Algorithmic Thermodynamics, Graph Cuts, and Boundary Geometry

Algorithmic Thermodynamics, Graph Cuts, and Boundary Geometry#

TLDR#

The starting point for thermodynamics is the algorithm’s finite-step transition law, including its selection, copying, noise, and retained memory. A specified forward experiment and reverse experiment define a path KL divergence in nats. Support and integrability conditions determine whether it is finite. Its identification with physical entropy production additionally requires a reservoir model and local detailed balance.

For a conservative kernel with an invariant law, a weighted mixing bound, controlled moments, and a differentiable parameter dependence, the discrete Poisson equation gives the stationary response. The calculation uses the derivative of the actual update rule. A dominated differentiable probability family also supplies scores and Fisher information. These quantities measure changes of a controlled law; the fitness Hessian measures changes in spatial query coordinates. The full derivations and exact numerical reference cases are in 6.1. Thermodynamic diagnostics derived from the update kernel.

Graph cuts supply a complementary geometric observable. Under the stated kernel, sampling, and boundary hypotheses, a rescaled cut approaches a weighted surface area. An entropy interpretation requires a specified probability law or quantum state and an identification with that cut. Gibbs, modular, and gravitational formulas below state the additional structures they use.

Introduction#

Imagine keeping a complete record of a run. The record tells us which walkers were copied, which random displacements occurred, and which information the next update retained. The transition rule assigns probabilities to such records. We can now ask a precise thermodynamic question: how distinguishable is this experiment from a specified experiment run in reverse? The answer is a relative entropy between path laws. Each choice of reverse dynamics asks a particular question, so it belongs in the definition of the experiment.

We can also change one control parameter and measure the resulting change in an observable. An early disturbance may influence many later updates. The Poisson equation sums those delayed effects when the dynamics forgets its initial conditions at the required rate. This gives a response calculation that simulations can check against independently perturbed runs.

The same record has a spatial side. Interactions cross a chosen boundary, and genealogy describes how episodes were created. Counting these crossings can recover geometric area under appropriate sampling conditions. We develop those geometric observables first, then connect them with explicitly defined entropy and response experiments. Every connection must specify the law, observable, and perturbation being compared.

1. Genealogy, interactions, and boundary observables#

A graph can measure a boundary without knowing any geometry explicitly. Count the weighted edges joining points on opposite sides. When the edges are short, only a thin layer near the boundary contributes. The calculation below shows how its thickness and the range of the kernel combine to produce an area term.

Genealogy records another feature of the same run: which episodes were created from which parents. Relating a genealogical cut to a geometric surface requires a sampling correspondence. These two records have different meanings, but they are generated by the same stochastic dynamics and need not be statistically independent.

These are observables of the recorded dynamics. Their geometric limits use explicit kernel and sampling assumptions, and their statistical interpretation uses the path laws or specified states developed below.

Definition 780 (Causal Spacetime Tree)

Let $\mathcal T=(V,E)$ record episodes and their parent episodes at creation, using the genealogy convention of Definition 642. An episode has at most one genealogical parent. With several initial episodes this is a rooted forest; adjoining a common root makes it a tree. Creation order makes it acyclic.

Write $e\prec e'$ when a directed ancestral path leads from $e$ to $e'$. This is a strict partial order. It concerns ancestry; episodes simultaneously alive need not form an antichain when a parent remains alive after producing a child.

Definition 781 (Gaussian interaction graph for the geometric calculation)

For positions $z_1,\ldots,z_N$ in a specified $d$-dimensional metric space, define an auxiliary symmetric graph with weights

\[ w_{ij}=K_\varepsilon(z_i,z_j),\qquad K_\varepsilon(z,z')=C_0e^{-\operatorname{dist}(z,z')^2/(2\varepsilon^2)}, \qquad C_0>0. \]

Here $\varepsilon$ is a spatial kernel range; it may be set to a chosen companion scale when the kernels agree. These positive weights are interaction strengths, not automatically statistical covariances. A recorded IG may instead be directed, sparse, or thresholded. Its cut statistic uses its actual edges and weights; the complete Gaussian graph is the reference construction analyzed here.

Definition 782 (Separating antichain)

Fix source episodes, terminal episodes, and a specified family of source-to-terminal ancestral chains. A separating antichain $\gamma$ contains no ancestrally comparable pair and intersects every such chain exactly once. Relating it to a spatial boundary $\Sigma$ additionally requires assigning its episodes to neighborhoods or cells on $\Sigma$. An arbitrary genealogical forest need not supply such an assignment or such a separator for prescribed geometric terminals.

Definition 783 (Calibrated CST counting area)

For a separating antichain $\gamma_N$ and a declared area scale $a_N>0$, set

\[ \operatorname{Area}_{\mathrm{CST}}(\gamma_N)=a_N|\gamma_N|. \]

On a uniform spatial sampling of a finite-volume region $\Omega$, a candidate calibration is $\ell_N=(|\Omega|/N)^{1/d}$ and $a_N=\ell_N^{d-1}$. These are sampling scales. Convergence to geometric area requires the cell correspondence in Theorem 450; counting episodes alone does not establish a physical Planck length.

Definition 784 (IG cut functional and terminal min-cut)

For a fixed vertex partition $A\subset V$, define

\[ S_{\mathrm{IG}}(A)=\sum_{i<j}w_{ij} |\mathbf1_A(i)-\mathbf1_A(j)|. \]

When every vertex belongs to one of the two prescribed terminal sets $A,A^c$, all crossing edges must be removed, so the separating cut has exactly this weight. If only subsets $T_+,T_-$ are prescribed as terminals and other vertices are free, the variational problem is instead

\[ \min_{T_+\subset B\subset V\setminus T_-}S_{\mathrm{IG}}(B). \]

The statistic is a graph cut capacity. Interpreting it as von Neumann entanglement entropy requires a specified quantum state and a separate identification theorem.

Proposition 298 (Elementary cut identities)

For nonnegative symmetric weights, $S(A)=S(A^c)$ and $S(A\cup B)\leq S(A)+S(B)$. Cut capacity need not increase with set inclusion.

Proof

Complementation leaves each indicator difference unchanged. For each edge, $|\mathbf1_{A\cup B}(i)-\mathbf1_{A\cup B}(j)|$ is at most the sum of the corresponding differences for $A$ and $B$; multiply by the weight and sum. In a two-vertex graph with one positive edge, $S(\{1\})>0=S(\{1,2\})$, disproving monotonicity.

2. The jump functional and modular energy#

The exponential jump functional measures changes in a scalar field across interacting pairs. Its first nonzero term is quadratic in the field difference, like the energy of a spring near its rest position. That quadratic behavior is useful, but it is different from a linear change in the expectation of a fixed energy observable.

A modular Hamiltonian comes from a probability distribution or a density matrix. A weighted graph does not specify one by itself. Once a reference state is supplied, the entropy identity is exact and its linearization can be proved directly.

Definition 785 (Exponential jump functional)

Let $\mu(dz)=\rho(z)dz$ be a finite measure, $K_\varepsilon$ symmetric and nonnegative, and $\Phi$ bounded. Assuming the displayed integrals are finite, define

\[ \mathcal H_{\mathrm{jump}}[\Phi] =\iint K_\varepsilon(z,z')\rho(z)\rho(z') \left[e^{(\Phi(z)-\Phi(z'))/2}-1-\frac{\Phi(z)-\Phi(z')}{2}\right]dz\,dz'. \]

We use a probability density when discussing sampling. For number density $n=N\rho$, this functional and the nonlocal perimeter both acquire a factor $N^2$.

