NONDUAL AUTOPOIETIC EXCITATIONS

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pdftitle = Nondual Autopoietic Excitations: Unified Variational--Dissipative Geometry on HK Law Space with RI/BV Events (Existence, Global EDI, and Conditional Metastability),
pdfauthor = K. Takahashi ,
pdfsubject = Gradient flows; Hellinger--Kantorovich geometry; rate-independent systems; energy--dissipation inequality ,
pdfkeywords= Autopoiesis, Hellinger--Kantorovich, unbalanced optimal transport, JKO, rate-independent, balanced viscosity, EDI, Bochner operator integrals, Modica--Mortola, Eyring--Kramers, nonduality ,
pdfcreator = LaTeX with hyperref ,
pdfproducer= pdflatex

theorem Theorem Theorems
theorem Theorem Theorems
assumption Assumption Assumptions
assumption Assumption Assumptions
proposition Proposition Propositions
proposition Proposition Propositions
lemma Lemma Lemmas
lemma Lemma Lemmas
remark Remark Remarks
remark Remark Remarks
definition Definition Definitions
definition Definition Definitions
equation Eq. Eqs.
equation Eq. Eqs.
section Section Sections
section Section Sections
appendix Appendix Appendices
appendix Appendix Appendices

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

ET
HK
R
P
D

Ent
Id

#1 _ H
#1 _ P_ H
#1,\,P^ -1 _ H \,#2

TITLE: -1em

Nondual Autopoietic Excitations:\
A Unified Variational--Dissipative Geometry on HK Law Space\
with RI/BV Events
[[EQ:eq0016]]

Regularities for weak solutions (standard): [[EQ:eq0031]] , [[EQ:eq0032]] with [[EQ:eq0033]] , [[EQ:eq0034]] , [[EQ:eq0035]] , [[EQ:eq0036]] , [[EQ:eq0037]] . BCs: Neumann for [[EQ:eq0038]] ; no-flux for the CH chemical potential; admissible [[EQ:eq0039]] BCs for [[EQ:eq0040]] ; natural no-flux for [[EQ:eq0041]] .

PARAGRAPH: Objective parameter space.

[[EQ:eq0042]] closed (Euclidean metric), enabling 1-homogeneous RI penalties on [[EQ:eq0043]] .

PARAGRAPH: Dictionary space and reference measure.

[[EQ:eq0044]] with reference [[EQ:eq0045]] , [[EQ:eq0046]] . Law measures [[EQ:eq0047]] are absolutely continuous: [[EQ:eq0048]] with [[EQ:eq0049]] . We always minimize over [[EQ:eq0050]] and abbreviate

[[EQ:eq0017]]

PARAGRAPH: HK geometry (dynamic form: measure-centric).

Among triplets [[EQ:eq0051]] solving in distributions

[[EQ:eq0018]]

the dynamic action is [[EQ:eq0052]] ; [[EQ:eq0053]] is its square-root infimum over admissible curves. Writing densities [[EQ:eq0054]] with [[EQ:eq0055]] , the continuity equation becomes

[[EQ:eq0019]]

Convention. Unit coefficients on [[EQ:eq0056]] and [[EQ:eq0057]] (time rescaling otherwise). The dynamic action satisfies the usual characterization [[EQ:eq0058]] , hence [[EQ:eq0059]] for any admissible curve LMS2016,LMSInvent2018,Chizat2018.

[Lebesgue reference by reweighting]
Equivalently, use Lebesgue densities [[EQ:eq0060]] and absorb [[EQ:eq0061]] into the potential via [[EQ:eq0062]] , yielding the same Gibbs form and JKO steps.

SECTION: Letters, Bochner Operator Integrals, and Fubini Conditions

PARAGRAPH: Letters.

Each [[EQ:eq0063]] yields: (i) a nonnegative density [[EQ:eq0064]] and (ii) a PSD co-mobility block [[EQ:eq0065]] on the physical Hilbert block.

[Bochner integrability, uniform PSD, Fubini/Tonelli]as:bochner
For a.e.\ [[EQ:eq0066]] , [[EQ:eq0067]] is measurable, polynomially bounded, and locally Lipschitz in [[EQ:eq0068]] uniformly in [[EQ:eq0069]] ; [[EQ:eq0070]] is strongly measurable with [[EQ:eq0071]] , [[EQ:eq0072]] , and [[EQ:eq0073]] . There exist [[EQ:eq0074]] and a fixed block-diagonal reference [[EQ:eq0075]] such that for all [[EQ:eq0076]] with [[EQ:eq0077]] ,

[[EQ:eq0001]]

Tonelli/Fubini apply whenever integrands are nonnegative or integrable as prescribed.

PARAGRAPH: Effective law and co-mobility (Bochner).

[[EQ:eq0002]]

[On uniform ellipticity]
The left inequality in eq:ellipticity for all [[EQ:eq0078]] implies [[EQ:eq0079]] for [[EQ:eq0080]] -a.e.\ [[EQ:eq0081]] . We keep this structural assumption for the core theory; relaxing it (e.g.\ restricting [[EQ:eq0082]] to be bounded away from [[EQ:eq0083]] ) is possible but not pursued here.

SECTION: Energy, Dissipation, and the Core (Entropic) Model

sec:energy

PARAGRAPH: Terminology.

``Co-mobility'' refers exclusively to the Hilbert block operator [[EQ:eq0084]] ; ``mobility'' denotes generic metric tensors when no dictionary dependence is implied.

PARAGRAPH: Separation of phase and dictionary energies.

The law-phase energy [[EQ:eq0085]] encodes phase selection on [[EQ:eq0086]] and is independent of [[EQ:eq0087]] . All [[EQ:eq0088]] -dependence is carried by [[EQ:eq0089]] in eq:lawpair and by the meta-functional on [[EQ:eq0090]] ; this avoids double counting.

PARAGRAPH: Physical and law-phase energy on [[EQ:eq0091]] .

Let [[EQ:eq0092]] , [[EQ:eq0093]] , [[EQ:eq0094]] an information Bregman divergence, and [[EQ:eq0095]] a gauge penalty with elliptic regularization. For the law-phase field [[EQ:eq0096]] ,

[[EQ:eq0003]]

with multiwell [[EQ:eq0097]] , [[EQ:eq0098]] , and simplex constraint via multiplier or barrier.
Each [[EQ:eq0099]] depends only on [[EQ:eq0100]] and does not depend on [[EQ:eq0101]] ; all dictionary averaging enters solely through [[EQ:eq0102]] in eq:lawpair.

PARAGRAPH: Meta functional on [[EQ:eq0103]] (entropic core, [[EQ:eq0104]] -measure).

For [[EQ:eq0105]] ,

[[EQ:eq0004]]

PARAGRAPH: Total functional.

[[EQ:eq0005]]

PARAGRAPH: Observation operators (coercivity/l.s.c.).

[[EQ:eq0106]] are (Fr\'echet-)continuous with [[EQ:eq0107]] , ensuring coercivity and lower semicontinuity of the data fidelity terms.

SECTION: Split Minimizing-Movements Scheme (Frozen Metric) and EDI

sec:split
Decompose [[EQ:eq0108]] into the Hilbert block [[EQ:eq0109]] and the RI block [[EQ:eq0110]] .

PARAGRAPH: Quadratic reference norms.

Define for increments on [[EQ:eq0111]] the frozen quadratic form

[[EQ:eq0006]]

where [[EQ:eq0112]] and

[[EQ:eq0020]]

Here [[EQ:eq0113]] is a fixed block-diagonal reference on the Hilbert block, and [[EQ:eq0114]] is a fixed reference on the RI block; both are time-independent. Moreover, [[EQ:eq0115]] is [[EQ:eq0116]] -homogeneous on [[EQ:eq0117]] (Appendix~app:ri).

PARAGRAPH: Notation for pre/post measures.

For brevity we write [[EQ:eq0118]] and [[EQ:eq0119]] .

PARAGRAPH: (A) Physical (RI/BV) step with frozen metric.

[[EQ:eq0007]]

PARAGRAPH: (B) Meta (HK--JKO) step with all [[EQ:eq0120]] -dependent energy.

[[EQ:eq0008]]

PARAGRAPH: (C) Birth step (augmented acceptance with HK gap).

For clarity the Step-(B) output is [[EQ:eq0121]] ; the post-acceptance value is [[EQ:eq0122]] . For each proposal center [[EQ:eq0123]] , define the [[EQ:eq0124]] -normalized bump

[[EQ:eq0021]]

Set

[[EQ:eq0022]]

Accept iff

[[EQ:eq0009]]

and set [[EQ:eq0125]] on acceptance, otherwise [[EQ:eq0126]] .
We allow at most one accepted birth per time-step (cf.\ lem:birth).

PARAGRAPH: Per-step dissipations and birth cost.

[[EQ:eq0010]]

[[EQ:eq0011]]

convention: if accepted then [[EQ:eq0127]] , else [[EQ:eq0128]] .

[Per-step EDI]thm:stepEDI
Under as:bochner and the coercivity/l.s.c.\ assumptions in :existence, each cycle (A)+(B)+(C) satisfies

[[EQ:eq0012]]

[Sketch]
Step-(B) minimality yields, for any competitor [[EQ:eq0129]] ,

[[EQ:eq0023]]

Choosing [[EQ:eq0130]] and rearranging gives exactly the [[EQ:eq0131]] HK gap inside [[EQ:eq0132]] in eq:birth-accept. With acceptance, [[EQ:eq0133]] , and adding (A)+(B)+(C) yields eq:stepEDI.

[Frozen-metric distortion vanishes in the limit]
Assume [[EQ:eq0134]] has bounded variation in operator norm and

[[EQ:eq0024]]

Then

[[EQ:eq0025]]

whose sum in [[EQ:eq0135]] is finite by the discrete EDI. Such an assumption is implied, for instance, if the Bochner kernel is [[EQ:eq0136]] –Lipschitz in [[EQ:eq0137]] and one has an estimate linking [[EQ:eq0138]] variations of [[EQ:eq0139]] to [[EQ:eq0140]] on bounded sets; lacking that, we simply take the [[EQ:eq0141]] -Lipschitz bound as a standing hypothesis.

SECTION: Existence of EDI Solutions, BV Scaling, and Parameter Stability

sec:existence
We work in the EDI (curves of maximal slope) framework; no EVI/contractivity claims are made. The frozen metric yields a time-dependent metric/energy setting; compactness and passage to the limit follow minimizing movements with time-dependent data and EDP-convergence.

PARAGRAPH: Proof roadmap (sketch).

(1) Discrete EDI: thm:stepEDI. (2) Compactness: bounds from eq:stepEDI yield tightness for [[EQ:eq0142]] and bounds for [[EQ:eq0143]] ; lem:birth controls the jump set. (3) Limit: time-dependent stability of [[EQ:eq0144]] and bounded-variation of [[EQ:eq0145]] ensure epi-convergence and stability of JKO minimizers; obtain a curve of maximal slope and the global EDI.

[Coercivity and HK geodesic convexity (core)]as:core
[[EQ:eq0146]] is coercive and l.s.c.\ in [[EQ:eq0147]] , uniformly in [[EQ:eq0148]] . For each frozen [[EQ:eq0149]] , [[EQ:eq0150]] in eq:meta is HK [[EQ:eq0151]] -geodesically convex and coercive along HK geodesics (entropic core).

[A sufficient condition for HK geodesic convexity]rem:HKconvex
If [[EQ:eq0152]] satisfies [[EQ:eq0153]] and [[EQ:eq0154]] , and the reference density [[EQ:eq0155]] of [[EQ:eq0156]] has semiconvex log-density [[EQ:eq0157]] , then [[EQ:eq0158]] is [[EQ:eq0159]] -geodesically convex along HK geodesics for some [[EQ:eq0160]] .

[BV scaling for RI (corrected)]as:bv
[[EQ:eq0161]] and [[EQ:eq0162]] as [[EQ:eq0163]] ; vanishing viscosity on [[EQ:eq0164]] selects balanced-viscosity limits MRS2015.

