":582,"end_line":591},{"kind":"readerroadmap","id":"L591","line":591,"end_line":591}]},{"number":26,"url":"read/paper-vi/atlas/v26.html","start_line":593,"end_line":618,"math_expressions":26,"math_sequence_sha256":"ac68df819cab23cdd19a9f296c4fa6f269c7aa6dcbb2828080b7cc7a2c36685a","nodes":[{"kind":"text","id":"L594","line":594,"end_line":604},{"kind":"figure","id":"L604","line":604,"end_line":604},{"kind":"readerroadmap","id":"L605","line":605,"end_line":605},{"kind":"section","id":"L606","line":606,"end_line":606},{"kind":"text","id":"L607","line":607,"end_line":617},{"kind":"readerroadmap","id":"L617","line":617,"end_line":617}]},{"number":27,"url":"read/paper-vi/atlas/v27.html","start_line":619,"end_line":642,"math_expressions":14,"math_sequence_sha256":"812483e19228e9f82916679df3e2ab225df7cb6408691eae807a85c0bdccde35","nodes":[{"kind":"text","id":"L620","line":620,"end_line":627},{"kind":"figure","id":"L627","line":627,"end_line":627},{"kind":"readerroadmap","id":"L628","line":628,"end_line":628},{"kind":"section","id":"L629","line":629,"end_line":629},{"kind":"text","id":"L630","line":630,"end_line":641},{"kind":"readerroadmap","id":"L641","line":641,"end_line":641}]},{"number":28,"url":"read/paper-vi/atlas/v28.html","start_line":643,"end_line":666,"math_expressions":0,"math_sequence_sha256":"4f53cda18c2baa0c0354bb5f9a3ecbe5ed12ab4d8e11ba873c2f11161202b945","nodes":[{"kind":"text","id":"L644","line":644,"end_line":647},{"kind":"section","id":"L647","line":647,"end_line":647},{"kind":"tabularx","id":"L648","line":648,"end_line":658},{"kind":"section","id":"L659","line":659,"end_line":659},{"kind":"text","id":"L660","line":660,"end_line":665},{"kind":"readerroadmap","id":"L665","line":665,"end_line":665}]},{"number":29,"url":"read/paper-vi/atlas/v29.html","start_line":667,"end_line":686,"math_expressions":23,"math_sequence_sha256":"3dfaef3037babffc38b98e9d9463ea13a9cbcfa5d0b0c2785e1a07cf952546ad","nodes":[{"kind":"text","id":"L668","line":668,"end_line":677},{"kind":"figure","id":"L677","line":677,"end_line":677},{"kind":"readerroadmap","id":"L678","line":678,"end_line":678},{"kind":"section","id":"L679","line":679,"end_line":679},{"kind":"text","id":"L680","line":680,"end_line":685},{"kind":"readerroadmap","id":"L685","line":685,"end_line":685}]},{"number":30,"url":"read/paper-vi/atlas/v30.html","start_line":687,"end_line":709,"math_expressions":15,"math_sequence_sha256":"f003fcd09ec600c6fe4dcc061df66d81089c1757a0b8b7f85209332854e9fd0a","nodes":[{"kind":"text","id":"L688","line":688,"end_line":694},{"kind":"figure","id":"L694","line":694,"end_line":694},{"kind":"readerroadmap","id":"L695","line":695,"end_line":695},{"kind":"section","id":"L696","line":696,"end_line":696},{"kind":"text","id":"L697","line":697,"end_line":708},{"kind":"readerroadmap","id":"L708","line":708,"end_line":708}]},{"number":31,"url":"read/paper-vi/atlas/v31.html","start_line":710,"end_line":731,"math_expressions":25,"math_sequence_sha256":"afd288e784c61ea7e0f9d6fed109121588f5c5eef87989f47a8d2fe369807fab","nodes":[{"kind":"text","id":"L711","line":711,"end_line":720},{"kind":"figure","id":"L720","line":720,"end_line":720},{"kind":"readerroadmap","id":"L721","line":721,"end_line":721},{"kind":"section","id":"L722","line":722,"end_line":722},{"kind":"text","id":"L723","line":723,"end_line":730},{"kind":"readerroadmap","id":"L730","line":730,"end_line":730}]},{"number":32,"url":"read/paper-vi/atlas/v32.html","start_line":732,"end_line":750,"math_expressions":25,"math_sequence_sha256":"bd76ec21f87c7d92358656bdba948ad803840d92f8d0ce54544ee74d44d3aa62","nodes":[{"kind":"text","id":"L733","line":733,"end_line":741},{"kind":"figure","id":"L741","line":741,"end_line":741},{"kind":"readerroadmap","id":"L742","line":742,"end_line":742},{"kind":"section","id":"L743","line":743,"end_line":743},{"kind":"text","id":"L744","line":744,"end_line":749},{"kind":"readerroadmap","id":"L749","line":749,"end_line":749}]},{"number":33,"url":"read/paper-vi/atlas/v33.html","start_line":751,"end_line":772,"math_expressions":28,"math_sequence_sha256":"c7b6dab8fc734bfe916f92df49fd5529fda7cfdd557b338583c5b20f455b8d92","nodes":[{"kind":"text","id":"L752","line":752,"end_line":758},{"kind":"figure","id":"L758","line":758,"end_line":758},{"kind":"readerroadmap","id":"L759","line":759,"end_line":759},{"kind":"section","id":"L760","line":760,"end_line":760},{"kind":"text","id":"L761","line":761,"end_line":771},{"kind":"readerroadmap","id":"L771","line":771,"end_line":771}]},{"number":34,"url":"read/paper-vi/atlas/v34.html","start_line":773,"end_line":797,"math_expressions":17,"math_sequence_sha256":"5aa4c3edb9e11466c23e3df82d70db965246a1360d4e7ac5d4c0cb6c3d341783","nodes":[{"kind":"text","id":"L774","line":774,"end_line":784},{"kind":"figure","id":"L784","line":784,"end_line":784},{"kind":"readerroadmap","id":"L785","line":785,"end_line":785},{"kind":"section","id":"L786","line":786,"end_line":786},{"kind":"text","id":"L787","line":787,"end_line":796},{"kind":"readerroadmap","id":"L796","line":796,"end_line":796}]},{"number":35,"url":"read/paper-vi/atlas/v35.html","start_line":798,"end_line":820,"math_expressions":15,"math_sequence_sha256":"a461018a05a70ca1ab048546a90bb786bc5e34207974ca39daa96e54b8a2d4c6","nodes":[{"kind":"text","id":"L799","line":799,"end_line":804},{"kind":"figure","id":"L804","line":804,"end_line":804},{"kind":"readerroadmap","id":"L805","line":805,"end_line":805},{"kind":"section","id":"L806","line":806,"end_line":806},{"kind":"text","id":"L807","line":807,"end_line":819},{"kind":"readerroadmap","id":"L819","line":819,"end_line":819}]},{"number":36,"url":"read/paper-vi/atlas/v36.html","start_line":821,"end_line":847,"math_expressions":23,"math_sequence_sha256":"31dd009ddc29d1e79f0303866f9d4bb66ce0f9565e069f9547c0db766bd9d909","nodes":[{"kind":"text","id":"L822","line":822,"end_line":828},{"kind":"figure","id":"L828","line":828,"end_line":828},{"kind":"readerroadmap","id":"L829","line":829,"end_line":829},{"kind":"section","id":"L830","line":830,"end_line":830},{"kind":"text","id":"L831","line":831,"end_line":846},{"kind":"readerroadmap","id":"L846","line":846,"end_line":846}]},{"number":37,"url":"read/paper-vi/atlas/v37.html","start_line":848,"end_line":867,"math_expressions":12,"math_sequence_sha256":"b945e5321d8e1cfd6ea2c48c26e69e1ba75600de0311f6879dac19b911605937","nodes":[{"kind":"text","id":"L849","line":849,"end_line":858},{"kind":"figure","id":"L858","line":858,"end_line":858},{"kind":"readerroadmap","id":"L859","line":859,"end_line":859},{"kind":"section","id":"L860","line":860,"end_line":860},{"kind":"text","id":"L861","line":861,"end_line":866},{"kind":"readerroadmap","id":"L866","line":866,"end_line":866}]},{"number":38,"url":"read/paper-vi/atlas/v38.html","start_line":868,"end_line":892,"math_expressions":24,"math_sequence_sha256":"015d1d0502f2e945d4401f893f03d0ca48fb45e56c738f5a79038b386acc2fa9","nodes":[{"kind":"text","id":"L869","line":869,"end_line":875},{"kind":"figure","id":"L875","line":875,"end_line":875},{"kind":"readerroadmap","id":"L876","line":876,"end_line":876},{"kind":"section","id":"L877","line":877,"end_line":877},{"kind":"text","id":"L878","line":878,"end_line":891},{"kind":"readerroadmap","id":"L891","line":891,"end_line":891}]},{"number":39,"url":"read/paper-vi/atlas/v39.html","start_line":893,"end_line":919,"math_expressions":22,"math_sequence_sha256":"92c714be8e8a5a89a4e0adfb8ef2a5d196bff3daa0caf3ff081d16d2c39067c6","nodes":[{"kind":"text","id":"L894","line":894,"end_line":903},{"kind":"figure","id":"L903","line":903,"end_line":903},{"kind":"readerroadmap","id":"L904","line":904,"end_line":904},{"kind":"section","id":"L905","line":905,"end_line":905},{"kind":"text","id":"L906","line":906,"end_line":918},{"kind":"readerroadmap","id":"L918","line":918,"end_line":918}]},{"number":40,"url":"read/paper-vi/atlas/v40.html","start_line":920,"end_line":940,"math_expressions":25,"math_sequence_sha256":"e6ce1973d78b3e5a980dd8008e3c59ebfed29348ee299c06243ded6e2654ec34","nodes":[{"kind":"text","id":"L921","line":921,"end_line":927},{"kind":"figure","id":"L927","line":927,"end_line":927},{"kind":"readerroadmap","id":"L928","line":928,"end_line":928},{"kind":"section","id":"L929","line":929,"end_line":929},{"kind":"text","id":"L930","line":930,"end_line":939},{"kind":"readerroadmap","id":"L939","line":939,"end_line":939}]},{"number":41,"url":"read/paper-vi/atlas/v41.html","start_line":941,"end_line":964,"math_expressions":20,"math_sequence_sha256":"7c84a0eeb1fd07b184dd93306b0feb7cb2e32d999e5649ae7b7d0c8482f35d8d","nodes":[{"kind":"text","id":"L942","line":942,"end_line":949},{"kind":"figure","id":"L949","line":949,"end_line":949},{"kind":"readerroadmap","id":"L950","line":950,"end_line":950},{"kind":"section","id":"L951","line":951,"end_line":951},{"kind":"text","id":"L952","line":952,"end_line":963},{"kind":"readerroadmap","id":"L963","line":963,"end_line":963}]},{"number":42,"url":"read/paper-vi/atlas/v42.html","start_line":965,"end_line":980,"math_expressions":0,"math_sequence_sha256":"4f53cda18c2baa0c0354bb5f9a3ecbe5ed12ab4d8e11ba873c2f11161202b945","nodes":[{"kind":"figure","id":"L966","line":966,"end_line":966},{"kind":"readerroadmap","id":"L967","line":967,"end_line":967},{"kind":"section","id":"L968","line":968,"end_line":968},{"kind":"text","id":"L969","line":969,"end_line":974},{"kind":"section","id":"L974","line":974,"end_line":974},{"kind":"text","id":"L975","line":975,"end_line":978},{"kind":"readerroadmap","id":"L978","line":978,"end_line":978}]}],"figures":{"three_views":{"name":"three_views","plate":1,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/three_views.tex","source_sha256":"1912b4ee138926e4a467705069719f8185b9c71bd4ec70757f80af8072685476","image":"assets/atlas/paper-vi/three_views.png","image_sha256":"481120c99437bbfc13b04efb343c730b782ec357e6e1b3e4235b1716c76a3e44","pdf_page":3,"crop_pdf_points":[60,196,552,365],"nodes":[{"text":"Keep the full state","source_text":"Keep the full state","position":[2.2,3.8],"line":4},{"text":"0","source_text":"0","position":[0.7,0.25],"line":6},{"text":"$\\frac{1}{2}$","source_text":"$\\frac{1}{2}$","position":[0.7,3.27],"line":7},{"text":"1","source_text":"1","position":[1.7,0.25],"line":9},{"text":"$0$","source_text":"$0$","position":[1.7,0.77],"line":10},{"text":"2","source_text":"2","position":[2.7,0.25],"line":12},{"text":"$\\frac{1}{4}$","source_text":"$\\frac{1}{4}$","position":[2.7,2.02],"line":13},{"text":"3","source_text":"3","position":[3.7,0.25],"line":15},{"text":"$\\frac{1}{4}$","source_text":"$\\frac{1}{4}$","position":[3.7,2.02],"line":16},{"text":"Retain only masses","source_text":"Retain only masses","position":[6.75,3.8],"line":18},{"text":"packet A","source_text":"packet A","position":[5.975,0.25],"line":20},{"text":"$\\frac{1}{2}$","source_text":"$\\frac{1}{2}$","position":[5.975,3.27],"line":21},{"text":"packet B","source_text":"packet B","position":[7.525,0.25],"line":23},{"text":"$\\frac{1}{2}$","source_text":"$\\frac{1}{2}$","position":[7.525,3.27],"line":24},{"text":"0","source_text":"0","position":[9.7,0.29],"line":29},{"text":"1","source_text":"1","position":[11.0,0.29],"line":32},{"text":"2","source_text":"2","position":[12.3,0.29],"line":35},{"text":"3","source_text":"3","position":[13.6,0.29],"line":38},{"text":"0","source_text":"0","position":[9.54,0.55],"line":40},{"text":"0.5","source_text":"0.5","position":[9.54,1.9],"line":42},{"text":"1","source_text":"1","position":[9.54,3.25],"line":44},{"text":"time","source_text":"time","position":[11.65,0.0],"line":47},{"text":"Eliminate a hidden state","source_text":"Eliminate a hidden state","position":[11.7,4.05],"line":50},{"text":"solid: exact; dashed: omit coupling","source_text":"solid: exact; dashed: omit coupling","position":[11.7,3.62],"line":51}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,22,23,21,14,15,16,17,18,19,20],"reading_aid":{"alt":"Full-state bars, coarse packet masses, and retained-state curves show three different reductions.","description":"Read left to right. The first two panels use the same distribution: the middle panel sums states into two packets and loses the within-packet shape. The separate right-hand model compares the exact retained evolution with the result of omitting its coupling. The solid and dashed curves start together and then separate. The curve panel is a different model from the bars.","blocks":[{"kind":"table","caption":0,"headers":["State","Probability"],"rows":[[1,2],[3,4],[5,6],[7,8]]},{"kind":"table","caption":9,"headers":["Packet","Mass"],"rows":[[10,11],[12,13]]},{"kind":"text","nodes":[22,23],"title":"Retained and hidden states"},{"kind":"text","nodes":[21,14,15,16,17],"title":"Horizontal axis and ticks"},{"kind":"text","nodes":[18,19,20],"title":"Vertical