PERSISTENCE AS CLOSURE: AN ASSUMPTION-TRANSPARENT MODULAR CORE FOR MOTION AND INTERNAL TIME

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

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TITLE: Persistence as Closure:\
An Assumption-Transparent Modular Core
for Motion and Internal Time

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

0009-0004-4273-3365

DATE:

We develop an assumption-transparent, fully deductive construction in which a single structural axiom---persistence as closure---together with a variational engine (existence of resolvent minimizers) yields: (i) closure with a stable layer, (ii) intrinsic motion via (generalized) minimizing movements, and (iii) an internal notion of time defined by the monotone decrease of a distance-to-stability functional. The analytical input is primarily the existence of resolvent minimizers; compactness/growth assumptions and, in strong models, standard convex-analytic tools yield upgrades (nonlinear semigroups, exact scaling, convergence to the stable layer). Stronger properties are provided as local add-ons with explicit hypotheses. No extra-mathematical premises are used.

SECTION: Foundations: Layers, Axiom, and Conventions

SUBSECTION: Layer O (Order)

Let [[EQ:eq0012]] be a poset.

[Closure and stable layer]def:closure
A closure on [[EQ:eq0013]] is a map [[EQ:eq0014]] that is extensive, monotone, and idempotent. The stable layer is [[EQ:eq0015]] .

[Persistence]ax:persistence
There exists a closure [[EQ:eq0016]] on [[EQ:eq0017]] with [[EQ:eq0018]] . Degenerate cases (e.g.\ [[EQ:eq0019]] ) satisfy the axiom but render [[EQ:eq0020]] ; our focus is the nontrivial regime [[EQ:eq0021]] where motion is meaningful.

SUBSECTION: Layer M (Metric)

Let [[EQ:eq0022]] be a complete metric space. When both layers are present we identify [[EQ:eq0023]] .

[Dual package]def:dual
Let [[EQ:eq0024]] be a poset and [[EQ:eq0025]] a metric space with [[EQ:eq0026]] .
An operator [[EQ:eq0027]] satisfies the dual package if
(i) [[EQ:eq0028]] is a closure on [[EQ:eq0029]] (extensive, monotone, idempotent) and
(ii) [[EQ:eq0030]] is [[EQ:eq0031]] -Lipschitz on [[EQ:eq0032]] .
We call [[EQ:eq0033]] the stable layer.

PARAGRAPH: Motivation.

Closure captures stabilization (idempotent persistence), while nonexpansiveness encodes no amplification of distance to stability; together they yield [[EQ:eq0034]] and interface the order and metric layers.

PARAGRAPH: Terminology.

We write stable layer for [[EQ:eq0035]] throughout. The symbol [[EQ:eq0036]] denotes a closure/persistent map (not expectation).

PARAGRAPH: Set-valued resolvents.

[[EQ:eq0037]] denotes the set of minimizers; we write [[EQ:eq0038]] for a selection. Identities involving [[EQ:eq0039]] are understood as inclusions unless single-valuedness is assumed or proved.

SUBSECTION: Universe Axioms (baseline and local add-ons)

[leftmargin=1.6em,labelsep=0.4em]
Standard ZFC; no extra-mathematical premises.
A poset [[EQ:eq0040]] and a closure [[EQ:eq0041]] with [[EQ:eq0042]] .
A complete metric space [[EQ:eq0043]] ; when [[EQ:eq0044]] , the same [[EQ:eq0045]] is [[EQ:eq0046]] -Lipschitz.
For each [[EQ:eq0047]] and [[EQ:eq0048]] , the minimization for [[EQ:eq0049]] has a solution.
[[EQ:eq0050]] proper and [[EQ:eq0051]] lsc with [[EQ:eq0052]] coercive (direct method).
Successive iterates in [[EQ:eq0053]] can be connected by constant-speed geodesics (or polygonal substitutes).
For given [[EQ:eq0054]] , [[EQ:eq0055]] is relatively compact.
[[EQ:eq0056]] such that [[EQ:eq0057]] (or [[EQ:eq0058]] ).
Bijective similarities [[EQ:eq0059]] with [[EQ:eq0060]] and multiplicative factor [[EQ:eq0061]] .
Orlicz- [[EQ:eq0063]] bounds or [[EQ:eq0064]] -homogeneity [[EQ:eq0065]] .
For each [[EQ:eq0066]] , [[EQ:eq0067]] admits an isotone and inflationary selection [[EQ:eq0068]] .

PARAGRAPH: Scope, novelty, and trade-offs (reader's map).

Our contribution is an assumption-transparent, modular synthesis: (i) an order--metric baseline where persistence is encoded as a closure with a stable layer; (ii) a variational engine stated at the weakest level (existence-only resolvents, no convexity); (iii) upgrade knobs (geodesicity, convexity, projection, [[EQ:eq0069]] , homogeneity) with explicit payoffs. Compared to AGS/BC frameworks, we trade sharpness for universality and traceability of dependencies. A comparison table is given in tab:depmap.

[t]

@ lLL@

& This paper (baseline) & Classical (AGS/BC)\

Energy/penalty & [[EQ:eq0070]] , [[EQ:eq0071]] coercive & [[EQ:eq0072]] -convex energies, quadratic penalties\
Resolvent & existence assumed & proximal/resolvent well-posed, single-valued\
Geometry & completeness (+ geodesic sublevels optional) & geodesic/CAT(0), often convexity\
Scaling & two-sided sandwich (inverse-image) \& exact ( [[EQ:eq0073]] -hom.) & exact (homogeneous) or model-specific\
Clock & [[EQ:eq0074]] (internal) & energy dissipation (external time)\

What we trade and what we gain: a modular dependency map.
tab:depmap

PARAGRAPH: At-a-glance dependency map.

@ LL@

Conclusion & Assumptions used \

Generated closure [[EQ:eq0075]] ; [[EQ:eq0076]] & Order-theoretic only \
Discrete minimizing movements: [[EQ:eq0077]] nonincreasing & UA2--UA3 \
GMM existence (continuous-time) & UA2--UA3--UA5--UA6--UA7 \
Scaling: two-sided sandwich (inverse-image form) & UA2--UA3--UA8--UA9 \
Scaling: forward inclusion [[EQ:eq0078]] & UA2--UA3--UA8--UA9 \
Exact scaling (equality under single-valuedness) & UA2--UA3--UA8--UA9 (+ single-valued [[EQ:eq0079]] ) \
Semigroup \& [[EQ:eq0080]] (model) & CAT(0)/Hilbert + projection [[EQ:eq0081]] + [[EQ:eq0082]] (+ Opial/compactness) \

SUBSECTION: Guiding examples

[leftmargin=1.2em]
- Convex safe zone (metric layer).
[[EQ:eq0083]] , [[EQ:eq0084]] closed convex; let [[EQ:eq0085]] be the metric projection (nonexpansive).
This example illustrates the metric layer; [[EQ:eq0086]] need not be a closure for an arbitrary ambient order.
If the order is chosen so that [[EQ:eq0087]] is an upper set and [[EQ:eq0088]] is inflationary and idempotent (e.g., in suitable lattice orders), then the dual package applies. With [[EQ:eq0089]] , [[EQ:eq0090]] and [[EQ:eq0091]] ; discrete MM yields layer-distance descent.
- Band projection on [[EQ:eq0092]] (order \& metric integrated).
Let [[EQ:eq0093]] , [[EQ:eq0094]] , ordered by the positive cone. Let [[EQ:eq0095]] be the band projection onto a sublattice.
Then [[EQ:eq0096]] is a closure (inflationary, monotone, idempotent) and [[EQ:eq0097]] -Lipschitz for the [[EQ:eq0098]] norm (contractive band projection).
With [[EQ:eq0099]] one gets strong scale compatibility and the scaling laws apply.
- Information-flavored model (caveat).
[[EQ:eq0100]] the probability simplex with total variation or Wasserstein metric; [[EQ:eq0101]] an equilibrium set; [[EQ:eq0102]] a (possibly approximate) projection onto [[EQ:eq0103]] . Here [[EQ:eq0104]] is a metric distance to equilibria. Approximate projections may violate invariance or strong scale compatibility; these lie outside our core unless explicitly assumed.