[Time-dependence and parameter stability]as:param
[[EQ:eq0165]] is measurable with locally uniform quadratic bounds; [[EQ:eq0166]] has bounded variation in operator norm. If [[EQ:eq0167]] then [[EQ:eq0168]] in [[EQ:eq0169]] with uniform growth. Uniform [[EQ:eq0170]] -geodesic convexity on compacts: for any compact [[EQ:eq0171]] and bounded [[EQ:eq0172]] in [[EQ:eq0173]] -space there exist [[EQ:eq0174]] and [[EQ:eq0175]] such that for all [[EQ:eq0176]] and [[EQ:eq0177]] , [[EQ:eq0178]] is [[EQ:eq0179]] -HK-geodesically convex and [[EQ:eq0180]] -coercive along HK geodesics. These ensure epi-convergence of JKO functionals and stability of minimizers under time-dependent data.

[Birth countability and action bound]lem:birth
Assume the per-step threshold [[EQ:eq0181]] in eq:birth-accept, [[EQ:eq0182]] with [[EQ:eq0183]] , and at most one accepted birth per time-step. Then on any [[EQ:eq0184]] : (i) the number of accepted births is at most countable and [[EQ:eq0185]] ; (ii) [[EQ:eq0186]] is HK-absolutely continuous on [[EQ:eq0187]] with jump set [[EQ:eq0188]] the accepted birth times; (iii) the discrete HK action is uniformly bounded on compact intervals.

[Existence and global EDI]thm:exist
Let [[EQ:eq0189]] , [[EQ:eq0190]] with [[EQ:eq0191]] , and viscosities [[EQ:eq0192]] , [[EQ:eq0193]] with as:bv. Under as:bochner,as:core,as:param, time-interpolants of the split scheme admit a subsequence converging to [[EQ:eq0194]] such that: (1) [[EQ:eq0195]] is an EDI solution of the physical problem with RI/BV on [[EQ:eq0196]] and co-mobility [[EQ:eq0197]] ; (2) [[EQ:eq0198]] is an HK gradient flow of eq:meta between countably many birth times; (3) for all [[EQ:eq0199]] ,

[[EQ:eq0013]]

with [[EQ:eq0200]] bounded below by the dynamic HK action.

[Degenerate mobilities]
Uniform ellipticity [[EQ:eq0201]] is structural to the core existence theory. Degenerate cases (mobilities vanishing on phases) require additional regularization and compensated compactness and lie beyond our scope.

SECTION: Law-Phase Transitions, RI/BV Jumps, and [[EQ:eq0202]]

Gamma-Limits sec:lawphase
Diffuse law-phase energy [[EQ:eq0203]] -converges (Modica--Mortola) as [[EQ:eq0204]] to a perimeter functional with surface tension given by the 1D heteroclinic profile. RI/BV jumps on [[EQ:eq0205]] satisfy a local selection rule.
[Local Maxwell selection]prop:Maxwell
Under complete separation of [[EQ:eq0206]] and [[EQ:eq0207]] and the Modica--Mortola limit [[EQ:eq0208]] , the jump cost for [[EQ:eq0209]] equals the sum of the RI cost and the perimeter term; admissible well-to-well transitions satisfy [[EQ:eq0210]] at the jump point.

Order of limits. The order [[EQ:eq0211]] (with [[EQ:eq0212]] ) then [[EQ:eq0213]] is adopted.

SECTION: Gauge/Relabeling Invariances and Diffuse Identity Balance

sec:inv
[Gauge covariance of letters]as:gauge
Letters depend only on gauge-covariant quantities, e.g.,
[[EQ:eq0214]] , and similarly for [[EQ:eq0215]] ; observables depend on gauge-covariant objects.

[Gauge invariance]
Under as:gauge and admissible BCs, [[EQ:eq0216]] and the EDI dynamics are invariant under [[EQ:eq0217]] : [[EQ:eq0218]] , [[EQ:eq0219]] .

[Relabeling invariance and diffuse identity balance]
Let [[EQ:eq0220]] be the Noether-type density/current for an internal relabeling symmetry with [[EQ:eq0221]] . For smooth [[EQ:eq0222]] ,

[[EQ:eq0014]]

which reduces to Reynolds transport in the sharp-interface limit.

SECTION: Metastability and the Monotone Effective Barrier Hypothesis

sec:mebh
Let [[EQ:eq0223]] denote an effective action barrier from a Galerkin reduction with Onsager matching, assembling potential gaps, RI/BV jump costs, law-phase interfacial costs, and HK meta costs along most-probable paths.
[MEBH (conditional)]def:mebh
Under: (i) slow motion (macroscopic time derivatives [[EQ:eq0224]] local relaxation rates); (ii) nondegenerate saddles (Eyring--Kramers regime); and (iii) HK [[EQ:eq0225]] -convexity of [[EQ:eq0226]] , we posit:

- On continuous segments, [[EQ:eq0227]] with [[EQ:eq0228]] in the slow-motion limit.
- At events (RI/BV jump, law-phase switch, accepted birth), [[EQ:eq0229]] .

[Eyring--Kramers upper bound]
Assuming MEBH and small noise [[EQ:eq0230]] , the exit hazard admits

[[EQ:eq0026]]

with slowly varying prefactor [[EQ:eq0231]] determined by local curvatures. Global monotonicity is not claimed.

SECTION: Statistical Mechanics on the Dictionary (Entropic Core)

sec:gibbs
For frozen [[EQ:eq0232]] , HK gradient flow admits entropic equilibria on [[EQ:eq0233]] :

[[EQ:eq0027]]

Here [[EQ:eq0234]] under the growth/semiconvexity of rem:HKconvex. This Gibbs form holds for the entropic core; with Huber/TV regularization, equilibria are altered and are not plain Gibbs densities.

SECTION: Minimal Working Model and Conditional Metastability

sec:minimal
Setting. [[EQ:eq0235]] ; [[EQ:eq0236]] obeys Allen--Cahn (Neumann), [[EQ:eq0237]] obeys Cahn--Hilliard (no-flux for the chemical potential); [[EQ:eq0238]] absent; [[EQ:eq0239]] has two wells ( [[EQ:eq0240]] ). [[EQ:eq0241]] with Gaussian [[EQ:eq0242]] ; [[EQ:eq0243]] constant PSD; [[EQ:eq0244]] and [[EQ:eq0245]] quadratic in [[EQ:eq0246]] and Lipschitz in [[EQ:eq0247]] .
Assume [[EQ:eq0248]] a.e., [[EQ:eq0249]] ; the Allen--Cahn operator satisfies a maximum principle, hence [[EQ:eq0250]] for all [[EQ:eq0251]] .
[Existence and global EDI, minimal model]thm:minimal
Under as:bv and the above setting, the split scheme converges (up to subsequences) to [[EQ:eq0252]] with: (i) [[EQ:eq0253]] an EDI solution with [[EQ:eq0254]] invariant; (ii) [[EQ:eq0255]] an HK gradient flow between countably many birth times; (iii) the global EDI holds with nonnegative event costs as in def:diss.

[Conditional MEBH, minimal model]
If additionally: (a) small additive noise acts on a finite-dimensional Galerkin reduction; (b) time scales are separated (slow motion); (c) saddles are nondegenerate; then MEBH holds and the Eyring--Kramers upper bound applies Bovier2004,BerglundGentz2010,BovierHollander2015,FreidlinWentzell2012.

SECTION: Discussion

The entropic core establishes existence and a global EDI for a nondual autopoietic geometry coupling continuous co-gradient evolution, RI/BV events, and HK law selection with [[EQ:eq0256]] -normalized births accepted by law-energy + HK gap decrease of order [[EQ:eq0257]] . Gauge/relabeling invariances render the construction physically meaningful. Metastability is framed through conditional barrier monotonicity and hazard upper bounds. Extensions (sparsity, atoms, renormalization tags, open-system coupling) can be added without altering the core existence/EDI, albeit with stronger assumptions or weaker conclusions.

PARAGRAPH: On [[EQ:eq0258]] .

The threshold [[EQ:eq0259]] is in the same free-energy units as [[EQ:eq0260]] (via [[EQ:eq0261]] ). Practically, one may set [[EQ:eq0262]] with [[EQ:eq0263]] to filter sub-informational (thermal) drops, or scale it to a characteristic curvature of [[EQ:eq0264]] on [[EQ:eq0265]] (minimum resolvable barrier decrement per added mass).

SECTION: Representative RI Penalties and BV Limits

app:ri
Typical 1-homogeneous penalties on [[EQ:eq0266]] include

[[EQ:eq0028]]

or their BV-time counterparts. Under as:bv, vanishing viscosity selects balanced-viscosity limits MRS2015.

SECTION: Sparsity Regularization on the Dictionary

app:sparsity
Adding a Huberized TV (e.g.\ [[EQ:eq0267]] ) promotes sparsity while admitting approximate HK geodesic convexity for small [[EQ:eq0268]] . Exact TV generally violates HK geodesic convexity; one can still obtain EDI (curves of maximal slope) with l.s.c.\ and coercivity along HK geodesics.

SECTION: Atomic Extensions

app:atomic
If atoms are allowed, write [[EQ:eq0269]] and replace the entropy by [[EQ:eq0270]] with a mixed reference. HK on mixed measures is well-defined; existence follows by tightness in the space of finite Radon measures with HK topology. Birth acceptance uses eq:birth-accept with measures throughout: replace [[EQ:eq0271]] by [[EQ:eq0272]] and [[EQ:eq0273]] by [[EQ:eq0274]] .

SECTION: Frozen Metric: Measurability and Uniform Bounds

app:frozen
By as:bochner, [[EQ:eq0275]] is strongly measurable and satisfies [[EQ:eq0276]] uniformly, ensuring well-posedness of eq:Astep and stability of distortions across steps.

SECTION: Law-Phase: Maxwell Selection and Locality

app:maxwell
At jumps [[EQ:eq0277]] , admissible transitions satisfy [[EQ:eq0278]] (equal depth) and minimize the RI cost; locality follows from BV compactness and [[EQ:eq0279]] -convergence (Modica--Mortola).

SECTION: Open-System Nondual Extension: Symmetric HK/ET Coupling

app:open
Introduce an environment law [[EQ:eq0280]] with density [[EQ:eq0281]] and define

[[EQ:eq0029]]

with [[EQ:eq0282]] of the same entropic core form as eq:meta and [[EQ:eq0283]] a symmetric HK/ET coupling, e.g.\ [[EQ:eq0284]] . A split scheme adds an HK--JKO step for [[EQ:eq0285]] :

[[EQ:eq0030]]

Then the per-step EDI gains a nonnegative environment dissipation and an interaction variation, while the global EDI reads

[[EQ:eq0015]]

This realizes nonduality in an open-system setting: exchanging [[EQ:eq0286]] leaves the interaction symmetric; boundary inputs are representation-dependent but enter only through [[EQ:eq0287]] .
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[[EQ:eq0016]]

[[EQ:eq0017]]

[[EQ:eq0018]]

[[EQ:eq0019]]

[[EQ:eq0020]]

[[EQ:eq0021]]

[[EQ:eq0022]]

[[EQ:eq0023]]

[[EQ:eq0024]]

[[EQ:eq0025]]

[[EQ:eq0026]]

[[EQ:eq0027]]

[[EQ:eq0028]]

[[EQ:eq0029]]

[[EQ:eq0030]]

[[EQ:eq0031]]

[[EQ:eq0032]]

[[EQ:eq0033]]

[[EQ:eq0034]]

[[EQ:eq0035]]

[[EQ:eq0036]]

[[EQ:eq0037]]

[[EQ:eq0038]]

[[EQ:eq0039]]

[[EQ:eq0040]]

[[EQ:eq0041]]

[[EQ:eq0042]]

[[EQ:eq0043]]

[[EQ:eq0044]]

[[EQ:eq0045]]

[[EQ:eq0046]]

[[EQ:eq0047]]

[[EQ:eq0048]]

[[EQ:eq0049]]

[[EQ:eq0050]]

[[EQ:eq0051]]

[[EQ:eq0052]]

[[EQ:eq0053]]

[[EQ:eq0054]]

[[EQ:eq0055]]

[[EQ:eq0056]]

[[EQ:eq0057]]

[[EQ:eq0058]]

[[EQ:eq0059]]

[[EQ:eq0060]]

[[EQ:eq0061]]

[[EQ:eq0062]]

[[EQ:eq0063]]

[[EQ:eq0064]]

[[EQ:eq0065]]

[[EQ:eq0066]]

[[EQ:eq0067]]

[[EQ:eq0068]]

[[EQ:eq0069]]

[[EQ:eq0070]]

[[EQ:eq0071]]

[[EQ:eq0072]]

