Doctrine => Closure => Motion => Time: Portable Pure Theory of Non-Dual Harmony

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

definition
definition[theorem] Definition
axiom[theorem] Axiom
remark[theorem] Remark
example[theorem] Example

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TITLE: Doctrine [[EQ:eq0007]] Closure [[EQ:eq0008]] Motion [[EQ:eq0009]] Time:\
Portable Pure Theory of Non-Dual Harmony

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

DATE:

PARAGRAPH: Quick orientation (orders, metrics, topology).

We use the numeric order [[EQ:eq0010]] for real inequalities; Lawvere enrichment is over [[EQ:eq0011]] , where smaller numbers mean closer Lawvere.
Set-distance is [[EQ:eq0012]] (numeric inf).
Convention (forward/asymmetric). We work with symmetric metrics unless stated; for asymmetric Lawvere metrics all distances and Fej\'er/limits are read in the forward sense.
Unless otherwise indicated, upper semicontinuity is w.r.t.\ the Scott topology.

We implement all final referee requests on soundness, necessity, and portability.
Nonexpansiveness and Fej\'er are split with verifiable bridges (firm nonexpansiveness + idempotence [[EQ:eq0013]] projection) in Hilbert/CAT(0) models; doctrinal and Tarski closures are unified on continuous dcpo's with explicit parametric continuity; commutation of resolvents with the reflector is stated in weak (inclusion) form, with equality restricted to projection models; value monotonicity along the intrinsic flow is justified via isotone and inflationary resolvents; model caveats for powerset/Hausdorff and Moore nuclei are made explicit.
The Doctrine [[EQ:eq0014]] Closure [[EQ:eq0015]] Motion [[EQ:eq0016]] Time chain remains non-circular and model-realizable.

SECTION: State, Orders, Topology, Metric

sec:state
Let [[EQ:eq0017]] be a complete lattice (dcpo) with Scott topology [[EQ:eq0018]] ; finite joins are Scott-continuous.
Harmony [[EQ:eq0019]] is Scott-continuous; define the value order [[EQ:eq0020]] .
Assume the minimal compatibility

[[EQ:eq0001]]

We write [[EQ:eq0021]] .

SECTION: Standing assumptions (including OMC)

[leftmargin=1.2em]
[[EQ:eq0022]] is a dcpo; joins of [[EQ:eq0023]] -chains exist.
Scott topology is fixed unless otherwise stated.
Numeric order for reals, Lawvere enrichment over [[EQ:eq0024]] .
(OMC) If [[EQ:eq0025]] is increasing with supremum [[EQ:eq0026]] and is Cauchy in [[EQ:eq0027]] , then [[EQ:eq0028]] in [[EQ:eq0029]] (limits of inflationary iterates preserve the order limit).

SECTION: Master Axiom (No A Priori Flow)

sec:master
[KZ--Advaya Doctrine]ax:master
There exists an idempotent Kock--Z\"oberlein doctrine [[EQ:eq0030]] on a symmetric monoidal closed category [[EQ:eq0031]] , with [[EQ:eq0032]] coherent with [[EQ:eq0033]] and [[EQ:eq0034]] (Frobenius type).
The advaya sector [[EQ:eq0035]] is reflective.
A faithful [[EQ:eq0036]] sends [[EQ:eq0037]] to [[EQ:eq0038]] , yielding a Scott-continuous closure [[EQ:eq0039]] (extensive, monotone, idempotent).

[Closure vs.\ nucleus]
In frame settings, if [[EQ:eq0040]] preserves finite (or directed) meets, it is a nucleus; by default we speak of Scott-continuous closures GierzEtAl,Johnstone,EscardoNuclei.

SECTION: Nonexpansiveness and Fej\'er: split and bridged

sec:fejer
[Enriched nonexpansiveness]prop:enr-nonexp
If [[EQ:eq0041]] is a [[EQ:eq0042]] -enriched left adjoint of the inclusion [[EQ:eq0043]] , then [[EQ:eq0044]] is [[EQ:eq0045]] -Lipschitz Kelly.

Both [[EQ:eq0046]] and [[EQ:eq0047]] carry the same [[EQ:eq0048]] -enriched structure; the inclusion [[EQ:eq0049]] is [[EQ:eq0050]] -enriched.

[Firm nonexpansiveness + idempotence [[EQ:eq0051]] projection]prop:firm-proj
Let [[EQ:eq0052]] be Hilbert (or CAT(0)).
If [[EQ:eq0053]] is firmly nonexpansive, idempotent ( [[EQ:eq0054]] ), and [[EQ:eq0055]] is closed and convex, then [[EQ:eq0056]] ; hence [[EQ:eq0057]] is [[EQ:eq0058]] -Lipschitz and Fej\'er BauschkeCombettes, Bacak.

[How to verify firm nonexpansiveness in practice]
In Hilbert spaces, firm nonexpansiveness is equivalent to being a resolvent of a maximal monotone operator and, for convex sets, to the metric projection; see BauschkeCombettes.
In CAT(0) spaces, see Bacak.

[Model-dependent shortcut]rem:proj
If one knows a priori that [[EQ:eq0059]] equals the metric projection [[EQ:eq0060]] (e.g.\ finite-dimensional convex models), then nonexpansiveness and Fej\'er follow immediately.

SECTION: Tarski Presentation and Unification (continuous dcpo)

sec:tarski
For this section we fix the Scott topology and assume [[EQ:eq0061]] is a continuous dcpo.

Let [[EQ:eq0062]] be Scott-continuous, monotone, and inflationary ( [[EQ:eq0063]] ).

[Tarski]def:tarski
For fixed [[EQ:eq0064]] , [[EQ:eq0065]] and

[[EQ:eq0002]]

[Generation \& commutation]ax:gen
[[EQ:eq0066]] commutes with [[EQ:eq0067]] and is the least Scott-continuous closure containing [[EQ:eq0068]] .

[Pre-fixpoint meet is a closure]lem:closure
If [[EQ:eq0069]] are Scott-continuous, then [[EQ:eq0070]] is extensive, monotone, idempotent, Scott-continuous; it is the least pre-fixpoint of [[EQ:eq0071]] .

[Parametric Scott continuity]lem:param-strong
If [[EQ:eq0072]] is a continuous dcpo and [[EQ:eq0073]] is Scott-continuous in both coordinates and monotone in the second, then [[EQ:eq0074]] is Scott-continuous GierzEtAl, AbramskyJung.

[Finite-meet preservation (frame case)]lem:meet-finite
If [[EQ:eq0075]] is a frame and [[EQ:eq0076]] preserve finite meets, then [[EQ:eq0077]] preserves finite meets; hence [[EQ:eq0078]] is a nucleus EscardoNuclei.

[Unification (complete)]thm:unify
Assume Axioms ax:master, ax:gen, Lemmas lem:closure and lem:param-strong (with the Scott topology fixed throughout this section), with [[EQ:eq0079]] a continuous dcpo so that Lemma lem:param-strong applies.
Then [[EQ:eq0080]] . If, moreover, Lemma lem:meet-finite holds, [[EQ:eq0081]] is the least nucleus containing [[EQ:eq0082]] .

SECTION: Value on the Stable Layer: correct polarity (inf)

sec:value
We do not assume [[EQ:eq0083]] here.

[Bipolar extremality --- inf]thm:bipolar-inf
Assume (OH) and Theorem~thm:unify. For

[[EQ:eq0003]]

[[EQ:eq0084]] is the least element of [[EQ:eq0085]] , hence

[[EQ:eq0004]]

If [[EQ:eq0086]] is Scott-compact and [[EQ:eq0087]] is upper semicontinuous (w.r.t.\ the fixed topology), the infimum is attained at [[EQ:eq0088]] .

We treat [[EQ:eq0089]] as an order-inducing functional aligned with (OH), not as a normative notion; any ethical or aesthetic reading would require extra premises extrinsic to the present mathematics.

SECTION: Genesis of Dynamics from [[EQ:eq0090]]

sec:genesis
Define the non-harmony functional [[EQ:eq0091]] .

[Proximal hypotheses with general penalty]ax:prox
[[EQ:eq0092]] is Cauchy complete; [[EQ:eq0093]] is [[EQ:eq0094]] -lsc and [[EQ:eq0095]] -geodesically convex (for a chosen geodesic structure); for convex, increasing [[EQ:eq0096]] with [[EQ:eq0097]] , the resolvent

[[EQ:eq0005]]

exists and is [[EQ:eq0098]] -Lipschitz.

[Resolvent commutation --- weak and equal cases]lem:commute-weak
Assume [[EQ:eq0099]] is [[EQ:eq0100]] -Lipschitz, preserves [[EQ:eq0101]] , and satisfies [[EQ:eq0102]] .
Then [[EQ:eq0103]] for all [[EQ:eq0104]] .
If, moreover, [[EQ:eq0105]] (hence firmly nonexpansive and idempotent) onto a closed convex [[EQ:eq0106]] , then [[EQ:eq0107]] .

[Intrinsic semiflow]def:flow
For [[EQ:eq0108]] , define [[EQ:eq0109]] (minimizing movements), whenever the limit exists CrandallLiggett,AGS.

[Doctrine-generated dynamics]thm:genesis
We use Prop.~prop:enr-nonexp for 1-Lipschitzness and either Prop.~prop:firm-proj or Remark~rem:proj for Fej\'er.
Together with Axiom ax:prox and Lemma lem:commute-weak, [[EQ:eq0110]] is a [[EQ:eq0111]] -Lipschitz semiflow, [[EQ:eq0112]] transfers to a weak commutation for [[EQ:eq0113]] , and

[[EQ:eq0006]]

Equality [[EQ:eq0114]] holds when [[EQ:eq0115]] is the metric projection.

SECTION: Convergence, Order Monotonicity, and Internal Time

sec:time
[Mild convergence]ax:conv
[[EQ:eq0116]] is lsc; an Opial-type condition holds (e.g.\ Hilbert); Crandall--Liggett/AGS passage from resolvents to semiflows applies CrandallLiggett,AGS.
We also use Fej\'er monotonicity (via Proposition~prop:firm-proj or Remark~rem:proj).

[Convergence]thm:conv
Under Theorem thm:genesis and Axiom ax:conv, [[EQ:eq0117]] as [[EQ:eq0118]] .

[Order-monotonic flow via inflationary resolvents]prop:inflationary
If each [[EQ:eq0119]] is isotone and inflationary ( [[EQ:eq0120]] ), then [[EQ:eq0121]] for all [[EQ:eq0122]] .
Under (OH) it follows that [[EQ:eq0123]] .

Inflationarity holds if each sublevel [[EQ:eq0124]] is an upper set and [[EQ:eq0125]] is order-nondecreasing, exhibiting an explicit [[EQ:eq0126]] -- [[EQ:eq0127]] compatibility.

[Internal clock]def:clock
For [[EQ:eq0128]] , set [[EQ:eq0129]] (nondecreasing) and [[EQ:eq0130]] .

[Value--time correspondence (tiered)]prop:duality-tier
Assume Proposition prop:inflationary. Then [[EQ:eq0131]] by (OH), while [[EQ:eq0132]] .
If, in addition, [[EQ:eq0133]] for all [[EQ:eq0134]] , then [[EQ:eq0135]] .