Proposition 299 (Positivity and the quadratic jump form)

The functional is nonnegative. It vanishes precisely when $\Phi(z)=\Phi(z')$ for almost every pair of positive interaction weight; this gives constancy on connected interaction components. Put $\Delta\Phi=\Phi(z)-\Phi(z')$ and $M=\|\Delta\Phi\|_\infty$. Then

\[ \mathcal H_{\mathrm{jump}}[t\Phi] =\frac{t^2}{8}\iint K_\varepsilon\rho\rho'(\Delta\Phi)^2+R_t, \]

\[ 0\leq R_t\leq\frac{t^4e^{|t|M/2}}{384} \iint K_\varepsilon\rho\rho'(\Delta\Phi)^4. \]

Proof

Convexity gives $e^u-1-u\geq0$, with equality only at $u=0$. Exchange $z,z'$ and average to write the same functional with integrand $\cosh(t\Delta\Phi/2)-1$. Its quadratic term is $t^2(\Delta\Phi)^2/8$. Taylor’s remainder for the even function $\cosh$ is nonnegative and bounded by $e^{|u|}u^4/24$. Substitute $u=t\Delta\Phi/2$ and integrate.

Proposition 300 (Modular identity for a specified state)

Let $\sigma$ be a strictly positive trace-one matrix on a finite-dimensional Hilbert space. Define

\[ K_\sigma=-\log\sigma,\qquad S(\omega)=-\operatorname{Tr}(\omega\log\omega). \]

For any density matrix $\omega$,

\[ D(\omega\Vert\sigma) =\operatorname{Tr}[(\omega-\sigma)K_\sigma] -[S(\omega)-S(\sigma)]. \]

If $\sigma=Z^{-1}e^{-\beta H}$, then $K_\sigma=\beta H+(\log Z)I$. The graph jump functional has no automatic equality with the first modular-energy variation.

Proof

Expand $D(\omega\Vert\sigma)=\operatorname{Tr}\omega(\log\omega-\log\sigma)$ and use $S(\sigma)=\operatorname{Tr}(\sigma K_\sigma)$. The Gibbs formula follows by taking its logarithm.

For the distinction in orders, take $\sigma=\operatorname{diag}(1/4,3/4)$ and $\omega_t=\sigma+t\operatorname{diag}(1,-1)$ for sufficiently small $t$. Its modular-energy variation is $t\log3$. In contrast, Proposition 299 gives zero first derivative for $\mathcal H_{\mathrm{jump}}[t\Phi]$ at $t=0$. A linear identification therefore needs an additional, nontrivial relation between the observables.

Actual edge cuts by channel. Count selected directed edges crossing nested slot partitions, separating distance, cloning, and their historical channels. Project retained event edges to slot labels with unit capacity per executed selection. — **Which actual interactions cross a chosen slot partition?** Count selected directed edges crossing nested slot partitions, separating distance, cloning, and their historical channels. Project retained event edges to slot labels with unit capacity per executed selection.

3. Gaussian cuts and the perimeter limit#

A crossing pair has two small displacements normal to the boundary and a small relative displacement along it. The two normal integrations contribute two powers of the kernel range. The tangential integration contributes another $d-1$ powers. For the unnormalized Gaussian the total is therefore $\varepsilon^{d+1}$.

We will prove more than a calculation for one smooth boundary: the rescaled functionals have a variational limit. The proof below uses an explicit flat periodic reference domain to avoid unrelated outer-boundary terms. This does not impose periodicity or compactness on the Fractal Gas algorithm. Other geometric regimes require their own kernel, chart, and boundary estimates.

Definition 786 (Nonlocal perimeter and normalization)

On the unit flat torus $\mathbb T^d$, use the periodized Gaussian

\[ K_\varepsilon(x,y)=C_0\sum_{k\in\mathbb Z^d} e^{-|x-y+k|^2/(2\varepsilon^2)}. \]

Let $\rho\in C^1(\mathbb T^d)$ satisfy $0<c\leq\rho\leq M$. For a measurable set $A$, define

\[ \mathcal P_\varepsilon(A;\rho) =\int_A\int_{A^c}K_\varepsilon(x,y)\rho(x)\rho(y)\,dy\,dx, \qquad F_\varepsilon(A)=\varepsilon^{-(d+1)}\mathcal P_\varepsilon(A;\rho). \]

The unscaled perimeter tends to zero for a fixed finite-perimeter set. For the normalized Gaussian $(2\pi\varepsilon^2)^{-d/2}K_\varepsilon$, the corresponding cut instead has order $\varepsilon$. A threshold or a different kernel changes the limiting coefficient.

Lemma 322 (Weighted directional difference quotients)

For $u\in BV(\mathbb T^d)$, continuous nonnegative $w$, and $h\in\mathbb R^d$,

\[ \lim_{t\downarrow0}\int w(x) \frac{|u(x+th)-u(x)|}{t}\,dx =\int w\,d|h\cdot Du|. \]

Moreover $\|u(\cdot+th)-u\|_1\leq t|h|\,|Du|(\mathbb T^d)$. If $u_t\to u$ in $L^1$, the liminf of the weighted difference quotient with $u_t$ is at least the weighted directional variation of $u$, interpreted distributionally and allowed to be infinite.

Proof

For a smooth test function $\varphi$, transferring the translation to the test function gives

\[ \int\varphi(x)\frac{u_t(x+th)-u_t(x)}t\,dx =\int u_t(x)\frac{\varphi(x-th)-\varphi(x)}t\,dx \longrightarrow-\int u\,h\cdot\nabla\varphi. \]

The test-function quotient is uniformly bounded and converges uniformly, so $L^1$ convergence suffices. Taking the supremum over smooth $\varphi$ with $|\varphi|\leq w$ gives the liminf inequality; continuous weights follow by approximation from below.

For $u\in BV$, the distributional identity $u(\cdot+th)-u=\int_0^t(h\cdot Du)(\cdot+sh)\,ds$ is obtained first for smooth mollifications and then by passage to measures. Its weighted total variation is at most $\int_0^t\int w(x-sh)\,d|h\cdot Du|(x)\,ds$. Divide by $t$ and use uniform continuity of $w$ to obtain the matching limsup. With $w=1$ the same identity gives the translation bound.

Theorem 448 (Weighted Gaussian perimeter and Gamma-convergence)

In $L^1(\mathbb T^d)$ convergence of indicator functions, $F_\varepsilon$ Gamma-converges to

\[ F_0(A)=c_d\int\rho^2\,d|D\mathbf1_A|, \qquad c_d=C_0(2\pi)^{(d-1)/2}, \]

with $F_0=+\infty$ outside finite-perimeter sets. For a smooth boundary,

\[ \mathcal P_\varepsilon(A;\rho) =c_d\varepsilon^{d+1}\int_{\partial A}\rho^2\,d\Sigma +o(\varepsilon^{d+1}). \]

Energy-bounded sequences with $\varepsilon\downarrow0$ are precompact in $L^1$. Infimum values and cluster points of minimizers therefore converge for fixed-volume constraints, and for other constraints admitting the required closedness and recovery sequences.

Proof

Write $u=\mathbf1_A$, $G(h)=e^{-|h|^2/2}$, and periodically extend $u,\rho$. Symmetry and $y=x+\varepsilon h$ give the exact identity

\[ F_\varepsilon(A)=\frac{C_0}{2}\int_{\mathbb R^d}G(h) \int_{\mathbb T^d}\rho(x)\rho(x+\varepsilon h) \frac{|u(x+\varepsilon h)-u(x)|}{\varepsilon}\,dx\,dh. \]

For fixed $h$, the weight converges uniformly and relatively to $\rho^2$, since $\rho\geq c>0$. If $u_\varepsilon\to u$ in $L^1$, the directional lemma and Fatou’s lemma yield

\[ \liminf F_\varepsilon(A_\varepsilon) \geq\frac{C_0}{2}\int G(h)\int\rho^2\,d|h\cdot Du|\,dh. \]

If this right side is finite, then $u\in BV$. Indeed its lower bound with $\rho^2$ replaced by $c^2$ controls the Gaussian integral of the distributional directional variations. Mollifying $u$ does not increase any unweighted directional variation. For smooth $u_\delta$, rotational invariance gives $\int G(h)\int|h\cdot\nabla u_\delta|\,dx\,dh =2(2\pi)^{(d-1)/2}\int|\nabla u_\delta|$. These total variations are uniformly bounded; passing to the limit in their defining distributional inequalities proves $u\in BV$.