[[EQ:eq0073]]

[[EQ:eq0074]]

[[EQ:eq0075]]

[[EQ:eq0076]]

[[EQ:eq0077]]

[[EQ:eq0078]]

[[EQ:eq0079]]

[[EQ:eq0080]]

[[EQ:eq0081]]

[[EQ:eq0082]]

[[EQ:eq0083]]

[[EQ:eq0084]]

[[EQ:eq0085]]

[[EQ:eq0086]]

[[EQ:eq0087]]

[[EQ:eq0088]]

[[EQ:eq0089]]

[[EQ:eq0090]]

[[EQ:eq0091]]

[[EQ:eq0092]]

[[EQ:eq0093]]

[[EQ:eq0094]]

[[EQ:eq0095]]

[[EQ:eq0096]]

[[EQ:eq0097]]

[[EQ:eq0098]]

[[EQ:eq0099]]

[[EQ:eq0100]]

[[EQ:eq0101]]

[[EQ:eq0102]]

[[EQ:eq0103]]

[[EQ:eq0104]]

[[EQ:eq0105]]

[[EQ:eq0106]]

[[EQ:eq0107]]

[[EQ:eq0108]]

[[EQ:eq0109]]

[[EQ:eq0110]]

[[EQ:eq0111]]

[[EQ:eq0112]]

[[EQ:eq0113]]

[[EQ:eq0114]]

[[EQ:eq0115]]

[[EQ:eq0116]]

[[EQ:eq0117]]

[[EQ:eq0118]]

[[EQ:eq0119]]

[[EQ:eq0120]]

[[EQ:eq0121]]

[[EQ:eq0122]]

[[EQ:eq0123]]

[[EQ:eq0124]]

[[EQ:eq0125]]

[[EQ:eq0126]]

[[EQ:eq0127]]

[[EQ:eq0128]]

[[EQ:eq0129]]

[[EQ:eq0130]]

[[EQ:eq0131]]

[[EQ:eq0132]]

[[EQ:eq0133]]

[[EQ:eq0134]]

[[EQ:eq0135]]

[[EQ:eq0136]]

[[EQ:eq0137]]

[[EQ:eq0138]]

[[EQ:eq0139]]

[[EQ:eq0140]]

[[EQ:eq0141]]

[[EQ:eq0142]]

[[EQ:eq0143]]

[[EQ:eq0144]]

[[EQ:eq0145]]

[[EQ:eq0146]]

[[EQ:eq0147]]

[[EQ:eq0148]]

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Introduction

Spaces, Variables, and HK Geometry

Letters, Bochner Operator Integrals, and Fubini Conditions

Energy, Dissipation, and the Core (Entropic) Model

Split Minimizing-Movements Scheme (Frozen Metric) and EDI

Existence of EDI Solutions, BV Scaling, and Parameter Stability

Law-Phase Transitions, RI/BV Jumps, and Γ

Gauge/Relabeling Invariances and Diffuse Identity Balance

Metastability and the Monotone Effective Barrier Hypothesis

Statistical Mechanics on the Dictionary (Entropic Core)

Minimal Working Model and Conditional Metastability

Discussion

Representative RI Penalties and BV Limits

Sparsity Regularization on the Dictionary

Atomic Extensions

Frozen Metric: Measurability and Uniform Bounds

Law-Phase: Maxwell Selection and Locality

Open-System Nondual Extension: Symmetric HK/ET Coupling

\begin{equation}\label{eq:ellipticity}
c_{\rm ell}\,\mathsf P_{\mathcal H}\ \preceq\ \mathbb K_{\mu,\mathcal H}(\Xi,\theta):=\int_{\Dcal} k_{\mathcal H}(\Xi,\theta;\xi)u(\xi)\,\dd\mu_0(\xi)\ \preceq\ C\,\mathsf P_{\mathcal H}.
\end{equation}

\label{eq:ellipticity} c_{\rm ell} \mathsf P_{\mathcal H}\ \preceq\ \mathbb K_{\mu,\mathcal H}(\Xi,\theta):=\int_{\Dcal} k_{\mathcal H}(\Xi,\theta;\xi)u(\xi) \dd\mu_0(\xi)\ \preceq\ C \mathsf P_{\mathcal H}.

\begin{align}
\mathcal L_\mu(\Xi,\theta)
  &:= \int_{\Dcal}\Big(\int_{\Ocal} f(x,\Xi,\theta;\xi)\,\dd x\Big)\,u(\xi)\,\dd\mu_0(\xi),\label{eq:lawpair}\\
\mathbb K_{\mu,\mathcal H}(\Xi,\theta)
  &:= \int_{\Dcal} k_{\mathcal H}(\Xi,\theta;\xi)\,u(\xi)\,\dd\mu_0(\xi).\nonumber
\end{align}

\mathcal L_\mu(\Xi,\theta) &:= \int_{\Dcal}\Big(\int_{\Ocal} f(x,\Xi,\theta;\xi) \dd x\Big) u(\xi) \dd\mu_0(\xi),\label{eq:lawpair}\\ \mathbb K_{\mu,\mathcal H}(\Xi,\theta) &:= \int_{\Dcal} k_{\mathcal H}(\Xi,\theta;\xi) u(\xi) \dd\mu_0(\xi).\nonumber

\begin{equation}\label{eq:Elaw}
E_{\rm law}(s)
  = \int_{\Ocal}\!\Big(W_{\rm law}(s)+\varepsilon_\ell^2|\nabla s|^2\Big)\dd x
    + \int_{\Ocal}\sum_{i=1}^{M} s_i(x)\, \mathcal L^{\rm eff}_i(\Xi,\theta)\,\dd x,
\end{equation}

\label{eq:Elaw} E_{\rm law}(s) = \int_{\Ocal} \Big(W_{\rm law}(s)+\varepsilon_\ell^2|\nabla s|^2\Big)\dd x + \int_{\Ocal}\sum_{i=1}^{M} s_i(x)  \mathcal L^{\rm eff}_i(\Xi,\theta) \dd x,

\begin{equation}\label{eq:meta}
\mathfrak L(u\mid \Xi,\theta)
  = \int_{\Dcal} \Phi(\Xi,\theta;\xi)\,u(\xi)\,\dd\mu_0(\xi)
    + \tau_{\rm ent}\!\int_{\Dcal} u(\xi)\log u(\xi)\,\dd\mu_0(\xi).
\end{equation}

\label{eq:meta} \mathfrak L(u\mid \Xi,\theta) = \int_{\Dcal} \Phi(\Xi,\theta;\xi) u(\xi) \dd\mu_0(\xi) + \tau_{\rm ent} \int_{\Dcal} u(\xi)\log u(\xi) \dd\mu_0(\xi).

\begin{align}\label{eq:Gtotal}
\mathcal G(z,u)
  &= \underbrace{\int_{\Ocal}\Big(\tfrac{\sigma}{2}|\nabla m|^2 + W(m)\Big)\dd x}_{\text{Allen--Cahn}}
   + \underbrace{F_{\rm CH}[\phi]}_{\text{Cahn--Hilliard}}
   + \underbrace{\gamma_0\,F + \kappa_0\,C}_{\text{information/gauge penalties}}\nonumber\\
  &\quad + \underbrace{U(\theta)}_{\text{objective penalty}}
   + \underbrace{\sum_{j=1}^{J}\frac{\lambda_j}{2}\int_{\Ocal}\!\big|h_j-\Pi_j[\Xi]\big|^2\dd x}_{\text{data fidelity}}\nonumber\\
  &\quad + \underbrace{E_{\rm law}(s)}_{\text{law-phase on }\Ocal}
   + \underbrace{\mathcal L_\mu(\Xi,\theta)}_{\text{dictionary effect on }\Ocal}
   + \underbrace{\mathfrak L(u\mid \Xi,\theta)}_{\text{meta on }\Dcal}.
\end{align}

\label{eq:Gtotal} \mathcal G(z,u) &= \underbrace{\int_{\Ocal}\Big(\tfrac{\sigma}{2}|\nabla m|^2 + W(m)\Big)\dd x}_{\text{Allen--Cahn}} + \underbrace{F_{\rm CH}[\phi]}_{\text{Cahn--Hilliard}} + \underbrace{\gamma_0 F + \kappa_0 C}_{\text{information/gauge penalties}}\nonumber\\ &\quad + \underbrace{U(\theta)}_{\text{objective penalty}} + \underbrace{\sum_{j=1}^{J}\frac{\lambda_j}{2}\int_{\Ocal} \big|h_j-\Pi_j[\Xi]\big|^2\dd x}_{\text{data fidelity}}\nonumber\\ &\quad + \underbrace{E_{\rm law}(s)}_{\text{law-phase on }\Ocal} + \underbrace{\mathcal L_\mu(\Xi,\theta)}_{\text{dictionary effect on }\Ocal} + \underbrace{\mathfrak L(u\mid \Xi,\theta)}_{\text{meta on }\Dcal}.

\begin{equation}\label{eq:norm}
\|z-z^{k-1}\|_{\mathbb K^{k-1}}^2
  := \ipPH{z_{\mathcal H}-z^{k-1}_{\mathcal H}}{\mathbb K^{k-1}_{\mathcal H}\,(z_{\mathcal H}-z^{k-1}_{\mathcal H})},
\end{equation}

\label{eq:norm} \|z-z^{k-1}\|_{\mathbb K^{k-1}}^2 := \ipPH{z_{\mathcal H}-z^{k-1}_{\mathcal H}}{\mathbb K^{k-1}_{\mathcal H} (z_{\mathcal H}-z^{k-1}_{\mathcal H})},

\begin{equation}\label{eq:Astep}
\begin{aligned}
z^{k}\in \operatorname*{arg\,min}_{z}\ \Big\{\,
&\mathcal G\big(z,\,u^{k-1,\mathrm{post}}\big)
+\frac{1}{2\tau}\big\|z-z^{k-1}\big\|_{\mathbb K^{k-1}}^{2}
+\mathcal R_{\rm RI}\!\big(z_{\mathcal R}-z^{k-1}_{\mathcal R}\big)\\
&\quad +\frac{\nu_{\mathcal H}(\tau)}{2}\big\|z_{\mathcal H}-z^{k-1}_{\mathcal H}\big\|_{\mathsf P_{\mathcal H}}^{2}
+\frac{\nu_{\rm RI}(\tau)}{2}\big\|z_{\mathcal R}-z^{k-1}_{\mathcal R}\big\|_{\mathsf P_{\mathcal R}}^{2}\, \Big\}.
\end{aligned}
\end{equation}

\label{eq:Astep} \begin{aligned} z^{k}\in \operatorname*{arg min}_{z}\ \Big\{  &\mathcal G\big(z, u^{k-1,\mathrm{post}}\big) +\frac{1}{2\tau}\big\|z-z^{k-1}\big\|_{\mathbb K^{k-1}}^{2} +\mathcal R_{\rm RI} \big(z_{\mathcal R}-z^{k-1}_{\mathcal R}\big)\\ &\quad +\frac{\nu_{\mathcal H}(\tau)}{2}\big\|z_{\mathcal H}-z^{k-1}_{\mathcal H}\big\|_{\mathsf P_{\mathcal H}}^{2} +\frac{\nu_{\rm RI}(\tau)}{2}\big\|z_{\mathcal R}-z^{k-1}_{\mathcal R}\big\|_{\mathsf P_{\mathcal R}}^{2}  \Big\}. \end{aligned}

\begin{align}\label{eq:Bstep}
u^{k,\mathrm{pre}} \in \operatorname*{arg\,min}_{\substack{u\ge0,\,\int u\dd\mu_0=1}}\
\Big\{
   &\int_{\Dcal}\!\Big[\Phi(\Xi^k,\theta^k;\xi)+\int_{\Ocal} f(x,\Xi^k,\theta^k;\xi)\,\dd x\Big]u(\xi)\,\dd\mu_0(\xi)\nonumber\\
   &\quad + \tau_{\rm ent}\!\int_{\Dcal} u\log u\,\dd\mu_0
     + \frac{1}{2\tau}\,\HK^2\!\big(u,\,u^{k-1,\mathrm{post}}\big)
\Big\}.
\end{align}

