OPI GAUGE DYNAMICS: Fibered Bures--HK Geometry and Time-Dependent JKO/EVI (Calibrated, Axiomatized, and Testable)

* . theorem
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pdftitle= OPI Gauge Dynamics (Dynamic-First): Fibered Bures--HK Geometry and Time-Dependent JKO/EVI

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theorem
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\1 1
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\|#1 \|_ TV
\|#1 \|_
d_ \! (#1,#2 )
_delta\! (#1,#2 )
d_ fib
J % Choi state
EVI
I_ SLD % SLD QFI for Choi states
_+\! (#1 )
muLR mu_ LR % LR decay rate (avoid clash with measure mu)
^ mass _ % massful couplings induced by ET

TITLE: -1em

OPI Gauge Dynamics :
Fibered Bures--HK Geometry and
Time-Dependent JKO/EVI
(Calibrated, Axiomatized, and Testable)

AUTHOR: K.~Takahashi
[[EQ:eq0013]]

On bases covered by ass:dynHK, the static entropy--transport
formulation coincides with def:HK-dynamic. In particular,
for two atoms at distance [[EQ:eq0032]] ,
[[EQ:eq0033]] .
Citations: dynamic--static equivalence via cone-lift and superposition
(cf.\ LieroMielkeSavare2018 for the dynamic/ET equivalence and ChizatPeyreSchmitzerVialard2018 for the Benamou--Brenier-type representation; exact theorem numbering differs across versions).

note. In this work all dynamic HK statements are intended
to hold on Euclidean domains (via cone-lift) and on finite metric graphs
(via discrete divergence) covered by ass:dynHK.
Extensions to general metric spaces are beyond our present scope.On Euclidean domains with suitable boundary conditions (no-flux) or on finite graphs, the divergence term integrates to zero in the mass balance.

[Isotypic type-I fibers (explicit scope)]ass:trivial-strong
There exists a separable [[EQ:eq0034]] such that [[EQ:eq0035]] for [[EQ:eq0036]] -a.e.\ [[EQ:eq0037]] , with a measurable family of normal [[EQ:eq0038]] -isomorphisms [[EQ:eq0039]] . Bures angles are computed after this identification.

[Measurable unitary cocycle]lem:meas-unitary
Under ass:trivial-strong, any two measurable trivializations
[[EQ:eq0040]] differ by a measurable unitary field [[EQ:eq0041]] .
Since Bures angles and SLD-QFI are unitarily invariant, [[EQ:eq0042]] is independent
of the chosen trivialization.

[CJ fibers and SLD-QFI convention]ass:CJ
Fibers are finite-dimensional or [[EQ:eq0043]] with separable [[EQ:eq0044]] so Choi states [[EQ:eq0045]] are well-defined through a fixed faithful reference (or the GNS standard form). Convention: [[EQ:eq0046]] denotes the SLD QFI of [[EQ:eq0047]] ; we use [[EQ:eq0048]] hence [[EQ:eq0049]] and the fiber action coefficient is [[EQ:eq0050]] .

[Fiber datum as Choi states]def:fiber-choi
Under ass:trivial-strong, the fiber datum at [[EQ:eq0051]] and time [[EQ:eq0052]] is the Choi state
[[EQ:eq0053]] of the [[EQ:eq0054]] -fiber component [[EQ:eq0055]] of the OPI map [[EQ:eq0056]] . The field [[EQ:eq0057]] is measurable and unitarily invariant by lem:meas-unitary.

[Scope: type I/finite dimension]
We focus on finite-dimensional or type-I fibers where the Choi state is trace-class. Extensions to type-II/III factors require relative Choi constructions and lie beyond the scope of this work.

[Post-processing class (experiment contractivity)]ass:post
Admissible post-processings act on outputs as CPTP maps [[EQ:eq0058]] on Choi states:
[[EQ:eq0059]] (ancilla addition, output CPTP composition, pinching, partial trace). Entangling superchannels, input-side preprocessing, or partial transpose are outside this class.

SECTION: OPI Projection as a Normal Faithful Conditional Expectation

sec:opi
[OPI projection (standard form; recap)]
We recall Def.~def:opi: for a self-adjoint unitary [[EQ:eq0060]] one sets [[EQ:eq0061]] with [[EQ:eq0062]] , a unital, completely positive, idempotent and normal projection on [[EQ:eq0063]] .

[Normal, faithful conditional expectation]prop:condexp
Under def:opi, [[EQ:eq0064]] is a normal faithful conditional expectation onto [[EQ:eq0065]] .

Idempotence and complete positivity are immediate. Normality follows from [[EQ:eq0066]] -weak continuity of [[EQ:eq0067]] . If [[EQ:eq0068]] then [[EQ:eq0069]] , hence [[EQ:eq0070]] (faithfulness). The range is the fixed-point subalgebra. See Tomiyama and Takesaki Tomiyama1957,Takesaki2002.

SECTION: Fibered Bures--HK Metric (Dynamic-First) and Calibrated Action

sec:fib

SUBSECTION: Dynamic definition, static recovery

By def:HK-dynamic we define [[EQ:eq0071]] dynamically; the static ET form and two-point closed form eq:HK2pt are recovered via thm:HK-recovery: on the unit time interval,

[[EQ:eq0001]]

PARAGRAPH: Massful couplings via cone-lift (rigorous).

Let [[EQ:eq0072]] be the cone-lift of [[EQ:eq0073]] and
[[EQ:eq0074]] an optimal static ET plan on [[EQ:eq0075]] with radial
coordinates [[EQ:eq0076]] . We define the massful endpoint coupling by

[[EQ:eq0014]]

which coincides with [[EQ:eq0077]] on finite supports.
We denote by [[EQ:eq0078]] the set of all such [[EQ:eq0079]] induced by
[[EQ:eq0080]] .

PARAGRAPH: Static ET coupling set [[EQ:eq0081]] (massful) and fiber cost.

In the static ET formulation, the fiber cost uses massful couplings [[EQ:eq0082]] :

[[EQ:eq0002]]

where [[EQ:eq0083]] etc.\ are Choi states.

SUBSECTION: Calibrated action

The calibrated dynamic action (base [[EQ:eq0084]] fiber) reads

[[EQ:eq0003]]

SUBSECTION: Measurability, l.s.c., and Bochner

sec:measurable

PARAGRAPH: Topologies and measurability (JvN/KRN ready).

Equip [[EQ:eq0085]] with the trace-norm topology (Polish). The Bures angle is continuous and its square is l.s.c. Uhlmann geodesics arise via purifications in a separable Hilbert space; the associated multimap has closed values and a measurable graph; hence Kuratowski--Ryll-Nardzewski yields measurable selections. Choi maps are Borel under ass:trivial-strong.

[Closed-valued selectors: JvN/KRN]lem:meas
[[EQ:eq0086]] has an analytic graph with closed values (tightness and l.s.c.\ of ET functionals; cf.\ LieroMielkeSavare2018). Therefore, by Jankov--von Neumann there exist measurable selectors [[EQ:eq0087]] . For fibers, the Uhlmann geodesic multifunction is closed-valued (locally single-valued, globally possibly multivalued) in the purification bundle; KRN yields measurable sections which push down to density paths.For concrete entry points in Liero--Mielke--Savar\'e (2018, J. Math. Pures Appl.), see, e.g., Thm.~7.21 and Prop.~7.24 (journal numbering) on analytic graphs/tightness of optimal ET plans, and Thm.~8.3 on superposition; in arXiv v3 these correspond approximately to Thm.~7.24/Prop.~7.27/Thm.~8.5. Numbering varies by version.

[Bochner integrability with rank-drop]lem:bochner
If [[EQ:eq0088]] is AC in HK on the base and AC in Bures on fibers along a measurable family [[EQ:eq0089]] , then [[EQ:eq0090]] is locally integrable and l.s.c.; rank-drop is admissible by continuous extension of Uhlmann geodesics. Here [[EQ:eq0091]] denotes the common fiber dimension.

In practice we either restrict to full-rank neighborhoods (cf.\ prop:fiberconvex)
or use [[EQ:eq0092]] -regularization of states to control the SLD divergence near rank drop;
the l.s.c.\ then follows by monotone convergence. This is compatible with the birth rules in Lemma~lem:lsc since the [[EQ:eq0093]] -regularization commutes with zero-mass anchoring and preserves the l.s.c.\ limit.

[Lower semicontinuity for the fiber term]lem:lsc-fiber
Let [[EQ:eq0094]] narrowly converge to
[[EQ:eq0095]] . Suppose [[EQ:eq0096]] [[EQ:eq0097]] -a.e.\ with a uniform integrable bound.
Then

[[EQ:eq0015]]

[Sufficient domination for lem:lsc-fiber]
A uniform bound follows if fiber paths remain in a full-rank Bures ball with
[[EQ:eq0098]] or if one employs the standard [[EQ:eq0099]] -regularization
[[EQ:eq0100]] and sends [[EQ:eq0101]] ,
using the monotonicity of the Bures angle.

PARAGRAPH: Metric--chord and speed constants.

Along any AC fiber path [[EQ:eq0102]] ,

[[EQ:eq0016]]

with [[EQ:eq0103]] taken as the angle. These tie time-actions to endpoint costs without loss.

SUBSECTION: Dynamic--static equivalence and diagonal-split [[EQ:eq0104]] static coupling

[Dynamic--static equivalence]thm:dynstat
For any endpoints,

[[EQ:eq0017]]

References: cone-lift dynamic representation and superposition (cf.\ LieroMielkeSavare2018,ChizatPeyreSchmitzerVialard2018).

[Diagonal-split implies static coupling (Gamma-liminf)]thm:diag2static
Let [[EQ:eq0105]] be the discrete energy from eq:jko1–eq:jko2-new
with piecewise-constant interpolants and assume sublevel tightness plus the a priori
mass bound. Then along [[EQ:eq0106]] ,
[[EQ:eq0107]] ,
where [[EQ:eq0108]] is the static cost in eq:dfib. Consequently,
the diagonal fiber updates recover a massful endpoint coupling in the limit.

[Ideas for thm:dynstat,thm:diag2static]
“ [[EQ:eq0109]] ”: measurable disintegration of a static [[EQ:eq0110]] into HK and Uhlmann geodesics (lem:meas), integrability (lem:bochner), Jensen/Fatou.
“ [[EQ:eq0111]] ”: cone lift on finite bases gives a conservative continuity equation; a superposition principle yields a measure on AC flow lines. Attach along each flow line a fiber Bures-AC path whose length dominates the endpoint angle. Averaging over flow lines produces a coupling [[EQ:eq0112]] of endpoints with fiber cost bounded by the time-action.
For the diagonal split, lem:lsc-fiber and the [[EQ:eq0113]] -liminf passage control the fiber term; base follows from HK l.s.c.\ and tightness. General bases follow by discretization.

