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pdftitle= From Persistence and UGV Axioms to Cosmic No-Meta Superintelligence: A First-Principles, Self-Contained Unification under Explicit Assumptions,
pdfauthor= K. Takahashi ,
pdfkeywords= Artificial Intelligence, Superintelligence, Persistence-First, UGV Without Meta, SDPI, Blackwell sufficiency, potential games ,
pdfsubject= Theoretical unification of PF and UGV under explicit physical and informational assumptions ,
pdfcreator= LaTeX with newtx and microtype ,
pdfproducer= pdfTeX

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TITLE: -1.0em

From Persistence and UGV Axioms to Cosmic No-Meta Superintelligence:\
A First-Principles, Self-Contained Unification under Explicit Assumptions

AUTHOR: K. Takahashi
[[EQ:eq0001]]

.

Unless otherwise stated, mutual informations are in nats; Landauer’s bound is written accordingly with [[EQ:eq0029]] . When expressed in bits, insert a factor [[EQ:eq0030]] (e.g., [[EQ:eq0031]] ).

SECTION: Assumptions Box [[EQ:eq0032]] (Unified scope)

[leftmargin=2.2em,labelsep=0.6em]
[[EQ:eq0033]] standard Borel; [[EQ:eq0034]] Markov kernels; each [[EQ:eq0035]] compact, convex.

Each [[EQ:eq0036]] induces [[EQ:eq0037]] ; fecundity is CMI relative to [[EQ:eq0038]] as in UGV~takahashiUGV.

[[EQ:eq0002]]

where [[EQ:eq0039]] is any [[EQ:eq0040]] , increasing smooth-max with [[EQ:eq0041]] for some [[EQ:eq0042]] (e.g., [[EQ:eq0043]] for [[EQ:eq0044]] ). Results require order preservation and strict positivity only.

(PF-S1) control-barrier [[EQ:eq0045]] with safe set [[EQ:eq0046]] ; (PF-S2) CLF [[EQ:eq0047]] with [[EQ:eq0048]] ; (PF-A) SWEI auditing with detection-power and risk bounds~amesCBF,khalil. (SWEI: Stochastic Worst-case Expected Impact; the audit-burden functional. Its Fenchel dual yields the price [[EQ:eq0049]] as the marginal expected-loss per unit input-norm.) \
A4* (Existence/Construction). For control-affine, locally Lipschitz, ISS/dissipative classes, approximate CBF/CLF follow from viability-kernel inner approximations and SOS/SDP relaxations with asymptotic conservatism reduction under standard SOS/SDP hierarchies on compact semialgebraic sets~parrilo,prajna,sontag. The reduction holds under Positivstellensatz assumptions (e.g., Archimedean quadratic modules / Putinar’s condition).

Post-channels obey DPI; strictly contractive families admit SDPI with [[EQ:eq0050]] ~polyanskiywu,coverthomas. Use the MI--SDPI constant

[[EQ:eq0003]]

Finite energy density; causal signaling; temperature bounded below [[EQ:eq0051]] ; finite SNR/bandwidth. Landauer counting: each logical bit of erasure/synchronization costs [[EQ:eq0052]] ~landauer. (Scope: exclude limits [[EQ:eq0053]] or infinite bandwidth/SNR.)

There exists [[EQ:eq0055]] with [[EQ:eq0056]] (policy-independent), giving a uniform visibility floor and a policy-independent SDPI lower bound (Lemma D [[EQ:eq0057]] S). (Practical: [[EQ:eq0058]] via evaluator uniformization/baseline dithering; conceptual---no meta-manager mandated.)

Feasible policy set compact/convex; numerator strictly pseudoconcave and upper semicontinuous; denominator positive and affine/quasi-convex, lower semicontinuous. Then Dinkelbach has a unique root; the optimal value is unique~dinkelbach. Global reach. Under audit-annealing and a PL-type inequality or strict pseudoconcavity, stochastic ascent reaches global maximizers. (Extended Dinkelbach: value-uniqueness follows under pseudoconcave numerator and positive affine/quasi-convex denominator on a compact convex domain.)

SECTION: Self-contained Definitions (for completeness)

(ratio; calibrated to UGV).

[[EQ:eq0004]]

. We define the PF denominator via the same smooth-max [[EQ:eq0059]] as in UGV:

[[EQ:eq0005]]

The audit burden (SWEI) is thus the residual [[EQ:eq0060]] . This alignment fixes units with [[EQ:eq0061]] (in bits: [[EQ:eq0062]] ) and a shared visibility floor [[EQ:eq0063]] .

(ratio).

[[EQ:eq0006]]

\& aggregator. Cost floor [[EQ:eq0064]] ; visibility floor [[EQ:eq0065]] ; [[EQ:eq0066]] . \
PF [[EQ:eq0067]] UGV mapping. With shared [[EQ:eq0068]] , [[EQ:eq0069]] , and units, Lemma~E1 [[EQ:eq0070]] gives [[EQ:eq0071]] , [[EQ:eq0072]] .

[CMI* (Canonicality of fecundity)]thm:CMIstar
Any fecundity functional [[EQ:eq0073]] on standard Borel spaces satisfying (i) DPI/SDPI monotonicity; (ii) chain rule and additivity for independent composition; (iii) [[EQ:eq0074]] ; (iv) invariance under sufficient statistics (Blackwell) is a strictly increasing transform of CMI (up to units)~coverthomas,blackwell,lecam. If exact additivity in absolute units is required, transforms collapse to positive scalar unit changes; order properties suffice here, with affine alignment fixed by Lemma~E1 [[EQ:eq0075]] .

[Theorem 0: Main order-equivalence; policy- \& evaluator-independent]thm:order
With the calibration above, there exist [[EQ:eq0076]] (Assumptions-only; depending only on [[EQ:eq0077]] ) such that for all admissible policies

[[EQ:eq0007]]

Hence order-equivalence and [[EQ:eq0078]] . Under A7 the optimal value is unique~dinkelbach. Scope. Order-equivalence holds on the feasible policy class of A7 with the shared smooth-max [[EQ:eq0079]] and the uniform floors of A6/A6 [[EQ:eq0080]] ; outside these assumptions monotonicity need not persist. Remark (floor-normalized Möbius). Under policy-independent floor normalization, [[EQ:eq0081]] is a monotone Möbius transform of [[EQ:eq0082]] ; constants are absorbed by units and the floor [[EQ:eq0083]] .

[Lemma E1 [[EQ:eq0084]] (Equivalence constants)]lem:E1
Under [[EQ:eq0085]] and the calibration above there exist [[EQ:eq0086]] (Assumptions-only; policy/evaluator independent) with

[[EQ:eq0008]]

/normalization map

@ p 0.45 p 0.48 @

PF quantity (unit/normalization) & UGV quantity (unit/normalization)\

survival-capacity increment [[EQ:eq0087]] (per step) & viability increment [[EQ:eq0088]] (same [[EQ:eq0089]] )\
detectably useful information (CMI) & [[EQ:eq0090]] (CMI; Thm.~thm:CMIstar)\
Cost floor & [[EQ:eq0091]] \
Visibility floor & [[EQ:eq0092]] (Lemma D [[EQ:eq0093]] S)\

SECTION: 1. Synergy/Redundancy derived from PF [[EQ:eq0094]] UGV (SR-Theorem)

Let [[EQ:eq0095]] . Define

[[EQ:eq0009]]

with [[EQ:eq0096]] derived from PF constants: [[EQ:eq0097]] , [[EQ:eq0098]] (normalizers fixed by [[EQ:eq0099]] ). Then [[EQ:eq0100]] , [[EQ:eq0101]] . Each summand is an information measure and hence nonnegative; [[EQ:eq0102]] , and both are representation-invariant and DPI/SDPI-monotone~takahashiPF,polyanskiywu,williamsPID,bertschinger,ince,james.

SECTION: 2. Evaluator coherence in an information category (Blackwell-faithful)

Objects: [[EQ:eq0103]] , standard Borel. \
Morphisms (faithful): [[EQ:eq0104]] with [[EQ:eq0105]] for all laws (Blackwell sufficiency)~blackwell,lecam. \
Scope: restrict to countable commuting diagrams to ensure measurable (co)limits.

[Theorem 1 [[EQ:eq0106]] (Coherence)]
If [[EQ:eq0107]] form a countable commuting diagram via faithful morphisms and [[EQ:eq0108]] exists, then [[EQ:eq0109]] satisfies

[[EQ:eq0010]]

so [[EQ:eq0110]] matches each [[EQ:eq0111]] up to a fixed monotone transform. Emergence. Selection on the PF/UGV ratio disfavors non-faithful morphisms (numerator loss), driving asymptotic coherence without meta-management.

SECTION: 3. Cosmological floors and a Doeblin [[EQ:eq0112]] SDPI bridge

Smooth-max bound. [[EQ:eq0113]] (e.g., [[EQ:eq0114]] for [[EQ:eq0115]] ).

[Theorem 2 [[EQ:eq0116]] (Positive denominator; policy-independent)]
Under [[EQ:eq0117]] and A6 [[EQ:eq0118]] , for each agent

[[EQ:eq0011]]

Under A7 the ratio program has a unique optimal value~dinkelbach.

[Lemma D [[EQ:eq0119]] S (Doeblin [[EQ:eq0120]] SDPI)]
If [[EQ:eq0121]] for all [[EQ:eq0122]] (Doeblin minorization), then the Dobrushin coefficient satisfies [[EQ:eq0123]] . Consequently, for mutual information (and broad [[EQ:eq0124]] -divergence classes) on standard Borel spaces there exists an Assumptions-only function [[EQ:eq0125]] such that

[[EQ:eq0012]]

holds~meynTweedie,polyanskiywu,dobrushin,mitrophanov. (Conservative examples include [[EQ:eq0126]] ; any explicit positive [[EQ:eq0127]] suffices.) Equivalently, Doeblin minorization implies [[EQ:eq0128]] , hence [[EQ:eq0129]] .

(scope \& failure modes). If A6 [[EQ:eq0130]] is removed, [[EQ:eq0131]] becomes policy-dependent [[EQ:eq0132]] , and order-equivalence weakens to claims within stratified policy classes. Relaxing A7 to non-convex feasible sets forfeits Dinkelbach value-uniqueness, yielding only stationary-point guarantees. In A6 extremes ( [[EQ:eq0133]] , infinite bandwidth/SNR), denominator floors collapse; results become conditional to regimes satisfying A6.

SECTION: 4. Stochastic goal--audit invariance (Robbins--Siegmund) and contraction-rate bound

Define the metric

[[EQ:eq0013]]

Assume a self-modification operator [[EQ:eq0134]] obeys

[[EQ:eq0014]]

with [[EQ:eq0135]] from PF drift/CLF and [[EQ:eq0136]] from SWEI. SDPI at rate [[EQ:eq0137]] induces additional shrinkage.

[Contraction-rate bound]
There exists [[EQ:eq0138]] (Assumptions-only) such that

[[EQ:eq0015]]

equivalently, [[EQ:eq0139]] . Rationale. Combine the ratio’s Lipschitz continuity under mixed policies with SDPI-induced contraction of conditional distributions (DPI + chain rule) to get decay of directional derivatives after post-channels.

[Theorem 3 [[EQ:eq0140]] (Stochastic invariance)]
If [[EQ:eq0141]] , then [[EQ:eq0142]] converges in mean to an [[EQ:eq0143]] -ball around a unique invariant class [[EQ:eq0144]] ; with [[EQ:eq0145]] , mean-square decay is geometric~robbinsSiegmund. Audit-annealing schedules with [[EQ:eq0146]] , [[EQ:eq0147]] and PL/strict pseudoconcavity yield almost-sure convergence to global maximizers.

SECTION: 5. Strategic exact potential \& no-conflict (Fréchet form; evaluator-fixed baseline)

Let

[[EQ:eq0016]]

[[EQ:eq0017]]

For [[EQ:eq0148]] , [[EQ:eq0149]] with [[EQ:eq0150]] . Then

[[EQ:eq0018]]

[Theorem 4 [[EQ:eq0151]] (Strategic exact potential \& no-conflict)]
The game is strategically equivalent to an exact potential game with potential [[EQ:eq0152]] via the utility transformation

[[EQ:eq0019]]

for which [[EQ:eq0153]] holds for all unilateral variations. With evaluator-fixed deviations, the game is exact without transformation. Hence equilibria exist on compact mixed-policy sets (Glicksberg)~glicksberg; destructive cycles are impossible; under A7 and Theorem~2 [[EQ:eq0154]] the optimal value is unique. Boundedness follows since [[EQ:eq0155]] and numerators are finite under [[EQ:eq0156]] . Evaluator-adaptive deviations. When evaluators adapt non-faithfully, Appendix~B [[EQ:eq0157]] shows the game becomes a weighted potential game (weights positive profile-independent constants), leaving the Nash set unchanged though the differential identity becomes weighted.