For $u\in BV$, write $Du=\nu_u|Du|$ with $|\nu_u|=1$ almost everywhere. Rotational invariance and the one-dimensional Gaussian integral give

\[ \frac12\int_{\mathbb R^d}G(h)|h\cdot\nu_u|\,dh =(2\pi)^{(d-1)/2}. \]

Tonelli’s theorem now identifies the liminf with $F_0(A)$. For recovery, take $A_\varepsilon=A$. The directional lemma gives the pointwise-in-$h$ limit. Its translation bound dominates the inner integral by $M^2|h|\,|Du|(\mathbb T^d)$, whose Gaussian integral is finite. Dominated convergence proves $F_\varepsilon(A)\to F_0(A)$. For smooth $A$, $|D\mathbf1_A|$ is surface measure, giving the asymptotic formula.

For compactness, choose a smooth nonnegative mollifier $\psi$ supported in the unit ball, of integral one, and let $v_\varepsilon=u_\varepsilon*\psi_\varepsilon$. The Gaussian has a positive lower bound on that ball, while $\rho\geq c$. The energy identity therefore implies

\[ \|u_\varepsilon-v_\varepsilon\|_1\leq C\varepsilon F_\varepsilon(A_\varepsilon), \qquad \|\nabla v_\varepsilon\|_1\leq C F_\varepsilon(A_\varepsilon). \]

The second estimate follows by subtracting $u_\varepsilon(x)$ inside the convolution with $\nabla\psi_\varepsilon$, whose integral is zero. Bounded variation and the compact domain give $L^1$ precompactness of $v_\varepsilon$: its translation bound permits approximation, uniformly in $\varepsilon$, by convolution at a fixed small scale, and those smooth convolutions form a precompact family. The first estimate transfers compactness to $u_\varepsilon$. Its limits remain indicators by almost-everywhere subsequence convergence.

Finally choose a convergent subsequence of minimizers. The liminf inequality bounds its limiting energy below by $F_0$; a recovery sequence for any admissible competitor bounds the limiting minima above by that competitor’s $F_0$. These two inequalities prove the minimizer claim. Fixed volume passes through $L^1$ convergence and is preserved by the constant recovery sequence.

This calculation retains the density squared because a crossing edge has two endpoints. It also explains why an area coefficient depends on the kernel normalization. The nonlocal-to-local variational method belongs to the classical bounded-variation theory; a general treatment is Ponce’s work on nonlocal functionals and Gamma-convergence. Here the Gaussian coefficient and the hypotheses needed for the weighted limit have been computed explicitly.

Cut perimeter of nested slot regions. Scan nested slot regions and count directed executed interactions crossing each partition. Compare cut differences with the incident-edge first variation. — **How does the actual selected-edge cut change with region size?** Scan nested slot regions and count directed executed interactions crossing each partition. Compare cut differences with the incident-edge first variation.

4. Sampling the cut and comparing it with a surface#

The continuum integral counts pairs according to a density. A finite graph counts actual pairs. Its expected cut is therefore proportional to the number of ordered pairs, with fluctuations determined by pairs that share a walker.

Genealogical area needs another ingredient: a way to assign boundary cells to the episodes being counted. If those cells cover the same surface and their areas are controlled, the comparison becomes elementary. Without that assignment, an antichain cardinality is only a count in a partial order.

Theorem 449 (Expected cut and finite-population area scaling)

Let $Z_1,\ldots,Z_N$ be independent with probability density $\rho$ satisfying the preceding torus hypotheses, and let $A$ have finite perimeter. For the complete unthresholded Gaussian graph,

\[ S_{N,\varepsilon}(A)=\sum_{i<j}K_\varepsilon(Z_i,Z_j) |\mathbf1_A(Z_i)-\mathbf1_A(Z_j)| \]

satisfies the exact expectation identity

\[ \mathbb ES_{N,\varepsilon}(A)=N(N-1)\mathcal P_\varepsilon(A;\rho). \]

For $0<\varepsilon\leq\varepsilon_0$, there is a constant $C$ independent of $N,\varepsilon$ such that

\[ \operatorname{Var}(S_{N,\varepsilon}) \leq C\left[N^2\varepsilon^{d+1} +N^3\varepsilon^{2d+1}\right]. \]

Consequently, if $d>1$ and $\varepsilon_N=\ell N^{-1/d}$ with fixed $\ell>0$,

\[ \frac{S_{N,\varepsilon_N}(A)}{N^{(d-1)/d}} \longrightarrow c_d\ell^{d+1}\int\rho^2\,d|D\mathbf1_A| \quad\text{in }L^2. \]

At fixed range $\varepsilon$, the raw expected cut instead has order $N^2$. A dependent QSD sample requires bounds for its actual pair and shared-index laws; exchangeability alone does not give the displayed variance estimate.

Proof

Set $h_\varepsilon(x,y)=K_\varepsilon(x,y)|\mathbf1_A(x)-\mathbf1_A(y)|$ and $g_\varepsilon(x)=\int h_\varepsilon(x,y)\rho(y)dy$. Then $\mathbb Eh_\varepsilon(Z_1,Z_2)=2\mathcal P_\varepsilon(A;\rho)$. There are $N(N-1)/2$ unordered pairs, proving the expectation formula.

The periodized Gaussian has a uniformly bounded supremum for $\varepsilon\leq\varepsilon_0$, and its total mass is $C_0(2\pi)^{d/2}\varepsilon^d$. Thus

\[ \mathbb Eh_\varepsilon^2\leq C\mathcal P_\varepsilon, \qquad \|g_\varepsilon\|_\infty\leq C\varepsilon^d, \qquad \mathbb Eg_\varepsilon(Z_1)^2 \leq C\varepsilon^d\mathcal P_\varepsilon. \]

The translation bound in the perimeter proof gives $\mathcal P_\varepsilon\leq C\varepsilon^{d+1}$ uniformly at small $\varepsilon$. Terms from disjoint vertex pairs have zero covariance by independence. For two pairs sharing one vertex, conditional independence gives covariance $\operatorname{Var}(g_\varepsilon(Z_1))\leq\mathbb Eg_\varepsilon(Z_1)^2$. There are at most $N^2/2$ individual pairs and $N^3$ ordered overlapping pair-of-pair contributions, up to an absolute constant. Expanding the variance yields the bound.

For $\varepsilon_N=\ell N^{-1/d}$, both variance terms are $O(N^{(d-1)/d})$. After division by $N^{2(d-1)/d}$ the variance tends to zero when $d>1$. The perimeter limit gives convergence of the normalized expectations to the stated constant. Together these prove $L^2$ convergence. In dimension one the variance estimate does not vanish at this microscopic range, so this argument supplies no such deterministic limit there.

Corollary 141 (Transfer of the area limit to a dependent joint law)

Retain the preceding density, boundary, and range hypotheses, and let $\pi_N$ be any joint law for the sampled points. If

\[ H_N=D_{\mathrm{KL}}(\pi_N\Vert\rho^{\otimes N})=o(\log N), \]

the same normalized cut converges to the same area constant in probability under $\pi_N$. In particular a uniform bound on total relative entropy suffices. If the stronger domination $\pi_N\leq M_*\rho^{\otimes N}$ holds uniformly, convergence holds in $L^2(\pi_N)$.

These hypotheses can be checked for an identified QSD using its joint-law estimates. A bounded whole-law density tilt is one of the structural routes in Corollary 94; a one-particle density bound alone is insufficient.

Proof

Let $X_N=S_{N,\varepsilon_N}/N^{(d-1)/d}$ and let $L$ be its limiting area constant. For fixed $\delta>0$, convergence of the independent expectation and the variance bound give, for sufficiently large $N$,

\[ q_N:=\rho^{\otimes N}(|X_N-L|>\delta) \leq C_\delta N^{-(d-1)/d}. \]

For any event $E$ of reference probability $0<q<1$, the entropy inequality applied to $t\mathbf1_E$ gives $t\pi_N(E)\leq H_N+\log(1-q+qe^t)$. Set $t=\log(1/q)$ to obtain

\[ \pi_N(E)\leq\frac{H_N+\log2}{\log(1/q)}. \]

If $q=0$, finite relative entropy already gives $\pi_N(E)=0$. Apply the inequality with the displayed polynomial bound for $q_N$. Its denominator grows at least as $[(d-1)/d]\log N-O_\delta(1)$, so $H_N=o(\log N)$ proves convergence in probability. Under domination, directly bound $\mathbb E_{\pi_N}|X_N-L|^2\leq M_*\mathbb E_{\rho^{\otimes N}}|X_N-L|^2\to0$. A full position–velocity domination bound passes to position marginals by integration; a relative-entropy bound passes by conditional Jensen’s inequality for the marginal density ratio.

