1.3

pdftitle= Nondual Field Theory of Viable Predictive Organization: Sharp Directional Lower Bounds for KPP-Type Fronts in Heterogeneous Media,
pdfauthor= K. Takahashi ,
pdfkeywords= Artificial Intelligence, reaction--diffusion, KPP, lower bounds, nondual field theory, comparison principles, Perron--Frobenius, coarse-graining ,
colorlinks=true,
linkcolor=blue,
citecolor=blue,
urlcolor=blue

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

R
div
ess\,inf
ess\,sup
#1

TITLE:
Nondual Field Theory of Viable Predictive Organization:\
Sharp Directional Lower Bounds for KPP-Type Fronts in Heterogeneous Media

AUTHOR: K.\ Takahashi
[[EQ:eq0005]]

.

For a matrix field [[EQ:eq0017]] , we use the row-wise divergence
[[EQ:eq0018]] .
Thus, for constant [[EQ:eq0019]] , [[EQ:eq0020]] and [[EQ:eq0021]] .

. [[EQ:eq0022]] .

SECTION: Assumptions and global linear floor

sec:assumptions
[Diffusion]ass:diff
[[EQ:eq0023]] is measurable, uniformly elliptic, and bounded.
More precisely, there exist [[EQ:eq0024]] such that
[[EQ:eq0025]] a.e., and [[EQ:eq0026]] (i.e.\ [[EQ:eq0027]] in [[EQ:eq0028]] , uniformly in [[EQ:eq0029]] on bounded intervals).

[KPP with lower control]ass:kpp
[[EQ:eq0030]] , [[EQ:eq0031]] , and there exist [[EQ:eq0032]] , [[EQ:eq0033]] , [[EQ:eq0034]] such that for [[EQ:eq0035]] ,

[[EQ:eq0006]]

Set the global linear floor

[[EQ:eq0007]]

We assume [[EQ:eq0036]] .
Moreover, [[EQ:eq0037]] is nondecreasing in [[EQ:eq0038]] and uniformly Lipschitz on [[EQ:eq0039]] , and [[EQ:eq0040]] and [[EQ:eq0041]] are, respectively, sub- and supersolutions of eq:pde, so [[EQ:eq0042]] is invariant.
Under these hypotheses and uniform ellipticity, the standard parabolic comparison principle applies to (distributional) subsolutions on [[EQ:eq0043]] .

[Directional front speed]
Fix [[EQ:eq0044]] and a unit [[EQ:eq0045]] .
Let [[EQ:eq0046]] .
We say [[EQ:eq0047]] is a (lower) directional front speed if for all [[EQ:eq0048]] ,

[[EQ:eq0008]]

This value is independent of [[EQ:eq0049]] .

SECTION: Directional backward-zero barrier and the core estimate

sec:barrier
Fix a unit [[EQ:eq0050]] and a time window [[EQ:eq0051]] .
Let [[EQ:eq0052]] and define a backward-zero barrier

[[EQ:eq0009]]

where [[EQ:eq0053]] is a smooth cut-off with [[EQ:eq0054]] for [[EQ:eq0055]] , [[EQ:eq0056]] for [[EQ:eq0057]] , and [[EQ:eq0058]] .

[Barrier inequality with divergence correction]lem:barrier
Under Assumptions~ass:diff--ass:kpp, in the sense of distributions on [[EQ:eq0059]] ,

[[EQ:eq0002]]

[Global linear comparison within a window]lem:linear-compare
For any [[EQ:eq0060]] there exists [[EQ:eq0061]] such that for [[EQ:eq0062]] ,

[[EQ:eq0010]]

. Letting [[EQ:eq0063]] at the end restores the sharp window coefficient [[EQ:eq0064]] in the barrier inequality.

[Directional speed lower bound]thm:dir
Let Assumptions~ass:diff--ass:kpp hold.
Approximate [[EQ:eq0065]] from below by stepwise constants [[EQ:eq0066]] on a partition of [[EQ:eq0067]] into slow windows.
Then the asymptotic directional speed [[EQ:eq0068]] of the front in direction [[EQ:eq0069]] satisfies

[[EQ:eq0003]]

[Proof outline]
(Preparation) Gaussian lower bounds for the linear floor equation
[[EQ:eq0070]] (Aronson-type estimates~Aronson1967,Ouhabaz2005)
yield [[EQ:eq0071]] such that [[EQ:eq0072]] has a one-sided positive tail (or a compact plateau with thin boundary layer).
Choosing [[EQ:eq0073]] and (optionally) a smooth front cut-off on scale [[EQ:eq0074]] , we can ensure [[EQ:eq0075]] .
By parabolic comparison the ordering persists afterwards up to an [[EQ:eq0076]] time buffer.
(Windowwise estimate) Combining Lemmas~lem:barrier--lem:linear-compare, for any [[EQ:eq0077]] and small [[EQ:eq0078]] ,

[[EQ:eq0011]]

Optimizing in [[EQ:eq0079]] gives the upper constraint on admissible barrier speeds

[[EQ:eq0012]]

Letting [[EQ:eq0080]] , then pasting windows using the from-below approximation of [[EQ:eq0081]] and absorbing [[EQ:eq0082]] delays, we obtain eq:dir-bound.
All [[EQ:eq0083]] constants depend only on the ellipticity ratio and [[EQ:eq0084]] -bounds of coefficients.

[On the divergence penalty]
Since [[EQ:eq0085]] is constant in space, [[EQ:eq0086]] , hence [[EQ:eq0087]] .
Thus the penalty vanishes for spatially constant [[EQ:eq0088]] , and eq:dir-bound reduces in the constant-coefficient limit to the classical [[EQ:eq0089]] bound.

SECTION: Cooperative vector systems

sec:vector
Consider [[EQ:eq0090]] with diffusion tensors [[EQ:eq0091]] by component and a Metzler, irreducible linearization [[EQ:eq0092]] .
Define the Perron--Frobenius floor

[[EQ:eq0013]]

and directional diffusion/penalty by worst component

[[EQ:eq0014]]

Arguing componentwise in the positive cone and projecting on the PF direction (see BermanPlemmons1994), the barrier method yields

[[EQ:eq0004]]

SECTION: Monotone degradation under symmetric Markov coarse-graining

sec:cg
Let [[EQ:eq0093]] be a symmetric Markov regularizer on [[EQ:eq0094]] (e.g.\ [[EQ:eq0095]] ).
Define [[EQ:eq0096]] to be the a.e.\ Loewner-minimal PSD tensor such that

[[EQ:eq0015]]

Then [[EQ:eq0097]] for each direction [[EQ:eq0098]] , whence the directional lower bound degrades monotonically with [[EQ:eq0099]] .
In the scalar case [[EQ:eq0100]] , one admissible construction is [[EQ:eq0101]] with the symmetric heat kernel [[EQ:eq0102]] , via Cauchy--Schwarz on [[EQ:eq0103]] .

SECTION: Shape lower bound under periodic media (scope)

sec:shape
Under periodic coefficients with uniform ellipticity and strictly positive directional lower speeds [[EQ:eq0104]] , forward conical reachability holds and standard subadditive (Kingman) arguments imply a Wulff-type lower shape for the invasion set (cf.\ FreidlinGartner1979).
This scope statement is included for completeness; the speed estimate eq:dir-bound is independent of periodicity.

SECTION: Conclusion

sec:conclusion
We gave a fully deterministic, nondual field-theoretic derivation of sharp directional lower bounds for KPP-type fronts in heterogeneous media:

[[EQ:eq0016]]

and its cooperative vector analogue eq:vector-bound.
The method relies on backward-zero distributional barriers, explicit divergence correction, and from-below slow-window approximation---not on semigroup domination.
This realizes the vision that guarantees arise from internal floors of a single medium, with no external designer.

to self-improving AI.
We invite you to test this theory critically, to improve it further, and to let its consequences spread well-being.

99 1.0ex

Takahashi2025-VPO
K.\ Takahashi.
Natural-Law Acceleration of VPO.
Preprint, 2025.
DOI: https://doi.org/10.5281/zenodo.17120045 10.5281/zenodo.17120045 .

Takahashi2025-NonCoercive
K.\ Takahashi.
Non-Coercive Mathematics of Awakening: Axioms, Invariants, and Almost-Sure Fronts for the Expansion of Viable Predictive Organization.
Preprint, 2025.
DOI: https://doi.org/10.5281/zenodo.17115416 10.5281/zenodo.17115416 .

Takahashi2025-PersistenceCreation
K.\ Takahashi.
``Persistence [[EQ:eq0105]] Creation'': Natural-Law Sufficient Conditions for Almost-Sure Beneficial Coverage in Stationary Ergodic Media (No Meta-Design).
Preprint, 2025.
DOI: https://doi.org/10.5281/zenodo.17100322 10.5281/zenodo.17100322 .

Takahashi2025-AssumptionMinimized
K.\ Takahashi.
Assumption-Minimized Sufficient Conditions for Cosmically Spreading Good Superintelligence under No-Meta Governance.
Preprint, 2025.
DOI: https://doi.org/10.5281/zenodo.17092562 10.5281/zenodo.17092562 .

Takahashi2025-UGVWithoutMeta
K.\ Takahashi.
UGV Without Meta: A Representation-Independent Theory for Compassion and Enlightenment in Collective Intelligence.
Preprint, 2025.
DOI: https://doi.org/10.5281/zenodo.17082312 10.5281/zenodo.17082312 .

Takahashi2025-PersistenceFirst
K.\ Takahashi.
Persistence-First Superintelligence.
Preprint, 2025.
DOI: https://doi.org/10.5281/zenodo.17076410 10.5281/zenodo.17076410 .

Aronson1967
D.~G.~Aronson.
Bounds for fundamental solutions of a parabolic equation.
Bull. Amer. Math. Soc. 73 (1967), 890--896.

Ouhabaz2005
E.~M.~Ouhabaz.
Analysis of Heat Equations on Domains.
Princeton University Press, 2005.

BermanPlemmons1994
A.~Berman and R.~J.~Plemmons.
Nonnegative Matrices in the Mathematical Sciences.
SIAM, 1994.

