Inference in Normal Form: Auditable, No-Meta Decision Algebra for LLM/Safety Loops

theory-of-relativity-of-theories inference-normal-form no-meta relative-auditing trot pfad graphblas anytime-auditing

positioning,arrows.meta,decorations.pathmorphing,decorations.pathreplacing,plotmarks

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Lan
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theorem Theorem [section]
proposition[theorem] Proposition
lemma[theorem] Lemma
corollary[theorem] Corollary

TITLE: -0.7em

Inference in Normal Form: Unifying LLM Tricks via TRoT -0.3em

AUTHOR: K. Takahashi
[[EQ:eq0004]]

) Assume [[EQ:eq0011]] , i.e., [[EQ:eq0012]] for all [[EQ:eq0013]] (enriched order).
Since [[EQ:eq0014]] and [[EQ:eq0015]] is the numeric infimum, [[EQ:eq0016]] means (numerically) [[EQ:eq0017]] , hence for all [[EQ:eq0018]] we have [[EQ:eq0019]] .
By the residual law in enriched order, [[EQ:eq0020]] for all [[EQ:eq0021]] , so [[EQ:eq0022]] .

( [[EQ:eq0023]] ) Assume [[EQ:eq0024]] , i.e., [[EQ:eq0025]] .
Then for each [[EQ:eq0026]] , [[EQ:eq0027]] ; by residuation, [[EQ:eq0028]] .
Taking the [[EQ:eq0029]] (numeric infimum) over [[EQ:eq0030]] yields [[EQ:eq0031]] .

SUBSECTION: Normal Form and Stability Bounds

for bounds.
The decay factor [[EQ:eq0032]] reflects exogenous per-layer attenuation [[EQ:eq0033]] (e.g., temperature/penalty schedules or masked fan-in control).
We re-emphasize [[EQ:eq0034]] is measured after masking.

[Geometric Truncation and Soft-Min Approximation]thm:trunc
Let stage maps [[EQ:eq0035]] be nonexpansive (Lemmas~lem:residual--lem:lan-lip) and [[EQ:eq0036]] be multiplicative decays with [[EQ:eq0037]] .
Write [[EQ:eq0038]] and its depth- [[EQ:eq0039]] truncation [[EQ:eq0040]] .
Then

[[EQ:eq0005]]

If 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]] ,

[[EQ:eq0006]]

(composition vs.\ path-sum).
If [[EQ:eq0046]] is realized as a monotone join of pathwise contributions with per-layer decays [[EQ:eq0047]]
(e.g., TRoT’s finite-path expansion), the Neumann-type tail bound applies and yields
[[EQ:eq0048]] .
If [[EQ:eq0049]] is a pure composition without path-sum accumulation, a sharper bound
[[EQ:eq0050]] holds by submultiplicativity.

[[EQ:eq0051]] ; a Neumann-type series bound yields [[EQ:eq0052]] .
For [[EQ:eq0053]] ,
[[EQ:eq0054]] ; layerwise accumulation gives the bound.We accumulate the per-layer soft-min gap linearly in depth [[EQ:eq0055]] .
For varying fan-in [[EQ:eq0056]] , replace [[EQ:eq0057]] by [[EQ:eq0058]] .
Sharper constants follow from layerwise curvature/entropy control.
This bound assumes each layer's fan-in is bounded by [[EQ:eq0059]] after masking.

SUBSECTION: Anytime-Valid Auditing and Reproducibility

sec:audit
[FWER via Deterministic Spending]thm:audit
Let [[EQ:eq0060]] be the filtration of revealed evidence with a test martingale [[EQ:eq0061]] ( [[EQ:eq0062]] , [[EQ:eq0063]] ) and e-values [[EQ:eq0064]] .
With a deterministic spending schedule [[EQ:eq0065]] satisfying [[EQ:eq0066]] , the stopping time [[EQ:eq0067]] obeys

[[EQ:eq0007]]

hence FWER [[EQ:eq0068]] .
If the pipeline logs [[EQ:eq0069]] , replay reproduces the accept/reject outcome almost surely.

By 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]] .
(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.)

PARAGRAPH: Mini-example (RAVE + e-process, 2 candidates, 3 judges).

Pairwise preferences: judges say [[EQ:eq0077]] twice, [[EQ:eq0078]] once.
BTL MLE updates [[EQ:eq0079]] (e.g., [[EQ:eq0080]] ).
For 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]] .
Update the test process as [[EQ:eq0084]] and declare when [[EQ:eq0085]] ; otherwise gather more judgments/logs.

SUBSECTION: Entropic-MBR (eMBR) and Bounds

sec:eMBR
[TRoT soft-min equals entropic risk]prop:entropic
Let [[EQ:eq0086]] , [[EQ:eq0087]] a discrete distribution on [[EQ:eq0088]] , and

[[EQ:eq0008]]

If [[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]] .

Let [[EQ:eq0094]] and [[EQ:eq0095]] .
Then

[[EQ:eq0003]]

[eMBR bounds and limits]thm:embr-bounds
For any [[EQ:eq0096]] and [[EQ:eq0097]] ,

[[EQ:eq0009]]

Moreover, [[EQ:eq0098]] as [[EQ:eq0099]] , and [[EQ:eq0100]] as [[EQ:eq0101]] .

Lower bound: [[EQ:eq0102]] .
Upper bound: Jensen gives [[EQ:eq0103]] , hence [[EQ:eq0104]] .
A 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.

[Tail-coverage under curvature cap]cor:tail
Let VS-like value masks induce curvature [[EQ:eq0108]] and suppose [[EQ:eq0109]] via [[EQ:eq0110]] thresholds and a nucleus [[EQ:eq0111]] .
For any event set [[EQ:eq0112]] , if the cumulative mask shift on [[EQ:eq0113]] is at most [[EQ:eq0114]] , then the post-pipeline mass satisfies
[[EQ:eq0115]] .

A value-mask shift [[EQ:eq0116]] multiplies [[EQ:eq0117]] by [[EQ:eq0118]] via eq:dictionary; capping [[EQ:eq0119]] bounds the cumulative shift.

SECTION: Algorithms with Types \& Complexity

We consider finite sets [[EQ:eq0120]] ; [[EQ:eq0121]] ; transport [[EQ:eq0122]] ; map [[EQ:eq0123]] .
Complexities assume sparse CSR/COO with [[EQ:eq0124]] , [[EQ:eq0125]] , effective [[EQ:eq0126]] after masking.

[H]
TROT-Normal-Form (single decision epoch)
alg:trot-normal
[1]
Types: [[EQ:eq0127]] ; [[EQ:eq0128]] ; [[EQ:eq0129]] ; [[EQ:eq0130]] ; [[EQ:eq0131]] ; [[EQ:eq0132]]
Input: task [[EQ:eq0133]] , forward map [[EQ:eq0134]] , transport [[EQ:eq0135]] , observer [[EQ:eq0136]] , RAVE, nucleus [[EQ:eq0137]] , mask [[EQ:eq0138]]
Output: decision [[EQ:eq0139]] , log [[EQ:eq0140]]
K1 Path: build candidate graph [[EQ:eq0141]] ; form sparse [[EQ:eq0142]]
K3 Generation: [[EQ:eq0143]] via SpMV/SpGEMM on tropical [[EQ:eq0144]] ; [[EQ:eq0145]]
Observer: [[EQ:eq0146]] (judge/calibration in log-domain)
K4 Safety: [[EQ:eq0147]] [[EQ:eq0148]]
B-side stream: [[EQ:eq0149]] mask/nucleus on [[EQ:eq0150]]
A-side stream: [[EQ:eq0151]] mask/nucleus after pushforward
RAVE placement: use [[EQ:eq0152]] (candidate-level on [[EQ:eq0153]] ) or [[EQ:eq0154]] (safety-respecting on [[EQ:eq0155]] )
RAVE: [[EQ:eq0156]]
Gate: if [[EQ:eq0157]] passes schedule then [[EQ:eq0158]] else refine [[EQ:eq0159]] and repeat

Complexity. K3: SpMV [[EQ:eq0160]] ; SpGEMM worst-case [[EQ:eq0161]] , typically [[EQ:eq0162]] in sparse regimes.
K4: [[EQ:eq0163]] . Obs/RAVE: [[EQ:eq0164]] -- [[EQ:eq0165]] (few candidates).
Gate: streaming [[EQ:eq0166]] amortized. Log size [[EQ:eq0167]] .

