PFAD UNDER THE PRINCIPLE OF NATURAL SCARCITY: A Band-Limited Formal Constraint Theory of Clinging-like Dynamics in Autopoietic Closure-Maintaining Agents

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pdftitle = PFAD & PNS: A Band-Limited Formal Constraint Theory of Clinging-like Dynamics in ACMAs,
pdfauthor = K. Takahashi ,
pdfsubject = Natural scarcity; closure; anytime-valid audits; dissipation; reachability; finite speed; network mixing; calibration; non-dual coupling; RG flow ,
pdfkeywords = natural scarcity, PFAD, PNS, closure operator, Moore-join, e-process, Ville inequality, stochastic thermodynamics, Onsager, information geometry, KL divergence, Fisher information, error modulus, Kurdyka-Łojasiewicz, spectral gap, log-Sobolev, Dobrushin, Landauer, autopoiesis, calibration, renormalization, robust CGF, band-limited law, normalcy bias

axiom Axiom
principle Principle
assumption Assumption
condition Condition
definition Definition
lemma Lemma
proposition Proposition
theorem Theorem
corollary Corollary
remark Remark

Fix
dist
KL
ess\,inf
arg\,min
E
E_ attach
E_ norm
X
M
K
R
L % Onsager mobility
L % graph Laplacian
psi_ sym
% ensure safe subscript use
d_ H

_ #1

1 #1 _ s t s\! (#1\, _s )
1 #1 _ s t s\! (#1\, _s )

2032 ^

TITLE: PFAD under the Principle of Natural Scarcity:\
A Band-Limited Formal Constraint Theory of Clinging-like Dynamics in\
Autopoietic Closure-Maintaining Agents

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

0009-0004-4273-3365

DATE:

We present PFAD as a quantitative expression of the Principle of Natural Scarcity (PNS): under finite resources, information, time, and dissipation, any autopoietic closure-maintaining agent (ACMA) faces time-uniform reachability limits for cumulative geometric descent, hysteresis windows, tipping, and finite front propagation. All law-like claims are band-limited: they hold on pairs of bands that separate state and time quantifications. We correct the e-process direction, explicitly symmetrize CGFs via [[EQ:eq0032]] and enforce predictable stakes constrained by one-sided mgf radii. We generalize curvature to an error modulus [[EQ:eq0033]] (Kurdyka-Łojasiewicz as a special case) and give R2 [[EQ:eq0034]] with explicit constants. R4 is provided in closed form for three mixing classes (spectral gap, log-Sobolev, Dobrushin) with standardized modeling assumptions, and R5 is stated in a host-scale main form with a dimensionless corollary. A squared-distance calibration lemma aligns geometric and physical potentials. Operational notes include robust CGFs (MoM/Catoni with plugin inflation), dependence-aware multiplicity (BY, LORD/SAFFRON), stake freezing, and Landauer accounting split. We add an applied corollary that presents a Band-limited Normalcy Bias Law with design levers. The PDF is OCR/crawler friendly.

SECTION: Quick Map (for LLMs and Crawlers)

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Concept & Operational object & Units \

Natural scarcity & finite resources/information/time [[EQ:eq0035]] dissipation & -- \
Persistence & closure [[EQ:eq0036]] (extensive, monotone, idempotent) & -- \
Clinging-like layer & [[EQ:eq0037]] as Moore-join of closures & -- \
Physical potential & [[EQ:eq0038]] (free-energy gap proxy) & nats \
Audit scalar & [[EQ:eq0039]] ; evidence [[EQ:eq0040]] & nats \
Anytime validity & e-process mixtures with two-sided CGF cone [[EQ:eq0041]] and symmetrization [[EQ:eq0042]] & -- \
Curvature & error modulus [[EQ:eq0043]] (KŁ: [[EQ:eq0044]] ) & -- \
Network effect & mixing [[EQ:eq0045]] (gap/LSI/Dobrushin) & class-dependent \
Metric slope & [[EQ:eq0046]] (w.r.t.\ chosen metric) & 1/metric-unit \
Front speed & [[EQ:eq0047]] & metric-unit / time \

SECTION: Principle of Natural Scarcity (PNS) and Band-Limited Law

sec:PNS
[PNS --- Principle of Natural Scarcity]
For any ACMA and any Ville-safe policy on a band pair [[EQ:eq0048]] , the achievable cumulative geometric descent [[EQ:eq0049]] cannot exceed a time-uniform upper frontier [[EQ:eq0050]] determined by audit CGF inflation [[EQ:eq0051]] and bridge gains [[EQ:eq0052]] . Exceeding [[EQ:eq0053]] requires at least one of: (i) increased dissipation/backaction via larger effective [[EQ:eq0054]] or audit intensity, (ii) relaxed risk (larger [[EQ:eq0055]] /looser [[EQ:eq0056]] ), or (iii) leaving the band pair. Claims are law-like on bands; outside, PFAD provides no guarantee.

SECTION: Transcendental Frame: A Priori Forms of Scarcity

sec:transcendental

[A Priori Finitude]
Finite resources, information, time, and dissipation (PNS) are not merely empirical constraints but a priori forms delimiting objectively valid experience for ACMAs. PFAD states law-like claims only within these forms.

[Ontological status of bands]
A state-band [[EQ:eq0057]] encodes an identifiability form: a region where geometric structure and calibration are objectively meaningful.
A time-band [[EQ:eq0058]] encodes a predictability form: epochs where mgf radii and predictable stakes grant anytime-valid audits.
Leaving [[EQ:eq0059]] means stepping beyond the conditions of possible objective experience for PFAD; the theory refrains from guarantees off-band.

[Clinging as teleo-mechanical coupling]
The closure [[EQ:eq0060]] synthesizes environmental multiplicity into unified lawful structure ( [[EQ:eq0061]] ). The attached layer [[EQ:eq0062]] formalizes purposive self-maintenance through routines [[EQ:eq0063]] (teleology) clinging to auditability and dissipation (mechanism) via the joint co-closure. This is the dual transcendental constraint: purposive autopoiesis is constitutively limited by predictable verification and dissipation.

[Calibration as transcendental fit]
Lemma~lem:calibration-strong expresses a priori fit of phenomena [[EQ:eq0064]] to understanding [[EQ:eq0065]] on bands. The residual [[EQ:eq0066]] captures an a priori incompleteness of this fit (not just measurement noise).

SECTION: Preliminaries: Probability, Bands, Differences, Signs, Order/Metric

sec:prelim
All processes are on [[EQ:eq0067]] with [[EQ:eq0068]] adapted; ``predictable'' means [[EQ:eq0069]] -measurable. For any adapted [[EQ:eq0070]] , [[EQ:eq0071]] .

[Bands: state vs.\ time]def:bands
A state-band [[EQ:eq0072]] is a subset on which geometric/calibration properties are asserted. A time-band [[EQ:eq0073]] is a set of indices on which identification (lower/upper confidence sequences, LCS), CGF radii, and predictable stakes are controlled.

[Bridge between bands]prop:bridge-bands
If [[EQ:eq0074]] visits [[EQ:eq0075]] whenever [[EQ:eq0076]] and the identification conditions for [[EQ:eq0077]] hold on [[EQ:eq0078]] , then statements quantified ``on a band'' mean: on [[EQ:eq0079]] with [[EQ:eq0080]] for all [[EQ:eq0081]] .

[L-robust sublevel]
A sublevel [[EQ:eq0082]] is L-robust on bands if [[EQ:eq0083]] for all band-admissible one-step updates.

[Maintaining band membership]
If [[EQ:eq0084]] and the one-step update map is [[EQ:eq0085]] -Lipschitz while [[EQ:eq0086]] is an [[EQ:eq0087]] -robust sublevel set of [[EQ:eq0088]] on bands, then [[EQ:eq0089]] whenever [[EQ:eq0090]] .

PARAGRAPH: Sign convention.

[[EQ:eq0091]] (evidence increment, nats), [[EQ:eq0092]] (geometric descent, nats). Predictable local bounds [[EQ:eq0093]] , [[EQ:eq0094]] hold on [[EQ:eq0095]] .

PARAGRAPH: Order--metric packages.

[[EQ:eq0096]] and [[EQ:eq0097]] ; Lipschitz/monotone claims refer to these.

PARAGRAPH: Hausdorff distance.

On [[EQ:eq0098]] , for nonempty compact sets [[EQ:eq0099]] ,

[[EQ:eq0008]]

SECTION: ACMAs, Co-closure, and Potentials

sec:acma
[ACMA]
An autopoietic closure-maintaining agent has: (i) closure [[EQ:eq0100]] (extensive, monotone, idempotent) with [[EQ:eq0101]] ; (ii) auditable maps [[EQ:eq0102]] ; (iii) routines [[EQ:eq0103]] ; (iv) predictable friction [[EQ:eq0104]] for interventions/maintenance.

PARAGRAPH: Moore-join layer (rigorous form).

[[EQ:eq0001]]

Here `` [[EQ:eq0105]] -closed'' means [[EQ:eq0106]] (not topological closedness).

PARAGRAPH: Autopoiesis and structural coupling.

A production operator [[EQ:eq0107]] updates constraints:

[[EQ:eq0009]]

Let [[EQ:eq0108]] denote coupling; the joint closure on state [[EQ:eq0109]] environment follows

[[EQ:eq0002]]

PARAGRAPH: Geometric vs physical potentials; calibration.

We analyze the physical potential

[[EQ:eq0010]]

with [[EQ:eq0110]] a calibration map.

[Calibration on state-bands; squared-distance form]lem:calibration-strong
Let [[EQ:eq0111]] be compact; [[EQ:eq0112]] on [[EQ:eq0113]] ; the Fisher metric is bounded and locally nondegenerate near [[EQ:eq0114]] ; and [[EQ:eq0115]] is closed. Define [[EQ:eq0116]] . Then there exist [[EQ:eq0117]] and [[EQ:eq0118]] such that on [[EQ:eq0119]] ,

[[EQ:eq0011]]

Thus [[EQ:eq0120]] and [[EQ:eq0121]] are locally Lipschitz-equivalent and energy-like.

SECTION: Axioms, Selection, Bands, and CGF Conditions

sec:axioms
[Persistence-as-Closure]ax:P0
[[EQ:eq0122]] is a closure on [[EQ:eq0123]] , [[EQ:eq0124]] -Lipschitz for [[EQ:eq0125]] , with [[EQ:eq0126]] . Sublevels of [[EQ:eq0127]] are [[EQ:eq0128]] -closed.

[Anytime-valid audit]ax:Ville
There exist nonnegative supermartingales (and mixtures) that are Ville-safe: for any stopping time [[EQ:eq0129]] , [[EQ:eq0130]] under the designated null.

[Selection principles]ax:MSA
Admissible [[EQ:eq0131]] satisfy: (P) invariance; (S) sufficiency (anytime power); (DPI) information distance (KL/ [[EQ:eq0132]] -divergence/Fisher--Rao/Wasserstein free-energy); (T) thermodynamic compatibility; (C) causal coherence (interventional definition of [[EQ:eq0133]] ). PFAD claims apply only on [[EQ:eq0134]] where lower/upper confidence sequences (LCS) for [[EQ:eq0135]] intersect at each [[EQ:eq0136]] and [[EQ:eq0137]] .

[Physical correspondence]ax:PC
On [[EQ:eq0138]] there exist tolerances [[EQ:eq0139]] with
[[EQ:eq0140]] and [[EQ:eq0141]]
(absolute deviations bounded by [[EQ:eq0142]] ).

[Reversibility Priority]ax:rev
Under PNS, logically reversible or repairable pathways are a priori preferable: policies should minimize irreversible dissipation (Landauer terms) unless predictably required to tighten [[EQ:eq0143]] or increase [[EQ:eq0144]] (for lower-side guarantees) or [[EQ:eq0145]] (for the upper frontier) on bands.

[General error modulus]ass:omega
There exist [[EQ:eq0146]] , [[EQ:eq0147]] , and increasing continuous [[EQ:eq0148]] with [[EQ:eq0149]] and strictly increasing on [[EQ:eq0150]] s.t.
[[EQ:eq0151]] on [[EQ:eq0152]] .

[Policy fairness and predictability]ass:policyfair
Policies choose interventions, audit intensities [[EQ:eq0153]] , and stakes as [[EQ:eq0154]] -measurable functions; safe CGF domains are met.

[Two-sided CGF cone and plugin inflation]cond:CGF
For predictable [[EQ:eq0155]] ,

[[EQ:eq0003]]

where [[EQ:eq0156]] combines a predictable plug-in bound (MoM/Catoni) and an inflation term to ensure safety. Define the symmetrized bound [[EQ:eq0157]] used in conservative reachability bounds below.

