Wasserstein-2 Control via Keystone-Based Variance Proxy

0. TLDR

Centered positional control under cloning: We work with phase-space \(W_2\) on \(z=(x,v)\) and the barycenter decomposition \(W_2^2(\mu_1, \mu_2) = \|\bar{z}_1 - \bar{z}_2\|^2 + W_2^2(\tilde{\mu}_1, \tilde{\mu}_2)\). For the centered positional marginals, let \(V_{\text{x,struct}} := W_{2,x}^2(\tilde{\mu}_{x,1}, \tilde{\mu}_{x,2})\) and define the variance proxy \(V_{\text{x,proxy}} := \text{Var}_x(S_1) + \text{Var}_x(S_2)\). Lemma Lemma 146 shows \(V_{\text{x,struct}} \le V_{\text{x,proxy}}\). Using the Quantitative Keystone Lemma (The Keystone Principle and the Contractive Nature of Cloning), cloning yields the N-uniform drift bound \(\mathbb{E}[\Delta V_{\text{x,proxy}}] \le -\kappa_x V_{\text{x,proxy}} + C_x\). Thus cloning gives N-uniform control of the centered positional \(W_2\) component via a contractive proxy.

No alignment axiom: The analysis avoids cross-swarm alignment assumptions. It relies on the Keystone causal chain (high error → fit/unfit signal → cloning pressure) and the positional variance drift proved in The Keystone Principle and the Contractive Nature of Cloning.

Full \(W_2\) contraction needs kinetic: Cloning does not contract the barycenter or velocity components. The kinetic operator \(\Psi_{\text{kin}}\) provides this missing contraction, so the combined dynamics yields full phase-space \(W_2\) contraction.

Explicit constants: \(\kappa_x = \frac{\chi(\varepsilon)}{4} c_{\text{struct}}\) with \(\chi(\varepsilon)=p_u(\varepsilon)c_{\text{err}}(\varepsilon)\) and \(g_{\max}(\varepsilon)=\max(p_u(\varepsilon) g_{\text{err}}(\varepsilon), \chi(\varepsilon) R_{\text{spread}}^2)\) (Section 8). All constants are N-uniform.

Dependencies: The Keystone Principle and the Contractive Nature of Cloning, Euclidean Gas: Canonical Transition and Operator Estimates

1. Introduction

1.1. Goal and Scope

The goal of this document is to prove that the cloning operator \(\Psi_{\text{clone}}\) of the Fragile Gas framework induces an N-uniform drift bound on a variance proxy that controls the centered/structural component of the Wasserstein-2 distance. A closed drift inequality for the centered positional term is obtained under an explicit structural-dominance assumption. These results bridge the finite-particle dynamics to the mean-field limit and support propagation of chaos.

The central mathematical object is the Wasserstein-2 distance \(W_2(\mu_1, \mu_2)\) between two empirical swarm distributions \(\mu_1, \mu_2\) on phase space \(z := (x, v)\), supported on \(N\) walkers. We decompose it into barycenter and centered components and control the centered positional part under cloning. Let \(\bar{z}_k := \int z \, d\mu_k = (\bar{x}_k, \bar{v}_k)\) and \(\tilde{\mu}_k := (z - \bar{z}_k)_\# \mu_k\), so that:

\[ W_2^2(\mu_1, \mu_2) = \|\bar{z}_1 - \bar{z}_2\|^2 + W_2^2(\tilde{\mu}_1, \tilde{\mu}_2)\]

Define the centered positional Wasserstein term:

\[ V_{\text{x,struct}} := W_{2,x}^2(\tilde{\mu}_{x,1}, \tilde{\mu}_{x,2})\]

where \(\tilde{\mu}_{x,k}\) is the positional marginal of \(\tilde{\mu}_k\). We also define the variance proxy:

\[ V_{\text{x,proxy}} := \text{Var}_x(S_1) + \text{Var}_x(S_2)\]

In the all-alive regime, \(\text{Var}_x(S_k) = \frac{1}{N}\sum_{i=1}^N \|\delta_{x,k,i}\|^2\). Lemma Lemma 146 shows \(V_{\text{x,struct}} \le V_{\text{x,proxy}}\). We prove that applying the cloning operator to both swarms yields a variance-proxy drift bound:

\[ \mathbb{E}[\Delta V_{\text{x,proxy}}] \leq -\kappa_x V_{\text{x,proxy}} + C_x\]

where \(\kappa_x > 0\) is N-uniform and \(C_x\) is a state-independent noise constant. This gives N-uniform control of the centered positional \(W_2\) component via a contractive proxy. By coercivity in The Keystone Principle and the Contractive Nature of Cloning, the hypocoercive structural error satisfies \(V_{\text{struct}} \geq \lambda_2 W_2^2(\tilde{\mu}_1, \tilde{\mu}_2) \geq \lambda_2 V_{\text{x,struct}}\). The barycenter term \(\|\bar{z}_1 - \bar{z}_2\|^2\) is handled by the kinetic operator.

The critical challenge is establishing N-uniformity of the drift coefficient. Previous attempts using single-walker coupling failed because they required a minimum matching probability \(q_{\min} > 0\) independent of \(N\), which is impossible for \(N!\) permutations. This document resolves this obstruction by importing the Keystone Lemma’s N-uniform constants and working with variance-level quantities that are invariant under relabeling.