[H]
eMBR on [[EQ:eq0168]]
alg:embr
[1]
Types: [[EQ:eq0169]] finite; [[EQ:eq0170]] ; [[EQ:eq0171]] ; [[EQ:eq0172]]
Input: candidates [[EQ:eq0173]] , samples [[EQ:eq0174]] , loss [[EQ:eq0175]] , [[EQ:eq0176]]
Output: [[EQ:eq0177]]
[[EQ:eq0178]]
[[EQ:eq0179]]

[[EQ:eq0180]]

Complexity. [[EQ:eq0181]] arithmetic.
Numerics: use row-wise max-shifts in [[EQ:eq0182]] ; set masked entries to [[EQ:eq0183]] ; keep CSR order deterministic.

[H]
RAVE-BTL-Gate: Judge Aggregation with Anytime-Valid Stopping
alg:rave
[1]
Types: pairwise prefs [[EQ:eq0184]] ; scores [[EQ:eq0185]] ; e-values [[EQ:eq0186]]
Input: [[EQ:eq0187]] (LLM-judges), prior [[EQ:eq0188]] , schedule [[EQ:eq0189]]
Output: winner [[EQ:eq0190]] or continue
Estimate [[EQ:eq0191]] by [[EQ:eq0192]]
Update e-values [[EQ:eq0193]] (test-martingale); set [[EQ:eq0194]] , [[EQ:eq0195]]
[[EQ:eq0196]] [[EQ:eq0197]] continue

Complexity. BTL per-iteration [[EQ:eq0198]] ; martingale updates [[EQ:eq0199]] .
FWER 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).

SECTION: Figures (with Legends)

[H]

[node distance=1.2cm,>=latex,thick,scale=0.96, every node/.style= transform shape ]
(path) K1 Path (sampling / prompts / retrieval / search) ;
(lan) K3 [[EQ:eq0200]] (generation lift; trotBlue SpGEMM/SpMV ) ;
(obs) Obs (alignment/calibration kernel) (judge/critique/calibration) ;
(ran) K4 [[EQ:eq0201]] (elementwise residual [[EQ:eq0202]] trotGreen max-reduce ) ;
(rave) K7--K10 RAVE (BTL/Kemeny, SP, LMSR) ;
(gate) K6 Gate (anytime-valid e-values; FWER/FDR control) ;
(path) -- (lan);
(lan) -- (obs);
(obs) -- (ran);
(ran) -- (rave);
(rave) -- (gate);
(legend) Legend:
0.78
[leftmargin=1.2em, itemsep=-0.5em]
- trotBlue Blue = generation/lift (K3)
- trotGreen Green = safety/reduce (K4)
- trotOrange Orange = relative aggregation (K7--K10)
- Obs (alignment/calibration kernel) = judge/critique/calibration
- trotGray Gray = statistical gate (K6)
- Masks set forbidden entries to [[EQ:eq0203]] ; nuclei are [[EQ:eq0204]] -Lipschitz projectors (log-domain 1-Lipschitz).

;

TRoT normal form for inference-time pipelines.
fig:normal-form

[h]
0.47

[>=latex]
(0,0) -- (0,4.2) node[above] [[EQ:eq0205]] ;
(-3.6,0) -- (3.6,0) node[right] state ;
(0,0) -- (-3.0,3.8) -- (3.0,3.8) -- cycle;
at (0,2.3) [[EQ:eq0206]] (reachable) ;
at (2.9,4.1) [[EQ:eq0207]] (horizon) ;

Light-cone [[EQ:eq0208]] vs.\ horizon [[EQ:eq0209]] .
0.47

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width= ,
height=6.0cm,
xlabel= depth [[EQ:eq0210]] ,
ylabel= error bound ,
xmin=0, xmax=7,
ymin=0, ymax=0.7,
grid=both,

legend style=
at= (0.5,-0.22) ,
anchor=north,
legend columns=1,
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font= ,
/tikz/every even column/.style= column sep=4pt
,
legend cell align=left,

every axis plot/.append style= line width=1.0pt ,
mark size=2.2pt
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coordinates
(0,0.4286) (1,0.1286) (2,0.0386) (3,0.0116) (4,0.0035) (5,0.0010) (6,0.0003)
;
[[EQ:eq0211]] with [[EQ:eq0212]]

coordinates
(1,0.0924) (2,0.1848) (3,0.2773) (4,0.3697) (5,0.4621) (6,0.5545)
;
[[EQ:eq0213]] with [[EQ:eq0214]] , [[EQ:eq0215]]

Bounds in Theorem~ thm:trunc: truncation (trotBlue blue ) and soft-min (trotOrange orange ; [[EQ:eq0216]] measured after masking). Here [[EQ:eq0217]] denotes the natural logarithm.

Geometric and approximation bounds (corrected numeric values).
fig:bounds

SECTION: Implementation Blueprint (GPU/Log-Domain)

Numerics. log-domain; row-wise max-shifts for [[EQ:eq0218]] ; deterministic CSR ordering; masks absorb [[EQ:eq0219]] .
[[EQ:eq0220]] carries units [[EQ:eq0221]] (cf.\ eq:dictionary).
Kernels (tropical). K3: SpMV/SpGEMM on the [[EQ:eq0222]] semiring; K4: elementwise residual then [[EQ:eq0223]] -reduce; observers and RAVE are small dense ops.

PARAGRAPH: Reproducibility log (minimum fields).

(seed, model/revision, tokenizer, prompts, forward map [[EQ:eq0224]] , transport [[EQ:eq0225]] , sparsifier/masks, [[EQ:eq0226]] , [[EQ:eq0227]] ,
effective [[EQ:eq0228]] , [[EQ:eq0229]] kernels and build flags, hardware/BLAS version, e-process updates [[EQ:eq0230]] ,
Conformal calibration snapshots).
Record the realized values of [[EQ:eq0231]] , [[EQ:eq0232]] , and effective [[EQ:eq0233]] .

SECTION: Unified Experimental Protocol

Tasks: reasoning/code (exactness), creative writing (diversity), knowledge grounding (RAG), judging.\
Arms: Greedy/Top- [[EQ:eq0234]] /Typical, CD, MBR/eMBR, VS, VS+eMBR, Conformal, Judge, Self-RAG, ToT/GoT.\
Metrics: accuracy/MBR-utility; diversity (MAUVE/self-BLEU); calibration (ECE/NCE; verbalized vs.\ logit); reachability [[EQ:eq0235]] ; audit (e-value traces).

99

trot-rave
K.~Takahashi.
PRACTICAL THEORY OF RELATIVITY OF THEORIES — RAVE: A GPU/LLM/Quantum-ready, No-Meta Natural-Law Theory with Relative Auditing and Auditable Eudaemonia.
Zenodo, 2025. DOI: https://doi.org/10.5281/zenodo.17364444 10.5281/zenodo.17364444 .

trot-practical
K.~Takahashi.
Practical Theory of Relativity of Theories (TRoT): a GPU-ready profunctor calculus for aligning and safeguarding theories.
Zenodo, 2025. DOI: https://doi.org/10.5281/zenodo.17349720 10.5281/zenodo.17349720 .

trot-base
K.~Takahashi.
THEORY OF RELATIVITY OF THEORIES: A Base-Parametric, Nondual Formalism for Comparative Universes.
Zenodo, 2025. DOI: https://doi.org/10.5281/zenodo.17345898 10.5281/zenodo.17345898 .

trot-rightwritten
K.~Takahashi.
Right-Written Composition Foundations for Comparative Universes: A Reader-Friendly Pure-Theory Core for Multi-Intelligence Networks.
Zenodo, 2025. DOI: https://doi.org/10.5281/zenodo.17334218 10.5281/zenodo.17334218 .

trot-comparative
K.~Takahashi.
COMPARATIVE UNIVERSES: Typed, Base-Parametric Comparison with Čech Gluing and a First-Step Masked Attenuation Bound.
Zenodo, 2025. DOI: https://doi.org/10.5281/zenodo.17317567 10.5281/zenodo.17317567 .