ticks"}]}},"clock":{"name":"clock","plate":3,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/clock.tex","source_sha256":"fd4f74aa1827e0c87e2177cdbd4be656d5f296d9d5fb6176751dadfae62be806","image":"assets/atlas/paper-vi/clock.png","image_sha256":"12a9ce7709862233a536cba3261fd09512f9aad27a22dafa720c686ba31bf2ca","pdf_page":5,"crop_pdf_points":[60,244,552,433],"nodes":[{"text":"$t=0$","source_text":"$t=0$","position":[2.25,4.15],"line":4},{"text":"state 0","source_text":"state 0","position":[1.375,0.5],"line":6},{"text":"$1$","source_text":"$1$","position":[1.375,3.62],"line":7},{"text":"state 1","source_text":"state 1","position":[3.125,0.5],"line":9},{"text":"$0$","source_text":"$0$","position":[3.125,1.02],"line":10},{"text":"$t=\\log(2)/2$","source_text":"$t=\\log(2)/2$","position":[6.75,4.15],"line":12},{"text":"state 0","source_text":"state 0","position":[5.875,0.5],"line":14},{"text":"$\\frac{3}{4}$","source_text":"$\\frac{3}{4}$","position":[5.875,2.97],"line":15},{"text":"state 1","source_text":"state 1","position":[7.625,0.5],"line":17},{"text":"$\\frac{1}{4}$","source_text":"$\\frac{1}{4}$","position":[7.625,1.67],"line":18},{"text":"$t=\\log(2)$","source_text":"$t=\\log(2)$","position":[11.25,4.15],"line":20},{"text":"state 0","source_text":"state 0","position":[10.375,0.5],"line":22},{"text":"$\\frac{5}{8}$","source_text":"$\\frac{5}{8}$","position":[10.375,2.645],"line":23},{"text":"state 1","source_text":"state 1","position":[12.125,0.5],"line":25},{"text":"$\\frac{3}{8}$","source_text":"$\\frac{3}{8}$","position":[12.125,1.995],"line":26},{"text":"Each bar pair is an exact probability vector","source_text":"Each bar pair is an exact probability vector","position":[7.0,4.6],"line":27}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"reading_aid":{"alt":"Three pairs of bars give the exact state probabilities at three elapsed times.","description":"Each panel uses the same two states and probability scale. Mass moves from state zero toward the uniform distribution. The time labels and every displayed bar value are transcribed in the three tables below.","blocks":[{"kind":"table","caption":0,"headers":["State","Probability"],"rows":[[1,2],[3,4]]},{"kind":"table","caption":5,"headers":["State","Probability"],"rows":[[6,7],[8,9]]},{"kind":"table","caption":10,"headers":["State","Probability"],"rows":[[11,12],[13,14]]},{"kind":"text","nodes":[15],"title":null}]}},"generator_modes":{"name":"generator_modes","plate":4,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/generator_modes.tex","source_sha256":"cde96a48afb1e5e6bdc8af307c983eefd835df4b7a80b6fdc65363f95b201ebc","image":"assets/atlas/paper-vi/generator_modes.png","image_sha256":"81116f0811672b89f1ba9ac6f3dd64d1def0efba926b4a81ef4ae5c3afb1f1ca","pdf_page":6,"crop_pdf_points":[60,247,552,426],"nodes":[{"text":"1","source_text":"1","position":[2.65,3.73],"line":6},{"text":"1","source_text":"1","position":[2.65,1.33],"line":8},{"text":"2","source_text":"2","position":[1.55,2.3],"line":10},{"text":"2","source_text":"2","position":[4.25,2.3],"line":12},{"text":"0","source_text":"0","position":[1.3,3.5],"line":14},{"text":"1","source_text":"1","position":[4.0,3.5],"line":16},{"text":"2","source_text":"2","position":[1.3,1.1],"line":18},{"text":"3","source_text":"3","position":[4.0,1.1],"line":20},{"text":"Off-diagonal rates of $Q$","source_text":"Off-diagonal rates of $Q$","position":[2.65,4.55],"line":21},{"text":"Top pair A; bottom pair B","source_text":"Top pair A; bottom pair B","position":[2.65,0.05],"line":22},{"text":"Mode signs (each divided by 2)","source_text":"Mode signs (each divided by 2)","position":[8.78,4.65],"line":23},{"text":"$1$","source_text":"$1$","position":[7.595,3.555],"line":25},{"text":"$1$","source_text":"$1$","position":[8.385,3.555],"line":27},{"text":"$1$","source_text":"$1$","position":[9.175,3.555],"line":29},{"text":"$1$","source_text":"$1$","position":[9.965,3.555],"line":31},{"text":"$1$","source_text":"$1$","position":[7.595,2.765],"line":33},{"text":"$-1$","source_text":"$-1$","position":[8.385,2.765],"line":35},{"text":"$1$","source_text":"$1$","position":[9.175,2.765],"line":37},{"text":"$-1$","source_text":"$-1$","position":[9.965,2.765],"line":39},{"text":"$1$","source_text":"$1$","position":[7.595,1.975],"line":41},{"text":"$1$","source_text":"$1$","position":[8.385,1.975],"line":43},{"text":"$-1$","source_text":"$-1$","position":[9.175,1.975],"line":45},{"text":"$-1$","source_text":"$-1$","position":[9.965,1.975],"line":47},{"text":"$1$","source_text":"$1$","position":[7.595,1.185],"line":49},{"text":"$-1$","source_text":"$-1$","position":[8.385,1.185],"line":51},{"text":"$-1$","source_text":"$-1$","position":[9.175,1.185],"line":53},{"text":"$1$","source_text":"$1$","position":[9.965,1.185],"line":55},{"text":"constant","source_text":"constant","position":[7.05,3.555],"line":56},{"text":"within","source_text":"within","position":[7.05,2.765],"line":57},{"text":"between","source_text":"between","position":[7.05,1.975],"line":58},{"text":"mixed","source_text":"mixed","position":[7.05,1.185],"line":59},{"text":"0","source_text":"0","position":[7.595,4.18],"line":60},{"text":"1","source_text":"1","position":[8.385,4.18],"line":61},{"text":"2","source_text":"2","position":[9.175,4.18],"line":62},{"text":"3","source_text":"3","position":[9.965,4.18],"line":63},{"text":"$0$","source_text":"$0$","position":[11.7,3.555],"line":64},{"text":"$-2$","source_text":"$-2$","position":[11.7,2.765],"line":65},{"text":"$-4$","source_text":"$-4$","position":[11.7,1.975],"line":66},{"text":"$-6$","source_text":"$-6$","position":[11.7,1.185],"line":67},{"text":"rate","source_text":"rate","position":[11.7,4.18],"line":68}],"node_order":[8,4,5,6,7,0,1,2,3,9,10,31,32,33,34,39,27,11,12,13,14,35,28,15,16,17,18,36,29,19,20,21,22,37,30,23,24,25,26,38],"reading_aid":{"alt":"A four-state rate graph is paired with four mode-sign rows and their decay rates.","description":"The left panel places states zero and one in the upper packet and states two and three in the lower packet. Horizontal and vertical bidirectional links have the rates listed below. The right panel gives each mode in state order, followed by its generator eigenvalue. Divide every sign entry by two, as the original caption specifies.","blocks":[{"kind":"table","caption":8,"headers":["Position","State"],"rows":[["Top left",4],["Top right",5],["Bottom left",6],["Bottom right",7]]},{"kind":"table","caption":"Bidirectional graph links","headers":["States joined","Rate"],"rows":[["0 and 1",0],["2 and 3",1],["0 and 2",2],["1 and 3",3]]},{"kind":"text","nodes":[9],"title":null},{"kind":"table","caption":10,"headers":["Mode",31,32,33,34,39],"rows":[[27,11,12,13,14,35],[28,15,16,17,18,36],[29,19,20,21,22,37],[30,23,24,25,26,38]]}]}},"lumpability":{"name":"lumpability","plate":5,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/lumpability.tex","source_sha256":"0168bb4d118842c4e96fa3c8f69a74c46048eff4e089ec98605b3130ef433505","image":"assets/atlas/paper-vi/lumpability.png","image_sha256":"8f4948b94e27eff5ef33d78957650dd602aec406396dea8eb27c6321f9ade833","pdf_page":7,"crop_pdf_points":[60,220,552,400],"nodes":[{"text":"$8K$","source_text":"$8K$","position":[2.17,4.45],"line":3},{"text":"$5$","source_text":"$5$","position":[1.03,3.37],"line":5},{"text":"$1$","source_text":"$1$","position":[1.79,3.37],"line":7},{"text":"$2$","source_text":"$2$","position":[2.55,3.37],"line":9},{"text":"$0$","source_text":"$0$","position":[3.31,3.37],"line":11},{"text":"$1$","source_text":"$1$","position":[1.03,2.61],"line":13},{"text":"$5$","source_text":"$5$","position":[1.79,2.61],"line":15},{"text":"$0$","source_text":"$0$","position":[2.55,2.61],"line":17},{"text":"$2$","source_text":"$2$","position":[3.31,2.61],"line":19},{"text":"$2$","source_text":"$2$","position":[1.03,1.85],"line":21},{"text":"$0$","source_text":"$0$","position":[1.79,1.85],"line":23},{"text":"$5$","source_text":"$5$","position":[2.55,1.85],"line":25},{"text":"$1$","source_text":"$1$","position":[3.31,1.85],"line":27},{"text":"$0$","source_text":"$0$","position":[1.03,1.09],"line":29},{"text":"$2$","source_text":"$2$","position":[1.79,1.09],"line":31},{"text":"$1$","source_text":"$1$","position":[2.55,1.09],"line":33},{"text":"$5$","source_text":"$5$","position":[3.31,1.09],"line":35},{"text":"0","source_text":"0","position":[0.5,3.37],"line":36},{"text":"1","source_text":"1","position":[0.5,2.61],"line":37},{"text":"2","source_text":"2","position":[0.5,1.85],"line":38},{"text":"3","source_text":"3","position":[0.5,1.09],"line":39},{"text":"0","source_text":"0","position":[1.03,3.98],"line":40},{"text":"1","source_text":"1","position":[1.79,3.98],"line":41},{"text":"2","source_text":"2","position":[2.55,3.98],"line":42},{"text":"3","source_text":"3","position":[3.31,3.98],"line":43},{"text":"$CK$: aggregate transition columns","source_text":"$CK$: aggregate transition columns","position":[7.56,4.35],"line":44},{"text":"$\\frac{3}{4}$","source_text":"$\\frac{3}{4}$","position":[5.94,3.11],"line":46},{"text":"$\\frac{3}{4}$","source_text":"$\\frac{3}{4}$","position":[7.02,3.11],"line":48},{"text":"$\\frac{1}{4}$","source_text":"$\\frac{1}{4}$","position":[8.1,3.11],"line":50},{"text":"$\\frac{1}{4}$","source_text":"$\\frac{1}{4}$","position":[9.18,3.11],"line":52},{"text":"$\\frac{1}{4}$","source_text":"$\\frac{1}{4}$","position":[5.94,2.03],"line":54},{"text":"$\\frac{1}{4}$","source_text":"$\\frac{1}{4}$","position":[7.02,2.03],"line":56},{"text":"$\\frac{3}{4}$","source_text":"$\\frac{3}{4}$","position":[8.1,2.03],"line":58},{"text":"$\\frac{3}{4}$","source_text":"$\\frac{3}{4}$","position":[9.18,2.03],"line":60},{"text":"A","source_text":"A","position":[5.25,3.11],"line":61},{"text":"B","source_text":"B","position":[5.25,2.03],"line":62},{"text":"0","source_text":"0","position":[5.94,3.88],"line":63},{"text":"1","source_text":"1","position":[7.02,3.88],"line":64},{"text":"2","source_text":"2","position":[8.1,3.88],"line":65},{"text":"3","source_text":"3","position":[9.18,3.88],"line":66},{"text":"Columns 0 and 1 agree; columns 2 and 3 agree.","source_text":"Columns 0 and 1 agree; columns 2 and 3 agree.","position":[7.55,0.6],"line":67},{"text":"$\\bar K=\\begin{pmatrix}3/4&1/4\\\\1/4&3/4\\end{pmatrix}$","source_text":"$\\bar K=\\begin{pmatrix}3/4&1/4\\\\1/4&3/4\\end{pmatrix}$","position":[12.0,3.55],"line":68},{"text":"Fine trajectories differ.\\newline Coarse predictions agree.","source_text":"Fine trajectories differ.\\\\Coarse predictions agree.","position":[12.0,1.6],"line":69}],"node_order":[0,21,22,23,24,17,1,2,3,4,18,5,6,7,8,19,9,10,11,12,20,13,14,15,16,25,36,37,38,39,34,26,27,28,29,35,30,31,32,33,40,41,42],"reading_aid":{"alt":"Fine transition columns differ, but aggregation makes columns agree within each coarse packet.","description":"The left table is eight times the fine transition matrix, with target states as rows and source states as columns. The middle table sums target rows into packets A and B. Its first two columns agree, as do its last two. The right-hand coarse matrix and the original comparison note follow the tables.","blocks":[{"kind":"table","caption":0,"headers":["Target / source",21,22,23,24],"rows":[[17,1,2,3,4],[18,5,6,7,8],[19,9,10,11,12],[20,13,14,15,16]]},{"kind":"table","caption":25,"headers":["Coarse packet / fine source",36,37,38,39],"rows":[[34,26,27,28,29],[35,30,31,32,33]]},{"kind":"text","nodes":[40,41,42],"title":null}]}},"defect_budget":{"name":"defect_budget","plate":6,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/defect_budget.tex","source_sha256":"56d45676764e423158ba8a9133a98830f309f243a563083a215a479867eac411","image":"assets/atlas/paper-vi/defect_budget.png","image_sha256":"585bb4dad4f416f2a040108dfdc0cc24e223c8c8dd09c5ff495b48bc0b5c4f5b","pdf_page":8,"crop_pdf_points":[60,218,552,402],"nodes":[{"text":"0","source_text":"0","position":[0.9,0.34],"line":5},{"text":"2","source_text":"2","position":[2.9,0.34],"line":8},{"text":"4","source_text":"4","position":[4.9,0.34],"line":11},{"text":"6","source_text":"6","position":[6.9,0.34],"line":14},{"text":"8","source_text":"8","position":[8.9,0.34],"line":17},{"text":"0","source_text":"0","position":[0.74,0.6],"line":19},{"text":"0.25","source_text":"0.25","position":[0.74,1.425],"line":21},{"text":"0.5","source_text":"0.5","position":[0.74,2.25],"line":23},{"text":"0.75","source_text":"0.75","position":[0.74,3.075],"line":25},{"text":"1","source_text":"1","position":[0.74,3.9],"line":27},{"text":"number of steps $n$","source_text":"number of steps $n$","position":[4.9,0.05],"line":30},{"text":"induced $\\ell^1$ operator norm","source_text":"induced $\\ell^1$ operator norm","position":[0.9,4.3],"line":31},{"text":"$\\|\\Delta\\|_1=1/8$","source_text":"$\\|\\Delta\\|_1=1/8$","position":[11.3,3.75],"line":43},{"text":"solid: exact defect\\newline