Euclidean Gas: Canonical Transition and Operator Estimates

Euclidean Gas: Canonical Transition and Operator Estimates#

0. TLDR#

The Euclidean Gas advances a complete marked population through one fixed sequence: sampled measurements, frozen acceptance decisions, connected-component collisions, BAOAB, position diffusion, smooth velocity compression, and terminal boundary classification. One shared orthogonal matrix rotates all relative velocities in a component. This gives exact component momentum conservation and multiplies its relative kinetic energy by the square of the restitution coefficient.

The comparison coordinates are bounded; the physical coordinates are retained. Dead slots use those retained coordinates when choosing revival donors and entering collisions. The primary population description is the discrete map $\mu_{n+1}=\mathcal F_h(\mu_n)$ derived in The Mean-Field Law of the Euclidean Gas.

1. Introduction#

Imagine stopping a simulation just before an update and writing down every random choice it is about to make. That list is the algorithm’s transition law. A theory of this gas must average that same list: which companion was measured, which donor was selected, which edge was accepted, and which other walkers joined the collision.

Two details make a large difference. First, acceptance acts on sampled fitness. An average measurement substituted before the nonlinear gate changes the law. Second, an accepted recipient can itself be a donor. The connected components of all accepted edges specify the collision groups without overlapping writes. A shared rotation preserves the sum of relative velocities; independent rotations do not.

We define the complete transition, prove its component identities, and derive kinetic bounds directly from its BAOAB stages. The final position has an exact Gaussian conditional law, which makes terminal survival calculations particularly transparent. The complete finite-population kernel is permutation equivariant and Feller. The population and chaos proofs are in The Mean-Field Law of the Euclidean Gas and The Discrete Population Limit and Propagation of Chaos.

        flowchart LR
    S["Frozen marked swarm"] --> M["Sample measurements and retain fitness"]
    M --> G["Draw donors and acceptance gates"]
    G --> C["Accepted-edge components"]
    C --> J["Frozen position copies and jitter"]
    C --> R["One shared orthogonal matrix per component"]
    J --> K["BAOAB"]
    R --> K
    K --> X["Position diffusion"]
    X --> V["Smooth velocity cap"]
    V --> A["Terminal alive/dead marks"]

2. Physical states and comparison geometry#

A bounded comparison feature is a way of measuring similarity, not a wall in physical space. Two distant positions may look similar after squashing, yet the force and the boundary still use their actual positions. We therefore keep both notions visible: the physical position–velocity metric for kinetic estimates, and the squashed metric for companion weights and bounded measurements.

The framework’s deterministic aggregation estimates can be used when their stated input bounds hold. Its independent-output arguments cannot be applied to the shared component collision. Our kernel proof below conditions on the whole accepted graph, and the population proof controls the graph seen by a tagged walker.

3. Definition: the Euclidean Gas#

A Euclidean Gas is the Fragile Swarm $\mathcal F_{\text{EG}}$ given by the tuple of environmental structures, parameters, operators, and noise measures below. It induces a Markov chain on the swarm state space via the Fragile Gas Algorithm $\mathcal{S}_{t+1}\!\sim\!\Psi_{\mathcal F_{\text{EG}}}(\mathcal S_t,\cdot)$ (Def. Fragile Gas Algorithm (Definition 483)). The canonical measurement pipeline uses global population variance with a positive quadratic regularization floor, followed by positive logistic rescaling. The detailed formulas below distinguish this choice from the explicitly scoped auxiliary standardization estimates.

3.1 Euclidean Gas algorithm (canonical pipeline)#

Freeze a photograph of the swarm at the beginning of the step. Every measurement, donor choice, and acceptance decision refers to this photograph. Draw an edge from each accepted recipient to its donor. If two collision groups overlap, they are one connected group: a walker has exactly one velocity destination. Rotate the whole group’s relative velocities with the same matrix. The vectors then still add to zero, which is the reason momentum is conserved.