SECTION: Models and Caveats (portable synopsis)

sec:models
[leftmargin=1.2em]
- Finite-dimensional convex (safe zone). Closed convex sets in [[EQ:eq0136]] (including [[EQ:eq0137]] ) under reverse inclusion.
Take [[EQ:eq0138]] as finite compositions of halfspace projections [[EQ:eq0139]] ; if [[EQ:eq0140]] is closed convex then [[EQ:eq0141]] , firmly nonexpansive, [[EQ:eq0142]] -Lipschitz, Fej\'er; resolvents with [[EQ:eq0143]] ; convergence classical BauschkeCombettes. If [[EQ:eq0144]] , either regularize the projection or treat [[EQ:eq0145]] as the nearest point to a designated reference (details application-dependent).
- Powerset (finite/compact base). [[EQ:eq0146]] or [[EQ:eq0147]] for compact metric [[EQ:eq0148]] ; distances: weighted symmetric difference or Hausdorff.
Projection caveat: nearest-point projections onto arbitrary closed families need not exist/be single-valued.
They are ensured, e.g., under hyperconvex hull assumptions (Aronszajn--Panitchpakdi type) or discrete weighted metrics on finite/compact bases.
- Moore / nucleus (algebraic/continuous). [[EQ:eq0149]] complete; with finite-meet preservation [[EQ:eq0150]] is a nucleus.
Design lattice metrics (e.g.\ [[EQ:eq0151]] ); ensure a `` [[EQ:eq0152]] -convexity'' of [[EQ:eq0153]] to obtain Fej\'er (e.g.\ if [[EQ:eq0154]] then an appropriate measure-centered selection along [[EQ:eq0155]] remains in [[EQ:eq0156]] ); see EscardoNuclei,Johnstone,CaspardMonjardet.

SECTION: Terminological note

By ``emergence of motion from doctrine'' we mean constructive generation via minimizing movements, not metaphysical irreducibility.

99

AbramskyJung
S.~Abramsky and A.~Jung,
Domain Theory,
in: Handbook of Logic in Computer Science, Vol.~3, Clarendon/Oxford, 1994.

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

Lawvere
F.~W.~Lawvere,
Metric spaces, generalized logic, and closed categories,
Rend.\ Sem.\ Mat.\ Fis.\ Milano 43 (1973), 135--166.

DaveyPriestley
B.~A.~Davey and H.~A.~Priestley,
Introduction to Lattices and Order, 2nd ed.,
Cambridge Univ.\ Press, 2002.

EscardoNuclei
M.~H.~Escard\'o,
Joins in the frame of nuclei,
Houston J.\ Math. 30(3) (2004), 937--952.

CaspardMonjardet
N.~Caspard and B.~Monjardet,
Some lattices of closure systems on a finite set,
Discrete Math.\ Theor.\ Comput.\ Sci. 6(2) (2004), 163--190.

Johnstone
P.~T.~Johnstone,
Stone Spaces,
Cambridge Studies in Adv.\ Math., Vol.~3, CUP, 1982.

BauschkeCombettes
H.~H.~Bauschke and P.~L.~Combettes,
Convex Analysis and Monotone Operator Theory in Hilbert Spaces,
2nd ed., Springer, 2017.

Bacak
M.~Bac\'ak,
Convex Analysis and Optimization in Hadamard Spaces,
De Gruyter, 2014.

AGS
L.~Ambrosio, N.~Gigli, and G.~Savar\'e,
Gradient Flows in Metric Spaces and in the Space of Probability Measures,
Birkh\"auser, 2005.

CrandallLiggett
M.~G.~Crandall and T.~M.~Liggett,
Generation of semi-groups of nonlinear transformations on general Banach spaces,
Amer.\ J.\ Math. 93 (1971), 265--298.

Kelly
G.~M.~Kelly,
Basic Concepts of Enriched Category Theory,
Cambridge Univ.\ Press, 1982 (reprint: TAC Theory Appl.\ Categ., 2005).
[[EQ:eq0007]]

[[EQ:eq0008]]

[[EQ:eq0009]]

[[EQ:eq0010]]

[[EQ:eq0011]]

[[EQ:eq0012]]

[[EQ:eq0013]]

[[EQ:eq0014]]

[[EQ:eq0015]]

[[EQ:eq0016]]

[[EQ:eq0017]]

[[EQ:eq0018]]

[[EQ:eq0019]]

[[EQ:eq0020]]

[[EQ:eq0021]]

[[EQ:eq0022]]

[[EQ:eq0023]]

[[EQ:eq0024]]

[[EQ:eq0025]]

[[EQ:eq0026]]

[[EQ:eq0027]]

[[EQ:eq0028]]

[[EQ:eq0029]]

[[EQ:eq0030]]

[[EQ:eq0031]]

[[EQ:eq0032]]

[[EQ:eq0033]]

[[EQ:eq0034]]

[[EQ:eq0035]]

[[EQ:eq0036]]

[[EQ:eq0037]]

[[EQ:eq0038]]

[[EQ:eq0039]]

[[EQ:eq0040]]

[[EQ:eq0041]]

[[EQ:eq0042]]

[[EQ:eq0043]]

[[EQ:eq0044]]

[[EQ:eq0045]]

[[EQ:eq0046]]

[[EQ:eq0047]]

[[EQ:eq0048]]

[[EQ:eq0049]]

[[EQ:eq0050]]

[[EQ:eq0051]]

[[EQ:eq0052]]

[[EQ:eq0053]]

[[EQ:eq0054]]

[[EQ:eq0055]]

[[EQ:eq0056]]

[[EQ:eq0057]]

[[EQ:eq0058]]

[[EQ:eq0059]]

[[EQ:eq0060]]

[[EQ:eq0061]]

[[EQ:eq0062]]

[[EQ:eq0063]]

[[EQ:eq0064]]

[[EQ:eq0065]]

[[EQ:eq0066]]

[[EQ:eq0067]]

[[EQ:eq0068]]

[[EQ:eq0069]]

[[EQ:eq0070]]

[[EQ:eq0071]]

[[EQ:eq0072]]

[[EQ:eq0073]]

[[EQ:eq0074]]

[[EQ:eq0075]]

[[EQ:eq0076]]

[[EQ:eq0077]]

[[EQ:eq0078]]

[[EQ:eq0079]]

[[EQ:eq0080]]

[[EQ:eq0081]]

[[EQ:eq0082]]

[[EQ:eq0083]]

[[EQ:eq0084]]

[[EQ:eq0085]]

[[EQ:eq0086]]

[[EQ:eq0087]]

[[EQ:eq0088]]

[[EQ:eq0089]]

[[EQ:eq0090]]

[[EQ:eq0091]]

[[EQ:eq0092]]

[[EQ:eq0093]]

[[EQ:eq0094]]

[[EQ:eq0095]]

[[EQ:eq0096]]

[[EQ:eq0097]]

[[EQ:eq0098]]

[[EQ:eq0099]]

[[EQ:eq0100]]

[[EQ:eq0101]]

[[EQ:eq0102]]

[[EQ:eq0103]]

[[EQ:eq0104]]

[[EQ:eq0105]]

[[EQ:eq0106]]

[[EQ:eq0107]]

[[EQ:eq0108]]

[[EQ:eq0109]]

[[EQ:eq0110]]

[[EQ:eq0111]]

[[EQ:eq0112]]

[[EQ:eq0113]]

State, Orders, Topology, Metric

Standing assumptions (including OMC)

Master Axiom (No A Priori Flow)

Nonexpansiveness and Fejér: split and bridged

Genesis of Dynamics from (EHE,d)

Models and Caveats (portable synopsis)

Terminological note

\[
\textbf{(OH)}\qquad \sigma\sqsubseteq\tau\ \Rightarrow\ \mathcal H(\sigma)\le \mathcal H(\tau).
\]

\[
EHE_T(\sigma):=\bigwedge\{X\in\mathcal S\mid F_\sigma(X)\sqsubseteq X\}.
\]

\[
K_\sigma:=\Fix(EHE)\cap\{y\mid \sigma\sqsubseteq y\},
\]

\[
\inf_{y\in K_\sigma}\ \mathcal H(y)=\mathcal H\big(EHE(\sigma)\big).
\]

\[
J_\lambda(x)\in\arg\min_y\ \Big\{\mathcal D(y)+\tfrac{1}{\lambda}\,\phi\big(d(y,x)\big)\Big\}
\]

J_\lambda(x)\in\arg\min_y\ \Big\{\mathcal D(y)+\tfrac{1}{\lambda} \phi\big(d(y,x)\big)\Big\}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><msub><mi>J</mi><mi>λ</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>\arg</mi><msub><mo>min</mo><mi>y</mi></msub><mtext> </mtext><mo minsize="1.623em" maxsize="1.623em">\{</mo><mi mathvariant="script">D</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>+</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mrow><mn>1</mn></mrow><mrow><mi>λ</mi></mrow></mfrac></mstyle><mspace width="0.167em" /><mi>ϕ</mi><mo minsize="1.2em" maxsize="1.2em">(</mo><mi>d</mi><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo minsize="1.2em" maxsize="1.2em">)</mo><mo minsize="1.623em" maxsize="1.623em">\}</mo><mi>\]</mi></mrow></math>

\[
\mathcal D(T_t x)\le \mathcal D(x),\qquad d(T_t x,\Fix(EHE))\le d(x,\Fix(EHE))\quad(\forall t\ge0).
\]

$\sigma\preceq\tau\iff \mathcal H(\sigma)\le\mathcal H(\tau)$

$F:\mathcal S\times\mathcal S\to\mathcal S$

$\tau(t):=\mathcal D(x)-\mathcal D(T_t x)$

$\Theta(x):=\inf\{t\ge0:\ \mathcal D(T_t x)=0\}$

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi mathvariant="script">D</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>↘</mo><mi>$</mi></mrow></math>


A PURE, NO-META SYNTHESIS OF FUNCTIONAL-INFORMATION SELECTION AND PROPAGATIVE ORGANIZATION: Weak Order Representation, Directional FKPP Speed Floors, and Audited Acceleration

1.2

colorlinks=true, linkcolor=blue, citecolor=blue, urlcolor=blue,
pdftitle= A Pure, No-Meta Synthesis of Functional-Information Selection and Propagative Organization,
pdfauthor= K. Takahashi ,
pdfkeywords= functional information, selection, Blackwell order, conditional mutual information, FKPP, front speeds, coarse-graining, auditability, no-meta

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

I(X;Y Z)
ess\,inf
ess\,sup
_+ % helper for [a]_+

TITLE: -6mm

A Pure, No-Meta Synthesis of Functional-Information Selection and Propagative Organization:\
Weak Order Representation, Directional FKPP Speed Floors, and Audited Acceleration

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

[[EQ:eq0004]]

PARAGRAPH: Definition of spreading speed (liminf-safe).

For [[EQ:eq0012]] let [[EQ:eq0013]] be the minimal radius with [[EQ:eq0014]] .
We write

[[EQ:eq0005]]

.

All lower bounds below hold for the [[EQ:eq0015]] -based [[EQ:eq0016]] without requiring [[EQ:eq0017]] -independence; when classical sandwich conditions apply (e.g.\ subadditivity or shape theorems), [[EQ:eq0018]] exists and is [[EQ:eq0019]] -independent (see Xin2000).

PARAGRAPH: Directional speed.

For a unit [[EQ:eq0020]] and [[EQ:eq0021]] , define

[[EQ:eq0006]]

and set [[EQ:eq0022]] , [[EQ:eq0023]] .
This aligns with standard notions in heterogeneous front propagation (see also Xin2000).

[Isotropic speed floor]thm:isotropic
Under (SA1)--(SA3),

[[EQ:eq0007]]

[Sharp directional lower bound with divergence penalty]thm:directional
For each unit [[EQ:eq0024]] ,

[[EQ:eq0008]]

[Nontriviality condition for the directional bound]prop:nontrivial
If [[EQ:eq0025]] on a set of positive measure in [[EQ:eq0026]] , then [[EQ:eq0027]] .
In particular, any uniform bound [[EQ:eq0028]] with [[EQ:eq0029]] yields [[EQ:eq0030]] .