dashed: $n/8$","source_text":"solid: exact defect\\\\dashed: $n/8$","position":[11.3,2.8],"line":44},{"text":"The bound is a guarantee,\\newline not a fitted trend.","source_text":"The bound is a guarantee,\\\\not a fitted trend.","position":[11.3,1.55],"line":45}],"node_order":[10,0,1,2,3,4,11,5,6,7,8,9,12,13,14],"reading_aid":{"alt":"A solid finite-horizon defect curve lies below a dashed linear certificate.","description":"The horizontal axis counts steps and the vertical axis uses the induced one-norm. The solid values come from the exact finite model. The dashed line is the bound based on the one-step defect; it is a guarantee and need not be sharp. The axis ticks and all original legend labels follow.","blocks":[{"kind":"text","nodes":[10,0,1,2,3,4],"title":"Horizontal axis and ticks"},{"kind":"text","nodes":[11,5,6,7,8,9],"title":"Vertical axis and ticks"},{"kind":"text","nodes":[12,13,14],"title":"One-step defect and legend"}]}},"weighted_projection":{"name":"weighted_projection","plate":7,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/weighted_projection.tex","source_sha256":"2fcaef6efb4d9669107379b1e06ed8bbfd7ee278050ded35cde26e83707dbc72","image":"assets/atlas/paper-vi/weighted_projection.png","image_sha256":"9e7aad8d01659ee3d098d532115c2c65f1e2c3591dff0f26e56cb4ce5171092c","pdf_page":9,"crop_pdf_points":[90,261,509,411],"nodes":[{"text":"Observable $f$","source_text":"Observable $f$","position":[3.25,3.85],"line":4},{"text":"0","source_text":"0","position":[1.1125,0.35],"line":6},{"text":"$0$","source_text":"$0$","position":[1.1125,0.87],"line":7},{"text":"1","source_text":"1","position":[2.5375,0.35],"line":9},{"text":"$3$","source_text":"$3$","position":[2.5375,2.34],"line":10},{"text":"2","source_text":"2","position":[3.9625,0.35],"line":12},{"text":"$2$","source_text":"$2$","position":[3.9625,1.85],"line":13},{"text":"3","source_text":"3","position":[5.3875,0.35],"line":15},{"text":"$5$","source_text":"$5$","position":[5.3875,3.32],"line":16},{"text":"Conditional expectation $Ef$","source_text":"Conditional expectation $Ef$","position":[10.45,3.85],"line":18},{"text":"0","source_text":"0","position":[8.3125,0.35],"line":20},{"text":"$2$","source_text":"$2$","position":[8.3125,1.85],"line":21},{"text":"1","source_text":"1","position":[9.7375,0.35],"line":23},{"text":"$2$","source_text":"$2$","position":[9.7375,1.85],"line":24},{"text":"2","source_text":"2","position":[11.1625,0.35],"line":26},{"text":"$\\frac{26}{7}$","source_text":"$\\frac{26}{7}$","position":[11.1625,2.69],"line":27},{"text":"3","source_text":"3","position":[12.5875,0.35],"line":29},{"text":"$\\frac{26}{7}$","source_text":"$\\frac{26}{7}$","position":[12.5875,2.69],"line":30},{"text":"Weights $\\pi=(1,2,3,4)/10$; the block values are 2 and $26/7$.","source_text":"Weights $\\pi=(1,2,3,4)/10$; the block values are 2 and $26/7$.","position":[7.0,4.45],"line":31}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],"reading_aid":{"alt":"An observable and its weighted conditional expectation are shown as bars.","description":"The left panel gives the four observable values. The right panel replaces values inside each block by its weighted average. The weights and the two resulting block values are written above the panels.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]}]}},"variance":{"name":"variance","plate":8,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/variance.tex","source_sha256":"ad8e2050542cf0bfb9f47b0dc594d0585c4edd2a17f2bef0e22bb683f16fdefe","image":"assets/atlas/paper-vi/variance.png","image_sha256":"f8778d0976d6e7506143a6aa7890e4bd0b54a374d46cf973b1eef945009d6fcf","pdf_page":10,"crop_pdf_points":[100,220,509,387],"nodes":[{"text":"mean\\newline $\\frac{256}{25}$","source_text":"mean\\\\$\\frac{256}{25}$","position":[3.0,3.73],"line":8},{"text":"between blocks\\newline $\\frac{108}{175}$","source_text":"between blocks\\\\$\\frac{108}{175}$","position":[9.2,3.73],"line":11},{"text":"within blocks\\newline $\\frac{15}{7}$","source_text":"within blocks\\\\$\\frac{15}{7}$","position":[12.25,3.73],"line":14},{"text":"Weighted squared norms: total $13$","source_text":"Weighted squared norms: total $13$","position":[7.0,4.55],"line":15},{"text":"$D_0f=\\left[\\begin{matrix}\\frac{16}{5}\\\\\\frac{16}{5}\\\\\\frac{16}{5}\\\\\\frac{16}{5}\\end{matrix}\\right]$","source_text":"$D_0f=\\left[\\begin{matrix}\\frac{16}{5}\\\\\\frac{16}{5}\\\\\\frac{16}{5}\\\\\\frac{16}{5}\\end{matrix}\\right]$","position":[2.3,0.7],"line":16},{"text":"$D_1f=\\left[\\begin{matrix}- \\frac{6}{5}\\\\- \\frac{6}{5}\\\\\\frac{18}{35}\\\\\\frac{18}{35}\\end{matrix}\\right]$","source_text":"$D_1f=\\left[\\begin{matrix}- \\frac{6}{5}\\\\- \\frac{6}{5}\\\\\\frac{18}{35}\\\\\\frac{18}{35}\\end{matrix}\\right]$","position":[6.9,0.7],"line":17},{"text":"$D_2f=\\left[\\begin{matrix}-2\\\\1\\\\- \\frac{12}{7}\\\\\\frac{9}{7}\\end{matrix}\\right]$","source_text":"$D_2f=\\left[\\begin{matrix}-2\\\\1\\\\- \\frac{12}{7}\\\\\\frac{9}{7}\\end{matrix}\\right]$","position":[11.5,0.7],"line":18}],"node_order":[0,1,2,3,4,5,6],"reading_aid":{"alt":"A weighted squared norm is split into mean, between-block, and within-block parts.","description":"Three adjacent segments represent the displayed orthogonal contributions. Their values appear above the bar, and the three component vectors appear below it. The source plate identifies the weighted inner product used for this decomposition.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6]}]}},"evidence":{"name":"evidence","plate":9,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/evidence.tex","source_sha256":"4f28d066163fff1688e6488d639107b97da71dccebf22d1fc85f35e741c93adb","image":"assets/atlas/paper-vi/evidence.png","image_sha256":"e8407a1c98c4a8d5dc67caba8f2b18722f4c9d21778abc949e9b4bb5c3b117ed","pdf_page":11,"crop_pdf_points":[97,235,503,385],"nodes":[{"text":"Prior $p$","source_text":"Prior $p$","position":[3.3,3.85],"line":4},{"text":"0","source_text":"0","position":[1.275,0.35],"line":6},{"text":"$\\frac{1}{4}$","source_text":"$\\frac{1}{4}$","position":[1.275,1.941875],"line":7},{"text":"1","source_text":"1","position":[2.625,0.35],"line":9},{"text":"$\\frac{1}{4}$","source_text":"$\\frac{1}{4}$","position":[2.625,1.941875],"line":10},{"text":"2","source_text":"2","position":[3.975,0.35],"line":12},{"text":"$\\frac{1}{4}$","source_text":"$\\frac{1}{4}$","position":[3.975,1.941875],"line":13},{"text":"3","source_text":"3","position":[5.325,0.35],"line":15},{"text":"$\\frac{1}{4}$","source_text":"$\\frac{1}{4}$","position":[5.325,1.941875],"line":16},{"text":"Posterior $\\mathcal B_\\lambda(p)$","source_text":"Posterior $\\mathcal B_\\lambda(p)$","position":[10.4,3.85],"line":18},{"text":"0","source_text":"0","position":[8.375,0.35],"line":20},{"text":"$\\frac{4}{7}$","source_text":"$\\frac{4}{7}$","position":[8.375,3.32],"line":21},{"text":"1","source_text":"1","position":[9.725,0.35],"line":23},{"text":"$\\frac{1}{7}$","source_text":"$\\frac{1}{7}$","position":[9.725,1.4825],"line":24},{"text":"2","source_text":"2","position":[11.075,0.35],"line":26},{"text":"$\\frac{1}{7}$","source_text":"$\\frac{1}{7}$","position":[11.075,1.4825],"line":27},{"text":"3","source_text":"3","position":[12.425,0.35],"line":29},{"text":"$\\frac{1}{7}$","source_text":"$\\frac{1}{7}$","position":[12.425,1.4825],"line":30},{"text":"$\\lambda=(4,1,1,1)$; evidence $Z=7/4$","source_text":"$\\lambda=(4,1,1,1)$; evidence $Z=7/4$","position":[7.0,4.45],"line":31}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],"reading_aid":{"alt":"A uniform prior is reweighted into a posterior.","description":"The left panel gives equal prior masses at four addresses. The right panel shows the posterior after the displayed likelihood is applied and normalized. The likelihood vector and total evidence are annotated above.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]}]}},"insufficiency":{"name":"insufficiency","plate":10,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/insufficiency.tex","source_sha256":"12199679d17805cdf52a2a089c06b7f0422140d236dcf4b9e9f5c09fc336edd5","image":"assets/atlas/paper-vi/insufficiency.png","image_sha256":"6c6ba011b91f6e1a434ab6344627c1b5a7920de6617b8e08a6eaa86e3839d436","pdf_page":12,"crop_pdf_points":[93,198,509,350],"nodes":[{"text":"Both coarse priors","source_text":"Both coarse priors","position":[2.25,3.85],"line":4},{"text":"A","source_text":"A","position":[1.375,0.35],"line":6},{"text":"$\\frac{1}{2}$","source_text":"$\\frac{1}{2}$","position":[1.375,2.095],"line":7},{"text":"B","source_text":"B","position":[3.125,0.35],"line":9},{"text":"$\\frac{1}{2}$","source_text":"$\\frac{1}{2}$","position":[3.125,2.095],"line":10},{"text":"From $p^{(a)}$","source_text":"From $p^{(a)}$","position":[6.9,3.85],"line":12},{"text":"A","source_text":"A","position":[6.025,0.35],"line":14},{"text":"$\\frac{4}{5}$","source_text":"$\\frac{4}{5}$","position":[6.025,2.83],"line":15},{"text":"B","source_text":"B","position":[7.775,0.35],"line":17},{"text":"$\\frac{1}{5}$","source_text":"$\\frac{1}{5}$","position":[7.775,1.36],"line":18},{"text":"From $p^{(b)}$","source_text":"From $p^{(b)}$","position":[11.55,3.85],"line":20},{"text":"A","source_text":"A","position":[10.675,0.35],"line":22},{"text":"$\\frac{1}{2}$","source_text":"$\\frac{1}{2}$","position":[10.675,2.095],"line":23},{"text":"B","source_text":"B","position":[12.425,0.35],"line":25},{"text":"$\\frac{1}{2}$","source_text":"$\\frac{1}{2}$","position":[12.425,2.095],"line":26},{"text":"Hidden redistribution changes the packet-A posterior from $1/2$ to $4/5$.","source_text":"Hidden redistribution changes the packet-A posterior from $1/2$ to $4/5$.","position":[7.0,4.5],"line":27}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"reading_aid":{"alt":"Two priors with the same coarse masses produce different coarse posteriors.","description":"The left panel gives their common two-packet prior. The middle and right panels give the two updated packet distributions. The note above attributes the difference to redistribution hidden inside the first packet.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]}]}},"loss_vector":{"name":"loss_vector","plate":11,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/loss_vector.tex","source_sha256":"805157c63250cf5fb2d0b653cb9495529d8ce7fec64fa76b07e0367c0d0fddca","image":"assets/atlas/paper-vi/loss_vector.png","image_sha256":"d0cea7904c8c1b1bae6317831713eec2dc7c49fdb44f323be9a77ae614495372","pdf_page":13,"crop_pdf_points":[110,227,503,360],"nodes":[{"text":"$(0,1)$","source_text":"$(0,1)$","position":[1.4,1.85],"line":4},{"text":"$(1,0)$","source_text":"$(1,0)$","position":[12.7,1.85],"line":5},{"text":"declared coarse update\\newline $(1/2,1/2)$","source_text":"declared coarse update\\\\$(1/2,1/2)$","position":[7.05,3.55],"line":9},{"text":"fine then coarse\\newline $(5/7,2/7)$","source_text":"fine then coarse\\\\$(5/7,2/7)$","position":[9.871429,1.0],"line":10},{"text":"$3/14$","source_text":"$3/14$","position":[8.260714,3.1],"line":11},{"text":"Simplex coordinate: posterior mass in packet A","source_text":"Simplex coordinate: posterior mass in packet A","position":[7.0,4.4],"line":12}],"node_order":[0,1,2,3,4,5],"reading_aid":{"alt":"Two update procedures land at different points on a two-packet simplex.","description":"The line represents posterior mass in the first packet. An open point marks the declared coarse update and a filled point marks updating first in the fine model. The arrow gives their displayed difference.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5]}]}},"odds":{"name":"odds","plate":12,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/odds.tex","source_sha256":"62ebefd60dac5d68631561b55dba1ab6f8ac3cd0663d05c37321d4e78e5ac1ae","image":"assets/atlas/paper-vi/odds.png","image_sha256":"67843fa34ed64dbc13605f93b3a3d3522f928b8e97be0787fddf208f1c6485db","pdf_page":14,"crop_pdf_points":[83,223,510,377],"nodes":[{"text":"0","source_text":"0","position":[0.8,0.34],"line":5},{"text":"1","source_text":"1","position":[2.975,0.34],"line":8},{"text":"2","source_text":"2","position":[5.15,0.34],"line":11},{"text":"3","source_text":"3","position":[7.325,0.34],"line":14},{"text":"4","source_text":"4","position":[9.5,0.34],"line":17},{"text":"0","source_text":"0","position":[0.64,0.6],"line":19},{"text":"0.5","source_text":"0.5","position":[0.64,2.25],"line":21},{"text":"1","source_text":"1","position":[0.64,3.9],"line":23},{"text":"time $t$","source_text":"time $t$","position":[5.15,0.05],"line":26},{"text":"probability","source_text":"probability","position":[0.8,4.3],"line":27},{"text":"$p_0(t)=\\dfrac{e^t}{1+e^t}$","source_text":"$p_0(t)=\\dfrac{e^t}{1+e^t}$","position":[11.6,3.7],"line":30},{"text":"$p_1(t)=\\dfrac{1}{1+e^t}$","source_text":"$p_1(t)=\\dfrac{1}{1+e^t}$","position":[11.6,2.7],"line":31},{"text":"$\\dot p_0=p_0(1-p_0)$","source_text":"$\\dot p_0=p_0(1-p_0)$","position":[11.6,1.5],"line":32},{"text":"Mass stays one.