\label{eq:Bstep} u^{k,\mathrm{pre}} \in \operatorname*{arg min}_{\substack{u\ge0, \int u\dd\mu_0=1}}\ \Big\{ &\int_{\Dcal} \Big[\Phi(\Xi^k,\theta^k;\xi)+\int_{\Ocal} f(x,\Xi^k,\theta^k;\xi) \dd x\Big]u(\xi) \dd\mu_0(\xi)\nonumber\\ &\quad + \tau_{\rm ent} \int_{\Dcal} u\log u \dd\mu_0 + \frac{1}{2\tau} \HK^2 \big(u, u^{k-1,\mathrm{post}}\big) \Big\}.

\begin{align}\label{eq:birth-accept}
\Delta_{\rm tot}
  &:= \Big[\mathcal L_{\tilde\mu}(\Xi^{k},\theta^{k})+\mathfrak L(\tilde u^{k}\mid \Xi^{k},\theta^{k})\Big]
   - \Big[\mathcal L_{\mu^{k,\mathrm{pre}}}(\Xi^{k},\theta^{k})+\mathfrak L(u^{k,\mathrm{pre}}\mid \Xi^{k},\theta^{k})\Big]\nonumber\\
  &\quad + \frac{1}{2\tau}\Big(\HK^2(\tilde u^{k},u^{k-1,\mathrm{post}})-\HK^2(u^{k,\mathrm{pre}},u^{k-1,\mathrm{post}})\Big)
  \ \le\ -\,c_{\rm birth}\,\varepsilon_b\sum_\ell\omega_\ell,
\end{align}

\label{eq:birth-accept} \Delta_{\rm tot} &:= \Big[\mathcal L_{\tilde\mu}(\Xi^{k},\theta^{k})+\mathfrak L(\tilde u^{k}\mid \Xi^{k},\theta^{k})\Big] - \Big[\mathcal L_{\mu^{k,\mathrm{pre}}}(\Xi^{k},\theta^{k})+\mathfrak L(u^{k,\mathrm{pre}}\mid \Xi^{k},\theta^{k})\Big]\nonumber\\ &\quad + \frac{1}{2\tau}\Big(\HK^2(\tilde u^{k},u^{k-1,\mathrm{post}})-\HK^2(u^{k,\mathrm{pre}},u^{k-1,\mathrm{post}})\Big) \ \le\ - c_{\rm birth} \varepsilon_b\sum_\ell\omega_\ell,

\begin{align}
\mathrm{Diss}^{k}_{\rm phys}
  &:= \frac{1}{2\tau}\|z^{k}-z^{k-1}\|^2_{\mathbb K^{k-1}}
      + \mathcal R_{\rm RI}(z^{k}_{\mathcal R}-z^{k-1}_{\mathcal R})
      + \frac{\nu_{\mathcal H}(\tau)}{2}\|z^{k}_{\mathcal H}-z^{k-1}_{\mathcal H}\|^2_{\mathsf P_{\mathcal H}}
      + \frac{\nu_{\rm RI}(\tau)}{2}\|z^{k}_{\mathcal R}-z^{k-1}_{\mathcal R}\|^2_{\mathsf P_{\mathcal R}}.
      \label{eq:diss-phys}
\end{align}

\begin{equation}\label{def:diss}
\mathrm{Diss}^{k}_{\rm HK}:=\frac{1}{2\tau}\,\HK^{2}\!\big(u^{k,\mathrm{post}},u^{k-1,\mathrm{post}}\big),
\qquad
\mathrm{Cost}^{k}_{\rm birth}:= -\,\Delta_{\rm tot}.
\end{equation}

\label{def:diss} \mathrm{Diss}^{k}_{\rm HK}:=\frac{1}{2\tau} \HK^{2} \big(u^{k,\mathrm{post}},u^{k-1,\mathrm{post}}\big), \qquad \mathrm{Cost}^{k}_{\rm birth}:= - \Delta_{\rm tot}.

\begin{equation}\label{eq:stepEDI}
\mathcal G(z^{k},u^{k,\mathrm{post}}) + \mathrm{Diss}^{k}_{\rm phys} + \mathrm{Diss}^{k}_{\rm HK} + \mathrm{Cost}^{k}_{\rm birth}
\ \le\ \mathcal G(z^{k-1},u^{k-1,\mathrm{post}}).
\end{equation}

\begin{equation*}
\begin{aligned}
\mathcal G\big(z(t_2),u(t_2)\big)
&+ \mathrm{Diss}_{\rm phys}[t_1,t_2]
+ \mathrm{Diss}_{\rm HK}[t_1,t_2]
+ \sum_{t\in(t_1,t_2]}\mathrm{Cost}_{\rm birth}(t)\\
&\le\ \mathcal G\big(z(t_1),u(t_1)\big),
\end{aligned}
\end{equation*}

\begin{equation}\label{eq:diffuse-identity}
\partial_t(h\rho_{\rm id})+\mathrm{div}(hJ_{\rm id})
  = h'(m)\,\partial_t m\,\rho_{\rm id} - J_{\rm id}\!\cdot\nabla h + hR,
\end{equation}

\label{eq:diffuse-identity} \partial_t(h\rho_{\rm id})+\mathrm{div}(hJ_{\rm id}) = h'(m) \partial_t m \rho_{\rm id} - J_{\rm id} \cdot\nabla h + hR,

\begin{equation*}
\begin{aligned}
\mathcal G\big(z(t_2),u(t_2),\nu(t_2)\big)
&+\mathrm{Diss}_{\rm phys}[t_1,t_2]
+\mathrm{Diss}_{\rm HK}\big[u; t_1,t_2\big]
+\mathrm{Diss}_{\rm HK}\big[\nu; t_1,t_2\big]
+\sum_{t\in(t_1,t_2]}\mathrm{Cost}_{\rm birth}(t)\\
&\le\ \mathcal G\big(z(t_1),u(t_1),\nu(t_1)\big)
+\int_{t_1}^{t_2}\!\mathcal P_{\rm bdry}(t)\,\dd t.
\end{aligned}
\end{equation*}

\begin{aligned} \mathcal G\big(z(t_2),u(t_2),\nu(t_2)\big) &+\mathrm{Diss}_{\rm phys}[t_1,t_2] +\mathrm{Diss}_{\rm HK}\big[u; t_1,t_2\big] +\mathrm{Diss}_{\rm HK}\big[\nu; t_1,t_2\big] +\sum_{t\in(t_1,t_2]}\mathrm{Cost}_{\rm birth}(t)\\ &\le\ \mathcal G\big(z(t_1),u(t_1),\nu(t_1)\big) +\int_{t_1}^{t_2} \mathcal P_{\rm bdry}(t) \dd t. \end{aligned}

\[0.25em]
\footnotesize (Existence, Global EDI, and Conditional Metastability)
\vspace{-0.25em}
}

\author{K.\ Takahashi\\{ORCID: \href{https://orcid.org/0009-0004-4273-3365}{0009-0004-4273-3365}}}
\date{September 30, 2025}

\begin{document}
\maketitle

\begin{abstract}
We develop a self-contained architecture where (i) continuous co-gradient evolution of physical fields and objective parameters, (ii) discontinuous rate-independent/balanced-viscosity (RI/BV) events on phase/criteria variables, and (iii) measure-valued selection of law components (the ``alphabet'') are unified by a single energy--dissipation principle. Law selection evolves on the Hellinger--Kantorovich (HK) geometry (rather than a naive $W_2$ + Fisher--Rao sum), ensuring JKO/gradient-flow consistency. Co-mobilities and effective energies are defined as Bochner operator integrals under explicit measurability and uniform-ellipticity hypotheses. A split minimizing-movements scheme is designed with: (A) a physical RI/BV step using a frozen metric acting only on the Hilbert block; (B) an HK--JKO step for the law measure that includes \emph{all} $u$-dependent energy; and (C) a \emph{law-energy + HK gap} acceptance rule for $\mu_0$-normalized births with $O(\varepsilon_b)$ thresholds. We prove existence of energy--dissipation inequality (EDI) solutions for the entropic core model and specify the BV scaling $\nu_{\rm RI}(\tau)/\tau\to0$. We give sufficient conditions for HK geodesic convexity of the meta-functional, establish gauge/relabeling invariances, separate phase vs.\ dictionary energies, and formulate a Monotone Effective Barrier Hypothesis (MEBH) under which Eyring--Kramers hazard upper bounds and eventwise nonnegative barrier increments hold. Optional sparsity (Huber/TV) and atomic extensions are in the appendices and do not affect the core existence/EDI results.
\end{abstract}

\section{Introduction}
Autopoietic individuality---the capacity to maintain and enact a self while exchanging with the surround---can be cast as a long-lived localized excitation of a \emph{nondual}\footnote{A nondual field is a field--process whose system/environment split is a gauge choice; swapping roles preserves the variational--dissipative structure (Appendix~\ref{app:open}).} field, where the agent/environment split is a representational gauge. We consolidate this into a rigorous architecture that:
\begin{itemize}
  \item integrates continuous co-gradient becoming and discontinuous creation (RI/BV jumps, law-phase switches, births of law letters) under a single variational--dissipative principle;
  \item evolves the ``alphabet'' of admissible laws as a measure on a dictionary space via the HK geometry of unbalanced optimal transport, guaranteeing JKO/gradient-flow consistency;
  \item delivers existence of global EDI solutions by a split scheme with a frozen metric on the physical step and a \emph{law-energy + HK gap} acceptance rule for births.
\end{itemize}

\paragraph{Nonduality, universality, and open-system stance.}
``Self'' and ``other'' are gauge-related descriptions of a single field-process. Universality means boundary choices and model partitions (agent vs.\ environment) are representational gauges rather than primitive cuts. Technically: (i) gauge/relabeling invariances (\S\ref{sec:inv}); (ii) an EDI framework with time-dependent data and boundary fluxes (\S\ref{sec:existence}); and (iii) an optional environment law measure $\nu_t$ on $\Dcal$ interacting with the system law $\mu_t$ via a symmetric HK/entropy-transport (\textbf{ET}) coupling (Appendix~\ref{app:open}).

\paragraph{Motivation and physical interpretation (nonduality made concrete).}
(i) \emph{Why HK?} Law selection entails transport (model revision) and reaction (birth/death of letters), naturally encoded by HK (unbalanced OT), not by a naive $W_2$ + Fisher--Rao sum.
(ii) \emph{Why RI/BV?} Autopoietic re-organization proceeds via rate-independent switches (criteria updates, phase switches), for which BV limits provide the correct event-selection principle.
(iii) \emph{Nonduality as gauge:} swapping system and environment corresponds to swapping $(\mu,\nu)$ in a symmetric HK/ET coupling (Appendix~\ref{app:open}).

\section{Spaces, Variables, and HK Geometry}\label{sec:spaces}
\paragraph{Physical domain and variables.}
$\Ocal\subset\R^d$ bounded Lipschitz ($d\in\{1,2,3\}$). Extended state $z=(\Xi,\theta,H,s)$ with
\[
\Xi=(m,\phi,\eta,\psi,A),\quad \theta\in\Theta,\quad H=(h_j)_{j=1}^J,\quad s:\Ocal\to\Delta^{M-1},\quad \sum_{i=1}^{M}s_i=1,\ s_i\ge0.
\]

\[
\partial_t\mu_t+\nabla_\xi\!\cdot(\mu_t v_t)=\alpha_t\,\mu_t,
\]

\partial_t\mu_t+\nabla_\xi \cdot(\mu_t v_t)=\alpha_t \mu_t,

\[
\partial_t u_t+\nabla_\xi\!\cdot(u_t v_t)+u_t\,v_t\!\cdot\nabla_\xi\log w=\alpha_t u_t.
\]

\partial_t u_t+\nabla_\xi \cdot(u_t v_t)+u_t v_t \cdot\nabla_\xi\log w=\alpha_t u_t.

c_{\rm ell} \|\cdot\|^2_{\mathsf P_{\mathcal H}} \ \le\ \|\cdot\|^2_{\mathbb K^{k-1}}\ \le\ C \|\cdot\|^2_{\mathsf P_{\mathcal H}}.

\[
\varpi_{\varepsilon,\xi_\ell}(\xi)
 := \frac{\rho_{\varepsilon}(\xi-\xi_\ell)}{\int_{\Dcal}\rho_{\varepsilon}(\zeta-\xi_\ell)\,\dd\mu_0(\zeta)},
 \qquad \int_{\Dcal}\varpi_{\varepsilon,\xi_\ell}\,\dd\mu_0=1.
\]