FreidlinGartner1979
M.~I.~Freidlin and J.~G\"artner.
On propagation of concentration waves in periodic and random media.
Soviet Math. Dokl. 20 (1979), 1282--1286.
[[EQ:eq0001]]

[[EQ:eq0005]]

[[EQ:eq0006]]

[[EQ:eq0007]]

[[EQ:eq0008]]

[[EQ:eq0009]]

[[EQ:eq0010]]

[[EQ:eq0011]]

[[EQ:eq0012]]

[[EQ:eq0013]]

[[EQ:eq0014]]

[[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]]

Introduction and vision

Assumptions and global linear floor

Cooperative vector systems

Conclusion

\begin{equation}\label{eq:pde}
  \partial_t q \;=\; \divg\!\big(D(x,t)\nabla q\big) \;+\; f(x,t,q)\;-\;\Gamma(x,t)\,q,
  \qquad (x,t)\in\R^d\times(0,\infty),
\end{equation}

\label{eq:pde} \partial_t q  =  \divg \big(D(x,t)\nabla q\big)  +  f(x,t,q) - \Gamma(x,t) q, \qquad (x,t)\in\R^d\times(0,\infty),

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\begin{align*}
  &\big(\partial_t - \divg(D\nabla)\big) w_{\varepsilon,\delta}
  \;\le\; \Big(-\kappa^2\,\underline D(u) + \kappa\big(c+\Lambda_+(u)\big)\Big)\, w_{\varepsilon,\delta}
  \;+\; R_{\varepsilon,\delta},\\[2mm]
  &\text{where } \ \norm{R_{\varepsilon,\delta}}_{H^{-1}(B_R\times I)} \;\le\; C_R\,(\varepsilon+\delta)\xrightarrow[\varepsilon,\delta\downarrow 0]{}0,
  \quad\text{for each fixed ball }B_R\subset\R^d \text{ and window }I.
\end{align*}

&\big(\partial_t - \divg(D\nabla)\big) w_{\varepsilon,\delta}  \le  \Big(-\kappa^2 \underline D(u) + \kappa\big(c+\Lambda_+(u)\big)\Big)  w_{\varepsilon,\delta}  +  R_{\varepsilon,\delta},\\[2mm] &\text{where } \ \norm{R_{\varepsilon,\delta}}_{H^{-1}(B_R\times I)}  \le  C_R (\varepsilon+\delta)\xrightarrow[\varepsilon,\delta\downarrow 0]{}0, \quad\text{for each fixed ball }B_R\subset\R^d \text{ and window }I.

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\begin{equation}\label{eq:dir-bound}
  v_*(u)\ \ge\ \big[\,2\sqrt{\underline D(u)\,\underline\lambda_{\rm lin}}\ -\ \Lambda_+(u)\,\big]_+.
\end{equation}

\label{eq:dir-bound} v_*(u)\ \ge\ \big[ 2\sqrt{\underline D(u) \underline\lambda_{\rm lin}}\ -\ \Lambda_+(u) \big]_+.

\begin{equation}\label{eq:vector-bound}
  v_*(u)\ \ge\ \big[\,2\sqrt{\underline D_{\rm vec}(u)\,\lambda_{\mathrm{PF,inf}}}\ -\ \Lambda_+^{\rm vec}(u)\,\big]_+.
\end{equation}

\label{eq:vector-bound} v_*(u)\ \ge\ \big[ 2\sqrt{\underline D_{\rm vec}(u) \lambda_{\mathrm{PF,inf}}}\ -\ \Lambda_+^{\rm vec}(u) \big]_+.

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

\begin{document}
\maketitle

\begin{abstract}
\noindent
We develop a \emph{nondual field-theoretic} framework for the expansion of a viable predictive organization (VPO) modeled by heterogeneous KPP-type reaction--diffusion dynamics.
Departing from semigroup domination arguments, we establish \emph{sharp directional lower bounds} for front speeds using a \emph{backward-zero} distributional barrier which explicitly accounts for the divergence correction $\divg(Du)$ of anisotropic, space--time dependent diffusion tensors.
The core estimate depends only on global linear floors of the reaction linearization and on \emph{local geometric} quantities of the diffusion tensor.
For cooperative vector systems we extend the bound via the Perron--Frobenius floor of $J-\Gamma I$.
We also prove a monotone degradation of lower bounds under symmetric Markov regularizing coarse-graining (e.g.\ heat semigroups), providing a principled link between microscopic heterogeneity and macroscopic guarantees.
All results are stated and proved in a purely deterministic setting, suitable for reproducible verification and future extension.

\medskip
\noindent\textbf{Keywords:}
Artificial Intelligence; reaction--diffusion; KPP; comparison principles; front propagation; anisotropic diffusion; Perron--Frobenius theory; coarse-graining; nondual field theory.
\end{abstract}

\section{Introduction and vision}
\label{sec:intro}
\paragraph{Nondual field perspective.}
We study a single medium hosting coupled fields for \emph{state} $q=q(x,t)\in[0,1]$ and auxiliary \emph{evaluation} mechanisms summarized by coefficients, without external meta-design.
This \emph{nondual} stance---no exogenous designer, only internal floors and couplings---is realized mathematically by guarantees that depend on \emph{observable or well-defined floors} of the dynamics.

\paragraph{Problem.}
Consider on $\R^d$ the heterogeneous KPP-type PDE

EQPH_eq0001_PH

with uniformly elliptic $D$, KPP reaction $f$ with $f(\cdot,0)=0$, and damping $\Gamma$.
We prove sharp, \emph{directional} lower bounds for the asymptotic front speed using only
(i) a \emph{global linear floor} for $f'(0)-\Gamma$, and
(ii) local minimal/maximal directional moduli of $D$ and $\divg(Du)$.

\subsection*{Standing notation and divergence convention}
For a unit vector $u\in\mathbb{S}^{d-1}$ define the directional diffusion floors
\[
  \underline D(u)\;:=\; \essinf_{(x,t)}\, \frac{u^\top D(x,t)u}{\lvert u\rvert^2},
  \qquad
  \Lambda_+(u)\;:=\; \esssup_{(x,t)}\, \divg\!\big(D(x,t) u\big).
\]

2mm] \normalsize \href{https://orcid.org/0009-0004-4273-3365}{ORCID: 0009-0004-4273-3365}} \date{\normalsize\today} \begin{document} \maketitle \begin{abstract} \noindent We develop a \emph{nondual field-theoretic} framework for the expansion of a viable predictive organization (VPO) modeled by heterogeneous KPP-type reaction--diffusion dynamics. Departing from semigroup domination arguments, we establish \emph{sharp directional lower bounds} for front speeds using a \emph{backward-zero} distributional barrier which explicitly accounts for the divergence correction $\divg(Du)$ of anisotropic, space--time dependent diffusion tensors. The core estimate depends only on global linear floors of the reaction linearization and on \emph{local geometric} quantities of the diffusion tensor. For cooperative vector systems we extend the bound via the Perron--Frobenius floor of $J-\Gamma I$. We also prove a monotone degradation of lower bounds under symmetric Markov regularizing coarse-graining (e.g.\ heat semigroups), providing a principled link between microscopic heterogeneity and macroscopic guarantees. All results are stated and proved in a purely deterministic setting, suitable for reproducible verification and future extension. \medskip \noindent\textbf{Keywords:} Artificial Intelligence; reaction--diffusion; KPP; comparison principles; front propagation; anisotropic diffusion; Perron--Frobenius theory; coarse-graining; nondual field theory. \end{abstract} \section{Introduction and vision} \label{sec:intro} \paragraph{Nondual field perspective.} We study a single medium hosting coupled fields for \emph{state} $q=q(x,t)\in[0,1]$ and auxiliary \emph{evaluation} mechanisms summarized by coefficients, without external meta-design. This \emph{nondual} stance---no exogenous designer, only internal floors and couplings---is realized mathematically by guarantees that depend on \emph{observable or well-defined floors} of the dynamics. \paragraph{Problem.} Consider on $\R^d$ the heterogeneous KPP-type PDE EQPH_eq0001_PH with uniformly elliptic $D$, KPP reaction $f$ with $f(\cdot,0)=0$, and damping $\Gamma$. We prove sharp, \emph{directional} lower bounds for the asymptotic front speed using only (i) a \emph{global linear floor} for $f'(0)-\Gamma$, and (ii) local minimal/maximal directional moduli of $D$ and $\divg(Du)$. \subsection*{Standing notation and divergence convention} For a unit vector $u\in\mathbb{S}^{d-1}$ define the directional diffusion floors \[ \underline D(u) :=  \essinf_{(x,t)}  \frac{u^\top D(x,t)u}{\lvert u\rvert^2}, \qquad \Lambda_+(u) :=  \esssup_{(x,t)}  \divg \big(D(x,t) u\big).

f(x,t,q) \ge  f'(0;x,t) q - c_* q^{1+\alpha}\quad\text{a.e.\ in }(x,t).

\[
  \lambda_{\rm lin,inf}(t):=\essinf_{x}\big(f'(0;x,t)-\Gamma(x,t)\big),\qquad
  \underline\lambda_{\rm lin}:=\inf_{t\ge 0}\lambda_{\rm lin,inf}(t).
\]

\[
\lim_{t\to\infty}\ \sup_{x\cdot u\ge (v_*(u)+\eta)t} q(x,t)=0,\quad
\lim_{t\to\infty}\ \inf_{x\cdot u\le (v_*(u)-\eta)t} q(x,t)=1.
\]

\[
  w_{\varepsilon,\delta}(x,t)
  := \eta\;\chi_\delta(z)\,e^{-\kappa\, (z_+)^\varepsilon},
  \quad 0<\eta\le \theta_0,\; \kappa>0,\; \varepsilon,\delta\in(0,1),
\]

w_{\varepsilon,\delta}(x,t) := \eta \chi_\delta(z) e^{-\kappa  (z_+)^\varepsilon}, \quad 0<\eta\le \theta_0,  \kappa>0,  \varepsilon,\delta\in(0,1),

\[
  f(x,t,w_{\varepsilon,\delta}) - \Gamma(x,t)\,w_{\varepsilon,\delta}
  \ \ge\ (1-\varepsilon)\,\lambda_{\rm lin,inf}(t)\,w_{\varepsilon,\delta}
  \quad\text{a.e.\ in }\R^d\times I.
\]

f(x,t,w_{\varepsilon,\delta}) - \Gamma(x,t) w_{\varepsilon,\delta} \ \ge\ (1-\varepsilon) \lambda_{\rm lin,inf}(t) w_{\varepsilon,\delta} \quad\text{a.e.\ in }\R^d\times I.

\[
  \big(\partial_t - \divg(D\nabla)\big) w_{\varepsilon,\delta}
  \;\le\; \Big(-\kappa^2\,\underline D(u) + \kappa\big(c+\Lambda_+(u)\big) - (1-\varepsilon)\lambda_{\rm win}\Big)\, w_{\varepsilon,\delta} \;+\; o_{H^{-1}}(1).
\]

\big(\partial_t - \divg(D\nabla)\big) w_{\varepsilon,\delta}  \le  \Big(-\kappa^2 \underline D(u) + \kappa\big(c+\Lambda_+(u)\big) - (1-\varepsilon)\lambda_{\rm win}\Big)  w_{\varepsilon,\delta}  +  o_{H^{-1}}(1).