After cloning, advance all slots through the kinetic stages and check the boundary at the end. A revived slot joins the collision using its retained input velocity. Its position comes from its selected donor, but its momentum has not been replaced by a second copy of the donor’s momentum.

Algorithm 3 (Euclidean Gas Update)

Given the complete marked swarm $S=((x_i,v_i,a_i))_{i=1}^N$, retain all coordinates, including those of dead slots.

If $M=\sum_i a_i=0$, return $S$ unchanged and stop the run.
Each alive row draws one measurement companion from the normalized Gaussian law on squashed phase-space distance, excluding itself when another alive row exists. Measure reward and the sampled separation, including the positive diversity floor, as specified in 6.2 Stage 2 — Single-shot measurement and frozen potentials.
Compute alive-population means and population variances. Standardize each channel using $\sqrt{\operatorname{Var}+\sigma_{\min}^2}$, apply its positive logistic map, and form the fitness product. Retain every realized sampled fitness throughout the cloning stage.
Each row draws one current cloning donor from the same distance-weighted law. For alive rows, accept with probability $\min\{1,[(V_{\mathrm{fit},j}-V_{\mathrm{fit},i})/(p_{\max}(V_{\mathrm{fit},i}+\varepsilon_{\mathrm{clone}}))]_+\}$. Accept every dead row whenever $M>0$. An alive singleton has no distinct donor and does not clone.
Form the undirected graph of accepted recipient–donor edges and compute its connected components. Copy accepted recipient positions from their frozen donors and add independent Gaussian jitter $\sigma_{\mathrm{clone}}\zeta_i$. For each nontrivial component $C$, independently draw $R_C$ from normalized Haar measure on $O(d)$ and set $$ \bar v_C=\frac1{|C|}\sum_{i\in C}v_i,\qquad \widetilde v_i=\bar v_C+\alpha_{\mathrm{restitution}}R_C(v_i-\bar v_C). $$ Uninvolved rows keep their input position and velocity. A donor’s position changes only if that donor is itself an accepted recipient. Every slot is alive after this stage.
Apply the BAOAB map of Definition 485 to each intermediate row, then independent Gaussian position diffusion and the smooth radial velocity cap. Use fresh independent kinetic noises across rows.
Apply terminal boundary classification once. Retain the positions and velocities of every terminally dead row in the output.

All coordinates used in Steps 2–5 are frozen input coordinates. In particular, velocity updates include donors and revived recipients, and no updated donor coordinate feeds another recipient during the same step.

Executed operator stages. B changes velocity, A changes position, and O damps velocity while adding Gaussian innovation. The final B uses the post-A position. — **At which positions are the two force evaluations made?** B changes velocity, A changes position, and O damps velocity while adding Gaussian innovation. The final B uses the post-A position.

3.2 Compiled configuration and experiment contract#

Definition 484 (Canonical Rust configuration)

GasConfig::euclidean(d, h) in the Rust algorithmic-gas library selects current-frame independent companions, Gaussian width $\epsilon_D=\epsilon_C=2$, squashing radii $R_x=R_v=2$, and phase-space weight $\lambda_v=1$. Both standardization floors are $0.1$, both logistic maps are $g(z)=2/(1+e^{-z})+0.1$, and both fitness exponents are $1$. The separation floor is $10^{-3}$; clone saturation is $p_{\max}=1$ and acceptance denominator floor is $10^{-6}$.

The collision has $\alpha_{\mathrm{restitution}}=0.5$ and $\sigma_{\mathrm{clone}}=0.1$. BAOAB uses unit mass, friction $1$, velocity diffusion factor $1$, final position diffusion $\sigma_x=0.1$, and velocity radius $V_{\mathrm{alg}}=2$. The absorbing box is $[-2,2]^d$, with terminal-only boundary classification. The caller supplies the objective and its potential gradient; the benchmark experiments use an explicit quadratic objective. Rust supports $1\le d\le256$ for the shared orthogonal sampler.