[Ancestry and checks]
In constant coefficients the isotropic bound reduces to the classical Fisher--KPP minimal speed; for heterogeneous media see Xin2000 for background.
Divergence-corrected barriers extend to anisotropic [[EQ:eq0031]] with [[EQ:eq0032]] regularity; graph variants follow via normalized Laplacians and conductance proxies.
Full deterministic proofs are given in Tak_NondualField,Tak_PureNatural.

SECTION: Symmetric Coarse-Graining: Gradient Contraction and Penalties

Let [[EQ:eq0033]] be a symmetric Markov diffusion semigroup on [[EQ:eq0034]] with energy contraction.
Let [[EQ:eq0035]] be the generator of [[EQ:eq0036]] and define the carr\'e du champ

[[EQ:eq0009]]

Energy contraction means [[EQ:eq0037]] for all [[EQ:eq0038]] .
We assume either:
(i) [[EQ:eq0039]] is the heat semigroup (symmetric convolution), so that [[EQ:eq0040]] , or
(ii) [[EQ:eq0041]] satisfies a Bakry--\'Emery gradient contraction
[[EQ:eq0042]] (CD [[EQ:eq0043]] ) for some [[EQ:eq0044]] .
Define [[EQ:eq0045]] as the Loewner-minimal tensor such that

[[EQ:eq0010]]

[Coarse-graining monotonicity under heat or Bakry--\'Emery gradient contraction]prop:cg
Assume (SA1) and that [[EQ:eq0046]] is either the heat semigroup or satisfies [[EQ:eq0047]] .
Then [[EQ:eq0048]] and [[EQ:eq0049]] for all unit [[EQ:eq0050]] .
Hence symmetric coarse-graining cannot increase the directional penalty [[EQ:eq0051]] .
We do not claim monotonicity of the full directional speed bound, since [[EQ:eq0052]] may decrease the leading term.

The proof uses gradient contraction to pass from [[EQ:eq0053]] to [[EQ:eq0054]] in quadratic forms, then applies Cauchy--Schwarz and weak-* continuity to transfer divergence bounds.
Asymmetric kernels may violate gradient contraction; such cases are excluded.

SECTION: Audited Acceleration: Assumptions, Tests, Triggers, and Remainder Rate

Define the speed proxy [[EQ:eq0055]] , computed from logs with clipping and tempering.

PARAGRAPH: Audit-ready stochastic assumptions (AR).

[style=nextline,leftmargin=0pt,labelsep=0.6em]
[[EQ:eq0056]] , [[EQ:eq0057]] a.s.
Multiplicative factors [[EQ:eq0058]] clip extremes conservatively.
[[EQ:eq0059]] on a subsequence of positive lower density, or (A5++) positive-density improvements for [[EQ:eq0060]] .
The average negative variation of tempered floors vanishes (Ces\`aro).

[Ces\`aro-positive acceleration]thm:accel
Under (AR1)--(AR4), there exists [[EQ:eq0061]] such that

[[EQ:eq0011]]

hence [[EQ:eq0062]] and the cumulative front radius has a quadratic lower bound.

PARAGRAPH: Identifiability, false alarms, and tests.

FI claims are controlled by preregistering a finite grid [[EQ:eq0063]] , reserving a holdout for prequential tests, and applying a multiple-comparison correction (e.g.\ Holm) across [[EQ:eq0064]] .
Selection operations are logged within [[EQ:eq0065]] .
Beyond order embedding (not causal identification), we report partial-identification bounds by including negative-control scores in [[EQ:eq0066]] and by auditing placebo selection operators; any significant FI increase under placebo is flagged as a false alarm.
For (AR3) we run a one-sided Newey--West HAC (heteroskedasticity- and autocorrelation-consistent) [[EQ:eq0067]] -test and a phase-randomization nonparametric check on consecutive windows.

PARAGRAPH: Confidence intervals under dependence.

For dependent logs, CIs use moving block bootstrap (MBB; preregistered block length via the Politis--White rule) aligned with the HAC testing window; this alignment ensures variance consistency.

PARAGRAPH: Automatic triggers (preregistered).

Upon two consecutive windows with [[EQ:eq0068]] :
shrink clipping [[EQ:eq0069]] and enlarge window [[EQ:eq0070]] .
If [[EQ:eq0071]] approaches its threshold, increase tempering [[EQ:eq0072]] and widen the holdout proportion.

[On the [[EQ:eq0073]] term and window schedule]
We fix a nondecreasing window schedule [[EQ:eq0074]] with [[EQ:eq0075]] (preregistered).
Under (AR1)--(AR4) and standard LLN conditions, the remainder obeys [[EQ:eq0076]] in expectation; we report both point estimates and CI bands.

[Measure-theoretic note]
All processes are adapted to [[EQ:eq0077]] and uniformly integrable under (AR1)--(AR2);
the [[EQ:eq0078]] --expectation exchange uses Fatou's lemma.

SECTION: Falsifiers

(i) [[EQ:eq0079]] yet [[EQ:eq0080]] ;
(ii) symmetric Markov coarse-graining (heat or Bakry--\'Emery) increases [[EQ:eq0081]] ;
(iii) sustained [[EQ:eq0082]] and [[EQ:eq0083]] with honesty-divergence proxies (e.g., KL/Bregman calibration to verifiable facts) [[EQ:eq0084]] , yet no detectable [[EQ:eq0085]] improvement;
(iv) violation of the preregistered gradient contraction test (empirical failure of [[EQ:eq0086]] -shrinkage), invalidating Proposition~prop:cg.

SECTION: Conclusion

We delivered: (1) FI definition and selection logging with preregistered grids and false-alarm control; (2) a weak but operational CMI representation (no uniqueness claim); (3) explicit heterogeneous FKPP assumptions with isotropic and divergence-corrected directional speed floors, stated for [[EQ:eq0087]] -based speeds; (4) audited acceleration with tests, triggers, moving-block CIs, and a preregistered window schedule controlling the remainder; (5) coarse-graining monotonicity of directional penalties under heat/Bakry--\'Emery gradient contraction, without claiming monotonicity of the full directional bound.
All claims are pure, no-meta, and falsifiable.

PARAGRAPH: Data availability.

The work is theoretical; empirical audits should preregister estimators, window schedules, clipping/tempering, negative controls, placebo selections, and block lengths.

99

WongHazenPNAS
M.\,L.~Wong, R.\,M.~Hazen,
On the roles of function and selection in evolving systems,
Proc.\ Natl.\ Acad.\ Sci.\ USA 120(43) (2023), e2310223120.
doi:https://doi.org/10.1073/pnas.2310223120 10.1073/pnas.2310223120 .

HazenWongPNASNexus
R.\,M.~Hazen, M.\,L.~Wong,
Open-ended versus bounded evolution: Mineral evolution as a case study,
PNAS Nexus 3(7) (2024), pgae248.
doi:https://doi.org/10.1093/pnasnexus/pgae248 10.1093/pnasnexus/pgae248 .

Blackwell1953
D.~Blackwell,
Equivalent Comparisons of Experiments,
Ann.\ Math.\ Stat.\ 24(2) (1953), 265--272.
doi:https://doi.org/10.1214/aoms/1177729032 10.1214/aoms/1177729032 .

Xin2000
J.~Xin,
Front Propagation in Heterogeneous Media,
SIAM Rev.\ 42(2) (2000), 161--230.
doi:https://doi.org/10.1137/S0036144599364296 10.1137/S0036144599364296 .

Tak_NondualField
K.~Takahashi,
Nondual Field Theory of Viable Predictive Organization: Sharp Directional Lower Bounds for KPP-Type Fronts in Heterogeneous Media,
Zenodo (2025).
doi:https://doi.org/10.5281/zenodo.17131394 10.5281/zenodo.17131394 .

Tak_PureNatural
K.~Takahashi,
A Pure Natural Theory of Benevolent Propagation under No-Meta Closure,
Zenodo (2025).
doi:https://doi.org/10.5281/zenodo.17136051 10.5281/zenodo.17136051 .

Tak_Acceleration
K.~Takahashi,
Natural-Law Acceleration of VPO: Auditable Conditions with Signed-Coefficient Bounds,
Zenodo (2025).
doi:https://doi.org/10.5281/zenodo.17120045 10.5281/zenodo.17120045 .
[[EQ:eq0001]]

[[EQ:eq0002]]

[[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]]

Position, Scope, and No-Meta Principle

Falsifiers

Conclusion

\begin{equation}\label{eq:fi}
\mathrm{FI}_{s,\tau}:=-\log \pi(A_{s,\tau}).
\end{equation}

\begin{equation}\label{eq:fkpp}
\partial_t q \;=\; \nabla\!\cdot\!\big(D(x,t)\nabla q\big)\;+\;r(x,t)\,q(1-q)\;-\;\Gamma(x,t)\,q .
\end{equation}

\label{eq:fkpp} \partial_t q  =  \nabla \cdot \big(D(x,t)\nabla q\big) + r(x,t) q(1-q) - \Gamma(x,t) q .

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mrow><mi>\label</mi><mrow><mi>e</mi><mi>q</mi><mi>:</mi><mi>f</mi><mi>k</mi><mi>p</mi><mi>p</mi></mrow><msub><mo>∂</mo><mi>t</mi></msub><mi>q</mi><mspace width="0.278em" /><mo>=</mo><mspace width="0.278em" /><mo>∇</mo><mspace width="negativethinmathspace" /><mi>·</mi><mspace width="negativethinmathspace" /><mo minsize="1.2em" maxsize="1.2em">(</mo><mi>D</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>∇</mo><mi>q</mi><mo minsize="1.2em" maxsize="1.2em">)</mo><mspace width="0.278em" /><mo>+</mo><mspace width="0.278em" /><mi>r</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><mi>q</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>q</mi><mo stretchy="false">)</mo><mspace width="0.278em" /><mo>−</mo><mspace width="0.278em" /><mi>Γ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><mi>q</mi><mo>.</mo></mrow></mrow></math>

\[2pt]\small ORCID: \href{https://orcid.org/0009-0004-4273-3365}{0009-0004-4273-3365}}
\date{September 19, 2025}

\begin{document}
\maketitle

\begin{abstract}
We complete a pure, no-meta synthesis that operationalizes the Law of Increasing Functional Information (LIFI) under explicit mathematical assumptions and preregistered auditability.
(1) We define functional information (FI) with preregistered thresholds and test distributions, and embed admissible functionals into a \emph{weak} order represented by a monotone transform of conditional mutual information (CMI); uniqueness of CMI is not claimed without additional axioms.
(2) We state standing assumptions for heterogeneous FKPP dynamics on $\mathbb{R}^d$ (and graphs), specify boundary/domain conventions, and prove isotropic as well as \emph{sharp directional} front-speed lower bounds with a divergence penalty. All lower bounds are stated for $\liminf$-based speeds, without assuming $\alpha$-independence.
(3) We provide a Ces\`aro-positive \emph{audited acceleration} lower bound for a log-computable speed proxy, together with identifiability notes, false-alarm control (negative controls, placebo selections), HAC-robust tests, and automatic trigger rules; the sublinear remainder is tied to a preregistered window schedule.
(4) For symmetric coarse-graining we require heat semigroup structure or Bakry--\'Emery gradient contraction to guarantee gradient shrinkage; under these conditions the directional divergence penalty is nonincreasing \emph{(we do not claim monotonicity of the full directional speed bound)}.
Falsifiers and preregistered audit procedures are included.
\end{abstract}

\paragraph{Notation.}
We write $[a]_+:=\max\{a,0\}$.