SECTION: Generated Closure on a Poset

sec:gen

[Closure lattice]def:cl-lattice
[[EQ:eq0105]] denotes the set of all closures on [[EQ:eq0106]] , ordered pointwise: [[EQ:eq0107]] for all [[EQ:eq0108]] . Then [[EQ:eq0109]] is a complete lattice; infima/suprema are pointwise.

[Closure lattice is complete]
[[EQ:eq0110]] is a complete lattice under pointwise order; arbitrary meets/joins are taken pointwise.

Classical; see Eilenberg (1944). See also Davey--Priestley Lattices and Order and Johnstone Stone Spaces for related nucleus/frame perspectives.

[Generated closure]def:gen
Let [[EQ:eq0111]] be inflationary and monotone (not assumed idempotent). Set [[EQ:eq0112]] and define the generated closure [[EQ:eq0113]] (pointwise).

[Least closure containing the interactions]thm:unification
Assume [[EQ:eq0114]] . Then [[EQ:eq0115]] exists and is the least closure with [[EQ:eq0116]] for all [[EQ:eq0117]] . Moreover,

[[EQ:eq0001]]

Equality may fail in general. It holds, e.g., when [[EQ:eq0118]] is pairwise commuting, or in frame/nucleus settings with finite-meet preservation (then [[EQ:eq0119]] is the least nucleus containing [[EQ:eq0120]] ). If a persistent [[EQ:eq0121]] from ax:persistence satisfies [[EQ:eq0122]] for all [[EQ:eq0123]] , then [[EQ:eq0124]] .

of [[EQ:eq0125]] .
If [[EQ:eq0126]] is a fixed point of every [[EQ:eq0127]] , then [[EQ:eq0128]] pointwise, hence [[EQ:eq0129]] .

[Idempotency of pointwise infimum]
If [[EQ:eq0130]] are closures, then [[EQ:eq0131]] (pointwise meet) is a closure. Indeed, [[EQ:eq0132]] is extensive and monotone; for idempotency,
[[EQ:eq0133]] by monotonicity and idempotency of each [[EQ:eq0134]] , while [[EQ:eq0135]] follows from extensivity and monotonicity.

SECTION: Weak Metric Framework and Distance to the Stable Layer

sec:weakM

[Weak metric framework ]ass:weakM
[[EQ:eq0136]] is complete. [[EQ:eq0137]] is nonexpansive with [[EQ:eq0138]] . Define [[EQ:eq0139]] (so [[EQ:eq0140]] is [[EQ:eq0141]] -Lipschitz and lsc). Let [[EQ:eq0142]] be lsc, strictly increasing, coercive, and [[EQ:eq0143]] . For each [[EQ:eq0144]] and [[EQ:eq0145]] , the problem

[[EQ:eq0002]]

admits at least one minimizer (no nonexpansiveness of [[EQ:eq0146]] is assumed).

Since [[EQ:eq0147]] is nonexpansive, it is continuous; hence [[EQ:eq0148]] is closed. Therefore [[EQ:eq0149]] is lower semicontinuous as an infimum of continuous functions.

[Direct-method existence ]prop:existence
If [[EQ:eq0150]] is proper, [[EQ:eq0151]] is lsc, and [[EQ:eq0152]] is lsc and coercive (\, [[EQ:eq0153]] and [[EQ:eq0154]] is closed by nonexpansiveness\,), then for every [[EQ:eq0155]] and [[EQ:eq0156]] the minimization defining [[EQ:eq0157]] admits at least one solution.

[Monotonicity of [[EQ:eq0158]] under [[EQ:eq0159]] Assumption~ass:weakM ]lem:Dmon
Under Assumption~ass:weakM, [[EQ:eq0160]] for all [[EQ:eq0161]] .

[Weak resolvent monotonicity ]lem:weak-comm-rev
With

[[EQ:eq0003]]

one has [[EQ:eq0162]] and, for every [[EQ:eq0163]] and [[EQ:eq0164]] ,

[[EQ:eq0004]]

[Commutation is model-dependent]
In general, neither inclusion [[EQ:eq0165]] nor the reverse need hold.
If, in addition, [[EQ:eq0166]] and [[EQ:eq0167]] commute (e.g., [[EQ:eq0168]] is the orthogonal projection onto a subspace
that leaves the energy [[EQ:eq0169]] invariant and [[EQ:eq0170]] is strictly convex; i.e.\ [[EQ:eq0171]] ), then
[[EQ:eq0172]] . We make no inclusion claim without such symmetry.

SECTION: Minimizing Movements: Discrete Core and Continuous Upgrade

sec:MM

[Discrete minimizing movements and layer-distance descent ]thm:discrete
For any partition [[EQ:eq0173]] and any choice [[EQ:eq0174]] with [[EQ:eq0175]] , the discrete trajectory satisfies

[[EQ:eq0005]]

for all [[EQ:eq0176]] . Hence [[EQ:eq0177]] and [[EQ:eq0178]] is nonincreasing and convergent (finite limit).

[GMM and topology ]def:GMM
Given a partition [[EQ:eq0179]] with mesh [[EQ:eq0180]] , set [[EQ:eq0181]] and define the piecewise-constant interpolation [[EQ:eq0182]] for [[EQ:eq0183]] . We also consider the piecewise-geodesic (or metric polygonal) interpolation [[EQ:eq0184]] joining [[EQ:eq0185]] to [[EQ:eq0186]] at constant speed on [[EQ:eq0187]] . Any limit [[EQ:eq0188]] of [[EQ:eq0189]] in [[EQ:eq0190]] for every [[EQ:eq0191]] is called a generalized minimizing movement (GMM) starting at [[EQ:eq0192]] .

PARAGRAPH: Geodesic availability on sublevels.

For the De Giorgi (piecewise geodesic) interpolation used in thm:GMM-exist, we assume that on each relevant sublevel [[EQ:eq0193]] any two successive iterates can be joined by a constant-speed geodesic (e.g., [[EQ:eq0194]] is a length space, or these sublevels are geodesic). When geodesics are unavailable, one may replace geodesic arcs by metric polygonal interpolations and reparameterizations that yield the same [[EQ:eq0195]] compactness; we keep the geodesic formulation for clarity. We apply the metric Arzelà--Ascoli/Helly compactness for [[EQ:eq0196]] curves on geodesic sublevels as in Ambrosio--Gigli--Savaré (AGS), Chapters~1--2.

[Compactness for passage to the limit ]ass:compact
For the given initial datum [[EQ:eq0197]] , the sublevel set [[EQ:eq0198]] is relatively compact in [[EQ:eq0199]] .

[Mild growth / equi-variational control ]ass:growth
There exist [[EQ:eq0200]] and [[EQ:eq0201]] such that [[EQ:eq0202]] for all [[EQ:eq0203]] (or, more generally, [[EQ:eq0204]] satisfies a [[EQ:eq0205]] condition).

[A priori [[EQ:eq0206]] bound ]lem:apriori
For a discrete scheme [[EQ:eq0207]] with mesh [[EQ:eq0208]] ,

[[EQ:eq0006]]

hence the piecewise-geodesic interpolants are equi- [[EQ:eq0209]] on compact intervals.

