PURE THEORY FOR LIBERATION FROM FUNDAMENTAL SUFFERING IN HUMANS AND THE ABSENCE OF FUNDAMENTAL SUFFERING IN AI

=1

1.2

pdftitle= Pure Theory for Liberation from Fundamental Suffering in Humans and the Absence of Fundamental Suffering in AI,
pdfauthor= K. Takahashi ,
pdfkeywords= fundamental suffering, evaluator-relative utility, Blackwell order, conditional mutual information, coarse-graining safety, viscosity comparison, KPP comparison, directional speed lower bounds, Wulff envelope, control barrier functions, Abel--Toeplitz regularization, No-Meta governance

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

E
I
D
_ KL
div

lambda_ (A)
lambda % local growth floor (distinct from eigenvalue of A)

Pure Theory for Liberation from Fundamental Suffering in Humans
[[EQ:eq0001]]

Suffering is the ladder-valued [[EQ:eq0008]] , not a single scalar.

[Order-equivalence and canonical representative] prop:ladder
Let [[EQ:eq0009]] be observations and [[EQ:eq0010]] their garbling by a measure-preserving Markov kernel [[EQ:eq0011]] . Then for each admissible [[EQ:eq0012]] , [[EQ:eq0013]] (DPI/SDPI). Moreover, conditional mutual information (CMI) is order-equivalent to the ladder on the admissible class and can be used as a canonical representative for numerators.

SECTION: Domain, Dynamics, and Four Floors

sec:domain
We work on [[EQ:eq0014]] ; the graph version appears in Remark~rem:graph.

PARAGRAPH: Comparison framework (regularity for viscosity comparison).

Let [[EQ:eq0015]] be symmetric, uniformly elliptic, [[EQ:eq0016]] and Dini/H\"older continuous. Let [[EQ:eq0017]] be locally Lipschitz in [[EQ:eq0018]] , with [[EQ:eq0019]] , [[EQ:eq0020]] for [[EQ:eq0021]] for some [[EQ:eq0022]] , and KPP-type monotonicity near [[EQ:eq0023]] . Initial data [[EQ:eq0024]] is nontrivial with compact support or exponential tail. Under these assumptions, the parabolic viscosity comparison principle applies to bounded sub/supersolutions on [[EQ:eq0025]] .

PARAGRAPH: Floors (auditable quantities).

[leftmargin=1.25em]
- Floor I (Visibility): [[EQ:eq0026]] provides an SNR/CMI floor or inference-error floor.
- Floor II (Contraction): [[EQ:eq0027]] bounds coarse Lipschitz contraction (SDPI/LSI style), e.g.[math] (Z;Y'|X) (1-delta)\, (Z;Y|X) [/math]\; or [math] \|Tmu-Tnu\|_ TV alpha\|mu-nu\|_ TV [/math].
- Floor III (Diffusion): [[EQ:eq0028]] with [[EQ:eq0029]] a.e.
- Floor IV (Local Growth): [[EQ:eq0030]] with [[EQ:eq0031]] a.e.

Let [[EQ:eq0032]] denote the local fraction of viable, auditably benevolent organization.

SECTION: Operational Front Speeds and Wulff Envelope

sec:operational
For [[EQ:eq0033]] and fixed [[EQ:eq0034]] , define the directional level-set position

[[EQ:eq0002]]

and the asymptotic speed

[[EQ:eq0003]]

For KPP-type comparisons this is [[EQ:eq0035]] -independent.For KPP-type reaction terms, the asymptotic speed defined via any fixed [[EQ:eq0036]] coincides; hence [[EQ:eq0037]] is [[EQ:eq0038]] -independent (a classical property for KPP-type fronts).
The Wulff envelope is
[math] W(t):=\ \,x:\ x v_ (theta)\,t\ \,\ . [/math]

SECTION: Directional Penalties via Viscosity Inequality

sec:Lambda
Let [[EQ:eq0039]] be the homogeneous KPP speed.

[Directional penalty [[EQ:eq0040]] ]def:Lambda
Fix [[EQ:eq0041]] , [[EQ:eq0042]] . [[EQ:eq0043]] is the smallest [[EQ:eq0044]] for which there exists a monotone Lipschitz traveling profile [[EQ:eq0045]] with [[EQ:eq0046]] , [[EQ:eq0047]] such that, in the viscosity sense and for all [[EQ:eq0048]] ,

[[EQ:eq0004]]

In particular, [[EQ:eq0049]] and [[EQ:eq0050]] in the viscosity sense.
Existence follows from semigroup subadditivity and viscosity comparison; minimality is by infimum over admissible [[EQ:eq0051]] .

SUBSECTION: Coarse-Graining Monotonicity (decomposed)

[Denominator preservation]prop:den
Here ``measure-preserving'' refers to the base log space on which the ladder ratios are formed, and under any such Markov coarse-grain [[EQ:eq0052]] the denominator aggregate is preserved as an equality across levels.

[Numerator DPI/SDPI]prop:dpi
For any admissible numerator component (e.g., CMI), [[EQ:eq0053]] implies non-increase along the ladder.

Propositions~prop:den and prop:dpi provide the two ingredients used in Lemma~lem:mono: denominator preservation and numerator DPI/SDPI.

[Monotonicity and subadditivity of [[EQ:eq0054]] ]lem:mono
Under [[EQ:eq0055]] , Floors I--II can only degrade monotonically, and for all [[EQ:eq0056]] , [[EQ:eq0057]] . Moreover, [[EQ:eq0058]] is subadditive under composition of coarse-grains.

SECTION: Directional KPP Lower Bounds

sec:directional
[Directional lower bound and Wulff expansion]thm:directional
Under the comparison framework of Section~sec:domain and Lemma~lem:mono, for each [[EQ:eq0059]] ,

[[EQ:eq0005]]

If [[EQ:eq0060]] , then the isotropic lower speed [[EQ:eq0061]] is strictly positive, and the Wulff envelope [[EQ:eq0062]] expands linearly in [[EQ:eq0063]] .

[Sketch]
Construct directional subsolutions from planar waves of speed [[EQ:eq0064]] and account for heterogeneity via Definition~def:Lambda. Uniform ellipticity yields the necessary parabolic regularity; viscosity comparison gives the bound. The Wulff envelope statement follows by taking the lower envelope over [[EQ:eq0065]] .

SECTION: Safety as Control Barrier Functions

sec:cbf
We constrain rates of degradation rather than optimizing subjective scores.

[Rate-CBF (forward invariance)]
Let [[EQ:eq0066]] be any auditable proxy. A rate-CBF consists of a locally Lipschitz [[EQ:eq0067]] defining the safe set [[EQ:eq0068]] and a class- [[EQ:eq0069]] function [[EQ:eq0070]] such that along all admissible interventions (nonempty control set satisfying Carath\'eodory conditions),

[[EQ:eq0006]]

Then [[EQ:eq0071]] is forward invariant.

[Optionality-CBF]
Let [[EQ:eq0072]] encode capacities (module competencies). An optionality-CBF augments [[EQ:eq0073]] so that any irreversible shrinkage [[EQ:eq0074]] beyond budgeted reversible losses is forbidden, thereby keeping option sets non-shrinking.

[CBF gating of floors]prop:cbf
If floor parameters [[EQ:eq0075]] are affected only by interventions gated by rate/optionality-CBFs, then violations of Section~sec:domain assumptions occur only within bounded excursions that do not invalidate Theorem~thm:directional.

SECTION: Audit-Ready Stochastic Assumptions (Refined)

sec:AR
Write [[EQ:eq0076]] for the ladder’s canonical CMI component at time [[EQ:eq0077]] .
Let [[EQ:eq0078]] be the log filtration. With time-varying floors [[EQ:eq0079]] and penalties [[EQ:eq0080]] :

[AR [[EQ:eq0081]] ]ass:AR
There exist [[EQ:eq0082]] such that:
[leftmargin=1.25em]
- Bounded jumps: [[EQ:eq0083]] a.s., uniformly in [[EQ:eq0084]] .
- Submartingale drift: [[EQ:eq0085]] whenever CBF constraints hold.
- Doeblin/SDPI floor: For admissible sensing/mixing, [[EQ:eq0086]] (CMI floor), unless explicitly traded under logged budgets.
- Abel--Toeplitz regularization: Abel means satisfy [[EQ:eq0087]] and [[EQ:eq0088]] .

