{"version":2,"work":"machine-theory-compendium","title":"Machine Theory Compendium","source_revision":"a12c6b2d9ac68b73031075180f51a8c98d38ee1e","sources":[{"repository_path":"monographs/machine-theory-compendium/README.md","file_url":"documents/files/7af439a3c2ddcef6a6903cb6342aa40c2d5c88bb/README.md","url":"read/machine-theory-compendium/readme-b42d11b18891.html","sha256":"7c5a2b13c6644b022cb4c697e8a8a902d05335793dd99301a316bf87c1c87702","git_blob_sha1":"7af439a3c2ddcef6a6903cb6342aa40c2d5c88bb","size":5431,"source_url":"read/machine-theory-compendium/readme-b42d11b18891-source.html","passages":0,"macros":{}},{"repository_path":"monographs/machine-theory-compendium/editorial/connected_synthesis.tex","file_url":"documents/files/3ebeaec1ef0bab91288e1e2b46dc341d4ba7e49c/connected_synthesis.tex","url":"read/machine-theory-compendium/connected-synthesis-54482cf6a750.html","sha256":"f4c2428de9c3ff2a46fb7abe0d37690aaa59eb9ffd48f0065a24346087b1ec5f","git_blob_sha1":"3ebeaec1ef0bab91288e1e2b46dc341d4ba7e49c","size":62722,"passages":132,"macros":{"headrulewidth":"0.3pt","F":"\\mathbb F","TV":"d_{\\mathrm{TV}}","im":"\\operatorname{im}","kerop":"\\operatorname{ker}"}},{"repository_path":"monographs/machine-theory-compendium/editorial/source_catalog.tex","file_url":"documents/files/d0ca8440581ed0069d566d9bbbc96b444462e9cb/source_catalog.tex","url":"read/machine-theory-compendium/source-catalog-13a4ff96a100.html","sha256":"81f9a6a7f4eb5cf2240bf44fc6edabf6033bcfa44af01e168fd5da93f963f0b3","git_blob_sha1":"d0ca8440581ed0069d566d9bbbc96b444462e9cb","size":8542,"passages":0,"macros":{}}],"pdfs":[{"repository_path":"monographs/machine-theory-compendium/readers/Komorebi_Machine_Theory_Compendium_v1.0.0.pdf","url":"read/machine-theory-compendium/komorebi-machine-theory-compendium-v1-0-0-8455252189f7.html","sha256":"4b6e3b03209e3d4a52705f7452c0435efcf792e8ce0189cb9b2294ec05f00842"},{"repository_path":"monographs/machine-theory-compendium/sources/Self_Improving_Intelligence_Visual_Guide_v1.0.0.pdf","url":"read/machine-theory-compendium/self-improving-intelligence-visual-guide-v1-0-0-84beeb2a1a3b.html","sha256":"e677300fd93661daedd4a429683a2be430564a89c13bdf0b9b9714e2c571eeca"},{"repository_path":"monographs/machine-theory-compendium/sources/AVOC_Lean_Companion_Final.pdf","url":"read/machine-theory-compendium/avoc-lean-companion-final-c8f8deab1ccb.html","sha256":"a45cbefb1aee2ee0227d823e765074fe05226c3942a3c35c7e628803e0efc878"}],"passages":[{"line":116,"start":11284,"end":11366,"expression":"\n\\mathfrak C=(\\Omega,\\Pi,\\mathcal Q,\\mathcal A,\\mathcal E,\\mathcal B,\\mathcal V).\n","context":0,"kind":"use","source":1},{"line":119,"start":11375,"end":11381,"expression":"\\Omega","context":0,"kind":"use","source":1},{"line":119,"start":11437,"end":11440,"expression":"\\Pi","context":0,"kind":"use","source":1},{"line":119,"start":11480,"end":11490,"expression":"\\mathcal Q","context":0,"kind":"use","source":1},{"line":119,"start":11518,"end":11528,"expression":"\\mathcal A","context":0,"kind":"use","source":1},{"line":119,"start":11567,"end":11577,"expression":"\\mathcal E","context":0,"kind":"use","source":1},{"line":119,"start":11603,"end":11613,"expression":"\\mathcal B","context":1,"kind":"use","source":1},{"line":119,"start":11643,"end":11653,"expression":"\\mathcal V","context":2,"kind":"use","source":1},{"line":121,"start":11886,"end":11889,"expression":"x_t","context":3,"kind":"use","source":1},{"line":121,"start":11916,"end":11919,"expression":"h_t","context":3,"kind":"use","source":1},{"line":121,"start":11951,"end":11954,"expression":"z_t","context":3,"kind":"use","source":1},{"line":139,"start":15131,"end":15132,"expression":"X","context":4,"kind":"use","source":1},{"line":139,"start":15164,"end":15167,"expression":"F_a","context":4,"kind":"use","source":1},{"line":139,"start":15191,"end":15192,"expression":"q","context":4,"kind":"use","source":1},{"line":140,"start":15204,"end":15303,"expression":"\nx\\sim y\\quad\\Longleftrightarrow\\quad\nq(F_wx)=q(F_wy)\\quad\\text{for every admitted finite word }w.\n","context":4,"kind":"use","source":1},{"line":146,"start":15777,"end":15781,"expression":"r(x)","context":5,"kind":"use","source":1},{"line":146,"start":15882,"end":15886,"expression":"r(x)","context":5,"kind":"use","source":1},{"line":149,"start":15970,"end":15985,"expression":"p:X\\to X/{\\sim}","context":6,"kind":"use","source":1},{"line":149,"start":16050,"end":16068,"expression":"i:r(X)\\to X/{\\sim}","context":7,"kind":"use","source":1},{"line":149,"start":16076,"end":16088,"expression":"i(r(x))=p(x)","context":8,"kind":"use","source":1},{"line":149,"start":16101,"end":16102,"expression":"x","context":9,"kind":"use","source":1},{"line":150,"start":16121,"end":16166,"expression":"\nr(x)=r(y)\\quad\\Longrightarrow\\quad x\\sim y.\n","context":10,"kind":"use","source":1},{"line":156,"start":16408,"end":16420,"expression":"i(r(x))=p(x)","context":11,"kind":"use","source":1},{"line":156,"start":16494,"end":16495,"expression":"x","context":12,"kind":"use","source":1},{"line":156,"start":16519,"end":16546,"expression":"\\overline F_a(p(x))=p(F_ax)","context":13,"kind":"use","source":1},{"line":156,"start":16664,"end":16683,"expression":"pF_w=\\overline F_wp","context":14,"kind":"use","source":1},{"line":156,"start":16718,"end":16719,"expression":"p","context":15,"kind":"use","source":1},{"line":166,"start":18131,"end":18139,"expression":"C:X\\to Z","context":16,"kind":"use","source":1},{"line":166,"start":18164,"end":18172,"expression":"A:X\\to X","context":16,"kind":"use","source":1},{"line":166,"start":18197,"end":18198,"expression":"B","context":16,"kind":"use","source":1},{"line":166,"start":18212,"end":18217,"expression":"CA=BC","context":16,"kind":"use","source":1},{"line":166,"start":18242,"end":18250,"expression":"\\kerop C","context":16,"kind":"use","source":1},{"line":166,"start":18272,"end":18273,"expression":"A","context":16,"kind":"use","source":1},{"line":166,"start":18390,"end":18391,"expression":"C","context":16,"kind":"use","source":1},{"line":169,"start":18605,"end":18728,"expression":"\n\\begin{pmatrix}z_{t+1}\\\\u_{t+1}\\end{pmatrix}\n=\n\\begin{pmatrix}A&B\\\\C&D\\end{pmatrix}\n\\begin{pmatrix}z_t\\\\u_t\\end{pmatrix}.\n","context":17,"kind":"use","source":1},{"line":175,"start":18740,"end":18743,"expression":"u_t","context":18,"kind":"use","source":1},{"line":175,"start":18773,"end":18777,"expression":"Bu_t","context":18,"kind":"use","source":1},{"line":178,"start":19252,"end":19263,"expression":"\\mathbb R^3","context":19,"kind":"use","source":1},{"line":179,"start":19268,"end":19381,"expression":"\nA_0=\\begin{pmatrix}1&1&0\\\\0&1&0\\\\0&0&2\\end{pmatrix},\\qquad\nA_1=\\begin{pmatrix}1&1&0\\\\0&1&0\\\\1&0&3\\end{pmatrix}.\n","context":20,"kind":"use","source":1},{"line":183,"start":19389,"end":19413,"expression":"p(x_1,x_2,x_3)=(x_1,x_2)","context":21,"kind":"use","source":1},{"line":183,"start":19438,"end":19446,"expression":"q(x)=x_1","context":21,"kind":"use","source":1},{"line":184,"start":19474,"end":19530,"expression":"\npA_i=Gp,\\qquad G=\\begin{pmatrix}1&1\\\\0&1\\end{pmatrix}.\n","context":22,"kind":"use","source":1},{"line":187,"start":19564,"end":19565,"expression":"k","context":23,"kind":"use","source":1},{"line":187,"start":19582,"end":19590,"expression":"x_1+kx_2","context":23,"kind":"use","source":1},{"line":194,"start":20710,"end":20718,"expression":"r:X\\to Z","context":24,"kind":"use","source":1},{"line":194,"start":20752,"end":20755,"expression":"G_a","context":24,"kind":"use","source":1},{"line":194,"start":20765,"end":20766,"expression":"Z","context":24,"kind":"use","source":1},{"line":194,"start":20778,"end":20786,"expression":"T:Z\\to W","context":24,"kind":"use","source":1},{"line":194,"start":20854,"end":20857,"expression":"H_a","context":24,"kind":"use","source":1},{"line":195,"start":20888,"end":20917,"expression":"\nrF_a=G_ar,\\qquad TG_a=H_aT.\n","context":24,"kind":"use","source":1},{"line":201,"start":21250,"end":21262,"expression":"q=\\bar q\\,Tr","context":25,"kind":"use","source":1},{"line":201,"start":21303,"end":21304,"expression":"w","context":25,"kind":"use","source":1},{"line":202,"start":21309,"end":21349,"expression":"\nTrF_w=H_wTr,\\qquad qF_w=\\bar q\\,H_wTr.\n","context":25,"kind":"use","source":1},{"line":207,"start":21403,"end":21420,"expression":"TrF_a=TG_ar=H_aTr","context":25,"kind":"use","source":1},{"line":207,"start":21567,"end":21573,"expression":"\\bar q","context":26,"kind":"use","source":1},{"line":210,"start":21855,"end":21856,"expression":"r","context":27,"kind":"use","source":1},{"line":210,"start":21862,"end":21863,"expression":"T","context":27,"kind":"use","source":1},{"line":219,"start":23501,"end":23517,"expression":"V\\subseteq\\Omega","context":28,"kind":"use","source":1},{"line":219,"start":23536,"end":23552,"expression":"V\\mapsto V\\cap E","context":28,"kind":"use","source":1},{"line":219,"start":23585,"end":23589,"expression":"t(V)","context":28,"kind":"use","source":1},{"line":220,"start":23632,"end":23684,"expression":"\n\\Sigma(V)=\\bigl(t(V\\cap E)\\bigr)_{E\\in\\mathcal E}.