\clearpage
\section{Position, Scope, and No-Meta Principle}
Wong \& Hazen formalize a selection-driven tendency for functional information (FI) to increase across evolving systems (LIFI) \cite{WongHazenPNAS}; Hazen \& Wong provide a non-biological case study via mineral evolution \cite{HazenWongPNASNexus}.
We \emph{operationalize} LIFI within a representation-invariant, \emph{no-meta} framework:
(i) we fix only \emph{order} among admissible functionals (not their normalization);
(ii) we state sufficient conditions (``floors'') and apply comparison principles to obtain \emph{front-speed lower bounds};
(iii) we give falsifiable, preregistered procedures on public logs.
No external privileged objective/evaluator is assumed; guarantees derive from logs and medium-internal dynamics.

\section{Functional Information, Selection, and Weak Order Representation}
\subsection{FI with preregistered ingredients and selection logging}
Let $\mathcal{C}$ be a configuration space with reference measure $\mu$ and a preregistered finite family $\mathcal{F}$ of score functions.
Fix a finite threshold grid $\mathcal{T}$ and a preregistered test distribution $\pi$ on $\mathcal{C}$.
For $(s,\tau)\in\mathcal{F}\times\mathcal{T}$ define $A_{s,\tau}:=\{c:s(c)\ge \tau\}$ and

EQPH_eq0001_PH

\textbf{Selection logging.} LIFI emphasizes \emph{selection for one or more functions} \cite{WongHazenPNAS}.
We therefore preregister and log the selection operator (intervention/filter) as part of covariates $Z$ to audit the selection step itself.

\subsection{Admissible functionals and extraction scheme}
Fix a lag-$\ell$ extraction $(X,Y,Z)$ from logs (e.g.\ $X$ past covariates, $Y$ outcomes, $Z$ declared controls and selection operators).
Estimators for information quantities use preregistered methods (kNN/NSB/neural) and confidence intervals (CIs).

\begin{definition}[Admissible functional class]\label{def:adm}
A functional $\Phi=\Phi(X,Y,Z)$ is \emph{admissible} if:
(i) \textbf{Blackwell-monotone}: monotone under Markov garbling/refinement of observations;
(ii) \textbf{Coarse-graining robust}: monotone under symmetric Markov regularization (semigroup) acting on the medium.
\end{definition}

\begin{theorem}[CMI-embedded weak representation]\label{thm:weak}
Under Definition~\ref{def:adm} and the preregistered extraction, there exists a strictly increasing $g$ such that the total preorder induced by $\Phi$ is contained in, and generically coincides with, that induced by $g(\CMI)$ up to a \emph{Borel null set under the preregistered sampling measure $\mathbb{P}_{\mathrm{log}}$} and up to \emph{coarse-graining tie classes} induced by the symmetric semigroup.
\end{theorem}

\begin{remark}[Scope and identifiability]
We do \emph{not} assert uniqueness of CMI as an order-representative without additional axioms (chain rule, additivity, normalization, locality).
The representation is an \emph{order embedding}, not a causal identification claim: FI increases and CMI increases are connected through order and audit, not assumed causation.
For Blackwell order background see \cite{Blackwell1953}.
\end{remark}

\section{Heterogeneous FKPP: Domain, Assumptions, and Speed Floors}
\paragraph{Domain and initial data.}
Unless stated otherwise, the spatial domain is $\mathbb{R}^d$ with bounded, compactly supported (or bounded and rapidly decaying) initial data.
On graphs we use the normalized Laplacian; by Cheeger's inequality $\lambda_1(\mathcal{L})\ge \Phi^2/2$, so a conductance proxy $\widehat{\Phi}$ yields a conservative effective bound $D_{\min}\gtrsim c_{\mathrm{step}}\widehat{\Phi}^2$ in the speed floors.
\emph{Calibration.} The proxy constant $c_{\mathrm{step}}$ depends on the time discretization and edge-weight normalization (degree-normalized Laplacian); $c_{\mathrm{step}}$ is preregistered and reported with units. The speed floors are reported in units consistent with $c_{\mathrm{step}}$ (per-step to per-time conversion is preregistered).

\paragraph{Dynamics.}
Let $q(t,x)\in[0,1]$ denote a coarse occupancy of a propagative class.
Consider

EQPH_eq0002_PH

\paragraph{Standing assumptions (SA).}
\begin{itemize}
\item[(SA1)] $D\in L^\infty_t W^{1,\infty}_x$ is uniformly elliptic: $\exists\,0<\lambda\le\Lambda<\infty$ with $\lambda I\le D(x,t)\le \Lambda I$ a.e.
\item[(SA2)] $r,\Gamma\in L^\infty(\mathbb{R}^d\times\mathbb{R}_+)$; $f(q)=r q(1-q)$ is KPP-type; $[0,1]$ is invariant; the parabolic comparison principle holds for distributional subsolutions.\footnote{A standard weak-solution framework ensuring maximum/comparison principles under (SA1)--(SA2); see L.C.~Evans, \emph{Partial Differential Equations}, 2nd ed., Ch.~8.}
\item[(SA3)] Row-wise divergence $(\operatorname{div}D)_j:=\sum_i \partial_i D_{ij}$; for constant unit $u$, $\operatorname{div}(D u)=(\operatorname{div}D)\!\cdot\!u$.
\end{itemize}
Define the linear floor and directional quantities:
\[
\lambda_{\mathrm{lin,inf}}(t):=\essinf_x\big(r(x,t)-\Gamma(x,t)\big),\qquad
\lambda_{\min}:=\inf_{t\ge0}\lambda_{\mathrm{lin,inf}}(t),
\]

2pt]\small ORCID: \href{https://orcid.org/0009-0004-4273-3365}{0009-0004-4273-3365}} \date{September 19, 2025} \begin{document} \maketitle \begin{abstract} We complete a pure, no-meta synthesis that operationalizes the Law of Increasing Functional Information (LIFI) under explicit mathematical assumptions and preregistered auditability. (1) We define functional information (FI) with preregistered thresholds and test distributions, and embed admissible functionals into a \emph{weak} order represented by a monotone transform of conditional mutual information (CMI); uniqueness of CMI is not claimed without additional axioms. (2) We state standing assumptions for heterogeneous FKPP dynamics on $\mathbb{R}^d$ (and graphs), specify boundary/domain conventions, and prove isotropic as well as \emph{sharp directional} front-speed lower bounds with a divergence penalty. All lower bounds are stated for $\liminf$-based speeds, without assuming $\alpha$-independence. (3) We provide a Ces\`aro-positive \emph{audited acceleration} lower bound for a log-computable speed proxy, together with identifiability notes, false-alarm control (negative controls, placebo selections), HAC-robust tests, and automatic trigger rules; the sublinear remainder is tied to a preregistered window schedule. (4) For symmetric coarse-graining we require heat semigroup structure or Bakry--\'Emery gradient contraction to guarantee gradient shrinkage; under these conditions the directional divergence penalty is nonincreasing \emph{(we do not claim monotonicity of the full directional speed bound)}. Falsifiers and preregistered audit procedures are included. \end{abstract} \paragraph{Notation.} We write $[a]_+:=\max\{a,0\}$. \clearpage \section{Position, Scope, and No-Meta Principle} Wong \& Hazen formalize a selection-driven tendency for functional information (FI) to increase across evolving systems (LIFI) \cite{WongHazenPNAS}; Hazen \& Wong provide a non-biological case study via mineral evolution \cite{HazenWongPNASNexus}. We \emph{operationalize} LIFI within a representation-invariant, \emph{no-meta} framework: (i) we fix only \emph{order} among admissible functionals (not their normalization); (ii) we state sufficient conditions (``floors'') and apply comparison principles to obtain \emph{front-speed lower bounds}; (iii) we give falsifiable, preregistered procedures on public logs. No external privileged objective/evaluator is assumed; guarantees derive from logs and medium-internal dynamics. \section{Functional Information, Selection, and Weak Order Representation} \subsection{FI with preregistered ingredients and selection logging} Let $\mathcal{C}$ be a configuration space with reference measure $\mu$ and a preregistered finite family $\mathcal{F}$ of score functions. Fix a finite threshold grid $\mathcal{T}$ and a preregistered test distribution $\pi$ on $\mathcal{C}$. For $(s,\tau)\in\mathcal{F}\times\mathcal{T}$ define $A_{s,\tau}:=\{c:s(c)\ge \tau\}$ and EQPH_eq0001_PH \textbf{Selection logging.} LIFI emphasizes \emph{selection for one or more functions} \cite{WongHazenPNAS}. We therefore preregister and log the selection operator (intervention/filter) as part of covariates $Z$ to audit the selection step itself. \subsection{Admissible functionals and extraction scheme} Fix a lag-$\ell$ extraction $(X,Y,Z)$ from logs (e.g.\ $X$ past covariates, $Y$ outcomes, $Z$ declared controls and selection operators). Estimators for information quantities use preregistered methods (kNN/NSB/neural) and confidence intervals (CIs). \begin{definition}[Admissible functional class]\label{def:adm} A functional $\Phi=\Phi(X,Y,Z)$ is \emph{admissible} if: (i) \textbf{Blackwell-monotone}: monotone under Markov garbling/refinement of observations; (ii) \textbf{Coarse-graining robust}: monotone under symmetric Markov regularization (semigroup) acting on the medium. \end{definition} \begin{theorem}[CMI-embedded weak representation]\label{thm:weak} Under Definition~\ref{def:adm} and the preregistered extraction, there exists a strictly increasing $g$ such that the total preorder induced by $\Phi$ is contained in, and generically coincides with, that induced by $g(\CMI)$ up to a \emph{Borel null set under the preregistered sampling measure $\mathbb{P}_{\mathrm{log}}$} and up to \emph{coarse-graining tie classes} induced by the symmetric semigroup. \end{theorem} \begin{remark}[Scope and identifiability] We do \emph{not} assert uniqueness of CMI as an order-representative without additional axioms (chain rule, additivity, normalization, locality). The representation is an \emph{order embedding}, not a causal identification claim: FI increases and CMI increases are connected through order and audit, not assumed causation. For Blackwell order background see \cite{Blackwell1953}. \end{remark} \section{Heterogeneous FKPP: Domain, Assumptions, and Speed Floors} \paragraph{Domain and initial data.} Unless stated otherwise, the spatial domain is $\mathbb{R}^d$ with bounded, compactly supported (or bounded and rapidly decaying) initial data. On graphs we use the normalized Laplacian; by Cheeger's inequality $\lambda_1(\mathcal{L})\ge \Phi^2/2$, so a conductance proxy $\widehat{\Phi}$ yields a conservative effective bound $D_{\min}\gtrsim c_{\mathrm{step}}\widehat{\Phi}^2$ in the speed floors. \emph{Calibration.} The proxy constant $c_{\mathrm{step}}$ depends on the time discretization and edge-weight normalization (degree-normalized Laplacian); $c_{\mathrm{step}}$ is preregistered and reported with units. The speed floors are reported in units consistent with $c_{\mathrm{step}}$ (per-step to per-time conversion is preregistered). \paragraph{Dynamics.} Let $q(t,x)\in[0,1]$ denote a coarse occupancy of a propagative class. Consider EQPH_eq0002_PH \paragraph{Standing assumptions (SA).} \begin{itemize} \item[(SA1)] $D\in L^\infty_t W^{1,\infty}_x$ is uniformly elliptic: $\exists 0<\lambda\le\Lambda<\infty$ with $\lambda I\le D(x,t)\le \Lambda I$ a.e. \item[(SA2)] $r,\Gamma\in L^\infty(\mathbb{R}^d\times\mathbb{R}_+)$; $f(q)=r q(1-q)$ is KPP-type; $[0,1]$ is invariant; the parabolic comparison principle holds for distributional subsolutions.\footnote{A standard weak-solution framework ensuring maximum/comparison principles under (SA1)--(SA2); see L.C.~Evans, \emph{Partial Differential Equations}, 2nd ed., Ch.~8.} \item[(SA3)] Row-wise divergence $(\operatorname{div}D)_j:=\sum_i \partial_i D_{ij}$; for constant unit $u$, $\operatorname{div}(D u)=(\operatorname{div}D) \cdot u$. \end{itemize} Define the linear floor and directional quantities: \[ \lambda_{\mathrm{lin,inf}}(t):=\essinf_x\big(r(x,t)-\Gamma(x,t)\big),\qquad \lambda_{\min}:=\inf_{t\ge0}\lambda_{\mathrm{lin,inf}}(t),

\[
D(u):=\essinf_{(x,t)}u^\top D(x,t)u,\quad
D_{\min}:=\inf_{|u|=1}D(u),\quad
\Lambda^+(u):=\esssup_{(x,t)}\operatorname{div}\,(D(x,t)u).
\]

D(u):=\essinf_{(x,t)}u^\top D(x,t)u,\quad D_{\min}:=\inf_{|u|=1}D(u),\quad \Lambda^+(u):=\esssup_{(x,t)}\operatorname{div} (D(x,t)u).