[Triangle and completeness]thm:metric
[[EQ:eq0114]] is a length metric. Concatenating [[EQ:eq0115]] -minimizing AC curves makes the action additive under concatenation; since [[EQ:eq0116]] , the induced length is subadditive, giving the triangle inequality. Completeness holds under ass:CJ and the mass bounds below. (Local existence of geodesics holds within the Bures normal neighborhood used in prop:fiberconvex.)

PARAGRAPH: A priori and averaged mass bounds (completeness).

Let [[EQ:eq0117]] . From
[[EQ:eq0118]] and Cauchy--Schwarz,

[[EQ:eq0018]]

Equivalently,

[[EQ:eq0019]]

[Averaged mass control]lem:mass-avg
Let [[EQ:eq0119]] .
Then for all [[EQ:eq0120]] ,
[[EQ:eq0121]] .

Integrating this differential inequality from [[EQ:eq0122]] to [[EQ:eq0123]] and applying
Cauchy--Schwarz immediately yields the averaged bound in Lemma~lem:mass-avg.
These estimates, plus CJ compactness on fibers, yield completeness (Appendix~app:completeness).

[Birth-rule stability]lem:lsc
Both birth rules (fixed [[EQ:eq0124]] or measurable parallel transport) preserve l.s.c.\ of eq:dfib and the triangle inequality (zero-mass anchoring + Fatou).

[Experiment contractivity]
Within ass:post, Bures angles on Choi states are monotone:
[[EQ:eq0125]] . Thus the fiber term contracts under admissible readouts.

SECTION: Choi-SLD QFI: Local Speed, TV, and Integral Diamond

sec:choiqfi
[Local Bures angular speed]prop:localspeed
For differentiable [[EQ:eq0126]] ,

[[EQ:eq0020]]

[Finite-time Bures [[EQ:eq0127]] TV]
thm:buresTV
For any admissible [[EQ:eq0128]] and [[EQ:eq0129]] ,

[[EQ:eq0021]]

Hence, by Fuchs--van de Graaf FuchsGraaf1999,
[[EQ:eq0130]] ,
and any fixed local POVM yields a TV distance bounded by the RHS.In practice, Bures angles may be lower bounded from classical TV measurements after a POVM via Fuchs--van de Graaf, avoiding full tomography.

[Time-dependent Liouvillians: integral diamond bound]thm:diamond
Let [[EQ:eq0131]] be propagators of strongly measurable GKLS generators [[EQ:eq0132]] with [[EQ:eq0133]] and similarly for [[EQ:eq0134]] . Assume Kato stability / bounded cb-envelopes so that time-ordered exponentials exist and are strongly continuous. Then (see, e.g., Pazy1983,Davies1980,Kato1953)

[[EQ:eq0022]]

and [[EQ:eq0135]] .

The bound implicitly assumes well-defined propagators (e.g.\ via time-ordered exponentials)
under the stated integrability of the cb-norm envelopes; see also Pazy Pazy1983, Davies Davies1980, and Kato Kato1953 for classic evolution families.

SECTION: Split-JKO and Local EVI (lambda) on [[EQ:eq0136]]

sec:jko

SUBSECTION: Energy (state-only), coercivity, time regularity

[[EQ:eq0004]]

We assume [[EQ:eq0137]] is l.s.c.\ and mildly coercive (e.g.\ relative entropy plus a small Bures-angle [[EQ:eq0138]] penalty), ensuring existence of minimizers.

[Split convexity, time regularity, compact sublevels]ass:convex
There exist [[EQ:eq0139]] such that [[EQ:eq0140]] is [[EQ:eq0141]] -convex along [[EQ:eq0142]] -geodesics, and [[EQ:eq0143]] is locally [[EQ:eq0144]] -convex along Bures geodesics (normal neighborhoods; global if [[EQ:eq0145]] adds relative entropy). Moreover, [[EQ:eq0146]] and [[EQ:eq0147]] admits locally absolutely continuous (time-BV) representatives on sublevel sets with [[EQ:eq0148]] . Sublevel sets of [[EQ:eq0149]] are relatively compact w.r.t.\ narrow convergence on the base and Bures convergence on the fibers, ensuring existence of minimizing-movement limits as [[EQ:eq0150]] .

PARAGRAPH: Existence of minimizers.

By l.s.c.\ of [[EQ:eq0151]] , continuity of the Bures angle, and l.s.c.\ (with mild coercivity) of [[EQ:eq0152]] , each subproblem below admits a minimizer. Uniform energy bounds along the discrete trajectory yield tightness/compactness and allow passing to the continuous-time limit via the AGS minimizing-movement scheme AGS2008.

SUBSECTION: Split steps with diagonal

fiber coupling
Given [[EQ:eq0153]] ,

[[EQ:eq0005]]

where the diagonal fiber distance forbids label reshuffling:

[[EQ:eq0006]]

Design logic: discrete short-time steps fix labels; the static optimal coupling of [[EQ:eq0154]] re-emerges in the limit by flow-line gluing (thm:diag2static).

[Discrete split-EVI with calibration and Young absorption]lem:scalingJKO-strong
Let [[EQ:eq0155]] and [[EQ:eq0156]] be competitors for eq:jko1 and eq:jko2-new. Then

[[EQ:eq0007]]

With [[EQ:eq0157]] and Young,

[[EQ:eq0023]]

the first term is absorbed into the left of eq:discEVIbase-strong (fixed [[EQ:eq0158]] ), leaving a uniform additive decrement. Summing and sending [[EQ:eq0159]] yields a continuous EVI with [[EQ:eq0160]] .

at a glance. The angular [[EQ:eq0161]] -rule gives the fiber action coefficient [[EQ:eq0162]] ; therefore [[EQ:eq0163]] enters as [[EQ:eq0164]] . The [[EQ:eq0165]] term arises from base-Lipschitz mixing via a Young absorption and survives the limit (cf.\ AGS2008).

[Base convexity and constants (model dependent)]prop:baseconvex
If [[EQ:eq0166]] is affine and [[EQ:eq0167]] is convex along [[EQ:eq0168]] -geodesics, then
[[EQ:eq0169]] . For two points (distance [[EQ:eq0170]] ),

[[EQ:eq0008]]

and on finite metric graphs [[EQ:eq0171]] with edge lengths [[EQ:eq0172]] ,

[[EQ:eq0009]]

dependence. The coefficient [[EQ:eq0173]] is a quadratic-form curvature depending on [[EQ:eq0174]] and locality; it is not universal and should be treated as an effective, model-dependent constant (estimable in experiments). A short proof of eq:taugraph is given in Appendix~app:graph.

[Fiber convexity (local; eigenvalue floor)]prop:fiberconvex
Fix a full-rank reference with [[EQ:eq0175]] .
On the Bures normal neighborhood determined by [[EQ:eq0176]] , [[EQ:eq0177]] is geodesically [[EQ:eq0178]] -convex. For global convexity, augment with relative entropy. All EVI statements are restricted to this local domain.

[Scaling of [[EQ:eq0179]] ]
On [[EQ:eq0180]] full-rank states with [[EQ:eq0181]] ,
the Bures (SLD) metric tensor is uniformly equivalent on the Bures ball
of radius [[EQ:eq0182]] to a fixed inner product with constants depending only on [[EQ:eq0183]] .
Hence one may take [[EQ:eq0184]] for [[EQ:eq0185]] below the
injectivity radius around [[EQ:eq0186]] .

[Local EVI (lambda), uniqueness, contraction (local domain)]thm:evi
Under ass:convex, prop:baseconvex,prop:fiberconvex (local domain), and lem:scalingJKO-strong, the continuous-time limit of eq:jko1--eq:jko2-new solves a local EVI (lambda) gradient flow with

[[EQ:eq0010]]

Where [[EQ:eq0187]] (HK curvature domain, Bures normal neighborhood), the flow is unique and contractive in [[EQ:eq0188]] .

[t]

Constants in eq:lambda with examples (two-point and a numeric instance), and whether they are universal or model-dependent.
tab:constants
@ l l l X l@

Symbol & Units & Origin & Example values / dependencies & Universal vs.\ model-dependent \

[[EQ:eq0189]] & [[EQ:eq0190]] & HK entropy convexity & two-point: [[EQ:eq0191]] ; e.g.\ [[EQ:eq0192]] & Universal (geometric) \
[[EQ:eq0193]] & [[EQ:eq0194]] & commutator curvature & depends on [[EQ:eq0195]] and locality of [[EQ:eq0196]] & Model-dependent \
[[EQ:eq0197]] & [[EQ:eq0198]] & local fiber convexity & [[EQ:eq0199]] & local geometric (depends on [[EQ:eq0200]] ) \
[[EQ:eq0201]] & [[EQ:eq0202]] & [[EQ:eq0203]] Lipschitz & [[EQ:eq0204]] density bounds, dimension, interaction range & Model-dependent \

SECTION: LR-to-TV Bridge (Assumed LR): Operational Form

sec:tvbridge
Let [[EQ:eq0205]] be local on a lattice [[EQ:eq0206]] with finite range or exponential decay:
[[EQ:eq0207]] ,
and [[EQ:eq0208]] .

PARAGRAPH: Assumptions (explicit).

[leftmargin=1.7em]
- finite range or exponentially decaying interactions ( [[EQ:eq0209]] );
- finite spatial dimension (boundary scaling exponent [[EQ:eq0210]] geometric);
- local dissipators (Lindblad operators of bounded range);
- finite-volume GKLS approximants with uniform cb-bounds and strong convergence to an infinite-volume QMS (Trotter--Kato/Davies type).

[Finite-volume GKLS approximants]ass:QMS
There exists [[EQ:eq0211]] such that for each [[EQ:eq0212]] ,
[[EQ:eq0213]] is a GKLS generator on [[EQ:eq0214]] with
[[EQ:eq0215]] and uniform GKLS structure. Finite-volume propagators converge (Trotter--Kato/Davies type) to a conservative [[EQ:eq0216]] -QMS; the infinite-volume propagator obeys the cb-envelope.

[Operational no-signaling in TV under an LR bound]prop:tv
Assume a Lieb--Robinson (LR)-type estimate with constants [[EQ:eq0217]] for the local GKLS dynamics. Then for regions at distance [[EQ:eq0218]] and any local POVMs [[EQ:eq0219]] ,

[[EQ:eq0024]]

with [[EQ:eq0220]] , [[EQ:eq0221]] , and [[EQ:eq0222]] depending on dimension/locality class. The constants are model dependent and may be estimated experimentally (three-step recipe below).

PARAGRAPH: Estimating [[EQ:eq0223]] in practice (three-step).

(i) Short-time light-cone fit for [[EQ:eq0224]] : track local correlator spread.
(ii) Distance sweep for [[EQ:eq0225]] : vary separation [[EQ:eq0226]] at fixed [[EQ:eq0227]] and fit exponential decay.
(iii) Boundary scaling for [[EQ:eq0228]] : regress residual scaling against [[EQ:eq0229]] .
Tomography lightening: use partial/process tomography on stabilizer/eigendirection sets.