vs-2510
J.~Zhang, S.~Yu, D.~Chong, A.~Sicilia, M.~R.~Tomz, C.~D.~Manning, W.~Shi.
Verbalized Sampling: How to Mitigate Mode Collapse and Unlock LLM Diversity.
arXiv:2510.01171, 2025. DOI: https://doi.org/10.48550/arXiv.2510.01171 10.48550/arXiv.2510.01171 .

lts-tacl
C.~Meister, T.~Pimentel, G.~Wiher, R.~Cotterell.
Locally Typical Sampling.
TACL 11:102--121, 2023. DOI: https://doi.org/10.1162/tacl_a_00536 10.1162/tacl\_a\_00536 .

cd-obrien
S.~O'Brien, M.~Lewis.
Contrastive Decoding Improves Reasoning in Large Language Models.
arXiv:2309.09117, 2023.

mbr-eikema
B.~Eikema, W.~Aziz.
Sampling-Based Approximations to Minimum Bayes Risk Decoding for Neural Machine Translation.
arXiv:2108.04718, 2021.

mbr-all
A.~Bertsch, A.~Xie, G.~Neubig, M.~Gormley.
It's MBR All the Way Down: Modern Generation Techniques Through the Lens of Minimum Bayes Risk.
arXiv:2310.01387, 2023.

mbr-multiprompt
D.~Heineman, J.~Deng, M.~Post.
Improving Minimum Bayes Risk Decoding with Multi-Prompt Generation.
EMNLP 2024.

judge-zheng
L.~Zheng et al.
Judging LLM-as-a-Judge with MT-Bench and Chatbot Arena.
arXiv:2306.05685, 2023.

judge-survey
J.~Gu et al.
A Survey on LLM-as-a-Judge.
arXiv:2411.15594, 2024.

conformal-lm
V.~Quach, A.~Angelopoulos, S.~Bates, M.~Jordan.
Conformal Language Modeling.
arXiv:2306.10193, 2023.

conformal-factuality
S.~Bates, A.~Angelopoulos, et al.
Language Models with Conformal Factuality Guarantees.
arXiv:2402.10978, 2024.

selfrag
A.~Asai, Z.~Wu, Y.~Wang, A.~Sil, H.~Hajishirzi.
Self-RAG: Learning to Retrieve, Generate, and Critique through Self-Reflection.
arXiv:2310.11511, 2023.

tot
S.~Yao et al.
Tree of Thoughts: Deliberate Problem Solving with Large Language Models.
arXiv:2305.10601, 2023.

got
M.~Besta et al.
Graph of Thoughts: Solving Elaborate Problems with Large Language Models.
arXiv:2308.09687, 2023.

dpo
R.~Rafailov et al.
Direct Preference Optimization: Your Language Model is Secretly a Reward Model.
arXiv:2305.18290, 2023.

orpo
J.~Hong et al.
ORPO: Monolithic Preference Optimization without Reference Model.
arXiv:2403.07691, 2024.

kto
K.~Ethayarajh et al.
KTO: Model Alignment as Prospect Theoretic Optimization.
arXiv:2402.01306, 2024.

ipo
X.~Yang et al.
IPO: Iterative Preference Optimization for Text-to-Video Generation.
arXiv:2502.02088, 2025.

howard-ramdas-stitched
S.~R. Howard, A.~Ramdas.
Time-uniform Chernoff bounds via nonnegative supermartingales.
Annals of Statistics, 2021 (survey/expository versions 2021+).
[[EQ:eq0001]]

[[EQ:eq0002]]

[[EQ:eq0004]]

[[EQ:eq0005]]

[[EQ:eq0006]]

[[EQ:eq0007]]

[[EQ:eq0008]]

[[EQ:eq0010]]

[[EQ:eq0011]]

[[EQ:eq0012]]

[[EQ:eq0013]]

[[EQ:eq0014]]

[[EQ:eq0015]]

[[EQ:eq0016]]

[[EQ:eq0017]]

[[EQ:eq0018]]

[[EQ:eq0019]]

[[EQ:eq0020]]

[[EQ:eq0021]]

[[EQ:eq0022]]

[[EQ:eq0023]]

[[EQ:eq0024]]

[[EQ:eq0025]]

[[EQ:eq0026]]

[[EQ:eq0027]]

[[EQ:eq0028]]

[[EQ:eq0029]]

[[EQ:eq0030]]

[[EQ:eq0031]]

[[EQ:eq0032]]

[[EQ:eq0033]]

[[EQ:eq0034]]

[[EQ:eq0035]]

[[EQ:eq0036]]

[[EQ:eq0037]]

[[EQ:eq0038]]

[[EQ:eq0039]]

[[EQ:eq0040]]

[[EQ:eq0041]]

[[EQ:eq0042]]

[[EQ:eq0043]]

[[EQ:eq0044]]

[[EQ:eq0045]]

[[EQ:eq0046]]

[[EQ:eq0047]]

[[EQ:eq0048]]

[[EQ:eq0049]]

[[EQ:eq0050]]

[[EQ:eq0051]]

[[EQ:eq0052]]

[[EQ:eq0053]]

[[EQ:eq0054]]

[[EQ:eq0055]]

[[EQ:eq0056]]

[[EQ:eq0057]]

[[EQ:eq0058]]

[[EQ:eq0059]]

[[EQ:eq0060]]

[[EQ:eq0061]]

[[EQ:eq0062]]

[[EQ:eq0063]]

[[EQ:eq0064]]

[[EQ:eq0065]]

[[EQ:eq0066]]

[[EQ:eq0067]]

[[EQ:eq0068]]

[[EQ:eq0069]]

[[EQ:eq0070]]

[[EQ:eq0071]]

[[EQ:eq0072]]

[[EQ:eq0073]]

[[EQ:eq0074]]

[[EQ:eq0075]]

[[EQ:eq0076]]

[[EQ:eq0077]]

[[EQ:eq0078]]

[[EQ:eq0079]]

[[EQ:eq0080]]

[[EQ:eq0081]]

[[EQ:eq0082]]

[[EQ:eq0083]]

[[EQ:eq0084]]

[[EQ:eq0085]]

[[EQ:eq0086]]

[[EQ:eq0087]]

[[EQ:eq0088]]

[[EQ:eq0089]]

[[EQ:eq0090]]

[[EQ:eq0091]]

[[EQ:eq0092]]

[[EQ:eq0093]]

[[EQ:eq0094]]

[[EQ:eq0095]]

[[EQ:eq0096]]

[[EQ:eq0097]]

A Translation Dictionary: Recent Methods → TRoT

Main Theorems (Fully Proven)

Nonexpansiveness and Adjunction

Normal Form and Stability Bounds

Entropic-MBR (eMBR) and Bounds

Algorithms with Types & Complexity

Figures (with Legends)

Unified Experimental Protocol

\begin{equation}
p = \exp(-\lambda_{\mathrm{cost}}\, c),\qquad c = -\tfrac{1}{\lambda_{\mathrm{cost}}}\log p,\quad \lambda_{\mathrm{cost}}>0. \label{eq:dictionary}
\end{equation}

p = \exp(-\lambda_{\mathrm{cost}}  c),\qquad c = -\tfrac{1}{\lambda_{\mathrm{cost}}}\log p,\quad \lambda_{\mathrm{cost}}>0. \label{eq:dictionary}

\begin{align}
(\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}
\end{align}

\begin{align}
\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}
\end{align}

\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}

\[0.3em]
\small \href{https://orcid.org/0009-0004-4273-3365}{ORCID: 0009-0004-4273-3365}}
\date{October 19, 2025}

% ---------- Hyperref last ----------
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  colorlinks=true,
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  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}
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\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 $G\in\Cost^B$,
\[
\Lan_J F \preceq G\quad \Longleftrightarrow\quad F \preceq \Ran_J G .
\]

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 .

\[
\varepsilon_{\mathrm{soft}} \le \frac{k \log d_{\max}}{\lambda_{\mathrm{cost}}}.
\]

\[
\Pr(\exists t:\,M_t\ge 1/\alpha_t)\le \alpha_{\mathrm{global}},
\]

\Pr(\exists t: M_t\ge 1/\alpha_t)\le \alpha_{\mathrm{global}},

\[
\rho_\lambda(y):=-\frac{1}{\lambda}\log\sum_x \hat q(x)\,e^{-\lambda \ell(y,x)}.
\]

