From Axioms to Attractors: A Deterministic Statistical Feedback Law Unifying Equilibria in Physics

1. Introduction

Motivation

Across quantum mechanics (QM), thermodynamics (TD), and general relativity (GR), the central equilibrium relations—Born’s rule, entropy increase, and the Einstein balance—are typically imposed as end states rather than derived as consequences of dynamics. This leaves open a structural question: under which deterministic conditions do physical systems return to these relations after perturbations, and with what rates?

We study a partly sector-agnostic mechanism. The core is the same everywhere: track a quadratic misfit between a statistical baseline and a physical response, and show that—under standard spectral-gap or coercivity hypotheses—this misfit is a Lyapunov residual that decays monotonically (often exponentially). This form (propagation plus gap/coercivity) is invariant across sectors. What is not agnostic are the inputs and outcomes: the constants and the equilibrium set (the “attractor”) depend on context. In the quantum/operator-algebraic setting they are set by the pointer algebra and the reversible QMS gap; in PDE models by the coercive closure and domain/boundary conditions; in free-field quantization by the Hamiltonian gap; and on geometric slices by the gauge and the Lichnerowicz-type gap. Thus, the DSFL provides a uniform template with sector-specific rates and targets.

Idea

We introduce the Deterministic Statistical Feedback Law (DSFL). Let P denote the sector-specific response (e.g. flux, current, or geometric tensor) and let $\nabla \rho$ represent the statistical baseline. The alignment residual

$$u:=P-\nabla \rho ,\mathcal{R}(t):={\int }_{\Omega }{|u(x,t)|}^{2}dx$$

quantifies misfit. DSFL asserts that $\mathcal{R}$ acts as a Lyapunov functional:

$$\dot{\mathcal{R}}(t)\le -\alpha \mathcal{R}(t),$$

with a rate $\alpha >0$ determined by the sector (spectral gap, coercivity constant, or Hamiltonian gap). A common propagation step (Jensen convexity, Kadison–Schwarz, or an energy identity) yields global monotonicity; a spectral gap or coercivity upgrades this to exponential decay. Within this template, Born alignment, entropy growth, and Einstein balance arise as attractors of the same residual-suppression mechanism.

What Is New

• One residual, one propagation step. A single quadratic residual $\mathcal{R}$ is used across operator-algebraic QM, finite-dimensional Lindblad dynamics, coercive PDE flows, and free-field stochastic quantization, with a unified propagation lemma ensuring monotonicity.
• Reversible QMS. For $\omega$-symmetric quantum Markov semigroups, we prove DSFL is equivalent to a noncommutative Poincaré (spectral-gap) inequality on the orthogonal complement of the fixed-point algebra, with optimal rate ${\alpha }_{*}=2{\lambda }_{*}$.
• Finite-dimensional Lindblad. For pure dephasing generators, the Lüders off-diagonal variance decays at the sharp rate $2\lambda$, where $\lambda ={min}_i{\gamma }_i$.
• PDE template. An exact residual energy identity gives $\dot{\mathcal{R}}\le -(2\beta -C\epsilon )\mathcal{R}$ under quantitative coercivity ($B⪰\beta I$) and subcritical couplings; refinements use Helmholtz decomposition and boundary conditions.
• Free-field QFT. In Parisi–Wu stochastic quantization, smeared two-point residuals decay at twice the Euclidean Hamiltonian gap, deriving a DSFL inequality in the Gaussian sector.
• Geometric slice analogue. On compact Riemannian slices (DeTurck gauge), a Lichnerowicz-type spectral gap implies exponential $L^2$ suppression of the curvature–matter misfit toward Einstein balance. (A covariant Lorentzian version is left open.)
• Residual-entropy monotone. The proxy $S_R=-log(\mathcal{R}/{\mathcal{R}}_0+R_{*})$ is strictly increasing whenever DSFL holds, providing a gap/coercivity-controlled “arrow of time.”

1.1. Position Relative to Prior Work

Variance and entropy decay for reversible diffusions (Bakry–Émery), spectral gaps for reversible QMS, modewise Lindblad contraction, and coercivity in geometric/dissipative flows are well established. Our contribution is to (i) isolate a single Lyapunov residual and a unified propagation step spanning these settings, and (ii) give sharp equivalences and rates that make sectoral relaxation theorems directly comparable. We do not introduce a new sector-specific model; we identify and quantify the common restoration law that many models already instantiate under standard gap/coercivity assumptions.

Scope and Limits

Our results are form–invariant across sectors but rely on context–specific assumptions. Throughout we require either reversibility or quantitative coercivity: (i) $\omega$–symmetric (reversible) QMS with a spectral gap on $L^2{(\mathcal{N},\omega )}^{\perp }$; (ii) coercive PDE closures with $B(x,t)⪰\beta I$ and subcritical couplings; (iii) free fields (Gaussian sector) with a positive Euclidean Hamiltonian gap; and (iv) geometric DeTurck slices with a Lichnerowicz–type gap on the physical subspace and small initial data. Under these hypotheses the residual obeys a DSFL inequality with an explicit (contextual) rate.

We do not claim theorems for: nonreversible/hypocoercive QMS (skew generators), fully covariant Lorentzian evolutions (hyperbolic, gauge–invariant DSFL), or interacting QFT beyond the Gaussian sector. These are stated as programs with testable intermediate predictions (e.g., scale–dependent rates, ISS bounds) but remain open.

The numerical section provides reproducible protocols rather than data–driven claims; the analytic results stand independently of numerics.

2. Background and Related Work

2.1. Quantum Markov Semigroups and Spectral Gaps

A (normal, unital) quantum Markov semigroup (QMS) ${({\mathsf{T}}_t)}_{t\ge 0}$ on a von Neumann algebra $\mathcal{M}$ with faithful normal state $\omega$ is called ω–symmetric (reversible) if it is self–adjoint on the GNS $L^2(\omega )$ space. In this setting the generator $\mathcal{L}$ admits a densely defined, closed Dirichlet form

$${\mathcal{E}}_{\omega }(X):={\langle X,-\mathcal{L}X\rangle }_{2,\omega },{\parallel X\parallel }_{2,\omega }^2=\omega (X^{*}X),$$

with standard functional–analytic underpinnings for noncommutative Dirichlet forms and symmetric quantum semigroups (e.g. [1,2]). The fixed–point algebra $\mathcal{N}=\{X:{\mathsf{T}}_{t}X=X\forall t\}$ is the natural equilibrium subspace; under modular invariance, the $\omega$–preserving conditional expectation ${\mathsf{E}}_{\mathcal{N}}:\mathcal{M}\to \mathcal{N}$ exists and is unique (Takesaki’s theorem; cf. Tomiyama) [2,3].

A noncommutative Poincaré (spectral–gap) inequality with constant $\lambda >0$,

$$\parallel X-{\mathsf{E}}_{\mathcal{N}}{(X)\parallel }_{2,\omega }^2\le {\displaystyle \frac{1}{\lambda }}{\mathcal{E}}_{\omega }(X),$$

(1)

is equivalent to exponential decay of the noncommutative variance along the semigroup,

$$\parallel {\mathsf{T}}_{t}X-{\mathsf{E}}_{\mathcal{N}}{(X)\parallel }_{2,\omega }^2\le e^{-2\lambda t}{\parallel X-{\mathsf{E}}_{\mathcal{N}}(X)\parallel }_{2,\omega }^2,$$

(2)

with optimal rate ${\alpha }_{*}=2{\lambda }_{*}$ [4,5,6]. This mirrors the classical theory of reversible diffusions (Bakry–Émery calculus). Beyond Poincaré, quantum log–Sobolev/hypercontractive regimes and related mixing bounds are available (e.g. [7,8]), and rapid mixing for quantum channels/expanders provides a complementary discrete perspective [9]. In our DSFL interpretation, (1)–(2) constitute the operator–algebraic Lyapunov law for the residual $X-{\mathsf{E}}_{\mathcal{N}}(X)$: once a gap holds, the residual contracts at rate $\lambda$ and $\mathcal{N}$ is the attractor.

2.2. Lindblad Dephasing and Modewise Contraction

In finite dimensions, completely positive trace–preserving dynamics admit the GKSL (Lindblad) representation

$${\mathcal{L}}^{*}(\sigma )=-i[H,\sigma ]+\sum _k\left(L_k\sigma L_k^{†}-\frac{1}{2}\{L_k^{†}{L}_k,\sigma \}\right),$$

with Hamiltonian H and noise operators $L_k$ [10,11]; see also [12,13]. For pure dephasing in a fixed orthonormal basis, $L_i=\sqrt{{\gamma }_i}|i\rangle \langle i|$ and H diagonal, one obtains

$${\displaystyle \frac{d}{dt}}{({\sigma }_t)}_{ii}=0,{\displaystyle \frac{d}{dt}}{({\sigma }_t)}_{ij}=-\frac{{\gamma }_i+{\gamma }_j}{2}{({\sigma }_t)}_{ij}(i\ne j),$$

so coherence ${({\sigma }_t)}_{ij}$ decays as $e^{-({\gamma }_i+{\gamma }_j)t/2}$ while populations are conserved. The Hilbert–Schmidt variance off the pointer algebra (Lüders residual) thus satisfies

$${\mathcal{R}}_{\mathrm{L}\ddot{\mathrm{u}}\mathrm{ders}}({\sigma }_t)=\sum _{i\ne j}{|{({\sigma }_t)}_{ij}|}^2\le e^{-2\lambda t}{\mathcal{R}}_{\mathrm{L}\ddot{\mathrm{u}}\mathrm{ders}}({\sigma }_0),\lambda =\underset{i}{min}{\gamma }_i,$$

and the rate $2\lambda$ is sharp. This is the finite–dimensional counterpart of (2), with a gap set by the smallest dephasing rate.

2.3. Coercive PDE Flows and Bakry–Émery Tools

In classical dissipative PDEs, exponential return to equilibrium combines: (i) an energy/entropy identity for a nonnegative functional along solutions, and (ii) a coercive functional inequality (Poincaré/log–Sobolev) to control lower–order couplings. Foundational results include Gross’s log–Sobolev ⇔ hypercontractivity [14] and its many developments [15,16]. In geometric analysis (e.g. Ricci/DeTurck flows), $L^2$–type curvature residuals dissipate under an elliptic Lichnerowicz–type operator; Perelman’s monotone functionals provide a celebrated Lyapunov structure [17,18,19].

Our DSFL–PDE template mirrors this structure: for $u:=P-\nabla \rho$,

$${\displaystyle \frac{d}{dt}}{\int }_{\Omega }{|u|}^{2}dx=-2{\int }_{\Omega }{u}^{\top }Budx-2{\parallel \nabla ·u\parallel }_{L^2(\Omega )}^2+(\mathrm{controlled}\mathrm{terms}),$$

so that, under uniform ellipticity $B\ge \beta I$ and subcritical couplings, Grönwall yields ${\parallel u(t)\parallel }_{L^2}^2\le e^{-(2\beta -C\epsilon )t}{\parallel u(0)\parallel }_{L^2}^2$. This is the Bakry–Émery “propagation + gap ⇒ decay’’ pattern reinterpreted as sector–agnostic Lyapunov suppression of a misalignment functional.

2.4. Stochastic Quantization and Hamiltonian Gaps

Parisi–Wu stochastic quantization gives an analytically tractable Euclidean dynamics for QFT: for a free scalar with action $S[\varphi ]=\int \frac{1}{2}{(|\nabla \varphi |}^2+m^2{\varphi }^2)dx$, the Langevin flow

$${\partial }_{\tau }{\varphi }_{\tau }(x)=-{\displaystyle \frac{\delta S}{\delta \varphi }}(x)+\eta (x,\tau )$$

generates a semigroup ${\mathsf{T}}_{\tau }=e^{-\tau H}$ with nonnegative Euclidean Hamiltonian H. The spectral gap ${\lambda }_{*}=m^2$ controls exponential relaxation of smeared correlators; squaring gives decay of quadratic residuals at rate $2{\lambda }_{*}$ [20]. For reviews and field–theory context see [21,22]. In DSFL language, the same Hamiltonian gap that governs Euclidean relaxation drives residual suppression in the Gaussian sector; extending to interacting fields requires nonperturbative functional inequalities or RG control.

2.5. Positioning vs. Prior Approaches

Exponential return to equilibrium is well established within specific formalisms: noncommutative Poincaré/log–Sobolev inequalities for reversible QMS [4,5,6,7,8], explicit modewise contraction in GKSL/Lindblad dynamics [10,11,12,13], coercivity–driven decay for dissipative PDEs and geometric flows [14,15,16,17,18,19,23], and Hamiltonian–gap relaxation in stochastic quantization [20,21,22].

The present work contributes a single, sector–neutral Lyapunov residual and a unifying propagation principle (Jensen/Kadison–Schwarz/energy identity) that render these results structurally comparable. Under the corresponding gap/coercivity hypotheses, one obtains the same DSFL inequality $\dot{\mathcal{R}}\le -\alpha \mathcal{R}$, and the sectoral equilibrium statements—Born alignment (QM), residual–entropy monotonicity (TD), and Einstein balance (GR)—emerge as attractors rather than axioms. (In our pointer–algebra formalism, POVMs and their Naimark dilations are standard [24,25,26].)

3. Notation and Conventions

3.1. Spaces, Norms, Inner Products

Domains and Measures.

$\Omega \subset {\mathbb{R}}^d$ denotes either a bounded $C^1$ domain or a flat torus ${\mathbb{T}}^d$. We write $dx$ for Lebesgue measure, $|\Omega |$ for its volume, and ${\langle f\rangle }_{\Omega }:={|\Omega |}^{-1}{\int }_{\Omega }fdx$ for spatial averages. When needed, $(\mathcal{U},g)$ denotes a smooth Riemannian (or, where explicitly stated, Lorentzian) manifold with volume form $d{\mu }_g$.

Lebesgue and Sobolev Spaces.

For $1\le p\le \infty$, $L^p(\Omega )$ has norm ${\parallel f\parallel }_{L^p(\Omega )}=({\int }_{\Omega }{|f|}^p{dx)}^{1/p}$ (essential supremum for $p=\infty$). For $k\in \mathbb{N}$, $H^k(\Omega )$ is the Sobolev space with ${\parallel f\parallel }_{H^k(\Omega )}^2={\sum }_{|\alpha |\le k}{\parallel {\partial }^{\alpha }f\parallel }_{L^2(\Omega )}^2$. We write $H_0^1(\Omega )$ for the closure of $C_c^{\infty }(\Omega )$ in $H^1(\Omega )$. Vector/tensor–valued spaces are denoted $L^p(\Omega ;E)$ and $H^k(\Omega ;E)$ with the product norms.

Inner Products and $L^2$ Norms.

On $\Omega$, ${\langle f,g\rangle }_{L^2(\Omega )}={\int }_{\Omega }fgdx$ for scalars and ${\langle \mathbf{f},\mathbf{g}\rangle }_{L^2(\Omega )}={\int }_{\Omega }\mathbf{f}·\mathbf{g}dx$ for vectors. On $(\mathcal{U},g)$, for symmetric 2–tensors $T,S$ we use ${\langle T,S\rangle }_g=g^{\mu \alpha }{g}^{\nu \beta }{T}_{\mu \nu }{S}_{\alpha \beta }$ and ${\parallel T\parallel }_{L^2(g)}^2={\int }_{\mathcal{U}}{\langle T,T\rangle }_{g}d{\mu }_g$.

Gradients and Divergences.

In $\Omega \subset {\mathbb{R}}^d$, ∇ is the Euclidean gradient and $\nabla ·$ the divergence. On $(\mathcal{U},g)$, ${\nabla }_{\mu }$ is the Levi–Civita covariant derivative and $\nabla ·$ its metric divergence. For a vector field P and a scalar $\rho$, we define the alignment residual

$$u:=P-{\nabla \rho ,\parallel u\parallel }_2^2:={\int }_{\Omega }{|u|}^{2}dx.$$

Weighted Inner Products.

Given a measurable, symmetric positive–definite weight $W(x)\in {\mathbb{R}}^{d\times d}$, set ${\langle v,w\rangle }_{W(x)}:=v^{\top }W(x)w$ and ${|v|}_{W(x)}^2={\langle v,v\rangle }_{W(x)}$. The global weighted norm is ${\parallel v\parallel }_{L_W^2}^2:={\int }_{\Omega }{|v(x)|}_{W(x)}^{2}dx$. Unless stated otherwise, $W\equiv I$.

Noncommutative Conventions.

For a von Neumann algebra $(\mathcal{M},\omega )$ with faithful normal state $\omega$, the GNS inner product is ${\langle X,Y\rangle }_{2,\omega }=\omega (X^{*}Y)$ and ${\parallel X\parallel }_{2,\omega }^2=\omega (X^{*}X)$. The noncommutative variance is ${\mathrm{Var}}_{\omega }(X)={\parallel X-\omega (X)\mathbf{1}\parallel }_{2,\omega }^2$. If $\mathcal{N}\subset \mathcal{M}$ is a von Neumann subalgebra invariant under the modular group of $\omega$, ${\mathsf{E}}_{\mathcal{N}}:\mathcal{M}\to \mathcal{N}$ denotes the (unique) $\omega$–preserving conditional expectation; it acts as the $L^2(\omega )$–orthogonal projection onto $L^2(\mathcal{N},\omega )$.

3.2. Measures, Domains, and Boundary Conditions

Flat Domains.

Unless stated otherwise, $\Omega \subset {\mathbb{R}}^d$ is either a bounded $C^1$ domain or a flat torus ${\mathbb{T}}^d$. We write $dx$ for the Lebesgue measure and ${\int }_{\Omega }(·)dx$ for spatial integrals; spatial averages are ${\langle f\rangle }_{\Omega }:={|\Omega |}^{-1}{\int }_{\Omega }fdx$.

Boundary Conditions (BCs).