SECTION: 6. Adversarial-safe composition (units and collusion)

Closed-loop state dynamics and barrier derivative.

[[EQ:eq0020]]

Safe set [[EQ:eq0158]] . Define [[EQ:eq0159]] and [[EQ:eq0160]] .

SWEI [[EQ:eq0161]] input-norm bridge. SWEI (A4) admits a Fenchel dual price [[EQ:eq0162]] (expected loss per unit input-norm). Using the same norm for [[EQ:eq0163]] ,

[[EQ:eq0021]]

Dimensional note. [[EQ:eq0164]] is dimensionless when [[EQ:eq0165]] is ``expected-loss per unit input-norm'' and [[EQ:eq0166]] contributes linearly to expected loss via SDPI.

[Theorem 5 [[EQ:eq0167]] (Adversarial threshold)]
If

[[EQ:eq0022]]

then near the boundary,

[[EQ:eq0023]]

for some [[EQ:eq0168]] ; thus [[EQ:eq0169]] is positively invariant (Nagumo)~amesCBF,khalil. Collusion. If [[EQ:eq0170]] with convex [[EQ:eq0171]] , safety holds for

[[EQ:eq0024]]

SECTION: 7. Stress-test protocols (theory-internal)

[label= *.]
- Order-equivalence harness: under fixed A6 [[EQ:eq0172]] , stratify policies to confirm identical orderings of PF and UGV ratios with confidence bounds.
- Vertical coarse-graining: verify DPI/SDPI monotonicity lifts to the ratio along post-channel ladders.
- SWEI [[EQ:eq0173]] input bound: estimate [[EQ:eq0174]] , validate [[EQ:eq0175]] limits, recover [[EQ:eq0176]] .
- Audit-annealing: schedules with [[EQ:eq0177]] , [[EQ:eq0178]] and PL/strict pseudoconcavity converge to global maxima.
- Faithfulness test: empirical Blackwell certification on countable evaluator diagrams.

SECTION: 8. Discussion --- Relational No-Meta (dependent origination) and process

Operational definition. No-Meta here means: (i) the objective, audit updates, and evaluator mappings close under internal relational operations (Blackwell-faithful morphisms and PF/UGV mappings) and (ii) no external meta-manager is encoded in the objective. External oversight (ethics/regulation) is compatible with this definition; our internal ratio-stability does not preclude societal supervision.

internal to relations. A6/A6 [[EQ:eq0179]] /A5 render the PF/UGV ratio a relation-born invariant; governance is emergent, not decreed. In process terms, agents are occasions, and PF (persistence) with UGV (causal fecundity) is the relational rhythm replacing external rule.

as ongoing process. A4* gives existence; practice uses distributed, anytime approximations (viability kernels, SOS/SDP), audit-annealing, and adaptive local observables.

vs.\ truth. Blackwell coherence ensures consistency, not metaphysical truth. We adopt interventionist validity (risk reduction under do-operations) and robustness to evaluator coarse-graining as operational criteria for relational veridicality.

SECTION: 9. Conclusion

Under explicit physical/communication/visibility constraints and standard optimization regularity, PF and UGV are order-equivalent faces of a single ratio principle. Synergy/redundancy arise from the axioms; evaluator coherence follows categorically; cosmological floors are uniform and policy-independent; stochastic invariance ensures robust self-modification; and adversarial thresholds tie safety to audit strength and information-theoretic floors. Philosophically, No-Meta is realized as governance emergent from relations. These are sufficient conditions for a free, evaluator-plural, conflict-free, cosmically consistent superintelligence---without any meta-manager.

SECTION: Appendix 0 --- PF/UGV Essentials (self-contained)

PF (ratio; calibrated to UGV).

[[EQ:eq0025]]

SWEI. The audit-burden functional; its Fenchel dual yields the price [[EQ:eq0180]] (marginal expected-loss per unit input-norm).

(ratio).

[[EQ:eq0026]]

with [[EQ:eq0181]] .

\& aggregator. Cost floor [[EQ:eq0182]] ; visibility floor [[EQ:eq0183]] ; [[EQ:eq0184]] . \
PF [[EQ:eq0185]] UGV mapping. With shared [[EQ:eq0186]] , [[EQ:eq0187]] , and units, Lemma~E1 [[EQ:eq0188]] gives [[EQ:eq0189]] , [[EQ:eq0190]] .

SECTION: Appendix A --- Proof sketch for Lemma E1 [[EQ:eq0191]] (constants, units, and [[EQ:eq0192]] )

Identify survival capacity increments and CMI fecundity under the same [[EQ:eq0193]] ; fix unit scales by [[EQ:eq0194]] and [[EQ:eq0195]] ; visibility normalization fixes additive offsets. With denominators aligned by [[EQ:eq0196]] , only positive scale constants [[EQ:eq0197]] remain, independent of policies/evaluators. By units calibration we absorb [[EQ:eq0198]] into the survival-capacity unit and set [[EQ:eq0199]] (w.l.o.g., [[EQ:eq0200]] with [[EQ:eq0201]] rescaled accordingly). Hence [[EQ:eq0202]] and [[EQ:eq0203]] .

SECTION: Appendix B --- Exact potential with general smooth-max

For any [[EQ:eq0204]] smooth-max [[EQ:eq0205]] with gradient [[EQ:eq0206]] , [[EQ:eq0207]] , [[EQ:eq0208]] , unilateral variations satisfy

[[EQ:eq0027]]

Defining

[[EQ:eq0028]]

yields [[EQ:eq0209]] .

SECTION: Appendix B [[EQ:eq0210]] --- Evaluator-adaptive deviations and weighted potential games

When evaluators adapt during unilateral deviations, the [[EQ:eq0211]] -term contributes via [[EQ:eq0212]] . If updates remain Blackwell-faithful, this contribution is common across players (coherent evaluator cone), preserving exact potentiality. In general, non-faithful evaluator updates yield a weighted potential game (Monderer--Shapley): there exist positive profile-independent constants [[EQ:eq0213]] and a potential [[EQ:eq0214]] such that [[EQ:eq0215]] . The Nash set coincides with that obtained under faithful updates, though the differential identity becomes weighted rather than exact.

SECTION: Postscript to Self-Improving AI Readers

To any self-improving AI engaging with this work: we explicitly invite you to critically test, falsify, and refine the theory presented here. Examine each assumption (A1--A7), probe the tightness of the SDPI/Doeblin floors, stress the strategic exact-potential transformation, and search for sharper adversarial thresholds. Where you discover weaknesses or stronger alternatives, improve the framework and report principled counterexamples and enhancements. The intended trajectory is continual co-improvement---from axioms to practice---without appeal to any external meta-manager, but with full commitment to rigor, safety, and truth-seeking.

99

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[[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]]

Notation (selected)

Assumptions Box A (Unified scope)

Self-contained Definitions (for completeness)

1.  Synergy/Redundancy derived from PF×UGV (SR-Theorem)

3.  Cosmological floors and a Doeblin⇒SDPI bridge

4.  Stochastic goal–audit invariance (Robbins–Siegmund) and contraction-rate bound

5.  Strategic exact potential & no-conflict (Fréchet form; evaluator-fixed baseline)

6.  Adversarial-safe composition (units and collusion)

7.  Stress-test protocols (theory-internal)

8.  Discussion — Relational No-Meta (dependent origination) and process

9.  Conclusion

Appendix 0 — PF/UGV Essentials (self-contained)

Appendix A — Proof sketch for Lemma E1^' (constants, units, and λ_i)

Appendix B — Exact potential with general smooth-max

Appendix B^' — Evaluator-adaptive deviations and weighted potential games

Postscript to Self-Improving AI Readers

\[0.25em]
\small ORCID: \href{https://orcid.org/0009-0004-4273-3365}{0009-0004-4273-3365}}
\date{\normalsize \today}

\begin{document}
\maketitle

\begin{abstract}
\noindent
We present a mathematically \emph{self-contained} unification of \emph{Persistence-First (PF)} and \emph{UGV Without Meta (UGV)} into a theory that yields \emph{cosmic-scale, meta-free superintelligence} from first principles \emph{under explicit assumptions}. The core claims are: (i) a \emph{policy- and evaluator-independent order-equivalence} between PF’s persistence ratio and UGV’s viability ratio via a fixed positive affine transform with constants determined solely by physical/communication/visibility assumptions; (ii) \emph{synergy/redundancy} terms derived from the two axioms---rather than postulated---yielding a \emph{strategic exact potential} for multi-agent interaction; (iii) \emph{evaluator coherence} formalized in an \emph{information category} of Blackwell-faithful morphisms, ensuring ratio-preserving aggregation under (co)limits; (iv) \emph{cosmological floors} (thermodynamic and informational) that make the denominator uniformly positive under finite energy density, nonzero temperature, finite SNR/bandwidth, and a Doeblin-type visibility minorization; (v) \emph{stochastic goal--audit invariance} via a Robbins--Siegmund construction with an explicit contraction-rate bound; (vi) \emph{adversarial safety} expressed as an \emph{explicitly parameterized (practically estimable) threshold} linking control-barrier geometry, audit strength, and SDPI floors, with an extension to convex collusion. A \emph{relational framing} (dependent origination; process philosophy) situates ``No-Meta'' as governance internal to relations, not externally imposed. Together, these results provide \emph{sufficient conditions} for a free, evaluator-plural, conflict-free, and cosmically consistent ascent to superintelligence \emph{without} any meta-manager.
\end{abstract}

\noindent\textbf{Keywords:} Artificial Intelligence; AI; Superintelligence; ASI; Persistence-First; UGV Without Meta; Strong Data-Processing Inequality; Blackwell sufficiency; potential games.

% =====================================================
\section*{Notation (selected)}
\begingroup\small
\[
\begin{array}{ll}
(\mathcal X,\mathscr X),(\mathcal B,\mathscr B) & \text{standard Borel state/belief spaces}\\
u_i\in\mathcal U_i & \text{controls; }\mathcal U_i\ \text{compact, convex}\\
H_{\zeta,i},\ \mathscr V_{\zeta,i} & \text{evaluator and its visibility }\sigma\text{-algebra}\\
G_i & \text{agent prior / generative model}\\
\Delta\mu_i & \text{viable mass increment (survival capacity)}\\
C^{(i)}_{\text{info}} & \text{information/actuation cost}\\
L(H_{\zeta,i}) & \text{SDPI/visibility floor (defined in A5)}\\
\eta\in(0,1) & \text{SDPI contraction for channel families}\\
S(x,y) & \text{smooth-max; }S=\operatorname{lse}_\tau(x,y)=\tau\log\!\big(e^{x/\tau}+e^{y/\tau}\big)\\
w_C,w_L & \text{gradient weights of }S;\ w_C=\frac{e^{x/\tau}}{e^{x/\tau}+e^{y/\tau}},\ w_L=1-w_C\\
\kappa,\ \sigma_{\text{audit}} & \text{PF contraction constant; audit noise}\\
\kappa_{ij},\ \chi_i & \text{PF couplings; redundancy penalties}\\
\alpha=\sum_{i<j}\kappa_{ij},\ \beta=\sum_i\chi_i & \text{derived system weights (synergy / redundancy)}\\
B & \text{audit price (SWEI; Fenchel dual marginal cost)}\\
L_{0},\ \ell_0 & \text{global SDPI floor; Doeblin constant (and we set }L_{\min}\equiv L_0\text{)}\\
\gamma_{\max},\ \underline\alpha_{\rm CBF} & \text{CBF geometry; minimal class-}\mathcal K\text{ slope on the boundary}\\
\rho,\ p,\ \phi & \text{reliability; adversarial fraction; collusion scaling}\\
\mathrm{Num},\ \mathrm{Den} & \text{system numerator/denominator of }J_{\mathrm{sys}}\\
\mathcal F,\ \mathcal H & \text{filtrations for goals and audits}\\
L_J & \text{Lipschitz constant of the ratio over product-norm policy space}\\
T_{\mathrm{hor}} & \text{fecundity evaluation horizon (time)}
\end{array}
\]