The mathematical formulas describe the real-arithmetic transition. Each interactive experiment executes the Rust engine and records its configuration, seed, accepted edges, component rotations, stage outputs, and terminal status. They do not run a separate Python trajectory implementation.

Uniform companions, uncapped kinetics, substep absorption, direct-copy cloning, historical donors, and additional force terms are separately configured library extensions. Statements for the canonical configuration apply to an extension only after its changed transition and proof hypotheses have been checked.

3.3 Position–velocity foundations and projection (Sasaki metric)#

Physical position space is $\mathbb R^d$. The canonical absorbing domain $D=\mathcal X_{\mathrm{valid}}$ is a closed box with nonempty interior. Its boundary has Lebesgue measure zero; smooth boundary is unnecessary here. Dead coordinates and intermediate Gaussian perturbations can lie outside $D$. The unbounded configuration takes $D=\mathbb R^d$ and requires physical moment and tail estimates separately.
Velocity radius $V_{\mathrm{alg}}\in(0,\infty)$ and velocity cap $\mathcal V_{\mathrm{alg}}:=\{v\in\mathbb R^d:\|v\|\le V_{\mathrm{alg}}\}$.
Positional radius $R_x\in(0,\infty)$, which sets the characteristic scale of the bounded algorithmic position space.
Walker state $w_i=(x_i,v_i,s_i)\in\mathcal X\times\mathbb R^d\times\{0,1\}$ collects position, velocity, and status.
Algorithmic space and Sasaki metric $(\mathcal Y,d_{\mathcal Y}^{\mathrm{Sasaki}})$ where the algorithmic space is the (Definition 446) closure of the projection image,

\[ \mathcal Y\;:=\;\overline{B(0,R_x)}\times\overline{B(0,V_{\mathrm{alg}})}\subset\mathbb R^d\times\mathbb R^d,\]

endowed with the Sasaki metric

\[ d_{\mathcal Y}^{\mathrm{Sasaki}}\bigl((y_x,y_v),(y'_x,y'_v)\bigr)^2:=\|y_x-y'_x\|^2+\lambda_v\|y_v-y'_v\|^2\]

for some fixed weight $\lambda_v>0$. For physical states we write $y=\varphi(x,v)$ and $y'=\varphi(x',v')$; the metric therefore measures differences between squashed coordinates. The compactness of $\mathcal Y$ ensures a finite algorithmic diameter.
Projection $\varphi:\mathbb R^d\times\mathbb R^d\to B(0,R_x)\times B(0,V_{\mathrm{alg}})$ given by $\varphi(x,v)=(\psi_x(x),\psi_v(v))$ with the smooth squashing maps

\[ \psi_x(x)\ :=\ R_x\,\frac{x}{R_x+\|x\|},\qquad \psi_v(v)\ :=\ V_{\mathrm{alg}}\,\frac{v}{V_{\mathrm{alg}}+\|v\|}.\]

Design Note

Each squashing map is $C^1$ globally and $C^\infty$ away from the origin. Its image is the open ball, so the final smooth cap does not create an atom on the velocity sphere. Minorization and any further continuum identification must use this particular map.

The projection $\varphi$ maps the physical state space $\mathbb R^d\times\mathbb R^d$ into the bounded product $B(0,R_x)\times B(0,V_{\mathrm{alg}})$. Its image has compact closure $\mathcal Y$, so the Axiom of Bounded Algorithmic Diameter (Axiom 23) holds by construction. Lemma Lemma 73 shows that each squashing map is $1$-Lipschitz, and Lemma Lemma 74 extends this to $\varphi$ under the Sasaki metric.
Algorithmic distance for companion selection (Definition 448). For intra-swarm measurements (companion selection for diversity and cloning), the algorithm uses the algorithmic distance between two walkers $i$ and $j$:

\[ d_{\text{alg}}(i,j)^2 := \|\psi_x(x_i)-\psi_x(x_j)\|^2 + \lambda_{\text{alg}}\|\psi_v(v_i)-\psi_v(v_j)\|^2\]