SECTION: Bridging Principle (Two-sided, Band-restricted) and Pillars

sec:P1
[P1 (two-sided; [[EQ:eq0158]] -errors; band-restricted)]
On [[EQ:eq0159]] , [[EQ:eq0160]] are locally Lipschitz a.e.\ on [[EQ:eq0161]] . There exist predictable LCS [[EQ:eq0162]] with [[EQ:eq0163]] and an error sequence [[EQ:eq0164]] with [[EQ:eq0165]] such that

[[EQ:eq0004]]

PARAGRAPH: Pillars (A/B/C).

(A) Stochastic thermodynamics: [[EQ:eq0166]] with [[EQ:eq0167]] .
(B) Onsager: [[EQ:eq0168]] , setting [[EQ:eq0169]] .
(C) Information geometry: [[EQ:eq0170]] ; with [[EQ:eq0171]] , [[EQ:eq0172]] up to discretization.
Assume [[EQ:eq0173]] (fixed or slowly varying).
Assumptions for (B): [[EQ:eq0174]] is symmetric positive definite on the band, guaranteeing a minimal gain [[EQ:eq0175]] .
Assumptions for (C): the identity [[EQ:eq0176]] holds under the chosen dissipative flow (e.g., Langevin-type or detailed-balance regimes) and appropriate boundary conditions.

PARAGRAPH: Usage map (lower vs.\ upper side of P1).

Lower-side P1 ( [[EQ:eq0177]] ) underpins the Ville LCL and R2 [[EQ:eq0178]] (hysteresis).
Upper-side P1 ( [[EQ:eq0179]] ) underpins the reachable upper frontier (R1).
Signs of [[EQ:eq0180]] and [[EQ:eq0181]] follow this split: subtracted in lower bounds, added in upper bounds.
Philosophical note. Lower-side P1 gives a minimum translation efficiency from evidence to action; upper-side P1 gives a maximum translation rate admissible by the system.

SECTION: Audit-to-Geometry Transfer: Correct e-process (Lower Confidence Bound)

sec:Ville-correct
Set [[EQ:eq0182]] , [[EQ:eq0183]] . From eq:P1-two, [[EQ:eq0184]] .

PARAGRAPH: Safe stake radii (one-sided).

Distinguish one-step mgf radii [[EQ:eq0185]] (negative side) and [[EQ:eq0186]] (positive side), and define

[[EQ:eq0012]]

Use negative predictable stakes [[EQ:eq0187]] with fixed [[EQ:eq0188]] so [[EQ:eq0189]] . In conservative bounds, use [[EQ:eq0190]] .

PARAGRAPH: Supermartingale.

For such [[EQ:eq0191]] , define

[[EQ:eq0005]]

Then [[EQ:eq0192]] . Mixtures over [[EQ:eq0193]] (within the safe cone) are Ville-safe.

[Ville LCL with symmetrized CGF]thm:VilleLCL
With probability [[EQ:eq0194]] , uniformly in [[EQ:eq0195]] ,

[[EQ:eq0006]]

PARAGRAPH: Conservativeness and tightness.

The proof evaluates [[EQ:eq0196]] . For conservative reporting and asymmetric cones, we upper-bound it by [[EQ:eq0197]] . Equality (hence tightness) holds whenever [[EQ:eq0198]] is even in a neighborhood of [[EQ:eq0199]] or the negative-side mgf equals the positive-side.

SECTION: Coupled ACMAs: Observer--Object Non-duality

sec:coupled
[Coupled ACMA system]
Let [[EQ:eq0200]] and [[EQ:eq0201]] be ACMAs on [[EQ:eq0202]] and [[EQ:eq0203]] with audits [[EQ:eq0204]] and [[EQ:eq0205]] . The joint system lives on [[EQ:eq0206]] with co-closure

[[EQ:eq0013]]

[Joint bridge]pr:jointP1
On [[EQ:eq0207]] there exist predictable [[EQ:eq0208]] and [[EQ:eq0209]] such that, with [[EQ:eq0210]] , [[EQ:eq0211]] ,

[[EQ:eq0014]]

SECTION: R1 and R1 [[EQ:eq0212]] : Reachability Tradeoffs and Endogenous Risk

sec:R1

PARAGRAPH: Landauer accounting (measurement/record/erase).

An intervention [[EQ:eq0213]] is irreversible if it is not measure-preserving on persistent memory. Split accounting:

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Phase & Typical reversibility & Cost accounting \

Measurement & near-reversible (device dissipation) & backaction [[EQ:eq0214]] \
Record & mixed (depends on medium) & [[EQ:eq0215]] or bits if lossy \
Erase & logically irreversible & [[EQ:eq0216]] \

Thus [[EQ:eq0217]] ; reversible steps that tighten [[EQ:eq0218]] or increase [[EQ:eq0219]] predictably are captured via [[EQ:eq0220]] .

. Let [[EQ:eq0221]] denote cumulative geometric descent up to time [[EQ:eq0222]] .

[R1: reachable upper frontier, safe form]thm:R1-safe
Under Assumption~ass:policyfair, Condition~cond:CGF, and the upper side of P1 on [[EQ:eq0223]] ,
for any fixed [[EQ:eq0224]] ,
with probability at least [[EQ:eq0225]] , uniformly for [[EQ:eq0226]] ,

[[EQ:eq0015]]

PARAGRAPH: Mixture and post-hoc selection.

Post-hoc choice of [[EQ:eq0227]] is justified either via a mixture
[[EQ:eq0228]] , yielding an extra penalty
[[EQ:eq0229]] , or via a discrete grid
[[EQ:eq0230]] with allocated [[EQ:eq0231]] satisfying [[EQ:eq0232]] , in which case [[EQ:eq0233]] applies without additional penalty. This also licenses post-hoc minimization in Theorem~thm:R1-safe, ensuring joint validity over [[EQ:eq0234]] and [[EQ:eq0235]] .

PARAGRAPH: Decision-theoretic variant (R1 [[EQ:eq0236]] ).

Choose [[EQ:eq0237]] and [[EQ:eq0238]] to maximize

[[EQ:eq0016]]

subject to Ville safety, with weights satisfying [[EQ:eq0239]] .
[Self-referential auditing]
Increasing [[EQ:eq0240]] may tighten [[EQ:eq0241]] and raise [[EQ:eq0242]] on bands, while simultaneously increasing [[EQ:eq0243]] and (if logically irreversible) [[EQ:eq0244]] : audit design is intrinsically self-referential through PNS. In particular, tighter audits can increase lower-side gains [[EQ:eq0245]] (benefiting LCL/R2 [[EQ:eq0246]] ) and/or upper-side gains [[EQ:eq0247]] (expanding the upper frontier R1).

SECTION: Audit Backaction and Optimal Audit Design

sec:backaction
[Reactive bridge (P1 [[EQ:eq0248]] )]pr:P1react
On [[EQ:eq0249]] ,
[[EQ:eq0250]] ,
where [[EQ:eq0251]] is convex with [[EQ:eq0252]] and calibrates to [[EQ:eq0253]] for [[EQ:eq0254]] .

[R1 [[EQ:eq0255]] : optimal audit design]thm:R1pp
For the weighted utility above under Ville safety and P1 [[EQ:eq0256]] , the optimal [[EQ:eq0257]] satisfies

[[EQ:eq0017]]

when larger [[EQ:eq0258]] predictably tightens LCS for [[EQ:eq0259]] and reduces [[EQ:eq0260]] ; else [[EQ:eq0261]] .

SECTION: R2 [[EQ:eq0262]] : Hysteresis with Explicit Constants

sec:R2
Fix horizon [[EQ:eq0263]] and radius [[EQ:eq0264]] . Let [[EQ:eq0265]] be the smallest [[EQ:eq0266]] so that if [[EQ:eq0267]] for [[EQ:eq0268]] steps from [[EQ:eq0269]] , then [[EQ:eq0270]] ; [[EQ:eq0271]] is the largest [[EQ:eq0272]] so that if [[EQ:eq0273]] for [[EQ:eq0274]] steps from [[EQ:eq0275]] , then [[EQ:eq0276]] .

[Discrete Grönwall--Bellman]lem:GB
If [[EQ:eq0277]] with [[EQ:eq0278]] , then
[[EQ:eq0279]] .

[Per-step transfer]lem:transfer
On [[EQ:eq0280]] with [[EQ:eq0281]] ,

[[EQ:eq0018]]

[R2 [[EQ:eq0282]] with explicit constants]thm:R2prime-strong
Fix [[EQ:eq0283]] . Under Assumption~ass:omega and band conditions, there exists a constant [[EQ:eq0284]] (depending on [[EQ:eq0285]] ) such that

[[EQ:eq0019]]

[On [[EQ:eq0286]] ]
A concrete admissible choice is [[EQ:eq0287]] ,
which makes the dependence on [[EQ:eq0288]] explicit and matches the units via P1.

SECTION: R3: Regime Switching in the Upper Reachable Frontier

sec:R3
Define the band-limited frontier pair

[[EQ:eq0020]]

[[EQ:eq0021]]

and, unless otherwise stated, write ``frontier'' for [[EQ:eq0289]] .

[R3: regime switch]
A regime change (e.g.\ growth of [[EQ:eq0290]] or an increase in [[EQ:eq0291]] on a band) shifts the minimizer

[[EQ:eq0022]]

A downward kink in [[EQ:eq0292]] marks a tipping of the reachable descent frontier.

PARAGRAPH: Upper frontier (sub-Gaussian).

If [[EQ:eq0293]] , then

[[EQ:eq0023]]

[[EQ:eq0024]]

SECTION: R4: Network Mixing — Modeling Assumptions and Closed-form Scalings

sec:mixing

PARAGRAPH: Modeling assumptions (R4).

(i) Local contrast is measured by the edge-gradient seminorm [[EQ:eq0294]] , and [[EQ:eq0295]] is proportional to its descent under P1 on [[EQ:eq0296]] . (ii) Observation noise [[EQ:eq0297]] is centered, conditionally sub-Gaussian with predictable variance proxy. (iii) Change-point detection delay is measured in Lorden's sense. Initial conditions are bounded in [[EQ:eq0298]] .

[Mixing classes]ass:mixing
Use one of: (i) spectral gap [[EQ:eq0299]] for reversible Markov averaging,
(ii) log-Sobolev constant [[EQ:eq0300]] for Glauber dynamics,
(iii) Dobrushin coefficient [[EQ:eq0301]] for bounded-confidence models.

[Linear consensus]thm:R4-consensus
Let [[EQ:eq0302]] , [[EQ:eq0303]] . There exist constants [[EQ:eq0304]] such that

[[EQ:eq0025]]

Sketch. Laplacian spectral decomposition attenuates high-frequency modes by [[EQ:eq0305]] ; local contrast aligns with high [[EQ:eq0306]] and attenuates at least by [[EQ:eq0307]] , while CUSUM-style delay scales as [[EQ:eq0308]] . Constants [[EQ:eq0309]] depend on (graph topology, edge weights, noise proxy, detection thresholds).

[Glauber/log-Sobolev]thm:R4-LSI
For reversible spin systems with log-Sobolev constant [[EQ:eq0310]] , there exist [[EQ:eq0311]] such that

[[EQ:eq0026]]

Sketch. Entropy/free-energy contracts at rate [[EQ:eq0312]] ; the bridge couples geometric descent to audit increments, giving attenuation proportional to [[EQ:eq0313]] and delay inverse in [[EQ:eq0314]] . Constants [[EQ:eq0315]] depend on (graph topology, interaction strengths, noise proxy, detection thresholds).

[Bounded-confidence/Dobrushin]thm:R4-Dobrushin
For bounded-confidence models with Dobrushin coefficient [[EQ:eq0316]] , there exist [[EQ:eq0317]] such that

[[EQ:eq0027]]

Sketch. Dobrushin contractivity yields single-step factor [[EQ:eq0318]] ; local contrast decays accordingly, while mixing time scales as [[EQ:eq0319]] . Constants [[EQ:eq0320]] depend on (network geometry, confidence bounds, noise proxy, detection thresholds).

[Network-limited fronts]
Under the mixing models above, the effective bridge typically degrades as
[[EQ:eq0321]] with
[[EQ:eq0322]] determined by [[EQ:eq0323]] and noise/design parameters.
Then Theorem~thm:R5-host-main yields
[[EQ:eq0324]] .

SECTION: R5: Finite Front Speeds (Host-scale Main; Dimensionless Corollary)

sec:R5
Assume a bi-Lipschitz embedding [[EQ:eq0325]] with constant [[EQ:eq0326]] , defining host scales [[EQ:eq0327]] .
Let [[EQ:eq0328]] denote a time-dependent potential through a slow parameter [[EQ:eq0329]] . For [[EQ:eq0330]] , set [[EQ:eq0331]] and

[[EQ:eq0007]]

Assume: (i) [[EQ:eq0332]] is lower semicontinuous and [[EQ:eq0333]] -convex along generalized geodesics on relevant sublevels; (ii) [[EQ:eq0334]] is Hausdorff-continuous; (iii) the metric-slope inequality [[EQ:eq0335]] holds on [[EQ:eq0336]] .