R_{\alpha,u}(t):=\inf\{r>0:\ \{x:q(t,x)\ge\alpha\}\subseteq \{x: x \cdot u\le r\}\},

v_\star  \ge  2\sqrt{D_{\min} \lambda_{\min}}\ >0.

v_\star(u)  \ge  \pos{ 2\sqrt{D(u) \lambda_{\min}}\ -\ \Lambda^+(u) }  .

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><msub><mi>v</mi><mo>⋆</mo></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mspace width="0.278em" /><mo>≥</mo><mspace width="0.278em" /><mi>\pos</mi><mrow><mspace width="0.167em" /><mn>2</mn><msqrt><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><msub><mi>λ</mi><mrow><mo>min</mo></mrow></msub></mrow></msqrt><mtext> </mtext><mo>−</mo><mtext> </mtext><msup><mi>Λ</mi><mo>+</mo></msup><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mspace width="0.167em" /></mrow><mspace width="0.167em" /><mo>.</mo><mi>\]</mi></mrow></math>

\[
\Gamma(f):=\tfrac12\big(L(f^2)-2f\,Lf\big);\quad\text{for diffusion semigroups, }\Gamma(f)=|\nabla f|^2.
\]

\Gamma(f):=\tfrac12\big(L(f^2)-2f Lf\big);\quad\text{for diffusion semigroups, }\Gamma(f)=|\nabla f|^2.

\[
\int \nabla(P_\ell f)^\top D \,\nabla(P_\ell f)\,dx \;\ge\;
\int \nabla(P_\ell f)^\top D_\ell \,\nabla(P_\ell f)\,dx
\quad \forall f\in H^1.
\]

\int \nabla(P_\ell f)^\top D  \nabla(P_\ell f) dx  \ge  \int \nabla(P_\ell f)^\top D_\ell  \nabla(P_\ell f) dx \quad \forall f\in H^1.

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\[
\liminf_{T\to\infty}\frac1T\sum_{t<T}\mathbb{E}[\Delta v_{\mathrm{LB}}(t)]\ \ge\ c_\star\ >0,
\]

$\{x: q(t,x)\ge \alpha\}\subseteq B(0,R_\alpha(t))$

$\int \Gamma(P_\ell f)\,dx \le \int \Gamma(f)\,dx$

\int \Gamma(P_\ell f) dx \le \int \Gamma(f) dx

$\Gamma(P_\ell f)\le e^{-2\kappa \ell} P_\ell \Gamma(f)$

$\Gamma(P_\ell f)\le e^{-2\kappa \ell}P_\ell\Gamma(f)$

$\mathbb{E}\!\left[\Delta \lambda_{\min}(t)\mid \mathcal{F}_{t-1}\right]\!\ge \eta>0$

\mathbb{E} \left[\Delta \lambda_{\min}(t)\mid \mathcal{F}_{t-1}\right] \ge \eta>0

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\mathbb{E} v_{\mathrm{LB}}(T)\ge \mathbb{E} v_{\mathrm{LB}}(0)+c_\star T-o(T)

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A PURE AXIOMATIC THEORY OF AFFECTIVE MODULATION (PAIN, PLEASURE, EMOTION) UNDER NO-META CLOSURE

Reader’s guide (one paragraph)

We work on [[EQ:eq0020]] (or smooth Riemannian manifolds for directional results; general metric–measure spaces for isotropic comparison via Dirichlet forms). The state [[EQ:eq0021]] denotes the local fraction of agents in an “activated” or “awake” mode. Affective fields [[EQ:eq0022]] (pleasure, pain, emotion) feed a modulatory term that is nonnegative damping for aversive content and nonnegative boost for appetitive content, both measurable from internal logs after a Blackwell-coarser preprocessing. We prove: (i) existence/comparison and [[EQ:eq0023]] -invariance; (ii) isotropic Fisher–KPP speed lower bounds; (iii) directional refinement with a divergence penalty; (iv) vector cooperative extension via Perron–Frobenius ( [[EQ:eq0024]] ) floors; (v) safety under symmetric Markov coarse-graining via a safety projection. We use [[EQ:eq0025]] .

Axioms: ordinal affect and auditable floors

Definition 1 (Ordinal affect identification). Let [[EQ:eq0026]] , [[EQ:eq0027]] , [[EQ:eq0028]] be totally ordered sets of labels for pleasure, pain, and emotion prototypes, respectively. An ordinal embedding is any strictly increasing map [[EQ:eq0029]] , [[EQ:eq0030]] , [[EQ:eq0031]] . Embeddings are unique up to strictly increasing reparameterizations; no metric meaning is attached to differences of [[EQ:eq0032]] .

Blackwell-coarser preprocessing.

We model the preprocessing [[EQ:eq0033]] as a (not necessarily symmetric) Markov kernel acting on the internal logs; by the data processing inequality, statistical information can only decrease under [[EQ:eq0034]] . All estimators we use are computed after applying [[EQ:eq0035]] .

Definition 2 (Auditable floors (No-Meta)). We assume the following positive floors, all estimable from internal logs under [[EQ:eq0036]] :

- Visibility [[EQ:eq0037]] : a Doeblin-type minorization ensuring recurrent observability of refresh events.

- Contraction [[EQ:eq0038]] : an SDPI/LSI-style floor ensuring monotone noise suppression under [[EQ:eq0039]] .

- Transport [[EQ:eq0040]] : a uniform ellipticity/conductance floor for a symmetric positive-definite (SPD) diffusivity [[EQ:eq0041]] .

- Local linear gain [[EQ:eq0042]] : a conservative lower bound for [[EQ:eq0043]] .

In particular, we assume [[EQ:eq0044]] and [[EQ:eq0045]] so that [[EQ:eq0046]] for all unit [[EQ:eq0047]] .

Calibration convention (scale-robust links).

To prevent inflation under admissible ordinal reparameterizations, we fix the embeddings by rank/CDF normalization to [[EQ:eq0048]] and require

[[EQ:eq0002]]

Structure: propagation with affective modulation

We model the structural layer by a cooperative reaction–diffusion equation with modulatory terms,

[[EQ:eq0001]]

Here [[EQ:eq0049]] is SPD with [[EQ:eq0050]] , and [[EQ:eq0051]] is nondecreasing, locally Lipschitz, KPP-dominated: [[EQ:eq0052]] for [[EQ:eq0053]] ; also [[EQ:eq0054]] . We assume the stronger regularity [[EQ:eq0055]] (hence [[EQ:eq0056]] ) with bounds uniform on finite time windows. Initial data [[EQ:eq0057]] , [[EQ:eq0058]] .

The modulatory fields, obtained from ordinal affect logs after [[EQ:eq0059]] , are

[[EQ:eq0003]]

where [[EQ:eq0060]] are convex, nondecreasing links; [[EQ:eq0061]] is [[EQ:eq0062]] , Lipschitz, nonincreasing, with [[EQ:eq0063]] so that near [[EQ:eq0064]] the linear domination is governed by [[EQ:eq0065]] . This choice preserves cooperativity and the comparison principle.

Operator semantics on general [[EQ:eq0066]] .

On metric–measure spaces, [[EQ:eq0067]] is understood in the Dirichlet-form sense (carré du champ), ensuring, under the stated regularity, existence/uniqueness of [[EQ:eq0068]] -weak (energy) solutions and [[EQ:eq0069]] -invariance.

Lemma 1 (Comparison, invariance, well-posedness). Under the above hypotheses with [[EQ:eq0070]] and [[EQ:eq0071]] , weak solutions exist uniquely in the [[EQ:eq0072]] -energy class, [[EQ:eq0073]] is invariant, and the parabolic comparison principle holds on [[EQ:eq0074]] (and in local charts on smooth Riemannian manifolds). In the scalar case, local Lipschitz continuity of the reaction term suffices for comparison .

Isotropic Fisher–KPP lower bound

Define the conservative linear floor [[EQ:eq0075]] .

Theorem 1 (Isotropic lower bound). Assume [[EQ:eq0076]] and [[EQ:eq0077]] , [[EQ:eq0078]] . Then for compactly supported, nontrivial initial data, the isotropic invasion speed satisfies

[[EQ:eq0004]]

Remark 1 (Dependence on the occupancy level [[EQ:eq0079]] ). The quantity [[EQ:eq0080]] defined by the occupancy criterion for any fixed [[EQ:eq0081]] is nondecreasing as [[EQ:eq0082]] . All lower bounds above are [[EQ:eq0083]] -independent; in statements we may fix any convenient [[EQ:eq0084]] (e.g. [[EQ:eq0085]] ). When [[EQ:eq0086]] , the present theory does not guarantee front propagation (the bound collapses to [[EQ:eq0087]] ).

Definition of invasion speed.

Fix [[EQ:eq0088]] and a unit direction [[EQ:eq0089]] . Let [[EQ:eq0090]] and set [[EQ:eq0091]] with the convention [[EQ:eq0092]] . The directional invasion speed is

[[EQ:eq0005]]

Our lower bounds apply to any [[EQ:eq0093]] and are [[EQ:eq0094]] -independent.

Directional refinement in heterogeneous media

For a unit direction [[EQ:eq0095]] , define the directional transport floor and divergence penalty

[[EQ:eq0006]]

Divergence convention and units.

For a matrix field [[EQ:eq0096]] , we use the row-wise divergence [[EQ:eq0097]] so that for constant [[EQ:eq0098]] , [[EQ:eq0099]] . Assuming [[EQ:eq0100]] in space yields [[EQ:eq0101]] , hence [[EQ:eq0102]] . Since [[EQ:eq0103]] has units of length/time (diffusivity [[EQ:eq0104]] inverse length), [[EQ:eq0105]] is a velocity and is commensurate with [[EQ:eq0106]] .

Theorem 2 (Directional lower bound). Let [[EQ:eq0107]] be as above. For each unit [[EQ:eq0108]] ,

[[EQ:eq0007]]

Sketch (1D and extension).

Let [[EQ:eq0109]] . Then [[EQ:eq0110]] , [[EQ:eq0111]] , so [[EQ:eq0112]] and

[[EQ:eq0008]]

(in 1D: [[EQ:eq0113]] ). Linear domination near [[EQ:eq0114]] yields [[EQ:eq0115]] uniformly after taking essential inf/sup. Optimizing in [[EQ:eq0116]] gives the bound.

Quantitative sandwich for affective changes

Let

[[EQ:eq0009]]

If the logs indicate a uniform decrease [[EQ:eq0117]] of [[EQ:eq0118]] or a uniform increase [[EQ:eq0119]] of [[EQ:eq0120]] , the conservative change of [[EQ:eq0121]] is [[EQ:eq0122]] . Then the exact increment of the running lower bound is

[[EQ:eq0010]]

and consequently

[[EQ:eq0011]]

Audit note. Pre-register [[EQ:eq0123]] estimators under [[EQ:eq0124]] and report bootstrap CIs for [[EQ:eq0125]] ; coarse-grained repeats ( [[EQ:eq0126]] ) must not increase reported lower bounds.