\varpi_{\varepsilon,\xi_\ell}(\xi) := \frac{\rho_{\varepsilon}(\xi-\xi_\ell)}{\int_{\Dcal}\rho_{\varepsilon}(\zeta-\xi_\ell) \dd\mu_0(\zeta)}, \qquad \int_{\Dcal}\varpi_{\varepsilon,\xi_\ell} \dd\mu_0=1.

\[
\tilde u^{k}=(1-\varepsilon_b)u^{k,\mathrm{pre}}+\varepsilon_b\sum_{\ell}\omega_\ell\,\varpi_{\varepsilon,\xi_\ell},
\qquad \sum_\ell\omega_\ell=1.
\]

\tilde u^{k}=(1-\varepsilon_b)u^{k,\mathrm{pre}}+\varepsilon_b\sum_{\ell}\omega_\ell \varpi_{\varepsilon,\xi_\ell}, \qquad \sum_\ell\omega_\ell=1.

\[
\mathcal L_{\mu^{k,\mathrm{pre}}}+\mathfrak L(u^{k,\mathrm{pre}}\mid \Xi^{k},\theta^{k})
+ \tfrac{1}{2\tau}\HK^2(u^{k,\mathrm{pre}},u^{k-1,\mathrm{post}})
\le
\mathcal L_{\mu}+\mathfrak L(u\mid \Xi^{k},\theta^{k})
+ \tfrac{1}{2\tau}\HK^2(u,u^{k-1,\mathrm{post}}).
\]

\[
\big\|\mathbb K_{\mu^{k,\mathrm{post}},\mathcal H}-\mathbb K_{\mu^{k-1,\mathrm{post}},\mathcal H}\big\|_{\mathrm{op}}
\ \le\ C\,\HK\!\big(u^{k,\mathrm{post}},u^{k-1,\mathrm{post}}\big).
\]

\big\|\mathbb K_{\mu^{k,\mathrm{post}},\mathcal H}-\mathbb K_{\mu^{k-1,\mathrm{post}},\mathcal H}\big\|_{\mathrm{op}} \ \le\ C \HK \big(u^{k,\mathrm{post}},u^{k-1,\mathrm{post}}\big).

\[
\lambda(t)\ \lesssim\ C(t)\,\exp\big(-\Delta\mathcal A_{\rm eff}(t)/\varepsilon\big),
\]

\lambda(t)\ \lesssim\ C(t) \exp\big(-\Delta\mathcal A_{\rm eff}(t)/\varepsilon\big),

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\[
\dd\mu^\star(\xi)=Z^{-1}\,e^{-\beta \Phi(\Xi,\theta;\xi)}\,\dd\mu_0(\xi),
\qquad
Z=\int_{\Dcal}e^{-\beta \Phi(\Xi,\theta;\xi)}\,\dd\mu_0(\xi),\quad \beta=1/\tau_{\rm ent}.
\]

\dd\mu^\star(\xi)=Z^{-1} e^{-\beta \Phi(\Xi,\theta;\xi)} \dd\mu_0(\xi), \qquad Z=\int_{\Dcal}e^{-\beta \Phi(\Xi,\theta;\xi)} \dd\mu_0(\xi),\quad \beta=1/\tau_{\rm ent}.

\[
\mathcal R_{\rm RI}(\Delta z_{\mathcal R})
  = \int_{\Ocal}\alpha_s(x)\,|\Delta s(x)|\,\dd x + \alpha_\theta\,|\Delta\theta|,
\]

\mathcal R_{\rm RI}(\Delta z_{\mathcal R}) = \int_{\Ocal}\alpha_s(x) |\Delta s(x)| \dd x + \alpha_\theta |\Delta\theta|,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><msub><mi mathvariant="script">R</mi><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">I</mi></mrow></msub><mo stretchy="false">(</mo><mi>Δ</mi><msub><mi>z</mi><mrow><mi mathvariant="script">R</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∫</mo><mrow><mi>\Ocal</mi></mrow></msub><msub><mi>α</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><mo stretchy="false">|</mo><mi>Δ</mi><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mspace width="0.167em" /><mi>ⅆ</mi><mi>x</mi><mo>+</mo><msub><mi>α</mi><mi>θ</mi></msub><mspace width="0.167em" /><mo stretchy="false">|</mo><mi>Δ</mi><mi>θ</mi><mo stretchy="false">|</mo><mo>,</mo><mi>\]</mi></mrow></math>

\[
\mathfrak L_{\rm tot}(u,\nu\mid \Xi,\theta)
  := \mathfrak L(u\mid \Xi,\theta)+\mathfrak L_{\rm env}(\nu\mid \Xi,\theta)+\mathcal I(\mu,\nu),
\]

\[
\nu^{k}\in\operatorname*{arg\,min}_{\substack{\nu\ge0,\,\int v\,\dd\mu_0=1}}\
\Big\{\mathfrak L_{\rm env}(\nu\mid \Xi^{k},\theta^{k})+\mathcal I(\mu^{k,\mathrm{post}},\nu)+\tfrac{1}{2\tau}\HK^2(\nu,\nu^{k-1})\Big\}.
\]

\nu^{k}\in\operatorname*{arg min}_{\substack{\nu\ge0, \int v \dd\mu_0=1}}\ \Big\{\mathfrak L_{\rm env}(\nu\mid \Xi^{k},\theta^{k})+\mathcal I(\mu^{k,\mathrm{post}},\nu)+\tfrac{1}{2\tau}\HK^2(\nu,\nu^{k-1})\Big\}.

u_t=\frac{\dd\mu_t}{\dd\mu_0}\in L^1(\mu_0)\cap L\log L(\mu_0)

$k_{\mathcal H}(\cdot;\xi)\succeq c_{\rm ell}\,\mathsf P_{\mathcal H}$

k_{\mathcal H}(\cdot;\xi)\succeq c_{\rm ell} \mathsf P_{\mathcal H}

$\mathbb K^{k-1}_{\mathcal H}:=\mathbb K_{\mu^{k-1,\mathrm{post}},\mathcal H}(\Xi^{k-1},\theta^{k-1})$

$\mathrm{Cost}^{k}_{\rm birth}\ge c_{\rm birth}\,\varepsilon_b\sum_\ell\omega_\ell$

\mathrm{Cost}^{k}_{\rm birth}\ge c_{\rm birth} \varepsilon_b\sum_\ell\omega_\ell

$\mathrm{Cost}^k_{\rm birth}=-\Delta_{\rm tot}\ge c_{\rm birth}\varepsilon_b\sum\omega_\ell$

$t\mapsto \mathbb K_{\mu(t),\mathcal H}$

$u\mapsto \mathfrak L(u\mid \Xi,\theta)$

$-\nabla^2\log w(\xi)\preceq \kappa_0 I$

$u\mapsto\int\Phi\,u\,\dd\mu_0+\tau_{\rm ent}\int u\log u\,\dd\mu_0$

u\mapsto\int\Phi u \dd\mu_0+\tau_{\rm ent}\int u\log u \dd\mu_0

$\sum_k \mathrm{Cost}^k_{\rm birth}\le \mathcal G(z^0,u^0)-\inf\mathcal G$

\sum_k \mathrm{Cost}^k_{\rm birth}\le \mathcal G(z^0,u^0)-\inf\mathcal G

$c_{\rm ell}\mathsf P_{\mathcal H}\preceq \mathbb K_{\mu,\mathcal H}$

c_{\rm ell}\mathsf P_{\mathcal H}\preceq \mathbb K_{\mu,\mathcal H}

$\mathcal L^{\rm eff}_i(\Xi,\theta)=\mathcal L^{\rm eff}_j(\Xi,\theta)$

\mathcal L^{\rm eff}_i(\Xi,\theta)=\mathcal L^{\rm eff}_j(\Xi,\theta)

\partial_t\rho_{\rm id}+\mathrm{div} J_{\rm id}=R

$\dot{\Delta\mathcal A}_{\rm eff}(t)\ge -\varepsilon(t)$

\dot{\Delta\mathcal A}_{\rm eff}(t)\ge -\varepsilon(t)

$\Delta\mathcal A_{\rm eff}(t^+)-\Delta\mathcal A_{\rm eff}(t^-)\ge c_{\rm event}\ge0$

\lambda_H\int \mathrm{Huber}_\delta(|\nabla u|) \dd\mu_0

$\int u_{\rm ac}\log u_{\rm ac}\,\dd\mu_0+\sum_i a_i(\log a_i-1)$

\int u_{\rm ac}\log u_{\rm ac} \dd\mu_0+\sum_i a_i(\log a_i-1)

$c_{\rm ell}\mathsf P_{\mathcal H}\preceq \mathbb K^{k-1}_{\mathcal H}\preceq C\mathsf P_{\mathcal H}$


NONDUAL DYNAMICAL QUANTUM GEOMETRY

1.3

colorlinks=true,
linkcolor=blue,
citecolor=blue,
urlcolor=blue,
pdftitle= Nondual Dynamical Quantum Geometry: OPI, Fibered Bures--HK, Law-Dependent Local GKLS, Quasi-Local Causality, and Falsifiable Inequalities,
pdfauthor= K. Takahashi (ORCID: 0009-0004-4273-3365) ,
pdfkeywords= OPI, Bures metric, Hellinger--Kantorovich, GKLS, Lieb--Robinson, quasi-locality, operational no-signalling, JKO, EVI, gradient flows, Gamma-convergence, Mosco convergence, quantum information geometry, nondual

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

HK
W_2
Spec
Tr
id
d
G
F
H
D
A
L
supp
arg\,min
KL

TITLE:
Nondual Dynamical Quantum Geometry:\
OPI, Fibered Bures--HK, Law-Dependent Local GKLS,\
Quasi-Local Causality, and Falsifiable Inequalities

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

0009-0004-4273-3365

DATE:

We advance a monistic (nondual) framework in which ``law'' and ``field'' are two projections of a single [[EQ:eq0016]] -state [[EQ:eq0017]] via an operational physical idealisation (OPI) fixed-point algebra. The law sector evolves on unbalanced Hellinger--Kantorovich (HK) geometry; the field sector evolves on fiber Bures geometry, giving a fibered Bures--HK distance [[EQ:eq0018]] and a dynamic action equivalent to the static entropy--transport (ET) cost. We construct law-dependent local GKLS generators with exponential locality and prove conservative QMS generation and diamond-norm stability; open-system Lieb--Robinson (LR) bounds imply operational no-signalling through channel quasi-locality and data processing. Time-dependent JKO/EVI is well-posed on the EVI domain and extends to a coupled minimizing movement for the hybrid flow under explicit Lipschitz--curvature conditions. Readout-stable inequalities (ringdown width, entanglement growth) yield falsifiable, one-sided tests with calibrated constants.

Dialectical stance. Claims are stated for the continuum (possibly type III); proofs are anchored on lattice (finite-range, bounded generators) and transported to the continuum by [[EQ:eq0019]] /Mosco convergence and quasi-locality.

A [[EQ:eq0020]] -convergence reduction from a quantum relative-entropy law sector to HK is rigorous in finite-dimensional all-rank prototypes; the infinite-dimensional noncommutative case is given under shielding assumptions. An explicitly illustrable [[EQ:eq0021]] OPI example is included. For conceptual antecedents and motivation, see also TakahashiZenodo2025.

SECTION: Introduction and Vision

SUBSECTION: Vision and Physical Motivation

as a testable organizing principle.
Nondual dynamical quantum geometry (NDQG) posits a single [[EQ:eq0022]] from which ``field'' and ``law''
are projections obtained by central disintegration and an OPI fixed-point construction. Operational background-independence is enforced by a [[EQ:eq0023]] environment--law swap and a conditional expectation [[EQ:eq0024]] ; there is no free ambient structure.

HK on the law?
Beyond gentle regimes, creation/annihilation of law mass is unavoidable (updates, accept/reject moves, non-equilibrium transitions).
Unbalanced HK co-mingles transport and reaction, embedding Bayesian updating, stochastic acceptance, and coarse-graining costs into geodesic length; pure [[EQ:eq0025]] misses reaction.

Bures/QFI on fibers?
Bures is the minimal monotone metric; any CPTP readout is nonexpansive. This turns readout stability into geometry,
controls entanglement growth through a Lipschitz constant [[EQ:eq0026]] , and ties device calibration to falsifiable one-sided bounds.

local GKLS and LR?
The field dynamics is open relative to law/environment. Local GKLS with exponential locality admits LR cones, recovers an emergent light-cone,
and implies operational no-signalling on the OPI-fixed algebra via channel quasi-locality and data processing.

closed test loop.