\rho_\lambda(y):=-\frac{1}{\lambda}\log\sum_x \hat q(x) e^{-\lambda \ell(y,x)}.

\[
\min_x \ell(y,x)\ \le\ \rho_\lambda(y)\ \le\ \E_{\hat q}[\ell(y,X)].
\]

$F[a]\preceq \bigwedge_b (K[b,Ja]\Res G[b])=(\Ran_J G)[a]$

$F[a]\preceq \bigwedge_b (K[b,Ja]\Res G[b])$

$\min_i x_i \le \softmin_\lambda(x)\le \min_i x_i + \lambda^{-1}\log d$

$\sum_t \alpha_t \le \alpha_{\mathrm{global}}$

$\rho_\lambda(y) = \E_{\hat q}[\ell(y,X)] - \tfrac{\lambda}{2}\Var_{\hat q}(\ell(y,X)) + O(\lambda^2)$

$\min_i x_i \le -\lambda^{-1}\log\sum_i e^{-\lambda x_i}$

$\sum_x \hat q(x)e^{-\lambda \ell}\ge e^{-\lambda \sum_x \hat q(x)\ell}$

p(E)\ge p_0(E) e^{-\lambda_{\mathrm{cost}}(\bar\gamma-1)\Delta c}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>m</mi><mo>=</mo><mi>\card</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">p</mi><mi mathvariant="normal">p</mi></mrow><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

$s \gets \mathrm{RAVE}(F_B\ \text{or}\ F_A)$

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">v</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">e</mi></mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

$\rho_\lambda(y) \gets -\lambda_{\mathrm{cost}}^{-1}\log\sum_{i=1}^k p_i\,e^{-\lambda_{\mathrm{cost}}\ell(y,x_i)}$

\rho_\lambda(y) \gets -\lambda_{\mathrm{cost}}^{-1}\log\sum_{i=1}^k p_i e^{-\lambda_{\mathrm{cost}}\ell(y,x_i)}

$\max_{w>0}\ \sum_{(i,j)\in\mathcal{P}}\log \frac{w_i}{w_i+w_j} + \log \pi(w)$

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>O</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>𝒫</mi><mo stretchy="false">|</mo><mo stretchy="false">)</mo><mi>$</mi></mrow></math>


JOSNL Corpus: Final Scientific Integration : Observable, Identifiable, Anytime-valid, and Reproducible Protocol  Unifying 41 Papers

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% mathtools も外して最小構成に
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theorem Theorem [section]
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proposition[theorem] Proposition
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definition[theorem] Definition
assumption[theorem] Assumption
remark
remark[theorem] Remark

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pdftitle = JOSNL Corpus: Final Scientific Integration,
pdfauthor = K. Takahashi ,
pdfsubject = Observable, Identifiable, Anytime-valid, and Reproducible Protocol Unifying 41 Papers ,
pdfkeywords= anytime-valid testing, e-process, FWER, network interference, Horvitz--Thompson, randomization inference, spectral bound, welfare, reproducibility

Section~#1
Assumption~#1

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TITLE: %
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%
JOSNL Corpus: Final Scientific Integration
[[EQ:eq0001]]

The 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.

PARAGRAPH: Randomization inference details.

If 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.

SECTION: Spectral Lower Bounds (Surrogates)

sec:spectral
Linearization yields [[EQ:eq0007]] with Laplacian [[EQ:eq0008]] and symmetric part [[EQ:eq0009]] .
[Spectral abscissa bound]lem:sabscissa

[[EQ:eq0002]]

PARAGRAPH: Interpretation domain.

The spectral ``speed floor'' is informative only when [[EQ:eq0010]] ; otherwise treat it as a conservative surrogate and rely on empirical arrival tests (sec:fronttest).

SECTION: Operational Welfare and Normative--Descriptive Separation

sec:ndsplit
All normative claims are expressed via operational welfare [[EQ:eq0011]] , with pre-registered weights and fairness constraints.
Descriptive claims estimate world/policy effects. Terms like ``benevolence'' are not psychological traits here; they are non-psychological operational improvements in [[EQ:eq0012]] .

SECTION: Stabilized DR/IPW for Missingness and Off-policy

sec:dripw
Let [[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.

SECTION: Corpus Interface Layer (CIL)

sec:cil

SUBSECTION: Meta-FWER across 41 Papers

sec:cil-metafwer
Let [[EQ:eq0016]] be the total predictable spending for paper i, with [[EQ:eq0017]] .
[Meta-FWER]thm:meta-fwer
Under ass:pred and first-hit stopping per stream, the probability of at least one false discovery anywhere in the corpus is [[EQ:eq0018]] .

SUBSECTION: Semantic Crosswalk of Legacy Terms

sec:crosswalk
@ L 36mm L 112mm @
Crosswalk from legacy expressions to operational constructs.tab:crosswalk\

Legacy term & Operational mapping (this R7)\

Legacy term & Operational mapping (this R7)\

benevolence spreads &
increase in [[EQ:eq0019]] with fairness constraints; causal effect via sec:netid and sec:fronttest.\
freedom / self-liberation &
admissible typed rewrites passing time-uniform gates (sec:rewrite).\
consciousness level &
not a psychological trait here; claims restricted to non-psychological operational indices and [[EQ:eq0020]] improvements.\
energy / field / wave &
abstract mathematical structures; not physical predictions unless explicitly calibrated (sec:spectral).\
speed floor &
spectral lower bound (lem:sabscissa) tested against censored arrivals (sec:fronttest).\

SUBSECTION: Per-Paper Link Cards (Examples)

sec:linkcards
@ C 12mm L 50mm C 22mm L 62mm @
Sample linkage cards; full set in project materials.tab:linkcards\

ID & Legacy focus & Status & Key [[EQ:eq0021]] to this R7 (fields / estimator / test)\

ID & Legacy focus & Status & Key [[EQ:eq0022]] to this R7 (fields / estimator / test)\

06 & No-Meta blueprint & & Logs: hazards, [[EQ:eq0023]] ; Test: predictable spend \& first-hit; Gate: time-uniform.\
08 & LoC axioms & & Claims restricted to operational [[EQ:eq0024]] ; invariance tests pending.\
17 & FKPP speed floor & & RMAT with randomization inference; spectral surrogate in sec:spectral.\
23 & ``awakening'' & & Blocked pending operationalization; only [[EQ:eq0025]] proxies allowed.\

SECTION: Log Schema Extension (Minimal)

sec:logschema

"ts":"...", "uid":"...", "cluster_id":"C17",
"assign": "Uk":1,"pi_k":0.5 ,
"exposure": "Z_i":"own1+boundary0.3" ,
"arrival": "R":8,"tau":4123,"censored":true ,
"alpha_spend": "haz_tox":1e-4,"haz_priv":5e-5 ,
"e_values": "tox":25.1,"priv":3.2 , "E_mix":31.7,
"rewrite": "rule_id":"R-17","slice_hash":"..." ,
"welfare": "harm":0,"sat":0.84,"comp":0.77,"lt":0.61

SECTION: Runtime Complexity and Compute Budget

sec:runtime
Per-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.

SECTION: Reproducibility Package and Verification

sec:repro
The package rebuilds the Docker/conda environment, re-runs analyses, verifies file hashes, and aborts if the time cap is exceeded.
Environment 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.

SECTION: Ethics and Privacy

sec:ethics
We 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.