\\newline Mixtures are not preserved.","source_text":"Mass stays one.\\\\Mixtures are not preserved.","position":[11.6,0.65],"line":33}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13],"reading_aid":{"alt":"Two posterior probabilities separate along a nonlinear Bayesian flow.","description":"The solid curve rises from one half while the dashed curve falls from one half. The expressions on the right give the probabilities and their differential equation. The note distinguishes conservation of total mass from preservation of mixtures.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10,11,12,13]}]}},"filter_order":{"name":"filter_order","plate":13,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/filter_order.tex","source_sha256":"c2c04e4904eada1a1bf45e8ec547bc31e76096192304b900d5edd46db995de50","image":"assets/atlas/paper-vi/filter_order.png","image_sha256":"1811727fc23bcb346760d53224609ae376ad54cf8ebf5338a493948d3a7907ac","pdf_page":15,"crop_pdf_points":[97,192,503,342],"nodes":[{"text":"Predict, then update","source_text":"Predict, then update","position":[3.3,3.9],"line":4},{"text":"0","source_text":"0","position":[1.95,0.4],"line":6},{"text":"$\\frac{6}{7}$","source_text":"$\\frac{6}{7}$","position":[1.95,3.02],"line":7},{"text":"1","source_text":"1","position":[4.65,0.4],"line":9},{"text":"$\\frac{1}{7}$","source_text":"$\\frac{1}{7}$","position":[4.65,1.27],"line":10},{"text":"Update, then predict","source_text":"Update, then predict","position":[10.4,3.9],"line":12},{"text":"0","source_text":"0","position":[9.05,0.4],"line":14},{"text":"$\\frac{3}{4}$","source_text":"$\\frac{3}{4}$","position":[9.05,2.7575],"line":15},{"text":"1","source_text":"1","position":[11.75,0.4],"line":17},{"text":"$\\frac{1}{4}$","source_text":"$\\frac{1}{4}$","position":[11.75,1.5325],"line":18},{"text":"Same prior $e_0$, transition $K_2$, and likelihood $(1,1/2)$","source_text":"Same prior $e_0$, transition $K_2$, and likelihood $(1,1/2)$","position":[7.0,4.5],"line":19}],"node_order":[0,1,2,3,4,5,6,7,8,9,10],"reading_aid":{"alt":"Two bar pairs compare prediction followed by update with the reverse order.","description":"Both panels start from the same prior, transition, and likelihood, as stated above them. The displayed probability vectors differ. The left panel uses prediction first; the right panel applies the observation first.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10]}]}},"beta":{"name":"beta","plate":14,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/beta.tex","source_sha256":"e271357f5db24ced616bb70f487555d457b5a2d27b7295d9521692870d2e771e","image":"assets/atlas/paper-vi/beta.png","image_sha256":"39a9d04a2a73ce32af13ecd30036b816c741d68fb7e93fab5c29bef4f646608d","pdf_page":16,"crop_pdf_points":[89,218,505,370],"nodes":[{"text":"0","source_text":"0","position":[0.8,0.39],"line":5},{"text":"0.25","source_text":"0.25","position":[3.0,0.39],"line":8},{"text":"0.5","source_text":"0.5","position":[5.2,0.39],"line":11},{"text":"0.75","source_text":"0.75","position":[7.4,0.39],"line":14},{"text":"1","source_text":"1","position":[9.6,0.39],"line":17},{"text":"0","source_text":"0","position":[0.64,0.65],"line":19},{"text":"1","source_text":"1","position":[0.64,2.004167],"line":21},{"text":"2","source_text":"2","position":[0.64,3.358333],"line":23},{"text":"$\\theta$","source_text":"$\\theta$","position":[5.2,0.1],"line":26},{"text":"density","source_text":"density","position":[0.8,4.3],"line":27},{"text":"Prior: $\\mathrm{Beta}(2,3)$","source_text":"Prior: $\\mathrm{Beta}(2,3)$","position":[11.65,3.75],"line":30},{"text":"Posterior: $\\mathrm{Beta}(4,4)$","source_text":"Posterior: $\\mathrm{Beta}(4,4)$","position":[11.65,2.95],"line":31},{"text":"ordered data: $3/35$\\newline count data: $9/35$","source_text":"ordered data: $3/35$\\\\count data: $9/35$","position":[11.65,1.8],"line":32},{"text":"Same posterior,\\newline different evidence.","source_text":"Same posterior,\\\\different evidence.","position":[11.65,0.6],"line":33}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13],"reading_aid":{"alt":"Prior and posterior beta densities are compared on the unit interval.","description":"The dashed curve is the prior and the solid curve is the posterior after the stated observation. The right-hand notes distinguish the evidence for ordered data from the evidence for count data, while retaining the same posterior.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10,11,12,13]}]}},"fractional":{"name":"fractional","plate":15,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/fractional.tex","source_sha256":"25fb8c3864cf5d80f937298a0c8ba7ac87b40fa0a7941b177c189317d628a934","image":"assets/atlas/paper-vi/fractional.png","image_sha256":"a5b8a2573956757e2ccd8eb3c472447b91ef452fdf2e48d8a8d998187c3fe8dc","pdf_page":17,"crop_pdf_points":[100,221,512,384],"nodes":[{"text":"0","source_text":"0","position":[1.0,3.6],"line":10},{"text":"1","source_text":"1","position":[4.0,3.6],"line":12},{"text":"2","source_text":"2","position":[1.0,1.0],"line":14},{"text":"3","source_text":"3","position":[4.0,1.0],"line":16},{"text":"Solid: $L$; dashed: new in $\\sqrt L$","source_text":"Solid: $L$; dashed: new in $\\sqrt L$","position":[2.5,4.5],"line":17},{"text":"0","source_text":"0","position":[6.6,0.39],"line":20},{"text":"0.5","source_text":"0.5","position":[7.92,0.39],"line":23},{"text":"1","source_text":"1","position":[9.24,0.39],"line":26},{"text":"1.5","source_text":"1.5","position":[10.56,0.39],"line":29},{"text":"2","source_text":"2","position":[11.88,0.39],"line":32},{"text":"2.5","source_text":"2.5","position":[13.2,0.39],"line":35},{"text":"0","source_text":"0","position":[6.44,0.65],"line":37},{"text":"0.5","source_text":"0.5","position":[6.44,2.05],"line":39},{"text":"1","source_text":"1","position":[6.44,3.45],"line":41},{"text":"time $t$","source_text":"time $t$","position":[9.9,0.1],"line":44},{"text":"fastest mode amplitude","source_text":"fastest mode amplitude","position":[6.6,3.85],"line":45},{"text":"solid: $e^{-t\\sqrt6}$; dashed: $e^{-6t}$","source_text":"solid: $e^{-t\\sqrt6}$; dashed: $e^{-6t}$","position":[10.0,4.45],"line":48}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"reading_aid":{"alt":"A changed generator graph is paired with changed mode decay.","description":"The left graph distinguishes the original couplings from the added fractional couplings. The right plot compares the two displayed decay factors for the fastest mode. The source retains the same spectral modes while changing their clocks.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]}]}},"gap":{"name":"gap","plate":16,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/gap.tex","source_sha256":"10837b33592f1bc3e7ce9f78fd04662976e74e0068c18e1cb04fd804b5c6c1f5","image":"assets/atlas/paper-vi/gap.png","image_sha256":"564cae24ede7cafe0612c2718dc719a5afe8b2365f8a69b965605891d93a3ee5","pdf_page":18,"crop_pdf_points":[78,236,511,388],"nodes":[{"text":"0","source_text":"0","position":[0.8,0.39],"line":5},{"text":"0.25","source_text":"0.25","position":[2.95,0.39],"line":8},{"text":"0.5","source_text":"0.5","position":[5.1,0.39],"line":11},{"text":"0.75","source_text":"0.75","position":[7.25,0.39],"line":14},{"text":"1","source_text":"1","position":[9.4,0.39],"line":17},{"text":"0","source_text":"0","position":[0.64,0.65],"line":19},{"text":"0.25","source_text":"0.25","position":[0.64,1.4625],"line":21},{"text":"0.5","source_text":"0.5","position":[0.64,2.275],"line":23},{"text":"0.75","source_text":"0.75","position":[0.64,3.0875],"line":25},{"text":"1","source_text":"1","position":[0.64,3.9],"line":27},{"text":"generator perturbation $\\varepsilon$","source_text":"generator perturbation $\\varepsilon$","position":[5.1,0.1],"line":30},{"text":"square-root difference","source_text":"square-root difference","position":[0.8,4.3],"line":31},{"text":"no gap: $\\sqrt{\\varepsilon}$","source_text":"no gap: $\\sqrt{\\varepsilon}$","position":[11.8,3.85],"line":35},{"text":"gap one: $\\sqrt{1+\\varepsilon}-1$","source_text":"gap one: $\\sqrt{1+\\varepsilon}-1$","position":[11.8,2.8],"line":36},{"text":"certificate: $\\varepsilon/2$","source_text":"certificate: $\\varepsilon/2$","position":[11.8,1.75],"line":37},{"text":"No uniform linear bound\\newline can hold at zero.","source_text":"No uniform linear bound\\\\can hold at zero.","position":[11.8,0.65],"line":38}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"reading_aid":{"alt":"Square-root perturbation curves are compared with and without a gap.","description":"The horizontal axis is generator perturbation. The upper curve is the ungapped difference; the lower curve is the gapped difference. A straight certificate is shown alongside the gapped case, and the note states the limitation at zero.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]}]}},"random_clock":{"name":"random_clock","plate":17,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/random_clock.tex","source_sha256":"c11fbbb95901ca541d68831be8c41388396bf7af29a5982c16aba95bb2b6242a","image":"assets/atlas/paper-vi/random_clock.png","image_sha256":"39c0486f12e488bc775da32ad8305c3233c633e592fe7ed650ea49f38c9dad19","pdf_page":19,"crop_pdf_points":[78,223,531,381],"nodes":[{"text":"0","source_text":"0","position":[0.8,0.39],"line":5},{"text":"0.5","source_text":"0.5","position":[2.975,0.39],"line":8},{"text":"1","source_text":"1","position":[5.15,0.39],"line":11},{"text":"1.5","source_text":"1.5","position":[7.325,0.39],"line":14},{"text":"2","source_text":"2","position":[9.5,0.39],"line":17},{"text":"0","source_text":"0","position":[0.64,0.65],"line":19},{"text":"0.25","source_text":"0.25","position":[0.64,1.316667],"line":21},{"text":"0.5","source_text":"0.5","position":[0.64,1.983333],"line":23},{"text":"0.75","source_text":"0.75","position":[0.64,2.65],"line":25},{"text":"1","source_text":"1","position":[0.64,3.316667],"line":27},{"text":"diffusion time $s$","source_text":"diffusion time $s$","position":[5.15,0.1],"line":30},{"text":"clock density $\\eta_1(s)$","source_text":"clock density $\\eta_1(s)$","position":[0.8,3.85],"line":31},{"text":"Clock mode at $s=1/6$","source_text":"Clock mode at $s=1/6$","position":[3.0,4.45],"line":34},{"text":"$\\displaystyle\\int_0^\\infty\\eta_t(s)\\,ds=1$","source_text":"$\\displaystyle\\int_0^\\infty\\eta_t(s)\\,ds=1$","position":[11.8,3.8],"line":35},{"text":"$\\displaystyle\\int_0^\\infty s\\eta_t(s)\\,ds=\\infty$","source_text":"$\\displaystyle\\int_0^\\infty s\\eta_t(s)\\,ds=\\infty$","position":[11.8,2.7],"line":36},{"text":"A probability clock\\newline need not have a finite mean.","source_text":"A probability clock\\\\need not have a finite mean.","position":[11.8,1.2],"line":37}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"reading_aid":{"alt":"A subordination-clock density is plotted against diffusion time.","description":"The curve rises to a marked mode and then has a long tail. The integrals at right distinguish its total mass from its first moment. The plate specifies the fixed clock parameter used for the drawing.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]}]}},"green":{"name":"green","plate":18,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/green.tex","source_sha256":"3098a3d930f5136e0da82c5582dae26c36357f535d7edecf29cdd261babcb618","image":"assets/atlas/paper-vi/green.png","image_sha256":"9aad76bb6cbf569ea2aa5faabf54e442dd6e8bc457cc3e2edcdc77c920c96892","pdf_page":20,"crop_pdf_points":[74,284,486,434],"nodes":[{"text":"0","source_text":"0","position":[0.8,0.39],"line":5},{"text":"0.25","source_text":"0.25","position":[2.625,0.39],"line":8},{"text":"0.5","source_text":"0.5","position":[4.45,0.39],"line":11},{"text":"0.75","source_text":"0.75","position":[6.275,0.39],"line":14},{"text":"1","source_text":"1","position":[8.1,0.39],"line":17},{"text":"0","source_text":"0","position":[0.64,0.65],"line":19},{"text":"0.125","source_text":"0.125","position":[0.64,1.983333],"line":21},{"text":"0.25","source_text":"0.25","position":[0.64,3.316667],"line":23},{"text":"$x$","source_text":"$x$","position":[4.45,0.1],"line":26},{"text":"$G(x,1/2)$","source_text":"$G(x,1/2)$","position":[0.8,4.25],"line":27},{"text":"$64 L_h^{-1}$, $h=1/4$","source_text":"$64 L_h^{-1}$, $h=1/4$","position":[11.29,4.05],"line":32},{"text":"$3$","source_text":"$3$","position":[10.43,2.92],"line":34},{"text":"$2$","source_text":"$2$","position":[11.29,2.92],"line":36},{"text":"$1$","source_text":"$1$","position":[12.15,2.92],"line":38},{"text":"$2$","source_text":"$2$","position":[10.43,2.06],"line":40},{"text":"$4$","source_text":"$4$","position":[11.29,2.06],"line":42},{"text":"$2$","source_text":"$2$","position":[12.15,2.06],"line":44},{"text":"$1$","source_text":"$1$","position":[10.43,1.2],"line":46},{"text":"$2$","source_text":"$2$","position":[11.29,1.2],"line":48},{"text":"$3$","source_text":"$3$","position":[12.15,1.2],"line":50},{"text":"$L_h^{-1}=hG_{\\rm nodes}$","source_text":"$L_h^{-1}=hG_{\\rm nodes}$","position":[11.3,0.4],"line":51}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],"reading_aid":{"alt":"A piecewise-linear Green function is paired with a scaled inverse matrix.","description":"The left graph samples one Green-function column at grid nodes. The matrix at right is the displayed scaled finite inverse. The equation beneath it keeps the grid-spacing factor between the nodal Green values and the inverse operator.