[Existence of GMM under compactness ]thm:GMM-exist
Under UA2--UA3--UA5--UA6--UA7, GMMs exist as [[EQ:eq0210]] limits of [[EQ:eq0211]] . Along any such [[EQ:eq0212]] , [[EQ:eq0213]] is nonincreasing and admits a right-continuous representative; thus [[EQ:eq0214]] exists (finite). Moreover, on every bounded time interval [[EQ:eq0215]] the family of De Giorgi interpolants is relatively compact in the uniform topology; hence a subsequence converges uniformly on [[EQ:eq0216]] to a curve [[EQ:eq0217]] which is [[EQ:eq0218]] on [[EQ:eq0219]] . The limit admits a right-continuous representative (see e.g.\ AGS, Ch.~1--2).

[Asymptotics --- split form]thm:asymp
(A) Under UA2, for any GMM [[EQ:eq0220]] (when it exists) the map [[EQ:eq0221]] is nonincreasing; hence [[EQ:eq0222]] exists (not necessarily [[EQ:eq0223]] ).\
(B) If, in addition, Fejér monotonicity with respect to [[EQ:eq0224]] holds (e.g., firmly nonexpansive proximal/projection iterations with convex [[EQ:eq0225]] in a uniformly convex space), then [[EQ:eq0226]] . In Hilbert spaces, with [[EQ:eq0227]] , firmly nonexpansive resolvents and the demiclosedness principle yield the same conclusion.

[Optional strong semigroup and commutation]
In CAT(0)/Hilbert with [[EQ:eq0228]] convex, [[EQ:eq0229]] a metric projector, and [[EQ:eq0230]] , the resolvent is firmly nonexpansive; the semigroup [[EQ:eq0231]] exists and may commute with [[EQ:eq0232]] under symmetry (e.g., orthogonal projection onto an invariant subspace). Under Opial/compactness or demiclosedness, [[EQ:eq0233]] .

SECTION: Scale Behavior: Strong and Exact Laws

sec:scale

[Strong scale compatibility ]ass:scale-strong
Each [[EQ:eq0234]] is a bijective similarity with [[EQ:eq0235]] and [[EQ:eq0236]] . The monoid law [[EQ:eq0237]] holds.

[Orlicz- [[EQ:eq0238]] scaling bounds ]ass:Delta2
For every [[EQ:eq0239]] there exist [[EQ:eq0240]] such that
[[EQ:eq0241]] for all [[EQ:eq0242]] .

[Scaled resolvents: two-sided sandwich under strong compatibility ]prop:scale-sandwich-strong
Under UA2--UA3--UA8--UA9, for [[EQ:eq0243]] ,

[[EQ:eq0007]]

[Forward inclusion under [[EQ:eq0244]] and strong compatibility ]prop:scale-incl
Assume ass:weakM,ass:scale-strong,ass:Delta2. For [[EQ:eq0245]] ,

[[EQ:eq0008]]

By prop:scale-sandwich-strong, for every [[EQ:eq0246]] ,

[[EQ:eq0009]]

Applying [[EQ:eq0247]] to both sides gives
[[EQ:eq0248]] , i.e.\ [[EQ:eq0249]] .

[Exact scaling under [[EQ:eq0250]] -homogeneity; set-valued ]thm:scale-exact
If [[EQ:eq0251]] , then

[[EQ:eq0010]]

If, in addition, [[EQ:eq0252]] is single-valued (e.g., [[EQ:eq0253]] strictly convex and [[EQ:eq0254]] geodesically convex), then equality holds (the argmin correspondence is transported isomorphically by [[EQ:eq0255]] ).

[Proof sketch]
Since [[EQ:eq0256]] and [[EQ:eq0257]] , we have
[[EQ:eq0258]] . If moreover [[EQ:eq0259]] ,

[[EQ:eq0011]]

which yields the inclusion (and equality under single-valuedness).

SECTION: Order Monotonicity, Value Monotonicity, Internal Time

sec:order-time

[Order-antitonicity of distance to a translated cone ]prop:compat
Let [[EQ:eq0260]] be a normed space ordered by a closed convex cone [[EQ:eq0261]] , and suppose [[EQ:eq0262]] for some [[EQ:eq0263]] . If [[EQ:eq0264]] then [[EQ:eq0265]] ; hence [[EQ:eq0266]] .

Since [[EQ:eq0267]] there exists [[EQ:eq0268]] with [[EQ:eq0269]] . For any [[EQ:eq0270]] , take [[EQ:eq0271]] with
[[EQ:eq0272]] . Then [[EQ:eq0273]] and
[[EQ:eq0274]] . Letting [[EQ:eq0275]] gives
[[EQ:eq0276]] .

[Order-monotone/inflationary resolvents ]ass:ordJ
For each [[EQ:eq0277]] , the resolvent map [[EQ:eq0278]] admits a selection [[EQ:eq0279]] that is isotone and inflationary.

Sufficient conditions for ass:ordJ hold under monotone comparative statics (e.g., Topkis supermodularity / Milgrom--Shannon single crossing) when the objective [[EQ:eq0280]] is sub/supermodular on a distributive lattice and [[EQ:eq0281]] is nondecreasing; then the argmin correspondence admits an increasing selection.

PARAGRAPH: Example (monotone selection via Topkis).

Let [[EQ:eq0282]] be a distributive lattice, [[EQ:eq0283]] be an [[EQ:eq0284]] -type metric induced by the lattice basis,
[[EQ:eq0285]] be nondecreasing, and suppose [[EQ:eq0286]] is submodular while [[EQ:eq0287]] is supermodular.
Then the objective is supermodular in [[EQ:eq0288]] and Topkis' theorem yields an increasing argmin selection [[EQ:eq0289]] .

[Discrete order monotonicity ]prop:order-discrete
Under ass:ordJ, define the discrete scheme by the monotone selection [[EQ:eq0290]] . Then [[EQ:eq0291]] holds for all [[EQ:eq0292]] .

[Passing order to the limit ]cor:order-limit
Equip [[EQ:eq0293]] with the product metric. If the order graph [[EQ:eq0294]] is closed (or [[EQ:eq0295]] is order-continuous) and [[EQ:eq0296]] pointwise for schemes built with [[EQ:eq0297]] , then [[EQ:eq0298]] for all [[EQ:eq0299]] .

PARAGRAPH: Internal/system time.

The clock [[EQ:eq0300]] is internal: it depends on the initial datum [[EQ:eq0301]] , it may stall if [[EQ:eq0302]] converges to a positive limit, and it is unrelated to any external absolute time. It measures dissipation of non-stability.

[Internal clock]def:clock
For a discrete trajectory [[EQ:eq0303]] define [[EQ:eq0304]] (nondecreasing). For a GMM [[EQ:eq0305]] (when it exists) define [[EQ:eq0306]] , which is nondecreasing and right-continuous.

Let [[EQ:eq0307]] be order-consistent ( [[EQ:eq0308]] ).

[Value monotonicity ]cor:value
If [[EQ:eq0309]] is order-consistent, then along schemes built by [[EQ:eq0310]] we have [[EQ:eq0311]] ; hence along any limit [[EQ:eq0312]] under cor:order-limit. Moreover, if [[EQ:eq0313]] for all [[EQ:eq0314]] (e.g., under a strong commuting model) and [[EQ:eq0315]] , then [[EQ:eq0316]] .

PARAGRAPH: On the meaning of time saturation.

If [[EQ:eq0317]] , the internal clock [[EQ:eq0318]] saturates at [[EQ:eq0319]] :
the system reaches a metastable regime relative to the chosen persistence structure [[EQ:eq0320]] .
In models with an external physical time, the two times may decouple; our construction isolates the
dissipated non-stability as the intrinsic time budget consumed along the motion.