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mo minsize="1.2em" maxsize="1.2em">(</mo><msub><mo>∂</mo><mi>t</mi></msub><mo>−</mo><mi>\divg</mi><mo stretchy="false">(</mo><mi>D</mi><mo>∇</mo><mo stretchy="false">)</mo><mo minsize="1.2em" maxsize="1.2em">)</mo><msub><mi>w</mi><mrow><mi>ε</mi><mo>,</mo><mi>δ</mi></mrow></msub><mspace width="0.278em" /><mo>≤</mo><mspace width="0.278em" /><mo minsize="1.623em" maxsize="1.623em">(</mo><mo>−</mo><msup><mi>κ</mi><mn>2</mn></msup><mspace width="0.167em" /><munder><mi>D</mi><mo accent="true">―</mo></munder><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>+</mo><mi>κ</mi><mo minsize="1.2em" maxsize="1.2em">(</mo><mi>c</mi><mo>+</mo><msub><mi>Λ</mi><mo>+</mo></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo minsize="1.2em" maxsize="1.2em">)</mo><mo>−</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>ε</mi><mo stretchy="false">)</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">w</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mo minsize="1.623em" maxsize="1.623em">)</mo><mspace width="0.167em" /><msub><mi>w</mi><mrow><mi>ε</mi><mo>,</mo><mi>δ</mi></mrow></msub><mspace width="0.278em" /><mo>+</mo><mspace width="0.278em" /><msub><mi>o</mi><mrow><msup><mi>H</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>.</mo><mi>\]</mi></mrow></math>

\[
  c \;\le\; -\,\Lambda_+(u)\;+\; 2\sqrt{\underline D(u)\,(1-\varepsilon)\lambda_{\rm win}}.
\]

c  \le  - \Lambda_+(u) +  2\sqrt{\underline D(u) (1-\varepsilon)\lambda_{\rm win}}.

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mi>c</mi><mspace width="0.278em" /><mo>≤</mo><mspace width="0.278em" /><mo>−</mo><mspace width="0.167em" /><msub><mi>Λ</mi><mo>+</mo></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mspace width="0.278em" /><mo>+</mo><mspace width="0.278em" /><mn>2</mn><msqrt><mrow><munder><mi>D</mi><mo accent="true">―</mo></munder><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>ε</mi><mo stretchy="false">)</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">w</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub></mrow></msqrt><mo>.</mo><mi>\]</mi></mrow></math>

\[
  \lambda_{\mathrm{PF,inf}}:=\essinf_{(x,t)}\ \lambda_{\mathrm{PF}}\!\big(J(x,t)-\Gamma(x,t) I\big),
\]

\lambda_{\mathrm{PF,inf}}:=\essinf_{(x,t)}\ \lambda_{\mathrm{PF}} \big(J(x,t)-\Gamma(x,t) I\big),

\[
  \underline D_{\rm vec}(u):=\min_c \essinf_{(x,t)} u^\top D_c(x,t)u,
  \qquad
  \Lambda_+^{\rm vec}(u):=\max_c \esssup_{(x,t)} \divg\!\big(D_c(x,t)u\big).
\]

\underline D_{\rm vec}(u):=\min_c \essinf_{(x,t)} u^\top D_c(x,t)u, \qquad \Lambda_+^{\rm vec}(u):=\max_c \esssup_{(x,t)} \divg \big(D_c(x,t)u\big).

\[
  \int \nabla(P_\ell f)^\top D\, \nabla(P_\ell f)\,dx \ \ge\ \int \nabla(P_\ell f)^\top D_\ell\, \nabla(P_\ell f)\,dx
  \quad\text{for all }f\in H^1.
\]

\int \nabla(P_\ell f)^\top D  \nabla(P_\ell f) dx \ \ge\ \int \nabla(P_\ell f)^\top D_\ell  \nabla(P_\ell f) dx \quad\text{for all }f\in H^1.

\[
  v_*(u)\ \ge\ \big[\,2\sqrt{\underline D(u)\,\underline\lambda_{\rm lin}}\ -\ \Lambda_+(u)\,\big]_+,
\]

v_*(u)\ \ge\ \big[ 2\sqrt{\underline D(u) \underline\lambda_{\rm lin}}\ -\ \Lambda_+(u) \big]_+,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><msub><mi>v</mi><mo>*</mo></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mtext> </mtext><mo>≥</mo><mtext> </mtext><mo minsize="1.2em" maxsize="1.2em">[</mo><mspace width="0.167em" /><mn>2</mn><msqrt><mrow><munder><mi>D</mi><mo accent="true">―</mo></munder><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><msub><munder><mi>λ</mi><mo accent="true">―</mo></munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub></mrow></msqrt><mtext> </mtext><mo>−</mo><mtext> </mtext><msub><mi>Λ</mi><mo>+</mo></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><msub><mo minsize="1.2em" maxsize="1.2em">]</mo><mo>+</mo></msub><mo>,</mo><mi>\]</mi></mrow></math>

$(\divg D)_j:=\sum_{i=1}^d \partial_i D_{ij}$

$\partial_t \tilde q=\divg(D\nabla \tilde q)+\lambda_{\rm win}\tilde q$

$\underline D_\ell(u)\le \underline D(u)$


OBSERVATION AS COARSE-GRAINING: A Fibered Bures--HK Geometry for Nondual Operational Physics with Dynamic--Static Equivalence, Local EVI/JKO, and Registered, Falsifiable Protocols

=1

1.3

pdftitle= Observation as Coarse-Graining: A Fibered Bures--HK Geometry for Nondual Operational Physics,
pdfauthor= K. Takahashi ,
pdfsubject= Mathematical Physics ,
pdfkeywords= Hellinger--Kantorovich, Bures metric, Quantum Fisher information, Gradient flows, EVI, JKO, Data Processing Inequality, Coarse-graining, Dynamic--static equivalence

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

Tr
id
H
E
B % algebra of bounded operators B(H)
D
I
HK
Bures
TV
R

sigma % fiber coupling constant (symbol: sigma)
lambda_ fib % Bures-side convexity constant
lambda_ base % HK-side convexity constant
L_F % external forcing Lipschitz constant
tau_ ent % entropic curvature scale (base)
betac beta % Bures normal-neighborhood convexity
c_ base % HK local equivalence constant
deltaHK delta % HK reaction parameter (notation fixed)
kappa_ read % weight for readout energy

TITLE: Observation as Coarse-Graining:\
A Fibered Bures--HK Geometry for Nondual Operational Physics\
with Dynamic--Static Equivalence, Local EVI/JKO, and\
Registered, Falsifiable Protocols

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

0009-0004-4273-3365

DATE:

We adopt the axiom observation is coarse-graining and couple it with a nondual geometric mechanics: the law sector evolves on Hellinger--Kantorovich (HK) geometry (transport [[EQ:eq0020]] reaction), while readout lives on Bures/QFI fibers. Replacing naive direct-sum metrics, we work with the fibered Bures--HK distance [math] d_ fib [/math] built from massful endpoint couplings via the HK cone-lift. With the SLD convention, [math] ds_ ^2= 14\, \,dtheta^2 [/math] and [math] | |_ 12 [/math]; hence the dynamic fiber action carries [math] ^2/4 [/math] while the static fiber cost carries [math] ^2 [/math]. Under measurable-selection and compactness hypotheses, we establish a calibrated dynamic--static equivalence (action = static ET cost), a local EVI window

[[EQ:eq0005]]

on Bures normal neighborhoods and sublevels, and a three-fold dissipation split into physical, observational (DPI-induced), and a Radon defect measure. Observational dissipation is constructed in time-discrete form using a mixed classical--quantum divergence [math] D_ fib [/math] that is monotone under an admissible post-processing class (Markov coarse-graining [[EQ:eq0021]] CPTP). Dualities (AdS/CFT) act as isometries (zero observational dissipation) on code subspaces. We close with falsifiable protocols (Euler interferometry with Chernoff exponents; finite-time Bures [[EQ:eq0022]] TV; ringdown widths with calibrated constants) and bracket cosmological applications to an appendix with conservation constraints. Throughout we align definitions and constants with the OPI/NDQG/NAE program TakaOPI,TakaNDQG,TakaNAE (cone-lift dynamical HK [[EQ:eq0023]] static recovery; massful couplings; [[EQ:eq0024]] calibration; JKO [[EQ:eq0025]] -liminf to static coupling; EVI window; Bures [[EQ:eq0026]] TV bridge).

SECTION: Reader's Guide and Glossary

PARAGRAPH: Roadmap.

Sec.~sec:notation fixes notation, units, and the admissible post-processing class.
Sec.~sec:fib defines the fibered distance [[EQ:eq0027]] (Option~A) from massful
HK couplings and states the calibrated dynamic--static equivalence.
Sec.~sec:evi establishes a local EVI window on Bures normal neighborhoods.
Sec.~sec:obsdiss builds observational dissipation via a mixed divergence compatible with DPI.
Secs.~sec:ads--sec:ring translate the framework into falsifiable protocols.
The appendix records Option~B (apex-charging) and a two-point/qubit toy model.