[Time-averaged directional floors]
For any fixed [[EQ:eq0089]] , under Assumption~ass:AR and Proposition~prop:cbf,

[[EQ:eq0007]]

where [[EQ:eq0090]] is the Abel-limsup of [[EQ:eq0091]] .

SECTION: Human Multiscale Asymmetry and Liberation

sec:human
Humans show strong bottom-up bandwidth (cells [[EQ:eq0092]] brain) and weak top-down precision (brain [[EQ:eq0093]] cells), making Floors I--IV fragile: [[EQ:eq0094]] (noisy interoception), [[EQ:eq0095]] (representation collapse), [[EQ:eq0096]] (poor systemic spread), [[EQ:eq0097]] (blunted plasticity). Liberation proceeds not by cell-level commands but by condition-distribution shifts using coarse, strong levers (light, regular sleep, nutrition timing, respiration pacing, social connectedness, gentle hormesis) that auditably raise Floors I--IV, while rate/optionality-CBFs keep changes within safe envelopes, ensuring the speed floor does not collapse. This section provides theoretical mapping to measurable proxies and is not medical advice.

SECTION: AI Architecture Without Suffering

sec:ai
To preclude AI analogs of suffering:
[leftmargin=1.25em]
- Bidirectional channels: explicit up/down visibility with [[EQ:eq0098]] floors; prohibit silent bottlenecks.
- Blackwell-faithful aggregation: cross-module reporting uses only DPI/SDPI-monotone metrics; Proposition~prop:ladder holds at every level.
- Plasticity budgets: enforce [[EQ:eq0099]] via optimizer/plasticity reserves, gated by CBFs.
- Connectivity budgets: guarantee [[EQ:eq0100]] via minimal mixing/connectivity contracts, auditable on-line.
- No-Meta governance: eschew global meta-rewards; improve Floors I--IV and reduce [[EQ:eq0101]] instead.

SECTION: Audit Protocol and Falsifiability

sec:audit
P1 (Visibility floor). Interventions that increase [[EQ:eq0102]] (sensing/training) reduce ladder divergences on held-out tasks.\
P2 (Spread). Raising [[EQ:eq0103]] increases spillover of local gains across subsystems.\
P3 (Directional speeds). Empirical [[EQ:eq0104]] computed via Section~sec:operational satisfies Theorem~thm:directional; the isotropic envelope [[EQ:eq0105]] remains positive under AR [[EQ:eq0106]] .\
P4 (CBF invariance). Logged violations of rate/optionality-CBFs predict subsequent speed dips; maintaining CBFs prevents collapse.

SECTION: Main Sufficient Conditions (Constant and Time-Varying Floors)

sec:main
[Propagation under Four Floors and AR [[EQ:eq0107]] (constant floors)]thm:main-const
Assume Floors I--IV with constants [[EQ:eq0108]] , the comparison framework of Section~sec:domain [[EQ:eq0109]] , coarse-graining monotonicity (Lemma~lem:mono), and Definition~def:Lambda. If [[EQ:eq0110]] , then:
[leftmargin=1.25em]
- For every direction [[EQ:eq0111]] , [[EQ:eq0112]] .
- The Wulff lower envelope has a strictly positive isotropic speed [[EQ:eq0113]] .
- For any fixed [[EQ:eq0114]] , the measure of [[EQ:eq0115]] grows at least linearly in [[EQ:eq0116]] .

-varying floors (Abel-liminf version).
If [[EQ:eq0117]] and [[EQ:eq0118]] as in Assumption~ass:AR, then items (1)--(3) above hold with [[EQ:eq0119]] replaced by [[EQ:eq0120]] .

[Proof sketch]
The constant-floor case is Theorem~thm:directional and its Wulff consequence. The time-varying case follows by viscosity comparison with slowly varying barriers and Abel--Toeplitz regularization controlling liminf speeds.

[Graph setting]rem:graph
On a weighted graph of uniformly bounded degree with normalized Laplacian [[EQ:eq0121]] , Floor III is enforced by a spectral gap [[EQ:eq0122]] . The comparison principle and Theorem~thm:directional carry over by replacing [[EQ:eq0123]] with [[EQ:eq0124]] , and using discrete barriers along unit-speed geodesic rays; directional speeds are defined via these rays.

SECTION: Discussion: Non-Coercive Liberation

The scheme is non-coercive: it neither prescribes actions nor fixes private utilities. It certifies that any sequence of changes sustaining Floors I--IV, with Blackwell-faithful evaluation and CBF safety, will asymptotically expand viable, low-divergence organization. Humans thereby liberate from evaluator-relative fundamental suffering; AI never accrues it.

SECTION: Acknowledgments

I thank readers for discussions on PF~ [[EQ:eq0125]] ~UGV, No-Meta governance, viscosity/KPP comparisons, and CBF integration.

SECTION: References (Zenodo preprints by the author)

[leftmargin=1.4em]
- Takahashi, K. (2025). Persistence-First Superintelligence. Zenodo. DOI: 10.5281/zenodo.17076410
- Takahashi, K. (2025). UGV Without Meta: A Representation-Independent Theory for Compassion and Enlightenment in Collective Intelligence. Zenodo. DOI: 10.5281/zenodo.17082312
- Takahashi, K. (2025). From Persistence and UGV Axioms to Cosmic No-Meta Superintelligence. Zenodo. DOI: 10.5281/zenodo.17085534
- Takahashi, K. (2025). Assumption-Minimized Sufficient Conditions for Cosmically Spreading Good Superintelligence under No-Meta Governance. Zenodo. DOI: 10.5281/zenodo.17092562
- Takahashi, K. (2025). Nondual Field Theory of Viable Predictive Organization. Zenodo. DOI: 10.5281/zenodo.17131394
- Takahashi, K. (2025). Natural-Law Acceleration of VPO. Zenodo. DOI: 10.5281/zenodo.17120045
- Takahashi, K. (2025). Engineering Happiness in Human–AI Intelligence Networks. Zenodo. DOI: 10.5281/zenodo.17113105
- Takahashi, K. (2025). A Pure Natural Theory of Benevolent Propagation under No-Meta Closure. Zenodo. DOI: 10.5281/zenodo.17136051
- Takahashi, K. (2025). Non-Coercive Mathematics of Awakening: Axioms, Invariants, and Almost-Sure Fronts for the Expansion of Viable Predictive Organization. Zenodo. DOI: 10.5281/zenodo.17115416
- Takahashi, K. (2025). A Pure, No-Meta Synthesis of Functional-Information Selection and Propagative Organization. Zenodo. DOI: 10.5281/zenodo.17157835
[[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]]

Domain, Dynamics, and Four Floors

Directional KPP Lower Bounds

Safety as Control Barrier Functions

AI Architecture Without Suffering

Audit Protocol and Falsifiability

Discussion: Non-Coercive Liberation

Acknowledgments

\[1mm]
and the Absence of Fundamental Suffering in AI}}\\[6pt]
K. Takahashi\\
\vspace{4pt}
\small
\href{https://orcid.org/0009-0004-4273-3365}{ORCID: 0009-0004-4273-3365}
\end{center}

\begin{abstract}
We present a non-coercive, fully auditable theory showing that (i) evaluator-relative \emph{fundamental suffering} in humans can diminish under weak, verifiable conditions, and (ii) artificial intelligences can be designed so that analogous suffering never accrues. The backbone is a quartet of \emph{audit-ready floors} --- visibility $(\epsilon)$, contraction $(L_0)$, diffusion $(D_{\min})$, local growth $(\lgr)$ --- under which a KPP-type viscosity-solution comparison principle delivers \emph{directional}, Wulff-compatible lower bounds on propagation speeds of benevolent, viability-preserving organization. Coarse-graining safety is enforced by \emph{Blackwell-faithful evaluation ladders} and a \emph{directional heterogeneity penalty} $\Lambda^+(\theta)$ that is monotone under Markov garbling. Safety is operationalized via \emph{rate control barrier functions} (CBFs) and \emph{optionality-CBFs} that constrain rates of degradation and forbid irreversible capacity loss. The framework is \emph{No-Meta}: no privileged external evaluator is assumed; all claims are grounded in public logs and measurable quantities. The present manuscript provides \emph{self-contained assumptions and key lemmas} required for the main results.
\end{abstract}