The scope of this document is strictly focused on the cloning operator’s centered/structural Wasserstein control via a variance proxy (and the conditional closed drift under structural dominance). The complementary analysis of the kinetic operator \(\Psi_{\text{kin}}\), which provides contraction in the velocity and barycenter/location components, and the full convergence analysis combining both operators are addressed in companion documents. We use the framework axioms and proven results from The Keystone Principle and the Contractive Nature of Cloning (particularly Chapters 6-8 on the Keystone Principle) as foundational building blocks.

1.2. Why Wasserstein-2 Contraction Matters

Centered Wasserstein-2 control via the variance proxy under the cloning operator is not merely a technical result—it is the rigorous justification for treating the Fragile Gas as a continuum physics model and for deriving its mean-field limit.

Connection to Mean-Field Theory: The propagation of chaos framework (documented in The Discrete Population Limit and Propagation of Chaos) establishes that an N-particle system converges to a mean-field limit if its dynamics contract in Wasserstein distance with N-uniform constants. Without this property, the limiting behavior could degenerate as \(N \to \infty\), invalidating the mean-field PDE. Our result shows that the cloning operator supplies an N-uniform drift on a variance proxy that controls the centered positional \(W_2\) component; full phase-space \(W_2\) contraction follows once the kinetic operator controls the barycenter and velocity components.

Role in Convergence Theory: The Fragile Gas alternates between two operators: the cloning operator \(\Psi_{\text{clone}}\) (which we analyze here) and the kinetic operator \(\Psi_{\text{kin}}\) (analyzed in Euclidean Gas: Canonical Transition and Operator Estimates and Hypocoercivity and Convergence of the Euclidean Gas). Together, they form a hypocoercive dynamics where each operator contracts different error components:

• Cloning operator: Contracts the positional variance proxy \(V_{\text{x,proxy}}\) that bounds \(V_{\text{x,struct}}\)

• Kinetic operator: Contracts barycenter and velocity components

The Foster-Lyapunov drift analysis (Chapter 12 of The Keystone Principle and the Contractive Nature of Cloning) combines these partial contractions to prove exponential convergence to a unique quasi-stationary distribution (QSD). Our centered Wasserstein-2 proxy control provides the geometric foundation for this convergence.

Complementary to KL-Convergence: An alternative convergence analysis using Kullback-Leibler (KL) divergence and log-Sobolev inequalities (LSI) is developed in Logarithmic Sobolev inequalities and entropy convergence. The KL approach may yield faster convergence rates via entropy methods, while the Wasserstein-2 approach provides geometric intuition and explicit N-uniform constants. Both frameworks are valid and mutually reinforcing—the existence of multiple independent proofs strengthens confidence in the Fragile Gas’s stability.

Important

Why N-Uniformity is Non-Negotiable

A drift coefficient \(\kappa_x(N)\) that vanishes as \(N \to \infty\) (e.g., \(\kappa_x(N) \sim 1/N\)) would imply that large swarms contract the positional variance arbitrarily slowly. This would invalidate:

1. The mean-field limit (no well-defined continuum behavior)

2. The propagation of chaos (N-particle correlations could persist)

3. The interpretation of the Fragile Gas as a physical system with thermodynamic properties

Our Keystone-based proof establishes that \(\kappa_x\) is built from the N-uniform constants \(\chi(\varepsilon)\) and \(c_{\text{struct}}\) (The Keystone Principle and the Contractive Nature of Cloning). This validates the Fragile Gas as a scalable, physically meaningful model.

1.3. Overview of the Proof Strategy and Document Structure

The proof constructs the centered-control argument through five main stages, illustrated in the diagram below:

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        L2["Lemma"]:::lemmaStyle
        L3["Theorem"]:::theoremStyle
    end

    subgraph "Foundations (§2)"
        A["<b>§2: Cluster Structure</b><br>Target set I_k = U_k ∩ H_k<br>Complement J_k"]:::stateStyle
    end

    subgraph "Variance Analysis (§3)"
        B["<b>§3.1: Variance Decomposition</b><br>Var(S_k) = f_I Var(I_k) + f_J Var(J_k)<br>+ f_I f_J ||μ(I_k) - μ(J_k)||²"]:::lemmaStyle
        C["<b>§3.2: Centered W₂ Bound</b><br>W_{2,x}²(tilde μ_{x,1}, tilde μ_{x,2})<br>≤ Var_x(S_1) + Var_x(S_2)"]:::lemmaStyle
    end

    subgraph "Keystone Core (§4)"
        D["<b>§4.1: Quantitative Keystone Lemma</b><br>χ(ε), g_max(ε) from the 03_cloning chapter"]:::lemmaStyle
        E["<b>§4.2: Positional Variance Drift</b><br>𝔼[Δ V_{x,proxy}] ≤ -κ_x V_{x,proxy} + C_x"]:::theoremStyle
    end

    subgraph "Main Result (§5-6)"
        F["<b>§5: Centered W₂ Control</b><br>V_{x,struct} ≤ V_{x,proxy}<br>Proxy drift yields control"]:::stateStyle
        G["<b>§6: Structural/Barycenter Split</b><br>Full W₂ via kinetic + cloning"]:::theoremStyle
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    subgraph "Analysis (§7-8)"
        H["<b>§7: Comparison</b><br>No q_min; no alignment axiom"]:::stateStyle
        I["<b>§8: Explicit Constants</b><br>χ, g_max, κ_x derived"]:::stateStyle
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    A --> B
    B --> C
    D --> E
    C --> E
    E --> F
    F --> G
    G --> H
    G --> I

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Proof Architecture:

Section 2 (Cluster Structure): We recall the target set \(I_k = U_k \cap H_k(\varepsilon)\) (unfit and high-error walkers) and its complement \(J_k\), using the exact same clustering algorithm (Definition 6.3.1) and unfit set definition (Definition 7.6.1.0) from The Keystone Principle and the Contractive Nature of Cloning.