[Terminology]
[[EQ:eq0321]] denotes a closure/persistent map (not expectation). By a Fejér cluster point with respect to [[EQ:eq0322]] we mean a cluster point of a sequence whose distances to [[EQ:eq0323]] form a nonincreasing sequence. Under (B) of thm:asymp, Fejér (or quasi-Fejér) monotonicity forces all cluster points into [[EQ:eq0324]] , hence [[EQ:eq0325]] .

SECTION: Optional Strong Models (Local Add-Ons)

sec:addons

PARAGRAPH: Case study (Hilbert + quadratic penalty).

Under UA2 with a Hilbert structure, [[EQ:eq0326]] , and convex [[EQ:eq0327]] with [[EQ:eq0328]] the metric projection,
[[EQ:eq0329]] is firmly nonexpansive and [[EQ:eq0330]] (nonlinear contraction semigroup).
With demiclosedness and Fejér/quasi-Fejér monotonicity one recovers [[EQ:eq0331]] ,
thus re-deriving a classical convergence statement from the UA-modular path.

PARAGRAPH: Frames/nuclei and powersets.

Layer O statements (sec:gen) hold verbatim. For Layer M, retain ass:weakM without identifying [[EQ:eq0332]] as a projector.

SECTION: Conclusion

From the single axiom persistence as closure and a modular metric framework, we deduced: a generated closure capturing interaction stabilization; a discrete intrinsic motion with layer-distance descent and an internal clock; and, under compactness and mild growth, continuous-time GMMs with well-posed asymptotics. Exact scale laws, semigroups, and convergence to the stable layer arise only when their natural local hypotheses are explicitly added. Thus the chain Closure [[EQ:eq0333]] Motion [[EQ:eq0334]] (Internal) Time is closed at a universal baseline and refineable in a modular way.

SECTION: References

[leftmargin=1.2em,itemsep=0.25em]
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- M.\ Bac\'ak, Convex Analysis and Optimization in Hadamard Spaces, De Gruyter, 2014.
- H.\ H.\ Bauschke and P.\ L.\ Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd ed., Springer, 2017.
- H.\ Brezis, Op\'erateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland, 1973.
- R.\ E.\ Bruck, ``Asymptotic convergence of nonexpansive mappings,'' Bull.\ Amer.\ Math.\ Soc. 79 (1973), 1258--1262.
- M.\ G.\ Crandall and T.\ M.\ Liggett, ``Generation of semigroups of nonlinear transformations on general Banach spaces,'' Amer.\ J.\ Math. 93 (1971), 265--298.
- B.\ A.\ Davey and H.\ A.\ Priestley, Lattices and Order, 2nd ed., Cambridge Univ.\ Press, 2002.
- E.\ De Giorgi, ``New problems on minimizing movements,'' in Boundary Value Problems for PDE and Applications, Springer, 1994, 81--98.
- S.\ Eilenberg, ``Closure operators and Galois theory in lattices,'' Trans.\ Amer.\ Math.\ Soc. 55 (1944), 1--23.
- K.\ Goebel and W.\ A.\ Kirk, Topics in Metric Fixed Point Theory, Cambridge Univ.\ Press, 1990.
- P.\ T.\ Johnstone, Stone Spaces, Cambridge Univ.\ Press, 1982.
- P.\ Milgrom and C.\ Shannon, ``Monotone comparative statics,'' Econometrica 62 (1994), 157--180.
- D.\ M.\ Topkis, ``Minimizing a submodular function on a lattice,'' SIAM J.\ Control Optim. 17 (1979), 787--793.
- J.-J.\ Moreau, ``Proximité et dualité dans un espace hilbertien,'' Bull.\ Soc.\ Math.\ France 93 (1965), 273--299.
- Z.\ Opial, ``Weak convergence of the sequence of successive approximations,'' Bull.\ Amer.\ Math.\ Soc. 73 (1967), 591--597.
- R.\ R.\ Phelps, Convex Functions, Monotone Operators and Differentiability, 2nd ed., Springer, 1993.
[[EQ:eq0012]]

[[EQ:eq0013]]

[[EQ:eq0014]]

[[EQ:eq0015]]

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[[EQ:eq0121]]

[[EQ:eq0122]]

[[EQ:eq0123]]

[[EQ:eq0124]]

[[EQ:eq0125]]

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Layer O (Order)

Layer M (Metric)

Guiding examples

Generated Closure on a Poset

Scale Behavior: Strong and Exact Laws

Optional Strong Models (Local Add-Ons)

Conclusion

References

\[
  \bigcap_{C\in\mathcal{C}}\Fix(C)\ \subseteq\ \Fix(\E_T).
\]

\[
  J_\lambda(x)\in\operatorname*{arg\,min}_{y\in X}\Bigl\{D(y)+\tfrac{1}{\lambda}\,\varphi\bigl(d(y,x)\bigr)\Bigr\}
\]

J_\lambda(x)\in\operatorname*{arg min}_{y\in X}\Bigl\{D(y)+\tfrac{1}{\lambda} \varphi\bigl(d(y,x)\bigr)\Bigr\}

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\[
  m(x):=\inf_y \bigl\{ D(y)+\lambda^{-1}\varphi(d(y,x))\bigr\},\qquad
  J^\varepsilon_\lambda(x):=\Bigl\{y:\ D(y)+\lambda^{-1}\varphi(d(y,x))\le m(x)+\varepsilon\Bigr\},
\]

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\[
  \E y\in J^\varepsilon_\lambda(\E x).
\]

\sum_k \frac{d(x_k,x_{k-1})^p}{\tau_k} \le \frac{1}{c} \big(D(x_0)-\inf D\big),

\[
  J_{\lambda/C_\kappa}(x)\ \subseteq\ S_\alpha^{-1}\!\big(J_\lambda(S_\alpha x)\big)\ \subseteq\ J_{\lambda/c_\kappa}(x).
\]

J_{\lambda/C_\kappa}(x)\ \subseteq\ S_\alpha^{-1} \big(J_\lambda(S_\alpha x)\big)\ \subseteq\ J_{\lambda/c_\kappa}(x).

\[
  J_\lambda\circ S_\alpha \ \subseteq\ S_\alpha\circ J_{\lambda/c_\kappa}.
\]

\[
  S_\alpha^{-1}\!\big(J_\lambda(S_\alpha x)\big)\ \subseteq\ J_{\lambda/c_\kappa}(x).
\]

\[
  J_\lambda\circ S_\alpha \ \subseteq\ S_\alpha\circ J_{\lambda/\kappa(\alpha)^{\,p-1}}.
\]

J_\lambda\circ S_\alpha \ \subseteq\ S_\alpha\circ J_{\lambda/\kappa(\alpha)^{ p-1}}.

\[
  D(S_\alpha y)+\lambda^{-1}\varphi(d(S_\alpha y,S_\alpha x))
  = \kappa(\alpha)\Big(D(y)+(\lambda/\kappa(\alpha)^{p-1})^{-1}\varphi(d(y,x))\Big),
\]

$J_\lambda S_\alpha\subseteq S_\alpha J_{\lambda/c_\kappa}$

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$\mathcal{C}:=\{C\in \mathrm{Cl}(S): Q_i\le C\ \forall i\}$

$C(Cx)=\bigwedge_i C_i(\bigwedge_j C_j(x))\le \bigwedge_i C_i(C_i(x))=\bigwedge_i C_i(x)=C(x)$

$\E(J_\lambda x)\subseteq J_\lambda(\E x)$

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d(S_\alpha x,S_\alpha y)=\kappa(\alpha) d(x,y)

$c_\kappa\,\varphi(r)\le \varphi(\kappa r)\le C_\kappa\,\varphi(r)$

c_\kappa \varphi(r)\le \varphi(\kappa r)\le C_\kappa \varphi(r)