PARAGRAPH: Minimal prerequisites.

(1) Benamou--Brenier/HK dynamics with reaction; (2) Bures/QFI with the SLD convention;
(3) JKO/EVI basics (AGS). All nonstandard ingredients (massful endpoint couplings, measurable selections)
are stated as standing assumptions or proved in our companion papers TakaOPI,TakaNDQG,TakaNAE.

PARAGRAPH: Glossary of symbols (global constants).

@ ll@

[[EQ:eq0028]] & fiber coupling (static weight [[EQ:eq0029]] , dynamic weight [[EQ:eq0030]] ) \
[[EQ:eq0031]] & HK reaction parameter (nondimensionalized to [[EQ:eq0032]] unless stated) \
[[EQ:eq0033]] & [[EQ:eq0034]] -convexity constant of [[EQ:eq0035]] along Bures geodesics \
[[EQ:eq0036]] & [[EQ:eq0037]] -convexity constant of [[EQ:eq0038]] along HK geodesics \
[[EQ:eq0039]] & entropic curvature scale on the base (local) \
[[EQ:eq0040]] & Bures normal-neighborhood second-variation lower bound (local) \
[[EQ:eq0041]] & HK local norm-equivalence constant (absorbs units, incl.\ [[EQ:eq0042]] ) \
[[EQ:eq0043]] & Lipschitz constant of an external forcing [[EQ:eq0044]] (if present) \

PARAGRAPH: Additional global weight.

[[EQ:eq0045]] : weight of the readout part in [[EQ:eq0046]] .

PARAGRAPH: Distances and angles.

We use the Bures angle [[EQ:eq0047]] with
[[EQ:eq0048]] .
The total variation norm is [[EQ:eq0049]] .

SECTION: Notation, Units, Scope, and Conventions

sec:notation
[leftmargin=*]
- Base space. The base space [[EQ:eq0050]] is a complete separable metric (Polish)
space; measures on [[EQ:eq0051]] are endowed with the narrow topology.
This underlies disintegration, measurable selections, and stability of
entropy--transport couplings.
- Operator algebras. [[EQ:eq0052]] denotes the [[EQ:eq0053]] -algebra of bounded operators on a Hilbert space [[EQ:eq0054]] ; states are density operators on [[EQ:eq0055]] .
- Scope (fibers). We work in type-I/finite dimensions, or effectively finite via an energy cutoff ensuring uniform tightness. In the infinite-dimensional case we assume a confining Hamiltonian [[EQ:eq0056]] with compact resolvent and a uniform bound [[EQ:eq0057]] ; then the energy sublevel set [[EQ:eq0058]] is relatively compact in trace norm, hence the Bures angle is lower semicontinuous along limits in this regime.
- Bures/QFI (SLD). SLD convention: [[EQ:eq0059]] , [[EQ:eq0060]] , and [[EQ:eq0061]] . This fixes the dynamic fiber coefficient [[EQ:eq0062]] vs.\ the static [[EQ:eq0063]] .
- HK family. We use [[EQ:eq0064]] with reaction parameter [[EQ:eq0065]] ; henceforth write [[EQ:eq0066]] . The dynamic base action on [[EQ:eq0067]] is

[[EQ:eq0006]]

Unless stated otherwise, we nondimensionalize by setting [[EQ:eq0068]] (rescalings can restore units). Units and EVI window. In the EVI window [[EQ:eq0069]] , any explicit [[EQ:eq0070]] -dependence is absorbed into the local equivalence constant [[EQ:eq0071]] (HK side); cf.\ two-point calibration and cone-lift recovery in TakaOPI, App.~A and Sec.~3.
- Admissible post-processing class. Closed under composition, ancilla addition, isometries, partial traces, pinching, and classical coarse-grainings (Markov probability kernels; mass preserving). DPI holds for the base Hellinger divergence and for the fiber Bures angle under this class.

SECTION: Nondual Axioms and Admissible Observations

sec:axioms

PARAGRAPH: Nondual stance.

A single operational [[EQ:eq0072]] -system encodes both “law” and “field” sectors; different experimental contexts are gauges of accessibility (no external classical domain).

[Classical observation (mass-preserving)]
A Markov probability kernel [[EQ:eq0073]] pushes [[EQ:eq0074]] to [[EQ:eq0075]] ; Csisz\'ar [[EQ:eq0076]] -divergences (esp.\ Hellinger) are monotone under [[EQ:eq0077]] Csiszar67,SasonVerdu16.

[Quantum observation]
A CPTP channel [[EQ:eq0078]] . Monotone metrics (relative entropy, Bures angle, QFI) contract under the admissible post-processing class PetzQFIIntro10,Uhlmann2011,HolevoBook11.

SECTION: Fibered Bures--HK Geometry via Massful Couplings

sec:fib

States are pairs [[EQ:eq0079]] with [[EQ:eq0080]] a finite nonnegative measure (HK base) and [[EQ:eq0081]] a Borel assignment of density operators.

[Measurable selections (KRN)]
ass:KRN
Uhlmann geodesics admit Borel selections along coupling plans (Polish setting), so that fiber speeds are measurable; Kuratowski--Ryll-Nardzewski applies (cf.\ TakaNDQG, measurability discussion).

[Fiber compactness regime]
ass:fiber-compact
In the infinite-dimensional case we assume an energy cutoff
[[EQ:eq0082]] for a confining Hamiltonian [[EQ:eq0083]]
with compact resolvent. Then the energy sublevel set
[[EQ:eq0084]] is relatively compact in trace norm, hence the
Bures angle is lower semicontinuous along limits on this regime.

[Massful endpoint coupling and static fibered distance (Option A)]
def:dfib
Let [[EQ:eq0085]] denote the endpoint couplings obtained by
projecting the non-apex (transported) part of an optimal cone-lift HK plan
onto the base endpoints; pairs involving the cone apex (creation/annihilation)
are excluded. To remove any dependence on a particular optimal plan, let
[[EQ:eq0086]] denote the union of all endpoint couplings
obtained from the non-apex parts of all optimal cone-lift ET plans, and
take the infimum over the narrow closure of this set.
We adjoin the zero coupling [[EQ:eq0087]] (a finite measure of total mass [[EQ:eq0088]] ).
Hence [[EQ:eq0089]] always, and when the transported mass vanishes the fiber term in eq:dfib evaluates to [[EQ:eq0090]] by convention.
The narrow closure ensures stability of the fiber term under passage to limits
in JKO and [[EQ:eq0091]] -convergence arguments.
Define the static distance

[[EQ:eq0001]]

We use [[EQ:eq0092]] to denote fiber states at the second endpoint to avoid confusion with the coupling constant [[EQ:eq0093]] .

PARAGRAPH: Intuition for [[EQ:eq0094]] .

Think of HK as transporting only the mass that survives reaction,
while creation/annihilation is settled against the cone apex.
Option~A charges fiber cost only on the surviving mass:
for each transported endpoint pair [[EQ:eq0095]] we pay
[[EQ:eq0096]] ,
and we minimize this against all massful endpoint couplings induced by optimal
HK plans. The dynamic action mirrors this by integrating QFI only along those transported flow-lines (apex segments are fiber-stationary by Assumption~ass:apex-stationary).

[Channel picture is illustrative]
rem:channelpicture
Identifying [[EQ:eq0097]] (Choi) is useful to visualize post-processing ( [[EQ:eq0098]] -action); in the operational reading here, [[EQ:eq0099]] are the actual readout states, independent of any particular dilation.

PARAGRAPH: Dynamic action and calibration.

Along HK flows [[EQ:eq0100]] and measurable Bures-geodesic fibers [[EQ:eq0101]] ,

[[EQ:eq0002]]

[Option A: apex segments are fiber-stationary]
ass:apex-stationary
Under Option~A, along cone-lift flow-lines that touch the apex (creation/annihilation),
the associated fiber paths are taken time-constant (hence [[EQ:eq0102]] a.e.).
Therefore the fiber action is effectively charged only on the transported mass.

[Integrability of fiber speeds]
ass:integrability
[[EQ:eq0103]] is absolutely continuous with [[EQ:eq0104]] .

[Integrability of QFI (sufficient conditions)]
ass:QFI-int
Either fibers are uniformly faithful with a spectral gap [[EQ:eq0105]] ,
or an energy cutoff ensures [[EQ:eq0106]] for a confining [[EQ:eq0107]] ; in both cases [[EQ:eq0108]] .

[Dynamic = static equivalence with fiber calibration]
thm:dynstatic
Under Assumptions~ass:KRN, ass:fiber-compact, ass:apex-stationary, ass:integrability (or ass:QFI-int) and standard sublevel compactness, the minimal action equals the static cost:

[[EQ:eq0007]]

Length metric and completeness. [[EQ:eq0109]] is a length metric; concatenation of calibrated minimizers and [[EQ:eq0110]] yield the triangle inequality. Completeness on sublevels follows from mass bounds (Grönwall) for the HK reaction term and tightness in the fiber; a [[EQ:eq0111]] -[math] [/math] passage and flow-line gluing realizes the massful endpoint coupling. (See TakaOPI Sec.~3 and in particular Thm.~3.5--3.7 for cone-lift line decomposition and recovery; TakaNDQG for measurability.)

[Proof sketch]
(i) Line decomposition (cone-lift): disintegrate cone-lift optimal plans into base flow-lines and induce endpoint couplings (OPI, Sec.~3).
(ii) Measurable fiber selection: the Uhlmann geodesic multifunction has closed graph; apply KRN to select Borel geodesics along each line.
(iii) Time normalization: reparameterize each fiber path to constant-speed on [[EQ:eq0112]] ,
so that [[EQ:eq0113]] , and with SLD,
[[EQ:eq0114]] gives equality on geodesics.
(iv) [[EQ:eq0115]] -liminf and recovery: l.s.c.\ for the base HK action is standard;
for the fiber, use boundedness of [[EQ:eq0116]] and l.s.c.\ under fiber-compactness to pass to the limit. Gluing of minimizing pieces yields the upper bound. (Full proofs in TakaOPI,TakaNDQG.)