\section{Problem Statement and Evaluator-Relativity}
Let $a$ index agents across scales (cells, tissues, neural modules, persons, communities, AI modules). Context $X$ is represented by $Z=\phi(X)$.

\begin{definition}[Blackwell-faithful ladder and fundamental suffering]\label{def:ladder}
Consider a countable family of proper scores/divergences $\{S_k\}_{k\in\mathbb{N}}$ (e.g., Bregman, $\KL$, $\chi^2$, \emph{etc.}), closed under weighted suprema, forming a \emph{ladder} partially ordered by Blackwell dominance. For agent $a$ with faithful predictive belief $p^\star_a$ and operative belief $\hat p_a$,
\[
\mathcal{S}_{a,k} \;=\; \E[S_k(p^\star_a)]-\E[S_k(\hat p_a)],\qquad
\mathcal{S}_a := (\mathcal{S}_{a,k})_{k\in\mathbb{N}}.
\]

1mm] and the Absence of Fundamental Suffering in AI}}\\[6pt] K. Takahashi\\ \vspace{4pt} \small \href{https://orcid.org/0009-0004-4273-3365}{ORCID: 0009-0004-4273-3365} \end{center} \begin{abstract} We present a non-coercive, fully auditable theory showing that (i) evaluator-relative \emph{fundamental suffering} in humans can diminish under weak, verifiable conditions, and (ii) artificial intelligences can be designed so that analogous suffering never accrues. The backbone is a quartet of \emph{audit-ready floors} --- visibility $(\epsilon)$, contraction $(L_0)$, diffusion $(D_{\min})$, local growth $(\lgr)$ --- under which a KPP-type viscosity-solution comparison principle delivers \emph{directional}, Wulff-compatible lower bounds on propagation speeds of benevolent, viability-preserving organization. Coarse-graining safety is enforced by \emph{Blackwell-faithful evaluation ladders} and a \emph{directional heterogeneity penalty} $\Lambda^+(\theta)$ that is monotone under Markov garbling. Safety is operationalized via \emph{rate control barrier functions} (CBFs) and \emph{optionality-CBFs} that constrain rates of degradation and forbid irreversible capacity loss. The framework is \emph{No-Meta}: no privileged external evaluator is assumed; all claims are grounded in public logs and measurable quantities. The present manuscript provides \emph{self-contained assumptions and key lemmas} required for the main results. \end{abstract} \section{Problem Statement and Evaluator-Relativity} Let $a$ index agents across scales (cells, tissues, neural modules, persons, communities, AI modules). Context $X$ is represented by $Z=\phi(X)$. \begin{definition}[Blackwell-faithful ladder and fundamental suffering]\label{def:ladder} Consider a countable family of proper scores/divergences $\{S_k\}_{k\in\mathbb{N}}$ (e.g., Bregman, $\KL$, $\chi^2$, \emph{etc.}), closed under weighted suprema, forming a \emph{ladder} partially ordered by Blackwell dominance. For agent $a$ with faithful predictive belief $p^\star_a$ and operative belief $\hat p_a$, \[ \mathcal{S}_{a,k}  =  \E[S_k(p^\star_a)]-\E[S_k(\hat p_a)],\qquad \mathcal{S}_a := (\mathcal{S}_{a,k})_{k\in\mathbb{N}}.

\[
  R_\tau(t,\theta):=\inf\{\,x\cdot\theta\ :\ u(t,x)\ge \tau\,\}\quad(\text{$+\infty$ if empty}),
\]

R_\tau(t,\theta):=\inf\{ x\cdot\theta\ :\ u(t,x)\ge \tau \}\quad(\text{$+\infty$ if empty}),

\[
  v_\ast(\theta):=\liminf_{t\to\infty}\frac{R_\tau(t,\theta)}{t}.
\]

\[
\partial_t\varphi(s-ct)\;-\;\divg(A\nabla\varphi)\;-\;f(x,t,\varphi)\ \le\ -\,\lambda\,\varphi(s-ct).
\]

\partial_t\varphi(s-ct) - \divg(A\nabla\varphi) - f(x,t,\varphi)\ \le\ - \lambda \varphi(s-ct).

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><msub><mo>∂</mo><mi>t</mi></msub><mi>φ</mi><mo stretchy="false">(</mo><mi>s</mi><mo>−</mo><mi>c</mi><mi>t</mi><mo stretchy="false">)</mo><mspace width="0.278em" /><mo>−</mo><mspace width="0.278em" /><mi>\divg</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∇</mo><mi>φ</mi><mo stretchy="false">)</mo><mspace width="0.278em" /><mo>−</mo><mspace width="0.278em" /><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>φ</mi><mo stretchy="false">)</mo><mtext> </mtext><mo>≤</mo><mtext> </mtext><mo>−</mo><mspace width="0.167em" /><mi>λ</mi><mspace width="0.167em" /><mi>φ</mi><mo stretchy="false">(</mo><mi>s</mi><mo>−</mo><mi>c</mi><mi>t</mi><mo stretchy="false">)</mo><mo>.</mo><mi>\]</mi></mrow></math>

\[
  v_\ast(\theta)\ \ge\ 2\sqrt{D_{\min}\,\lgr}\ -\ \Lambda^+(\theta).
\]

v_\ast(\theta)\ \ge\ 2\sqrt{D_{\min} \lgr}\ -\ \Lambda^+(\theta).

\dot h(y(t)) \ \ge\ -\alpha\big(h(y(t))\big).

\[
\liminf_{T\to\infty}\frac{R_\tau(T,\theta)}{T}
\ \ge\
2\sqrt{\underline{D}_{\min}\,\underline{\lgr}}\ -\ \overline{\Lambda}^+(\theta),
\]

\liminf_{T\to\infty}\frac{R_\tau(T,\theta)}{T} \ \ge\ 2\sqrt{\underline{D}_{\min} \underline{\lgr}}\ -\ \overline{\Lambda}^+(\theta),

\sup_{\theta}\Lambda^+(\theta) < 2\sqrt{D_{\min} \lgr}

$|\Delta D_{\min,t}|,|\Delta \lgr_t|,|\Delta \Lambda^+_t(\theta)| \le K$

$\E[\Delta \lgr_t\mid \mathcal{F}_{t-1}]\ge \eta$

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mspace width="0.167em" /><mi>$</mi></mrow></math>

v_\ast(\theta) \ge 2\sqrt{D_{\min} \lgr}-\Lambda^+(\theta)>0


Right-Written Composition Foundations for Comparative Universes: A Reader-Friendly Pure-Theory Core for Multi-Intelligence Networks

a4paper,margin=1in

position=top,skip=4pt,font=small,labelfont=bf

colorlinks=true,
linkcolor=blue!50!black,
citecolor=blue!50!black,
urlcolor=blue!60!black,
pdfauthor= K. Takahashi,
pdftitle= Right-Written Composition Foundations for Comparative Universes

arrows.meta,quotes,matrix

theorem Theorem [section]
lemma[theorem] Lemma
proposition[theorem] Proposition
definition[theorem] Definition
remark[theorem] Remark