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$J_\lambda(S_\alpha x)\subseteq S_\alpha J_{\lambda/c_\kappa}(x)$

$J_\lambda\circ S_\alpha \subseteq S_\alpha\circ J_{\lambda/c_\kappa}$

d(S_\alpha y,S_\alpha z)=\kappa(\alpha) d(y,z)

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PERSISTENCE-FIRST EMERGENCE OF RELATIONAL BENEVOLENCE: Creation and Propagation as Natural-Law--Style Asymptotic Regularities without External Meta-Governance

margin=1in

hidelinks

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

E % primitive closure / persistor
Fix % fixed-point set
D % geometric potential
H_ RB % RB state in lattice M
T % minimizing movement semigroup
J % resolvent
M % generic supermartingale
W % (log-)wealth
F % friction (predictable, nondecreasing)
kappaunder % predictable bridge coefficient

%
^ _
h_ RB
^#1 %
_ #1 h_ RB ^ #1
#1_#2 %
h_ RB,#2 ^ #1
_#1 %
^ #1 h_ RB,#1
#1^#2 %
h_ RB,#1 ^ #2

TITLE: Persistence-First Emergence of Relational Benevolence:\
Creation and Propagation as Natural-Law--Style Asymptotic Regularities without External Meta-Governance

AUTHOR: K.~Takahashi
[[EQ:eq0003]]

Let [math] F [/math] be the family of all intersections of members of [math] F [/math].
The Moore family generated by [math] F [/math] induces a closure [math] E_T [/math] with
[math] (E_T)= _ S FS [/math]
and [math] Q_i E_T [/math] for all [math] i [/math].

Closure operators and closed-set systems are in antitone correspondence (Davey--Priestley; Gierz et al.); [math] Cl(X) [/math] is a complete lattice.

SUBSECTION: Fixed-point aggregation and audit scalars

[lfp existence]ass:lfp
Either (Tarski) [math] ( M, ) [/math] is a complete lattice and [math] F_ C [/math] is monotone; or (Kleene) [math] ( M, ) [/math] is a dcpo with bottom and [math] F_ C [/math] is Scott-continuous. Then [math] ^ =lfp(F_ C ) [/math] exists.

[Audit state and scalar evaluations]def:state-eval
Let [math] ^ :=lfp(F_ C ) M [/math] be the RB state, and set [math] ^ :=L( ^ ) [/math] (constant).
For any adapted trajectory [math] (x_t) [/math] define the scalar audit process

[[EQ:eq0004]]

All difference statements and e-process constructions apply to [math] ( _t)_t [/math].

SECTION: Nondual bridge (P1, P2, P2')

SUBSECTION: P1: predictable signed error-bound

ax:P1
[P1]ass:P1
There exist monotone 1-Lipschitz maps [math] :X M [/math] and [math] L: M R [/math], a predictable coefficient [math] kappaunder_t>0 [/math], and [math] _t 0 [/math] such that

[[EQ:eq0001]]

[Sufficient conditions for P1 in a Hilbert model]prop:P1-sufficient
Let [math] X= R^d [/math], [math] C X [/math] closed convex, [math] =P_C [/math], [math] (x)=dist(x,C) [/math].
Let [math] g:X R [/math] be convex, monotone (product order), [math] L [/math]-Lipschitz; set [math] _t=g(x_t) [/math].
Pick any subgradient [math] g_t g(x_t) [/math].
If there exists [math] gamma (0,1] [/math] with

[[EQ:eq0005]]

then for predictable [math] kappaunder_t:=gamma/L [/math] and some mesh error [math] _t 0 [/math], inequality eq:P1 holds.

(Product order:) [math] x y [/math] iff [math] x_i y_i [/math] for all coordinates [math] i [/math].

[Anytime-valid estimation of [math] kappaunder_t [/math]]
Regress [math] Y_t:=- _t [/math] on [math] X_t:=- _t [/math] using mixture e-process stakes; publish a lower confidence sequence [math] kappaunder_t [/math] satisfying [math] P( t:kappa^ <kappaunder_t) alpha [/math].

SUBSECTION: P2: Galois/adjoint naturality

[P2]ass:P2
There exists [math] : M X [/math], monotone and 1-Lipschitz, such that [math] [/math] (i.e. [math] (x) m x (m) [/math]). On [math] (E_T) [/math]:

[[EQ:eq0006]]

[Sufficient conditions for P2]prop:suff
If [math] M [/math] is the cone of monotone 1-Lipschitz functionals on [math] X [/math] with sup-norm, [math] [/math] the characteristic embedding, and [math] [/math] the residuation (or an [math] [/math]-nearest selection), then [math] , [/math] are 1-Lipschitz and [math] [/math] (weak). If [math] F_ C [/math] is Scott-continuous and preserves fixed points of [math] E_T [/math] (i.e., [math] x (E_T) x F_ C x [/math]), then the two inclusions above hold.

When [math] [/math] is defined via [math] [/math]-nearest selections, we assume a measurable selection, which holds under standard properness and closed-image conditions.

SUBSECTION: P2': adjoint preservation ([math] [/math]-inclusion)

[P2']ass:P2prime
There exist errors [math] epsilon_t 0 [/math] such that

[[EQ:eq0007]]

in [math] [/math]-inclusion; metric relaxation:

[[EQ:eq0008]]

When [math] M [/math] is the functional-cone model, the distance above is the sup-norm on functionals.

SECTION: Resolvents, [[EQ:eq0018]]

epsilon-commutation, and minimizing movements

SUBSECTION: Weak proximal scheme and existence (AGS)

For [math] lambda>0 [/math], define the resolvent

[[EQ:eq0009]]

where [math] [/math] is convex, increasing, lower semicontinuous, superlinear, and [math] (0)=0 [/math].
On proper CAT(0)/Hilbert spaces with [math] [/math] lower semicontinuous and bounded below,
generalized minimizing movements exist (Ambrosio--Gigli--Savar\'e).

SUBSECTION: Quantitative [[EQ:eq0019]]

epsilon-commutation
Note that for any idempotent [math] :X X [/math], we have [math] x ( ) [/math] and hence [math] D D [/math] automatically.
[ [[EQ:eq0020]] epsilon -commutation rate]lem:eps
If [math] [/math] is [math] 1 [/math]-Lipschitz and [math] [/math], then for any [math] lambda>0 [/math] and any [math] y _lambda(x) [/math],

[[EQ:eq0010]]

where [math] C_1,C_2>0 [/math] and [math] alpha,beta (0,1] [/math] depend only on local regularity of [math] , [/math].

[Rate origin]
Nonexpansiveness of [math] [/math], triangle inequalities, [math] [/math], and resolvent optimality yield the rate. If [math] [/math] satisfies a local Kurdyka--Łojasiewicz inequality with exponent [math] theta (0,1] [/math] near [math] ( ) [/math], then [math] omega(r) r^theta [/math]; with [math] (r)= 12 r^2 [/math] one may take [math] alpha=1 [/math].

SUBSECTION: Minimizing movements and internal potential time

Discrete scheme: [math] (x_k)+tau_k^ -1 (d(x_k,x_ k-1 )) (x_ k-1 ) [/math]. Interpolants converge (subsequence) to a minimizing movement [math] u [/math]. Define internal potential time [math] tau_x(t):= (x)- (u(t)) [/math] (nondecreasing).
[Units: potential vs physical time]rem:units
[math] tau_x [/math] measures dissipated geometric potential, not physical time. With uniform mesh, residuals decompose as [math] r_t^ ( ) = _ s t _s [/math], [math] r_t^ (omega) C _ s t omega(tau_s) [/math], [math] r_t^ (mesh) C' _ s t tau_s [/math].