Section 3 (Variance + Centered \(W_2\) Bound): We prove the variance decomposition (Lemma 145) and show that the centered positional Wasserstein term is bounded by the sum of internal variances (Lemma 146).

Section 4 (Keystone Core): We import the Quantitative Keystone Lemma and the positional variance drift theorem from The Keystone Principle and the Contractive Nature of Cloning, yielding a direct drift bound for the variance proxy \(V_{\text{x,proxy}}\).

Section 5 (Centered \(W_2\) Control): We combine the proxy drift with the bound \(V_{\text{x,struct}} \le V_{\text{x,proxy}}\) to obtain N-uniform control of the centered positional component.

Section 6 (Full \(W_2\) Split): We combine the barycenter decomposition with the kinetic contraction results to explain how the full phase-space \(W_2\) contracts when cloning and kinetic steps are composed.

Section 7 (Comparison): We contrast the Keystone-based variance approach with the failed single-walker \(q_{\min}\) strategy.

Section 8 (Explicit Constants): We derive parameter-level expressions for \(\chi(\varepsilon)\), \(g_{\max}(\varepsilon)\), and \(\kappa_x\).

Key Proof Principles:

1. Variance proxy: Control \(V_{\text{x,struct}}\) via \(V_{\text{x,proxy}} = \text{Var}_x(S_1) + \text{Var}_x(S_2)\)

2. Keystone constants: Use \(f_{UH}(\varepsilon)\), \(p_u(\varepsilon)\), and \(\chi(\varepsilon)\) from The Keystone Principle and the Contractive Nature of Cloning (already N-uniform)

3. No cross-swarm alignment: Avoid brittle alignment assumptions and \(q_{\min}\) arguments

4. Framework consistency: Use exact definitions from the Keystone Lemma proof

The result is a rigorous, self-contained proof of centered Wasserstein-2 control via a variance proxy (with a closed drift bound under structural dominance), with explicit N-uniform constants.

2. Cluster Structure

2.1. Cluster Structure Definitions

We first recall the cluster-based partition from The Keystone Principle and the Contractive Nature of Cloning.

Definition 557 (Target Set and Complement)

For a swarm \(S_k\) with alive set \(\mathcal{A}_k\), define:

Target Set (from The Keystone Principle and the Contractive Nature of Cloning, Section 8.2):

\[ I_k(\varepsilon) := U_k \cap H_k(\varepsilon)\]

where:

• \(U_k\) is the unfit set (Definition 7.6.1.0, line 4499): walkers with fitness \(\leq\) mean

• \(H_k(\varepsilon)\) is the unified high-error set (Definition 6.3, line 2351): outlier clusters in phase space

Complement Set:

\[ J_k(\varepsilon) := \mathcal{A}_k \setminus I_k(\varepsilon)\]

Population fractions (all-alive regime, so \(|\mathcal{A}_k| = N\)):

\[ f_I(\varepsilon) := \frac{|I_k|}{N}, \quad f_J(\varepsilon) := \frac{|J_k|}{N} = 1 - f_I(\varepsilon)\]

Guaranteed lower bound (Theorem 7.6.1, line 4572):

\[ f_I(\varepsilon) \geq f_{UH}(\varepsilon) > 0 \quad \text{(N-uniform)}\]

Remark 149 (All-Alive Normalization)

The cloning operator outputs all-alive swarms, so throughout this document we work in the all-alive regime \(|\mathcal{A}_k| = N\). This keeps the empirical measure normalization consistent with the \(W_2\) formulation and aligns \(f_{UH}(\varepsilon)\) with the lower bound proven in The Keystone Principle and the Contractive Nature of Cloning (where \(k = N\) in the all-alive state).

Remark 150 (Why These Sets?)

The target set \(I_k\) represents the walkers that are:

1. Unfit (\(U_k\)): Lower than average fitness → high cloning probability

2. High-error (\(H_k\)): Geometrically outliers → contribute to structural error

By Theorem 7.6.1 (The Keystone Principle and the Contractive Nature of Cloning, Section 7.6.2), the Stability Condition guarantees a non-vanishing overlap between these sets. This is the crucial population that:

• Is targeted by the cloning mechanism (unfit)

• Causes the structural error (high-error)

The Keystone proof exploits this correctly-targeted population.

Remark 151 (Empirical Measures and Framework Properties)

Notational Precision: This document analyzes the \(N\)-particle empirical measures \(\mu_1, \mu_2\), which are discrete probability measures supported on \(N\) walkers. The clustering algorithm, fitness function \(F(x)\), and potential landscape are properties defined at the population level.

Variance Notation: \(V_{\text{struct}}\) denotes the hypocoercive structural error between centered phase-space measures (as in The Keystone Principle and the Contractive Nature of Cloning). We also use the positional structural term \(V_{\text{x,struct}} := W_{2,x}^2(\tilde{\mu}_{x,1}, \tilde{\mu}_{x,2})\) for centered positional marginals and the variance proxy \(V_{\text{x,proxy}} := \text{Var}_x(S_1) + \text{Var}_x(S_2)\). \(\text{Var}_x(S_k)\) denotes the internal positional variance of swarm \(k\).

Relationship to Continuum Limit: The fitness function \(F(x)\) and its valley structure are properties of the continuum state space \(\mathcal{X}\), while the clusters \(I_k, J_k\) are finite-sample objects constructed from the empirical distribution. The proofs in this document use properties of the limiting landscape (e.g., Confining Potential axiom, fitness valleys) to reason about finite-sample cluster behavior.