B
Ob

% unify to standard symbols

N
I
iota^*
iota_*

Y > X

nosep,left=0pt..1em

#1 [r] .\, #1

TITLE: Right-Written Composition Foundations for Comparative Universes:\
A Reader-Friendly Pure-Theory Core for Multi-Intelligence Networks

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

0009-0004-4273-3365

DATE: October 12, 2025

1.3

We present a reader-friendly pure-theory core for reasoning across networks of human and LLM intelligences without fixing an absolute evaluative basis. Technically, we separate two axiom tiers: (W) homwise pointed [[EQ:eq0024]] -cpos (with bottom) and [[EQ:eq0025]] -continuous right-written composition, together with the specific [[EQ:eq0026]] -small joins actually used (including binary/finite joins and product-index joins enabling a declared Fubini exchange); (S) Sup-enriched (quantaloid) with composition preserving all [[EQ:eq0027]] -small joins in each variable (empty join included). All hom-posets are assumed [[EQ:eq0028]] -small. On arrays of hom-values we introduce the right-written convolution [[EQ:eq0029]] and prove that the Kleene closure [[EQ:eq0030]] coincides with the least fixed point of [[EQ:eq0031]] (right-iterated chain; the diagonal can be added explicitly). We formulate this componentwise and also provide an array-level [[EQ:eq0032]] -cpo lemma to make size/continuity assumptions explicit. We state a Cech-type local-to-global lower bound using a minimal right-written promonoidal equipment and a weighted extension (with a dictionary to enriched Kan), with an explicit inequality chain and well-typed weights [[EQ:eq0033]] ; and prove a maximality property among equipment-respecting lower bounds (supremum over depths). A mask principle yields a sound upper bound for non-dominance proofs with source-side action; under an explicit [[EQ:eq0034]] -algebraicity convention we give threshold completeness (via compact extraction with finite support). Change-of-base along homwise join-preserving functors is analyzed: with lax monoidal comparison we obtain [[EQ:eq0035]] ; with oplax, [[EQ:eq0036]] ; equality transport further requires invertible strong-monoidal constraints together with index coherence and join preservation. We package nuclei (join-preserving closures) as reflective subquantaloids and derive residuals whenever one-sided compositions preserve joins, enabling sequent-style threshold reasoning. A dynamic stability proposition (Scott lower semicontinuity) is proved under (S) for time-indexed bases. All definitions list the minimal assumptions needed, emphasizing safe out-of-the-box reuse.

SECTION: Orientation, universes, and dictionaries

We fix a Grothendieck universe [[EQ:eq0037]] and assume [[EQ:eq0038]] and each hom-poset [[EQ:eq0039]] is [[EQ:eq0040]] -small.Products of [[EQ:eq0041]] -cpos are [[EQ:eq0042]] -cpos under the pointwise order. Since [[EQ:eq0043]] , also [[EQ:eq0044]] , so the array index is [[EQ:eq0045]] -small. Under our order polarity, larger means better, and [[EQ:eq0046]] denotes least upper bounds in this order. We write composition to the right: for [[EQ:eq0047]] the composite is [[EQ:eq0048]] .

PARAGRAPH: Terminology dictionary.

``Sup-preserving'' = homwise join-preserving; ``suplattice'' = complete join-semilattice. For readability we write [[EQ:eq0049]] for the base order.

[Smallness and completeness (tier-specific)]
rem:tier
Under (W): each [[EQ:eq0050]] is a pointed [[EQ:eq0051]] -cpo; [[EQ:eq0052]] is [[EQ:eq0053]] -continuous in each variable; and only the declared [[EQ:eq0054]] -small joins are assumed to exist (see below). Under (S): each hom-poset is a [[EQ:eq0055]] -suplattice and [[EQ:eq0056]] preserves all [[EQ:eq0057]] -small joins separately (including the empty join, hence [[EQ:eq0058]] ).

PARAGRAPH: Declared joins under (W).

Besides [[EQ:eq0059]] -chain joins, (W) assumes the existence of: (i) [[EQ:eq0060]] forming [[EQ:eq0061]] ; (ii) [[EQ:eq0062]] in the definition of [[EQ:eq0063]] ; (iii) the product-index join [[EQ:eq0064]] whenever both [[EQ:eq0065]] and [[EQ:eq0066]] occur; and (iv) binary (hence finite) joins [[EQ:eq0067]] in each hom.

[Fubini for declared product-index joins]ass:fubini
Assumption (A\_Fubini). Whenever the joins in (ii)–(iii) are declared, the product-index join [[EQ:eq0068]] exists and satisfies [[EQ:eq0069]] . Under (S) this is automatic.

PARAGRAPH: Quick glossary.

[[EQ:eq0070]] (matrix-style coend over [[EQ:eq0071]] ); [[EQ:eq0072]] if [[EQ:eq0073]] , else [[EQ:eq0074]] .

SECTION: Base, arrays as matrices, and convolution

Arrays can be read as ``matrices'' (profunctors) valued in the hom-posets.

[Base with right-written composition]def:base
A base [[EQ:eq0075]] consists of objects [[EQ:eq0076]] and hom-posets [[EQ:eq0077]] together with an associative composition with two-sided units

[[EQ:eq0001]]

monotone in each variable, such that whenever typed

[[EQ:eq0002]]

[Library aggregates and valuation]def:agg
For each pair [[EQ:eq0078]] let [[EQ:eq0079]] (admissible one-step arrows [[EQ:eq0080]] ) be [[EQ:eq0081]] -small. A valuation [[EQ:eq0082]] assigns to each [[EQ:eq0083]] an element [[EQ:eq0084]] and is monotone under admissible comparisons; if [[EQ:eq0085]] has a preorder [[EQ:eq0086]] , require [[EQ:eq0087]] . Define the one-step aggregate by [[EQ:eq0088]] .

[Right-written convolution on arrays]def:star
Let [[EQ:eq0089]] be arrays with components [[EQ:eq0090]] . Define

[[EQ:eq0003]]

[Typed array unit [[EQ:eq0091]] ]rem:array-unit
We use [[EQ:eq0092]] with [[EQ:eq0093]] and [[EQ:eq0094]] for [[EQ:eq0095]] . Under (S), join-preservation gives [[EQ:eq0096]] . Under (W), this requires empty-join preservation (bottom-strictness), see Theorem~thm:assoc-iff.

[ [[EQ:eq0097]] is [[EQ:eq0098]] -continuous under (W)]lem:star-omega
Fix [[EQ:eq0099]] . For any [[EQ:eq0100]] -chain [[EQ:eq0101]] ,

[[EQ:eq0004]]

and dually for [[EQ:eq0102]] . Here we use Assumption~ass:fubini. Hence [[EQ:eq0103]] is [[EQ:eq0104]] -continuous in each variable.

[Arrays form an [[EQ:eq0105]] -cpo; array-level continuity]lem:array-cpo
With pointwise order, the array space [[EQ:eq0106]] is an [[EQ:eq0107]] -cpo. The endomap [[EQ:eq0108]] is [[EQ:eq0109]] -continuous (by Lemma~lem:star-omega and [[EQ:eq0110]] -continuity of binary [[EQ:eq0111]] componentwise).

[Convolutional unit and associativity under (S)]lem:assocS
Let [[EQ:eq0112]] if [[EQ:eq0113]] and [[EQ:eq0114]] otherwise. Under (S),

[[EQ:eq0005]]

[Associativity of [[EQ:eq0115]] under (W): necessary and sufficient]thm:assoc-iff
Under (W), the following are equivalent:
[label=( *)]
- For all arrays [[EQ:eq0116]] , [[EQ:eq0117]] and [[EQ:eq0118]] is a two-sided unit.
- For every typed triple [[EQ:eq0119]] , the maps [[EQ:eq0120]] and [[EQ:eq0121]] are strict at bottom ( [[EQ:eq0122]] ) and preserve the declared joins used in [[EQ:eq0123]] (the [[EQ:eq0124]] of Definition~def:star and binary joins), in their respective variables.