SECTION: Auditing: e-processes and supermartingale bridge transfer

SUBSECTION: Local CGF and cone domain

ass:cgf
Proxy model: [math] _t= _t+b_t+xi_t [/math] with [math] E[xi_t F_ t-1 ]=0 [/math], [math] b_t F_ t-1 [/math]. There exists a cone [math] R_+ [/math] and a convex function [math] psi [/math] such that for all predictable stakes [math] s_t kappaunder_t [/math],

[[EQ:eq0002]]

In addition, when using [math] Z_t(lambda) [/math] built from the proxy [math] h_ RB [/math], assume
[math] E[ \ -lambdaxi_t\ F_ t-1 ] psi(lambda) [/math];
then [math] E[Z_t(lambda) F_ t-1 ] 1 [/math] follows.

In short, [math] Z_t(lambda) [/math] isolates the noise [math] xi_t [/math] (treating [math] - _t [/math] as predictable drift), while the bridge transfer uses the CGF bound eq:CGF directly on [math] - _t [/math] with stakes [math] s_t=lambdakappaunder_t [/math].

SUBSECTION: Conservation--Friction (product form)

Single-step factors:

[[EQ:eq0011]]

Then [math] _ s t Z_s(lambda) [/math] is a nonnegative supermartingale; mixtures [math] _t= _ s t Z_s(lambda)\,nu(dlambda) [/math] are Ville-safe. There exists predictable nondecreasing friction [math] _t [/math] such that

[[EQ:eq0012]]

Here [math] c_0:= _0 [/math] captures the initial normalization (we may set [math] _0=1 [/math] so that [math] c_0=0 [/math]).
. The inequality above comes from a Doob--Meyer-type accounting where predictable spending and rebates are collected into [math] _t [/math].

Equality (exact conservation) requires tight CGF, symmetry-invariant spending, and exact rebate accounting.

SUBSECTION: Bridge transfer to geometry (supermartingale form)

[Bridge transfer to [math] D [/math]]thm:bridge-transfer
Assume P1 and eq:CGF for all predictable stakes [math] s_t _t [/math]. Then, for every [math] lambda [/math],

[[EQ:eq0013]]

is a nonnegative supermartingale, and its mixture [math] _t(lambda)\,nu(dlambda) [/math] is Ville-safe. Therefore, with probability at least [math] 1-alpha [/math] (uniformly over all [math] t [/math]),

[[EQ:eq0014]]

For sub-Gaussian noise with variance proxy [math] sigma^2 [/math], one may take [math] psi(lambdakappaunder_s)= 12sigma^2lambda^2kappaunder_s^2 [/math], yielding a quadratic bound inside the infimum.

SECTION: Main results

PARAGRAPH: Assumptions at a glance (unless stated otherwise):

P0 and dual package; weak proximal scheme (AGS); lfp existence (Tarski/Kleene);
P1--P2--P2'; local CGF and cone domain; bounded one-step increments for [math] ( _t) [/math] and [math] ( _t) [/math], and [math] _t _t< [/math].
Equalities are confined to the projection model with indicator potential; otherwise [math] [/math]-inclusion.

SUBSECTION: Necessity: persistence implies RB

[Necessity]thm:necessity
If any of C1--C6 is persistently violated, there exists a stationary ergodic environment such that

[[EQ:eq0015]]

[Sketch]
Gate frequency (C6) and Ville safety with P1 preclude geometric descent without audit descent; ergodicity forces [math] _t [/math].

SUBSECTION: Quasi-sufficiency: RB implies persistence

[Quasi-sufficiency]thm:qsuff
If C1--C6 (thresholded) hold, [math] F_ C [/math] is Scott-continuous (order-continuous), and the bridge/[math] [/math]-commutation conditions hold, then [math] _t [/math] is Ville-safely nonincreasing (i.e., admits an anytime-valid upper confidence sequence certifying nonincrease; via Assumption~ass:cgf and C6), [math] (x_t) [/math] decreases up to [math] (kappaunder_t, _t) [/math], and the minimizing movement [math] u(t) [/math] converges weakly to an attractor [math] A ( ) [/math]. In the indicator projection model, [math] u(t) C [/math] for all [math] t [/math] (hence [math] (u(t)) 0 [/math]).

SUBSECTION: Charter-Accumulation (endogenous creation)

[Charter-Accumulation]thm:charter
Let [math] E^ (0) = [/math]. Whenever [math] _ T _T/T delta>0 [/math], add a minimal corrective closure [math] E^ (r) [/math] supported on observed violations and set [math] E^ (r+1) :=E^ (r) E^ (r) [/math]. Then [math] E^ (r) [/math] is nondecreasing and [math] (E^ (r) ) (E_T) [/math]. Under P1--P2 and a KŁ-type error bound (e.g., semi-algebraic/analytic/o-minimal), the number of revisions is finite and [math] _T/T 0 [/math].

SUBSECTION: Bi-criteria: stability and vanishing friction

[Stability implies vanishing friction]thm:bi-forward
If [math] (u(t)) 0 [/math], then [math] _T/T 0 [/math].

[Converse under observable coercivity]prop:bi-conv
If [math] kappaunder:= _t kappaunder_t>0 [/math] and there exists [math] c(delta)>0 [/math] with

[[EQ:eq0016]]

then [math] _T/T 0 (u(t)) 0 [/math].

SUBSECTION: Propagation (Lipschitz bound; conditional finite speed)

[Lipschitz-propagation bound]thm:prop
Synchronized schemes with the same mesh and penalty, and the same predictable [math] kappaunder_t [/math], satisfy

[[EQ:eq0017]]

with residual [math] r_t = _ s t _s + C _ s t omega(tau_s) + C' _ s t tau_s [/math].
If [math] omega(r)=Lr [/math], [math] kappaunder:= _skappaunder_s>0 [/math], and [math] _ s t (- ^ (x) _s) eta t [/math], then a finite-speed form holds with [math] v_ L/kappaunder [/math].

SUBSECTION: Statistical inconsistency and stress trigger

[Statistical inconsistency w.r.t.\ P0]def:stress
We say [math] ( M,F_ C ) [/math] is inconsistent with P0 if any Ville-safe e-process predicted by Theorem~thm:bridge-transfer triggers infinitely many pause--revise events with positive lower frequency.

SECTION: Examples (order--metric compatibility)

[Projection model with indicator potential: exact commutation]ex:proj
Let [math] X [/math] be Hilbert (or Hadamard), [math] =P_C [/math] the metric projection onto a closed convex [math] C [/math].
Take [math] =iota_C [/math] (indicator of [math] C [/math]), hence [math] _lambda=prox_ iota_C =P_C [/math]. Then
[math] _lambda= _lambda [/math] and [math] _t= _t [/math].
This exact-commutation model is for intuition; most results use real-valued [math] D [/math] and rely on quantitative [math] [/math]-inclusion.

[Functional-cone model: general media]
[math] M [/math] is the cone of monotone 1-Lipschitz functions with sup-norm; [math] (x) [/math] is the evaluation vector; [math] [/math] an [math] [/math]-nearest representative (measurable selection). Only [math] [/math]-inclusion is claimed; transfer and equivalence still hold.

SECTION: Scope and host robustness

Exact equalities are confined to Example~ex:proj; elsewhere use the quantitative [math] [/math]-inclusion with published rates. In powerset/Hausdorff settings, nearest-point selections may be set-valued; measurable selection is assumed. Under bi-Lipschitz host transitions with constant [math] L [/math], nonexpansiveness and moduli transport up to [math] L [/math]-dependent factors; [math] kappaunder_t^ ( ) kappaunder_t/poly(L) [/math]. Report empirical exponents in host-transition experiments.