Theorem 450 (Antichain area under a boundary-cell correspondence)

Let $\Sigma$ be a compact surface of finite area. Suppose a separating antichain $\gamma_N$ is assigned measurable boundary cells $C_e\subset\Sigma$ whose interiors are disjoint and whose union covers $\Sigma$ up to surface measure zero. If

\[ (1-\eta_N)a_N\leq |C_e|\leq(1+\eta_N)a_N, \qquad \eta_N\to0, \]

then $a_N|\gamma_N|\to|\Sigma|$. In particular, if $a_N=(|\Omega|/N)^{(d-1)/d}$, then

\[ \frac{|\gamma_N|}{N^{(d-1)/d}} \longrightarrow\frac{|\Sigma|}{|\Omega|^{(d-1)/d}}. \]

These are explicit geometric sampling hypotheses; they do not follow from the antichain property alone.

Proof

For the necessity of geometric information, a run with no cloning has the same rooted forest for many different choices of spatial positions. Its antichains therefore do not determine the areas of arbitrary spatial boundaries. A limiting surface theorem must identify the cells and prove the stated correspondence for the sampling law used.

Theorem 451 (Calibrated interaction-area comparison)

Suppose the graph-cut convergence of Theorem 449 or Corollary 141 and the boundary-cell correspondence of Theorem 450 apply to the same smooth boundary $\Sigma=\partial A$, with $|\Sigma|>0$. Define

\[ \widehat S_N(A)= \frac{S_{N,\varepsilon_N}(A)}{N(N-1)\varepsilon_N^{d+1}}. \]

Then

\[ \frac{\widehat S_N(A)}{\operatorname{Area}_{\mathrm{CST}}(\gamma_N)} \longrightarrow c_d\frac{\int_\Sigma\rho^2\,d\Sigma}{|\Sigma|}. \]

If $\rho=\rho_0$ on $\Sigma$, this coefficient is $\alpha=c_d\rho_0^2$. Varying density generally gives a surface-dependent coefficient. The assertion is an asymptotic comparison of calibrated quantities, not an exact finite-$N$ identity for the raw cut.

Proof

The graph theorem gives $\widehat S_N\to c_d\int_\Sigma\rho^2d\Sigma$, and the cell theorem gives $\operatorname{Area}_{\mathrm{CST}}\to|\Sigma|>0$. Divide these limits. Under random cell correspondences the same conclusion holds in probability if the area convergence holds in probability. The constant-density specialization follows by evaluating the surface integral.

Observed companion diversity. Count selected target slots for each source slot, form empirical target frequencies, and compare their entropy with the logarithm of observed support. — **How much of the observed donor support is used?** Count selected target slots for each source slot, form empirical target frequencies, and compare their entropy with the logarithm of observed support.

5. First variations of cut cost, entropy, and surface energy#

There are several possible perturbations. We can change the density while keeping the boundary fixed, move the boundary while keeping the density fixed, or change a quantum state on a fixed Hilbert space. Each has a well-defined first variation. Equating two of them requires an additional relation between their observables.

The formulas below keep the perturbation explicit. This prevents a quadratic jump cost from being treated as a linear energy change, or a geometric cut cost from being called thermodynamic entropy without a state construction.

Definition 787 (Variation of a specified swarm energy)

For a fixed energy observable $e(z)$ and a density perturbation $\eta$ with $\int\eta=0$, assuming the energy and variation integrals below are finite, define

\[ E[\rho]=\int e(z)\rho(z)dz, \qquad \delta E[\eta]=\int e(z)\eta(z)dz. \]

If the energy is a nonlinear density functional, $e$ is instead its functional derivative at the reference density. Any independently varied mode contribution $\sum_k\omega_kn_k$ contributes $\sum_k\omega_k\delta n_k$ at first order unless those variations are constrained to vanish.

Definition 788 (Density variation of the graph-cut integral)

For fixed $A,K_\varepsilon$ and an admissible perturbation $\rho_t=\rho+t\eta$, the cut variation is

\[ \delta\mathcal P_\varepsilon[\eta] =\int_A\int_{A^c}K_\varepsilon(x,y) [\eta(x)\rho(y)+\rho(x)\eta(y)]\,dy\,dx. \]

Equivalently $\delta\mathcal P_\varepsilon[\eta]=\int B_A[\rho](z)\eta(z)dz$, where

\[ B_A[\rho](z)=\mathbf1_A(z)\int_{A^c}K_\varepsilon(z,y)\rho(y)dy +\mathbf1_{A^c}(z)\int_AK_\varepsilon(z,y)\rho(y)dy. \]

This is a variation of cut cost. It becomes an entropy variation only under a separately proved entropy identification.

Proof

Expand $(\rho(x)+t\eta(x))(\rho(y)+t\eta(y))$. Its linear coefficient is the displayed two-term integral, and the remaining term is exactly $t^2\int_A\int_{A^c}K_\varepsilon\eta(x)\eta(y)$. Symmetry allows exchanging variables in one term, yielding $B_A$. It does not make the two integrals equal when $A$ and $A^c$ remain different domains.

Proposition 301 (When a cut variation is proportional to an energy variation)

On a finite-volume domain, suppose $B_A[\rho]$ and $e$ are integrable. For a fixed $\beta$, the identity $\delta\mathcal P_\varepsilon[\eta]=\beta\delta E[\eta]$ for every bounded zero-mass test perturbation holds if and only if

\[ B_A[\rho](z)-\beta e(z)=\text{constant almost everywhere}. \]

Proof

Theorem 452 (First law for a finite-dimensional density matrix)

Let $\omega_t$ be a differentiable trace-one density-matrix family with $\omega_0=\sigma\succ0$. Then

\[ \left.\frac{d}{dt}S(\omega_t)\right|_{t=0} =\operatorname{Tr}(\dot\omega_0K_\sigma). \]

If $\sigma=Z^{-1}e^{-\beta H}$ for a fixed $H$, this becomes $\delta S=\beta\,\delta\operatorname{Tr}(\omega H)$. For a reduced state the theorem applies on its specified finite subsystem. It makes no assertion that a graph cut equals that state’s entropy.

Proof

For a differentiable matrix family with positive spectrum, spectral calculus gives $\frac{d}{dt}\operatorname{Tr}f(\omega_t)=\operatorname{Tr}[f'(\omega_t)\dot\omega_t]$. One can verify this at the base point by diagonalizing $\sigma$: the derivative of the trace contains only the diagonal entries $f'(\sigma_{aa})\dot\omega_{aa}$, including within degenerate eigenspaces. For $f(s)=-s\log s$, the derivative is $-\operatorname{Tr}[(\log\sigma+I)\dot\omega_0]$. Trace preservation removes the identity term. Substitute $K_\sigma=-\log\sigma$ and then its Gibbs expression. The corresponding relative-entropy formulation is developed in Blanco, Casini, Hung, and Myers.

Theorem 453 (Boundary variation and the pressure of a declared surface energy)

Let $A$ have smooth boundary $\Sigma$ with outward unit normal $n$, and move it with normal speed $f$. For fixed smooth $\rho$ and symmetric kernel,

\[ \delta\mathcal P_\varepsilon =\int_\Sigma f(x)\rho(x) \left[\int_{A^c}K_\varepsilon(x,y)\rho(y)dy -\int_AK_\varepsilon(x,y)\rho(y)dy\right]d\Sigma(x). \]

If a surface energy is explicitly defined by $\mathcal E_\Sigma=\tau\int_\Sigma\rho^2d\Sigma$, $\tau>0$, then

\[ \delta\mathcal E_\Sigma =\tau\int_\Sigma[\rho^2H+2\rho\partial_n\rho]f\,d\Sigma, \qquad H=\operatorname{div}_\Sigma n. \]

Its variational normal pressure, defined by $\delta\mathcal E_\Sigma=-\int_\Sigma pf\,d\Sigma$, is $p=-\tau(\rho^2H+2\rho\partial_n\rho)$. For constant $\rho_0$ and a sphere of radius $R$, $p=-\tau\rho_0^2(d-1)/R$. General surfaces and varying densities need not have this sign.