\n","context":28,"kind":"use","source":1},{"line":240,"start":26754,"end":26757,"expression":"\\mu","context":29,"kind":"use","source":1},{"line":240,"start":26764,"end":26775,"expression":"\\widehat\\mu","context":30,"kind":"use","source":1},{"line":240,"start":26840,"end":26866,"expression":"\\TV(\\mu,\\widehat\\mu)\\leq e","context":31,"kind":"use","source":1},{"line":240,"start":26917,"end":26954,"expression":"d(y_\\omega,\\widehat y_\\omega)\\leq d_0","context":32,"kind":"use","source":1},{"line":240,"start":26989,"end":27006,"expression":"\\ell(a,y)\\in[0,1]","context":33,"kind":"use","source":1},{"line":240,"start":27012,"end":27013,"expression":"K","context":34,"kind":"use","source":1},{"line":240,"start":27083,"end":27084,"expression":"a","context":35,"kind":"use","source":1},{"line":241,"start":27089,"end":27138,"expression":"\n|R(a)-\\widehat R(a)|\\leq\\eta:=\\min\\{1,e+Kd_0\\},\n","context":36,"kind":"use","source":1},{"line":244,"start":27148,"end":27190,"expression":"R(a)=\\sum_\\omega\\mu_\\omega\\ell(a,y_\\omega)","context":37,"kind":"use","source":1},{"line":244,"start":27197,"end":27265,"expression":"\\widehat R(a)=\\sum_\\omega\\widehat\\mu_\\omega\\ell(a,\\widehat y_\\omega)","context":38,"kind":"use","source":1},{"line":244,"start":27290,"end":27300,"expression":"\\widehat R","context":39,"kind":"use","source":1},{"line":244,"start":27322,"end":27327,"expression":"2\\eta","context":40,"kind":"use","source":1},{"line":244,"start":27357,"end":27358,"expression":"R","context":41,"kind":"use","source":1},{"line":247,"start":27472,"end":27510,"expression":"\\sum\\widehat\\mu_\\omega\\ell(a,y_\\omega)","context":42,"kind":"use","source":1},{"line":247,"start":27546,"end":27547,"expression":"e","context":43,"kind":"use","source":1},{"line":247,"start":27578,"end":27583,"expression":"[0,1]","context":44,"kind":"use","source":1},{"line":247,"start":27609,"end":27613,"expression":"Kd_0","context":45,"kind":"use","source":1},{"line":247,"start":27641,"end":27646,"expression":"[0,1]","context":46,"kind":"use","source":1},{"line":247,"start":27669,"end":27679,"expression":"\\widehat a","context":47,"kind":"use","source":1},{"line":247,"start":27686,"end":27689,"expression":"a^*","context":48,"kind":"use","source":1},{"line":247,"start":27753,"end":27838,"expression":"R(\\widehat a)\\leq\\widehat R(\\widehat a)+\\eta\\leq\\widehat R(a^*)+\\eta\\leq R(a^*)+2\\eta","context":49,"kind":"use","source":1},{"line":298,"start":36110,"end":36293,"expression":"\n\\text{acquire}\\ \\longrightarrow\\ \\text{represent}\\ \\longrightarrow\\ \\text{select}\\ \\longrightarrow\\ \\text{verify}\\ \\longrightarrow\\ \\text{execute}\\ \\longrightarrow\\ \\text{maintain}.\n","context":50,"kind":"use","source":1},{"line":304,"start":36585,"end":36605,"expression":"\\Delta\\in\\mathbb R^d","context":51,"kind":"use","source":1},{"line":304,"start":36684,"end":36701,"expression":"w^\\top\\Delta\\geq0","context":51,"kind":"use","source":1},{"line":304,"start":36714,"end":36737,"expression":"w\\in\\mathbb R^d_{\\geq0}","context":51,"kind":"use","source":1},{"line":304,"start":36775,"end":36781,"expression":"\\Delta","context":51,"kind":"use","source":1},{"line":310,"start":37413,"end":37430,"expression":"(1552,1552,-1868)","context":52,"kind":"use","source":1},{"line":322,"start":38448,"end":38479,"expression":"\n\\omega=(b,\\theta,u)\\in\\F_2^3.\n","context":53,"kind":"use","source":1},{"line":325,"start":38491,"end":38492,"expression":"b","context":53,"kind":"use","source":1},{"line":325,"start":38551,"end":38557,"expression":"\\theta","context":53,"kind":"use","source":1},{"line":325,"start":38601,"end":38602,"expression":"u","context":53,"kind":"use","source":1},{"line":325,"start":38698,"end":38713,"expression":"s=b\\oplus\\theta","context":53,"kind":"use","source":1},{"line":325,"start":38739,"end":38745,"expression":"\\theta","context":53,"kind":"use","source":1},{"line":325,"start":38774,"end":38775,"expression":"b","context":53,"kind":"use","source":1},{"line":325,"start":38787,"end":38793,"expression":"\\theta","context":53,"kind":"use","source":1},{"line":325,"start":38800,"end":38801,"expression":"u","context":53,"kind":"use","source":1},{"line":325,"start":38824,"end":38827,"expression":"R_i","context":54,"kind":"use","source":1},{"line":325,"start":38854,"end":38857,"expression":"b=i","context":55,"kind":"use","source":1},{"line":330,"start":39279,"end":39282,"expression":"z=s","context":56,"kind":"use","source":1},{"line":330,"start":39328,"end":39336,"expression":"G(z)=1-z","context":56,"kind":"use","source":1},{"line":330,"start":39387,"end":39388,"expression":"b","context":56,"kind":"use","source":1},{"line":330,"start":39395,"end":39401,"expression":"\\theta","context":56,"kind":"use","source":1},{"line":335,"start":39873,"end":39876,"expression":"R_b","context":57,"kind":"use","source":1},{"line":335,"start":39920,"end":39927,"expression":"(0,0,u)","context":57,"kind":"use","source":1},{"line":335,"start":39934,"end":39941,"expression":"(1,1,u)","context":57,"kind":"use","source":1},{"line":336,"start":40111,"end":40129,"expression":"\nb=s\\oplus\\theta.\n","context":57,"kind":"use","source":1},{"line":339,"start":40214,"end":40220,"expression":"\\theta","context":58,"kind":"use","source":1},{"line":344,"start":40720,"end":40730,"expression":"(s,\\theta)","context":59,"kind":"use","source":1},{"line":344,"start":40752,"end":40762,"expression":"(b,\\theta)","context":59,"kind":"use","source":1},{"line":345,"start":40782,"end":40818,"expression":"\nT(s,\\theta)=(s\\oplus\\theta,\\theta)\n","context":59,"kind":"use","source":1},{"line":348,"start":40901,"end":40902,"expression":"T","context":59,"kind":"use","source":1},{"line":350,"start":41182,"end":41183,"expression":"s","context":60,"kind":"use","source":1},{"line":358,"start":41832,"end":41842,"expression":"(b,\\theta)","context":61,"kind":"use","source":1},{"line":358,"start":41900,"end":41901,"expression":"u","context":61,"kind":"use","source":1},{"line":358,"start":42075,"end":42081,"expression":"2^4=16","context":61,"kind":"use","source":1},{"line":358,"start":42116,"end":42123,"expression":"2^8=256","context":62,"kind":"use","source":1},{"line":360,"start":42289,"end":42290,"expression":"b","context":63,"kind":"use","source":1},{"line":365,"start":43063,"end":43068,"expression":"(0,0)","context":64,"kind":"use","source":1},{"line":365,"start":43075,"end":43080,"expression":"(1,1)","context":64,"kind":"use","source":1},{"line":365,"start":43112,"end":43117,"expression":"(0,1)","context":64,"kind":"use","source":1},{"line":365,"start":43124,"end":43129,"expression":"(1,0)","context":64,"kind":"use","source":1},{"line":365,"start":43199,"end":43206,"expression":"\\{0,1\\}","context":64,"kind":"use","source":1},{"line":367,"start":43293,"end":43301,"expression":"\\theta=0","context":65,"kind":"use","source":1},{"line":367,"start":43326,"end":43329,"expression":"b=0","context":65,"kind":"use","source":1},{"line":367,"start":43353,"end":43356,"expression":"b=1","context":65,"kind":"use","source":1},{"line":370,"start":43935,"end":43936,"expression":"b","context":66,"kind":"use","source":1},{"line":382,"start":45682,"end":45685,"expression":"R_*","context":67,"kind":"use","source":1},{"line":382,"start":45717,"end":45718,"expression":"b","context":67,"kind":"use","source":1},{"line":382,"start":45767,"end":45768,"expression":"b","context":67,"kind":"use","source":1},{"line":382,"start":45914,"end":45917,"expression":"R_*","context":67,"kind":"use","source":1},{"line":384,"start":46221,"end":46224,"expression":"R_*","context":68,"kind":"use","source":1}],"entries":[{"id":"symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","occurrences":[80,81,127,130,131]},{"id":"symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"use","occurrences":[34,43,68,81]},{"id":"symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"use","occurrences":[0,14,19,21,22,24,25,30,34,38,39,40,41,49,50,52,53,60,68,69,70,88,92,98,99,100,106,110,115,116,123,124,125]},{"id":"symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"use","occurrences":[28,30,32,34,38,41]},{"id":"symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"use","occurrences":[0,4]},{"id":"symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","occurrences":[12,24,45,48,49,53,65,67,68,69,70,74,80,81]},{"id":"symbol-3c08d2cf77d38259","tex":"a^*","aliases":"a^*","display":"a^*","kind":"use","occurrences":[80,81]},{"id":"symbol-993220b6cfa76019","tex":"A_0","aliases":"A_0","display":"A_0","kind":"use","occurrences":[38]},{"id":"symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"use","occurrences":[38]},{"id":"symbol-574cc7908da88025","tex":"A_i","aliases":"A_i","display":"A_i","kind":"use","occurrences":[41]},{"id":"symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","occurrences":[29,30,34,36]},{"id":"symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"use","occurrences":[0,6]},{"id":"symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","occurrences":[88,89,92,94,98,101,103,106,109,113,117,124,125,126,128,129]},{"id":"symbol-4f135c4d5c93e6a2","tex":"\\bar