SECTION: Reproducibility protocol (operational checklist)

[leftmargin=1.25em]
- Publish [math] Q_i [/math]; construct [math] E_T [/math] via Moore; visualize [math] (E^ (r) ) (E_T) [/math].
- Log lfp iterates (Kleene) or Tarski conditions; document chain convergence in [math] M [/math].
- Release [math] ( _t, _t, _t) [/math]; verify supermartingale inequalities and predictable, nondecreasing [math] _t [/math].
- Test [math] [/math]-commutation: sample [math] ( _lambda x) _lambda^ (lambda,x) ( x) [/math]; estimate [math] (alpha,beta) [/math] with bands.
- Report internal potential time [math] tau_x(t) [/math] and monotonicity of [math] [/math].
- Calibrate predictable lower bands [math] kappaunder_t [/math] via anytime-valid regression of [math] - _t [/math] on [math] - _t [/math] (mixture e-process).
- Track adjoint-preservation error [math] delta_ adj (t)=dist( _t, F_ C ^ \,t ) [/math] with anytime-valid bands.
- Track parameter drifts for [math] ( , ,L) [/math]; report long-run average meta-action (should decay toward zero).
- Publish gate frequencies for C2/C3/C6 with anytime confidence sequences.
- Host-transition experiments: measure [math] kappaunder [/math] degradation and friction change under bi-Lipschitz transforms.

SECTION: Symbol glossary (plain text)

[leftmargin=1.25em]
- [math] [/math]: primitive closure (persistor), 1-Lipschitz.
- [math] ( ) [/math]: stable layer (closed set).
- [math] (x) [/math]: distance to [math] ( ) [/math] (or indicator in Example~ex:proj); geometric potential.
- [math] E_T [/math]: Moore-generated closure from chartered interactions [math] \ Q_i\ [/math].
- [math] F_ C [/math]: monotone / Scott-continuous aggregator on [math] ( M, ) [/math]; [math] ^ =lfp(F_ C ) [/math].
- [math] , [/math]: evaluation embedding and (possibly [math] [/math]-)residuation; [math] [/math] (weak adjunction).
- [math] _t [/math]: scalar audit process [math] (L )(x_t) [/math]; improvement [math] _t 0 [/math].
- [math] kappaunder_t [/math]: predictable bridge coefficient (anytime-valid lower band).
- [math] _t, _t, _t [/math]: mixture supermartingale, (log-)wealth, predictable nondecreasing friction.

SECTION: Conclusion

Starting from persistence as closure, a dual order/metric package yields motion and an internal potential time. The RB charter is necessary for persistence and quasi-sufficient under mild regularity. A nondual bridge---predictable signed error-bounds and adjoint naturality/preservation---transports Ville-safe guarantees from audit to geometry and aligns stabilization with asymptotic non-stress. Charter-Accumulation realizes endogenous creation of RB; a Lipschitz-propagation bound (with conditional finite speed) documents media-agnostic spread. Exact equalities are confined to the projection model with indicator potential; elsewhere, quantitative [math] [/math]-inclusion suffices. The kernel frames creation and propagation as testable asymptotic regularities without external meta-governance; “No-meta” is interpreted as the empirical decay of meta-intervention averages.

99

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H. H. Bauschke, P. L. Combettes.
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SECTION: Minimal verifiable model (sketch)

Space [math] X= R^d [/math], closed convex [math] C [/math], [math] =P_C [/math], [math] (x)=dist(x,C) [/math].
Audit [math] g [/math] convex, monotone, [math] L [/math]-Lipschitz; [math] _t=g(x_t) [/math].
P1 via subgradient angle lower bound and Lipschitz ratio [math] kappaunder_t:=gamma/L [/math].
Resolvents and [math] [/math]-rates as in Lemma~lem:eps.
e-process: sub-Gaussian noise, mixture over [math] lambda [/math].
Quasi-sufficiency and propagation follow by discrete Gr\"onwall with residual control.
[[EQ:eq0003]]

[[EQ:eq0004]]

[[EQ:eq0005]]

[[EQ:eq0006]]

[[EQ:eq0007]]

[[EQ:eq0008]]

[[EQ:eq0009]]

[[EQ:eq0010]]

[[EQ:eq0018]]

[[EQ:eq0019]]

[[EQ:eq0020]]

First principle and structural setup

Persistence as closure (P0)

Dual order/metric package

Measure basis

Observable RB clauses (C1–C6)

Moore-generated closure E_T

Nondual bridge (P1, P2, P2')

P1: predictable signed error-bound

P2: Galois/adjoint naturality

P2': adjoint preservation (ε-inclusion)

Resolvents, ε

Quantitative ε

Local CGF and cone domain

Conservation–Friction (product form)

Main results

Necessity: persistence implies RB

Examples (order–metric compatibility)

Scope and host robustness

Symbol glossary (plain text)

Conclusion

Minimal verifiable model (sketch)

\begin{equation}\label{eq:P1}
\Delta \D_t \ \le\ \kappaunder_t\,\Delta \rbrisk_t\ +\ \varepsilon_t
\quad\text{equivalently}\quad
-\Delta \D_t \ \ge\ \kappaunder_t(-\Delta \rbrisk_t)-\varepsilon_t.
\end{equation}

\label{eq:P1} \Delta \D_t \ \le\ \kappaunder_t \Delta \rbrisk_t\ +\ \varepsilon_t \quad\text{equivalently}\quad -\Delta \D_t \ \ge\ \kappaunder_t(-\Delta \rbrisk_t)-\varepsilon_t.

\begin{equation}\label{eq:CGF}
\log \mathbb E\!\left[\exp\{s_t(-\Delta \rbrisk_t)\}\,\big|\, \mathcal F_{t-1}\right]\ \le\ \psi(s_t).
\end{equation}

\label{eq:CGF} \log \mathbb E \left[\exp\{s_t(-\Delta \rbrisk_t)\} \big|  \mathcal F_{t-1}\right]\ \le\ \psi(s_t).

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\[2pt]
\small ORCID: \href{https://orcid.org/0009-0004-4273-3365}{0009\texttt{-}0004\texttt{-}4273\texttt{-}3365}}
\date{\today}

\begin{document}
\maketitle

\begin{abstract}
We take persistence as closure (P0) as a first principle. From a dual order/metric package we obtain intrinsic motion via minimizing movements and define an internal potential time as the decay of the geometric potential \(D\) (distance to the stable layer \(\Fix(\clos)\)). A charter of relational benevolence (RB)---non-harm, consent, accountability, reversibility, repairability, and testability---is encoded (i) as a Moore-generated closure \(E_T\) from chartered interactions and (ii) as a least fixed-point (lfp) state \(\RBstate^\star=\mathrm{lfp}(F_{\mathcal C})\) on a complete lattice (or, alternatively, on a dcpo under Scott-continuity) \((\mathcal M,\preceq)\).

