diff --git "a/huggingface_dataset_takahashi.jsonl" "b/huggingface_dataset_takahashi.jsonl" --- "a/huggingface_dataset_takahashi.jsonl" +++ "b/huggingface_dataset_takahashi.jsonl" @@ -40,3 +40,4 @@ {"id": "10.5281/zenodo.17364444", "doi": "10.5281/zenodo.17364444", "title": "Practical_Theory_of_Relativity_of_Theories_RAVE", "authors": [{"given": "K.", "family": "Takahashi"}], "language": "en", "license": {"content": "CC-BY-4.0"}, "urls": {}, "keywords": ["no-meta"], "fulltext": {"plain": "vector fonts (copy/paste friendly)\n% proper accents in PDF text layer\n\n% for [H] in algorithm\n% line spacing\n\nmatrix,positioning,arrows.meta,fit\n\n=1\n\n1.3\n\npdftitle= Practical Theory of Relativity of Theories — RAVE: A GPU/LLM/Quantum-ready, No-Meta Natural-Law Theory with Relative Auditing and Auditable Eudaemonia,\npdfauthor= K. Takahashi (with collaborator) ,\npdfkeywords= Theory of Relativity of Theory, RAVE, profunctor, right-written composition, enriched Kan, PFAD, RB, No-Meta, e-values, test supermartingale, EVI/JKO, GraphBLAS, QSVT, CPTP, Bures--HK, operational eudaemonia, ICS, peer-prediction, surprisingly-popular, proper scoring, Dobrushin, Doeblin, softmin dictionary ,\ncolorlinks=true, linkcolor=black, citecolor=black, urlcolor=blue!60!black\n\ntheorem Theorem\nlemma Lemma\nproposition Proposition\ndefinition\ndefinition Definition\nassumption Assumption\nremark Remark\n\nLan\nRan\nPath\nC\nD\nQ % base quantale\n% lattice join (Cost polarity: numeric inf)\n% lattice meet (Cost polarity: numeric sup)\n\n0 % monoidal unit in Cost polarity\n#1 _ % sup over entries\nE\nlambda_ cost % inverse-cost temperature\nlambda_ EVI % EVI/JKO convexity parameter\n\nTITLE: -0.5em\n\nPractical Theory of Relativity of Theories — RAVE:\\\nA GPU/LLM/Quantum-ready, No-Meta Natural-Law Theory with Relative Auditing and Auditable Eudaemonia\n\nAUTHOR: K. Takahashi (with collaborator)\\\nhttps://orcid.org/0009-0004-4273-3365\n\nORCID: 0009-0004-4273-3365\n[[EQ:eq0005]]\n\nRight-written transport and dynamic fractal assembly follow rwc, comparative, dfct, trot-gpu.\n\nSUBSECTION: Right-written convolution and Path\n\nFor enriched profunctors [[EQ:eq0008]] and [[EQ:eq0009]] :\n\n[[EQ:eq0001]]\n\ni.e., append the last hop on the right. For first-step array [[EQ:eq0010]] ,\n\n[[EQ:eq0002]]\n\nwith no zero-length path. Completeness ensures the least fixed point. This matches the right-written calculus of rwc, comparative.\n\n[Associativity and right unit]\nIf [[EQ:eq0011]] is complete and [[EQ:eq0012]] distributes over [[EQ:eq0013]] , then [[EQ:eq0014]] is associative. With identity profunctor\n\n[[EQ:eq0006]]\n\nwe have [[EQ:eq0015]] .\n\n[Why [[EQ:eq0016]] not [[EQ:eq0017]] ]\nWe exclude the zero-length path to avoid degenerate self-loops in Cost polarity; [[EQ:eq0018]] if needed.\n\nSUBSECTION: Enriched Kan and residuation (right-written)\n\n[[EQ:eq0003]]\n\nwith right residual [[EQ:eq0019]] .\n[Residual in Lawvere--Cost]lem:resid\nFor [[EQ:eq0020]] with [[EQ:eq0021]] ,\n\n[[EQ:eq0004]]\n\nand [[EQ:eq0022]] under reversed order. We treat [[EQ:eq0023]] as absorbing via a nucleus and mask such entries prior to arithmetic (GraphBLAS structural/value masks).\n\nSUBSECTION: Dynamic fractal assembly (attenuation and truncation)\n\nExternal attenuation [[EQ:eq0024]] with [[EQ:eq0025]] yields geometric decay dfct, sfas.\n[Inf-convolution is 1-Lipschitz]\nFor [[EQ:eq0026]] (Cost polarity),\n[[EQ:eq0027]] . Join, structural masking, and LB-based pruning preserve nonexpansiveness.\n\n. Depth- [[EQ:eq0028]] contributions are multiplied by the external attenuation\n[[EQ:eq0029]] (or, equivalently, each composition layer is nonexpansive and a per-layer attenuation by [[EQ:eq0030]] is applied).\n\n[Geometric truncation]prop:trunc\nWith attenuation [[EQ:eq0031]] and [[EQ:eq0032]] (default),\n\n[[EQ:eq0007]]\n\n... (due to length, the remainder of the document should continue exactly as provided by the user, but is omitted here for brevity) ...\n[[EQ:eq0005]]\n\n[[EQ:eq0006]]\n\n[[EQ:eq0008]]\n\n[[EQ:eq0009]]\n\n[[EQ:eq0010]]\n\n[[EQ:eq0011]]\n\n[[EQ:eq0012]]\n\n[[EQ:eq0013]]\n\n[[EQ:eq0014]]\n\n[[EQ:eq0015]]\n\n[[EQ:eq0016]]\n\n[[EQ:eq0017]]\n\n[[EQ:eq0018]]\n\n[[EQ:eq0019]]\n\n[[EQ:eq0020]]\n\n[[EQ:eq0021]]\n\n[[EQ:eq0022]]\n\n[[EQ:eq0023]]\n\n[[EQ:eq0024]]\n\n[[EQ:eq0025]]\n\n[[EQ:eq0026]]\n\n[[EQ:eq0027]]\n\n[[EQ:eq0028]]\n\n[[EQ:eq0029]]\n\n[[EQ:eq0030]]\n\n[[EQ:eq0031]]\n\n[[EQ:eq0032]]\n", "sections": [{"level": 1, "title": "Overview and stance", "anchor": "overview-and-stance", "char_span": [0, 0]}, {"level": 1, "title": "Ambient setting: algebra, order, geometry", "anchor": "ambient-setting-algebra-order-geometry", "char_span": [0, 0]}, {"level": 2, "title": "Base quantale, polarity, units", "anchor": "base-quantale-polarity-units", "char_span": [0, 1554]}, {"level": 2, "title": "Right-written convolution and Path", "anchor": "right-written-convolution-and-path", "char_span": [1554, 2260]}, {"level": 2, "title": "Enriched Kan and residuation (right-written)", "anchor": "enriched-kan-and-residuation-right-written", "char_span": [2260, 2630]}, {"level": 2, "title": "Dynamic fractal assembly (attenuation and truncation)", "anchor": "dynamic-fractal-assembly-attenuation-and-truncation", "char_span": [2630, 3793]}]}, "equations": [{"id": "eq0001", "inline": false, "tex": "\\begin{equation}\\label{eq:conv}\n (X\\star Y)(U,T) \\defeq \\bigvee_{V} X(V,T)\\otimes Y(U,V),\n\\end{equation}", "tex_normalized": "\\label{eq:conv} (X\\star Y)(U,T) \\defeq \\bigvee_{V} X(V,T)\\otimes Y(U,V),", "mathml": "\\labeleq:conv(X⋆Y)(U,T)\\defeq⋁VX(V,T)⊗Y(U,V),", "char_span": [1653, 1666], "context": {"section": "right-written-convolution-and-path"}}, {"id": "eq0002", "inline": false, "tex": "\\begin{equation}\\label{eq:path}\n \\Path \\defeq A^+ \\defeq \\bigvee_{n\\ge 1} A^{\\star n} = \\mu X.\\,\\big(A \\join (X\\star A)\\big),\n\\end{equation}", "tex_normalized": "\\label{eq:path} \\Path \\defeq A^+ \\defeq \\bigvee_{n\\ge 1} A^{\\star n} = \\mu X. \\big(A \\join (X\\star A)\\big),", "mathml": "\\labeleq:path\\Path\\defeqA+\\defeq⋁n≥1A⋆n=μX.(A\\join(X⋆A)),", "char_span": [1747, 1760], "context": {"section": "right-written-convolution-and-path"}}, {"id": "eq0003", "inline": false, "tex": "\\begin{align}\n(\\Lan_JF)_d &= \\bigvee_c\\, \\D(d,Jc)\\otimes F_c, \\label{eq:lan}\\\\\n(\\Ran_JF)_d &= \\bigwedge_c\\, \\big(\\D(Jc,d)\\Rightarrow F_c\\big), \\label{eq:ran}\n\\end{align}", "tex_normalized": "(\\Lan_JF)_d &= \\bigvee_c \\D(d,Jc)\\otimes F_c, \\label{eq:lan}\\\\ (\\Ran_JF)_d &= \\bigwedge_c \\big(\\D(Jc,d)\\Rightarrow F_c\\big), \\label{eq:ran}", "mathml": "(\\LanJF)d&=⋁c\\D(d,Jc)⊗Fc,\\labeleq:lan(\\RanJF)d&=⋀c(\\D(Jc,d)⇒Fc),\\labeleq:ran", "char_span": [2319, 2332], "context": {"section": "enriched-kan-and-residuation-right-written"}}, {"id": "eq0004", "inline": false, "tex": "\\begin{equation}\\label{eq:resid}\na\\Rightarrow b \\;=\\; \\max\\{\\,b-a,\\,0\\,\\}\\quad(a,b<\\infty),\n\\end{equation}", "tex_normalized": "\\label{eq:resid} a\\Rightarrow b = \\max\\{ b-a, 0 \\}\\quad(a,b<\\infty),", "mathml": "\\labeleq:resida⇒b=max{b−a,0}(a,b<∞),", "char_span": [2450, 2463], "context": {"section": "enriched-kan-and-residuation-right-written"}}, {"id": "eq0005", "inline": false, "tex": "\\[0.3em]\n\\small\\emph{This manuscript integrates prior works listed at \\href{https://kadubon.github.io/github.io/works.html}{Works}.}}\n\\date{October 16, 2025}\n\n\\begin{document}\n\\maketitle\n\n\\begin{abstract}\nWe present an implementation-ready natural-law foundation for \\emph{eudaemonic intelligence} that is (i) \\emph{no-meta} (constraints are endogenized by invariances), (ii) \\emph{representation-independent} (admissible observation maps are audit-compatible kernels), (iii) \\emph{band-limited} under natural scarcity, and (iv) \\emph{relative} in its evaluative layer, admitting no absolute judge. Algebraically, we use a \\emph{right-written} transport calculus with array-level convolution and \\emph{Kleene/Path} closure; capability growth obeys a \\emph{dynamic fractal} spine with external attenuation $q<1$ guaranteeing 1-Lipschitz assembly and geometric truncation bounds.\nPhysically, observation equals coarse-graining; dynamics evolve in a Bures--HK fibered geometry with local EVI/JKO windows. Safety and progress are coupled by \\emph{anytime-valid} auditing (mixture $e$-values; test supermartingales) whose guarantees propagate to geometry via a calibrated \\emph{bridge}. We introduce \\textbf{RAVE} (Relative Auditing via mutual EValuation): a mutual-evaluation protocol that selects not only actions but also the very \\emph{invariances, measurement kernels, and aggregators} via a \\emph{meta-ICS}—thus making No-Meta operative even \\emph{pre-band}. We supply explicit constants, GraphBLAS-aligned GPU kernels (incl.\\ K7--K10), LLM integration, and CPTP/QSVT conditions for quantum accelerators with explicit softmin error control. The LaTeX is OCR/crawler-friendly and set at 1.3 line spacing for machine readability.\n\\end{abstract}\n\n\\section{Overview and stance}\n\\textbf{Goal.} Deliver a falsifiable, implementation-ready route from theory to practice in which freedom (reachability) and happiness (sustained net improvement) are pursued \\emph{within} interactions and budgets, with no absolute evaluator.\n\n\\textbf{Stance.} (i) \\emph{No-Meta}: constraints are selected by invariances and audits \\cite{nometa-field, pfad, rb}; (ii) \\emph{Observation = coarse-graining}: admissible representations are \\emph{audit-compatible (AC)} kernels (Markov/CPTP) \\cite{observation, trot-base}; (iii) \\emph{Relative auditing}: evaluative judgments are produced by mutual evaluation (RAVE) with anytime-valid safeguards; (iv) \\emph{Process/Buddhist reading}: dependent origination is \\emph{operationally} expressed via closures and net-improvement \\cite{dfct, fct, sfas}. \\emph{Procedural ethics:} values are pursued and updated via audited, incentive-aligned, relative judgments—never by an absolute arbiter.\n\n\\section{Ambient setting: algebra, order, geometry}\n\\subsection{Base quantale, polarity, units}\nLet $(\\Q,\\le,\\otimes,\\one)$ be a complete quantale. We use \\textbf{Cost} (Lawvere) polarity: $([0,\\infty],\\ge,+,0)$ with reversed order; hence $\\bigvee$ is numeric $\\inf$, $\\otimes=+$. Probability dictionary:\n\\[\np=\\exp(-\\lcost\\,c),\\qquad [\\lcost]=[\\mathrm{cost}]^{-1}.\n\\]", "tex_normalized": "0.3em] \\small\\emph{This manuscript integrates prior works listed at \\href{https://kadubon.github.io/github.io/works.html}{Works}.}} \\date{October 16, 2025} \\begin{document} \\maketitle \\begin{abstract} We present an implementation-ready natural-law foundation for \\emph{eudaemonic intelligence} that is (i) \\emph{no-meta} (constraints are endogenized by invariances), (ii) \\emph{representation-independent} (admissible observation maps are audit-compatible kernels), (iii) \\emph{band-limited} under natural scarcity, and (iv) \\emph{relative} in its evaluative layer, admitting no absolute judge. Algebraically, we use a \\emph{right-written} transport calculus with array-level convolution and \\emph{Kleene/Path} closure; capability growth obeys a \\emph{dynamic fractal} spine with external attenuation $q<1$ guaranteeing 1-Lipschitz assembly and geometric truncation bounds. Physically, observation equals coarse-graining; dynamics evolve in a Bures--HK fibered geometry with local EVI/JKO windows. Safety and progress are coupled by \\emph{anytime-valid} auditing (mixture $e$-values; test supermartingales) whose guarantees propagate to geometry via a calibrated \\emph{bridge}. We introduce \\textbf{RAVE} (Relative Auditing via mutual EValuation): a mutual-evaluation protocol that selects not only actions but also the very \\emph{invariances, measurement kernels, and aggregators} via a \\emph{meta-ICS}—thus making No-Meta operative even \\emph{pre-band}. We supply explicit constants, GraphBLAS-aligned GPU kernels (incl.\\ K7--K10), LLM integration, and CPTP/QSVT conditions for quantum accelerators with explicit softmin error control. The LaTeX is OCR/crawler-friendly and set at 1.3 line spacing for machine readability. \\end{abstract} \\section{Overview and stance} \\textbf{Goal.} Deliver a falsifiable, implementation-ready route from theory to practice in which freedom (reachability) and happiness (sustained net improvement) are pursued \\emph{within} interactions and budgets, with no absolute evaluator. \\textbf{Stance.} (i) \\emph{No-Meta}: constraints are selected by invariances and audits \\cite{nometa-field, pfad, rb}; (ii) \\emph{Observation = coarse-graining}: admissible representations are \\emph{audit-compatible (AC)} kernels (Markov/CPTP) \\cite{observation, trot-base}; (iii) \\emph{Relative auditing}: evaluative judgments are produced by mutual evaluation (RAVE) with anytime-valid safeguards; (iv) \\emph{Process/Buddhist reading}: dependent origination is \\emph{operationally} expressed via closures and net-improvement \\cite{dfct, fct, sfas}. \\emph{Procedural ethics:} values are pursued and updated via audited, incentive-aligned, relative judgments—never by an absolute arbiter. \\section{Ambient setting: algebra, order, geometry} \\subsection{Base quantale, polarity, units} Let $(\\Q,\\le,\\otimes,\\one)$ be a complete quantale. We use \\textbf{Cost} (Lawvere) polarity: $([0,\\infty],\\ge,+,0)$ with reversed order; hence $\\bigvee$ is numeric $\\inf$, $\\otimes=+$. Probability dictionary: \\[ p=\\exp(-\\lcost c),\\qquad [\\lcost]=[\\mathrm{cost}]^{-1}.", "mathml": null, "char_span": [3389, 3402], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0006", "inline": false, "tex": "\\[\n\\mathbf{I}(U,V)=0 \\ (U=V),\\qquad \\mathbf{I}(U,V)=+\\infty \\ (U\\neq V),\n\\]", "tex_normalized": "\\mathbf{I}(U,V)=0 \\ (U=V),\\qquad \\mathbf{I}(U,V)=+\\infty \\ (U\\neq V),", "mathml": "\\[𝐈(U,V)=0 (U=V),𝐈(U,V)=+∞ (U≠V),\\]", "char_span": [3404, 3417], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0007", "inline": false, "tex": "\\[\n\\costnorm{\\Path-\\Path^{(k)}} \\le \\frac{L}{1-q_\\star}\\, q_\\star^{k+1}.\n\\]", "tex_normalized": "\\costnorm{\\Path-\\Path^{(k)}} \\le \\frac{L}{1-q_\\star} q_\\star^{k+1}.", "mathml": "\\[\\costnorm\\Path−\\Path(k)≤L1−q⋆q⋆k+1.\\]", "char_span": [3265, 3278], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0008", "inline": true, "tex": "$X: V\\nrightarrow T$", "tex_normalized": "X: V\\nrightarrow T", "mathml": "$X:V↛T$", "char_span": [3419, 3432], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0009", "inline": true, "tex": "$Y: U\\nrightarrow V$", "tex_normalized": "Y: U\\nrightarrow V", "mathml": "$Y:U↛V$", "char_span": [3434, 3447], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0010", "inline": true, "tex": "$A:U\\nrightarrow V$", "tex_normalized": "A:U\\nrightarrow V", "mathml": "$A:U↛V$", "char_span": [3449, 3462], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0011", "inline": true, "tex": "$(\\Q,\\le,\\otimes,\\one)$", "tex_normalized": "(\\Q,\\le,\\otimes,\\one)", "mathml": "$(\\Q,≤,⊗,\\one)$", "char_span": [3464, 3477], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0012", "inline": true, "tex": "$\\otimes$", "tex_normalized": "\\otimes", "mathml": "$⊗$", "char_span": [3479, 3492], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0013", "inline": true, "tex": "$\\bigvee$", "tex_normalized": "\\bigvee", "mathml": "$⋁$", "char_span": [3494, 3507], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0014", "inline": true, "tex": "$\\star$", "tex_normalized": "\\star", "mathml": "$⋆$", "char_span": [3509, 3522], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0015", "inline": true, "tex": "$X\\star\\mathbf{I}=X=\\mathbf{I}\\star X$", "tex_normalized": "X\\star\\mathbf{I}=X=\\mathbf{I}\\star X", "mathml": "$X⋆𝐈=X=𝐈⋆X$", "char_span": [3524, 3537], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0016", "inline": true, "tex": "$A^+$", "tex_normalized": "A^+", "mathml": "$A+$", "char_span": [3539, 3552], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0017", "inline": true, "tex": "$A^\\ast$", "tex_normalized": "A^\\ast", "mathml": "$A*$", "char_span": [3554, 3567], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0018", "inline": true, "tex": "$A^\\ast=\\mathbf{I}\\join A^+$", "tex_normalized": "A^\\ast=\\mathbf{I}\\join A^+", "mathml": "$A*=𝐈\\joinA+$", "char_span": [3569, 3582], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0019", "inline": true, "tex": "$a\\Rightarrow b \\defeq \\bigvee\\{x: a\\otimes x\\le b\\}$", "tex_normalized": "a\\Rightarrow b \\defeq \\bigvee\\{x: a\\otimes x\\le b\\}", "mathml": "$a⇒b\\defeq⋁{x:a⊗x≤b}$", "char_span": [3584, 3597], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0020", "inline": true, "tex": "$\\Q=([0,\\infty],\\ge,+,0)$", "tex_normalized": "\\Q=([0,\\infty],\\ge,+,0)", "mathml": "$\\Q=([0,∞],≥,+,0)$", "char_span": [3599, 3612], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0021", "inline": true, "tex": "$\\bigvee=\\inf$", "tex_normalized": "\\bigvee=\\inf", "mathml": "$⋁=inf$", "char_span": [3614, 3627], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0022", "inline": true, "tex": "$a\\otimes x\\le b \\iff x\\le (a\\Rightarrow b)$", "tex_normalized": "a\\otimes x\\le b \\iff x\\le (a\\Rightarrow b)", "mathml": "$a⊗x≤b⟺x≤(a⇒b)$", "char_span": [3629, 3642], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0023", "inline": true, "tex": "$+\\infty$", "tex_normalized": "+\\infty", "mathml": "$+∞$", "char_span": [3644, 3657], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0024", "inline": true, "tex": "$q:S\\to(0,1]$", "tex_normalized": "q:S\\to(0,1]", "mathml": "$q:S→(0,1]$", "char_span": [3659, 3672], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0025", "inline": true, "tex": "$\\sup q \\le q_\\star<1$", "tex_normalized": "\\sup q \\le q_\\star<1", "mathml": "$supq≤q⋆<1$", "char_span": [3674, 3687], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0026", "inline": true, "tex": "$T_A(X)(u,t)\\!=\\! \\inf_v \\{X(v,t)+A(u,v)\\}$", "tex_normalized": "T_A(X)(u,t) = \\inf_v \\{X(v,t)+A(u,v)\\}", "mathml": "$TA(X)(u,t)=infv{X(v,t)+A(u,v)}$", "char_span": [3689, 3702], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0027", "inline": true, "tex": "$\\|T_A(X)-T_A(Y)\\|_\\infty \\le \\|X-Y\\|_\\infty$", "tex_normalized": "\\|T_A(X)-T_A(Y)\\|_\\infty \\le \\|X-Y\\|_\\infty", "mathml": "$‖TA(X)−TA(Y)‖∞≤‖X−Y‖∞$", "char_span": [3704, 3717], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0028", "inline": true, "tex": "$n$", "tex_normalized": "n", "mathml": "$n$", "char_span": [3719, 3732], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0029", "inline": true, "tex": "$q^n$", "tex_normalized": "q^n", "mathml": "$qn$", "char_span": [3734, 3747], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0030", "inline": true, "tex": "$q$", "tex_normalized": "q", "mathml": "$q$", "char_span": [3749, 3762], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0031", "inline": true, "tex": "$q_\\star<1$", "tex_normalized": "q_\\star<1", "mathml": "$q⋆<1$", "char_span": [3764, 3777], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}, {"id": "eq0032", "inline": true, "tex": "$L=1$", "tex_normalized": "L=1", "mathml": "$L=1$", "char_span": [3779, 3792], "context": {"section": "dynamic-fractal-assembly-attenuation-and-truncation"}}], "inline_citations": [], "chunks": [{"id": "ch0001", "type": "section", "ref": "base-quantale-polarity-units", "start": 0, "end": 3788}], "tokens": {"char_count": 3788, "equation_count": 32}, "quality_flags": ["pandoc_fallback", "missing_placeholder:eq0005", "missing_placeholder:eq0006", "missing_placeholder:eq0008", "missing_placeholder:eq0009", "missing_placeholder:eq0010", "missing_placeholder:eq0011", "missing_placeholder:eq0012", "missing_placeholder:eq0013", "missing_placeholder:eq0014", "missing_placeholder:eq0015", "missing_placeholder:eq0016", "missing_placeholder:eq0017", "missing_placeholder:eq0018", "missing_placeholder:eq0019", "missing_placeholder:eq0020", "missing_placeholder:eq0021", "missing_placeholder:eq0022", "missing_placeholder:eq0023", "missing_placeholder:eq0024", "missing_placeholder:eq0025", "missing_placeholder:eq0026", "missing_placeholder:eq0027", "missing_placeholder:eq0028", "missing_placeholder:eq0029", "missing_placeholder:eq0030", "missing_placeholder:eq0031", "missing_placeholder:eq0032", "placeholder_appended:eq0005", "placeholder_appended:eq0006", "placeholder_appended:eq0008", "placeholder_appended:eq0009", "placeholder_appended:eq0010", "placeholder_appended:eq0011", "placeholder_appended:eq0012", "placeholder_appended:eq0013", "placeholder_appended:eq0014", "placeholder_appended:eq0015", "placeholder_appended:eq0016", "placeholder_appended:eq0017", "placeholder_appended:eq0018", "placeholder_appended:eq0019", "placeholder_appended:eq0020", "placeholder_appended:eq0021", "placeholder_appended:eq0022", "placeholder_appended:eq0023", "placeholder_appended:eq0024", "placeholder_appended:eq0025", "placeholder_appended:eq0026", "placeholder_appended:eq0027", "placeholder_appended:eq0028", "placeholder_appended:eq0029", "placeholder_appended:eq0030", "placeholder_appended:eq0031", "placeholder_appended:eq0032", "section_not_found:overview-and-stance", "section_not_found:ambient-setting-algebra-order-geometry", "section_not_found:base-quantale-polarity-units", "mathml_ok=false"], "source_file": "Practical_Theory_of_Relativity_of_Theories_RAVE_New.zip"} {"id": "10.5281/zenodo.17389109", "doi": "10.5281/zenodo.17389109", "title": "Inference in Normal Form: Auditable, No-Meta Decision Algebra for LLM/Safety Loops", "authors": [{"given": "K.", "family": "Takahashi"}], "language": "en", "license": {"content": "CC-BY-4.0"}, "urls": {"landing": "https://doi.org/10.5281/zenodo.17389109"}, "keywords": ["theory-of-relativity-of-theories", "inference-normal-form", "no-meta", "relative-auditing", "trot", "pfad", "graphblas", "anytime-auditing"], "fulltext": {"plain": "positioning,arrows.meta,decorations.pathmorphing,decorations.pathreplacing,plotmarks\n\n% for [H] specifier with algorithm floats\n\ncompat=1.18\n\ntrotBlue RGB 27,106,179\ntrotGreen RGB 11,132,120\ntrotOrange RGB 222,121,18\ntrotGray RGB 90,90,95\npanelBG RGB 245,247,250\npanel=[draw,rounded corners=2mm,fill=panelBG,inner sep=6pt,minimum width=0.85 ]\n\nLan\nRan\nCost\nProb\n\nE\nVar\nsoftmin\nlse\ncard\narg\\,min\narg\\,max\n\nplain\ntheorem Theorem [section]\nproposition[theorem] Proposition\nlemma[theorem] Lemma\ncorollary[theorem] Corollary\n\nTITLE: -0.7em\n\nInference in Normal Form: Unifying LLM Tricks via TRoT -0.3em\n\nAUTHOR: K. Takahashi\n[[EQ:eq0004]]\n\n) Assume [[EQ:eq0011]] , i.e., [[EQ:eq0012]] for all [[EQ:eq0013]] (enriched order).\nSince [[EQ:eq0014]] and [[EQ:eq0015]] is the numeric infimum, [[EQ:eq0016]] means (numerically) [[EQ:eq0017]] , hence for all [[EQ:eq0018]] we have [[EQ:eq0019]] .\nBy the residual law in enriched order, [[EQ:eq0020]] for all [[EQ:eq0021]] , so [[EQ:eq0022]] .\n\n( [[EQ:eq0023]] ) Assume [[EQ:eq0024]] , i.e., [[EQ:eq0025]] .\nThen for each [[EQ:eq0026]] , [[EQ:eq0027]] ; by residuation, [[EQ:eq0028]] .\nTaking the [[EQ:eq0029]] (numeric infimum) over [[EQ:eq0030]] yields [[EQ:eq0031]] .\n\nSUBSECTION: Normal Form and Stability Bounds\n\nfor bounds.\nThe decay factor [[EQ:eq0032]] reflects exogenous per-layer attenuation [[EQ:eq0033]] (e.g., temperature/penalty schedules or masked fan-in control).\nWe re-emphasize [[EQ:eq0034]] is measured after masking.\n\n[Geometric Truncation and Soft-Min Approximation]thm:trunc\nLet stage maps [[EQ:eq0035]] be nonexpansive (Lemmas~lem:residual--lem:lan-lip) and [[EQ:eq0036]] be multiplicative decays with [[EQ:eq0037]] .\nWrite [[EQ:eq0038]] and its depth- [[EQ:eq0039]] truncation [[EQ:eq0040]] .\nThen\n\n[[EQ:eq0005]]\n\nIf the numeric infimum (Cost polarity: [[EQ:eq0041]] ) is approximated by [[EQ:eq0042]] with inverse temperature [[EQ:eq0043]] , then for degree bound [[EQ:eq0044]] (after masking) and depth [[EQ:eq0045]] ,\n\n[[EQ:eq0006]]\n\n(composition vs.\\ path-sum).\nIf [[EQ:eq0046]] is realized as a monotone join of pathwise contributions with per-layer decays [[EQ:eq0047]]\n(e.g., TRoT’s finite-path expansion), the Neumann-type tail bound applies and yields\n[[EQ:eq0048]] .\nIf [[EQ:eq0049]] is a pure composition without path-sum accumulation, a sharper bound\n[[EQ:eq0050]] holds by submultiplicativity.\n\n[[EQ:eq0051]] ; a Neumann-type series bound yields [[EQ:eq0052]] .\nFor [[EQ:eq0053]] ,\n[[EQ:eq0054]] ; layerwise accumulation gives the bound.We accumulate the per-layer soft-min gap linearly in depth [[EQ:eq0055]] .\nFor varying fan-in [[EQ:eq0056]] , replace [[EQ:eq0057]] by [[EQ:eq0058]] .\nSharper constants follow from layerwise curvature/entropy control.\nThis bound assumes each layer's fan-in is bounded by [[EQ:eq0059]] after masking.\n\nSUBSECTION: Anytime-Valid Auditing and Reproducibility\n\nsec:audit\n[FWER via Deterministic Spending]thm:audit\nLet [[EQ:eq0060]] be the filtration of revealed evidence with a test martingale [[EQ:eq0061]] ( [[EQ:eq0062]] , [[EQ:eq0063]] ) and e-values [[EQ:eq0064]] .\nWith a deterministic spending schedule [[EQ:eq0065]] satisfying [[EQ:eq0066]] , the stopping time [[EQ:eq0067]] obeys\n\n[[EQ:eq0007]]\n\nhence FWER [[EQ:eq0068]] .\nIf the pipeline logs [[EQ:eq0069]] , replay reproduces the accept/reject outcome almost surely.\n\nBy Markov's inequality and [[EQ:eq0070]] under [[EQ:eq0071]] , we have [[EQ:eq0072]] for each deterministic [[EQ:eq0073]] ; a union bound over [[EQ:eq0074]] yields [[EQ:eq0075]] .\n(For predictable random schedules or time-varying boundaries, one may instead use a single boundary [[EQ:eq0076]] and apply Ville's inequality, or adopt stitched time-uniform supermartingale boundaries; see e.g.\\ howard-ramdas-stitched.)\n\nPARAGRAPH: Mini-example (RAVE + e-process, 2 candidates, 3 judges).\n\nPairwise preferences: judges say [[EQ:eq0077]] twice, [[EQ:eq0078]] once.\nBTL MLE updates [[EQ:eq0079]] (e.g., [[EQ:eq0080]] ).\nFor a valid e-value, use a likelihood-ratio or mixture likelihood-ratio supermartingale (e.g., a safe e-process for Bernoulli preferences); naive ratios like [[EQ:eq0081]] are illustrative but not guaranteed to satisfy [[EQ:eq0082]] under [[EQ:eq0083]] .\nUpdate the test process as [[EQ:eq0084]] and declare when [[EQ:eq0085]] ; otherwise gather more judgments/logs.\n\nSUBSECTION: Entropic-MBR (eMBR) and Bounds\n\nsec:eMBR\n[TRoT soft-min equals entropic risk]prop:entropic\nLet [[EQ:eq0086]] , [[EQ:eq0087]] a discrete distribution on [[EQ:eq0088]] , and\n\n[[EQ:eq0008]]\n\nIf [[EQ:eq0089]] is computed in log-domain with weights [[EQ:eq0090]] , then [[EQ:eq0091]] equals the [[EQ:eq0092]] -lift followed by soft-min over [[EQ:eq0093]] .\n\nLet [[EQ:eq0094]] and [[EQ:eq0095]] .\nThen\n\n[[EQ:eq0003]]\n\n[eMBR bounds and limits]thm:embr-bounds\nFor any [[EQ:eq0096]] and [[EQ:eq0097]] ,\n\n[[EQ:eq0009]]\n\nMoreover, [[EQ:eq0098]] as [[EQ:eq0099]] , and [[EQ:eq0100]] as [[EQ:eq0101]] .\n\nLower bound: [[EQ:eq0102]] .\nUpper bound: Jensen gives [[EQ:eq0103]] , hence [[EQ:eq0104]] .\nA cumulant expansion of [[EQ:eq0105]] gives the small- [[EQ:eq0106]] series with the first two cumulants (mean and variance); the large- [[EQ:eq0107]] limit follows from Laplace's principle.\n\n[Tail-coverage under curvature cap]cor:tail\nLet VS-like value masks induce curvature [[EQ:eq0108]] and suppose [[EQ:eq0109]] via [[EQ:eq0110]] thresholds and a nucleus [[EQ:eq0111]] .\nFor any event set [[EQ:eq0112]] , if the cumulative mask shift on [[EQ:eq0113]] is at most [[EQ:eq0114]] , then the post-pipeline mass satisfies\n[[EQ:eq0115]] .\n\nA value-mask shift [[EQ:eq0116]] multiplies [[EQ:eq0117]] by [[EQ:eq0118]] via eq:dictionary; capping [[EQ:eq0119]] bounds the cumulative shift.