q","aliases":"bar q bar","display":"\\bar q","kind":"use","occurrences":[50,52,54]},{"id":"symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","occurrences":[27,30,31,33,34]},{"id":"symbol-ab8bde9a479f56a0","tex":"\\mathfrak C","aliases":"C mathfrak","display":"\\mathfrak C","kind":"use","occurrences":[0]},{"id":"symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","occurrences":[34]},{"id":"symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"use","occurrences":[64,68,77,83,85]},{"id":"symbol-86bc40a9dfa0a3bf","tex":"d_0","aliases":"d_0","display":"d_0","kind":"use","occurrences":[64,68,77]},{"id":"symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","occurrences":[58,60]},{"id":"symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"use","occurrences":[0,5,60]},{"id":"symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","occurrences":[63,68,75]},{"id":"symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","occurrences":[12,14,24,25,49,52,53]},{"id":"symbol-366584274fbf8700","tex":"\\F","aliases":"F mathbb","display":"{\\mathbb F}","kind":"use","occurrences":[88]},{"id":"symbol-62198a97093af5d0","tex":"\\F_2^3","aliases":"F_2^3 F mathbb","display":"{\\mathbb F}_2^3","kind":"use","occurrences":[88]},{"id":"symbol-ff53476e6fc124f9","tex":"F_a","aliases":"F_a","display":"F_a","kind":"use","occurrences":[12,24,49,53]},{"id":"symbol-52bb9700d630070f","tex":"F_w","aliases":"F_w","display":"F_w","kind":"use","occurrences":[14,25,52]},{"id":"symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","occurrences":[41,45,49,53,100]},{"id":"symbol-8d428b98976a03b3","tex":"G_a","aliases":"G_a","display":"G_a","kind":"use","occurrences":[45,49,53]},{"id":"symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","occurrences":[48,49,52,53]},{"id":"symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","occurrences":[9]},{"id":"symbol-047a2d7f6191ab3d","tex":"H_a","aliases":"H_a","display":"H_a","kind":"use","occurrences":[48,49,53]},{"id":"symbol-480c877d455d2c2b","tex":"h_t","aliases":"h_t","display":"h_t","kind":"use","occurrences":[9]},{"id":"symbol-bc0529a14c9530dd","tex":"H_w","aliases":"H_w","display":"H_w","kind":"use","occurrences":[52]},{"id":"symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","occurrences":[18,19,22,41,97,98]},{"id":"symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","occurrences":[66,68,77]},{"id":"symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","occurrences":[42,43]},{"id":"symbol-6f4dd664f5dbbd2b","tex":"\\kerop","aliases":"kerop ker operatorname","display":"{\\operatorname{ker}}","kind":"use","occurrences":[31]},{"id":"symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","occurrences":[14]},{"id":"symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","occurrences":[21]},{"id":"symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","occurrences":[68]},{"id":"symbol-d472066b913c5d84","tex":"\\overline F","aliases":"overline F overline","display":"\\overline F","kind":"use","occurrences":[24,25]},{"id":"symbol-cd1227e8872d5282","tex":"\\overline F_a","aliases":"overline F_a overline","display":"\\overline F_a","kind":"use","occurrences":[24]},{"id":"symbol-6d0e1a1d5f7803a0","tex":"\\overline F_w","aliases":"overline F_w overline","display":"\\overline F_w","kind":"use","occurrences":[25]},{"id":"symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"use","occurrences":[17,19,22,24,25,26,39,41]},{"id":"symbol-ffc41bdcb0951586","tex":"\\mathcal Q","aliases":"Q mathcal","display":"\\mathcal Q","kind":"use","occurrences":[0,3]},{"id":"symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","occurrences":[13,14,40,50,52]},{"id":"symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","occurrences":[68,69,73,81,97,103,127,130,131]},{"id":"symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"use","occurrences":[37,83,85]},{"id":"symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","occurrences":[15,16,18,19,21,22,44,49,50,52,53,55]},{"id":"symbol-15bd74f3637fba8d","tex":"\\mathbb R^3","aliases":"R^3 mathbb","display":"\\mathbb R^3","kind":"use","occurrences":[37]},{"id":"symbol-855b51431c459bf3","tex":"\\mathbb R^d","aliases":"R^d mathbb","display":"\\mathbb R^d","kind":"use","occurrences":[83]},{"id":"symbol-39ee75563d2546f5","tex":"\\mathbb R^d_{\\geq0}","aliases":"R^d_≥0 mathbb geq","display":"\\mathbb R^d_{\\geq0}","kind":"use","occurrences":[85]},{"id":"symbol-236f80d54e55c576","tex":"R_*","aliases":"R_*","display":"R_*","kind":"use","occurrences":[127,130,131]},{"id":"symbol-baf9ce2c27891e4e","tex":"R_b","aliases":"R_b","display":"R_b","kind":"use","occurrences":[103]},{"id":"symbol-00c00f8247bf728a","tex":"R_i","aliases":"R_i","display":"R_i","kind":"use","occurrences":[97]},{"id":"symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","occurrences":[92,99,106,108,110,112]},{"id":"symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","occurrences":[47,49,50,52,53,56,110,111]},{"id":"symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","occurrences":[8,9,10,34,35,36,59,60]},{"id":"symbol-985b4ee40a8bb177","tex":"\\top","aliases":"top","display":"\\top","kind":"use","occurrences":[84]},{"id":"symbol-b3f5e05f76a79869","tex":"\\TV","aliases":"TV d_TV mathrm","display":"{d_{\\mathrm{TV}}}","kind":"use","occurrences":[63]},{"id":"symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","occurrences":[34,35,36,88,91,96,104,105,114]},{"id":"symbol-a6ab3f6998c3dac0","tex":"u_t","aliases":"u_t","display":"u_t","kind":"use","occurrences":[34,35,36]},{"id":"symbol-bb8ce9f4ae8d655a","tex":"u_{t+1}","aliases":"u_t+1","display":"u_{t+1}","kind":"use","occurrences":[34]},{"id":"symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","occurrences":[57,58,59,60]},{"id":"symbol-6c8c1adb5fb270ca","tex":"\\mathcal V","aliases":"V mathcal","display":"\\mathcal V","kind":"use","occurrences":[0,7]},{"id":"symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","occurrences":[47]},{"id":"symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","occurrences":[14,25,51,52,84,85]},{"id":"symbol-25a84a8fd44f7f54","tex":"w^\\top","aliases":"w^top top","display":"w^\\top","kind":"use","occurrences":[84]},{"id":"symbol-1b7b2107b59db7f3","tex":"\\widehat a","aliases":"widehat a widehat","display":"\\widehat a","kind":"use","occurrences":[79,81]},{"id":"symbol-967c63ebf4cd78be","tex":"\\widehat R","aliases":"widehat R widehat","display":"\\widehat R","kind":"use","occurrences":[68,70,71,81]},{"id":"symbol-51fdd71f6572f7c6","tex":"\\widehat y","aliases":"widehat y widehat","display":"\\widehat y","kind":"use","occurrences":[64,70]},{"id":"symbol-e1c6d4c93a294c7e","tex":"\\widehat y_\\omega","aliases":"widehat y_ω widehat omega","display":"\\widehat y_\\omega","kind":"use","occurrences":[64,70]},{"id":"symbol-b186d96cd6172b72","tex":"\\widehat\\mu","aliases":"widehatμ widehat mu","display":"\\widehat\\mu","kind":"use","occurrences":[62,63,70,74]},{"id":"symbol-dac00a99d5864526","tex":"\\widehat\\mu_\\omega","aliases":"widehatμ_ω widehat mu omega","display":"\\widehat\\mu_\\omega","kind":"use","occurrences":[70,74]},{"id":"symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","occurrences":[11,17,18,27,28,44]},{"id":"symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","occurrences":[8,14,15,16,19,20,21,22,23,24,39,40,43]},{"id":"symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","occurrences":[39,40,43]},{"id":"symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","occurrences":[39,43]},{"id":"symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","occurrences":[39]},{"id":"symbol-1029cb8ad51ed272","tex":"x_t","aliases":"x_t","display":"x_t","kind":"use","occurrences":[8]},{"id":"symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","occurrences":[14,21,64,65,69,74]},{"id":"symbol-38f55764f5bb6924","tex":"y_\\omega","aliases":"y_ω omega","display":"y_\\omega","kind":"use","occurrences":[64,69,74]},{"id":"symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","occurrences":[27,44,46,47]},{"id":"symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","occurrences":[10,34,99,100]},{"id":"symbol-8b24cd3e78c875c1","tex":"z_t","aliases":"z_t","display":"z_t","kind":"use","occurrences":[10,34]},{"id":"symbol-8e5d0000abb9cf07","tex":"z_{t+1}","aliases":"z_t+1","display":"z_{t+1}","kind":"use","occurrences":[34]},{"id":"symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","occurrences":[68]},{"id":"symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","occurrences":[83,84,86]},{"id":"symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","occurrences":[68,72,81]},{"id":"symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","occurrences":[88,90,92,93,95,102,106,107,108,109,110,113,123]},{"id":"symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","occurrences":[61,63,69]},{"id":"symbol-3e08c12818fdd261","tex":"\\mu_\\omega","aliases":"μ_ω mu