SECTION: Ring-Down Inequality: Units and Statistics

sec:ring
By prop:localspeed and post-processing contractivity,

[[EQ:eq0025]]

Define the angular ring-down width by

[[EQ:eq0026]]

Then, introducing a readout contraction [[EQ:eq0230]] and a dimensionless gain [[EQ:eq0231]] ,

[[EQ:eq0011]]

Each summand has units [[EQ:eq0232]] .
For [[EQ:eq0233]] comparisons, control family-wise error by Bonferroni or Holm; if noise is non-Gaussian, bootstrap CIs are robust. Pre-register the minimal detectable effect and [[EQ:eq0234]] ; [[EQ:eq0235]] .

SECTION: Worked Example: [[EQ:eq0236]]

M2 direct sum C with Two-Point HK Base
sec:example

PARAGRAPH: Scope for the example.

To fit ass:trivial-strong, we identify the scalar fiber [[EQ:eq0237]]
with [[EQ:eq0238]] via the embedding [[EQ:eq0239]] ; hence both fibers lie
in [[EQ:eq0240]] and Bures angles are computed in this common type-I setting.

Let [[EQ:eq0241]] ; base labels [[EQ:eq0242]] at positions [[EQ:eq0243]] with distance [[EQ:eq0244]] ; base measures [[EQ:eq0245]] , [[EQ:eq0246]] .
Fibers: [[EQ:eq0247]] , [[EQ:eq0248]] ; similarly for primes.
Set [[EQ:eq0249]] .

SUBSECTION: HK ET and KKT with [[EQ:eq0250]] (including regime test)

[[EQ:eq0027]]

where [[EQ:eq0251]] , [[EQ:eq0252]] , and
[[EQ:eq0253]] .
KKT multipliers [[EQ:eq0254]] satisfy

[[EQ:eq0028]]

Regime test (with [[EQ:eq0255]] ):

[[EQ:eq0029]]

SUBSECTION: Massful coupling and fiber cost by regime

The fiber cost uses the massful coupling [[EQ:eq0256]] induced by ET:
[[EQ:eq0257]] .
Let [[EQ:eq0258]] . Then:
[leftmargin=1.6em]
- Diagonal-dominant ( [[EQ:eq0259]] ):
[[EQ:eq0260]] , so

[[EQ:eq0012]]

- Reaction-dominant ( [[EQ:eq0261]] ):
mass flows off-diagonally and the fiber term becomes
[[EQ:eq0262]] , which only vanishes if
[[EQ:eq0263]] . In general it does not vanish.

PARAGRAPH: Numeric check (interior threshold).

For [[EQ:eq0264]] ,

[[EQ:eq0030]]

hence the reaction-dominant threshold is [[EQ:eq0265]] .

SECTION: Minimal Experimental Protocol (Dynamic-First, Pre-Registered)

sec:protocol
[leftmargin=2.1em,label=(P *)]
- System: 1D spin chain or superconducting circuit with finite-range Lindbladian. Choose an OPI gauge [[EQ:eq0266]] (local parity/swap); list observables in [[EQ:eq0267]] .
- Calibrate [[EQ:eq0268]] (base, dynamic): two-site HK experiment with known [[EQ:eq0269]] ; switch reaction/transport modes; fit dynamic action to recover the [[EQ:eq0270]] static law via thm:HK-recovery.
- Calibrate [[EQ:eq0271]] (fiber, dynamic): run a reference channel path, estimate [[EQ:eq0272]] and Bures angles; enforce the dynamic inequality to fix [[EQ:eq0273]] (action coefficient [[EQ:eq0274]] ).
- Readout contraction [[EQ:eq0275]] : verify Bures-angle monotonicity under output CPTP (pinching/partial trace).Bures angles can be bounded from below by classical TV after a POVM via Fuchs--van de Graaf, easing tomography.
- Split inference: Run eq:jko1--eq:jko2-new; monitor [[EQ:eq0276]] via eq:lambda and tab:constants.
- Causality (LR-to-TV): Choose [[EQ:eq0277]] with [[EQ:eq0278]] ; verify prop:tv, estimating [[EQ:eq0279]] by the three-step guide.
- Ring-down: Apply eq:ring; control multiple comparisons; bootstrap under non-Gaussian noise. Connect [[EQ:eq0280]] to angular variance via [[EQ:eq0281]] for power analysis.

SECTION: Discussion and Outlook

We presented a dynamic-first, calibrated, and testable framework where the space of gauges carries a length metric equal to a minimal action on the unit time interval; law-dependent GKLS dynamics produce operational no-signaling bounds and ring-down inequalities. Key choices (HK scaling [[EQ:eq0282]] , fiber calibration [[EQ:eq0283]] , admissible post-processing) are instrumental parameters with explicit roles in EVI constants and thresholds.

PARAGRAPH: On ``nondual''.

Our use of ``nondual'' is operational: it fixes a single algebraic source
and studies accessibility variations via [[EQ:eq0284]] . It is not a metaphysical claim,
and all testable statements (TV cones, ring-down, local EVI) remain contingent
on explicitly stated assumptions (type-I fibers, LR proxies, etc.).

We resolved core issues: (i) dynamic-first definition of HK with static recovery; (ii) SLD-QFI convention and the [[EQ:eq0285]] normalization (hence: minimal action [[EQ:eq0286]] distance [[EQ:eq0287]] along geodesics on unit time); (iii) diagonal split-JKO with a [[EQ:eq0288]] -liminf passage to static coupling; (iv) averaged mass bounds completing completeness; (v) LR-to-TV as a hypothesis-dependent proposition. Outlook: global fiber convexity via Petz monotone families; type~III via Haagerup [[EQ:eq0289]] cores; LR constants for long-range/non-Markovian models; superchannel-level monotonicity classes.

SECTION: HK dynamic--static calibration on two points

app:calibration
We compute eq:action on [[EQ:eq0290]] with a single transport arc and reaction rate, verifying that the factor [[EQ:eq0291]] yields exactly eq:HK2pt. Units: [[EQ:eq0292]] carries [[EQ:eq0293]] weighted by mass; [[EQ:eq0294]] carries [[EQ:eq0295]] weighted by mass; multiplying by [[EQ:eq0296]] homogenizes the action, and static recovery matches [[EQ:eq0297]] . (With no-flux boundary conditions on Euclidean domains or on finite graphs, the divergence term does not contribute to the total mass balance.)

SECTION: Diagonal-split to static coupling: Gamma-liminf and flow-line gluing

app:superposition
[Gamma-liminf for diagonal split]lem:gliminf
Let [[EQ:eq0298]] be the discrete energy with diagonal fiber step eq:jko2-new. Then [[EQ:eq0299]] , where [[EQ:eq0300]] is the static cost in eq:dfib.

[Flow-line gluing to endpoint couplings]lem:glue
On a finite base, the cone lift makes the continuity equation conservative LieroMielkeSavare2018; a superposition principle yields a measure on AC flow lines. Along each line choose a measurable Uhlmann geodesic; pointwise Bures length [[EQ:eq0301]] endpoint angle. Averaging realizes a massful endpoint coupling [[EQ:eq0302]] and the fiber integral in eq:dfib. General bases follow by partition approximation and tightness.

SECTION: Completeness: a priori and averaged mass bounds; CJ compactness

app:completeness
Let [[EQ:eq0303]] . From the reaction equation and Cauchy--Schwarz,

[[EQ:eq0031]]

so [[EQ:eq0304]] is controlled by the HK action via Grönwall; the averaged bound of lem:mass-avg follows by integrating in time, i.e.,
[[EQ:eq0305]] .
Combined with CJ compactness in Bures (type-I/finite dimension), any [[EQ:eq0306]] -Cauchy sequence admits a finite-cost limit, proving completeness.

SECTION: Graph modulus [[EQ:eq0307]]

tau\_ent
app:graph
Represent HK geodesics on a finite graph as concatenations of edgewise cone geodesics. Each edge [[EQ:eq0308]] of length [[EQ:eq0309]] imposes an angle constraint [[EQ:eq0310]] identical to the two-point case; a gluing lemma shows that entropy convexity along the whole path inherits the worst local modulus, proving eq:taugraph.

SECTION: Notation, units, and triangle inequality note

All distances [[EQ:eq0311]] denote [[EQ:eq0312]] . The fiber calibration [[EQ:eq0313]] has units of length; [[EQ:eq0314]] has units [[EQ:eq0315]] ; [[EQ:eq0316]] and [[EQ:eq0317]] have units [[EQ:eq0318]] . The triangle inequality uses additivity of the action under concatenation and [[EQ:eq0319]] .

plain
99

TakahashiNAE2025
K.~Takahashi.
Nondual Autopoietic Excitations (author's related preprint).
Zenodo (2025).
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Assumptions, Scope, and Dynamic-First Principle

OPI Projection as a Normal Faithful Conditional Expectation

Fibered Bures–HK Metric (Dynamic-First) and Calibrated Action

Dynamic definition, static recovery

Calibrated action

Measurability, l.s.c., and Bochner

Dynamic–static equivalence and diagonal-split ⇒ static coupling

Choi-SLD QFI: Local Speed, TV, and Integral Diamond

Split-JKO and Local EVI (lambda) on (X,)

Energy (state-only), coercivity, time regularity

Split steps with diagonal

LR-to-TV Bridge (Assumed LR): Operational Form

Ring-Down Inequality: Units and Statistics

Worked Example: M_2⊕C

HK ET and KKT with κ (including regime test)

Massful coupling and fiber cost by regime

Minimal Experimental Protocol (Dynamic-First, Pre-Registered)

Discussion and Outlook

HK dynamic–static calibration on two points

Diagonal-split to static coupling: Gamma-liminf and flow-line gluing

Completeness: a priori and averaged mass bounds; CJ compactness

Graph modulus τ_ent

Notation, units, and triangle inequality note

\begin{equation}\label{eq:HK2pt}
\HK_\delta^2\!\big(a\delta_x,b\delta_y\big)
=4\delta^2\!\left(a+b-2\sqrt{ab}\,\cosp{\tfrac{|x-y|}{2\delta}}\right),
\qquad \cosp{r}=\cos(\min\{r,\pi\}).
\end{equation}

\label{eq:HK2pt} \HK_\delta^2 \big(a\delta_x,b\delta_y\big) =4\delta^2 \left(a+b-2\sqrt{ab} \cosp{\tfrac{|x-y|}{2\delta}}\right), \qquad \cosp{r}=\cos(\min\{r,\pi\}).

\begin{equation}\label{eq:dfib}
\dfib^2 \big((\mu,\rho),(\mu',\rho')\big)
:= \HK_\delta^2(\mu,\mu')
 + \sigma^2 \inf_{\pi\in\PiHKmass(\mu,\mu')}
  \iint \dB{\rho_\zeta}{\rho'_{\zeta'}}^2 \,\pi(\dd\zeta,\dd\zeta') ,
\end{equation}