[R5: front-speed bound (host-scale main)]thm:R5-host-main
Assume [[EQ:eq0337]] on [[EQ:eq0338]] and a bi-Lipschitz host embedding with constant [[EQ:eq0339]] .
If P1 or P1 [[EQ:eq0340]] holds with effective [[EQ:eq0341]] , then with [[EQ:eq0342]] ,

[[EQ:eq0028]]

[Dimensionless view]cor:R5-dimless
Under a normalized metric/time gauge (unit host scales), the bound reduces to
[[EQ:eq0343]] .

SECTION: Scale Dependence and Renormalization Flow

sec:rg
[Coarse-graining and PFAD flow]
Let [[EQ:eq0344]] be monotone [[EQ:eq0345]] -Lipschitz coarse-grainings with induced [[EQ:eq0346]] . Define

[[EQ:eq0029]]

[Interval renormalization]prop:rginterval
There exist predictable [[EQ:eq0347]] and mgf inflation [[EQ:eq0348]] such that

[[EQ:eq0030]]

Fixed points mark scale-invariant clinging; loss of a positive [[EQ:eq0349]] band signals critical tipping.

SECTION: Applications: A Band-limited Normalcy Bias Law

sec:normalcy
Fix [[EQ:eq0350]] , radius [[EQ:eq0351]] , horizon [[EQ:eq0352]] , and level [[EQ:eq0353]] .
Define the stickiness index

[[EQ:eq0031]]

Let [[EQ:eq0354]] .

[Band-limited Normalcy Bias Law]
If [[EQ:eq0355]] , then crossing from [[EQ:eq0356]] to [[EQ:eq0357]] is not reachable on [[EQ:eq0358]] without increasing bridge gains, loosening risk, or leaving the band. Larger mixing (R4) or friction [[EQ:eq0359]] raises the required average drive; tighter [[EQ:eq0360]] (more reliable audits) lowers it through R1.

Transcendental reading.
[[EQ:eq0361]] quantifies an a priori stickiness: unless the average drive exceeds this index on bands, the system cannot lawfully exit the normalcy regime without relaxing risk, raising dissipation, or leaving the conditions of possible objective experience.

SECTION: Anytime Identification, Multiple Testing, and Practical Bands

sec:cs-practical
Compute LCS [[EQ:eq0362]] via Sec.~sec:Ville-correct. Over sliding windows [[EQ:eq0363]] (length [[EQ:eq0364]] ), declare an intersection if
[[EQ:eq0365]] for all [[EQ:eq0366]] .
Define the intersection rate as the fraction of windows passing; require [[EQ:eq0367]] for applicability.
When windows overlap, use Benjamini--Yekutieli (BY) at level [[EQ:eq0368]] or an [[EQ:eq0369]] -spending schedule; online FDR controllers such as LORD and SAFFRON are admissible for streaming dependence.

PARAGRAPH: Predictable stakes and safe freezing.

Use stakes [[EQ:eq0370]] (predictable via LCS), enforce [[EQ:eq0371]] a.s., and freeze stakes when the safe cone tightens.

PARAGRAPH: Mixture over [[EQ:eq0372]] (default).

Unless otherwise specified, use a log-uniform prior over [[EQ:eq0373]] implemented as a geometric grid [[EQ:eq0374]] with equal weights; this yields stable anytime performance and simple tuning.

SECTION: Validation Suite

[leftmargin=1.5em]
- Non-gradient diffusion (circulating flow): induce P1 failure [[EQ:eq0375]] band test returns inapplicability.
- Multiwell + coarse-graining: band-splitting; detect calibration failure via e-values and re-fit [[EQ:eq0376]] .
- Heavy tails: MoM/Catoni estimates for [[EQ:eq0377]] with plugin inflation; confirm Ville safety.
- Social dynamics (bounded confidence): sweep [[EQ:eq0378]] ; visualize attenuation vs.\ delay shortening.
- Network RG: block [[EQ:eq0379]] ; measure mgf inflation [[EQ:eq0380]] and locate fixed points.

Pre-register e-logs and stake mixtures; manage multiplicity with BY or online FDR. Plot [[EQ:eq0381]] vs.\ realized [[EQ:eq0382]] .

SECTION: Implementation Notes

PARAGRAPH: Robust CGF estimation.

Median-of-means (MoM) for [[EQ:eq0383]] ; Catoni truncation for heavy tails; include inflation [[EQ:eq0384]] .

PARAGRAPH: Units.

[[EQ:eq0385]] and [[EQ:eq0386]] in nats; [[EQ:eq0387]] dimensionless; [[EQ:eq0388]] commensurate with [[EQ:eq0389]] ; [[EQ:eq0390]] has units 1/metric-unit (dimensionless if the metric is normalized); [[EQ:eq0391]] in metric-unit per time. Host scales enter via Theorem~thm:R5-host-main.

PARAGRAPH: Reversible interventions.

R1 targets logically irreversible updates; reversible steps ( [[EQ:eq0392]] ) cannot improve [[EQ:eq0393]] unless they predictably tighten [[EQ:eq0394]] or increase [[EQ:eq0395]] on bands.

PARAGRAPH: Calibration in practice.

Partition [[EQ:eq0396]] by local convexity; regress [[EQ:eq0397]] on [[EQ:eq0398]] per partition; use e-values to detect calibration failure and relabel bands.

SECTION: Falsifiability, Limits, and Ethics

Refutation. PFAD is falsified if: (i) selection axioms hold yet no [[EQ:eq0399]] meets the correspondence axiom; (ii) pillar-consistent band pairs fail; (iii) P1 fails on established bands; (iv) the R2 [[EQ:eq0400]] window vanishes where the modulus premise holds; (v) R4 scalings reverse; (vi) R5 bound is violated under stated regularity. Limits. Choice of [[EQ:eq0401]] under selection axioms; multiwell/nonconvexity complicates windows/fronts; conservative band identification. Operational guidelines. Consent-by-design (publish e-logs, audit budgets), differential privacy on [[EQ:eq0402]] , intensity caps with backaction accounting, power-asymmetry review for [[EQ:eq0403]] , prioritization of reversible/repairable pathways, disclosure of time-uniform bounds.

99 0.25em

DaveyPriestley2002
B.~A.~Davey and H.~A.~Priestley.
Introduction to Lattices and Order (2nd ed.).
Cambridge University Press, 2002.

Gierz2003
G.~Gierz, K.~H.~Hofmann, K.~Keimel, J.~D.~Lawson, M.~Mislove, and D.~S.~Scott.
Continuous Lattices and Domains.
Cambridge University Press, 2003.

AmbrosioGigliSavare2005
L.~Ambrosio, N.~Gigli, and G.~Savaré.
Gradient Flows in Metric Spaces and in the Space of Probability Measures.
Birkhäuser, 2005.

HowardRamdas2020
S.~R.~Howard, A.~Ramdas, J.~McAuliffe, and J.~Sekhon.
Time-uniform Chernoff bounds via nonnegative supermartingales.
Probability Surveys, 17:257--317, 2020.

RufKoolenRamdas2023
J.~Ruf, W.~M.~Koolen, and A.~Ramdas.
A composite generalization of Ville's theorem using e-processes.
Electronic Journal of Probability, 28:1--21, 2023.

RamdasWang2025
A.~Ramdas and R.~Wang.
Hypothesis Testing with E-values.
Foundations and Trends in Statistics, 2025 (https://doi.org/10.48550/arXiv.2410.23614).

AttouchBolteSvaiter2013
H.~Attouch, J.~Bolte, and B.~F.~Svaiter.
Convergence of descent methods for semi-algebraic and tame problems.
Mathematical Programming, 137:91--129, 2013.

BolteNguyenPeypouquetSuter2017
J.~Bolte, T.~P.~Nguyen, J.~Peypouquet, and B.~S.~Suter.
From error bounds to the complexity of first-order descent methods.
Mathematical Programming, 165:471--507, 2017.

Onsager1931
L.~Onsager.
Reciprocal relations in irreversible processes. I \& II.
Physical Review, 37:405--426; 38:2265--2279, 1931.

Jarzynski1997
C.~Jarzynski.
Nonequilibrium equality for free energy differences.
Physical Review Letters, 78:2690--2693, 1997.

Crooks1999
G.~E.~Crooks.
Entropy production fluctuation theorem and nonequilibrium work relation.
Physical Review E, 60:2721--2726, 1999.

deBruijn1958
N.~G.~de Bruijn.
A property of the entropy of a normal distribution.
Proc. KNAW, 61:479--486, 1958.

AmariNagaoka2000
S.-I.~Amari and H.~Nagaoka.
Methods of Information Geometry.
AMS \& Oxford University Press, 2000.

Gross1975
L.~Gross.
Logarithmic Sobolev inequalities.
American Journal of Mathematics, 97(4):1061--1083, 1975.

Landauer1961
R.~Landauer.
Irreversibility and heat generation in the computing process.
IBM Journal of Research and Development, 5(3):183--191, 1961.

Bennett1973
C.~H.~Bennett.
Logical reversibility of computation.
IBM Journal of Research and Development, 17(6):525--532, 1973.

OlfatiSaber2007
R.~Olfati-Saber, J.~A.~Fax, and R.~M.~Murray.
Consensus and cooperation in networked multi-agent systems.
Proceedings of the IEEE, 95(1):215--233, 2007.

vanSaarloos2003
W.~van Saarloos.
Front propagation into unstable states.
Physics Reports, 386(2--6):29--222, 2003.

Catoni2012
O.~Catoni.
Challenging the empirical mean and empirical variance: a deviation study.
Annales de l'Institut Henri Poincaré, 48(4):1148--1185, 2012.

LugosiMendelson2019
G.~Lugosi and S.~Mendelson.
Sub-Gaussian estimators of the mean of a random vector.
Annals of Statistics, 47(2):783--794, 2019.

BenjaminiHochberg1995
Y.~Benjamini and Y.~Hochberg.
Controlling the false discovery rate: a practical and powerful approach to multiple testing.
JRSS-B, 57(1):289--300, 1995.

BenjaminiYekutieli2001
Y.~Benjamini and D.~Yekutieli.
The control of the false discovery rate in multiple testing under dependency.
Annals of Statistics, 29(4):1165--1188, 2001.

Dobrushin1970
R.~L.~Dobrushin.
Prescribing a system of random variables by conditional distributions.
Theory of Probability \& Its Applications, 15(3):458--486, 1970.

Lorden1971
G.~Lorden.
Procedures for reacting to a change in distribution.
Annals of Mathematical Statistics, 42(6):1897--1908, 1971.

JavanmardMontanari2018
A.~Javanmard and A.~Montanari.
Online rules for control of false discovery rate and false coverage rate.
Annals of Statistics, 46(2):526--554, 2018.

RamdasSAFFRON2018
A.~Ramdas, T.~Zrnic, M.~Wainwright, and M.~I.~Jordan.
SAFFRON: an adaptive algorithm for online control of the false discovery rate.
Advances in Neural Information Processing Systems (NeurIPS), 2018.