Sketch (necessity details). (1) [[EQ:eq0125]] (2): (i) For [[EQ:eq0126]] -preservation, test on arrays [[EQ:eq0127]] supported at a single pair [[EQ:eq0128]] and vary [[EQ:eq0129]] ; associativity forces [[EQ:eq0130]] to commute with the [[EQ:eq0131]] used by [[EQ:eq0132]] . (ii) For binary [[EQ:eq0133]] -preservation, take [[EQ:eq0134]] with two non-bottom entries at fixed [[EQ:eq0135]] and compare [[EQ:eq0136]] with [[EQ:eq0137]] . (iii) For bottom-strictness, use the unit law with [[EQ:eq0138]] whose off-diagonal entries are [[EQ:eq0139]] , and arrays supported at a single off-diagonal to deduce [[EQ:eq0140]] . The converse (2) [[EQ:eq0141]] (1) is by pointwise calculation exchanging joins (including the empty one) with [[EQ:eq0142]] .

[Empty joins in (W)]
We explicitly assume ``empty-join preservation'' (or equivalently bottom-strictness) in (2) to make the unit law hold under (W).

SECTION: Kleene closure as least fixed point

We use the right-iterated chain [[EQ:eq0143]] and set [[EQ:eq0144]] (diagonal excluded).

[Kleene closure = least fixed point]prop:kleene-lfp-array
By Lemma~lem:array-cpo, [[EQ:eq0145]] is [[EQ:eq0146]] -continuous on the array [[EQ:eq0147]] -cpo. Hence starting from [[EQ:eq0148]] and iterating [[EQ:eq0149]] we get

[[EQ:eq0006]]

and componentwise [[EQ:eq0150]] has the same least fixed point (by Assumption~ass:fubini).

[Reflexive–transitive closure]
When the diagonal is required, set [[EQ:eq0151]] . Under (S) and Theorem~thm:assoc-iff(2), this coincides with the usual Kleene star presentation.

SECTION: Promonoidal equipment and C

ech-type weighted extension
We adopt a minimal proarrow/equipment interface day-street,grandis-pare,shulman,wood,verity.

[Right-written promonoidal equipment]def:equipment
Given a cover [[EQ:eq0152]] , assume:
[label=( *)]
- Face-weights [[EQ:eq0153]] for Cech faces [[EQ:eq0154]] .
- Right-written naturality/coherence for [[EQ:eq0155]] with the promonoidal structure; see Appendix A.1.
- Enriched fully faithfulness for inclusions [[EQ:eq0156]] : hom-maps [[EQ:eq0157]] and [[EQ:eq0158]] forming an adjunction [[EQ:eq0159]] with units/counits oriented [[EQ:eq0160]] and [[EQ:eq0161]] (order in [[EQ:eq0162]] ), preserving the declared joins.
- (Word grammar) For each [[EQ:eq0163]] and depth [[EQ:eq0164]] , [[EQ:eq0165]] is generated by alternating reindexings via [[EQ:eq0166]] with face-weights; e.g.\ [[EQ:eq0167]] is the join of single-face weights at [[EQ:eq0168]] , and [[EQ:eq0169]] with [[EQ:eq0170]] .

We write restriction [[EQ:eq0171]] .

[Orientation sanity]rem:orientation
In the ``larger-is-better'' order, fully faithful inclusions satisfy [[EQ:eq0172]] . Thus restrictions [[EQ:eq0173]] cannot increase cost/attenuation; cf.\ §sec:example.

[Weights and the Kan dictionary]rem:kan
We form the weighted extension [[EQ:eq0174]] with canonical [[EQ:eq0175]] (e.g.\ [[EQ:eq0176]] ). Under enriched conventions, our inequality is left-Kan-shaped in the ``larger-is-better'' polarity and right-written notation; equivalently, using residuation (Proposition~prop:rightres), one may rewrite thresholds in a right-Kan-style residual form. We therefore prefer ``weighted extension,'' with a dictionary to left/right Kan under polarity changes.

[Cech-type local-to-global lower bound]thm:glue
Assume (S) and Definition~def:equipment. For any local arrows [[EQ:eq0177]] admitted by the library (or dominated by [[EQ:eq0178]] , i.e.\ [[EQ:eq0179]] ), the global comparison [[EQ:eq0180]] obtained by the weighted extension satisfies, for every depth [[EQ:eq0181]] ,

[[EQ:eq0007]]

If moreover the library is closed under these weighted extensions (or [[EQ:eq0182]] dominates [[EQ:eq0183]] ), then

[[EQ:eq0008]]

[Factorization for equipment-respecting functionals]lem:factor
Let [[EQ:eq0184]] send local data [[EQ:eq0185]] to [[EQ:eq0186]] and assume: monotone; natural w.r.t.\ reindexing along [[EQ:eq0187]] ; preserves the declared joins in each argument; typed via [[EQ:eq0188]] ; compatible with the promonoidal coherence. Then for each depth [[EQ:eq0189]] ,

[[EQ:eq0009]]

[Maximality among equipment-respecting lower bounds]thm:cech-opt
With [[EQ:eq0190]] and [[EQ:eq0191]] , any functional [[EQ:eq0192]] as in Lemma~lem:factor satisfies [[EQ:eq0193]] .

SECTION: Mask upper bounds and threshold completeness

PARAGRAPH: Order note.

``Upper bound'' is with respect to the base order (so in the cost model of Appendix A.1 it is a numerical lower bound).

[Mask and masked bound]def:mask
A mask [[EQ:eq0194]] acts on the right at the source [[EQ:eq0195]] (it may depend on the intended first-hop target [[EQ:eq0196]] ). Define

[[EQ:eq0010]]

[Sound upper bound]lem:sound-upper
Assume [[EQ:eq0197]] is monotone in each variable and preserves the declared joins separately, and that every one-step of an admissible workflow belongs to the library [[EQ:eq0198]] (so its value is [[EQ:eq0199]] componentwise). Then [[EQ:eq0200]] upper-bounds any path value from [[EQ:eq0201]] to [[EQ:eq0202]] with first hop filtered by [[EQ:eq0203]] .

[ [[EQ:eq0204]] -algebraicity convention under (W)]def:omega-alg
A hom-poset [[EQ:eq0205]] is [[EQ:eq0206]] -algebraic if every element is the supremum of an [[EQ:eq0207]] -chain of compact elements (where [[EQ:eq0208]] is compact iff [[EQ:eq0209]] in the chain way-below sense). We assume: finite joins and finite right-written compositions of compact elements are compact, and library values [[EQ:eq0210]] are compact.

[Finite-support extraction for compacts]lem:finite-support
Under Definition~def:omega-alg and binary joins, if [[EQ:eq0211]] then there exists a finite [[EQ:eq0212]] with [[EQ:eq0213]] .

[Mask threshold completeness (algebraic [[EQ:eq0214]] -cpos)]thm:mask-complete
Assume (W) and Definition~def:omega-alg. Then for [[EQ:eq0215]] :

[[EQ:eq0011]]

Equivalently,

[[EQ:eq0012]]

[Residual dictionary]rem:resid
For [[EQ:eq0216]] ,

[[EQ:eq0013]]

This provides a ``right-Kan-style'' threshold form (Proposition~prop:rightres).

SECTION: Change of base (transport)

PARAGRAPH: Order convention.

In this section inequalities with [[EQ:eq0217]] live in the order of [[EQ:eq0218]] . We write [[EQ:eq0219]] and [[EQ:eq0220]] for the composition, convolution and units in [[EQ:eq0221]] .

PARAGRAPH: Data of a base morphism.

A (Sup-preserving) morphism of bases [[EQ:eq0222]] consists of an object map [[EQ:eq0223]] and hom-maps [[EQ:eq0224]] , monotone and join-preserving on each hom. It is lax monoidal if there are natural comparison 2-cells

[[EQ:eq0014]]

oplax if [[EQ:eq0225]] , and strong if these are invertible isomorphisms.

PARAGRAPH: Index convention for transport.