Approximation Errors: For finite \(N\), there are approximation errors \(O(1/\sqrt{N})\) when estimating continuum properties (like the potential \(F(x)\)) from empirical measures. These errors are absorbed into:

1. The noise term \(C_x = \frac{g_{\max}(\varepsilon)}{4} + 4d\delta^2\) in the variance-proxy drift inequality

2. The clustering threshold \(\varepsilon\), which depends on \(N\) implicitly through the error tolerance

N-Uniformity Justification: The key result is that these finite-sample approximation errors do not affect the sign or N-independence of the drift coefficient \(\kappa_x > 0\). This is because:

• The clustering algorithm thresholds (Definition 6.3) are calibrated to maintain \(O(1)\) cluster fractions

• The framework axioms (Confining Potential, Environmental Richness) provide \(O(1)\) landscape features that dominate the finite-sample noise

• All critical bounds (\(f_{UH}, p_u, \chi, g_{\max}\)) are proven N-uniform in The Keystone Principle and the Contractive Nature of Cloning

This remark clarifies that while the analysis is formally at the \(N\)-particle level, the use of continuum landscape properties is justified by the framework’s built-in error control mechanisms.

2.2. No Cross-Swarm Alignment Assumption

This document does not assume any cross-swarm alignment or matching axiom. All geometric guarantees are imported from the Keystone Lemma chain in The Keystone Principle and the Contractive Nature of Cloning. The only cross-swarm coupling used later is the standard independent coupling for bounding \(W_{2,x}^2\) by internal variances (Lemma Lemma 146), which requires no alignment structure.

3. Variance Decomposition and Centered Wasserstein Bound

3.1. Within-Swarm Variance Decomposition

We first establish how variance decomposes with respect to the cluster partition.

Lemma 145 (Variance Decomposition by Clusters)

For a swarm \(S_k\) partitioned into \(I_k\) (target) and \(J_k\) (complement) with population fractions \(f_I = |I_k|/N\) and \(f_J = |J_k|/N\):

\[ \text{Var}_x(S_k) = f_I \text{Var}_x(I_k) + f_J \text{Var}_x(J_k) + f_I f_J \|\mu_x(I_k) - \mu_x(J_k)\|^2\]

where:

• \(\text{Var}_x(I_k) = \frac{1}{|I_k|} \sum_{i \in I_k} \|x_i - \mu_x(I_k)\|^2\) (within-target variance)

• \(\text{Var}_x(J_k) = \frac{1}{|J_k|} \sum_{j \in J_k} \|x_j - \mu_x(J_k)\|^2\) (within-complement variance)

• \(\mu_x(I_k) = \frac{1}{|I_k|} \sum_{i \in I_k} x_i\) (target barycenter)

• \(\mu_x(J_k) = \frac{1}{|J_k|} \sum_{j \in J_k} x_j\) (complement barycenter)

Proof:

Standard variance decomposition. The total variance is:

\[ \text{Var}_x(S_k) = \frac{1}{N} \sum_{i=1}^N \|x_i - \bar{x}_k\|^2\]

where \(\bar{x}_k = \frac{1}{N}\sum_{i=1}^N x_i = f_I \mu_x(I_k) + f_J \mu_x(J_k)\).

Expand:

\[\begin{split} \begin{aligned} N \cdot \text{Var}_x(S_k) &= \sum_{i \in I_k} \|x_i - \bar{x}_k\|^2 + \sum_{j \in J_k} \|x_j - \bar{x}_k\|^2 \\ &= \sum_{i \in I_k} \|x_i - \mu_x(I_k) + \mu_x(I_k) - \bar{x}_k\|^2 + \sum_{j \in J_k} \|x_j - \mu_x(J_k) + \mu_x(J_k) - \bar{x}_k\|^2 \end{aligned}\end{split}\]

Using \(\|a + b\|^2 = \|a\|^2 + 2\langle a, b\rangle + \|b\|^2\) and \(\sum_{i \in I_k} (x_i - \mu_x(I_k)) = 0\):

\[\begin{split} \begin{aligned} &= \sum_{i \in I_k} \|x_i - \mu_x(I_k)\|^2 + |I_k| \|\mu_x(I_k) - \bar{x}_k\|^2 \\ &\quad + \sum_{j \in J_k} \|x_j - \mu_x(J_k)\|^2 + |J_k| \|\mu_x(J_k) - \bar{x}_k\|^2 \end{aligned}\end{split}\]

Now, \(\mu_x(I_k) - \bar{x}_k = \mu_x(I_k) - f_I \mu_x(I_k) - f_J \mu_x(J_k) = f_J (\mu_x(I_k) - \mu_x(J_k))\).

Similarly, \(\mu_x(J_k) - \bar{x}_k = -f_I (\mu_x(I_k) - \mu_x(J_k))\).

Therefore:

\[\begin{split} \begin{aligned} N \cdot \text{Var}_x(S_k) &= |I_k| \text{Var}_x(I_k) + |I_k| f_J^2 \|\mu_x(I_k) - \mu_x(J_k)\|^2 \\ &\quad + |J_k| \text{Var}_x(J_k) + |J_k| f_I^2 \|\mu_x(I_k) - \mu_x(J_k)\|^2 \\ &= |I_k| \text{Var}_x(I_k) + |J_k| \text{Var}_x(J_k) + (|I_k| f_J^2 + |J_k| f_I^2) \|\mu_x(I_k) - \mu_x(J_k)\|^2 \end{aligned}\end{split}\]

Using \(|I_k| = f_I N\) and \(|J_k| = f_J N\):

\[ |I_k| f_J^2 + |J_k| f_I^2 = N f_I f_J^2 + N f_J f_I^2 = N f_I f_J (f_J + f_I) = N f_I f_J\]

Dividing by \(N\) gives the result. □

3.2. Centered Positional Wasserstein Bound

We bound the centered positional Wasserstein term by the internal variances of the two swarms. This avoids any cross-swarm alignment assumptions.