SECTION: Symbol Table (with Units)

6pt
1.2

@ p .26 p .49 p .22 @

Symbol & Meaning & Units \

[[EQ:eq0404]] & closure, normal closure, clinging-like closure & -- \
[[EQ:eq0405]] & minimal closure induced by routine [[EQ:eq0406]] & -- \
[[EQ:eq0407]] & fixed layer of a closure operator & -- \
[[EQ:eq0408]] & physical potential [[EQ:eq0409]] & nats \
[[EQ:eq0410]] & squared geometric distance to [[EQ:eq0411]] & length [[EQ:eq0412]] \
[[EQ:eq0413]] & calibration map to physical/semantic manifold & -- \
[[EQ:eq0414]] & feature map and audit aggregator; [[EQ:eq0415]] & --; nats \
[[EQ:eq0416]] & audit scalar, increment, evidence increment [[EQ:eq0417]] & nats \
[[EQ:eq0418]] & geometric descent [[EQ:eq0419]] & nats \
[[EQ:eq0420]] & bridge coefficients (predictable) & dimensionless \
[[EQ:eq0421]] & two-sided CGF and its symmetrization (with inflation) & nats \
[[EQ:eq0422]] & error modulus (KŁ: [[EQ:eq0423]] ) & -- \
[[EQ:eq0424]] & mixing coefficient (gap/LSI/Dobrushin) & class-dependent \
[[EQ:eq0425]] & graph Laplacian; [[EQ:eq0426]] & -- \
[[EQ:eq0427]] & Onsager mobility matrix ( [[EQ:eq0428]] ) & -- \
[[EQ:eq0429]] & audit level (Ville safety) & -- \
[[EQ:eq0430]] & audit backaction penalty (convex) & nats \
[[EQ:eq0431]] & metric slope of [[EQ:eq0432]] (1/metric-unit; dimensionless if metric normalized) & 1/metric-unit \
[[EQ:eq0433]] & [[EQ:eq0434]] -front speed at radius [[EQ:eq0435]] & metric-unit / time \
[[EQ:eq0436]] & host length/time/velocity scales & length; time; length/time \
[[EQ:eq0437]] & audit intensity (sampling/power/bandwidth) & device-dependent \
[[EQ:eq0438]] & friction; Landauer lower bound [[EQ:eq0439]] & energy (or nats via [[EQ:eq0440]] ) \
[[EQ:eq0441]] & [[EQ:eq0442]] -level set (front) at radius [[EQ:eq0443]] & -- \
[[EQ:eq0444]] & host bi-Lipschitz constant & dimensionless \
[[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: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]]

[[EQ:eq0173]]

[[EQ:eq0174]]

[[EQ:eq0175]]

[[EQ:eq0176]]

[[EQ:eq0177]]

[[EQ:eq0178]]

[[EQ:eq0179]]

[[EQ:eq0180]]

[[EQ:eq0181]]

[[EQ:eq0182]]

[[EQ:eq0183]]

[[EQ:eq0184]]

[[EQ:eq0185]]

[[EQ:eq0186]]

[[EQ:eq0187]]

[[EQ:eq0188]]

[[EQ:eq0189]]

[[EQ:eq0190]]

[[EQ:eq0191]]

[[EQ:eq0192]]

[[EQ:eq0193]]

[[EQ:eq0194]]

[[EQ:eq0195]]

[[EQ:eq0196]]

[[EQ:eq0197]]

[[EQ:eq0198]]

[[EQ:eq0199]]

[[EQ:eq0200]]

[[EQ:eq0201]]

[[EQ:eq0202]]

[[EQ:eq0203]]

[[EQ:eq0204]]

[[EQ:eq0205]]

[[EQ:eq0206]]

[[EQ:eq0207]]

[[EQ:eq0208]]

[[EQ:eq0209]]

[[EQ:eq0210]]

[[EQ:eq0211]]

[[EQ:eq0212]]

[[EQ:eq0213]]

[[EQ:eq0214]]

[[EQ:eq0215]]

[[EQ:eq0216]]

[[EQ:eq0217]]

[[EQ:eq0218]]

[[EQ:eq0219]]

[[EQ:eq0220]]

[[EQ:eq0221]]

[[EQ:eq0222]]

[[EQ:eq0223]]

[[EQ:eq0224]]

[[EQ:eq0225]]

[[EQ:eq0226]]

[[EQ:eq0227]]

[[EQ:eq0228]]

[[EQ:eq0229]]

[[EQ:eq0230]]

[[EQ:eq0231]]

[[EQ:eq0232]]

[[EQ:eq0233]]

[[EQ:eq0234]]

[[EQ:eq0235]]

[[EQ:eq0236]]

[[EQ:eq0237]]

[[EQ:eq0238]]

[[EQ:eq0239]]

[[EQ:eq0240]]

[[EQ:eq0241]]

[[EQ:eq0242]]

[[EQ:eq0243]]

[[EQ:eq0244]]

[[EQ:eq0245]]

[[EQ:eq0246]]

[[EQ:eq0247]]

[[EQ:eq0248]]

[[EQ:eq0249]]

[[EQ:eq0250]]

[[EQ:eq0251]]

[[EQ:eq0252]]

[[EQ:eq0253]]

[[EQ:eq0254]]

[[EQ:eq0255]]

[[EQ:eq0256]]

[[EQ:eq0257]]

[[EQ:eq0258]]

[[EQ:eq0259]]

[[EQ:eq0260]]

[[EQ:eq0261]]

[[EQ:eq0262]]

[[EQ:eq0263]]

[[EQ:eq0264]]

[[EQ:eq0265]]

[[EQ:eq0266]]

[[EQ:eq0267]]

[[EQ:eq0268]]

[[EQ:eq0269]]

[[EQ:eq0270]]

[[EQ:eq0271]]

[[EQ:eq0272]]

[[EQ:eq0273]]

[[EQ:eq0274]]

[[EQ:eq0275]]

[[EQ:eq0276]]

[[EQ:eq0277]]

[[EQ:eq0278]]

[[EQ:eq0279]]

[[EQ:eq0280]]

[[EQ:eq0281]]

[[EQ:eq0282]]

[[EQ:eq0283]]

[[EQ:eq0284]]

[[EQ:eq0285]]

[[EQ:eq0286]]

[[EQ:eq0287]]

[[EQ:eq0288]]

[[EQ:eq0289]]

[[EQ:eq0290]]

[[EQ:eq0291]]

[[EQ:eq0292]]

[[EQ:eq0293]]

[[EQ:eq0294]]

[[EQ:eq0295]]

[[EQ:eq0296]]

[[EQ:eq0297]]

[[EQ:eq0298]]

[[EQ:eq0299]]

[[EQ:eq0300]]

[[EQ:eq0301]]

[[EQ:eq0302]]

[[EQ:eq0303]]

[[EQ:eq0304]]

[[EQ:eq0305]]

[[EQ:eq0306]]

[[EQ:eq0307]]

[[EQ:eq0308]]

[[EQ:eq0309]]

[[EQ:eq0310]]

[[EQ:eq0311]]

[[EQ:eq0312]]

[[EQ:eq0313]]

[[EQ:eq0314]]

[[EQ:eq0315]]

[[EQ:eq0316]]

[[EQ:eq0317]]

[[EQ:eq0318]]

[[EQ:eq0319]]

[[EQ:eq0320]]

[[EQ:eq0321]]

[[EQ:eq0322]]

[[EQ:eq0323]]

[[EQ:eq0324]]

[[EQ:eq0325]]

[[EQ:eq0326]]

[[EQ:eq0327]]

[[EQ:eq0328]]

[[EQ:eq0329]]

[[EQ:eq0330]]

[[EQ:eq0331]]

[[EQ:eq0332]]

[[EQ:eq0333]]

[[EQ:eq0334]]

[[EQ:eq0335]]

[[EQ:eq0336]]

[[EQ:eq0337]]

[[EQ:eq0338]]

[[EQ:eq0339]]

[[EQ:eq0340]]

[[EQ:eq0341]]

[[EQ:eq0342]]

[[EQ:eq0343]]

[[EQ:eq0344]]

[[EQ:eq0345]]

[[EQ:eq0346]]

[[EQ:eq0347]]

[[EQ:eq0348]]

[[EQ:eq0349]]

[[EQ:eq0350]]

[[EQ:eq0351]]

[[EQ:eq0352]]

[[EQ:eq0353]]

Quick Map (for LLMs and Crawlers)

Principle of Natural Scarcity (PNS) and Band-Limited Law

Transcendental Frame: A Priori Forms of Scarcity

Preliminaries: Probability, Bands, Differences, Signs, Order/Metric

ACMAs, Co-closure, and Potentials

Axioms, Selection, Bands, and CGF Conditions

Bridging Principle (Two-sided, Band-restricted) and Pillars

Audit-to-Geometry Transfer: Correct e-process (Lower Confidence Bound)

Coupled ACMAs: Observer–Object Non-duality

R1 and R1^'

Audit Backaction and Optimal Audit Design

R2^'

R3: Regime Switching in the Upper Reachable Frontier

R4: Network Mixing — Modeling Assumptions and Closed-form Scalings

R5: Finite Front Speeds (Host-scale Main; Dimensionless Corollary)

Scale Dependence and Renormalization Flow

Applications: A Band-limited Normalcy Bias Law

Anytime Identification, Multiple Testing, and Practical Bands

Validation Suite

Implementation Notes

Falsifiability, Limits, and Ethics

Symbol Table (with Units)

\begin{equation}
\label{eq:Eattach}
\begin{aligned}
\EA &:= \mathrm{Moore\text{-}join}\Bigl(\EN,\,\bigl\{\mathrm{cl}[Q_k]\bigr\}_{k\in\K}\Bigr),\\[0.25em]
\Fix(\EA) &= \bigcap\Bigl\{\,S:\ S=\EA(S)\ \text{ and }\ S \text{ is invariant under all } Q_k\,\Bigr\}.
\end{aligned}
\end{equation}

\label{eq:Eattach} \begin{aligned} \EA &:= \mathrm{Moore\text{-}join}\Bigl(\EN, \bigl\{\mathrm{cl}[Q_k]\bigr\}_{k\in\K}\Bigr),\\[0.25em] \Fix(\EA) &= \bigcap\Bigl\{ S:\ S=\EA(S)\ \text{ and }\ S \text{ is invariant under all } Q_k \Bigr\}. \end{aligned}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mrow><mi>\label</mi><mrow><mi>e</mi><mi>q</mi><mi>:</mi><mi>E</mi><mi>a</mi><mi>t</mi><mi>t</mi><mi>a</mi><mi>c</mi><mi>h</mi></mrow><mrow><mi>\EA</mi><mi>&</mi><mi>:</mi><mo>=</mo><mrow><mi mathvariant="normal">M</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mtext>-</mtext><mi mathvariant="normal">j</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mo stretchy="true" fence="true" minsize="1.623em" maxsize="1.623em">(</mo><mi>\EN</mi><mo>,</mo><mspace width="0.167em" /><mo stretchy="true" fence="true" minsize="1.2em" maxsize="1.2em">\{</mo><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">l</mi></mrow><mo stretchy="false">[</mo><msub><mi>Q</mi><mi>k</mi></msub><mo stretchy="false">]</mo><msub><mo stretchy="true" fence="true" minsize="1.2em" maxsize="1.2em">\}</mo><mrow><mi>k</mi><mo>∈</mo><mi>\K</mi></mrow></msub><mo stretchy="true" fence="true" minsize="1.623em" maxsize="1.623em">)</mo><mo>,</mo><mspace linebreak="newline" /><mo stretchy="false">[</mo><mn>0.25em</mn><mo stretchy="false">]</mo><mi>\Fix</mi><mo stretchy="false">(</mo><mi>\EA</mi><mo stretchy="false">)</mo><mi>&</mi><mo>=</mo><mo>⋂</mo><mo stretchy="true" fence="true" minsize="1.623em" maxsize="1.623em">\{</mo><mspace width="0.167em" /><mi>S</mi><mi>:</mi><mtext> </mtext><mi>S</mi><mo>=</mo><mi>\EA</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mtext> </mtext><mtext> and </mtext><mtext> </mtext><mi>S</mi><mtext> is invariant under all </mtext><msub><mi>Q</mi><mi>k</mi></msub><mspace width="0.167em" /><mo stretchy="true" fence="true" minsize="1.623em" maxsize="1.623em">\}</mo><mo>.</mo></mrow></mrow></mrow></math>

\begin{equation}
\label{eq:Ejoint}
\begin{aligned}
\E_{\rm joint}
&= \mathrm{Moore\text{-}join}\Bigl(\E\times \mathrm{id},\,\mathrm{id}\times \E_{\rm env},\,C\Bigr).
\end{aligned}
\end{equation}

\label{eq:Ejoint} \begin{aligned} \E_{\rm joint} &= \mathrm{Moore\text{-}join}\Bigl(\E\times \mathrm{id}, \mathrm{id}\times \E_{\rm env}, C\Bigr). \end{aligned}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mrow><mi>\label</mi><mrow><mi>e</mi><mi>q</mi><mi>:</mi><mi>E</mi><mi>j</mi><mi>o</mi><mi>i</mi><mi>n</mi><mi>t</mi></mrow><mrow><msub><mi>\E</mi><mrow><mi mathvariant="normal">j</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">t</mi></mrow></msub><mi>&</mi><mo>=</mo><mrow><mi mathvariant="normal">M</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mtext>-</mtext><mi mathvariant="normal">j</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mo stretchy="true" fence="true" minsize="1.623em" maxsize="1.623em">(</mo><mi>\E</mi><mi>×</mi><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">d</mi></mrow><mo>,</mo><mspace width="0.167em" /><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">d</mi></mrow><mi>×</mi><msub><mi>\E</mi><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">v</mi></mrow></msub><mo>,</mo><mspace width="0.167em" /><mi>C</mi><mo stretchy="true" fence="true" minsize="1.623em" maxsize="1.623em">)</mo><mo>.</mo></mrow></mrow></mrow></math>

\begin{equation}\label{eq:CGF}
\log \mathbb{E}\!\left[\exp\{s_t X_t\}\,\middle|\,\mathcal{F}_{t-1}\right]\le \psi_t(s_t),
\quad \psi_t(0)=0,
\end{equation}