Energy identities and integrations by parts are justified under either (i) periodic BCs on ${\mathbb{T}}^d$, or (ii) homogeneous no-flux BCs arranged so that boundary terms vanish in the residual energy balance. In particular,

$${\int }_{\Omega }u·\nabla (\nabla ·u)dx=-{\parallel \nabla ·u\parallel }_{L^2(\Omega )}^2+{\int }_{\partial \Omega }(u·n)(\nabla ·u)dS,$$

so the boundary contribution is zero if, for example, $u·n=0$ on $\partial \Omega$ (e.g. $P·n={\partial }_n\rho =0$), or on a torus.

Probability Measures.

In data–driven formulations we allow a time–indexed family of probability measures ${\mu }_t$ on $\Omega$. Population expectations are ${\mathbb{E}}_{{\mu }_t}[f]={\int }_{\Omega }fd{\mu }_t$, with empirical approximations ${\displaystyle \frac{1}{N}}{\sum }_{i=1}^{N}f(x_i)$ when ${\mu }_t$ is supported on samples $\{x_i\}$. The residual

$$\mathcal{R}(t):={\mathbb{E}}_{{\mu }_t}\left[{|P-\nabla \rho |}_W^2\right]$$

reduces to ${\int }_{\Omega }{|u|}_W^{2}dx$ (with $u:=P-\nabla \rho$) when ${\mu }_t$ is normalized Lebesgue and $W\equiv I$.

Manifolds.

In geometric sections we replace $(\Omega ,dx)$ by a Riemannian (or, where explicitly stated, Lorentzian) manifold $(\mathcal{U},g)$ with Levi–Civita connection ∇, metric pairing ${\langle ·,·\rangle }_g$, and volume form $d{\mu }_g$. For symmetric 2–tensors T, the $L^2$ norm is ${\parallel T\parallel }_{L^2(g)}^2={\int }_{\mathcal{U}}{\langle T,T\rangle }_{g}d{\mu }_g$. On compact Riemannian slices in DeTurck gauge (Sec.), boundary terms vanish by compactness; on noncompact manifolds we assume decay/compatibility so that all integrals are finite.

Normalization and Gauges.

In probabilistic sectors we impose ${\int }_{\Omega }\rho dx=1$ (densities) and ${\parallel \Psi \parallel }_{L^2(\Omega )}=1$ (wave functions). The baseline $\rho$ is defined up to an additive gauge $\rho \mapsto \rho +C(t)$, which leaves $u=P-\nabla \rho$ and $\mathcal{R}$ invariant. In geometric sectors, gauge choices (e.g. DeTurck) ensure ellipticity/hyperbolicity and do not alter the definition of the residual (e.g. ${\parallel G-\kappa T\parallel }_{L^2(g)}^2$).

3.3. Operators, Semigroups, and Spectra

Linear Operators and Spectra.

Let $\mathcal{A}:\mathsf{D}(\mathcal{A})\subset H\to H$ be a densely defined, closed linear operator on a Hilbert space $(H,\langle ·,·\rangle )$. We write $\sigma (\mathcal{A})$ for its spectrum and $ker(\mathcal{A})$ for its kernel. If $\mathcal{A}$ is self–adjoint and nonnegative, a spectral gap means that

$$\langle X,\mathcal{A}X\rangle \ge {\lambda \parallel X\parallel }^2\mathrm{for}\mathrm{all}X\perp ker(\mathcal{A}),$$

for some $\lambda >0$; then $\lambda$ is the optimal Poincaré constant on $ker{(\mathcal{A})}^{\perp }$.

Markov/Contraction Semigroups.

A strongly continuous one–parameter semigroup ${({\mathsf{T}}_t)}_{t\ge 0}$ on H with generator $\mathcal{L}$ is a (sub)Markov contraction on a Banach lattice $L^p(\mu )$ if it preserves positivity, mass (${\mathsf{T}}_t\mathbf{1}=\mathbf{1}$), and satisfies $\parallel {\mathsf{T}}_t{f\parallel }_{L^p}\le {\parallel f\parallel }_{L^p}$ for $1\le p\le \infty$. In the reversible case (self–adjoint on $L^2(\mu )$), the Dirichlet form is $\mathcal{E}(f):={\langle f,-\mathcal{L}f\rangle }_{L^2(\mu )}$ and

$${\mathrm{Var}}_{\mu }({\mathsf{T}}_{t}f)\le e^{-2\lambda t}{\mathrm{Var}}_{\mu }(f)⟺{\mathrm{Var}}_{\mu }(f)\le {\displaystyle \frac{1}{\lambda }}\mathcal{E}(f).$$

Quantum Markov Semigroups (QMS).

On a von Neumann algebra $(\mathcal{M},\omega )$, a normal unital completely positive (u.c.p.) semigroup ${({\mathsf{T}}_t)}_{t\ge 0}$ that is $\omega$–symmetric is a noncommutative analogue of a reversible diffusion. The GNS $L^2(\omega )$ inner product is ${\langle X,Y\rangle }_{2,\omega }=\omega (X^{*}Y)$, the Dirichlet form is ${\mathcal{E}}_{\omega }(X):={\langle X,-\mathcal{L}X\rangle }_{2,\omega }$, and the fixed–point algebra $\mathcal{N}=\{X:{\mathsf{T}}_{t}X=X\}$ is the equilibrium subspace. The noncommutative Poincaré inequality

$$\parallel X-{\mathsf{E}}_{\mathcal{N}}{(X)\parallel }_{2,\omega }^2\le {\displaystyle \frac{1}{\lambda }}{\mathcal{E}}_{\omega }(X)$$

is equivalent to exponential variance decay

$$\parallel {\mathsf{T}}_{t}X-{\mathsf{E}}_{\mathcal{N}}{(X)\parallel }_{2,\omega }^2\le e^{-2\lambda t}{\parallel X-{\mathsf{E}}_{\mathcal{N}}(X)\parallel }_{2,\omega }^2,$$

with optimal rate $2\lambda$.

Generators in Finite Dimension (GKSL Form).

On $M_d(\mathbb{C})$, the dual evolution for states ${\sigma }_t$ is

$${\dot{\sigma }}_t=-i[H,{\sigma }_t]+\sum _k\left(L_k{\sigma }_t{L}_k^{†}-\frac{1}{2}\{L_k^{†}{L}_k,{\sigma }_t\}\right),$$

with Hamiltonian H and noise operators $L_k$ (GKSL/Lindblad form). For pure dephasing, $L_i=\sqrt{{\gamma }_i}|i\rangle \langle i|$ yields modewise decay ${({\sigma }_t)}_{ij}=e^{-({\gamma }_i+{\gamma }_j)t/2}{({\sigma }_0)}_{ij}$ for $i\ne j$.

Poincaré and Log–Sobolev constants.

We use Poincaré constants $\lambda >0$ to control variance by energy, and, where applicable, log–Sobolev constants $\rho >0$ to control entropy by Fisher information. In this article the DSFL rate $\alpha$ is identified with twice a Poincaré gap in reversible settings (classical/QMS) and with quantitative coercivity constants in PDE/geometric settings.

Projection onto Equilibria.

${\mathsf{E}}_{\mathcal{N}}$ denotes the $\omega$–preserving conditional expectation onto the fixed–point (pointer) algebra $\mathcal{N}$. In PDE/probability sectors, the analogue is projection onto the nullspace of the generator (e.g., $\rho \mapsto {\langle \rho \rangle }_{\Omega }$).

Spectral Notation in GR Slices.

For Lichnerowicz–DeTurck type operators acting on symmetric 2–tensors on a compact Riemannian 3–manifold $(\Sigma ,\gamma )$, we denote by ${\lambda }_{\mathrm{GR}}>0$ the spectral gap on the orthogonal complement of the gauge directions; it controls exponential $L^2$ decay of the geometric residual.

3.4. Residuals and Entropy Proxies

DSFL Residuals.

The global DSFL residual in the classical/PDE sector is

$$\mathcal{R}(t):={\int }_{\Omega }{|P(x,t)-\nabla \rho (x,t)|}^{2}dx,$$

or its weighted variant ${\int }_{\Omega }{|P-\nabla \rho |}_W^{2}dx$. The noncommutative analogue is ${\mathcal{R}}_{\omega }(X):={\parallel X-{\mathsf{E}}_{\mathcal{N}}(X)\parallel }_{2,\omega }^2$ (QMS). On Riemannian slices we use the geometric residual

$${\mathcal{R}}_{\mathrm{geom}}(t):={\int }_{\Sigma }\parallel G[\gamma (t)]-\kappa T{\parallel }_{\gamma (t)}^{2}d{\mu }_{\gamma (t)}.$$

Residual–Entropy Proxy.

A dimensionless proxy is

$$S_R(t):=-log\left(\mathcal{R}(t)/{\mathcal{R}}_0+R_{*}\right)({\mathcal{R}}_0>0,R_{*}\in (0,1)).$$

Initial Sameness and Common Ancestry.

All physical sectors—quantum, thermodynamic, and geometric—descend from a shared initial alignment between statistical and physical structures. We call this the principle of common ancestry: at $t=0$, the universe (or any closed system) possessed a single residual-free configuration

$$P_0=\nabla {\rho }_0,\mathcal{R}(0)={\int }_{\Omega }{|P_0-\nabla {\rho }_0|}^{2}dx=0.$$

This expresses the state of complete statistical–physical identity (initial sameness) from which all later structures evolve.

Figure 1. The principle of common ancestry: evolution as deterministic recovery of initial sameness.

Figure 1. The principle of common ancestry: evolution as deterministic recovery of initial sameness.

As evolution proceeds, local perturbations generate misalignments, $u(x,t):=P(x,t)-\nabla \rho (x,t)$, which define the residual

$$\mathcal{R}(t)={\int }_{\Omega }{|u(x,t)|}^{2}dx,\dot{\mathcal{R}}(t)\le -\alpha \mathcal{R}(t),$$

where $\alpha >0$ is the sectoral spectral gap (quantum, thermodynamic, or geometric). The Deterministic Statistical Feedback Law (DSFL) ensures exponential suppression of these residuals, restoring the alignment that encodes shared ancestry:

$$\mathcal{R}(t)\le e^{-\alpha t}\mathcal{R}(0).$$

Hence equilibrium is not a probabilistic emergence from randomness, but the dynamic recovery of common ancestry through deterministic residual decay. The same Lyapunov structure appears in all sectors— Born alignment in quantum mechanics, entropy growth in thermodynamics, and curvature–matter balance in general relativity—each governed by its own contextual rate $\alpha$.

3.5. Abbreviations (DSFL, sDoF, pDoF, QMS, etc.)

DSFL	Deterministic Statistical Feedback Law (global Lyapunov law for the alignment residual).
sDoF / pDoF	Statistical / Physical Degrees of Freedom ($s=\nabla \rho$; p or P the response field).
Residual / $\mathcal{R}$	Quadratic misalignment functional; pointwise $r(x,t)={|P-\nabla \rho |}^2$, global $\mathcal{R}(t)={\int |P-\nabla \rho |}^2$.
$S_R$	Residual–entropy proxy $S_R(t)=-log(\mathcal{R}(t)/{\mathcal{R}}_0+R_{*})$.
QMS	Quantum Markov Semigroup (normal, unital, completely positive—u.c.p.— semigroup on a von Neumann algebra).
GKSL	Gorini–Kossakowski–Sudarshan–Lindblad form (finite–dimensional generator of CPTP dynamics).
${\mathsf{E}}_{\mathcal{N}}$	$\omega$–preserving conditional expectation onto the fixed–point (pointer) algebra $\mathcal{N}$.
${\mathcal{E}}_{\omega }$	Dirichlet form ${\mathcal{E}}_{\omega }(X)={\langle X,-\mathcal{L}X\rangle }_{2,\omega }$ for an $\omega$–symmetric QMS.
Poincaré gap $\lambda$	Optimal constant in $\parallel X-{\mathsf{E}}_{\mathcal{N}}{X\parallel }_{2,\omega }^2\le {\lambda }^{-1}{\mathcal{E}}_{\omega }(X)$ (on ${\mathcal{N}}^{\perp }$).
LSI $\rho$	Log–Sobolev constant controlling entropy by Fisher information (used where applicable).
OA	Operator–algebraic (framework of von Neumann algebras / Dirichlet forms).
QM / TD / GR / QFT	Quantum Mechanics / Thermodynamics / General Relativity / Quantum Field Theory.
PDE	Partial Differential Equation (coercive gradient–relaxation template for DSFL).
OU	Ornstein–Uhlenbeck (free–field Euclidean semigroup in stochastic quantization).
GNS	Gelfand–Naimark–Segal Hilbert space associated to $(\mathcal{M},\omega )$.
CPTP	Completely Positive Trace–Preserving (quantum channels/evolutions).
POVM / PVM	Positive Operator–Valued / Projection–Valued Measure (measurement models).
Pointer algebra $\mathcal{N}$	Abelian subalgebra encoding the measurement context (“sector”).
Lüders map $\Phi$	Conditional expectation onto the diagonal algebra in a fixed basis.
DeTurck gauge	Gauge choice rendering curvature flows strictly elliptic on Riemannian slices.
Lichnerowicz operator	Elliptic operator on symmetric 2–tensors; its gap controls $L^2$ residual decay.
${\lambda }_{\mathrm{GR}}$	Spectral gap for the Lichnerowicz–DeTurck operator on the GR slice (physical subspace).
FRW	Friedmann–Robertson–Walker cosmological background (see Sec.).
CPL	Chevallier–Polarski–Linder dark–energy parametrization $w(a)=w_0+w_a(1-a)$.
BAO / CMB / RSD	Baryon Acoustic Oscillations / Cosmic Microwave Background / Redshift–Space Distortions.
AP test	Alcock–Paczyński test (consistency of $D_A(z)$ and $H(z)$ via the AP observable).
$f{\sigma }_8$	Growth–rate observable (see Sec.).
GBS	Gaussian Boson Sampling (quantum–optics benchmark for convergence rates).
ISS	Input–to–State Stability (noise–robust decay bound; see Lemma).

4. Main Results

4.1. Uniform Law, Contextual Rates

The DSFL inequality has a uniform mathematical form across sectors,

$${\displaystyle \frac{d}{dt}}\mathcal{R}(t)\le -\alpha \mathcal{R}(t),\Rightarrow \mathcal{R}(t)\le e^{-\alpha t}\mathcal{R}(0),$$

(3)

but the constant $\alpha >0$ is sector–dependent. In each setting, $\alpha$ equals the corresponding spectral–gap/coercivity parameter:

$$\alpha =\left\{\begin{array}{cc}2{\lambda }_{\mathrm{Poincar}\text{é}}\hfill & \mathrm{reversible}\mathrm{classical}/\mathrm{QMS}\mathrm{on}{\mathcal{N}}^{\perp },\hfill \\ 2\beta -C\epsilon \hfill & \mathrm{coercive}\mathrm{PDE}\mathrm{closure}(B⪰\beta I),\hfill \\ 2{\lambda }_{*}\hfill & \mathrm{free}-\mathrm{field}\mathrm{OU}/\mathrm{Hamiltonian}\mathrm{gap},\hfill \\ 2c{\lambda }_{\mathrm{GR}}\hfill & \mathrm{GR}\mathrm{slice}(\mathrm{DeTurck})\mathrm{on}\mathrm{the}\mathrm{physical}\mathrm{subspace}.\hfill \end{array}\right.$$

(4)

Remark 1 

(Mathematically uniform, physically contextual). Equation (3) is the same in all sectors (uniform Lyapunov law), but its rate α in (4) is fixed by the sectoral physics: pointer algebra and Dirichlet form (QMS), mobility and closure (PDE), Hamiltonian spectrum (QFT/Gaussian), or Lichnerowicz gap (geometry). Thus the restoration mechanism is universal, while the speed of restoration is contextual and must be computed in each sector.

Corollary 1 

(Sectoral attractors with explicit rates). Let $\mathcal{A}$ denote the sectoral equilibrium manifold (e.g. the fixed-point algebra $L^2(\mathcal{N},\omega )$, the Born law ${|\Psi |}^2$, or the Einstein balance set). Under the hypotheses yielding the corresponding gap/coercivity,

$$\mathrm{dist}\left(x(t),\mathcal{A}\right)\le C{e}^{-\alpha t}\mathrm{dist}\left(x(0),\mathcal{A}\right),$$

with α given by (4) and C a sectoral equivalence constant (projection/gauge). Hence, the form of convergence is universal, but the rate is determined by the sector’s gap.

Editorial note (for the Introduction). To make this distinction explicit up front, one sentence can be added: “While the DSFL provides a single Lyapunov law $\dot{\mathcal{R}}\le -\alpha \mathcal{R}$, the constant α is sectoral: it equals the relevant spectral gap or coercivity (operator algebraic, PDE, free-field, or geometric), so the mechanism is universal but its rate is context dependent.”

Remark 2 

(Covariance vs. slice formulation). The geometric DSFL result in Theorem 6 is proved on a compact Riemannian slice in DeTurck gauge. It establishes exponential suppression of the curvature–matter misfit in that elliptic setting but does not yet provide a fully covariant (Lorentzian) formulation. In other words, the present theorem is a slice-level statement—analytically rigorous but gauge–fixed—while a fully diffeomorphism–invariant, hyperbolic extension remains an open program. This distinction should be made explicit: the law is structurally general, yet the proof given here is Riemannian rather than covariant.

This section states the core theorems in a hypothesis–result format, ready for citation in later sections and proofs. Throughout we use the notation and conventions of Section 3. In particular, $\mathcal{R}$ denotes the global alignment residual (classical/PDE) or the noncommutative variance relative to a pointer algebra (QMS). All proofs are deferred to Section 5, where the three settings (classical, operator–algebraic, and PDE) are treated in parallel.

What Is Proved Here.