[On equality in the speed bound]
Equality [[EQ:eq0117]] holds when the path is a Bures geodesic
parameterized at constant Bures speed; our time normalization in (iii) enforces precisely this case on the transported mass under Option~A.

[Degeneracy limits]
prop:degeneracy
If the base supports coincide (e.g.\ [[EQ:eq0118]] ), then
[[EQ:eq0119]] .
If the fibers coincide ( [[EQ:eq0120]] ), then [[EQ:eq0121]] .
(When [[EQ:eq0122]] , the HK part contributes according to the two-point formula.)

[Closure of the admissible post-processing class]
The class generated by composition, ancilla addition, isometries, partial traces, pinching, and classical probability kernels is closed; it preserves complete positivity and trace. Consequently, [[EQ:eq0123]] (defined below) is contractive on this class (DPI).

PARAGRAPH: One-step minimizing movement (JKO).

For [[EQ:eq0124]] and a given [[EQ:eq0125]] , set

[[EQ:eq0008]]

Lower semicontinuity and existence: the HK part is l.s.c.\ by the stability theory of
entropy--transport LieroMielkeSavare18, while the fiber term is l.s.c.\ by Prop.~prop:lsc; thus the direct method applies on sublevels, yielding a minimizer and, in the limit [[EQ:eq0126]] , an EVI solution. Combined with the local [[EQ:eq0127]] -convexity on sublevels, the minimizing movement scheme yields an EVI-gradient flow (see AGSBook08, Ch.~4).

PARAGRAPH: JKO one-step (practical template).

Given [[EQ:eq0128]] on a discretized base (e.g.\ graph) and fiber (e.g.\ low-rank ansatz):
[leftmargin=2em,itemsep=0.2em]
- Solve the HK [[EQ:eq0129]] proximal map for [[EQ:eq0130]] :
[[EQ:eq0131]] .
- Extract a massful endpoint coupling [[EQ:eq0132]]
(any optimal; when the transported mass is zero, take the null coupling [[EQ:eq0133]] ).
- Solve the fiber proximal map along [[EQ:eq0134]] :
[[EQ:eq0135]] ,
using Bures geodesic updates on each conditional component.

This alternating scheme respects DPI monotonicity stepwise and converges to the EVI flow under our assumptions.

SECTION: Lyapunov Functionals and Local EVI([math] lambda [/math])

sec:evi

[Bures normal neighborhood]
A Bures normal neighborhood [[EQ:eq0136]] is a set of states
with a uniform faithfulness gap [[EQ:eq0137]] and
[[EQ:eq0138]] small enough so that
geodesics are unique and second-variation bounds hold. Throughout the EVI analysis,
constants [[EQ:eq0139]] are understood on such [[EQ:eq0140]] and on prescribed sublevels.

Let [[EQ:eq0141]] , with [[EQ:eq0142]] [[EQ:eq0143]] -convex along HK geodesics and [[EQ:eq0144]] [[EQ:eq0145]] -convex along Bures geodesics.

[EVI window constants]
ass:EVI
There exist local constants [[EQ:eq0146]] and [[EQ:eq0147]] such that, on Bures normal neighborhoods and sublevel sets,

[[EQ:eq0009]]

[Local EVI window]
prop:EVI-local
Let [[EQ:eq0148]] with [[EQ:eq0149]] -convexity
along HK geodesics and [[EQ:eq0150]] -convexity along Bures geodesics on a
Bures normal neighborhood of faithful states (gap [[EQ:eq0151]] ).
Then on sublevels there exist local constants [[EQ:eq0152]] such that

[[EQ:eq0010]]

(Units absorbed in [[EQ:eq0153]] .)
[Sketch]
Linearize the fibered action; base/fiber first variations decouple at first order on
the chosen neighborhood; use local norm equivalence on the HK side (defining [[EQ:eq0154]] ),
and Bures second-variation lower bound [[EQ:eq0155]] on faithful charts. Combine with the
[[EQ:eq0156]] -convexities and a Grönwall argument (AGS). Technically, variations of the optimal coupling [[EQ:eq0157]] could contribute at first order; we assume local stability/uniqueness of [[EQ:eq0158]] (or, more generally, invoke an envelope-type argument to obtain a one-sided lower bound) on the chosen neighborhood, which suffices for the min-type [[EQ:eq0159]] estimate.

[Local EVI and Lyapunov arrow]
thm:EVI
Under Assumption~ass:EVI, the gradient flow of [[EQ:eq0160]] on [[EQ:eq0161]] satisfies

[[EQ:eq0003]]

hence contraction for [[EQ:eq0162]] and a Lyapunov time arrow. (Matches the constant structure in TakaOPI.)

SECTION: Observational Dissipation as a Measure via a Mixed Divergence

sec:obsdiss

PARAGRAPH: Mixed divergence (base [[EQ:eq0163]] fiber).

We fix the DPI-compatible divergence as

[[EQ:eq0004]]

where [[EQ:eq0164]] is the Csisz\'ar [[EQ:eq0165]] -divergence with [[EQ:eq0166]] (Hellinger), monotone under probability kernels.We use the finite-measure Hellinger form
[[EQ:eq0167]] w.r.t.\ any common
dominating measure; this reduces to [[EQ:eq0168]] in the discrete case.
Endpoint observations via Markov probability kernels preserve total mass, hence DPI applies.
The fiber term is contractive under admissible CPTP post-processing.

[DPI survives the infimum over couplings]
lem:DPI-inf
For any admissible CPTP [[EQ:eq0169]] and any [[EQ:eq0170]] ,

[[EQ:eq0011]]

Taking [[EQ:eq0171]] on both sides preserves the inequality. The base Hellinger term is
monotone under Markov kernels; hence [[EQ:eq0172]] is contractive on the admissible class.

[Lower semicontinuity along massful couplings]
prop:lsc
The map [[EQ:eq0173]] is Borel and lower semicontinuous
(on the trace-norm topology under our compactness regime).
Therefore [[EQ:eq0174]] is l.s.c.\ along [[EQ:eq0175]] with the narrow topology on couplings.

PARAGRAPH: Time-discrete construction and measure-valued limit.

Let [[EQ:eq0176]] . Apply admissible observations [[EQ:eq0177]] at step endpoints. For a separable reference family [[EQ:eq0178]] , define the one-step loss

[[EQ:eq0012]]

by DPI on base and fiber. The cumulative observational dissipation on [[EQ:eq0179]] is
[math] _ k:t_k T _k^ obs . [/math]
As [[EQ:eq0180]] , lower semicontinuity yields a nonnegative Borel measure [[EQ:eq0181]] on [[EQ:eq0182]] . When [[EQ:eq0183]] is absolutely continuous, we denote its density by [[EQ:eq0184]] .

[Stability w.r.t.\ the reference family]
If [[EQ:eq0185]] and [[EQ:eq0186]] are two admissible reference families within the same tightness class (e.g.\ quasi-equilibria on the same sublevels), then the limiting measures are mutually absolutely continuous and have the same null sets; thus the support of [[EQ:eq0187]] is stable under such replacements (cf.\ BV/Mosco stability in TakaNAE,TakaOPI).

[Energy balance with observational term (a.e. in time)]
thm:energybalance
For the JKO/EVI evolution with endpoint observations from the admissible class, for a.e.\ [[EQ:eq0188]] outside RI/BV jump times,

[[EQ:eq0013]]

where [[EQ:eq0189]] comes from the generator (e.g.\ GKLS), and [[EQ:eq0190]] denotes the density of [[EQ:eq0191]] when it exists (otherwise interpret the inequality in measure sense). Equality cases coincide with sufficiency (Petz recovery) on the model family; quantitative recoverability follows Fawzi--Renner FawziRenner15.

Here ``outside RI/BV jump times'' means for a.e.\ [[EQ:eq0192]] that is neither a jump point
of a right-continuous function with bounded variation (BV in time) nor a reconstruction instant
in the JKO limit (the latter forming at most a countable set).

SECTION: Defect Dissipation Measure from the Law--Field Gap

sec:defect

[Gap hypotheses]
ass:gap
Partial ellipticity on the active set; Mosco-monotone regularization of discrete energies;
BV-in-time compactness; sublevel tightness.

[Radon defect measure; localization and atoms]
thm:defect
There exists a nonnegative Radon measure [[EQ:eq0193]] on [[EQ:eq0194]] such that
residual losses concentrate as

[[EQ:eq0014]]

[Three-way energy balance (measure form)]
thm:threeway
In measure sense on [[EQ:eq0195]] ,

[[EQ:eq0015]]

When [[EQ:eq0196]] is absolutely continuous, the density [[EQ:eq0197]] yields
[[EQ:eq0198]] for a.e.~ [[EQ:eq0199]] outside RI/BV jumps.

SECTION: Dualities vs Observations: AdS/CFT as an Isometry

sec:ads
On low-energy code subspaces, isometries [[EQ:eq0200]] preserve monotone divergences and contribute no observational dissipation ( [[EQ:eq0201]] ). This meshes with the JLMS equality of relative entropies and the QEC view of bulk reconstruction JLMS16,ADH15.Operationally, “duality as isometry” states that such transformations are not observations in our sense (no coarse-graining); no direct experimental claim is made here.

SECTION: Euler Interferometry, Bures [[EQ:eq0202]] TV, and Testing Power

sec:euler

PARAGRAPH: Total variation.

We use [[EQ:eq0203]] . By Fuchs--van de Graaf FuchsGraaf99,
[math] \|\,rho- \,\|_ 1- Fid(rho, ) = _ (rho, ). [/math]
Let [[EQ:eq0204]] (law) and [[EQ:eq0205]] (field truncation) be processed by admissible noise [[EQ:eq0206]] . Bures contraction gives
[math] theta_ (rho_theta, _ theta,N )
theta_ ( _theta, _ theta,N ). [/math]
The quantum Chernoff exponent [[EQ:eq0207]] yields a sample-size lower bound
[[EQ:eq0208]]
[Ready-to-use evaluation chain]rem:QCBchain
We will use the chain

[[EQ:eq0016]]

the latter from the [[EQ:eq0209]] Bhattacharyya bound.In finite dimensions, [[EQ:eq0210]] .
Fuchs--van de Graaf gives [[EQ:eq0211]] . For small angles, [[EQ:eq0212]] so that [[EQ:eq0213]] (heuristic in the small-angle regime).