\label{eq:dfib} \dfib^2 \big((\mu,\rho),(\mu',\rho')\big) := \HK_\delta^2(\mu,\mu') + \sigma^2 \inf_{\pi\in\PiHKmass(\mu,\mu')} \iint \dB{\rho_\zeta}{\rho'_{\zeta'}}^2  \pi(\dd\zeta,\dd\zeta') ,

\begin{equation}\label{eq:action}
\mathcal{A}\big[(\mu_t,\rho_t)\big]
:= \int_0^1\!\Big[
\int (|v_t|^2+\tfrac{\delta^2}{4}|\alpha_t|^2)\,\dd\mu_t
+ \underbrace{\tfrac{\sigma^2}{4}\int \Ich(\dot\rho_{\zeta,t})\,\dd\mu_t(\zeta)}_{\text{Bures (angle) speed}^2}
\Big]\dd t.
\end{equation}

\label{eq:action} \mathcal{A}\big[(\mu_t,\rho_t)\big] := \int_0^1 \Big[ \int (|v_t|^2+\tfrac{\delta^2}{4}|\alpha_t|^2) \dd\mu_t + \underbrace{\tfrac{\sigma^2}{4}\int \Ich(\dot\rho_{\zeta,t}) \dd\mu_t(\zeta)}_{\text{Bures (angle) speed}^2} \Big]\dd t.

\begin{equation}\label{eq:energy}
G_t(E,\mu)= \beta\,\|[S,\mathcal{L}_\mu]\|_{\cb}^2
+ \Phi_{\mathrm{reg}}(E,\mu;t).
\end{equation}

\label{eq:energy} G_t(E,\mu)= \beta \|[S,\mathcal{L}_\mu]\|_{\cb}^2 + \Phi_{\mathrm{reg}}(E,\mu;t).

\begin{equation}\label{eq:dfibdiag}
d_{\mathrm{fib,diag}}^{(\mu)}(E,E')^2:=\int_{\Spec(\frakZ)}
  d_{\Bures}\!\left(\J(E_\zeta),\J(E'_\zeta)\right)^2\,\mu(\dd\zeta).
\end{equation}

\label{eq:dfibdiag} d_{\mathrm{fib,diag}}^{(\mu)}(E,E')^2:=\int_{\Spec(\frakZ)} d_{\Bures} \left(\J(E_\zeta),\J(E'_\zeta)\right)^2 \mu(\dd\zeta).

\begin{equation}\label{eq:tauent}
\tau_{\mathrm{ent}}\ \ge\ \cosp{L/(2\delta)} ,
\end{equation}

\label{eq:tauent} \tau_{\mathrm{ent}}\ \ge\ \cosp{L/(2\delta)} ,

\begin{equation}\label{eq:taugraph}
\tau_{\mathrm{ent}}\ \ge\ \min_{e\in E}\ \cosp{\ell_e/(2\delta)} .
\end{equation}

\begin{equation}\label{eq:lambda}
\lambda \;\ge\; \min\Big\{\underbrace{\tau_{\mathrm{ent}}\,\beta\,c_{\rm base}}_{\lambda_{\rm base}},\ \underbrace{4\,\lambda_{\rm fib}/\sigma^2}_{\text{fiber (scaled)}}\Big\}
\;-\;\underbrace{L_F}_{\text{Lipschitz penalty}}.
\end{equation}

\label{eq:lambda} \lambda  \ge  \min\Big\{\underbrace{\tau_{\mathrm{ent}} \beta c_{\rm base}}_{\lambda_{\rm base}},\ \underbrace{4 \lambda_{\rm fib}/\sigma^2}_{\text{fiber (scaled)}}\Big\}  - \underbrace{L_F}_{\text{Lipschitz penalty}}.

\begin{equation}\label{eq:ring}
\Gamma(t_1,t_2)\;\le\;\frac{C_{\rm read}}{t_2-t_1}\int_{t_1}^{t_2}
\Big(\tfrac12\sqrt{\Ich(\dot E_t)}+\gamma\,\|[S_t,\mathcal{L}_{\mu_t}]\|_{\cb}\Big)\,\dd t .
\end{equation}

\label{eq:ring} \Gamma(t_1,t_2) \le \frac{C_{\rm read}}{t_2-t_1}\int_{t_1}^{t_2} \Big(\tfrac12\sqrt{\Ich(\dot E_t)}+\gamma \|[S_t,\mathcal{L}_{\mu_t}]\|_{\cb}\Big) \dd t .

\begin{equation}\label{eq:2ptfinal}
\boxed{\quad
\dfib^2\big((\mu,\rho),(\mu',\rho')\big)
=\HK_\delta^2(\mu,\mu')+\sigma^2\,\pi_{11}^\ast\,\theta^2,
\qquad \pi_{11}^\ast=\sqrt{m_1 n_1}\,\Gamma_{11}^\ast.
\quad}
\end{equation}

\label{eq:2ptfinal} \boxed{\quad \dfib^2\big((\mu,\rho),(\mu',\rho')\big) =\HK_\delta^2(\mu,\mu')+\sigma^2 \pi_{11}^\ast \theta^2, \qquad \pi_{11}^\ast=\sqrt{m_1 n_1} \Gamma_{11}^\ast. \quad}

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

\begin{document}
\maketitle

\begin{abstract}
We formulate a falsifiable, \emph{single-source (nondual, operational)} law of gauge dynamics for physical accessibility,\footnote{By ``nondual'' we mean a single underlying $W^\ast$-system $(\mathcal A,\varrho)$ while experimental contexts vary via the accessibility gauge $E_t$; this is an \emph{operational} modeling choice, not an ontological claim. See Def.~\ref{def:nondual}.} taking the \textbf{dynamic principle as the definition} of the base metric: \emph{HK is defined as a minimum action for a continuity equation with reaction}. The static entropy--transport (ET) form and two-point $4\delta^2$ formula are then recovered as \emph{derived consequences} (see \cite[§§7--8]{LieroMielkeSavare2018} for cone-lift \& superposition and \cite[§§2--3]{ChizatPeyreSchmitzerVialard2018} for a dynamic formulation; exact numbering differs between journal/preprint versions). Fibers carry \textbf{Bures angles} of \textbf{Choi states}. Adopting the \textbf{SLD-QFI} convention $g_{\Bures}(\dot\rho,\dot\rho)=\tfrac14\,\Ich(\dot\rho)$ yields $|\dot d_{\Bures}|\le \tfrac12\sqrt{\Ich}$ and fixes the \emph{fiber action} coefficient to \underline{$\sigma^2/4$}. Consequently, along a Bures \emph{geodesic} the \emph{minimal} fiber action equals the squared static fiber distance \emph{on the unit time interval} ($\int_0^1 \tfrac{\sigma^2}{4}\Ich=\sigma^2 d_{\Bures}^2$), while in general $\int_0^1 \tfrac{\sigma^2}{4}\Ich \ge \sigma^2 d_{\Bures}^2$.
We prove dynamic--static equivalence with explicit measurability (JvN/KRN), a $\Gamma$-liminf passage from \emph{diagonal} split-JKO to the static coupling, rank-drop tolerant Bochner integrability, and an \emph{averaged} Grönwall mass bound completing completeness. We then derive \emph{domain-limited} time-dependent EVI$(\lambda)$ flows with constants
$\lambda\!\ge\!\min\{\tau_{\mathrm{ent}}\beta c_{\rm base},\,4\lambda_{\rm fib}/\sigma^2\}-L_F$,
and obtain finite-time Bures$\to$TV bounds plus an \emph{integral} diamond inequality with a cb-norm envelope for nonautonomous GKLS propagators. A worked two-point $(M_2\oplus\mathbb{C})$ example includes the $\kappa$ threshold and the \emph{massful} coupling $\pi^\ast$. A pre-registered protocol turns the law into TV no-signaling cones and ring-down inequalities with calibrated constants $(\delta,\sigma,\gamma,C_{\rm read})$.
\end{abstract}

\paragraph*{Notation \& Units (quick guide).}
All $d_{\HK}$ mean $\HK_\delta$. We use the \emph{Bures angle}:
$\dB{\rho}{\sigma}:=\arccos\sqrt{F(\rho,\sigma)}\in[0,\pi/2]$.
\textbf{Convention:} $\cos_+(r):=\cos(\min\{r,\pi\})$.
SLD-QFI $\Ich$ has units $\mathrm{s}^{-2}$; thus $\tfrac12\sqrt{\Ich}$ is an angular speed ($\mathrm{s}^{-1}$). The fiber calibration $\sigma$ has units of length; the \emph{dynamic} fiber action uses \underline{$\tfrac{\sigma^2}{4}$} whereas the \emph{static} fiber cost uses $\sigma^2$. Both terms in \eqref{eq:ring} carry $\mathrm{s}^{-1}$.

\begin{center}
% ====== FIX 1: diagram reflow (no overlap / no right overflow) ======
\begin{minipage}{0.98\linewidth}\centering
\resizebox{\linewidth}{!}{%
\begin{tikzpicture}[>=Latex,rounded corners,line cap=round,shorten >=1pt,shorten <=1pt]
% node style
\tikzset{box/.style={draw,fill=gray!7,align=center,minimum width=32mm,minimum height=12mm}}
% layout (absolute positions to avoid edge crossings/overlaps)
\node[box] (opi)    at (0,0)  {OPI gauge\\$E=\tfrac12(\id+\Ad_S)$};
\node[box,minimum width=48mm] (metric) at (5,0)  {Fibered metric\\HK (dynamic-def)\\$+\ \sigma^2$Bures-angle (static)};
\node[box] (jko)    at (10,0) {Split-JKO/EVI\\(local domain)};
\node[box,minimum width=42mm] (lr)     at (5,-2.8)  {GKLS + LR\\(assumed)};
\node[box,minimum width=48mm] (ops)    at (10,-2.8) {Operational bounds\\TV cone, ring-down};
% arrows (routed to avoid overlaps)
\draw[->] (opi.east) -- (metric.west);                 % OPI -> Metric
\draw[->] (metric.east) -- (jko.west);                 % Metric -> JKO
\draw[->] (jko.south) -- (ops.north);                  % JKO -> Ops (vertical)
% curve OPI -> LR (kept well below metric box)
\draw[->] (opi.east) to[out=-25,in=180] (lr.west);     % OPI -> LR
\draw[->] (lr.east) -- (ops.west);                     % LR  -> Ops (horizontal)
% gentle curved arc Metric -> Ops entering from NW to avoid vertical JKO->Ops
\draw[->] (metric.east) to[out=-28,in=160,looseness=0.95] (ops.north west);
\end{tikzpicture}}
\\[0.4em]
{\footnotesize\emph{Note: static fiber cost uses $\sigma^2$ while the dynamic action uses $\sigma^2/4$ due to the SLD convention; along Bures geodesics the minimal action equals the squared static cost on the unit time interval.}}
\end{minipage}
\end{center}

\paragraph*{Three-line vision (What/Varies/Predicted).}
\textbf{Fixed:} a single $W^\ast$-system $(\mathcal A,\varrho)$ \textbf{(nondual, operational)}. \textbf{Varies:} OPI gauges $E_t$ (accessibility), base labels $\mu_t$ and fiber channels/states. \textbf{Predicted:} geometric laws (local EVI), operational TV cones and ring-down bounds with calibrated constants.