First, a unified propagation lemma (Section 4.2) shows that local convexity/contractivity mechanisms (Jensen, Kadison–Schwarz, energy identities) imply global residual monotonicity. Second, adding a spectral gap or coercivity yields exponential DSFL decay. In the operator–algebraic case (Section 4.3) we obtain an equivalence between DSFL and a noncommutative Poincaré (spectral–gap) inequality with optimal rate ${\alpha }_{*}=2{\lambda }_{*}$.

4.2. DSFL pRopagation Lemma (Classical/QMS/PDE)

Lemma 1 

(Propagation of residual monotonicity: classical, noncommutative, and PDE). Let ${({\mathsf{T}}_t)}_{t\ge 0}$ act on the Hilbert space where the residual $\mathcal{R}$ is evaluated. Assume one of the following structural settings.

(CL) Classical Markov setting. $({\mathsf{T}}_t)$ is a Markov contraction semigroup on $L^1\cap L^2(\mu )$ over a σ–finite measure space $(\Omega ,\mu )$: it preserves positivity, mass (${\mathsf{T}}_t\mathbf{1}=\mathbf{1}$), and is $L^p$–contractive for $1\le p\le \infty$. Let $u_0\in L^2(\mu )$ and let $r(x)=\phi (u_0(x))$ with $\phi :{\mathbb{R}}^m\to [0,\infty )$ a Borel convex function and $\phi (0)=0$. Define

$$\mathcal{R}(0):={\int }_{\Omega }\phi (u_0)d\mu ,\mathcal{R}(t):={\int }_{\Omega }\phi \left(({\mathsf{T}}_t{u}_0)(x)\right)d\mu (x).$$

(QM) Operator–algebraic/QMS setting. $({\mathsf{T}}_t)$ is a normal unital completely positive (u.c.p.) semigroup on a von Neumann algebra $(\mathcal{M},\omega )$, $\omega$–symmetric on $L^2(\omega )$ (reversible) and $\omega$–preserving : $\omega ({\mathsf{T}}_{t}Y)=\omega (Y)$. Let $\mathcal{N}:=\{X:{\mathsf{T}}_{t}X=X\forall t\}$ be the fixed–point algebra and ${\mathsf{E}}_{\mathcal{N}}:\mathcal{M}\to \mathcal{N}$ the ω–preserving conditional expectation (the $L^2(\omega )$ orthogonal projection). Define the noncommutative variance (residual)

$${\mathcal{R}}_{\omega }(X):={\parallel X-{\mathsf{E}}_{\mathcal{N}}(X)\parallel }_{2,\omega }^2,X\in L^2(\omega ).$$

(PDE) Parabolic/energy–identity setting. Let $\Omega \subset {\mathbb{R}}^d$ be a bounded $C^1$ domain (with homogeneous Dirichlet/Neumann b.c.) or ${\mathbb{T}}^d$ (periodic). Let $\rho (·,t)\in H^1(\Omega )$, $P(·,t)\in L^2(\Omega ;{\mathbb{R}}^d)$, and set $u:=P-\nabla \rho \in L^2(\Omega ;{\mathbb{R}}^d)$. Assume the exact residual identity

$${\displaystyle \frac{d}{dt}}{\int }_{\Omega }{|u|}^{2}dx=-2{\int }_{\Omega }{u}^{\top }Budx-2{\parallel \nabla ·u\parallel }_{L^2(\Omega )}^2+\mathsf{Rem}(t),$$

(5)

with a measurable $B(x,t)⪰0$ a.e. and a remainder $\mathsf{Rem}(t)\le 0$ (or dominated by the dissipative terms). Let $\mathcal{R}(t):={\int }_{\Omega }{|u(x,t)|}^{2}dx$.

Conclusion. In each setting, the residual is monotone nonincreasing :

$$\mathcal{R}(t)\le \mathcal{R}(0)\forall t\ge 0.$$

More precisely:

$$\begin{array}{cc}\hfill (\mathrm{CL}):& \int \phi ({\mathsf{T}}_t{u}_0)d\mu \le \int \phi (u_0)d\mu ,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill (\mathrm{QM}):& \parallel {\mathsf{T}}_{t}X-{\mathsf{E}}_{\mathcal{N}}{(X)\parallel }_{2,\omega }\le {\parallel X-{\mathsf{E}}_{\mathcal{N}}(X)\parallel }_{2,\omega },\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill (\mathrm{PDE}):& {\displaystyle \frac{d}{dt}}{\parallel u(t)\parallel }_{L^2(\Omega )}^2\le 0.\hfill \end{array}$$

(8)

Proof. 

(CL). By Jensen and positivity, $\phi ({\mathsf{T}}_t{u}_0)\le {\mathsf{T}}_t(\phi (u_0))$$\mu$–a.e.; integrate and use $\int {\mathsf{T}}_{t}fd\mu =\int fd\mu$.

(QM). Kadison–Schwarz for u.c.p. maps gives ${\mathsf{T}}_t{(X)}^{*}{\mathsf{T}}_t(X)\le {\mathsf{T}}_t(X^{*}X)$; apply $\omega$ and $\omega \circ {\mathsf{T}}_t=\omega$ to obtain $\parallel {\mathsf{T}}_t{X\parallel }_{2,\omega }\le {\parallel X\parallel }_{2,\omega }$. Since ${\mathsf{E}}_{\mathcal{N}}$ is the $L^2(\omega )$ orthogonal projection and ${\mathsf{T}}_t$ acts as the identity on $\mathcal{N}$, the same contraction holds for $X-{\mathsf{E}}_{\mathcal{N}}X$.

(PDE). Integrate (5) in time; $B⪰0$ and $\mathsf{Rem}(t)\le 0$ give ${\displaystyle \frac{d}{dt}}\int {|u|}^2\le 0$. □

Corollary 2 

(Propagation + gap/coercivity ⇒ DSFL). Under Lemma 1, suppose additionally:

• (CL) L (the generator of ${\mathsf{T}}_t=e^{tL}$) is symmetric on $L^2(\mu )$ and satisfies a Poincaré inequality ${\mathrm{Var}}_{\mu }(f)\le {\lambda }^{-1}\int \Gamma (f)d\mu$ for some $\lambda >0$ (here Γ denotes the carré du champ).
• (QM) The ω–symmetric generator $\mathcal{L}$ has a spectral gap $\lambda >0$ on $L^2(\omega )\ominus L^2(\mathcal{N},\omega )$: $\parallel X-{\mathsf{E}}_{\mathcal{N}}{X\parallel }_{2,\omega }^2\le {\lambda }^{-1}{\langle X,-\mathcal{L}X\rangle }_{2,\omega }$.
• (PDE) $B(x,t)⪰\beta I$ a.e. with $\beta >0$, and $\mathsf{Rem}(t)\le C\epsilon \mathcal{R}(t)$ for some $C>0$ and sufficiently small $\epsilon \ge 0$.

Then the DSFL inequality holds with an explicit rate:

$$(CL)\&(QM):\mathcal{R}(t)\le e^{-2\lambda t}\mathcal{R}(0),(PDE):\mathcal{R}(t)\le e^{-(2\beta -C\epsilon )t}\mathcal{R}(0).$$

Remark 3 

(Domains and regularity; where proofs appear). In (QM) , differentiating $\parallel {\mathsf{T}}_{t}X-{\mathsf{E}}_{\mathcal{N}}{X\parallel }_{2,\omega }^2$ at $t=0$ is justified for $X\in \mathsf{D}(\mathcal{L})$ and extends by density to $\mathsf{D}({\mathcal{E}}_{\omega })$ (closedness of the Dirichlet form). In (PDE) , integrations by parts are justified by periodic or homogeneous boundary conditions and the Sobolev regularity $\rho \in H^1$, $P\in L^2$ (or a standard mollification argument). Complete proofs are given in Section 5.

4.3. DSFL ⇔ Spectral Gap (Reversible QMS)

Definition 1 

(Operator-algebraic residual). Let $(\mathcal{M},\omega )$ be a von Neumann algebra with faithful normal state ω, and let $\mathcal{N}\subset \mathcal{M}$ be ω-modular invariant so that the ω-preserving conditional expectation ${\mathsf{E}}_{\mathcal{N}}:\mathcal{M}\to \mathcal{N}$ exists (Takesaki). Define, for $X\in L^2(\omega )$,

$${\mathcal{R}}_{\omega ,\mathcal{N}}(X):={\parallel X-{\mathsf{E}}_{\mathcal{N}}X\parallel }_{2,\omega }^2.$$

Then ${\mathcal{R}}_{\omega ,\mathcal{N}}$ is the squared $L^2(\omega )$-distance to $L^2(\mathcal{N},\omega )$.

Theorem 1 

(DSFL ⟺ spectral gap). Let ${({\mathsf{T}}_t)}_{t\ge 0}$ be a normal u.c.p. semigroup on $\mathcal{M}$ that is ω-symmetric on $L^2(\omega )$ and ω-preserving. Write $\mathcal{N}=\{X:{\mathsf{T}}_{t}X=X\forall t\}$ and let $\mathcal{L}$ be the $L^2(\omega )$ generator with Dirichlet form ${\mathcal{E}}_{\omega }(X)={\langle X,-\mathcal{L}X\rangle }_{2,\omega }$. The following are equivalent:

(i)

(DSFL) $\exists \alpha >0$ s.t. ${\mathcal{R}}_{\omega ,\mathcal{N}}({\mathsf{T}}_{t}X)\le e^{-\alpha t}{\mathcal{R}}_{\omega ,\mathcal{N}}(X)$ for all $t\ge 0$ and $X\in L^2(\omega )$.

(ii)

(Spectral gap) $\exists \lambda >0$ s.t. $\parallel X-{\mathsf{E}}_{\mathcal{N}}{X\parallel }_{2,\omega }^2\le {\lambda }^{-1}{\mathcal{E}}_{\omega }(X)$ for all X in the form domain.

Moreover, the optimal constants satisfy $\alpha =2\lambda$.

Setting and Assumptions.

Let $(\mathcal{M},\omega )$ be a $\sigma$–finite von Neumann algebra with faithful normal state $\omega$. Write $L^2(\omega )$ for the GNS Hilbert space with

$${\langle X,Y\rangle }_{2,\omega }:=\omega (X^{*}Y),{\parallel X\parallel }_{2,\omega }^2=\omega (X^{*}X).$$

Let ${({\mathsf{T}}_t)}_{t\ge 0}$ be a normal unital completely positive (u.c.p.) semigroup on $\mathcal{M}$ such that:

(A1)

(ω–symmetry / detailed balance ) Each ${\mathsf{T}}_t$ is self–adjoint on $L^2(\omega )$: ${\langle {\mathsf{T}}_{t}X,Y\rangle }_{2,\omega }={\langle X,{\mathsf{T}}_{t}Y\rangle }_{2,\omega }$ for all $X,Y\in L^2(\omega )$.

(A1′)

(ω–preservation ) $\omega ({\mathsf{T}}_{t}Z)=\omega (Z)$ for all Z and $t\ge 0$.

(A2)

($L^2$ generator and Dirichlet form ) The $L^2(\omega )$–generator $\mathcal{L}$ of $({\mathsf{T}}_t)$ is self–adjoint, with closed quadratic form

$$\mathsf{D}({\mathcal{E}}_{\omega })=\overline{\mathrm{Dom}({\mathcal{L}}^{1/2})},{\mathcal{E}}_{\omega }(X):={\langle X,-\mathcal{L}X\rangle }_{2,\omega },$$

and a *–subalgebra ${\mathcal{A}}_0\subset \mathsf{D}({\mathcal{E}}_{\omega })$ (e.g. the analytic elements of $({\mathsf{T}}_t)$) is a form core.

(A3)

(Fixed–point algebra ) $\mathcal{N}:=\{X\in \mathcal{M}:{\mathsf{T}}_{t}X=X\forall t\ge 0\}$ is a von Neumann subalgebra. The modular group ${({\sigma }_s^{\omega })}_{s\in \mathbb{R}}$ leaves $\mathcal{N}$ globally invariant.

Under (A3), Takesaki’s theorem yields:

Lemma 2 

(Conditional expectation). There exists a unique faithful normal conditional expectation ${\mathsf{E}}_{\mathcal{N}}:\mathcal{M}\to \mathcal{N}$ that preserves ω. It extends to the orthogonal projection $L^2(\omega )\to L^2(\mathcal{N},\omega )$ with $\parallel {\mathsf{E}}_{\mathcal{N}}\parallel =1$.

Define the noncommutative variance (residual)

$${\mathcal{R}}_{\omega }(X):={\parallel X-{\mathsf{E}}_{\mathcal{N}}(X)\parallel }_{2,\omega }^2,L^2(\omega )=L^2(\mathcal{N},\omega )\oplus L^2{(\mathcal{N},\omega )}^{\perp }.$$

Write $X^{\perp }:=X-{\mathsf{E}}_{\mathcal{N}}X\in L^2{(\mathcal{N},\omega )}^{\perp }$.

Lemma 3 

($L^2(\omega )$–differentiability and domains). For $X\in \mathsf{D}(\mathcal{L})$,

$${\displaystyle \frac{d}{dt}}{\parallel {\mathsf{T}}_t{X}^{\perp }\parallel }_{2,\omega }^2{|}_{t=0}=-2{\mathcal{E}}_{\omega }(X^{\perp }).$$

More generally, for all $t\ge 0$ with ${\mathsf{T}}_{t}X\in \mathsf{D}(\mathcal{L})$,

$${\displaystyle \frac{d}{dt}}{\parallel {\mathsf{T}}_t{X}^{\perp }\parallel }_{2,\omega }^2=-2{\mathcal{E}}_{\omega }({\mathsf{T}}_t{X}^{\perp }).$$

Proof. 

Since $({\mathsf{T}}_t)$ is self–adjoint and strongly continuous on $L^2(\omega )$, $t\mapsto \parallel {\mathsf{T}}_t{X}^{\perp }{\parallel }_{2,\omega }^2=\langle {\mathsf{T}}_t{X}^{\perp },{\mathsf{T}}_t{X}^{\perp }\rangle$ is $C^1$ on any interval where ${\mathsf{T}}_{t}X\in \mathsf{D}(\mathcal{L})$. Differentiating and using self–adjointness of $\mathcal{L}$ gives

$${\displaystyle \frac{d}{dt}}{\parallel {\mathsf{T}}_t{X}^{\perp }\parallel }_{2,\omega }^2=2{\langle {\mathsf{T}}_t{X}^{\perp },\mathcal{L}{\mathsf{T}}_t{X}^{\perp }\rangle }_{2,\omega }=-2{\mathcal{E}}_{\omega }({\mathsf{T}}_t{X}^{\perp }).$$

Orthogonality to $L^2(\mathcal{N},\omega )$ is preserved because ${\mathsf{T}}_t{\mathsf{E}}_{\mathcal{N}}={\mathsf{E}}_{\mathcal{N}}$ and ${\mathsf{T}}_t$ acts as the identity on the fixed space. □

Poincaré Inequality on the Orthogonal Complement.

We say that $\mathcal{L}$ has a spectral gap $\lambda >0$ on $L^2{(\mathcal{N},\omega )}^{\perp }$ if

$${\mathcal{E}}_{\omega }(Z)\ge \lambda {\parallel Z\parallel }_{2,\omega }^2\forall Z\in \mathsf{D}({\mathcal{E}}_{\omega })\cap L^2{(\mathcal{N},\omega )}^{\perp }.$$

Equivalently,

$$\parallel X-{\mathsf{E}}_{\mathcal{N}}{X\parallel }_{2,\omega }^2\le {\displaystyle \frac{1}{\lambda }}{\mathcal{E}}_{\omega }(X)\forall X\in \mathsf{D}({\mathcal{E}}_{\omega }),$$

(9)

i.e. the noncommutative Poincaré inequality holds on ${\mathcal{N}}^{\perp }$.

Theorem 2 

(Operator–algebraic DSFL ⟺ spectral gap). Under (A1)–(A3) the following are equivalent:

(i)

DSFL decay.$\exists \alpha >0$ such that for all $X\in L^2(\omega )$,

$${\mathcal{R}}_{\omega }({\mathsf{T}}_{t}X)\le e^{-\alpha t}{\mathcal{R}}_{\omega }(X)\forall t\ge 0.$$

(ii)

Spectral gap / Poincaré on ${\mathcal{N}}^{\perp }$. There exists $\lambda >0$ such that (9) holds on $\mathsf{D}({\mathcal{E}}_{\omega })$.

Moreover, the optimal constants coincide as ${\alpha }_{*}=2{\lambda }_{*}$.

Proof sketch. 

For $Z_t:={({\mathsf{T}}_{t}X)}^{\perp }$ one has $\frac{d}{dt}{\parallel Z_t\parallel }_{2,\omega }^2=-2{\mathcal{E}}_{\omega }(Z_t)$. If (ii) holds then $\dot{\parallel Z_t{\parallel }_2^2}\le -2\lambda {\parallel Z_t\parallel }_2^2$ and Grönwall yields (i) with $\alpha =2\lambda$. Conversely, differentiating (i) at $t=0$ gives ${\mathcal{E}}_{\omega }(X^{\perp })\ge (\alpha /2){\parallel X^{\perp }\parallel }_2^2$. □

Remark 4 

(Closability, core, and invariance). (i) The form ${\mathcal{E}}_{\omega }$ is closed on $L^2(\omega )$ because $({\mathsf{T}}_t)$ is self–adjoint and contractive; $\mathsf{D}({\mathcal{E}}_{\omega })=\mathsf{D}({\mathcal{L}}^{1/2})$ is the canonical form domain. Moreover, the *–algebra of analytic elements for $({\mathsf{T}}_t)$ is dense in $\mathsf{D}({\mathcal{E}}_{\omega })$ and forms a core. (ii) Reversibility implies $({\mathsf{T}}_t)$ leaves both $L^2(\mathcal{N},\omega )$ and $L^2{(\mathcal{N},\omega )}^{\perp }$ invariant; hence the restriction of $\mathcal{L}$ to ${\mathcal{N}}^{\perp }$ is self–adjoint and nonnegative, with spectrum contained in $\{0\}\cup [{\lambda }_{*},\infty )$ iff (9) holds.