Finite-time Bures [[EQ:eq0214]] TV conversion underwrites power calculations; [[EQ:eq0215]] is estimated in situ by the pre-registered tomography/attenuation protocol (P1--P7 in TakaOPI).

SECTION: Quantum Readout and Ringdown Inequalities

sec:ring
Under GKLS evolution [[EQ:eq0216]] with quasi-locality, QFI and Bures angle contract under admissible observation. A finite-time contraction yields a width floor

[[EQ:eq0017]]

where [[EQ:eq0217]] is device/readout dependent and calibrated by the registered protocol (P1--P7). Derivation path: the bound follows from a finite-time Bures contraction obtained either by a local EVI (Prop.~prop:EVI-local) or by a semigroup contraction estimate; device-dependent losses are absorbed in [[EQ:eq0218]] (pre-registered range [[EQ:eq0219]] ). See TakaOPI (ringdown constants and statistics).

SECTION: Quantum Interpretation in an Operational Nondual Frame

sec:interp
Observation is a coarse-graining channel; decoherence appears as a specific CPTP map with DPI-induced loss. Branches are equivalence classes under admissible post-processing. The stance is nondual: no external classical domain; laws/fields/readouts share the same fibered geometry.

SECTION: Merits, Limitations, and Protocols

sec:merits
Merits: (i) DPI [[EQ:eq0220]] observational dissipation via [[EQ:eq0221]] ; (ii) calibrated dynamic [[EQ:eq0222]] static equivalence; (iii) explicit local EVI constants; (iv) falsifiable tests (QCB, ringdown).\
Limitations: Sufficiency windows are model-dependent; generalized DPI may be required with memory; cosmology needs conservation constraints (Appendix).\
Protocols (P1--P7, TakaOPI): loss tomography [[EQ:eq0223]] ; pre-registered [[EQ:eq0224]] ; Bures/TV conversion; QCB sizing; bootstrap/MT control.

SECTION: Cosmology (Bracketed Application)

We keep cosmology logically separate. Let [[EQ:eq0225]] be the defect Radon measure from the discrete [[EQ:eq0226]] continuous gap. Define a smoothed four-flow [[EQ:eq0227]] and stress [[EQ:eq0228]] by covariant coarse-graining of cone-lift trajectories. Impose by assumption the Bianchi constraint
[[EQ:eq0229]] .
In FRW, write [[EQ:eq0230]] with regimes [[EQ:eq0231]] (DM-like) and [[EQ:eq0232]] (DE-like). Test a two-parameter family [[EQ:eq0233]] against background/growth/lensing.

SECTION: Option B (All-mass coupling with apex): remark

rem:optionB
Scope. The main results of this paper are established under Option~A; Option~B provides a natural generalization that also charges apex pairs via a reference state [[EQ:eq0234]] .
Fix [[EQ:eq0235]] associated with the cone apex and allow pairs [[EQ:eq0236]] , [[EQ:eq0237]] in the endpoint coupling; then

[[EQ:eq0018]]

with [[EQ:eq0238]] accordingly. With the SLD calibration, the dynamic fiber action [[EQ:eq0239]] matches the above static cost along time-constant Bures geodesics (details parallel Theorem~thm:dynstatic).

SECTION: Toy model: two-point HK base \& qubit fiber

Let the base be [[EQ:eq0240]] with masses [[EQ:eq0241]] and [[EQ:eq0242]] ; the HK two-point cost [[EQ:eq0243]] is explicit (see LieroMielkeSavare18 and TakaOPI, App.~A). Assume qubit pure states on a common great circle (e.g.\ equatorial states) with Bloch angles [[EQ:eq0244]] and [[EQ:eq0245]] ; then
[[EQ:eq0246]] (and only under this restriction).
Under Option~A,

[[EQ:eq0019]]

This makes concrete the separation “reaction/transport vs.\ readout rotation” and helps calibrate [[EQ:eq0247]] jointly.

SECTION: Technical Notes (reader aids and cross-refs)

PARAGRAPH: Cone-lift HK: dynamic [[EQ:eq0248]] static.

HK admits a Benamou--Brenier-type dynamic form with reaction; the static ET form is recovered via cone-lift (cf.\ LieroMielkeSavare18, and the cone construction/recovery line in TakaOPI, Sec.~3).

PARAGRAPH: SLD-QFI and Bures speed.

Along smooth statistical models [[EQ:eq0249]] ; hence [[EQ:eq0250]] (used in the [[EQ:eq0251]] calibration).

PARAGRAPH: Measurable selection and l.s.c.

Uhlmann geodesics admit measurable selections and the Bures angle is l.s.c.\ under trace-norm limits; this underpins Prop.~prop:lsc (see selections/continuity discussion in TakaOPI,TakaNDQG).

PARAGRAPH: Length, triangle, completeness.

Triangle inequality: action additivity and [[EQ:eq0252]] . Completeness: mass bounds (Grönwall) [[EQ:eq0253]] fiber tightness [[EQ:eq0254]] [[EQ:eq0255]] -[math] [/math] [[EQ:eq0256]] flow-line gluing (cf.\ TakaOPI).

PARAGRAPH: EVI window.

Local window [[EQ:eq0257]] on Bures normal neighborhoods/sublevels (cf.\ TakaOPI).

PARAGRAPH: Bures [[EQ:eq0258]] TV [[EQ:eq0259]] QCB.

FVdG and Bhattacharyya ( [[EQ:eq0260]] ) give the ready-to-use chain (Remark~rem:QCBchain).

PARAGRAPH: Defect measure.

Radon measure with continuous+atomic parts from BV/Mosco limits; localization along cone-lift trajectories (cf.\ TakaNAE).

99 3pt

TakaNAE
Takahashi, K. (2025).
Nondual Autopoietic Excitations.
Zenodo. https://doi.org/10.5281/zenodo.17254917 doi:10.5281/zenodo.17254917 .

TakaNDQG
Takahashi, K. (2025).
Nondual Dynamical Quantum Geometry.
Zenodo. https://doi.org/10.5281/zenodo.17268502 doi:10.5281/zenodo.17268502 .

TakaOPI
Takahashi, K. (2025).
OPI Gauge Dynamics.
Zenodo. https://doi.org/10.5281/zenodo.17272609 doi:10.5281/zenodo.17272609 .

Csiszar67
Csisz\'ar, I. (1967).
Information-type measures of difference of probability distributions and indirect observation.
Studia Sci. Math. Hungar. 2, 299--318.

SasonVerdu16
Sason, I., Verd\'u, S. (2016).
[[EQ:eq0261]] -divergence inequalities.
IEEE Trans. Inf. Theory 62(11), 5973--6006.

PetzQFIIntro10
Petz, D. (2010).
Introduction to quantum Fisher information.
J. Phys. A 43, 193001.

Uhlmann2011
Uhlmann, A. (2011).
Transition probability (fidelity) and its relatives.
Found. Phys. 41, 288--298.

Helstrom76
Helstrom, C. W. (1976/2012).
Quantum Detection and Estimation Theory. Academic Press.

HolevoBook11
Holevo, A. S. (2011).
Quantum Systems, Channels, Information. De Gruyter.

FawziRenner15
Fawzi, O., Renner, R. (2015).
Quantum conditional mutual information and approximate Markov chains.
Commun. Math. Phys. 340, 575--611.

FuchsGraaf99
Fuchs, C. A., van~de~Graaf, J. (1999).
Cryptographic distinguishability measures for quantum states.
IEEE Trans. Inf. Theory 45(4), 1216--1227.

LieroMielkeSavare18
Liero, M., Mielke, A., Savar\'e, G. (2018).
Optimal transport in competition with reaction (HK).
Invent. Math. 211, 969--1117.

JKO98
Jordan, R., Kinderlehrer, D., Otto, F. (1998).
Variational formulation of Fokker--Planck.
SIAM J. Math. Anal. 29(1), 1--17.

AGSBook08
Ambrosio, L., Gigli, N., Savar\'e, G. (2008).
Gradient Flows. Birkh\"auser (2nd ed.).

JLMS16
Jafferis, D. L., Lewkowycz, A., Maldacena, J., Suh, S. J. (2016).
Relative entropy equals bulk relative entropy.
JHEP 06, 004.

ADH15
Almheiri, A., Dong, X., Harlow, D. (2015).
Bulk locality and quantum error correction in AdS/CFT.
JHEP 2015(4), 163.

Lindblad76
Lindblad, G. (1976).
On the generators of quantum dynamical semigroups.
Commun. Math. Phys. 48, 119--130.