omega","display":"\\mu_\\omega","kind":"use","occurrences":[69]},{"id":"symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"use","occurrences":[0,2]},{"id":"symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"use","occurrences":[60]},{"id":"symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","occurrences":[0,1,57]},{"id":"symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","occurrences":[64,69,70,74,88]},{"id":"symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","occurrences":[65,69,70,74]},{"id":"symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","occurrences":[17,18,27,28,44,47]},{"id":"symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","occurrences":[58]},{"id":"symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"use","occurrences":[60,65,83,85,88]},{"id":"symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","occurrences":[69,70,74]},{"id":"symbol-ea8a48c6147135b3","tex":"\\sum_\\omega","aliases":"∑_ω sum omega","display":"\\sum_\\omega","kind":"use","occurrences":[69,70]},{"id":"symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","occurrences":[58,60]},{"id":"symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","occurrences":[14,17,18,21]},{"id":"symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"use","occurrences":[63,64,68,81]},{"id":"symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"use","occurrences":[84,85]},{"id":"symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","occurrences":[57]},{"id":"symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","occurrences":[92,106,110]},{"id":"symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","occurrences":[82]}],"contexts":["\\section{A common contract, not a universal architecture}\nFor editorial purposes, write a task contract as\n\\[\n\\mathfrak C=(\\Omega,\\Pi,\\mathcal Q,\\mathcal A,\\mathcal E,\\mathcal B,\\mathcal V).\n\\]\nHere $\\Omega$ is the declared family of possible worlds or models, $\\Pi$ the permitted observation protocols, $\\mathcal Q$ the protected questions, $\\mathcal A$ the admitted actions or operations, $\\mathcal E$ the evidence language, $\\mathcal B$ the resource envelope, and $\\mathcal V$ the validation and authority rules. This tuple is an editorial interface schema. It is not proposed as the unique primitive definition used by every source.","not a universal architecture}\nFor editorial purposes, write a task contract as\n\\[\n\\mathfrak C=(\\Omega,\\Pi,\\mathcal Q,\\mathcal A,\\mathcal E,\\mathcal B,\\mathcal V).\n\\]\nHere $\\Omega$ is the declared family of possible worlds or models, $\\Pi$ the permitted observation protocols, $\\mathcal Q$ the protected questions, $\\mathcal A$ the admitted actions or operations, $\\mathcal E$ the evidence language, $\\mathcal B$ the resource envelope, and $\\mathcal V$ the validation and authority rules. This tuple is an editorial interface schema. It is not proposed as the unique primitive definition used by every source.","ial purposes, write a task contract as\n\\[\n\\mathfrak C=(\\Omega,\\Pi,\\mathcal Q,\\mathcal A,\\mathcal E,\\mathcal B,\\mathcal V).\n\\]\nHere $\\Omega$ is the declared family of possible worlds or models, $\\Pi$ the permitted observation protocols, $\\mathcal Q$ the protected questions, $\\mathcal A$ the admitted actions or operations, $\\mathcal E$ the evidence language, $\\mathcal B$ the resource envelope, and $\\mathcal V$ the validation and authority rules. This tuple is an editorial interface schema. It is not proposed as the unique primitive definition used by every source.","A realization of the contract may carry physical or computational state $x_t$, an observation history $h_t$, a retained predictive state $z_t$, a candidate set or probability record, an operator library, and applicable certificates. An institution adds offices, permissions, claims, resource rights, and revision procedures. The representation of one layer is not automatically the complete state of the others.","Suppose a deterministic state space $X$ has admitted transformations $F_a$ and protected output $q$. Define\n\\[\nx\\sim y\\quad\\Longleftrightarrow\\quad\nq(F_wx)=q(F_wy)\\quad\\text{for every admitted finite word }w.\n\\]\nThe empty word is included. When the operation domain is invariant, this equivalence supports well-defined quotient updates. It identifies which distinctions are needed for all specified continuations, not which distinctions are needed to reproduce every hidden detail. This principle occurs in Sufficient Resolution, Machine Understanding, and Stratified Universality under their respective hypotheses. \\src{SR}\\src{MU}\\src{SU}","The missing link is initialization. Let $r(x)$ be the observer's actual initial record. A quotient may exist mathematically while a decoder from $r(x)$ does not.","\\begin{connection}[Evidence must initialize the predictive class]\nLet $p:X\\to X/{\\sim}$ be the behavioral quotient above. There exists an initializer $i:r(X)\\to X/{\\sim}$ with $i(r(x))=p(x)$ for every $x$ if and only if\n\\[\nr(x)=r(y)\\quad\\Longrightarrow\\quad x\\sim y.\n\\]\nWhen this holds, the initialized quotient predicts every declared finite continuation using its induced updates.\n\\end{connection}\n\\begin{proof}\nNecessity follows because a function has one value on an equal input. For sufficiency, assign $i(r(x))=p(x)$; the hypothesis makes the assignment independent of the representative $x$. The quotient update $\\overline F_a(p(x))=p(F_ax)$ is well-defined because behavioral equivalence is preserved by prefixing an action. Induction on word length gives $pF_w=\\overline F_wp$,","\\begin{connection}[Evidence must initialize the predictive class]\nLet $p:X\\to X/{\\sim}$ be the behavioral quotient above. There exists an initializer $i:r(X)\\to X/{\\sim}$ with $i(r(x))=p(x)$ for every $x$ if and only if\n\\[\nr(x)=r(y)\\quad\\Longrightarrow\\quad x\\sim y.\n\\]\nWhen this holds, the initialized quotient predicts every declared finite continuation using its induced updates.\n\\end{connection}\n\\begin{proof}\nNecessity follows because a function has one value on an equal input. For sufficiency, assign $i(r(x))=p(x)$; the hypothesis makes the assignment independent of the representative $x$. The quotient update $\\overline F_a(p(x))=p(F_ax)$ is well-defined because behavioral equivalence is preserved by prefixing an action. Induction on word length gives $pF_w=\\overline F_wp$, and the output factors through $p$. Thus the initialized quotient returns the prot","\\begin{connection}[Evidence must initialize the predictive class]\nLet $p:X\\to X/{\\sim}$ be the behavioral quotient above. There exists an initializer $i:r(X)\\to X/{\\sim}$ with $i(r(x))=p(x)$ for every $x$ if and only if\n\\[\nr(x)=r(y)\\quad\\Longrightarrow\\quad x\\sim y.\n\\]\nWhen this holds, the initialized quotient predicts every declared finite continuation using its induced updates.\n\\end{connection}\n\\begin{proof}\nNecessity follows because a function has one value on an equal input. For sufficiency, assign $i(r(x))=p(x)$; the hypothesis makes the assignment independent of the representative $x$. The quotient update $\\overline F_a(p(x))=p(F_ax)$ is well-defined because behavioral equivalence is preserved by prefixing an action. Induction on word length gives $pF_w=\\overline F_wp$, and the output factors through $p$. Thus the initialized quotient returns the protected output after e","\\begin{connection}[Evidence must initialize the predictive class]\nLet $p:X\\to X/{\\sim}$ be the behavioral quotient above. There exists an initializer $i:r(X)\\to X/{\\sim}$ with $i(r(x))=p(x)$ for every $x$ if and only if\n\\[\nr(x)=r(y)\\quad\\Longrightarrow\\quad x\\sim y.\n\\]\nWhen this holds, the initialized quotient predicts every declared finite continuation using its induced updates.\n\\end{connection}\n\\begin{proof}\nNecessity follows because a function has one value on an equal input. For sufficiency, assign $i(r(x))=p(x)$; the hypothesis makes the assignment independent of the representative $x$. The quotient update $\\overline F_a(p(x))=p(F_ax)$ is well-defined because behavioral equivalence is preserved by prefixing an action. Induction on word length gives $pF_w=\\overline F_wp$, and the output factors through $p$. Thus the initialized quotient returns the protected output after every word.\n\\en","\\begin{connection}[Evidence must initialize the predictive class]\nLet $p:X\\to X/{\\sim}$ be the behavioral quotient above. There exists an initializer $i:r(X)\\to X/{\\sim}$ with $i(r(x))=p(x)$ for every $x$ if and only if\n\\[\nr(x)=r(y)\\quad\\Longrightarrow\\quad x\\sim y.\n\\]\nWhen this holds, the initialized quotient predicts every declared finite continuation using its induced updates.\n\\end{connection}\n\\begin{proof}\nNecessity follows because a function has one value on an equal input. For sufficiency, assign $i(r(x))=p(x)$; the hypothesis makes the assignment independent of the representative $x$. The quotient update $\\overline F_a(p(x))=p(F_ax)$ is well-defined because behavioral equivalence is preserved by prefixing an action. Induction on word length gives $pF_w=\\overline F_wp$, and the output factors through $p$. Thus the initialized quotient returns the protected output after every word.