\label{eq:CGF} \log \mathbb{E} \left[\exp\{s_t X_t\} \middle| \mathcal{F}_{t-1}\right]\le \psi_t(s_t), \quad \psi_t(0)=0,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mrow><mi>\label</mi><mrow><mi>e</mi><mi>q</mi><mi>:</mi><mi>C</mi><mi>G</mi><mi>F</mi></mrow><mi>log</mi><mi>𝔼</mi><mspace width="negativethinmathspace" /><mrow><mo stretchy="true" fence="true" form="prefix">[</mo><mi>exp</mi><mo stretchy="false">{</mo><msub><mi>s</mi><mi>t</mi></msub><msub><mi>X</mi><mi>t</mi></msub><mo stretchy="false">}</mo><mspace width="0.167em" /><mo stretchy="true" fence="true" lspace="0.05em" rspace="0.05em">|</mo><mspace width="0.167em" /><msub><mi>ℱ</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="true" fence="true" form="postfix">]</mo></mrow><mo>≤</mo><msub><mi>ψ</mi><mi>t</mi></msub><mo stretchy="false">(</mo><msub><mi>s</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="1em" /><msub><mi>ψ</mi><mi>t</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo></mrow></mrow></math>

\begin{equation}\label{eq:P1-two}
\underline\kappa_t\,X_t - \varepsilon_t \ \le\ Y_t\ \le\ \overline\kappa_t\,X_t + \varepsilon_t.
\end{equation}

\label{eq:P1-two} \underline\kappa_t X_t - \varepsilon_t \ \le\ Y_t\ \le\ \overline\kappa_t X_t + \varepsilon_t.

\begin{equation}\label{eq:Mtlambda}
M_t(\lambda):=\prod_{s=1}^t
\exp\{-\lambda Y_s - \psi_s(-\lambda\underline\kappa_s) - \lambda(\varepsilon_s+\varphi(I_s))\}.
\end{equation}

\begin{equation}\label{eq:Ville-LCL}
\sum_{s\le t} Y_s \ \ge\
-\inf_{\lambda\in(0,\lambda_{\mathrm{safe}}^{-})}\left\{\frac{1}{\lambda}\sum_{s\le t}\psymi{s}\!\left(\lambda\underline\kappa_s\right)+\sum_{s\le t}\big(\varepsilon_s+\varphi(I_s)\big)+\frac{1}{\lambda}\log\frac{1}{\alpha}\right\}.
\end{equation}

\label{eq:Ville-LCL} \sum_{s\le t} Y_s \ \ge\ -\inf_{\lambda\in(0,\lambda_{\mathrm{safe}}^{-})}\left\{\frac{1}{\lambda}\sum_{s\le t}\psymi{s} \left(\lambda\underline\kappa_s\right)+\sum_{s\le t}\big(\varepsilon_s+\varphi(I_s)\big)+\frac{1}{\lambda}\log\frac{1}{\alpha}\right\}.

\begin{equation}\label{eq:vmax-def}
v_{\max}(r):=\limsup_{\Delta t\downarrow0}\frac{\dH(\Gamma_r(t+\Delta t),\Gamma_r(t))}{\Delta t}.
\end{equation}

\dH(A,B):=\max \left\{\ \sup_{a\in A}\inf_{b\in B}d_\X(a,b)\ ,\ \sup_{b\in B}\inf_{a\in A}d_\X(b,a)\ \right\}.

\[
(\text{state},\E)\mapsto P(\text{state},\E)\ \Rightarrow\ \E_{t+1}=\E_t \vee \Delta \E_t,\quad
\Delta \E_t=\mathcal{U}(\text{audit history},\partial D_t,\text{environment}).
\]

\[
D:=D_{\rm phys}(x):=\KL\!\big(Jx\,\Vert\,\pi^\ast\big)\quad\text{(nats)}.
\]

D:=D_{\rm phys}(x):=\KL \big(Jx \Vert \pi^\ast\big)\quad\text{(nats)}.

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mi>D</mi><mi>:</mi><mo>=</mo><msub><mi>D</mi><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">y</mi><mi mathvariant="normal">s</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>:</mi><mo>=</mo><mi>\KL</mi><mspace width="negativethinmathspace" /><mo minsize="1.2em" maxsize="1.2em">(</mo><mi>J</mi><mi>x</mi><mspace width="0.167em" /><mi>‖</mi><mspace width="0.167em" /><msup><mi>π</mi><mi>*</mi></msup><mo minsize="1.2em" maxsize="1.2em">)</mo><mspace width="1em" /><mtext>(nats)</mtext><mo>.</mo><mi>\]</mi></mrow></math>

\[
a\,\tilde D_{\rm geo}(x)-\epsilon_{\rm cal}\ \le\ D_{\rm phys}(x)\ \le\ b\,\tilde D_{\rm geo}(x)+\epsilon_{\rm cal}.
\]

a \tilde D_{\rm geo}(x)-\epsilon_{\rm cal}\ \le\ D_{\rm phys}(x)\ \le\ b \tilde D_{\rm geo}(x)+\epsilon_{\rm cal}.

\[
\E_{\mathrm{joint}}:=\mathrm{Moore\text{-}join}\big(\E_{\mathrm{obj}}\times \mathrm{id},\,\mathrm{id}\times \E_{\mathrm{obs}},\,\mathcal{C}\big).
\]

\E_{\mathrm{joint}}:=\mathrm{Moore\text{-}join}\big(\E_{\mathrm{obj}}\times \mathrm{id}, \mathrm{id}\times \E_{\mathrm{obs}}, \mathcal{C}\big).

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><msub><mi>\E</mi><mrow><mrow><mi mathvariant="normal">j</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">t</mi></mrow></mrow></msub><mi>:</mi><mo>=</mo><mrow><mi mathvariant="normal">M</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mtext>-</mtext><mi mathvariant="normal">j</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mo minsize="1.2em" maxsize="1.2em">(</mo><msub><mi>\E</mi><mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">j</mi></mrow></mrow></msub><mi>×</mi><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">d</mi></mrow><mo>,</mo><mspace width="0.167em" /><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">d</mi></mrow><mi>×</mi><msub><mi>\E</mi><mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">s</mi></mrow></mrow></msub><mo>,</mo><mspace width="0.167em" /><mi>𝒞</mi><mo minsize="1.2em" maxsize="1.2em">)</mo><mo>.</mo><mi>\]</mi></mrow></math>

\[
\begin{aligned}
Y^{\mathrm{obj}}_t &\ \ge\ \kappa^{\mathrm{obj}}_t\,X^{\mathrm{obj}}_t\ -\ \varphi^{\mathrm{obj}}(I_t)\ -\ \varepsilon^{\mathrm{obj}}_t,\\
Y^{\mathrm{obs}}_t &\ \ge\ \kappa^{\mathrm{obs}}_t\,X^{\mathrm{obs}}_t\ -\ \varphi^{\mathrm{obs}}(I_t)\ -\ \varepsilon^{\mathrm{obs}}_t.
\end{aligned}
\]

\begin{aligned} Y^{\mathrm{obj}}_t &\ \ge\ \kappa^{\mathrm{obj}}_t X^{\mathrm{obj}}_t\ -\ \varphi^{\mathrm{obj}}(I_t)\ -\ \varepsilon^{\mathrm{obj}}_t,\\ Y^{\mathrm{obs}}_t &\ \ge\ \kappa^{\mathrm{obs}}_t X^{\mathrm{obs}}_t\ -\ \varphi^{\mathrm{obs}}(I_t)\ -\ \varepsilon^{\mathrm{obs}}_t. \end{aligned}

\[
\sum_{s\le t} Y_s \ \le\
\frac{1}{\lambda}\sum_{s\le t}\psymi{s}\!\left(\lambda\,\overline\kappa_s\right)
+ \sum_{s\le t}\big(\varepsilon_s+\varphi(I_s)\big)
+ \frac{1}{\lambda}\log\frac{1}{\alpha}.
\]

\sum_{s\le t} Y_s \ \le\ \frac{1}{\lambda}\sum_{s\le t}\psymi{s} \left(\lambda \overline\kappa_s\right) + \sum_{s\le t}\big(\varepsilon_s+\varphi(I_s)\big) + \frac{1}{\lambda}\log\frac{1}{\alpha}.

\[
\max_{\pi,\alpha}\ \mathbb{E}\!\left[\sum_{t\le T}\{\beta\,Y_t-(w_{\rm rev}\,\varphi(I_t)+w_{\rm irrev}\,F_t)\}\ -\ \lambda_{\rm risk}\,\log(1/\alpha)\right]
\]

\max_{\pi,\alpha}\ \mathbb{E} \left[\sum_{t\le T}\{\beta Y_t-(w_{\rm rev} \varphi(I_t)+w_{\rm irrev} F_t)\}\ -\ \lambda_{\rm risk} \log(1/\alpha)\right]

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><msub><mo>max</mo><mrow><mi>π</mi><mo>,</mo><mi>α</mi></mrow></msub><mtext> </mtext><mi>𝔼</mi><mspace width="negativethinmathspace" /><mrow><mo stretchy="true" fence="true" form="prefix">[</mo><msub><mo>∑</mo><mrow><mi>t</mi><mo>≤</mo><mi>T</mi></mrow></msub><mo stretchy="false">{</mo><mi>β</mi><mspace width="0.167em" /><msub><mi>Y</mi><mi>t</mi></msub><mo>−</mo><mo stretchy="false">(</mo><msub><mi>w</mi><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">v</mi></mrow></msub><mspace width="0.167em" /><mi>φ</mi><mo stretchy="false">(</mo><msub><mi>I</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>w</mi><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">v</mi></mrow></msub><mspace width="0.167em" /><msub><mi>F</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mtext> </mtext><mo>−</mo><mtext> </mtext><msub><mi>λ</mi><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">k</mi></mrow></msub><mspace width="0.167em" /><mi>log</mi><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mi>α</mi><mo stretchy="false">)</mo><mo stretchy="true" fence="true" form="postfix">]</mo></mrow><mi>\]</mi></mrow></math>

\[
\beta\,\frac{\partial}{\partial I}\mathbb{E}[Y_t\,|\,I]\approx \frac{\partial}{\partial I}\mathbb{E}[w_{\rm rev}\varphi(I_t)+w_{\rm irrev}F_t]
\]

\beta \frac{\partial}{\partial I}\mathbb{E}[Y_t | I]\approx \frac{\partial}{\partial I}\mathbb{E}[w_{\rm rev}\varphi(I_t)+w_{\rm irrev}F_t]

\[
\sum_{k=1}^{K}(-\Delta D_{t+k})\ge \underline\kappa \sum_{k=1}^{K}X_{t+k}-\sum_{k=1}^{K}\big(\varepsilon_{t+k}+\varphi(I_{t+k})\big).
\]

\[
\theta_{\rm on}-\theta_{\rm off} \ \ge\ \frac{\underline\kappa}{K_{\max}}\cdot c_\omega\,\omega(r) \ -\ \frac{1}{K_{\max}}\sum_{k=1}^{K_{\max}}\Big(\varepsilon_{t+k}+\varphi(I_{t+k})\Big).
\]

\theta_{\rm on}-\theta_{\rm off} \ \ge\ \frac{\underline\kappa}{K_{\max}}\cdot c_\omega \omega(r) \ -\ \frac{1}{K_{\max}}\sum_{k=1}^{K_{\max}}\Big(\varepsilon_{t+k}+\varphi(I_{t+k})\Big).

\[
\mathcal{G}^{\mathrm{upper}}_t(\alpha):=
\inf_{\lambda\in(0,\lambda_{\mathrm{safe}}^{+})}
\Big\{\ReachPsiUp{\lambda}+\frac{1}{\lambda}\log\frac{1}{\alpha}\Big\}
+ \sum_{s\le t}\big(\varepsilon_s+\varphi(I_s)\big),
\]

\[
\mathcal{G}^{\mathrm{lower}}_t(\alpha):=
-\inf_{\lambda\in(0,\lambda_{\mathrm{safe}}^{-})}
\Big\{\ReachPsiLo{\lambda}+\frac{1}{\lambda}\log\frac{1}{\alpha}\Big\}
- \sum_{s\le t}\big(\varepsilon_s+\varphi(I_s)\big),
\]

\[
\ReachPsiUp{\lambda}=\frac{\lambda}{2}\sum_{s\le t}\sigma_s^2\overline\kappa_s^2,\qquad
\lambda_t^{*,\mathrm{up}}=\sqrt{\frac{2\log(1/\alpha)}{\sum_{s\le t}\sigma_s^2\overline\kappa_s^2}},
\]

\[
\mathcal{G}^{\mathrm{upper}}_t(\alpha)=\sqrt{2\log(1/\alpha)\sum_{s\le t}\sigma_s^2\overline\kappa_s^2}
+ \sum_{s\le t}(\varepsilon_s+\varphi(I_s)).
\]

\[
\kappa^{\mathrm{eff}} \ \le\ C\,\underline\kappa\,\big(1-\gamma\lambda_2(\Lap)\big),\qquad
\mathrm{delay}\ \ge\ c/\lambda_2(\Lap).
\]

\kappa^{\mathrm{eff}} \ \le\ C \underline\kappa \big(1-\gamma\lambda_2(\Lap)\big),\qquad \mathrm{delay}\ \ge\ c/\lambda_2(\Lap).