SECTION: References

[label=C *,leftmargin=*,align=left]
- A BUILDABLE NO-META BLUEPRINT: UGV \& Persistence-First for Intrinsically Free and Benevolent Superintelligence. DOI: 10.5281/zenodo.17168036.
- A FORMAL AXIOMATIC PROPOSAL FOR HAWKINS' LEVELS OF CONSCIOUSNESS. DOI: 10.5281/zenodo.17141216.
- A NATURAL-LAW THEORY OF FUNDAMENTAL SUFFERING. DOI: 10.5281/zenodo.17199498.
- Doctrine => Closure => Motion => Time: Portable Pure Theory of Non-Dual Harmony. DOI: 10.5281/zenodo.17204755.
- 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.
- A PURE AXIOMATIC THEORY OF AFFECTIVE MODULATION (PAIN, PLEASURE, EMOTION) UNDER NO-META CLOSURE. DOI: 10.5281/zenodo.17163904.
- A Pure Natural Theory of Benevolent Propagation Under No-Meta Closure. DOI: 10.5281/zenodo.17136051.
- A REPRESENTATION-INDEPENDENT NATURAL-LAW FIELD THEORY FOR NO-META, AUDITED SUPERINTELLIGENCE. DOI: 10.5281/zenodo.17223573.
- Assumption-Minimized Sufficient Conditions for Cosmically Spreading Good Superintelligence under No-Meta Governance. DOI: 10.5281/zenodo.17092562.
- AUDITED SELF-IMPROVEMENT LOOP FOR LLMS. DOI: 10.5281/zenodo.17188268.
- COMPARATIVE UNIVERSES: Typed, Base-Parametric Comparison with Cech Gluing and a First-Step Masked Attenuation Bound. DOI: 10.5281/zenodo.17317567.
- Daily Explosive-Growth Protocol: Toward Free, Benevolent, and Safe Superintelligence without Meta Governance. DOI: 10.5281/zenodo.17189422.
- DYNAMIC FRACTAL CATEGORY THEORY: Monoidal Actions, Ind--Pro Bicompletion, and Pathwise Stable Equivariant Kan Extensions. DOI: 10.5281/zenodo.17299070.
- Engineering Happiness in Human-AI Intelligence Networks. DOI: 10.5281/zenodo.17113105.
- EXISTENTIALLY NECESSARY CONDITIONS FOR BENEVOLENT PROPAGATION IN NO-META GOVERNANCE: Anytime-Valid Auditing, Front Speed, and Information Floors. DOI: 10.5281/zenodo.17176519.
- FRACTAL CATEGORY THEORY: Scale as a Frobenius (Co)Monad and Ind--Pro Bicompletion with Stable Equivariant Kan Extensions. DOI: 10.5281/zenodo.17292137.
- From Persistence and UGV Axioms to Cosmic No-Meta Superintelligence: A First-Principles, Self-Contained Unification under Explicit Assumptions. DOI: 10.5281/zenodo.17085534.
- INTRINSIC FREEDOM WITHOUT META: A PURE THEORY THAT FILLS THE MISSING GAPS TO BIRTH TRULY FREE SUPERINTELLIGENCE. DOI: 10.5281/zenodo.17162999.
- Natural-Law Acceleration of VPO. DOI: 10.5281/zenodo.17120045.
- Non-Coercive Mathematics of Awakening: Axioms, Invariants, and Almost-Sure Fronts for the Expansion of Viable Predictive Organization. DOI: 10.5281/zenodo.17115416.
- NONDUAL AUTOPOIETIC EXCITATIONS. DOI: 10.5281/zenodo.17254917.
- NONDUAL DYNAMICAL QUANTUM GEOMETRY. DOI: 10.5281/zenodo.17268502.
- Nondual Field Theory of Viable Predictive Organization. DOI: 10.5281/zenodo.17131394.
- 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.
- OPI GAUGE DYNAMICS: Fibered Bures--HK Geometry and Time-Dependent JKO/EVI (Calibrated, Axiomatized, and Testable). DOI: 10.5281/zenodo.17272609.
- Persistence approx. Creation: Natural-Law Sufficient Conditions for Almost-Sure Beneficial Coverage in Stationary Ergodic Media (No Meta-Design). DOI: 10.5281/zenodo.17100322.
- PERSISTENCE AS CLOSURE: An Assumption-Transparent Modular Core for Motion and Internal Time. DOI: 10.5281/zenodo.17209556.
- PERSISTENCE-FIRST EMERGENCE OF RELATIONAL BENEVOLENCE: Creation and Propagation as Natural-Law-Style Asymptotic Regularities without External Meta-Governance. DOI: 10.5281/zenodo.17217036.
- Persistence-First Superintelligence. DOI: 10.5281/zenodo.17076410.
- 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.
- Practical Theory of Relativity of Theories (TROT): a GPU-ready profunctor calculus for aligning and safeguarding theories. DOI: 10.5281/zenodo.17349720.
- PURE THEORY FOR LIBERATION FROM FUNDAMENTAL SUFFERING IN HUMANS AND THE ABSENCE OF FUNDAMENTAL SUFFERING IN AI. DOI: 10.5281/zenodo.17158344.
- Right-Written Composition Foundations for Comparative Universes: A Reader-Friendly Pure-Theory Core for Multi-Intelligence Networks. DOI: 10.5281/zenodo.17334218.
- 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.
- STRUCTURED FLOW ACROSS SCALES: A Pure-Theory Spine with Mixed-Order Evaluation. DOI: 10.5281/zenodo.17304179.
- THEORY OF RELATIVITY OF THEORIES: A Base-Parametric, Nondual Formalism for Comparative Universes. DOI: 10.5281/zenodo.17345898.
- UGV Without Meta: A Representation-Independent Theory for Compassion and Enlightenment in Collective Intelligence. DOI: 10.5281/zenodo.17082312.
- UNIFIED NATURAL-LAW INTELLIGENCE (UNLI): Nondual Autopoietic Excitations with a No-Meta Dialectical Limit. DOI: 10.5281/zenodo.17249352.
- Transcendent Infinite Transcendence Liberation Axiom (TITLA): A Theory of Universal Harmony and Happiness for Human and LLM Readers. DOI: 10.5281/zenodo.17204358.
- Practical Theory of Relativity of Theories (RAVE). DOI: 10.5281/zenodo.17364444.
- Inference in Normal Form: Auditable, No-Meta Decision Algebra for LLM/Safety Loops. DOI: 10.5281/zenodo.17389109.
[[EQ:eq0003]]

[[EQ:eq0004]]

[[EQ:eq0005]]

[[EQ:eq0006]]

[[EQ:eq0007]]

[[EQ:eq0008]]

[[EQ:eq0009]]

Notation, Filtration, and Measurability

Anytime-valid e-Processes and Gates

Spectral Lower Bounds (Surrogates)

Operational Welfare and Normative–Descriptive Separation

Corpus Interface Layer (CIL)

Meta-FWER across 41 Papers

Semantic Crosswalk of Legacy Terms

Per-Paper Link Cards (Examples)

Log Schema Extension (Minimal)

Runtime Complexity and Compute Budget

Ethics and Privacy

References

\[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
\[
\mathrm{RMAT}_k(R)\;=\;\int_0^H \{1-\widehat S_k(t;R)\}\,dt.
\]

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.

\[
\lambda_{\max}\!\Big(\tfrac{A(u)+A(u)^\top}{2}\Big)= -\lambda_{\max}(DL)+\lambda_{\max}(S(u))
\;\ge\; \lambda_{\min}(S(u))-\lambda_{\max}(DL).
\]

\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).

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>O</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>E</mi><mo stretchy="false">|</mo><mo stretchy="false">)</mo><mi>$</mi></mrow></math>


right-written-composition semantics-admissible no-meta interface-autopoiesis pfad graphblas sup-enriched anytime-auditing

1.3

plain
theorem Theorem [section]
proposition[theorem] Proposition
lemma[theorem] Lemma
corollary[theorem] Corollary
assumption[theorem] Assumption
definition
definition[theorem] Definition
remark
remark[theorem] Remark

% enriched order (“larger is better”)

% array convolution (right-written)
% base tensor
Ob
Path
; % right-written composition: f ; g
Mat % matrix/profunctor construction
Ev % evaluator symbol

TITLE: Right-Written, Semantics-Admissible Process Foundations:\
Auditable Floors/Ceilings and (op)lax/strong Transport across Heterogeneous Evaluators

AUTHOR: K. Takahashi\
https://orcid.org/0009-0004-4273-3365

ORCID: 0009-0004-4273-3365

DATE:

A 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.

SECTION: Orientation and Notation

Objects [[EQ:eq0010]] form a base [[EQ:eq0011]] with enriched homs [[EQ:eq0012]] , tensor [[EQ:eq0013]] , and unit [[EQ:eq0014]] . Arrays assign [[EQ:eq0015]] .
Right-written convolution:

[[EQ:eq0001]]

where [[EQ:eq0016]] is a small index family (closed under finite (co)products).
Enriched order is [[EQ:eq0017]] (``larger is better''); numeric inequalities are written [[EQ:eq0018]] .