\n\\end{proof}","tient above. There exists an initializer $i:r(X)\\to X/{\\sim}$ with $i(r(x))=p(x)$ for every $x$ if and only if\n\\[\nr(x)=r(y)\\quad\\Longrightarrow\\quad x\\sim y.\n\\]\nWhen this holds, the initialized quotient predicts every declared finite continuation using its induced updates.\n\\end{connection}\n\\begin{proof}\nNecessity follows because a function has one value on an equal input. For sufficiency, assign $i(r(x))=p(x)$; the hypothesis makes the assignment independent of the representative $x$. The quotient update $\\overline F_a(p(x))=p(F_ax)$ is well-defined because behavioral equivalence is preserved by prefixing an action. Induction on word length gives $pF_w=\\overline F_wp$, and the output factors through $p$. Thus the initialized quotient returns the protected output after every word.\n\\end{proof}","every $x$ if and only if\n\\[\nr(x)=r(y)\\quad\\Longrightarrow\\quad x\\sim y.\n\\]\nWhen this holds, the initialized quotient predicts every declared finite continuation using its induced updates.\n\\end{connection}\n\\begin{proof}\nNecessity follows because a function has one value on an equal input. For sufficiency, assign $i(r(x))=p(x)$; the hypothesis makes the assignment independent of the representative $x$. The quotient update $\\overline F_a(p(x))=p(F_ax)$ is well-defined because behavioral equivalence is preserved by prefixing an action. Induction on word length gives $pF_w=\\overline F_wp$, and the output factors through $p$. Thus the initialized quotient returns the protected output after every word.\n\\end{proof}","\\[\nr(x)=r(y)\\quad\\Longrightarrow\\quad x\\sim y.\n\\]\nWhen this holds, the initialized quotient predicts every declared finite continuation using its induced updates.\n\\end{connection}\n\\begin{proof}\nNecessity follows because a function has one value on an equal input. For sufficiency, assign $i(r(x))=p(x)$; the hypothesis makes the assignment independent of the representative $x$. The quotient update $\\overline F_a(p(x))=p(F_ax)$ is well-defined because behavioral equivalence is preserved by prefixing an action. Induction on word length gives $pF_w=\\overline F_wp$, and the output factors through $p$. Thus the initialized quotient returns the protected output after every word.\n\\end{proof}","induced updates.\n\\end{connection}\n\\begin{proof}\nNecessity follows because a function has one value on an equal input. For sufficiency, assign $i(r(x))=p(x)$; the hypothesis makes the assignment independent of the representative $x$. The quotient update $\\overline F_a(p(x))=p(F_ax)$ is well-defined because behavioral equivalence is preserved by prefixing an action. Induction on word length gives $pF_w=\\overline F_wp$, and the output factors through $p$. Thus the initialized quotient returns the protected output after every word.\n\\end{proof}","sity follows because a function has one value on an equal input. For sufficiency, assign $i(r(x))=p(x)$; the hypothesis makes the assignment independent of the representative $x$. The quotient update $\\overline F_a(p(x))=p(F_ax)$ is well-defined because behavioral equivalence is preserved by prefixing an action. Induction on word length gives $pF_w=\\overline F_wp$, and the output factors through $p$. Thus the initialized quotient returns the protected output after every word.\n\\end{proof}","For a linear map $C:X\\to Z$ and a known operation $A:X\\to X$, an induced operation $B$ satisfying $CA=BC$ exists precisely when $\\kerop C$ is invariant under $A$, with the codomain restricted to the attained image if necessary. The condition says that two inputs collapsed by $C$ remain collapsed after the operation. If it fails, the discarded distinction can return through the retained state.","This failure has a concrete block form. In coordinates separating retained and hidden state,\n\\[\n\\begin{pmatrix}z_{t+1}\\\\u_{t+1}\\end{pmatrix}\n=\n\\begin{pmatrix}A&B\\\\C&D\\end{pmatrix}\n\\begin{pmatrix}z_t\\\\u_t\\end{pmatrix}.\n\\]\nEven if $u_t$ has small norm, the effect $Bu_t$ may be large. Retaining less state is therefore not justified by omitted magnitude alone. A predictive error bound needs a sensitivity or propagation bound. Sufficient Resolution studies hidden-return defects and exact memory alternatives. Large Signal Model Foundations places those mechanisms inside its refinement and predictive-closure account. \\src{SR}\\src{LSM}","u_t\\end{pmatrix}.\n\\]\nEven if $u_t$ has small norm, the effect $Bu_t$ may be large. Retaining less state is therefore not justified by omitted magnitude alone. A predictive error bound needs a sensitivity or propagation bound. Sufficient Resolution studies hidden-return defects and exact memory alternatives. Large Signal Model Foundations places those mechanisms inside its refinement and predictive-closure account. \\src{SR}\\src{LSM}","\\subsection{One common quotient across uncertain dynamics}\nConsider two possible linear realizations on $\\mathbb R^3$,\n\\[\nA_0=\\begin{pmatrix}1&1&0","\\subsection{One common quotient across uncertain dynamics}\nConsider two possible linear realizations on $\\mathbb R^3$,\n\\[\nA_0=\\begin{pmatrix}1&1&0\\\\0&1&0\\\\0&0&2\\end{pmatrix},\\qquad\nA_1=\\begin{pmatrix}1&1&0\\\\0&1&0\\\\1&0&3\\end{pmatrix}.\n\\]\nLet $p(x_1,x_2,x_3)=(x_1,x_2)$ and let the output be $q(x)=x_1$. Both operators satisfy\n\\[\npA_i=Gp,\\qquad G=\\begin{pmatrix}1&1","1&0&3\\end{pmatrix}.\n\\]\nLet $p(x_1,x_2,x_3)=(x_1,x_2)$ and let the output be $q(x)=x_1$. Both operators satisfy\n\\[\npA_i=Gp,\\qquad G=\\begin{pmatrix}1&1","1&0&3\\end{pmatrix}.\n\\]\nLet $p(x_1,x_2,x_3)=(x_1,x_2)$ and let the output be $q(x)=x_1$. Both operators satisfy\n\\[\npA_i=Gp,\\qquad G=\\begin{pmatrix}1&1\\\\0&1\\end{pmatrix}.\n\\]\nConsequently the output after $k$ operations is $x_1+kx_2$ for either fixed model. In this particular example it also holds for arbitrary switching, since the same local intertwining equation holds for both operations. The uncertain third-coordinate dynamics remain irrelevant to the protected output.","0&1\\end{pmatrix}.\n\\]\nConsequently the output after $k$ operations is $x_1+kx_2$ for either fixed model. In this particular example it also holds for arbitrary switching, since the same local intertwining equation holds for both operations. The uncertain third-coordinate dynamics remain irrelevant to the protected output.","For a common invariant domain, let $r:X\\to Z$ encode the relevant state, let $G_a$ act on $Z$, and let $T:Z\\to W$ translate that state into another representation with operations $H_a$. The desired equations are\n\\[\nrF_a=G_ar,\\qquad TG_a=H_aT.\n\\]\nThe first connects the source process to its representation. The second connects that representation to the recipient interface.","\\begin{connection}[Two local bridges compose over finite continuations]\nAssume the domains used by both equations are preserved under all admitted operations and that a protected output has the form $q=\\bar q\\,Tr$. Then, for every admitted finite word $w$,\n\\[\nTrF_w=H_wTr,\\qquad qF_w=\\bar q\\,H_wTr.\n\\]\n\\end{connection}\n\\begin{proof}\nFor one operation, $TrF_a=TG_ar=H_aTr$. The empty word gives the identity. Applying the one-operation relation successively proves the first equation by induction, and composing with $\\bar q$ proves the second. Domain invariance ensures that every successive application is licensed.\n\\end{proof}","preserved under all admitted operations and that a protected output has the form $q=\\bar q\\,Tr$. Then, for every admitted finite word $w$,\n\\[\nTrF_w=H_wTr,\\qquad qF_w=\\bar q\\,H_wTr.\n\\]\n\\end{connection}\n\\begin{proof}\nFor one operation, $TrF_a=TG_ar=H_aTr$. The empty word gives the identity. Applying the one-operation relation successively proves the first equation by induction, and composing with $\\bar q$ proves the second. Domain invariance ensures that every successive application is licensed.\n\\end{proof}","The result is a composition of already familiar intertwining arguments. It states exactly what would be preserved by a correct learned semantic bridge. It does not construct $r$ or $T$, authenticate causal use of the selected carrier, or bound the translator's complexity. Those remain separate obligations. A powerful interpreter that solves the problem after receiving a trivial label has not supplied a free representation of the source mechanism. Machine Understanding makes recipient programs and their installed information explicit for this reason. \\src{MU}","For exact candidate elimination, a current set $V\\subseteq\\Omega$ is corrected by $V\\mapsto V\\cap E$. If the target observation is $t(V)$, a future-correction signature records\n\\[\n\\Sigma(V)=\\bigl(t(V\\cap E)\\bigr)_{E\\in\\mathcal E}.