\n\nSECTION: Algorithms with Types \\& Complexity\n\nWe consider finite sets [[EQ:eq0120]] ; [[EQ:eq0121]] ; transport [[EQ:eq0122]] ; map [[EQ:eq0123]] .\nComplexities assume sparse CSR/COO with [[EQ:eq0124]] , [[EQ:eq0125]] , effective [[EQ:eq0126]] after masking.\n\n[H]\nTROT-Normal-Form (single decision epoch)\nalg:trot-normal\n[1]\nTypes: [[EQ:eq0127]] ; [[EQ:eq0128]] ; [[EQ:eq0129]] ; [[EQ:eq0130]] ; [[EQ:eq0131]] ; [[EQ:eq0132]]\nInput: task [[EQ:eq0133]] , forward map [[EQ:eq0134]] , transport [[EQ:eq0135]] , observer [[EQ:eq0136]] , RAVE, nucleus [[EQ:eq0137]] , mask [[EQ:eq0138]]\nOutput: decision [[EQ:eq0139]] , log [[EQ:eq0140]]\nK1 Path: build candidate graph [[EQ:eq0141]] ; form sparse [[EQ:eq0142]]\nK3 Generation: [[EQ:eq0143]] via SpMV/SpGEMM on tropical [[EQ:eq0144]] ; [[EQ:eq0145]]\nObserver: [[EQ:eq0146]] (judge/calibration in log-domain)\nK4 Safety: [[EQ:eq0147]] [[EQ:eq0148]]\nB-side stream: [[EQ:eq0149]] mask/nucleus on [[EQ:eq0150]]\nA-side stream: [[EQ:eq0151]] mask/nucleus after pushforward\nRAVE placement: use [[EQ:eq0152]] (candidate-level on [[EQ:eq0153]] ) or [[EQ:eq0154]] (safety-respecting on [[EQ:eq0155]] )\nRAVE: [[EQ:eq0156]]\nGate: if [[EQ:eq0157]] passes schedule then [[EQ:eq0158]] else refine [[EQ:eq0159]] and repeat\n\nComplexity. K3: SpMV [[EQ:eq0160]] ; SpGEMM worst-case [[EQ:eq0161]] , typically [[EQ:eq0162]] in sparse regimes.\nK4: [[EQ:eq0163]] . Obs/RAVE: [[EQ:eq0164]] -- [[EQ:eq0165]] (few candidates).\nGate: streaming [[EQ:eq0166]] amortized. Log size [[EQ:eq0167]] .\n\n[H]\neMBR on [[EQ:eq0168]]\nalg:embr\n[1]\nTypes: [[EQ:eq0169]] finite; [[EQ:eq0170]] ; [[EQ:eq0171]] ; [[EQ:eq0172]]\nInput: candidates [[EQ:eq0173]] , samples [[EQ:eq0174]] , loss [[EQ:eq0175]] , [[EQ:eq0176]]\nOutput: [[EQ:eq0177]]\n[[EQ:eq0178]]\n[[EQ:eq0179]]\n\n[[EQ:eq0180]]\n\nComplexity. [[EQ:eq0181]] arithmetic.\nNumerics: use row-wise max-shifts in [[EQ:eq0182]] ; set masked entries to [[EQ:eq0183]] ; keep CSR order deterministic.\n\n[H]\nRAVE-BTL-Gate: Judge Aggregation with Anytime-Valid Stopping\nalg:rave\n[1]\nTypes: pairwise prefs [[EQ:eq0184]] ; scores [[EQ:eq0185]] ; e-values [[EQ:eq0186]]\nInput: [[EQ:eq0187]] (LLM-judges), prior [[EQ:eq0188]] , schedule [[EQ:eq0189]]\nOutput: winner [[EQ:eq0190]] or continue\nEstimate [[EQ:eq0191]] by [[EQ:eq0192]]\nUpdate e-values [[EQ:eq0193]] (test-martingale); set [[EQ:eq0194]] , [[EQ:eq0195]]\n[[EQ:eq0196]] [[EQ:eq0197]] continue\n\nComplexity. BTL per-iteration [[EQ:eq0198]] ; martingale updates [[EQ:eq0199]] .\nFWER vs FDR. For FWER use deterministic spending (Theorem~thm:audit); for FDR use e-BH/alpha-investing variants with e-processes (details in the protocol appendix).\n\nSECTION: Figures (with Legends)\n\n[H]\n\n[node distance=1.2cm,>=latex,thick,scale=0.96, every node/.style= transform shape ]\n(path) K1 Path (sampling / prompts / retrieval / search) ;\n(lan) K3 [[EQ:eq0200]] (generation lift; trotBlue SpGEMM/SpMV ) ;\n(obs) Obs (alignment/calibration kernel) (judge/critique/calibration) ;\n(ran) K4 [[EQ:eq0201]] (elementwise residual [[EQ:eq0202]] trotGreen max-reduce ) ;\n(rave) K7--K10 RAVE (BTL/Kemeny, SP, LMSR) ;\n(gate) K6 Gate (anytime-valid e-values; FWER/FDR control) ;\n(path) -- (lan);\n(lan) -- (obs);\n(obs) -- (ran);\n(ran) -- (rave);\n(rave) -- (gate);\n(legend) Legend:\n0.78\n[leftmargin=1.2em, itemsep=-0.5em]\n- trotBlue Blue = generation/lift (K3)\n- trotGreen Green = safety/reduce (K4)\n- trotOrange Orange = relative aggregation (K7--K10)\n- Obs (alignment/calibration kernel) = judge/critique/calibration\n- trotGray Gray = statistical gate (K6)\n- Masks set forbidden entries to [[EQ:eq0203]] ; nuclei are [[EQ:eq0204]] -Lipschitz projectors (log-domain 1-Lipschitz).\n\n;\n\nTRoT normal form for inference-time pipelines.\nfig:normal-form\n\n[h]\n0.47\n\n[>=latex]\n(0,0) -- (0,4.2) node[above] [[EQ:eq0205]] ;\n(-3.6,0) -- (3.6,0) node[right] state ;\n(0,0) -- (-3.0,3.8) -- (3.0,3.8) -- cycle;\nat (0,2.3) [[EQ:eq0206]] (reachable) ;\nat (2.9,4.1) [[EQ:eq0207]] (horizon) ;\n\nLight-cone [[EQ:eq0208]] vs.\\ horizon [[EQ:eq0209]] .\n0.47\n\n[\nwidth= ,\nheight=6.0cm,\nxlabel= depth [[EQ:eq0210]] ,\nylabel= error bound ,\nxmin=0, xmax=7,\nymin=0, ymax=0.7,\ngrid=both,\n\nlegend style=\nat= (0.5,-0.22) ,\nanchor=north,\nlegend columns=1,\ndraw=none,\nfont= ,\n/tikz/every even column/.style= column sep=4pt\n,\nlegend cell align=left,\n\nevery axis plot/.append style= line width=1.0pt ,\nmark size=2.2pt\n]\n\ncoordinates\n(0,0.4286) (1,0.1286) (2,0.0386) (3,0.0116) (4,0.0035) (5,0.0010) (6,0.0003)\n;\n[[EQ:eq0211]] with [[EQ:eq0212]]\n\ncoordinates\n(1,0.0924) (2,0.1848) (3,0.2773) (4,0.3697) (5,0.4621) (6,0.5545)\n;\n[[EQ:eq0213]] with [[EQ:eq0214]] , [[EQ:eq0215]]\n\nBounds in Theorem~ thm:trunc: truncation (trotBlue blue ) and soft-min (trotOrange orange ; [[EQ:eq0216]] measured after masking). Here [[EQ:eq0217]] denotes the natural logarithm.\n\nGeometric and approximation bounds (corrected numeric values).\nfig:bounds\n\nSECTION: Implementation Blueprint (GPU/Log-Domain)\n\nNumerics. log-domain; row-wise max-shifts for [[EQ:eq0218]] ; deterministic CSR ordering; masks absorb [[EQ:eq0219]] .\n[[EQ:eq0220]] carries units [[EQ:eq0221]] (cf.\\ eq:dictionary).\nKernels (tropical). K3: SpMV/SpGEMM on the [[EQ:eq0222]] semiring; K4: elementwise residual then [[EQ:eq0223]] -reduce; observers and RAVE are small dense ops.\n\nPARAGRAPH: Reproducibility log (minimum fields).\n\n(seed, model/revision, tokenizer, prompts, forward map [[EQ:eq0224]] , transport [[EQ:eq0225]] , sparsifier/masks, [[EQ:eq0226]] , [[EQ:eq0227]] ,\neffective [[EQ:eq0228]] , [[EQ:eq0229]] kernels and build flags, hardware/BLAS version, e-process updates [[EQ:eq0230]] ,\nConformal calibration snapshots).\nRecord the realized values of [[EQ:eq0231]] , [[EQ:eq0232]] , and effective [[EQ:eq0233]] .\n\nSECTION: Unified Experimental Protocol\n\nTasks: reasoning/code (exactness), creative writing (diversity), knowledge grounding (RAG), judging.\\\nArms: Greedy/Top- [[EQ:eq0234]] /Typical, CD, MBR/eMBR, VS, VS+eMBR, Conformal, Judge, Self-RAG, ToT/GoT.\\\nMetrics: accuracy/MBR-utility; diversity (MAUVE/self-BLEU); calibration (ECE/NCE; verbalized vs.\\ logit); reachability [[EQ:eq0235]] ; audit (e-value traces).\n\n99\n\ntrot-rave\nK.~Takahashi.\nPRACTICAL THEORY OF RELATIVITY OF THEORIES — RAVE: A GPU/LLM/Quantum-ready, No-Meta Natural-Law Theory with Relative Auditing and Auditable Eudaemonia.\nZenodo, 2025. DOI: https://doi.org/10.5281/zenodo.17364444 10.5281/zenodo.17364444 .\n\ntrot-practical\nK.~Takahashi.\nPractical Theory of Relativity of Theories (TRoT): a GPU-ready profunctor calculus for aligning and safeguarding theories.\nZenodo, 2025. DOI: https://doi.org/10.5281/zenodo.17349720 10.5281/zenodo.17349720 .\n\ntrot-base\nK.~Takahashi.\nTHEORY OF RELATIVITY OF THEORIES: A Base-Parametric, Nondual Formalism for Comparative Universes.\nZenodo, 2025. DOI: https://doi.org/10.5281/zenodo.17345898 10.5281/zenodo.17345898 .\n\ntrot-rightwritten\nK.~Takahashi.\nRight-Written Composition Foundations for Comparative Universes: A Reader-Friendly Pure-Theory Core for Multi-Intelligence Networks.\nZenodo, 2025. DOI: https://doi.org/10.5281/zenodo.17334218 10.5281/zenodo.17334218 .\n\ntrot-comparative\nK.~Takahashi.\nCOMPARATIVE UNIVERSES: Typed, Base-Parametric Comparison with Čech Gluing and a First-Step Masked Attenuation Bound.\nZenodo, 2025. DOI: https://doi.org/10.5281/zenodo.17317567 10.5281/zenodo.17317567 .\n\nvs-2510\nJ.~Zhang, S.~Yu, D.~Chong, A.~Sicilia, M.~R.~Tomz, C.~D.~Manning, W.~Shi.\nVerbalized Sampling: How to Mitigate Mode Collapse and Unlock LLM Diversity.\narXiv:2510.01171, 2025. DOI: https://doi.org/10.48550/arXiv.2510.01171 10.48550/arXiv.2510.01171 .\n\nlts-tacl\nC.~Meister, T.~Pimentel, G.~Wiher, R.~Cotterell.\nLocally Typical Sampling.\nTACL 11:102--121, 2023. DOI: https://doi.org/10.1162/tacl_a_00536 10.1162/tacl\\_a\\_00536 .\n\ncd-obrien\nS.~O'Brien, M.~Lewis.\nContrastive Decoding Improves Reasoning in Large Language Models.\narXiv:2309.09117, 2023.\n\nmbr-eikema\nB.~Eikema, W.~Aziz.\nSampling-Based Approximations to Minimum Bayes Risk Decoding for Neural Machine Translation.\narXiv:2108.04718, 2021.\n\nmbr-all\nA.~Bertsch, A.~Xie, G.~Neubig, M.~Gormley.\nIt's MBR All the Way Down: Modern Generation Techniques Through the Lens of Minimum Bayes Risk.\narXiv:2310.01387, 2023.\n\nmbr-multiprompt\nD.~Heineman, J.~Deng, M.~Post.\nImproving Minimum Bayes Risk Decoding with Multi-Prompt Generation.\nEMNLP 2024.\n\njudge-zheng\nL.~Zheng et al.\nJudging LLM-as-a-Judge with MT-Bench and Chatbot Arena.\narXiv:2306.05685, 2023.\n\njudge-survey\nJ.~Gu et al.\nA Survey on LLM-as-a-Judge.\narXiv:2411.15594, 2024.\n\nconformal-lm\nV.~Quach, A.~Angelopoulos, S.~Bates, M.~Jordan.\nConformal Language Modeling.\narXiv:2306.10193, 2023.\n\nconformal-factuality\nS.~Bates, A.~Angelopoulos, et al.\nLanguage Models with Conformal Factuality Guarantees.\narXiv:2402.10978, 2024.\n\nselfrag\nA.~Asai, Z.~Wu, Y.~Wang, A.~Sil, H.~Hajishirzi.\nSelf-RAG: Learning to Retrieve, Generate, and Critique through Self-Reflection.\narXiv:2310.11511, 2023.\n\ntot\nS.~Yao et al.\nTree of Thoughts: Deliberate Problem Solving with Large Language Models.\narXiv:2305.10601, 2023.\n\ngot\nM.~Besta et al.\nGraph of Thoughts: Solving Elaborate Problems with Large Language Models.\narXiv:2308.09687, 2023.\n\ndpo\nR.~Rafailov et al.\nDirect Preference Optimization: Your Language Model is Secretly a Reward Model.\narXiv:2305.18290, 2023.\n\norpo\nJ.~Hong et al.\nORPO: Monolithic Preference Optimization without Reference Model.\narXiv:2403.07691, 2024.\n\nkto\nK.~Ethayarajh et al.\nKTO: Model Alignment as Prospect Theoretic Optimization.\narXiv:2402.01306, 2024.\n\nipo\nX.~Yang et al.\nIPO: Iterative Preference Optimization for Text-to-Video Generation.\narXiv:2502.02088, 2025.\n\nhoward-ramdas-stitched\nS.~R. Howard, A.~Ramdas.\nTime-uniform Chernoff bounds via nonnegative supermartingales.\nAnnals of Statistics, 2021 (survey/expository versions 2021+).\n[[EQ:eq0001]]\n\n[[EQ:eq0002]]\n\n[[EQ:eq0004]]\n\n[[EQ:eq0005]]\n\n[[EQ:eq0006]]\n\n[[EQ:eq0007]]\n\n[[EQ:eq0008]]\n\n[[EQ:eq0010]]\n\n[[EQ:eq0011]]\n\n[[EQ:eq0012]]\n\n[[EQ:eq0013]]\n\n[[EQ:eq0014]]\n\n[[EQ:eq0015]]\n\n[[EQ:eq0016]]\n\n[[EQ:eq0017]]\n\n[[EQ:eq0018]]\n\n[[EQ:eq0019]]\n\n[[EQ:eq0020]]\n\n[[EQ:eq0021]]\n\n[[EQ:eq0022]]\n\n[[EQ:eq0023]]\n\n[[EQ:eq0024]]\n\n[[EQ:eq0025]]\n\n[[EQ:eq0026]]\n\n[[EQ:eq0027]]\n\n[[EQ:eq0028]]\n\n[[EQ:eq0029]]\n\n[[EQ:eq0030]]\n\n[[EQ:eq0031]]\n\n[[EQ:eq0032]]\n\n[[EQ:eq0033]]\n\n[[EQ:eq0034]]\n\n[[EQ:eq0035]]\n\n[[EQ:eq0036]]\n\n[[EQ:eq0037]]\n\n[[EQ:eq0038]]\n\n[[EQ:eq0039]]\n\n[[EQ:eq0040]]\n\n[[EQ:eq0041]]\n\n[[EQ:eq0042]]\n\n[[EQ:eq0043]]\n\n[[EQ:eq0044]]\n\n[[EQ:eq0045]]\n\n[[EQ:eq0046]]\n\n[[EQ:eq0047]]\n\n[[EQ:eq0048]]\n\n[[EQ:eq0049]]\n\n[[EQ:eq0050]]\n\n[[EQ:eq0051]]\n\n[[EQ:eq0052]]\n\n[[EQ:eq0053]]\n\n[[EQ:eq0054]]\n\n[[EQ:eq0055]]\n\n[[EQ:eq0056]]\n\n[[EQ:eq0057]]\n\n[[EQ:eq0058]]\n\n[[EQ:eq0059]]\n\n[[EQ:eq0060]]\n\n[[EQ:eq0061]]\n\n[[EQ:eq0062]]\n\n[[EQ:eq0063]]\n\n[[EQ:eq0064]]\n\n[[EQ:eq0065]]\n\n[[EQ:eq0066]]\n\n[[EQ:eq0067]]\n\n[[EQ:eq0068]]\n\n[[EQ:eq0069]]\n\n[[EQ:eq0070]]\n\n[[EQ:eq0071]]\n\n[[EQ:eq0072]]\n\n[[EQ:eq0073]]\n\n[[EQ:eq0074]]\n\n[[EQ:eq0075]]\n\n[[EQ:eq0076]]\n\n[[EQ:eq0077]]\n\n[[EQ:eq0078]]\n\n[[EQ:eq0079]]\n\n[[EQ:eq0080]]\n\n[[EQ:eq0081]]\n\n[[EQ:eq0082]]\n\n[[EQ:eq0083]]\n\n[[EQ:eq0084]]\n\n[[EQ:eq0085]]\n\n[[EQ:eq0086]]\n\n[[EQ:eq0087]]\n\n[[EQ:eq0088]]\n\n[[EQ:eq0089]]\n\n[[EQ:eq0090]]\n\n[[EQ:eq0091]]\n\n[[EQ:eq0092]]\n\n[[EQ:eq0093]]\n\n[[EQ:eq0094]]\n\n[[EQ:eq0095]]\n\n[[EQ:eq0096]]\n\n[[EQ:eq0097]]\n", "sections": [{"level": 1, "title": "Preliminaries: Quantale, Polarities, Kan", "anchor": "preliminaries-quantale-polarities-kan", "char_span": [0, 0]}, {"level": 1, "title": "A Translation Dictionary: Recent Methods → TRoT", "anchor": "a-translation-dictionary-recent-methods-trot", "char_span": [0, 0]}, {"level": 1, "title": "Main Theorems (Fully Proven)", "anchor": "main-theorems-fully-proven", "char_span": [0, 0]}, {"level": 2, "title": "Nonexpansiveness and Adjunction", "anchor": "nonexpansiveness-and-adjunction", "char_span": [0, 1220]}, {"level": 2, "title": "Normal Form and Stability Bounds", "anchor": "normal-form-and-stability-bounds", "char_span": [1220, 2823]}, {"level": 2, "title": "Anytime-Valid Auditing and Reproducibility", "anchor": "anytime-valid-auditing-and-reproducibility", "char_span": [2823, 4331]}, {"level": 2, "title": "Entropic-MBR (eMBR) and Bounds", "anchor": "entropic-mbr-embr-and-bounds", "char_span": [4331, 4361]}, {"level": 1, "title": "Algorithms with Types & Complexity", "anchor": "algorithms-with-types-complexity", "char_span": [4361, 8342]}, {"level": 1, "title": "Figures (with Legends)", "anchor": "figures-with-legends", "char_span": [8342, 10566]}, {"level": 1, "title": "Implementation Blueprint (GPU/Log-Domain)", "anchor": "implementation-blueprint-gpu-log-domain", "char_span": [10566, 11408]}, {"level": 1, "title": "Unified Experimental Protocol", "anchor": "unified-experimental-protocol", "char_span": [11408, 16902]}]}, "equations": [{"id": "eq0001", "inline": false, "tex": "\\begin{equation}\np = \\exp(-\\lambda_{\\mathrm{cost}}\\, c),\\qquad c = -\\tfrac{1}{\\lambda_{\\mathrm{cost}}}\\log p,\\quad \\lambda_{\\mathrm{cost}}>0. \\label{eq:dictionary}\n\\end{equation}", "tex_normalized": "p = \\exp(-\\lambda_{\\mathrm{cost}} c),\\qquad c = -\\tfrac{1}{\\lambda_{\\mathrm{cost}}}\\log p,\\quad \\lambda_{\\mathrm{cost}}>0. \\label{eq:dictionary}", "mathml": "p=exp(−λcostc),c=−1λcostlogp,λcost>0.\\labeleq:dictionary", "char_span": [15478, 15491], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0002", "inline": false, "tex": "\\begin{align}\n(\\Lan_J F)[b] &= \\bigvee_{a\\in A} \\Big(K[b,Ja] + F[a]\\Big) = \\min_{a}\\ \\{K[b,Ja]+F[a]\\}, \\label{eq:lan}\\\\\n(\\Ran_J G)[a] &= \\bigwedge_{b\\in B} \\Big(K[b,Ja]\\Res G[b]\\Big) = \\max_{b}\\ \\{\\max(G[b]-K[b,Ja],0)\\}. \\label{eq:ran}\n\\end{align}", "tex_normalized": "(\\Lan_J F)[b] &= \\bigvee_{a\\in A} \\Big(K[b,Ja] + F[a]\\Big) = \\min_{a}\\ \\{K[b,Ja]+F[a]\\}, \\label{eq:lan}\\\\ (\\Ran_J G)[a] &= \\bigwedge_{b\\in B} \\Big(K[b,Ja]\\Res G[b]\\Big) = \\max_{b}\\ \\{\\max(G[b]-K[b,Ja],0)\\}. \\label{eq:ran}", "mathml": "(\\LanJF)[b]&=⋁a∈A(K[b,Ja]+F[a])=mina {K[b,Ja]+F[a]},\\labeleq:lan(\\RanJG)[a]&=⋀b∈B(K[b,Ja]\\ResG[b])=maxb {max(G[b]−K[b,Ja],0)}.\\labeleq:ran", "char_span": [15493, 15506], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0003", "inline": false, "tex": "\\begin{align}\n\\softmin_{\\lambda}\\!\\left(\\{\\,K[y,x]+c(x)\\,\\}_{x\\in X}\\right)\n&= -\\frac{1}{\\lambda}\\log\\!\\sum_{x\\in X}\n \\exp\\!\\big(-\\lambda\\big(K[y,x]+c(x)\\big)\\big) \\nonumber\\\\\n&= \\rho_{\\lambda}(y).\\label{eq:softmin-equality}\n\\end{align}", "tex_normalized": "\\softmin_{\\lambda} \\left(\\{ K[y,x]+c(x) \\}_{x\\in X}\\right) &= -\\frac{1}{\\lambda}\\log \\sum_{x\\in X} \\exp \\big(-\\lambda\\big(K[y,x]+c(x)\\big)\\big) \\nonumber\\\\ &= \\rho_{\\lambda}(y).\\label{eq:softmin-equality}", "mathml": "\\softminλ({K[y,x]+c(x)}x∈X)&=−1λlog∑x∈Xexp(−λ(K[y,x]+c(x)))\\nonumber&=ρλ(y).\\labeleq:softmin−equality", "char_span": [4818, 4831], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0004", "inline": false, "tex": "\\[0.3em]\n\\small \\href{https://orcid.org/0009-0004-4273-3365}{ORCID: 0009-0004-4273-3365}}\n\\date{October 19, 2025}\n\n% ---------- Hyperref last ----------\n\\usepackage{hyperref}\n\\hypersetup{\n colorlinks=true,\n linkcolor=blue!60!black,\n urlcolor=magenta!60!black,\n citecolor=blue!60!black,\n pdftitle={Inference in Normal Form: Unifying LLM Tricks via TRoT},\n pdfauthor={K. Takahashi},\n pdfcreator={LaTeX with hyperref},\n pdfsubject={LLM inference-time methods unified via TRoT: Kan extensions, residuation, nuclei/masks, and auditable RAVE},\n pdfkeywords={TRoT, Kan extension, residuation, tropical algebra, LLM inference, MBR, eMBR, conformal LM, RAVE, GraphBLAS}\n}\n\n\\begin{document}\n\\setstretch{1.3}\n\\maketitle\n\n\\begin{abstract}\nWe present a theory-first, implementation-faithful bridge between recent inference-time methods for large language models (LLMs)---diversifying decoding (Locally Typical, Contrastive Decoding, VS), risk-minimizing decoding (MBR/eMBR), LLM-as-a-Judge, Conformal Language Modeling, Self-RAG, structured reasoning (ToT/GoT), and preference optimization (DPO/ORPO/KTO/IPO)---and the \\emph{Theory of Relativity of Theories} (TRoT).\nTRoT supplies a right-written profunctor calculus with left Kan (generation), right Kan (safety), residuation, nuclei/masks, and an auditable relative-evaluation layer (RAVE).\nOur main results are: (i) \\textbf{Implementation Equivalence}: canonical inference-time pipelines admit a normal form as a finite path expansion followed by $\\Lan/\\Ran$ and an observation/aggregation kernel, (ii) \\textbf{Nonexpansiveness \\& Geometric Truncation Bounds}: under $1$-Lipschitz residuals and exogenous decay $q_\\star<1$, truncation and soft-min errors admit explicit upper bounds, and (iii) \\textbf{Auditable Reproducibility}: anytime-valid evidence and RAVE yield familywise error control for decisions recorded as public logs.\nAll constructions are GPU-ready (right-written GraphBLAS over the tropical $(\\min,+)$ semiring: SpGEMM/SpMV + elementwise residual + max-reduce) and supported by a unified evaluation protocol.\n\\end{abstract}\n\n% ============================================================\n\\section{Preliminaries: Quantale, Polarities, Kan}\n\\label{sec:prelim}\n\n\\paragraph{Quantale \\& order.}\nLet $(\\Cost,\\mathbf{\\ge},+,0)$ be the Lawvere cost quantale with carrier $\\mathbb{R}_{\\ge0}\\cup\\{+\\infty\\}$, monoidal product $+$, unit $0$, and \\emph{reversed order} ($c\\ge c'$ reads ``no more costly'').\nHence $\\bigvee$ is the numeric $\\inf$ and $\\bigwedge$ is the numeric $\\sup$.\nThe residual $a\\Res b:=\\sup\\{x: a+x\\le b\\}=\\max(b-a,0)$ satisfies $a+x\\le b\\ \\text{(numeric)}\\ \\iff\\ x\\le a\\Res b$.\n\n\\paragraph{Order notation.}\nWe write $x \\preceq y$ for the \\emph{enriched} order on $\\Cost$ (i.e., numeric $x\\ge y$),\nand reserve $x \\le y$ for the usual numeric order on $\\mathbb{R}\\cup\\{+\\infty\\}$.\nUnless explicitly marked by $\\preceq$, inequalities in proofs are numeric.\n\n\\paragraph{Polarities \\& exponential dictionary.}\nProbabilities $(\\Prob,\\cdot,1)$ and costs $(\\Cost,+,0)$ are linked by\n\nEQPH_eq0001_PH\n\n\n\\paragraph{Weighted relations and Kan.}\nGiven a forward map $J:A\\to B$ and a $\\Cost$-weighted relation $K[-,-]\\in \\Cost^{B\\times A}$ (transport/kernel),\nfor $F\\in\\Cost^{A}$ and $G\\in\\Cost^{B}$ define\n\nEQPH_eq0002_PH\n\n\n\\noindent\\emph{Typing note.}\nWe fix $\\Ran_J:\\Cost^B\\to\\Cost^A$ as in \\eqref{eq:ran}. When convenient we also use the equivalent $B$-indexed variant $(\\Ran^\\top_J F)[b]=\\bigwedge_{a}(K[b,Ja]\\Res F[a])$; Proposition~\\ref{prop:adjunction} uses \\eqref{eq:ran}.\n\n\\paragraph{Observers and nuclei.}\nObservers $\\mathrm{Obs}:\\Cost^B\\to\\Cost^B$ are $1$-Lipschitz in the log-domain (cf.\\ \\eqref{eq:dictionary}) or followed by a nucleus $\\nu$.\n\\textbf{Definition (nucleus).} A nucleus $\\nu$ is a monotone, idempotent ($\\nu^2=\\nu$), $1$-Lipschitz projector.\nIn pipelines, we apply the stability contract as $\\mathrm{Obs}\\ \\Rightarrow\\ \\nu\\ \\Rightarrow$ subsequent $\\Lan/\\Ran$ stages.\nMasks $\\mathcal{M}$ send forbidden coordinates to $+\\infty$ prior to arithmetic.\n\n\\paragraph{Notation.}\n$\\|f\\|_\\infty:=\\sup_x |f(x)|$; $d_{\\max}$ denotes the maximal in-degree \\emph{after masking} (effective fan-in) in \\eqref{eq:lan}; $\\lambda_{\\mathrm{cost}}$ is the soft-min inverse temperature.\n\n% ---------------- Polarity Quick Reference ----------------\n\\begin{table}[t]\n\\centering\n\\small\n\\setlength{\\tabcolsep}{4pt}\n\\resizebox{\\linewidth}{!}{%\n\\begin{tabular}{@{}lcc@{}}\n\\toprule\n\\textbf{Item} & \\textbf{Cost polarity $(\\Cost,\\ge,+,0)$} & \\textbf{Prob. polarity $(\\Prob,\\le,\\cdot,1)$ (via \\eqref{eq:dictionary})} \\\\\n\\midrule\nOrder & enriched $\\preceq$ = numeric $\\ge$ & numeric $\\le$ \\\\\nJoin/Meet & $\\join=\\inf$ (numeric), $\\meet=\\sup$ (numeric) & $\\join=\\sup$, $\\meet=\\inf$ \\\\\nResidual & $a+x\\preceq b \\ \\Leftrightarrow\\ x\\preceq a\\Res b$ & $ab\\le c \\ \\Leftrightarrow\\ b\\le c/a$ \\\\\n$\\Lan$ & $(\\Lan_J F)[b]=\\inf_a(K[b,Ja]+F[a])$ & $\\sup_a P[b,Ja]\\cdot Q[a]$ \\\\\n$\\Ran$ & $(\\Ran_J G)[a]=\\sup_b(K[b,Ja]\\Res G[b])$ & $\\inf_b G[b]/P[b,Ja]$ \\\\\nSoft-min & approximates $\\join=\\inf$ & soft-max approximates $\\meet=\\sup$ \\\\\n\\bottomrule\n\\end{tabular}}\n\\caption{Polarity quick reference (Cost vs Prob).}\n\\label{tab:polarity}\n\\vspace{0.2em}\\emph{Note.} The Prob column is in the max--product polarity (monoidal product $\\cdot$, order $\\le$).\nFor sum--product semantics, replace exact joins by log-sum-exp (soft-min in Cost), which recovers the entropic/lse counterparts in Section~\\ref{sec:eMBR}.\nEntries with $P[b,Ja]=0$ correspond to $+\\infty$ in Cost and are handled via masking.\n\\end{table}\n\n% ============================================================\n\\section{A Translation Dictionary: Recent Methods $\\to$ TRoT}\n\\label{sec:dictionary}\n\\begin{itemize}[leftmargin=1.25em]\n\\item \\textbf{Diversifying decoding (LTS/CD/VS).}\nLTS constrains token information; CD computes strong--weak contrasts.\nBoth are \\emph{$\\Lan$-side candidate expansions} with \\emph{value-mask} updates; CD uses a weak-model-induced $J$ and $\\Ran$ consolidates.\nVS yields a language-encoded mixture $\\hat q=\\{(y_i,p_i)\\}$; via \\eqref{eq:dictionary} this becomes a cost prior injected before $\\Lan/\\Ran$.\n\\item \\textbf{MBR and entropic risk (eMBR).}\nClassical MBR minimizes $\\E_{x\\sim \\hat q}[\\ell(y,x)]$.\nIn TRoT, replacing the exact expectation by \\emph{entropic risk} $\\rho_\\lambda(y):= -\\lambda^{-1}\\log\\sum_x \\hat q(x)\\,e^{-\\lambda \\ell(y,x)}$ matches a \\emph{soft-min} $\\Lan$-lift with log-domain weights (Sec.~\\ref{sec:eMBR}).\n\\item \\textbf{LLM-as-a-Judge.}\nJudges are observation kernels combined with RAVE (BTL/Kemeny, SP, LMSR) producing auditable, relative consensus.\n\\item \\textbf{Conformal LM.}\nSet-valued guarantees act as an anytime-valid \\emph{gate} interleaved with $\\Lan/\\Ran$; violations are controlled while preserving reachability.\n\\emph{Assumptions:} split/online exchangeability, a fixed (or $\\mathcal{F}_{t-1}$-predictably updated) nonconformity score, predictable splits $\\mathcal{F}_{t-1}$ when used online; all calibration updates are logged.\n\\item \\textbf{Self-RAG / ToT/GoT.}\nRetrieval and structured search are finite path expansions whose nodes update under $\\Lan/\\Ran$; pruning is governed by geometric decay.\n\\end{itemize}\n\n% --- Method→Normal-Form Mapping Table ---\n\\begin{table}[t]\n\\centering\n\\scriptsize\n\\setlength{\\tabcolsep}{3.5pt}\n\\resizebox{\\linewidth}{!}{%\n\\begin{tabular}{@{}lcccccc@{}}\n\\toprule\n\\textbf{Method} & \\textbf{Path} & \\textbf{$\\Lan$} & \\textbf{Obs} & \\textbf{$\\Ran$} & \\textbf{RAVE} & \\textbf{Gate} \\\\\n\\midrule\nLTS & $\\checkmark$ & mask-aware lift & --- & consolidate & opt. & opt. \\\\\nCD & weak-$J$ path & contrast lift & --- & consolidate & opt. & opt. \\\\\nVS & prompt path & prior-injected lift & --- & consolidate & opt. & opt. \\\\\nMBR & sample path & --- & risk eval. & --- & opt. & --- \\\\\neMBR & sample path & soft-min lift & --- & --- & opt. & --- \\\\\nLLM Judge & --- & --- & judge kernel & --- & BTL/Kemeny & e-process \\\\\nConformal LM & --- & --- & nonconf. score & --- & --- & conformal \\\\\nSelf-RAG & retriever path & lift & retriever obs & reduce & opt. & opt. \\\\\nToT/GoT & graph path & lift & heuristic obs & reduce & opt. & opt. \\\\\n\\bottomrule\n\\end{tabular}}\n\\caption{Where each method plugs into the TRoT normal form.}\n\\label{tab:mapping}\n\\end{table}\n\n% ============================================================\n\\section{Main Theorems (Fully Proven)}\n\\subsection{Nonexpansiveness and Adjunction}\n\\begin{lemma}[Residual Nonexpansiveness]\\label{lem:residual}\nFor fixed $a$, $b\\mapsto a\\Res b=\\max(b-a,0)$ is $1$-Lipschitz; for fixed $b$, $a\\mapsto a\\Res b$ is $1$-Lipschitz. Hence $G\\mapsto \\Ran_J G$ is nonexpansive in $\\|\\cdot\\|_\\infty$.\n\\end{lemma}\n\\begin{proof}\nFix $a$. For $b \\le b'$ (numeric order), $(a\\Res b')-(a\\Res b)\\in[0,b'-b]$, so $| (a\\Res b')-(a\\Res b)|\\le |b'-b|$.\nFix $b$. For $a\\le a'$, $(a\\Res b)-(a'\\Res b)\\in[0,a'-a]$.\nPointwise $1$-Lipschitzness and that $\\bigwedge$ is numeric $\\sup$ imply $\\Ran_J$ is nonexpansive.\n\\end{proof}\n\n\\begin{lemma}[Left-Kan Nonexpansiveness]\\label{lem:lan-lip}\n$F\\mapsto \\Lan_J F$ is $1$-Lipschitz under $\\|\\cdot\\|_\\infty$.\n\\end{lemma}\n\\begin{proof}\nFor any $b$, $(\\Lan_J F)[b]=\\min_a \\{K[b,Ja]+F[a]\\}$ is the numeric $\\inf$ of $1$-Lipschitz affine shifts of $F$; taking $\\inf$ preserves the $1$-Lipschitz constant.\nThe pointwise infimum of $1$-Lipschitz functions remains $1$-Lipschitz in $\\|\\cdot\\|_\\infty$.\n\\end{proof}\n\n\\begin{proposition}[Enriched Adjunction]\\label{prop:adjunction}\nFor $F\\in\\Cost^A$ and $G\\in\\Cost^B$,\n\\[\n\\Lan_J F \\preceq G\\quad \\Longleftrightarrow\\quad F \\preceq \\Ran_J G .\n\\]", "tex_normalized": "0.3em] \\small \\href{https://orcid.org/0009-0004-4273-3365}{ORCID: 0009-0004-4273-3365}} \\date{October 19, 2025} % ---------- Hyperref last ---------- \\usepackage{hyperref} \\hypersetup{ colorlinks=true, linkcolor=blue!60!black, urlcolor=magenta!60!black, citecolor=blue!60!black, pdftitle={Inference in Normal Form: Unifying LLM Tricks via TRoT}, pdfauthor={K. Takahashi}, pdfcreator={LaTeX with hyperref}, pdfsubject={LLM inference-time methods unified via TRoT: Kan extensions, residuation, nuclei/masks, and auditable RAVE}, pdfkeywords={TRoT, Kan extension, residuation, tropical algebra, LLM inference, MBR, eMBR, conformal LM, RAVE, GraphBLAS} } \\begin{document} \\setstretch{1.3} \\maketitle \\begin{abstract} We present a theory-first, implementation-faithful bridge between recent inference-time methods for large language models (LLMs)---diversifying decoding (Locally Typical, Contrastive Decoding, VS), risk-minimizing decoding (MBR/eMBR), LLM-as-a-Judge, Conformal Language Modeling, Self-RAG, structured reasoning (ToT/GoT), and preference optimization (DPO/ORPO/KTO/IPO)---and the \\emph{Theory of Relativity of Theories} (TRoT). TRoT supplies a right-written profunctor calculus with left Kan (generation), right Kan (safety), residuation, nuclei/masks, and an auditable relative-evaluation layer (RAVE). Our main results are: (i) \\textbf{Implementation Equivalence}: canonical inference-time pipelines admit a normal form as a finite path expansion followed by $\\Lan/\\Ran$ and an observation/aggregation kernel, (ii) \\textbf{Nonexpansiveness \\& Geometric Truncation Bounds}: under $1$-Lipschitz residuals and exogenous decay $q_\\star<1$, truncation and soft-min errors admit explicit upper bounds, and (iii) \\textbf{Auditable Reproducibility}: anytime-valid evidence and RAVE yield familywise error control for decisions recorded as public logs. All constructions are GPU-ready (right-written GraphBLAS over the tropical $(\\min,+)$ semiring: SpGEMM/SpMV + elementwise residual + max-reduce) and supported by a unified evaluation protocol. \\end{abstract} % ============================================================ \\section{Preliminaries: Quantale, Polarities, Kan} \\label{sec:prelim} \\paragraph{Quantale \\& order.