[[EQ:eq0001]]

Slow-law phases recover GR+EFT; critical-law spurts transiently enlarge HK speed and tilt inequalities for ringdown widths and entanglement growth.

SUBSECTION: Dialectical resolution: continuum claims, lattice footing, transport

We explicitly separate claims (continuum/type III admissible) from footing (finite-range lattice, bounded generators).
Rigour is first obtained on the lattice and then transported to the continuum through [[EQ:eq0027]] /Mosco convergence of energies and open-system quasi-locality; see Theorem~thm:transfer.

SECTION: Monism and the Operational Physical Algebra

sec:monism

[Monism and central disintegration]ass:monism
[[EQ:eq0028]] is a separable [[EQ:eq0029]] -algebra with a normal faithful state [[EQ:eq0030]] and center [[EQ:eq0031]] .
There exist a probability measure [[EQ:eq0032]] on [[EQ:eq0033]] and a measurable field of normal states
[[EQ:eq0034]] such that [[EQ:eq0035]] .
We write [[EQ:eq0036]] for the effective center used in OPI.
We assume [[EQ:eq0037]] is standard Borel, so that disintegrations and measurable selections used below are available.

[Measurability convention]
Throughout, ``measurable selector'' means ``universally measurable selector'' unless stated otherwise.

[OPI-fixed physical algebra]def:OPI
Let [[EQ:eq0038]] denote the environment--law swap unitary. Define

[[EQ:eq0002]]

Then [[EQ:eq0039]] is a normal faithful conditional expectation onto [[EQ:eq0040]] (Lemma~lem:inner-normal).

[Continuum/net with split and standard form]ass:split
Let [[EQ:eq0041]] be a Haag--Kastler net on a separable Hilbert space with a faithful normal state [[EQ:eq0042]] in standard form. Assume the split property for [[EQ:eq0043]] , i.e.\ there exists a type I factor [[EQ:eq0044]] with
[[EQ:eq0045]] .

[Central direct integral and semifinite core]ass:core
There is a central direct integral [[EQ:eq0046]] .
For each fiber, let [[EQ:eq0047]] be the Haagerup semifinite core with faithful normal semifinite trace [[EQ:eq0048]] (see Haagerup1979 for the standard construction of the semifinite core).

[Inner implementability and normality]
lem:inner-normal
Under Assumptions~ass:split--ass:core, the environment--law swap [[EQ:eq0049]] is implemented fiberwise by measurable unitaries; hence
[[EQ:eq0050]] is a normal [[EQ:eq0051]] -automorphism. Therefore
[[EQ:eq0052]] is a normal faithful conditional expectation onto [[EQ:eq0053]] .

[Fiberwise flip via split or core]prop:flip
Under Assumptions~ass:split--ass:core, for [[EQ:eq0054]] -a.e.\ [[EQ:eq0055]] there is a spatial decomposition
[[EQ:eq0056]]
(locally via split) or inside the core [[EQ:eq0057]] .
Hence the flip [[EQ:eq0058]] is a normal [[EQ:eq0059]] -automorphism (core level), descending to [[EQ:eq0060]] on a conull set. Therefore [[EQ:eq0061]] is normal on [[EQ:eq0062]] , and [[EQ:eq0063]] is a normal faithful conditional expectation onto [[EQ:eq0064]] .

[No-Free-Background]
prop:nofree
On [[EQ:eq0065]] , any observable not invariant under the environment--law exchange is unphysical (undefined/preparation-dependent) outside a [[EQ:eq0066]] --null set on [[EQ:eq0067]] . Thus the physical algebra is the [[EQ:eq0068]] fixed-point algebra of the swap.

SECTION: Fibered Bures--HK Geometry

sec:fibered

[Base geometry]
ass:base-geometry
[[EQ:eq0069]] is a complete separable geodesic metric space and [[EQ:eq0070]] is Borel.
We assume [[EQ:eq0071]] up to null-sets, so the pullback cost separates points.

[Integrability and tightness]ass:tight
There exists a measurable purification section; the base has a finite second moment
[[EQ:eq0072]] for some [[EQ:eq0073]] , and the fibers have a finite Bures second moment
[[EQ:eq0074]]
for a fixed reference [[EQ:eq0075]] .
Base marginals are tight and fiber Bures-second-moment sublevels are uniformly bounded
on the energy sublevels considered below.

[Infinite-dimensional fibers and Bures/QFI]ass:BuresID
Fibers may be infinite-dimensional and non-faithful. Bures distance is defined via Uhlmann fidelity in the standard form (Tomita--Takesaki), hence independent of representations. The QFI speed is the lower semicontinuous relaxation (for the trace-norm topology on states) of the SLD speed computed on [[EQ:eq0076]] ; faithful regularization [[EQ:eq0077]] yields an [[EQ:eq0078]] -uniform integrable bound on sublevels (by [[EQ:eq0079]] ), ensuring domination and lower semicontinuity in the trace-norm topology.

[State space and equivalence]def:state-space
[[EQ:eq0080]] is the set of classes [[EQ:eq0081]] where [[EQ:eq0082]] is a measurable family of density operators; two representatives are identified if they coincide [[EQ:eq0083]] -a.e.

[QFI speed at rank-deficiency]def:QFI-rankdef
Let [[EQ:eq0084]] denote the symmetric logarithmic derivative (SLD) map; define

[[EQ:eq0003]]

using the Moore--Penrose pseudoinverse on [[EQ:eq0085]] .

[Support and infinite speed]
If [[EQ:eq0086]] has a component orthogonal to [[EQ:eq0087]] , then
[[EQ:eq0088]] , consistently penalising rank-creating motion unless accommodated by the birth rule.

[Static HK via entropy--transport]def:HK-static
With ground cost [[EQ:eq0089]] and unit penalties,

[[EQ:eq0004]]

[Unbalanced fibered Bures--HK (static)]def:fibmetric
Let [[EQ:eq0090]] be the set of optimal ET couplings for HK. Then

[[EQ:eq0005]]

[Existence of HK-optimal couplings]
Under Assumption~ass:base-geometry and l.s.c.\ of the cost, the ET problem admits minimizers, hence [[EQ:eq0091]] .

[Measurable selection for HK-optimal couplings]
lem:measurable-selection
Let [[EQ:eq0092]] be the set-valued map of ET-optimal couplings on a complete separable metric base (hence Polish).
Then there exists a universally measurable selector
[[EQ:eq0093]] .

[Selection via the base]
Equivalently, select an optimal ET plan [[EQ:eq0094]] on the Polish base [[EQ:eq0095]]
(by Kuratowski--Ryll-Nardzewski/Jankov--von Neumann), and then lift it to a selector on [[EQ:eq0096]] along a Borel version of the regular conditional probabilities of [[EQ:eq0097]] ; this uses the standard Borel property of [[EQ:eq0098]] . Disintegrations are taken with respect to [[EQ:eq0099]] and [[EQ:eq0100]] .

[Birth rule preserves l.s.c.]lem:birth-lsc
Assign newborn HK mass either (i) a fixed [[EQ:eq0101]] , or (ii) Bures parallel transport of the nearest-neighbor along a measurable geodesic. Then [[EQ:eq0102]] is l.s.c.\ under narrow/weak [[EQ:eq0103]] convergence, and the triangle inequality for [[EQ:eq0104]] is preserved.

[Bures contraction for basic readouts]lem:bures-contract
For partial traces and pinching maps [[EQ:eq0105]] ,
[[EQ:eq0106]] .
In particular [[EQ:eq0107]] for these readouts.

[On Bures contraction]
By fidelity monotonicity under CPTP maps Petz1996,NielsenChuang, Bures distance is contractive; we record this standard fact for completeness Uhlmann1976,Hubner1992.

[Identity of indiscernibles for [[EQ:eq0108]] ]lem:ioi
If [[EQ:eq0109]] , then [[EQ:eq0110]] and
[[EQ:eq0111]] for [[EQ:eq0112]] -a.e.\ [[EQ:eq0113]] (modulo Definition~def:state-space).

[Proof sketch]
From [[EQ:eq0114]] we have [[EQ:eq0115]] , hence [[EQ:eq0116]] .
Let [[EQ:eq0117]] be optimal with cost [[EQ:eq0118]] ; it is supported on the diagonal
(by [[EQ:eq0119]] from Assumption~ass:base-geometry) a.e.
Moreover, the fiber term satisfies
[[EQ:eq0120]] ,
so [[EQ:eq0121]] for [[EQ:eq0122]] -a.e.\ [[EQ:eq0123]] .

[Dynamic--static equivalence in the continuum]thm:dynstat
Under Assumptions~ass:base-geometry, ass:tight, and ass:BuresID with the fixed-birth rule [[EQ:eq0124]] ,
the fibered dynamic action equals the static ET cost.
Sketch. Lower bound: faithful approximation [[EQ:eq0125]] with uniform domination by finite Bures second moments and Fatou;
Upper bound: universally measurable selector [[EQ:eq0126]] and measurable Bures geodesics;
Fubini/Jensen give the reverse inequality. See Chizat2018,LieroMielkeSavare2018.

[Birth rule variants]
The same dynamic--static equality holds for measurable nearest-neighbor parallel-transport birth, provided the selector is universally measurable and the induced QFI energy is l.s.c.; the proof uses the same faithful regularization and Jensen convexity.

[Metricity, completeness, geodesicity]
thm:metricity
Under Assumptions~ass:base-geometry--ass:BuresID, [[EQ:eq0127]] is a metric on [[EQ:eq0128]] ; [[EQ:eq0129]] is complete and separable, and admits constant-speed geodesics whose speed equals the static cost.

SECTION: Time-Dependent JKO/EVI and the Hybrid Flow

sec:evi

Let [[EQ:eq0130]] be a proper l.s.c.\ functional on [[EQ:eq0131]] ; by the AGS theory of gradient flows AGS2008, split convexity along the base (HK) and fibers (Bures) yields EVI with parameter [[EQ:eq0132]] .

[Field-side metric induced by mobility]def:DistK
For fixed [[EQ:eq0133]] , the length distance on field variables [[EQ:eq0134]] associated with a positive-definite mobility tensor [[EQ:eq0135]] is

[[EQ:eq0006]]

with [[EQ:eq0136]] , [[EQ:eq0137]] .

[Split convexity, proof sketch]lem:split
Take product geodesics built by measurable HK--geodesics on the base and Bures geodesics on fibers along an HK-optimal coupling. The action is convex along each component; Fubini and measurability give additivity of curvature bounds, hence [[EQ:eq0138]] --convexity.

[Coupled minimizing movement and existence]
rem:coupledMM
Fix [[EQ:eq0139]] and define the alternating implicit scheme

[[EQ:eq0007]]

If [[EQ:eq0140]] and the [[EQ:eq0141]] -dependence of [[EQ:eq0142]] (denoted abstractly by [[EQ:eq0143]] )
are locally Lipschitz in [[EQ:eq0144]] with constants dominated by the EVI curvature [[EQ:eq0145]] on a common sublevel set, the step map is a contraction for small [[EQ:eq0146]] . Hence there exists a coupled curve of maximal slope [[EQ:eq0147]] obtained as [[EQ:eq0148]] , which solves the hybrid EDI and is unique on the EVI domain. A concrete sufficient condition is

[[EQ:eq0008]]

SECTION: Law-Dependent Local GKLS, Quasi-Locality and Operational Causality

sec:constraints

[Exponential locality and local cb--Lipschitz in the law]ass:local-cb
There exist local terms [[EQ:eq0149]] such that

[[EQ:eq0009]]

and, for some [[EQ:eq0150]] and all [[EQ:eq0151]] in the relevant sublevel set,

[[EQ:eq0010]]

[Bounded approximants and trace preservation]
ass:bounded-approx
There exist finite-volume, finite-range, exponentially local bounded generators
[[EQ:eq0152]] with [[EQ:eq0153]] ,
each generating a conservative QMS (CPTP). They satisfy
[[EQ:eq0154]] (uniform on compact times) and
[[EQ:eq0155]] in form sense on a common invariant core [[EQ:eq0156]] ;
if an unbounded Hamiltonian part [[EQ:eq0157]] is present, assume [[EQ:eq0158]] --relative bound [[EQ:eq0159]] on [[EQ:eq0160]] .