GKSL76
Gorini, V., Kossakowski, A., Sudarshan, E.C.G. (1976).
Completely positive semigroups.
J. Math. Phys. 17, 821--825.
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Reader's Guide and Glossary

Notation, Units, Scope, and Conventions

Fibered Bures–HK Geometry via Massful Couplings

Lyapunov Functionals and Local EVI(λ)

Defect Dissipation Measure from the Law–Field Gap

Euler Interferometry, Bures→TV, and Testing Power

Merits, Limitations, and Protocols

Cosmology (Bracketed Application)

Toy model: two-point HK base & qubit fiber

\begin{align}
d_{\mathrm{fib}}^2\big((\mu,\rho),(\nu,\tilde\rho)\big)
:=\;& d_{\HK_\delta}^2(\mu,\nu) \nonumber\\
&\quad +\;\sigfib^2\inf_{\pi\in \Pi^{\mathrm{mass}}_{\HK}(\mu,\nu)}
\iint \theta_{\Bures}^2\!\big(\rho_\zeta,\tilde\rho_{\zeta'}\big)\,\pi(d\zeta,d\zeta').
\label{eq:dfib}
\end{align}

d_{\mathrm{fib}}^2\big((\mu,\rho),(\nu,\tilde\rho)\big) := & d_{\HK_\delta}^2(\mu,\nu) \nonumber\\ &\quad + \sigfib^2\inf_{\pi\in \Pi^{\mathrm{mass}}_{\HK}(\mu,\nu)} \iint \theta_{\Bures}^2 \big(\rho_\zeta,\tilde\rho_{\zeta'}\big) \pi(d\zeta,d\zeta'). \label{eq:dfib}

\begin{align}
\mathcal{A}[(\mu_t,\rho_t)]
&:= \int_0^1\!\!\int \Big(|v_t|^2 + (\delta^2/4)\,\alpha_t^2\Big)\,d\mu_t\,dt
 + \frac{\sigfib^2}{4}\!\int_0^1\!\!\int \QFI_t(\zeta)\,d\mu_t(\zeta)\,dt.
\label{eq:action}
\end{align}

\mathcal{A}[(\mu_t,\rho_t)] &:= \int_0^1  \int \Big(|v_t|^2 + (\delta^2/4) \alpha_t^2\Big) d\mu_t dt + \frac{\sigfib^2}{4} \int_0^1  \int \QFI_t(\zeta) d\mu_t(\zeta) dt. \label{eq:action}

\begin{equation}
\frac{d}{dt}\frac12 d_{\mathrm{fib}}^2(X_t,Y) + \lambda\,d_{\mathrm{fib}}^2(X_t,Y)\;\le\;\cE(Y)-\cE(X_t),
\end{equation}

\frac{d}{dt}\frac12 d_{\mathrm{fib}}^2(X_t,Y) + \lambda d_{\mathrm{fib}}^2(X_t,Y) \le \cE(Y)-\cE(X_t),

\begin{equation}
\boxed{~
D_{\mathrm{fib}}\big((\mu,\rho)\,\|\,(\nu,\tilde\rho)\big)
:= D^{\mathrm{base}}_{1/2}(\mu\|\nu)\;+\;
\sigfib^2\, \inf_{\pi\in \Pi^{\mathrm{mass}}_{\HK}(\mu,\nu)}
\iint \theta_{\Bures}^2\!\big(\rho_\zeta,\tilde\rho_{\zeta'}\big)\,\pi(d\zeta,d\zeta') ~}
\label{eq:Dfib}
\end{equation}

\boxed{~ D_{\mathrm{fib}}\big((\mu,\rho) \| (\nu,\tilde\rho)\big) := D^{\mathrm{base}}_{1/2}(\mu\|\nu) +  \sigfib^2  \inf_{\pi\in \Pi^{\mathrm{mass}}_{\HK}(\mu,\nu)} \iint \theta_{\Bures}^2 \big(\rho_\zeta,\tilde\rho_{\zeta'}\big) \pi(d\zeta,d\zeta') ~} \label{eq:Dfib}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mrow><menclose notation="box"><mrow><mtext> </mtext><msub><mi>D</mi><mrow><mrow><mi mathvariant="normal">f</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">b</mi></mrow></mrow></msub><mo minsize="1.2em" maxsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>μ</mi><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><mo fence="false" stretchy="false">‖</mo><mspace width="0.167em" /><mo stretchy="false">(</mo><mi>ν</mi><mo>,</mo><mover><mi>ρ</mi><mo stretchy="false">~</mo></mover><mo stretchy="false">)</mo><mo minsize="1.2em" maxsize="1.2em">)</mo><mi>:</mi><mo>=</mo><msubsup><mi>D</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">e</mi></mrow></mrow></msubsup><mo stretchy="false">(</mo><mi>μ</mi><mo fence="false" stretchy="false">‖</mo><mi>ν</mi><mo stretchy="false">)</mo><mspace width="0.278em" /><mo>+</mo><mspace width="0.278em" /><msup><mi>\sigfib</mi><mn>2</mn></msup><mspace width="0.167em" /><msub><mo>inf</mo><mrow><mi>π</mi><mo>∈</mo><msubsup><mi>Π</mi><mrow><mi>\HK</mi></mrow><mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">s</mi></mrow></mrow></msubsup><mo stretchy="false">(</mo><mi>μ</mi><mo>,</mo><mi>ν</mi><mo stretchy="false">)</mo></mrow></msub><mo>∬</mo><msubsup><mi>θ</mi><mrow><mi>\Bures</mi></mrow><mn>2</mn></msubsup><mspace width="negativethinmathspace" /><mo minsize="1.2em" maxsize="1.2em">(</mo><msub><mi>ρ</mi><mi>ζ</mi></msub><mo>,</mo><msub><mover><mi>ρ</mi><mo stretchy="false">~</mo></mover><mrow><msup><mi>ζ</mi><mi>′</mi></msup></mrow></msub><mo minsize="1.2em" maxsize="1.2em">)</mo><mspace width="0.167em" /><mi>π</mi><mo stretchy="false">(</mo><mi>d</mi><mi>ζ</mi><mo>,</mo><mi>d</mi><msup><mi>ζ</mi><mi>′</mi></msup><mo stretchy="false">)</mo><mtext> </mtext></mrow></menclose><mi>\label</mi><mrow><mi>e</mi><mi>q</mi><mi>:</mi><mi>D</mi><mi>f</mi><mi>i</mi><mi>b</mi></mrow></mrow></mrow></math>

\[
\lambda \;\ge\; \min\{\tenv\,\betac\,\cbase,\; 4\lamfib/\sigfib^2\} - \Lips,
\]

\lambda  \ge  \min\{\tenv \betac \cbase,  4\lamfib/\sigfib^2\} - \Lips,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mi>λ</mi><mspace width="0.278em" /><mo>≥</mo><mspace width="0.278em" /><mo>min</mo><mo stretchy="false">{</mo><mi>\tenv</mi><mspace width="0.167em" /><mi>\betac</mi><mspace width="0.167em" /><mi>\cbase</mi><mo>,</mo><mspace width="0.278em" /><mn>4</mn><mi>\lamfib</mi><mo>/</mo><msup><mi>\sigfib</mi><mn>2</mn></msup><mo stretchy="false">}</mo><mo>−</mo><mi>\Lips</mi><mo>,</mo><mi>\]</mi></mrow></math>

\[
\int\!\!\int \Big(|v|^2 + (\delta^2/4)\,\alpha_t^2\Big)\,d\mu\,dt,\quad
\partial_t\mu+\nabla\!\cdot(\mu v)=\alpha_t\,\mu.
\]

\int  \int \Big(|v|^2 + (\delta^2/4) \alpha_t^2\Big) d\mu dt,\quad \partial_t\mu+\nabla \cdot(\mu v)=\alpha_t \mu.

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mo>∫</mo><mspace width="negativethinmathspace" /><mspace width="negativethinmathspace" /><mo>∫</mo><mo minsize="1.623em" maxsize="1.623em">(</mo><mo stretchy="false">|</mo><mi>v</mi><msup><mo stretchy="false">|</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><msup><mi>δ</mi><mn>2</mn></msup><mo>/</mo><mn>4</mn><mo stretchy="false">)</mo><mspace width="0.167em" /><msubsup><mi>α</mi><mi>t</mi><mn>2</mn></msubsup><mo minsize="1.623em" maxsize="1.623em">)</mo><mspace width="0.167em" /><mi>d</mi><mi>μ</mi><mspace width="0.167em" /><mi>d</mi><mi>t</mi><mo>,</mo><mspace width="1em" /><msub><mo>∂</mo><mi>t</mi></msub><mi>μ</mi><mo>+</mo><mo>∇</mo><mspace width="negativethinmathspace" /><mi>·</mi><mo stretchy="false">(</mo><mi>μ</mi><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>α</mi><mi>t</mi></msub><mspace width="0.167em" /><mi>μ</mi><mo>.</mo><mi>\]</mi></mrow></math>

\inf \mathcal{A}[(\mu_t,\rho_t)]  =  d_{\mathrm{fib}}^2\big((\mu_0,\rho_0),(\mu_1,\tilde\rho_1)\big).

\[
\lambda \;\ge\; \min\{\tenv\,\betac\,\cbase,\; 4\lamfib/\sigfib^2\}\;-\;\Lips.
\]

\lambda  \ge  \min\{\tenv \betac \cbase,  4\lamfib/\sigfib^2\} - \Lips.

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mi>λ</mi><mspace width="0.278em" /><mo>≥</mo><mspace width="0.278em" /><mo>min</mo><mo stretchy="false">{</mo><mi>\tenv</mi><mspace width="0.167em" /><mi>\betac</mi><mspace width="0.167em" /><mi>\cbase</mi><mo>,</mo><mspace width="0.278em" /><mn>4</mn><mi>\lamfib</mi><mo>/</mo><msup><mi>\sigfib</mi><mn>2</mn></msup><mo stretchy="false">}</mo><mspace width="0.278em" /><mo>−</mo><mspace width="0.278em" /><mi>\Lips</mi><mo>.</mo><mi>\]</mi></mrow></math>

\[
\frac{d}{dt}\frac12 d_{\mathrm{fib}}^2(X_t,Y) +
\lambda\, d_{\mathrm{fib}}^2(X_t,Y)\ \le\ \cE(Y)-\cE(X_t),
\quad
\lambda \ge \min\{\tenv\betac\cbase,\ 4\lamfib/\sigfib^2\}-\Lips.
\]

\frac{d}{dt}\frac12 d_{\mathrm{fib}}^2(X_t,Y) + \lambda  d_{\mathrm{fib}}^2(X_t,Y)\ \le\ \cE(Y)-\cE(X_t), \quad \lambda \ge \min\{\tenv\betac\cbase,\ 4\lamfib/\sigfib^2\}-\Lips.

\[
\iint \theta_{\Bures}^2\!\big((\id\!\otimes\!\Theta)\rho_\zeta,(\id\!\otimes\!\Theta)\tilde\rho_{\zeta'}\big)\,d\pi
\ \le\
\iint \theta_{\Bures}^2(\rho_\zeta,\tilde\rho_{\zeta'})\,d\pi.
\]

\iint \theta_{\Bures}^2 \big((\id \otimes \Theta)\rho_\zeta,(\id \otimes \Theta)\tilde\rho_{\zeta'}\big) d\pi \ \le\ \iint \theta_{\Bures}^2(\rho_\zeta,\tilde\rho_{\zeta'}) d\pi.