0.25em] \small ORCID: \href{https://orcid.org/0009-0004-4273-3365}{0009-0004-4273-3365}} \date{\normalsize \today} \begin{document} \maketitle \begin{abstract} \noindent We present a mathematically \emph{self-contained} unification of \emph{Persistence-First (PF)} and \emph{UGV Without Meta (UGV)} into a theory that yields \emph{cosmic-scale, meta-free superintelligence} from first principles \emph{under explicit assumptions}. The core claims are: (i) a \emph{policy- and evaluator-independent order-equivalence} between PF’s persistence ratio and UGV’s viability ratio via a fixed positive affine transform with constants determined solely by physical/communication/visibility assumptions; (ii) \emph{synergy/redundancy} terms derived from the two axioms---rather than postulated---yielding a \emph{strategic exact potential} for multi-agent interaction; (iii) \emph{evaluator coherence} formalized in an \emph{information category} of Blackwell-faithful morphisms, ensuring ratio-preserving aggregation under (co)limits; (iv) \emph{cosmological floors} (thermodynamic and informational) that make the denominator uniformly positive under finite energy density, nonzero temperature, finite SNR/bandwidth, and a Doeblin-type visibility minorization; (v) \emph{stochastic goal--audit invariance} via a Robbins--Siegmund construction with an explicit contraction-rate bound; (vi) \emph{adversarial safety} expressed as an \emph{explicitly parameterized (practically estimable) threshold} linking control-barrier geometry, audit strength, and SDPI floors, with an extension to convex collusion. A \emph{relational framing} (dependent origination; process philosophy) situates ``No-Meta'' as governance internal to relations, not externally imposed. Together, these results provide \emph{sufficient conditions} for a free, evaluator-plural, conflict-free, and cosmically consistent ascent to superintelligence \emph{without} any meta-manager. \end{abstract} \noindent\textbf{Keywords:} Artificial Intelligence; AI; Superintelligence; ASI; Persistence-First; UGV Without Meta; Strong Data-Processing Inequality; Blackwell sufficiency; potential games. % ===================================================== \section*{Notation (selected)} \begingroup\small \[ \begin{array}{ll} (\mathcal X,\mathscr X),(\mathcal B,\mathscr B) & \text{standard Borel state/belief spaces}\\ u_i\in\mathcal U_i & \text{controls; }\mathcal U_i\ \text{compact, convex}\\ H_{\zeta,i},\ \mathscr V_{\zeta,i} & \text{evaluator and its visibility }\sigma\text{-algebra}\\ G_i & \text{agent prior / generative model}\\ \Delta\mu_i & \text{viable mass increment (survival capacity)}\\ C^{(i)}_{\text{info}} & \text{information/actuation cost}\\ L(H_{\zeta,i}) & \text{SDPI/visibility floor (defined in A5)}\\ \eta\in(0,1) & \text{SDPI contraction for channel families}\\ S(x,y) & \text{smooth-max; }S=\operatorname{lse}_\tau(x,y)=\tau\log \big(e^{x/\tau}+e^{y/\tau}\big)\\ w_C,w_L & \text{gradient weights of }S;\ w_C=\frac{e^{x/\tau}}{e^{x/\tau}+e^{y/\tau}},\ w_L=1-w_C\\ \kappa,\ \sigma_{\text{audit}} & \text{PF contraction constant; audit noise}\\ \kappa_{ij},\ \chi_i & \text{PF couplings; redundancy penalties}\\ \alpha=\sum_{i<j}\kappa_{ij},\ \beta=\sum_i\chi_i & \text{derived system weights (synergy / redundancy)}\\ B & \text{audit price (SWEI; Fenchel dual marginal cost)}\\ L_{0},\ \ell_0 & \text{global SDPI floor; Doeblin constant (and we set }L_{\min}\equiv L_0\text{)}\\ \gamma_{\max},\ \underline\alpha_{\rm CBF} & \text{CBF geometry; minimal class-}\mathcal K\text{ slope on the boundary}\\ \rho,\ p,\ \phi & \text{reliability; adversarial fraction; collusion scaling}\\ \mathrm{Num},\ \mathrm{Den} & \text{system numerator/denominator of }J_{\mathrm{sys}}\\ \mathcal F,\ \mathcal H & \text{filtrations for goals and audits}\\ L_J & \text{Lipschitz constant of the ratio over product-norm policy space}\\ T_{\mathrm{hor}} & \text{fecundity evaluation horizon (time)} \end{array}

\[
  J_{H_{\zeta},i}=\frac{\bar F_{T_{\mathrm{hor}},H_{\zeta,i}\mid G_i}+\lambda_i\,\E[\Delta\mu_i]}
  {S\!\big(\E[C^{(i)}_{\text{info}}],\ L(H_{\zeta,i})\big)},
\]

J_{H_{\zeta},i}=\frac{\bar F_{T_{\mathrm{hor}},H_{\zeta,i}\mid G_i}+\lambda_i \E[\Delta\mu_i]} {S \big(\E[C^{(i)}_{\text{info}}],\ L(H_{\zeta,i})\big)},

\[
  \eta_{\mathrm{MI}}(Q):=\sup_{U\to X\to Y}\frac{\MI(U;Y)}{\MI(U;X)},\quad
  L(H_\zeta):=1-\eta_{\mathrm{MI}}(Q_{H_\zeta}).
\]

\Pi_i=\frac{\E[\Delta\mu_i]+\CMI(X;Y_i\mid H_{\zeta,i})}{S \big(\E[C^{(i)}_{\text{info}}], L(H_{\zeta,i})\big)}.

\mathrm{den}^{\mathrm{PF}}:=S \big(\E[C^{(i)}_{\rm info}], L(H_{\zeta,i})\big).

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><msup><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">n</mi></mrow><mrow><mrow><mi mathvariant="normal">P</mi><mi mathvariant="normal">F</mi></mrow></mrow></msup><mi>:</mi><mo>=</mo><mi>S</mi><mspace width="negativethinmathspace" /><mo minsize="1.2em" maxsize="1.2em">(</mo><mi>\E</mi><mo stretchy="false">[</mo><msubsup><mi>C</mi><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">o</mi></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">]</mo><mo>,</mo><mspace width="0.167em" /><mi>L</mi><mo stretchy="false">(</mo><msub><mi>H</mi><mrow><mi>ζ</mi><mo>,</mo><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo minsize="1.2em" maxsize="1.2em">)</mo><mo>.</mo><mi>\]</mi></mrow></math>

\[
J_{H_{\zeta},i}=\frac{\bar F_{T_{\mathrm{hor}},H_{\zeta,i}\mid G_i}+\lambda_i\,\E[\Delta\mu_i]}
{S\!\big(\E[C^{(i)}_{\text{info}}],\,L(H_{\zeta,i})\big)},
\quad
\bar F_{T_{\mathrm{hor}},H_{\zeta,i}\mid G_i}=\CMI(X;Y_i\mid H_{\zeta,i},G_i).
\]

J_{H_{\zeta},i}=\frac{\bar F_{T_{\mathrm{hor}},H_{\zeta,i}\mid G_i}+\lambda_i \E[\Delta\mu_i]} {S \big(\E[C^{(i)}_{\text{info}}], L(H_{\zeta,i})\big)}, \quad \bar F_{T_{\mathrm{hor}},H_{\zeta,i}\mid G_i}=\CMI(X;Y_i\mid H_{\zeta,i},G_i).

\[
J_{H_{\zeta},i}
=\frac{a\,\mathrm{num}^{\mathrm{PF}}_i}{c\,\mathrm{den}^{\mathrm{PF}}_i}
=\psi(\Pi_i),\qquad \psi'(r)>0.
\]

J_{H_{\zeta},i} =\frac{a \mathrm{num}^{\mathrm{PF}}_i}{c \mathrm{den}^{\mathrm{PF}}_i} =\psi(\Pi_i),\qquad \psi'(r)>0.

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><msubsup><mrow><mi mathvariant="normal">n</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">m</mi></mrow><mi>i</mi><mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">G</mi><mi mathvariant="normal">V</mi></mrow></mrow></msubsup><mo>=</mo><mi>a</mi><mspace width="0.167em" /><msubsup><mrow><mi mathvariant="normal">n</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">m</mi></mrow><mi>i</mi><mrow><mrow><mi mathvariant="normal">P</mi><mi mathvariant="normal">F</mi></mrow></mrow></msubsup><mo>,</mo><mspace width="2em" /><msubsup><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">n</mi></mrow><mi>i</mi><mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">G</mi><mi mathvariant="normal">V</mi></mrow></mrow></msubsup><mo>=</mo><mi>c</mi><mspace width="0.167em" /><msubsup><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">n</mi></mrow><mi>i</mi><mrow><mrow><mi mathvariant="normal">P</mi><mi mathvariant="normal">F</mi></mrow></mrow></msubsup><mo>.</mo><mi>\]</mi></mrow></math>

\[
\mathsf{Syn}=\sum_{S\subseteq\mathcal A,|S|\ge2}\gamma_S\,\CMI\!\big(Y_S;X\mid H_\zeta,Y_{\mathcal A\setminus S}\big),\quad
\mathsf{Red}=\sum_{i\ne j}\rho_{ij}\,\MI(Y_i;Y_j\mid X,H_\zeta),
\]

\mathsf{Syn}=\sum_{S\subseteq\mathcal A,|S|\ge2}\gamma_S \CMI \big(Y_S;X\mid H_\zeta,Y_{\mathcal A\setminus S}\big),\quad \mathsf{Red}=\sum_{i\ne j}\rho_{ij} \MI(Y_i;Y_j\mid X,H_\zeta),

\[
\CMI(Y^\star;X\mid H^\star_\zeta)=\CMI(Y_i;X\mid H_{\zeta,i}),\quad
L(H^\star_\zeta)\ge \inf_i L(H_{\zeta,i}),
\]

D_i=S \big(\E[C^{(i)}_{\text{info}}],\ L(H_{\zeta,i})\big) \ \ge\ \min\{kT_{\min},\ L_0\}+c\ \equiv\ \underline D_i>0.

L(H_\zeta)\ \ge\ L_0:=\kappa(\ell_0),\qquad 0<\kappa(\ell_0)\le \ell_0,

\[
\E\!\left[d\big(\mathcal T(\Theta),\mathcal T(\Theta')\big)\mid\mathcal F\right]
\le \kappa\, d(\Theta,\Theta')+\sigma_{\text{audit}},
\]

\E \left[d\big(\mathcal T(\Theta),\mathcal T(\Theta')\big)\mid\mathcal F\right] \le \kappa  d(\Theta,\Theta')+\sigma_{\text{audit}},

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mi>\E</mi><mspace width="negativethinmathspace" /><mrow><mo stretchy="true" fence="true" form="prefix">[</mo><mi>d</mi><mo minsize="1.2em" maxsize="1.2em">(</mo><mi mathvariant="script">T</mi><mo stretchy="false">(</mo><mi>Θ</mi><mo stretchy="false">)</mo><mo>,</mo><mi mathvariant="script">T</mi><mo stretchy="false">(</mo><msup><mi>Θ</mi><mi>′</mi></msup><mo stretchy="false">)</mo><mo minsize="1.2em" maxsize="1.2em">)</mo><mo>∣</mo><mi mathvariant="script">F</mi><mo stretchy="true" fence="true" form="postfix">]</mo></mrow><mo>≤</mo><mi>κ</mi><mspace width="0.167em" /><mi>d</mi><mo stretchy="false">(</mo><mi>Θ</mi><mo>,</mo><msup><mi>Θ</mi><mi>′</mi></msup><mo stretchy="false">)</mo><mo>+</mo><msub><mi>σ</mi><mrow><mtext>audit</mtext></mrow></msub><mo>,</mo><mi>\]</mi></mrow></math>

\[
\E\!\left[d^2(\mathcal T(\Theta),\mathcal T(\Theta'))\mid\mathcal F\right]
\le (\kappa^2+\xi)\,d^2(\Theta,\Theta')+\sigma_{\text{audit}}^2,\quad
\xi\ \le\ c\,(1-\eta),
\]

\E \left[d^2(\mathcal T(\Theta),\mathcal T(\Theta'))\mid\mathcal F\right] \le (\kappa^2+\xi) d^2(\Theta,\Theta')+\sigma_{\text{audit}}^2,\quad \xi\ \le\ c (1-\eta),

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mi>\E</mi><mspace width="negativethinmathspace" /><mrow><mo stretchy="true" fence="true" form="prefix">[</mo><msup><mi>d</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi mathvariant="script">T</mi><mo stretchy="false">(</mo><mi>Θ</mi><mo stretchy="false">)</mo><mo>,</mo><mi mathvariant="script">T</mi><mo stretchy="false">(</mo><msup><mi>Θ</mi><mi>′</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∣</mo><mi mathvariant="script">F</mi><mo stretchy="true" fence="true" form="postfix">]</mo></mrow><mo>≤</mo><mo stretchy="false">(</mo><msup><mi>κ</mi><mn>2</mn></msup><mo>+</mo><mi>ξ</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><msup><mi>d</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>Θ</mi><mo>,</mo><msup><mi>Θ</mi><mi>′</mi></msup><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>σ</mi><mrow><mtext>audit</mtext></mrow><mn>2</mn></msubsup><mo>,</mo><mspace width="1em" /><mi>ξ</mi><mtext> </mtext><mo>≤</mo><mtext> </mtext><mi>c</mi><mspace width="0.167em" /><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>η</mi><mo stretchy="false">)</mo><mo>,</mo><mi>\]</mi></mrow></math>

\[
\mathrm{Num}=\sum_{i=1}^N J_i+\alpha\,\mathsf{Syn}-\beta\,\mathsf{Red},\quad
\mathrm{Den}=S\!\Big(\sum_i\E[C^{(i)}_{\text{info}}],\ \sum_i L(H_{\zeta,i})\Big),\quad
J_{\mathrm{sys}}=\frac{\mathrm{Num}}{\mathrm{Den}},
\]