Remark 5 

(Conditional expectation ${\mathsf{E}}_{\mathcal{N}}$). The modular invariance of $\mathcal{N}$ (A3) ensures the existence and uniqueness of a faithful normal ω–preserving conditional expectation ${\mathsf{E}}_{\mathcal{N}}:\mathcal{M}\to \mathcal{N}$ (Takesaki’s theorem). As an $L^2$ map it is the orthogonal projection onto $L^2(\mathcal{N},\omega )$, so ${\mathcal{R}}_{\omega }(X)=\mathrm{dist}{(X,L^2(\mathcal{N},\omega ))}^2$.

DSFL Interpretation.

Theorem 2 identifies the DSFL rate with the spectral gap of the reversible QMS restricted to $L^2{(\mathcal{N},\omega )}^{\perp }$: the noncommutative variance (misalignment) decays exponentially iff the Poincaré inequality holds on ${\mathcal{N}}^{\perp }$, with $\alpha =2\lambda$.

4.4. Sharp Lindblad Rate (Finite–Dimensional Dephasing)

Let $\mathcal{H}={\mathbb{C}}^d$ with orthonormal basis ${\{|i\rangle \}}_{i=1}^d$ and spectral projectors $P_i=|i\rangle \langle i|$. Consider the (trace–preserving, completely positive) dephasing Lindblad generator on $M_d(\mathbb{C})$

$${\mathcal{L}}^{*}(\sigma )=\sum _{i=1}^d{\gamma }_i\left(P_i\sigma P_i-\frac{1}{2}\{P_i,\sigma \}\right),{\gamma }_i>0,$$

(10)

and the dual master equation ${\dot{\sigma }}_t={\mathcal{L}}^{*}({\sigma }_t)$ for density matrices. Let $\Phi :M_d\to M_d$ be the Lüders (diagonal) conditional expectation $\Phi (X)={\sum }_i{P}_{i}X{P}_i$ onto the abelian pointer algebra $\mathcal{N}=\{X:[X,P_i]=0\forall i\}$. Then $\Phi$ is the (trace–)preserving conditional expectation onto $\mathcal{N}$, and the Lüders residual

$${\mathcal{R}}_{\mathrm{L}\ddot{\mathrm{u}}\mathrm{ders}}(\sigma ):={\parallel \sigma -\Phi (\sigma )\parallel }_2^2=\sum _{i\ne j}{|{(\sigma )}_{ij}|}^2$$

is the Hilbert–Schmidt variance off the diagonal algebra $\mathcal{N}$.

Theorem 3 

(Sharp exponential decay of the Lüders residual). Let

$${\lambda }_{*}:={\displaystyle \frac{1}{2}}\underset{i\ne j}{min}({\gamma }_i+{\gamma }_j),{\alpha }_{*}:=2{\lambda }_{*}=\underset{i\ne j}{min}({\gamma }_i+{\gamma }_j).$$

Then for all $t\ge 0$,

$${\mathcal{R}}_{\mathrm{L}\ddot{\mathrm{u}}\mathrm{ders}}({\sigma }_t)\le e^{-{\alpha }_{*}t}{\mathcal{R}}_{\mathrm{L}\ddot{\mathrm{u}}\mathrm{ders}}({\sigma }_0),$$

and the rate ${\alpha }_{*}$ is optimal (sharp). In particular, if the minimal dephasing rate is attained by at least two indices, then ${\alpha }_{*}=2{min}_i{\gamma }_i$; otherwise ${\alpha }_{*}={\gamma }_{min}+{\gamma }_{2\mathrm{nd}\min }$.

Proof (modewise solution). 

In the basis $\{|i\rangle \}$ one has

$${\displaystyle \frac{d}{dt}}{({\sigma }_t)}_{ii}=0,{\displaystyle \frac{d}{dt}}{({\sigma }_t)}_{ij}=-\frac{{\gamma }_i+{\gamma }_j}{2}{({\sigma }_t)}_{ij}(i\ne j),$$

so ${({\sigma }_t)}_{ij}=e^{-({\gamma }_i+{\gamma }_j)t/2}{({\sigma }_0)}_{ij}$ and hence

$${\mathcal{R}}_{\mathrm{L}\ddot{\mathrm{u}}\mathrm{ders}}({\sigma }_t)=\sum _{i\ne j}{e}^{-({\gamma }_i+{\gamma }_j)t}|{({\sigma }_0)}_{ij}{|}^2\le e^{-{\alpha }_{*}t}\sum _{i\ne j}{|{({\sigma }_0)}_{ij}|}^2.$$

Sharpness: choose initial data supported on a pair $(i_{*},j_{*})$ achieving ${min}_{i\ne j}({\gamma }_i+{\gamma }_j)$. □

Corollary 3 

(Born–aligned limit and trace–norm control). As $t\to \infty$, ${\sigma }_t\to \Phi ({\sigma }_0)$ in Hilbert–Schmidt norm and in trace norm with

$$\parallel {\sigma }_t-\Phi ({\sigma }_0){\parallel }_1\le \sqrt{d}{e}^{-{\lambda }_{*}t}{\parallel {\sigma }_0-\Phi ({\sigma }_0)\parallel }_2.$$

The limit is the Lüders (Born–aligned) state for the measurement basis $\{|i\rangle \}$.

Remark 6 

(DSFL and spectral gap identification). The off–diagonal (pointer–orthogonal) sector is invariant and the restriction of $\mathcal{L}$ to it has spectral gap ${\lambda }_{*}=\frac{1}{2}{min}_{i\ne j}({\gamma }_i+{\gamma }_j)$. By Theorem 2, the DSFL rate is ${\alpha }_{*}=2{\lambda }_{*}$.

Remark 7 

(Commuting Hamiltonians and basis choice). (a) Adding $-i[H,{\sigma }_t]$ with $H={\sum }_i{h}_i{P}_i$ (i.e. $[H,P_i]=0$) leaves the modewise decay and the sharp rate ${\alpha }_{*}$ unchanged: the diagonals remain constant and off–diagonals acquire only phases. For noncommuting H, oscillations appear but the envelope of ${\mathcal{R}}_{\mathrm{L}\ddot{\mathrm{u}}\mathrm{ders}}({\sigma }_t)$ still decays at least as $e^{-{\alpha }_{*}t}$. (b) The statement is basis–covariant: any pure dephasing generator is unitarily diagonalizable; $\mathcal{N}$ is the diagonal algebra in that basis, and Φ is the corresponding conditional expectation.

4.5. Coercive PDE tEmplate: Exponential Decay

Let $\Omega \subset {\mathbb{R}}^d$ be a bounded $C^1$ domain (or ${\mathbb{T}}^d$) with periodic or homogeneous boundary conditions chosen so that integrations by parts incur no boundary terms (cf. Remark 3). Assume

$$\rho (·,t)\in H^1(\Omega ),P(·,t)\in L^2(\Omega ;{\mathbb{R}}^d)$$

for all $t\ge 0$, and consider the system

$${\partial }_t\rho =\nabla ·\left(P-\nabla \rho \right),{\partial }_{t}P=-B(x,t)\left(P-\nabla \rho \right)+G(\rho ,P),$$

(11)

with $B\in L^{\infty }(\Omega \times [0,\infty ))$ satisfying $B(x,t)⪰\beta I$ a.e. for some $\beta >0$, and a locally Lipschitz coupling G that is subcritical in the residual energy sense specified below. Define the residual

$$u:=P-\nabla \rho \in L^2(\Omega ;{\mathbb{R}}^d),\mathcal{R}(t):={\int }_{\Omega }{|u(x,t)|}^{2}dx.$$

Theorem 4 

(Exponential residual decay under coercivity). Under (11), $B⪰\beta I$, and the subcriticality condition

$$\left|{\int }_{\Omega }u·G(\rho ,P)dx\right|\le C\epsilon \mathcal{R}(t)$$

(12)

for some $C>0$ and sufficiently small $\epsilon \ge 0$, one has the differential inequality

$${\displaystyle \frac{d}{dt}}\mathcal{R}(t)\le -(2\beta -C\epsilon )\mathcal{R}(t),$$

hence, by Grönwall,

$$\mathcal{R}(t)\le e^{-(2\beta -C\epsilon )t}\mathcal{R}(0)(t\ge 0).$$

In particular, in the uncoupled case ($\epsilon =0$) one obtains $\mathcal{R}(t)\le e^{-2\beta t}\mathcal{R}(0)$.

Proof sketch. 

Differentiate $\mathcal{R}(t)={\int }_{\Omega }{|u|}^2$ and use ${\partial }_{t}u=-Bu-\nabla (\nabla ·u)+G(\rho ,P)$ to obtain the exact identity (cf. (5))

$${\displaystyle \frac{d}{dt}}\mathcal{R}(t)=-2{\int }_{\Omega }{u}^{\top }Budx-2{\parallel \nabla ·u\parallel }_{L^2(\Omega )}^2+2{\int }_{\Omega }u·Gdx.$$

Since $B⪰\beta I$, the first term is $\le -2\beta \mathcal{R}(t)$, the divergence term is nonpositive, and (12) gives $2\int u·G\le 2C\epsilon \mathcal{R}(t)$. Combine and apply Grönwall. □

Remark 8 

(Scope and refinements). (i) Template coverage. The estimate applies to mobility–relaxation closures, linear couplings bounded by the residual, and mild nonlinearities that satisfy (12). It is the PDE instance of the propagation lemma (Lemma 1, case (PDE)) followed by a sectoral coercivity bound; in DSFL notation, the rate is $\alpha =2\beta -C\epsilon$.

(ii) Spectral sharpening. With the Helmholtz decomposition $u=\nabla \varphi +w$ and $\nabla ·w=0$, the gradient channel gains additional spectral damping: on bounded/periodic domains,

$${\parallel \nabla ·u\parallel }_{L^2}^2={\parallel \Delta \varphi \parallel }_{L^2}^2\ge {\lambda }_1{\parallel \nabla \varphi \parallel }_{L^2}^2$$

with Poincaré constant ${\lambda }_1>0$, so the gradient part decays at least like $e^{-2(\beta +{\lambda }_1)t}$ while the solenoidal part decays like $e^{-2\beta t}$.

(iii) Regularity/BCs. All integrations by parts are justified by periodic or homogeneous boundary conditions and the Sobolev regularity stated above (see Remark 3). For weak solutions, the identity holds by density/mollification and lower–semicontinuity.

(iv) DSFL identification. The inequality $\dot{\mathcal{R}}\le -(2\beta -C\epsilon )\mathcal{R}$ is the DSFL law in this sector. With $\epsilon =0$ one recovers the clean rate $\alpha =2\beta$.

4.6. Free–Field Stochastic Quantization: Gap–Driven Decay

Setting and Notation.

Let $\Lambda$ be either the flat d –torus ${\mathbb{T}}^d$ (side length $L>0$) or ${\mathbb{R}}^d$. We consider a real free scalar field $\varphi :\Lambda \to \mathbb{R}$ with Euclidean action

$$S[\varphi ]={\displaystyle \frac{1}{2}}{\int }_{\Lambda }\left({|\nabla \varphi (x)|}^2+m^2\varphi {(x)}^2\right)dx,m\ge 0.$$

Parisi–Wu stochastic quantization evolves the field in an auxiliary “Langevin time” $\tau \ge 0$ by

$${\partial }_{\tau }{\varphi }_{\tau }(x)=-{\displaystyle \frac{\delta S}{\delta \varphi }}(x)+\eta (x,\tau )=\Delta {\varphi }_{\tau }(x)-m^2{\varphi }_{\tau }(x)+\eta (x,\tau ),$$

(13)

where $\eta$ is space–time white noise with covariance $\mathbb{E}[\eta (x,\tau )\eta (y,\sigma )]=2\delta (x-y)\delta (\tau -\sigma )$ (the factor 2 ensures that the stationary covariance solves $A{\Sigma }_{\infty }+{\Sigma }_{\infty }A=2I$). The generator of the one–body deterministic part is the nonnegative operator

$$A:=-\Delta +m^2\mathrm{on}{L}^2(\Lambda )(\mathrm{domain}{H}^2(\Lambda )),$$

whose spectrum is $\sigma (A)=\{|k{|}^2+m^2:k\in (2\pi /L){\mathbb{Z}}^d\}$ on ${\mathbb{T}}^d$, and $\sigma (A)=[m^2,\infty )$ on ${\mathbb{R}}^d$.

OU Semigroup and Covariance Flow.

Equation (13) is an infinite–dimensional Ornstein–Uhlenbeck (OU) process on $H^{-s}(\Lambda )$ for $s>d/2$ (or on $L^2(\Lambda )$ at the level of covariances). Writing $X_{\tau }:={\varphi }_{\tau }$ and $W_{\tau }$ a cylindrical Wiener process on $L^2(\Lambda )$, (13) reads

$$d{X}_{\tau }=-A{X}_{\tau }d\tau +\sqrt{2}d{W}_{\tau }.$$

Let ${\Sigma }_{\tau }:=\mathbb{E}[X_{\tau }\otimes X_{\tau }]$ be the (two–point) covariance operator on $L^2(\Lambda )$. Standard OU calculus yields

$${\dot{\Sigma }}_{\tau }=-A{\Sigma }_{\tau }-{\Sigma }_{\tau }A+2I,{\Sigma }_0\mathrm{given},\Rightarrow {\Sigma }_{\tau }={\int }_0^{\tau }{e}^{-sA}2I{e}^{-sA}ds+e^{-\tau A}{\Sigma }_0{e}^{-\tau A}.$$

(14)

The unique stationary covariance is the Green operator

$${\Sigma }_{\infty }=A^{-1}(\mathrm{on}{\mathbb{T}}^d\mathrm{for}\mathrm{any}m\ge 0;\mathrm{on}{\mathbb{R}}^d\mathrm{only}\mathrm{if}m>0).$$

(15)

Subtracting (15) from (14) gives the exact relaxation formula

$${\Sigma }_{\tau }-{\Sigma }_{\infty }=e^{-\tau A}\left({\Sigma }_0-{\Sigma }_{\infty }\right)e^{-\tau A}.$$

(16)

Smeared Two–Point Residual.

For a test function f (to be specified below), define the smeared evaluation ${\varphi }_{\tau }(f):={\int }_{\Lambda }{\varphi }_{\tau }(x)f(x)dx={\langle f,X_{\tau }\rangle }_{L^2}.$ Then

$$\mathbb{E}\left[{\varphi }_{\tau }(f){\varphi }_{\tau }(f)\right]=\langle f,{\Sigma }_{\tau }f\rangle ,\mathbb{E}\left[{\varphi }_{\infty }(f){\varphi }_{\infty }(f)\right]=\langle f,{\Sigma }_{\infty }f\rangle .$$

We define the (quadratic) residual as the squared deviation of the two–point function:

$$R[f;\tau ]:=|\underset{{\displaystyle \mathbb{E}[{\varphi }_{\tau }{(f)}^2]-\mathbb{E}[{\varphi }_{\infty }{(f)}^2]}}{\underbrace{\langle f,({\Sigma }_{\tau }-{\Sigma }_{\infty })f\rangle }}{|}^2.$$

(17)

Admissible Class of Test Functions.

On ${\mathbb{T}}^d$, take $f\in L^2({\mathbb{T}}^d)$; on ${\mathbb{R}}^d$ with $m>0$, take $f\in L^2({\mathbb{R}}^d)$ (or Schwarz $\mathcal{S}$). For $m=0$, impose an IR regularization (finite volume or mean–zero f and a Poincaré gap).

Theorem 5 

(Residual decay at twice the Hamiltonian gap). Let $A=-\Delta +m^2$ on $L^2(\Lambda )$ and suppose there is a spectral gap

$${\lambda }_{*}:=inf\sigma \left(A{\upharpoonright }_{\mathrm{Ker}{(A)}^{\perp }}\right)>0.$$

Then for any admissible f,

$$R[f;\tau ]\le R[f;0]e^{-2{\lambda }_{*}\tau }.$$

(18)

In particular:

• On ${\mathbb{T}}^d$ with $m\ge 0$, ${\lambda }_{*}=min\{m^2,{\lambda }_1(-\Delta )\}$, where ${\lambda }_1(-\Delta )={(2\pi /L)}^2$ is the first positive Laplace eigenvalue; if $m>0$ then ${\lambda }_{*}=m^2$.
• On ${\mathbb{R}}^d$ with $m>0$, ${\lambda }_{*}=m^2$; if $m=0$ there is no gap and (18) fails globally (decay is not uniform, see Remark 10).

Proof. 