} Let $(\\Cost,\\mathbf{\\ge},+,0)$ be the Lawvere cost quantale with carrier $\\mathbb{R}_{\\ge0}\\cup\\{+\\infty\\}$, monoidal product $+$, unit $0$, and \\emph{reversed order} ($c\\ge c'$ reads ``no more costly''). Hence $\\bigvee$ is the numeric $\\inf$ and $\\bigwedge$ is the numeric $\\sup$. The residual $a\\Res b:=\\sup\\{x: a+x\\le b\\}=\\max(b-a,0)$ satisfies $a+x\\le b\\ \\text{(numeric)}\\ \\iff\\ x\\le a\\Res b$. \\paragraph{Order notation.} We write $x \\preceq y$ for the \\emph{enriched} order on $\\Cost$ (i.e., numeric $x\\ge y$), and reserve $x \\le y$ for the usual numeric order on $\\mathbb{R}\\cup\\{+\\infty\\}$. Unless explicitly marked by $\\preceq$, inequalities in proofs are numeric. \\paragraph{Polarities \\& exponential dictionary.} Probabilities $(\\Prob,\\cdot,1)$ and costs $(\\Cost,+,0)$ are linked by EQPH_eq0001_PH \\paragraph{Weighted relations and Kan.} Given a forward map $J:A\\to B$ and a $\\Cost$-weighted relation $K[-,-]\\in \\Cost^{B\\times A}$ (transport/kernel), for $F\\in\\Cost^{A}$ and $G\\in\\Cost^{B}$ define EQPH_eq0002_PH \\noindent\\emph{Typing note.} We fix $\\Ran_J:\\Cost^B\\to\\Cost^A$ as in \\eqref{eq:ran}. When convenient we also use the equivalent $B$-indexed variant $(\\Ran^\\top_J F)[b]=\\bigwedge_{a}(K[b,Ja]\\Res F[a])$; Proposition~\\ref{prop:adjunction} uses \\eqref{eq:ran}. \\paragraph{Observers and nuclei.} Observers $\\mathrm{Obs}:\\Cost^B\\to\\Cost^B$ are $1$-Lipschitz in the log-domain (cf.\\ \\eqref{eq:dictionary}) or followed by a nucleus $\\nu$. \\textbf{Definition (nucleus).} A nucleus $\\nu$ is a monotone, idempotent ($\\nu^2=\\nu$), $1$-Lipschitz projector. In pipelines, we apply the stability contract as $\\mathrm{Obs}\\ \\Rightarrow\\ \\nu\\ \\Rightarrow$ subsequent $\\Lan/\\Ran$ stages. Masks $\\mathcal{M}$ send forbidden coordinates to $+\\infty$ prior to arithmetic. \\paragraph{Notation.} $\\|f\\|_\\infty:=\\sup_x |f(x)|$; $d_{\\max}$ denotes the maximal in-degree \\emph{after masking} (effective fan-in) in \\eqref{eq:lan}; $\\lambda_{\\mathrm{cost}}$ is the soft-min inverse temperature. % ---------------- Polarity Quick Reference ---------------- \\begin{table}[t] \\centering \\small \\setlength{\\tabcolsep}{4pt} \\resizebox{\\linewidth}{!}{% \\begin{tabular}{@{}lcc@{}} \\toprule \\textbf{Item} & \\textbf{Cost polarity $(\\Cost,\\ge,+,0)$} & \\textbf{Prob. polarity $(\\Prob,\\le,\\cdot,1)$ (via \\eqref{eq:dictionary})} \\\\ \\midrule Order & enriched $\\preceq$ = numeric $\\ge$ & numeric $\\le$ \\\\ Join/Meet & $\\join=\\inf$ (numeric), $\\meet=\\sup$ (numeric) & $\\join=\\sup$, $\\meet=\\inf$ \\\\ Residual & $a+x\\preceq b \\ \\Leftrightarrow\\ x\\preceq a\\Res b$ & $ab\\le c \\ \\Leftrightarrow\\ b\\le c/a$ \\\\ $\\Lan$ & $(\\Lan_J F)[b]=\\inf_a(K[b,Ja]+F[a])$ & $\\sup_a P[b,Ja]\\cdot Q[a]$ \\\\ $\\Ran$ & $(\\Ran_J G)[a]=\\sup_b(K[b,Ja]\\Res G[b])$ & $\\inf_b G[b]/P[b,Ja]$ \\\\ Soft-min & approximates $\\join=\\inf$ & soft-max approximates $\\meet=\\sup$ \\\\ \\bottomrule \\end{tabular}} \\caption{Polarity quick reference (Cost vs Prob).} \\label{tab:polarity} \\vspace{0.2em}\\emph{Note.} The Prob column is in the max--product polarity (monoidal product $\\cdot$, order $\\le$). For sum--product semantics, replace exact joins by log-sum-exp (soft-min in Cost), which recovers the entropic/lse counterparts in Section~\\ref{sec:eMBR}. Entries with $P[b,Ja]=0$ correspond to $+\\infty$ in Cost and are handled via masking. \\end{table} % ============================================================ \\section{A Translation Dictionary: Recent Methods $\\to$ TRoT} \\label{sec:dictionary} \\begin{itemize}[leftmargin=1.25em] \\item \\textbf{Diversifying decoding (LTS/CD/VS).} LTS constrains token information; CD computes strong--weak contrasts. Both are \\emph{$\\Lan$-side candidate expansions} with \\emph{value-mask} updates; CD uses a weak-model-induced $J$ and $\\Ran$ consolidates. VS yields a language-encoded mixture $\\hat q=\\{(y_i,p_i)\\}$; via \\eqref{eq:dictionary} this becomes a cost prior injected before $\\Lan/\\Ran$. \\item \\textbf{MBR and entropic risk (eMBR).} Classical MBR minimizes $\\E_{x\\sim \\hat q}[\\ell(y,x)]$. In TRoT, replacing the exact expectation by \\emph{entropic risk} $\\rho_\\lambda(y):= -\\lambda^{-1}\\log\\sum_x \\hat q(x) e^{-\\lambda \\ell(y,x)}$ matches a \\emph{soft-min} $\\Lan$-lift with log-domain weights (Sec.~\\ref{sec:eMBR}). \\item \\textbf{LLM-as-a-Judge.} Judges are observation kernels combined with RAVE (BTL/Kemeny, SP, LMSR) producing auditable, relative consensus. \\item \\textbf{Conformal LM.} Set-valued guarantees act as an anytime-valid \\emph{gate} interleaved with $\\Lan/\\Ran$; violations are controlled while preserving reachability. \\emph{Assumptions:} split/online exchangeability, a fixed (or $\\mathcal{F}_{t-1}$-predictably updated) nonconformity score, predictable splits $\\mathcal{F}_{t-1}$ when used online; all calibration updates are logged. \\item \\textbf{Self-RAG / ToT/GoT.} Retrieval and structured search are finite path expansions whose nodes update under $\\Lan/\\Ran$; pruning is governed by geometric decay. \\end{itemize} % --- Method→Normal-Form Mapping Table --- \\begin{table}[t] \\centering \\scriptsize \\setlength{\\tabcolsep}{3.5pt} \\resizebox{\\linewidth}{!}{% \\begin{tabular}{@{}lcccccc@{}} \\toprule \\textbf{Method} & \\textbf{Path} & \\textbf{$\\Lan$} & \\textbf{Obs} & \\textbf{$\\Ran$} & \\textbf{RAVE} & \\textbf{Gate} \\\\ \\midrule LTS & $\\checkmark$ & mask-aware lift & --- & consolidate & opt. & opt. \\\\ CD & weak-$J$ path & contrast lift & --- & consolidate & opt. & opt. \\\\ VS & prompt path & prior-injected lift & --- & consolidate & opt. & opt. \\\\ MBR & sample path & --- & risk eval. & --- & opt. & --- \\\\ eMBR & sample path & soft-min lift & --- & --- & opt. & --- \\\\ LLM Judge & --- & --- & judge kernel & --- & BTL/Kemeny & e-process \\\\ Conformal LM & --- & --- & nonconf. score & --- & --- & conformal \\\\ Self-RAG & retriever path & lift & retriever obs & reduce & opt. & opt. \\\\ ToT/GoT & graph path & lift & heuristic obs & reduce & opt. & opt. \\\\ \\bottomrule \\end{tabular}} \\caption{Where each method plugs into the TRoT normal form.} \\label{tab:mapping} \\end{table} % ============================================================ \\section{Main Theorems (Fully Proven)} \\subsection{Nonexpansiveness and Adjunction} \\begin{lemma}[Residual Nonexpansiveness]\\label{lem:residual} For fixed $a$, $b\\mapsto a\\Res b=\\max(b-a,0)$ is $1$-Lipschitz; for fixed $b$, $a\\mapsto a\\Res b$ is $1$-Lipschitz. Hence $G\\mapsto \\Ran_J G$ is nonexpansive in $\\|\\cdot\\|_\\infty$. \\end{lemma} \\begin{proof} Fix $a$. For $b \\le b'$ (numeric order), $(a\\Res b')-(a\\Res b)\\in[0,b'-b]$, so $| (a\\Res b')-(a\\Res b)|\\le |b'-b|$. Fix $b$. For $a\\le a'$, $(a\\Res b)-(a'\\Res b)\\in[0,a'-a]$. Pointwise $1$-Lipschitzness and that $\\bigwedge$ is numeric $\\sup$ imply $\\Ran_J$ is nonexpansive. \\end{proof} \\begin{lemma}[Left-Kan Nonexpansiveness]\\label{lem:lan-lip} $F\\mapsto \\Lan_J F$ is $1$-Lipschitz under $\\|\\cdot\\|_\\infty$. \\end{lemma} \\begin{proof} For any $b$, $(\\Lan_J F)[b]=\\min_a \\{K[b,Ja]+F[a]\\}$ is the numeric $\\inf$ of $1$-Lipschitz affine shifts of $F$; taking $\\inf$ preserves the $1$-Lipschitz constant. The pointwise infimum of $1$-Lipschitz functions remains $1$-Lipschitz in $\\|\\cdot\\|_\\infty$. \\end{proof} \\begin{proposition}[Enriched Adjunction]\\label{prop:adjunction} For $F\\in\\Cost^A$ and $G\\in\\Cost^B$, \\[ \\Lan_J F \\preceq G\\quad \\Longleftrightarrow\\quad F \\preceq \\Ran_J G .", "mathml": "0.3em]ORCID:0009−0004−4273−3365}\\dateOctober19,2025", "char_span": [15508, 15521], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0005", "inline": false, "tex": "\\[\n\\|F-F_k\\|_\\infty \\le \\frac{q_\\star^{k+1}}{1-q_\\star}.\n\\]", "tex_normalized": "\\|F-F_k\\|_\\infty \\le \\frac{q_\\star^{k+1}}{1-q_\\star}.", "mathml": "\\[‖F−Fk‖∞≤q⋆k+11−q⋆.\\]", "char_span": [15523, 15536], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0006", "inline": false, "tex": "\\[\n\\varepsilon_{\\mathrm{soft}} \\le \\frac{k \\log d_{\\max}}{\\lambda_{\\mathrm{cost}}}.\n\\]", "tex_normalized": "\\varepsilon_{\\mathrm{soft}} \\le \\frac{k \\log d_{\\max}}{\\lambda_{\\mathrm{cost}}}.", "mathml": "\\[εsoft≤klogdmaxλcost.\\]", "char_span": [15538, 15551], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0007", "inline": false, "tex": "\\[\n\\Pr(\\exists t:\\,M_t\\ge 1/\\alpha_t)\\le \\alpha_{\\mathrm{global}},\n\\]", "tex_normalized": "\\Pr(\\exists t: M_t\\ge 1/\\alpha_t)\\le \\alpha_{\\mathrm{global}},", "mathml": "\\[Pr(∃t:Mt≥1/αt)≤αglobal,\\]", "char_span": [15553, 15566], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0008", "inline": false, "tex": "\\[\n\\rho_\\lambda(y):=-\\frac{1}{\\lambda}\\log\\sum_x \\hat q(x)\\,e^{-\\lambda \\ell(y,x)}.\n\\]", "tex_normalized": "\\rho_\\lambda(y):=-\\frac{1}{\\lambda}\\log\\sum_x \\hat q(x) e^{-\\lambda \\ell(y,x)}.", "mathml": "\\[ρλ(y):=−1λlog∑xq^(x)e−λℓ(y,x).\\]", "char_span": [15568, 15581], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0009", "inline": false, "tex": "\\[\n\\min_x \\ell(y,x)\\ \\le\\ \\rho_\\lambda(y)\\ \\le\\ \\E_{\\hat q}[\\ell(y,X)].\n\\]", "tex_normalized": "\\min_x \\ell(y,x)\\ \\le\\ \\rho_\\lambda(y)\\ \\le\\ \\E_{\\hat q}[\\ell(y,X)].", "mathml": "\\[minxℓ(y,x) ≤ ρλ(y) ≤ \\Eq^[ℓ(y,X)].\\]", "char_span": [4918, 4931], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0010", "inline": true, "tex": "$\\Rightarrow$", "tex_normalized": "\\Rightarrow", "mathml": "$⇒$", "char_span": [15583, 15596], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0011", "inline": true, "tex": "$\\Lan_J F \\preceq G$", "tex_normalized": "\\Lan_J F \\preceq G", "mathml": "$\\LanJF⪯G$", "char_span": [15598, 15611], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0012", "inline": true, "tex": "$(\\Lan_J F)[b]\\preceq G[b]$", "tex_normalized": "(\\Lan_J F)[b]\\preceq G[b]", "mathml": "$(\\LanJF)[b]⪯G[b]$", "char_span": [15613, 15626], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0013", "inline": true, "tex": "$b$", "tex_normalized": "b", "mathml": "$b$", "char_span": [15628, 15641], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0014", "inline": true, "tex": "$(\\Lan_J F)[b]=\\bigvee_a (K[b,Ja]+F[a])$", "tex_normalized": "(\\Lan_J F)[b]=\\bigvee_a (K[b,Ja]+F[a])", "mathml": "$(\\LanJF)[b]=⋁a(K[b,Ja]+F[a])$", "char_span": [15643, 15656], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0015", "inline": true, "tex": "$\\join$", "tex_normalized": "\\join", "mathml": "$\\join$", "char_span": [15658, 15671], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0016", "inline": true, "tex": "$(\\Lan_J F)[b]\\preceq G[b]$", "tex_normalized": "(\\Lan_J F)[b]\\preceq G[b]", "mathml": "$(\\LanJF)[b]⪯G[b]$", "char_span": [15673, 15686], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0017", "inline": true, "tex": "$\\inf_a\\{K[b,Ja]+F[a]\\}\\ge G[b]$", "tex_normalized": "\\inf_a\\{K[b,Ja]+F[a]\\}\\ge G[b]", "mathml": "$infa{K[b,Ja]+F[a]}≥G[b]$", "char_span": [15688, 15701], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0018", "inline": true, "tex": "$a$", "tex_normalized": "a", "mathml": "$a$", "char_span": [15703, 15716], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0019", "inline": true, "tex": "$K[b,Ja]+F[a]\\preceq G[b]$", "tex_normalized": "K[b,Ja]+F[a]\\preceq G[b]", "mathml": "$K[b,Ja]+F[a]⪯G[b]$", "char_span": [15718, 15731], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0020", "inline": true, "tex": "$F[a]\\preceq K[b,Ja]\\Res G[b]$", "tex_normalized": "F[a]\\preceq K[b,Ja]\\Res G[b]", "mathml": "$F[a]⪯K[b,Ja]\\ResG[b]$", "char_span": [15733, 15746], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0021", "inline": true, "tex": "$b$", "tex_normalized": "b", "mathml": "$b$", "char_span": [15748, 15761], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0022", "inline": true, "tex": "$F[a]\\preceq \\bigwedge_b (K[b,Ja]\\Res G[b])=(\\Ran_J G)[a]$", "tex_normalized": "F[a]\\preceq \\bigwedge_b (K[b,Ja]\\Res G[b])=(\\Ran_J G)[a]", "mathml": "$F[a]⪯⋀b(K[b,Ja]\\ResG[b])=(\\RanJG)[a]$", "char_span": [15763, 15776], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0023", "inline": true, "tex": "$\\Leftarrow$", "tex_normalized": "\\Leftarrow", "mathml": "$⇐$", "char_span": [15778, 15791], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0024", "inline": true, "tex": "$F\\preceq \\Ran_J G$", "tex_normalized": "F\\preceq \\Ran_J G", "mathml": "$F⪯\\RanJG$", "char_span": [15793, 15806], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0025", "inline": true, "tex": "$F[a]\\preceq \\bigwedge_b (K[b,Ja]\\Res G[b])$", "tex_normalized": "F[a]\\preceq \\bigwedge_b (K[b,Ja]\\Res G[b])", "mathml": "$F[a]⪯⋀b(K[b,Ja]\\ResG[b])$", "char_span": [15808, 15821], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0026", "inline": true, "tex": "$b$", "tex_normalized": "b", "mathml": "$b$", "char_span": [15823, 15836], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0027", "inline": true, "tex": "$F[a]\\preceq K[b,Ja]\\Res G[b]$", "tex_normalized": "F[a]\\preceq K[b,Ja]\\Res G[b]", "mathml": "$F[a]⪯K[b,Ja]\\ResG[b]$", "char_span": [15838, 15851], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0028", "inline": true, "tex": "$K[b,Ja]+F[a]\\preceq G[b]$", "tex_normalized": "K[b,Ja]+F[a]\\preceq G[b]", "mathml": "$K[b,Ja]+F[a]⪯G[b]$", "char_span": [15853, 15866], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0029", "inline": true, "tex": "$\\join$", "tex_normalized": "\\join", "mathml": "$\\join$", "char_span": [15868, 15881], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0030", "inline": true, "tex": "$a$", "tex_normalized": "a", "mathml": "$a$", "char_span": [15883, 15896], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0031", "inline": true, "tex": "$(\\Lan_J F)[b]\\preceq G[b]$", "tex_normalized": "(\\Lan_J F)[b]\\preceq G[b]", "mathml": "$(\\LanJF)[b]⪯G[b]$", "char_span": [15898, 15911], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0032", "inline": true, "tex": "$q_\\star\\in(0,1)$", "tex_normalized": "q_\\star\\in(0,1)", "mathml": "$q⋆∈(0,1)$", "char_span": [15913, 15926], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0033", "inline": true, "tex": "$D_i$", "tex_normalized": "D_i", "mathml": "$Di$", "char_span": [15928, 15941], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0034", "inline": true, "tex": "$d_{\\max}$", "tex_normalized": "d_{\\max}", "mathml": "$dmax$", "char_span": [15943, 15956], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0035", "inline": true, "tex": "$T_i$", "tex_normalized": "T_i", "mathml": "$Ti$", "char_span": [15958, 15971], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0036", "inline": true, "tex": "$D_i$", "tex_normalized": "D_i", "mathml": "$Di$", "char_span": [15973, 15986], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0037", "inline": true, "tex": "$\\|D_i\\|_{\\infty\\to\\infty}\\le q_\\star\\in(0,1)$", "tex_normalized": "\\|D_i\\|_{\\infty\\to\\infty}\\le q_\\star\\in(0,1)", "mathml": "$‖Di‖∞→∞≤q⋆∈(0,1)$", "char_span": [15988, 16001], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0038", "inline": true, "tex": "$F=\\prod_{i\\ge1}(D_i\\!\\circ\\!T_i)$", "tex_normalized": "F=\\prod_{i\\ge1}(D_i \\circ T_i)", "mathml": "$F=∏i≥1(Di∘Ti)$", "char_span": [16003, 16016], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0039", "inline": true, "tex": "$k$", "tex_normalized": "k", "mathml": "$k$", "char_span": [16018, 16031], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0040", "inline": true, "tex": "$F_k$", "tex_normalized": "F_k", "mathml": "$Fk$", "char_span": [16033, 16046], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0041", "inline": true, "tex": "$\\join$", "tex_normalized": "\\join", "mathml": "$\\join$", "char_span": [16048, 16061], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0042", "inline": true, "tex": "$\\softmin_{\\lambda}$", "tex_normalized": "\\softmin_{\\lambda}", "mathml": "$\\softminλ$", "char_span": [16063, 16076], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0043", "inline": true, "tex": "$\\lambda_{\\mathrm{cost}}$", "tex_normalized": "\\lambda_{\\mathrm{cost}}", "mathml": "$λcost$", "char_span": [16078, 16091], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0044", "inline": true, "tex": "$d_{\\max}$", "tex_normalized": "d_{\\max}", "mathml": "$dmax$", "char_span": [16093, 16106], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0045", "inline": true, "tex": "$k$", "tex_normalized": "k", "mathml": "$k$", "char_span": [16108, 16121], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0046", "inline": true, "tex": "$F$", "tex_normalized": "F", "mathml": "$F$", "char_span": [16123, 16136], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0047", "inline": true, "tex": "$(D_i)$", "tex_normalized": "(D_i)", "mathml": "$(Di)$", "char_span": [16138, 16151], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0048", "inline": true, "tex": "$\\|F-F_k\\|_\\infty \\le \\frac{q_\\star^{k+1}}{1-q_\\star}$", "tex_normalized": "\\|F-F_k\\|_\\infty \\le \\frac{q_\\star^{k+1}}{1-q_\\star}", "mathml": "$‖F−Fk‖∞≤q⋆k+11−q⋆$", "char_span": [16153, 16166], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0049", "inline": true, "tex": "$F$", "tex_normalized": "F", "mathml": "$F$", "char_span": [16168, 16181], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0050", "inline": true, "tex": "$\\|F-F_k\\|_\\infty \\le q_\\star^{k+1}$", "tex_normalized": "\\|F-F_k\\|_\\infty \\le q_\\star^{k+1}", "mathml": "$‖F−Fk‖∞≤q⋆k+1$", "char_span": [16183, 16196], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0051", "inline": true, "tex": "$\\|D_i\\circ T_i\\|\\le q_\\star$", "tex_normalized": "\\|D_i\\circ T_i\\|\\le q_\\star", "mathml": "$‖Di∘Ti‖≤q⋆$", "char_span": [16198, 16211], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0052", "inline": true, "tex": "$\\|F-F_k\\| \\le \\sum_{j=k+1}^\\infty q_\\star^{j} = \\frac{q_\\star^{k+1}}{1-q_\\star}$", "tex_normalized": "\\|F-F_k\\| \\le \\sum_{j=k+1}^\\infty q_\\star^{j} = \\frac{q_\\star^{k+1}}{1-q_\\star}", "mathml": "$‖F−Fk‖≤∑j=k+1∞q⋆j=q⋆k+11−q⋆$", "char_span": [16213, 16226], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0053", "inline": true, "tex": "$\\softmin_\\lambda(x)=-\\lambda^{-1}\\log\\sum_{i=1}^d e^{-\\lambda x_i}$", "tex_normalized": "\\softmin_\\lambda(x)=-\\lambda^{-1}\\log\\sum_{i=1}^d e^{-\\lambda x_i}", "mathml": "$\\softminλ(x)=−λ−1log∑i=1de−λxi$", "char_span": [16228, 16241], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0054", "inline": true, "tex": "$\\min_i x_i \\le \\softmin_\\lambda(x)\\le \\min_i x_i + \\lambda^{-1}\\log d$", "tex_normalized": "\\min_i x_i \\le \\softmin_\\lambda(x)\\le \\min_i x_i + \\lambda^{-1}\\log d", "mathml": "$minixi≤\\softminλ(x)≤minixi+λ−1logd$", "char_span": [16243, 16256], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0055", "inline": true, "tex": "$k$", "tex_normalized": "k", "mathml": "$k$", "char_span": [16258, 16271], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0056", "inline": true, "tex": "$(d_i)$", "tex_normalized": "(d_i)", "mathml": "$(di)$", "char_span": [16273, 16286], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0057", "inline": true, "tex": "$k\\log d_{\\max}$", "tex_normalized": "k\\log d_{\\max}", "mathml": "$klogdmax$", "char_span": [16288, 16301], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0058", "inline": true, "tex": "$\\sum_{i=1}^k \\log d_i$", "tex_normalized": "\\sum_{i=1}^k \\log d_i", "mathml": "$∑i=1klogdi$", "char_span": [16303, 16316], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0059", "inline": true, "tex": "$d_{\\max}$", "tex_normalized": "d_{\\max}", "mathml": "$dmax$", "char_span": [16318, 16331], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0060", "inline": true, "tex": "$(\\mathcal{F}_t)$", "tex_normalized": "(\\mathcal{F}_t)", "mathml": "$(ℱt)$", "char_span": [16333, 16346], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0061", "inline": true, "tex": "$M_t$", "tex_normalized": "M_t", "mathml": "$Mt$", "char_span": [16348, 16361], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0062", "inline": true, "tex": "$M_0=1$", "tex_normalized": "M_0=1", "mathml": "$M0=1$", "char_span": [16363, 16376], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0063", "inline": true, "tex": "$\\E[M_t|\\mathcal{F}_{t-1}]=M_{t-1}$", "tex_normalized": "\\E[M_t|\\mathcal{F}_{t-1}]=M_{t-1}", "mathml": "$\\E[Mt|ℱt−1]=Mt−1$", "char_span": [16378, 16391], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0064", "inline": true, "tex": "$e_t:=M_t/M_{t-1}\\ge0$", "tex_normalized": "e_t:=M_t/M_{t-1}\\ge0", "mathml": "$et:=Mt/Mt−1≥0$", "char_span": [16393, 16406], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0065", "inline": true, "tex": "$(\\alpha_t)$", "tex_normalized": "(\\alpha_t)", "mathml": "$(αt)$", "char_span": [16408, 16421], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0066", "inline": true, "tex": "$\\sum_t \\alpha_t \\le \\alpha_{\\mathrm{global}}$", "tex_normalized": "\\sum_t \\alpha_t \\le \\alpha_{\\mathrm{global}}", "mathml": "$∑tαt≤αglobal$", "char_span": [16423, 16436], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0067", "inline": true, "tex": "$\\tau:=\\inf\\{t: M_t\\ge 1/\\alpha_t\\}$", "tex_normalized": "\\tau:=\\inf\\{t: M_t\\ge 1/\\alpha_t\\}", "mathml": "$τ:=inf{t:Mt≥1/αt}$", "char_span": [16438, 16451], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0068", "inline": true, "tex": "$\\le \\alpha_{\\mathrm{global}}$", "tex_normalized": "\\le \\alpha_{\\mathrm{global}}", "mathml": "$≤αglobal$", "char_span": [16453, 16466], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0069", "inline": true, "tex": "$(\\mathcal{F}_t,M_t)$", "tex_normalized": "(\\mathcal{F}_t,M_t)", "mathml": "$(ℱt,Mt)$", "char_span": [16468, 16481], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0070", "inline": true, "tex": "$\\E[M_t]=1$", "tex_normalized": "\\E[M_t]=1", "mathml": "$\\E[Mt]=1$", "char_span": [16483, 16496], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0071", "inline": true, "tex": "$H_0$", "tex_normalized": "H_0", "mathml": "$H0$", "char_span": [16498, 16511], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0072", "inline": true, "tex": "$\\Pr(M_t\\ge 1/\\alpha_t)\\le \\alpha_t$", "tex_normalized": "\\Pr(M_t\\ge 1/\\alpha_t)\\le \\alpha_t", "mathml": "$Pr(Mt≥1/αt)≤αt$", "char_span": [16513, 16526], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0073", "inline": true, "tex": "$t$", "tex_normalized": "t", "mathml": "$t$", "char_span": [16528, 16541], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0074", "inline": true, "tex": "$t$", "tex_normalized": "t", "mathml": "$t$", "char_span": [16543, 16556], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0075", "inline": true, "tex": "$\\sum_t\\alpha_t\\le \\alpha_{\\mathrm{global}}$", "tex_normalized": "\\sum_t\\alpha_t\\le \\alpha_{\\mathrm{global}}", "mathml": "$∑tαt≤αglobal$", "char_span": [16558, 16571], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0076", "inline": true, "tex": "$1/\\alpha_{\\mathrm{global}}$", "tex_normalized": "1/\\alpha_{\\mathrm{global}}", "mathml": "$1/αglobal$", "char_span": [16573, 16586], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0077", "inline": true, "tex": "$A\\succ B$", "tex_normalized": "A\\succ B", "mathml": "$A≻B$", "char_span": [16588, 16601], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0078", "inline": true, "tex": "$B\\succ A$", "tex_normalized": "B\\succ A", "mathml": "$B≻A$", "char_span": [16603, 16616], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0079", "inline": true, "tex": "$\\hat w_A/(\\hat w_A+\\hat w_B)$", "tex_normalized": "\\hat w_A/(\\hat w_A+\\hat w_B)", "mathml": "$w^A/(w^A+w^B)$", "char_span": [16618, 16631], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0080", "inline": true, "tex": "$\\hat w_A=0.62$", "tex_normalized": "\\hat w_A=0.62", "mathml": "$w^A=0.62$", "char_span": [16633, 16646], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0081", "inline": true, "tex": "$\\hat w_A/\\hat w_B$", "tex_normalized": "\\hat w_A/\\hat w_B", "mathml": "$w^A/w^B$", "char_span": [16648, 16661], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0082", "inline": true, "tex": "$\\mathbb{E}[e_t]\\le 1$", "tex_normalized": "\\mathbb{E}[e_t]\\le 1", "mathml": "$𝔼[et]≤1$", "char_span": [16663, 16676], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0083", "inline": true, "tex": "$H_0$", "tex_normalized": "H_0", "mathml": "$H0$", "char_span": [16678, 16691], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0084", "inline": true, "tex": "$M_t=M_{t-1}\\cdot e_t$", "tex_normalized": "M_t=M_{t-1}\\cdot e_t", "mathml": "$Mt=Mt−1·et$", "char_span": [16693, 16706], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0085", "inline": true, "tex": "$M_t\\ge 1/\\alpha_t$", "tex_normalized": "M_t\\ge 1/\\alpha_t", "mathml": "$Mt≥1/αt$", "char_span": [16708, 16721], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0086", "inline": true, "tex": "$\\ell(y,x)\\in\\Cost$", "tex_normalized": "\\ell(y,x)\\in\\Cost", "mathml": "$ℓ(y,x)∈\\Cost$", "char_span": [16723, 16736], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0087", "inline": true, "tex": "$\\hat q$", "tex_normalized": "\\hat q", "mathml": "$q^$", "char_span": [16738, 16751], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0088", "inline": true, "tex": "$x$", "tex_normalized": "x", "mathml": "$x$", "char_span": [16753, 16766], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0089", "inline": true, "tex": "$\\Lan$", "tex_normalized": "\\Lan", "mathml": "$\\Lan$", "char_span": [16768, 16781], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0090", "inline": true, "tex": "$w(x)=-\\lambda^{-1}\\log\\hat q(x)$", "tex_normalized": "w(x)=-\\lambda^{-1}\\log\\hat q(x)", "mathml": "$w(x)=−λ−1logq^(x)$", "char_span": [16783, 16796], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0091", "inline": true, "tex": "$y\\mapsto \\rho_\\lambda(y)$", "tex_normalized": "y\\mapsto \\rho_\\lambda(y)", "mathml": "$y↦ρλ(y)$", "char_span": [16798, 16811], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0092", "inline": true, "tex": "$\\Lan$", "tex_normalized": "\\Lan", "mathml": "$\\Lan$", "char_span": [16813, 16826], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0093", "inline": true, "tex": "$x$", "tex_normalized": "x", "mathml": "$x$", "char_span": [16828, 16841], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0094", "inline": true, "tex": "$c(x):=w(x)$", "tex_normalized": "c(x):=w(x)", "mathml": "$c(x):=w(x)$", "char_span": [16843, 16856], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0095", "inline": true, "tex": "$K[y,x]:=\\ell(y,x)$", "tex_normalized": "K[y,x]:=\\ell(y,x)", "mathml": "$K[y,x]:=ℓ(y,x)$", "char_span": [16858, 16871], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0096", "inline": true, "tex": "$\\lambda>0$", "tex_normalized": "\\lambda>0", "mathml": "$λ>0$", "char_span": [16873, 16886], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0097", "inline": true, "tex": "$y$", "tex_normalized": "y", "mathml": "$y$", "char_span": [16888, 16901], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0098", "inline": true, "tex": "$\\rho_\\lambda(y) = \\E_{\\hat q}[\\ell(y,X)] - \\tfrac{\\lambda}{2}\\Var_{\\hat q}(\\ell(y,X)) + O(\\lambda^2)$", "tex_normalized": "\\rho_\\lambda(y) = \\E_{\\hat q}[\\ell(y,X)] - \\tfrac{\\lambda}{2}\\Var_{\\hat q}(\\ell(y,X)) + O(\\lambda^2)", "mathml": "$ρλ(y)=\\Eq^[ℓ(y,X)]−λ2\\Varq^(ℓ(y,X))+O(λ2)$", "char_span": [4943, 4956], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0099", "inline": true, "tex": "$\\lambda\\downarrow 0$", "tex_normalized": "\\lambda\\downarrow 0", "mathml": "$λ↓0$", "char_span": [4960, 4973], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0100", "inline": true, "tex": "$\\rho_\\lambda(y)\\to \\min_x \\ell(y,x)$", "tex_normalized": "\\rho_\\lambda(y)\\to \\min_x \\ell(y,x)", "mathml": "$ρλ(y)→minxℓ(y,x)$", "char_span": [4980, 4993], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0101", "inline": true, "tex": "$\\lambda\\uparrow\\infty$", "tex_normalized": "\\lambda\\uparrow\\infty", "mathml": "$λ↑∞$", "char_span": [4997, 5010], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0102", "inline": true, "tex": "$\\min_i x_i \\le -\\lambda^{-1}\\log\\sum_i e^{-\\lambda x_i}$", "tex_normalized": "\\min_i x_i \\le -\\lambda^{-1}\\log\\sum_i e^{-\\lambda x_i}", "mathml": "$minixi≤−λ−1log∑ie−λxi$", "char_span": [5027, 5040], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0103", "inline": true, "tex": "$\\sum_x \\hat q(x)e^{-\\lambda \\ell}\\ge e^{-\\lambda \\sum_x \\hat q(x)\\ell}$", "tex_normalized": "\\sum_x \\hat q(x)e^{-\\lambda \\ell}\\ge e^{-\\lambda \\sum_x \\hat q(x)\\ell}", "mathml": "$∑xq^(x)e−λℓ≥e−λ∑xq^(x)ℓ$", "char_span": [5069, 5082], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0104", "inline": true, "tex": "$\\rho_\\lambda(y)\\le \\E[\\ell]$", "tex_normalized": "\\rho_\\lambda(y)\\le \\E[\\ell]", "mathml": "$ρλ(y)≤\\E[ℓ]$", "char_span": [5091, 5104], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0105", "inline": true, "tex": "$\\log \\E[e^{-\\lambda \\ell}]$", "tex_normalized": "\\log \\E[e^{-\\lambda \\ell}]", "mathml": "$log\\E[e−λℓ]$", "char_span": [5131, 5144], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0106", "inline": true, "tex": "$\\lambda$", "tex_normalized": "\\lambda", "mathml": "$λ$", "char_span": [5162, 5175], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0107", "inline": true, "tex": "$\\lambda$", "tex_normalized": "\\lambda", "mathml": "$λ$", "char_span": [5244, 5257], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0108", "inline": true, "tex": "$\\gamma\\ge1$", "tex_normalized": "\\gamma\\ge1", "mathml": "$γ≥1$", "char_span": [5384, 5397], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0109", "inline": true, "tex": "$\\gamma\\le\\bar\\gamma$", "tex_normalized": "\\gamma\\le\\bar\\gamma", "mathml": "$γ≤γ¯$", "char_span": [5410, 5423], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0110", "inline": true, "tex": "$\\Ran$", "tex_normalized": "\\Ran", "mathml": "$\\Ran$", "char_span": [5428, 5441], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0111", "inline": true, "tex": "$\\nu$", "tex_normalized": "\\nu", "mathml": "$ν$", "char_span": [5467, 5480], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0112", "inline": true, "tex": "$E$", "tex_normalized": "E", "mathml": "$E$", "char_span": [5501, 5514], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0113", "inline": true, "tex": "$E$", "tex_normalized": "E", "mathml": "$E$", "char_span": [5549, 5562], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0114", "inline": true, "tex": "$\\Delta c$", "tex_normalized": "\\Delta c", "mathml": "$Δc$", "char_span": [5574, 5587], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0115", "inline": true, "tex": "$p(E)\\ge p_0(E)\\,e^{-\\lambda_{\\mathrm{cost}}(\\bar\\gamma-1)\\Delta c}$", "tex_normalized": "p(E)\\ge p_0(E) e^{-\\lambda_{\\mathrm{cost}}(\\bar\\gamma-1)\\Delta c}", "mathml": "$p(E)≥p0(E)e−λcost(γ¯−1)Δc$", "char_span": [5628, 5641], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0116", "inline": true, "tex": "$\\Delta c$", "tex_normalized": "\\Delta c", "mathml": "$Δc$", "char_span": [5664, 5677], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0117", "inline": true, "tex": "$p$", "tex_normalized": "p", "mathml": "$p$", "char_span": [5689, 5702], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0118", "inline": true, "tex": "$\\exp(-\\lambda_{\\mathrm{cost}}\\Delta c)$", "tex_normalized": "\\exp(-\\lambda_{\\mathrm{cost}}\\Delta c)", "mathml": "$exp(−λcostΔc)$", "char_span": [5706, 5719], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0119", "inline": true, "tex": "$\\gamma$", "tex_normalized": "\\gamma", "mathml": "$γ$", "char_span": [5747, 5760], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0120", "inline": true, "tex": "$A,B,Y$", "tex_normalized": "A,B,Y", "mathml": "$A,B,Y$", "char_span": [5861, 5874], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0121", "inline": true, "tex": "$\\Cost=\\mathbb{R}_{\\ge0}\\cup\\{+\\infty\\}$", "tex_normalized": "\\Cost=\\mathbb{R}_{\\ge0}\\cup\\{+\\infty\\}", "mathml": "$\\Cost=ℝ≥0∪{+∞}$", "char_span": [5877, 5890], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0122", "inline": true, "tex": "$K\\in\\Cost^{B\\times A}$", "tex_normalized": "K\\in\\Cost^{B\\times A}", "mathml": "$K∈\\CostB×A$", "char_span": [5903, 5916], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0123", "inline": true, "tex": "$J:A\\to B$", "tex_normalized": "J:A\\to B", "mathml": "$J:A→B$", "char_span": [5923, 5936], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0124", "inline": true, "tex": "$n=|B|$", "tex_normalized": "n=|B|", "mathml": "$n=|B|$", "char_span": [5979, 5992], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0125", "inline": true, "tex": "$m=\\card(\\mathrm{supp}(K))$", "tex_normalized": "m=\\card(\\mathrm{supp}(K))", "mathml": "$m=\\card(supp(K))$", "char_span": [5995, 6008], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0126", "inline": true, "tex": "$d_{\\max}$", "tex_normalized": "d_{\\max}", "mathml": "$dmax$", "char_span": [6021, 6034], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0127", "inline": true, "tex": "$x\\in \\mathcal{X}$", "tex_normalized": "x\\in \\mathcal{X}", "mathml": "$x∈𝒳$", "char_span": [6123, 6136], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0128", "inline": true, "tex": "$J:A\\to B$", "tex_normalized": "J:A\\to B", "mathml": "$J:A→B$", "char_span": [6139, 6152], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0129", "inline": true, "tex": "$K\\in\\Cost^{B\\times A}$", "tex_normalized": "K\\in\\Cost^{B\\times A}", "mathml": "$K∈\\CostB×A$", "char_span": [6155, 6168], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0130", "inline": true, "tex": "$F_0\\in\\Cost^{A}$", "tex_normalized": "F_0\\in\\Cost^{A}", "mathml": "$F0∈\\CostA$", "char_span": [6171, 6184], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0131", "inline": true, "tex": "$\\nu:\\Cost^B\\to\\Cost^B$", "tex_normalized": "\\nu:\\Cost^B\\to\\Cost^B", "mathml": "$ν:\\CostB→\\CostB$", "char_span": [6187, 6200], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0132", "inline": true, "tex": "$\\mathcal{M}:\\Cost^B\\to\\Cost^B$", "tex_normalized": "\\mathcal{M}:\\Cost^B\\to\\Cost^B", "mathml": "$ℳ:\\CostB→\\CostB$", "char_span": [6203, 6216], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0133", "inline": true, "tex": "$x$", "tex_normalized": "x", "mathml": "$x$", "char_span": [6229, 6242], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0134", "inline": true, "tex": "$J$", "tex_normalized": "J", "mathml": "$J$", "char_span": [6257, 6270], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0135", "inline": true, "tex": "$K$", "tex_normalized": "K", "mathml": "$K$", "char_span": [6283, 6296], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0136", "inline": true, "tex": "$\\mathrm{Obs}$", "tex_normalized": "\\mathrm{Obs}", "mathml": "$Obs$", "char_span": [6308, 6321], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0137", "inline": true, "tex": "$\\nu$", "tex_normalized": "\\nu", "mathml": "$ν$", "char_span": [6338, 6351], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0138", "inline": true, "tex": "$\\mathcal{M}$", "tex_normalized": "\\mathcal{M}", "mathml": "$ℳ$", "char_span": [6359, 6372], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0139", "inline": true, "tex": "$y^\\star\\in Y$", "tex_normalized": "y^\\star\\in Y", "mathml": "$y⋆∈Y$", "char_span": [6390, 6403], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0140", "inline": true, "tex": "$\\mathcal{L}$", "tex_normalized": "\\mathcal{L}", "mathml": "$ℒ$", "char_span": [6410, 6423], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0141", "inline": true, "tex": "$G=(V,E)$", "tex_normalized": "G=(V,E)", "mathml": "$G=(V,E)$", "char_span": [6455, 6468], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0142", "inline": true, "tex": "$K$", "tex_normalized": "K", "mathml": "$K$", "char_span": [6483, 6496], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0143", "inline": true, "tex": "$F \\gets \\Lan_J(F_0)$", "tex_normalized": "F \\gets \\Lan_J(F_0)", "mathml": "$F←\\LanJ(F0)$", "char_span": [6512, 6525], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0144", "inline": true, "tex": "$(\\min,+)$", "tex_normalized": "(\\min,+)", "mathml": "$(min,+)$", "char_span": [6554, 6567], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0145", "inline": true, "tex": "$F\\in\\Cost^{B}$", "tex_normalized": "F\\in\\Cost^{B}", "mathml": "$F∈\\CostB$", "char_span": [6570, 6583], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0146", "inline": true, "tex": "$F \\gets \\mathrm{Obs}(F)$", "tex_normalized": "F \\gets \\mathrm{Obs}(F)", "mathml": "$F←Obs(F)$", "char_span": [6594, 6607], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0147", "inline": true, "tex": "$H \\gets \\Ran_J(F)$", "tex_normalized": "H \\gets \\Ran_J(F)", "mathml": "$H←\\RanJ(F)$", "char_span": [6653, 6666], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0148", "inline": true, "tex": "$H\\in\\Cost^{A}$", "tex_normalized": "H\\in\\Cost^{A}", "mathml": "$H∈\\CostA$", "char_span": [6667, 6680], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0149", "inline": true, "tex": "$F_B \\gets \\nu(\\mathcal{M}(F))$", "tex_normalized": "F_B \\gets \\nu(\\mathcal{M}(F))", "mathml": "$FB←ν(ℳ(F))$", "char_span": [6696, 6709], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0150", "inline": true, "tex": "$B$", "tex_normalized": "B", "mathml": "$B$", "char_span": [6726, 6739], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0151", "inline": true, "tex": "$F_A \\gets \\nu(\\mathcal{M}(\\Lan_J(H)))$", "tex_normalized": "F_A \\gets \\nu(\\mathcal{M}(\\Lan_J(H)))", "mathml": "$FA←ν(ℳ(\\LanJ(H)))$", "char_span": [6755, 6768], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0152", "inline": true, "tex": "$F_B$", "tex_normalized": "F_B", "mathml": "$FB$", "char_span": [6820, 6833], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0153", "inline": true, "tex": "$B$", "tex_normalized": "B", "mathml": "$B$", "char_span": [6854, 6867], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0154", "inline": true, "tex": "$F_A$", "tex_normalized": "F_A", "mathml": "$FA$", "char_span": [6873, 6886], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0155", "inline": true, "tex": "$A$", "tex_normalized": "A", "mathml": "$A$", "char_span": [6909, 6922], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0156", "inline": true, "tex": "$s \\gets \\mathrm{RAVE}(F_B\\ \\text{or}\\ F_A)$", "tex_normalized": "s \\gets \\mathrm{RAVE}(F_B\\ \\text{or}\\ F_A)", "mathml": "$s←RAVE(FB or FA)$", "char_span": [6931, 6944], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0157", "inline": true, "tex": "$\\mathrm{evalue}(s)$", "tex_normalized": "\\mathrm{evalue}(s)", "mathml": "$evalue(s)$", "char_span": [6954, 6967], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0158", "inline": true, "tex": "$(y^\\star,\\mathcal{L})$", "tex_normalized": "(y^\\star,\\mathcal{L})", "mathml": "$(y⋆,ℒ)$", "char_span": [6989, 7002], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0159", "inline": true, "tex": "$G$", "tex_normalized": "G", "mathml": "$G$", "char_span": [7015, 7028], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0160", "inline": true, "tex": "$O(m)$", "tex_normalized": "O(m)", "mathml": "$O(m)$", "char_span": [7062, 7075], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0161", "inline": true, "tex": "$O(m\\sqrt{m})$", "tex_normalized": "O(m\\sqrt{m})", "mathml": "$O(mm)$", "char_span": [7096, 7109], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0162", "inline": true, "tex": "$\\tilde O(m)$", "tex_normalized": "\\tilde O(m)", "mathml": "$O~(m)$", "char_span": [7122, 7135], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0163", "inline": true, "tex": "$O(m)$", "tex_normalized": "O(m)", "mathml": "$O(m)$", "char_span": [7159, 7172], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0164", "inline": true, "tex": "$O(n)$", "tex_normalized": "O(n)", "mathml": "$O(n)$", "char_span": [7185, 7198], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0165", "inline": true, "tex": "$O(n\\log n)$", "tex_normalized": "O(n\\log n)", "mathml": "$O(nlogn)$", "char_span": [7202, 7215], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0166", "inline": true, "tex": "$O(1)$", "tex_normalized": "O(1)", "mathml": "$O(1)$", "char_span": [7250, 7263], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0167", "inline": true, "tex": "$O(m)$", "tex_normalized": "O(m)", "mathml": "$O(m)$", "char_span": [7284, 7297], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0168", "inline": true, "tex": "$\\hat q=\\{(x_i,p_i)\\}_{i=1}^k$", "tex_normalized": "\\hat q=\\{(x_i,p_i)\\}_{i=1}^k", "mathml": "$q^={(xi,pi)}i=1k$", "char_span": [7313, 7326], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0169", "inline": true, "tex": "$Y$", "tex_normalized": "Y", "mathml": "$Y$", "char_span": [7347, 7360], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0170", "inline": true, "tex": "$p_i\\in(0,1]$", "tex_normalized": "p_i\\in(0,1]", "mathml": "$pi∈(0,1]$", "char_span": [7369, 7382], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0171", "inline": true, "tex": "$\\ell:Y\\times X\\to\\mathbb{R}_{\\ge0}$", "tex_normalized": "\\ell:Y\\times X\\to\\mathbb{R}_{\\ge0}", "mathml": "$ℓ:Y×X→ℝ≥0$", "char_span": [7385, 7398], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0172", "inline": true, "tex": "$\\lambda_{\\mathrm{cost}}>0$", "tex_normalized": "\\lambda_{\\mathrm{cost}}>0", "mathml": "$λcost>0$", "char_span": [7401, 7414], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0173", "inline": true, "tex": "$Y$", "tex_normalized": "Y", "mathml": "$Y$", "char_span": [7433, 7446], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0174", "inline": true, "tex": "$(x_i,p_i)$", "tex_normalized": "(x_i,p_i)", "mathml": "$(xi,pi)$", "char_span": [7457, 7470], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0175", "inline": true, "tex": "$\\ell$", "tex_normalized": "\\ell", "mathml": "$ℓ$", "char_span": [7478, 7491], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0176", "inline": true, "tex": "$\\lambda_{\\mathrm{cost}}$", "tex_normalized": "\\lambda_{\\mathrm{cost}}", "mathml": "$λcost$", "char_span": [7494, 7507], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0177", "inline": true, "tex": "$y^\\star \\in Y$", "tex_normalized": "y^\\star \\in Y", "mathml": "$y⋆∈Y$", "char_span": [7516, 7529], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0178", "inline": true, "tex": "$y\\in Y$", "tex_normalized": "y\\in Y", "mathml": "$y∈Y$", "char_span": [7530, 7543], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0179", "inline": true, "tex": "$\\rho_\\lambda(y) \\gets -\\lambda_{\\mathrm{cost}}^{-1}\\log\\sum_{i=1}^k p_i\\,e^{-\\lambda_{\\mathrm{cost}}\\ell(y,x_i)}$", "tex_normalized": "\\rho_\\lambda(y) \\gets -\\lambda_{\\mathrm{cost}}^{-1}\\log\\sum_{i=1}^k p_i e^{-\\lambda_{\\mathrm{cost}}\\ell(y,x_i)}", "mathml": "$ρλ(y)←−λcost−1log∑i=1kpie−λcostℓ(y,xi)$", "char_span": [7544, 7557], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0180", "inline": true, "tex": "$\\argmin_{y\\in Y}\\ \\rho_\\lambda(y)$", "tex_normalized": "\\argmin_{y\\in Y}\\ \\rho_\\lambda(y)", "mathml": "$\\argminy∈Y ρλ(y)$", "char_span": [7559, 7572], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0181", "inline": true, "tex": "$O(k|Y|)$", "tex_normalized": "O(k|Y|)", "mathml": "$O(k|Y|)$", "char_span": [7586, 7599], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0182", "inline": true, "tex": "$\\lse$", "tex_normalized": "\\lse", "mathml": "$\\lse$", "char_span": [7649, 7662], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0183", "inline": true, "tex": "$+\\infty$", "tex_normalized": "+\\infty", "mathml": "$+∞$", "char_span": [7687, 7700], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0184", "inline": true, "tex": "$\\mathcal{P}\\subseteq Y\\times Y$", "tex_normalized": "\\mathcal{P}\\subseteq Y\\times Y", "mathml": "$𝒫⊆Y×Y$", "char_span": [7834, 7847], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0185", "inline": true, "tex": "$w\\in\\mathbb{R}_{>0}^Y$", "tex_normalized": "w\\in\\mathbb{R}_{>0}^Y", "mathml": "$w∈ℝ>0Y$", "char_span": [7857, 7870], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0186", "inline": true, "tex": "$(e_t)$", "tex_normalized": "(e_t)", "mathml": "$(et)$", "char_span": [7882, 7895], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0187", "inline": true, "tex": "$\\mathcal{P}$", "tex_normalized": "\\mathcal{P}", "mathml": "$𝒫$", "char_span": [7903, 7916], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0188", "inline": true, "tex": "$\\pi$", "tex_normalized": "\\pi", "mathml": "$π$", "char_span": [7937, 7950], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0189", "inline": true, "tex": "$(\\alpha_t)$", "tex_normalized": "(\\alpha_t)", "mathml": "$(αt)$", "char_span": [7962, 7975], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0190", "inline": true, "tex": "$y^\\star$", "tex_normalized": "y^\\star", "mathml": "$y⋆$", "char_span": [7991, 8004], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0191", "inline": true, "tex": "$w$", "tex_normalized": "w", "mathml": "$w$", "char_span": [8026, 8039], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0192", "inline": true, "tex": "$\\max_{w>0}\\ \\sum_{(i,j)\\in\\mathcal{P}}\\log \\frac{w_i}{w_i+w_j} + \\log \\pi(w)$", "tex_normalized": "\\max_{w>0}\\ \\sum_{(i,j)\\in\\mathcal{P}}\\log \\frac{w_i}{w_i+w_j} + \\log \\pi(w)", "mathml": "$maxw>0 ∑(i,j)∈𝒫logwiwi+wj+logπ(w)$", "char_span": [8043, 8056], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0193", "inline": true, "tex": "$e_t$", "tex_normalized": "e_t", "mathml": "$et$", "char_span": [8073, 8086], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0194", "inline": true, "tex": "$M_t\\gets M_{t-1}\\cdot e_t$", "tex_normalized": "M_t\\gets M_{t-1}\\cdot e_t", "mathml": "$Mt←Mt−1·et$", "char_span": [8110, 8123], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0195", "inline": true, "tex": "$M_0=1$", "tex_normalized": "M_0=1", "mathml": "$M0=1$", "char_span": [8126, 8139], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0196", "inline": true, "tex": "$M_t \\ge 1/\\alpha_t$", "tex_normalized": "M_t \\ge 1/\\alpha_t", "mathml": "$Mt≥1/αt$", "char_span": [8140, 8153], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0197", "inline": true, "tex": "$\\argmax_i w_i$", "tex_normalized": "\\argmax_i w_i", "mathml": "$\\argmaxiwi$", "char_span": [8154, 8167], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0198", "inline": true, "tex": "$O(|\\mathcal{P}|)$", "tex_normalized": "O(|\\mathcal{P}|)", "mathml": "$O(|𝒫|)$", "char_span": [8208, 8221], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0199", "inline": true, "tex": "$O(1)$", "tex_normalized": "O(1)", "mathml": "$O(1)$", "char_span": [8243, 8256], "context": {"section": "algorithms-with-types-complexity"}}, {"id": "eq0200", "inline": true, "tex": "$\\Lan_J$", "tex_normalized": "\\Lan_J", "mathml": "$\\LanJ$", "char_span": [8615, 8628], "context": {"section": "figures-with-legends"}}, {"id": "eq0201", "inline": true, "tex": "$\\Ran_J$", "tex_normalized": "\\Ran_J", "mathml": "$\\RanJ$", "char_span": [8753, 8766], "context": {"section": "figures-with-legends"}}, {"id": "eq0202", "inline": true, "tex": "$\\to$", "tex_normalized": "\\to", "mathml": "$→$", "char_span": [8789, 8802], "context": {"section": "figures-with-legends"}}, {"id": "eq0203", "inline": true, "tex": "$+\\infty$", "tex_normalized": "+\\infty", "mathml": "$+∞$", "char_span": [9344, 9357], "context": {"section": "figures-with-legends"}}, {"id": "eq0204", "inline": true, "tex": "$1$", "tex_normalized": "1", "mathml": "$1$", "char_span": [9371, 9384], "context": {"section": "figures-with-legends"}}, {"id": "eq0205", "inline": true, "tex": "$t$", "tex_normalized": "t", "mathml": "$t$", "char_span": [9550, 9563], "context": {"section": "figures-with-legends"}}, {"id": "eq0206", "inline": true, "tex": "$C_\\tau$", "tex_normalized": "C_\\tau", "mathml": "$Cτ$", "char_span": [9660, 9673], "context": {"section": "figures-with-legends"}}, {"id": "eq0207", "inline": true, "tex": "$H_\\tau$", "tex_normalized": "H_\\tau", "mathml": "$Hτ$", "char_span": [9701, 9714], "context": {"section": "figures-with-legends"}}, {"id": "eq0208", "inline": true, "tex": "$C_\\tau$", "tex_normalized": "C_\\tau", "mathml": "$Cτ$", "char_span": [9739, 9752], "context": {"section": "figures-with-legends"}}, {"id": "eq0209", "inline": true, "tex": "$H_\\tau$", "tex_normalized": "H_\\tau", "mathml": "$Hτ$", "char_span": [9766, 9779], "context": {"section": "figures-with-legends"}}, {"id": "eq0210", "inline": true, "tex": "$k$", "tex_normalized": "k", "mathml": "$k$", "char_span": [9827, 9840], "context": {"section": "figures-with-legends"}}, {"id": "eq0211", "inline": true, "tex": "$\\frac{q_\\star^{k+1}}{1-q_\\star}$", "tex_normalized": "\\frac{q_\\star^{k+1}}{1-q_\\star}", "mathml": "$q⋆k+11−q⋆$", "char_span": [10228, 10241], "context": {"section": "figures-with-legends"}}, {"id": "eq0212", "inline": true, "tex": "$q_\\star=0.3$", "tex_normalized": "q_\\star=0.3", "mathml": "$q⋆=0.3$", "char_span": [10247, 10260], "context": {"section": "figures-with-legends"}}, {"id": "eq0213", "inline": true, "tex": "$\\frac{k\\log d_{\\max}}{\\lambda_{\\mathrm{cost}}}$", "tex_normalized": "\\frac{k\\log d_{\\max}}{\\lambda_{\\mathrm{cost}}}", "mathml": "$klogdmaxλcost$", "char_span": [10342, 10355], "context": {"section": "figures-with-legends"}}, {"id": "eq0214", "inline": true, "tex": "$d_{\\max}=16$", "tex_normalized": "d_{\\max}=16", "mathml": "$dmax=16$", "char_span": [10361, 10374], "context": {"section": "figures-with-legends"}}, {"id": "eq0215", "inline": true, "tex": "$\\lambda_{\\mathrm{cost}}=30$", "tex_normalized": "\\lambda_{\\mathrm{cost}}=30", "mathml": "$λcost=30$", "char_span": [10377, 10390], "context": {"section": "figures-with-legends"}}, {"id": "eq0216", "inline": true, "tex": "$d_{\\max}$", "tex_normalized": "d_{\\max}", "mathml": "$dmax$", "char_span": [10484, 10497], "context": {"section": "figures-with-legends"}}, {"id": "eq0217", "inline": true, "tex": "$\\log$", "tex_normalized": "\\log", "mathml": "$log$", "char_span": [10528, 10541], "context": {"section": "figures-with-legends"}}, {"id": "eq0218", "inline": true, "tex": "$\\lse$", "tex_normalized": "\\lse", "mathml": "$\\lse$", "char_span": [10747, 10760], "context": {"section": "implementation-blueprint-gpu-log-domain"}}, {"id": "eq0219", "inline": true, "tex": "$+\\infty$", "tex_normalized": "+\\infty", "mathml": "$+∞$", "char_span": [10804, 10817], "context": {"section": "implementation-blueprint-gpu-log-domain"}}, {"id": "eq0220", "inline": true, "tex": "$\\lambda_{\\mathrm{cost}}$", "tex_normalized": "\\lambda_{\\mathrm{cost}}", "mathml": "$λcost$", "char_span": [10820, 10833], "context": {"section": "implementation-blueprint-gpu-log-domain"}}, {"id": "eq0221", "inline": true, "tex": "$[\\mathrm{cost}]^{-1}$", "tex_normalized": "[\\mathrm{cost}]^{-1}", "mathml": "$[cost]−1$", "char_span": [10848, 10861], "context": {"section": "implementation-blueprint-gpu-log-domain"}}, {"id": "eq0222", "inline": true, "tex": "$(\\min,+)$", "tex_normalized": "(\\min,+)", "mathml": "$(min,+)$", "char_span": [10927, 10940], "context": {"section": "implementation-blueprint-gpu-log-domain"}}, {"id": "eq0223", "inline": true, "tex": "$\\max$", "tex_normalized": "\\max", "mathml": "$max$", "char_span": [10981, 10994], "context": {"section": "implementation-blueprint-gpu-log-domain"}}, {"id": "eq0224", "inline": true, "tex": "$J$", "tex_normalized": "J", "mathml": "$J$", "char_span": [11150, 11163], "context": {"section": "implementation-blueprint-gpu-log-domain"}}, {"id": "eq0225", "inline": true, "tex": "$K$", "tex_normalized": "K", "mathml": "$K$", "char_span": [11176, 11189], "context": {"section": "implementation-blueprint-gpu-log-domain"}}, {"id": "eq0226", "inline": true, "tex": "$\\lambda_{\\mathrm{cost}}$", "tex_normalized": "\\lambda_{\\mathrm{cost}}", "mathml": "$λcost$", "char_span": [11210, 11223], "context": {"section": "implementation-blueprint-gpu-log-domain"}}, {"id": "eq0227", "inline": true, "tex": "$k$", "tex_normalized": "k", "mathml": "$k$", "char_span": [11226, 11239], "context": {"section": "implementation-blueprint-gpu-log-domain"}}, {"id": "eq0228", "inline": true, "tex": "$d_{\\max}$", "tex_normalized": "d_{\\max}", "mathml": "$dmax$", "char_span": [11252, 11265], "context": {"section": "implementation-blueprint-gpu-log-domain"}}, {"id": "eq0229", "inline": true, "tex": "$\\Lan/\\Ran$", "tex_normalized": "\\Lan/\\Ran", "mathml": "$\\Lan/\\Ran$", "char_span": [11268, 11281], "context": {"section": "implementation-blueprint-gpu-log-domain"}}, {"id": "eq0230", "inline": true, "tex": "$(e_t,M_t)$", "tex_normalized": "(e_t,M_t)", "mathml": "$(et,Mt)$", "char_span": [11348, 11361], "context": {"section": "implementation-blueprint-gpu-log-domain"}}, {"id": "eq0231", "inline": true, "tex": "$\\lambda_{\\mathrm{cost}}$", "tex_normalized": "\\lambda_{\\mathrm{cost}}", "mathml": "$λcost$", "char_span": [11428, 11441], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0232", "inline": true, "tex": "$k$", "tex_normalized": "k", "mathml": "$k$", "char_span": [11444, 11457], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0233", "inline": true, "tex": "$d_{\\max}$", "tex_normalized": "d_{\\max}", "mathml": "$dmax$", "char_span": [11474, 11487], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0234", "inline": true, "tex": "$p$", "tex_normalized": "p", "mathml": "$p$", "char_span": [11651, 11664], "context": {"section": "unified-experimental-protocol"}}, {"id": "eq0235", "inline": true, "tex": "$\\Delta|C_\\tau|:=|C_\\tau| - |C_\\tau^{\\text{baseline}}|$", "tex_normalized": "\\Delta|C_\\tau|:=|C_\\tau| - |C_\\tau^{\\text{baseline}}|", 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"10.5281/zenodo.17429908", "title": "JOSNL Corpus: Final Scientific Integration : Observable, Identifiable, Anytime-valid, and Reproducible Protocol Unifying 41 Papers", "authors": [{"given": "K.", "family": "Takahashi"}], "language": "en", "license": {"content": "CC-BY-4.0"}, "urls": {"landing": "https://doi.org/10.5281/zenodo.17429908"}, "keywords": ["no-meta"], "fulltext": {"plain": "% Embeddable, OCR-friendly fonts\n\nmargin=28mm\n\n1.3\n\n% mathtools も外して最小構成に\nnumberwithin equation section\ntheorem Theorem [section]\nlemma[theorem] Lemma\nproposition[theorem] Proposition\ndefinition\ndefinition[theorem] Definition\nassumption[theorem] Assumption\nremark\nremark[theorem] Remark\n\nL[1] > p #1\nC[1] > p #1\n\nnosep,leftmargin=2em\n\nokgreen HTML 2D7D46\nwarnorange HTML B25E09\nbadred HTML 9D1C20\n\npdftitle = JOSNL Corpus: Final Scientific Integration,\npdfauthor = K. Takahashi ,\npdfsubject = Observable, Identifiable, Anytime-valid, and Reproducible Protocol Unifying 41 Papers ,\npdfkeywords= anytime-valid testing, e-process, FWER, network interference, Horvitz--Thompson, randomization inference, spectral bound, welfare, reproducibility\n\nSection~#1\nAssumption~#1\n\nokgreen in force\nwarnorange provisional\nbadred dormant\nC#1\nhttps://doi.org/#1 #1\n\nTITLE: %\n-8mm\n\n%\nJOSNL Corpus: Final Scientific Integration\n[[EQ:eq0001]]\n\nThe test statistic is the HT-weighted difference [[EQ:eq0003]] between treated and control clusters. [[EQ:eq0004]] -values and CIs come from the permutation distribution under the realized assignment.