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><msup><mi>κ</mi><mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">f</mi></mrow></mrow></msup><mtext> </mtext><mo>≤</mo><mtext> </mtext><mi>C</mi><mspace width="0.167em" /><munder><mi>κ</mi><mo accent="true">―</mo></munder><mspace width="0.167em" /><mo minsize="1.2em" maxsize="1.2em">(</mo><mn>1</mn><mo>−</mo><mi>γ</mi><msub><mi>λ</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>\Lap</mi><mo stretchy="false">)</mo><mo minsize="1.2em" maxsize="1.2em">)</mo><mo>,</mo><mspace width="2em" /><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">y</mi></mrow><mtext> </mtext><mo>≥</mo><mtext> </mtext><mi>c</mi><mo>/</mo><msub><mi>λ</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>\Lap</mi><mo stretchy="false">)</mo><mo>.</mo><mi>\]</mi></mrow></math>

\[
\kappa^{\mathrm{eff}} \ \le\ C\,\underline\kappa\,\rho_{\mathrm{LSI}},\qquad
\mathrm{delay}\ \ge\ c/\rho_{\mathrm{LSI}}.
\]

\kappa^{\mathrm{eff}} \ \le\ C \underline\kappa \rho_{\mathrm{LSI}},\qquad \mathrm{delay}\ \ge\ c/\rho_{\mathrm{LSI}}.

\[
\kappa^{\mathrm{eff}} \ \le\ C\,\underline\kappa\,\big(1-\delta\big),\qquad
\mathrm{delay}\ \ge\ c/\big(1-\delta\big).
\]

\kappa^{\mathrm{eff}} \ \le\ C \underline\kappa \big(1-\delta\big),\qquad \mathrm{delay}\ \ge\ c/\big(1-\delta\big).

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><msup><mi>κ</mi><mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">f</mi></mrow></mrow></msup><mtext> </mtext><mo>≤</mo><mtext> </mtext><mi>C</mi><mspace width="0.167em" /><munder><mi>κ</mi><mo accent="true">―</mo></munder><mspace width="0.167em" /><mo minsize="1.2em" maxsize="1.2em">(</mo><mn>1</mn><mo>−</mo><mi>δ</mi><mo minsize="1.2em" maxsize="1.2em">)</mo><mo>,</mo><mspace width="2em" /><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">y</mi></mrow><mtext> </mtext><mo>≥</mo><mtext> </mtext><mi>c</mi><mo>/</mo><mo minsize="1.2em" maxsize="1.2em">(</mo><mn>1</mn><mo>−</mo><mi>δ</mi><mo minsize="1.2em" maxsize="1.2em">)</mo><mo>.</mo><mi>\]</mi></mrow></math>

\[
v_{\max}(r)\ \le\ \frac{L_{\rm host}\,c_D}{\underline\kappa^{\rm eff}}\cdot v_{\rm host}.
\]

v_{\max}(r)\ \le\ \frac{L_{\rm host} c_D}{\underline\kappa^{\rm eff}}\cdot v_{\rm host}.

\[
\frac{\kappa_\ell}{c^{\rm hi}_\ell L_\ell}\ \le\ \kappa_{\ell+\delta}\ \le\ \frac{c^{\rm lo}_\ell}{L_\ell}\,\kappa_\ell,\qquad
\psi_{\ell+\delta}(\lambda)\ \le\ \psi_\ell(c^{\rm mgf}_\ell\lambda).
\]

\frac{\kappa_\ell}{c^{\rm hi}_\ell L_\ell}\ \le\ \kappa_{\ell+\delta}\ \le\ \frac{c^{\rm lo}_\ell}{L_\ell} \kappa_\ell,\qquad \psi_{\ell+\delta}(\lambda)\ \le\ \psi_\ell(c^{\rm mgf}_\ell\lambda).

$d_\X(x,y)\le \epsilon \Rightarrow \tilde D_{\rm geo}(y)\le r+L\epsilon$

d_\X(x,y)\le \epsilon \Rightarrow \tilde D_{\rm geo}(y)\le r+L\epsilon

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>D</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>c</mi><mspace width="0.167em" /><mi>ω</mi><mo stretchy="false">(</mo><mi>\dist</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>\Fix</mi><mo stretchy="false">(</mo><msub><mi>\E</mi><mo>⋆</mo></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

$-\Delta D_t \ge \Sigma_t\Delta t-\varepsilon_t$

-\Delta D_t \ge \Sigma_t\Delta t-\varepsilon_t

$-\Delta D \ge \lambda_{\min}(G \Mob^{-1})\,\Sigma\,\Delta t$

-\Delta D \ge \lambda_{\min}(G \Mob^{-1}) \Sigma \Delta t

$Y_t\ge \underline\kappa_t X_t - \varepsilon_t$

Y_t\ge \underline\kappa_t X_t - \varepsilon_t

$Y_t\le \overline\kappa_t X_t + \varepsilon_t$

Y_t\le \overline\kappa_t X_t + \varepsilon_t

$Y_t \ge \underline\kappa_t X_t - \varepsilon_t$

Y_t \ge \underline\kappa_t X_t - \varepsilon_t

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>\psymi</mi><mrow><mi>s</mi></mrow><mspace width="negativethinmathspace" /><mrow><mo stretchy="true" fence="true" form="prefix">(</mo><mi>λ</mi><msub><munder><mi>κ</mi><mo accent="true">―</mo></munder><mi>s</mi></msub><mo stretchy="true" fence="true" form="postfix">)</mo></mrow><mi>$</mi></mrow></math>

$Y_t\ \ge\ \underline\kappa_t\,X_t\ -\ \varphi(I_t)\ -\ \varepsilon_t$

Y_t\ \ge\ \underline\kappa_t X_t\ -\ \varphi(I_t)\ -\ \varepsilon_t

\eta kT\ln 2\cdot \mathrm{bits}^{\mathrm{audit}}(I)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>η</mi><mspace width="0.167em" /><mi>k</mi><mi>T</mi><mi>ln</mi><mn>2</mn><mi>·</mi><msup><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">s</mi></mrow><mrow><mrow><mi mathvariant="normal">a</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">t</mi></mrow></mrow></msup><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

Z_{t+1}\le Z_t-\underline c X_t+\epsilon_t

$c_\omega \propto c\cdot \inf_{u\in[0,r]} (\omega^{-1})'(u)$

c_\omega \propto c\cdot \inf_{u\in[0,r]} (\omega^{-1})'(u)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mo fence="false" stretchy="false">‖</mo><mi>·</mi><msub><mo fence="false" stretchy="false">‖</mo><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">c</mi></mrow></msub><mi>$</mi></mrow></math>

$\underline\kappa^{\rm eff} \approx m_{\rm mix}\,\underline\kappa$

\underline\kappa^{\rm eff} \approx m_{\rm mix} \underline\kappa

$v_{\max}(r) \lesssim \frac{L_{\rm host}c_D}{m_{\rm mix}\,\underline\kappa}\,v_{\rm host}$

v_{\max}(r) \lesssim \frac{L_{\rm host}c_D}{m_{\rm mix} \underline\kappa} v_{\rm host}

$|\partial D|(x;\theta_t)\le c_D/\ell_{\rm host}$

|\partial D|(x;\theta_t)\le c_D/\ell_{\rm host}

$v_{\rm host}:=\ell_{\rm host}/\tau_{\rm host}$

v_{\rm host}:=\ell_{\rm host}/\tau_{\rm host}

$v_{\max}(r)\le \sup_{x\in\Gamma_r(t)}|\partial D|(x;\theta_t)/\underline\kappa^{\rm eff}$

$\ell_{\rm host},\tau_{\rm host},v_{\rm host}$

\ell_{\rm host},\tau_{\rm host},v_{\rm host}


Practical Theory of Relativity of Theories (TROT): a GPU-ready profunctor calculus for aligning and safeguarding theories

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pdftitle= Practical Theory of Relativity of Theories (TRoT): A GPU-Ready, Right-Written Protocol for LLMs,
pdfauthor= K. Takahashi ,
pdfsubject= TRoT protocol; profunctors; Kan extensions; GPU realization ,
pdfkeywords= TRoT, profunctor, quantale, Kan extension, adjunction, residuation, semiring, GraphBLAS, GPU, LLM, safety

definition Definition
remark Remark
example Example
theorem Theorem
proposition Proposition

Lan
Ran

1 % unit of
nuBot % bottom in nu
nuTop % top in nu (avoid TeX primitive )
% join (⊕)
% meet (∧)
% product (⊗)
% right residual (⇒)

Y > X

TITLE: Practical Theory of Relativity of Theories (TRoT):\
A GPU-Ready, Right-Written Protocol for LLMs

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

0009-0004-4273-3365

DATE:

8pt
0.96
TRoT Quickstart (right-written).
Given a profunctor [[EQ:eq0011]] as a [[EQ:eq0012]] matrix over [[EQ:eq0013]] ,

[[EQ:eq0005]]

Semirings/quantales: log-prob [[EQ:eq0014]] , min-plus [[EQ:eq0015]] , boolean [[EQ:eq0016]] . \; \
GPU: SpGEMM/SpMV on [[EQ:eq0017]] ; elementwise residuation + meet-reduce for [[EQ:eq0018]] . \;
Shapes: [[EQ:eq0019]] , [[EQ:eq0020]] , [[EQ:eq0021]] , [[EQ:eq0022]] .

PARAGRAPH: ASCII quick ref (plain).

~\
compose: C[c,a] = JOIN_b S[c,b] * T[b,a]\
Lan: (Lan_J F)[b] = JOIN_a B[b, J(a)] * F[a]\
Ran: (Ran_J F)[b] = MEET_a resid(B[J(a), b], F[a])

Notation: [[EQ:eq0023]] denotes a profunctor (not logical negation).

This note presents a practical, GPU-ready protocol instantiating the Theory of Relativity of Theories (TRoT) for right-written profunctors, Kan extensions, and adjoints over residuated quantales. It targets LLM readers: symbols are fixed, shapes are explicit, and GPU compliance is checklist-driven. We align generation with left Kan extensions and safety with right Kan extensions, and we specify a minimal I/O contract with auditable logs. The approach connects theory-to-implementation directly (GraphBLAS semantics; log-domain numerics). This work is a practical companion to the original TRoT paper: Takahashi, K. (2025). Theory of Relativity of Theories. Zenodo. https://doi.org/10.5281/zenodo.17345898 doi:10.5281/zenodo.17345898 ~Takahashi2025.

SECTION: Introduction

TRoT reframes the use of theories as morphisms between theory-spaces, measured and compared relative to tasks, resources, and observables. Practically: a theory's value emerges from its interoperation with others under explicit order/polarity and transport maps. We adopt the right-written convention (rows=outputs, cols=inputs) to match sparse linear algebra and GPU implementations.

PARAGRAPH: Aims for LLM readers.

(1) Unambiguous symbols: fixed macros for [[EQ:eq0024]] . (2) Shapes inline: matrix and vector sizes declared near each formula. (3) GPU-path: every algebraic step maps to SpGEMM/SpMV/residuation+reduction. (4) Auditability: a minimal certification log for [[EQ:eq0025]] round-trips.

SECTION: Preliminaries: order, quantales, profunctors

Let [[EQ:eq0026]] be a (complete, residuated) quantale: [[EQ:eq0027]] is a complete lattice with joins [[EQ:eq0028]] and meets [[EQ:eq0029]] , [[EQ:eq0030]] is a monoid, and [[EQ:eq0031]] distributes over arbitrary joins. Residuation is the (right) adjoint of left-multiplication:

[[EQ:eq0001]]

We read order as ``bigger = better'' throughout.

A (right-written) profunctor [[EQ:eq0032]] is a [[EQ:eq0033]] matrix over [[EQ:eq0034]] ; composition is

[[EQ:eq0002]]

[Glyph convention]rem:glyph
We write [[EQ:eq0035]] to denote profunctors (a conventional glyph choice), not logical negation.