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10]},{"kind":"table","caption":"Drawing grid 1 · rows top to bottom, columns left to right","headers":["Row / column","Column 1","Column 2","Column 3"],"rows":[["Row 1",11,12,13],["Row 2",14,15,16],["Row 3",17,18,19]]},{"kind":"text","nodes":[20]}]}},"mesh":{"name":"mesh","plate":19,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/mesh.tex","source_sha256":"e1d23798defaaa4353379b5be62db2836254d171a0623d4813ea36b41a531013","image":"assets/atlas/paper-vi/mesh.png","image_sha256":"39657010f44dc9ca1dfacf7ef23d7d78e42180b7988ec2335cc70158d4a0dc7b","pdf_page":21,"crop_pdf_points":[89,252,533,399],"nodes":[{"text":"0","source_text":"0","position":[0.8,0.39],"line":5},{"text":"0.25","source_text":"0.25","position":[2.7,0.39],"line":8},{"text":"0.5","source_text":"0.5","position":[4.6,0.39],"line":11},{"text":"0.75","source_text":"0.75","position":[6.5,0.39],"line":14},{"text":"1","source_text":"1","position":[8.4,0.39],"line":17},{"text":"0","source_text":"0","position":[0.64,0.65],"line":19},{"text":"$x$","source_text":"$x$","position":[4.6,0.1],"line":22},{"text":"$|u_h-u|=h^2x(1-x)$","source_text":"$|u_h-u|=h^2x(1-x)$","position":[0.8,4.15],"line":23},{"text":"grid spacing $h$\\qquad maximum error","source_text":"grid spacing $h$\\qquad maximum error","position":[11.35,3.95],"line":27},{"text":"$\\frac{1}{4}$\\qquad $\\frac{1}{64}$","source_text":"$\\frac{1}{4}$\\qquad $\\frac{1}{64}$","position":[11.35,3.0],"line":28},{"text":"$\\frac{1}{8}$\\qquad $\\frac{1}{256}$","source_text":"$\\frac{1}{8}$\\qquad $\\frac{1}{256}$","position":[11.35,2.0],"line":29},{"text":"$\\frac{1}{16}$\\qquad $\\frac{1}{1024}$","source_text":"$\\frac{1}{16}$\\qquad $\\frac{1}{1024}$","position":[11.35,1.0],"line":30}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11],"reading_aid":{"alt":"Three finite-difference error curves shrink as the mesh is refined.","description":"The curves give the stated quartic test problem at three grid spacings. The table on the right pairs each spacing with its maximum error. The formula above the plot states the exact error profile for this example.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10,11]}]}},"paths":{"name":"paths","plate":20,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/paths.tex","source_sha256":"4d87ca7d289bc3b114af6ed113fe7e20843753f81e7187c5c98499e262ab7281","image":"assets/atlas/paper-vi/paths.png","image_sha256":"feb2405f98530b075019891b2aea41415ce7818ca9d8e20f49cc37f38dcf7359","pdf_page":22,"crop_pdf_points":[97,243,512,409],"nodes":[{"text":"Path","source_text":"Path","position":[1.6,4.4],"line":3},{"text":"probability","source_text":"probability","position":[4.4,4.4],"line":4},{"text":"weight","source_text":"weight","position":[7.0,4.4],"line":5},{"text":"terminal","source_text":"terminal","position":[9.5,4.4],"line":6},{"text":"contribution","source_text":"contribution","position":[12.0,4.4],"line":7},{"text":"\\texttt{0000}","source_text":"\\texttt{0000}","position":[1.6,3.8],"line":8},{"text":"$27/64$","source_text":"$27/64$","position":[4.4,3.8],"line":9},{"text":"$1$","source_text":"$1$","position":[7.0,3.8],"line":10},{"text":"$2$","source_text":"$2$","position":[9.5,3.8],"line":11},{"text":"$27/32$","source_text":"$27/32$","position":[12.0,3.8],"line":12},{"text":"\\texttt{0001}","source_text":"\\texttt{0001}","position":[1.6,3.33],"line":14},{"text":"$9/64$","source_text":"$9/64$","position":[4.4,3.33],"line":15},{"text":"$1$","source_text":"$1$","position":[7.0,3.33],"line":16},{"text":"$1$","source_text":"$1$","position":[9.5,3.33],"line":17},{"text":"$9/64$","source_text":"$9/64$","position":[12.0,3.33],"line":18},{"text":"\\texttt{0010}","source_text":"\\texttt{0010}","position":[1.6,2.86],"line":19},{"text":"$3/64$","source_text":"$3/64$","position":[4.4,2.86],"line":20},{"text":"$1/2$","source_text":"$1/2$","position":[7.0,2.86],"line":21},{"text":"$2$","source_text":"$2$","position":[9.5,2.86],"line":22},{"text":"$3/64$","source_text":"$3/64$","position":[12.0,2.86],"line":23},{"text":"\\texttt{0011}","source_text":"\\texttt{0011}","position":[1.6,2.39],"line":25},{"text":"$9/64$","source_text":"$9/64$","position":[4.4,2.39],"line":26},{"text":"$1/2$","source_text":"$1/2$","position":[7.0,2.39],"line":27},{"text":"$1$","source_text":"$1$","position":[9.5,2.39],"line":28},{"text":"$9/128$","source_text":"$9/128$","position":[12.0,2.39],"line":29},{"text":"\\texttt{0100}","source_text":"\\texttt{0100}","position":[1.6,1.92],"line":30},{"text":"$3/64$","source_text":"$3/64$","position":[4.4,1.92],"line":31},{"text":"$1/2$","source_text":"$1/2$","position":[7.0,1.92],"line":32},{"text":"$2$","source_text":"$2$","position":[9.5,1.92],"line":33},{"text":"$3/64$","source_text":"$3/64$","position":[12.0,1.92],"line":34},{"text":"\\texttt{0101}","source_text":"\\texttt{0101}","position":[1.6,1.45],"line":36},{"text":"$1/64$","source_text":"$1/64$","position":[4.4,1.45],"line":37},{"text":"$1/2$","source_text":"$1/2$","position":[7.0,1.45],"line":38},{"text":"$1$","source_text":"$1$","position":[9.5,1.45],"line":39},{"text":"$1/128$","source_text":"$1/128$","position":[12.0,1.45],"line":40},{"text":"\\texttt{0110}","source_text":"\\texttt{0110}","position":[1.6,0.98],"line":41},{"text":"$3/64$","source_text":"$3/64$","position":[4.4,0.98],"line":42},{"text":"$1/4$","source_text":"$1/4$","position":[7.0,0.98],"line":43},{"text":"$2$","source_text":"$2$","position":[9.5,0.98],"line":44},{"text":"$3/128$","source_text":"$3/128$","position":[12.0,0.98],"line":45},{"text":"\\texttt{0111}","source_text":"\\texttt{0111}","position":[1.6,0.51],"line":47},{"text":"$9/64$","source_text":"$9/64$","position":[4.4,0.51],"line":48},{"text":"$1/4$","source_text":"$1/4$","position":[7.0,0.51],"line":49},{"text":"$1$","source_text":"$1$","position":[9.5,0.51],"line":50},{"text":"$9/256$","source_text":"$9/256$","position":[12.0,0.51],"line":51},{"text":"Total probability $=1$; weighted terminal expectation $=311/256$.","source_text":"Total probability $=1$; weighted terminal expectation $=311/256$.","position":[7.0,-0.22],"line":52}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],"reading_aid":{"alt":"Eight finite paths are listed with their probabilities, weights, and contributions.","description":"Read each row across from the path address to its probability, departure-state weight, terminal value, and weighted contribution. The bottom note gives the total path probability and weighted terminal expectation. The selectable transcript retains every original table label.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45]}]}},"resolvent":{"name":"resolvent","plate":21,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/resolvent.tex","source_sha256":"49380ec490922939b5dd9f12d7be26cbe14f0e371d9b809e345070221b209cdb","image":"assets/atlas/paper-vi/resolvent.png","image_sha256":"0caccc58f29fbb6da558751af7f7b6ed2453f16d7021238996d64093b65244e8","pdf_page":23,"crop_pdf_points":[89,277,522,428],"nodes":[{"text":"0","source_text":"0","position":[0.8,0.39],"line":5},{"text":"1","source_text":"1","position":[2.925,0.39],"line":8},{"text":"2","source_text":"2","position":[5.05,0.39],"line":11},{"text":"3","source_text":"3","position":[7.175,0.39],"line":14},{"text":"4","source_text":"4","position":[9.3,0.39],"line":17},{"text":"0","source_text":"0","position":[0.64,0.65],"line":19},{"text":"1","source_text":"1","position":[0.64,1.733333],"line":21},{"text":"2","source_text":"2","position":[0.64,2.816667],"line":23},{"text":"3","source_text":"3","position":[0.64,3.9],"line":25},{"text":"$z>0$","source_text":"$z>0$","position":[5.05,0.1],"line":28},{"text":"resolvent coefficient","source_text":"resolvent coefficient","position":[0.8,4.3],"line":29},{"text":"exact: $\\dfrac{z+1}{z(z+2)}$","source_text":"exact: $\\dfrac{z+1}{z(z+2)}$","position":[11.7,3.7],"line":32},{"text":"omit hidden state: $\\dfrac1{z+1}$","source_text":"omit hidden state: $\\dfrac1{z+1}$","position":[11.7,2.5],"line":33},{"text":"The pole at zero survives\\newline through hidden feedback.","source_text":"The pole at zero survives\\\\through hidden feedback.","position":[11.7,1.0],"line":34}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13],"reading_aid":{"alt":"An exact retained resolvent is compared with deletion of the hidden state.","description":"The solid and dashed curves show the two expressions written at right for positive spectral parameter. The source note identifies the pole retained through hidden feedback. The full system and the deleted model use different resolvent coefficients.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10,11,12,13]}]}},"memory":{"name":"memory","plate":22,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/memory.tex","source_sha256":"78564b37027b76ce424345a5a9ebb1e09d0182398d621af258914a76e40625db","image":"assets/atlas/paper-vi/memory.png","image_sha256":"4efcde6e9cf7f7c61caea7d5d4b79e843096cb2dd359d93186c37427e57f23cb","pdf_page":24,"crop_pdf_points":[83,307,524,458],"nodes":[{"text":"0","source_text":"0","position":[0.8,0.39],"line":5},{"text":"1","source_text":"1","position":[2.925,0.39],"line":8},{"text":"2","source_text":"2","position":[5.05,0.39],"line":11},{"text":"3","source_text":"3","position":[7.175,0.39],"line":14},{"text":"4","source_text":"4","position":[9.3,0.39],"line":17},{"text":"0","source_text":"0","position":[0.64,0.65],"line":19},{"text":"0.5","source_text":"0.5","position":[0.64,2.25],"line":21},{"text":"1","source_text":"1","position":[0.64,3.85],"line":23},{"text":"time $t$","source_text":"time $t$","position":[5.05,0.1],"line":26},{"text":"retained state $x(t)$","source_text":"retained state $x(t)$","position":[0.8,4.25],"line":27},{"text":"exact: $\\dfrac{1+e^{-2t}}2$","source_text":"exact: $\\dfrac{1+e^{-2t}}2$","position":[11.7,3.75],"line":30},{"text":"naive deletion: $e^{-t}$","source_text":"naive deletion: $e^{-t}$","position":[11.7,2.7],"line":31},{"text":"$\\displaystyle K(t)=e^{-t}$\\newline returns hidden excursions.","source_text":"$\\displaystyle K(t)=e^{-t}$\\\\returns hidden excursions.","position":[11.7,1.25],"line":32}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12],"reading_aid":{"alt":"Exact retained evolution is compared with naive deletion.","description":"The two curves start together and then separate: the exact retained state approaches a nonzero limit, while the deletion curve decays toward zero. The formulas and memory kernel are written to the right.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10,11,12]}]}},"jordan":{"name":"jordan","plate":23,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/jordan.tex","source_sha256":"8e8c8e566306d98dfe26b36a2213a98ef30f7bda91cc967e0237008e6283d8bb","image":"assets/atlas/paper-vi/jordan.png","image_sha256":"f8e1efe4216b926a8ce8fcff5dca99473449e35ef903e9a66fb3349e56800bb3","pdf_page":25,"crop_pdf_points":[83,265,517,416],"nodes":[{"text":"0","source_text":"0","position":[0.8,0.39],"line":5},{"text":"0.5","source_text":"0.5","position":[2.216667,0.39],"line":8},{"text":"1","source_text":"1","position":[3.633333,0.39],"line":11},{"text":"1.5","source_text":"1.5","position":[5.05,0.39],"line":14},{"text":"2","source_text":"2","position":[6.466667,0.39],"line":17},{"text":"2.5","source_text":"2.5","position":[7.883333,0.39],"line":20},{"text":"3","source_text":"3","position":[9.3,0.39],"line":23},{"text":"0","source_text":"0","position":[0.64,0.65],"line":25},{"text":"0.1","source_text":"0.1","position":[0.64,2.15],"line":27},{"text":"0.2","source_text":"0.2","position":[0.64,3.65],"line":29},{"text":"time $t$","source_text":"time $t$","position":[5.05,0.1],"line":32},{"text":"memory coefficient $t e^{-2t}$","source_text":"memory coefficient $t e^{-2t}$","position":[0.8,4.2],"line":33},{"text":"$J=-2I+N,\\quad N^2=0$","source_text":"$J=-2I+N,\\quad N^2=0$","position":[11.7,3.9],"line":36},{"text":"$e^{tJ}=e^{-2t}(I+tN)$","source_text":"$e^{tJ}=e^{-2t}(I+tN)$","position":[11.7,2.8],"line":37},{"text":"$\\widehat K(z)=\\dfrac1{(z+2)^2}$","source_text":"$\\widehat K(z)=\\dfrac1{(z+2)^2}$","position":[11.7,1.35],"line":38},{"text":"A repeated pole carries a jet.","source_text":"A repeated pole carries a jet.","position":[11.7,0.4],"line":39}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"reading_aid":{"alt":"A polynomial-exponential memory coefficient is paired with a repeated-pole formula.","description":"The curve rises from zero, reaches a marked maximum, and decays. The annotations retain the nilpotent Jordan part, its exponential, and the transform with a repeated pole. The note identifies the derivative information carried by that pole.