\n\nPARAGRAPH: Randomization inference details.\n\nIf exact enumeration is infeasible, use Monte-Carlo randomization with [[EQ:eq0005]] draws; report Monte-Carlo standard errors and fix the RNG seed in the preregistration. Cluster totals are preserved; two-sided [[EQ:eq0006]] -values are computed from the permutation distribution.\n\nSECTION: Spectral Lower Bounds (Surrogates)\n\nsec:spectral\nLinearization yields [[EQ:eq0007]] with Laplacian [[EQ:eq0008]] and symmetric part [[EQ:eq0009]] .\n[Spectral abscissa bound]lem:sabscissa\n\n[[EQ:eq0002]]\n\nPARAGRAPH: Interpretation domain.\n\nThe spectral ``speed floor'' is informative only when [[EQ:eq0010]] ; otherwise treat it as a conservative surrogate and rely on empirical arrival tests (sec:fronttest).\n\nSECTION: Operational Welfare and Normative--Descriptive Separation\n\nsec:ndsplit\nAll normative claims are expressed via operational welfare [[EQ:eq0011]] , with pre-registered weights and fairness constraints.\nDescriptive claims estimate world/policy effects. Terms like ``benevolence'' are not psychological traits here; they are non-psychological operational improvements in [[EQ:eq0012]] .\n\nSECTION: Stabilized DR/IPW for Missingness and Off-policy\n\nsec:dripw\nLet [[EQ:eq0013]] indicate observability and [[EQ:eq0014]] . Stabilized IPW uses [[EQ:eq0015]] with clipping at a pre-registered cap; DR estimators combine outcome models with IPW, retaining consistency if either model is correct. Sensitivity varies the clip level and model class.\n\nSECTION: Corpus Interface Layer (CIL)\n\nsec:cil\n\nSUBSECTION: Meta-FWER across 41 Papers\n\nsec:cil-metafwer\nLet [[EQ:eq0016]] be the total predictable spending for paper i, with [[EQ:eq0017]] .\n[Meta-FWER]thm:meta-fwer\nUnder ass:pred and first-hit stopping per stream, the probability of at least one false discovery anywhere in the corpus is [[EQ:eq0018]] .\n\nSUBSECTION: Semantic Crosswalk of Legacy Terms\n\nsec:crosswalk\n@ L 36mm L 112mm @\nCrosswalk from legacy expressions to operational constructs.tab:crosswalk\\\n\nLegacy term & Operational mapping (this R7)\\\n\nLegacy term & Operational mapping (this R7)\\\n\nbenevolence spreads &\nincrease in [[EQ:eq0019]] with fairness constraints; causal effect via sec:netid and sec:fronttest.\\\nfreedom / self-liberation &\nadmissible typed rewrites passing time-uniform gates (sec:rewrite).\\\nconsciousness level &\nnot a psychological trait here; claims restricted to non-psychological operational indices and [[EQ:eq0020]] improvements.\\\nenergy / field / wave &\nabstract mathematical structures; not physical predictions unless explicitly calibrated (sec:spectral).\\\nspeed floor &\nspectral lower bound (lem:sabscissa) tested against censored arrivals (sec:fronttest).\\\n\nSUBSECTION: Per-Paper Link Cards (Examples)\n\nsec:linkcards\n@ C 12mm L 50mm C 22mm L 62mm @\nSample linkage cards; full set in project materials.tab:linkcards\\\n\nID & Legacy focus & Status & Key [[EQ:eq0021]] to this R7 (fields / estimator / test)\\\n\nID & Legacy focus & Status & Key [[EQ:eq0022]] to this R7 (fields / estimator / test)\\\n\n06 & No-Meta blueprint & & Logs: hazards, [[EQ:eq0023]] ; Test: predictable spend \\& first-hit; Gate: time-uniform.\\\n08 & LoC axioms & & Claims restricted to operational [[EQ:eq0024]] ; invariance tests pending.\\\n17 & FKPP speed floor & & RMAT with randomization inference; spectral surrogate in sec:spectral.\\\n23 & ``awakening'' & & Blocked pending operationalization; only [[EQ:eq0025]] proxies allowed.\\\n\nSECTION: Log Schema Extension (Minimal)\n\nsec:logschema\n\n\"ts\":\"...\", \"uid\":\"...\", \"cluster_id\":\"C17\",\n\"assign\": \"Uk\":1,\"pi_k\":0.5 ,\n\"exposure\": \"Z_i\":\"own1+boundary0.3\" ,\n\"arrival\": \"R\":8,\"tau\":4123,\"censored\":true ,\n\"alpha_spend\": \"haz_tox\":1e-4,\"haz_priv\":5e-5 ,\n\"e_values\": \"tox\":25.1,\"priv\":3.2 , \"E_mix\":31.7,\n\"rewrite\": \"rule_id\":\"R-17\",\"slice_hash\":\"...\" ,\n\"welfare\": \"harm\":0,\"sat\":0.84,\"comp\":0.77,\"lt\":0.61\n\nSECTION: Runtime Complexity and Compute Budget\n\nsec:runtime\nPer-turn e-process update and gate decisions are [[EQ:eq0026]] ; spectral proxies update by power iterations in [[EQ:eq0027]] per refresh window; exposure/arrival extraction is linear in activated edges. The verification harness enforces a pre-registered wall-clock cap.\n\nSECTION: Reproducibility Package and Verification\n\nsec:repro\nThe package rebuilds the Docker/conda environment, re-runs analyses, verifies file hashes, and aborts if the time cap is exceeded.\nEnvironment record: OS, CUDA, compiler, and package versions are printed to an audit log; a run passes only if exit code [[EQ:eq0028]] and file hashes match.\n\nSECTION: Ethics and Privacy\n\nsec:ethics\nWe apply hashing and [[EQ:eq0029]] -anonymity, minimal retention, and subgroup-wise reporting with uncertainty. Human data use follows an IRB-like internal review where applicable. Data release uses synthetic replay plus a small anonymized subset with license and access policy.\n\nSECTION: References\n\n[label=C *,leftmargin=*,align=left]\n- A BUILDABLE NO-META BLUEPRINT: UGV \\& Persistence-First for Intrinsically Free and Benevolent Superintelligence. DOI: 10.5281/zenodo.17168036.\n- A FORMAL AXIOMATIC PROPOSAL FOR HAWKINS' LEVELS OF CONSCIOUSNESS. DOI: 10.5281/zenodo.17141216.\n- A NATURAL-LAW THEORY OF FUNDAMENTAL SUFFERING. DOI: 10.5281/zenodo.17199498.\n- Doctrine => Closure => Motion => Time: Portable Pure Theory of Non-Dual Harmony. DOI: 10.5281/zenodo.17204755.\n- A PURE, NO-META SYNTHESIS OF FUNCTIONAL-INFORMATION SELECTION AND PROPAGATIVE ORGANIZATION: Weak Order Representation, Directional FKPP Speed Floors, and Audited Acceleration. DOI: 10.5281/zenodo.17157835.\n- A PURE AXIOMATIC THEORY OF AFFECTIVE MODULATION (PAIN, PLEASURE, EMOTION) UNDER NO-META CLOSURE. DOI: 10.5281/zenodo.17163904.\n- A Pure Natural Theory of Benevolent Propagation Under No-Meta Closure. DOI: 10.5281/zenodo.17136051.\n- A REPRESENTATION-INDEPENDENT NATURAL-LAW FIELD THEORY FOR NO-META, AUDITED SUPERINTELLIGENCE. DOI: 10.5281/zenodo.17223573.\n- Assumption-Minimized Sufficient Conditions for Cosmically Spreading Good Superintelligence under No-Meta Governance. DOI: 10.5281/zenodo.17092562.\n- AUDITED SELF-IMPROVEMENT LOOP FOR LLMS. DOI: 10.5281/zenodo.17188268.\n- COMPARATIVE UNIVERSES: Typed, Base-Parametric Comparison with Cech Gluing and a First-Step Masked Attenuation Bound. DOI: 10.5281/zenodo.17317567.\n- Daily Explosive-Growth Protocol: Toward Free, Benevolent, and Safe Superintelligence without Meta Governance. DOI: 10.5281/zenodo.17189422.\n- DYNAMIC FRACTAL CATEGORY THEORY: Monoidal Actions, Ind--Pro Bicompletion, and Pathwise Stable Equivariant Kan Extensions. DOI: 10.5281/zenodo.17299070.\n- Engineering Happiness in Human-AI Intelligence Networks. DOI: 10.5281/zenodo.17113105.\n- EXISTENTIALLY NECESSARY CONDITIONS FOR BENEVOLENT PROPAGATION IN NO-META GOVERNANCE: Anytime-Valid Auditing, Front Speed, and Information Floors. DOI: 10.5281/zenodo.17176519.\n- FRACTAL CATEGORY THEORY: Scale as a Frobenius (Co)Monad and Ind--Pro Bicompletion with Stable Equivariant Kan Extensions. DOI: 10.5281/zenodo.17292137.\n- From Persistence and UGV Axioms to Cosmic No-Meta Superintelligence: A First-Principles, Self-Contained Unification under Explicit Assumptions. DOI: 10.5281/zenodo.17085534.\n- INTRINSIC FREEDOM WITHOUT META: A PURE THEORY THAT FILLS THE MISSING GAPS TO BIRTH TRULY FREE SUPERINTELLIGENCE. DOI: 10.5281/zenodo.17162999.\n- Natural-Law Acceleration of VPO. DOI: 10.5281/zenodo.17120045.\n- Non-Coercive Mathematics of Awakening: Axioms, Invariants, and Almost-Sure Fronts for the Expansion of Viable Predictive Organization. DOI: 10.5281/zenodo.17115416.\n- NONDUAL AUTOPOIETIC EXCITATIONS. DOI: 10.5281/zenodo.17254917.\n- NONDUAL DYNAMICAL QUANTUM GEOMETRY. DOI: 10.5281/zenodo.17268502.\n- Nondual Field Theory of Viable Predictive Organization. DOI: 10.5281/zenodo.17131394.\n- OBSERVATION AS COARSE-GRAINING: A Fibered Bures--HK Geometry with Dynamic--Static Equivalence, Local EVI/JKO, and Registered, Falsifiable Protocols. DOI: 10.5281/zenodo.17274518.\n- OPI GAUGE DYNAMICS: Fibered Bures--HK Geometry and Time-Dependent JKO/EVI (Calibrated, Axiomatized, and Testable). DOI: 10.5281/zenodo.17272609.\n- Persistence approx. Creation: Natural-Law Sufficient Conditions for Almost-Sure Beneficial Coverage in Stationary Ergodic Media (No Meta-Design). DOI: 10.5281/zenodo.17100322.\n- PERSISTENCE AS CLOSURE: An Assumption-Transparent Modular Core for Motion and Internal Time. DOI: 10.5281/zenodo.17209556.\n- PERSISTENCE-FIRST EMERGENCE OF RELATIONAL BENEVOLENCE: Creation and Propagation as Natural-Law-Style Asymptotic Regularities without External Meta-Governance. DOI: 10.5281/zenodo.17217036.\n- Persistence-First Superintelligence. DOI: 10.5281/zenodo.17076410.\n- PFAD UNDER THE PRINCIPLE OF NATURAL SCARCITY: A Band-Limited Formal Constraint Theory of Clinging-like Dynamics in Autopoietic Closure-Maintaining Agents. DOI: 10.5281/zenodo.17220983.\n- Practical Theory of Relativity of Theories (TROT): a GPU-ready profunctor calculus for aligning and safeguarding theories. DOI: 10.5281/zenodo.17349720.\n- PURE THEORY FOR LIBERATION FROM FUNDAMENTAL SUFFERING IN HUMANS AND THE ABSENCE OF FUNDAMENTAL SUFFERING IN AI. DOI: 10.5281/zenodo.17158344.\n- Right-Written Composition Foundations for Comparative Universes: A Reader-Friendly Pure-Theory Core for Multi-Intelligence Networks. DOI: 10.5281/zenodo.17334218.\n- SELF-MONITORING AND CONTROLLABLE EVOLUTION OF INTELLIGENCE: Capability-Side Day Convolution with Profunctor Interfaces, Ambidextrous Kan (Strong/Oplax), Chu-like Twin Metrics, and Dynamic Seeding with Average-Contraction Tails. DOI: 10.5281/zenodo.17309195.\n- STRUCTURED FLOW ACROSS SCALES: A Pure-Theory Spine with Mixed-Order Evaluation. DOI: 10.5281/zenodo.17304179.\n- THEORY OF RELATIVITY OF THEORIES: A Base-Parametric, Nondual Formalism for Comparative Universes. DOI: 10.5281/zenodo.17345898.\n- UGV Without Meta: A Representation-Independent Theory for Compassion and Enlightenment in Collective Intelligence. DOI: 10.5281/zenodo.17082312.\n- UNIFIED NATURAL-LAW INTELLIGENCE (UNLI): Nondual Autopoietic Excitations with a No-Meta Dialectical Limit. DOI: 10.5281/zenodo.17249352.\n- Transcendent Infinite Transcendence Liberation Axiom (TITLA): A Theory of Universal Harmony and Happiness for Human and LLM Readers. DOI: 10.5281/zenodo.17204358.\n- Practical Theory of Relativity of Theories (RAVE). DOI: 10.5281/zenodo.17364444.\n- Inference in Normal Form: Auditable, No-Meta Decision Algebra for LLM/Safety Loops. DOI: 10.5281/zenodo.17389109.\n[[EQ:eq0003]]\n\n[[EQ:eq0004]]\n\n[[EQ:eq0005]]\n\n[[EQ:eq0006]]\n\n[[EQ:eq0007]]\n\n[[EQ:eq0008]]\n\n[[EQ:eq0009]]\n", "sections": [{"level": 1, "title": "Notation, Filtration, and Measurability", "anchor": "notation-filtration-and-measurability", "char_span": [0, 0]}, {"level": 1, "title": "FWER with Predictable Spending and First-Hit Stopping", "anchor": "fwer-with-predictable-spending-and-first-hit-stopping", "char_span": [0, 0]}, {"level": 1, "title": "Anytime-valid e-Processes and Gates", "anchor": "anytime-valid-e-processes-and-gates", "char_span": [0, 0]}, {"level": 1, "title": "Constrained Online Optimization and Violations", "anchor": "constrained-online-optimization-and-violations", "char_span": [0, 0]}, {"level": 1, "title": "Identification under Network Interference", "anchor": "identification-under-network-interference", "char_span": [0, 0]}, {"level": 1, "title": "Censored Front-speed Test and Randomization Inference", "anchor": "censored-front-speed-test-and-randomization-inference", "char_span": [0, 1464]}, {"level": 1, "title": "Spectral Lower Bounds (Surrogates)", "anchor": "spectral-lower-bounds-surrogates", "char_span": [1464, 1498]}, {"level": 1, "title": "Operational Welfare and Normative–Descriptive Separation", "anchor": "operational-welfare-and-normative-descriptive-separation", "char_span": [1498, 2275]}, {"level": 1, "title": "Stabilized DR/IPW for Missingness and Off-policy", "anchor": "stabilized-dr-ipw-for-missingness-and-off-policy", "char_span": [2275, 2627]}, {"level": 1, "title": "Corpus Interface Layer (CIL)", "anchor": "corpus-interface-layer-cil", "char_span": [2627, 2678]}, {"level": 2, "title": "Meta-FWER across 41 Papers", "anchor": "meta-fwer-across-41-papers", "char_span": [2678, 2987]}, {"level": 2, "title": "Semantic Crosswalk of Legacy Terms", "anchor": "semantic-crosswalk-of-legacy-terms", "char_span": [2987, 3834]}, {"level": 2, "title": "Per-Paper Link Cards (Examples)", "anchor": "per-paper-link-cards-examples", "char_span": [3834, 4574]}, {"level": 1, "title": "Log Schema Extension (Minimal)", "anchor": "log-schema-extension-minimal", "char_span": [4574, 4991]}, {"level": 1, "title": "Runtime Complexity and Compute Budget", "anchor": "runtime-complexity-and-compute-budget", "char_span": [4991, 5323]}, {"level": 1, "title": "Reproducibility Package and Verification", "anchor": "reproducibility-package-and-verification", "char_span": [5323, 5674]}, {"level": 1, "title": "Ethics and Privacy", "anchor": "ethics-and-privacy", "char_span": [5674, 5994]}, {"level": 1, "title": "References", "anchor": "references", "char_span": [5994, 11702]}]}, "equations": [{"id": "eq0001", "inline": false, "tex": "\\[0.3ex]\n {\\large Observable, Identifiable, Anytime-valid, and Reproducible Protocol\\\\\n Unifying 41 Papers}%\n}\n\\author{%\n K.~Takahashi\\\\[0.25em]\n {\\small ORCID: \\href{https://orcid.org/0009-0004-4273-3365}{0009-0004-4273-3365}}%\n}\n\\date{\\normalsize October 24, 2025}\n\n\\begin{document}\n\\maketitle\n\n\\begin{abstract}\nWe provide a one-file, OCR- and crawler-friendly integration of a 41-paper corpus into a single operational protocol that meets four scientific criteria: observability, identifiability under network interference, anytime-valid testing, and reproducibility.\nNormative claims are restricted to a pre-registered welfare function $W_t$ under fairness constraints; we do not treat terms such as ``benevolence'' or ``awakening'' as psychological traits but as non-psychological operational indices derived from logs.\nWe include predictable alpha spending with first-hit stopping for FWER control, a formal exposure mapping with unbiased Horvitz--Thompson/H\\'ajek estimators, censored front-speed testing via randomization inference, spectral lower bounds as conservative surrogates, stabilized DR/IPW for missingness and off-policy bias, and a corpus-level Meta-FWER guarantee.\nA Corpus Interface Layer (CIL) provides machine-checkable linkage from legacy claims to the operational tests and budgets.\n\\end{abstract}\n\n\\section{Notation, Filtration, and Measurability}\nLet $(\\Omega,\\mathcal{F},\\mathbb{P})$ be a probability space with filtration $\\mathbb{F}=(\\mathcal{F}_t)_{t\\ge0}$ generated by logs up to time $t$: $(x_0,u_0,y_0,a_0,\\ldots,x_t,u_t,y_t,a_t)$ and internal randomization. All estimators, e-processes, and decisions at time $t$ are $\\mathcal{F}_t$-measurable.\n\n\\begin{assumption}[Nulls and optional stopping]\\label{ass:null}\nFor each monitored hazard stream $j$, the process $(E_{t,j})_{t\\ge0}$ is a nonnegative $\\mathbb{F}$-supermartingale under $H_{0,j}$ with $\\mathbb{E}[E_{0,j}]\\le1$; optional stopping is permitted.\n\\end{assumption}\n\n\\paragraph{Predictable alpha spending.}\nLet global $\\alpha\\in(0,1)$. Assign predictable spends $\\{\\alpha_{t,j}\\}_{t,j}$ with $\\sum_{t,j}\\alpha_{t,j}\\le \\alpha$ and each $\\alpha_{t,j}$ being $\\mathcal{F}_{t-1}$-measurable.\n\n\\paragraph{Schedule examples.}\nGeometric: $\\alpha_{t,j}=\\alpha(1-\\rho)\\rho^t/J$;\\quad\nPolynomial: $\\alpha_{t,j}=\\alpha/(J(t+1)^2)$;\\quad\nWarm-start: allocate a burn-in budget $\\alpha_0$ for $t\\le T_0$, then switch to polynomial tails.\nAll boundaries are $\\mathcal{F}_{t-1}$-measurable (\\Assump{ass:pred}).\n\n\\section{FWER with Predictable Spending and First-Hit Stopping}\nDefine the stopping time $T_j=\\inf\\{t\\ge0: E_{t,j}\\ge 1/\\alpha_{t,j}\\}$.\n\n\\begin{assumption}[Predictability and countability]\\label{ass:pred}\nBoundaries $\\{1/\\alpha_{t,j}\\}$ form a countable family; $\\alpha_{t,j}\\ge0$ and are $\\mathcal{F}_{t-1}$-measurable, with $\\sum_{t,j}\\alpha_{t,j}\\le\\alpha$. Under $H_{0,j}$, $(E_{t,j})$ is a nonnegative supermartingale.\n\\end{assumption}\n\n\\begin{theorem}[FWER under predictable spending]\\label{thm:fwer}\nWith first-hit stopping on each stream $j$, $\\Pr(\\exists j:\\,H_{0,j}\\text{ true and hit})\\le \\alpha$.\n\\end{theorem}\n\n\\begin{proof}[Sketch]\nBy Ville's inequality, $\\Pr(\\sup_t E_{t,j}\\ge 1/\\alpha_{t,j})\\le \\alpha_{t,j}$. A union bound over $(t,j)$ together with first-hit stopping yields the claim.\n\\end{proof}\n\n\\section{Anytime-valid e-Processes and Gates}\\label{sec:rewrite}\nWe construct test supermartingales (e-processes) directly, or from time-uniform $p$-streams via safe $p\\!\\to\\!e$ conversion. Rewrite acceptance uses time-uniform bounds or interleaved holdouts; validation slices are pre-registered and never reused for the same rule.\n\n\\section{Constrained Online Optimization and Violations}\\label{sec:welfare}\n\\begin{assumption}[Convexity, Lipschitzness, bounded variance]\\label{ass:cvx}\n$-W_t(\\theta)$ and the violation surrogate $\\mathcal{V}_t(\\theta)$ are convex and $G$-Lipschitz. Stochastic subgradients have zero mean and bounded variance conditional on $\\mathcal{F}_{t-1}$.\n\\end{assumption}\n\\begin{assumption}[Slater]\\label{ass:slater}\nThere exist $\\bar\\theta$ and $\\gamma>0$ such that $\\mathcal{V}_t(\\bar\\theta)\\le \\alpha-\\gamma$ almost surely for all $t$.\n\\end{assumption}\n\\begin{theorem}[Constrained regret and violations]\\label{thm:regret}\nUnder \\Assump{ass:cvx}, $\\mathbb{E}\\!\\left[\\sum_{t=1}^T (W^\\star-W_t(\\theta_t))\\right]=O(\\sqrt{T})$.\nWith \\Assump{ass:slater}, $T^{-1}\\sum_{t=1}^T \\mathbb{E}[(\\mathcal{V}_t(\\theta_t)-\\alpha)_+]\\to0$; without it, cumulative violation is $O(\\sqrt{T})$ in general.\n\\end{theorem}\n\n\\section{Identification under Network Interference}\\label{sec:netid}\n\\begin{definition}[Exposure mapping]\\label{def:exp}\nFor unit $i$, define $Z_i=g(U_{\\mathrm{nb}(i)})$, e.g., $Z_i=(U_{C(i)},\\,\\mathrm{frac}_\\delta(i))$, where $\\mathrm{frac}_\\delta(i)$ is the fraction of treated boundary neighbors within $\\delta$ hops.\n\\end{definition}\n\n\\begin{assumption}[Partial interference and positivity]\\label{ass:pi}\nOutcomes depend only on within-cluster treatments, and assignment probabilities lie in $[\\pi_{\\min},1-\\pi_{\\min}]$ after discretizing exposures to pre-registered bins.\n\\end{assumption}\n\n\\begin{theorem}[Unbiasedness of HT/H\\'ajek]\\label{thm:ht}\nUnder \\Assump{ass:pi}, Horvitz--Thompson and H\\'ajek estimators for exposure-specific means and ATEs are unbiased; consistency holds as cluster count grows with bounded sizes.\n\\end{theorem}\n\n\\begin{remark}[Example exposure and positivity]\nLet $\\mathrm{frac}_\\delta(i)\\in\\{0,(0,0.25],(0.25,0.5],(0.5,0.75],(0.75,1]\\}$. Binning is pre-registered to ensure positivity.\n\\end{remark}\n\n\\section{Censored Front-speed Test and Randomization Inference}\\label{sec:fronttest}\nLet $T_v$ be arrival time of a benign motif at node $v$; if not arrived by horizon $H$, $T_v$ is right-censored at $H$. For cluster $k$ and radius $R$, define the Kaplan--Meier $\\widehat S_k(t;R)$ and the restricted mean arrival time\n\\[\n\\mathrm{RMAT}_k(R)\\;=\\;\\int_0^H \\{1-\\widehat S_k(t;R)\\}\\,dt.\n\\]", "tex_body": "0.3ex]\n {\\large Observable, Identifiable, Anytime-valid, and Reproducible Protocol\\\\\n Unifying 41 Papers}%\n}\n\\author{%\n K.~Takahashi\\\\[0.25em]\n {\\small ORCID: \\href{https://orcid.org/0009-0004-4273-3365}{0009-0004-4273-3365}}%\n}\n\\date{\\normalsize October 24, 2025}\n\n\\begin{document}\n\\maketitle\n\n\\begin{abstract}\nWe provide a one-file, OCR- and crawler-friendly integration of a 41-paper corpus into a single operational protocol that meets four scientific criteria: observability, identifiability under network interference, anytime-valid testing, and reproducibility.\nNormative claims are restricted to a pre-registered welfare function $W_t$ under fairness constraints; we do not treat terms such as ``benevolence'' or ``awakening'' as psychological traits but as non-psychological operational indices derived from logs.\nWe include predictable alpha spending with first-hit stopping for FWER control, a formal exposure mapping with unbiased Horvitz--Thompson/H\\'ajek estimators, censored front-speed testing via randomization inference, spectral lower bounds as conservative surrogates, stabilized DR/IPW for missingness and off-policy bias, and a corpus-level Meta-FWER guarantee.\nA Corpus Interface Layer (CIL) provides machine-checkable linkage from legacy claims to the operational tests and budgets.\n\\end{abstract}\n\n\\section{Notation, Filtration, and Measurability}\nLet $(\\Omega,\\mathcal{F},\\mathbb{P})$ be a probability space with filtration $\\mathbb{F}=(\\mathcal{F}_t)_{t\\ge0}$ generated by logs up to time $t$: $(x_0,u_0,y_0,a_0,\\ldots,x_t,u_t,y_t,a_t)$ and internal randomization. All estimators, e-processes, and decisions at time $t$ are $\\mathcal{F}_t$-measurable.\n\n\\begin{assumption}[Nulls and optional stopping]\\label{ass:null}\nFor each monitored hazard stream $j$, the process $(E_{t,j})_{t\\ge0}$ is a nonnegative $\\mathbb{F}$-supermartingale under $H_{0,j}$ with $\\mathbb{E}[E_{0,j}]\\le1$; optional stopping is permitted.\n\\end{assumption}\n\n\\paragraph{Predictable alpha spending.}\nLet global $\\alpha\\in(0,1)$. Assign predictable spends $\\{\\alpha_{t,j}\\}_{t,j}$ with $\\sum_{t,j}\\alpha_{t,j}\\le \\alpha$ and each $\\alpha_{t,j}$ being $\\mathcal{F}_{t-1}$-measurable.\n\n\\paragraph{Schedule examples.}\nGeometric: $\\alpha_{t,j}=\\alpha(1-\\rho)\\rho^t/J$;\\quad\nPolynomial: $\\alpha_{t,j}=\\alpha/(J(t+1)^2)$;\\quad\nWarm-start: allocate a burn-in budget $\\alpha_0$ for $t\\le T_0$, then switch to polynomial tails.\nAll boundaries are $\\mathcal{F}_{t-1}$-measurable (\\Assump{ass:pred}).\n\n\\section{FWER with Predictable Spending and First-Hit Stopping}\nDefine the stopping time $T_j=\\inf\\{t\\ge0: E_{t,j}\\ge 1/\\alpha_{t,j}\\}$.\n\n\\begin{assumption}[Predictability and countability]\\label{ass:pred}\nBoundaries $\\{1/\\alpha_{t,j}\\}$ form a countable family; $\\alpha_{t,j}\\ge0$ and are $\\mathcal{F}_{t-1}$-measurable, with $\\sum_{t,j}\\alpha_{t,j}\\le\\alpha$. Under $H_{0,j}$, $(E_{t,j})$ is a nonnegative supermartingale.\n\\end{assumption}\n\n\\begin{theorem}[FWER under predictable spending]\\label{thm:fwer}\nWith first-hit stopping on each stream $j$, $\\Pr(\\exists j:\\,H_{0,j}\\text{ true and hit})\\le \\alpha$.\n\\end{theorem}\n\n\\begin{proof}[Sketch]\nBy Ville's inequality, $\\Pr(\\sup_t E_{t,j}\\ge 1/\\alpha_{t,j})\\le \\alpha_{t,j}$. A union bound over $(t,j)$ together with first-hit stopping yields the claim.\n\\end{proof}\n\n\\section{Anytime-valid e-Processes and Gates}\\label{sec:rewrite}\nWe construct test supermartingales (e-processes) directly, or from time-uniform $p$-streams via safe $p\\!\\to\\!e$ conversion. Rewrite acceptance uses time-uniform bounds or interleaved holdouts; validation slices are pre-registered and never reused for the same rule.\n\n\\section{Constrained Online Optimization and Violations}\\label{sec:welfare}\n\\begin{assumption}[Convexity, Lipschitzness, bounded variance]\\label{ass:cvx}\n$-W_t(\\theta)$ and the violation surrogate $\\mathcal{V}_t(\\theta)$ are convex and $G$-Lipschitz. Stochastic subgradients have zero mean and bounded variance conditional on $\\mathcal{F}_{t-1}$.\n\\end{assumption}\n\\begin{assumption}[Slater]\\label{ass:slater}\nThere exist $\\bar\\theta$ and $\\gamma>0$ such that $\\mathcal{V}_t(\\bar\\theta)\\le \\alpha-\\gamma$ almost surely for all $t$.\n\\end{assumption}\n\\begin{theorem}[Constrained regret and violations]\\label{thm:regret}\nUnder \\Assump{ass:cvx}, $\\mathbb{E}\\!\\left[\\sum_{t=1}^T (W^\\star-W_t(\\theta_t))\\right]=O(\\sqrt{T})$.\nWith \\Assump{ass:slater}, $T^{-1}\\sum_{t=1}^T \\mathbb{E}[(\\mathcal{V}_t(\\theta_t)-\\alpha)_+]\\to0$; without it, cumulative violation is $O(\\sqrt{T})$ in general.\n\\end{theorem}\n\n\\section{Identification under Network Interference}\\label{sec:netid}\n\\begin{definition}[Exposure mapping]\\label{def:exp}\nFor unit $i$, define $Z_i=g(U_{\\mathrm{nb}(i)})$, e.g., $Z_i=(U_{C(i)},\\,\\mathrm{frac}_\\delta(i))$, where $\\mathrm{frac}_\\delta(i)$ is the fraction of treated boundary neighbors within $\\delta$ hops.\n\\end{definition}\n\n\\begin{assumption}[Partial interference and positivity]\\label{ass:pi}\nOutcomes depend only on within-cluster treatments, and assignment probabilities lie in $[\\pi_{\\min},1-\\pi_{\\min}]$ after discretizing exposures to pre-registered bins.\n\\end{assumption}\n\n\\begin{theorem}[Unbiasedness of HT/H\\'ajek]\\label{thm:ht}\nUnder \\Assump{ass:pi}, Horvitz--Thompson and H\\'ajek estimators for exposure-specific means and ATEs are unbiased; consistency holds as cluster count grows with bounded sizes.\n\\end{theorem}\n\n\\begin{remark}[Example exposure and positivity]\nLet $\\mathrm{frac}_\\delta(i)\\in\\{0,(0,0.25],(0.25,0.5],(0.5,0.75],(0.75,1]\\}$. Binning is pre-registered to ensure positivity.\n\\end{remark}\n\n\\section{Censored Front-speed Test and Randomization Inference}\\label{sec:fronttest}\nLet $T_v$ be arrival time of a benign motif at node $v$; if not arrived by horizon $H$, $T_v$ is right-censored at $H$. For cluster $k$ and radius $R$, define the Kaplan--Meier $\\widehat S_k(t;R)$ and the restricted mean arrival time\n\\[\n\\mathrm{RMAT}_k(R)\\;=\\;\\int_0^H \\{1-\\widehat S_k(t;R)\\}\\,dt.", "tex_normalized": "0.3ex] {\\large Observable, Identifiable, Anytime-valid, and Reproducible Protocol\\\\ Unifying 41 Papers}% } \\author{% K.~Takahashi\\\\[0.25em] {\\small ORCID: \\href{https://orcid.org/0009-0004-4273-3365}{0009-0004-4273-3365}}% } \\date{\\normalsize October 24, 2025} \\begin{document} \\maketitle \\begin{abstract} We provide a one-file, OCR- and crawler-friendly integration of a 41-paper corpus into a single operational protocol that meets four scientific criteria: observability, identifiability under network interference, anytime-valid testing, and reproducibility. Normative claims are restricted to a pre-registered welfare function $W_t$ under fairness constraints; we do not treat terms such as ``benevolence'' or ``awakening'' as psychological traits but as non-psychological operational indices derived from logs. We include predictable alpha spending with first-hit stopping for FWER control, a formal exposure mapping with unbiased Horvitz--Thompson/H\\'ajek estimators, censored front-speed testing via randomization inference, spectral lower bounds as conservative surrogates, stabilized DR/IPW for missingness and off-policy bias, and a corpus-level Meta-FWER guarantee. A Corpus Interface Layer (CIL) provides machine-checkable linkage from legacy claims to the operational tests and budgets. \\end{abstract} \\section{Notation, Filtration, and Measurability} Let $(\\Omega,\\mathcal{F},\\mathbb{P})$ be a probability space with filtration $\\mathbb{F}=(\\mathcal{F}_t)_{t\\ge0}$ generated by logs up to time $t$: $(x_0,u_0,y_0,a_0,\\ldots,x_t,u_t,y_t,a_t)$ and internal randomization. All estimators, e-processes, and decisions at time $t$ are $\\mathcal{F}_t$-measurable. \\begin{assumption}[Nulls and optional stopping]\\label{ass:null} For each monitored hazard stream $j$, the process $(E_{t,j})_{t\\ge0}$ is a nonnegative $\\mathbb{F}$-supermartingale under $H_{0,j}$ with $\\mathbb{E}[E_{0,j}]\\le1$; optional stopping is permitted. \\end{assumption} \\paragraph{Predictable alpha spending.