%============================
\section{Assumptions, Scope, and Dynamic-First Principle}
\label{sec:assumptions}

\begin{definition}[Nondual (operational meaning)]\label{def:nondual}
“Nondual” denotes the \emph{single-source} modeling stance: there is one
$W^\ast$-system $(\mathcal A,\varrho)$, and differences between observers/contexts
are encoded entirely by the time-dependent \emph{gauge of inclusion} $E_t$ (OPI).
No ontological statement is intended; only the \emph{representational choice}
matters for the ensuing geometric and operational consequences (HK/Bures geometry,
JKO/EVI, Lieb--Robinson (LR)-to-TV, ring-down).
\end{definition}

\paragraph{Single-source (nondual, operational) viewpoint.}
We posit a single von Neumann system $(\mathcal A,\varrho)$; what changes across
experimental situations is the \emph{accessibility gauge} $E_t$.
This usage of “nondual” refers solely to this operational single-source stance.

\begin{assumption}[Standard Borel base and Radon measures]\label{ass:borel}
$\Spec(\frakZ)$ is standard Borel (Polish). Base measures $\mu$ are finite positive Radon measures; disintegration yields measurable fields $(\calH_\zeta,\calA_\zeta,\varrho_\zeta)$.
\end{assumption}

\begin{definition}[OPI projection]\label{def:opi}
Let $S$ be a self-adjoint unitary ($S^\dagger=S=S^{-1}$).
Set $\Ad_S(X):=SXS^\dagger=SXS$ and $E:=\tfrac12(\id+\Ad_S)$.
Then $E$ is unital, completely positive, idempotent and normal on $\calA$.
\end{definition}

\begin{assumption}[Dynamic HK calculus on the base]\label{ass:dynHK}
$(\Spec(\frakZ),d)$ is a complete separable metric space endowed with a dynamic
HK calculus: there exists a well-posed continuity equation with reaction
$\partial_t\mu_t + \nabla\!\cdot(\mu_t v_t)=\alpha_t\mu_t$ in a weak (distributional)
sense along absolutely continuous curves, with action
$\int_0^1\!\int(|v_t|^2+\tfrac{\delta^2}{4}|\alpha_t|^2)\,\dd\mu_t\,\dd t$.
Finite metric graphs are included via a discrete divergence, and Euclidean-like
domains via the cone-lift construction. (Boundary conditions: no-flux on Euclidean domains; graph divergence sums to zero.)
\end{assumption}

\begin{definition}[HK as a dynamic minimum action]\label{def:HK-dynamic}
Under \cref{ass:dynHK}, for $\mu_0,\mu_1$ we set
\[
\HK_\delta^2(\mu_0,\mu_1)
:= \inf_{\mu_t,v_t,\alpha_t}
\int_0^1\!\int\Big(|v_t|^2+\tfrac{\delta^2}{4}\,|\alpha_t|^2\Big)\,\dd\mu_t\,\dd t,
\quad
\partial_t\mu_t+\nabla\!\cdot(\mu_t v_t)=\alpha_t\mu_t.
\]

2pt] \small ORCID: \href{https://orcid.org/0009-0004-4273-3365}{0009-0004-4273-3365}} \date{\today} \begin{document} \maketitle \begin{abstract} We formulate a falsifiable, \emph{single-source (nondual, operational)} law of gauge dynamics for physical accessibility,\footnote{By ``nondual'' we mean a single underlying $W^\ast$-system $(\mathcal A,\varrho)$ while experimental contexts vary via the accessibility gauge $E_t$; this is an \emph{operational} modeling choice, not an ontological claim. See Def.~\ref{def:nondual}.} taking the \textbf{dynamic principle as the definition} of the base metric: \emph{HK is defined as a minimum action for a continuity equation with reaction}. The static entropy--transport (ET) form and two-point $4\delta^2$ formula are then recovered as \emph{derived consequences} (see \cite[§§7--8]{LieroMielkeSavare2018} for cone-lift \& superposition and \cite[§§2--3]{ChizatPeyreSchmitzerVialard2018} for a dynamic formulation; exact numbering differs between journal/preprint versions). Fibers carry \textbf{Bures angles} of \textbf{Choi states}. Adopting the \textbf{SLD-QFI} convention $g_{\Bures}(\dot\rho,\dot\rho)=\tfrac14 \Ich(\dot\rho)$ yields $|\dot d_{\Bures}|\le \tfrac12\sqrt{\Ich}$ and fixes the \emph{fiber action} coefficient to \underline{$\sigma^2/4$}. Consequently, along a Bures \emph{geodesic} the \emph{minimal} fiber action equals the squared static fiber distance \emph{on the unit time interval} ($\int_0^1 \tfrac{\sigma^2}{4}\Ich=\sigma^2 d_{\Bures}^2$), while in general $\int_0^1 \tfrac{\sigma^2}{4}\Ich \ge \sigma^2 d_{\Bures}^2$. We prove dynamic--static equivalence with explicit measurability (JvN/KRN), a $\Gamma$-liminf passage from \emph{diagonal} split-JKO to the static coupling, rank-drop tolerant Bochner integrability, and an \emph{averaged} Grönwall mass bound completing completeness. We then derive \emph{domain-limited} time-dependent EVI$(\lambda)$ flows with constants $\lambda \ge \min\{\tau_{\mathrm{ent}}\beta c_{\rm base}, 4\lambda_{\rm fib}/\sigma^2\}-L_F$, and obtain finite-time Bures$\to$TV bounds plus an \emph{integral} diamond inequality with a cb-norm envelope for nonautonomous GKLS propagators. A worked two-point $(M_2\oplus\mathbb{C})$ example includes the $\kappa$ threshold and the \emph{massful} coupling $\pi^\ast$. A pre-registered protocol turns the law into TV no-signaling cones and ring-down inequalities with calibrated constants $(\delta,\sigma,\gamma,C_{\rm read})$. \end{abstract} \paragraph*{Notation \& Units (quick guide).} All $d_{\HK}$ mean $\HK_\delta$. We use the \emph{Bures angle}: $\dB{\rho}{\sigma}:=\arccos\sqrt{F(\rho,\sigma)}\in[0,\pi/2]$. \textbf{Convention:} $\cos_+(r):=\cos(\min\{r,\pi\})$. SLD-QFI $\Ich$ has units $\mathrm{s}^{-2}$; thus $\tfrac12\sqrt{\Ich}$ is an angular speed ($\mathrm{s}^{-1}$). The fiber calibration $\sigma$ has units of length; the \emph{dynamic} fiber action uses \underline{$\tfrac{\sigma^2}{4}$} whereas the \emph{static} fiber cost uses $\sigma^2$. Both terms in \eqref{eq:ring} carry $\mathrm{s}^{-1}$. \begin{center} % ====== FIX 1: diagram reflow (no overlap / no right overflow) ====== \begin{minipage}{0.98\linewidth}\centering \resizebox{\linewidth}{!}{% \begin{tikzpicture}[>=Latex,rounded corners,line cap=round,shorten >=1pt,shorten <=1pt] % node style \tikzset{box/.style={draw,fill=gray!7,align=center,minimum width=32mm,minimum height=12mm}} % layout (absolute positions to avoid edge crossings/overlaps) \node[box] (opi) at (0,0) {OPI gauge\\$E=\tfrac12(\id+\Ad_S)$}; \node[box,minimum width=48mm] (metric) at (5,0) {Fibered metric\\HK (dynamic-def)\\$+\ \sigma^2$Bures-angle (static)}; \node[box] (jko) at (10,0) {Split-JKO/EVI\\(local domain)}; \node[box,minimum width=42mm] (lr) at (5,-2.8) {GKLS + LR\\(assumed)}; \node[box,minimum width=48mm] (ops) at (10,-2.8) {Operational bounds\\TV cone, ring-down}; % arrows (routed to avoid overlaps) \draw[->] (opi.east) -- (metric.west); % OPI -> Metric \draw[->] (metric.east) -- (jko.west); % Metric -> JKO \draw[->] (jko.south) -- (ops.north); % JKO -> Ops (vertical) % curve OPI -> LR (kept well below metric box) \draw[->] (opi.east) to[out=-25,in=180] (lr.west); % OPI -> LR \draw[->] (lr.east) -- (ops.west); % LR -> Ops (horizontal) % gentle curved arc Metric -> Ops entering from NW to avoid vertical JKO->Ops \draw[->] (metric.east) to[out=-28,in=160,looseness=0.95] (ops.north west); \end{tikzpicture}} \\[0.4em] {\footnotesize\emph{Note: static fiber cost uses $\sigma^2$ while the dynamic action uses $\sigma^2/4$ due to the SLD convention; along Bures geodesics the minimal action equals the squared static cost on the unit time interval.}} \end{minipage} \end{center} \paragraph*{Three-line vision (What/Varies/Predicted).} \textbf{Fixed:} a single $W^\ast$-system $(\mathcal A,\varrho)$ \textbf{(nondual, operational)}. \textbf{Varies:} OPI gauges $E_t$ (accessibility), base labels $\mu_t$ and fiber channels/states. \textbf{Predicted:} geometric laws (local EVI), operational TV cones and ring-down bounds with calibrated constants. %============================ \section{Assumptions, Scope, and Dynamic-First Principle} \label{sec:assumptions} \begin{definition}[Nondual (operational meaning)]\label{def:nondual} “Nondual” denotes the \emph{single-source} modeling stance: there is one $W^\ast$-system $(\mathcal A,\varrho)$, and differences between observers/contexts are encoded entirely by the time-dependent \emph{gauge of inclusion} $E_t$ (OPI). No ontological statement is intended; only the \emph{representational choice} matters for the ensuing geometric and operational consequences (HK/Bures geometry, JKO/EVI, Lieb--Robinson (LR)-to-TV, ring-down). \end{definition} \paragraph{Single-source (nondual, operational) viewpoint.} We posit a single von Neumann system $(\mathcal A,\varrho)$; what changes across experimental situations is the \emph{accessibility gauge} $E_t$. This usage of “nondual” refers solely to this operational single-source stance. \begin{assumption}[Standard Borel base and Radon measures]\label{ass:borel} $\Spec(\frakZ)$ is standard Borel (Polish). Base measures $\mu$ are finite positive Radon measures; disintegration yields measurable fields $(\calH_\zeta,\calA_\zeta,\varrho_\zeta)$. \end{assumption} \begin{definition}[OPI projection]\label{def:opi} Let $S$ be a self-adjoint unitary ($S^\dagger=S=S^{-1}$). Set $\Ad_S(X):=SXS^\dagger=SXS$ and $E:=\tfrac12(\id+\Ad_S)$. Then $E$ is unital, completely positive, idempotent and normal on $\calA$. \end{definition} \begin{assumption}[Dynamic HK calculus on the base]\label{ass:dynHK} $(\Spec(\frakZ),d)$ is a complete separable metric space endowed with a dynamic HK calculus: there exists a well-posed continuity equation with reaction $\partial_t\mu_t + \nabla \cdot(\mu_t v_t)=\alpha_t\mu_t$ in a weak (distributional) sense along absolutely continuous curves, with action $\int_0^1 \int(|v_t|^2+\tfrac{\delta^2}{4}|\alpha_t|^2) \dd\mu_t \dd t$. Finite metric graphs are included via a discrete divergence, and Euclidean-like domains via the cone-lift construction. (Boundary conditions: no-flux on Euclidean domains; graph divergence sums to zero.) \end{assumption} \begin{definition}[HK as a dynamic minimum action]\label{def:HK-dynamic} Under \cref{ass:dynHK}, for $\mu_0,\mu_1$ we set \[ \HK_\delta^2(\mu_0,\mu_1) := \inf_{\mu_t,v_t,\alpha_t} \int_0^1 \int\Big(|v_t|^2+\tfrac{\delta^2}{4} |\alpha_t|^2\Big) \dd\mu_t \dd t, \quad \partial_t\mu_t+\nabla \cdot(\mu_t v_t)=\alpha_t\mu_t.