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]}]}},"survival":{"name":"survival","plate":24,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/survival.tex","source_sha256":"c3eca6560f4d8d995cdf26eae1e8e8d8f291f7f4eaa4f234b56a7c745a101e4b","image":"assets/atlas/paper-vi/survival.png","image_sha256":"e81c5fcd0ab1b434e7fd5c54a5a22b2b8666294ce000ecf5e94932c2ea187666","pdf_page":26,"crop_pdf_points":[83,290,516,440],"nodes":[{"text":"0","source_text":"0","position":[0.8,0.39],"line":5},{"text":"1","source_text":"1","position":[2.5,0.39],"line":8},{"text":"2","source_text":"2","position":[4.2,0.39],"line":11},{"text":"3","source_text":"3","position":[5.9,0.39],"line":14},{"text":"4","source_text":"4","position":[7.6,0.39],"line":17},{"text":"5","source_text":"5","position":[9.3,0.39],"line":20},{"text":"0","source_text":"0","position":[0.64,0.65],"line":22},{"text":"0.5","source_text":"0.5","position":[0.64,2.25],"line":24},{"text":"1","source_text":"1","position":[0.64,3.85],"line":26},{"text":"time $t$","source_text":"time $t$","position":[5.05,0.1],"line":29},{"text":"probability or rate","source_text":"probability or rate","position":[0.8,4.25],"line":30},{"text":"solid: survival $S(t)$","source_text":"solid: survival $S(t)$","position":[11.7,3.8],"line":34},{"text":"dashed: absorbed mass","source_text":"dashed: absorbed mass","position":[11.7,2.8],"line":35},{"text":"dotted: hazard $h(t)$","source_text":"dotted: hazard $h(t)$","position":[11.7,1.8],"line":36},{"text":"$h(0)=0$, but eventual exit\\newline has probability one.","source_text":"$h(0)=0$, but eventual exit\\\\has probability one.","position":[11.7,0.6],"line":37}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"reading_aid":{"alt":"Survival, absorbed mass, and hazard are plotted as different curves.","description":"The solid survival curve decreases, the dashed absorbed mass increases, and the dotted hazard starts at zero. Their meanings and the eventual-exit statement are written beside the plot. The plate keeps probability and rate distinct.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]}]}},"naturality":{"name":"naturality","plate":25,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/naturality.tex","source_sha256":"7de9c5fb3bba8efdb62f0fe31aceeabe379625c0a01e3e3ac8654bc81926bae3","image":"assets/atlas/paper-vi/naturality.png","image_sha256":"0c8cf2615ea3bcc9fe621a3364961864cffa686e8851d239d45d5ee4cc6bf9cc","pdf_page":27,"crop_pdf_points":[103,286,512,413],"nodes":[{"text":"$A_f$","source_text":"$A_f$","position":[2.18,4.3],"line":3},{"text":"$2$","source_text":"$2$","position":[1.26,3.14],"line":5},{"text":"$1$","source_text":"$1$","position":[2.18,3.14],"line":7},{"text":"$0$","source_text":"$0$","position":[3.1,3.14],"line":9},{"text":"$0$","source_text":"$0$","position":[1.26,2.22],"line":11},{"text":"$2$","source_text":"$2$","position":[2.18,2.22],"line":13},{"text":"$0$","source_text":"$0$","position":[3.1,2.22],"line":15},{"text":"$0$","source_text":"$0$","position":[1.26,1.3],"line":17},{"text":"$0$","source_text":"$0$","position":[2.18,1.3],"line":19},{"text":"$9$","source_text":"$9$","position":[3.1,1.3],"line":21},{"text":"$F(A_f)$ for $F(z)=(1+z)^{-1}$","source_text":"$F(A_f)$ for $F(z)=(1+z)^{-1}$","position":[6.78,4.3],"line":22},{"text":"$\\frac{1}{3}$","source_text":"$\\frac{1}{3}$","position":[5.86,3.14],"line":24},{"text":"$- \\frac{1}{9}$","source_text":"$- \\frac{1}{9}$","position":[6.78,3.14],"line":26},{"text":"$0$","source_text":"$0$","position":[7.7,3.14],"line":28},{"text":"$0$","source_text":"$0$","position":[5.86,2.22],"line":30},{"text":"$\\frac{1}{3}$","source_text":"$\\frac{1}{3}$","position":[6.78,2.22],"line":32},{"text":"$0$","source_text":"$0$","position":[7.7,2.22],"line":34},{"text":"$0$","source_text":"$0$","position":[5.86,1.3],"line":36},{"text":"$0$","source_text":"$0$","position":[6.78,1.3],"line":38},{"text":"$\\frac{1}{10}$","source_text":"$\\frac{1}{10}$","position":[7.7,1.3],"line":40},{"text":"$F(2)=1/3$","source_text":"$F(2)=1/3$","position":[11.7,3.6],"line":42},{"text":"$F'(2)=-1/9$","source_text":"$F'(2)=-1/9$","position":[11.7,2.65],"line":43},{"text":"The dashed block is\\newline $F(A_c)$, not just $F(2)I$.","source_text":"The dashed block is\\\\$F(A_c)$, not just $F(2)I$.","position":[11.7,1.25],"line":44}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],"reading_aid":{"alt":"A block operator and its rational function are shown as matrices.","description":"The dashed block in the function matrix corresponds to the non-diagonal repeated-eigenvalue block. The right-hand annotations give the scalar function and its derivative at that eigenvalue. The note distinguishes the whole block function from scalar substitution alone.","blocks":[{"kind":"text","nodes":[0]},{"kind":"table","caption":"Drawing grid 1 · rows top to bottom, columns left to right","headers":["Row / column","Column 1","Column 2","Column 3"],"rows":[["Row 1",1,2,3],["Row 2",4,5,6],["Row 3",7,8,9]]},{"kind":"text","nodes":[10]},{"kind":"table","caption":"Drawing grid 2 · rows top to bottom, columns left to right","headers":["Row / column","Column 1","Column 2","Column 3"],"rows":[["Row 1",11,12,13],["Row 2",14,15,16],["Row 3",17,18,19]]},{"kind":"text","nodes":[20,21,22]}]}},"integrated_filter":{"name":"integrated_filter","plate":26,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/integrated_filter.tex","source_sha256":"372f70fa322c1426035476c42177d8b9de45bb11940cc2c8f2cc299d98df00c3","image":"assets/atlas/paper-vi/integrated_filter.png","image_sha256":"f5ce0dc9202a8b04ddffdc2cc5af8ae52ba7509fa7030afb477cbdec383357b4","pdf_page":28,"crop_pdf_points":[90,248,516,400],"nodes":[{"text":"First posterior","source_text":"First posterior","position":[2.5,3.8],"line":4},{"text":"0","source_text":"0","position":[0.925,0.3],"line":6},{"text":"$\\frac{3}{4}$","source_text":"$\\frac{3}{4}$","position":[0.925,3.27],"line":7},{"text":"1","source_text":"1","position":[1.975,0.3],"line":9},{"text":"$\\frac{3}{20}$","source_text":"$\\frac{3}{20}$","position":[1.975,1.31],"line":10},{"text":"2","source_text":"2","position":[3.025,0.3],"line":12},{"text":"$\\frac{1}{10}$","source_text":"$\\frac{1}{10}$","position":[3.025,1.146667],"line":13},{"text":"3","source_text":"3","position":[4.075,0.3],"line":15},{"text":"$0$","source_text":"$0$","position":[4.075,0.82],"line":16},{"text":"Second posterior","source_text":"Second posterior","position":[7.3,3.8],"line":18},{"text":"0","source_text":"0","position":[5.725,0.3],"line":20},{"text":"$\\frac{41}{104}$","source_text":"$\\frac{41}{104}$","position":[5.725,2.107821],"line":21},{"text":"1","source_text":"1","position":[6.775,0.3],"line":23},{"text":"$\\frac{15}{104}$","source_text":"$\\frac{15}{104}$","position":[6.775,1.291154],"line":24},{"text":"2","source_text":"2","position":[7.825,0.3],"line":26},{"text":"$\\frac{5}{13}$","source_text":"$\\frac{5}{13}$","position":[7.825,2.07641],"line":27},{"text":"3","source_text":"3","position":[8.875,0.3],"line":29},{"text":"$\\frac{1}{13}$","source_text":"$\\frac{1}{13}$","position":[8.875,1.071282],"line":30},{"text":"Coarse posterior","source_text":"Coarse posterior","position":[11.9,3.8],"line":32},{"text":"A","source_text":"A","position":[11.1,0.3],"line":34},{"text":"$\\frac{7}{13}$","source_text":"$\\frac{7}{13}$","position":[11.1,2.578974],"line":35},{"text":"B","source_text":"B","position":[12.7,0.3],"line":37},{"text":"$\\frac{6}{13}$","source_text":"$\\frac{6}{13}$","position":[12.7,2.327692],"line":38},{"text":"Evidence $Z_1=5/8$, $Z_2=13/20$, and $Z_1Z_2=13/32$","source_text":"Evidence $Z_1=5/8$, $Z_2=13/20$, and $Z_1Z_2=13/32$","position":[7.0,4.45],"line":39}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],"reading_aid":{"alt":"Two fine posteriors and one coarse posterior complete a two-observation example.","description":"Read the three bar panels from left to right. The first two use the four fine states; the third aggregates into the two declared packets. The two evidence values and their product are written above.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]}]}},"summability":{"name":"summability","plate":27,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/summability.tex","source_sha256":"fe5650347d22340e9181db1897f162389b71359a5eac26e9ea56e4d8c18c740a","image":"assets/atlas/paper-vi/summability.png","image_sha256":"cb93f6859b3e7beef75b78ccd19e70a9d9945b30f6db11c5429e42ee3d0a5084","pdf_page":29,"crop_pdf_points":[89,244,516,392],"nodes":[{"text":"0","source_text":"0","position":[0.8,0.39],"line":5},{"text":"4","source_text":"4","position":[2.925,0.39],"line":8},{"text":"8","source_text":"8","position":[5.05,0.39],"line":11},{"text":"12","source_text":"12","position":[7.175,0.39],"line":14},{"text":"16","source_text":"16","position":[9.3,0.39],"line":17},{"text":"0","source_text":"0","position":[0.64,0.65],"line":19},{"text":"1","source_text":"1","position":[0.64,1.564286],"line":21},{"text":"2","source_text":"2","position":[0.64,2.478571],"line":23},{"text":"3","source_text":"3","position":[0.64,3.392857],"line":25},{"text":"partial-sum cutoff $N$","source_text":"partial-sum cutoff $N$","position":[5.05,0.1],"line":28},{"text":"sum of defect norms","source_text":"sum of defect norms","position":[0.8,4.25],"line":29},{"text":"solid: $\\sum_{n\\le N}n^{-1}$","source_text":"solid: $\\sum_{n\\le N}n^{-1}$","position":[11.7,3.95],"line":33},{"text":"dashed: $\\sum_{n\\le N}n^{-2}$","source_text":"dashed: $\\sum_{n\\le N}n^{-2}$","position":[11.7,2.95],"line":34},{"text":"dotted: $\\frac12\\sum_{n\\le N}n^{-2}$","source_text":"dotted: $\\frac12\\sum_{n\\le N}n^{-2}$","position":[11.7,1.95],"line":35},{"text":"The infinite-series conclusion\\newline uses the exact formulas.","source_text":"The infinite-series conclusion\\\\uses the exact formulas.","position":[11.7,0.55],"line":36}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"reading_aid":{"alt":"Three partial sums distinguish accumulated defects from a small one-step value.","description":"The solid, dashed, and dotted curves are the three sums listed at right. The source note reserves the infinite-series conclusion for the exact formulas. The horizontal axis is the refinement cutoff.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]}]}},"ext_paths":{"name":"ext_paths","plate":29,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/ext_paths.tex","source_sha256":"5aacf0e75aebbf5f1329924f1afa3b25e5c186d415198b4522e758889e0ace50","image":"assets/atlas/paper-vi/ext_paths.png","image_sha256":"5a4230c103e452b8bd71b4824601870e4645f357f6fddd064bcad2e4f7993995","pdf_page":31,"crop_pdf_points":[78,266,529,412],"nodes":[{"text":"packet 0","source_text":"packet 0","position":[0.0,1.3],"line":5},{"text":"packet 1","source_text":"packet 1","position":[3.6,1.3],"line":5},{"text":"packet 2","source_text":"packet 2","position":[7.2,1.3],"line":5},{"text":"$2$","source_text":"$2$","position":null,"line":6},{"text":"$3$","source_text":"$3$","position":null,"line":6},{"text":"$1$","source_text":"$1$","position":[0.0,0.0],"line":7},{"text":"$2t$","source_text":"$2t$","position":[3.6,0.0],"line":7},{"text":"$3t^2$","source_text":"$3t^2$","position":[7.2,0.0],"line":7},{"text":"At $t=1$: $(1,2,3)$\\quad\\quad At $t=2$: $(1,4,12)$","source_text":"At $t=1$: $(1,2,3)$\\quad\\quad At $t=2$: $(1,4,12)$","position":[3.6,-0.9],"line":8}],"node_order":[0,1,2,3,4,5,6,7,8],"reading_aid":{"alt":"Three directed packets have only finitely many forward paths.","description":"Two arrows lead from packet zero to packet one and then packet two, with their transfer weights written above. The time-dependent components are written below the packets, followed by two exact-time examples.