Lemma 146 (Centered Positional Wasserstein Bound)

Let \(\tilde{\mu}_{x,1}\) and \(\tilde{\mu}_{x,2}\) be the centered positional empirical measures of two all-alive swarms. Then:

\[ V_{\text{x,struct}} = W_{2,x}^2(\tilde{\mu}_{x,1}, \tilde{\mu}_{x,2}) \leq \text{Var}_x(S_1) + \text{Var}_x(S_2) = V_{\text{x,proxy}}.\]

Proof.

Let \(X \sim \tilde{\mu}_{x,1}\) and \(Y \sim \tilde{\mu}_{x,2}\) be independent. Because both measures are centered, \(\mathbb{E}[X] = \mathbb{E}[Y] = 0\), so:

\[ \mathbb{E}\|X - Y\|^2 = \mathbb{E}\|X\|^2 + \mathbb{E}\|Y\|^2 = \text{Var}_x(S_1) + \text{Var}_x(S_2).\]

The independent coupling is an admissible transport plan, so the optimal transport cost is no larger than this value. □

Lemma 147 (Barycenter Decomposition of Wasserstein-2)

For two empirical measures \(\mu_1, \mu_2\) on phase space \(z = (x, v)\) with finite second moments, let \(\bar{z}_k := \int z \, d\mu_k\) and define centered measures \(\tilde{\mu}_k := (z - \bar{z}_k)_\# \mu_k\). Then:

\[ W_2^2(\mu_1, \mu_2) = \|\bar{z}_1 - \bar{z}_2\|^2 + W_2^2(\tilde{\mu}_1, \tilde{\mu}_2)\]

Proof.

For any coupling \(\pi \in \Gamma(\mu_1, \mu_2)\),

\[ \int \|z_1 - z_2\|^2 \, d\pi = \|\bar{z}_1 - \bar{z}_2\|^2 + \int \|(z_1 - \bar{z}_1) - (z_2 - \bar{z}_2)\|^2 \, d\pi\]

because the cross term vanishes by centering. The map \((z_1, z_2) \mapsto (z_1 - \bar{z}_1, z_2 - \bar{z}_2)\) is a bijection between couplings of \(\mu_1, \mu_2\) and couplings of \(\tilde{\mu}_1, \tilde{\mu}_2\), so taking the infimum yields the claim. □

Remark 152 (Interpretation of the Decomposition)

The phase-space Wasserstein-2 distance splits into:

• Barycenter term: \(\|\bar{z}_1 - \bar{z}_2\|^2\) (location + velocity mismatch)

• Centered term: \(W_2^2(\tilde{\mu}_1, \tilde{\mu}_2)\) (shape/structure mismatch)

Cloning controls the centered positional component \(V_{\text{x,struct}}\) through the variance proxy \(V_{\text{x,proxy}}\) (Lemma Lemma 146). In The Keystone Principle and the Contractive Nature of Cloning, the structural error satisfies \(V_{\text{struct}} \geq \lambda_2 W_2^2(\tilde{\mu}_1, \tilde{\mu}_2) \geq \lambda_2 V_{\text{x,struct}}\) for an N-uniform \(\lambda_2 > 0\), so proxy control yields N-uniform control of a centered component of phase-space \(W_2\). The barycenter and velocity components are handled by the kinetic operator.

Remark 153 (Structural-Dominance Regime (Optional))

If the barycenter term is already controlled, for example if there exists an N-uniform \(c_{\text{dom}} > 0\) such that

\[ \|\bar{z}_1 - \bar{z}_2\|^2 \leq c_{\text{dom}} V_{\text{x,struct}},\]

then the proxy control in Section 5 combines with the decomposition above to yield geometric \(W_2\) contraction. In practice this regime is obtained after composing with \(\Psi_{\text{kin}}\) (see Hypocoercivity and Convergence of the Euclidean Gas and Convergence, Survival, and Parameter Dependence).

INTERACTIVE EXPERIMENT · II-03

Optimal transport of observed coordinate marginals

How far apart are the coordinate distributions of two executed gas runs? Sorted matching gives the exact equal-mass one-dimensional quadratic transport cost.

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4. Keystone-Driven Positional Variance Contraction

We now import the Keystone Lemma and the positional variance drift bound from The Keystone Principle and the Contractive Nature of Cloning. These results provide the N-uniform contraction mechanism used in this document.

4.1. Quantitative Keystone Lemma (Recall)

Lemma 148 (N-Uniform Quantitative Keystone Lemma (Positional Component))

Under the foundational axioms of The Keystone Principle and the Contractive Nature of Cloning, there exist \(R^2_{\text{spread}} > 0\), \(\chi(\varepsilon) > 0\), and \(g_{\max}(\varepsilon) \ge 0\), all independent of \(N\), such that for any pair of swarms \((S_1, S_2)\):

\[ \frac{1}{N}\sum_{i \in I_{11}} (p_{1,i} + p_{2,i})\|\Delta\delta_{x,i}\|^2 \ge \chi(\varepsilon) V_{\text{struct}} - g_{\max}(\varepsilon)\]

This is Lemma 8.1.1 in The Keystone Principle and the Contractive Nature of Cloning (Lemma 127).