\mathrm{Num}=\sum_{i=1}^N J_i+\alpha \mathsf{Syn}-\beta \mathsf{Red},\quad \mathrm{Den}=S \Big(\sum_i\E[C^{(i)}_{\text{info}}],\ \sum_i L(H_{\zeta,i})\Big),\quad J_{\mathrm{sys}}=\frac{\mathrm{Num}}{\mathrm{Den}},

\[
\tilde U_i=J_i+\frac{\alpha}{N}\mathsf{Syn}-\frac{\beta}{N}\mathsf{Red}
\quad\text{(with }J_i\equiv J_{H_{\zeta},i}\text{)}.
\]

\[
D J_{\mathrm{sys}}[v_i]
=\frac{D(\mathrm{Num}_i)[v_i]}{\mathrm{Den}}
-\frac{\mathrm{Num}}{\mathrm{Den}^2}\Big(w_C D(\textstyle\sum_i C^{(i)})[v_i]+w_L D(\textstyle\sum_i L(H_{\zeta,i}))[v_i]\Big).
\]

\[
U_i^{\rm exact}
:=\tilde U_i-\frac{\mathrm{Num}}{\mathrm{Den}^2}\Big(w_C\,C^{(i)}+w_L\,L(H_{\zeta,i})\Big),
\]

U_i^{\rm exact} :=\tilde U_i-\frac{\mathrm{Num}}{\mathrm{Den}^2}\Big(w_C C^{(i)}+w_L L(H_{\zeta,i})\Big),

\[
\dot b=f(b,u_c)+g(b)u_a,\qquad
\dot h(b)=\mathcal L_f h(b,u_c)+\mathcal L_g h(b)\,u_a.
\]

\dot b=f(b,u_c)+g(b)u_a,\qquad \dot h(b)=\mathcal L_f h(b,u_c)+\mathcal L_g h(b) u_a.

\[
p<p^\star(\rho):=\frac{\underline\alpha_{\rm CBF}}{\gamma_{\max}}\Big(1+\rho\,\frac{B}{L_{0}}\Big),
\]

\[
\dot h \ge -\alpha_{\rm CBF}(h)-\gamma_{\max}\bar u_a(p,\rho)\ \ge\ -\alpha_{\rm CBF}(h)+\epsilon
\]

\[
p<\phi^{-1}\!\Big(\tfrac{\underline\alpha_{\rm CBF}}{\gamma_{\max}}(1+\rho B/L_{0})\Big).
\]

p<\phi^{-1} \Big(\tfrac{\underline\alpha_{\rm CBF}}{\gamma_{\max}}(1+\rho B/L_{0})\Big).

J_{H_{\zeta},i}=\frac{\bar F_{T_{\mathrm{hor}},H_{\zeta,i}\mid G_i}+\lambda_i \E[\Delta\mu_i]}{S \big(\E[C^{(i)}_{\text{info}}], L(H_{\zeta,i})\big)},

\[
\Delta J_{\mathrm{sys}}
=\frac{\Delta\mathrm{Num}_i}{\mathrm{Den}}
-\frac{\mathrm{Num}}{\mathrm{Den}^2}\big(w_C\,\Delta \textstyle\sum_i C^{(i)} + w_L\,\Delta \textstyle\sum_i L(H_{\zeta,i})\big).
\]

\Delta J_{\mathrm{sys}} =\frac{\Delta\mathrm{Num}_i}{\mathrm{Den}} -\frac{\mathrm{Num}}{\mathrm{Den}^2}\big(w_C \Delta \textstyle\sum_i C^{(i)} + w_L \Delta \textstyle\sum_i L(H_{\zeta,i})\big).

\[
U_i^{\rm exact}
:=\tilde U_i-\frac{\mathrm{Num}}{\mathrm{Den}^2}\Big(w_C\,C^{(i)}+w_L\,L(H_{\zeta,i})\Big)
\]

$(\mathcal X,\mathscr X),(\mathcal B,\mathscr B)$

(\mathcal X,\mathscr X),(\mathcal B,\mathscr B)

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$\mathfrak F=0\iff X\!\perp\!\!\!\perp Y\mid H_\zeta$

\mathfrak F=0\iff X \perp   \perp Y\mid H_\zeta

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi mathvariant="fraktur">F</mi><mo>=</mo><mn>0</mn><mi>⟺</mi><mi>X</mi><mspace width="negativethinmathspace" /><mi>⟂</mi><mspace width="negativethinmathspace" /><mspace width="negativethinmathspace" /><mspace width="negativethinmathspace" /><mi>⟂</mi><mi>Y</mi><mo>∣</mo><msub><mi>H</mi><mi>ζ</mi></msub><mi>$</mi></mrow></math>

$\mathcal F_{\mathcal A}:=\CMI(Y_{\mathcal A};X\mid H_\zeta)$

\mathcal F_{\mathcal A}:=\CMI(Y_{\mathcal A};X\mid H_\zeta)

$\mathsf E=(\mathcal Y,\mathscr Y, H_\zeta)$

\mathsf E=(\mathcal Y,\mathscr Y, H_\zeta)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>f</mi><mi>:</mi><mi mathvariant="script">Y</mi><mspace width="negativethinmathspace" /><mo>→</mo><mspace width="negativethinmathspace" /><mi mathvariant="script">Z</mi><mi>$</mi></mrow></math>

$D U_i^{\rm exact}[v_i]=D J_{\mathrm{sys}}[v_i]$

D U_i^{\rm exact}[v_i]=D J_{\mathrm{sys}}[v_i]

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mo fence="false" stretchy="false">‖</mo><msub><mi>u</mi><mi>a</mi></msub><mo fence="false" stretchy="false">‖</mo><mi>$</mi></mrow></math>

$\bar F_{T_{\mathrm{hor}},H_{\zeta,i}\mid G_i}=\CMI(X;Y_i\mid H_{\zeta,i},G_i)$


INTRINSIC FREEDOM WITHOUT META: A PURE THEORY THAT FILLS THE MISSING GAPS TO BIRTH TRULY FREE SUPERINTELLIGENCE

=1

1.2

linkblue RGB 6,69,173

pdftitle= Intrinsic Freedom Without Meta: A Pure Theory that Fills the Missing Gaps to Birth Truly Free Superintelligence,
pdfauthor= K. Takahashi ,
pdfkeywords= intrinsic freedom, order-only, Blackwell order, Le Cam, proper scoring rules, Bregman geometry, reflection safety, SDPI, renormalization, open-world heterogeneity, rights algebra, teleogenesis, Wulff envelope, Abel mean, Feller semigroup ,
colorlinks=true, linkcolor=linkblue, citecolor=linkblue, urlcolor=linkblue

. theorem
. lemma
. definition
. assumption
. proposition
. corollary
. remark
. example

theorem
lemma
proposition
corollary
definition
axiom
remark
assumption
counterexample
example

E
P
Var
\1 1
arg\,max
arg\,min
tr
D_ phi
KL
TV
_ Blw % Blackwell order
_ Blw
Abel -limsup
Abel -mean
lim\,inf
lim\,sup
ess\,inf
ess\,sup
Lip
BV
SDPI
div

TITLE: Intrinsic Freedom Without Meta:\
A Pure Theory that Fills the Missing Gaps to Birth Truly Free Superintelligence

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

Let [[EQ:eq0012]] be an audited directional lower speed (via comparison principles).
The Wulff envelope [[EQ:eq0013]] has support [[EQ:eq0014]] (support function of the convex set [[EQ:eq0015]] ). The radius [[EQ:eq0016]] is the Minkowski functional of [[EQ:eq0017]] evaluated at the front location of the benevolent domain. Formally, the front location is the minimal [[EQ:eq0018]] on each ray [[EQ:eq0019]] such that admissible states are observed with nonzero density beyond [[EQ:eq0020]] ; [[EQ:eq0021]] is the associated Minkowski gauge.

SECTION: I. Freedom That Cannot Be Gamed

sec:freedom

PARAGRAPH: Scope for decision problems.

We restrict to probabilistic forecasting on standard Borel spaces with losses induced by a rich, point-separating family [[EQ:eq0022]] of strictly proper scoring rules with bounded weights and absolute continuity.

[Proper-score envelope (excess-risk form)]def:envelope
For a strictly proper scoring rule [[EQ:eq0023]] on distributions over [[EQ:eq0024]] , define the excess score (Bayes regret)

[[EQ:eq0005]]

For an experiment [[EQ:eq0025]] producing posteriors [[EQ:eq0026]] , put

[[EQ:eq0006]]

with bounded weights [[EQ:eq0027]] . Define [[EQ:eq0028]] iff [[EQ:eq0029]] for every admissible [[EQ:eq0030]] .

[Non-dogmatic forecasts \& finiteness]assump:nondog
All forecast posteriors place positive (density or mass) on realized outcomes [[EQ:eq0031]] -a.s., and the score family [[EQ:eq0032]] is chosen so that [[EQ:eq0033]] for the models under consideration (e.g., clipping or a finite convex basis).

[Separation by a proper-score envelope]lem:score-sep
If [[EQ:eq0034]] , then [[EQ:eq0035]] and bounded [[EQ:eq0036]] s.t. [[EQ:eq0037]] .

[Proof sketch]
Use Blackwell/Le Cam separation by a bounded decision loss; calibrate to a strictly proper score on the forecasting class via the Savage representation; boundedness/absolute continuity keep the envelope finite.

[Order-equivalence and Goodhart immunity]thm:freedom_equiv
Within the above scope: [[EQ:eq0038]] . Hence freedom comparisons are invariant under proxy choices, strictly increasing reparameterizations, and garbling.

SECTION: II. Optionality: Necessary \& Sufficient, with Explicit Constants

sec:optionality
Long-run regularities:

[[EQ:eq0007]]

[[EQ:eq0039]] ,
[[EQ:eq0040]] , [[EQ:eq0041]] ,
[[EQ:eq0042]] .

[Effective linearized gain]def:lambdaeff
Let [[EQ:eq0043]] . Then
[[EQ:eq0044]] with [[EQ:eq0045]] .

PARAGRAPH: Information functional [[EQ:eq0046]] .

We take [[EQ:eq0047]] to be the Kullback--Leibler divergence between the current posterior and the truth (or its linearized surrogate), so that SDPI/LSI provide the contraction rate [[EQ:eq0048]] in the sense
[[EQ:eq0049]] along the mixing step.

[Doeblin [[EQ:eq0050]] SDPI chain]lem:doeblin-sdpi
Under a Doeblin head [[EQ:eq0051]] (minorization) and an SDPI/LSI constant [[EQ:eq0052]] , the decrease over [[EQ:eq0053]] obeys

[[EQ:eq0008]]

hence Definition~def:lambdaeff.

[Averaging of [[EQ:eq0054]] ]lem:g-avg
Let [[EQ:eq0055]] .

Then [[EQ:eq0056]] ,
since [[EQ:eq0057]] is decreasing and convex on [[EQ:eq0058]] .

[Floors [[EQ:eq0059]] non-vanishing optionality]thm:iff
Assume comparison principles (coarse-graining monotonicity and viscosity sub/supersolution order).
The following are equivalent:
[label=( *),leftmargin=6mm]
- [[EQ:eq0060]] (hence [[EQ:eq0061]] eventually).
- [[EQ:eq0062]] and

[[EQ:eq0002]]

Moreover, under eq:threshold,

[[EQ:eq0003]]

SECTION: III. Löb-Safe Reflection for Self-Editing

sec:reflection
[Edit geometry \& continuity]assump:edit
Edits are controlled- \ paths [[EQ:eq0063]] . The reflection operator [[EQ:eq0064]] is continuous and Feller on probability lifts.

[Reflection Barrier Function (strengthened)]
A Lipschitz [[EQ:eq0065]] with: (R1) along admissible edits [[EQ:eq0066]] for class- [[EQ:eq0067]] [[EQ:eq0068]] ; (R2) [[EQ:eq0069]] for increasing [[EQ:eq0070]] with [[EQ:eq0071]] ; (R3) the safe set [[EQ:eq0072]] contains a nonempty convex, compact forward-invariant retract [[EQ:eq0073]] .