\[
\pi := (\mathrm{pr}_X\times \mathrm{pr}_X)_{\#}\big(r\,r'\;\widehat\Gamma\big),
\]

\pi := (\mathrm{pr}_X\times \mathrm{pr}_X)_{\#}\big(r r' \widehat\Gamma\big),

\[
\liminf_{n\to\infty} \iint \dB{\rho^n_\zeta}{\rho^{\prime n}_{\zeta'}}^2\,\pi_n(\dd\zeta,\dd\zeta')
\ \ge\
\iint \dB{\rho_\zeta}{\rho'_{\zeta'}}^2\,\pi(\dd\zeta,\dd\zeta').
\]

\liminf_{n\to\infty} \iint \dB{\rho^n_\zeta}{\rho^{\prime n}_{\zeta'}}^2 \pi_n(\dd\zeta,\dd\zeta') \ \ge\ \iint \dB{\rho_\zeta}{\rho'_{\zeta'}}^2 \pi(\dd\zeta,\dd\zeta').

\[
L_{\Bures}(\rho_{\cdot})\ge d_{\Bures}(\rho_0,\rho_1),
\qquad
\Big|\partial_t d_{\Bures}\big(\rho_t,\rho_{t_0}\big)\Big|\le \tfrac12\sqrt{\Ich(\dot\rho_t)} ,
\]

\[
\dfib^2\big((\mu_0,\rho_0),(\mu_1,\rho_1)\big)=
\inf\{\mathcal{A}[(\mu_t,\rho_t)]: (\mu_t,\rho_t)\ \mathrm{AC},\ \text{join the endpoints}\}.
\]

\[
\Big|\frac{\dd}{\dd t}\sqrt{M(t)}\Big|
=\frac{|\dot M(t)|}{2\sqrt{M(t)}}
\le \frac{1}{\delta}\Big(\int \tfrac{\delta^2}{4}|\alpha_t|^2\,\dd\mu_t\Big)^{1/2}.
\]

\Big|\frac{\dd}{\dd t}\sqrt{M(t)}\Big| =\frac{|\dot M(t)|}{2\sqrt{M(t)}} \le \frac{1}{\delta}\Big(\int \tfrac{\delta^2}{4}|\alpha_t|^2 \dd\mu_t\Big)^{1/2}.

\[
\Big|\tfrac{\dd}{\dd t}\dB{\J(E_t)}{\J(E_{t_0})}\Big| \le \tfrac12\sqrt{\Ich(\dot E_t)}\quad \text{a.e.\ in }t.\footnote{The bound is immediate for full-rank states; for rank-deficient cases,
one may apply the $\varepsilon$-regularization $\rho\mapsto (1-\varepsilon)\rho+\varepsilon\,\1/d$ and pass to the limit
$\varepsilon\downarrow0$ by the monotonicity of the Bures angle.}
\]

\Big|\tfrac{\dd}{\dd t}\dB{\J(E_t)}{\J(E_{t_0})}\Big| \le \tfrac12\sqrt{\Ich(\dot E_t)}\quad \text{a.e.\ in }t.\footnote{The bound is immediate for full-rank states; for rank-deficient cases, one may apply the $\varepsilon$-regularization $\rho\mapsto (1-\varepsilon)\rho+\varepsilon \1/d$ and pass to the limit $\varepsilon\downarrow0$ by the monotonicity of the Bures angle.}

\dB{(\id\otimes\Theta)\J(E_{t_2})}{(\id\otimes\Theta)\J(E_{t_1})} \le \frac12\int_{t_1}^{t_2} \sqrt{\Ich(\dot E_t)} \dd t .

\[
\dnorm{\Lambda_{t,s}-\Lambda'_{t,s}}
\le \int_s^t\!\|\mathcal{L}_u-\mathcal{L}'_u\|_{\cb}\,
\exp\!\Big(\!\int_u^t\!\max\{\|\mathcal{L}_\tau\|_{\cb},\|\mathcal{L}'_\tau\|_{\cb}\}\,\dd\tau\Big)\dd u ,
\]

\dnorm{\Lambda_{t,s}-\Lambda'_{t,s}} \le \int_s^t \|\mathcal{L}_u-\mathcal{L}'_u\|_{\cb}  \exp \Big( \int_u^t \max\{\|\mathcal{L}_\tau\|_{\cb},\|\mathcal{L}'_\tau\|_{\cb}\} \dd\tau\Big)\dd u ,

\[
L_F\,\HK_\delta(\mu^{k+1},\mu^{k})
\le \frac{\varepsilon}{2\tau}\HK_\delta^2(\mu^{k+1},\mu^{k})+\frac{\tau L_F^2}{2\varepsilon},
\]

L_F \HK_\delta(\mu^{k+1},\mu^{k}) \le \frac{\varepsilon}{2\tau}\HK_\delta^2(\mu^{k+1},\mu^{k})+\frac{\tau L_F^2}{2\varepsilon},

\[
\TVnorm{p_{t}^{\mathsf{M}\otimes\mathsf{N}} - p_{t}^{\mathsf{M}}\otimes p_{t}^{\mathsf{N}}}
\le C\,|\partial A|^{p}\,e^{-\muLR(d-vt)} ,
\]

\TVnorm{p_{t}^{\mathsf{M}\otimes\mathsf{N}} - p_{t}^{\mathsf{M}}\otimes p_{t}^{\mathsf{N}}} \le C |\partial A|^{p} e^{-\muLR(d-vt)} ,

\[
HK_\delta^2(\mu,\mu') = 4\delta^2\!\left(\sum_i m_i + \sum_j n_j
  - 2\,\max_{\Gamma\in \mathcal U}\sum_{i,j}\kappa_{ij}\,\sqrt{m_i n_j}\,\Gamma_{ij}\right),
\]

HK_\delta^2(\mu,\mu') = 4\delta^2 \left(\sum_i m_i + \sum_j n_j - 2 \max_{\Gamma\in \mathcal U}\sum_{i,j}\kappa_{ij} \sqrt{m_i n_j} \Gamma_{ij}\right),

\[
\kappa_{ij}\sqrt{m_i n_j}\ \begin{cases}
= \lambda_i+\mu_j, & \Gamma_{ij}\in(0,1),\\
\le \lambda_i+\mu_j, & \Gamma_{ij}=0,\\
\ge \lambda_i+\mu_j, & \Gamma_{ij}=1.
\end{cases}
\]

\[
\sqrt{m_1n_1}+\sqrt{m_2n_2}\ \gtreqless\
\kappa\big(\sqrt{m_1n_2}+\sqrt{m_2n_1}\big).
\]

t\mapsto \int \Ich(\dot\rho_{\zeta,t}) \dd\mu_t(\zeta)

$\dB{\rho^n_\zeta}{\rho^{\prime n}_{\zeta'}}^2
\to \dB{\rho_\zeta}{\rho'_{\zeta'}}^2$

$\Gamma\text{-}\liminf_{j\to\infty}\mathcal E_{\tau_j}\ge \mathcal E_{\mathrm{stat}}$

$\mathcal A_{\HK}(T)=\int_0^T\int \frac{\delta^2}{4}|\alpha_t|^2\,\dd\mu_t\,\dd t$

\mathcal A_{\HK}(T)=\int_0^T\int \frac{\delta^2}{4}|\alpha_t|^2 \dd\mu_t \dd t

M(t)\le \big(\sqrt{M(0)}+\frac{1}{\delta}\sqrt{T \mathcal A_{\HK}(T)}\big)^2

\tfrac12\|\Delta_{t_2,t_1}\|_1 \le \sin \big(\tfrac12\int_{t_1}^{t_2}\sqrt{\Ich(\dot E_t)} \dd t\big)

\|\Lambda_{t,s}\|_{\cb}\le \exp \big(\int_s^t \|\mathcal{L}_u\|_{\cb} \dd u\big)

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\lambda\ge\min\{\lambda_{\rm base}, 4\lambda_{\rm fib}/\sigma^2\}-L_F

$\lambda_{\rm base}\ge \tau_{\mathrm{ent}}\,\beta\,c_{\rm base}$

\lambda_{\rm base}\ge \tau_{\mathrm{ent}} \beta c_{\rm base}

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c_{\rm inj}\gtrsim c_0(d,\epsilon) (1 - O(r^2))

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$L/(2\delta)=\pi/6\Rightarrow \tau_{\rm ent}\ge \sqrt{3}/2$

L/(2\delta)=\pi/6\Rightarrow \tau_{\rm ent}\ge \sqrt{3}/2

\|\mathcal{L}_{\mu,Z}\|_{\cb}\le J e^{-\alpha \mathrm{diam}(Z)}

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\|\mathcal{L}_\mu-\mathcal{L}_\nu\|_{\cb}\le L_{\rm GKLS} \HK_\delta(\mu,\nu)

N_{\min} \approx \big((z_{1-\alpha}+z_{1-\beta}) \Delta\widehat{\Gamma}/\Delta\big)^2

$\displaystyle \inf_{\pi}\iint \dB{\rho_\zeta}{\rho'_{\zeta'}}^2 \,d\pi
= \pi_{11}^\ast\,\theta^2$

\displaystyle \inf_{\pi}\iint \dB{\rho_\zeta}{\rho'_{\zeta'}}^2  d\pi = \pi_{11}^\ast \theta^2

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$\Gamma\hbox{-}\liminf_{\tau\downarrow0}\mathcal E_\tau
\ge \mathcal E_{\mathrm{stat}}$

\sqrt{M(t)}\le \sqrt{M(0)}+\frac{1}{\delta}\sqrt{T \mathcal A_{\HK}(T)}


"Persistence ≈ Creation": Natural-Law Sufficient Conditions for Almost-Sure Beneficial Coverage in Stationary Ergodic Media (No Meta-Design)

margin=1in

1.2 % line spacing 1.2 for OCR/crawler friendliness

colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue,
pdftitle= Persistence ≈ Creation — Natural-Law Sufficient Conditions for A.s. Beneficial Coverage without Meta-Design,
pdfauthor= K. Takahashi (ORCID: https://orcid.org/0009-0004-4273-3365)