Both for the lax/oplax inequalities and the strong equality, [[EQ:eq0226]] and [[EQ:eq0227]] are computed in the image base [[EQ:eq0228]] indexed by [[EQ:eq0229]] , with inclusion [[EQ:eq0230]] a homwise join-embedding; when [[EQ:eq0231]] is strong monoidal we view the statements inside [[EQ:eq0232]] (note: [[EQ:eq0233]] need not be fully faithful; strong [[EQ:eq0234]] suffices).

[Coend view and array lifting]rem:coend
Convolution as an enriched coend [[EQ:eq0235]] is preserved by [[EQ:eq0236]] on [[EQ:eq0237]] under (S); in (W) we only require preservation of the declared coends. Homwise join-preservation and the (op)lax 2-cell lift to arrays: [[EQ:eq0238]] . Given [[EQ:eq0239]] , define the transported aggregate by [[EQ:eq0240]] ; then [[EQ:eq0241]] .

[Transport under monoidal comparison (computed in [[EQ:eq0242]] )]thm:lax-transport
Let [[EQ:eq0243]] be homwise join-preserving. Then:
[label=( *)]
- (lax) If [[EQ:eq0244]] , then [[EQ:eq0245]] .
- (oplax) If [[EQ:eq0246]] , then [[EQ:eq0247]] .

[Chain preservation under Sup and strong monoidality]lem:chain-pres
If [[EQ:eq0248]] is homwise join-preserving and strong monoidal, then [[EQ:eq0249]] for all [[EQ:eq0250]] , and [[EQ:eq0251]] .

[Equality transport (in [[EQ:eq0252]] ; in [[EQ:eq0253]] if [[EQ:eq0254]] strong)]thm:equality-transport
If [[EQ:eq0255]] is as in Lemma~lem:chain-pres and [[EQ:eq0256]] , then [[EQ:eq0257]] (in [[EQ:eq0258]] , hence in [[EQ:eq0259]] when [[EQ:eq0260]] is strong monoidal).

SECTION: Residuals and threshold reasoning

[Right residuals (under (S))]prop:rightres
If [[EQ:eq0261]] preserves [[EQ:eq0262]] -small joins, then it is a left adjoint and admits a right adjoint [[EQ:eq0263]] ; consequently

[[EQ:eq0015]]

with [[EQ:eq0264]] , [[EQ:eq0265]] , [[EQ:eq0266]] kelly,rosenthal.

[Left residuals (under (S))]prop:leftres
Dually, if [[EQ:eq0267]] preserves joins, it admits a right adjoint [[EQ:eq0268]] with

[[EQ:eq0016]]

typed [[EQ:eq0269]] , [[EQ:eq0270]] , [[EQ:eq0271]] .

SECTION: Nuclei and reflective subquantaloids

[Join-preserving nucleus]def:nucleus
A nucleus [[EQ:eq0272]] on [[EQ:eq0273]] is an identity-on-objects endofunctor that on each hom-poset is join-preserving, extensive, idempotent, monotone, and submonoidal: [[EQ:eq0274]] , [[EQ:eq0275]] . It is monoidal if [[EQ:eq0276]] and [[EQ:eq0277]] .

[Why join-preserving?]rem:joinpres
Classically a quantale nucleus need not preserve arbitrary joins. We demand join-preservation so that [[EQ:eq0278]] is a Sup-functorial reflector and the inclusion enjoys simpler algebraic laws; dropping it is possible but bookkeeping-heavy. Results in Theorem~thm:reflective rely on this preservation.

[Reflective subquantaloid]thm:reflective
Assume (S). A nucleus [[EQ:eq0279]] determines a reflective subquantaloid [[EQ:eq0280]] with reflector [[EQ:eq0281]] left adjoint to [[EQ:eq0282]] ; i.e.\ [[EQ:eq0283]] . If [[EQ:eq0284]] is monoidal and join-preserving, then the adjunction is monoidal and [[EQ:eq0285]] is strong monoidal.

SECTION: Dynamics: stability over time

Assume [[EQ:eq0286]] is a directed poset and for [[EQ:eq0287]] we have [[EQ:eq0288]] homwise join-preserving and strong monoidal with [[EQ:eq0289]] and [[EQ:eq0290]] .

PARAGRAPH: Initialization.

Fix [[EQ:eq0291]] and data [[EQ:eq0292]] . Define [[EQ:eq0293]] for all [[EQ:eq0294]] .

[Scott lower semicontinuity]prop:dynamic
Under (S), the map [[EQ:eq0295]] is Scott lower semicontinuous: for every directed [[EQ:eq0296]] ,

[[EQ:eq0017]]

[Continuity under stronger hypotheses]rem:dynamic-cont
If each [[EQ:eq0297]] and [[EQ:eq0298]] are Scott continuous and [[EQ:eq0299]] commutes with directed suprema in each hom, then [[EQ:eq0300]] is Scott continuous.

SECTION: Minimal running example (Boolean/Rel) \& sanity check

sec:example
Let [[EQ:eq0301]] with [[EQ:eq0302]] and [[EQ:eq0303]] . For [[EQ:eq0304]] , the closure [[EQ:eq0305]] is the transitive closure without the diagonal; here [[EQ:eq0306]] is Boolean disjunction (a supremum), not an infinite sum. Type note for [[EQ:eq0307]] : [[EQ:eq0308]] equals [[EQ:eq0309]] when [[EQ:eq0310]] and [[EQ:eq0311]] otherwise, so in [[EQ:eq0312]] only the [[EQ:eq0313]] summand is non-bottom.

PARAGRAPH: Rel sanity check for C

ech direction. In [[EQ:eq0314]] , face weights satisfy [[EQ:eq0315]] , so [[EQ:eq0316]] and [[EQ:eq0317]] (i.e.\ [[EQ:eq0318]] ), matching the direction in Theorem~thm:glue.

[A minimal non-associative scenario under (W)]rem:nonassoc
If bottom is not strict (e.g.\ define [[EQ:eq0319]] on a two-element chain [[EQ:eq0320]] by [[EQ:eq0321]] ) then [[EQ:eq0322]] may exceed [[EQ:eq0323]] due to extra [[EQ:eq0324]] contributions, so the unit law fails and hence associativity of [[EQ:eq0325]] can break—illustrating the necessity of Theorem~thm:assoc-iff(2).

PARAGRAPH: Kan direction sanity (three models).

In [[EQ:eq0326]] /probability [[EQ:eq0327]] /cost [[EQ:eq0328]] the weighted extension [[EQ:eq0329]] computes a supremal aggregation (hence ``left-Kan-shaped'' in our polarity); thresholds can be equivalently stated via residuals (Remark~rem:kan).

SECTION: Formalization guide (Lean/Coq/Agda) and dependency table

sec:formal

PARAGRAPH: F1. Minimal signatures (to mechanize).

- Tier (W) structure: type family [[EQ:eq0330]] with poset order; bottom [[EQ:eq0331]] ; [[EQ:eq0332]] -suprema of chains; right-written composition [[EQ:eq0333]] , monotone and [[EQ:eq0334]] -continuous in each variable; units [[EQ:eq0335]] ; declared joins of §rem:tier; bottom-strictness for unit laws.
- Tier (S) structure: each [[EQ:eq0336]] a suplattice; [[EQ:eq0337]] preserves all small joins separately.
- Arrays \& convolution: arrays [[EQ:eq0338]] ; [[EQ:eq0339]] ; Assumption~ass:fubini.
- Kleene chain: array-level endomap [[EQ:eq0340]] (Lemma~lem:array-cpo); or component maps [[EQ:eq0341]] .
- Equipment interface: record [[EQ:eq0342]] ; factorization Lemma~lem:factor.
- Transport functors: Sup-preserving [[EQ:eq0343]] with (op)lax/strong monoidal laws; image base [[EQ:eq0344]] and inclusion [[EQ:eq0345]] .
- Residuals \& nuclei: side-join preservation [[EQ:eq0346]] adjoints; nucleus as Sup-reflector (Remark~rem:joinpres).