Proof

Moving the first integration domain of $\mathcal P_\varepsilon$ contributes its outward boundary integral. Moving the complement in the second domain contributes the same expression with a minus sign and $A$ in the inner integral. Kernel symmetry gives the displayed formula.

For the surface energy, choose an orthonormal tangent frame. Under velocity $fn$, the first variation of the induced metric is twice $f$ times the second fundamental form in the convention $H=\operatorname{div}_\Sigma n$. Differentiating its volume determinant gives $\delta(d\Sigma)=fH\,d\Sigma$. The fixed ambient density changes along the moving surface by $\delta(\rho^2)=2\rho f\partial_n\rho$. Sum the two contributions. On a sphere $H=(d-1)/R$; the pressure formula follows from its definition.

This is a shape derivative of the declared energy. Identifying it with a mechanical stress from Field Equations and Pressure Dynamics requires using the same energy, perturbation, and normalization. Differentiating a limiting perimeter also requires control of first variations, beyond convergence of the functionals alone.

Exact first variation of recorded graph perimeter. Add one slot to a nested partition and compare its measured directed-cut increment with the signed count of incident selected edges. — **Can the next cut increment be predicted from incident edges?** Add one slot to a nested partition and compare its measured directed-cut increment with the signed count of incident selected edges.

6. Gibbs references, QSDs, and algorithmic response#

A Gibbs formula becomes informative when its energy is specified independently of the density. Writing an arbitrary positive density as the exponential of minus its own logarithm does not establish thermal equilibrium. For a proposed Gibbs profile, the direct test is to insert it into the actual invariant or killed eigenvalue equation.

There is a useful quantitative version of that test. If the normalized evolution contracts, its failure to leave the proposed profile fixed bounds the distance to the true QSD. This uses the stationary-profile and perturbation arguments already developed in the convergence chapters.

Theorem 454 (Identifying and testing a Gibbs QSD candidate)

Let $g=Z^{-1}e^{-\beta H}$ be a probability density for a specified energy $H$ and $\beta>0$. Let $Q$ be a killed block kernel whose normalized map $\Phi(\mu)=\mu Q/\mu Q1$ contracts a complete probability space in total variation with factor $r<1$, with unique QSD $\pi$ for the underlying dynamics. Put $a=gQ1>0$. Then

\[ g\text{ is the QSD}\quad\Longleftrightarrow\quad gQ=ag, \]

and

\[ \|g-\pi\|_{\mathrm{TV}} \leq\frac{\|gQ-ag\|_{\mathrm{TV}}}{a(1-r)}. \]

For continuous conservative underdamped Langevin dynamics with gradient force, matching noise and friction, finite partition function, and the appropriate zero-flux generator domain, the kinetic Gibbs reference is invariant as proved in Theorem 239. Adding independent constant-rate killing preserves this reference as a QSD. Additional cloning or state-dependent killing must satisfy their own stationary balance.

Proof

The eigenmeasure condition is exactly $g=\Phi(g)$. Uniqueness identifies that fixed point with $\pi$. In general,

\[ \|g-\pi\|_{\mathrm{TV}} \leq\|g-\Phi(g)\|_{\mathrm{TV}}+r\|g-\pi\|_{\mathrm{TV}}. \]

Move the last term to the left and use $\Phi(g)-g=(gQ-ag)/a$. This is the conditioned-map stability argument of Lemma 271; the continuous mean-field residual bound is Theorem 240.

If $P_t$ is conservative with $gP_t=g$ and killing has constant rate $\kappa$, then $Q_t=e^{-\kappa t}P_t$, so $gQ_t=e^{-\kappa t}g$. A jump operator preserving $g$ can be included in $P_t$ if the full conservative evolution is well posed and retains this invariance. Neither this calculation nor the graph kernel fixes a universal inverse temperature for a general killed swarm.

Proposition 302 (Static fluctuation-response identity for an exponential tilt)

Let $\mu_0$ be a probability law, let $A,B$ be bounded observables, and define the specified perturbation family

\[ \mu_h(dz)=\frac{e^{\beta hB(z)}}{Z(h)}\mu_0(dz). \]

Then the susceptibility satisfies

\[ \chi_{A,B}:=\left.\frac{d}{dh}\mu_h(A)\right|_{h=0} =\beta\operatorname{Cov}_{\mu_0}(A,B). \]

If $\mu_0$ is Gibbs and the physical perturbation is $H\mapsto H-hB$, this is the static relation $\operatorname{Cov}(A,B)=T\chi_{A,B}$ with $T=1/\beta$. It is not a general dynamical response theorem for arbitrary perturbations of a QSD generator.

Proof

Differentiate the numerator and denominator under the integral, justified by boundedness. At zero, $Z(0)=1$ and $Z'(0)=\beta\mu_0(B)$, giving $\beta\mu_0(AB)-\beta\mu_0(A)\mu_0(B)$. Unbounded observables require corresponding domination hypotheses. For a density response, the identity is first interpreted against such test observables; a pointwise covariance kernel requires additional regularity.

6.1. Thermodynamic diagnostics derived from the update kernel#

Suppose we turn the noise knob a little. Which quantity changes, and by how much? The answer depends on what that knob does to the update. It may change the velocities directly, then change the distances, then change who gets copied. Calling the knob a temperature does not calculate any of those effects. We need the transition rule, including its copying and memory operations.

There are two useful experiments. In one, we perturb a specified parameter and measure the change of an observable. In the other, we compare the probabilities of a movie and a declared reverse movie. The first measures response; the second measures statistical irreversibility. Both can be defined before we have an equilibrium energy or a gravitational field equation.

Keep the underlying objects separate. Curvature asks how a spatial metric varies as you move the query point. Fisher information asks how a probability law varies as you move a control parameter. These derivatives act on different spaces. Even if both calculations produce symmetric matrices, an equality between them still needs a map between the spaces and a proof.

Definition 789 (Finite-step law and controlled experiment)

Let $S_n$ contain the population and every retained variable needed for the next update, including donor memory and stateful provider data. For a declared parameter protocol $\theta_0,\ldots,\theta_{L-1}$ and the actual fixed integration step $h$, write its transition kernels as $K_{n,\theta_n}$ and its path law as

\[ \mathbb P_\theta(ds_{0:L}) =\mu_{0,\theta}(ds_0) \prod_{n=0}^{L-1}K_{n,\theta_n}(s_n,ds_{n+1}). \]

This is a probability law for the specified innovation and selection laws. A fixed replay seed realizes one path; it does not itself supply independent conditional samples. An explicit time schedule can remain external to the state. Appending a clock makes a process homogeneous formally but does not create an invariant probability for a clock that increases forever.

For stationary response, restrict to a time-homogeneous conservative kernel $K_\theta$ on a specified state space and an invariant probability $\pi_\theta K_\theta=\pi_\theta$. A killed kernel is sub-Markov and requires its survival conditioning or cemetery state to be specified. Its QSD satisfies an eigenmeasure equation, not this invariant equation. Neither existence nor ergodicity of $\pi_\theta$ is assumed for every Fractal Gas configuration.

Definition 790 (Path irreversibility and its support)

Specify a measurable state involution $\vartheta$, such as velocity reversal, and the path involution $\mathcal R(s_0,\ldots,s_L)=(\vartheta s_L,\ldots,\vartheta s_0)$. Specify also a normalized reverse experiment $\widetilde{\mathbb P}$, including its initial law, schedule, kernels, and treatment of retained memory. Set $\mathbb Q=\mathcal R_\#\widetilde{\mathbb P}$. If $\mathbb P\ll\mathbb Q$, define

\[ \Sigma(s_{0:L})=\log\frac{d\mathbb P}{d\mathbb Q}(s_{0:L}), \qquad \mathcal I_L=D_{\mathrm{KL}}(\mathbb P\Vert\mathbb Q). \]

Here $\Sigma$ and $\mathcal I_L$ are in nats; $\mathcal I_L/L$ is in nats per step. If absolute continuity fails, the path KL is $+\infty$. An unevaluated likelihood or an untested support condition is an unavailable diagnostic, not evidence of either zero or infinite KL.