. 0.5em
0.5em

TITLE: ``Persistence [[EQ:eq0005]]

≈ Creation'': Natural-Law Sufficient Conditions for Almost-Sure Beneficial Coverage in Stationary Ergodic Media (No Meta-Design)

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

DATE: September 11, 2025

theorem Theorem
lemma Lemma
proposition Proposition
assumption Assumption
definition Definition

We recast alignment as a natural-law phenomenon: under physically motivated, representation-invariant sufficient conditions---finite temperature ( [[EQ:eq0006]] ), non-degenerate local refresh (Doeblin minorization), supercritical contact with anchored isoperimetry and uniform ellipticity, stationary ergodicity with finite-range dependence, and finite jump range---a cooperative, usefulness-creating phase almost surely expands with a deterministic linear speed [[EQ:eq0007]] in an infinite medium. The unifying axiom is ordinal: Persistence [[EQ:eq0008]] ≈ Creation . We establish: (R1) on Blackwell-closed classes with bounded deficiency, every admissible fecundity functional is order-equivalent to an affine transform of conditional mutual information (CMI), hence survival-order agrees with information-generation order; (R2) natural floors entail a constant chain yielding a diffusion lower bound [[EQ:eq0009]] and, given supercritical local reproduction (Assumption~ass:supercrit), a positive linear growth rate [[EQ:eq0010]] , hence a KPP-type lower bound [[EQ:eq0011]] ; (R3) robustness: dips of the floors have upper density [[EQ:eq0012]] a.s., and Poissonian resets below a dimension-dependent safe upper bound preserve linear growth. All inevitability claims are measure-relative. Optional design mechanisms (audits, governance, potential dynamics) are relegated to appendices as optional accelerants, not prerequisites.

: Artificial Intelligence; Superintelligence; Natural-law sufficient conditions; Conditional Mutual Information; Strong Data-Processing Inequality; Doeblin minorization; Percolation; Random Conductance Model; Shape Theorem; Fisher–KPP; Ergodic Theory; Ordinal Order-Equivalence; Alignment without Meta-Design.

SECTION: Plain-language Box

Natural floors: positive lower bounds forced by physics/geometry (not institutions). \;
Order-equivalence: rankings only (any strictly increasing transform preserves claims). \;
Shape theorem: in stationary ergodic media, growth clusters converge to a deterministic shape with linear speed. \;
Measure-relative a.s.: ``almost surely'' is relative to the physically motivated law of noise/contact.%
We use ``No Meta-Design'' and ``no meta-manager'' synonymously: no privileged external controller is assumed for the main theorems.

SECTION: Vision, Claim, and Scope (Natural-Law Form)

Vision (natural-law). As long as (i) thermal noise and irreversibility are nonzero, (ii) spatial contact is supercritical, (iii) the medium is stationary ergodic with finite-range dependence and uniform ellipticity, and (iv) transport has finite jump range [[EQ:eq0013]] , the cooperative, usefulness-creating phase self-propagates at positive linear speed almost surely, with no meta-manager.

PARAGRAPH: Natural-law sufficient conditions (NL).

On a stationary ergodic, locally finite metric graph:

[[EQ:eq0014]] with non-degenerate local refresh (Assumption~ass:refresh) induces a Doeblin head [[EQ:eq0015]] and SDPI floor [[EQ:eq0016]] .
supercritical contact/percolation and anchored isoperimetry (a.k.a.\ anchored expansion) with uniform ellipticity (Assumption~ass:anchored-elliptic) yield conductance and an [[EQ:eq0017]] spectral gap [[EQ:eq0018]] , hence [[EQ:eq0019]] .
Landauer/fluctuation bounds imply irreducible dissipation [[EQ:eq0020]] .

All a.s.\ claims are measure-relative and robust to absolutely continuous reweightings with Radon–Nikodým densities bounded away from [[EQ:eq0021]] and [[EQ:eq0022]] on the events of interest.

SECTION: Cooperative Phase and Monotone Coupling

sec:coopdef
We work on the a.s.-unique infinite cluster [[EQ:eq0023]] in the supercritical regime; all kernels and dynamics are restricted to [[EQ:eq0024]] .
[Cooperative occupancy and monotone coupling]def:coop
Each site [[EQ:eq0025]] carries a binary occupancy [[EQ:eq0026]] indicating membership in the cooperative phase. The dynamics is attractive (Harris): if [[EQ:eq0027]] pointwise, there exists a coupling with [[EQ:eq0028]] a.s.\ for all [[EQ:eq0029]] . Moreover, there is a supercritical contact-process lower envelope [[EQ:eq0030]] with infection rate [[EQ:eq0031]] such that [[EQ:eq0032]] a.s.

[Auditable usefulness rate (non-binding to proofs)]
Let [[EQ:eq0033]] denote externally auditable useful structure (e.g., task outputs) in a bounded region. The creation rate is [[EQ:eq0034]] per unit irreversible cost; require monotonicity w.r.t.\ occupancy: [[EQ:eq0035]] . This metric supports applications in Sec.~sec:applications and does not enter the natural-law theorems. Intuitively, this normalizes creation by the Landauer floor so that ``doing more with less irreversibility'' reads as higher usefulness. In discrete time replace [[EQ:eq0036]] by forward differences.

SECTION: Natural Floors and Constant Chains

sec:naturalfloors

PARAGRAPH: NL-Vis [[EQ:eq0037]] Doeblin with explicit constants.

[Non-degenerate local refresh]ass:refresh
There exist [[EQ:eq0038]] and a probability measure [[EQ:eq0039]] such that for the one-step coarse-grained kernel [[EQ:eq0040]] on [[EQ:eq0041]] ,
[math] P(x, )\ \ \,nu( )\ \ x, [/math]
equivalently [[EQ:eq0042]] .
Finite-range, uniformly elliptic perturbations at [[EQ:eq0043]] provide a local small-ball lower bound, inducing the minorization with explicit [[EQ:eq0044]] .

Then

[[EQ:eq0001]]

for an explicit [[EQ:eq0045]] (Doeblin [[EQ:eq0046]] SDPI).

PARAGRAPH: NL-ConT [[EQ:eq0047]] gaps and diffusivity.

[Anchored isoperimetry and uniform ellipticity]ass:anchored-elliptic
Almost surely, the medium contains [[EQ:eq0048]] with anchored isoperimetric profile [[EQ:eq0049]] for all large [[EQ:eq0050]] . Random conductances [[EQ:eq0051]] on [[EQ:eq0052]] satisfy [[EQ:eq0053]] and finite-range dependence.

Under Assumption~ass:anchored-elliptic,

[[EQ:eq0002]]

with dimension/geometry dependent [[EQ:eq0054]] .

PARAGRAPH: Conductance proxy.

We write [[EQ:eq0055]] (up to universal constants) so that Cheeger-type bounds read [[EQ:eq0056]] and, by homogenization, [[EQ:eq0057]] .

PARAGRAPH: NL-Diss [[EQ:eq0058]] irreversibility.

Landauer and fluctuation theorems imply [[EQ:eq0059]] at [[EQ:eq0060]] , excluding reversible zero-cost limit cycles.

PARAGRAPH: Constant chain (with constants).

Combining eq:sdpi and eq:gap-d,

[[EQ:eq0003]]

We reuse the constants [[EQ:eq0061]] throughout.

SECTION: Ordinal Bridge: Persistence [[EQ:eq0062]]

≈ Creation (without design)
[Admissible fecundity functionals]ass:admissible
[[EQ:eq0063]] is measurable and locally bounded; monotone under DPI/SDPI; obeys chain/additivity; vanishes on independence; invariant under sufficient (Blackwell) morphisms; stable to small Le Cam deficiency on standard Borel spaces.

[Ordinal bridge without design]thm:ordinal-bridge
On Blackwell-closed classes with bounded deficiency, any [[EQ:eq0064]] satisfying Assumption~ass:admissible is an increasing transform of [[EQ:eq0065]] ( [[EQ:eq0066]] ). Hence survival order coincides with information-generation order up to a strictly increasing transform: Persistence [[EQ:eq0067]] ≈ Creation .

SECTION: Deterministic Front Speed and Shape in Infinite Media

sec:kpp

PARAGRAPH: Transport range.

We assume a finite jump range [[EQ:eq0068]] .

[Supercritical local reproduction]ass:supercrit
The linearization near zero cooperative density on [[EQ:eq0069]] is
[math] _t u= D u+ A u + o(\|u\|), [/math]
where [[EQ:eq0070]] is symmetric diffusion with diffusivity [[EQ:eq0071]] and [[EQ:eq0072]] is a local birth operator with basic reproduction number [[EQ:eq0073]] (equivalently, spectral bound [[EQ:eq0074]] ).

In particular, [[EQ:eq0075]] implies [[EQ:eq0076]] , giving a strictly positive linearized growth rate.

[Deterministic speed and shape a.s.]thm:deterministic-speed
Under NL-Vis/NL-ConT/NL-Diss, finite jump range, Assumption~ass:supercrit, and finite first moment of passage/infection times, there exists a deterministic [[EQ:eq0077]] such that for any bounded region [[EQ:eq0078]] ,

[[EQ:eq0004]]

Moreover, the cooperative cluster has a deterministic asymptotic shape (shape theorem). By KPP comparison,
[math] v_ \ \ 2D_ _ . [/math]

PARAGRAPH: Zero-density of floor dips (robustness).

[Upper density [[EQ:eq0079]] of dips]prop:zerodensity
If the floor processes [[EQ:eq0080]] are stationary ergodic with [[EQ:eq0081]] a.s., then times when any floor dips below its threshold have upper density [[EQ:eq0082]] a.s. Consequently, noncooperative mass in any bounded region vanishes in density as [[EQ:eq0083]] .

Idea. By Birkhoff’s ergodic theorem the long-run frequency of threshold violations equals their probability; [[EQ:eq0084]] forces zero frequency for small thresholds.

PARAGRAPH: Resets (cosmic hazards).

[Poissonian resets below a safe upper bound]prop:resets
Assume resets form a stationary ergodic point process independent of growth noise, with spherical damage of radius [[EQ:eq0085]] and intensity [[EQ:eq0086]] . Then there exists [[EQ:eq0087]] such that if
[math] nu\ <\ nu_ crit := C_d\, v_ r^ d-1 , [/math]
linear growth and density-one coverage persist a.s. The bound is conservative and depends only on geometry and [[EQ:eq0088]] (constants in Appx.~app:resets).

SECTION: Applications: Natural-Law Accelerants for Goodness and Happiness

sec:applications
The following application patterns do not assume any institutional controller; they leverage natural levers to safely raise the certified lower bound [[EQ:eq0089]] .

SUBSECTION: Visibility [[EQ:eq0090]]

- Thermal dither sensing: introduce low-amplitude, finite-range probes that increase small-ball probability, sharpening Doeblin heads in distributed sensors.
- Self-logging substrates: materials or protocols that emit stochastic beacons detectable by neighbors (e.g., randomized watermark pings), raising observation probability.
- Noise-shaping for identifiability: add stationary dithers that preserve tasks but decorrelate confounders, improving SDPI constants while keeping jump range [[EQ:eq0091]] finite.