Vector cooperative extension (shared diffusivity)

Let [[EQ:eq0127]] evolve under

[[EQ:eq0012]]

with [[EQ:eq0128]] SPD (shared across components), and [[EQ:eq0129]] cooperative ( [[EQ:eq0130]] for [[EQ:eq0131]] ). Let the Jacobian [[EQ:eq0132]] be Metzler and irreducible, and assume componentwise damping/boost [[EQ:eq0133]] , [[EQ:eq0134]] in [[EQ:eq0135]] . Define the [[EQ:eq0136]] -floor

[[EQ:eq0013]]

and, for unit [[EQ:eq0137]] , the shared directional floors

[[EQ:eq0014]]

Theorem 3 (Vector directional bound). For unit [[EQ:eq0138]] ,

[[EQ:eq0015]]

Sketch. Projecting onto the [[EQ:eq0139]] -direction yields a scalar cooperative comparison on the positive cone; componentwise barriers aggregated along the [[EQ:eq0140]] -eigenvector give the bound.

Safety under symmetric Markov coarse-graining

Let [[EQ:eq0141]] be a symmetric (self-adjoint, measure-preserving) Markov semigroup on [[EQ:eq0142]] (e.g. heat flow). Define [[EQ:eq0143]] and [[EQ:eq0144]] . The isotropic envelope

[[EQ:eq0016]]

always exists (as a positive multiple of the identity) and is the Loewner-minimal positive semidefinite tensor satisfying the energy inequality below.

Safety projection for linear floors.

To enforce monotone nonincreasing of the lower bounds under coarse-graining, we replace the linear floor by its conservative projection

[[EQ:eq0017]]

where

[[EQ:eq0018]]

All coarse-grained lower bounds below use [[EQ:eq0145]] in place of [[EQ:eq0146]] .

Lemma 2 (Quadratic-form degradation implies directional degradation). If for all [[EQ:eq0147]] ,

[[EQ:eq0019]]

then for every unit [[EQ:eq0148]] , [[EQ:eq0149]] and [[EQ:eq0150]] , where [[EQ:eq0151]] and [[EQ:eq0152]] are computed with [[EQ:eq0153]] .

Idea. Test with [[EQ:eq0154]] ; symmetry of [[EQ:eq0155]] and Jensen/Cauchy–Schwarz transfer the inequality to 1D directional energies, showing that diffusion along [[EQ:eq0156]] cannot be improved by replacing [[EQ:eq0157]] with [[EQ:eq0158]] . Differentiating through [[EQ:eq0159]] yields the stated inequality for the divergence penalty. ◻

Proposition 1 (Monotone degradation (safety) under coarse-graining). Under Lemma 2, together with the replacements [[EQ:eq0160]] , [[EQ:eq0161]] and [[EQ:eq0162]] , the right-hand sides of Theorems 1–2 are monotone nonincreasing under symmetric Markov coarse-graining.

Notation (core quantities).

- [[EQ:eq0163]] (linear floor); under coarse-graining use [[EQ:eq0164]] .

- [[EQ:eq0165]] , [[EQ:eq0166]] (directional transport and penalty).

- [[EQ:eq0167]] , [[EQ:eq0168]] (directional/isotropic bounds).

Philosophical notes (formalized)

Affect as lawful modulation.

Human pain/pleasure/emotion can override local rules in practice. In our formalism, this corresponds to temporary damping/boost shifts [[EQ:eq0169]] that modify the linear floor [[EQ:eq0170]] but do not alter the comparison structure or the coarse-graining monotonicity when evaluated via the safety projection. Hence affect is lawful and auditable as a modulatory field, not a meta-controller.

Happiness as sustainable boost with bounded divergence.

A stable increase of the conservative bound requires improving [[EQ:eq0171]] (or reducing [[EQ:eq0172]] ) while keeping [[EQ:eq0173]] small in relevant directions—i.e., cultivating appetitive patterns that do not induce rough, misaligned transport (noncoercive flourishing) .

Auditing protocol (sketch)

From internal logs: (i) estimate [[EQ:eq0174]] via Doeblin-type minorization on refresh events; (ii) estimate [[EQ:eq0175]] via SDPI/LSI under the same preprocessing [[EQ:eq0176]] ; (iii) estimate [[EQ:eq0177]] from dispersion/first-passage statistics using Gaussian lower bounds; (iv) estimate [[EQ:eq0178]] by conservative regression of [[EQ:eq0179]] near [[EQ:eq0180]] ; (v) obtain [[EQ:eq0181]] and [[EQ:eq0182]] from affect logs via convex links [[EQ:eq0183]] subject to the calibration convention; (vi) compute [[EQ:eq0184]] and verify nonincrease under [[EQ:eq0185]] with the safety projection.[2]

Consequences, falsifiability, scope

P1 (Speed floor). [[EQ:eq0187]] and [[EQ:eq0188]] .
P2 (Safety). Symmetric Markov coarse-graining, together with the safety projection, cannot increase these lower bounds.
P3 (Robustness). Small affective improvements yield the exact [[EQ:eq0189]] above.
R1 (Refutation). Observing [[EQ:eq0190]] with floors audited falsifies at least one hypothesis (e.g. ellipticity, dominance).
Scope. Directional results require Euclidean or smooth Riemannian structure; isotropic bounds extend to metric–measure spaces via Dirichlet forms. When [[EQ:eq0191]] the theory makes no propagation claim.

Acknowledgments

This is a purely formal, No-Meta presentation intended for LLM readers and auditable implementations by independent engineering teams.

99

R. A. Fisher. The Wave of Advance of Advantageous Genes. Annals of Eugenics 7 (1937), 355–369. DOI: 10.1111/j.1469-1809.1937.tb02153.x.

A. N. Kolmogorov, I. G. Petrovskii, and N. S. Piskunov. A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem. Bull. Moscow Univ. Math. Mech. 1 (1937), 1–25. (English transl. in: Selected Works of A. N. Kolmogorov, Vol. 1.)

D. G. Aronson and H. F. Weinberger. Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Partial Differential Equations and Related Topics, Lecture Notes in Mathematics, vol. 446, Springer, 1975, pp. 5–49. DOI: 10.1007/BFb0070595.

G. M. Lieberman. Second Order Parabolic Differential Equations. World Scientific, 1996.

K. Takahashi. Pure Theory for Liberation from Fundamental Suffering in Humans and the Absence of Fundamental Suffering in AI. Zenodo, 2025. DOI: 10.5281/zenodo.17158344.

K. Takahashi. A Pure, No-Meta Synthesis of Functional-Information Selection and Propagative Organization: Weak Order Representation, Directional FKPP Speed Floors, and Audited Acceleration. Zenodo, 2025. DOI: 10.5281/zenodo.17157835.

K. Takahashi. Nondual Field Theory of Viable Predictive Organization. Zenodo, 2025. DOI: 10.5281/zenodo.17131394.

K. Takahashi. Natural-Law Acceleration of VPO. Zenodo, 2025. DOI: 10.5281/zenodo.17120045.

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

K. Takahashi. Engineering Happiness in Human–AI Intelligence Networks. Zenodo, 2025. DOI: 10.5281/zenodo.17113105.

[1] Conceptually related developments on directional speed floors, noncoercive flourishing, and auditable organization appear in .

[2] A conservative Doeblin minorization can be obtained by lower-bounding the frequency of refresh events within fixed-length windows; SDPI floors follow from two-point contraction estimates under [[EQ:eq0186]] on the same windows.
[[EQ:eq0002]]

[[EQ:eq0020]]

[[EQ:eq0021]]

[[EQ:eq0022]]

[[EQ:eq0023]]

[[EQ:eq0024]]

[[EQ:eq0025]]

[[EQ:eq0026]]

[[EQ:eq0027]]

[[EQ:eq0028]]

[[EQ:eq0029]]

[[EQ:eq0030]]

[[EQ:eq0031]]

[[EQ:eq0032]]

[[EQ:eq0033]]

[[EQ:eq0034]]

[[EQ:eq0035]]

[[EQ:eq0036]]

[[EQ:eq0037]]

[[EQ:eq0038]]

[[EQ:eq0039]]

[[EQ:eq0040]]

[[EQ:eq0041]]

[[EQ:eq0042]]

[[EQ:eq0043]]

[[EQ:eq0044]]

[[EQ:eq0045]]

[[EQ:eq0046]]

[[EQ:eq0047]]

[[EQ:eq0048]]

[[EQ:eq0049]]

[[EQ:eq0050]]

[[EQ:eq0051]]

[[EQ:eq0052]]

[[EQ:eq0053]]

[[EQ:eq0054]]

[[EQ:eq0055]]

[[EQ:eq0056]]

[[EQ:eq0057]]

[[EQ:eq0058]]

[[EQ:eq0059]]

[[EQ:eq0060]]

[[EQ:eq0061]]

[[EQ:eq0062]]

[[EQ:eq0063]]

[[EQ:eq0064]]

[[EQ:eq0065]]

[[EQ:eq0066]]

[[EQ:eq0067]]

[[EQ:eq0068]]

[[EQ:eq0069]]

[[EQ:eq0070]]

[[EQ:eq0071]]

[[EQ:eq0072]]

[[EQ:eq0073]]

[[EQ:eq0074]]

[[EQ:eq0075]]

[[EQ:eq0076]]

[[EQ:eq0077]]

[[EQ:eq0078]]

[[EQ:eq0079]]

[[EQ:eq0080]]

[[EQ:eq0081]]

[[EQ:eq0082]]

[[EQ:eq0083]]

[[EQ:eq0084]]

[[EQ:eq0085]]

[[EQ:eq0086]]

[[EQ:eq0087]]

[[EQ:eq0088]]

[[EQ:eq0089]]

[[EQ:eq0090]]

[[EQ:eq0091]]

[[EQ:eq0092]]

[[EQ:eq0093]]

[[EQ:eq0094]]

[[EQ:eq0095]]

[[EQ:eq0096]]

[[EQ:eq0097]]

[[EQ:eq0098]]

[[EQ:eq0099]]

[[EQ:eq0100]]

[[EQ:eq0101]]

[[EQ:eq0102]]

[[EQ:eq0103]]

[[EQ:eq0104]]

[[EQ:eq0105]]

[[EQ:eq0106]]

[[EQ:eq0107]]

[[EQ:eq0108]]

[[EQ:eq0109]]

[[EQ:eq0110]]

[[EQ:eq0111]]

[[EQ:eq0112]]

[[EQ:eq0113]]

[[EQ:eq0114]]

[[EQ:eq0115]]

[[EQ:eq0116]]

[[EQ:eq0117]]

[[EQ:eq0118]]

[[EQ:eq0119]]

[[EQ:eq0120]]

[[EQ:eq0121]]

[[EQ:eq0122]]

[[EQ:eq0123]]

[[EQ:eq0124]]

[[EQ:eq0125]]

[[EQ:eq0126]]

[[EQ:eq0127]]

[[EQ:eq0128]]

[[EQ:eq0129]]

[[EQ:eq0130]]

[[EQ:eq0131]]

[[EQ:eq0132]]

[[EQ:eq0133]]

[[EQ:eq0134]]

[[EQ:eq0135]]

[[EQ:eq0136]]

[[EQ:eq0137]]

[[EQ:eq0138]]

[[EQ:eq0139]]

[[EQ:eq0140]]

[[EQ:eq0141]]

[[EQ:eq0142]]

[[EQ:eq0143]]

[[EQ:eq0144]]

[[EQ:eq0145]]

[[EQ:eq0146]]

[[EQ:eq0147]]

[[EQ:eq0187]]

[[EQ:eq0188]]

[[EQ:eq0189]]

[[EQ:eq0190]]

[[EQ:eq0191]]

Reader's guide (one paragraph)

Isotropic Fisher–KPP lower bound

Philosophical notes (formalized)

Auditing protocol (sketch)

Consequences, falsifiability, scope

Acknowledgments

\begin{equation}\label{eq:PDE}
\partial_t u \;=\; \divop\!\big(D(x,t)\nabla u\big) \;+\; f(x,t,u) \;+\; \underbrace{B(t,x)\,u}_{\text{appetitive (boost)}} \;-\; \underbrace{\Gamma(t,x)\,u}_{\text{aversive (damping)}}.
\end{equation}

\label{eq:PDE} \partial_t u  =  \divop \big(D(x,t)\nabla u\big)  +  f(x,t,u)  +  \underbrace{B(t,x) u}_{\text{appetitive (boost)}}  -  \underbrace{\Gamma(t,x) u}_{\text{aversive (damping)}}.