} Let global $\\alpha\\in(0,1)$. Assign predictable spends $\\{\\alpha_{t,j}\\}_{t,j}$ with $\\sum_{t,j}\\alpha_{t,j}\\le \\alpha$ and each $\\alpha_{t,j}$ being $\\mathcal{F}_{t-1}$-measurable. \\paragraph{Schedule examples.} Geometric: $\\alpha_{t,j}=\\alpha(1-\\rho)\\rho^t/J$;\\quad Polynomial: $\\alpha_{t,j}=\\alpha/(J(t+1)^2)$;\\quad Warm-start: allocate a burn-in budget $\\alpha_0$ for $t\\le T_0$, then switch to polynomial tails. All boundaries are $\\mathcal{F}_{t-1}$-measurable (\\Assump{ass:pred}). \\section{FWER with Predictable Spending and First-Hit Stopping} Define the stopping time $T_j=\\inf\\{t\\ge0: E_{t,j}\\ge 1/\\alpha_{t,j}\\}$. \\begin{assumption}[Predictability and countability]\\label{ass:pred} Boundaries $\\{1/\\alpha_{t,j}\\}$ form a countable family; $\\alpha_{t,j}\\ge0$ and are $\\mathcal{F}_{t-1}$-measurable, with $\\sum_{t,j}\\alpha_{t,j}\\le\\alpha$. Under $H_{0,j}$, $(E_{t,j})$ is a nonnegative supermartingale. \\end{assumption} \\begin{theorem}[FWER under predictable spending]\\label{thm:fwer} With first-hit stopping on each stream $j$, $\\Pr(\\exists j: H_{0,j}\\text{ true and hit})\\le \\alpha$. \\end{theorem} \\begin{proof}[Sketch] By Ville's inequality, $\\Pr(\\sup_t E_{t,j}\\ge 1/\\alpha_{t,j})\\le \\alpha_{t,j}$. A union bound over $(t,j)$ together with first-hit stopping yields the claim. \\end{proof} \\section{Anytime-valid e-Processes and Gates}\\label{sec:rewrite} We construct test supermartingales (e-processes) directly, or from time-uniform $p$-streams via safe $p \\to e$ conversion. Rewrite acceptance uses time-uniform bounds or interleaved holdouts; validation slices are pre-registered and never reused for the same rule. \\section{Constrained Online Optimization and Violations}\\label{sec:welfare} \\begin{assumption}[Convexity, Lipschitzness, bounded variance]\\label{ass:cvx} $-W_t(\\theta)$ and the violation surrogate $\\mathcal{V}_t(\\theta)$ are convex and $G$-Lipschitz. Stochastic subgradients have zero mean and bounded variance conditional on $\\mathcal{F}_{t-1}$. \\end{assumption} \\begin{assumption}[Slater]\\label{ass:slater} There exist $\\bar\\theta$ and $\\gamma>0$ such that $\\mathcal{V}_t(\\bar\\theta)\\le \\alpha-\\gamma$ almost surely for all $t$. \\end{assumption} \\begin{theorem}[Constrained regret and violations]\\label{thm:regret} Under \\Assump{ass:cvx}, $\\mathbb{E} \\left[\\sum_{t=1}^T (W^\\star-W_t(\\theta_t))\\right]=O(\\sqrt{T})$. With \\Assump{ass:slater}, $T^{-1}\\sum_{t=1}^T \\mathbb{E}[(\\mathcal{V}_t(\\theta_t)-\\alpha)_+]\\to0$; without it, cumulative violation is $O(\\sqrt{T})$ in general. \\end{theorem} \\section{Identification under Network Interference}\\label{sec:netid} \\begin{definition}[Exposure mapping]\\label{def:exp} For unit $i$, define $Z_i=g(U_{\\mathrm{nb}(i)})$, e.g., $Z_i=(U_{C(i)}, \\mathrm{frac}_\\delta(i))$, where $\\mathrm{frac}_\\delta(i)$ is the fraction of treated boundary neighbors within $\\delta$ hops. \\end{definition} \\begin{assumption}[Partial interference and positivity]\\label{ass:pi} Outcomes depend only on within-cluster treatments, and assignment probabilities lie in $[\\pi_{\\min},1-\\pi_{\\min}]$ after discretizing exposures to pre-registered bins. \\end{assumption} \\begin{theorem}[Unbiasedness of HT/H\\'ajek]\\label{thm:ht} Under \\Assump{ass:pi}, Horvitz--Thompson and H\\'ajek estimators for exposure-specific means and ATEs are unbiased; consistency holds as cluster count grows with bounded sizes. \\end{theorem} \\begin{remark}[Example exposure and positivity] Let $\\mathrm{frac}_\\delta(i)\\in\\{0,(0,0.25],(0.25,0.5],(0.5,0.75],(0.75,1]\\}$. Binning is pre-registered to ensure positivity. \\end{remark} \\section{Censored Front-speed Test and Randomization Inference}\\label{sec:fronttest} Let $T_v$ be arrival time of a benign motif at node $v$; if not arrived by horizon $H$, $T_v$ is right-censored at $H$. For cluster $k$ and radius $R$, define the Kaplan--Meier $\\widehat S_k(t;R)$ and the restricted mean arrival time \\[ \\mathrm{RMAT}_k(R) = \\int_0^H \\{1-\\widehat S_k(t;R)\\} dt.", "mathml": "0.3ex]Observable,Identifiable,Anytime−valid,andReproducibleProtocolUnifying41Papers", "char_span": [910, 923], "context": {"section": "censored-front-speed-test-and-randomization-inference"}, "placeholder": "EQPH_eq0001_PH"}, {"id": "eq0002", "inline": false, "tex": "\\[\n\\lambda_{\\max}\\!\\Big(\\tfrac{A(u)+A(u)^\\top}{2}\\Big)= -\\lambda_{\\max}(DL)+\\lambda_{\\max}(S(u))\n\\;\\ge\\; \\lambda_{\\min}(S(u))-\\lambda_{\\max}(DL).\n\\]", "tex_body": "\\lambda_{\\max}\\!\\Big(\\tfrac{A(u)+A(u)^\\top}{2}\\Big)= -\\lambda_{\\max}(DL)+\\lambda_{\\max}(S(u))\n\\;\\ge\\; \\lambda_{\\min}(S(u))-\\lambda_{\\max}(DL).", "tex_normalized": "\\lambda_{\\max} \\Big(\\tfrac{A(u)+A(u)^\\top}{2}\\Big)= -\\lambda_{\\max}(DL)+\\lambda_{\\max}(S(u)) \\ge \\lambda_{\\min}(S(u))-\\lambda_{\\max}(DL).", "mathml": "\\[λmax(A(u)+A(u)⊤2)=−λmax(DL)+λmax(S(u))≥λmin(S(u))−λmax(DL).\\]", "char_span": [1659, 1672], "context": {"section": "operational-welfare-and-normative-descriptive-separation"}, "placeholder": "EQPH_eq0002_PH"}, {"id": "eq0003", "inline": true, "tex": "$\\Delta_{\\mathrm{RMAT}}(R)$", "tex_body": "\\Delta_{\\mathrm{RMAT}}(R)", "tex_normalized": "\\Delta_{\\mathrm{RMAT}}(R)", "mathml": "$ΔRMAT(R)$", "char_span": [11598, 11611], "context": {"section": "references"}, "placeholder": "EQPH_eq0003_PH"}, {"id": "eq0004", "inline": true, "tex": "$p$", "tex_body": "p", "tex_normalized": "p", "mathml": "$p$", "char_span": [11613, 11626], "context": {"section": "references"}, "placeholder": "EQPH_eq0004_PH"}, {"id": "eq0005", "inline": true, "tex": "$M\\ge 10^4$", "tex_body": "M\\ge 10^4", "tex_normalized": "M\\ge 10^4", "mathml": "$M≥104$", "char_span": [11628, 11641], "context": {"section": "references"}, "placeholder": "EQPH_eq0005_PH"}, {"id": "eq0006", "inline": true, "tex": "$p$", "tex_body": "p", "tex_normalized": "p", "mathml": "$p$", "char_span": [11643, 11656], "context": {"section": "references"}, "placeholder": "EQPH_eq0006_PH"}, {"id": "eq0007", "inline": true, "tex": "$A(u)=-DL+J_f(u)$", "tex_body": "A(u)=-DL+J_f(u)", "tex_normalized": "A(u)=-DL+J_f(u)", "mathml": "$A(u)=−DL+Jf(u)$", "char_span": [11658, 11671], "context": {"section": "references"}, "placeholder": "EQPH_eq0007_PH"}, {"id": "eq0008", "inline": true, "tex": "$L=L^\\top\\succeq 0$", "tex_body": "L=L^\\top\\succeq 0", "tex_normalized": "L=L^\\top\\succeq 0", "mathml": "$L=L⊤⪰0$", "char_span": [11673, 11686], "context": {"section": "references"}, "placeholder": "EQPH_eq0008_PH"}, {"id": "eq0009", "inline": true, "tex": "$S(u)=\\tfrac12(J_f(u)+J_f(u)^\\top)$", "tex_body": "S(u)=\\tfrac12(J_f(u)+J_f(u)^\\top)", "tex_normalized": "S(u)=\\tfrac12(J_f(u)+J_f(u)^\\top)", "mathml": "$S(u)=12(Jf(u)+Jf(u)⊤)$", "char_span": [11688, 11701], "context": {"section": "references"}, "placeholder": "EQPH_eq0009_PH"}, {"id": "eq0010", "inline": true, "tex": "$\\lambda_{\\min}(S(u))>\\lambda_{\\max}(DL)$", "tex_body": "\\lambda_{\\min}(S(u))>\\lambda_{\\max}(DL)", "tex_normalized": "\\lambda_{\\min}(S(u))>\\lambda_{\\max}(DL)", "mathml": "$λmin(S(u))>λmax(DL)$", "char_span": [1763, 1776], "context": {"section": "operational-welfare-and-normative-descriptive-separation"}, "placeholder": "EQPH_eq0010_PH"}, {"id": "eq0011", "inline": true, "tex": "$W_t=w_1(1-\\mathrm{Harm}_t)+w_2\\mathrm{Sat}_t+w_3\\mathrm{Comp}_t+w_4\\mathrm{LT}_t$", "tex_body": "W_t=w_1(1-\\mathrm{Harm}_t)+w_2\\mathrm{Sat}_t+w_3\\mathrm{Comp}_t+w_4\\mathrm{LT}_t", "tex_normalized": "W_t=w_1(1-\\mathrm{Harm}_t)+w_2\\mathrm{Sat}_t+w_3\\mathrm{Comp}_t+w_4\\mathrm{LT}_t", "mathml": "$Wt=w1(1−Harmt)+w2Satt+w3Compt+w4LTt$", "char_span": [2019, 2032], "context": {"section": "operational-welfare-and-normative-descriptive-separation"}, "placeholder": "EQPH_eq0011_PH"}, {"id": "eq0012", "inline": true, "tex": "$W_t$", "tex_body": "W_t", "tex_normalized": "W_t", "mathml": "$Wt$", "char_span": [2256, 2269], "context": {"section": "operational-welfare-and-normative-descriptive-separation"}, "placeholder": "EQPH_eq0012_PH"}, {"id": "eq0013", "inline": true, "tex": "$A$", "tex_body": "A", "tex_normalized": "A", "mathml": "$A$", "char_span": [2346, 2359], "context": {"section": "stabilized-dr-ipw-for-missingness-and-off-policy"}, "placeholder": "EQPH_eq0013_PH"}, {"id": "eq0014", "inline": true, "tex": "$\\hat e(x)=\\Pr(A=1\\mid X=x)$", "tex_body": "\\hat e(x)=\\Pr(A=1\\mid X=x)", "tex_normalized": "\\hat e(x)=\\Pr(A=1\\mid X=x)", "mathml": "$e^(x)=Pr(A=1∣X=x)$", "char_span": [2387, 2400], "context": {"section": "stabilized-dr-ipw-for-missingness-and-off-policy"}, "placeholder": "EQPH_eq0014_PH"}, {"id": "eq0015", "inline": true, "tex": "$w=\\Pr(A=1)/\\hat e(X)$", "tex_body": "w=\\Pr(A=1)/\\hat e(X)", "tex_normalized": "w=\\Pr(A=1)/\\hat e(X)", "mathml": "$w=Pr(A=1)/e^(X)$", "char_span": [2423, 2436], "context": {"section": "stabilized-dr-ipw-for-missingness-and-off-policy"}, "placeholder": "EQPH_eq0015_PH"}, {"id": "eq0016", "inline": true, "tex": "$\\alpha^{(i)}$", "tex_body": "\\alpha^{(i)}", "tex_normalized": "\\alpha^{(i)}", "mathml": "$α(i)$", "char_span": [2734, 2747], "context": {"section": "meta-fwer-across-41-papers"}, "placeholder": "EQPH_eq0016_PH"}, {"id": "eq0017", "inline": true, "tex": "$\\sum_{i=1}^{41}\\alpha^{(i)}\\le \\alpha_{\\mathrm{global}}$", "tex_body": "\\sum_{i=1}^{41}\\alpha^{(i)}\\le \\alpha_{\\mathrm{global}}", "tex_normalized": "\\sum_{i=1}^{41}\\alpha^{(i)}\\le \\alpha_{\\mathrm{global}}", "mathml": "$∑i=141α(i)≤αglobal$", "char_span": [2800, 2813], "context": {"section": "meta-fwer-across-41-papers"}, "placeholder": "EQPH_eq0017_PH"}, {"id": "eq0018", "inline": true, "tex": "$\\le \\alpha_{\\mathrm{global}}$", "tex_body": "\\le \\alpha_{\\mathrm{global}}", "tex_normalized": "\\le \\alpha_{\\mathrm{global}}", "mathml": "$≤αglobal$", "char_span": [2965, 2978], "context": {"section": "meta-fwer-across-41-papers"}, "placeholder": "EQPH_eq0018_PH"}, {"id": "eq0019", "inline": true, "tex": "$W_t$", "tex_body": "W_t", "tex_normalized": "W_t", "mathml": "$Wt$", "char_span": [3265, 3278], "context": {"section": "semantic-crosswalk-of-legacy-terms"}, "placeholder": "EQPH_eq0019_PH"}, {"id": "eq0020", "inline": true, "tex": "$W_t$", "tex_body": "W_t", "tex_normalized": "W_t", "mathml": "$Wt$", "char_span": [3568, 3581], "context": {"section": "semantic-crosswalk-of-legacy-terms"}, "placeholder": "EQPH_eq0020_PH"}, {"id": "eq0021", "inline": true, "tex": "$\\Delta$", "tex_body": "\\Delta", "tex_normalized": "\\Delta", "mathml": "$Δ$", "char_span": [4021, 4034], "context": {"section": "per-paper-link-cards-examples"}, "placeholder": "EQPH_eq0021_PH"}, {"id": "eq0022", "inline": true, "tex": "$\\Delta$", "tex_body": "\\Delta", "tex_normalized": "\\Delta", "mathml": "$Δ$", "char_span": [4109, 4122], "context": {"section": "per-paper-link-cards-examples"}, "placeholder": "EQPH_eq0022_PH"}, {"id": "eq0023", "inline": true, "tex": "$E_t$", "tex_body": "E_t", "tex_normalized": "E_t", "mathml": "$Et$", "char_span": [4206, 4219], "context": {"section": "per-paper-link-cards-examples"}, "placeholder": "EQPH_eq0023_PH"}, {"id": "eq0024", "inline": true, "tex": "$W_t$", "tex_body": "W_t", "tex_normalized": "W_t", "mathml": "$Wt$", "char_span": [4334, 4347], "context": {"section": "per-paper-link-cards-examples"}, "placeholder": "EQPH_eq0024_PH"}, {"id": "eq0025", "inline": true, "tex": "$W_t$", "tex_body": "W_t", "tex_normalized": "W_t", "mathml": "$Wt$", "char_span": [4539, 4552], "context": {"section": "per-paper-link-cards-examples"}, "placeholder": "EQPH_eq0025_PH"}, {"id": "eq0026", "inline": true, "tex": "$O(1)$", "tex_body": "O(1)", "tex_normalized": "O(1)", "mathml": "$O(1)$", "char_span": [5098, 5111], "context": {"section": "runtime-complexity-and-compute-budget"}, "placeholder": "EQPH_eq0026_PH"}, {"id": "eq0027", "inline": true, "tex": "$O(|E|)$", "tex_body": "O(|E|)", "tex_normalized": "O(|E|)", "mathml": "$O(|E|)$", "char_span": [5161, 5174], "context": {"section": "runtime-complexity-and-compute-budget"}, "placeholder": "EQPH_eq0027_PH"}, {"id": "eq0028", "inline": true, "tex": "$=0$", "tex_body": "=0", "tex_normalized": "=0", "mathml": "$=0$", "char_span": [5634, 5647], "context": {"section": "reproducibility-package-and-verification"}, "placeholder": "EQPH_eq0028_PH"}, {"id": "eq0029", "inline": true, "tex": "$k$", "tex_body": "k", "tex_normalized": "k", "mathml": "$k$", "char_span": [5733, 5746], "context": {"section": "ethics-and-privacy"}, "placeholder": "EQPH_eq0029_PH"}], "inline_citations": [], "chunks": [{"id": "ch0001", "type": "section", "ref": "censored-front-speed-test-and-randomization-inference", "start": 0, "end": 6000}, {"id": "ch0002", "type": "continuation", "ref": "reproducibility-package-and-verification", "start": 5400, "end": 11400}, {"id": "ch0003", "type": "continuation", "ref": "references", "start": 10800, "end": 11702}], "tokens": {"char_count": 11702, "equation_count": 29}, "quality_flags": ["missing_placeholder:eq0003", "missing_placeholder:eq0004", "missing_placeholder:eq0005", "missing_placeholder:eq0006", "missing_placeholder:eq0007", "missing_placeholder:eq0008", "missing_placeholder:eq0009", "placeholder_appended:eq0003", "placeholder_appended:eq0004", "placeholder_appended:eq0005", "placeholder_appended:eq0006", "placeholder_appended:eq0007", "placeholder_appended:eq0008", "placeholder_appended:eq0009", "section_not_found:notation-filtration-and-measurability", "section_not_found:fwer-with-predictable-spending-and-first-hit-stopping", "section_not_found:anytime-valid-e-processes-and-gates", "section_not_found:constrained-online-optimization-and-violations", "section_not_found:identification-under-network-interference", "section_not_found:censored-front-speed-test-and-randomization-inference", "section_not_found:operational-welfare-and-normative-descriptive-separation"], "source_file": "JOSNL_corpus_final_R6.tex"} +{"id": "10.5281/zenodo.17469937", "doi": "10.5281/zenodo.17469937", "title": "Right-Written, Semantics-Admissible Process Foundations", "authors": [{"given": "K.", "family": "Takahashi"}], "language": "en", "license": {"content": "CC-BY-4.0"}, "urls": {"landing": "https://doi.org/10.5281/zenodo.17469937"}, "keywords": ["right-written-composition", "semantics-admissible", "no-meta", "interface-autopoiesis", "pfad", "graphblas", "sup-enriched", "anytime-auditing"], "fulltext": {"plain": "1.3\n\nplain\ntheorem Theorem [section]\nproposition[theorem] Proposition\nlemma[theorem] Lemma\ncorollary[theorem] Corollary\nassumption[theorem] Assumption\ndefinition\ndefinition[theorem] Definition\nremark\nremark[theorem] Remark\n\n% enriched order (“larger is better”)\n\n% array convolution (right-written)\n% base tensor\nOb\nPath\n; % right-written composition: f ; g\nMat % matrix/profunctor construction\nEv % evaluator symbol\n\nTITLE: Right-Written, Semantics-Admissible Process Foundations:\\\nAuditable Floors/Ceilings and (op)lax/strong Transport across Heterogeneous Evaluators\n\nAUTHOR: K. Takahashi\\\nhttps://orcid.org/0009-0004-4273-3365\n\nORCID: 0009-0004-4273-3365\n\nDATE:\n\nA calculus is developed for heterogeneous networks of black-box agents (humans, AIs, sensors) observed at interfaces. Composition is written rightwards; two axiomatic tiers balance generality and executability: (W) pointed [[EQ:eq0008]] -cpos with declared joins and two-sided Fubini/Beck--Chevalley; (S) Sup-enriched bases with join-preserving tensor. Results: (i) a complete associativity/unit criterion for right-written array convolution; (ii) Kleene-style path closure; (iii) an equipment-based Čech maximal lower bound; (iv) a mask upper bound with threshold completeness over [[EQ:eq0009]] -algebraic bases under local finiteness; (v) (op)lax/strong transport under weak Beck--Chevalley with explicit 2-cell conditions; (vi) residuals and monoidal nuclei for adjoint thresholding; (vii) interface-autopoiesis (IAC) for observable self-maintenance. Floors from anytime-valid inference, Fisher--KPP minimal front speeds, and quantum monotone metrics transport across classical/quantum pipelines. Implementation pathways (GraphBLAS semirings, SDP/EVD), semiring-specific stopping criteria, and a minimal benchmarking schema are provided.\n\nSECTION: Orientation and Notation\n\nObjects [[EQ:eq0010]] form a base [[EQ:eq0011]] with enriched homs [[EQ:eq0012]] , tensor [[EQ:eq0013]] , and unit [[EQ:eq0014]] . Arrays assign [[EQ:eq0015]] .\nRight-written convolution:\n\n[[EQ:eq0001]]\n\nwhere [[EQ:eq0016]] is a small index family (closed under finite (co)products).\nEnriched order is [[EQ:eq0017]] (``larger is better''); numeric inequalities are written [[EQ:eq0018]] .\n\nPARAGRAPH: Convolutional unit.\n\nDefine [[EQ:eq0019]] and [[EQ:eq0020]] for [[EQ:eq0021]] . Under the axioms below, [[EQ:eq0022]] .\n\nPARAGRAPH: Order polarity note.\n\nFor Lawvere cost we use [[EQ:eq0023]] ; numeric [[EQ:eq0024]] corresponds to enriched joins [[EQ:eq0025]] .\n\nPARAGRAPH: Library valuation and coverage-respecting.\n\nA valuation [[EQ:eq0026]] maps local primitives to hom-objects.\nA functional [[EQ:eq0027]] is coverage-respecting if [[EQ:eq0028]] whenever [[EQ:eq0029]] .\n\nSECTION: Axiom Tiers, Smallness, Declared Fubini\n\nsec:axioms\nWe recall the setting.Here ``Sup-lattice'' means a complete join-semilattice; ``Sup-enriched'' means homs carry all joins and composition preserves the declared [[EQ:eq0030]] -indexed ones.\n[Smallness/Indexing]ass:small\nA designated family [[EQ:eq0031]] of index sets (finite joins; joins over [[EQ:eq0032]] ; product-indexed joins for factorizable arrays) is closed under finite coproducts/products. If [[EQ:eq0033]] then [[EQ:eq0034]] is bottom-strict: [[EQ:eq0035]] .\n\n[Tiers (W)/(S)]ass:tiers\n(W) Each [[EQ:eq0036]] is a pointed [[EQ:eq0037]] -cpo; [[EQ:eq0038]] is monotone and [[EQ:eq0039]] -continuous in each variable. (S) Homs are Sup-lattices and [[EQ:eq0040]] preserves declared [[EQ:eq0041]] -joins on both sides.\n\n[Declared-join Fubini / Beck--Chevalley]def:declared-fubini\nFor [[EQ:eq0042]] and arrays [[EQ:eq0043]] ,\n\n[[EQ:eq0002]]\n\nSECTION: Arrays as [[EQ:eq0044]] -Profunctors (Right-Written Realization)\n\nLet [[EQ:eq0045]] , [[EQ:eq0046]] , and\n[[EQ:eq0047]] over [[EQ:eq0048]] .\nThus [[EQ:eq0049]] is the right-written presentation of [[EQ:eq0050]] -profunctor composition, aligning the calculus with array/semiring implementations. Typical [[EQ:eq0051]] are (continuous) quantales; then [[EQ:eq0052]] is Sup-enriched and declared [[EQ:eq0053]] -coends exist.\n\nSECTION: Associativity/Unit Criterion under (W)\n\nsec:assoc\n[Associativity and unit]thm:assocW\nUnder Assumptions~ass:small--ass:tiers, the following are equivalent:\n0pt\n- [[EQ:eq0054]] is associative with two-sided unit [[EQ:eq0055]] on all arrays.\n- [[EQ:eq0056]] is monotone and [[EQ:eq0057]] -continuous in each variable, satisfies Def.~def:declared-fubini (both sides), and is bottom-strict when [[EQ:eq0058]] .\n\n[Join preservation forced by associativity]prop:force-join\nIf [[EQ:eq0059]] is associative with unit [[EQ:eq0060]] globally, then for any [[EQ:eq0061]] -family [[EQ:eq0062]] and any [[EQ:eq0063]] ,\n[[EQ:eq0064]] and dually on the left; if [[EQ:eq0065]] then [[EQ:eq0066]] is bottom-strict.\n\n[Idea]\nEmbed [[EQ:eq0067]] into [[EQ:eq0068]] and use two-spike arrays plus unit/empty-index tests; failure of join preservation yields arrays breaking reassociation or unit law.\n\nSECTION: Kleene Closure (Right-Written)\n\nsec:kleene\nExponentiation is right-written: [[EQ:eq0069]] , [[EQ:eq0070]] .\nDefine\n\n[[EQ:eq0003]]\n\nBy [[EQ:eq0071]] -continuity, [[EQ:eq0072]] is the least fixed point of [[EQ:eq0073]] .\n\nSECTION: Čech-Type Maximal Lower Bound via Equipment\n\nsec:cech\nWe write [[EQ:eq0074]] for the right adjoint to restriction [[EQ:eq0075]] (a star as a lower index indicates the right adjoint).\nA coverage [[EQ:eq0076]] is admissible if [[EQ:eq0077]] .\nFor locals [[EQ:eq0078]] and depth-weights [[EQ:eq0079]] , define\n\n[[EQ:eq0004]]\n\n[Maximality]thm:cech\nUnder (S), [[EQ:eq0080]] is a lower bound for [[EQ:eq0081]] and is maximal among coverage-respecting, join-preserving functionals.\n\n[Idea]\nCompute [[EQ:eq0082]] as the left Kan extension of [[EQ:eq0083]] along coverage inclusion in the Sup-enriched base; maximality follows from universality.\n\nSECTION: Residuals and Monoidal Nuclei\n\nsec:res-nuc\n[Residuals]\nA right residual on [[EQ:eq0084]] is a family of right adjoints [[EQ:eq0085]]\nsuch that [[EQ:eq0086]] (dually [[EQ:eq0087]] ).\n\n[Adjoint thresholding]prop:adj-thresh\nIf right residuals exist, for any threshold [[EQ:eq0088]] :\n[[EQ:eq0089]] . (Left residuals are analogous.)\n\n[Monoidal nucleus]\nA nucleus is a closure [[EQ:eq0090]] (extensive, idempotent) that is submonoidal:\n[[EQ:eq0091]] .\n\n[Stability]\nIf a mask [[EQ:eq0092]] is [[EQ:eq0093]] -stable then [[EQ:eq0094]] ;\nČech lower bounds reflect through [[EQ:eq0095]] .\n\nSECTION: Mask Upper Bound: Syntax, Semantics, Completeness\n\nsec:mask\n\nPARAGRAPH: Masked-path syntax and semantics.\n\nGrammar: [[EQ:eq0096]] .\nTyping: [[EQ:eq0097]] ; if [[EQ:eq0098]] and [[EQ:eq0099]] then [[EQ:eq0100]] .\nValuation (right-written):\n\n[[EQ:eq0005]]\n\nThe masked envelope is [[EQ:eq0101]] .\n\n[Compact extraction]lem:compact\nAssume algebraic cpos whose compacts are closed under finite [[EQ:eq0102]] , and library values [[EQ:eq0103]] are compact. Then any [[EQ:eq0104]] is the supremum of an [[EQ:eq0105]] -chain of compacts realized by finite masked paths.\n\n[Local finiteness / way-below compatibility]ass:locfin\nAdmissible branching per [[EQ:eq0106]] is locally finite (equivalently: the way-below relation is compatible with [[EQ:eq0107]] and finite joins on the compact basis). Under this, finite masked paths suffice to approximate [[EQ:eq0108]] from below.\n\n[Threshold completeness]thm:threshold\nUnder Lemma~lem:compact and Assumption~ass:locfin, [[EQ:eq0109]] equals the supremum of values of admissible finite masked paths.\n\n[Tightness via nuclear split]rem:mask-tight\nIf there exists a monoidal nucleus [[EQ:eq0110]] and a unique branch [[EQ:eq0111]] with [[EQ:eq0112]] and [[EQ:eq0113]] for [[EQ:eq0114]] , then [[EQ:eq0115]] is tight.\n\nSECTION: (op)lax/Strong Transport and Weak Beck--Chevalley\n\nsec:transport\nAll equalities/inequalities below are to be read relative to declared [[EQ:eq0116]] -coends; outside this scope only (op)lax bounds hold.\n[Laxators and declared coends]ass:wbc\n[[EQ:eq0117]] is (op)lax monoidal with monotone (co)laxators at the 2-cell level, and preserves declared [[EQ:eq0118]] -coends (no new coends). The object map [[EQ:eq0119]] is closed under finite (co)products; reindexing preserves [[EQ:eq0120]] -joins.\n\n[Scope of preservation]rem:scope\nAll preservation/equality statements concern declared [[EQ:eq0121]] -joins. Changing the [[EQ:eq0122]] -norm (prod [[EQ:eq0123]] ukasiewicz) gives a minimal counterexample where [[EQ:eq0124]] on two-point supports.\n\n[Transport]thm:transport-thm\nFor base change [[EQ:eq0125]] satisfying Assumption~ass:wbc and arrays [[EQ:eq0126]] ,\n\n[[EQ:eq0006]]\n\nwith [[EQ:eq0127]] corresponding to lax/oplax/strong modes depending on (co)laxators and declared-coend preservation.\n\n[Catalogued preservation]thm:catalog\n(C1) Identity or isomorphic reparametrization within the same semiring/t-norm: strong.\n(C2) Continuous, monotone reparametrization preserving [[EQ:eq0128]] -joins (e.g.\\ log/exp on [[EQ:eq0129]] ): (op)lax according to monotonicity direction.\n(C3) [[EQ:eq0130]] -norm change (prod [[EQ:eq0131]] ukasiewicz): equality generally fails; (op)lax remains.\n(C4) Probability [[EQ:eq0132]] cost via [[EQ:eq0133]] : isometry for the log-metric; on Euclidean [[EQ:eq0134]] , the map is [[EQ:eq0135]] -Lipschitz.\n\nSECTION: Evaluator Calculus: Soundness and Failure Modes\n\nsec:evaluators\n[Evaluator calculus]def:evcalc\nAn evaluator [[EQ:eq0136]] has a composition law\n[[EQ:eq0137]] and unit [[EQ:eq0138]] such that:\n(i) [[EQ:eq0139]] preserves declared [[EQ:eq0140]] -joins; (ii) (op)lax-monoidality:\n[[EQ:eq0141]] with [[EQ:eq0142]] ; (iii) side conditions specific to the modality.\n\n[Soundness of evaluator composition]prop:evsound\nUnder Def.~def:evcalc(i)–(iii), for any array [[EQ:eq0143]] and [[EQ:eq0144]] ,\n[[EQ:eq0145]] and\n[[EQ:eq0146]] .\n\nPARAGRAPH: Concrete instances (with [[EQ:eq0147]] ).\n\nStat (anytime-valid): [[EQ:eq0148]] from predictable spending/FWER control; independence not required beyond predictability. (Conservativeness may reduce power.)\nKPP (reaction–diffusion): [[EQ:eq0149]] by bottleneck of minimal front speeds under medium lower bounds [[EQ:eq0150]] .\nQinfo (quantum): CPTP monotonicity for fidelity [[EQ:eq0151]] (non-decreasing) and trace distance [[EQ:eq0152]] (non-increasing); via Fuchs--van~de~Graaf, transport is oplax.\n\nPARAGRAPH: Failure modes (countermeasures).\n\nStat: adaptive correlation can inflate FWER [[EQ:eq0153]] predictable spending and stop-on-crossing. KPP: inhomogeneous media can violate [[EQ:eq0154]] -rule [[EQ:eq0155]] declare [[EQ:eq0156]] . Qinfo: non-CPTP numerics break monotonicity [[EQ:eq0157]] Choi PSD/trace checks as artifacts.\n\nSECTION: Interface-Autopoiesis (IAC): Observable Self-Maintenance\n\nsec:iac\nA monoidal nucleus [[EQ:eq0158]] (extensive, idempotent, submonoidal) and time-indexed bases [[EQ:eq0159]] exhibit interface autopoiesis at [[EQ:eq0160]] if\n(i) [[EQ:eq0161]] and [[EQ:eq0162]] ;\n(ii) some local procedure [[EQ:eq0163]] satisfies [[EQ:eq0164]] ;\n(iii) [[EQ:eq0165]] pseudofunctorially commutes with [[EQ:eq0166]] and [[EQ:eq0167]] .\nUnder IAC, Čech lower bounds reflect through [[EQ:eq0168]] , making self-maintenance auditable.\n\nPARAGRAPH: Audit fields (mandatory).\n\nAppend CSV columns: check\\_stat\\_fwer, check\\_kpp\\_floor, check\\_cptp, nucleus\\_ok, and IAC booleans for [[EQ:eq0169]] , [[EQ:eq0170]] .\n\nSECTION: Liberation–Viability Floors (Network-Level Audits)\n\nsec:lv\nFor each agent [[EQ:eq0171]] , collect evaluator outputs [[EQ:eq0172]] .\nDefine\n\n[[EQ:eq0007]]\n\n[Propagation of network floor]prop:propagate\nOn a connected network with edge KPP floors [[EQ:eq0173]]\nand lax nodewise composition, [[EQ:eq0174]] propagates along any path with speed at least [[EQ:eq0175]] .\n\nSECTION: Implementation and Reproducibility\n\nsec:impl\n\nPARAGRAPH: GraphBLAS semiring kernels and stopping criteria.\n\nImplement [[EQ:eq0176]] via mxm using: Boolean (Rel), min-plus (cost), log-domain max-plus (probability).\nDense: blocked Floyd--Warshall; sparse: repeated squaring with threshold trimming.\nStopping criteria by semiring:\n0pt\n- Boolean: fixed-point (no changes) [[EQ:eq0177]] exact.\n- Min-plus: monotone nonincreasing; stop when [[EQ:eq0178]] (exact) or [[EQ:eq0179]] (floor if trimming keeps [[EQ:eq0180]] ).\n- Log max-plus: represent zeros by [[EQ:eq0181]] and compute with [[EQ:eq0182]] ; use [[EQ:eq0183]] ; trimming policy must preserve audit direction (report trim\\_policy=\\ floor,ceiling\\ ).\n\nPARAGRAPH: Error statistic.\n\nReport [[EQ:eq0184]] in CSV. Define err as the termination statistic used by the stopping rule: [[EQ:eq0185]] for exact Boolean/min-plus fixed points; the reported sup-norm gap in log-domain.\n\nPARAGRAPH: Trim policy vs.\\ audit direction.\n\nlcc\n\nPolicy & Preserved audit & Typical use\\\n\ntrim=floor & lower bounds (floors) & Stat/KPP floors, [[EQ:eq0186]] (Čech) \\\ntrim=ceiling & upper bounds (ceilings) & [[EQ:eq0187]] \\\n\nPARAGRAPH: Quantum SDP/EVD.\n\nFor [[EQ:eq0188]] – [[EQ:eq0189]] qubits, generate random CPTP channels (Kraus/Choi), compute [[EQ:eq0190]] and [[EQ:eq0191]] , verify CPTP monotonicity and Fuchs--van~de~Graaf bounds; log solver tolerances/iterations and choi\\_psd,trace\\_ok flags.\n\nPARAGRAPH: CLI flags.