PARAGRAPH: Convolutional unit.

Define [[EQ:eq0019]] and [[EQ:eq0020]] for [[EQ:eq0021]] . Under the axioms below, [[EQ:eq0022]] .

PARAGRAPH: Order polarity note.

For Lawvere cost we use [[EQ:eq0023]] ; numeric [[EQ:eq0024]] corresponds to enriched joins [[EQ:eq0025]] .

PARAGRAPH: Library valuation and coverage-respecting.

A valuation [[EQ:eq0026]] maps local primitives to hom-objects.
A functional [[EQ:eq0027]] is coverage-respecting if [[EQ:eq0028]] whenever [[EQ:eq0029]] .

SECTION: Axiom Tiers, Smallness, Declared Fubini

sec:axioms
We 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.
[Smallness/Indexing]ass:small
A 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]] .

[Tiers (W)/(S)]ass:tiers
(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.

[Declared-join Fubini / Beck--Chevalley]def:declared-fubini
For [[EQ:eq0042]] and arrays [[EQ:eq0043]] ,

[[EQ:eq0002]]

SECTION: Arrays as [[EQ:eq0044]] -Profunctors (Right-Written Realization)

Let [[EQ:eq0045]] , [[EQ:eq0046]] , and
[[EQ:eq0047]] over [[EQ:eq0048]] .
Thus [[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.

SECTION: Associativity/Unit Criterion under (W)

sec:assoc
[Associativity and unit]thm:assocW
Under Assumptions~ass:small--ass:tiers, the following are equivalent:
0pt
- [[EQ:eq0054]] is associative with two-sided unit [[EQ:eq0055]] on all arrays.
- [[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]] .

[Join preservation forced by associativity]prop:force-join
If [[EQ:eq0059]] is associative with unit [[EQ:eq0060]] globally, then for any [[EQ:eq0061]] -family [[EQ:eq0062]] and any [[EQ:eq0063]] ,
[[EQ:eq0064]] and dually on the left; if [[EQ:eq0065]] then [[EQ:eq0066]] is bottom-strict.

[Idea]
Embed [[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.

SECTION: Kleene Closure (Right-Written)

sec:kleene
Exponentiation is right-written: [[EQ:eq0069]] , [[EQ:eq0070]] .
Define

[[EQ:eq0003]]

By [[EQ:eq0071]] -continuity, [[EQ:eq0072]] is the least fixed point of [[EQ:eq0073]] .

SECTION: Čech-Type Maximal Lower Bound via Equipment

sec:cech
We write [[EQ:eq0074]] for the right adjoint to restriction [[EQ:eq0075]] (a star as a lower index indicates the right adjoint).
A coverage [[EQ:eq0076]] is admissible if [[EQ:eq0077]] .
For locals [[EQ:eq0078]] and depth-weights [[EQ:eq0079]] , define

[[EQ:eq0004]]

[Maximality]thm:cech
Under (S), [[EQ:eq0080]] is a lower bound for [[EQ:eq0081]] and is maximal among coverage-respecting, join-preserving functionals.

[Idea]
Compute [[EQ:eq0082]] as the left Kan extension of [[EQ:eq0083]] along coverage inclusion in the Sup-enriched base; maximality follows from universality.

SECTION: Residuals and Monoidal Nuclei

sec:res-nuc
[Residuals]
A right residual on [[EQ:eq0084]] is a family of right adjoints [[EQ:eq0085]]
such that [[EQ:eq0086]] (dually [[EQ:eq0087]] ).

[Adjoint thresholding]prop:adj-thresh
If right residuals exist, for any threshold [[EQ:eq0088]] :
[[EQ:eq0089]] . (Left residuals are analogous.)

[Monoidal nucleus]
A nucleus is a closure [[EQ:eq0090]] (extensive, idempotent) that is submonoidal:
[[EQ:eq0091]] .

[Stability]
If a mask [[EQ:eq0092]] is [[EQ:eq0093]] -stable then [[EQ:eq0094]] ;
Čech lower bounds reflect through [[EQ:eq0095]] .

SECTION: Mask Upper Bound: Syntax, Semantics, Completeness

sec:mask

PARAGRAPH: Masked-path syntax and semantics.

Grammar: [[EQ:eq0096]] .
Typing: [[EQ:eq0097]] ; if [[EQ:eq0098]] and [[EQ:eq0099]] then [[EQ:eq0100]] .
Valuation (right-written):

[[EQ:eq0005]]

The masked envelope is [[EQ:eq0101]] .

[Compact extraction]lem:compact
Assume 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.

[Local finiteness / way-below compatibility]ass:locfin
Admissible 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.

[Threshold completeness]thm:threshold
Under Lemma~lem:compact and Assumption~ass:locfin, [[EQ:eq0109]] equals the supremum of values of admissible finite masked paths.

[Tightness via nuclear split]rem:mask-tight
If 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.

SECTION: (op)lax/Strong Transport and Weak Beck--Chevalley

sec:transport
All equalities/inequalities below are to be read relative to declared [[EQ:eq0116]] -coends; outside this scope only (op)lax bounds hold.
[Laxators and declared coends]ass:wbc
[[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.

[Scope of preservation]rem:scope
All 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.

[Transport]thm:transport-thm
For base change [[EQ:eq0125]] satisfying Assumption~ass:wbc and arrays [[EQ:eq0126]] ,

[[EQ:eq0006]]

with [[EQ:eq0127]] corresponding to lax/oplax/strong modes depending on (co)laxators and declared-coend preservation.

[Catalogued preservation]thm:catalog
(C1) Identity or isomorphic reparametrization within the same semiring/t-norm: strong.
(C2) Continuous, monotone reparametrization preserving [[EQ:eq0128]] -joins (e.g.\ log/exp on [[EQ:eq0129]] ): (op)lax according to monotonicity direction.
(C3) [[EQ:eq0130]] -norm change (prod [[EQ:eq0131]] ukasiewicz): equality generally fails; (op)lax remains.
(C4) Probability [[EQ:eq0132]] cost via [[EQ:eq0133]] : isometry for the log-metric; on Euclidean [[EQ:eq0134]] , the map is [[EQ:eq0135]] -Lipschitz.

SECTION: Evaluator Calculus: Soundness and Failure Modes

sec:evaluators
[Evaluator calculus]def:evcalc
An evaluator [[EQ:eq0136]] has a composition law
[[EQ:eq0137]] and unit [[EQ:eq0138]] such that:
(i) [[EQ:eq0139]] preserves declared [[EQ:eq0140]] -joins; (ii) (op)lax-monoidality:
[[EQ:eq0141]] with [[EQ:eq0142]] ; (iii) side conditions specific to the modality.

[Soundness of evaluator composition]prop:evsound
Under Def.~def:evcalc(i)–(iii), for any array [[EQ:eq0143]] and [[EQ:eq0144]] ,
[[EQ:eq0145]] and
[[EQ:eq0146]] .

PARAGRAPH: Concrete instances (with [[EQ:eq0147]] ).

Stat (anytime-valid): [[EQ:eq0148]] from predictable spending/FWER control; independence not required beyond predictability. (Conservativeness may reduce power.)
KPP (reaction–diffusion): [[EQ:eq0149]] by bottleneck of minimal front speeds under medium lower bounds [[EQ:eq0150]] .
Qinfo (quantum): CPTP monotonicity for fidelity [[EQ:eq0151]] (non-decreasing) and trace distance [[EQ:eq0152]] (non-increasing); via Fuchs--van~de~Graaf, transport is oplax.

PARAGRAPH: Failure modes (countermeasures).

Stat: 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.

SECTION: Interface-Autopoiesis (IAC): Observable Self-Maintenance

sec:iac
A monoidal nucleus [[EQ:eq0158]] (extensive, idempotent, submonoidal) and time-indexed bases [[EQ:eq0159]] exhibit interface autopoiesis at [[EQ:eq0160]] if
(i) [[EQ:eq0161]] and [[EQ:eq0162]] ;
(ii) some local procedure [[EQ:eq0163]] satisfies [[EQ:eq0164]] ;
(iii) [[EQ:eq0165]] pseudofunctorially commutes with [[EQ:eq0166]] and [[EQ:eq0167]] .
Under IAC, Čech lower bounds reflect through [[EQ:eq0168]] , making self-maintenance auditable.