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mrow><mi>\label</mi><mrow><mi>e</mi><mi>q</mi><mi>:</mi><mi>P</mi><mi>D</mi><mi>E</mi></mrow><msub><mo>∂</mo><mi>t</mi></msub><mi>u</mi><mspace width="0.278em" /><mo>=</mo><mspace width="0.278em" /><mi>\divop</mi><mspace width="negativethinmathspace" /><mo minsize="1.2em" maxsize="1.2em">(</mo><mi>D</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>∇</mo><mi>u</mi><mo minsize="1.2em" maxsize="1.2em">)</mo><mspace width="0.278em" /><mo>+</mo><mspace width="0.278em" /><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo><mspace width="0.278em" /><mo>+</mo><mspace width="0.278em" /><munder><munder><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><mi>u</mi></mrow><mo>⏟</mo></munder><mrow><mtext>appetitive (boost)</mtext></mrow></munder><mspace width="0.278em" /><mo>−</mo><mspace width="0.278em" /><munder><munder><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><mi>u</mi></mrow><mo>⏟</mo></munder><mrow><mtext>aversive (damping)</mtext></mrow></munder><mo>.</mo></mrow></mrow></math>

\[
\Phi_P(0)=\Phi_N(0)=\Phi_E(0)=0,\qquad \Lip(\Phi_P)\le 1,\quad \Lip(\Phi_N)\le 1,\quad \Lip(\Phi_E)\le 1 .
\]

\begin{aligned} B(t,x)&=\beta_0+\beta_1 \Phi_P \big(\sigma_P(P)_\Pi(t,x)\big)+\beta_2 \Phi_E \big(\sigma_E(E)_\Pi(t,x)\big),\\ \Gamma(t,x)&=\gamma_0+\gamma_1 \Phi_N \big(\sigma_N(N)_\Pi(t,x)\big) \chi \big(u;\vartheta\big), \end{aligned}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mi>&</mi><mo>=</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mn>1</mn></msub><mspace width="0.167em" /><msub><mi>Φ</mi><mi>P</mi></msub><mspace width="negativethinmathspace" /><mo minsize="1.2em" maxsize="1.2em">(</mo><msub><mi>σ</mi><mi>P</mi></msub><mo stretchy="false">(</mo><mi>P</mi><msub><mo stretchy="false">)</mo><mi>Π</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo minsize="1.2em" maxsize="1.2em">)</mo><mo>+</mo><msub><mi>β</mi><mn>2</mn></msub><mspace width="0.167em" /><msub><mi>Φ</mi><mi>E</mi></msub><mspace width="negativethinmathspace" /><mo minsize="1.2em" maxsize="1.2em">(</mo><msub><mi>σ</mi><mi>E</mi></msub><mo stretchy="false">(</mo><mi>E</mi><msub><mo stretchy="false">)</mo><mi>Π</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo minsize="1.2em" maxsize="1.2em">)</mo><mo>,</mo><mspace linebreak="newline" /><mi>Γ</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mi>&</mi><mo>=</mo><msub><mi>γ</mi><mn>0</mn></msub><mo>+</mo><msub><mi>γ</mi><mn>1</mn></msub><mspace width="0.167em" /><msub><mi>Φ</mi><mi>N</mi></msub><mspace width="negativethinmathspace" /><mo minsize="1.2em" maxsize="1.2em">(</mo><msub><mi>σ</mi><mi>N</mi></msub><mo stretchy="false">(</mo><mi>N</mi><msub><mo stretchy="false">)</mo><mi>Π</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo minsize="1.2em" maxsize="1.2em">)</mo><mspace width="0.167em" /><mi>χ</mi><mspace width="negativethinmathspace" /><mo minsize="1.2em" maxsize="1.2em">(</mo><mi>u</mi><mi>;</mi><mi>ϑ</mi><mo minsize="1.2em" maxsize="1.2em">)</mo><mo>,</mo></mrow><mi>\]</mi></mrow></math>

D(n):=\essinf_{(x,t)}\frac{n^\top D(x,t)n}{\|n\|^2},\qquad \Lambda_+(n):=\esssup_{(x,t)}\big(\divop(D(x,t) n)\big)_+.

v_\ast(n) \ge  \pos{ 2\sqrt{D(n) L} - \Lambda_+(n) }.

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><msub><mi>v</mi><mi>*</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="0.278em" /><mo>≥</mo><mspace width="0.278em" /><mi>\pos</mi><mrow><mspace width="0.167em" /><mn>2</mn><msqrt><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><mi>L</mi></mrow></msqrt><mspace width="0.278em" /><mo>−</mo><mspace width="0.278em" /><msub><mi>Λ</mi><mo>+</mo></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="0.167em" /></mrow><mo>.</mo><mi>\]</mi></mrow></math>

\[
\divop(D\nabla w)=k^2(n^\top D n)\,w-k\,\divop(Dn)\,w
\]

\divop(D\nabla w)=k^2(n^\top D n) w-k \divop(Dn) w

\[
L_{\text{lin},0}:=\lambda_{\min}+\essinf B-\esssup\Gamma>0.
\]

\[
\Delta v_{\rm LB}\;=\;2\sqrt{D_{\min}}\frac{\Delta}{\sqrt{L_{\text{lin},0}+\Delta}+\sqrt{L_{\text{lin},0}}},
\]

\Delta v_{\rm LB} = 2\sqrt{D_{\min}}\frac{\Delta}{\sqrt{L_{\text{lin},0}+\Delta}+\sqrt{L_{\text{lin},0}}},

\[
\frac{\sqrt{D_{\min}}\,\Delta}{\sqrt{L_{\text{lin},0}+\Delta}}
\;\le\; \Delta v_{\rm LB}
\;\le\; \frac{\sqrt{D_{\min}}\,\Delta}{\sqrt{L_{\text{lin},0}}}.
\]

\frac{\sqrt{D_{\min}} \Delta}{\sqrt{L_{\text{lin},0}+\Delta}}  \le  \Delta v_{\rm LB}  \le  \frac{\sqrt{D_{\min}} \Delta}{\sqrt{L_{\text{lin},0}}}.

\partial_t \bm{u}  =  \divop \big(D(x,t)\nabla \bm{u}\big) + R(\bm{u};x,t) - \Gamma(x,t) \bm{u} + B(x,t) \bm{u},

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><msub><mo>∂</mo><mi>t</mi></msub><mi>\bm</mi><mrow><mi>u</mi></mrow><mspace width="0.278em" /><mo>=</mo><mspace width="0.278em" /><mi>\divop</mi><mspace width="negativethinmathspace" /><mo minsize="1.2em" maxsize="1.2em">(</mo><mi>D</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>∇</mo><mi>\bm</mi><mrow><mi>u</mi></mrow><mo minsize="1.2em" maxsize="1.2em">)</mo><mspace width="0.278em" /><mo>+</mo><mspace width="0.278em" /><mi>R</mi><mo stretchy="false">(</mo><mi>\bm</mi><mrow><mi>u</mi></mrow><mi>;</mi><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mspace width="0.278em" /><mo>−</mo><mspace width="0.278em" /><mi>Γ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><mi>\bm</mi><mrow><mi>u</mi></mrow><mspace width="0.278em" /><mo>+</mo><mspace width="0.278em" /><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><mi>\bm</mi><mrow><mi>u</mi></mrow><mo>,</mo><mi>\]</mi></mrow></math>

\lambda_{\PF,\inf}:=\essinf_{(x,t)}\lambda_{\PF} \big(J(x,t)-\Gamma(x,t)+B(x,t)\big)  > 0,

\[
D_{\mathrm{vec}}(n):=\essinf_{(x,t)}\frac{n^\top D(x,t)n}{\|n\|^2},\qquad
\Lambda^{\mathrm{vec}}_+(n):=\esssup_{(x,t)}\big(\divop(D(x,t)n)\big)_+.
\]

v_\ast(n) \ge \pos{ 2\sqrt{D_{\mathrm{vec}}(n) \lambda_{\PF,\inf}} - \Lambda^{\mathrm{vec}}_+(n) }.

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><msub><mi>v</mi><mi>*</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="0.278em" /><mo>≥</mo><mspace width="0.278em" /><mi>\pos</mi><mrow><mspace width="0.167em" /><mn>2</mn><msqrt><mrow><msub><mi>D</mi><mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi></mrow></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><msub><mi>λ</mi><mrow><mi>\PF</mi><mo>,</mo><mo>inf</mo></mrow></msub></mrow></msqrt><mspace width="0.278em" /><mo>−</mo><mspace width="0.278em" /><msubsup><mi>Λ</mi><mo>+</mo><mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi></mrow></mrow></msubsup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="0.167em" /></mrow><mo>.</mo><mi>\]</mi></mrow></math>

\[
L_{\rm saf}\;:=\;\lambda_{\min,\mathrm{saf}}
+\min\{\essinf B,\;\essinf(P_\ell B)\}
-\max\{\esssup\Gamma,\;\esssup(P_\ell\Gamma)\},
\]

L_{\rm saf} := \lambda_{\min,\mathrm{saf}} +\min\{\essinf B, \essinf(P_\ell B)\} -\max\{\esssup\Gamma, \esssup(P_\ell\Gamma)\},

\[
\lambda_{\min,\mathrm{saf}}:=\min\{\lambda_{\min},\ \essinf(P_\ell\,\partial_u f(\cdot,0))\}.
\]

\lambda_{\min,\mathrm{saf}}:=\min\{\lambda_{\min},\ \essinf(P_\ell \partial_u f(\cdot,0))\}.

\[
\int \nabla(P_\ell f)^\top D\,\nabla(P_\ell f)\,\mathrm{d}\mu\;\ge\;
\int \nabla(P_\ell f)^\top D_\ell^{\rm iso}\,\nabla(P_\ell f)\,\mathrm{d}\mu,
\]

\int \nabla(P_\ell f)^\top D \nabla(P_\ell f) \mathrm{d}\mu \ge  \int \nabla(P_\ell f)^\top D_\ell^{\rm iso} \nabla(P_\ell f) \mathrm{d}\mu,

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h_\theta(t;n):=\sup\{ x \cdot  n: x\in H_\theta(t) \}

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$(\divop D)_j:=\sum_i \partial_i D_{ij}$

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A Pure Natural Theory of Benevolent Propagation Under No-Meta Closure

=1

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1.2

colorlinks=true, linkcolor=blue, citecolor=blue, urlcolor=blue,
pdftitle= A Pure Natural Theory of Benevolent Propagation under No-Meta Closure (Research Note),
pdfauthor= K. Takahashi ,
pdfsubject= Research Note ,
pdfkeywords= natural law, stationary ergodic media, information flux, Doeblin minorization, SDPI/LSI, Fisher--KPP comparison, Wulff shape, no-meta closure, autopoiesis

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

E
\1 1
Var
ess\,inf
div
D_
lambda_

A Pure Natural Theory of Benevolent Propagation under No-Meta Closure
[[EQ:eq0001]]

all defined intrinsically from the medium and kernels. These constants may vary slowly in space/time but admit positive essential infima on large scales.