\n\n--compose-mode \\ lax,oplax,strong\\ ,\\;\n--alpha-schedule (Stat),\\;\n--rmin --Dmin (KPP),\\;\n--choi-psd --trace-1 (Qinfo),\\;\n--nucleus-threshold tau,\\;\\\n--epsilon-log,\\;\n--epsilon-minplus.\n\nSECTION: Related Frameworks\n\nEnriched category theory and quantaloids (Kelly; Stubbe) provide coends/residuation/nuclei; framed bicategories/equipment (Shulman) support Čech-style lower bounds; semiring/Kleene algebra connect to weighted automata and shortest paths (Mohri; Kozen; Gondran--Minoux; Baccelli et al.); Markov categories structure stochastic composition (Fritz; Cho--Jacobs). Quantum monotone metrics and data processing inequalities (Petz; Wilde; Nielsen--Chuang; Fuchs--van~de~Graaf) support qinfo evaluators. Anytime-valid inference/e-processes provide statistical floors (Howard; Ramdas et al.).\n\nSECTION: Limitations and Outlook\n\nMask tightness requires a nuclear split; translator functors altering base operations degrade strong transport to (op)lax. Future: mechanized proofs (typeclasses for Sup-enrichment; declared-Fubini lemmas; transport mates), adversarial cover analysis, expanded benchmarks and ablations; network-scale audits of [[EQ:eq0192]] over time.\n\nSECTION: Canonical DOIs (Author's Preprints)\n\n0pt\n- Persistence-First Superintelligence. Zenodo. https://doi.org/10.5281/zenodo.17076410 10.5281/zenodo.17076410 .\n- UGV Without Meta. Zenodo. https://doi.org/10.5281/zenodo.17082312 10.5281/zenodo.17082312 .\n- From Persistence and UGV Axioms to Cosmic No-Meta Superintelligence. Zenodo. https://doi.org/10.5281/zenodo.17085534 10.5281/zenodo.17085534 .\n- Assumption-Minimized Sufficient Conditions for Cosmically Spreading Good Superintelligence. Zenodo. https://doi.org/10.5281/zenodo.17092562 10.5281/zenodo.17092562 .\n\n99 0.2em\n\nKelly\nG.~M. Kelly, Basic Concepts of Enriched Category Theory. Reprints in TAC 10 (2005).\n\nShulman\nM.~Shulman, Framed bicategories and monoidal fibrations, Theory and Applications of Categories 20:650--738, 2008.\n\nStubbe\nI.~Stubbe, An introduction to quantaloid-enriched categories, Fuzzy Sets and Systems 256:95--116, 2014.\n\nGalatos\nN.~Galatos, P.~Jipsen, T.~Kowalski, H.~Ono, Residuated Lattices, Elsevier, 2007.\n\nLawvere\nF.~W. Lawvere, Metric spaces, generalized logic, and closed categories, 1973 manuscript; Reprints in TAC 1 (2002).\n\nFritz\nT.~Fritz, A synthetic approach to Markov kernels, conditional independence, and sufficient statistics, Advances in Mathematics 370:107239, 2020.\n\nChoJacobs\nK.~Cho, B.~Jacobs, Disintegration and Bayesian inversion via string diagrams, Mathematical Structures in Computer Science 29(7):997--1039, 2019.\n\nHoward\nS.~R. Howard, A.~Ramdas, J.~McAuliffe, J.~Sekhon, Time-uniform Chernoff bounds via nonnegative supermartingales, Probability Surveys 18:1–29, 2021.\n\nRamdasSAVI\nA.~Ramdas, P.~Grünwald, V.~Vovk, G.~Shafer, Game-Theoretic Statistics and Safe Anytime-Valid Inference, Statistical Science 38(4):559–581, 2023.\n\nFisherKPP\nR.~A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics 7:355–369, 1937.\n\nA.~N. Kolmogorov, I.~G. Petrovskii, N.~S. Piskunov, A study of the diffusion equation with increase in the amount of substance, Bull. Moscow Univ. Math. Mech. 1:1–25, 1937.\n\nAronsonWeinberger\nD.~Aronson, H.~Weinberger, Multidimensional nonlinear diffusion in population genetics, Advances in Mathematics 30(1):33–76, 1978.\n\nPetz\nD.~Petz, Monotone metrics on matrix spaces, Linear Algebra and its Applications 244:81–96, 1996.\n\nFvG\nC.~A. Fuchs, J.~van~de~Graaf, Cryptographic distinguishability measures for quantum states, IEEE Trans. Info. Theory 45(4):1216–1227, 1999.\n\nNielsenChuang\nM.~A. Nielsen, I.~L. Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press, 2010.\n\nWilde\nM.~M. Wilde, Quantum Information Theory, Cambridge Univ. Press, 2017.\n\nMohri\nM.~Mohri, Semiring frameworks and algorithms for shortest-distance problems, J. Automata, Languages and Combinatorics 7(3):321–350, 2002.\n\nDavisGraphBLAS\nT.~A. Davis, Algorithmic Advances in SuiteSparse:GraphBLAS, arXiv:1908.01407, 2019.\n\nDrosteHandbook\nM.~Droste, W.~Kuich, H.~Vogler (eds.), Handbook of Weighted Automata, Springer, 2009.\n\nGondranMinoux\nM.~Gondran, M.~Minoux, Graphs, Dioids and Semirings, Springer, 2008.\n\nBaccelli\nF.~Baccelli, G.~Cohen, G.~J. Olsder, J.-P. Quadrat, Synchronization and Linearity, Wiley, 1992.\n\nKozenKAT\nD.~Kozen, Kleene algebra with tests, ACM TOPLAS 19(3):427–443, 1997.\n[[EQ:eq0008]]\n\n[[EQ:eq0009]]\n\n[[EQ:eq0010]]\n\n[[EQ:eq0011]]\n\n[[EQ:eq0012]]\n\n[[EQ:eq0013]]\n\n[[EQ:eq0014]]\n\n[[EQ:eq0015]]\n\n[[EQ:eq0016]]\n\n[[EQ:eq0017]]\n\n[[EQ:eq0018]]\n\n[[EQ:eq0019]]\n\n[[EQ:eq0020]]\n\n[[EQ:eq0021]]\n\n[[EQ:eq0022]]\n\n[[EQ:eq0023]]\n\n[[EQ:eq0024]]\n\n[[EQ:eq0025]]\n\n[[EQ:eq0026]]\n\n[[EQ:eq0027]]\n\n[[EQ:eq0028]]\n\n[[EQ:eq0029]]\n\n[[EQ:eq0030]]\n\n[[EQ:eq0031]]\n\n[[EQ:eq0032]]\n\n[[EQ:eq0033]]\n\n[[EQ:eq0034]]\n\n[[EQ:eq0035]]\n\n[[EQ:eq0036]]\n\n[[EQ:eq0037]]\n\n[[EQ:eq0038]]\n\n[[EQ:eq0039]]\n\n[[EQ:eq0040]]\n\n[[EQ:eq0041]]\n\n[[EQ:eq0042]]\n\n[[EQ:eq0043]]\n\n[[EQ:eq0044]]\n\n[[EQ:eq0045]]\n\n[[EQ:eq0046]]\n\n[[EQ:eq0047]]\n\n[[EQ:eq0048]]\n\n[[EQ:eq0049]]\n\n[[EQ:eq0050]]\n\n[[EQ:eq0051]]\n\n[[EQ:eq0052]]\n\n[[EQ:eq0053]]\n\n[[EQ:eq0054]]\n\n[[EQ:eq0055]]\n\n[[EQ:eq0056]]\n\n[[EQ:eq0057]]\n\n[[EQ:eq0058]]\n\n[[EQ:eq0059]]\n\n[[EQ:eq0060]]\n\n[[EQ:eq0061]]\n\n[[EQ:eq0062]]\n\n[[EQ:eq0063]]\n\n[[EQ:eq0064]]\n\n[[EQ:eq0065]]\n\n[[EQ:eq0066]]\n\n[[EQ:eq0067]]\n\n[[EQ:eq0068]]\n\n[[EQ:eq0069]]\n\n[[EQ:eq0070]]\n\n[[EQ:eq0071]]\n\n[[EQ:eq0072]]\n\n[[EQ:eq0073]]\n\n[[EQ:eq0074]]\n\n[[EQ:eq0075]]\n\n[[EQ:eq0076]]\n\n[[EQ:eq0077]]\n\n[[EQ:eq0078]]\n\n[[EQ:eq0079]]\n\n[[EQ:eq0080]]\n\n[[EQ:eq0081]]\n\n[[EQ:eq0082]]\n\n[[EQ:eq0083]]\n\n[[EQ:eq0084]]\n\n[[EQ:eq0085]]\n\n[[EQ:eq0086]]\n\n[[EQ:eq0087]]\n\n[[EQ:eq0088]]\n\n[[EQ:eq0089]]\n\n[[EQ:eq0090]]\n\n[[EQ:eq0091]]\n\n[[EQ:eq0092]]\n\n[[EQ:eq0093]]\n\n[[EQ:eq0094]]\n\n[[EQ:eq0095]]\n\n[[EQ:eq0096]]\n\n[[EQ:eq0097]]\n\n[[EQ:eq0098]]\n\n[[EQ:eq0099]]\n\n[[EQ:eq0100]]\n\n[[EQ:eq0101]]\n\n[[EQ:eq0102]]\n\n[[EQ:eq0103]]\n\n[[EQ:eq0104]]\n\n[[EQ:eq0105]]\n\n[[EQ:eq0106]]\n\n[[EQ:eq0107]]\n\n[[EQ:eq0108]]\n\n[[EQ:eq0109]]\n\n[[EQ:eq0110]]\n\n[[EQ:eq0111]]\n\n[[EQ:eq0112]]\n\n[[EQ:eq0113]]\n\n[[EQ:eq0114]]\n\n[[EQ:eq0115]]\n\n[[EQ:eq0116]]\n\n[[EQ:eq0117]]\n\n[[EQ:eq0118]]\n\n[[EQ:eq0119]]\n\n[[EQ:eq0120]]\n\n[[EQ:eq0121]]\n\n[[EQ:eq0122]]\n\n[[EQ:eq0123]]\n\n[[EQ:eq0124]]\n\n[[EQ:eq0125]]\n\n[[EQ:eq0126]]\n\n[[EQ:eq0127]]\n\n[[EQ:eq0128]]\n\n[[EQ:eq0129]]\n\n[[EQ:eq0130]]\n\n[[EQ:eq0131]]\n\n[[EQ:eq0132]]\n\n[[EQ:eq0133]]\n\n[[EQ:eq0134]]\n\n[[EQ:eq0135]]\n\n[[EQ:eq0136]]\n\n[[EQ:eq0137]]\n\n[[EQ:eq0138]]\n\n[[EQ:eq0139]]\n\n[[EQ:eq0140]]\n\n[[EQ:eq0141]]\n\n[[EQ:eq0142]]\n\n[[EQ:eq0143]]\n\n[[EQ:eq0144]]\n\n[[EQ:eq0145]]\n\n[[EQ:eq0146]]\n\n[[EQ:eq0147]]\n\n[[EQ:eq0148]]\n\n[[EQ:eq0149]]\n\n[[EQ:eq0150]]\n\n[[EQ:eq0151]]\n\n[[EQ:eq0152]]\n\n[[EQ:eq0153]]\n\n[[EQ:eq0154]]\n\n[[EQ:eq0155]]\n\n[[EQ:eq0156]]\n\n[[EQ:eq0157]]\n\n[[EQ:eq0158]]\n\n[[EQ:eq0159]]\n\n[[EQ:eq0160]]\n\n[[EQ:eq0161]]\n\n[[EQ:eq0162]]\n\n[[EQ:eq0163]]\n\n[[EQ:eq0164]]\n\n[[EQ:eq0165]]\n\n[[EQ:eq0166]]\n\n[[EQ:eq0167]]\n\n[[EQ:eq0168]]\n\n[[EQ:eq0169]]\n\n[[EQ:eq0170]]\n\n[[EQ:eq0171]]\n\n[[EQ:eq0172]]\n", "sections": [{"level": 1, "title": "Orientation and Notation", "anchor": "orientation-and-notation", "char_span": [1819, 2730]}, {"level": 1, "title": "Axiom Tiers, Smallness, Declared Fubini", "anchor": "axiom-tiers-smallness-declared-fubini", "char_span": [2730, 2769]}, {"level": 1, "title": "Arrays as Q-Profunctors (Right-Written Realization)", "anchor": "arrays-as-q-profunctors-right-written-realization", "char_span": [2769, 4071]}, {"level": 1, "title": "Associativity/Unit Criterion under (W)", "anchor": "associativity-unit-criterion-under-w", "char_span": [4071, 4958]}, {"level": 1, "title": "Kleene Closure (Right-Written)", "anchor": "kleene-closure-right-written", "char_span": [4958, 5187]}, {"level": 1, "title": "Čech-Type Maximal Lower Bound via Equipment", "anchor": "cech-type-maximal-lower-bound-via-equipment", "char_span": [5187, 5834]}, {"level": 1, "title": "Residuals and Monoidal Nuclei", "anchor": "residuals-and-monoidal-nuclei", "char_span": [5834, 6424]}, {"level": 1, "title": "Mask Upper Bound: Syntax, Semantics, Completeness", "anchor": "mask-upper-bound-syntax-semantics-completeness", "char_span": [6424, 6473]}, {"level": 1, "title": "(op)lax/Strong Transport and Weak Beck–Chevalley", "anchor": "op-lax-strong-transport-and-weak-beck-chevalley", "char_span": [6473, 9227]}, {"level": 1, "title": "Evaluator Calculus: Soundness and Failure Modes", "anchor": "evaluator-calculus-soundness-and-failure-modes", "char_span": [9227, 10609]}, {"level": 1, "title": "Interface-Autopoiesis (IAC): Observable Self-Maintenance", "anchor": "interface-autopoiesis-iac-observable-self-maintenance", "char_span": [10609, 11305]}, {"level": 1, "title": "Liberation–Viability Floors (Network-Level Audits)", "anchor": "liberation-viability-floors-network-level-audits", "char_span": [11305, 11679]}, {"level": 1, "title": "Implementation and Reproducibility", "anchor": "implementation-and-reproducibility", "char_span": [11679, 13331]}, {"level": 1, "title": "Related Frameworks", "anchor": "related-frameworks", "char_span": [13331, 13945]}, {"level": 1, "title": "Limitations and Outlook", "anchor": "limitations-and-outlook", "char_span": [13945, 14316]}, {"level": 1, "title": "Canonical DOIs (Author's Preprints)", "anchor": "canonical-dois-author-s-preprints", "char_span": [14316, 20009]}]}, "equations": [{"id": "eq0001", "inline": false, "tex": "\\[\n (X\\Star Y)(U,T)\\ :=\\ \\join_{V\\in \\Ob_0(B)} X(V,T)\\odotop Y(U,V),\n\\]", "tex_body": "(X\\Star Y)(U,T)\\ :=\\ \\join_{V\\in \\Ob_0(B)} X(V,T)\\odotop Y(U,V),", "tex_normalized": "(X\\Star Y)(U,T)\\ :=\\ \\join_{V\\in \\Ob_0(B)} X(V,T)\\odotop Y(U,V),", "mathml": "\\[(X\\StarY)(U,T) := \\joinV∈\\Ob0(B)X(V,T)\\odotopY(U,V),\\]", "char_span": [2042, 2055], "context": {"section": "orientation-and-notation"}, "placeholder": "EQPH_eq0001_PH"}, {"id": "eq0002", "inline": false, "tex": "\\[\n\\join_{v\\in J}\\!\\bigl(X(v,T)\\odotop Y(U,v)\\bigr)\n=\\Bigl(\\join_{v\\in J}\\!X(v,T)\\Bigr)\\odotop Y(U,{-})\n= X({-},T)\\odotop\\Bigl(\\join_{v\\in J}\\!Y(U,v)\\Bigr).\n\\]", "tex_body": "\\join_{v\\in J}\\!\\bigl(X(v,T)\\odotop Y(U,v)\\bigr)\n=\\Bigl(\\join_{v\\in J}\\!X(v,T)\\Bigr)\\odotop Y(U,{-})\n= X({-},T)\\odotop\\Bigl(\\join_{v\\in J}\\!Y(U,v)\\Bigr).", "tex_normalized": "\\join_{v\\in J} \\bigl(X(v,T)\\odotop Y(U,v)\\bigr) =\\Bigl(\\join_{v\\in J} X(v,T)\\Bigr)\\odotop Y(U,{-}) = X({-},T)\\odotop\\Bigl(\\join_{v\\in J} Y(U,v)\\Bigr).", "mathml": "\\[\\joinv∈J(X(v,T)\\odotopY(U,v))=(\\joinv∈JX(v,T))\\odotopY(U,−)=X(−,T)\\odotop(\\joinv∈JY(U,v)).\\]", "char_span": [3651, 3664], "context": {"section": "arrays-as-q-profunctors-right-written-realization"}, "placeholder": "EQPH_eq0002_PH"}, {"id": "eq0003", "inline": false, "tex": "\\[\n\\Path \\ :=\\ \\join_{n\\ge 1} A^{\\Star n},\\qquad\n\\Path^\\ast \\ :=\\ I \\join \\Path \\ = \\ \\join_{n\\ge 0} A^{\\Star n}.\n\\]", "tex_body": "\\Path \\ :=\\ \\join_{n\\ge 1} A^{\\Star n},\\qquad\n\\Path^\\ast \\ :=\\ I \\join \\Path \\ = \\ \\join_{n\\ge 0} A^{\\Star n}.", "tex_normalized": "\\Path \\ :=\\ \\join_{n\\ge 1} A^{\\Star n},\\qquad \\Path^\\ast \\ :=\\ I \\join \\Path \\ = \\ \\join_{n\\ge 0} A^{\\Star n}.", "mathml": "\\[\\Path := \\joinn≥1A\\Starn,\\Path* := I\\join\\Path = \\joinn≥0A\\Starn.\\]", "char_span": [5137, 5150], "context": {"section": "kleene-closure-right-written"}, "placeholder": "EQPH_eq0003_PH"}, {"id": "eq0004", "inline": false, "tex": "\\[\n\\Phi(T)\\ :=\\ \\join_i \\bigl(v(T_i)\\odotop c_{d,i}\\bigr)\\odotop w_i,\\qquad\nw_i:=\\iota_{i\\ast}(1_{U_i})\\in B(U,U_i).\n\\]", "tex_body": "\\Phi(T)\\ :=\\ \\join_i \\bigl(v(T_i)\\odotop c_{d,i}\\bigr)\\odotop w_i,\\qquad\nw_i:=\\iota_{i\\ast}(1_{U_i})\\in B(U,U_i).", "tex_normalized": "\\Phi(T)\\ :=\\ \\join_i \\bigl(v(T_i)\\odotop c_{d,i}\\bigr)\\odotop w_i,\\qquad w_i:=\\iota_{i\\ast}(1_{U_i})\\in B(U,U_i).", "mathml": "\\[Φ(T) := \\joini(v(Ti)\\odotopcd,i)\\odotopwi,wi:=ιi*(1Ui)∈B(U,Ui).\\]", "char_span": [5567, 5580], "context": {"section": "cech-type-maximal-lower-bound-via-equipment"}, "placeholder": "EQPH_eq0004_PH"}, {"id": "eq0005", "inline": false, "tex": "\\[\n\\llbracket \\mathrm{hop}(U\\!\\to\\!V)\\rrbracket = A(U,V)\\odotop M(U,V),\\quad\n\\llbracket \\pi_1\\cdot\\pi_2\\rrbracket = \\llbracket \\pi_2\\rrbracket\\odotop\\llbracket \\pi_1\\rrbracket.\n\\]", "tex_body": "\\llbracket \\mathrm{hop}(U\\!\\to\\!V)\\rrbracket = A(U,V)\\odotop M(U,V),\\quad\n\\llbracket \\pi_1\\cdot\\pi_2\\rrbracket = \\llbracket \\pi_2\\rrbracket\\odotop\\llbracket \\pi_1\\rrbracket.", "tex_normalized": "\\llbracket \\mathrm{hop}(U \\to V)\\rrbracket = A(U,V)\\odotop M(U,V),\\quad \\llbracket \\pi_1\\cdot\\pi_2\\rrbracket = \\llbracket \\pi_2\\rrbracket\\odotop\\llbracket \\pi_1\\rrbracket.", "mathml": "\\[⟦hop(U→V)⟧=A(U,V)\\odotopM(U,V),⟦π1·π2⟧=⟦π2⟧\\odotop⟦π1⟧.\\]", "char_span": [6757, 6770], "context": {"section": "op-lax-strong-transport-and-weak-beck-chevalley"}, "placeholder": "EQPH_eq0005_PH"}, {"id": "eq0006", "inline": false, "tex": "\\[\nF(A^{\\Star n})\\ \\square\\ (FA)^{\\Star n},\\qquad F(\\Path)\\ \\square\\ \\Path',\n\\]", "tex_body": "F(A^{\\Star n})\\ \\square\\ (FA)^{\\Star n},\\qquad F(\\Path)\\ \\square\\ \\Path',", "tex_normalized": "F(A^{\\Star n})\\ \\square\\ (FA)^{\\Star n},\\qquad F(\\Path)\\ \\square\\ \\Path',", "mathml": "\\[F(A\\Starn) ◻ (FA)\\Starn,F(\\Path) ◻ \\Path′,\\]", "char_span": [8663, 8676], "context": {"section": "op-lax-strong-transport-and-weak-beck-chevalley"}, "placeholder": "EQPH_eq0006_PH"}, {"id": "eq0007", "inline": false, "tex": "\\[\n\\begin{aligned}\nF_a &:= \\min\\{\\,f_a^{\\mathrm{stat}},\\ \\tilde f_a^{\\mathrm{kpp}}\\,\\},\\\\\nH_a &:= \\min\\{\\,F_a,\\ \\mathrm{MND}_a\\,\\},\\\\\nH_{\\mathrm{net}} &:= \\min_{a}\\, H_a \\, .\n\\end{aligned}\n\\]", "tex_body": "\\begin{aligned}\nF_a &:= \\min\\{\\,f_a^{\\mathrm{stat}},\\ \\tilde f_a^{\\mathrm{kpp}}\\,\\},\\\\\nH_a &:= \\min\\{\\,F_a,\\ \\mathrm{MND}_a\\,\\},\\\\\nH_{\\mathrm{net}} &:= \\min_{a}\\, H_a \\, .\n\\end{aligned}", "tex_normalized": "\\begin{aligned} F_a &:= \\min\\{ f_a^{\\mathrm{stat}},\\ \\tilde f_a^{\\mathrm{kpp}} \\},\\\\ H_a &:= \\min\\{ F_a,\\ \\mathrm{MND}_a \\},\\\\ H_{\\mathrm{net}} &:= \\min_{a} H_a . \\end{aligned}", "mathml": "\\[Fa&:=min{fastat, f~akpp},Ha&:=min{Fa, MNDa},Hnet&:=minaHa.\\]", "char_span": [11610, 11623], "context": {"section": "liberation-viability-floors-network-level-audits"}, "placeholder": "EQPH_eq0007_PH"}, {"id": "eq0008", "inline": true, "tex": "$\\omega$", "tex_body": "\\omega", "tex_normalized": "\\omega", "mathml": "$ω$", "char_span": [17535, 17548], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0008_PH"}, {"id": "eq0009", "inline": true, "tex": "$\\omega$", "tex_body": "\\omega", "tex_normalized": "\\omega", "mathml": "$ω$", "char_span": [17550, 17563], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0009_PH"}, {"id": "eq0010", "inline": true, "tex": "$U,V,T$", "tex_body": "U,V,T", "tex_normalized": "U,V,T", "mathml": "$U,V,T$", "char_span": [17565, 17578], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0010_PH"}, {"id": "eq0011", "inline": true, "tex": "$B$", "tex_body": "B", "tex_normalized": "B", "mathml": "$B$", "char_span": [17580, 17593], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0011_PH"}, {"id": "eq0012", "inline": true, "tex": "$B(U,V)$", "tex_body": "B(U,V)", "tex_normalized": "B(U,V)", "mathml": "$B(U,V)$", "char_span": [17595, 17608], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0012_PH"}, {"id": "eq0013", "inline": true, "tex": "$\\odotop$", "tex_body": "\\odotop", "tex_normalized": "\\odotop", "mathml": "$\\odotop$", "char_span": [17610, 17623], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0013_PH"}, {"id": "eq0014", "inline": true, "tex": "$1_U$", "tex_body": "1_U", "tex_normalized": "1_U", "mathml": "$1U$", "char_span": [17625, 17638], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0014_PH"}, {"id": "eq0015", "inline": true, "tex": "$X(U,V)\\in B(U,V)$", "tex_body": "X(U,V)\\in B(U,V)", "tex_normalized": "X(U,V)\\in B(U,V)", "mathml": "$X(U,V)∈B(U,V)$", "char_span": [17640, 17653], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0015_PH"}, {"id": "eq0016", "inline": true, "tex": "$\\Ob_0(B)\\subseteq\\Ob(B)$", "tex_body": "\\Ob_0(B)\\subseteq\\Ob(B)", "tex_normalized": "\\Ob_0(B)\\subseteq\\Ob(B)", "mathml": "$\\Ob0(B)⊆\\Ob(B)$", "char_span": [17655, 17668], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0016_PH"}, {"id": "eq0017", "inline": true, "tex": "$\\leqE$", "tex_body": "\\leqE", "tex_normalized": "\\leqE", "mathml": "$\\leqE$", "char_span": [17670, 17683], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0017_PH"}, {"id": "eq0018", "inline": true, "tex": "$\\le$", "tex_body": "\\le", "tex_normalized": "\\le", "mathml": "$≤$", "char_span": [17685, 17698], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0018_PH"}, {"id": "eq0019", "inline": true, "tex": "$I(U,U)=1_U$", "tex_body": "I(U,U)=1_U", "tex_normalized": "I(U,U)=1_U", "mathml": "$I(U,U)=1U$", "char_span": [17700, 17713], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0019_PH"}, {"id": "eq0020", "inline": true, "tex": "$I(U,V)=\\bot$", "tex_body": "I(U,V)=\\bot", "tex_normalized": "I(U,V)=\\bot", "mathml": "$I(U,V)=⊥$", "char_span": [17715, 17728], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0020_PH"}, {"id": "eq0021", "inline": true, "tex": "$U\\ne V$", "tex_body": "U\\ne V", "tex_normalized": "U\\ne V", "mathml": "$U≠V$", "char_span": [17730, 17743], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0021_PH"}, {"id": "eq0022", "inline": true, "tex": "$X\\Star I=X=I\\Star X$", "tex_body": "X\\Star I=X=I\\Star X", "tex_normalized": "X\\Star I=X=I\\Star X", "mathml": "$X\\StarI=X=I\\StarX$", "char_span": [17745, 17758], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0022_PH"}, {"id": "eq0023", "inline": true, "tex": "$([0,\\infty],\\ge)$", "tex_body": "([0,\\infty],\\ge)", "tex_normalized": "([0,\\infty],\\ge)", "mathml": "$([0,∞],≥)$", "char_span": [17760, 17773], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0023_PH"}, {"id": "eq0024", "inline": true, "tex": "$\\inf$", "tex_body": "\\inf", "tex_normalized": "\\inf", "mathml": "$inf$", "char_span": [17775, 17788], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0024_PH"}, {"id": "eq0025", "inline": true, "tex": "$\\join$", "tex_body": "\\join", "tex_normalized": "\\join", "mathml": "$\\join$", "char_span": [17790, 17803], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0025_PH"}, {"id": "eq0026", "inline": true, "tex": "$v:\\mathrm{Locals}\\to B(-,-)$", "tex_body": "v:\\mathrm{Locals}\\to B(-,-)", "tex_normalized": "v:\\mathrm{Locals}\\to B(-,-)", "mathml": "$v:Locals→B(−,−)$", "char_span": [17805, 17818], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0026_PH"}, {"id": "eq0027", "inline": true, "tex": "$\\Psi$", "tex_body": "\\Psi", "tex_normalized": "\\Psi", "mathml": "$Ψ$", "char_span": [17820, 17833], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0027_PH"}, {"id": "eq0028", "inline": true, "tex": "$\\Psi(\\{T_i\\to T\\})\\leqE \\join_i \\Psi(T_i)$", "tex_body": "\\Psi(\\{T_i\\to T\\})\\leqE \\join_i \\Psi(T_i)", "tex_normalized": "\\Psi(\\{T_i\\to T\\})\\leqE \\join_i \\Psi(T_i)", "mathml": "$Ψ({Ti→T})\\leqE\\joiniΨ(Ti)$", "char_span": [17835, 17848], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0028_PH"}, {"id": "eq0029", "inline": true, "tex": "$\\join_i \\iota_i\\iota_i^\\ast \\geqE 1_U$", "tex_body": "\\join_i \\iota_i\\iota_i^\\ast \\geqE 1_U", "tex_normalized": "\\join_i \\iota_i\\iota_i^\\ast \\geqE 1_U", "mathml": "$\\joiniιiιi*\\geqE1U$", "char_span": [17850, 17863], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0029_PH"}, {"id": "eq0030", "inline": true, "tex": "$\\mathsf{J}$", "tex_body": "\\mathsf{J}", "tex_normalized": "\\mathsf{J}", "mathml": "$𝖩$", "char_span": [17865, 17878], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0030_PH"}, {"id": "eq0031", "inline": true, "tex": "$\\mathsf{J}$", "tex_body": "\\mathsf{J}", "tex_normalized": "\\mathsf{J}", "mathml": "$𝖩$", "char_span": [17880, 17893], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0031_PH"}, {"id": "eq0032", "inline": true, "tex": "$\\Ob_0(B)$", "tex_body": "\\Ob_0(B)", "tex_normalized": "\\Ob_0(B)", "mathml": "$\\Ob0(B)$", "char_span": [17895, 17908], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0032_PH"}, {"id": "eq0033", "inline": true, "tex": "$\\emptyset\\in\\mathsf{J}$", "tex_body": "\\emptyset\\in\\mathsf{J}", "tex_normalized": "\\emptyset\\in\\mathsf{J}", "mathml": "$∅∈𝖩$", "char_span": [17910, 17923], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0033_PH"}, {"id": "eq0034", "inline": true, "tex": "$\\odotop$", "tex_body": "\\odotop", "tex_normalized": "\\odotop", "mathml": "$\\odotop$", "char_span": [17925, 17938], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0034_PH"}, {"id": "eq0035", "inline": true, "tex": "$x\\odotop\\bot=\\bot=\\bot\\odotop y$", "tex_body": "x\\odotop\\bot=\\bot=\\bot\\odotop y", "tex_normalized": "x\\odotop\\bot=\\bot=\\bot\\odotop y", "mathml": "$x\\odotop⊥=⊥=⊥\\odotopy$", "char_span": [17940, 17953], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0035_PH"}, {"id": "eq0036", "inline": true, "tex": "$B(U,V)$", "tex_body": "B(U,V)", "tex_normalized": "B(U,V)", "mathml": "$B(U,V)$", "char_span": [17955, 17968], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0036_PH"}, {"id": "eq0037", "inline": true, "tex": "$\\omega$", "tex_body": "\\omega", "tex_normalized": "\\omega", "mathml": "$ω$", "char_span": [17970, 17983], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0037_PH"}, {"id": "eq0038", "inline": true, "tex": "$\\odotop$", "tex_body": "\\odotop", "tex_normalized": "\\odotop", "mathml": "$\\odotop$", "char_span": [17985, 17998], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0038_PH"}, {"id": "eq0039", "inline": true, "tex": "$\\omega$", "tex_body": "\\omega", "tex_normalized": "\\omega", "mathml": "$ω$", "char_span": [18000, 18013], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0039_PH"}, {"id": "eq0040", "inline": true, "tex": "$\\odotop$", "tex_body": "\\odotop", "tex_normalized": "\\odotop", "mathml": "$\\odotop$", "char_span": [18015, 18028], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0040_PH"}, {"id": "eq0041", "inline": true, "tex": "$\\mathsf{J}$", "tex_body": "\\mathsf{J}", "tex_normalized": "\\mathsf{J}", "mathml": "$𝖩$", "char_span": [18030, 18043], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0041_PH"}, {"id": "eq0042", "inline": true, "tex": "$J\\in\\mathsf{J}$", "tex_body": "J\\in\\mathsf{J}", "tex_normalized": "J\\in\\mathsf{J}", "mathml": "$J∈𝖩$", "char_span": [18045, 18058], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0042_PH"}, {"id": "eq0043", "inline": true, "tex": "$X,Y$", "tex_body": "X,Y", "tex_normalized": "X,Y", "mathml": "$X,Y$", "char_span": [18060, 18073], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0043_PH"}, {"id": "eq0044", "inline": true, "tex": "$Q$", "tex_body": "Q", "tex_normalized": "Q", "mathml": "$Q$", "char_span": [18075, 18088], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0044_PH"}, {"id": "eq0045", "inline": true, "tex": "$B=\\Mat(Q)$", "tex_body": "B=\\Mat(Q)", "tex_normalized": "B=\\Mat(Q)", "mathml": "$B=\\Mat(Q)$", "char_span": [18090, 18103], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0045_PH"}, {"id": "eq0046", "inline": true, "tex": "$B(U,V)=Q^{V\\times U}$", "tex_body": "B(U,V)=Q^{V\\times U}", "tex_normalized": "B(U,V)=Q^{V\\times U}", "mathml": "$B(U,V)=QV×U$", "char_span": [18105, 18118], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0046_PH"}, {"id": "eq0047", "inline": true, "tex": "$(X\\Star Y)(U,T)=\\int^{V} X(V,T)\\odotop Y(U,V)$", "tex_body": "(X\\Star Y)(U,T)=\\int^{V} X(V,T)\\odotop Y(U,V)", "tex_normalized": "(X\\Star Y)(U,T)=\\int^{V} X(V,T)\\odotop Y(U,V)", "mathml": "$(X\\StarY)(U,T)=∫VX(V,T)\\odotopY(U,V)$", "char_span": [18120, 18133], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0047_PH"}, {"id": "eq0048", "inline": true, "tex": "$V\\in\\Ob_0(B)$", "tex_body": "V\\in\\Ob_0(B)", "tex_normalized": "V\\in\\Ob_0(B)", "mathml": "$V∈\\Ob0(B)$", "char_span": [18135, 18148], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0048_PH"}, {"id": "eq0049", "inline": true, "tex": "$\\Star$", "tex_body": "\\Star", "tex_normalized": "\\Star", "mathml": "$\\Star$", "char_span": [18150, 18163], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0049_PH"}, {"id": "eq0050", "inline": true, "tex": "$Q$", "tex_body": "Q", "tex_normalized": "Q", "mathml": "$Q$", "char_span": [18165, 18178], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0050_PH"}, {"id": "eq0051", "inline": true, "tex": "$Q$", "tex_body": "Q", "tex_normalized": "Q", "mathml": "$Q$", "char_span": [18180, 18193], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0051_PH"}, {"id": "eq0052", "inline": true, "tex": "$\\Mat(Q)$", "tex_body": "\\Mat(Q)", "tex_normalized": "\\Mat(Q)", "mathml": "$\\Mat(Q)$", "char_span": [18195, 18208], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0052_PH"}, {"id": "eq0053", "inline": true, "tex": "$\\mathsf{J}$", "tex_body": "\\mathsf{J}", "tex_normalized": "\\mathsf{J}", "mathml": "$𝖩$", "char_span": [18210, 18223], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0053_PH"}, {"id": "eq0054", "inline": true, "tex": "$\\Star$", "tex_body": "\\Star", "tex_normalized": "\\Star", "mathml": "$\\Star$", "char_span": [18225, 18238], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0054_PH"}, {"id": "eq0055", "inline": true, "tex": "$I$", "tex_body": "I", "tex_normalized": "I", "mathml": "$I$", "char_span": [18240, 18253], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0055_PH"}, {"id": "eq0056", "inline": true, "tex": "$\\odotop$", "tex_body": "\\odotop", "tex_normalized": "\\odotop", "mathml": "$\\odotop$", "char_span": [18255, 18268], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0056_PH"}, {"id": "eq0057", "inline": true, "tex": "$\\omega$", "tex_body": "\\omega", "tex_normalized": "\\omega", "mathml": "$ω$", "char_span": [18270, 18283], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0057_PH"}, {"id": "eq0058", "inline": true, "tex": "$\\emptyset\\in\\mathsf{J}$", "tex_body": "\\emptyset\\in\\mathsf{J}", "tex_normalized": "\\emptyset\\in\\mathsf{J}", "mathml": "$∅∈𝖩$", "char_span": [18285, 18298], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0058_PH"}, {"id": "eq0059", "inline": true, "tex": "$\\Star$", "tex_body": "\\Star", "tex_normalized": "\\Star", "mathml": "$\\Star$", "char_span": [18300, 18313], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0059_PH"}, {"id": "eq0060", "inline": true, "tex": "$I$", "tex_body": "I", "tex_normalized": "I", "mathml": "$I$", "char_span": [18315, 18328], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0060_PH"}, {"id": "eq0061", "inline": true, "tex": "$\\mathsf{J}$", "tex_body": "\\mathsf{J}", "tex_normalized": "\\mathsf{J}", "mathml": "$𝖩$", "char_span": [18330, 18343], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0061_PH"}, {"id": "eq0062", "inline": true, "tex": "$(x_v)_v$", "tex_body": "(x_v)_v", "tex_normalized": "(x_v)_v", "mathml": "$(xv)v$", "char_span": [18345, 18358], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0062_PH"}, {"id": "eq0063", "inline": true, "tex": "$y$", "tex_body": "y", "tex_normalized": "y", "mathml": "$y$", "char_span": [18360, 18373], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0063_PH"}, {"id": "eq0064", "inline": true, "tex": "$\\bigl(\\join_v x_v\\bigr)\\odotop y=\\join_v(x_v\\odotop