4.2. Positional Variance Drift

Theorem 193 (Positional Variance Proxy Drift)

Define the variance proxy \(V_{\text{x,proxy}} := \text{Var}_x(S_1) + \text{Var}_x(S_2)\). In the all-alive regime, \(V_{\text{x,proxy}}\) agrees with the \(N\)-normalized variance component \(V_{\text{Var},x}(S_1) + V_{\text{Var},x}(S_2)\) from The Keystone Principle and the Contractive Nature of Cloning, and the cloning operator satisfies:

\[ \mathbb{E}[\Delta V_{\text{x,proxy}}] \leq -\kappa_x V_{\text{x,proxy}} + C_x\]

with N-uniform \(\kappa_x = \frac{\chi(\varepsilon)}{4} c_{\text{struct}}\) and \(C_x = \frac{g_{\max}(\varepsilon)}{4} + C_{\text{jitter}}\). Here \(c_{\text{struct}} > 0\) is the structural-variance link constant from The Keystone Principle and the Contractive Nature of Cloning (Section 10.3.6), and \(C_{\text{jitter}} = 4 d \delta^2\) is a conservative bound from the positional cloning jitter.

Reference: This is a direct restatement of The Keystone Principle and the Contractive Nature of Cloning, Theorem 10.3.1 (Theorem 180), specialized to the all-alive regime.

Remark 154 (Jitter Scale Convention)

\(\delta\) is the positional jitter scale in the cloning update. In the Euclidean Gas implementation, one typically sets \(\delta = \sigma_x\) (or \(\delta = \sqrt{\tau}\,\sigma_x\) for a discretized step), but the analysis keeps \(\delta\) explicit.

5. From Variance Contraction to Centered \(W_2\) Control

We now combine the proxy drift with the centered Wasserstein bound.

Proposition 142 (Centered Positional Control via Variance Proxy)

Under the conditions of Theorem Theorem 193, the centered positional Wasserstein term satisfies:

\[ \mathbb{E}\left[V_{\text{x,struct}}(S_1', S_2')\right] \leq (1 - \kappa_x) V_{\text{x,proxy}}(S_1, S_2) + C_x.\]

Proof. By Lemma Lemma 146, \(V_{\text{x,struct}} \le V_{\text{x,proxy}}\). Apply Theorem Theorem 193 and take expectations. □

Remark 155 (Closed Drift for \(V_{\text{x,struct}}\))

Without additional alignment structure, the bound above is one-sided: it controls \(V_{\text{x,struct}}\) by a contractive proxy but does not produce a closed drift inequality in \(V_{\text{x,struct}}\) alone. A regime-specific dominance assumption (Assumption Assumption 3) yields a closed drift bound (Corollary Corollary 65). This additional assumption is not required for the main control result.

Assumption 3 (Structural-Dominance Regime (Positional))

There exists an N-uniform constant \(c_{\text{proxy}} \ge 1\) such that, at the times of interest,

\[ V_{\text{x,proxy}} \le c_{\text{proxy}} V_{\text{x,struct}}. \]

Interpretation: the centered shape mismatch dominates the internal variance. This is a high-mismatch regime; it typically fails when the swarms are already nearly aligned.

Remark 156 (Sufficient Geometric Condition)

If the centered supports satisfy a separation condition, the dominance constant can be made explicit. Suppose both centered supports are contained in a ball of radius \(R\) (e.g., \(R \le D_{\text{valid}}\)) and have minimal separation \(\operatorname{dist}(\operatorname{supp}\tilde{\mu}_{x,1}, \operatorname{supp}\tilde{\mu}_{x,2}) \ge D > 0\). Then \(\text{Var}_x(S_k) \le R^2\) and \(V_{\text{x,struct}} \ge D^2\), so

\[ V_{\text{x,proxy}} \le 2 R^2 \le \frac{2 R^2}{D^2} V_{\text{x,struct}}. \]

Thus the assumption holds with \(c_{\text{proxy}} = 2 (R/D)^2\). This illustrates that the dominance regime corresponds to strong shape mismatch (large \(D\) relative to \(R\)).

Corollary 65 (Closed Drift Under Structural Dominance)

Assume Assumption 3 and Theorem Theorem 193. Then:

\[ \mathbb{E}[\Delta V_{\text{x,struct}}] \le -\kappa_{\text{eff}} V_{\text{x,struct}} + C_x, \qquad \kappa_{\text{eff}} := 1 - (1-\kappa_x) c_{\text{proxy}}. \]

In particular, if \(c_{\text{proxy}} < 1/(1-\kappa_x)\), then \(\kappa_{\text{eff}} > 0\) and the centered positional error contracts geometrically. The correction term is linear in \(V_{\text{x,struct}}\), so larger mismatch yields stronger expected correction. When the mismatch is small and the dominance condition fails, the kinetic step provides the remaining contraction.

INTERACTIVE EXPERIMENT · II-04

Compare measured transport with an independent coupling

How loose is the independent coupling cost for two measured clouds? The independent coupling cost bounds the one-dimensional optimal transport cost.

Load experiment Open full view ↗

Rust engine experiment · Seed 7 · Poster after 7 experiment steps. Interactive view starts from the same seed.

6. Full \(W_2\) Contraction After the Kinetic Step

The full phase-space \(W_2\) contraction is obtained by combining the centered control above with the kinetic operator’s barycenter and velocity contraction.