where $\lambda_{\text{alg}}=\lambda_v$ in the canonical configuration. Thus this is exactly the Sasaki distance between squashed features. The physical phase-space metric instead uses unsquashed coordinates; the two metrics have different quantitative bounds.
Reward $R:\mathcal X_{\mathrm{valid}}\times\mathcal V_{\mathrm{alg}}\to\mathbb R$ couples the position potential with a kinetic regularizer:

\[ R(x,v):=R_{\mathrm{pos}}(x)-\lambda_{\mathrm{vel}}\|v\|^2,\]

where $R_{\mathrm{pos}}=-U$ for a minimized objective $U$. The canonical preset permits $\lambda_{\mathrm{vel}}=0$; a positive kinetic regularizer is an additional objective choice. The acceleration $F=-\nabla U$ must be defined on all of $\mathbb R^d$, because jitter and BAOAB drift can leave $D$ before the terminal check. For the estimates below assume $F$ is globally Lipschitz with constant $L_F$, so $\|F(x)\|\le B_F+L_F\|x\|$, where $B_F=\|F(0)\|$. Bounded comparison features do not supply physical confinement.

Lemma 73 (Properties of smooth radial squashing maps)

For any constant $C>0$ define $\psi_C: \mathbb R^d\to B(0,C)$ by $\psi_C(z):=C\,z/(C+\|z\|)$. The map $\psi_C$ satisfies:

$\psi_C$ is $1$-Lipschitz on $\mathbb R^d$.
$\psi_C\in C^{\infty}(\mathbb R^d\setminus\{0\})$.
$\psi_C(\mathbb R^d)\subset B(0,C)$.

Both the positional squashing map $\psi_x$ and the velocity squashing map $\psi_v$ are obtained by setting $C=R_x$ and $C=V_{\mathrm{alg}}$, respectively, so they inherit the 1-Lipschitz and smoothness properties of Lemma Lemma 73.

Lemma 74 (Lipschitz continuity of the projection $\varphi$)

For $(x,v),(x',v')\in\mathbb R^d\times\mathbb R^d$ the projection $\varphi(x,v)=(\psi_x(x),\psi_v(v))$ satisfies

\[ d_{\mathcal Y}^{\mathrm{Sasaki}}\bigl(\varphi(x,v),\varphi(x',v')\bigr)\le\sqrt{\|x-x'\|^2+\lambda_v\|v-v'\|^2}.\]

That is, $\varphi$ is $1$-Lipschitz when both domain and codomain carry the Sasaki metric with the same weight $\lambda_v$.

This bound compares projected and physical displacements. The physical kinetic moment estimate is proved in Lemma 80.

The projection is injective on finite coordinates and gives bounded comparison features. Its inverse is not uniformly Lipschitz near the boundary of the feature image. Consequently physical moment estimates and local changes of metric remain separate parts of the analysis.

3.4 Swarm distance and canonical operators#

We measure dispersion in the Sasaki metric and retain the canonical aggregation pipeline:

Dispersion distance. For swarms $\mathcal S_1,\mathcal S_2$ write

\[ d_{\mathrm{Disp},\mathcal Y}^{\mathrm{Sasaki}}(\mathcal S_1,\mathcal S_2)^2 := \frac{1}{N}\sum_{i=1}^{N} d_{\mathcal Y}^{\mathrm{Sasaki}}\big(\varphi(x_{1,i},v_{1,i}),\varphi(x_{2,i},v_{2,i})\big)^2 + \frac{\lambda_{\mathrm{status}}}{N}\sum_{i=1}^{N}(s_{1,i}-s_{2,i})^2,\]

with status penalty $\lambda_{\mathrm{status}}>0$ as in the canonical framework. Because the Sasaki metric adds a velocity term, Section 4.3 re-validates every deterministic Lipschitz bound against $d_{\mathrm{Disp},\mathcal Y}^{\mathrm{Sasaki}}$.
Distinction: Algorithmic Distance vs. Sasaki Metric