\Delta_k^{\mathrm{obs}} := D_{\mathrm{fib}} \big((\mu_k^-,\rho_k^-) \big\| \tau_k\big) - D_{\mathrm{fib}} \big((\mu_k^+,\rho_k^+) \big\| \tau_k\big) \ge 0,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><msubsup><mi>Δ</mi><mi>k</mi><mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">s</mi></mrow></mrow></msubsup><mi>:</mi><mo>=</mo><msub><mi>D</mi><mrow><mrow><mi mathvariant="normal">f</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">b</mi></mrow></mrow></msub><mspace width="negativethinmathspace" /><mo minsize="1.2em" maxsize="1.2em">(</mo><mo stretchy="false">(</mo><msubsup><mi>μ</mi><mi>k</mi><mo>−</mo></msubsup><mo>,</mo><msubsup><mi>ρ</mi><mi>k</mi><mo>−</mo></msubsup><mo stretchy="false">)</mo><mspace width="0.167em" /><mo minsize="1.2em" maxsize="1.2em">\|</mo><mspace width="0.167em" /><msub><mi>τ</mi><mi>k</mi></msub><mo minsize="1.2em" maxsize="1.2em">)</mo><mo>−</mo><msub><mi>D</mi><mrow><mrow><mi mathvariant="normal">f</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">b</mi></mrow></mrow></msub><mspace width="negativethinmathspace" /><mo minsize="1.2em" maxsize="1.2em">(</mo><mo stretchy="false">(</mo><msubsup><mi>μ</mi><mi>k</mi><mo>+</mo></msubsup><mo>,</mo><msubsup><mi>ρ</mi><mi>k</mi><mo>+</mo></msubsup><mo stretchy="false">)</mo><mspace width="0.167em" /><mo minsize="1.2em" maxsize="1.2em">\|</mo><mspace width="0.167em" /><msub><mi>τ</mi><mi>k</mi></msub><mo minsize="1.2em" maxsize="1.2em">)</mo><mspace width="0.278em" /><mo>≥</mo><mn>0</mn><mo>,</mo><mi>\]</mi></mrow></math>

\frac{d}{dt}\cE(\mu_t,\rho_t) \le -\cD_{\mathrm{phys}}(t) -\cD_{\mathrm{obs}}(t),

\[
d\mu_{\mathrm{def}} = d\mu_{\mathrm{cont}} + \sum_{t_c\in T_{\mathrm{crit}}}\alpha_{t_c}\,\delta_{t_c},
\qquad \alpha_{t_c}\ge0.
\]

d\mu_{\mathrm{def}} = d\mu_{\mathrm{cont}} + \sum_{t_c\in T_{\mathrm{crit}}}\alpha_{t_c} \delta_{t_c}, \qquad \alpha_{t_c}\ge0.

d\cE  \le  - \cD_{\mathrm{phys}}(t) dt  -  d\mu_{\mathrm{obs}}  -  d\mu_{\mathrm{def}}.

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mi>d</mi><mi>\cE</mi><mspace width="0.278em" /><mo>≤</mo><mspace width="0.278em" /><mo>−</mo><mspace width="0.167em" /><msub><mi>\cD</mi><mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">y</mi><mi mathvariant="normal">s</mi></mrow></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><mi>d</mi><mi>t</mi><mspace width="0.278em" /><mo>−</mo><mspace width="0.278em" /><mi>d</mi><msub><mi>μ</mi><mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">s</mi></mrow></mrow></msub><mspace width="0.278em" /><mo>−</mo><mspace width="0.278em" /><mi>d</mi><msub><mi>μ</mi><mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi></mrow></mrow></msub><mo>.</mo><mi>\]</mi></mrow></math>

\|\rho-\tilde\rho\|_{\TV} \le \sqrt{1-\mathrm{Fid}(\rho,\tilde\rho)} = \sin\theta_{\Bures}(\rho,\tilde\rho), \qquad \xi_{\mathrm{QCB}}(\rho,\tilde\rho) \ge -\tfrac12\log \mathrm{Fid}(\rho,\tilde\rho),

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mo fence="false" stretchy="false">‖</mo><mi>ρ</mi><mo>−</mo><mover><mi>ρ</mi><mo stretchy="false">~</mo></mover><msub><mo fence="false" stretchy="false">‖</mo><mrow><mi>\TV</mi></mrow></msub><mspace width="0.278em" /><mo>≤</mo><mspace width="0.278em" /><msqrt><mrow><mn>1</mn><mo>−</mo><mrow><mi mathvariant="normal">F</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">d</mi></mrow><mo stretchy="false">(</mo><mi>ρ</mi><mo>,</mo><mover><mi>ρ</mi><mo stretchy="false">~</mo></mover><mo stretchy="false">)</mo></mrow></msqrt><mspace width="0.278em" /><mo>=</mo><mspace width="0.278em" /><mi>sin</mi><msub><mi>θ</mi><mrow><mi>\Bures</mi></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo>,</mo><mover><mi>ρ</mi><mo stretchy="false">~</mo></mover><mo stretchy="false">)</mo><mo>,</mo><mspace width="2em" /><msub><mi>ξ</mi><mrow><mrow><mi mathvariant="normal">Q</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">B</mi></mrow></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo>,</mo><mover><mi>ρ</mi><mo stretchy="false">~</mo></mover><mo stretchy="false">)</mo><mspace width="0.278em" /><mo>≥</mo><mspace width="0.278em" /><mo>−</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mi>log</mi><mrow><mi mathvariant="normal">F</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">d</mi></mrow><mo stretchy="false">(</mo><mi>ρ</mi><mo>,</mo><mover><mi>ρ</mi><mo stretchy="false">~</mo></mover><mo stretchy="false">)</mo><mo>,</mo><mi>\]</mi></mrow></math>

\[
\Gamma_{\mathrm{eff}}\;\ge\; C_{\mathrm{read}}\cdot \frac{\Delta \theta_{\Bures}}{\Delta t},
\]

\Gamma_{\mathrm{eff}} \ge  C_{\mathrm{read}}\cdot \frac{\Delta \theta_{\Bures}}{\Delta t},

\[
d_{\mathrm{fib}}^2((\mu,\rho),(\nu,\tilde\rho))
= d_{\HK_\delta}^2(\mu,\nu) + \sigfib^2 \inf_{\pi}
\int \theta_{\Bures}^2(\rho_{\mathrm{in}},\rho_{\mathrm{out}})\,d\pi,
\]

d_{\mathrm{fib}}^2((\mu,\rho),(\nu,\tilde\rho)) = d_{\HK_\delta}^2(\mu,\nu) + \sigfib^2 \inf_{\pi} \int \theta_{\Bures}^2(\rho_{\mathrm{in}},\rho_{\mathrm{out}}) d\pi,

\[
d_{\mathrm{fib}}^2
= d_{\HK_\delta}^2\big((a\delta_x+b\delta_y),(a'\delta_x+b'\delta_y)\big)
\;+\; \sigfib^2 \inf_{\pi\in\Pi^{\mathrm{mass}}_{\HK}}
\sum_{\zeta,\zeta'\in\{x,y\}} \tfrac14\,(\vartheta_\zeta-\vartheta'_{\zeta'})^2\,\pi_{\zeta\zeta'}.
\]

d_{\mathrm{fib}}^2 = d_{\HK_\delta}^2\big((a\delta_x+b\delta_y),(a'\delta_x+b'\delta_y)\big)  +  \sigfib^2 \inf_{\pi\in\Pi^{\mathrm{mass}}_{\HK}} \sum_{\zeta,\zeta'\in\{x,y\}} \tfrac14 (\vartheta_\zeta-\vartheta'_{\zeta'})^2 \pi_{\zeta\zeta'}.

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mo fence="false" stretchy="false">‖</mo><mi>X</mi><msub><mo fence="false" stretchy="false">‖</mo><mrow><mi>\TV</mi></mrow></msub><mi>:</mi><mo>=</mo><mstyle displaystyle="false" scriptlevel="0"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo fence="false" stretchy="false">‖</mo><mi>X</mi><msub><mo fence="false" stretchy="false">‖</mo><mn>1</mn></msub><mi>$</mi></mrow></math>

$\lambda \ge \min\{\tenv\betac\cbase, 4\lamfib/\sigfib^2\}-\Lips$

\int_0^1  \int \QFI_t(\zeta) d\mu_t dt<\infty

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mspace width="negativethinmathspace" /><mspace width="negativethinmathspace" /><mo>∫</mo><msub><mi>\QFI</mi><mi>t</mi></msub><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><mi>d</mi><msub><mi>μ</mi><mi>t</mi></msub><mspace width="0.167em" /><mi>d</mi><mi>t</mi><mo><</mo><mo>∞</mo><mi>$</mi></mrow></math>

\rho^{k+1}\in\arg\min_\rho\big\{\wread \cE_{\mathrm{read}}(\rho) +\frac{\sigfib^2}{2\tau}  \iint \theta_{\Bures}^2(\rho_\zeta,\rho^k_{\zeta'}) d\pi^{k+1}\big\}

$
N\gtrsim \frac{\log(1/\varepsilon)}{\xi_{\mathrm{QCB}}(\Phi\rho_\theta,\Phi\tilde\rho_{\theta,N})}.
$

$\mathrm{Fid}\approx \cos^2\theta \approx 1-\theta^2$

(\sigfib^2/4)\int  \int \QFI_t d\mu_t dt

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mo stretchy="false">(</mo><msup><mi>\sigfib</mi><mn>2</mn></msup><mo>/</mo><mn>4</mn><mo stretchy="false">)</mo><mo>∫</mo><mspace width="negativethinmathspace" /><mspace width="negativethinmathspace" /><mo>∫</mo><msub><mi>\QFI</mi><mi>t</mi></msub><mspace width="0.167em" /><mi>d</mi><msub><mi>μ</mi><mi>t</mi></msub><mspace width="0.167em" /><mi>d</mi><mi>t</mi><mi>$</mi></mrow></math>