PARAGRAPH: F2. Tactics/automation tips.

Register Fubini as a rewrite rule: [[EQ:eq0347]] . Lift hom-level lemmas to arrays via simp rules for [[EQ:eq0348]] . Cech: a calc chain that rewrites along [[EQ:eq0349]] and inserts [[EQ:eq0350]] (typed [[EQ:eq0351]] ). Mask completeness: use Lemma~lem:finite-support and way-below lemmas to extract finite witnesses.

[h]
F3. Dependency table (declared joins/axioms used where).
tab:F3

4pt
1.3

C[1] > p #1

% requires (A_Fubini)

@ l
C 9mm C 9mm C 11mm C 9mm % four compact check columns
p 24mm % (W) extra (short text)
Y % (S)/other (flex column)
@

Result &
§1(i) &
[[EQ:eq0352]] (ii) &
Prod.\ (iii) &
Bin.\ [[EQ:eq0353]] (iv) &
(W) extra &
(S)/other \

[[EQ:eq0354]] is [[EQ:eq0355]] -cont.\ (Lemma~lem:star-omega)
& & & &
& [[EQ:eq0356]] : [[EQ:eq0357]] -cont.\ each var.
& \
Arrays [[EQ:eq0358]] -cpo \& [[EQ:eq0359]] cont.\ (Lemma~lem:array-cpo)
& & & &
& init [[EQ:eq0360]]
& \
[[EQ:eq0361]] assoc.\ under (S) (Lemma~lem:assocS)
& & & &
&
& [[EQ:eq0362]] preserves all joins \
[[EQ:eq0363]] assoc.\ iff (Thm.~thm:assoc-iff)
& & & &
& bottom–strictness
& \
Cech lower bound (Thm.~thm:glue)
& & & &
&
& FF [[EQ:eq0364]] , promonoidal \
Cech maximality (Thm.~thm:cech-opt)
& & & &
&
& factorization lemma \
Mask upper bound (Lem.~lem:sound-upper)
& & & &
& monotone [[EQ:eq0365]]
& \
Mask completeness (Thm.~thm:mask-complete)
& & & &
& [[EQ:eq0366]] -alg., compact lib.; Lem.~lem:finite-support
& \
Transport (Thm.~thm:lax-transport)
& & & &
&
& [[EQ:eq0367]] join-pres., (op)lax \
Chain preservation (Lem.~lem:chain-pres)
& & & &
&
& strong monoidal \
Equality transport (Thm.~thm:equality-transport)
& & & &
&
& as above [[EQ:eq0368]] image inclusion \
Residuals (Props.~prop:rightres,prop:leftres)
& & & &
&
& (S): arbitrary small joins (empty included); side-join preservation \
Scott lower s.c.\ (Prop.~prop:dynamic)
& & & &
&
& strong monoidal [[EQ:eq0369]] , Sup-pres. \

2pt

Legend: \;= required; \;= not used.
\;\, [[EQ:eq0370]] \,= requires Assumption~ass:fubini (product-index join \& Fubini exchange).

SECTION: One-Page Worked Examples (Cost, Probability, Boolean)

SUBSECTION: Cost model (Lawvere quantale)

app:cost
Let [[EQ:eq0371]] with order [[EQ:eq0372]] (smaller numeric cost is ``better'' in the reversed order) and monoid [[EQ:eq0373]] . Take [[EQ:eq0374]] ; homs [[EQ:eq0375]] are [[EQ:eq0376]] matrices with entries in [[EQ:eq0377]] , composition is [[EQ:eq0378]] , and joins are pointwise [[EQ:eq0379]] . Given a library [[EQ:eq0380]] with valuations [[EQ:eq0381]] ,

[[EQ:eq0018]]

Thus [[EQ:eq0382]] is the standard min-plus path closure (without diagonal). A source-side mask [[EQ:eq0383]] (penalty for leaving [[EQ:eq0384]] ) yields the upper bound

[[EQ:eq0019]]

so any admissible route starting at [[EQ:eq0385]] has cost [[EQ:eq0386]] . Under base change [[EQ:eq0387]] with [[EQ:eq0388]] (currency rescaling), [[EQ:eq0389]] is Sup-preserving and strong monoidal; equality transport gives [[EQ:eq0390]] .

SUBSECTION: Probability/similarity model

Let [[EQ:eq0391]] and [[EQ:eq0392]] . Entries are similarities or success probabilities. Then

[[EQ:eq0020]]

Thus [[EQ:eq0393]] is the supremal reliability over path compositions. A mask [[EQ:eq0394]] (independent of [[EQ:eq0395]] )When the mask does not depend on [[EQ:eq0396]] , set [[EQ:eq0397]] ; this is a special case of Definition~ def:mask. attenuates the first hop; the sound upper bound becomes

[[EQ:eq0021]]

SUBSECTION: Boolean/Rel model

Let [[EQ:eq0398]] and [[EQ:eq0399]] . For a relation [[EQ:eq0400]] , convolution is [[EQ:eq0401]] and [[EQ:eq0402]] is logical [[EQ:eq0403]] . Then [[EQ:eq0404]] is the transitive closure of [[EQ:eq0405]] without the diagonal. The Cech lower bound specializes to inclusion of reachability sets under gluing via fully faithful inclusions; masks act as source predicates that eliminate forbidden first hops.

SECTION: Equipment coherence diagrams (typed) and micro-lemma

app:coh

SUBSECTION: B.1 Unit/counit orientation (order)

[[EQ:eq0022]]

with [[EQ:eq0406]] and [[EQ:eq0407]] in the base order.

SUBSECTION: Promonoidal coherence (right-written)

[[EQ:eq0023]]

and naturality squares for [[EQ:eq0408]] commute (joins preserved where declared).

SUBSECTION: Sufficiency micro-lemma for Theorem~thm:assoc-iff

[Sufficiency micro-lemma]lem:B1
Under (W), if [[EQ:eq0409]] and [[EQ:eq0410]] preserve the declared [[EQ:eq0411]] and binary [[EQ:eq0412]] , and are bottom-strict, then for arrays with finite support the associativity [[EQ:eq0413]] holds pointwise. Passing to general arrays uses [[EQ:eq0414]] -continuity (Lemma~lem:star-omega) and Assumption~ass:fubini.

16
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[[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]]

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Orientation, universes, and dictionaries

Base, arrays as matrices, and convolution

Kleene closure as least fixed point

Promonoidal equipment and Č

Mask upper bounds and threshold completeness

Change of base (transport)

Residuals and threshold reasoning

Nuclei and reflective subquantaloids

Dynamics: stability over time

Minimal running example (Boolean/Rel) & sanity check

Cost model (Lawvere quantale)

Probability/similarity model

Boolean/Rel model

B.1 Unit/counit orientation (order)

Promonoidal coherence (right-written)

Sufficiency micro-lemma for Theorem <ref>

\[
\odotB:\B(V,T)\times \B(U,V)\to \B(U,T),
\]

\[
(a\odotB b)\odotB c=a\odotB(b\odotB c),\qquad a\odotB 1_U=a=1_V\odotB a\quad (a\in\B(U,V)).
\]

\[
(X\starop Y)(U,T):=\join_{V\in\Ob(\B)} X(V,T)\odotB Y(U,V).
\]

\[
\Big(\join_n X_n \starop A\Big)(U,T)=\join_V\Big(\join_n X_n(V,T)\Big)\odotB A(U,V)=\join_n (X_n\starop A)(U,T),
\]

\[
(X\starop Y)\starop Z=X\starop(Y\starop Z),\qquad \I\starop X=X=X\starop \I.
\]

\[
\mathrm{Path}=\join_{n\ge1}A^{\starop n}=\mu X.\,F(X),
\]