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8]}]}},"ext_closure":{"name":"ext_closure","plate":30,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/ext_closure.tex","source_sha256":"2b0ec5d60a80744690e6515341d26006795a3b3698a73ca434cd52800ad22f5a","image":"assets/atlas/paper-vi/ext_closure.png","image_sha256":"e56c05298b3951543ca93ec485c5f400277f72800eb9d7a00d83e4e1050a579c","pdf_page":32,"crop_pdf_points":[84,231,530,360],"nodes":[{"text":"$(1,0)$","source_text":"$(1,0)$","position":null,"line":7},{"text":"$(0,1)$","source_text":"$(0,1)$","position":null,"line":7},{"text":"$(-1,0)$","source_text":"$(-1,0)$","position":null,"line":7},{"text":"full two-step return: $-1$","source_text":"full two-step return: $-1$","position":[5.1,1.2],"line":8},{"text":"project after each step: $0$","source_text":"project after each step: $0$","position":[5.1,-0.2],"line":9}],"node_order":[0,1,2,3,4],"reading_aid":{"alt":"A rotating vector returns through a hidden direction.","description":"The semicircle follows the two-step excursion from the horizontal axis through the vertical direction to the opposite horizontal point. The boxes compare the full two-step return with inserting the retained projector after each step.","blocks":[{"kind":"text","nodes":[0,1,2,3,4]}]}},"ext_tails":{"name":"ext_tails","plate":31,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/ext_tails.tex","source_sha256":"4b6ad501889a243b6f274b1c0275864a106f9b5097715fc57d3e927e405551e9","image":"assets/atlas/paper-vi/ext_tails.png","image_sha256":"7e615bfee30865832175b68e746c79e880fcd5fe421519e2fb61b4ac416382e1","pdf_page":33,"crop_pdf_points":[84,251,523,438],"nodes":[{"text":"$n=0$","source_text":"$n=0$","position":[0.35,-0.35],"line":5},{"text":"$1$","source_text":"$1$","position":[0.35,2.4],"line":5},{"text":"$n=1$","source_text":"$n=1$","position":[1.8,-0.35],"line":7},{"text":"$1/2$","source_text":"$1/2$","position":[1.8,1.2],"line":7},{"text":"$n=2$","source_text":"$n=2$","position":[3.25,-0.35],"line":9},{"text":"$1/4$","source_text":"$1/4$","position":[3.25,0.6],"line":9},{"text":"$n=3$","source_text":"$n=3$","position":[4.7,-0.35],"line":11},{"text":"$1/8$","source_text":"$1/8$","position":[4.7,0.3],"line":11},{"text":"$n=4$","source_text":"$n=4$","position":[6.15,-0.35],"line":13},{"text":"$1/16$","source_text":"$1/16$","position":[6.15,0.15],"line":13},{"text":"$n=5$","source_text":"$n=5$","position":[7.6,-0.35],"line":15},{"text":"$1/32$","source_text":"$1/32$","position":[7.6,0.075],"line":15}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11],"reading_aid":{"alt":"A sequence of halving tail bounds is shown as bars.","description":"The horizontal labels give successive refinement levels. Each bar is half the preceding one, and its exact value is written above it. The plate explains the stated scalar completion and connecting maps.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10,11]}]}},"ext_obstruction":{"name":"ext_obstruction","plate":32,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/ext_obstruction.tex","source_sha256":"735bd2f888f5e26cea84783f2ba6c8cceabeaa7ecf7eec202f0268d1b9369cc6","image":"assets/atlas/paper-vi/ext_obstruction.png","image_sha256":"d35415bc6341ce4adc4ae9fa236d7ccafc90797272695d24f4b9727b1692b58a","pdf_page":34,"crop_pdf_points":[78,259,530,428],"nodes":[{"text":"$w=e_0-e_1$","source_text":"$w=e_0-e_1$","position":[0.0,1.5],"line":5},{"text":"$K_\\varepsilon w$","source_text":"$K_\\varepsilon w$","position":[6.1,1.5],"line":5},{"text":"$Cw=0$","source_text":"$Cw=0$","position":[0.0,-0.2],"line":6},{"text":"$CK_\\varepsilon w=(-1,1)^T/16$","source_text":"$CK_\\varepsilon w=(-1,1)^T/16$","position":[6.1,-0.2],"line":6},{"text":"$K_\\varepsilon$","source_text":"$K_\\varepsilon$","position":null,"line":7},{"text":"$C$","source_text":"$C$","position":null,"line":7},{"text":"$C$","source_text":"$C$","position":null,"line":7},{"text":"?","source_text":"?","position":null,"line":8},{"text":"No linear coarse map can make this step: $0$ must map to $0$.","source_text":"No linear coarse map can make this step: $0$ must map to $0$.","position":[3.05,-1.15],"line":9}],"node_order":[0,1,2,3,4,5,6,7,8],"reading_aid":{"alt":"An aggregation diagram exposes a forgotten-direction witness.","description":"The upper row applies the fine transition to a vector in the aggregation kernel. The vertical arrows aggregate before and after that transition. The dashed lower arrow and its question mark show the attempted coarse step, with the obstruction written beneath.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8]}]}},"ext_hermite":{"name":"ext_hermite","plate":33,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/ext_hermite.tex","source_sha256":"ff4a9993f0c0ec717bed7f8187969712893cb048dc02bd39a1adb99c23709100","image":"assets/atlas/paper-vi/ext_hermite.png","image_sha256":"2acc55cf693ae2c6cca85e335aec4a04fe3880b2564978a7e23ea84edd907533","pdf_page":35,"crop_pdf_points":[85,207,527,364],"nodes":[{"text":"$t=0$","source_text":"$t=0$","position":[-0.25,1.7],"line":5},{"text":"$t=\\log2$","source_text":"$t=\\log2$","position":[-0.25,0.5],"line":5},{"text":"$1$","source_text":"$1$","position":[1.0,1.7],"line":7},{"text":"$2$","source_text":"$2$","position":[4.0,1.7],"line":7},{"text":"$1/4+1/8=3/8$","source_text":"$1/4+1/8=3/8$","position":[1.1,0.5],"line":9},{"text":"Dark: degree 1 energy\\qquad Light: degree 2 energy","source_text":"Dark: degree 1 energy\\qquad Light: degree 2 energy","position":[3.0,-0.45],"line":10}],"node_order":[0,1,2,3,4,5],"reading_aid":{"alt":"Two Gaussian-degree energies decay at different rates.","description":"The upper stacked bar gives the initial degree-one and degree-two energies. The smaller lower bar gives them at the displayed later time. The legend distinguishes the two degrees and the note gives their total remaining energy.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5]}]}},"ext_profiles":{"name":"ext_profiles","plate":34,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/ext_profiles.tex","source_sha256":"0b23f4e62d247233b4a3d65f10d761f0218bc7c413b52b42bc5f4183778afd2b","image":"assets/atlas/paper-vi/ext_profiles.png","image_sha256":"405c94a591f12b021075c26b002bdf64aafc8cba8d1af1a46c1eb9d078be4179","pdf_page":36,"crop_pdf_points":[78,279,524,431],"nodes":[{"text":"state 0\\newline $(2,1)$","source_text":"state 0\\\\$(2,1)$","position":[0.0,1.3],"line":5},{"text":"state 1\\newline $(2,1)$","source_text":"state 1\\\\$(2,1)$","position":[2.0,1.3],"line":5},{"text":"state 2\\newline $(1,3)$","source_text":"state 2\\\\$(1,3)$","position":[4.8,1.3],"line":6},{"text":"state 3\\newline $(1,1)$","source_text":"state 3\\\\$(1,1)$","position":[7.2,1.3],"line":6},{"text":"one packet","source_text":"one packet","position":null,"line":7},{"text":"separate packet","source_text":"separate packet","position":[4.8,0.1],"line":8},{"text":"separate packet","source_text":"separate packet","position":[7.2,0.1],"line":8},{"text":"Joint likelihood profile $=(\\lambda^{(1)}_i,\\lambda^{(2)}_i)$","source_text":"Joint likelihood profile $=(\\lambda^{(1)}_i,\\lambda^{(2)}_i)$","position":[3.6,-0.75],"line":9}],"node_order":[0,1,2,3,4,5,6,7],"reading_aid":{"alt":"Four states are grouped by two-coordinate likelihood profiles.","description":"The first two states share their displayed profile and are joined by a brace. The remaining states have separate profiles and are labelled as separate packets. The bottom formula identifies the joint profile being compared.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7]}]}},"ext_information":{"name":"ext_information","plate":35,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/ext_information.tex","source_sha256":"f8c56111e493c557cefc876faf1507a0726805ebfe8d5d37aed681a9c11f00fe","image":"assets/atlas/paper-vi/ext_information.png","image_sha256":"8ac79ca0de008a82c5ded0bff3cd5b293e05fb86f92bb3a3756faf59ddb6fb75","pdf_page":37,"crop_pdf_points":[78,196,523,331],"nodes":[{"text":"$D(p\\Vert q)=\\frac14\\log2$","source_text":"$D(p\\Vert q)=\\frac14\\log2$","position":[4.0,0.85],"line":6},{"text":"coarse information\\newline $\\frac34\\log3-\\log2$","source_text":"coarse information\\\\$\\frac34\\log3-\\log2$","position":[3.01955001,-0.6],"line":7},{"text":"conditional information\\newline $\\frac54\\log2-\\frac34\\log3$","source_text":"conditional information\\\\$\\frac54\\log2-\\frac34\\log3$","position":[7.01955001,-0.6],"line":8}],"node_order":[0,1,2],"reading_aid":{"alt":"Relative entropy is split into coarse and conditional parts.","description":"The two segments of one bar represent the terms labelled below them. The total is displayed above the bar. The source plate states the positivity assumptions for the reference masses.","blocks":[{"kind":"text","nodes":[0,1,2]}]}},"ext_cumulants":{"name":"ext_cumulants","plate":36,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/ext_cumulants.tex","source_sha256":"35e6613119108ccdaef070a8b8296fd503b934bc4547bc59435a37d81befbf56","image":"assets/atlas/paper-vi/ext_cumulants.png","image_sha256":"1456171967c5643ebc88d4bbbcc11d0a739c42bfe900d19a37d14a5a20ee85bb","pdf_page":38,"crop_pdf_points":[78,205,523,405],"nodes":[{"text":"score","source_text":"score","position":[7.8,0.0],"line":5},{"text":"$0$","source_text":"$0$","position":[1.0,0.0],"line":7},{"text":"$1$","source_text":"$1$","position":[3.5,0.0],"line":7},{"text":"$2$","source_text":"$2$","position":[6.0,0.0],"line":7},{"text":"$1/7$","source_text":"$1/7$","position":[1.0,0.6],"line":8},{"text":"$2/7$","source_text":"$2/7$","position":[3.5,1.2],"line":8},{"text":"$4/7$","source_text":"$4/7$","position":[6.0,2.4],"line":8},{"text":"At $t=\\log2$: mean $10/7$, variance $26/49$","source_text":"At $t=\\log2$: mean $10/7$, variance $26/49$","position":[4.0,-0.7],"line":9}],"node_order":[0,1,2,3,4,5,6,7],"reading_aid":{"alt":"A tilted three-score distribution is shown as a bar chart.","description":"The horizontal addresses are the three score values, and each bar is labelled by its posterior probability. The annotation below gives the time, mean, and variance used by the worked example.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7]}]}},"ext_selection":{"name":"ext_selection","plate":37,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/ext_selection.tex","source_sha256":"f24f9d92acdb710477896d6ae372a74767c15dd0c957e917403c65646d2f8840","image":"assets/atlas/paper-vi/ext_selection.png","image_sha256":"351353c61933b1325391d133f213711797d6e3adadcc24e9a425b2b1cb17c2fe","pdf_page":39,"crop_pdf_points":[84,252,524,388],"nodes":[{"text":"rate 2, Jordan","source_text":"rate 2, Jordan","position":[0.0,1.9],"line":5},{"text":"$2$","source_text":"$2$","position":[6.35,1.9],"line":5},{"text":"rate 1","source_text":"rate 1","position":[0.0,0.9],"line":6},{"text":"$1/4$","source_text":"$1/4$","position":[1.1,0.9],"line":6},{"text":"rate 3","source_text":"rate 3","position":[0.0,-0.1],"line":7},{"text":"$1/9$","source_text":"$1/9$","position":[0.7,-0.1],"line":7}],"node_order":[0,1,2,3,4,5],"reading_aid":{"alt":"Three memory blocks are ranked by certified contributions.","description":"Each horizontal bar is labelled by its decay rate and contribution. The longest bar is the Jordan block; the smaller bars represent the two other rates. The plate states the cost and block-count budget for this selection problem.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5]}]}},"ext_trajectory":{"name":"ext_trajectory","plate":38,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/ext_trajectory.tex","source_sha256":"7de12380f33098ed63be017061b64f7e5197336a21213f1589d45a7d9bdedae8","image":"assets/atlas/paper-vi/ext_trajectory.png","image_sha256":"9b78800db03385eb832095edb02dae38ae85ffcb52bf0935f2134dd5a53bdbb3","pdf_page":40,"crop_pdf_points":[83,237,530,398],"nodes":[{"text":"$t$","source_text":"$t$","position":null,"line":5},{"text":"error","source_text":"error","position":null,"line":5},{"text":"$T$","source_text":"$T$","position":[4.0,0.0],"line":7},{"text":"$T^2/2$","source_text":"$T^2/2$","position":[4.15,1.6],"line":7},{"text":"interval certificate\\newline $T^2e^{T^2}$","source_text":"interval certificate\\\\$T^2e^{T^2}$","position":[7.3,1.35],"line":8},{"text":"Exact error curve: $t^2/2$","source_text":"Exact error curve: $t^2/2$","position":[2.0,-0.65],"line":9}],"node_order":[0,1,2,3,4,5],"reading_aid":{"alt":"A trajectory-error curve is paired with a finite-horizon certificate.","description":"The curve rises from the origin to the marked time horizon. Its exact expression is written beside and below the plot. A separate box contains the interval certificate, which the source derives under common initial data and forcing.