It is critical to distinguish two different distance metrics used in this document:
1. Algorithmic distance (Definition 448) $d_{\text{alg}}(i,j)$: Used by the algorithm itself for intra-swarm companion selection (diversity measurement and cloning). This defines how the algorithm “perceives” proximity between walkers within the same swarm.
2. Sasaki metric $d_{\mathcal Y}^{\mathrm{Sasaki}}$: Used by the analysis to measure inter-swarm dispersion and derive continuity bounds. This is an analytical tool for proving convergence properties.
For the Euclidean Gas, we set $\lambda_{\text{alg}} = \lambda_v$ so that these metrics coincide in their functional form, simplifying the connection between algorithmic behavior and analytical properties. However, they serve conceptually different roles: the algorithmic distance is intrinsic to the algorithm’s design, while the Sasaki metric is extrinsic to the convergence analysis.
Walkers: $N\ge1$, with the explicit singleton convention; the empirical reward and distance aggregators (Definition 449) keep their canonical formulas. Lemma Lemma 87 supplies Sasaki-specific error moduli, and Lemma Lemma 95 applies them to the regularized standard deviation and logistic rescale operators.
Dynamics weights: $\alpha,\beta\ge 0$ with $\alpha+\beta>0$. The preset takes both exponents equal to one; any stronger amplification inequality used by a convergence theorem is an additional condition.

Reward and mechanical motion. Each replica executes the chosen reward and force configuration; endpoint means and recorded reward traces measure their combined effect. — **How do separate reward and force choices affect independent gas runs?** Each replica executes the chosen reward and force configuration; endpoint means and recorded reward traces measure their combined effect.

3.5 Kinetic Langevin perturbations with velocity capping#

Keep track of where the force is evaluated. The first half-kick uses the post-cloning position. The second uses the position after the second drift. Then position diffusion moves the walker once more, and the velocity cap acts once. Reordering any of these operations changes the transition law.

The cap is smooth radial compression: even a small nonzero velocity is reduced. It is therefore an order-one operation per update when its radius is held fixed. We study the population limit at fixed step size before asking whether any differential equation describes a further limit.

Definition 485 (BAOAB, position diffusion, smooth cap, and terminal classification)

Write $h=\tau>0$, $\gamma\ge0$, $F=-\nabla U$, and $$ c=e^{-\gamma h},\qquad q^2=\sigma_v^2\begin{cases}(1-e^{-2\gamma h})/(2\gamma),&\gamma>0,\\ h,&\gamma=0.\end{cases} $$ From the post-collision row $(x,v)$ draw independent standard Gaussian vectors $\xi_v,\xi_x$ and apply $$ \begin{aligned} v_1&=v+\tfrac h2F(x), &x_1&=x+\tfrac h2v_1,\\ v_2&=c v_1+q\xi_v, &x_2&=x_1+\tfrac h2v_2,\\ v_3&=v_2+\tfrac h2F(x_2), &x^+&=x_2+\sigma_x\sqrt h\xi_x,\\ v^+&=\psi_v(v_3)=\frac{V_{\mathrm{alg}}v_3}{V_{\mathrm{alg}}+\|v_3\|}, &a^+&=\mathbf1_D(x^+). \end{aligned} $$ The force is an acceleration; a nonunit physical mass is incorporated into $F$. The canonical thermostat is centered at zero. No boundary operation occurs at the clone, B, A, or O stages. The cap changes velocities only, and terminal absorption changes the mark only. In particular, $|v^+|<V_{\mathrm{alg}}$ for every finite input.

Clone jitter uses the separate parameter $\sigma_{\mathrm{clone}}$ and independent $\zeta_i$ for every accepted row, including revived rows. These jitters and the kinetic noises are independent across rows and independent of the graph. Rotations are independent across components, with one rotation shared inside each component. Conditional collision outputs are consequently correlated.