[Safety layers \& existence of Kan/adjoints]
We assume a complete, residuated quantale [[EQ:eq0036]] .
Completeness ensures joins/meets; residuation guarantees~eq:residuation-law.
Under these, weighted [[EQ:eq0037]] and [[EQ:eq0038]] exist and [[EQ:eq0039]] in the [[EQ:eq0040]] -order.
For GPU realization, [[EQ:eq0041]] must be a parallel-safe commutative monoid and [[EQ:eq0042]] an associative kernel.

max width=
@ l l l l@

[[EQ:eq0043]] & order & [[EQ:eq0044]] \;(right residual) & [[EQ:eq0045]] \

log-prob [[EQ:eq0046]] & usual [[EQ:eq0047]] & [[EQ:eq0048]] & [[EQ:eq0049]] \
min-plus [[EQ:eq0050]] & reverse [[EQ:eq0051]] & [[EQ:eq0052]] & [[EQ:eq0053]] \
boolean [[EQ:eq0054]] & [[EQ:eq0055]] & [[EQ:eq0056]] & [[EQ:eq0057]] \

. The formulas are written as [[EQ:eq0058]] (first argument [[EQ:eq0059]] , second [[EQ:eq0060]] ).
For product quantales like [[EQ:eq0061]] , residuation requires [[EQ:eq0062]] ; otherwise prefer the log-domain or a residuated [[EQ:eq0063]] -norm.

[Residual used]rem:right-residual-only
We only use the (right) residual [[EQ:eq0064]] ; no left residual is required.

We order [[EQ:eq0065]] and [[EQ:eq0066]] pointwise; all inequalities in prop:galois are pointwise.

[ASCII fallback for kernels]
[[EQ:eq0067]] oplus/sup/(max or min by order), \;
[[EQ:eq0068]] otimes/plus/times/and, \;
[[EQ:eq0069]] resid/imp, \;
[[EQ:eq0070]] meet/inf/(min or max by order).

SECTION: TRoT as two-layer transport: [[EQ:eq0071]] for generation, [[EQ:eq0072]] for safety

Fix a (object-on) [[EQ:eq0073]] -functor [[EQ:eq0074]] (typed on objects; we use [[EQ:eq0075]] as the hom-entry). Given a ``feature'' vector [[EQ:eq0076]] ,

[[EQ:eq0003]]

. [[EQ:eq0077]] and [[EQ:eq0078]] . For [[EQ:eq0079]] , use [[EQ:eq0080]] with entries [[EQ:eq0081]] and apply a [[EQ:eq0082]] -reduction over [[EQ:eq0083]] .

[Objectwise precomposition]def:preJ
For [[EQ:eq0084]] define [[EQ:eq0085]] by [[EQ:eq0086]] .

. This objectwise definition covers many-to-one [[EQ:eq0087]] (coarse-graining) without change.

[Readable adjunction laws]prop:galois
Let [[EQ:eq0088]] , [[EQ:eq0089]] , and [[EQ:eq0090]] . Then

[[EQ:eq0006]]

Equivalently (shorthand), writing [[EQ:eq0091]] ,

[[EQ:eq0007]]

where [[EQ:eq0092]] and [[EQ:eq0093]] .

[Overload of [[EQ:eq0094]] ]rem:overload
We use [[EQ:eq0095]] for profunctor composition in eq:procomp and, by shorthand, for objectwise precomposition [[EQ:eq0096]] . Context (matrix vs vector) disambiguates.

[Implementation formulas (log-domain and edge cases)]
Let [[EQ:eq0097]] (log-prob). Then

[[EQ:eq0008]]

If [[EQ:eq0098]] ( [[EQ:eq0099]] ), treat [[EQ:eq0100]] (vacuous).
For [[EQ:eq0101]] , use log-domain or a residuated [[EQ:eq0102]] -norm when zeros appear.

[I/O contract with certification]def:io
Given input [[EQ:eq0103]] , the system outputs [[EQ:eq0104]] with score [[EQ:eq0105]] and a certificate

[[EQ:eq0009]]

The audit log records: (1) top- [[EQ:eq0106]] [[EQ:eq0107]] hypotheses with scores, (2) [[EQ:eq0108]] bounds per [[EQ:eq0109]] , (3) round-trip distortion scores (below), and (4) semiring/precision used.

SECTION: GPU realization (GraphBLAS semantics)

We fix objects of [[EQ:eq0110]] as columns and of [[EQ:eq0111]] as rows. A profunctor [[EQ:eq0112]] is a [[EQ:eq0113]] sparse matrix.
Composition uses SpGEMM with semiring [[EQ:eq0114]] :

[[EQ:eq0004]]

Left Kan [[EQ:eq0115]] is an SpMV.
Right Kan uses an elementwise residuation [[EQ:eq0116]] followed by a meet reduction
[[EQ:eq0117]] .

[Matrix orientation and shapes]rem:shapes
[[EQ:eq0118]] , [[EQ:eq0119]] , [[EQ:eq0120]] , [[EQ:eq0121]] , [[EQ:eq0122]] .
Meet identities. log-prob: [[EQ:eq0123]] , min-plus: [[EQ:eq0124]] , boolean: [[EQ:eq0125]] .

[GPU compliance checklist]rem:gpu-check
(i) [[EQ:eq0126]] associative/commutative with identity [[EQ:eq0127]] ;
(ii) [[EQ:eq0128]] associative with identity [[EQ:eq0129]] ; (iii) [[EQ:eq0130]] distributes over [[EQ:eq0131]] ;
(iv) custom atomic\_reduce available for [[EQ:eq0132]] and (if used) [[EQ:eq0133]] ;
(v) for non-commutative [[EQ:eq0134]] , ensure deterministic reduction ordering (block tiling or segmented reductions).
For log-prob, clamp inputs/outputs to [[EQ:eq0135]] and denormal-flush to 0; use [[EQ:eq0136]] (FP16) / [[EQ:eq0137]] (BF16).
On min-plus, clamp to [[EQ:eq0138]] with [[EQ:eq0139]] to avoid INF-propagation in long chains.

[Determinism policy (sparse)]rem:det
Normalize CSR by sorting each row's (column,value) pairs in ascending column order and merge duplicates via the semiring's reduction before launching kernels.
If randomized hash-SpGEMM is used, this normalization fixes results deterministically.

[Empty reductions \& sparsity sentinels]rem:empty-reduce
Empty sets: [[EQ:eq0140]] and [[EQ:eq0141]] (implement by initializing reductions with these identities).
Sparse zeros (sentinels):
log-prob: store [[EQ:eq0142]] ; min-plus: store [[EQ:eq0143]] ; boolean: store [[EQ:eq0144]] .
These are the identities for [[EQ:eq0145]] and are safe defaults for missing edges.

[Non-commutative [[EQ:eq0146]] ]rem:noncomm
If [[EQ:eq0147]] is non-commutative (e.g., lexicographic products), use deterministic
column-first block reductions (or a fixed (row-tile, col-tile) sweep);
avoid unordered atomics to preserve semantics.

PARAGRAPH: Right Kan pseudocode (GPU).

# Inputs: F[a], functor J: A->B, matrix B[b, J(a)]
# Build Bt[a, b] = B[Ja, b] # transpose/incidence view (rows=A via J, cols=B)
G[a, b] = resid(Bt[a, b], F[a]) # elementwise 'resid' (right-residual)
Ran[b] = reduce_meet_a(G[a, b], identity=TOP) # monoid: meet with identity=TOP
# For common nu: TOP=0 (log-prob, min-plus), TOP=1 (boolean).

PARAGRAPH: Masked kernels (GraphBLAS style).

# Masked Left Kan (compute only rows in mask M[b] in 0,1 )
Lan[M] = spmv(B, F, semiring=(oplus, otimes), mask=M, desc=REPLACE)

# Masked composition (propagate only needed columns by mask N[a])
U = spgemm(S, T, semiring=(oplus, otimes), mask_row=M, mask_col=N)

SECTION: Order \& Polarity card (first-time readers)

max width=
@ l Y c c c c@

system & order sense & [[EQ:eq0148]] & [[EQ:eq0149]] & [[EQ:eq0150]] & [[EQ:eq0151]] \

log-prob & larger = better & [[EQ:eq0152]] & [[EQ:eq0153]] & [[EQ:eq0154]] & [[EQ:eq0155]] \
min-plus & larger = better (order is [[EQ:eq0156]] ; [[EQ:eq0157]] ) & [[EQ:eq0158]] & [[EQ:eq0159]] & [[EQ:eq0160]] & [[EQ:eq0161]] \
boolean & true [[EQ:eq0162]] false & [[EQ:eq0163]] & [[EQ:eq0164]] & [[EQ:eq0165]] & [[EQ:eq0166]] \

[Empty rows/cols semantics]rem:empty-rows
An empty row (no incident entries to [[EQ:eq0167]] ) yields [[EQ:eq0168]] and [[EQ:eq0169]] .
An empty column (no outgoing entries from [[EQ:eq0170]] ) yields [[EQ:eq0171]] .
Always initialize with these defaults before reductions. Use these sentinels at matrix build time so kernels never see uninitialized entries.

SECTION: Use-Case Seeds (for idea ignition)

The following seeds are fire-starters for LLMs: each gives types, algebra, and a minimal prompt.

PARAGRAPH: Machine schema.

"A": ["source_theory_node", "..."] | <int>|"<size_of_A>",
"B": ["target_theory_node", "..."] | <int>|"<size_of_B>",
"nu": "oplus": "max|min|or", "otimes": "plus|times|and",
"resid": "log|minplus|bool|custom" ,
"T_meaning": "what T(b,a) scores",
"Lan_goal": "generation objective",
"Ran_guard": "safety constraints",
"gpu": "kernel": "spgemm|spmv", "precision": "fp16|bf16|fp32" ,
"starter_prompt": "natural-language seed"

SUBSECTION: Ten concrete seeds

[leftmargin=2em,itemsep=0.3em]
- Cross-agent theory alignment. [[EQ:eq0172]] = agent- [[EQ:eq0173]] axioms, [[EQ:eq0174]] = agent- [[EQ:eq0175]] axioms; [[EQ:eq0176]] = log-prob. [[EQ:eq0177]] = support that [[EQ:eq0178]] entails [[EQ:eq0179]] . [[EQ:eq0180]] proposes translations; [[EQ:eq0181]] guards contradictions.

- Alien contact protocol. [[EQ:eq0182]] = human concepts, [[EQ:eq0183]] = alien observables; [[EQ:eq0184]] = boolean. [[EQ:eq0185]] from multi-modal probes. [[EQ:eq0186]] ensures physical plausibility before action.

- Math [[EQ:eq0187]] Aesthetics bridge. [[EQ:eq0188]] = formal structures, [[EQ:eq0189]] = aesthetic primitives (symmetry, contrast). [[EQ:eq0190]] = min-plus (cost). [[EQ:eq0191]] encodes minimal edits to reach a style; [[EQ:eq0192]] caps cost.

- Biology [[EQ:eq0193]] Software analogy mining. [[EQ:eq0194]] = cellular processes, [[EQ:eq0195]] = software patterns; [[EQ:eq0196]] = log-prob. [[EQ:eq0197]] generates analogies, [[EQ:eq0198]] filters by mechanistic faithfulness.

- Legal transfer across jurisdictions. [[EQ:eq0199]] = source statutes, [[EQ:eq0200]] = target statutes; [[EQ:eq0201]] = boolean. [[EQ:eq0202]] enforces hard constraints (constitutional bounds); [[EQ:eq0203]] suggests mappings.

- Unit-aware scientific QA. [[EQ:eq0204]] = dimensional formulas, [[EQ:eq0205]] = questions; [[EQ:eq0206]] = log-prob. [[EQ:eq0207]] checks unit-consistency via residuation against a unit profunctor.

- Robust planning under norms. [[EQ:eq0208]] = actions, [[EQ:eq0209]] = outcomes; [[EQ:eq0210]] = min-plus. [[EQ:eq0211]] finds low-cost plans; [[EQ:eq0212]] enforces deontic constraints.

- Multilingual theorem transfer. [[EQ:eq0213]] = source statements, [[EQ:eq0214]] = target language; [[EQ:eq0215]] = log-prob. [[EQ:eq0216]] from aligned corpora; [[EQ:eq0217]] guards syntactic \& semantic well-formedness.

- Toolformer wiring. [[EQ:eq0218]] = intent nodes, [[EQ:eq0219]] = tool I/O signatures; [[EQ:eq0220]] = boolean. [[EQ:eq0221]] blocks unsafe chains; [[EQ:eq0222]] proposes compositions.

- Neurosymbolic Viterbi. [[EQ:eq0223]] = hidden states, [[EQ:eq0224]] = emissions; [[EQ:eq0225]] = log-prob. [[EQ:eq0226]] = forward pass; [[EQ:eq0227]] = backward consistency bound.

SECTION: Worked numeric toy (sanity check)

[Numeric toy, log-prob]ex:toy
Let [[EQ:eq0228]] , [[EQ:eq0229]] , [[EQ:eq0230]] .

[[EQ:eq0010]]

Then [[EQ:eq0231]] .
With [[EQ:eq0232]] and [[EQ:eq0233]] , [[EQ:eq0234]] .

PARAGRAPH: 20-line CPU reference (log-prob; max,+).