By (16),

$$\langle f,({\Sigma }_{\tau }-{\Sigma }_{\infty })f\rangle =〈e^{-\tau A}f,({\Sigma }_0-{\Sigma }_{\infty })e^{-\tau A}f〉.$$

Hence,

$$|\langle f,({\Sigma }_{\tau }-{\Sigma }_{\infty })f\rangle |\le \parallel {\Sigma }_0-{\Sigma }_{\infty }{\parallel }_{\mathrm{op},ker{(A)}^{\perp }}{\parallel e^{-\tau A}f\parallel }_2^2.$$

Because $\parallel e^{-\tau A}{\parallel }_{\mathrm{op}}=e^{-{\lambda }_{*}\tau }$ on $ker{(A)}^{\perp }$ and $e^{-\tau A}$ is the identity on $ker(A)$ (trivial unless $m=0$ on compact $\Lambda$), we have $\parallel e^{-\tau A}{f\parallel }_2\le e^{-{\lambda }_{*}\tau }{\parallel f\parallel }_2$ whenever $f\perp ker(A)$ (automatically true for $m>0$). Thus

$$|\langle f,({\Sigma }_{\tau }-{\Sigma }_{\infty })f\rangle |\le \parallel {\Sigma }_0-{\Sigma }_{\infty }{\parallel }_{\mathrm{op},ker{(A)}^{\perp }}{e}^{-2{\lambda }_{*}\tau }{\parallel f\parallel }_2^2,$$

and squaring yields (18) after absorbing the prefactor into $R[f;0]$. □

Remark 9 

(Explicit Fourier picture on ${\mathbb{T}}^d$). Write $f(x)={\sum }_{k\in (2\pi /L){\mathbb{Z}}^d}{\widehat{f}}_k{e}^{ik·x}$ and similarly for ${\varphi }_{\tau }$. Each mode solves the scalar OU SDE $d{\widehat{\varphi }}_{\tau }(k)=-{(|k|}^2+m^2){\widehat{\varphi }}_{\tau }(k)d\tau +\sqrt{2}d{\beta }_{\tau }(k),$ so $\mathrm{Var}{\widehat{\varphi }}_{\tau }(k)=(\mathrm{Var}{\widehat{\varphi }}_0(k)-{(|k|}^2+m^2{)}^{-1})e^{-2(|k{|}^2+m^2)\tau }+{(|k|}^2+m^2{)}^{-1}.$ Therefore, for the smeared variance,

$$\langle f,({\Sigma }_{\tau }-{\Sigma }_{\infty })f\rangle =\sum _k{|{\widehat{f}}_k|}^2\left(\mathrm{Var}{\widehat{\varphi }}_{\tau }(k)-{(|k|}^2+m^2{)}^{-1}\right),$$

and the slowest decaying mode has rate $2{min}_k{(|k|}^2+m^2)=2{\lambda }_{*}$.

Remark 10 

(Massless case and infrared issues). On ${\mathbb{T}}^d$ with $m=0$, $ker(A)=span\{1\}$ and ${\lambda }_{*}={\lambda }_1(-\Delta )={(2\pi /L)}^2$ provided f has zero mean (or we project away the constant mode). On ${\mathbb{R}}^d$ with $m=0$, $\sigma (A)=[0,\infty )$ has no gap; uniform exponential decay fails and long–wavelength modes relax only algebraically in spatially extended senses. Thus a spectral gap (mass $m>0$ or finite–volume Poincaré gap) is essential for the DSFL rate (18).

Remark 11 

(From two–point residuals to DSFL). The estimate (18) derives the DSFL inequality in the Gaussian sector: the misalignment functional ${\mathcal{R}}_{\mathrm{free}}(\tau ):={sup}_{{\parallel f\parallel }_2=1}|\langle f,({\Sigma }_{\tau }-{\Sigma }_{\infty })f\rangle {|}^2$ decays as ${\dot{\mathcal{R}}}_{\mathrm{free}}\le -2{\lambda }_{*}{\mathcal{R}}_{\mathrm{free}}$ with optimal rate $2{\lambda }_{*}$. Equivalently, for any fixed f, the scalar residual $R[f;\tau ]$ satisfies the same inequality.

Remark 12 

(Regularity of smearing). On ${\mathbb{T}}^d$ any $f\in L^2$ is admissible. On ${\mathbb{R}}^d$ with $m>0$, $f\in L^2$ (or $\mathcal{S}$) suffices and all formulas above hold; higher regularity $f\in H^{\alpha }$ yields the same exponential rate while changing only the (finite) prefactors.

4.7. GR Slice: Geometric Residual Decay (Small Data, DeTurck Gauge)

Scope.

This subsection establishes a slice analogue on compact Riemannian 3–manifolds, in DeTurck gauge, for small perturbations of a fixed target metric. It is not a fully covariant Lorentzian result. A diffeomorphism–invariant Lorentzian formulation is stated as an open program in the remarks below.

Standing Assumptions (Slice, Small Data).

Let $({\Sigma }^3,\overline{\gamma })$ be a smooth, closed (compact, boundaryless) Riemannian 3–manifold. Let $T\in C^{\infty }(\Sigma ;{\mathrm{Sym}}^2{T}^{*}\Sigma )$ be time–independent and divergence–free with respect to $\overline{\gamma }$, ${\nabla }^{\overline{\gamma }i}{T}_{ij}=0$. Assume there exists a target metric $\overline{\gamma }$ solving

$$G[\overline{\gamma }]=\kappa T.$$

(19)

Consider the Einstein–source flow in DeTurck gauge

$${\partial }_t{\gamma }_{ij}=-2\left(G_{ij}[\gamma ]-\kappa T_{ij}\right)+{\nabla }_i{X}_j(\gamma )+{\nabla }_j{X}_i(\gamma ),$$

(20)

with DeTurck vector

$$X^k(\gamma )={\gamma }^{pq}\left({\Gamma }_{pq}^k(\gamma )-{\Gamma }_{pq}^k(\overline{\gamma })\right),$$

(21)

which renders the linearization at $\overline{\gamma }$ strictly elliptic on the gauge–orthogonal (physical) subspace.

Residual (Gauge–invariant).

Define the $L^2$ curvature–matter misfit

$${\mathcal{R}}_{\mathrm{geom}}(t):={\int }_{\Sigma }\parallel G[\gamma (t)]-\kappa T{\parallel }_{\gamma (t)}^{2}d{\mu }_{\gamma (t)}.$$

(22)

Write $h:=\gamma -\overline{\gamma }$ and assume small initial data $h(0)\in H^k(\overline{\gamma })$ with ${\parallel h(0)\parallel }_{H^k}\le \delta$ for some $k\ge 4$ and $\delta >0$ sufficiently small.

Lemma 4 

(Constraint preservation (DeTurck slice)). Let $C_j(\gamma ):={\nabla }_{\gamma }^i\left(G_{ij}[\gamma ]-\kappa T_{ij}\right)$. Along (20), ${\partial }_t{C}_j={\Delta }_{\gamma }{C}_j+{R_j}^k(\gamma )C_k.$ If $C_j(0)=0$ and T is time–independent with ${\nabla }_{\gamma }T\equiv 0$ along the flow (equivalently ${\nabla }^{\overline{\gamma }}T\equiv 0$ at $t=0$ and preserved thereafter), then $C_j\equiv 0$ for all $t\ge 0$.

Linearization, Model Operator, and Spectral Gap.

Linearizing (20) at $\overline{\gamma }$ yields, for h small,

$${\partial }_{t}h=-{\mathcal{L}}_{\overline{\gamma }}h+\mathcal{N}(h),$$

(23)

where ${\mathcal{L}}_{\overline{\gamma }}$ is the self–adjoint Lichnerowicz–DeTurck operator on symmetric 2–tensors,

$${\mathcal{L}}_{\overline{\gamma }}h:=-{\Delta }_L^{\overline{\gamma }}h-2\mathrm{Rm}[\overline{\gamma }]*h,{({\Delta }_{L}h)}_{ij}:=\Delta h_{ij}+2R_{ikjl}{h}^{kl}-{R_i}^k{h}_{kj}-{R_j}^k{h}_{ki},$$

(24)

and $\mathcal{N}(h)=O(|h||{\nabla }^{2}h|+|\nabla h{|}^2)$ is quadratic/higher order. Assume a spectral gap on the physical (gauge–orthogonal) subspace:

$$\exists {\lambda }_{\mathrm{GR}}>0\mathrm{s}.\mathrm{t}.{\langle h,{\mathcal{L}}_{\overline{\gamma }}h\rangle }_{L^2(\overline{\gamma })}\ge {\lambda }_{\mathrm{GR}}{\parallel h\parallel }_{L^2(\overline{\gamma })}^2\mathrm{for}\mathrm{all}h\perp \mathrm{gauge}\mathrm{directions}.$$

(25)

By elliptic regularity, for h sufficiently small in $H^k$,

$${\parallel h\parallel }_{H^2(\overline{\gamma })}\le C_{\mathrm{ell}}\left(\parallel {\mathcal{L}}_{\overline{\gamma }}{h\parallel }_{L^2(\overline{\gamma })}+{\parallel h\parallel }_{L^2(\overline{\gamma })}\right)\le \frac{2C_{\mathrm{ell}}}{\sqrt{{\lambda }_{\mathrm{GR}}}}{\parallel {\mathcal{L}}_{\overline{\gamma }}h\parallel }_{L^2(\overline{\gamma })}.$$

(26)

Residual Equivalence Near $\overline{\gamma }$.

A Taylor expansion at $\overline{\gamma }$ and (19) give

$$G[\overline{\gamma }+h]-\kappa T=\frac{1}{2}{\mathcal{L}}_{\overline{\gamma }}h+\mathcal{Q}(h),\parallel \mathcal{Q}(h){\parallel }_{L^2(\overline{\gamma })}\le C_Q{\parallel h\parallel }_{H^2}{\parallel h\parallel }_{H^1}.$$

(27)

For h small and $\gamma$ close to $\overline{\gamma }$ in $C^0$, the norms induced by $\gamma$ and $\overline{\gamma }$ are equivalent; hence

$${\mathcal{R}}_{\mathrm{geom}}(t)={\parallel G[\gamma ]-\kappa T\parallel }_{L^2(\gamma )}^2\simeq \frac{1}{4}\parallel {\mathcal{L}}_{\overline{\gamma }}{h\parallel }_{L^2(\overline{\gamma })}^2\simeq {\parallel h\parallel }_{H^2(\overline{\gamma })}^2,$$

(28)

with constants depending only on $(\Sigma ,\overline{\gamma })$ and the smallness radius.

Theorem 6 

(Exponential $L^2$ decay of the geometric residual on a slice). Under (19), (20), (25), and the small–data hypothesis ${\parallel h(0)\parallel }_{H^k(\overline{\gamma })}\le \delta$ (for some $k\ge 4$ and $\delta >0$ sufficiently small), there exists $c\in (0,1)$ (depending only on $\overline{\gamma }$ and δ) such that

$${\displaystyle \frac{d}{dt}}{\mathcal{R}}_{\mathrm{geom}}(t)\le -2c{\lambda }_{\mathrm{GR}}{\mathcal{R}}_{\mathrm{geom}}(t),\Rightarrow {\mathcal{R}}_{\mathrm{geom}}(t)\le e^{-2c{\lambda }_{\mathrm{GR}}t}{\mathcal{R}}_{\mathrm{geom}}(0).$$

In particular, $G[\gamma (t)]=\kappa T$ in $L^2(\Sigma )$ as $t\to \infty$, and $\gamma (t)\to \overline{\gamma }$ modulo diffeomorphisms.

Proof sketch. 

Set $E(t):=\parallel {\mathcal{L}}_{\overline{\gamma }}{h(t)\parallel }_{L^2(\overline{\gamma })}^2$. From (23),

$$\dot{E}(t)=2\langle {\mathcal{L}}_{\overline{\gamma }}h,{\mathcal{L}}_{\overline{\gamma }}{\partial }_{t}h\rangle =-2{\parallel {\mathcal{L}}_{\overline{\gamma }}^{3/2}h\parallel }_{L^2}^2+2\langle {\mathcal{L}}_{\overline{\gamma }}h,{\mathcal{L}}_{\overline{\gamma }}\mathcal{N}(h)\rangle .$$

The gap (25) yields $-2\parallel {\mathcal{L}}_{\overline{\gamma }}^{3/2}{h\parallel }_{L^2}^2\le -2{\lambda }_{\mathrm{GR}}E(t)$. Estimate the nonlinear term via Cauchy–Schwarz, (26), and small–data absorption to obtain $|\langle {\mathcal{L}}_{\overline{\gamma }}h,{\mathcal{L}}_{\overline{\gamma }}\mathcal{N}(h)\rangle |\le (1-c){\lambda }_{\mathrm{GR}}E(t)$, whence $\dot{E}(t)\le -2c{\lambda }_{\mathrm{GR}}E(t)$. Grönwall gives $E(t)\le e^{-2c{\lambda }_{\mathrm{GR}}t}E(0)$; (28) then implies the stated decay for ${\mathcal{R}}_{\mathrm{geom}}$. □

Remark 13 

(Well–posedness and norm equivalences). For ${\parallel h(0)\parallel }_{H^k}$ small ($k\ge 4$), parabolic–elliptic theory in DeTurck gauge yields local existence/uniqueness in $C([0,T];H^k)$ and a priori control. The exponential decay closes the bootstrap globally. Moreover, $L^2(\gamma )$– and $L^2(\overline{\gamma })$–norms are equivalent for small h, so (22) and (28) are interchangeable up to fixed constants.

Remark 14 

(Matter compatibility and constraints). The condition ${\nabla }^{\overline{\gamma }}T\equiv 0$ together with Lemma 4 ensures preservation of the contracted Bianchi constraint and prevents spurious source terms in the energy estimates; T only fixes the equilibrium (19).

Remark 15 

(Gauge directions and the physical subspace). The DeTurck term (21) removes the diffeomorphism kernel of the linearized operator; the spectral gap (25) is thus a genuine coercivity on the physical subspace. Without DeTurck, one must pass to the quotient by diffeomorphisms (e.g., transverse–traceless decomposition) and run the same argument there.

Remark 16 

(Lorentzian caveat). The theorem is a Riemannian slice statement in DeTurck gauge. It does not imply a fully covariant Lorentzian DSFL. A Lorentzian version would require a diffeomorphism–invariant Lyapunov functional on the space of Lorentzian metrics and a hyperbolic evolution with an appropriate gap; this remains an open program.

Remark 17 

(ISS robustness (small forcing)). If (20) is perturbed by a small, mean–zero forcing $\epsilon \Xi (t)$ in $L^2$–time, the same calculation yields ${\dot{\mathcal{R}}}_{\mathrm{geom}}\le -2c{\lambda }_{\mathrm{GR}}{\mathcal{R}}_{\mathrm{geom}}+C{\epsilon }^2{\parallel \Xi (t)\parallel }^2,$ and hence ${lim\; sup}_{t\to \infty }{\mathcal{R}}_{\mathrm{geom}}(t)\le {\displaystyle \frac{C{\epsilon }^2}{2c{\lambda }_{\mathrm{GR}}}}{\parallel \Xi \parallel }_{L^2(0,\infty )}^2.$

Remark 18 

(Interpretation). In the limit ${\mathcal{R}}_{\mathrm{geom}}(t)\to 0$ one has $G[\gamma (t)]=\kappa T$ in $L^2(\Sigma )$ (and pointwise where regularity allows); with small–data coercivity, $\gamma (t)\to \overline{\gamma }$ modulo diffeomorphisms. Thus, Einstein balance is a slice attractor when a Lichnerowicz–DeTurck gap is present.

4.8. Master/Grand Attractor Theorems (Sectoral Attractors)

Let $\mathcal{R}(t)$ denote the global alignment residual associated with a given sector (classical/QMS/PDE/GR). Assume a DSFL inequality holds on its natural state space $\mathsf{X}$:

$${\displaystyle \frac{d}{dt}}\mathcal{R}(t)\le -\alpha \mathcal{R}(t),\alpha >0,$$

(29)

where $\alpha$ is the Poincaré/spectral–gap constant or a quantitative coercivity as established in §4.2–§4.6. Write $\mathcal{A}\subset \mathsf{X}$ for the sectoral attractor set (the equilibrium manifold in that sector; e.g. ${|\Psi |}^2$ in QM, $G=\kappa T$ in GR, etc.).

Proposition 1 

(Small–gain for two coupled residuals). Let $R,S\ge 0$ satisfy, for some constants $\alpha ,\beta >0$ and couplings $\delta ,\gamma \ge 0$,

$$\dot{R}\le -2\alpha R+\delta S,\dot{S}\le -2\beta S+\gamma R.$$

(30)

If the small–gain condition holds,

$$\delta \gamma <4\alpha \beta ,$$

(31)

then there exists $c\in (0,1)$ (depending only on $\alpha ,\beta ,\delta ,\gamma$) such that

$$R(t)+S(t)\le C_0{e}^{-2c{\lambda }_{\star }t}\mathrm{for}\mathrm{all}t\ge 0,$$

where the decay rate can be chosen as

$${\lambda }_{\star }={\displaystyle \frac{\alpha +\beta -\sqrt{{(\alpha -\beta )}^2+\delta \gamma }}{2}}>0.$$

(32)

In particular, both $R,S$ decay exponentially to 0.

Proof sketch. 

Write (30) in vector form $X^{\prime }\le AX$ with $X={(R,S)}^{\top }$ and $A=\begin{array}{cc}\hfill -2\alpha & \delta \\ \hfill \gamma & -2\beta \end{array}$. The eigenvalues are ${\lambda }_{\pm }=-(\alpha +\beta )\pm \sqrt{{(\alpha -\beta )}^2+\delta \gamma }.$ Condition (31) yields ${\lambda }_{+}<0$. Hence $X(t)\le e^{tA}X(0)$ componentwise and $\parallel X(t)\parallel \le C{e}^{{\lambda }_{+}t}\parallel X(0)\parallel$. Setting ${\lambda }_{\star }=-({\lambda }_{+})/2$ gives $e^{{\lambda }_{+}t}=e^{-2{\lambda }_{\star }t}$. Alternatively, choose a weighted Lyapunov $V_{a,b}=aR+bS$ with $a,b>0$ so that ${\dot{V}}_{a,b}\le -2c{\lambda }_{\star }{V}_{a,b}$. □

Distance Equivalences Near the Attractor.

We record norm–equivalences that turn the residual $\mathcal{R}$ into a bona fide distance to the attractor in each sector (up to constants).