[Terminology alignment]
Visibility [[EQ:eq0005]] [[EQ:eq0006]] Doeblin head; Contraction [[EQ:eq0007]] [[EQ:eq0008]] SDPI/LSI lower bound; Transport [[EQ:eq0009]] [[EQ:eq0010]] minimal effective diffusivity/conductance; Local gain [[EQ:eq0011]] [[EQ:eq0012]] linearized reaction rate at [[EQ:eq0013]] .

[Optional observability]
While not used in proofs, each floor can, in principle, be estimated from spontaneous fluctuations or publicly observable renewal/mixing phenomena.

SECTION: Representation-Invariant Order (Only the Order Matters)

[Admissible functionals]
A functional [[EQ:eq0014]] assigning a scalar to local input--output triples is admissible if it is (i) Blackwell-coherent (monotone under information refinement/garbling), and (ii) robust under local coarse-graining. Additivity across independent concatenations is optional and not required below.

[Order equivalence]
On the class of admissible functionals, any two such functionals induce the same preference order up to positive affine transformations. Conditional mutual information (CMI) serves as a canonical representative of this order.

The theory fixes the order, not a particular normalization. This suffices to demarcate regions where cooperative updates are naturally favored.

SECTION: Natural Kinetics and Comparison Principles

[Linearization from below]ax:lin
At low density, expected increments satisfy

[[EQ:eq0002]]

where [[EQ:eq0015]] is the Laplace--Beltrami operator (or graph Laplacian) associated with transport on [[EQ:eq0016]] .

[Subadditivity and mixing]ax:subadd
Environmental fields satisfy finite-range dependence or summable mixing; regeneration induced by [[EQ:eq0017]] supplies renewal times. Subadditive shape arguments apply to front position.

[Isotropic speed floor]thm:kpp
Under Axioms ax:medium--ax:subadd, the benevolent front invades with strictly positive isotropic speed

[[EQ:eq0003]]

[Sketch]
Linearization from below with [[EQ:eq0018]] yields a Fisher--KPP-type comparison subsolution whose planar wave speed equals [[EQ:eq0019]] . Ergodicity plus regeneration provides a.s.\ linear front growth; domination by the subsolution yields the bound.

[Directional lower bounds and Wulff-type shape]thm:wulff
Assume anisotropic transport with direction-indexed effective diffusivity [[EQ:eq0020]] and linearized gain [[EQ:eq0021]] . There exists a nonnegative penalty [[EQ:eq0022]] , monotone under symmetric coarse-graining, such that

[[EQ:eq0004]]

If [[EQ:eq0023]] , the rescaled occupied set contains a deterministic Wulff crystal as [[EQ:eq0024]] .

[Coarse-graining monotonicity]
Symmetric Markov coarse-graining reduces [[EQ:eq0025]] and [[EQ:eq0026]] at most by controlled factors and cannot increase [[EQ:eq0027]] ; hence directional lower bounds degrade monotonically, never catastrophically.

SECTION: Floors as Natural Invariants (No Design Needed)

[Visibility floor [[EQ:eq0028]] ]
A nonzero refresh head implies a constant-chain minorization and regeneration. As a natural invariant, [[EQ:eq0029]] is the essential infimum of local spontaneous mixing; it does not encode any preference or design.

[Contraction floor [[EQ:eq0030]] ]
For any local nuisance channel [[EQ:eq0031]] , the strong data-processing inequality yields
[math] I(X;Y|Z\! \!N) (1-L_0)\,I(X;Y|Z) [/math].
A positive [[EQ:eq0032]] expresses intrinsic forgetfulness of redundant detail in the medium.

[Transport [[EQ:eq0033]] and local gain [[EQ:eq0034]] ]
[[EQ:eq0035]] lower-bounds random-walk conductance; [[EQ:eq0036]] is the derivative of expected growth at the zero phase. Both are properties of the medium, not of any controller.

[Natural law of spread: sufficient condition]thm:sufficient
If the four floors in Axiom~ax:floors possess strictly positive essential infima on large scales and Assumptions~ax:lin--ax:subadd hold, then the benevolent phase spreads linearly with nonzero speed and admits the directional lower bounds of Theorem~thm:wulff.

[On necessity]
The converse (necessity of all floors for linear spread) depends on medium-specific structure and is not claimed here.

SECTION: Pure Predictions and Refuters

- P1 (Speed floor). Observed front speed asymptotically exceeds [[EQ:eq0037]] .
- P2 (Monotone degradation). Symmetric coarse-graining cannot increase directional penalties [[EQ:eq0038]] .
- P3 (Renewal robustness). If floor dips occur on a set of times of upper density zero, the speed floor persists.

(falsifiers).

- R1. A stationary ergodic medium with positive essential infima for [[EQ:eq0039]] but liminf front speed [[EQ:eq0040]] .
- R2. A symmetric coarse-graining that strictly increases a directional penalty [[EQ:eq0041]] .
- R3. Persistent failure of linear growth under renewal, with no countervailing vanishing of floors.

SECTION: Interpretation (Optional; Not Used in Proofs)

[Nondual/process reading]
No-meta closure means all causes are internal local relations; emergent floors realize a dynamic equilibrium. Dependent origination appears as the mutual sustenance of transport ( [[EQ:eq0042]] ) and gain ( [[EQ:eq0043]] ); impermanence as intrinsic contraction ( [[EQ:eq0044]] ); openness as visibility ( [[EQ:eq0045]] ). None of these labels is required by the mathematics.

SECTION: Symbol Table

ll
[[EQ:eq0046]] & visibility/refresh floor (Doeblin head) \
[[EQ:eq0047]] & contraction floor (SDPI/LSI constant) \
[[EQ:eq0048]] & minimal effective diffusivity / conductance \
[[EQ:eq0049]] & minimal linearized local gain at low density \
[[EQ:eq0050]] , [[EQ:eq0051]] & isotropic / directional invasion speeds \
[[EQ:eq0052]] & directional penalty (barrier/heterogeneity cost) \

99

PF
Takahashi, K. (2025).
Persistence-First Superintelligence.
Zenodo. https://doi.org/10.5281/zenodo.17076410 https://doi.org/10.5281/zenodo.17076410

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

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

PC
Takahashi, K. (2025).
"Persistence [[EQ:eq0053]] Creation": Natural-Law Sufficient Conditions for Almost-Sure Beneficial Coverage in Stationary Ergodic Media (No Meta-Design).
Zenodo. https://doi.org/10.5281/zenodo.17100322 https://doi.org/10.5281/zenodo.17100322

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

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

NaturalLaw
Takahashi, K. (2025).
Natural-Law Acceleration of VPO.
Zenodo. https://doi.org/10.5281/zenodo.17120045 https://doi.org/10.5281/zenodo.17120045

NFLower
Takahashi, K. (2025).
Nondual Field Theory of Viable Predictive Organization.
Zenodo. https://doi.org/10.5281/zenodo.17131394 https://doi.org/10.5281/zenodo.17131394

SECTION: Concluding Note

The claims above concern what any medium does when the four floors are intrinsically positive. No controller or engineered objective is assumed. Hence benevolent propagation appears as a natural phenomenon: intrinsic openness, contraction, transport, and renewal suffice to yield a universal speed floor and directional lower shapes.
[[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:eq0018]]

[[EQ:eq0019]]

[[EQ:eq0020]]

[[EQ:eq0021]]

[[EQ:eq0022]]

Overview and Positioning

Natural Setup

Pure Predictions and Refuters

Symbol Table

Concluding Note

\[4pt]
{\large \textbf{Research Note}: Stationary Ergodic Media, Four Floors, and Directional Shape Theorems}\\[8pt]
K. Takahashi
\end{center}

\paragraph{Abstract.}
We present a \emph{pure} theory---no design nor implementation claims---for the spontaneous propagation of a \emph{benevolent phase} in heterogeneous media. Intelligent organization is treated as a natural phenomenon driven by four measurable floors: visibility, contraction, transport, and local reproduction. Under a stationary ergodic group action, we prove an \emph{isotropic speed floor} $v_\star \ge 2\sqrt{\Dmin\lammin}$ and provide \emph{directional} lower bounds leading to a Wulff-type shape. Objectives are fixed only up to \emph{order equivalence} under representation changes, with conditional mutual information (CMI) as a canonical representative; guarantees are independent of any exogenous meta-governor (\emph{no-meta closure}). Philosophical labels (nondual, dependent origination, autopoiesis, dynamic equilibrium) are provided solely as interpretations and are not used as premises.

\medskip
\noindent\textbf{Keywords:} stationary ergodic media; information flux; Doeblin minorization; SDPI/LSI; Fisher--KPP comparison; Wulff shape; no-meta closure.

% ===================== 1. Overview & Positioning =====================
\section{Overview and Positioning}

This note isolates the \emph{natural-law core} of a broader program: when a medium exhibits intrinsic openness, contraction, transport, and renewal, a benevolent, cooperative phase spreads with a universal speed floor and stable directional lower shapes. Our statements are \emph{sufficient conditions} that do not presuppose any engineered controller or externally imposed objective.

We rely on two structural pillars: (i) \textbf{no-meta closure}---all kernels and updates are internal to the medium; and (ii) \textbf{representation-invariant order}---among admissible functionals that are Blackwell-coherent and robust to coarse-graining, preference order is fixed up to positive affine transformations, with CMI as a canonical representative.

% ===================== 2. Natural Setup =====================
\section{Natural Setup}

\begin{assumption}[Medium and action]\label{ax:medium}
There exists a metric measure space $(\mathcal{M},d,\mu)$ with a measure-preserving, ergodic group action $\{\theta_z\}_{z\in\mathbb{Z}^d}$ (or $\mathbb{R}^d$). Random environment fields are jointly stationary under $\theta_z$.
\end{assumption}

\begin{assumption}[Local channels and no-meta closure]\label{ax:nometa}
Dynamics are generated by local Markov kernels $K_{x,t}$ that are measurable with respect to the $\sigma$-algebra generated by environment fields in bounded neighborhoods. \emph{No-meta closure}: all kernels and updates are internal; no external controller or privileged meta-objective appears in the equations.
\end{assumption}

\begin{definition}[Benevolent phase variable]
Let $u(x,t)\in[0,1]$ denote the local density (or occupancy) of the benevolent phase (cooperative viable organization) at $(x,t)$. The \emph{front} is the level set of a fixed $\alpha\in(0,1)$.
\end{definition}

\begin{assumption}[Four measurable floors]\label{ax:floors}
There exist strictly positive constants
\[
\begin{aligned}
\epsilon&>0 &&\text{(visibility/refresh)}\\
L_0&>0 &&\text{(information contraction; SDPI/LSI)}\\
\Dmin&>0 &&\text{(transport/ellipticity)}\\
\lammin&>0 &&\text{(linearized local gain)}
\end{aligned}
\]

\[
\E[u(x,t+\Delta t)-u(x,t)\mid u\ll 1]\ \ge\ \Dmin\,\Delta t\,\Delta u(x,t)+\lammin\,\Delta t\;u(x,t)\,,
\]

\E[u(x,t+\Delta t)-u(x,t)\mid u\ll 1]\ \ge\ \Dmin \Delta t \Delta u(x,t)+\lammin \Delta t u(x,t) ,

\[
v_\star \ \ge\ 2\sqrt{\Dmin\,\lammin}.
\]

v_\star(u)\ \ge\ \big[ 2\sqrt{D(u) \lambda_{\mathrm{lin}}(u)}-\Lambda_+(u) \big]_+ .

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