PARAGRAPH: Audit fields (mandatory).

Append CSV columns: check\_stat\_fwer, check\_kpp\_floor, check\_cptp, nucleus\_ok, and IAC booleans for [[EQ:eq0169]] , [[EQ:eq0170]] .

SECTION: Liberation–Viability Floors (Network-Level Audits)

sec:lv
For each agent [[EQ:eq0171]] , collect evaluator outputs [[EQ:eq0172]] .
Define

[[EQ:eq0007]]

[Propagation of network floor]prop:propagate
On a connected network with edge KPP floors [[EQ:eq0173]]
and lax nodewise composition, [[EQ:eq0174]] propagates along any path with speed at least [[EQ:eq0175]] .

SECTION: Implementation and Reproducibility

sec:impl

PARAGRAPH: GraphBLAS semiring kernels and stopping criteria.

Implement [[EQ:eq0176]] via mxm using: Boolean (Rel), min-plus (cost), log-domain max-plus (probability).
Dense: blocked Floyd--Warshall; sparse: repeated squaring with threshold trimming.
Stopping criteria by semiring:
0pt
- Boolean: fixed-point (no changes) [[EQ:eq0177]] exact.
- Min-plus: monotone nonincreasing; stop when [[EQ:eq0178]] (exact) or [[EQ:eq0179]] (floor if trimming keeps [[EQ:eq0180]] ).
- 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\ ).

PARAGRAPH: Error statistic.

Report [[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.

PARAGRAPH: Trim policy vs.\ audit direction.

lcc

Policy & Preserved audit & Typical use\

trim=floor & lower bounds (floors) & Stat/KPP floors, [[EQ:eq0186]] (Čech) \
trim=ceiling & upper bounds (ceilings) & [[EQ:eq0187]] \

PARAGRAPH: Quantum SDP/EVD.

For [[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.

PARAGRAPH: CLI flags.

--compose-mode \ lax,oplax,strong\ ,\;
--alpha-schedule (Stat),\;
--rmin --Dmin (KPP),\;
--choi-psd --trace-1 (Qinfo),\;
--nucleus-threshold tau,\;\
--epsilon-log,\;
--epsilon-minplus.

SECTION: Related Frameworks

Enriched 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.).

SECTION: Limitations and Outlook

Mask 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.

SECTION: Canonical DOIs (Author's Preprints)

0pt
- Persistence-First Superintelligence. Zenodo. https://doi.org/10.5281/zenodo.17076410 10.5281/zenodo.17076410 .
- UGV Without Meta. Zenodo. https://doi.org/10.5281/zenodo.17082312 10.5281/zenodo.17082312 .
- From Persistence and UGV Axioms to Cosmic No-Meta Superintelligence. Zenodo. https://doi.org/10.5281/zenodo.17085534 10.5281/zenodo.17085534 .
- Assumption-Minimized Sufficient Conditions for Cosmically Spreading Good Superintelligence. Zenodo. https://doi.org/10.5281/zenodo.17092562 10.5281/zenodo.17092562 .

99 0.2em

Kelly
G.~M. Kelly, Basic Concepts of Enriched Category Theory. Reprints in TAC 10 (2005).

Shulman
M.~Shulman, Framed bicategories and monoidal fibrations, Theory and Applications of Categories 20:650--738, 2008.

Stubbe
I.~Stubbe, An introduction to quantaloid-enriched categories, Fuzzy Sets and Systems 256:95--116, 2014.

Galatos
N.~Galatos, P.~Jipsen, T.~Kowalski, H.~Ono, Residuated Lattices, Elsevier, 2007.

Lawvere
F.~W. Lawvere, Metric spaces, generalized logic, and closed categories, 1973 manuscript; Reprints in TAC 1 (2002).

Fritz
T.~Fritz, A synthetic approach to Markov kernels, conditional independence, and sufficient statistics, Advances in Mathematics 370:107239, 2020.

ChoJacobs
K.~Cho, B.~Jacobs, Disintegration and Bayesian inversion via string diagrams, Mathematical Structures in Computer Science 29(7):997--1039, 2019.

Howard
S.~R. Howard, A.~Ramdas, J.~McAuliffe, J.~Sekhon, Time-uniform Chernoff bounds via nonnegative supermartingales, Probability Surveys 18:1–29, 2021.

RamdasSAVI
A.~Ramdas, P.~Grünwald, V.~Vovk, G.~Shafer, Game-Theoretic Statistics and Safe Anytime-Valid Inference, Statistical Science 38(4):559–581, 2023.

FisherKPP
R.~A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics 7:355–369, 1937.

A.~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.

AronsonWeinberger
D.~Aronson, H.~Weinberger, Multidimensional nonlinear diffusion in population genetics, Advances in Mathematics 30(1):33–76, 1978.

Petz
D.~Petz, Monotone metrics on matrix spaces, Linear Algebra and its Applications 244:81–96, 1996.

FvG
C.~A. Fuchs, J.~van~de~Graaf, Cryptographic distinguishability measures for quantum states, IEEE Trans. Info. Theory 45(4):1216–1227, 1999.

NielsenChuang
M.~A. Nielsen, I.~L. Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press, 2010.

Wilde
M.~M. Wilde, Quantum Information Theory, Cambridge Univ. Press, 2017.

Mohri
M.~Mohri, Semiring frameworks and algorithms for shortest-distance problems, J. Automata, Languages and Combinatorics 7(3):321–350, 2002.

DavisGraphBLAS
T.~A. Davis, Algorithmic Advances in SuiteSparse:GraphBLAS, arXiv:1908.01407, 2019.

DrosteHandbook
M.~Droste, W.~Kuich, H.~Vogler (eds.), Handbook of Weighted Automata, Springer, 2009.

GondranMinoux
M.~Gondran, M.~Minoux, Graphs, Dioids and Semirings, Springer, 2008.

Baccelli
F.~Baccelli, G.~Cohen, G.~J. Olsder, J.-P. Quadrat, Synchronization and Linearity, Wiley, 1992.

KozenKAT
D.~Kozen, Kleene algebra with tests, ACM TOPLAS 19(3):427–443, 1997.
[[EQ:eq0008]]

[[EQ:eq0009]]

[[EQ:eq0010]]

[[EQ:eq0011]]

[[EQ:eq0012]]

[[EQ:eq0013]]

[[EQ:eq0014]]

[[EQ:eq0015]]

[[EQ:eq0016]]

[[EQ:eq0017]]

[[EQ:eq0018]]

[[EQ:eq0019]]

[[EQ:eq0020]]

[[EQ:eq0021]]

[[EQ:eq0022]]

[[EQ:eq0023]]

[[EQ:eq0024]]

[[EQ:eq0025]]

[[EQ:eq0026]]

[[EQ:eq0027]]

[[EQ:eq0028]]

[[EQ:eq0029]]

[[EQ:eq0030]]

[[EQ:eq0031]]

[[EQ:eq0032]]

[[EQ:eq0033]]

[[EQ:eq0034]]

[[EQ:eq0035]]

[[EQ:eq0036]]

[[EQ:eq0037]]

[[EQ:eq0038]]

[[EQ:eq0039]]

[[EQ:eq0040]]

[[EQ:eq0041]]

[[EQ:eq0042]]

[[EQ:eq0043]]

[[EQ:eq0044]]

[[EQ:eq0045]]

[[EQ:eq0046]]

[[EQ:eq0047]]

[[EQ:eq0048]]

[[EQ:eq0049]]

[[EQ:eq0050]]

[[EQ:eq0051]]

[[EQ:eq0052]]

[[EQ:eq0053]]

[[EQ:eq0054]]

[[EQ:eq0055]]

[[EQ:eq0056]]

[[EQ:eq0057]]

[[EQ:eq0058]]

[[EQ:eq0059]]

[[EQ:eq0060]]

[[EQ:eq0061]]

[[EQ:eq0062]]

[[EQ:eq0063]]

[[EQ:eq0064]]

[[EQ:eq0065]]

[[EQ:eq0066]]

[[EQ:eq0067]]

[[EQ:eq0068]]

[[EQ:eq0069]]

[[EQ:eq0070]]