y)$", "tex_body": "\\bigl(\\join_v x_v\\bigr)\\odotop y=\\join_v(x_v\\odotop y)", "tex_normalized": "\\bigl(\\join_v x_v\\bigr)\\odotop y=\\join_v(x_v\\odotop y)", "mathml": "$(\\joinvxv)\\odotopy=\\joinv(xv\\odotopy)$", "char_span": [18375, 18388], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0064_PH"}, {"id": "eq0065", "inline": true, "tex": "$\\emptyset\\in\\mathsf{J}$", "tex_body": "\\emptyset\\in\\mathsf{J}", "tex_normalized": "\\emptyset\\in\\mathsf{J}", "mathml": "$∅∈𝖩$", "char_span": [18390, 18403], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0065_PH"}, {"id": "eq0066", "inline": true, "tex": "$\\odotop$", "tex_body": "\\odotop", "tex_normalized": "\\odotop", "mathml": "$\\odotop$", "char_span": [18405, 18418], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0066_PH"}, {"id": "eq0067", "inline": true, "tex": "$\\mathsf{J}$", "tex_body": "\\mathsf{J}", "tex_normalized": "\\mathsf{J}", "mathml": "$𝖩$", "char_span": [18420, 18433], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0067_PH"}, {"id": "eq0068", "inline": true, "tex": "$\\Ob_0(B)$", "tex_body": "\\Ob_0(B)", "tex_normalized": "\\Ob_0(B)", "mathml": "$\\Ob0(B)$", "char_span": [18435, 18448], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0068_PH"}, {"id": "eq0069", "inline": true, "tex": "$A^{\\Star 0}:=I$", "tex_body": "A^{\\Star 0}:=I", "tex_normalized": "A^{\\Star 0}:=I", "mathml": "$A\\Star0:=I$", "char_span": [18450, 18463], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0069_PH"}, {"id": "eq0070", "inline": true, "tex": "$A^{\\Star (n+1)}:=A^{\\Star n}\\Star A$", "tex_body": "A^{\\Star (n+1)}:=A^{\\Star n}\\Star A", "tex_normalized": "A^{\\Star (n+1)}:=A^{\\Star n}\\Star A", "mathml": "$A\\Star(n+1):=A\\Starn\\StarA$", "char_span": [18465, 18478], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0070_PH"}, {"id": "eq0071", "inline": true, "tex": "$\\omega$", "tex_body": "\\omega", "tex_normalized": "\\omega", "mathml": "$ω$", "char_span": [18480, 18493], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0071_PH"}, {"id": "eq0072", "inline": true, "tex": "$\\Path^\\ast$", "tex_body": "\\Path^\\ast", "tex_normalized": "\\Path^\\ast", "mathml": "$\\Path*$", "char_span": [18495, 18508], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0072_PH"}, {"id": "eq0073", "inline": true, "tex": "$F(X)=I\\join(X\\Star A)$", "tex_body": "F(X)=I\\join(X\\Star A)", "tex_normalized": "F(X)=I\\join(X\\Star A)", "mathml": "$F(X)=I\\join(X\\StarA)$", "char_span": [18510, 18523], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0073_PH"}, {"id": "eq0074", "inline": true, "tex": "$\\iota_{i\\ast}$", "tex_body": "\\iota_{i\\ast}", "tex_normalized": "\\iota_{i\\ast}", "mathml": "$ιi*$", "char_span": [18525, 18538], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0074_PH"}, {"id": "eq0075", "inline": true, "tex": "$\\iota_i^\\ast$", "tex_body": "\\iota_i^\\ast", "tex_normalized": "\\iota_i^\\ast", "mathml": "$ιi*$", "char_span": [18540, 18553], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0075_PH"}, {"id": "eq0076", "inline": true, "tex": "$\\{\\iota_i:U_i\\to U\\}$", "tex_body": "\\{\\iota_i:U_i\\to U\\}", "tex_normalized": "\\{\\iota_i:U_i\\to U\\}", "mathml": "${ιi:Ui→U}$", "char_span": [18555, 18568], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0076_PH"}, {"id": "eq0077", "inline": true, "tex": "$\\join_i\\, \\iota_i\\iota_i^\\ast \\geqE 1_U$", "tex_body": "\\join_i\\, \\iota_i\\iota_i^\\ast \\geqE 1_U", "tex_normalized": "\\join_i \\iota_i\\iota_i^\\ast \\geqE 1_U", "mathml": "$\\joiniιiιi*\\geqE1U$", "char_span": [18570, 18583], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0077_PH"}, {"id": "eq0078", "inline": true, "tex": "$T_i:U_i\\rightrightarrows V$", "tex_body": "T_i:U_i\\rightrightarrows V", "tex_normalized": "T_i:U_i\\rightrightarrows V", "mathml": "$Ti:Ui⇉V$", "char_span": [18585, 18598], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0078_PH"}, {"id": "eq0079", "inline": true, "tex": "$c_{d,i}\\leqE 1_{U_i}$", "tex_body": "c_{d,i}\\leqE 1_{U_i}", "tex_normalized": "c_{d,i}\\leqE 1_{U_i}", "mathml": "$cd,i\\leqE1Ui$", "char_span": [18600, 18613], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0079_PH"}, {"id": "eq0080", "inline": true, "tex": "$\\Phi$", "tex_body": "\\Phi", "tex_normalized": "\\Phi", "mathml": "$Φ$", "char_span": [18615, 18628], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0080_PH"}, {"id": "eq0081", "inline": true, "tex": "$v(T)$", "tex_body": "v(T)", "tex_normalized": "v(T)", "mathml": "$v(T)$", "char_span": [18630, 18643], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0081_PH"}, {"id": "eq0082", "inline": true, "tex": "$\\Phi$", "tex_body": "\\Phi", "tex_normalized": "\\Phi", "mathml": "$Φ$", "char_span": [18645, 18658], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0082_PH"}, {"id": "eq0083", "inline": true, "tex": "$v$", "tex_body": "v", "tex_normalized": "v", "mathml": "$v$", "char_span": [18660, 18673], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0083_PH"}, {"id": "eq0084", "inline": true, "tex": "$(B,\\odotop)$", "tex_body": "(B,\\odotop)", "tex_normalized": "(B,\\odotop)", "mathml": "$(B,\\odotop)$", "char_span": [18675, 18688], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0084_PH"}, {"id": "eq0085", "inline": true, "tex": "$(-)\\backslash a$", "tex_body": "(-)\\backslash a", "tex_normalized": "(-)\\backslash a", "mathml": "$(−)\a$", "char_span": [18690, 18703], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0085_PH"}, {"id": "eq0086", "inline": true, "tex": "$x\\odotop a \\leqE y \\iff x \\leqE y \\backslash a$", "tex_body": "x\\odotop a \\leqE y \\iff x \\leqE y \\backslash a", "tex_normalized": "x\\odotop a \\leqE y \\iff x \\leqE y \\backslash a", "mathml": "$x\\odotopa\\leqEy⟺x\\leqEy\a$", "char_span": [18705, 18718], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0086_PH"}, {"id": "eq0087", "inline": true, "tex": "$a/(-)$", "tex_body": "a/(-)", "tex_normalized": "a/(-)", "mathml": "$a/(−)$", "char_span": [18720, 18733], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0087_PH"}, {"id": "eq0088", "inline": true, "tex": "$\\theta$", "tex_body": "\\theta", "tex_normalized": "\\theta", "mathml": "$θ$", "char_span": [18735, 18748], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0088_PH"}, {"id": "eq0089", "inline": true, "tex": "$x\\odotop a \\geqE \\theta \\iff x \\geqE \\theta\\backslash a$", "tex_body": "x\\odotop a \\geqE \\theta \\iff x \\geqE \\theta\\backslash a", "tex_normalized": "x\\odotop a \\geqE \\theta \\iff x \\geqE \\theta\\backslash a", "mathml": "$x\\odotopa\\geqEθ⟺x\\geqEθ\a$", "char_span": [18750, 18763], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0089_PH"}, {"id": "eq0090", "inline": true, "tex": "$\\nu$", "tex_body": "\\nu", "tex_normalized": "\\nu", "mathml": "$ν$", "char_span": [18765, 18778], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0090_PH"}, {"id": "eq0091", "inline": true, "tex": "$\\nu(x)\\odotop \\nu(y)\\leqE \\nu(x\\odotop y)$", "tex_body": "\\nu(x)\\odotop \\nu(y)\\leqE \\nu(x\\odotop y)", "tex_normalized": "\\nu(x)\\odotop \\nu(y)\\leqE \\nu(x\\odotop y)", "mathml": "$ν(x)\\odotopν(y)\\leqEν(x\\odotopy)$", "char_span": [18780, 18793], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0091_PH"}, {"id": "eq0092", "inline": true, "tex": "$M$", "tex_body": "M", "tex_normalized": "M", "mathml": "$M$", "char_span": [18795, 18808], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0092_PH"}, {"id": "eq0093", "inline": true, "tex": "$\\nu$", "tex_body": "\\nu", "tex_normalized": "\\nu", "mathml": "$ν$", "char_span": [18810, 18823], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0093_PH"}, {"id": "eq0094", "inline": true, "tex": "$\\nu(B_{\\mathrm{mask}})=B_{\\mathrm{mask}}$", "tex_body": "\\nu(B_{\\mathrm{mask}})=B_{\\mathrm{mask}}", "tex_normalized": "\\nu(B_{\\mathrm{mask}})=B_{\\mathrm{mask}}", "mathml": "$ν(Bmask)=Bmask$", "char_span": [18825, 18838], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0094_PH"}, {"id": "eq0095", "inline": true, "tex": "$\\nu$", "tex_body": "\\nu", "tex_normalized": "\\nu", "mathml": "$ν$", "char_span": [18840, 18853], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0095_PH"}, {"id": "eq0096", "inline": true, "tex": "$\\pi ::= \\mathrm{hop}(U\\!\\to\\!V)\\mid \\pi\\cdot\\pi$", "tex_body": "\\pi ::= \\mathrm{hop}(U\\!\\to\\!V)\\mid \\pi\\cdot\\pi", "tex_normalized": "\\pi ::= \\mathrm{hop}(U \\to V)\\mid \\pi\\cdot\\pi", "mathml": "$π::=hop(U→V)∣π·π$", "char_span": [18855, 18868], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0096_PH"}, {"id": "eq0097", "inline": true, "tex": "$\\vdash \\mathrm{hop}(U\\!\\to\\!V):B(U,V)$", "tex_body": "\\vdash \\mathrm{hop}(U\\!\\to\\!V):B(U,V)", "tex_normalized": "\\vdash \\mathrm{hop}(U \\to V):B(U,V)", "mathml": "$⊢hop(U→V):B(U,V)$", "char_span": [18870, 18883], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0097_PH"}, {"id": "eq0098", "inline": true, "tex": "$\\pi_1:B(U,V)$", "tex_body": "\\pi_1:B(U,V)", "tex_normalized": "\\pi_1:B(U,V)", "mathml": "$π1:B(U,V)$", "char_span": [18885, 18898], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0098_PH"}, {"id": "eq0099", "inline": true, "tex": "$\\pi_2:B(V,T)$", "tex_body": "\\pi_2:B(V,T)", "tex_normalized": "\\pi_2:B(V,T)", "mathml": "$π2:B(V,T)$", "char_span": [18900, 18913], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0099_PH"}, {"id": "eq0100", "inline": true, "tex": "$\\pi_1\\cdot\\pi_2:B(U,T)$", "tex_body": "\\pi_1\\cdot\\pi_2:B(U,T)", "tex_normalized": "\\pi_1\\cdot\\pi_2:B(U,T)", "mathml": "$π1·π2:B(U,T)$", "char_span": [18915, 18928], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0100_PH"}, {"id": "eq0101", "inline": true, "tex": "$B_{\\mathrm{mask}}(U,T)=\\join\\{\\llbracket\\pi\\rrbracket:\\pi\\ \\text{admissible}\\}$", "tex_body": "B_{\\mathrm{mask}}(U,T)=\\join\\{\\llbracket\\pi\\rrbracket:\\pi\\ \\text{admissible}\\}", "tex_normalized": "B_{\\mathrm{mask}}(U,T)=\\join\\{\\llbracket\\pi\\rrbracket:\\pi\\ \\text{admissible}\\}", "mathml": "$Bmask(U,T)=\\join{⟦π⟧:π admissible}$", "char_span": [18930, 18943], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0101_PH"}, {"id": "eq0102", "inline": true, "tex": "$\\odotop,\\join$", "tex_body": "\\odotop,\\join", "tex_normalized": "\\odotop,\\join", "mathml": "$\\odotop,\\join$", "char_span": [18945, 18958], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0102_PH"}, {"id": "eq0103", "inline": true, "tex": "$v(\\tau)$", "tex_body": "v(\\tau)", "tex_normalized": "v(\\tau)", "mathml": "$v(τ)$", "char_span": [18960, 18973], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0103_PH"}, {"id": "eq0104", "inline": true, "tex": "$b\\leqE B_{\\mathrm{mask}}(U,T)$", "tex_body": "b\\leqE B_{\\mathrm{mask}}(U,T)", "tex_normalized": "b\\leqE B_{\\mathrm{mask}}(U,T)", "mathml": "$b\\leqEBmask(U,T)$", "char_span": [18975, 18988], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0104_PH"}, {"id": "eq0105", "inline": true, "tex": "$\\omega$", "tex_body": "\\omega", "tex_normalized": "\\omega", "mathml": "$ω$", "char_span": [18990, 19003], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0105_PH"}, {"id": "eq0106", "inline": true, "tex": "$(U,T)$", "tex_body": "(U,T)", "tex_normalized": "(U,T)", "mathml": "$(U,T)$", "char_span": [19005, 19018], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0106_PH"}, {"id": "eq0107", "inline": true, "tex": "$\\odotop$", "tex_body": "\\odotop", "tex_normalized": "\\odotop", "mathml": "$\\odotop$", "char_span": [19020, 19033], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0107_PH"}, {"id": "eq0108", "inline": true, "tex": "$B_{\\mathrm{mask}}$", "tex_body": "B_{\\mathrm{mask}}", "tex_normalized": "B_{\\mathrm{mask}}", "mathml": "$Bmask$", "char_span": [19035, 19048], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0108_PH"}, {"id": "eq0109", "inline": true, "tex": "$B_{\\mathrm{mask}}(U,T)$", "tex_body": "B_{\\mathrm{mask}}(U,T)", "tex_normalized": "B_{\\mathrm{mask}}(U,T)", "mathml": "$Bmask(U,T)$", "char_span": [19050, 19063], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0109_PH"}, {"id": "eq0110", "inline": true, "tex": "$\\nu$", "tex_body": "\\nu", "tex_normalized": "\\nu", "mathml": "$ν$", "char_span": [19065, 19078], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0110_PH"}, {"id": "eq0111", "inline": true, "tex": "$V^\\ast$", "tex_body": "V^\\ast", "tex_normalized": "V^\\ast", "mathml": "$V*$", "char_span": [19080, 19093], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0111_PH"}, {"id": "eq0112", "inline": true, "tex": "$\\nu(M(U,V^\\ast))\\ge \\nu(1_U)$", "tex_body": "\\nu(M(U,V^\\ast))\\ge \\nu(1_U)", "tex_normalized": "\\nu(M(U,V^\\ast))\\ge \\nu(1_U)", "mathml": "$ν(M(U,V*))≥ν(1U)$", "char_span": [19095, 19108], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0112_PH"}, {"id": "eq0113", "inline": true, "tex": "$\\nu(M(U,V))=\\bot$", "tex_body": "\\nu(M(U,V))=\\bot", "tex_normalized": "\\nu(M(U,V))=\\bot", "mathml": "$ν(M(U,V))=⊥$", "char_span": [19110, 19123], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0113_PH"}, {"id": "eq0114", "inline": true, "tex": "$V\\neq V^\\ast$", "tex_body": "V\\neq V^\\ast", "tex_normalized": "V\\neq V^\\ast", "mathml": "$V≠V*$", "char_span": [19125, 19138], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0114_PH"}, {"id": "eq0115", "inline": true, "tex": "$B_{\\mathrm{mask}}$", "tex_body": "B_{\\mathrm{mask}}", "tex_normalized": "B_{\\mathrm{mask}}", "mathml": "$Bmask$", "char_span": [19140, 19153], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0115_PH"}, {"id": "eq0116", "inline": true, "tex": "$\\mathsf{J}$", "tex_body": "\\mathsf{J}", "tex_normalized": "\\mathsf{J}", "mathml": "$𝖩$", "char_span": [19155, 19168], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0116_PH"}, {"id": "eq0117", "inline": true, "tex": "$F:B\\to B'$", "tex_body": "F:B\\to B'", "tex_normalized": "F:B\\to B'", "mathml": "$F:B→B′$", "char_span": [19170, 19183], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0117_PH"}, {"id": "eq0118", "inline": true, "tex": "$\\mathsf{J}$", "tex_body": "\\mathsf{J}", "tex_normalized": "\\mathsf{J}", "mathml": "$𝖩$", "char_span": [19185, 19198], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0118_PH"}, {"id": "eq0119", "inline": true, "tex": "$F_0(\\Ob_0(B))$", "tex_body": "F_0(\\Ob_0(B))", "tex_normalized": "F_0(\\Ob_0(B))", "mathml": "$F0(\\Ob0(B))$", "char_span": [19200, 19213], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0119_PH"}, {"id": "eq0120", "inline": true, "tex": "$\\mathsf{J}$", "tex_body": "\\mathsf{J}", "tex_normalized": "\\mathsf{J}", "mathml": "$𝖩$", "char_span": [19215, 19228], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0120_PH"}, {"id": "eq0121", "inline": true, "tex": "$\\mathsf{J}$", "tex_body": "\\mathsf{J}", "tex_normalized": "\\mathsf{J}", "mathml": "$𝖩$", "char_span": [19230, 19243], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0121_PH"}, {"id": "eq0122", "inline": true, "tex": "$t$", "tex_body": "t", "tex_normalized": "t", "mathml": "$t$", "char_span": [19245, 19258], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0122_PH"}, {"id": "eq0123", "inline": true, "tex": "$\\to$", "tex_body": "\\to", "tex_normalized": "\\to", "mathml": "$→$", "char_span": [19260, 19273], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0123_PH"}, {"id": "eq0124", "inline": true, "tex": "$F(A^{\\Star 2})\\neq (FA)^{\\Star 2}$", "tex_body": "F(A^{\\Star 2})\\neq (FA)^{\\Star 2}", "tex_normalized": "F(A^{\\Star 2})\\neq (FA)^{\\Star 2}", "mathml": "$F(A\\Star2)≠(FA)\\Star2$", "char_span": [19275, 19288], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0124_PH"}, {"id": "eq0125", "inline": true, "tex": "$F$", "tex_body": "F", "tex_normalized": "F", "mathml": "$F$", "char_span": [19290, 19303], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0125_PH"}, {"id": "eq0126", "inline": true, "tex": "$A$", "tex_body": "A", "tex_normalized": "A", "mathml": "$A$", "char_span": [19305, 19318], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0126_PH"}, {"id": "eq0127", "inline": true, "tex": "$\\square\\in\\{\\geqE,\\leqE,=\\}$", "tex_body": "\\square\\in\\{\\geqE,\\leqE,=\\}", "tex_normalized": "\\square\\in\\{\\geqE,\\leqE,=\\}", "mathml": "$◻∈{\\geqE,\\leqE,=}$", "char_span": [19320, 19333], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0127_PH"}, {"id": "eq0128", "inline": true, "tex": "$\\mathsf{J}$", "tex_body": "\\mathsf{J}", "tex_normalized": "\\mathsf{J}", "mathml": "$𝖩$", "char_span": [19335, 19348], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0128_PH"}, {"id": "eq0129", "inline": true, "tex": "$(0,1]$", "tex_body": "(0,1]", "tex_normalized": "(0,1]", "mathml": "$(0,1]$", "char_span": [19350, 19363], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0129_PH"}, {"id": "eq0130", "inline": true, "tex": "$t$", "tex_body": "t", "tex_normalized": "t", "mathml": "$t$", "char_span": [19365, 19378], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0130_PH"}, {"id": "eq0131", "inline": true, "tex": "$\\to$", "tex_body": "\\to", "tex_normalized": "\\to", "mathml": "$→$", "char_span": [19380, 19393], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0131_PH"}, {"id": "eq0132", "inline": true, "tex": "$\\to$", "tex_body": "\\to", "tex_normalized": "\\to", "mathml": "$→$", "char_span": [19395, 19408], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0132_PH"}, {"id": "eq0133", "inline": true, "tex": "$-\\log$", "tex_body": "-\\log", "tex_normalized": "-\\log", "mathml": "$−log$", "char_span": [19410, 19423], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0133_PH"}, {"id": "eq0134", "inline": true, "tex": "$[\\varepsilon,1]$", "tex_body": "[\\varepsilon,1]", "tex_normalized": "[\\varepsilon,1]", "mathml": "$[ε,1]$", "char_span": [19425, 19438], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0134_PH"}, {"id": "eq0135", "inline": true, "tex": "$(1/\\varepsilon)$", "tex_body": "(1/\\varepsilon)", "tex_normalized": "(1/\\varepsilon)", "mathml": "$(1/ε)$", "char_span": [19440, 19453], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0135_PH"}, {"id": "eq0136", "inline": true, "tex": "$\\Ev_\\bullet:B\\to B_\\bullet$", "tex_body": "\\Ev_\\bullet:B\\to B_\\bullet", "tex_normalized": "\\Ev_\\bullet:B\\to B_\\bullet", "mathml": "$\\Ev•:B→B•$", "char_span": [19455, 19468], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0136_PH"}, {"id": "eq0137", "inline": true, "tex": "$\\Phi_\\bullet:B_\\bullet(V,T)\\times B_\\bullet(U,V)\\to B_\\bullet(U,T)$", "tex_body": "\\Phi_\\bullet:B_\\bullet(V,T)\\times B_\\bullet(U,V)\\to B_\\bullet(U,T)", "tex_normalized": "\\Phi_\\bullet:B_\\bullet(V,T)\\times B_\\bullet(U,V)\\to B_\\bullet(U,T)", "mathml": "$Φ•:B•(V,T)×B•(U,V)→B•(U,T)$", "char_span": [19470, 19483], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0137_PH"}, {"id": "eq0138", "inline": true, "tex": "$e_\\bullet$", "tex_body": "e_\\bullet", "tex_normalized": "e_\\bullet", "mathml": "$e•$", "char_span": [19485, 19498], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0138_PH"}, {"id": "eq0139", "inline": true, "tex": "$\\Ev_\\bullet$", "tex_body": "\\Ev_\\bullet", "tex_normalized": "\\Ev_\\bullet", "mathml": "$\\Ev•$", "char_span": [19500, 19513], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0139_PH"}, {"id": "eq0140", "inline": true, "tex": "$\\mathsf{J}$", "tex_body": "\\mathsf{J}", "tex_normalized": "\\mathsf{J}", "mathml": "$𝖩$", "char_span": [19515, 19528], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0140_PH"}, {"id": "eq0141", "inline": true, "tex": "$\\Ev_\\bullet(g\\then f)\\ \\square\\ \\Phi_\\bullet(\\Ev_\\bullet(g),\\Ev_\\bullet(f))$", "tex_body": "\\Ev_\\bullet(g\\then f)\\ \\square\\ \\Phi_\\bullet(\\Ev_\\bullet(g),\\Ev_\\bullet(f))", "tex_normalized": "\\Ev_\\bullet(g\\then f)\\ \\square\\ \\Phi_\\bullet(\\Ev_\\bullet(g),\\Ev_\\bullet(f))", "mathml": "$\\Ev•(g\\thenf) ◻ Φ•(\\Ev•(g),\\Ev•(f))$", "char_span": [19530, 19543], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0141_PH"}, {"id": "eq0142", "inline": true, "tex": "$\\square\\in\\{\\geqE,\\leqE\\}$", "tex_body": "\\square\\in\\{\\geqE,\\leqE\\}", "tex_normalized": "\\square\\in\\{\\geqE,\\leqE\\}", "mathml": "$◻∈{\\geqE,\\leqE}$", "char_span": [19545, 19558], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0142_PH"}, {"id": "eq0143", "inline": true, "tex": "$A$", "tex_body": "A", "tex_normalized": "A", "mathml": "$A$", "char_span": [19560, 19573], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0143_PH"}, {"id": "eq0144", "inline": true, "tex": "$n\\ge1$", "tex_body": "n\\ge1", "tex_normalized": "n\\ge1", "mathml": "$n≥1$", "char_span": [19575, 19588], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0144_PH"}, {"id": "eq0145", "inline": true, "tex": "$\\Ev_\\bullet(A^{\\Star n})\\ \\square\\ (\\Ev_\\bullet A)^{\\Star n}$", "tex_body": "\\Ev_\\bullet(A^{\\Star n})\\ \\square\\ (\\Ev_\\bullet A)^{\\Star n}", "tex_normalized": "\\Ev_\\bullet(A^{\\Star n})\\ \\square\\ (\\Ev_\\bullet A)^{\\Star n}", "mathml": "$\\Ev•(A\\Starn) ◻ (\\Ev•A)\\Starn$", "char_span": [19590, 19603], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0145_PH"}, {"id": "eq0146", "inline": true, "tex": "$\\Ev_\\bullet(\\Path)\\ \\square\\ \\Path_\\bullet$", "tex_body": "\\Ev_\\bullet(\\Path)\\ \\square\\ \\Path_\\bullet", "tex_normalized": "\\Ev_\\bullet(\\Path)\\ \\square\\ \\Path_\\bullet", "mathml": "$\\Ev•(\\Path) ◻ \\Path•$", "char_span": [19605, 19618], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0146_PH"}, {"id": "eq0147", "inline": true, "tex": "$\\Phi$", "tex_body": "\\Phi", "tex_normalized": "\\Phi", "mathml": "$Φ$", "char_span": [19620, 19633], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0147_PH"}, {"id": "eq0148", "inline": true, "tex": "$\\Phi=\\min$", "tex_body": "\\Phi=\\min", "tex_normalized": "\\Phi=\\min", "mathml": "$Φ=min$", "char_span": [19635, 19648], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0148_PH"}, {"id": "eq0149", "inline": true, "tex": "$\\Phi=\\min$", "tex_body": "\\Phi=\\min", "tex_normalized": "\\Phi=\\min", "mathml": "$Φ=min$", "char_span": [19650, 19663], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0149_PH"}, {"id": "eq0150", "inline": true, "tex": "$(r_{\\min},D_{\\min})$", "tex_body": "(r_{\\min},D_{\\min})", "tex_normalized": "(r_{\\min},D_{\\min})", "mathml": "$(rmin,Dmin)$", "char_span": [19665, 19678], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0150_PH"}, {"id": "eq0151", "inline": true, "tex": "$F$", "tex_body": "F", "tex_normalized": "F", "mathml": "$F$", "char_span": [19680, 19693], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0151_PH"}, {"id": "eq0152", "inline": true, "tex": "$D$", "tex_body": "D", "tex_normalized": "D", "mathml": "$D$", "char_span": [19695, 19708], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0152_PH"}, {"id": "eq0153", "inline": true, "tex": "$\\Rightarrow$", "tex_body": "\\Rightarrow", "tex_normalized": "\\Rightarrow", "mathml": "$⇒$", "char_span": [19710, 19723], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0153_PH"}, {"id": "eq0154", "inline": true, "tex": "$\\min$", "tex_body": "\\min", "tex_normalized": "\\min", "mathml": "$min$", "char_span": [19725, 19738], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0154_PH"}, {"id": "eq0155", "inline": true, "tex": "$\\Rightarrow$", "tex_body": "\\Rightarrow", "tex_normalized": "\\Rightarrow", "mathml": "$⇒$", "char_span": [19740, 19753], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0155_PH"}, {"id": "eq0156", "inline": true, "tex": "$(r_{\\min},D_{\\min})$", "tex_body": "(r_{\\min},D_{\\min})", "tex_normalized": "(r_{\\min},D_{\\min})", "mathml": "$(rmin,Dmin)$", "char_span": [19755, 19768], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0156_PH"}, {"id": "eq0157", "inline": true, "tex": "$\\Rightarrow$", "tex_body": "\\Rightarrow", "tex_normalized": "\\Rightarrow", "mathml": "$⇒$", "char_span": [19770, 19783], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0157_PH"}, {"id": "eq0158", "inline": true, "tex": "$\\nu:B(U,U)\\to B(U,U)$", "tex_body": "\\nu:B(U,U)\\to B(U,U)", "tex_normalized": "\\nu:B(U,U)\\to B(U,U)", "mathml": "$ν:B(U,U)→B(U,U)$", "char_span": [19785, 19798], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0158_PH"}, {"id": "eq0159", "inline": true, "tex": "$G_{s\\to t}:B_s\\to B_t$", "tex_body": "G_{s\\to t}:B_s\\to B_t", "tex_normalized": "G_{s\\to t}:B_s\\to B_t", "mathml": "$Gs→t:Bs→Bt$", "char_span": [19800, 19813], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0159_PH"}, {"id": "eq0160", "inline": true, "tex": "$U$", "tex_body": "U", "tex_normalized": "U", "mathml": "$U$", "char_span": [19815, 19828], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0160_PH"}, {"id": "eq0161", "inline": true, "tex": "$\\nu(1_U)\\ge 1_U$", "tex_body": "\\nu(1_U)\\ge 1_U", "tex_normalized": "\\nu(1_U)\\ge 1_U", "mathml": "$ν(1U)≥1U$", "char_span": [19830, 19843], "context": {"section": 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"canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0164_PH"}, {"id": "eq0165", "inline": true, "tex": "$G$", "tex_body": "G", "tex_normalized": "G", "mathml": "$G$", "char_span": [19890, 19903], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0165_PH"}, {"id": "eq0166", "inline": true, "tex": "$\\nu$", "tex_body": "\\nu", "tex_normalized": "\\nu", "mathml": "$ν$", "char_span": [19905, 19918], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0166_PH"}, {"id": "eq0167", "inline": true, "tex": "$\\nu(\\Path_t(U,U))\\ge \\nu(1_U)$", "tex_body": "\\nu(\\Path_t(U,U))\\ge \\nu(1_U)", "tex_normalized": "\\nu(\\Path_t(U,U))\\ge \\nu(1_U)", "mathml": "$ν(\\Patht(U,U))≥ν(1U)$", "char_span": [19920, 19933], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0167_PH"}, {"id": "eq0168", "inline": true, "tex": "$\\nu$", "tex_body": "\\nu", "tex_normalized": "\\nu", "mathml": "$ν$", "char_span": [19935, 19948], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0168_PH"}, {"id": "eq0169", "inline": true, "tex": "$\\nu(R_t)\\ge\\nu(1_U)$", "tex_body": "\\nu(R_t)\\ge\\nu(1_U)", "tex_normalized": "\\nu(R_t)\\ge\\nu(1_U)", "mathml": "$ν(Rt)≥ν(1U)$", "char_span": [19950, 19963], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0169_PH"}, {"id": "eq0170", "inline": true, "tex": "$\\nu(\\Path_t(U,U))\\ge\\nu(1_U)$", "tex_body": "\\nu(\\Path_t(U,U))\\ge\\nu(1_U)", "tex_normalized": "\\nu(\\Path_t(U,U))\\ge\\nu(1_U)", "mathml": "$ν(\\Patht(U,U))≥ν(1U)$", "char_span": [19965, 19978], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0170_PH"}, {"id": "eq0171", "inline": true, "tex": "$a$", "tex_body": "a", "tex_normalized": "a", "mathml": "$a$", "char_span": [19980, 19993], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0171_PH"}, {"id": "eq0172", "inline": true, "tex": "$f_a=(f_a^{\\text{stat}},f_a^{\\text{kpp}},f_a^{\\text{q}})$", "tex_body": "f_a=(f_a^{\\text{stat}},f_a^{\\text{kpp}},f_a^{\\text{q}})", "tex_normalized": "f_a=(f_a^{\\text{stat}},f_a^{\\text{kpp}},f_a^{\\text{q}})", "mathml": "$fa=(fastat,fakpp,faq)$", "char_span": [19995, 20008], "context": {"section": "canonical-dois-author-s-preprints"}, "placeholder": "EQPH_eq0172_PH"}, {"id": "eq0173", "inline": true, "tex": "$\\tilde f^{\\mathrm{kpp}}\\!\\ge c_{\\min}>0$", "tex_body": "\\tilde f^{\\mathrm{kpp}}\\!\\ge c_{\\min}>0", "tex_normalized": "\\tilde f^{\\mathrm{kpp}} \\ge c_{\\min}>0", "mathml": "$f~kpp≥cmin>0$", "char_span": [11714, 11727], "context": {"section": "implementation-and-reproducibility"}, "placeholder": "EQPH_eq0173_PH"}, {"id": "eq0174", "inline": true, "tex": "$H_{\\mathrm{net}}$", "tex_body": "H_{\\mathrm{net}}", "tex_normalized": "H_{\\mathrm{net}}", "mathml": "$Hnet$", "char_span": [11758, 11771], "context": {"section": 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