Remark 108 (Continuous-time interpretation)

BAOAB discretizes underdamped Langevin dynamics before the final diffusion, cap, and selection operations are composed with it. The complete canonical kernel is the composition just defined. Its stationary law is not asserted to be a Gibbs law. Holding the cap radius and order-one cloning rule fixed while $h\downarrow0$ need not yield a finite continuous-time generator; The Mean-Field Law of the Euclidean Gas analyzes this identification using the actual one-step map.

4. Axiom-by-axiom validation (Sasaki formulation)#

The estimates in this section state the transition and metric to which they apply. The physical phase-space metric is $$ d_{\mathrm{phys}}((x,v),(x',v'))^2=\|x-x'\|^2+\lambda_v\|v-v'\|^2. $$ The squashed comparison metric has the same topology on finite physical states, but a global physical Lipschitz estimate cannot be inferred from bounded feature distance. On each compact physical set the inverse squashing map is Lipschitz, so local constants transfer between these metrics.

4.1 Revival and terminal survival#

Lemma 75 (Scheduled revival)

If at least one slot is alive, every dead slot draws an eligible donor using its retained position and velocity in the Gaussian weights and is accepted with probability one. Thus the post-cloning population has $N$ alive slots. If no slot is alive, the swarm is absorbing. This conclusion does not require an inequality relating the fitness floor to the clone acceptance denominator.

Proof. The Gaussian donor weights are positive for all finite coordinates. Their sum over the nonempty alive pool is positive. The dead-row branch of the cloning rule accepts the selected donor deterministically. The component transform and jitter are then applied before any terminal classification. $\square$

Lemma 76 (Lipschitz property of the kinetic position map)

For the canonical kinetic step, let $b=h(1+c)/2$ and $$ M_h(x,v)=x+b\bigl(v+\tfrac h2F(x)\bigr),\qquad s_h^2=\tfrac{h^2q^2}{4}+h\sigma_x^2. $$ Then $x^+$ has law $\mathcal N(M_h(x,v),s_h^2 I_d)$, and under identical innovations $$ \|x^+-x'^+\|\le L_{\mathrm{flow}}d_{\mathrm{phys}}((x,v),(x',v')), \qquad L_{\mathrm{flow}}=1+\tfrac{bhL_F}{2}+\frac b{\sqrt{\lambda_v}}. $$

Proof. Substitute $v_2=cv_1+q\xi_v$ into $x_2=x+h(v_1+v_2)/2$. This gives $x^+=M_h+(hq/2)\xi_v+\sigma_x\sqrt h\xi_x$. The noises are independent and their covariance is $s_h^2I_d$. In a synchronous coupling they cancel. The Lipschitz bound on $F$ and the two coordinate bounds supplied by $d_{\mathrm{phys}}$ give the result. The B2 kick and cap do not change $x^+$. $\square$

Lemma 77 (Lipschitz continuity of the death probability)

For any Borel domain $D$, define $p_{\mathrm{dead}}(x,v)=\mathbb P(x^+\notin D)$. If $s_h>0$, then $$ |p_{\mathrm{dead}}(x,v)-p_{\mathrm{dead}}(x',v')| \le \frac{L_{\mathrm{flow}}}{\sqrt{2\pi}s_h}\, d_{\mathrm{phys}}((x,v),(x',v')). $$ For a domain with Lebesgue-null boundary, $\mathbb P(x^+\in\partial D)=0$.

Proof. Two Gaussians with covariance $s_h^2I$ and mean separation $r$ have total variation distance $2\Phi(r/(2s_h))-1$. To see this, divide their densities: the region where the first is larger is the half-space through their midpoint perpendicular to the mean difference. Integrating over that half-space reduces the distance to the displayed one-dimensional expression. Since $\Phi'$ is at most $1/\sqrt{2\pi}$, this distance is at most $r/(\sqrt{2\pi}s_h)$. Apply the bound to the exit event and then use Lemma 76. The null-boundary assertion follows from the Gaussian density. $\square$

Finite algorithmic diameter. Section 3.3 built $(\mathcal Y,d_{\mathcal Y}^{\mathrm{Sasaki}})$ from the capped velocities and showed that the projection $\varphi$ is $1$-Lipschitz. Consequently $\operatorname{diam}_{d_{\mathcal Y}^{\mathrm{Sasaki}}}(\mathcal Y)<\infty$, meeting the Axiom of Bounded Algorithmic Diameter (Axiom 23).