A nondual bridge connects geometry and audit: P1 (predictable signed error-bound) links geometric descent \(\Delta \D_t\) to audit improvement \(\Delta \rbrisk_t\); P2 (Galois/adjoint naturality) and P2' (adjoint preservation under \(\varepsilon\)-inclusion) align generated closure with fixed-point aggregation. We establish: (i) Bridge Transfer of Ville-safe supermartingale guarantees from audit to geometry; (ii) necessity of the RB charter for persistence; (iii) quasi-sufficiency (under mild regularity); (iv) Charter-Accumulation showing endogenous emergence of RB (finite revisions under Kurdyka--Łojasiewicz (KŁ)-type bounds); (v) a Lipschitz-propagation bound (and conditional finite speed). Exact equalities are confined to the projection model with indicator potential; otherwise, quantitative \(\varepsilon\)-inclusion suffices. Creation and propagation thus appear as testable asymptotic regularities (natural-law style) conditioned on stated observables and calibration protocols, without external meta-governance.
\end{abstract}

\section{First principle and structural setup}

\subsection{Persistence as closure (P0)}
\begin{assumption}[Persistence as closure]\label{ax:P0}
There exists a closure \(\clos:X\to X\) that is extensive, monotone, idempotent, with nonempty stable layer \(\Fix(\clos):=\{x:\clos x=x\}\neq\varnothing\).
\end{assumption}

\subsection{Dual order/metric package}
\begin{assumption}[Dual package]\label{ass:dual}
On a poset/metric pair \((X,\preceq,d)\), \(\clos\) is \(1\)-Lipschitz for \(d\). Define the geometric potential \(\D(x):=\mathrm{dist}(x,\Fix(\clos))\), which is \(1\)-Lipschitz and lower semicontinuous. When needed we use the linear modulus \(\omega(r)=r\).
\end{assumption}

\subsection{Probability, measurability, and differences}
All stochastic statements are on a filtered probability space \((\Omega,\mathcal F,(\mathcal F_t)_{t\ge 0},\mathbb P)\).
Trajectories \((x_t)\) are \((\mathcal F_t)\)-adapted; predictable means \(\mathcal F_{t-1}\)-measurable. Acceptance gates occur at stopping times \((\tau_k)\).
For any adapted process \(X_t\), define \(\Delta X_t:=X_t-X_{t-1}\).
\paragraph{Sign convention.} Audit improvement means \(\Delta \rbrisk_t \le 0\). Geometric descent means \(\Delta \D_t \le 0\). We model randomness via centered noise \(\xi_t\); conditionally on \(\mathcal F_{t-1}\), \(-\Delta \rbrisk_t\) acts as a predictable drift in e-process constructions.

\paragraph{Pointwise order on self-maps.}
For self-maps \(Q,C:X\to X\) on \((X,\preceq)\), write \(Q\le C\) iff \(Q(x)\preceq C(x)\) for all \(x\in X\) (pointwise order). We use this in Definition~\ref{def:moore}.

\subsection{Measure basis}
For essential infimum/supremum: Lebesgue measure on \(\mathbb R^n\times\mathbb R_+\) for PDE media; counting measure on node-time pairs for graphs.

\section{RB charter: generated closure and fixed-point aggregation}

\subsection{Observable RB clauses (C1--C6)}
We specify predictable or bounded-increment indicators enabling anytime-valid frequency estimation:
\begin{itemize}[leftmargin=1.25em]
\item C1 Non-harm: a harm indicator \(H^{\mathrm{harm}}\) is nonincreasing at acceptance points.
\item C2 Consent: \(\limsup_{t\to\infty} p_{\neg\mathrm{consent}}(t)\le \varepsilon_{\mathrm{cons}}\) (declared tolerance).
\item C3 Accountability: reason coverage \(c_{\mathrm{reason}}(t)\to 1\).
\item C4 Reversibility: rollback success probability \(p_{\mathrm{rb}}\ge \eta>0\) uniformly.
\item C5 Repairability: \(\mathbb E[T_{\mathrm{repair}}]<\infty\).
\item C6 Testability: in risk zones, Ville-safe gates fire with lower frequency \(\rho>0\).
\end{itemize}

\subsection{Moore-generated closure \(E_T\)}
\begin{definition}[Generated closure by chartered interactions]\label{def:moore}
Let \(\{Q_i:X\to X\}\) encode the charter C1--C6 as inflationary, monotone interactions. Define
\[
\mathcal F:=\{\Fix(C):\ C\in\mathrm{Cl}(X),\ Q_i\le C\ \forall i\}.
\]

\langle g_t,  x_t-P_C x_t\rangle \ \ge\ \gamma \|g_t\| \|x_t-P_C x_t\|,

\[
\Phi\circ E_T\ \preceq\ F_{\mathcal C}\circ \Phi,\qquad
E_T\circ \Pi\ \preceq\ \Pi\circ F_{\mathcal C}.
\]

\[
\Phi\circ \T_t \ \preceq\ F_{\mathcal C}^{\,t}\circ \Phi,\qquad
\T_t\circ \Pi \ \succeq\ \Pi\circ F_{\mathcal C}^{\,t}
\]

\Phi\circ \T_t \ \preceq\ F_{\mathcal C}^{ t}\circ \Phi,\qquad \T_t\circ \Pi \ \succeq\ \Pi\circ F_{\mathcal C}^{ t}

\[
d\big(\Phi\circ \T_t,\ F_{\mathcal C}^{\,t}\circ \Phi\big)\ \le\ \epsilon_t,
\qquad
\epsilon_t=\sum_{s\le t}\varepsilon(\tau_s,x_{s-1}) \ \text{(from Lemma~\ref{lem:eps})}.
\]

d\big(\Phi\circ \T_t,\ F_{\mathcal C}^{ t}\circ \Phi\big)\ \le\ \epsilon_t, \qquad \epsilon_t=\sum_{s\le t}\varepsilon(\tau_s,x_{s-1}) \ \text{(from Lemma~\ref{lem:eps})}.

\[
\J_\lambda(x)\in \arg\min_{y\in X}\ \D(y)+\lambda^{-1}\,\varphi\big(d(y,x)\big),
\]

\J_\lambda(x)\in \arg\min_{y\in X}\ \D(y)+\lambda^{-1} \varphi\big(d(y,x)\big),

\[
\clos(y)\in \J_\lambda^{\varepsilon(\lambda,x)}(\clos x),
\qquad
\varepsilon(\lambda,x)\ \le\ C_1\,\lambda^{-\alpha}+C_2\,\omega(\D(x))^{\beta},
\]

\clos(y)\in \J_\lambda^{\varepsilon(\lambda,x)}(\clos x), \qquad \varepsilon(\lambda,x)\ \le\ C_1 \lambda^{-\alpha}+C_2 \omega(\D(x))^{\beta},

\[
Z_t(\lambda):=\exp\{\lambda(-\Delta \widehat \rbrisk_t+b_t)-\psi(\lambda)\},
\qquad \mathbb E[Z_t(\lambda)\mid\mathcal F_{t-1}]\le 1.
\]

\[
\Wealth_t \ \le\ c_0+\log \Mart_t - \Fric_t.
\]

\[
\widetilde \Mart_t(\lambda):=\prod_{s=1}^t \exp\{\lambda(-\Delta \D_s+\varepsilon_s)-\psi(\lambda \kappaunder_s)\}
\]

\[
\sum_{s\le t}(-\Delta \D_s)
\ \le\
\inf_{\lambda\in\Lambda}
\Bigg\{
\frac{1}{\lambda}\sum_{s\le t}\psi(\lambda \kappaunder_s)
\ +\ \sum_{s\le t}\varepsilon_s
\ +\ \frac{1}{\lambda}\log\frac{1}{\alpha}
\Bigg\}.
\]

\[
\liminf_{t\to\infty}\Pr\big(u(t)\in\Fix(\clos)\big)=0,
\qquad
\lim_{t\to\infty}\Fric_t=+\infty \quad \text{a.s.}
\]

\[
\D(u(t))>\delta \ \Rightarrow\ \mathbb E[-\Delta \rbrisk_t\mid\mathcal F_{t-1}]\ \ge\ c(\delta),
\]

\[
\D(y_t)\ \le\ \D(x_t)\ +\ \omega(d(x,y))\ -\ \sum_{s\le t}\kappaunder_s\big(\Delta \rbrisk^{(x)}_s-\Delta \rbrisk^{(y)}_s\big)\ +\ r_t,
\]