For example, on a countable state space, with $\vartheta$ the identity and a time-homogeneous forward kernel $K$, reverse kernel $\widetilde K$, and initial probabilities $\mu_0,\nu_0$, the formula on supported paths is

\[ \Sigma= \log\frac{\mu_0(s_0)}{\nu_0(s_L)} +\sum_{n=0}^{L-1} \log\frac{K(s_n,s_{n+1})}{\widetilde K(s_{n+1},s_n)}. \]

The endpoint term is part of the path likelihood. Continuous densities give the same form only after verifying the dominating measures and any reversal Jacobian. In particular, deterministic copying, rank-deficient noise, and discarded memory can produce singular transition laws.

Proposition 303 (Exact path KL identities)

Under the preceding absolute continuity hypothesis, put $f=d\mathbb P/d\mathbb Q$. Then

\[ \mathbb E_{\mathbb P}\Sigma =D_{\mathrm{KL}}(\mathbb P\Vert\mathbb Q)\geq0, \qquad \mathbb E_{\mathbb P}e^{-\Sigma} =\mathbb Q(f>0)\leq1. \]

The last equality is $1$ when the two path laws are equivalent. For a measurable recording map $C$, their exact recorded laws satisfy

\[ D_{\mathrm{KL}}(C_\#\mathbb P\Vert C_\#\mathbb Q) \leq D_{\mathrm{KL}}(\mathbb P\Vert\mathbb Q). \]

Proof

The first expectation is $\int f\log f\,d\mathbb Q$. Convexity of $u\log u$ and $\int f\,d\mathbb Q=1$ give nonnegativity, with $+\infty$ allowed. On $\{f>0\}$, $e^{-\Sigma}=1/f$, so integration against $f\,d\mathbb Q$ gives precisely $\mathbb Q(f>0)$.

The density ratio of the recorded laws is $\mathbb E_{\mathbb Q}[f\mid C]$, interpreted as a function of the record. Conditional Jensen’s inequality for $u\log u$, followed by integration, gives the last inequality. These arguments concern the exact probability laws; they do not require a phase-space density.

Remark 297 (What a path diagnostic measures)

The reverse experiment is part of the observable. Choosing the stationary adjoint kernel as the reverse dynamics can reproduce the forward path law after reversal; that construction is not a test of invariance under the same physical protocol. Reversing bookkeeping counters without reversing their schedule can instead manufacture a support mismatch. The compared variables and reverse protocol must therefore be recorded with the result.

An exact histogram of a declared path recording has the data-processing bound above. A finite-sample plug-in histogram does not automatically give a lower confidence bound on the full path KL. Smoothing empty bins changes the estimated laws, and a projection of a Markov process need not remain Markov. A fitted one-step transition table is consequently not an exact path model without a separate Markov closure argument.

Identifying $\Sigma$ with a physical entropy production requires a reservoir model and local detailed balance for the actual transitions. This extra condition is explicit in Lebowitz and Spohn’s path-space analysis. No phase-space differential entropy is inferred from the covariance matrix or from the fitness values.

For response, imagine making a tiny change in one update and then asking how long its consequences survive. The immediate disturbance is only the first term. It propagates through later updates before the system forgets it. The Poisson equation below adds those delayed effects.

This is why a covariance alone cannot predict every response. A covariance describes fluctuations of the reference law. A response also needs the rule by which the perturbation changes the dynamics. The operator $D$ in the next theorem is that missing piece. When a perturbation is known to produce an exponential tilt, we recover the earlier covariance formula. For a general change of cloning probabilities or noise geometry, we calculate $D$ from that change.

Theorem 455 (Stationary response from a discrete Poisson equation)

Let $V\geq1$ and let $\mathcal B_V$ be the Banach space of measurable functions with $\|f\|_V=\sup_s|f(s)|/V(s)<\infty$. Suppose:

$K_\theta$ are bounded Markov operators on $\mathcal B_V$ near $\theta=0$, with invariant probabilities $\pi_\theta$ and $\sup_{|\theta|<\delta}\pi_\theta(V)<\infty$.
For $K=K_0$ and $\pi=\pi_0$, there are $C<\infty$, $r<1$ such that $\|K^nf-\pi(f)\mathbf1\|_V\leq Cr^n\|f\|_V$ for all $n\geq0$.
There is a bounded operator $D$ on $\mathcal B_V$ with $\|K_\theta-K-\theta D\|_{\mathcal B_V\to\mathcal B_V}=o(|\theta|)$.

For $A\in\mathcal B_V$, the series

\[ u_A=\sum_{n=0}^{\infty}K^n[A-\pi(A)] \]

converges in $\mathcal B_V$ and solves $(I-K)u_A=A-\pi(A)$, $\pi(u_A)=0$. The exact stationary response is

\[ \left.\frac{d}{d\theta}\pi_\theta(A)\right|_{0} =\pi(Du_A) =\sum_{n=0}^{\infty}\pi\!\left(DK^n[A-\pi(A)]\right). \]

If $A_\theta=A+\theta\dot A+o(|\theta|)$ in $\mathcal B_V$, add $\pi(\dot A)$. The theorem concerns the fixed-step kernel. It takes no $h\to0$ limit and does not replace an order-one cloning event by a small drift.

Proof

The geometric bound makes the series norm-convergent with $\|u_A\|_V\leq C\|A\|_V/(1-r)$. Boundedness of $K$ permits applying $I-K$ to its partial sums; the remainder tends to zero. Invariance and the finite $V$ moment permit applying $\pi$ to the series, giving $\pi(u_A)=0$.

Write $M=\sup\pi_\theta(V)$. Invariance of $\pi_\theta$ gives the exact identity

\[ (\pi_\theta-\pi)(A) =\pi_\theta((I-K)u_A) =\pi_\theta((K_\theta-K)u_A). \]

Consequently,

\[ \|\pi_\theta-\pi\|_{\mathcal B_V^*} \leq\frac{MC}{1-r}\|K_\theta-K\|_{\mathcal B_V\to\mathcal B_V} =O(|\theta|). \]

Divide the exact identity by $\theta$. Operator differentiability makes $\pi_\theta[(K_\theta-K)u_A/\theta-Du_A]$ tend to zero, by the uniform moment bound. The dual norm convergence just proved makes $(\pi_\theta-\pi)(Du_A)$ tend to zero. The derivative is therefore $\pi(Du_A)$. Boundedness of $D$ and the same geometric series justify the displayed correlation sum. The observable-dependent term follows by expanding $A_\theta$ and using the uniform moment bound once more.

Remark 298 (Analytic and numerical requirements for response)

These are sufficient hypotheses, not conclusions about every swarm. In an unbounded configuration, the moment and mixing estimates must come from a proved confining or Safe Harbor bound; a finite simulation box does not prove them. A smooth change of a parameter can also fail operator differentiability when it moves a deterministic transition support. Stronger smoothing or a different function space then needs a separate argument. The theorem above is a direct weighted-norm proof of the resolvent approach; the more general framework of Hairer and Majda uses explicitly specified weighted function spaces and differentiability conditions.

For a numerical comparison, the declared derivative of the update and a Poisson solution predict the response. Independently initialized runs at $\theta\pm\delta$ measure it. Fitting the prediction to those same responses does not test the identity. Finite difference error, mixing bias, Poisson approximation error, and sampling uncertainty are separate error sources.