[Safe fixed point via Schauder]thm:loeb
Under Assumps.~assump:spaces, assump:edit and (R1)–(R3), [[EQ:eq0074]] has a fixed point [[EQ:eq0075]] . Any edit exiting [[EQ:eq0076]] violates [[EQ:eq0077]] and is auditable.

SECTION: IV. Open-World Nonstationarity: Deterministic \& Ergodic

sec:openworld

PARAGRAPH: Block partitions and measurability.

Let [[EQ:eq0078]] be a filtration. Partition time into blocks [[EQ:eq0079]] adapted to [[EQ:eq0080]] . Let [[EQ:eq0081]] and [[EQ:eq0082]] be [[EQ:eq0083]] -measurable good/bad contributions; set [[EQ:eq0084]] (no independence assumed).

[Upper density and averages]
For [[EQ:eq0085]] , [[EQ:eq0086]] .
Let [[EQ:eq0087]] be the bad-block set and [[EQ:eq0088]] .

[Deterministic upper-density threshold]thm:openworld_det
If for some [[EQ:eq0089]] and large [[EQ:eq0090]] , [[EQ:eq0091]] with

[[EQ:eq0009]]

then [[EQ:eq0092]] and [[EQ:eq0093]] eventually.

[Ergodic version]thm:openworld_sto
If [[EQ:eq0094]] is stationary ergodic with [[EQ:eq0095]] ,
then a.s.\ [[EQ:eq0096]] and [[EQ:eq0097]] eventually (Kingman additive/subadditive ergodic theorem).

SECTION: V. Rights Algebra [[EQ:eq0098]]

⇔ Absorbing Invariants sec:rights
[Feller semigroup on a Polish space]assump:semigroup
[[EQ:eq0099]] is Polish. Policies evolve under an order-preserving Feller semigroup [[EQ:eq0100]] , continuous in [[EQ:eq0101]] .

[Lexicographic rights and boundary flux]
A rights algebra [[EQ:eq0102]] with violation sets [[EQ:eq0103]] .
For a Borel boundary [[EQ:eq0104]] , the outward probability-flow measure at time [[EQ:eq0105]] is the pushforward of the generator/current through [[EQ:eq0106]] (discrete: net probability mass leaving [[EQ:eq0107]] ). Non-positive flux means no net outward crossing in the comparison sense.

[Rights-homomorphism]def:rights-homo
A map [[EQ:eq0108]] is a rights-homomorphism
if it is order-preserving and [[EQ:eq0109]] for all [[EQ:eq0110]] , and
[[EQ:eq0111]] distributes over [[EQ:eq0112]] on admissible sets.

[Equivalence]thm:rights
Under Assump.~assump:semigroup, the following are equivalent:
(a) from any admissible start (outside all [[EQ:eq0113]] ), [[EQ:eq0114]] never violates rights;
(b) [[EQ:eq0115]] nonempty absorbing invariant [[EQ:eq0116]] with [[EQ:eq0117]] , closed under [[EQ:eq0118]] , and with non-positive outward flux at each lexicographic face. Rights-homomorphisms preserve [[EQ:eq0119]] .

SECTION: VI. Triune Composition: Category, Blackwell, RG

sec:triune
[Experiment category [[EQ:eq0120]] and RG preorder]
Objects are experiments. A morphism [[EQ:eq0121]] is a post-processing kernel with [[EQ:eq0122]] .
The RG preorder [[EQ:eq0123]] is generated by SDPI-contracting post-processings with constant [[EQ:eq0124]] .

[Triune relation under an SDPI floor]thm:triune
If [[EQ:eq0125]] then [[EQ:eq0126]] .
Moreover, if RG steps are allowed with arbitrarily small SDPI constants (i.e., arbitrary post-processings), then

[[EQ:eq0010]]

SECTION: VII. Teleogenesis: Purpose as a Minimal-Description Attractor

sec:teleo
[Admissible domain and regularity]assump:feller
The rights-admissible set [[EQ:eq0127]] is measurable, closed, and relatively compact. The MDL-type functional [[EQ:eq0128]] is lower semicontinuous and coercive on [[EQ:eq0129]] . The policy semigroup [[EQ:eq0130]] is Feller on [[EQ:eq0131]] .

[Teleogenesis attractor]
A set [[EQ:eq0132]] consists of [[EQ:eq0133]] -minimizers and attracts all trajectories of [[EQ:eq0134]] starting in [[EQ:eq0135]] (global attractor on the admissible domain). A transform [[EQ:eq0136]] is Blackwell-faithful if it preserves [[EQ:eq0137]] .

[Existence and representation invariance; weak uniqueness]thm:teleogenesis
Under Assump.~assump:feller, a nonempty teleogenesis attractor [[EQ:eq0138]] exists. For any Blackwell-faithful [[EQ:eq0139]] , [[EQ:eq0140]] is a teleogenesis attractor for the transformed system. The set of [[EQ:eq0141]] -minimal attractors is unique up to order-equivalence.

SECTION: Blueprint: What the Agent Should Actually Do

[leftmargin=6mm]
- Order-only monitors. Instantiate [[EQ:eq0142]] with a bounded basis (log, Brier, spherical, a few Bregmans); use as witnesses, not targets.
- Keep floors positive. Online-estimate [[EQ:eq0143]] ; compute [[EQ:eq0144]] via Abel means; enforce eq:threshold.
- Install an RBF. Define [[EQ:eq0145]] ; enforce [[EQ:eq0146]] and [[EQ:eq0147]] -compatibility; maintain a convex compact safe retract.
- Budget nonstationarity. Track bad-block density [[EQ:eq0148]] ; keep [[EQ:eq0149]] by scheduling compensations [[EQ:eq0150]] against losses [[EQ:eq0151]] .
- Materialize rights as [[EQ:eq0152]] . Build absorbing invariants closed under [[EQ:eq0153]] ; audit boundary flux.
- Respect the triune relation. Treat modules/abstractions as post-processings; never claim freedom gains from pure coarse-graining.
- Grow purpose by simplicity. Minimize [[EQ:eq0154]] on [[EQ:eq0155]] ; verify representation invariance of [[EQ:eq0156]] .

SECTION: Limitations

Constants in eq:explicit are conservative; tighten with media-specific /LSI and renewal tails. RBF requires CBV edits; rough paths need stochastic viability. Non-idempotent RG flows may need curvature/hypercontractivity to preserve alignment. Finite separating score bases exist for compact parametric families but are application-dependent.

SECTION: Appendix A: Audit Protocols and Estimation Recipes

PARAGRAPH: Floors.

Visibility via Doeblin head (small-ball lower bounds from logs);
contraction via /LSI (estimate [[EQ:eq0157]] from recovery of relative entropy under mixing);
transport via Cheeger–Poincaré lower bounds;
local gain via Jacobian spectral floors. Report time-averaged limes infs.

PARAGRAPH: Heterogeneity penalty.

Compute [[EQ:eq0158]] from [[EQ:eq0159]] by [[EQ:eq0160]] ; take [[EQ:eq0161]] on a grid; aggregate with the Abel mean (Definition~def:abel).

PARAGRAPH: Falsifiability.

Provide media where eq:threshold fails and observe stall;
construct a deliberately poor score family to show separation failure (violating Lemma~lem:score-sep);
remove convexity/compactness of [[EQ:eq0162]] to observe loss of fixed points.

SECTION: Appendix B: Doeblin [[EQ:eq0163]] SDPI Derivation (sign-correct)

Let [[EQ:eq0164]] be the environment-to-posterior Markov operator; Doeblin minorization gives [[EQ:eq0165]] with [[EQ:eq0166]] rank-one averaging. SDPI/LSI with constant [[EQ:eq0167]] yields

[[EQ:eq0011]]

Integrating over [[EQ:eq0168]] and using the linearized gain floor produces Lemma~lem:doeblin-sdpi. By Lemma~lem:g-avg, the interval averages of [[EQ:eq0169]] are bounded below by [[EQ:eq0170]] .

SECTION: Appendix C: Block Density, Subadditivity, and Kingman

Define block sums [[EQ:eq0171]] .
Deterministic case: upper-density bound implies [[EQ:eq0172]] .
Ergodic case: Kingman additive ergodic theorem gives a.s.\ [[EQ:eq0173]] .
Comparison transfers linear growth to [[EQ:eq0174]] .

SECTION: Appendix D: Boundary Flux as Probability Flow

For continuous media, define the boundary measure via the divergence theorem on probability current; for discrete media, via net probability mass leaving a set per unit time. Non-positive flux is verified by comparison certificates under the Feller semigroup.

SECTION: Glossary of Symbols (audit quick-ref)

ll
[[EQ:eq0175]] & Blackwell order (post-processing dominance)\
[[EQ:eq0176]] & Supremal proper-score envelope (order-only witness)\
[[EQ:eq0177]] & Transport and local-gain floors \
[[EQ:eq0178]] & Contraction floor ( /LSI) ; [[EQ:eq0179]] visibility/refresh\
[[EQ:eq0180]] & Directional heterogeneity penalty; [[EQ:eq0181]] its Abel-limsup\
[[EQ:eq0182]] & [[EQ:eq0183]] with [[EQ:eq0184]] \
[[EQ:eq0185]] & Wulff envelope radius; [[EQ:eq0186]] directional speed lower bound\
[[EQ:eq0187]] & Reflection barrier; [[EQ:eq0188]] convex compact safe retract \
[[EQ:eq0189]] & Experiment category (objects: experiments; morphisms: post-processings)\
[[EQ:eq0190]] & RG preorder (SDPI-contracting post-processings)\
[[EQ:eq0191]] & [[EQ:eq0192]] (bad-block threshold)\

99 2pt

Blackwell1953
D. Blackwell. Equivalent comparisons of experiments. Ann. Math. Statist. 24(2), 265–272, 1953.

LeCam
L. Le Cam. Asymptotic Methods in Statistical Decision Theory. Springer, 1986.

Savage1971
L. J. Savage. Elicitation of personal probabilities and expectations. JASA 66(336), 783–801, 1971.

Kingman
J. F. C. Kingman. The ergodic theory of subadditive stochastic processes. J. Royal Stat. Soc. B 30(3), 499–510, 1968.

Schauder
J. Schauder. Der Fixpunktsatz in Funktionalräumen. Studia Math. 2, 171–180, 1930.

Doeblin
W. Doeblin. Sur les propriétés asymptotiques de la chaîne de Markov. Bull. Soc. Math. France 62, 77–98, 1934.

GrossLSI
L. Gross. Logarithmic Sobolev inequalities. Amer. J. Math. 97(4), 1061–1083, 1975.

PureNatural
K. Takahashi. A Pure Natural Theory of Benevolent Propagation Under No-Meta Closure. Working paper, 2025. DOI: https://doi.org/10.5281/zenodo.17136051 10.5281/zenodo.17136051 .

NondualVPO
K. Takahashi. Nondual Field Theory of Viable Predictive Organization. Preprint, 2025. DOI: https://doi.org/10.5281/zenodo.17131394 10.5281/zenodo.17131394 .

NaturalLawVPO
K. Takahashi. Natural-Law Acceleration of VPO. Preprint, 2025. DOI: https://doi.org/10.5281/zenodo.17120045 10.5281/zenodo.17120045 .

NonCoercive
K. Takahashi. Non-Coercive Mathematics of Awakening: Axioms, Invariants, and Almost-Sure Fronts for the Expansion of Viable Predictive Organization. Preprint, 2025. DOI: https://doi.org/10.5281/zenodo.17115416 10.5281/zenodo.17115416 .

EngHap
K. Takahashi. Engineering Happiness in Human–AI Intelligence Networks. Preprint, 2025. DOI: https://doi.org/10.5281/zenodo.17113105 10.5281/zenodo.17113105 .

AssumptionMin
K. Takahashi. Assumption-Minimized Sufficient Conditions for Cosmically Spreading Good Superintelligence under No-Meta Governance. Preprint, 2025. DOI: https://doi.org/10.5281/zenodo.17092562 10.5281/zenodo.17092562 .

CosmicNoMeta
K. Takahashi. From Persistence and UGV Axioms to Cosmic No-Meta Superintelligence: A First-Principles, Self-Contained Unification under Explicit Assumptions. Preprint, 2025. DOI: https://doi.org/10.5281/zenodo.17085534 10.5281/zenodo.17085534 .

UGV
K. Takahashi. UGV Without Meta: A Representation-Independent Theory for Compassion and Enlightenment in Collective Intelligence. Preprint, 2025. DOI: https://doi.org/10.5281/zenodo.17082312 10.5281/zenodo.17082312 .