\n\\]\nUnder the source's closure hypotheses, this captures the distinctions needed to answer after every permitted filter. The empty set matters: recognizing that all candidates have been contradicted is part of a correct update, not an exceptional case to omit from the theorem. \\src{GA}","\\begin{connection}[A common decision bound]\nLet $\\mu$ and $\\widehat\\mu$ be probability measures on the same finite candidate set with $\\TV(\\mu,\\widehat\\mu)\\leq e$. Suppose true and approximate responses satisfy $d(y_\\omega,\\widehat y_\\omega)\\leq d_0$ for all retained candidates. If $\\ell(a,y)\\in[0,1]$ is $K$-Lipschitz in its second argument, then for every admissible action $a$,\n\\[\n|R(a)-\\widehat R(a)|\\leq\\eta:=\\min\\{1,e+Kd_0\\},\n\\]\nwhere $R(a)=\\sum_\\omega\\mu_\\omega\\ell(a,y_\\omega)$ and $\\widehat R(a)=\\sum_\\omega\\widehat\\mu_\\omega\\ell(a,\\widehat y_\\omega)$. An action minimizing $\\widehat R$ has regret at most $2\\eta$ relative to a minimizer of $R$ when the action set is finite and nonempty.\n\\end{connection}\n\\begin{proof}\nInsert the intermediat","\\begin{connection}[A common decision bound]\nLet $\\mu$ and $\\widehat\\mu$ be probability measures on the same finite candidate set with $\\TV(\\mu,\\widehat\\mu)\\leq e$. Suppose true and approximate responses satisfy $d(y_\\omega,\\widehat y_\\omega)\\leq d_0$ for all retained candidates. If $\\ell(a,y)\\in[0,1]$ is $K$-Lipschitz in its second argument, then for every admissible action $a$,\n\\[\n|R(a)-\\widehat R(a)|\\leq\\eta:=\\min\\{1,e+Kd_0\\},\n\\]\nwhere $R(a)=\\sum_\\omega\\mu_\\omega\\ell(a,y_\\omega)$ and $\\widehat R(a)=\\sum_\\omega\\widehat\\mu_\\omega\\ell(a,\\widehat y_\\omega)$. An action minimizing $\\widehat R$ has regret at most $2\\eta$ relative to a minimizer of $R$ when the action set is finite and nonempty.\n\\end{connection}\n\\begin{proof}\nInsert the intermediate expectation $\\su","\\begin{connection}[A common decision bound]\nLet $\\mu$ and $\\widehat\\mu$ be probability measures on the same finite candidate set with $\\TV(\\mu,\\widehat\\mu)\\leq e$. Suppose true and approximate responses satisfy $d(y_\\omega,\\widehat y_\\omega)\\leq d_0$ for all retained candidates. If $\\ell(a,y)\\in[0,1]$ is $K$-Lipschitz in its second argument, then for every admissible action $a$,\n\\[\n|R(a)-\\widehat R(a)|\\leq\\eta:=\\min\\{1,e+Kd_0\\},\n\\]\nwhere $R(a)=\\sum_\\omega\\mu_\\omega\\ell(a,y_\\omega)$ and $\\widehat R(a)=\\sum_\\omega\\widehat\\mu_\\omega\\ell(a,\\widehat y_\\omega)$. An action minimizing $\\widehat R$ has regret at most $2\\eta$ relative to a minimizer of $R$ when the action set is finite and nonempty.\n\\end{connection}\n\\begin{proof}\nInsert the intermediate expectation $\\sum\\widehat\\mu_\\omega\\ell(a,y_\\omega)$. The first difference is at most $e$ because the integ","\\begin{connection}[A common decision bound]\nLet $\\mu$ and $\\widehat\\mu$ be probability measures on the same finite candidate set with $\\TV(\\mu,\\widehat\\mu)\\leq e$. Suppose true and approximate responses satisfy $d(y_\\omega,\\widehat y_\\omega)\\leq d_0$ for all retained candidates. If $\\ell(a,y)\\in[0,1]$ is $K$-Lipschitz in its second argument, then for every admissible action $a$,\n\\[\n|R(a)-\\widehat R(a)|\\leq\\eta:=\\min\\{1,e+Kd_0\\},\n\\]\nwhere $R(a)=\\sum_\\omega\\mu_\\omega\\ell(a,y_\\omega)$ and $\\widehat R(a)=\\sum_\\omega\\widehat\\mu_\\omega\\ell(a,\\widehat y_\\omega)$. An action minimizing $\\widehat R$ has regret at most $2\\eta$ relative to a minimizer of $R$ when the action set is finite and nonempty.\n\\end{connection}\n\\begin{proof}\nInsert the intermediate expectation $\\sum\\widehat\\mu_\\omega\\ell(a,y_\\omega)$. The first difference is at most $e$ because the integrand is in $[0,1]$. The second is at most $Kd_0$. Risks themselves lie in $[0,1]$, givin","\\begin{connection}[A common decision bound]\nLet $\\mu$ and $\\widehat\\mu$ be probability measures on the same finite candidate set with $\\TV(\\mu,\\widehat\\mu)\\leq e$. Suppose true and approximate responses satisfy $d(y_\\omega,\\widehat y_\\omega)\\leq d_0$ for all retained candidates. If $\\ell(a,y)\\in[0,1]$ is $K$-Lipschitz in its second argument, then for every admissible action $a$,\n\\[\n|R(a)-\\widehat R(a)|\\leq\\eta:=\\min\\{1,e+Kd_0\\},\n\\]\nwhere $R(a)=\\sum_\\omega\\mu_\\omega\\ell(a,y_\\omega)$ and $\\widehat R(a)=\\sum_\\omega\\widehat\\mu_\\omega\\ell(a,\\widehat y_\\omega)$. An action minimizing $\\widehat R$ has regret at most $2\\eta$ relative to a minimizer of $R$ when the action set is finite and nonempty.\n\\end{connection}\n\\begin{proof}\nInsert the intermediate expectation $\\sum\\widehat\\mu_\\omega\\ell(a,y_\\omega)$. The first difference is at most $e$ because the integrand is in $[0,1]$. The second is at most $Kd_0$. Risks themselves lie in $[0,1]$, giving the cap. If $\\widehat a$ and $a^*$ minimize the ap","\\begin{connection}[A common decision bound]\nLet $\\mu$ and $\\widehat\\mu$ be probability measures on the same finite candidate set with $\\TV(\\mu,\\widehat\\mu)\\leq e$. Suppose true and approximate responses satisfy $d(y_\\omega,\\widehat y_\\omega)\\leq d_0$ for all retained candidates. If $\\ell(a,y)\\in[0,1]$ is $K$-Lipschitz in its second argument, then for every admissible action $a$,\n\\[\n|R(a)-\\widehat R(a)|\\leq\\eta:=\\min\\{1,e+Kd_0\\},\n\\]\nwhere $R(a)=\\sum_\\omega\\mu_\\omega\\ell(a,y_\\omega)$ and $\\widehat R(a)=\\sum_\\omega\\widehat\\mu_\\omega\\ell(a,\\widehat y_\\omega)$. An action minimizing $\\widehat R$ has regret at most $2\\eta$ relative to a minimizer of $R$ when the action set is finite and nonempty.\n\\end{connection}\n\\begin{proof}\nInsert the intermediate expectation $\\sum\\widehat\\mu_\\omega\\ell(a,y_\\omega)$. The first difference is at most $e$ because the integrand is in $[0,1]$. The second is at most $Kd_0$. Risks themselves lie in $[0,1]$, giving the cap. If $\\widehat a$ and $a^*$ minimize the approxima","\\begin{connection}[A common decision bound]\nLet $\\mu$ and $\\widehat\\mu$ be probability measures on the same finite candidate set with $\\TV(\\mu,\\widehat\\mu)\\leq e$. Suppose true and approximate responses satisfy $d(y_\\omega,\\widehat y_\\omega)\\leq d_0$ for all retained candidates. If $\\ell(a,y)\\in[0,1]$ is $K$-Lipschitz in its second argument, then for every admissible action $a$,\n\\[\n|R(a)-\\widehat R(a)|\\leq\\eta:=\\min\\{1,e+Kd_0\\},\n\\]\nwhere $R(a)=\\sum_\\omega\\mu_\\omega\\ell(a,y_\\omega)$ and $\\widehat R(a)=\\sum_\\omega\\widehat\\mu_\\omega\\ell(a,\\widehat y_\\omega)$. An action minimizing $\\widehat R$ has regret at most $2\\eta$ relative to a minimizer of $R$ when the action set is finite and nonempty.\n\\end{connection}\n\\begin{proof}\nInsert the intermediate expectation $\\sum\\widehat\\mu_\\omega\\ell(a,y_\\omega)$. The first difference is at most $e$ because the integrand is in $[0,1]$. The second is at most $Kd_0$. Risks themselves lie in $[0,1]$, giving the cap. If $\\widehat a$ and $a^*$ minimize the approximate and exact risks, respectively, then $R(\\widehat a)\\leq\\widehat R(\\wi","\\begin{connection}[A common decision bound]\nLet $\\mu$ and $\\widehat\\mu$ be probability measures on the same finite candidate set with $\\TV(\\mu,\\widehat\\mu)\\leq e$. Suppose true and approximate responses satisfy $d(y_\\omega,\\widehat y_\\omega)\\leq d_0$ for all retained candidates. If $\\ell(a,y)\\in[0,1]$ is $K$-Lipschitz in its second argument, then for every admissible action $a$,\n\\[\n|R(a)-\\widehat R(a)|\\leq\\eta:=\\min\\{1,e+Kd_0\\},\n\\]\nwhere $R(a)=\\sum_\\omega\\mu_\\omega\\ell(a,y_\\omega)$ and $\\widehat R(a)=\\sum_\\omega\\widehat\\mu_\\omega\\ell(a,\\widehat y_\\omega)$. An action minimizing $\\widehat R$ has regret at most $2\\eta$ relative to a minimizer of $R$ when the action set is finite and nonempty.\n\\end{connection}\n\\begin{proof}\nInsert the intermediate expectation $\\sum\\widehat\\mu_\\omega\\ell(a,y_\\omega)$. The first difference is at most $e$ because the integrand is in $[0,1]$. The second is at most $Kd_0$. Risks themselves lie in $[0,1]$, giving the cap. If $\\widehat a$ and $a^*$ minimize the approximate and exact risks, respectively, then $R(\\widehat a)\\leq\\widehat R(\\widehat a)+\\eta\\leq\\wid…","Let $\\mu$ and $\\widehat\\mu$ be probability measures on the same finite candidate set with $\\TV(\\mu,\\widehat\\mu)\\leq e$. Suppose true and approximate responses satisfy $d(y_\\omega,\\widehat y_\\omega)\\leq d_0$ for all retained candidates. If $\\ell(a,y)\\in[0,1]$ is $K$-Lipschitz in its second argument, then for every admissible action $a$,\n\\[\n|R(a)-\\widehat R(a)|\\leq\\eta:=\\min\\{1,e+Kd_0\\},\n\\]\nwhere $R(a)=\\sum_\\omega\\mu_\\omega\\ell(a,y_\\omega)$ and $\\widehat R(a)=\\sum_\\omega\\widehat\\mu_\\omega\\ell(a,\\widehat y_\\omega)$. An action minimizing $\\widehat R$ has regret at most $2\\eta$ relative to a minimizer of $R$ when the action set is finite and nonempty.\n\\end{connection}\n\\begin{proof}\nInsert the intermediate expectation $\\sum\\widehat\\mu_\\omega\\ell(a,y_\\omega)$. The first difference is at most $e$ because the integrand is in $[0,1]$. The second is at most $Kd_0$. Risks themselves lie in $[0,1]$, giving the cap. If $\\widehat a$ and $a^*$ minimize the approximate and exact risks, respectively, then $R(\\widehat a)\\leq\\widehat R(\\widehat a)+\\eta\\leq\\widehat R(a^*)+\\eta\\leq R(a^*)+2\\eta$.