[Generation and conservative limit QMS]
prop:qms-limit
Under Assumptions~ass:bounded-approx and ass:local-cb,
[[EQ:eq0161]] generates a conservative [[EQ:eq0162]] --QMS. Moreover, for [[EQ:eq0163]] in a compact set,
[[EQ:eq0164]] .

[cb vs.\ diamond norms]
In finite dimensions [[EQ:eq0165]] .
We use [[EQ:eq0166]] for locality bounds of generators and
[[EQ:eq0167]] for operational stability of propagators EngelNagel,BratteliRobinson.

[Open-system quasi-locality]ass:QL
Finite-volume CPTP semigroups [[EQ:eq0168]] obey an open-system LR-type quasi-locality: for observables [[EQ:eq0169]] supported in [[EQ:eq0170]] and channels localized in [[EQ:eq0171]] with [[EQ:eq0172]] ,

[[EQ:eq0011]]

Uniformity holds along a refining net [[EQ:eq0173]] (bounded range, exponential decay) BarthelKliesch2012,SwekeEtAl2019.

[Operational no-signalling via quasi-locality]thm:opNS
Let [[EQ:eq0174]] be local CPTP readouts in spacelike-separated regions [[EQ:eq0175]] .
Then the joint outcome law [[EQ:eq0176]] satisfies the total-variation bound

[[EQ:eq0012]]

Idea. Approximate global channels by localized ones using Assumption~ass:QL, and apply the data processing inequality for statistical distances (contractivity of trace distance under CPTP maps NielsenChuang); pass to the thermodynamic limit. Since classicalization of a CPTP instrument is itself CPTP, the same bound holds for the trace distance of the corresponding classical states, and [[EQ:eq0177]] equals the trace distance for classical laws. For [[EQ:eq0178]] -type constants compare also Csisz\'ar--Kullback--Pinsker Csiszar1967.

[Explicit scaling to [[EQ:eq0179]] ]
Let lattice spacing be [[EQ:eq0180]] , local strength [[EQ:eq0181]] and localization rate [[EQ:eq0182]] . Standard open-system LR estimates give [[EQ:eq0183]] after unit/dimension matching.

SECTION: Quantum--to--HK Reduction on the Center

sec:qlaw

[Topology and tightness for [[EQ:eq0184]] -limit]
ass:Gamma-topology
[[EQ:eq0185]] are equi-coercive on the narrow topology of measures on [[EQ:eq0186]] , and sublevel sets are tight uniformly in [[EQ:eq0187]] on compact intervals.

[Discrete-time [[EQ:eq0188]] -convergence on a commutative center]
thm:qLDP-comm
Let [[EQ:eq0189]] be commutative and let quantum free-energy functionals [[EQ:eq0190]] satisfy
[[EQ:eq0191]] ,
where [[EQ:eq0192]] is the classical ET/KL functional on the center. Then the time-discrete scheme
[[EQ:eq0193]]
converges (as [[EQ:eq0194]] ) to the HK gradient flow of [[EQ:eq0195]] .

[Finite-dimensional, all-rank prototype]
In finite dimensions with full-rank law states, the central marginal of the quantum relative entropy [[EQ:eq0196]] [[EQ:eq0197]] -converges to the classical [[EQ:eq0198]] on [[EQ:eq0199]] ; thus the above time-discrete scheme converges to HK. See also monotonicity/l.s.c.\ of relative entropy under CPTP maps HiaiPetz1991.

[Central marginal: narrow l.s.c. and tightness]
lem:central-lsc
Let [[EQ:eq0200]] in weak [[EQ:eq0201]] topology with uniformly integrable central marginals.
Then the induced measures on [[EQ:eq0202]] converge narrowly and the central marginal of quantum relative entropy is l.s.c.\ along this convergence. Under Assumption~ass:Gamma-topology, equi-coercivity implies tightness.

[Counterexamples and shielding assumptions]
rem:shield
To avoid pathologies in infinite dimension: assume standard Borel central spectrum, hyperfinite or semifinite class for fibers, and faithful reference states on sublevels. These ensure the central marginal map is Borel and entropy is well-behaved.

SECTION: Lattice-to-Continuum Transport

sec:transport

[Lattice-to-continuum transfer]thm:transfer
Let [[EQ:eq0203]] be a mesh- [[EQ:eq0204]] discretization of [[EQ:eq0205]] with canonical embeddings.
Assume (i) the associated action functionals (equivalently, the squared distances)
[[EQ:eq0206]] [[EQ:eq0207]] -converge to [[EQ:eq0208]] on energy sublevels;
(ii) [[EQ:eq0209]] Mosco-converges to [[EQ:eq0210]] with common coercivity;
(iii) LR parameters [[EQ:eq0211]] under [[EQ:eq0212]] scaling;
(iv) generators satisfy Assumptions~ass:local-cb--ass:bounded-approx uniformly in [[EQ:eq0213]] ; and
(v) well-prepared initial data: [[EQ:eq0214]] in [[EQ:eq0215]] with
[[EQ:eq0216]] .
Then JKO/EVI solutions and quasi-locality/operational no-signalling bounds transfer to the continuum limit.

SECTION: Hybrid Energy, Dissipation, and Readout Inequalities

sec:inequalities

[Separated multi-birth safety constants]def:multi-birth
Fix a separation radius [[EQ:eq0217]] and a maximal local overlap [[EQ:eq0218]] for proposed newborn atoms.
Let [[EQ:eq0219]] denote the Poisson--disk packing constant giving a uniform
lower bound on the HK action increase per birth. Then any separated multi-birth move with weights [[EQ:eq0220]]
contributes at least [[EQ:eq0221]] to the discrete decrement.

[Dissipation functionals]
def:dissipations
For an admissible path [[EQ:eq0222]] with base velocity--reaction [[EQ:eq0223]] ,

[[EQ:eq0013]]

If [[EQ:eq0224]] satisfies GNS-detailed balance w.r.t.\ a steady state [[EQ:eq0225]] ,

[[EQ:eq0014]]

Otherwise we use the EDI surrogate
[[EQ:eq0226]] .

[Nonnegativity of [[EQ:eq0227]] ]
Under GNS-detailed balance, [[EQ:eq0228]] is a Dirichlet form and hence nonnegative.
In the EDI surrogate, convexity of the entropy yields
$ _ tau 0 (rho_t _t)- (rho_ t+tau _ t+tau ) tau 0 [[EQ:eq0229]] [[EQ:eq0230]] [[EQ:eq0231]] v_t 0 [[EQ:eq0232]] _t 0 [[EQ:eq0233]] D_ , 0 [[EQ:eq0234]] D_ , [[EQ:eq0235]] \ lambda>0\ [[EQ:eq0236]] alpha_mu>0 [[EQ:eq0237]] C_ read [[EQ:eq0238]] alpha_mu [[EQ:eq0239]] S_ EE (t) C_ read alpha_mu ( D_ ,mu (t)+ D_ , (t) ) [[EQ:eq0240]] C_ jump (t) 0 [[EQ:eq0241]] ( _ clock , _ EP ,C_ read ,alpha_mu) [[EQ:eq0242]] w [[EQ:eq0243]] _ (mu,mu') [[EQ:eq0244]] [[EQ:eq0245]] d_ Bures [[EQ:eq0246]] nu_ cen [[EQ:eq0247]] (Z_ eff ) [[EQ:eq0248]] [[EQ:eq0249]] (lambda_ int /2)\, ^2(mu, ) [[EQ:eq0250]] D_ , [[EQ:eq0251]] D_ , 0 [[EQ:eq0252]] D [[EQ:eq0253]] D_ ,mu [[EQ:eq0254]] D_ , [[EQ:eq0255]] AC([0,1]) [[EQ:eq0256]] H_0 [[EQ:eq0257]] H_0 [[EQ:eq0258]] <1 [[EQ:eq0259]] gamma(mu) [[EQ:eq0260]] [[EQ:eq0261]] 2 2 [[EQ:eq0262]] A=M_2( C) M_2( C) [[EQ:eq0263]] sigma_ x,y,z [[EQ:eq0264]] S(\,X Y\,)=Y X [[EQ:eq0265]] E_ OPI (A)= 12(A+ S A S^ ) [[EQ:eq0266]] E_ OPI [[EQ:eq0267]] A_ phys [[EQ:eq0268]] L^p [[EQ:eq0269]] ^ $-algebra,
Rep. Math. Phys. 9 (1976), 273--279.

Villani2003
C.~Villani, Topics in Optimal Transportation, AMS, 2003.
[[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]]

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Introduction and Vision

Vision and Physical Motivation

Dialectical resolution: continuum claims, lattice footing, transport

Monism and the Operational Physical Algebra

Fibered Bures–HK Geometry

Time-Dependent JKO/EVI and the Hybrid Flow

Law-Dependent Local GKLS, Quasi-Locality and Operational Causality

Quantum–to–HK Reduction on the Center

Lattice-to-Continuum Transport

Hybrid Energy, Dissipation, and Readout Inequalities

Notation summary (excerpt)

Appendix A: Measurable selection and gluing (technical)

Appendix B: Dynamic–static equivalence and triangle inequality

Appendix C: Generation examples for local GKLS

Appendix D: Unbalanced HK — existence, geodesics, l.s.c. (pointers)

Appendix E: A 2×2 illustrative OPI example

\[
\text{OPI + Bures--HK}\ \Rightarrow\ \text{GKLS}\oplus\HK
\ \Rightarrow\ \text{Lyapunov/Readout inequalities}\ \Rightarrow\ \text{calibrated, falsifiable tests}.
\]

\[
\mathfrak A_{\rm phys}:=\{A\in\mathfrak A:\ \mathsf S A \mathsf S^\dagger=A\},\qquad
E_{\rm OPI}(X):=\tfrac12\big(X+\mathsf S X \mathsf S^\dagger\big).
\]

\[
\|\dot\rho\|_{\rm QFI}^2=\tfrac14\,\Tr\!\big(\dot\rho\,\mathcal L_\rho^{-1}[\dot\rho]\big)
\]

\|\dot\rho\|_{\rm QFI}^2=\tfrac14 \Tr \big(\dot\rho \mathcal L_\rho^{-1}[\dot\rho]\big)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mo fence="false" stretchy="false">‖</mo><mover><mi>ρ</mi><mo>˙</mo></mover><msubsup><mo fence="false" stretchy="false">‖</mo><mrow><mi mathvariant="normal">Q</mi><mi mathvariant="normal">F</mi><mi mathvariant="normal">I</mi></mrow><mn>2</mn></msubsup><mo>=</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mspace width="0.167em" /><mi>\Tr</mi><mspace width="negativethinmathspace" /><mo minsize="1.2em" maxsize="1.2em">(</mo><mover><mi>ρ</mi><mo>˙</mo></mover><mspace width="0.167em" /><msubsup><mi mathvariant="script">L</mi><mi>ρ</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">[</mo><mover><mi>ρ</mi><mo>˙</mo></mover><mo stretchy="false">]</mo><mo minsize="1.2em" maxsize="1.2em">)</mo><mi>\]</mi></mrow></math>

\[
\|K_\mu^{-1}-K_\nu^{-1}\|_{\rm op}\le L_K\,\HK(\mu,\nu),\quad
|\mathcal F_\mu[z]-\mathcal F_\nu[z]|\le L_F\,\HK(\mu,\nu),\quad
L_K+L_F<\lambda .
\]

\|K_\mu^{-1}-K_\nu^{-1}\|_{\rm op}\le L_K \HK(\mu,\nu),\quad |\mathcal F_\mu[z]-\mathcal F_\nu[z]|\le L_F \HK(\mu,\nu),\quad L_K+L_F<\lambda .

\[
\mathcal L_\mu=\sum_X \mathcal L_{\mu,X},\qquad
\sum_X e^{\kappa_{\rm LR}\,\mathrm{diam}(X)}\,\|\mathcal L_{\mu,X}\|_{\rm cb}<\infty,
\]

\mathcal L_\mu=\sum_X \mathcal L_{\mu,X},\qquad \sum_X e^{\kappa_{\rm LR} \mathrm{diam}(X)} \|\mathcal L_{\mu,X}\|_{\rm cb}<\infty,

\[
\sum_X e^{\kappa_{\rm LR}\,\mathrm{diam}(X)}\,\|\mathcal L_{\mu,X}-\mathcal L_{\nu,X}\|_{\rm cb}
\ \le\ L_{\rm loc}\,\HK(\mu,\nu).
\]

\sum_X e^{\kappa_{\rm LR} \mathrm{diam}(X)} \|\mathcal L_{\mu,X}-\mathcal L_{\nu,X}\|_{\rm cb} \ \le\ L_{\rm loc} \HK(\mu,\nu).