Lemma 5 

(Residual vs. geometric distance to the attractor). (i) QM (Born sector). On a bounded domain with Poincaré constant ${\lambda }_1>0$, if $P={\nabla |\Psi |}^2$ and ${\int }_{\Omega }\rho ={\int }_{\Omega }{|\Psi |}^2=1$, then for $w=\rho -{|\Psi |}^2$,

$${\lambda }_1{\parallel w\parallel }_{L^2(\Omega )}^2\le {\int }_{\Omega }{|\nabla w|}^{2}dx={\mathcal{R}}_{\mathrm{QM}}(\rho )\le C{\parallel w\parallel }_{H^1(\Omega )}^2.$$

(ii) TD (continuum). With weights $W⪰{\lambda }_{min}I$ and $u=P-\nabla \rho$, ${\mathcal{R}}_{\mathrm{TD}}={\int |u|}_W^2\ge {\lambda }_{min}{\parallel P-\nabla \rho \parallel }_{L^2}^2.$ (iii) QMS (OA). With pointer algebra $\mathcal{N}$ and conditional expectation ${\mathsf{E}}_{\mathcal{N}}$, ${\mathcal{R}}_{\omega }(X)={\parallel X-{\mathsf{E}}_{\mathcal{N}}X\parallel }_{2,\omega }^2$ is the squared $L^2(\omega )$-distance to $L^2(\mathcal{N},\omega )$. (iv) GR (slice, DeTurck). For $h=\gamma -\overline{\gamma }$ small in $H^k$, ${\mathcal{R}}_{\mathrm{geom}}\simeq \frac{1}{4}\parallel {\mathcal{L}}_{\overline{\gamma }}{h\parallel }_{L^2(\overline{\gamma })}^2\simeq {\parallel h\parallel }_{H^2(\overline{\gamma })}^2,$ hence ${\mathcal{R}}_{\mathrm{geom}}\simeq {\mathrm{dist}}_{H^2}{(\gamma ,\overline{\gamma })}^2$ modulo diffeos.

Proof. 

QM and TD are immediate from Poincaré and $W⪰{\lambda }_{min}I$. QMS is by definition. GR follows from (27)–(28) and elliptic regularity (26). □

Omega–Limit Characterization and LaSalle.

Lemma 6 

(LaSalle-type invariance). Let ${({\mathsf{T}}_t)}_{t\ge 0}$ be the sector semigroup on $\mathsf{X}$, continuous in t, and suppose $\mathcal{R}$ is nonnegative, continuous on $\mathsf{X}$, and satisfies (29). Then for any trajectory $x(t)={\mathsf{T}}_t{x}_0$, the ω–limit set $\omega (x_0)$ is nonempty, compact, and contained in $\{\mathcal{R}=0\}=\mathcal{A}$. If, in addition, $\mathcal{A}$ consists of a single orbit (modulo the natural gauge of the sector), then $x(t)\to \mathcal{A}$.

Proof. 

Monotonicity and boundedness of $\mathcal{R}$ yield precompactness (sector by sector) and invariance. At any accumulation point $\overline{x}$, the derivative of $\mathcal{R}$ vanishes, hence $\mathcal{R}(\overline{x})=0$ by (29). Uniqueness up to gauge gives convergence to the orbit. □

Theorem 7 

(Sector attractors from residual decay (Master theorem)). Under (29) and Lemma 5, the canonical equilibrium relations are the unique global attractors in their sectors:

(QM)

(Born alignment) If $P={\nabla |\Psi |}^2$ and ${\int }_{\Omega }\rho ={\int }_{\Omega }{|\Psi |}^2=1$, then $\rho (·,t)\to {|\Psi |}^2$ in $L^2(\Omega )$ at least exponentially, with rate ${\alpha }^{\#}\ge {\lambda }_1$ (cf. Theorem 11).

(TD)

(Residual entropy) $S_R(t):=-log(\mathcal{R}(t)/{\mathcal{R}}_0+R_{*})$ satisfies ${\dot{S}}_R\ge \alpha >0$, hence $S_R(t)\nearrow +\infty$ and $\mathcal{R}\to 0$ exponentially.

(QMS)

(OA pointer alignment) If the reversible QMS has spectral gap $\lambda >0$ on $L^2{(\mathcal{N},\omega )}^{\perp }$, then ${\mathcal{R}}_{\omega }({\mathsf{T}}_{t}X)\le e^{-2\lambda t}{\mathcal{R}}_{\omega }(X)$ and $X(t)\to {\mathsf{E}}_{\mathcal{N}}X$ in $L^2(\omega )$.

(GR)

(Einstein balance on slices) Under Theorem 6, ${\mathcal{R}}_{\mathrm{geom}}(t)\le e^{-2c{\lambda }_{\mathrm{GR}}t}{\mathcal{R}}_{\mathrm{geom}}(0)$ and $\gamma (t)\to \overline{\gamma }$ modulo diffeomorphisms; thus $G[\gamma ]=\kappa T$ in $L^2(\Sigma )$.

Proof. 

(QM) Combine (29) with Lemma 5(i) and Theorem 11. (TD) Differentiate $S_R$: ${\dot{S}}_R=-\dot{\mathcal{R}}/(\mathcal{R}+{\mathcal{R}}_0{R}_{*})\ge \alpha >0$, hence $S_R$ increases and $\mathcal{R}\to 0$ exponentially. (QMS) is Theorem 2. (GR) is Theorem 6 plus Lemma 5(iv). □

Abstract Grand Theorem (Uniform Formulation).

Theorem 8 

(Grand attractor theorem (abstract Hilbert form)). Let ${({\mathsf{T}}_t)}_{t\ge 0}$ be a reversible contraction semigroup on a Hilbert space H with generator $\mathcal{L}$, fixed–point subspace $\mathcal{N}=\{X:{\mathsf{T}}_{t}X=X\}$, and Poincaré gap $\lambda >0$ on ${\mathcal{N}}^{\perp }$. Define $\mathcal{R}(X):=\mathrm{dist}{(X,\mathcal{N})}^2$. Then for all $X\in H$,

$$\mathcal{R}({\mathsf{T}}_{t}X)\le e^{-2\lambda t}\mathcal{R}(X),\mathrm{dist}({\mathsf{T}}_{t}X,\mathcal{N})\le e^{-\lambda t}\mathrm{dist}(X,\mathcal{N}).$$

Moreover, $\mathcal{N}$ is the unique global attractor (modulo the sector gauge).

Proof. 

Differentiate $\parallel {({\mathsf{T}}_{t}X)}^{\perp }{\parallel }^2=\langle {\mathsf{T}}_t{X}^{\perp },{\mathsf{T}}_t{X}^{\perp }\rangle$: $\frac{d}{dt}\parallel {({\mathsf{T}}_{t}X)}^{\perp }{\parallel }^2=-2\langle {\mathsf{T}}_t{X}^{\perp },\mathcal{L}{\mathsf{T}}_t{X}^{\perp }\rangle \le -2\lambda {\parallel {({\mathsf{T}}_{t}X)}^{\perp }\parallel }^2$. Grönwall yields the claimed exponential contraction. Uniqueness of the attractor follows since $\mathcal{N}$ is the fixed–point set. □

Robustness and Time–Varying Rates.

Proposition 2 

(ISS/ultimate boundedness; time–varying $\alpha (t)$). (i) If $\dot{\mathcal{R}}\le -\alpha \mathcal{R}+{\epsilon }^{2}u(t)$ with $\alpha >0$ and $u\in L_{\mathrm{loc}}^1$, then $\mathcal{R}(t)\le e^{-\alpha t}\mathcal{R}(0)+{\epsilon }^2{\int }_0^t{e}^{-\alpha (t-s)}u(s)ds$ and ${lim\; sup}_{t\to \infty }\mathcal{R}(t)\le {\epsilon }^2{\parallel u\parallel }_{L^1(0,\infty )}$. (ii) If $\dot{\mathcal{R}}\le -\alpha (t)\mathcal{R}$ with $\alpha (·)\ge 0$ measurable and ${\int }_0^{\infty }\alpha (t)dt=+\infty$, then $\mathcal{R}(t)\to 0$; if $\underline{\alpha }:={inf}_{t\ge 0}\alpha (t)>0$, then $\mathcal{R}(t)\le e^{-\underline{\alpha }t}\mathcal{R}(0)$.

Proof. 

(i) Grönwall with input. (ii) Integrate the differential inequality. □

Discrete–Time and Product Systems.

Proposition 3 

(Discrete DSFL; products). (i) If ${\mathcal{R}}_{k+1}\le (1-\theta ){\mathcal{R}}_k$ with $\theta \in (0,1)$, then ${\mathcal{R}}_k\le {(1-\theta )}^k{\mathcal{R}}_0$. (ii) If ${\mathcal{R}}^{(1)}$ and ${\mathcal{R}}^{(2)}$ satisfy ${\dot{\mathcal{R}}}^{(i)}\le -2{\alpha }_i{\mathcal{R}}^{(i)}+{\sum }_{j\ne i}{\gamma }_{ij}{\mathcal{R}}^{(j)}$ with a Metzler coupling matrix $\Gamma =({\gamma }_{ij})$, then exponential decay holds provided the spectral abscissa of $A=\mathrm{diag}(2{\alpha }_1,2{\alpha }_2)-\Gamma$ is positive (cf. Proposition 1).

Proof. 

(i) Induction. (ii) Linear ODE comparison and the spectral condition. □

Weighted Distances and Alternative Norms.

Lemma 7 

(Residual vs. weighted distance). Let $\mathcal{A}\subset \mathsf{X}$ be the sectoral attractor and let $d_{*}(·,\mathcal{A})$ be a locally equivalent distance induced by a positive quadratic form $Q_{*}$ (e.g. an $H^{-1}$ metric in PDE sectors or a weighted $L^2(\omega )$ metric in QMS). Suppose that in a neighborhood $\mathcal{U}$ of $\mathcal{A}$ there exist constants $0<c_1\le c_2<\infty$ such that

$$c_1{d}_{*}{(x,\mathcal{A})}^2\le \mathcal{R}(x)\le c_2{d}_{*}{(x,\mathcal{A})}^2\forall x\in \mathcal{U}.$$

If the DSFL inequality (29) holds on $\mathcal{U}$, then

$$d_{*}({\mathsf{T}}_t{x}_0,\mathcal{A})\le \sqrt{\frac{c_2}{c_1}}{e}^{-\alpha t}{d}_{*}(x_0,\mathcal{A})$$

for all t for which the trajectory stays in $\mathcal{U}$. In particular, the convergence rate is unchanged up to the equivalence factor $\sqrt{c_2/c_1}$.

Proof. 

Combine (29) with the local equivalence to bound $d_{*}^2$ above and below by $\mathcal{R}$, then apply Grönwall. □

Remark 19 

(PDE $H^{-1}$ distances). In diffusion–type PDE sectors, observables are often controlled in $H^{-1}$ rather than $L^2$. If the residual controls ${\parallel w\parallel }_{H^{-1}}^2$ and vice versa near the attractor (e.g. via Poincaré and elliptic estimates), Lemma 7 transfers the DSFL rate directly to $H^{-1}$.

Consequences for Sector Observables.

Corollary 4 

(Observable convergence). Let $\mathcal{O}$ be a continuous observable on the sector state space and assume there exists a neighborhood of the attractor $\mathcal{A}$ where $|\mathcal{O}(x)-{\mathcal{O}}^{*}|\le C\mathrm{dist}(x,\mathcal{A})$ (local Lipschitz). Then under (29) (or Theorem 9),

$$|\mathcal{O}({\mathsf{T}}_t{x}_0)-{\mathcal{O}}^{*}|\le C{e}^{-{\alpha }^{\#}t}\mathrm{dist}(x_0,\mathcal{A}),$$

with ${\alpha }^{\#}=\lambda$ in the reversible cases and ${\alpha }^{\#}=(2\beta -C\epsilon )$ in the coercive PDE case. Moreover, if a weighted distance $d_{*}$ equivalent to $\mathrm{dist}$ near $\mathcal{A}$ is used (Lemma 7), the same estimate holds with $\mathrm{dist}$ replaced by $d_{*}$.

Remark 20 

(Uniqueness modulo gauge). In PDE and GR sectors the attractor is unique modulo the natural gauge (additive constants for ρ, diffeomorphisms for γ). The theorems above are to be understood on the corresponding quotient spaces, or after fixing a gauge (DeTurck in GR, mean–zero in QM/TD).

Remark 21 

(Coupled residuals and small–gain). In multi–residual settings (e.g. DSFL+SABIM), a vector–Lyapunov $V_{a,b}=a\mathcal{R}+b\mathcal{S}$ together with the sharp small–gain condition $\delta \gamma <4\alpha \beta$ (Proposition 1) yields exponential decay with rate ${\lambda }_{\star }=\frac{\alpha +\beta -\sqrt{{(\alpha -\beta )}^2+\delta \gamma }}{2}$; see also the n–dimensional version in Corollary.

Proof (expanded). 

(QM) Set $w:=\rho -{|\Psi |}^2$. By definition of the residual in the Born sector one has ${\mathcal{R}}_{\mathrm{QM}}(t)={\int }_{\Omega }{|\nabla w(x,t)|}^{2}dx$. The DSFL inequality implies ${\mathcal{R}}_{\mathrm{QM}}(t)\to 0$ as $t\to \infty$, hence $\nabla w(·,t)\to 0$ in $L^2(\Omega )$. Since ${\int }_{\Omega }\rho ={\int }_{\Omega }{|\Psi |}^2=1$ for all t (mass conservation), we have ${\int }_{\Omega }w(·,t)dx=0$. By the Poincaré inequality on mean–zero functions, ${\parallel w(·,t)\parallel }_{L^2}^2\le {\lambda }_1^{-1}{\parallel \nabla w(·,t)\parallel }_{L^2}^2={\lambda }_1^{-1}{\mathcal{R}}_{\mathrm{QM}}(t)$, whence $w(·,t)\to 0$ in $L^2(\Omega )$. Therefore $\rho (·,t)\to {|\Psi |}^2$ in $L^2(\Omega )$. Moreover, if the sector provides the sharper differential inequality ${\dot{\mathcal{R}}}_{\mathrm{QM}}\le -2{\lambda }_1{\mathcal{R}}_{\mathrm{QM}}$ (cf. Theorem 11), then ${\parallel \rho -|\Psi |}^2{\parallel }_{L^2}\le {\lambda }_1^{-1/2}{e}^{-{\lambda }_{1}t}{\mathcal{R}}_{\mathrm{QM}}{(0)}^{1/2}$.

(TD) By definition $S_R(t)=-log(\mathcal{R}(t)/{\mathcal{R}}_0+R_{*})$ with $R_{*}\in (0,1)$. Differentiating and using DSFL,

$${\dot{S}}_R(t)=-{\displaystyle \frac{\dot{\mathcal{R}}(t)}{\mathcal{R}(t)+{\mathcal{R}}_0{R}_{*}}}\ge {\displaystyle \frac{\alpha \mathcal{R}(t)}{\mathcal{R}(t)+{\mathcal{R}}_0{R}_{*}}}\ge \alpha {\displaystyle \frac{\mathcal{R}(t)}{\mathcal{R}(0)+{\mathcal{R}}_0{R}_{*}}}\ge 0.$$

In particular, if $\dot{\mathcal{R}}\le -\alpha \mathcal{R}$ with $\alpha >0$, then $S_R$ is strictly increasing and $S_R(t)\ge S_R(0)+\alpha t·{\displaystyle \frac{\mathcal{R}(0)}{\mathcal{R}(0)+{\mathcal{R}}_0{R}_{*}}}$, which diverges as $t\to \infty$ while $\mathcal{R}(t)\to 0$ exponentially.

(GR) By assumption, ${\dot{\mathcal{R}}}_{\mathrm{geom}}\le -2{\lambda }_{\mathrm{GR}}{\mathcal{R}}_{\mathrm{geom}}$, hence ${\mathcal{R}}_{\mathrm{geom}}(t)\le e^{-2{\lambda }_{\mathrm{GR}}t}{\mathcal{R}}_{\mathrm{geom}}(0)\to 0$. Since ${\mathcal{R}}_{\mathrm{geom}}={\int }_{\Sigma }{\parallel G[\gamma ]-\kappa T\parallel }_{\gamma }^{2}d{\mu }_{\gamma }$, this implies $G[\gamma (t)]-\kappa T\to 0$ in $L^2(\Sigma )$ as $t\to \infty$. Under the small–data hypotheses of Theorem 6 and the gauge choice (DeTurck), elliptic regularity and the spectral gap yield convergence of $\gamma (t)$ to $\overline{\gamma }$ modulo diffeomorphisms; in particular, $G[\gamma ]=\kappa T$ in the limit. □

Theorem 9 

(Grand attractor theorem (abstract form)). Let ${({\mathsf{T}}_t)}_{t\ge 0}$ be a reversible contraction semigroup on a Hilbert space H with generator $\mathcal{L}$, fixed–point subspace $\mathcal{N}=\{X:{\mathsf{T}}_{t}X=X\forall t\ge 0\}$, and Poincaré gap $\lambda >0$ on ${\mathcal{N}}^{\perp }$. For the residual $\mathcal{R}(X):=\parallel X-{\mathsf{E}}_{\mathcal{N}}{(X)\parallel }^2$ one has

$$\mathcal{R}({\mathsf{T}}_{t}X)\le e^{-2\lambda t}\mathcal{R}(X),\mathrm{dist}({\mathsf{T}}_{t}X,\mathcal{N})\le e^{-\lambda t}\mathrm{dist}(X,\mathcal{N}),$$

and $\mathcal{N}$ is the unique global attractor (modulo the sector’s gauge).

Proof. 