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5]}]}},"ext_auxiliary":{"name":"ext_auxiliary","plate":39,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/ext_auxiliary.tex","source_sha256":"fd5368da5dcc873399e7aa957f38f449145ec3ad6291bac1412de47ed960891f","image":"assets/atlas/paper-vi/ext_auxiliary.png","image_sha256":"e7461103ac7fd0f1bc49c100866ac478a9747de64dce4cd6a84c740a3945aa66","pdf_page":41,"crop_pdf_points":[78,244,523,414],"nodes":[{"text":"$x$","source_text":"$x$","position":[0.0,1.0],"line":5},{"text":"$y_2$","source_text":"$y_2$","position":[3.7,1.0],"line":5},{"text":"$y_1$","source_text":"$y_1$","position":[7.4,1.0],"line":5},{"text":"input $1$","source_text":"input $1$","position":null,"line":6},{"text":"link $1$","source_text":"link $1$","position":null,"line":6},{"text":"feedback $1$","source_text":"feedback $1$","position":null,"line":7},{"text":"decay rate $2$","source_text":"decay rate $2$","position":[3.7,2.0],"line":8},{"text":"decay rate $2$","source_text":"decay rate $2$","position":[7.4,2.0],"line":8}],"node_order":[0,1,2,3,4,5,6,7],"reading_aid":{"alt":"Two auxiliary states and one retained state form a feedback realization.","description":"The top arrows pass the retained input through the two auxiliary states. The lower arrow feeds the second auxiliary output back to the retained state. The input, link, feedback, and decay-rate labels are all retained.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7]}]}},"ext_risks":{"name":"ext_risks","plate":40,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/ext_risks.tex","source_sha256":"5b7080fe65f4d76e47e7e66e23b91f476bbe36981c01e9f9384e42f98e624d7a","image":"assets/atlas/paper-vi/ext_risks.png","image_sha256":"e32505b90fcc4ff3e714601c47d60051020a0709ea4ab19d40cb2eeba6b15a6e","pdf_page":42,"crop_pdf_points":[78,212,529,352],"nodes":[{"text":"$3/8$","source_text":"$3/8$","position":[1.5,0.5],"line":6},{"text":"$1/4$","source_text":"$1/4$","position":[4.0,0.5],"line":6},{"text":"$3/8$","source_text":"$3/8$","position":[6.5,0.5],"line":6},{"text":"surviving","source_text":"surviving","position":[1.5,-0.5],"line":7},{"text":"cause 1 absorbed","source_text":"cause 1 absorbed","position":[4.0,-0.5],"line":7},{"text":"cause 2 absorbed","source_text":"cause 2 absorbed","position":[6.5,-0.5],"line":7},{"text":"At $t=\\log2$: surviving mixture $(2/3,1/3)$","source_text":"At $t=\\log2$: surviving mixture $(2/3,1/3)$","position":[4.0,1.55],"line":8}],"node_order":[0,1,2,3,4,5,6],"reading_aid":{"alt":"A population is split into survivors and two absorbed causes.","description":"The three adjacent segments give the surviving mass and the two cumulative absorbed masses at the displayed time. The note above states the mixture among survivors. The labels distinguish that mixture from the cumulative cause totals.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6]}]}},"ext_substitution":{"name":"ext_substitution","plate":41,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/ext_substitution.tex","source_sha256":"9544e259567e742f78f67c4d93e555baecf8affac9efca536eb2f4340f0e4bb5","image":"assets/atlas/paper-vi/ext_substitution.png","image_sha256":"297e1c9c20b297d28a7036e5708c1f8c4d44bbef52bfd2abc22b427145408acb","pdf_page":43,"crop_pdf_points":[78,233,530,372],"nodes":[{"text":"operator defect\\newline $\\Delta_{13}=7\\eta$","source_text":"operator defect\\\\$\\Delta_{13}=7\\eta$","position":[0.0,1.2],"line":5},{"text":"rational-calculus defect\\newline $-7\\eta/30$","source_text":"rational-calculus defect\\\\$-7\\eta/30$","position":[6.2,1.2],"line":6},{"text":"$F(z)=(1+z)^{-1}$","source_text":"$F(z)=(1+z)^{-1}$","position":null,"line":7},{"text":"Left inverse factor $1/3$\\quad Right inverse factor $1/10$\\quad Sign $-$","source_text":"Left inverse factor $1/3$\\quad Right inverse factor $1/10$\\quad Sign $-$","position":[3.1,-0.15],"line":8},{"text":"At $\\eta=0$: both defects vanish exactly","source_text":"At $\\eta=0$: both defects vanish exactly","position":[3.1,-0.85],"line":9}],"node_order":[0,1,2,3,4],"reading_aid":{"alt":"An operator defect is transported through a rational function.","description":"The arrow maps the displayed original defect to the rational-calculus defect. The notes below give the two inverse factors, the sign, and the zero-defect control. The plate compares this rectangular substitution with its core naturality result.","blocks":[{"kind":"text","nodes":[0,1,2,3,4]}]}},"ext_evidence":{"name":"ext_evidence","plate":42,"source":"papers/paper-vi/current/editable/traditional/visual_graphics/ext_evidence.tex","source_sha256":"506644a5f07b9d6205624a0f0309f5e7e5e09f1276e61446e677c110029c3b74","image":"assets/atlas/paper-vi/ext_evidence.png","image_sha256":"9a3a7c91dade91a05dffeba5f92aee880f7d8645bc7cbc26d3f225047c5ba1f6","pdf_page":44,"crop_pdf_points":[78,128,530,246],"nodes":[{"text":"inherited formal\\newline 33 scoped results\\newline saved Lean logs","source_text":"inherited formal\\\\33 scoped results\\\\saved Lean logs","position":[0.0,1.0],"line":5},{"text":"fresh examples\\newline rational checks\\newline false controls","source_text":"fresh examples\\\\rational checks\\\\false controls","position":[3.45,1.0],"line":6},{"text":"fresh edition\\newline PDF build\\newline page review","source_text":"fresh edition\\\\PDF build\\\\page review","position":[6.9,1.0],"line":7},{"text":"Separate claims, separate evidence records.\\quad No combined proof count.","source_text":"Separate claims, separate evidence records.\\quad No combined proof count.","position":[3.45,-0.5],"line":8}],"node_order":[0,1,2,3],"reading_aid":{"alt":"Three boxes separate inherited formal scope, example checks, and edition records.","description":"The boxes describe the retained source edition: saved formal logs, its rational example checks and false controls, and its PDF build and page review. The bottom note keeps these evidence categories separate and rejects a combined proof count.","blocks":[{"kind":"text","nodes":[0,1,2,3]}]}}},"manuscript_references":{"1":[["Section 1","read/paper-vi/core-the-transfer-semigroup-realization.html#L296"],["Section 2","read/paper-vi/core-packet-blocks-of-a-semigroup.html#L355"],["Section 3","read/paper-vi/core-refinement-defects-and-the-duhamel-identity.html#L441"],["Section 8","read/paper-vi/core-exact-bayesian-refinement-and-information-loss-packets.html#L827"],["Section 13","read/paper-vi/core-exact-elimination-of-latent-packets.html#L1479"]],"2":[["Section 2","read/paper-vi/core-packet-blocks-of-a-semigroup.html#L355"],["Section 4","read/paper-vi/core-markov-seeds-and-stochastic-transfer.html#L538"],["Section 6","read/paper-vi/core-conditional-expectation-as-a-stochastic-projector.html#L642"],["Section 7","read/paper-vi/core-bayesian-conditioning-as-a-normalized-nonlinear-operator.html#L723"],["Section 12","read/paper-vi/core-finite-state-feynman-kac-transfer.html#L1439"]],"3":[["Theorem 1.2","read/paper-vi/core-the-transfer-semigroup-realization.html#L332"],["Theorem 4.1","read/paper-vi/core-markov-seeds-and-stochastic-transfer.html#L545"]],"4":[["Theorem 2.1","read/paper-vi/core-packet-blocks-of-a-semigroup.html#L386"],["Corollary 2.2","read/paper-vi/core-packet-blocks-of-a-semigroup.html#L409"]],"5":[["Theorem 5.1","read/paper-vi/core-markov-lumping-as-exact-refinement.html#L595"]],"6":[["Theorem 3.1","read/paper-vi/core-refinement-defects-and-the-duhamel-identity.html#L460"],["Theorem 3.3","read/paper-vi/core-refinement-defects-and-the-duhamel-identity.html#L520"],["Theorem 5.2","read/paper-vi/core-markov-lumping-as-exact-refinement.html#L610"]],"7":[["Theorem 6.1","read/paper-vi/core-conditional-expectation-as-a-stochastic-projector.html#L656"]],"8":[["Theorem 6.2","read/paper-vi/core-conditional-expectation-as-a-stochastic-projector.html#L682"]],"9":[["Theorem 7.1","read/paper-vi/core-bayesian-conditioning-as-a-normalized-nonlinear-operator.html#L739"],["Theorem 7.2","read/paper-vi/core-bayesian-conditioning-as-a-normalized-nonlinear-operator.html#L773"],["Theorem 7.3","read/paper-vi/core-bayesian-conditioning-as-a-normalized-nonlinear-operator.html#L803"]],"10":[["Theorem 8.2","read/paper-vi/core-exact-bayesian-refinement-and-information-loss-packets.html#L878"]],"11":[["Theorem 8.1","read/paper-vi/core-exact-bayesian-refinement-and-information-loss-packets.html#L852"],["Theorem 8.4","read/paper-vi/core-exact-bayesian-refinement-and-information-loss-packets.html#L924"],["Corollary 8.6","read/paper-vi/core-exact-bayesian-refinement-and-information-loss-packets.html#L987"]],"12":[["Theorem 7.1","read/paper-vi/core-bayesian-conditioning-as-a-normalized-nonlinear-operator.html#L739"],["Theorem 9.1","read/paper-vi/core-bayesian-flows-conjugacy-and-hidden-states.html#L1039"]],"13":[["Proposition 9.3","read/paper-vi/core-bayesian-flows-conjugacy-and-hidden-states.html#L1098"]],"14":[["Example 9.2","read/paper-vi/core-bayesian-flows-conjugacy-and-hidden-states.html#L1070"]],"15":[["Theorem 10.1","read/paper-vi/core-fractional-powers-as-spectral-packet-transfer.html#L1134"]],"16":[["Theorem 10.3","read/paper-vi/core-fractional-powers-as-spectral-packet-transfer.html#L1200"],["Corollary 10.4","read/paper-vi/core-fractional-powers-as-spectral-packet-transfer.html#L1294"]],"17":[["Theorem 10.2","read/paper-vi/core-fractional-powers-as-spectral-packet-transfer.html#L1164"]],"18":[["Theorem 11.1","read/paper-vi/core-green-operators-and-the-poisson-seed.html#L1377"]],"19":[["Theorem 11.1","read/paper-vi/core-green-operators-and-the-poisson-seed.html#L1377"],["Theorem 11.2","read/paper-vi/core-green-operators-and-the-poisson-seed.html#L1415"]],"20":[["Theorem 12.1","read/paper-vi/core-finite-state-feynman-kac-transfer.html#L1450"]],"21":[["Theorem 13.1","read/paper-vi/core-exact-elimination-of-latent-packets.html#L1494"]],"22":[["Theorem 13.2","read/paper-vi/core-exact-elimination-of-latent-packets.html#L1544"]],"23":[["Theorem 13.2","read/paper-vi/core-exact-elimination-of-latent-packets.html#L1544"],["Section 15","read/paper-vi/core-one-common-operator-language.html#L1628"]],"24":[["Theorem 14.1","read/paper-vi/core-survival-and-hazard-semigroups.html#L1596"]],"25":[["Theorem 15.1","read/paper-vi/core-one-common-operator-language.html#L1675"]],"26":[["Proposition 9.3","read/paper-vi/core-bayesian-flows-conjugacy-and-hidden-states.html#L1098"],["Theorem 5.1","read/paper-vi/core-markov-lumping-as-exact-refinement.html#L595"],["Theorem 8.2","read/paper-vi/core-exact-bayesian-refinement-and-information-loss-packets.html#L878"]],"27":[["Corollary 10.5","read/paper-vi/core-fractional-powers-as-spectral-packet-transfer.html#L1327"],["Section 17","read/paper-vi/core-remaining-conjectural-extensions.html#L1756"]],"28":[],"29":[["Theorem E2.8","read/paper-vi/extended-packet-resolved-semigroups.html#L2233"],["Corollary E2.9","read/paper-vi/extended-packet-resolved-semigroups.html#L2267"]],"30":[["Theorem E4.8","read/paper-vi/extended-refinement-towers-and-limiting-evolution.html#L2647"],["Example E4.10","read/paper-vi/extended-refinement-towers-and-limiting-evolution.html#L2687"]],"31":[["Theorem E4.5","read/paper-vi/extended-refinement-towers-and-limiting-evolution.html#L2541"],["Corollary E4.7","read/paper-vi/extended-refinement-towers-and-limiting-evolution.html#L2617"]],"32":[["Theorem E5.3","read/paper-vi/extended-finite-markov-transfer-and-lumping.html#L2862"],["Corollary E5.4","read/paper-vi/extended-finite-markov-transfer-and-lumping.html#L2895"]],"33":[["Theorem E7.1","read/paper-vi/extended-gaussian-transfer-and-hermite-packets.html#L3137"],["Corollary E7.2","read/paper-vi/extended-gaussian-transfer-and-hermite-packets.html#L3192"],["Corollary E7.4","read/paper-vi/extended-gaussian-transfer-and-hermite-packets.html#L3269"],["Corollary E7.5","read/paper-vi/extended-gaussian-transfer-and-hermite-packets.html#L3300"]],"34":[["Theorem E9.3","read/paper-vi/extended-bayesian-coarsening-and-finite-sufficiency.html#L3542"]],"35":[["Theorem E9.8","read/paper-vi/extended-bayesian-coarsening-and-finite-sufficiency.html#L3646"],["Corollary E9.9","read/paper-vi/extended-bayesian-coarsening-and-finite-sufficiency.html#L3683"]],"36":[["Theorem E10.1","read/paper-vi/extended-bayesian-flows-and-hidden-markov-recursion.html#L3774"],["Theorem E10.2","read/paper-vi/extended-bayesian-flows-and-hidden-markov-recursion.html#L3798"]],"37":[["Theorem E15.1","read/paper-vi/extended-memory-modes-and-controlled-compression.html#L4643"],["Corollary E15.2","read/paper-vi/extended-memory-modes-and-controlled-compression.html#L4725"]],"38":[["Theorem E15.6","read/paper-vi/extended-memory-modes-and-controlled-compression.html#L4818"]],"39":[["Corollary E15.4","read/paper-vi/extended-memory-modes-and-controlled-compression.html#L4780"],["Corollary E15.5","read/paper-vi/extended-memory-modes-and-controlled-compression.html#L4795"]],"40":[["Proposition E16.4","read/paper-vi/extended-survival-moments-and-competing-risks.html#L5017"],["Theorem E16.5","read/paper-vi/extended-survival-moments-and-competing-risks.html#L5036"]],"41":[["Theorem E3.4","read/paper-vi/extended-generator-defects-and-functional-calculus.html#L2377"],["Theorem 15.1","read/paper-vi/core-one-common-operator-language.html#L1675"]],"42":[]},"new_proof_review":false,"new_lean_execution":false,"new_mathematical_example_execution":false,"new_tex_build":false}