4.2 Environmental axioms#

The physical state remains Euclidean, and the retained dead coordinates need not belong to the valid box. Reward statistics use alive rows. Global force growth, rather than a bound restricted to the alive box, controls intermediate kinetic states.

Lemma 78 (Reward regularity in the Sasaki metric)

The reward function $R(x,v)=R_{\mathrm{pos}}(x)-\lambda_{\mathrm{vel}}\|v\|^2$ is continuous in physical coordinates and Lipschitz on each compact physical set. Expressed in squashed coordinates it is Lipschitz on the image of each such compact set. This gives the reward-regularity bound on the alive box with capped velocities. It does not assert a bounded continuous extension to the boundary of the full feature-space compactification.

Lemma 79 (Reward variation and a quantitative richness condition)

Let $\pi_B$ be the reference probability on a specified local region $B$ used in a richness assertion. Suppose two measurable subsets $B_1,B_2\subset B$ satisfy $\pi_B(B_i)\ge p_i>0$ and $$ \inf_{z\in B_1,z'\in B_2}|R(z)-R(z')|\ge\Delta>0. $$ Then $\operatorname{Var}_{\pi_B}R\ge p_1p_2\Delta^2$.

Proof. If $Z,Z'$ are independent with law $\pi_B$, then $\operatorname{Var}R=\frac12\mathbb E[(R(Z)-R(Z'))^2]$. The two ordered events $B_1\times B_2$ and $B_2\times B_1$ contribute at least $2p_1p_2\Delta^2$. $\square$

A nonzero kinetic penalty creates local reward variation, but a pair of distinct reward values alone does not provide the probability factors $p_1,p_2$. A uniform richness axiom requires these factors and the gap to be bounded uniformly over the specified regions and reference law. The canonical fixed-step population proofs use positive regularization floors and do not assume this additional richness statement. For a constant objective with zero kinetic penalty, reward variance is exactly zero while the regularized algorithm remains defined.

4.3 Algorithmic & operator axioms#

Valid noise measure (kinetic perturbation). Lemma Lemma 80 provides a quadratic-growth second-moment bound and the Feller property for the capped kinetic kernel.

Non-compact moment interpretation

On an unbounded domain we cannot demand a uniform moment bound. Instead, the kinetic axiom tracks the squared Sasaki increment through a Lyapunov-style control that grows at most quadratically in $\|x\|$ and $\|v\|$. The following lemma establishes this controlled growth together with the requisite Feller property.

Lemma 80 (Perturbation second moment in physical phase space)

For the canonical BAOAB step, let $b,s_h$ be as in Lemma 76, and assume $\|F(x)\|\le B_F+L_F\|x\|$. Then $$ \mathbb E d_{\mathrm{phys}}((x,v),(x^+,v^+))^2 \le C_x\|x\|^2+C_v\|v\|^2+C_0, $$ where $$ C_x=\frac{3b^2h^2L_F^2}{4},\qquad C_v=3b^2+2\lambda_v,\qquad C_0=\frac{3b^2h^2B_F^2}{4}+d s_h^2+2\lambda_v V_{\mathrm{alg}}^2. $$ The unmarked kinetic kernel maps bounded continuous functions to bounded continuous functions.