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mo fence="false" stretchy="false">‖</mo><msubsup><mi>T</mi><mi>t</mi><mrow><mo stretchy="false">(</mo><mi>Λ</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>−</mo><msubsup><mi>T</mi><mi>t</mi><mrow><mo stretchy="false">(</mo><msup><mi>Λ</mi><mi>′</mi></msup><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo fence="false" stretchy="false">‖</mo><mo>≤</mo><mi>C</mi><mspace width="0.167em" /><mo fence="false" stretchy="false">‖</mo><mi>A</mi><mo fence="false" stretchy="false">‖</mo><mspace width="0.167em" /><msup><mi>e</mi><mrow><mo>−</mo><mi>μ</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mi>v</mi><mi>t</mi><mo stretchy="false">)</mo></mrow></msup><mo>.</mo><mi>\]</mi></mrow></math>

\|\mathsf P_t-\mathsf P_t^{(1)}\otimes \mathsf P_t^{(2)}\|_{\rm TV}\ \le\ C e^{-\mu(d-vt)}.

\[
\mathcal D_{\HK,\mu}(t):=\int (|v_t|^2+|\alpha_t|^2)\,\dd\mu_t,\qquad
\mathcal D_{\HK,\bar\mu}(t):=\int (|\bar v_t|^2+|\bar\alpha_t|^2)\,\dd\bar\mu_t.
\]

\mathcal D_{\HK,\mu}(t):=\int (|v_t|^2+|\alpha_t|^2) \dd\mu_t,\qquad \mathcal D_{\HK,\bar\mu}(t):=\int (|\bar v_t|^2+|\bar\alpha_t|^2) \dd\bar\mu_t.

\[
\mathcal E_\mu(f,f)\ \text{ is the Dirichlet form and }\
\mathcal D_{\mathrm{phys}}(t):=\mathcal E_{\mu(t)}\big(\sqrt{\rho_t}\big).
\]

\[
\mathfrak A_{\rm phys}
=\{A: \mathsf S A \mathsf S^\dagger=A\}
=\mathrm{span}\{\,I\otimes I,\ \sigma_x\otimes\sigma_x,\ \sigma_y\otimes\sigma_y,\ \sigma_z\otimes\sigma_z\,\},
\]

\mathfrak A_{\rm phys} =\{A: \mathsf S A \mathsf S^\dagger=A\} =\mathrm{span}\{ I\otimes I,\ \sigma_x\otimes\sigma_x,\ \sigma_y\otimes\sigma_y,\ \sigma_z\otimes\sigma_z \},

\varrho(\cdot)=\int \varrho_\zeta(\cdot) \nu_{\rm cen}(\dd\zeta)

$\{\mathfrak A(\mathcal O)\}_{\mathcal O}$

\{\mathfrak A(\mathcal O)\}_{\mathcal O}

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$\mathfrak A(\mathcal O_1)\subset \mathfrak N\subset \mathfrak A(\mathcal O_2)'$

\mathfrak A(\mathcal O_1)\subset \mathfrak N\subset \mathfrak A(\mathcal O_2)'

$\mathfrak A\cong\int^\oplus \mathfrak A(\zeta)\,\nu_{\rm cen}(\dd\zeta)$

\mathfrak A\cong\int^\oplus \mathfrak A(\zeta) \nu_{\rm cen}(\dd\zeta)

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$\widetilde{\mathfrak A}(\zeta)=\mathfrak A(\zeta)\rtimes_{\sigma^\varrho}\mathbb R$

\widetilde{\mathfrak A}(\zeta)=\mathfrak A(\zeta)\rtimes_{\sigma^\varrho}\mathbb R

$E_{\rm OPI}(X)=\tfrac12(X+\mathsf S X \mathsf S^\dagger)$

E_{\rm OPI}(X)=\tfrac12(X+\mathsf S X \mathsf S^\dagger)

$\mathfrak A(\zeta)\simeq \mathfrak M_{\rm env}(\zeta)\ \overline\otimes\ \mathfrak M_{\rm law}(\zeta)$

$\mathsf S_\zeta(X\otimes Y)=Y\otimes X$

\int d_{\rm Bures}^2(\rho_\zeta,\rho_\star) \mu(\dd\zeta)<\infty

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">p</mi><mi mathvariant="normal">p</mi></mrow><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

t\mapsto \int\|\dot\rho_{\zeta,t}\|_{\rm QFI}^2 \mu_t(\dd\zeta)

$d_{\rm Bures}(\mathsf R(\rho),\mathsf R(\sigma)) \le d_{\rm Bures}(\rho,\sigma)$

d_{\rm Bures}(\mathsf R(\rho),\mathsf R(\sigma)) \le d_{\rm Bures}(\rho,\sigma)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><msub><mi>d</mi><mrow><mi mathvariant="normal">B</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="sans-serif">R</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>,</mo><mi mathvariant="sans-serif">R</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≤</mo><msub><mi>d</mi><mrow><mi mathvariant="normal">B</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

\iint d_{\rm Bures}^2(\rho_\zeta,\rho'_{\zeta'}) \pi^\star(\dd\zeta,\dd\zeta')=0

$\sum_X e^{\kappa_{\rm LR} \mathrm{diam}(X)}\|{\cal L}^{(\Lambda,n)}_{\mu,X}\|_{\rm cb}\!\le\! C$

\sum_X e^{\kappa_{\rm LR} \mathrm{diam}(X)}\|{\cal L}^{(\Lambda,n)}_{\mu,X}\|_{\rm cb} \le  C

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mo fence="false" stretchy="false">‖</mo><msubsup><mrow><mi mathvariant="script">L</mi></mrow><mi>μ</mi><mrow><mo stretchy="false">(</mo><mi>Λ</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>−</mo><msubsup><mrow><mi mathvariant="script">L</mi></mrow><mi>μ</mi><mrow><mo stretchy="false">(</mo><mi>Λ</mi><mo stretchy="false">)</mo></mrow></msubsup><msub><mo fence="false" stretchy="false">‖</mo><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">b</mi></mrow></msub><mspace width="negativethinmathspace" /><mo>→</mo><mspace width="negativethinmathspace" /><mn>0</mn><mi>$</mi></mrow></math>

{\cal L}^{(\Lambda)}_\mu \to {\cal L}_\mu

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$v_{\max}(a)\!\asymp\! c_0 J_0/\kappa_0 \to c_{\mathrm{light}}$

v_{\max}(a) \asymp  c_0 J_0/\kappa_0 \to c_{\mathrm{light}}

$\Gamma\!-\!\lim_{\varepsilon\downarrow0}\mathcal F^\varepsilon_t=\mathcal F^{\rm cl}_t$

\Gamma - \lim_{\varepsilon\downarrow0}\mathcal F^\varepsilon_t=\mathcal F^{\rm cl}_t

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

\rho^\varepsilon_{\rm law} \to \rho_{\rm law}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><msubsup><mi>ρ</mi><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">w</mi></mrow><mi>ε</mi></msubsup><mspace width="negativethinmathspace" /><mo>→</mo><mspace width="negativethinmathspace" /><msub><mi>ρ</mi><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">w</mi></mrow></msub><mi>$</mi></mrow></math>

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c_{\rm birth} \varepsilon_b\sum_\ell \omega_\ell

$. See also \cite{FagnolaUmanita,CarlenMaas2017}.
\end{remark}

\begin{remark}[Reference-flow convention for $

$]
Unless stated otherwise, the reference law $

] Unless stated otherwise, the reference law

$ is exogenous.
For a static reference we set $

is exogenous. For a static reference we set

$.
When calibration requires a time-dependent reference (e.g.\ device or environment drift),
$

. When calibration requires a time-dependent reference (e.g.\ device or environment drift),

$ is computed along the chosen reference flow, selectable
universally measurably on the complete separable geodesic base.
\end{remark}

\begin{theorem}[Entanglement growth bound on $

$]
\label{thm:EE}
Assuming the log-Sobolev prerequisite (uniform $

] \label{thm:EE} Assuming the log-Sobolev prerequisite (uniform

$ on calibrated sublevels) and for readouts in Lemma~\ref{lem:bures-contract},
the entanglement growth admits a one-sided bound controlled by $

$ (details omitted).
\end{theorem}

\begin{remark}[Representative form]
A typical instance reads
$

$,
with constants calibrated as in Section~\ref{sec:inequalities}.
\end{remark}

\begin{remark}[Jump cost sign]
Under the separated multi-birth proposal and Definition~\ref{def:multi-birth}, the continuum limit satisfies
$

$ almost everywhere. Consequently, the Lyapunov decay is strict when births occur.
\end{remark}

\begin{remark}[Calibration and testing protocol]
Fix calibration runs to estimate $

$ with bootstrap CIs; lock hyperparameters before evaluation runs. For the ringdown width and EE tests, control FPR at 1--5\% via pre-registered windows $

$ and one-sided tests with multiplicity correction across overtones.
\end{remark}
\clearpage
% =========================================================
\section*{Notation summary (excerpt)}
\begin{itemize}
\item $

$ (set of optimal ET couplings for HK); $

$: reference law; appears in the free-energy coupling $

: reference law; appears in the free-energy coupling

$: common invariant core for (unbounded) generators; not to be confused with
dissipations $

: common invariant core for (unbounded) generators; not to be confused with dissipations

$: absolutely continuous curves on the unit interval.
\end{itemize}

% =========================================================
\appendix

\section{Appendix A: Measurable selection and gluing (technical)}
We use Kuratowski--Ryll-Nardzewski on standard Borel bases (closed-valued multifunctions) and Jankov--von Neumann for analytic graphs \cite{KuratowskiRyll1965,Kechris1995,CastaingValadier}. Bures geodesic selectors are universally measurable in finite dimension via continuous Uhlmann horizontal lifts followed by projection; nearest-neighbor selectors are Borel on Polish bases, and ties are resolved lexicographically.

\section{Appendix B: Dynamic--static equivalence and triangle inequality}
Lower bound: faithful approximation and Fatou l.s.c.\ for the QFI energy; upper bound: HK dynamic formulation \cite{Chizat2018,LieroMielkeSavare2018} plus measurable fiber geodesics and Jensen convexity. Triangle inequality: measurable gluing of optimal couplings and concatenation.

\section{Appendix C: Generation examples for local GKLS}
On a cubic lattice with $

$ is ensured by Kato--Rellich \cite{ReedSimonII}; cb--Lipschitz dependence follows if the rates $

$ are HK-Lipschitz. Bounded approximants: finite volume $

are HK-Lipschitz. Bounded approximants: finite volume

$, finite range, coefficient truncations; convergence by Trotter--Kato \cite{KatoBook1976,EngelNagel}. For dilations of time-dependent generators, see HP-QSDE \cite{HudsonParthasarathy1984} and time-dependent existence results \cite{LindsayWills2007}.

\section{Appendix D: Unbalanced HK --- existence, geodesics, l.s.c. (pointers)}
\noindent\emph{Pointers.} In \cite{LieroMielkeSavare2018}, see Thm.~3.3 (existence of minimizers), Prop.~5.4 (geodesics), Thm.~7.20 (l.s.c.). In \cite{Chizat2018}, see Thm.~3.1 (dynamic formulation), Thm.~5.6 (equivalence with ET), Cor.~5.9 (geodesic structure).

\section{Appendix E: A $

$.
Then the fixed-point algebra is

EQPH_eq0015_PH

the Bell-basis fixed-point. Here $

. Then the fixed-point algebra is EQPH_eq0015_PH the Bell-basis fixed-point. Here

$ is an explicit normal faithful conditional expectation onto $

is an explicit normal faithful conditional expectation onto

$.

% =========================================================
\begin{thebibliography}{99}\setlength{\itemsep}{2pt}

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Quasi-locality and efficient simulation of Markovian quantum dynamics,
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E.~A.~Carlen and J.~Maas,
Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance,
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C.~Castaing and M.~Valadier,
\emph{Convex Analysis and Measurable Multifunctions}, Springer, 1977.

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K.-J.~Engel and R.~Nagel,
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