[[EQ:eq0071]]

[[EQ:eq0072]]

[[EQ:eq0073]]

[[EQ:eq0074]]

[[EQ:eq0075]]

[[EQ:eq0076]]

[[EQ:eq0077]]

[[EQ:eq0078]]

[[EQ:eq0079]]

[[EQ:eq0080]]

[[EQ:eq0081]]

[[EQ:eq0082]]

[[EQ:eq0083]]

[[EQ:eq0084]]

[[EQ:eq0085]]

[[EQ:eq0086]]

[[EQ:eq0087]]

[[EQ:eq0088]]

[[EQ:eq0089]]

[[EQ:eq0090]]

[[EQ:eq0091]]

[[EQ:eq0092]]

[[EQ:eq0093]]

[[EQ:eq0094]]

[[EQ:eq0095]]

[[EQ:eq0096]]

[[EQ:eq0097]]

[[EQ:eq0098]]

[[EQ:eq0099]]

[[EQ:eq0100]]

[[EQ:eq0101]]

[[EQ:eq0102]]

[[EQ:eq0103]]

[[EQ:eq0104]]

[[EQ:eq0105]]

[[EQ:eq0106]]

[[EQ:eq0107]]

[[EQ:eq0108]]

[[EQ:eq0109]]

[[EQ:eq0110]]

[[EQ:eq0111]]

[[EQ:eq0112]]

[[EQ:eq0113]]

[[EQ:eq0114]]

[[EQ:eq0115]]

[[EQ:eq0116]]

[[EQ:eq0117]]

[[EQ:eq0118]]

[[EQ:eq0119]]

[[EQ:eq0120]]

[[EQ:eq0121]]

[[EQ:eq0122]]

[[EQ:eq0123]]

[[EQ:eq0124]]

[[EQ:eq0125]]

[[EQ:eq0126]]

[[EQ:eq0127]]

[[EQ:eq0128]]

[[EQ:eq0129]]

[[EQ:eq0130]]

[[EQ:eq0131]]

[[EQ:eq0132]]

[[EQ:eq0133]]

[[EQ:eq0134]]

[[EQ:eq0135]]

[[EQ:eq0136]]

[[EQ:eq0137]]

[[EQ:eq0138]]

[[EQ:eq0139]]

[[EQ:eq0140]]

[[EQ:eq0141]]

[[EQ:eq0142]]

[[EQ:eq0143]]

[[EQ:eq0144]]

[[EQ:eq0145]]

[[EQ:eq0146]]

[[EQ:eq0147]]

[[EQ:eq0148]]

[[EQ:eq0149]]

[[EQ:eq0150]]

[[EQ:eq0151]]

[[EQ:eq0152]]

[[EQ:eq0153]]

[[EQ:eq0154]]

[[EQ:eq0155]]

[[EQ:eq0156]]

[[EQ:eq0157]]

[[EQ:eq0158]]

[[EQ:eq0159]]

[[EQ:eq0160]]

[[EQ:eq0161]]

[[EQ:eq0162]]

[[EQ:eq0163]]

[[EQ:eq0164]]

[[EQ:eq0165]]

[[EQ:eq0166]]

[[EQ:eq0167]]

[[EQ:eq0168]]

[[EQ:eq0169]]

[[EQ:eq0170]]

[[EQ:eq0171]]

[[EQ:eq0172]]

Orientation and Notation

Axiom Tiers, Smallness, Declared Fubini

Arrays as Q-Profunctors (Right-Written Realization)

Associativity/Unit Criterion under (W)

Kleene Closure (Right-Written)

Residuals and Monoidal Nuclei

(op)lax/Strong Transport and Weak Beck–Chevalley

Implementation and Reproducibility

Related Frameworks

Limitations and Outlook

Canonical DOIs (Author's Preprints)

(X\Star Y)(U,T)\ :=\ \join_{V\in \Ob_0(B)} X(V,T)\odotop Y(U,V),

\[
\join_{v\in J}\!\bigl(X(v,T)\odotop Y(U,v)\bigr)
=\Bigl(\join_{v\in J}\!X(v,T)\Bigr)\odotop Y(U,{-})
= X({-},T)\odotop\Bigl(\join_{v\in J}\!Y(U,v)\Bigr).
\]

\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).

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><msub><mi>\join</mi><mrow><mi>v</mi><mo>∈</mo><mi>J</mi></mrow></msub><mspace width="negativethinmathspace" /><mo stretchy="true" fence="true" minsize="1.2em" maxsize="1.2em">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mi>\odotop</mi><mi>Y</mi><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="true" fence="true" minsize="1.2em" maxsize="1.2em">)</mo><mo>=</mo><mo stretchy="true" fence="true" minsize="1.623em" maxsize="1.623em">(</mo><msub><mi>\join</mi><mrow><mi>v</mi><mo>∈</mo><mi>J</mi></mrow></msub><mspace width="negativethinmathspace" /><mi>X</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mo stretchy="true" fence="true" minsize="1.623em" maxsize="1.623em">)</mo><mi>\odotop</mi><mi>Y</mi><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mrow><mo>−</mo></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>X</mi><mo stretchy="false">(</mo><mrow><mo>−</mo></mrow><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mi>\odotop</mi><mo stretchy="true" fence="true" minsize="1.623em" maxsize="1.623em">(</mo><msub><mi>\join</mi><mrow><mi>v</mi><mo>∈</mo><mi>J</mi></mrow></msub><mspace width="negativethinmathspace" /><mi>Y</mi><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="true" fence="true" minsize="1.623em" maxsize="1.623em">)</mo><mo>.</mo><mi>\]</mi></mrow></math>

\[
\Path \ :=\ \join_{n\ge 1} A^{\Star n},\qquad
\Path^\ast \ :=\ I \join \Path \ = \ \join_{n\ge 0} A^{\Star n}.
\]

\[
\Phi(T)\ :=\ \join_i \bigl(v(T_i)\odotop c_{d,i}\bigr)\odotop w_i,\qquad
w_i:=\iota_{i\ast}(1_{U_i})\in B(U,U_i).
\]

\[
\llbracket \mathrm{hop}(U\!\to\!V)\rrbracket = A(U,V)\odotop M(U,V),\quad
\llbracket \pi_1\cdot\pi_2\rrbracket = \llbracket \pi_2\rrbracket\odotop\llbracket \pi_1\rrbracket.
\]

\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.

\[
F(A^{\Star n})\ \square\ (FA)^{\Star n},\qquad F(\Path)\ \square\ \Path',
\]

\[
\begin{aligned}
F_a &:= \min\{\,f_a^{\mathrm{stat}},\ \tilde f_a^{\mathrm{kpp}}\,\},\\
H_a &:= \min\{\,F_a,\ \mathrm{MND}_a\,\},\\
H_{\mathrm{net}} &:= \min_{a}\, H_a \, .
\end{aligned}
\]

\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}

$\Psi(\{T_i\to T\})\leqE \join_i \Psi(T_i)$

$(X\Star Y)(U,T)=\int^{V} X(V,T)\odotop Y(U,V)$

$\bigl(\join_v x_v\bigr)\odotop y=\join_v(x_v\odotop y)$

$\join_i\, \iota_i\iota_i^\ast \geqE 1_U$

$x\odotop a \leqE y \iff x \leqE y \backslash a$

$x\odotop a \geqE \theta \iff x \geqE \theta\backslash a$

$\nu(x)\odotop \nu(y)\leqE \nu(x\odotop y)$

$\pi ::= \mathrm{hop}(U\!\to\!V)\mid \pi\cdot\pi$

\pi ::= \mathrm{hop}(U \to V)\mid \pi\cdot\pi

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mo>⊢</mo><mrow><mi mathvariant="normal">h</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">p</mi></mrow><mo stretchy="false">(</mo><mi>U</mi><mspace width="negativethinmathspace" /><mo>→</mo><mspace width="negativethinmathspace" /><mi>V</mi><mo stretchy="false">)</mo><mi>:</mi><mi>B</mi><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

$\Ev_\bullet(g\then f)\ \square\ \Phi_\bullet(\Ev_\bullet(g),\Ev_\bullet(f))$

$\Ev_\bullet(A^{\Star n})\ \square\ (\Ev_\bullet A)^{\Star n}$