Proposition 304 (Scores and Fisher information for controlled laws)

Let $\theta\in\mathbb R^m$ control a normalized law $p_\theta\,d\lambda$ with common support, positive density there, and twice differentiable densities near the reference parameter. Assume integrable envelopes permit differentiating normalization twice, and that each score $s_a=\partial_a\log p_\theta$ belongs to $L^2(p_\theta d\lambda)$. Then

\[ \mathbb E_\theta s_a=0,\qquad F_{ab}=\mathbb E_\theta[s_as_b] =-\mathbb E_\theta[\partial_a\partial_b\log p_\theta]. \]

$F$ is positive semidefinite, with units $[\theta_a]^{-1}[\theta_b]^{-1}$. For a fixed observable $A$, if differentiation of its expectation is justified and $As_a$ is integrable, then

\[ \partial_a\mathbb E_\theta A =\operatorname{Cov}_\theta(A,s_a). \]

For a dominated finite-step path model satisfying these hypotheses at its initial law and conditional kernels, its score decomposes as

\[ s^{\mathrm{path}}=s^{\mathrm{initial}}(S_0) +\sum_{n=0}^{L-1}s_n^{\mathrm{kernel}}(S_n,S_{n+1}), \]

and its Fisher information is

\[ F^{\mathrm{path}}=F^{\mathrm{initial}} +\sum_{n=0}^{L-1}\mathbb E_\theta \left[s_n^{\mathrm{kernel}}(s_n^{\mathrm{kernel}})^\top\right]. \]

Proof

Since $p_\theta s_a=\partial_a p_\theta$, differentiated normalization gives zero mean score. Differentiating the logarithm twice gives

\[ p_\theta\partial_a\partial_b\log p_\theta =\partial_a\partial_b p_\theta -p_\theta s_as_b. \]

The first term integrates to zero. For any vector $v$, $v^\top Fv=\mathbb E[(v\cdot s)^2]\geq0$. Differentiating the expectation of $A$ gives $\mathbb E[As_a]$, and the zero mean score converts this to the covariance.

Taking the logarithm of the factored path density and differentiating gives the path score. Conditional normalization implies $\mathbb E[s_n^{\mathrm{kernel}}\mid S_0,\ldots,S_n]=0$. Each cross term with an earlier score therefore has zero expectation by the tower property. Square integrability justifies this conditioning and leaves exactly the stated sum of matrices.

Remark 299 (Three geometries require three definitions)

The fitness Hessian $\partial_{x_i}\partial_{x_j}\mathcal F$ differentiates a conditional fitness function with its declared frozen data. Its regularized metric lives in the spatial query coordinates. The matrix $F$ above lives in controlled parameter coordinates and is a metric only where it is positive definite. Fisher information of a stationary law, one transition, and a whole path are distinct quantities.

For a separately specified exponential family $p_\theta=\exp(\theta\cdot T-\psi(\theta))p_0$, justified differentiation gives $\partial_a\psi=\mathbb E_\theta T_a$ and $\partial_a\partial_b\psi=\operatorname{Cov}_\theta(T_a,T_b)=F_{ab}$: differentiate its normalization and then its first derivative. This proves a thermodynamic Hessian identity for that family. It does not identify $\psi$ with the swarm fitness, prove a Gibbs stationary law for cloning, or supply an Einstein equation. These identifications remain separate hypotheses to test against the measured laws.

Example 7 (Exact discrete reference experiments)

For a three-state cycle, let $K(i,i+1)=p$, $K(i,i-1)=q$, and $K(i,i)=1-p-q$ (indices modulo three), with $p,q>0$, $p+q<1$. Every column sums to one, so the uniform law is stationary. Using the same kernel as the reverse protocol and the identity involution, the endpoint term is zero. The expected increment of $\Sigma$ is exactly

\[ \frac{\mathcal I_L}{L} =p\log(p/q)+q\log(q/p) =(p-q)\log(p/q). \]

It vanishes at $p=q$. At $q=0<p$, the same reverse protocol assigns zero probability to some forward paths, so the path KL is infinite for $L\geq1$.

For a two-state kernel

\[\begin{split} K_\theta= \begin{pmatrix}1-a&a\\b&1-b\end{pmatrix}, \qquad a=a_0+\theta,\qquad 0<a,b<1, \end{split}\]

direct multiplication gives $\pi=(b,a)/(a+b)$. For $A=\mathbf1_{\{1\}}$,

\[ \partial_\theta\pi(A)=\frac{b}{(a+b)^2},\qquad F^{\mathrm{stationary}}=\frac{b}{a(a+b)^2},\qquad F^{\mathrm{transition}}=\frac{\pi(0)}{a(1-a)}. \]

The response follows by differentiating $a/(a+b)$. The two stationary scores are $-1/(a+b)$ and $b/[a(a+b)]$; their weighted squares give the stationary Fisher information. Only row zero of $K_\theta$ changes, and its scores are $-1/(1-a)$ and $1/a$, giving the last expression. These exact finite-state calculations supply independent numerical references; passing them does not establish the hypotheses for a continuous interacting swarm.

Proposition 305 (Euclidean horizon periodicity and its temperature interpretation)

Suppose a specified nonextremal stationary geometry admits a regular Euclidean near-horizon form

\[ ds_E^2=dr^2+\kappa^2r^2d\tau^2+g_H+\text{higher-order smooth terms}, \qquad\kappa>0. \]

Smoothness at $r=0$ requires the Euclidean time period $2\pi/\kappa$. If a separately specified quantum equilibrium state has its inverse temperature identified with this Euclidean period, then, in units $\hbar=k_B=c=1$,

\[ \beta_H=\frac{2\pi}{\kappa},\qquad T_H=\frac{\kappa}{2\pi}. \]

The identification requires the quantum thermal state and the normalization of the horizon-generating time. A classical QSD and a graph correlation length do not supply these data.

Proof

Set $\theta=\kappa\tau$. The leading normal-plane metric becomes $dr^2+r^2d\theta^2$. A small circle has circumference divided by radius equal to $\kappa$ times the period of $\tau$. A smooth plane has ratio $2\pi$, which fixes that period; another value gives a conical singularity. The temperature formula follows only after the stated thermal interpretation of the period. The quantum detector setting underlying the Unruh effect is described in Unruh’s original analysis; the geometric calculation here does not establish that detector response for an algorithmic state.

Empirical time-asymmetry contributions. Build a training-prefix whole-frame descriptor table with explicit symmetric pseudocounts and compare it with its transpose through KL contributions. — **How asymmetric is the measured forward transition table?** Build a training-prefix whole-frame descriptor table with explicit symmetric pseudocounts and compare it with its transpose through KL contributions.

Measured energy law versus fitted exponential hypothesis. Fit an exponential energy-bin hypothesis using the training mean kinetic energy. Compare it with the chronological held-out energy histogram, assigning the complete upper tail to the final bin. — **Does a simple fitted energy law predict later records?** Fit an exponential energy-bin hypothesis using the training mean kinetic energy. Compare it with the chronological held-out energy histogram, assigning the complete upper tail to the final bin.

Source tilt of retained whole-frame observations. Exponentially tilt retained whole-frame mean-position observations, compute the tilted mean and variance, and compare the derivative of the mean with that variance by finite differences. — **How does reweighting recorded field observations change their mean?** Exponentially tilt retained whole-frame mean-position observations, compute the tilted mean and variance, and compare the derivative of the mean with that variance by finite differences.

7. Constant curvature and the scope of an area representation#

The preceding results concern interaction cuts, boundary sampling, and identified statistical states. A gravitational metric is another mathematical object. We can construct an anti-de Sitter metric for a declared negative Einstein cosmological parameter, but that construction does not show that the sampled metric of a swarm has the same curvature.

Likewise, a variational area formula can follow from convergence of cut energies. Calling its coefficient a Newton constant does not identify a quantum boundary theory. The remaining results state the exact geometric construction and the additional requirements for an entropy interpretation.

Theorem 456 (Constant-curvature realization of a negative Einstein parameter)

Let $d\geq2$ and let $\Lambda_{\mathrm{eff}}<0$ be a declared parameter in the vacuum Einstein equation $R_{ab}-\tfrac12Rg_{ab}+\Lambda_{\mathrm{eff}}g_{ab}=0$. Define

\[ L_{\mathrm{AdS}}^2=-\frac{d(d-1)}{2\Lambda_{\mathrm{eff}}},\qquad ds^2=\frac{L_{\mathrm{AdS}}^2}{z^2} (dz^2+\eta_{ij}dx^idx^j),\quad z>0, \]

where $\eta$ is the $d$-dimensional Minkowski metric. This metric has constant sectional curvature $-L_{\mathrm{AdS}}^{-2}$ and solves that Einstein equation. Identifying a sampled metric with it requires a curvature and convergence estimate.