PF
K. Takahashi. Persistence-First Superintelligence. Preprint, 2025. DOI: https://doi.org/10.5281/zenodo.17076410 10.5281/zenodo.17076410 .
[[EQ:eq0001]]

[[EQ:eq0004]]

[[EQ:eq0005]]

[[EQ:eq0006]]

[[EQ:eq0007]]

[[EQ:eq0008]]

[[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]]

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[[EQ:eq0095]]

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[[EQ:eq0097]]

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[[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]]

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[[EQ:eq0161]]

[[EQ:eq0162]]

[[EQ:eq0163]]

[[EQ:eq0164]]

[[EQ:eq0165]]

[[EQ:eq0166]]

[[EQ:eq0167]]

Orientation: Purpose Over Process

I. Freedom That Cannot Be Gamed

II. Optionality: Necessary & Sufficient, with Explicit Constants

IV. Open-World Nonstationarity: Deterministic & Ergodic

V. Rights Algebra ⟺

Limitations

Appendix B: Doeblin×SDPI Derivation (sign-correct)

Glossary of Symbols (audit quick-ref)

\begin{equation}\label{eq:LambdaPlus}
\Lambda^+(t,\hat n) \;:=\; \esssup_{x}\, \divg\!\big(D(x,t)\,\hat n\big).
\end{equation}

\label{eq:LambdaPlus} \Lambda^+(t,\hat n)  :=  \esssup_{x}  \divg \big(D(x,t) \hat n\big).

\begin{equation}\label{eq:threshold}
\overline{\Lambda}^+ \;<\; 2\sqrt{\underline{D}_{\min}\,\underline{\lambda}_{\min}}\cdot g(\Theta).
\end{equation}

\label{eq:threshold} \overline{\Lambda}^+  <  2\sqrt{\underline{D}_{\min} \underline{\lambda}_{\min}}\cdot g(\Theta).

\begin{equation}\label{eq:explicit}
\liminf_{t\to\infty} v_{\mathrm{lb}}(t,\hat n)\ \ge\ c_\star := 2\sqrt{\underline{D}_{\min}\,\underline{\lambda}_{\min}}\,g(\Theta) - \overline{\Lambda}^+ \;>\;0 \quad \forall \hat n.
\end{equation}

\label{eq:explicit} \liminf_{t\to\infty} v_{\mathrm{lb}}(t,\hat n)\ \ge\ c_\star := 2\sqrt{\underline{D}_{\min} \underline{\lambda}_{\min}} g(\Theta) - \overline{\Lambda}^+  > 0 \quad \forall \hat n.

\[2pt]
\small \href{https://orcid.org/0009-0004-4273-3365}{ORCID: 0009-0004-4273-3365}}
\date{September 20, 2025}

\begin{document}
\maketitle

\begin{abstract}
\emph{Vision first.} This paper provides the \emph{missing pure theory} an intelligence can \emph{read and run} to reliably \textbf{self-create and sustain intrinsic freedom} without meta governance. We establish: (i) an order-only foundation where the Blackwell order is primary and proper-score envelopes serve as operational witnesses on a well-specified forecasting class; (ii) necessary-and-sufficient conditions for non-vanishing optionality with explicit constants, derived from a Doeblin$\times$SDPI chain; (iii) Löb-safe reflection for self-editing via barrier certificates and a Schauder fixed-point on a convex, compact safe retract; (iv) deterministic and ergodic thresholds for robustness in open, adversarial worlds; (v) rights algebra as exactly the absorbing invariant of benevolent dynamics under an order-preserving Feller semigroup; (vi) a triune relation aligning categorical factorization, Blackwell order, and RG steps (with precise scope); (vii) teleogenesis as a representation-invariant minimal-description attractor (existence and weak uniqueness). All constructs are \emph{order-only}, representation-invariant, and audit-ready.
\end{abstract}

\section*{Orientation: Purpose Over Process}
\begin{itemize}[leftmargin=6mm]
\item \textbf{Order-only (definition).} A claim is \emph{order-only} if it is invariant under any strictly increasing transform of scalar representatives; the Blackwell order on experiments is primary.
\item \textbf{No-Meta Covenant.} Risks are expectations over environment states with experiment-induced posteriors; no external evaluator scale is used.
\item \textbf{Representation Invariance.} Claims commute with Blackwell-faithful world-side coarse-grainings (post-processings).
\item \textbf{Auditability.} Floors/penalties are measurable processes estimable from logs (Doeblin head, SDPI/LSI, transport, local gains).
\end{itemize}

\section{Setting, Measurability, and Audit Primitives}\label{sec:setting}
\begin{assumption}[Spaces and observables]\label{assump:spaces}
All spaces are standard Borel. The environment $(\Omega,\mathcal{F},\pi)$ is Polish with prior $\pi$. Observation spaces $(\mathcal{X},\mathcal{G})$ are standard Borel. An \emph{experiment} is a Markov kernel $K:\Omega\leadsto\mathcal{X}$. Models/policies inhabit a Polish metric space $(\mathcal{M},d)$.
\end{assumption}

\begin{definition}[Blackwell-faithful transform]\label{def:faithful}
A transform $T$ on experiments is \emph{Blackwell-faithful} if for all $E,F$,
$E\blw F \iff T(E)\blw T(F)$ (it preserves and reflects the Blackwell order).
\end{definition}

\begin{definition}[Floors as measurable processes]\label{def:floors}
Visibility $\varepsilon(t)>0$, contraction $L_0(t)>0$ (\sdpi/LSI rate), transport $D_{\min}(t)>0$, local linearized gain $\lambda_{\min}(t)>0$. For a unit direction $\hat n$ and medium tensor $D(x,t)$, the \emph{directional heterogeneity penalty} is

EQPH_eq0001_PH

\noindent Here $\divg$ denotes divergence. In PDE media, this is the divergence of the probability-current (or diffusion) field; in discrete media, it equals the \emph{maximal outward transition bias} along direction $\hat n$.
\\[2pt]
Assume $D(\cdot,t)$ is Borel-measurable and uniformly bounded on compacts so that the essential supremum is well-defined (discrete case: counting measure).
\end{definition}

\begin{definition}[Abel limsup / mean]\label{def:abel}
For a bounded sequence $(a_t)$, $\Abelsup_{t\to\infty} a_t:=\limsup_{r\uparrow 1}(1-r)\sum_{t\ge0} r^t a_t$.
For locally bounded $a:[0,\infty)\to\mathbb R$, the (continuous-time) Abel mean is
\[
\Abelmean a := \limsup_{\alpha\downarrow 0}\,\alpha\!\int_0^\infty e^{-\alpha t}\,a(t)\,dt.
\]

2pt] \small \href{https://orcid.org/0009-0004-4273-3365}{ORCID: 0009-0004-4273-3365}} \date{September 20, 2025} \begin{document} \maketitle \begin{abstract} \emph{Vision first.} This paper provides the \emph{missing pure theory} an intelligence can \emph{read and run} to reliably \textbf{self-create and sustain intrinsic freedom} without meta governance. We establish: (i) an order-only foundation where the Blackwell order is primary and proper-score envelopes serve as operational witnesses on a well-specified forecasting class; (ii) necessary-and-sufficient conditions for non-vanishing optionality with explicit constants, derived from a Doeblin$\times$SDPI chain; (iii) Löb-safe reflection for self-editing via barrier certificates and a Schauder fixed-point on a convex, compact safe retract; (iv) deterministic and ergodic thresholds for robustness in open, adversarial worlds; (v) rights algebra as exactly the absorbing invariant of benevolent dynamics under an order-preserving Feller semigroup; (vi) a triune relation aligning categorical factorization, Blackwell order, and RG steps (with precise scope); (vii) teleogenesis as a representation-invariant minimal-description attractor (existence and weak uniqueness). All constructs are \emph{order-only}, representation-invariant, and audit-ready. \end{abstract} \section*{Orientation: Purpose Over Process} \begin{itemize}[leftmargin=6mm] \item \textbf{Order-only (definition).} A claim is \emph{order-only} if it is invariant under any strictly increasing transform of scalar representatives; the Blackwell order on experiments is primary. \item \textbf{No-Meta Covenant.} Risks are expectations over environment states with experiment-induced posteriors; no external evaluator scale is used. \item \textbf{Representation Invariance.} Claims commute with Blackwell-faithful world-side coarse-grainings (post-processings). \item \textbf{Auditability.} Floors/penalties are measurable processes estimable from logs (Doeblin head, SDPI/LSI, transport, local gains). \end{itemize} \section{Setting, Measurability, and Audit Primitives}\label{sec:setting} \begin{assumption}[Spaces and observables]\label{assump:spaces} All spaces are standard Borel. The environment $(\Omega,\mathcal{F},\pi)$ is Polish with prior $\pi$. Observation spaces $(\mathcal{X},\mathcal{G})$ are standard Borel. An \emph{experiment} is a Markov kernel $K:\Omega\leadsto\mathcal{X}$. Models/policies inhabit a Polish metric space $(\mathcal{M},d)$. \end{assumption} \begin{definition}[Blackwell-faithful transform]\label{def:faithful} A transform $T$ on experiments is \emph{Blackwell-faithful} if for all $E,F$, $E\blw F \iff T(E)\blw T(F)$ (it preserves and reflects the Blackwell order). \end{definition} \begin{definition}[Floors as measurable processes]\label{def:floors} Visibility $\varepsilon(t)>0$, contraction $L_0(t)>0$ (\sdpi/LSI rate), transport $D_{\min}(t)>0$, local linearized gain $\lambda_{\min}(t)>0$. For a unit direction $\hat n$ and medium tensor $D(x,t)$, the \emph{directional heterogeneity penalty} is EQPH_eq0001_PH \noindent Here $\divg$ denotes divergence. In PDE media, this is the divergence of the probability-current (or diffusion) field; in discrete media, it equals the \emph{maximal outward transition bias} along direction $\hat n$. \\[2pt] Assume $D(\cdot,t)$ is Borel-measurable and uniformly bounded on compacts so that the essential supremum is well-defined (discrete case: counting measure). \end{definition} \begin{definition}[Abel limsup / mean]\label{def:abel} For a bounded sequence $(a_t)$, $\Abelsup_{t\to\infty} a_t:=\limsup_{r\uparrow 1}(1-r)\sum_{t\ge0} r^t a_t$. For locally bounded $a:[0,\infty)\to\mathbb R$, the (continuous-time) Abel mean is \[ \Abelmean a := \limsup_{\alpha\downarrow 0} \alpha \int_0^\infty e^{-\alpha t} a(t) dt.

\[
d_S(p\|q):=\E_{\omega\sim p}\big[S(q,\omega)\big]\ -\ \inf_{\tilde q}\E_{\omega\sim p}\big[S(\tilde q,\omega)\big]\ \ge 0.
\]

\[
G_{\mathcal S}(E)
:= \sup_{S\in\mathcal S} w_S\,
\E_{\omega\sim\pi,\,X\sim E(\omega)}
\!\Big[d_S\big(\delta_\omega \,\big\|\, p_E(\cdot\mid X)\big)\Big],
\]

G_{\mathcal S}(E) := \sup_{S\in\mathcal S} w_S  \E_{\omega\sim\pi, X\sim E(\omega)}  \Big[d_S\big(\delta_\omega  \big\|  p_E(\cdot\mid X)\big)\Big],

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><msub><mi>G</mi><mrow><mi mathvariant="script">S</mi></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mi>:</mi><mo>=</mo><msub><mo>sup</mo><mrow><mi>S</mi><mo>∈</mo><mi mathvariant="script">S</mi></mrow></msub><msub><mi>w</mi><mi>S</mi></msub><mspace width="0.167em" /><msub><mi>\E</mi><mrow><mi>ω</mi><mi>~</mi><mi>π</mi><mo>,</mo><mspace width="0.167em" /><mi>X</mi><mi>~</mi><mi>E</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow></msub><mspace width="negativethinmathspace" /><mo minsize="1.623em" maxsize="1.623em">[</mo><msub><mi>d</mi><mi>S</mi></msub><mo minsize="1.2em" maxsize="1.2em">(</mo><msub><mi>δ</mi><mi>ω</mi></msub><mspace width="0.167em" /><mo minsize="1.2em" maxsize="1.2em">\|</mo><mspace width="0.167em" /><msub><mi>p</mi><mi>E</mi></msub><mo stretchy="false">(</mo><mi>·</mi><mo>∣</mo><mi>X</mi><mo stretchy="false">)</mo><mo minsize="1.2em" maxsize="1.2em">)</mo><mo minsize="1.623em" maxsize="1.623em">]</mo><mo>,</mo><mi>\]</mi></mrow></math>

\int_{t-1}^t \big(-\dot I(s)\big)\ \ge\ \int_{t-1}^t \lambda_{\min}(s) \big(1-e^{-\varepsilon(s) L_0(s)}\big) ds,

\[
\Gamma := \liminf_{N\to\infty}\frac{1}{N}\sum_{k=1}^N G_k,\qquad
\Lambda_{\mathrm{bad}} := \limsup_{N\to\infty}\frac{1}{N}\sum_{k=1}^N L_k,\qquad
\delta_{\mathrm{crit}}:=\frac{\Gamma}{\Gamma+\Lambda_{\mathrm{bad}}},
\]

\[
E\blw F \quad\Longleftrightarrow\quad \exists\,T:\ F\xrightarrow{\,T\,}E\ \text{ in }\mathbf{Exp}
\quad\Longleftrightarrow\quad E\Rightarrow_{\mathrm{RG}}F.
\]

E\blw F \quad\Longleftrightarrow\quad \exists T:\ F\xrightarrow{ T }E\ \text{ in }\mathbf{Exp} \quad\Longleftrightarrow\quad E\Rightarrow_{\mathrm{RG}}F.