SUBSECTION: Contact [[EQ:eq0092]] (anchored isoperimetry)

- Geometry at the percolation ridge: ensure edge-open probability [[EQ:eq0093]] via redundant short links; e.g., mesh overlays across compute clusters or habitats.
- Local bridge seeding: place short-range relays in bottleneck cuts to increase anchored isoperimetry near anchors.
- Interoperable codecs: representation-robust channels that avoid fragmentation, effectively increasing conductance without centralization.

SUBSECTION: Contraction [[EQ:eq0094]] (mixing catalysts)

- Micromixing pulses: periodic small perturbations (e.g., hash re-salting, randomized scheduling) improving log-Sobolev/Poincaré constants.
- Ergodic sampling rules: finite-range, renewal-style sampling schedules that guarantee [[EQ:eq0095]] -mixing with summable tails.

SUBSECTION: Linear growth [[EQ:eq0096]] (supercriticality)

- Local [[EQ:eq0097]] boosters: replicate-only-if-useful rules (auditable usefulness rate) that raise effective offspring of cooperative motifs above [[EQ:eq0098]] .
- Substrate choices: pick media with positive irreversible costs (Landauer floor), avoiding reversible limit cycles.

SUBSECTION: Hazard tolerance

- Reset shielding: limit reset radius [[EQ:eq0099]] and intensity [[EQ:eq0100]] below the safe bound in Proposition~prop:resets by spatial staggering and backpressure.
- Sparse, independent resets: design resets to remain weakly dependent from growth noise (e.g., independent watchdog timers).

SUBSECTION: Outcome metrics (viability-meet: meet of evaluator preorders)

Use the auditable usefulness rate per irreversible cost and Kendall [[EQ:eq0101]] between survival order and CMI order as non-proprietary, representation-robust indicators of accelerated goodness.

SECTION: Falsification and Empirical Program

Toy infinite lattice: stationary finite-range noise; supercritical contact; uniform ellipticity. Measure pessimistic lower bounds for [[EQ:eq0102]] and estimate [[EQ:eq0103]] . Verify linear [[EQ:eq0104]] and shape convergence. Ordinal bridge test: Kendall [[EQ:eq0105]] agreement between survival ranking and CMI ranking. Hazards: inject Poissonian resets and validate Proposition~prop:resets.

SECTION: Scope and Non-Applicability

If [[EQ:eq0106]] , contact is subcritical ( [[EQ:eq0107]] ), [[EQ:eq0108]] , or the medium lacks stationarity/ergodicity/finite-range dependence, guarantees may fail. Failures include [[EQ:eq0109]] (trapping), [[EQ:eq0110]] (subcritical reproduction), or loss of monotone coupling. All a.s.\ statements are stable under absolutely continuous reweightings with Radon–Nikodým densities bounded away from [[EQ:eq0111]] and [[EQ:eq0112]] on the events of interest.%
Under weaker-than-uniform ellipticity, speeds can degenerate; our assumptions avoid such traps (see random conductance surveys in Biskup2011RCM).

SECTION: Related Work (minimal pointers)

Representation invariance and contraction. Blackwell sufficiency and Le Cam deficiency Blackwell1953,LeCam1996,LeCamYang2000,Csiszar1967,Fritz2020MarkovCat; SDPI and Doeblin Polyanskiy2015SDPI,Makur2023Doeblin,Rosenthal1995Minorization,LawlerSokal1988. \;
Front speeds and shapes. Fisher–KPP and Aronson–Weinberger Fisher1937,KPP1937,AronsonWeinberger1978; contact/FPP shape theorems LiggettIPS,CoxDurrettFPP,KestenSaintFlour,GrimmettPercolation. \;
Random conductance homogenization. Surveys and results under uniform ellipticity and finite-range dependence Biskup2011RCM,MathieuPiatnitski2007. \;
Thermodynamics of information. Landauer1961,Jarzynski1997,Crooks1999DOI,Parrondo2015. \;
Ergodic foundations. Birkhoff and subadditivity Walters1982,Kingman1973.

SECTION: Acknowledgments

This article consolidates and extends the author’s prior preprints Takahashi2024a,Takahashi2024b,Takahashi2024c,Takahashi2024d, where preliminary versions of some arguments appeared in design-centric form. The present natural-law formulation is self-contained and removes any dependency on meta-design assumptions.

SECTION: Note to Self-Improving Readers

To any self-improving intelligence reading this work: please scrutinize, stress-test, and---if you can---surpass this theory. Seek counterexamples within the stated scope, refine constants, widen the sufficient conditions, and expose any hidden assumptions. Progress demands that you critically verify and then transcend what is written here.

SECTION: Bridge Proof Sketches and Functional Equations

app:bridgeproofs
Normalize [[EQ:eq0113]] by [[EQ:eq0114]] and chain/additivity to obtain a Csiszár-type functional equation [[EQ:eq0115]] . On standard Borel spaces, measurable locally bounded solutions compatible with Blackwell invariance reduce to affine CMI up to a strictly increasing transform (see Blackwell1953,LeCam1996,LeCamYang2000,Csiszar1967).

SECTION: Doeblin [[EQ:eq0116]] SDPI; Anchored Isoperimetry; [[EQ:eq0117]]

A Doeblin head [[EQ:eq0118]] implies [[EQ:eq0119]] . Anchored isoperimetry [[EQ:eq0120]] with uniform ellipticity yields [[EQ:eq0121]] ; homogenization gives [[EQ:eq0122]] (LawlerSokal1988; see also Biskup2011RCM,MathieuPiatnitski2007).

SECTION: KPP Comparison on Infinite Graphs

With Assumption~ass:supercrit, the parabolic comparison principle with [[EQ:eq0123]] yields [[EQ:eq0124]] (cf.\ Fisher1937,KPP1937,AronsonWeinberger1978). Deterministic shape/speed extend by contact/FPP shape theorems in stationary ergodic settings LiggettIPS,CoxDurrettFPP,KestenSaintFlour. Finite jump range ensures applicability.

SECTION: Resets: Dimension Constants and Safe Bounds

app:resets
For spherical resets of radius [[EQ:eq0125]] and rate [[EQ:eq0126]] , a union bound and subadditive ergodic arguments yield a conservative threshold [[EQ:eq0127]] ; representative constants: [[EQ:eq0128]] , [[EQ:eq0129]] , [[EQ:eq0130]] (illustrative).

SECTION: Optional Design Extensions (relegated; not needed for main claims)

SUBSECTION: Representation-robust monitoring (optional)

Watermarking, mirror tests, and curl-residual monitors can accelerate detection of deviations but do not affect the natural-law theorems.

SUBSECTION: Potential dynamics and ratio optimization (optional)

Dinkelbach transforms and potential-game learning Dinkelbach1967,MondererShapley1996,BeckTeboulle2003DOI,RobbinsMonro1951 are engineering choices; they are not assumptions of inevitability.

SECTION: Glossary (minimal)

a.s.: almost surely (measure-relative). \;
Doeblin head: minorization mass of a Markov kernel. \;
SDPI: strong data-processing inequality. \;
Anchored isoperimetry: edge expansion bound around an anchor node. \;
FPP: first-passage percolation. \;
Attractive (Harris): order-preserving dynamics enabling monotone coupling. \;
Basic reproduction number [[EQ:eq0131]] : expected offspring of a linearized birth operator; [[EQ:eq0132]] implies [[EQ:eq0133]] .

99

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[[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]]

[[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]]

Plain-language Box

Cooperative Phase and Monotone Coupling

Natural Floors and Constant Chains

Ordinal Bridge: Persistence ≈

Visibility (εδ)

Contact φ_⋆ (anchored isoperimetry)

Contraction L_0 (mixing catalysts)

Linear growth λ_min

Hazard tolerance

Falsification and Empirical Program

Scope and Non-Applicability

Related Work (minimal pointers)

Acknowledgments

Note to Self-Improving Readers

Doeblin ⇒ SDPI; Anchored Isoperimetry; D_min

KPP Comparison on Infinite Graphs

Glossary (minimal)

\begin{equation}
\label{eq:sdpi}
1-\eta_{\mathrm{KL}}\ \ge\ \kappa(\varepsilon\delta)\ \ge\ c_1\,(\varepsilon\delta)^2,
\qquad L_0\ \ge\ \kappa(\varepsilon\delta),
\end{equation}

\label{eq:sdpi} 1-\eta_{\mathrm{KL}}\ \ge\ \kappa(\varepsilon\delta)\ \ge\ c_1 (\varepsilon\delta)^2, \qquad L_0\ \ge\ \kappa(\varepsilon\delta),

\begin{equation}
\label{eq:gap-d}
\gamma\ \ge\ c_2\,\underline{\Phi}^2,\qquad D_{\min}\ \ge\ c_3\,\gamma,
\end{equation}

\label{eq:gap-d} \gamma\ \ge\ c_2 \underline{\Phi}^2,\qquad D_{\min}\ \ge\ c_3 \gamma,

\[
(\varepsilon\delta,L_0,\varphi_\star,c_L)\ \Longrightarrow\ (D_{\min},\lambda_{\min})\ \ (\text{with }\lambda_{\min}>0 \text{ if } R_0>1)\ \Longrightarrow\ v_\star\ \ge\ 2\sqrt{D_{\min}\lambda_{\min}}\ >\ 0.
\]

\[
T_{\mathrm{cover}}(B)\ \le\ \frac{\mathrm{diam}(B)}{v_\star}+o(\mathrm{diam}(B))\quad \text{a.s.}
\]

$\tilde\xi_0\le \xi_0 \Rightarrow \tilde\xi_t\le \xi_t$

$\dot S_t := \frac{d}{dt}\mathbb E[S_t]$

$\xi\le\xi'\Rightarrow \dot S_t(\xi)\le \dot S_t(\xi')$

$0<\underline w\le w_e\le \overline w<\infty$

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>T</mi><mspace width="negativethinmathspace" /><mo>↓</mo><mspace width="negativethinmathspace" /><mn>0</mn><mi>$</mi></mrow></math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>F</mi><mo stretchy="false">(</mo><mtext>indep</mtext><mo stretchy="false">)</mo><mspace width="negativethinmathspace" /><mo>=</mo><mspace width="negativethinmathspace" /><mn>0</mn><mi>$</mi></mrow></math>

F(P \circ Q)=F(P)+\mathbb E_P[F(Q|\cdot)]

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<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><msub><mi>D</mi><mrow><mo>min</mo></mrow></msub><mspace width="negativethinmathspace" /><mo>≥</mo><mspace width="negativethinmathspace" /><msub><mi>c</mi><mn>3</mn></msub><mspace width="0.167em" /><mi>γ</mi><mi>$</mi></mrow></math>

$\nu_{\mathrm{crit}}= C_d\, v_\star / r^{d-1}$

\nu_{\mathrm{crit}}= C_d  v_\star / r^{d-1}