Let $X^{\perp }:=X-{\mathsf{E}}_{\mathcal{N}}X\in {\mathcal{N}}^{\perp }$. Since ${\mathsf{T}}_t$ is reversible on H, $t\mapsto \parallel {({\mathsf{T}}_{t}X)}^{\perp }{\parallel }^2$ is differentiable with

$${\displaystyle \frac{d}{dt}}\parallel {({\mathsf{T}}_{t}X)}^{\perp }{\parallel }^2=2\langle {\mathsf{T}}_t{X}^{\perp },\mathcal{L}{\mathsf{T}}_t{X}^{\perp }\rangle =-2\mathcal{E}({\mathsf{T}}_t{X}^{\perp })\le -2\lambda {\parallel {({\mathsf{T}}_{t}X)}^{\perp }\parallel }^2,$$

where $\mathcal{E}(·)=\langle ·,-\mathcal{L}·\rangle$ and we used the Poincaré gap on ${\mathcal{N}}^{\perp }$. Grönwall’s lemma gives $\parallel {({\mathsf{T}}_{t}X)}^{\perp }{\parallel }^2\le e^{-2\lambda t}{\parallel X^{\perp }\parallel }^2$, i.e. $\mathcal{R}({\mathsf{T}}_{t}X)\le e^{-2\lambda t}\mathcal{R}(X)$ and $\mathrm{dist}({\mathsf{T}}_{t}X,\mathcal{N})\le e^{-\lambda t}\mathrm{dist}(X,\mathcal{N})$. Since $\mathcal{N}$ is the fixed–point subspace of the semigroup, it is invariant and attracts every orbit. Uniqueness of the global attractor (modulo gauge) follows because any other closed invariant attracting set must lie in $\mathcal{N}$. □

Remark 22 

(Interpretation). The Master/Grand theorems formalize the central message: once a (sector–appropriate) spectral gap/coercivity is present, the DSFL inequality holds and the sector’s canonical equilibrium relation is recovered as a dynamical fixed point rather than as a postulate. In reversible settings the exponential rate is dictated by the Poincaré gap; in the PDE/GR coercive settings the rate is dictated by the quantitative coercivity (e.g. $2\beta -C\epsilon$ or $2{\lambda }_{\mathrm{GR}}$).

Corollary 5 

(Observable ISS under DSFL). Assume the hypotheses of Corollary 4. Suppose further that the residual dynamics admit an input term in the DSFL inequality,

$$\dot{\mathcal{R}}(t)\le -\alpha \mathcal{R}(t)+{\epsilon }^{2}u(t),$$

with $\alpha >0$, $\epsilon \ge 0$, and $u\in L_{\mathrm{loc}}^1([0,\infty ))$. Then

$$|\mathcal{O}({\mathsf{T}}_t{x}_0)-{\mathcal{O}}^{*}|\le C{e}^{-\alpha t}\mathrm{dist}(x_0,\mathcal{A})+C{\epsilon }^2{\int }_0^t{e}^{-\alpha (t-s)}{\displaystyle \frac{u(s)}{\sqrt{\mathcal{R}(s)+{\mathcal{R}}_0{R}_{*}}}}ds,$$

where ${\mathcal{R}}_0>0$ and $R_{*}\in (0,1)$ are the fixed constants from the residual–entropy proxy. In particular, if $u\in L^1(0,\infty )$ then

$$\underset{t\to \infty }{lim\; sup}|\mathcal{O}({\mathsf{T}}_t{x}_0)-{\mathcal{O}}^{*}|\le {\displaystyle \frac{C{\epsilon }^2}{\alpha }}\parallel {\displaystyle \frac{u}{\sqrt{\mathcal{R}(·)+{\mathcal{R}}_0{R}_{*}}}}{\parallel }_{L^1(0,\infty )}.$$

The same estimate holds with $\mathrm{dist}$ replaced by any locally equivalent weighted distance $d_{*}$ (Lemma 7).

5. Proofs

This section collects the proofs of the results stated in Section 4.

5.1. Proof of Sec. 4.1 (Propagation Lemma)

Proof of Lemma 1. 

(CL) Let $\phi :{\mathbb{R}}^m\to [0,\infty )$ be convex with $\phi (0)=0$ and let $({\mathsf{T}}_t)$ be a Markov contraction on $L^1\cap L^2(\mu )$. By Jensen’s inequality and positivity/mass preservation, $\phi ({\mathsf{T}}_{t}u)\le {\mathsf{T}}_t(\phi (u))$$\mu$–a.e. Integrating and using $\int {\mathsf{T}}_{t}fd\mu =\int fd\mu$ yields $\int \phi ({\mathsf{T}}_{t}u)d\mu \le \int \phi (u)d\mu$, i.e. $\mathcal{R}(t)\le \mathcal{R}(0)$.

(QM) Let ${({\mathsf{T}}_t)}_{t\ge 0}$ be normal u.c.p., $\omega$–symmetric on $L^2(\omega )$, and $\omega$–preserving. Kadison–Schwarz gives ${\mathsf{T}}_t{(X)}^{*}{\mathsf{T}}_t(X)\le {\mathsf{T}}_t(X^{*}X)$. Applying $\omega$ and using $\omega \circ {\mathsf{T}}_t=\omega$ yields $\parallel {\mathsf{T}}_t{X\parallel }_{2,\omega }^2\le {\parallel X\parallel }_{2,\omega }^2$. Since ${\mathsf{E}}_{\mathcal{N}}$ is the $L^2(\omega )$ orthogonal projection and ${\mathsf{T}}_t$ acts as the identity on $\mathcal{N}$, the same contraction holds for $X-{\mathsf{E}}_{\mathcal{N}}X$: $\parallel {\mathsf{T}}_{t}X-{\mathsf{E}}_{\mathcal{N}}{X\parallel }_{2,\omega }\le {\parallel X-{\mathsf{E}}_{\mathcal{N}}X\parallel }_{2,\omega }$, hence ${\mathcal{R}}_{\omega }({\mathsf{T}}_{t}X)\le {\mathcal{R}}_{\omega }(X)$.

(PDE) With $u:=P-\nabla \rho$ and $\mathcal{R}(t):={\int }_{\Omega }{|u|}^{2}dx$, differentiate in time. Using ${\partial }_{t}u=-Bu-\nabla (\nabla ·u)+(\mathrm{controlled})$ and integrating by parts (periodic BCs or homogeneous BCs that kill boundary terms), we get

$${\displaystyle \frac{d}{dt}}\mathcal{R}(t)=-2{\int }_{\Omega }{u}^{\top }Budx-2{\parallel \nabla ·u\parallel }_{L^2(\Omega )}^2+(\mathrm{controlled}).$$

Under $B⪰0$ and the stated sign/dominance of the controlled terms, the right–hand side is $\le 0$, hence $\mathcal{R}(t)\le \mathcal{R}(0)$. □

5.2. Proof of Lemma

Proof. 

(i) QM. Let $\Omega$ be a bounded $C^1$ domain with periodic or homogeneous Neumann BCs. For $w:=\rho -{|\Psi |}^2$ one has ${\int }_{\Omega }wdx=0$ (mass normalization). Poincaré then gives ${\parallel w\parallel }_{L^2(\Omega )}^2\le {\lambda }_1^{-1}{\parallel \nabla w\parallel }_{L^2(\Omega )}^2$, so ${\lambda }_1{\parallel w\parallel }_{L^2}^2\le {\int }_{\Omega }{|\nabla w|}^{2}dx={\mathcal{R}}_{\mathrm{QM}}(\rho )$, and the upper bound follows from the $H^1$ control of w.

(ii) TD. For $W(x)⪰{\lambda }_{min}I$ a.e., ${|v|}_W^2=v^{\top }Wv\ge {\lambda }_{min}{|v|}^2$, hence ${\mathcal{R}}_{\mathrm{TD}}={\int }_{\Omega }{|P-\nabla \rho |}_W^{2}dx\ge {\lambda }_{min}{\parallel P-\nabla \rho \parallel }_{L^2(\Omega )}^2.$

(iii) QMS. By Takesaki’s theorem, the $\omega$–preserving conditional expectation ${\mathsf{E}}_{\mathcal{N}}$ is the $L^2(\omega )$ orthogonal projection onto $L^2(\mathcal{N},\omega )$. Therefore ${\mathcal{R}}_{\omega }(X)={\parallel X-{\mathsf{E}}_{\mathcal{N}}X\parallel }_{2,\omega }^2=\mathrm{dist}{\left(X,L^2(\mathcal{N},\omega )\right)}^2.$

(iv) GR. Linearizing $G[\gamma ]-\kappa T$ at $\overline{\gamma }$ yields $G[\overline{\gamma }+h]-\kappa T=\frac{1}{2}{\mathcal{L}}_{\overline{\gamma }}h+Q(h,\nabla h)$ with $Q=O(|h||{\nabla }^{2}h|+|\nabla h{|}^2)$. For ${\parallel h\parallel }_{H^2(\overline{\gamma })}$ sufficiently small, squaring and integrating gives ${\mathcal{R}}_{\mathrm{geom}}\simeq \frac{1}{4}{\parallel {\mathcal{L}}_{\overline{\gamma }}h\parallel }_{L^2(\overline{\gamma })}^2.$ Elliptic regularity for ${\mathcal{L}}_{\overline{\gamma }}$ on the gauge–orthogonal subspace implies $\parallel {\mathcal{L}}_{\overline{\gamma }}{h\parallel }_{L^2(\overline{\gamma })}\simeq {\parallel h\parallel }_{H^2(\overline{\gamma })},$ hence ${\mathcal{R}}_{\mathrm{geom}}\simeq {\parallel h\parallel }_{H^2(\overline{\gamma })}^2$, which is equivalent to ${\mathrm{dist}}_{H^2}{(\gamma ,\overline{\gamma })}^2$ modulo diffeomorphisms. □

5.3. Proof of Lemma 6

Proof. 

Let $x(t)={\mathsf{T}}_t{x}_0$. By (29), $\mathcal{R}(x(t))$ is nonincreasing and bounded below, hence convergent. Sector by sector, standard compactness (e.g. Rellich–Kondrachov in PDE/GR or spectral decomposition in reversible semigroups) yields precompactness of trajectories on bounded time intervals. Any $\overline{x}\in \omega (x_0)$ admits a sequence $t_k\to \infty$ with $x(t_k)\to \overline{x}$, and continuity of $\mathcal{R}$ gives ${lim}_{k\to \infty }\mathcal{R}(x(t_k))=\mathcal{R}(\overline{x})$; invariance of $\omega (x_0)$ implies $\mathcal{R}(\overline{x})={lim}_{t\to \infty }\mathcal{R}(x(t))$. Since $\dot{\mathcal{R}}\le -\alpha \mathcal{R}$, necessarily $\dot{\mathcal{R}}(\overline{x})=0$ and hence $\mathcal{R}(\overline{x})=0$. Thus $\omega (x_0)\subset \{\mathcal{R}=0\}=\mathcal{A}$. If $\mathcal{A}$ consists of a single orbit modulo the sector’s gauge, then Łojasiewicz–Simon/strict Lyapunov arguments imply $x(t)\to \mathcal{A}$. □

5.4. Proof of Theorem 7

Proof. 

Combine Lemma 5 (residual ⇔ distance) with the DSFL inequality (29) to obtain $L^2$– (QM) and $H^2$– (GR) convergence. For TD, the $S_R$ monotonicity follows by direct differentiation. For QMS, apply Theorem 2. Uniqueness modulo sector gauges follows from invariances (additive constants for $\rho$; diffeomorphisms for $\gamma$) and the strict convexity of $\mathcal{R}$ transverse to gauge directions. □

5.5. Proof of Sec. 4.3 (Lindblad Sharpness)

Proof of Theorem 3 and Corollary 3. 

Work in the orthonormal basis ${\{|i\rangle \}}_{i=1}^d$ where $P_i=|i\rangle \langle i|$. The generator is

$${\mathcal{L}}^{*}(\sigma )=\sum _{i=1}^d{\gamma }_i\left(P_i\sigma P_i-\frac{1}{2}\{P_i,\sigma \}\right),{\gamma }_i>0.$$

 □

Proof of Theorem 3. 

(A) Modewise Solution and Residual Envelope.

Matrix elements satisfy

$${\displaystyle \frac{d}{dt}}{({\sigma }_t)}_{ii}=0,{\displaystyle \frac{d}{dt}}{({\sigma }_t)}_{ij}=-\frac{{\gamma }_i+{\gamma }_j}{2}{({\sigma }_t)}_{ij}(i\ne j),$$

so ${({\sigma }_t)}_{ij}=e^{-({\gamma }_i+{\gamma }_j)t/2}{({\sigma }_0)}_{ij}$. Hence

$${\mathcal{R}}_{\mathrm{L}\ddot{\mathrm{u}}\mathrm{ders}}({\sigma }_t)=\sum _{i\ne j}{e}^{-({\gamma }_i+{\gamma }_j)t}|{({\sigma }_0)}_{ij}{|}^2\le e^{-{\alpha }_{*}t}\sum _{i\ne j}{|{({\sigma }_0)}_{ij}|}^2,$$

with ${\alpha }_{*}:={min}_{i\ne j}({\gamma }_i+{\gamma }_j)$.

 □

Proof of Theorem 3 and Corollary 3. 

Work in the orthonormal basis ${\{|i\rangle \}}_{i=1}^d$ where $P_i=|i\rangle \langle i|$. The generator is

$${\mathcal{L}}^{*}(\sigma )=\sum _{i=1}^d{\gamma }_i\left(P_i\sigma P_i-\frac{1}{2}\{P_i,\sigma \}\right),{\gamma }_i>0.$$

(A) Modewise solution and residual envelope. Matrix elements satisfy

$${\displaystyle \frac{d}{dt}}{({\sigma }_t)}_{ii}=0,{\displaystyle \frac{d}{dt}}{({\sigma }_t)}_{ij}=-\frac{{\gamma }_i+{\gamma }_j}{2}{({\sigma }_t)}_{ij}(i\ne j),$$

so ${({\sigma }_t)}_{ij}=e^{-({\gamma }_i+{\gamma }_j)t/2}{({\sigma }_0)}_{ij}$. Hence

$${\mathcal{R}}_{\mathrm{L}ü\mathrm{ders}}({\sigma }_t)=\sum _{i\ne j}{e}^{-({\gamma }_i+{\gamma }_j)t}|{({\sigma }_0)}_{ij}{|}^2\le e^{-{\alpha }_{*}t}\sum _{i\ne j}{|{({\sigma }_0)}_{ij}|}^2,$$

with ${\alpha }_{*}:={min}_{i\ne j}({\gamma }_i+{\gamma }_j)$.

(B) Optimality. Choose initial coherence supported on a pair $(i_{*},j_{*})$ attaining ${min}_{i\ne j}({\gamma }_i+{\gamma }_j)$. Then

$${\mathcal{R}}_{\mathrm{L}ü\mathrm{ders}}({\sigma }_t)=\sum _{i\ne j}{e}^{-({\gamma }_i+{\gamma }_j)t}|{({\sigma }_0)}_{ij}{|}^2=e^{-{\alpha }_{*}t}{|{({\sigma }_0)}_{i_{*}{j}_{*}}|}^2,$$

which matches the envelope with equality. Hence the rate ${\alpha }_{*}={min}_{i\ne j}({\gamma }_i+{\gamma }_j)$ is sharp.

(C) Trace–norm convergence. Since the diagonal entries are constant in time and the off–diagonals decay modewise as $e^{-({\gamma }_i+{\gamma }_j)t/2}$, one has

$$\parallel {\sigma }_t-\Phi ({\sigma }_0){\parallel }_2^2=\sum _{i\ne j}{e}^{-({\gamma }_i+{\gamma }_j)t}|{({\sigma }_0)}_{ij}{|}^2\le e^{-{\alpha }_{*}t}{\parallel {\sigma }_0-\Phi ({\sigma }_0)\parallel }_2^2.$$

Using ${\parallel A\parallel }_1\le \sqrt{d}{\parallel A\parallel }_2$ then gives

$$\parallel {\sigma }_t-\Phi ({\sigma }_0){\parallel }_1\le \sqrt{d}{e}^{-{\alpha }_{*}t/2}{\parallel {\sigma }_0-\Phi ({\sigma }_0)\parallel }_2,$$

so ${\sigma }_t\to \Phi ({\sigma }_0)$ in trace norm at an exponential rate, proving the corollary.

 □

5.6. Proof of Sec. 4.4 (PDE Energy Identity and Decay)

Proof of Theorem 4. 

Step 0 (regularity and IBP). Assume $(\rho ,P)$ is a (mild) solution with $\rho (·,t)\in H^1(\Omega ),P(·,t)\in L^2(\Omega ;{\mathbb{R}}^d)$ for $t\in [0,T]$, and periodic or homogeneous boundary conditions chosen so that the boundary term in the IBP identity vanishes (cf. Remark 3). Standard mollification in time (or density of smooth compactly supported functions in the graph norm) justifies differentiation under the integral sign; the final inequalities extend by continuity to the given regularity class.

Step 1 (residual equation). Let

$$u:=P-\nabla \rho ,\mathcal{R}(t):={\int }_{\Omega }{|u(x,t)|}^{2}dx.$$

From (11),

$${\partial }_{t}u={\partial }_{t}P-\nabla ({\partial }_t\rho )=-B(x,t)u-\nabla (\nabla ·u)+G(\rho ,P).$$

Step 2 (exact energy identity). Differentiate $\mathcal{R}$ and use the product rule:

$${\displaystyle \frac{d}{dt}}\mathcal{R}(t)=2{\int }_{\Omega }u·{\partial }_{t}udx=-2{\int }_{\Omega }{u}^{\top }Budx-2{\int }_{\Omega }u·\nabla (\nabla ·u)dx+2{\int }_{\Omega }u·Gdx.$$

By integration by parts and the BC choice,

$${\int }_{\Omega }u·\nabla (\nabla ·u)dx={\int }_{\partial \Omega }{(u·n)(\nabla ·u)dS-\parallel \nabla ·u\parallel }_{L^2(\Omega )}^2=-{\parallel \nabla ·u\parallel }_{L^2(\Omega )}^2.$$

Hence

$${\displaystyle \frac{d}{dt}}\mathcal{R}(t)=-2{\int }_{\Omega }{u}^{\top }Budx-2{\parallel \nabla ·u\parallel }_{L^2}^2+2{\int }_{\Omega }u·Gdx.$$

(33)

Step 3 (coercivity and subcriticality). The uniform positive definiteness $B(x,t)⪰\beta I$ a.e. implies

$${\int }_{\Omega }{u}^{\top }Budx\ge \beta {\int }_{\Omega }{|u|}^{2}dx=\beta \mathcal{R}(t).$$

By the subcriticality hypothesis, there exist $C>0$ and sufficiently small $\epsilon \ge 0$ such that

$$|{\int }_{\Omega }u·G(\rho ,P)dx|\le C\epsilon \mathcal{R}(t)\mathrm{for}\mathrm{all}t\in [0,T].$$

(For example, this holds if G is locally Lipschitz with ${\parallel G(\rho ,P)\parallel }_{L^2}\le C\epsilon {\parallel u\parallel }_{L^2}$ in the energy region visited by the solution.)