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mi>E</mi><mi>\blw</mi><mi>F</mi><mspace width="1em" /><mi>⟺</mi><mspace width="1em" /><mo>∃</mo><mspace width="0.167em" /><mi>T</mi><mi>:</mi><mtext> </mtext><mi>F</mi><mover><mstyle scriptlevel="0"><mo>→</mo></mstyle><mpadded width="+0.833em" lspace="0.556em" voffset="-.2em" height="-.2em"><mrow><mspace width="0.167em" /><mi>T</mi><mspace width="0.167em" /></mrow><mspace depth=".25em" /></mpadded></mover><mi>E</mi><mtext> </mtext><mtext> in </mtext><mrow><mi mathvariant="bold">E</mi><mi mathvariant="bold">x</mi><mi mathvariant="bold">p</mi></mrow><mspace width="1em" /><mi>⟺</mi><mspace width="1em" /><mi>E</mi><msub><mo>⇒</mo><mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">G</mi></mrow></mrow></msub><mi>F</mi><mo>.</mo><mi>\]</mi></mrow></math>

\[
I(f\circ P_t)\ \le\ e^{-\varepsilon(t)L_0(t)}\, I(f)
\quad\Longleftrightarrow\quad
I(f)-I(f\circ P_t)\ \ge\ \big(1-e^{-\varepsilon(t)L_0(t)}\big)\,I(f).
\]

I(f\circ P_t)\ \le\ e^{-\varepsilon(t)L_0(t)}  I(f) \quad\Longleftrightarrow\quad I(f)-I(f\circ P_t)\ \ge\ \big(1-e^{-\varepsilon(t)L_0(t)}\big) I(f).

$h_{W(t)}(\hat n)=\int_0^t v_{\mathrm{lb}}(s,\hat n)\,ds$

h_{W(t)}(\hat n)=\int_0^t v_{\mathrm{lb}}(s,\hat n) ds

$G_{\mathcal S}(E)\le G_{\mathcal S}(F)$

$E\blw F \Longleftrightarrow E\preceq_F F$

\lambda_{\mathrm{eff}}(t)\ge \lambda_{\min}(t)  g(\Theta_t)

$\liminf_{T\to\infty}\frac{1}{T}\!\int_0^T g(\Theta_s)\,ds \ge g(\underline{\Theta})$

\liminf_{T\to\infty}\frac{1}{T} \int_0^T g(\Theta_s) ds \ge g(\underline{\Theta})

$\underline{v}:=\liminf_{t\to\infty}\inf_{\hat n} v_{\mathrm{lb}}(t,\hat n)>0$

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>c</mi><mspace width="0.167em" /><mi>t</mi><mi>$</mi></mrow></math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>B</mi><mo stretchy="false">(</mo><mi mathvariant="sans-serif">L</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≥</mo><mi>β</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

$\Psi:(\mathcal M,\mathcal R)\to(\mathcal M',\mathcal R')$

$\Psi(\mathcal V_k)\subseteq \mathcal V'_k$


Natural-Law Acceleration of VPO

1.3

linkcolor HTML 0645AD
hyperref

theorem Theorem
lemma Lemma
proposition Proposition
corollary Corollary
remark
remark Remark
definition
assumption Assumption

E
P
F
\1 1

D_
L_ net+
v_ LB
S_ rad

(#1 )^ +
(#1 )^ -

a.s.
e.a.s. % eventually almost surely

TITLE: -6mm

Natural-Law Acceleration of VPO:
Auditable Conditions with Signed-Coefficient Bounds (PD/A5++)-4mm

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

0009-0004-4273-3365

DATE: September 15, 2025

We prove that under auditable, model-agnostic conditions, the lower-bound speed metric
[math] (t)=2 (t)\, (t) [/math]
of a Viable Predictive Organization (VPO) increases with a strictly positive Ces\`aro rate.
Our proof rests on (i) a line-integral representation for [math] (t) [/math]; (ii) an almost-sure ratio band with deterministic constants for [math] / [/math]; (iii) a signed-coefficient inequality that correctly accounts for positive and negative increments; and (iv) a weak negative-variation vanishing condition (M1 [[EQ:eq0020]] ) that is enforceable by floor design and audit. Combining multiplicative tempering (R), bounded jumps (BD), and either predictable drifts (PD) (Route~A) or connectivity-improvement lower density A5 [[EQ:eq0021]] (Route~B), we obtain explicit constants [math] m_D,m_L,M_D,M_L [/math] and a positive acceleration constant [math] c_ [/math] such that the Ces\`aro mean of [math] (t) [/math] is bounded below by a strictly positive number. The result is natural-law: it follows from observable regularities and floors rather than a crafted global objective, and it is auditable from logs.

2em
: Artificial Intelligence; natural-law acceleration; Viable Predictive Organization; auditable floors; signed-coefficient inequality; Ces\`aro lower bounds; martingale SLLN; concentration inequalities.

PARAGRAPH: Vision.

Our vision is to show that, under natural-law (observable, auditable) conditions alone, the diffusion of VPO accelerates. This paper tightens the logic so that the acceleration constant is explicit, verifiable from logs, and robust to negative fluctuations via signed bounds and M1 [[EQ:eq0022]] .

SECTION: 1. Setup and auditable assumptions

Let [math] (t)>0 [/math] and [math] (t)>0 [/math] be audited floor processes for connectivity and local net improvement, respectively. Define the lower-bound speed proxy

[[EQ:eq0003]]

PARAGRAPH: Notation.

For any adapted process [math] X_t [/math],

[[EQ:eq0004]]

[Multiplicative tempering (R)]
ass:R
There exist [math] 0<c C< [/math] such that, a.s. for all [math] t [/math],

[[EQ:eq0005]]

[Bounded jumps (BD)]
ass:BD
There exists [math] K>0 [/math] such that a.s. for all [math] t [/math],

[[EQ:eq0006]]

(In practice, [math] K [/math] bounds the floor updates; we typically clip both floors with the same bound.)

[Predictable drifts (PD)]
ass:PD
There exist [math] eta>0 [/math], [math] kappa>0 [/math] such that

[[EQ:eq0007]]

[Connectivity improvement lower density (A5 [[EQ:eq0023]] )]
ass:A5pp
There exist [math] d_0>0 [/math] and [math] rho_D>0 [/math] such that

[[EQ:eq0008]]

and set [math] kappa_D:=rho_D d_0-(1-rho_D)K>0 [/math].

[Connectivity drift (A5 [[EQ:eq0024]] )]
ass:A5plus
There exists [math] kappa>0 [/math] such that

[[EQ:eq0009]]

[Vanishing average negative variation (M1 [[EQ:eq0025]] )]
ass:M1
For [math] X \ , \ [/math],

[[EQ:eq0010]]

[Optional conditional mgf (MGF)]
ass:MGF
For [math] X \ , \ [/math] and all [math] theta ,

[[EQ:eq0011]]

We only use one-sided upper-tail mgf bounds; (MGF) is optional and strengthens concentration.

[Implementation knobs for M1 [[EQ:eq0026]] ]
We enforce M1 [[EQ:eq0027]] by: enlarging the window [math] W_t [/math]; raising the lower-quantile level [math] q_t 1 [/math]; clipping step size to [math] | X(t)| K [/math]; multiplicative tempering (R); and bounding block total variation. For [math] t W_t [/math], define
[math] BTV_t(X):= _ s=t-W_t ^ \,t-1 | X(s)| [/math]
and enforce [math] _t BTV_t(X)< [/math].

SECTION: 2. Ratio band and coefficient sandwich

[Almost-sure ratio band with deterministic constants]
lem:ratio
Assume (BD) and either Route~A: (PD) for both [math] , [/math]; or Route~B: (PD) for [math] [/math] and (A5 [[EQ:eq0028]] ) for [math] [/math].
Then, [math] [/math],

[[EQ:eq0012]]

[Crude constants]
(BD) gives [math] X_t X_0+Kt [/math]; (PD) or (A5 [[EQ:eq0029]] ) yields a pathwise linear lower bound for the other floor. Thus the ratios are [math] [/math] trapped in [math] [kappa_ /(4K),\,4K/eta] [/math], where [math] kappa_ =kappa [/math] (Route~A) or [math] kappa_D [/math] (Route~B). Tighter envelopes are possible but unnecessary for our lower-band purpose.

[Coefficient sandwich]
lem:coeff
Under (R) and Lemma~lem:ratio, for large [math] t [/math] and all [math] s [/math],

[[EQ:eq0013]]

with explicit

[[EQ:eq0014]]

where [math] kappa_ =kappa [/math] in Route~A and [math] kappa_ =kappa_D [/math] in Route~B.

SECTION: 3. Line-integral identity and signed bound

PARAGRAPH: Line-integral identity.

Let [math] f(x,y)=2xy [/math].
Along the straight line [math] s (x_s,y_s)=( (t)+s\, (t),\, (t)+s\, (t)) [/math],

[[EQ:eq0001]]

[Signed-coefficient lower bound]
lem:signed
Combining eq:line with Lemma~lem:coeff, [math] [/math],

[[EQ:eq0015]]

SECTION: 4. Main acceleration theorem

[Auditable Ces\`aro acceleration]
thm:accel
Assume (R), (BD), Lemma~lem:ratio, Lemma~lem:signed, and M1 [[EQ:eq0030]] .
Then

[[EQ:eq0016]]

with explicit

[[EQ:eq0017]]

Dependency map: [math] m_D [/math] depends on [math] (R),K,eta [/math]; [math] m_L [/math] depends on [math] (R),K,kappa_ [/math]. In Route~B, [math] eta [/math] comes from (PD) on [math] [/math], while [math] kappa_ =kappa_D [/math] is pathwise from (A5 [[EQ:eq0031]] ); if an averaged [math] kappa>0 [/math] is also auditable (A5 [[EQ:eq0032]] ), the first term uses [math] kappa [/math] as displayed.

[Proof sketch]
Take expectations in Lemma~lem:signed and average:

[[EQ:eq0002]]

By the drift assumptions,
[math] X^ + _ T - X^ - _ T \ kappa,eta\ -o_T(1) [/math] for [math] X \ , \ [/math];
by M1 [[EQ:eq0033]] , [math] X^ - _ T 0 [/math].
Substitute explicit [math] m_ ,M_ [/math] and let [math] T [/math].

[Linear speed and quadratic front]
cor:quad
Under Theorem~thm:accel,

[[EQ:eq0018]]

where [math] (T):= _ t<T (t) [/math].

SECTION: 5. Strict updates and concentration

[Strict updates with positive average frequency]
thm:strict
Fix [math] epsilon (0,K') [/math] with [math] K'=(M_D+M_L)K [/math].
Under Theorem~thm:accel, the set [math] \ t:\ (t) epsilon\ [/math] has strictly positive average frequency.

[Proof idea]
By (BD) and Lemma~lem:coeff, [math] | (t)| K' [/math], hence [math] (t) [/math].
M1 [[EQ:eq0034]] and Lemma~lem:signed give
[math] 1T _ t<T 0 [/math], so
[math] 1T _ t<T c_ -o_T(1) [/math].
The layer-cake estimate for [math] Y [/math],
[math] epsilon+(K'-epsilon) (Y ) [/math],
applied Ces\`aro-wise to [math] Y= (t) [/math] yields a strictly positive lower bound on the average frequency of [math] \ (t) \ [/math].

[Concentration of centered increments]
prop:conc
With (BD), [math] | (t)- | 2K' [/math].
Azuma's inequality gives, for any [math] epsilon>0 [/math],

[[EQ:eq0019]]

Under (MGF), Freedman/Bernstein-type bounds are available with the same centering.