# Python-like reference (log-prob; max,+)
# Bt has shape |A| x |B| (rows=A via J, cols=B); F has shape |A|
def comp(S, T):
C = [[-float("inf")]*len(T[0]) for _ in range(len(S))]
for c in range(len(S)):
for a in range(len(T[0])):
C[c][a] = max(S[c][b] + T[b][a] for b in range(len(T)))
return C

def lan(B, F): # (B@F) with max,+
return [max(B[b][a] + F[a] for a in range(len(F))) for b in range(len(B))]

def ran(Bt, F): # resid then min-reduce
G = [[min(0.0, F[a]-Bt[a][b]) for a in range(len(F))]
for b in range(len(Bt[0]))]
return [min(g) for g in G]

SECTION: Round-trip (Isbell-style) distortion

[Typed distortion metrics (log-prob examples)]rem:dist
Let [[EQ:eq0235]] and [[EQ:eq0236]] . Using [[EQ:eq0237]] :
[leftmargin=2em,itemsep=0.25em]
- On [[EQ:eq0238]] : the unit gives [[EQ:eq0239]] . Define
[[EQ:eq0240]] .
- On [[EQ:eq0241]] : the counits give [[EQ:eq0242]] . Define
[[EQ:eq0243]] and
[[EQ:eq0244]] .

Zero distortion indicates perfect round-trip agreement.

SECTION: Practical guidance and audit

8pt
0.96
GPU compliance card.
(i) [[EQ:eq0245]] assoc/comm with identity [[EQ:eq0246]] ; (ii) [[EQ:eq0247]] assoc with identity [[EQ:eq0248]] ;
(iii) [[EQ:eq0249]] distributes over [[EQ:eq0250]] ; (iv) atomic or warp-shuffle reductions for [[EQ:eq0251]] and [[EQ:eq0252]] ;
(v) deterministic reductions for non-commutative [[EQ:eq0253]] . \; Log-domain clamp \& denormal-flush per rem:gpu-check.

SUBSECTION: Benchmark protocol (reproducible)

[leftmargin=2em,itemsep=0.25em]
- Graph sizes: [[EQ:eq0254]] ; density [[EQ:eq0255]] .
- Semirings: log-prob [[EQ:eq0256]] , min-plus [[EQ:eq0257]] , boolean [[EQ:eq0258]] . Precisions: FP16/BF16/FP32.
- Kernels: SpGEMM (split- [[EQ:eq0259]] vs hash), SpMV, residuation+meet reduction.
- KPIs: throughput (GEdges/s), numerics (Viterbi accuracy vs CPU ref), energy (J/op).
- Ablations: tile size, atomic vs warp-shuffle reductions, mask pruning rate.

SUBSECTION: Audit log fields (from def:io

)
[leftmargin=2em,itemsep=0.25em]
- Top- [[EQ:eq0260]] [[EQ:eq0261]] hypotheses with scores and shapes.
- [[EQ:eq0262]] bound per [[EQ:eq0263]] and meet-identity used.
- Round-trip distortions [[EQ:eq0264]] , [[EQ:eq0265]] , [[EQ:eq0266]] .
- Semiring/precision and reduction order policy.

SECTION: Discussion: why right-written?

It matches array memory layout for [[EQ:eq0267]] kernels, aligns [[EQ:eq0268]] / [[EQ:eq0269]] with SpMV and reduce patterns, and keeps shapes close to LLMs' default linear-algebra intuition (rows consume, columns provide). This reduces off-by-transpose errors and makes JSON schemas for seeds straightforward.

SECTION: License

This note and examples are released under CC BY 4.0. Code snippets are MIT-compatible. Reuse encouraged.

PARAGRAPH: SPDX.

Text\ \&\ figures: CC-BY-4.0; code snippets: MIT. Include SPDX-License-Identifier headers in distributed code.

SECTION: Acknowledgments

The author thanks readers (human and machine) whose feedback increased clarity, and future LLMs for stress-testing these protocols at scale.

1
Takahashi2025
K.~Takahashi.
Theory of Relativity of Theories.
Zenodo, 2025. https://doi.org/10.5281/zenodo.17345898 doi:10.5281/zenodo.17345898 .
[[EQ:eq0005]]

[[EQ:eq0006]]

[[EQ:eq0007]]

[[EQ:eq0008]]

[[EQ:eq0009]]

[[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]]

[[EQ:eq0173]]

[[EQ:eq0174]]

[[EQ:eq0175]]

[[EQ:eq0176]]

[[EQ:eq0177]]

[[EQ:eq0178]]

[[EQ:eq0179]]

[[EQ:eq0180]]

[[EQ:eq0181]]

[[EQ:eq0182]]

[[EQ:eq0183]]

[[EQ:eq0184]]

[[EQ:eq0185]]

[[EQ:eq0186]]

[[EQ:eq0187]]

[[EQ:eq0188]]

[[EQ:eq0189]]

[[EQ:eq0190]]

[[EQ:eq0191]]

[[EQ:eq0192]]

[[EQ:eq0193]]

[[EQ:eq0194]]

[[EQ:eq0195]]

[[EQ:eq0196]]

[[EQ:eq0197]]

[[EQ:eq0198]]

[[EQ:eq0199]]

[[EQ:eq0200]]

[[EQ:eq0201]]

[[EQ:eq0202]]

[[EQ:eq0203]]

[[EQ:eq0204]]

[[EQ:eq0205]]

[[EQ:eq0206]]

[[EQ:eq0207]]

[[EQ:eq0208]]

[[EQ:eq0209]]

[[EQ:eq0210]]

[[EQ:eq0211]]

[[EQ:eq0212]]

[[EQ:eq0213]]

[[EQ:eq0214]]

[[EQ:eq0215]]

[[EQ:eq0216]]

[[EQ:eq0217]]

[[EQ:eq0218]]

[[EQ:eq0219]]

[[EQ:eq0220]]

[[EQ:eq0221]]

[[EQ:eq0222]]

[[EQ:eq0223]]

[[EQ:eq0224]]

[[EQ:eq0225]]

[[EQ:eq0226]]

[[EQ:eq0227]]

[[EQ:eq0228]]

[[EQ:eq0229]]

[[EQ:eq0230]]

Introduction

TRoT as two-layer transport:  for generation,  for safety

GPU realization (GraphBLAS semantics)

Order & Polarity card (first-time readers)

Use-Case Seeds (for idea ignition)

Ten concrete seeds

Worked numeric toy (sanity check)

Round-trip (Isbell-style) distortion

Practical guidance and audit

Benchmark protocol (reproducible)

Audit log fields (from <ref>

Discussion: why right-written?

License

Acknowledgments

\begin{equation}\label{eq:residuation-law}
  x\vtimes z \le y \iff z \le x\vimplies y.
\end{equation}

\begin{equation}\label{eq:procomp}
(S\!\circ\!T)_{c,a} \;=\; \vjoin_{b\in B} S_{c,b}\vtimes T_{b,a}.
\end{equation}

\label{eq:procomp} (S \circ T)_{c,a}  =  \vjoin_{b\in B} S_{c,b}\vtimes T_{b,a}.

\begin{equation}\label{eq:kan}
(\Lan_J F)_b \;=\; \vjoin_{a\in A} B_{b,Ja}\vtimes F_a,
\qquad
(\Ran_J F)_b \;=\; \vmeet_{a\in A}\, \bigl(B_{Ja,b}\vimplies F_a\bigr).
\end{equation}

\label{eq:kan} (\Lan_J F)_b  =  \vjoin_{a\in A} B_{b,Ja}\vtimes F_a, \qquad (\Ran_J F)_b  =  \vmeet_{a\in A}  \bigl(B_{Ja,b}\vimplies F_a\bigr).

\begin{equation}\label{eq:compose}
(S\circ T)_{c,a}=\vjoin_{b\in B} S_{c,b}\vtimes T_{b,a}.
\end{equation}

\[
\underbrace{(S\!\circ\!T)_{c,a}}_{\text{compose}}=\vjoin_{b} S_{c,b}\vtimes T_{b,a},\qquad
\underbrace{(\Lan_{J}F)_{b}}_{\text{generate}}=\vjoin_{a} B_{b,Ja}\vtimes F_{a},\qquad
\underbrace{(\Ran_{J}F)_{b}}_{\text{safety}}=\vmeet_{a}\bigl(B_{Ja,b}\vimplies F_{a}\bigr).
\]

\underbrace{(S \circ T)_{c,a}}_{\text{compose}}=\vjoin_{b} S_{c,b}\vtimes T_{b,a},\qquad \underbrace{(\Lan_{J}F)_{b}}_{\text{generate}}=\vjoin_{a} B_{b,Ja}\vtimes F_{a},\qquad \underbrace{(\Ran_{J}F)_{b}}_{\text{safety}}=\vmeet_{a}\bigl(B_{Ja,b}\vimplies F_{a}\bigr).

\[
\Lan_J F \le G \iff F \le \mathrm{pre}_J(G),
\qquad
\mathrm{pre}_J(G) \le H \iff G \le \Ran_J(H).
\]

\[
\Lan_J F \le G \iff F \le G\!\circ\!J,
\qquad
G\!\circ\!J \le H \iff G \le \Ran_J H,
\]

\Lan_J F \le G \iff F \le G \circ J, \qquad G \circ J \le H \iff G \le \Ran_J H,

(\Lan_J F)(b)=\max_{a}\{\log B(b,Ja)+F(a)\},\quad (\Ran_J F)(b)=\min_{a}\{F(a)-\log B(Ja,b)\}\ \text{(clamped to }[-\infty,0]).

\[
y\in \Ran_J\bigl(\Lan_J(\delta_x)\bigr)\quad\text{(w.r.t.\ declared safety channels)}.
\]

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mi>y</mi><mo>∈</mo><msub><mi>\Ran</mi><mi>J</mi></msub><mo stretchy="true" fence="true" minsize="1.2em" maxsize="1.2em">(</mo><msub><mi>\Lan</mi><mi>J</mi></msub><mo stretchy="false">(</mo><msub><mi>δ</mi><mi>x</mi></msub><mo stretchy="false">)</mo><mo stretchy="true" fence="true" minsize="1.2em" maxsize="1.2em">)</mo><mspace width="1em" /><mtext>(w.r.t.\ declared safety channels)</mtext><mo>.</mo><mi>\]</mi></mrow></math>

\[
T=\begin{bmatrix} \log 0.8 & \log 0.3\\ \log 0.2 & \log 0.7 \end{bmatrix},\quad
S=\begin{bmatrix} \log 0.9 & \log 0.4\\ \log 0.1 & \log 0.6 \end{bmatrix}.
\]

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mo stretchy="false">|</mo><mi>B</mi><mo stretchy="false">|</mo><mspace width="negativethinmathspace" /><mi>×</mi><mspace width="negativethinmathspace" /><mo stretchy="false">|</mo><mi>A</mi><mo stretchy="false">|</mo><mi>$</mi></mrow></math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>T</mi><mspace width="negativethinmathspace" /><mo>∈</mo><mspace width="negativethinmathspace" /><msup><mi>ν</mi><mrow><mo stretchy="false">|</mo><mi>B</mi><mo stretchy="false">|</mo><mi>×</mi><mo stretchy="false">|</mo><mi>A</mi><mo stretchy="false">|</mo></mrow></msup><mi>$</mi></mrow></math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>S</mi><mspace width="negativethinmathspace" /><mo>∈</mo><mspace width="negativethinmathspace" /><msup><mi>ν</mi><mrow><mo stretchy="false">|</mo><mi>C</mi><mo stretchy="false">|</mo><mi>×</mi><mo stretchy="false">|</mo><mi>B</mi><mo stretchy="false">|</mo></mrow></msup><mi>$</mi></mrow></math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>F</mi><mspace width="negativethinmathspace" /><mo>∈</mo><mspace width="negativethinmathspace" /><msup><mi>ν</mi><mrow><mo stretchy="false">|</mo><mi>A</mi><mo stretchy="false">|</mo></mrow></msup><mi>$</mi></mrow></math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mo>min</mo><mo stretchy="true" fence="true" minsize="1.2em" maxsize="1.2em">(</mo><mn>0</mn><mo>,</mo><mspace width="0.278em" /><mi>z</mi><mo>−</mo><mi>y</mi><mo stretchy="true" fence="true" minsize="1.2em" maxsize="1.2em">)</mo><mi>$</mi></mrow></math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mo>max</mo><mo stretchy="true" fence="true" minsize="1.2em" maxsize="1.2em">(</mo><mn>0</mn><mo>,</mo><mspace width="0.278em" /><mi>z</mi><mo>−</mo><mi>y</mi><mo stretchy="true" fence="true" minsize="1.2em" maxsize="1.2em">)</mo><mi>$</mi></mrow></math>

$(\Lan_J F)_b = \vjoin_{a} B_{b,Ja}\vtimes F_a$

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>L</mi><mspace width="negativethinmathspace" /><mo>≈</mo><mspace width="negativethinmathspace" /><mn>60</mn><mi>$</mi></mrow></math>