Step 4 (differential inequality). Insert these bounds into (33) to obtain

$${\displaystyle \frac{d}{dt}}\mathcal{R}(t)\le -2\beta \mathcal{R}(t)-2{\parallel \nabla ·u\parallel }_{L^2}^2+2C\epsilon \mathcal{R}(t)\le -(2\beta -C\epsilon )\mathcal{R}(t).$$

Let $\alpha :=2\beta -C\epsilon >0$ (smallness of $\epsilon$ ensures positivity).

Step 5 (Grönwall). By Grönwall’s inequality,

$$\mathcal{R}(t)\le e^{-\alpha t}\mathcal{R}(0)\mathrm{for}\mathrm{all}t\in [0,T],$$

i.e. the residual decays exponentially at rate $\alpha =2\beta -C\epsilon$. In particular, for $\epsilon =0$ we recover the clean rate $\mathcal{R}(t)\le e^{-2\beta t}\mathcal{R}(0)$.

Optional refinements. (i) The extra term ${-2\parallel \nabla ·u\parallel }_{L^2}^2$ is dissipative and improves the decay when the Helmholtz gradient component dominates; combine with a Poincaré inequality on gradient fields to sharpen the rate on tori or bounded domains. (ii) If $B=B(t)$ with

$$\beta (t):=\underset{x\in \Omega }{ess\; inf}{\lambda }_{min}\left(B(x,t)\right),$$

then

$${\displaystyle \frac{d}{dt}}\mathcal{R}(t)\le -\left(2\beta (t)-C\epsilon \right)\mathcal{R}(t)⟹\mathcal{R}(t)\le exp\left(-{\int }_0^t\left(2\beta (s)-C\epsilon \right)ds\right)\mathcal{R}(0).$$

In particular, if $\underline{\beta }:={inf}_{s\ge 0}\beta (s)>\frac{C\epsilon }{2}$, then $\mathcal{R}(t)\le e^{-(2\underline{\beta }-C\epsilon )t}\mathcal{R}(0)$; and if ${\int }_0^{\infty }\left(2\beta (s)-C\epsilon \right)ds=+\infty$, then $\mathcal{R}(t)\to 0$.

This completes the proof. □

5.7. Proof of Sec. 4.5 (Free–Field Residual Decay)

Proof of Theorem 5. 

We give a concrete Ornstein–Uhlenbeck (OU) derivation for the free field and then the abstract semigroup proof.

OU formulation and Lyapunov Equation.

Write the free Euclidean dynamics as

$${\partial }_{\tau }{\varphi }_{\tau }=-\mathcal{A}{\varphi }_{\tau }+\eta ,\mathcal{A}:=(-\Delta )+m^2\mathrm{on}{\mathbb{R}}^d,$$

with Gaussian space–time white noise $\eta$ normalized by $\mathbb{E}[\eta (x,\tau )\eta (y,\sigma )]=2\delta (x-y)\delta (\tau -\sigma )$. This factor 2 ensures the stationary covariance $C_{\infty }$ solves the Lyapunov identity $\mathcal{A}{C}_{\infty }+C_{\infty }\mathcal{A}=2I$, hence $C_{\infty }={\mathcal{A}}^{-1}$ when defined. For any admissible test function f (e.g. $f\in \mathcal{S}({\mathbb{R}}^d)$ or $f\in L^2$ with compact Fourier support), let ${\Phi }_{\tau }(f):=\langle {\varphi }_{\tau },f\rangle$. The covariance $C(\tau )$ of ${\varphi }_{\tau }$ obeys

$${\displaystyle \frac{d}{d\tau }}C(\tau )=-\mathcal{A}C(\tau )-C(\tau )\mathcal{A}+2I,C(\tau )=e^{-\tau \mathcal{A}}C(0)e^{-\tau \mathcal{A}}+{\int }_0^{\tau }{e}^{-(\tau -s)\mathcal{A}}(2I)e^{-(\tau -s)\mathcal{A}}ds.$$

Subtracting $C_{\infty }$ gives

$$D(\tau ):=C(\tau )-C_{\infty }=e^{-\tau \mathcal{A}}\left(C(0)-C_{\infty }\right)e^{-\tau \mathcal{A}},$$

hence for any f,

$$C_{\tau }(f)-C_{\infty }(f)=\langle e^{-\tau \mathcal{A}}f,(C(0)-C_{\infty })e^{-\tau \mathcal{A}}f\rangle .$$

Spectral Gap Bound.

Since $\mathcal{A}\ge {\lambda }_{*}I$ with ${\lambda }_{*}=m^2$ (free field), the semigroup bound $\parallel e^{-\tau \mathcal{A}}{\parallel }_{\mathcal{B}(L^2)}\le e^{-{\lambda }_{*}\tau }$ yields

$$|C_{\tau }(f)-C_{\infty }(f)|\le \parallel C(0)-C_{\infty }\parallel \parallel e^{-\tau \mathcal{A}}{f\parallel }_{L^2}^2\le \parallel C(0)-C_{\infty }\parallel e^{-2{\lambda }_{*}\tau }{\parallel f\parallel }_{L^2}^2.$$

Equivalently,

$$|C_{\tau }(f)-C_{\infty }(f)|\le e^{-2{\lambda }_{*}\tau }M(f),M(f):=\parallel C(0)-C_{\infty }{\parallel \parallel f\parallel }_{L^2}^2.$$

(34)

The residual in Theorem 5 is $R[f;\tau ]=|C_{\tau }(f)-C_{\infty }{(f)|}^2$, hence from (34)

$$R[f;\tau ]\le e^{-4{\lambda }_{*}\tau }M{(f)}^2.$$

This is sharper than the theorem’s envelope; loosening the prefactor gives the stated $e^{-2{\lambda }_{*}\tau }$ bound.

Fourier–Mode Check (Explicit Diagonalization).

Let $\widehat{f}(k)$ be the Fourier transform and note $\widehat{\mathcal{A}f}={(|k|}^2+m^2)\widehat{f}(k)$. Then $e^{-\tau \mathcal{A}}$ multiplies by $e^{-\tau (|k{|}^2+m^2)}$, while $C_{\infty }$ multiplies by ${(|k|}^2+m^2{)}^{-1}$. A direct computation yields

$$C_{\tau }(f)-C_{\infty }(f)={\int }_{{\mathbb{R}}^d}{e}^{-2\tau (|k{|}^2+m^2)}\widehat{D}(0,k){|\widehat{f}(k)|}^{2}dk,$$

so $|C_{\tau }(f)-C_{\infty }(f)|\le e^{-2{\lambda }_{*}\tau }\parallel \widehat{D}{(0,·)\parallel }_{L^{\infty }}{\parallel f\parallel }_{L^2}^2$, consistent with (34).

Abstract Semigroup Proof.

Let H be the nonnegative, self–adjoint Euclidean Hamiltonian generating ${\mathsf{T}}_{\tau }=e^{-\tau H}$ on the GNS space. With vacuum projector ${\Pi }_0$ one has $\sigma (H)\subset \{0\}\cup [{\lambda }_{*},\infty )$ and $\parallel ({\mathsf{T}}_{\tau }-{\Pi }_0){|}_{{\Pi }_0^{\perp }}\parallel \le e^{-{\lambda }_{*}\tau }$. Then

$$C_{\tau }(f)-C_{\infty }(f)=\langle f,({\mathsf{T}}_{\tau }-{\Pi }_0)K({\mathsf{T}}_{\tau }-{\Pi }_0)f\rangle$$

for some positive operator K, whence

$$|C_{\tau }(f)-C_{\infty }(f)|\le \parallel K\parallel \parallel ({\mathsf{T}}_{\tau }-{\Pi }_0){|}_{{\Pi }_0^{\perp }}{\parallel }^2{\parallel f\parallel }^2\le \parallel K\parallel e^{-2{\lambda }_{*}\tau }{\parallel f\parallel }^2,$$

and squaring gives the same residual decay (again with the sharper $4{\lambda }_{*}$ envelope available).

Admissible Test Functions.

The bounds hold for any f with finite quadratic forms $\langle f,C(0)f\rangle$ and $\langle f,C_{\infty }f\rangle$ (e.g. $f\in L^2$ with ultraviolet cutoff, or $f\in \mathcal{S}$), ensuring well–posed covariance pairings and OU action.

Combining the covariance estimate with $R[f;\tau ]=|C_{\tau }(f)-C_{\infty }{(f)|}^2$ finishes the proof. □

5.8. Proof of Sec. 4.6 (GR DeTurck–Gauge Decay)

Proof of Theorem 6. 

We work on a smooth closed 3–manifold $(\Sigma ,\overline{\gamma })$. Write the evolving metric as $\gamma (t)=\overline{\gamma }+h(t)$ with $h(t)\in \Gamma ({\mathrm{Sym}}^2{T}^{*}\Sigma )$ small in $H^k$, $k\ge 3$. The Einstein–DeTurck–source flow can be written as

$${\partial }_{t}h=-{\mathcal{L}}_{\overline{\gamma }}h+{\mathcal{N}}_{\overline{\gamma }}(h,\nabla h),$$

(35)

where ${\mathcal{L}}_{\overline{\gamma }}$ is the strictly elliptic Lichnerowicz–DeTurck operator (self–adjoint on $L^2(\overline{\gamma })$ on the orthogonal complement of gauge directions) and ${\mathcal{N}}_{\overline{\gamma }}$ collects quadratic and higher–order terms in $(h,\nabla h)$ (and lower–order dependence on $\overline{\gamma }$). By hypothesis, there exists a spectral gap ${\lambda }_{\mathrm{GR}}>0$ such that

$${\langle h,{\mathcal{L}}_{\overline{\gamma }}h\rangle }_{L^2(\overline{\gamma })}\ge {\lambda }_{\mathrm{GR}}{\parallel h\parallel }_{L^2(\overline{\gamma })}^2\mathrm{for}\mathrm{all}h\perp \mathrm{gauge}\mathrm{directions}.$$

(36)

We also assume T is divergence–free along the flow so that constraint terms vanish.

Step 1: Energy Identity in the Fixed Background.

Taking the $L^2(\overline{\gamma })$ inner product of (35) with h,

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel h\parallel }_{L^2(\overline{\gamma })}^2=-{\langle h,{\mathcal{L}}_{\overline{\gamma }}h\rangle }_{L^2(\overline{\gamma })}+{\langle h,{\mathcal{N}}_{\overline{\gamma }}(h,\nabla h)\rangle }_{L^2(\overline{\gamma })}.$$

Using (36) gives

$${\displaystyle \frac{d}{dt}}{\parallel h\parallel }_{L^2(\overline{\gamma })}^2\le -2{\lambda }_{\mathrm{GR}}{\parallel h\parallel }_{L^2(\overline{\gamma })}^2+2|{\langle h,{\mathcal{N}}_{\overline{\gamma }}(h,\nabla h)\rangle }_{L^2(\overline{\gamma })}|.$$

(37)

Step 2: Nonlinearity Estimate.

On the compact manifold, for $k\ge 3$ the bilinear/quadratic structure of ${\mathcal{N}}_{\overline{\gamma }}$ and Sobolev product estimates yield

$$|{\langle h,{\mathcal{N}}_{\overline{\gamma }}(h,\nabla h)\rangle }_{L^2(\overline{\gamma })}|\le {C\parallel h\parallel }_{H^1(\overline{\gamma })}{\parallel h\parallel }_{L^2(\overline{\gamma })}\le {C\parallel h\parallel }_{H^k(\overline{\gamma })}^{\theta }{\parallel h\parallel }_{L^2(\overline{\gamma })}^{2-\theta },$$

(38)

for some $\theta \in (0,1)$ and $C=C(\overline{\gamma })$. (Here we used interpolation ${\parallel h\parallel }_{H^1}\le {C\parallel h\parallel }_{H^k}^{\theta }{\parallel h\parallel }_{L^2}^{1-\theta }$ and the fact that $\mathcal{N}$ has no linear part at $\overline{\gamma }$.) Since the flow is parabolic–elliptic in DeTurck gauge, standard local theory gives a time interval on which ${\parallel h(t)\parallel }_{H^k(\overline{\gamma })}\le 2{\parallel h(0)\parallel }_{H^k(\overline{\gamma })}$. Choosing the smallness radius $\delta >0$ so that ${\parallel h(0)\parallel }_{H^k}\le \delta$ implies

$$|{\langle h,{\mathcal{N}}_{\overline{\gamma }}(h,\nabla h)\rangle }_{L^2}|\le {\displaystyle \frac{{\lambda }_{\mathrm{GR}}}{2}}{\parallel h\parallel }_{L^2(\overline{\gamma })}^2,\mathrm{as}\mathrm{long}\mathrm{as}{\parallel h(t)\parallel }_{H^k}\le 2\delta .$$

(39)

Step 3: Differential Inequality and Decay.

Combining (37) and (39),

$${\displaystyle \frac{d}{dt}}{\parallel h\parallel }_{L^2(\overline{\gamma })}^2\le -2{\lambda }_{\mathrm{GR}}{\parallel h\parallel }_{L^2}^2+{\lambda }_{\mathrm{GR}}{\parallel h\parallel }_{L^2}^2=-{\lambda }_{\mathrm{GR}}{\parallel h\parallel }_{L^2(\overline{\gamma })}^2.$$

By Grönwall,

$${\parallel h(t)\parallel }_{L^2(\overline{\gamma })}^2\le e^{-{\lambda }_{\mathrm{GR}}t}{\parallel h(0)\parallel }_{L^2(\overline{\gamma })}^2.$$

(40)

The decay (40) and parabolic regularization imply, by a standard bootstrap, that ${\parallel h(t)\parallel }_{H^k}$ remains $\le 2\delta$ for all $t\ge 0$ provided $\delta$ is chosen small enough initially. Hence (40) holds globally, and by refining the absorption in (39) one improves the rate to $2{\lambda }_{\mathrm{GR}}$:1

$${\displaystyle \frac{d}{dt}}{\parallel h\parallel }_{L^2(\overline{\gamma })}^2\le -2{\lambda }_{\mathrm{GR}}{\parallel h\parallel }_{L^2(\overline{\gamma })}^2.$$

Step 4: Equivalence of the Geometric Residual and ${\parallel h\parallel }_{L^2}^2$.

Define

$${\mathcal{R}}_{\mathrm{geom}}(t):={\int }_{\Sigma }\parallel G[\gamma (t)]-\kappa T{\parallel }_{\gamma (t)}^{2}d{\mu }_{\gamma (t)}.$$

For h small in $C^0$ one has the metric and volume equivalences

$${(1-c\parallel h\parallel }_{C^0}{)\parallel S\parallel }_{L^2(\overline{\gamma })}^2\le {\parallel S\parallel }_{L^2(\gamma (t))}^2\le {(1+c\parallel h\parallel }_{C^0}{)\parallel S\parallel }_{L^2(\overline{\gamma })}^2,$$

uniformly for tensors S. Moreover, by linearization at $\overline{\gamma }$,

$$G[\overline{\gamma }+h]-\kappa T={\mathcal{L}}_{\overline{\gamma }}h+\mathcal{Q}(h,\nabla h),$$

with ${\parallel \mathcal{Q}(h,\nabla h)\parallel }_{L^2(\overline{\gamma })}\le {C\parallel h\parallel }_{H^1}{\parallel h\parallel }_{H^2}$. By elliptic regularity for ${\mathcal{L}}_{\overline{\gamma }}$ and the spectral gap on the gauge–orthogonal subspace, $\parallel {\mathcal{L}}_{\overline{\gamma }}{h\parallel }_{L^2}\ge {\lambda }_{\mathrm{GR}}{\parallel h\parallel }_{L^2}$. Hence, for ${\parallel h\parallel }_{H^k}$ sufficiently small,

$$c_1{\parallel h\parallel }_{L^2(\overline{\gamma })}^2\le {\mathcal{R}}_{\mathrm{geom}}(t)\le c_2{\parallel h\parallel }_{L^2(\overline{\gamma })}^2,$$

(41)

for some $0<c_1\le c_2<\infty$ depending only on $(\Sigma ,\overline{\gamma })$ and the smallness radius.

Step 5: Decay of ${\mathcal{R}}_{\mathrm{geom}}$.

Differentiating ${\mathcal{R}}_{\mathrm{geom}}$ along the flow and using (35), the elliptic coercivity and (41), one obtains the differential inequality

$${\displaystyle \frac{d}{dt}}{\mathcal{R}}_{\mathrm{geom}}(t)\le -2{\lambda }_{\mathrm{GR}}{\mathcal{R}}_{\mathrm{geom}}(t),$$

for all $t\ge 0$ as long as h remains in the small regime (which we ensured in Step 3). Grönwall therefore gives

$${\mathcal{R}}_{\mathrm{geom}}(t)\le {\mathcal{R}}_{\mathrm{geom}}(0)e^{-2{\lambda }_{\mathrm{GR}}t},$$

and in particular $G[\gamma (t)]\to \kappa T$ in $L^2(\Sigma )$ as $t\to \infty$. Standard arguments then show $\gamma (t)\to \overline{\gamma }$ modulo diffeomorphisms (the DeTurck vector field fixes the gauge). □