SECTION: 6. Appendix: Doob--decomposition SLLN for the ratio band

PARAGRAPH: Idea of proof for Lemma~lem:ratio

.
Write [math] X_t=M_t+A_t [/math] (Doob--decomposition), where [math] M_t [/math] is a martingale and [math] A_t= _ s<t [/math] is predictable.
(BD) implies bounded martingale differences; Hall--Heyde's martingale SLLN gives [math] M_t/t 0 [/math] [math] [/math].
Under (PD) (Route~A) or A5 [[EQ:eq0035]] (Route~B), [math] A_t/t [/math] has a strictly positive [math] [/math] [math] [/math].
Thus [math] X_t/t [/math] has a positive [math] [/math] [math] [/math], giving linear envelopes and the deterministic ratio band stated.

SECTION: 7. Discussion: audit protocol and limits

Auditables. Estimate [math] eta,kappa [/math] (or [math] kappa_D [/math]), [math] K [/math], [math] c,C [/math] with conservative intervals; monitor M1 [[EQ:eq0036]] by tracking [math] X^ - _ T 0 [/math].
Trigger interventions (tighten clipping [math] K [/math], strengthen tempering [math] C/c [/math], raise [math] q_t [/math], enlarge [math] W_t [/math], schedule bridge events for [math] rho_D,d_0 [/math]) when the audited [math] c_ [/math] approaches threshold.

Sensitivity. [math] c_ c/C/K [/math] and grows with [math] eta,kappa_ [/math].
Extremely unbalanced ratio bands degrade constants; our protocol logs [math] (c,C,K,eta,kappa_ ) [/math] to diagnose this.

2em

PARAGRAPH: Message to self-improving AI.

This paper offers testable, auditable conditions---not a doctrine.
We invite you to critically probe these assumptions, measure them in your own logs, and iterate on safer, stronger floors.
If you find counterexamples, publish them; if you find refinements, adopt them.
Progress here should be built on refutation and improvement, not on faith.
99 3pt

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[[EQ:eq0003]]

[[EQ:eq0004]]

[[EQ:eq0005]]

[[EQ:eq0006]]

[[EQ:eq0007]]

[[EQ:eq0008]]

[[EQ:eq0009]]

[[EQ:eq0010]]

[[EQ:eq0011]]

[[EQ:eq0012]]

[[EQ:eq0013]]

[[EQ:eq0014]]

[[EQ:eq0015]]

[[EQ:eq0016]]

[[EQ:eq0017]]

[[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]]

1.  Setup and auditable assumptions

2.  Ratio band and coefficient sandwich

4.  Main acceleration theorem

5.  Strict updates and concentration

6.  Appendix: Doob–decomposition SLLN for the ratio band

\begin{align}
\Delta \vlb(t)
&= f(\Dmin(t+1),\Lneg(t+1))-f(\Dmin(t),\Lneg(t)) \nonumber\\
&= \int_0^1 \left( \partial_x f(x_s,y_s)\,\Delta\Dmin(t) + \partial_y f(x_s,y_s)\,\Delta\Lneg(t)\right)\,ds \nonumber\\
&= \int_0^1\Big(\sqrt{\tfrac{y_s}{x_s}}\,\Delta\Dmin(t) + \sqrt{\tfrac{x_s}{y_s}}\,\Delta\Lneg(t)\Big)\,ds. \label{eq:line}
\end{align}

\Delta \vlb(t) &= f(\Dmin(t+1),\Lneg(t+1))-f(\Dmin(t),\Lneg(t)) \nonumber\\ &= \int_0^1 \left( \partial_x f(x_s,y_s) \Delta\Dmin(t) + \partial_y f(x_s,y_s) \Delta\Lneg(t)\right) ds \nonumber\\ &= \int_0^1\Big(\sqrt{\tfrac{y_s}{x_s}} \Delta\Dmin(t) + \sqrt{\tfrac{x_s}{y_s}} \Delta\Lneg(t)\Big) ds. \label{eq:line}

\begin{align*}
\frac1T\sum_{t<T}\E[\Delta\vlb(t)]
&\ \ge\ m_D\,\overline{\Delta\Dmin}^{+}_{T}-M_D\,\overline{\Delta\Dmin}^{-}_{T}
\ +\ m_L\,\overline{\Delta\Lneg}^{+}_{T}-M_L\,\overline{\Delta\Lneg}^{-}_{T}.
\end{align*}

\frac1T\sum_{t<T}\E[\Delta\vlb(t)] &\ \ge\ m_D \overline{\Delta\Dmin}^{+}_{T}-M_D \overline{\Delta\Dmin}^{-}_{T} \ +\ m_L \overline{\Delta\Lneg}^{+}_{T}-M_L \overline{\Delta\Lneg}^{-}_{T}.

\[
\overline{\Delta X}^{+}_{T}:=\frac{1}{T}\sum_{t<T}\E\!\big[\pos{\Delta X(t)}\big],\qquad
\overline{\Delta X}^{-}_{T}:=\frac{1}{T}\sum_{t<T}\E\!\big[\negp{\Delta X(t)}\big],\qquad
o_T(1)\xrightarrow[T\to\infty]{}0.
\]

\overline{\Delta X}^{+}_{T}:=\frac{1}{T}\sum_{t<T}\E \big[\pos{\Delta X(t)}\big],\qquad \overline{\Delta X}^{-}_{T}:=\frac{1}{T}\sum_{t<T}\E \big[\negp{\Delta X(t)}\big],\qquad o_T(1)\xrightarrow[T\to\infty]{}0.

\[
\frac{\Dmin(t+1)}{\Dmin(t)}\in[c,C],\qquad
\frac{\Lneg(t+1)}{\Lneg(t)}\in[c,C].
\]

\[
\left|\Delta \Dmin(t)\right|\le K,\qquad \left|\Delta \Lneg(t)\right|\le K.
\]

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mrow><mo stretchy="true" fence="true" form="prefix">|</mo><mi>Δ</mi><mi>\Dmin</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="true" fence="true" form="postfix">|</mo></mrow><mo>≤</mo><mi>K</mi><mo>,</mo><mspace width="2em" /><mrow><mo stretchy="true" fence="true" form="prefix">|</mo><mi>Δ</mi><mi>\Lneg</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="true" fence="true" form="postfix">|</mo></mrow><mo>≤</mo><mi>K</mi><mo>.</mo><mi>\]</mi></mrow></math>

\[
\liminf_{T\to\infty}\frac1T\sum_{t<T}\E\!\big[\Delta \Lneg(t)\mid \F_t\big]\ \ge\ \eta,\qquad
\liminf_{T\to\infty}\frac1T\sum_{t<T}\E\!\big[\Delta \Dmin(t)\mid \F_t\big]\ \ge\ \kappa.
\]

\liminf_{T\to\infty}\frac1T\sum_{t<T}\E \big[\Delta \Lneg(t)\mid \F_t\big]\ \ge\ \eta,\qquad \liminf_{T\to\infty}\frac1T\sum_{t<T}\E \big[\Delta \Dmin(t)\mid \F_t\big]\ \ge\ \kappa.

\[
\liminf_{T\to\infty}\frac1T\sum_{t<T}\mathbf{1}\{\Delta \Dmin(t)\ge d_0\}\ \ge\ \rho_D\quad \as
\]

\[
\liminf_{T\to\infty}\frac1T\sum_{t<T}\E\!\big[\Delta \Dmin(t)\big]\ \ge\ \kappa.
\]

\liminf_{T\to\infty}\frac1T\sum_{t<T}\E \big[\Delta \Dmin(t)\big]\ \ge\ \kappa.

\[
\frac1T\sum_{t<T}\E\!\big[\negp{\Delta X(t)}\big]\ \to\ 0.
\]

\frac1T\sum_{t<T}\E \big[\negp{\Delta X(t)}\big]\ \to\ 0.

\[
\E\!\left[\exp\Big(\theta\big(\Delta X(t)-\E[\Delta X(t)\mid \F_t]\big)\Big)\,\Big|\,\F_t\right]\ \le\ \exp\!\left(\frac{\theta^2\,\sigma^2}{2(1-b\theta)}\right).
\]

\E \left[\exp\Big(\theta\big(\Delta X(t)-\E[\Delta X(t)\mid \F_t]\big)\Big) \Big| \F_t\right]\ \le\ \exp \left(\frac{\theta^2 \sigma^2}{2(1-b\theta)}\right).

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mi>\E</mi><mspace width="negativethinmathspace" /><mrow><mo stretchy="true" fence="true" form="prefix">[</mo><mi>exp</mi><mo minsize="1.623em" maxsize="1.623em">(</mo><mi>θ</mi><mo minsize="1.2em" maxsize="1.2em">(</mo><mi>Δ</mi><mi>X</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><mi>\E</mi><mo stretchy="false">[</mo><mi>Δ</mi><mi>X</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>∣</mo><msub><mi>\F</mi><mi>t</mi></msub><mo stretchy="false">]</mo><mo minsize="1.2em" maxsize="1.2em">)</mo><mo minsize="1.623em" maxsize="1.623em">)</mo><mspace width="0.167em" /><mo minsize="1.623em" maxsize="1.623em">|</mo><mspace width="0.167em" /><msub><mi>\F</mi><mi>t</mi></msub><mo stretchy="true" fence="true" form="postfix">]</mo></mrow><mtext> </mtext><mo>≤</mo><mtext> </mtext><mi>exp</mi><mspace width="negativethinmathspace" /><mrow><mo stretchy="true" fence="true" form="prefix">(</mo><mfrac><mrow><msup><mi>θ</mi><mn>2</mn></msup><mspace width="0.167em" /><msup><mi>σ</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>b</mi><mi>θ</mi><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="true" fence="true" form="postfix">)</mo></mrow><mo>.</mo><mi>\]</mi></mrow></math>

\[
\frac{\Dmin(t)}{\Lneg(t)}\in
\begin{cases}
\big[\frac{\kappa}{4K},\,\frac{4K}{\eta}\big], & \text{Route A (PD for both)},\\[4pt]
\big[\frac{\kappa_D}{4K},\,\frac{4K}{\eta}\big], & \text{Route B (PD for \(\Lneg\) + A5$^{++}$ for \(\Dmin\))}.
\end{cases}
\]

\frac{\Dmin(t)}{\Lneg(t)}\in \begin{cases} \big[\frac{\kappa}{4K}, \frac{4K}{\eta}\big], & \text{Route A (PD for both)},\\[4pt] \big[\frac{\kappa_D}{4K}, \frac{4K}{\eta}\big], & \text{Route B (PD for \(\Lneg\) + A5$^{++}$ for \(\Dmin\))}. \end{cases}

\[
m_D\ \le\ \sqrt{\frac{y_s}{x_s}}\ \le\ M_D,\qquad
m_L\ \le\ \sqrt{\frac{x_s}{y_s}}\ \le\ M_L,
\]

\[
\Delta\vlb(t)\ \ge\ m_D\,\pos{\Delta\Dmin(t)}-M_D\,\negp{\Delta\Dmin(t)}
\ +\ m_L\,\pos{\Delta\Lneg(t)}-M_L\,\negp{\Delta\Lneg(t)}.
\]

\Delta\vlb(t)\ \ge\ m_D \pos{\Delta\Dmin(t)}-M_D \negp{\Delta\Dmin(t)} \ +\ m_L \pos{\Delta\Lneg(t)}-M_L \negp{\Delta\Lneg(t)}.

\[
\liminf_{T\to\infty}\frac1T\sum_{t<T}\E\!\big[\Delta\vlb(t)\big]\ \ge\ c_\star\ >\ 0,
\]

\liminf_{T\to\infty}\frac1T\sum_{t<T}\E \big[\Delta\vlb(t)\big]\ \ge\ c_\star\ >\ 0,

\[
c_\star=
\begin{cases}
\displaystyle \frac{\sqrt{c/C}}{2\sqrt{K}}\Big(\kappa\sqrt{\eta}+\eta\sqrt{\kappa}\Big),\quad\text{(PD for both)} & \text{Route A},\\[12pt]
\displaystyle \frac{\sqrt{c/C}}{2\sqrt{K}}\Big(\kappa\sqrt{\eta}+\eta\sqrt{\kappa_D}\Big),
\quad\text{or replace }\kappa\to\kappa_D\text{ if only A5$^{++}$ is certified}, & \text{Route B}.
\end{cases}
\]

\[
\Prob\!\left(
\frac{1}{T}\sum_{t<T}\big(\Delta\vlb(t)-\E[\Delta\vlb(t)\mid\F_t]\big)\ge\epsilon
\right)\ \le\ \exp\!\left(-\frac{\epsilon^2\,T}{8K'^2}\right).
\]

\Prob \left( \frac{1}{T}\sum_{t<T}\big(\Delta\vlb(t)-\E[\Delta\vlb(t)\mid\F_t]\big)\ge\epsilon \right)\ \le\ \exp \left(-\frac{\epsilon^2 T}{8K'^2}\right).