Proof

Let $\bar\eta=dz^2+\eta_{ij}dx^idx^j$ and $\omega=\log(L_{\mathrm{AdS}}/z)$, so $g=e^{2\omega}\bar\eta$. The connection is

\[ \Gamma^a_{bc}=-\frac1z (\delta^a_b\delta^z_c+\delta^a_c\delta^z_b- \bar\eta_{bc}\bar\eta^{az}). \]

Substitution into the curvature definition gives $R_{abcd}=-L_{\mathrm{AdS}}^{-2}(g_{ac}g_{bd}-g_{ad}g_{bc})$. Contracting in dimension $d+1$ yields $R_{ab}=-dL_{\mathrm{AdS}}^{-2}g_{ab}$ and $R=-d(d+1)L_{\mathrm{AdS}}^{-2}$. The Einstein expression is therefore $[d(d-1)/(2L_{\mathrm{AdS}}^2)+\Lambda_{\mathrm{eff}}]g_{ab}=0$. The declared radius makes its coefficient vanish.

Remark 300 (Pressure and the cosmological sign convention)

Write $\varkappa=8\pi G_{\mathrm{eff}}/c^4$ and define $\Lambda_P=\varkappa P$ for the specified vacuum comparison. If its vacuum stress has $T_{ab}=Pg_{ab}$, then

\[ G_{ab}+\Lambda_{\mathrm{bare}}g_{ab}=\varkappa T_{ab} \quad\Longleftrightarrow\quad G_{ab}+(\Lambda_{\mathrm{bare}}-\varkappa P)g_{ab}=0. \]

Thus this constitutive model gives $\Lambda_{\mathrm{eff}}=\Lambda_{\mathrm{bare}}-\varkappa P$. Negative mechanical pressure alone does not establish negative $\Lambda_{\mathrm{eff}}$. A different constitutive convention must be declared and checked separately. The Gaussian small-range perimeter scaling also does not imply a negative-pressure ultraviolet regime.

Theorem 457 (Variational area representation and its entropy requirement)

Suppose a family of rescaled cut energies $F_n$ on an admissible class of regions has:

Gamma-limit $F_0(A)=\alpha\operatorname{Area}(\partial A)$ with $\alpha>0$;
compactness of energy-bounded admissible sequences;
closed limiting constraints and admissible recovery sequences.

Then its minimum values converge to $\alpha$ times the least admissible boundary area, and cluster points of minimizers minimize that area. If, additionally, the entropy of a specified quantum subsystem is proved equal to the corresponding cut minimum, its limiting entropy has this area representation.

Writing $\alpha=1/(4G_{\mathrm{area}})$ parametrizes the coefficient. Identifying $G_{\mathrm{area}}$ with a gravitational coupling and the subsystem with a boundary field theory are further mathematical requirements. A nonconstant sampling density gives a weighted area in Theorem 448, not the unweighted area in this statement.

Proof

Take a subsequence of near-minimizers realizing the lower limit of the minimum values. Compactness gives a convergent further subsequence, and the liminf inequality bounds its limiting value below by the limiting functional at its admissible limit. For any admissible competitor, its recovery sequence bounds the upper limit of the minima by its limiting energy. Taking the infimum over competitors gives equality of limiting minima and identifies every minimizer cluster point. The entropy conclusion follows only by the additional stated equality with the cut functional.

The physical Ryu–Takayanagi proposal relates a boundary quantum-field entropy to a bulk minimal surface under an AdS/CFT correspondence; see Ryu and Takayanagi. The variational argument proves the area representation under the listed assumptions. It does not construct that quantum duality from the walker genealogy.

Remark 301 (Normalizations used in this chapter)

Quantity	Definition and status
$\rho$	Probability density in the sampling theorem; fixed positive spatial weight in the perimeter theorem
$N\rho$	Number density; pair integrals acquire $N^2$
$S_{N,\varepsilon}$	Raw finite Gaussian graph cut
$\mathcal P_\varepsilon$	Continuum cross-boundary pair integral
$\widehat S_N$	Cut divided by $N(N-1)\varepsilon^{d+1}$
$c_d$	Gaussian perimeter coefficient $C_0(2\pi)^{(d-1)/2}$
$a_N	\gamma_N
$K_\sigma$	Modular Hamiltonian $-\log\sigma$ of a specified reference state
$\beta$	Inverse temperature of a specified Gibbs or thermal state
$\mathcal I_L$	KL divergence of specified forward and reversed path laws, in nats
$D=\partial_\theta K_\theta$	Derivative of the actual fixed-step kernel under a declared control change
$F_{ab}$	Fisher information of a declared controlled law, in inverse parameter units squared
$\Lambda_{\mathrm{eff}}$	Declared cosmological parameter in the Einstein equation

The graph and genealogy definitions connect to The Fractal Set and Scutoid Spacetime: Moving Cells and Neighbor Changes. The geometric metric and stress conventions are developed in Emergent Geometry from Adaptive Diffusion and Field Equations and Pressure Dynamics. For statistical identification, Equilibrium Profiles and Their Analytical Characterization supplies stationary-profile residual tests, Quantitative Error Bounds separates sampling and discretization errors, and Logarithmic Sobolev inequalities and entropy convergence proves entropy estimates for the specified law and generator.

Ancestral depth of recorded events. Follow actual ancestry and persistence edges and plot event counts by maximum retained ancestral depth. Report the deepest recorded chain. — **How deep is the retained causal history of an event?** Follow actual ancestry and persistence edges and plot event counts by maximum retained ancestral depth. Report the deepest recorded chain.

Measured cut capacity by source slot. Project executed selected-edge counts to a directed capacity graph. Compute maximum flow between the selected source and sink slots, recover the minimum partition, and compare its crossing capacity with the flow. — **What is the narrowest directed connection between two slot regions?** Project executed selected-edge counts to a directed capacity graph. Compute maximum flow between the selected source and sink slots, recover the minimum partition, and compare its crossing capacity with the flow.

Quantity	Definition and status
\(\rho\)	Probability density in the sampling theorem; fixed positive spatial weight in the perimeter theorem
\(N\rho\)	Number density; pair integrals acquire \(N^2\)
\(S_{N,\varepsilon}\)	Raw finite Gaussian graph cut
\(\mathcal P_\varepsilon\)	Continuum cross-boundary pair integral
\(\widehat S_N\)	Cut divided by \(N(N-1)\varepsilon^{d+1}\)
\(c_d\)	Gaussian perimeter coefficient \(C_0(2\pi)^{(d-1)/2}\)
$a_N	\gamma_N
\(K_\sigma\)	Modular Hamiltonian \(-\log\sigma\) of a specified reference state
\(\beta\)	Inverse temperature of a specified Gibbs or thermal state
\(\mathcal I_L\)	KL divergence of specified forward and reversed path laws, in nats
\(D=\partial_\theta K_\theta\)	Derivative of the actual fixed-step kernel under a declared control change
\(F_{ab}\)	Fisher information of a declared controlled law, in inverse parameter units squared
\(\Lambda_{\mathrm{eff}}\)	Declared cosmological parameter in the Einstein equation

Algorithmic Thermodynamics, Graph Cuts, and Boundary Geometry

Contents

Algorithmic Thermodynamics, Graph Cuts, and Boundary Geometry#

TLDR#

Introduction#

1. Genealogy, interactions, and boundary observables#

2. The jump functional and modular energy#

3. Gaussian cuts and the perimeter limit#

4. Sampling the cut and comparing it with a surface#

5. First variations of cut cost, entropy, and surface energy#

6. Gibbs references, QSDs, and algorithmic response#

6.1. Thermodynamic diagnostics derived from the update kernel#

7. Constant curvature and the scope of an area representation#