\n\\end{pro…","res on the same finite candidate set with $\\TV(\\mu,\\widehat\\mu)\\leq e$. Suppose true and approximate responses satisfy $d(y_\\omega,\\widehat y_\\omega)\\leq d_0$ for all retained candidates. If $\\ell(a,y)\\in[0,1]$ is $K$-Lipschitz in its second argument, then for every admissible action $a$,\n\\[\n|R(a)-\\widehat R(a)|\\leq\\eta:=\\min\\{1,e+Kd_0\\},\n\\]\nwhere $R(a)=\\sum_\\omega\\mu_\\omega\\ell(a,y_\\omega)$ and $\\widehat R(a)=\\sum_\\omega\\widehat\\mu_\\omega\\ell(a,\\widehat y_\\omega)$. An action minimizing $\\widehat R$ has regret at most $2\\eta$ relative to a minimizer of $R$ when the action set is finite and nonempty.\n\\end{connection}\n\\begin{proof}\nInsert the intermediate expectation $\\sum\\widehat\\mu_\\omega\\ell(a,y_\\omega)$. The first difference is at most $e$ because the integrand is in $[0,1]$. The second is at most $Kd_0$. Risks themselves lie in $[0,1]$, giving the cap. If $\\widehat a$ and $a^*$ minimize the approximate and exact risks, respectively, then $R(\\widehat a)\\leq\\widehat R(\\widehat a)+\\eta\\leq\\widehat R(a^*)+\\eta\\leq R(a^*)+2\\eta$.\n\\end{proof}","oximate responses satisfy $d(y_\\omega,\\widehat y_\\omega)\\leq d_0$ for all retained candidates. If $\\ell(a,y)\\in[0,1]$ is $K$-Lipschitz in its second argument, then for every admissible action $a$,\n\\[\n|R(a)-\\widehat R(a)|\\leq\\eta:=\\min\\{1,e+Kd_0\\},\n\\]\nwhere $R(a)=\\sum_\\omega\\mu_\\omega\\ell(a,y_\\omega)$ and $\\widehat R(a)=\\sum_\\omega\\widehat\\mu_\\omega\\ell(a,\\widehat y_\\omega)$. An action minimizing $\\widehat R$ has regret at most $2\\eta$ relative to a minimizer of $R$ when the action set is finite and nonempty.\n\\end{connection}\n\\begin{proof}\nInsert the intermediate expectation $\\sum\\widehat\\mu_\\omega\\ell(a,y_\\omega)$. The first difference is at most $e$ because the integrand is in $[0,1]$. The second is at most $Kd_0$. Risks themselves lie in $[0,1]$, giving the cap. If $\\widehat a$ and $a^*$ minimize the approximate and exact risks, respectively, then $R(\\widehat a)\\leq\\widehat R(\\widehat a)+\\eta\\leq\\widehat R(a^*)+\\eta\\leq R(a^*)+2\\eta$.\n\\end{proof}","omega,\\widehat y_\\omega)\\leq d_0$ for all retained candidates. If $\\ell(a,y)\\in[0,1]$ is $K$-Lipschitz in its second argument, then for every admissible action $a$,\n\\[\n|R(a)-\\widehat R(a)|\\leq\\eta:=\\min\\{1,e+Kd_0\\},\n\\]\nwhere $R(a)=\\sum_\\omega\\mu_\\omega\\ell(a,y_\\omega)$ and $\\widehat R(a)=\\sum_\\omega\\widehat\\mu_\\omega\\ell(a,\\widehat y_\\omega)$. An action minimizing $\\widehat R$ has regret at most $2\\eta$ relative to a minimizer of $R$ when the action set is finite and nonempty.\n\\end{connection}\n\\begin{proof}\nInsert the intermediate expectation $\\sum\\widehat\\mu_\\omega\\ell(a,y_\\omega)$. The first difference is at most $e$ because the integrand is in $[0,1]$. The second is at most $Kd_0$. Risks themselves lie in $[0,1]$, giving the cap. If $\\widehat a$ and $a^*$ minimize the approximate and exact risks, respectively, then $R(\\widehat a)\\leq\\widehat R(\\widehat a)+\\eta\\leq\\widehat R(a^*)+\\eta\\leq R(a^*)+2\\eta$.\n\\end{proof}","or all retained candidates. If $\\ell(a,y)\\in[0,1]$ is $K$-Lipschitz in its second argument, then for every admissible action $a$,\n\\[\n|R(a)-\\widehat R(a)|\\leq\\eta:=\\min\\{1,e+Kd_0\\},\n\\]\nwhere $R(a)=\\sum_\\omega\\mu_\\omega\\ell(a,y_\\omega)$ and $\\widehat R(a)=\\sum_\\omega\\widehat\\mu_\\omega\\ell(a,\\widehat y_\\omega)$. An action minimizing $\\widehat R$ has regret at most $2\\eta$ relative to a minimizer of $R$ when the action set is finite and nonempty.\n\\end{connection}\n\\begin{proof}\nInsert the intermediate expectation $\\sum\\widehat\\mu_\\omega\\ell(a,y_\\omega)$. The first difference is at most $e$ because the integrand is in $[0,1]$. The second is at most $Kd_0$. Risks themselves lie in $[0,1]$, giving the cap. If $\\widehat a$ and $a^*$ minimize the approximate and exact risks, respectively, then $R(\\widehat a)\\leq\\widehat R(\\widehat a)+\\eta\\leq\\widehat R(a^*)+\\eta\\leq R(a^*)+2\\eta$.\n\\end{proof}","le action $a$,\n\\[\n|R(a)-\\widehat R(a)|\\leq\\eta:=\\min\\{1,e+Kd_0\\},\n\\]\nwhere $R(a)=\\sum_\\omega\\mu_\\omega\\ell(a,y_\\omega)$ and $\\widehat R(a)=\\sum_\\omega\\widehat\\mu_\\omega\\ell(a,\\widehat y_\\omega)$. An action minimizing $\\widehat R$ has regret at most $2\\eta$ relative to a minimizer of $R$ when the action set is finite and nonempty.\n\\end{connection}\n\\begin{proof}\nInsert the intermediate expectation $\\sum\\widehat\\mu_\\omega\\ell(a,y_\\omega)$. The first difference is at most $e$ because the integrand is in $[0,1]$. The second is at most $Kd_0$. Risks themselves lie in $[0,1]$, giving the cap. If $\\widehat a$ and $a^*$ minimize the approximate and exact risks, respectively, then $R(\\widehat a)\\leq\\widehat R(\\widehat a)+\\eta\\leq\\widehat R(a^*)+\\eta\\leq R(a^*)+2\\eta$.\n\\end{proof}","$R(a)=\\sum_\\omega\\mu_\\omega\\ell(a,y_\\omega)$ and $\\widehat R(a)=\\sum_\\omega\\widehat\\mu_\\omega\\ell(a,\\widehat y_\\omega)$. An action minimizing $\\widehat R$ has regret at most $2\\eta$ relative to a minimizer of $R$ when the action set is finite and nonempty.\n\\end{connection}\n\\begin{proof}\nInsert the intermediate expectation $\\sum\\widehat\\mu_\\omega\\ell(a,y_\\omega)$. The first difference is at most $e$ because the integrand is in $[0,1]$. The second is at most $Kd_0$. Risks themselves lie in $[0,1]$, giving the cap. If $\\widehat a$ and $a^*$ minimize the approximate and exact risks, respectively, then $R(\\widehat a)\\leq\\widehat R(\\widehat a)+\\eta\\leq\\widehat R(a^*)+\\eta\\leq R(a^*)+2\\eta$.\n\\end{proof}","(a,y_\\omega)$ and $\\widehat R(a)=\\sum_\\omega\\widehat\\mu_\\omega\\ell(a,\\widehat y_\\omega)$. An action minimizing $\\widehat R$ has regret at most $2\\eta$ relative to a minimizer of $R$ when the action set is finite and nonempty.\n\\end{connection}\n\\begin{proof}\nInsert the intermediate expectation $\\sum\\widehat\\mu_\\omega\\ell(a,y_\\omega)$. The first difference is at most $e$ because the integrand is in $[0,1]$. The second is at most $Kd_0$. Risks themselves lie in $[0,1]$, giving the cap. If $\\widehat a$ and $a^*$ minimize the approximate and exact risks, respectively, then $R(\\widehat a)\\leq\\widehat R(\\widehat a)+\\eta\\leq\\widehat R(a^*)+\\eta\\leq R(a^*)+2\\eta$.\n\\end{proof}",")=\\sum_\\omega\\widehat\\mu_\\omega\\ell(a,\\widehat y_\\omega)$. An action minimizing $\\widehat R$ has regret at most $2\\eta$ relative to a minimizer of $R$ when the action set is finite and nonempty.\n\\end{connection}\n\\begin{proof}\nInsert the intermediate expectation $\\sum\\widehat\\mu_\\omega\\ell(a,y_\\omega)$. The first difference is at most $e$ because the integrand is in $[0,1]$. The second is at most $Kd_0$. Risks themselves lie in $[0,1]$, giving the cap. If $\\widehat a$ and $a^*$ minimize the approximate and exact risks, respectively, then $R(\\widehat a)\\leq\\widehat R(\\widehat a)+\\eta\\leq\\widehat R(a^*)+\\eta\\leq R(a^*)+2\\eta$.\n\\end{proof}","ell(a,\\widehat y_\\omega)$. An action minimizing $\\widehat R$ has regret at most $2\\eta$ relative to a minimizer of $R$ when the action set is finite and nonempty.\n\\end{connection}\n\\begin{proof}\nInsert the intermediate expectation $\\sum\\widehat\\mu_\\omega\\ell(a,y_\\omega)$. The first difference is at most $e$ because the integrand is in $[0,1]$. The second is at most $Kd_0$. Risks themselves lie in $[0,1]$, giving the cap. If $\\widehat a$ and $a^*$ minimize the approximate and exact risks, respectively, then $R(\\widehat a)\\leq\\widehat R(\\widehat a)+\\eta\\leq\\widehat R(a^*)+\\eta\\leq R(a^*)+2\\eta$.\n\\end{proof}","n action minimizing $\\widehat R$ has regret at most $2\\eta$ relative to a minimizer of $R$ when the action set is finite and nonempty.\n\\end{connection}\n\\begin{proof}\nInsert the intermediate expectation $\\sum\\widehat\\mu_\\omega\\ell(a,y_\\omega)$. The first difference is at most $e$ because the integrand is in $[0,1]$. The second is at most $Kd_0$. Risks themselves lie in $[0,1]$, giving the cap. If $\\widehat a$ and $a^*$ minimize the approximate and exact risks, respectively, then $R(\\widehat a)\\leq\\widehat R(\\widehat a)+\\eta\\leq\\widehat R(a^*)+\\eta\\leq R(a^*)+2\\eta$.\n\\end{proof}","ng $\\widehat R$ has regret at most $2\\eta$ relative to a minimizer of $R$ when the action set is finite and nonempty.\n\\end{connection}\n\\begin{proof}\nInsert the intermediate expectation $\\sum\\widehat\\mu_\\omega\\ell(a,y_\\omega)$. The first difference is at most $e$ because the integrand is in $[0,1]$. The second is at most $Kd_0$. Risks themselves lie in $[0,1]$, giving the cap. If $\\widehat a$ and $a^*$ minimize the approximate and exact risks, respectively, then $R(\\widehat a)\\leq\\widehat R(\\widehat a)+\\eta\\leq\\widehat R(a^*)+\\eta\\leq R(a^*)+2\\eta$.