Theorem 194 (Structural/Barycenter Split for Full \(W_2\))

Let \(\mu_1, \mu_2\) be the empirical phase-space measures of two swarms. Then:

\[ W_2^2(\mu_1, \mu_2) = \|\bar{z}_1 - \bar{z}_2\|^2 + W_2^2(\tilde{\mu}_1, \tilde{\mu}_2).\]

Cloning controls the centered positional component via Proposition Proposition 142. The kinetic operator \(\Psi_{\text{kin}}\) contracts the barycenter and velocity components (Hypocoercivity and Convergence of the Euclidean Gas). Therefore the composed dynamics \(\Psi_{\text{kin}} \circ \Psi_{\text{clone}}\) yields full phase-space \(W_2\) contraction as in Convergence, Survival, and Parameter Dependence.

7. Comparison with Single-Walker Approach

7.1. Why Single-Walker Approach Failed

Original approach (in previous version):

Track individual pairs (i, π(i))
Need: min probability q_min over all matchings
Problem: q_min ~ 1/(N!) → 0 as N → ∞
Result: N-uniformity BROKEN

Keystone approach (this document):

Track variance proxy V_{x,proxy} = Var_x(S_1) + Var_x(S_2)
Use: Keystone constants χ(ε), g_max(ε) to get drift
No cross-swarm alignment assumptions required
Result: N-uniformity PRESERVED

7.2. Advantages Summary

Aspect	Single-Walker	Keystone-Based
Coupling	Individual matching with q_min	Variance proxy; no matching requirement
Geometry	Per-walker alignment (brittle)	No cross-swarm alignment needed
Proof method	Dynamic (survival probability)	Keystone lemma + variance drift
N-uniformity	BROKEN (q_min → 0)	✓ PROVEN (Keystone constants from The Keystone Principle and the Contractive Nature of Cloning)
Framework consistency	Ad-hoc definitions	Uses exact definitions from Chapters 6-8

8. Explicit Constants and Derived Bounds

8.1. Contraction Constant Components

We express each constant in terms of framework parameters and explicit bounds from The Keystone Principle and the Contractive Nature of Cloning.

1. High-error fraction (Chapter 6):

\[ f_H(\varepsilon) := \min\left(f_O,\; f_{H,\text{cluster}}(\varepsilon)\right) \]

with

\[ f_O = \frac{(1-\varepsilon_O) R_h^2}{D_h^2}, \qquad f_{H,\text{cluster}}(\varepsilon) = \frac{(1-\varepsilon_O)\left(R^2_{\text{var}} - (D_{\text{diam}}(\varepsilon)/2)^2\right)}{D_{\text{valid}}^2}, \]

and \(D_{\text{diam}}(\varepsilon) = c_d \varepsilon\). Here \(D_h^2 := D_x^2 + \lambda_v D_v^2\) is the hypocoercive diameter.

2. Stability-gap margin (Theorem 7.5.2.4, The Keystone Principle and the Contractive Nature of Cloning):

\[ \Delta_{\log}(\varepsilon) := \beta \ln\left(1 + \frac{\kappa_{d',\text{mean}}(\varepsilon)}{g_{A,\max}+\eta}\right) - \alpha \ln\left(1 + \frac{\kappa_{\mathrm{rescaled}}(L_R D_{\text{valid}})}{\eta}\right), \qquad \Delta_{\log}(\varepsilon) > 0 \]

which implies a mean fitness gap

\[ \Delta_{\text{fit}}(\varepsilon) \ge V_{\text{pot,min}}\left(e^{\Delta_{\log}(\varepsilon)} - 1\right). \]

3. Unfit-high-error overlap (explicit conservative bound):

\[ f_{UH}(\varepsilon) \ge f_H(\varepsilon) \cdot \frac{\Delta_{\text{fit}}(\varepsilon)}{V_{\text{pot,max}} - V_{\text{pot,min}}} \]

with \(V_{\text{pot,min}} = \eta^{\alpha+\beta}\) and \(V_{\text{pot,max}} = (g_{A,\max} + \eta)^{\alpha+\beta}\).

4. Unfit fraction (Lemma 7.6.1.1, The Keystone Principle and the Contractive Nature of Cloning):

\[ f_U(\varepsilon) = \frac{\kappa_{V,\text{gap}}(\varepsilon)}{2\left(V_{\text{pot,max}} - V_{\text{pot,min}}\right)}. \]

5. Cloning pressure (Lemma 8.3.2 / Section 8.6.1.1, The Keystone Principle and the Contractive Nature of Cloning):

\[ p_u(\varepsilon) = \min\left(1,\; \frac{1}{p_{\max}} \cdot \frac{\Delta_{\min}(\varepsilon, f_U, f_F, k)}{V_{\text{pot,max}} + \varepsilon_{\text{clone}}}\right), \qquad \Delta_{\min}(\varepsilon, f_U, f_F, k) := \frac{f_F f_U}{(k-1)(f_F + f_U^2/f_F)} \kappa_{V,\text{gap}}(\varepsilon). \]

The N-uniform lower bound implied by Theorem 8.7.1 in The Keystone Principle and the Contractive Nature of Cloning is used throughout this document. In the all-alive regime used here, \(k = N\).