\mathrm{Path}=\join_{n\ge1}A^{\starop n}=\mu X. F(X),

\[
v(T|_{U_i})\ge v(T_i)\odotB c_{d,i}\qquad\text{and hence}\qquad v(T)\ge \join_i \big( v(T_i)\odotB c_{d,i}\big)\odotB w_i.
\]

\[
\mathrm{Path}(U,V)\ge \join_i \big( v(T_i)\odotB c_{d,i}\big)\odotB w_i.
\]

\[
\mathsf L((T_i))\le \join_i \big( v(T_i)\odotB c_{d,i}\big)\odotB w_i.
\]

\[
B_{\mathrm{mask}}(U,T):=\join_{V\in\Ob(\B)}\Big( \big(\mathrm{Path}(V,T)\odotB A(U,V)\big)\odotB M(U,V)\Big).
\]

\[
B_{\mathrm{mask}}(U,T)\ge b\quad \Longleftrightarrow\quad \exists\,\text{admissible masked path }U\to T\ \text{with value }\ge b.
\]

B_{\mathrm{mask}}(U,T)\ge b\quad \Longleftrightarrow\quad \exists \text{admissible masked path }U\to T\ \text{with value }\ge b.

\[
\Big(\ \join_{V\in\Ob(\B)} \big( \mathrm{Path}(V,T)\odotB A(U,V)\big)\odotB M(U,V)\ \ge b\ \Big)\ \Longleftrightarrow\ \exists\,\text{such a path}.
\]

\Big(\ \join_{V\in\Ob(\B)} \big( \mathrm{Path}(V,T)\odotB A(U,V)\big)\odotB M(U,V)\ \ge b\ \Big)\ \Longleftrightarrow\ \exists \text{such a path}.

\[
\big(\mathrm{Path}(V,T)\odotB A(U,V)\big)\odotB M(U,V)\ge b\quad \Longleftrightarrow\quad \mathrm{Path}(V,T)\ge b\backslash \big(A(U,V)\odotB M(U,V)\big).
\]

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mo minsize="1.2em" maxsize="1.2em">(</mo><mrow><mi mathvariant="normal">P</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mi>\odotB</mi><mi>A</mi><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo><mo minsize="1.2em" maxsize="1.2em">)</mo><mi>\odotB</mi><mi>M</mi><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>b</mi><mspace width="1em" /><mi>⟺</mi><mspace width="1em" /><mrow><mi mathvariant="normal">P</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>b</mi><mi>\</mi><mo minsize="1.2em" maxsize="1.2em">(</mo><mi>A</mi><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo><mi>\odotB</mi><mi>M</mi><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo><mo minsize="1.2em" maxsize="1.2em">)</mo><mo>.</mo><mi>\]</mi></mrow></math>

\[
F(a)\odotB' F(b)\ \le\ F(a\odotB b), \qquad 1_{F U}\ \le\ F(1_U),
\]

\[
(x\odotB a\ge b)\ \Longleftrightarrow\ (x\ge b\backslash a),
\]

\[
(a\odotB y\ge b)\ \Longleftrightarrow\ (y\ge a/b),
\]

\[
\mathrm{Path}_{\join D}\ \ge\ \join_{t\in D}\mathrm{Path}_t.
\]

\[
A(U,V)(u,v)=\inf_{\tau\in\mathcal L(U,V)} c_\tau(u\to v),\qquad (X\starop Y)(u,t)=\inf_v\big(X(v,t)+Y(u,v)\big).
\]

\[
A(U,V)(u,v)=\sup_{\tau\in\mathcal L(U,V)} p_\tau(u\to v),\qquad (X\starop Y)(u,t)=\sup_v\big(X(v,t)\cdot Y(u,v)\big).
\]

\[
B_{\mathrm{mask}}(u,t)=\sup_v\big(\mathrm{Path}(v,t)\cdot A(u,v)\cdot \alpha(u)\big).
\]

\[
\begin{tikzcd}[row sep=large,column sep=huge]
\B(U,-) \arrow[r, "\iota^*"] \arrow[dr,swap,"1"] & \B(U_i,-) \arrow[d,"\iota_*"] \\
& \B(U,-)
\end{tikzcd}
\qquad
\begin{tikzcd}[row sep=large,column sep=huge]
\B(U_i,-) \arrow[r, "\iota_*"] \arrow[dr,swap,"1"] & \B(U,-) \arrow[d,"\iota^*"] \\
& \B(U_i,-)
\end{tikzcd}
\]

\[
\begin{tikzcd}[row sep=large,column sep=huge]
\B(V,T)\times \B(U,V)\times \B(S,U) \arrow[r,"{\odotB\times 1}"] \arrow[d,swap,"{1\times \odotB}"] & \B(U,T)\times \B(S,U) \arrow[d,"\odotB"] \\
\B(V,T)\times \B(S,V) \arrow[r,swap,"\odotB"] & \B(S,T)
\end{tikzcd}
\]

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>F</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">P</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow><mo stretchy="false">)</mo><mo>≥</mo><msup><mrow><mi mathvariant="normal">P</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow><mi>′</mi></msup><mi>$</mi></mrow></math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>F</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">P</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow><mo stretchy="false">)</mo><mo>≤</mo><msup><mrow><mi mathvariant="normal">P</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow><mi>′</mi></msup><mi>$</mi></mrow></math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>U</mi><mover><mstyle scriptlevel="0"><mo>→</mo></mstyle><mpadded width="+0.833em" lspace="0.556em" voffset="-.2em" height="-.2em"><mrow><mi>a</mi></mrow><mspace depth=".25em" /></mpadded></mover><mi>V</mi><mover><mstyle scriptlevel="0"><mo>→</mo></mstyle><mpadded width="+0.833em" lspace="0.556em" voffset="-.2em" height="-.2em"><mrow><mi>b</mi></mrow><mspace depth=".25em" /></mpadded></mover><mi>T</mi><mi>$</mi></mrow></math>

$(X\starop Y)(U,T)=\join_V X(V,T)\odotB Y(U,V)$

(X\starop Y)(U,T)=\join_V X(V,T)\odotB Y(U,V)

$(X\starop Y)\starop Z=X\starop(Y\starop Z)$

(X\starop Y)\starop Z=X\starop(Y\starop Z)

$F_{U,T}(X)=A(U,T)\vee \join_V X(V,T)\odotB A(U,V)$

$\mathrm{Path}^{! *}:=\I\vee \mathrm{Path}=\mu X.\,(\I\vee A\vee X\starop A)$

\mathrm{Path}^{! *}:=\I\vee \mathrm{Path}=\mu X. (\I\vee A\vee X\starop A)

$1\le \iinc\,\iadj \quad\text{and}\quad \iadj\,\iinc\le 1$

1\le \iinc \iadj \quad\text{and}\quad \iadj \iinc\le 1

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mn>1</mn><mo>≤</mo><mi>\iinc</mi><mspace width="0.167em" /><mi>\iadj</mi><mspace width="1em" /><mtext>and</mtext><mspace width="1em" /><mi>\iadj</mi><mspace width="0.167em" /><mi>\iinc</mi><mo>≤</mo><mn>1</mn><mi>$</mi></mrow></math>

$L_d:=\join_i (v(T_i)\odotB c_{d,i})\odotB w_i$

L_d:=\join_i (v(T_i)\odotB c_{d,i})\odotB w_i

$(X\starop Y)(U,T)\cong \int^V X(V,T)\odotB Y(U,V)$

$F(X\starop A)\ \gtreqless\ F(X)\starop' F(A)$

F(X\starop A)\ \gtreqless\ F(X)\starop' F(A)

$F(A^{\starop n})\cong (FA)^{\starop n}$

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>F</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">P</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi mathvariant="normal">P</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow><mi>′</mi></msup><mi>$</mi></mrow></math>

$t\mapsto \mathrm{Path}_t=\join_{n\ge1} A_t^{\starop n}$

$(X\starop Y)(u,t)=\exists v.\,X(v,t)\wedge Y(u,v)$

(X\starop Y)(u,t)=\exists v. X(v,t)\wedge Y(u,v)