\n\\end{proof}","of $R$ when the action set is finite and nonempty.\n\\end{connection}\n\\begin{proof}\nInsert the intermediate expectation $\\sum\\widehat\\mu_\\omega\\ell(a,y_\\omega)$. The first difference is at most $e$ because the integrand is in $[0,1]$. The second is at most $Kd_0$. Risks themselves lie in $[0,1]$, giving the cap. If $\\widehat a$ and $a^*$ minimize the approximate and exact risks, respectively, then $R(\\widehat a)\\leq\\widehat R(\\widehat a)+\\eta\\leq\\widehat R(a^*)+\\eta\\leq R(a^*)+2\\eta$.\n\\end{proof}","The combined lesson is neither that every representation change speeds computation nor that compression is futile. It is that the complete path must be accounted for:\n\\[\n\\text{acquire}\\ \\longrightarrow\\ \\text{represent}\\ \\longrightarrow\\ \\text{select}\\ \\longrightarrow\\ \\text{verify}\\ \\longrightarrow\\ \\text{execute}\\ \\longrightarrow\\ \\text{maintain}.\n\\]\nThe arrows can have different units. Model observations, interpreter visits, physical time, peak memory, and stored records should not be summed without an explicit conversion model.","\\begin{connection}[Resource gains belong to a declared ledger]\nFor one fixed workflow and workload, let $\\Delta\\in\\mathbb R^d$ be old-minus-new event counts after all charged setup and maintenance. Then $w^\\top\\Delta\\geq0$ for every $w\\in\\mathbb R^d_{\\geq0}$ if and only if every coordinate of $\\Delta$ is nonnegative. For prices restricted to the convex hull of a finite set, the smallest priced saving is attained at one of its listed points.\n\\end{connection}\n\\begin{proof}\nNonnegative coordinates give a nonnegative scalar product. A negative coordinate is exposed by assigning positive price only to that coordinate. A linear functional at a convex combination is the same combination of its values, which cannot be below the smallest listed value. A listed minimizer attains that value.\n\\end{proof}","This restates the accounting principle in Self-Improving Intelligence. For its recorded weighted-search comparison, the vector $(1552,1552,-1868)$ counts saved model reads, saved additions, and added comparisons. Equal prices give a positive margin; other prices give a loss. Semantic validity does not change with that choice. Neither does a correct price calculation imply that the price is a measured property of a particular processor. \\src{SII}","Let the possible world be\n\\[\n\\omega=(b,\\theta,u)\\in\\F_2^3.\n\\]\nThe bit $b$ determines which of two repairs is appropriate. The bit $\\theta$ is a sensor convention. The nuisance bit $u$ affects none of the admitted observations, transitions, or repairs. A normal reading returns $s=b\\oplus\\theta$. A calibration returns $\\theta$. A public operation flips $b$, leaving $\\theta$ and $u$ unchanged. A repair $R_i$ succeeds precisely when $b=i$.","the possible world be\n\\[\n\\omega=(b,\\theta,u)\\in\\F_2^3.\n\\]\nThe bit $b$ determines which of two repairs is appropriate. The bit $\\theta$ is a sensor convention. The nuisance bit $u$ affects none of the admitted observations, transitions, or repairs. A normal reading returns $s=b\\oplus\\theta$. A calibration returns $\\theta$. A public operation flips $b$, leaving $\\theta$ and $u$ unchanged. A repair $R_i$ succeeds precisely when $b=i$.","a=(b,\\theta,u)\\in\\F_2^3.\n\\]\nThe bit $b$ determines which of two repairs is appropriate. The bit $\\theta$ is a sensor convention. The nuisance bit $u$ affects none of the admitted observations, transitions, or repairs. A normal reading returns $s=b\\oplus\\theta$. A calibration returns $\\theta$. A public operation flips $b$, leaving $\\theta$ and $u$ unchanged. A repair $R_i$ succeeds precisely when $b=i$.","\\subsection{First: a two-state predictor is enough for the display}\nFor the original question family, the task is only to predict the next display after a finite number of flips. The retained state $z=s$ has two possible values, and its update is $G(z)=1-z$. Every future display is determined even though $b$ and $\\theta$ are not separately known. The nuisance bit is irrelevant throughout.","\\subsection{Second: the repair question changes the required distinction}\nWhen the task becomes selecting $R_b$, the old display is insufficient. Worlds $(0,0,u)$ and $(1,1,u)$ have the same display and different successful repairs. Repeating the normal reading changes no information. Calibration supplies the missing convention, after which\n\\[\nb=s\\oplus\\theta.\n\\]\nThe refinement is a genuine information acquisition. Merely adding a field named $\\theta$ to the retained record would not initialize it.","the required distinction}\nWhen the task becomes selecting $R_b$, the old display is insufficient. Worlds $(0,0,u)$ and $(1,1,u)$ have the same display and different successful repairs. Repeating the normal reading changes no information. Calibration supplies the missing convention, after which\n\\[\nb=s\\oplus\\theta.\n\\]\nThe refinement is a genuine information acquisition. Merely adding a field named $\\theta$ to the retained record would not initialize it.","\\subsection{Third: a translation can preserve the method}\nOne recipient stores $(s,\\theta)$ and another stores $(b,\\theta)$. The translator\n\\[\nT(s,\\theta)=(s\\oplus\\theta,\\theta)\n\\]\nis invertible. Both coordinate systems have a well-defined flip operation, and $T$ commutes with those operations. Both decode the same required repair. The exact correspondence gives an elementary realization of the method-transport contract.","The translator still has an access requirement and an execution cost. It cannot translate a record containing only $s$ into the missing pair. It does not infer a sensor convention from a matching variable name. This is the distinction between a valid map on its domain and a procedure that can obtain the map's input.","\\section{Memory, messages, and a changed task}\nFor candidate sets in the eight-world model, evidence depends only on $(b,\\theta)$. There are four nonempty cells, each differing only in $u$. Restrict the evidence language to exactly the unions of those four cells. The occupancy record stores whether the current candidate set meets each cell. It therefore has $2^4=16$ possible states rather than the $2^8=256$ possible complete candidate sets.","ection{Memory, messages, and a changed task}\nFor candidate sets in the eight-world model, evidence depends only on $(b,\\theta)$. There are four nonempty cells, each differing only in $u$. Restrict the evidence language to exactly the unions of those four cells. The occupancy record stores whether the current candidate set meets each cell. It therefore has $2^4=16$ possible states rather than the $2^8=256$ possible complete candidate sets.","That occupancy is sufficient to filter by every event in the declared evidence algebra and to return the possible target values $b$, including contradiction. It is also necessary across all candidate subsets for this contract. If two occupancies differ on one cell, filtering by that cell gives a nonempty target set in one case and the empty set in the other. This is the partition-evidence special case developed in Grounded Abstraction. \\src{GA}","\\subsection{Separate marginals can erase the next correction}\nAfter display zero, the possible pairs are $(0,0)$ and $(1,1)$. After display one, they are $(0,1)$ and $(1,0)$. In both cases the separate target and sensor possibility sets are $\\{0,1\\}$. A memory retaining only those marginals identifies the two histories.","Now observe $\\theta=0$. In the first history $b=0$, while in the second $b=1$. The same future evidence must lead to different corrections. The lost relation was not restored by the arrival of that evidence. Retaining the joint occupancy, or the relevant original display with its scope, avoids the error.","\\subsection{Two agents and one action}\nLet agent A obtain the normal reading and agent B obtain the calibration. Neither has the complete repair information alone. B can send its one-bit calibration result together with the declared provenance required by the communication protocol. Under truthful, correctly associated delivery, A can determine $b$ and request the correct repair.","\\section{Revision can change the proof obligation}\nSuppose a newly admitted repair $R_*$ succeeds for either value of $b$. Then the task no longer requires identifying $b$ before acting. The old statement that both information channels are necessary applied to the previous repair language. It does not apply after $R_*$ is introduced.","A well-formed revision record preserves the sensor identities and the old theorem while invalidating its application to the new task. The display predictor remains valid. The coordinate translator remains valid. The old minimum-information claim does not. A new controller can execute $R_*$ immediately under its declared cost and authority conditions. The old performance comparison must be recomputed if it referred to the former observation-and-repair workflow."]}