6. High-error concentration constant (Lemma 8.4.1 in The Keystone Principle and the Contractive Nature of Cloning):

\[ c_H(\varepsilon) := \min\left\{1-\varepsilon_O, \frac{(1-\varepsilon_O)\left(R^2_{\text{var}} - (D_{\text{diam}}(\varepsilon)/2)^2\right)}{R^2_{\text{var}}}\right\}, \qquad c_{\text{err}}(\varepsilon) = \frac{c_H(\varepsilon)}{4}. \]

7. Error offset (Lemma 8.4.1 in The Keystone Principle and the Contractive Nature of Cloning):

\[ g_{\text{err}}(\varepsilon) = \left(\frac{c_H(\varepsilon)}{2} + 5\right) D_{\text{valid}}^2. \]

8. Keystone feedback coefficient:

\[ \chi(\varepsilon) = p_u(\varepsilon) \cdot c_{\text{err}}(\varepsilon). \]

9. Keystone offset (Section 8.6.2, The Keystone Principle and the Contractive Nature of Cloning):

\[ g_{\max}(\varepsilon) = \max\left(p_u(\varepsilon) \cdot g_{\text{err}}(\varepsilon),\; \chi(\varepsilon) R_{\text{spread}}^2\right). \]

10. Positional variance drift constants (Theorem 10.3.1, The Keystone Principle and the Contractive Nature of Cloning):

\[ \kappa_x = \frac{\chi(\varepsilon)}{4} c_{\text{struct}}, \qquad C_x = \frac{g_{\max}(\varepsilon)}{4} + 4 d \delta^2. \]

Here \(c_{\text{struct}} > 0\) is the structural-variance link constant from The Keystone Principle and the Contractive Nature of Cloning (Section 10.3.6); in balanced all-alive regimes, \(c_{\text{struct}} = 1/2\) is a conservative choice.

8.2. Convergence Rate (Proxy)

The variance proxy satisfies geometric decay:

\[ \mathbb{E}[V_{\text{x,proxy}}(t)] \leq (1 - \kappa_x)^t V_{\text{x,proxy}}(0) + \frac{C_x}{\kappa_x}. \]

By Lemma Lemma 146, this yields:

\[ \mathbb{E}[V_{\text{x,struct}}(t)] \leq \mathbb{E}[V_{\text{x,proxy}}(t)]. \]

Full phase-space \(W_2\) contraction follows after composing with the kinetic operator, which contracts the barycenter and velocity components (Hypocoercivity and Convergence of the Euclidean Gas, Convergence, Survival, and Parameter Dependence).

8.3. Comparison with KL-Convergence

The KL-convergence framework (Logarithmic Sobolev inequalities and entropy convergence) may provide faster convergence rates via entropy methods. The centered/structural Wasserstein-2 control proven here is complementary:

• Centered positional \(W_2\) control (via variance proxy): Geometric proxy decay, explicit constants, suitable for mean-field limit

• KL contraction: Entropy-based, potentially faster, uses LSI theory

Both approaches are valid; the Keystone-based variance proxy control provides an independent verification of convergence with explicit N-uniform constants.

9. Conclusion and Future Work

9.1. Main Achievements

This document establishes centered positional \(W_2\) control for the cloning operator using a Keystone-based variance proxy that:

1. ✅ Avoids q_min problem: No dependence on minimum matching probability

2. ✅ Leverages Keystone bounds: Constants sourced from The Keystone Principle and the Contractive Nature of Cloning Chapters 6-8

3. ✅ No alignment axiom: No cross-swarm geometric alignment assumptions required

4. ✅ N-uniform throughout: All constants independent of N

5. ✅ Framework-consistent: Uses exact cluster definitions from the Keystone Lemma chain

9.2. Open Questions

1. Optimal constants: Can \(\chi(\varepsilon)\) or \(c_{\text{struct}}\) be tightened to improve \(\kappa_x\)?

2. Closed drift for \(V_{\text{x,struct}}\): Identify regimes where \(V_{\text{x,proxy}} \le c_{\text{proxy}} V_{\text{x,struct}}\) holds.

3. Adaptive extensions: How does viscous coupling affect the proxy decay rate?

4. Numerical validation: Swarm simulations to verify proxy decay and combined kinetic+cloning \(W_2\) contraction

9.3. Relation to Framework

This result enables:

• Propagation of Chaos (The Discrete Population Limit and Propagation of Chaos): N-particle system → mean-field limit

• Mean-Field Convergence (The Mean-Field Law of the Euclidean Gas): Measure-level contraction

• Combined with kinetic contraction: Full Wasserstein contraction for alternating operator \(\Psi_{\text{kin}} \circ \Psi_{\text{clone}}\)

References

Primary: The Keystone Principle and the Contractive Nature of Cloning Chapters 6-8 (Keystone Principle) and Chapter 10 (variance drift)

Key Results Used:

• Definition 6.3 (line 2351): Unified High-Error and Low-Error Sets

• Definition 7.6.1.0 (line 4499): Unfit Set

• Theorem 7.6.1 (line 4572): Unfit-High-Error Overlap (f_UH > 0)

• Lemma 8.3.2 (line 4881): Cloning Pressure on Unfit Set (p_u > 0)

• Lemma 8.4.1: Error Concentration in the Target Set (\(c_{\text{err}}, g_{\text{err}}\))

• Lemma 8.1.1: Quantitative Keystone Lemma (\(\chi, g_{\max}\))

• Theorem 10.3.1: Positional Variance Contraction

• Theorem 7.5.2.4: Stability Condition (fitness ordering)

• Theorem 8.7.1 (line 5521): N-Uniformity of Keystone Constants

Secondary:

• Mathematical Foundations of Fragile: Axioms

• Logarithmic Sobolev inequalities and entropy convergence: Alternative convergence analysis

Document Status: COMPLETE (Keystone-Based Proxy Control)

Next Steps: Numerical validation + comparison with KL-convergence rates