Proof. The Gaussian position formula gives $$ \mathbb E\|x^+-x\|^2=b^2\|v+\tfrac h2F(x)\|^2+d s_h^2 \le3b^2\|v\|^2+\tfrac{3b^2h^2}{4}(L_F^2\|x\|^2+B_F^2)+d s_h^2. $$ The smooth cap gives $|v^±v|^2\le2V_{\mathrm{alg}}^2+2|v|^2$. Sum these inequalities with velocity weight $\lambda_v$. Every BAOAB substep and the cap are continuous in the input for each fixed pair of innovations. Dominated convergence proves the asserted kernel continuity. The bound for projected displacement follows from the squashing maps' 1-Lipschitz property. $\square$

Lemma 81 (Kinetic drift, positional covariance, and local phase-space nondegeneracy)

For the same transition, $$ \mathbb E(x^+-x)=b(v+\tfrac h2F(x)),\qquad \operatorname{Cov}(x^+)=s_h^2I_d. $$ Hence the positional covariance condition number is exactly $1$, and the phase-space mean displacement has magnitude at most $\sqrt{C_x|x|^2+C_v|v|^2+C_0}$. The full position–velocity covariance is not generally isotropic.

If $F$ is $C^1$, $q>0$, $\sigma_x>0$, and $h^2L_F/4<1$, then the full physical phase-space covariance is positive definite. Its condition number is bounded on every compact set of inputs, with constants depending on that set and on $h$.

Proof. The first two identities follow directly from the Gaussian position formula; Jensen’s inequality and Lemma 80 give the drift bound. For the final assertion, condition on $x_1$. The map from the O-stage velocity $w$ to the B2 velocity is $$ T(w)=w+\tfrac h2F(x_1+\tfrac h2w). $$ Its perturbation of the identity has Lipschitz constant $h^2L_F/4<1$. For every target $y$, the equation $w=y-\tfrac h2F(x_1+hw/2)$ has a unique solution by geometric convergence of this contraction. Its derivative is invertible, so $T$ is a $C^1$ bijection with continuous inverse. Composing with $\psi_v$ maps onto $B(0,V_{\mathrm{alg}})$ with a positive density. Independent final position noise then gives a positive joint density on $\mathbb R^d\times B(0,V_{\mathrm{alg}})$.

A nonzero linear functional cannot be constant on this open set, so its variance is positive. Covariance entries depend continuously on the initial state by Gaussian moment domination and the velocity bound. Compactness therefore supplies a positive minimum eigenvalue and a finite maximum eigenvalue on each compact input set. Their ratio is the stated local condition-number bound. $\square$

For the unit quadratic potential used by the canonical experiments, $F(x)=-x$ and $L_F=1$, so the positive-definiteness condition is verified whenever $0<h<2$, including the experiment default $h=0.04$. There is an exact obstruction at $h=2$: $x_2=x_1+v_2$ and $v_3=v_2-x_2=-x_1$, so the final velocity is deterministic conditional on the kinetic input, despite positive thermostat noise. Its velocity covariance is zero. The final position still has the displayed nondegenerate Gaussian law.

These are fixed-step drift and covariance statements. The mean force drift, the full covariance, and contraction of the selection–collision kernel are different quantities.

Auxiliary distance continuity. The following inequalities isolate the geometric and scalar-array calculations used when their sampling law and input bounds apply.

2.3.3 Continuity of the Expected Raw Distance Vector ($k \ge 2$ Regime)#

Remark 109 (Scope of the auxiliary uniform-companion estimates)

The expected-distance estimates Lemma 83 and Theorem 159 below are for the explicitly defined extension with uniform independent companions. Their constants do not cover the canonical finite-width Gaussian donor probabilities. The deterministic aggregation and standardization inequalities remain usable for actual input arrays satisfying their stated bounds. Replacing a sampled distance by its expectation before nonlinear standardization or acceptance is not licensed by an expected-distance estimate.

For the canonical weighted law, use the normalized-kernel denominator bounds, marked measurement law, component exploration estimates, and one-step consistency proof in The Mean-Field Law of the Euclidean Gas and The Discrete Population Limit and Propagation of Chaos. Pairing without replacement is another configuration, not the canonical independent single-companion law.

Definition 486 (Notation for auxiliary finite-swarm continuity estimates)

For two swarms $\mathcal S_r$ let $\mathcal A_r$ be their alive sets, $k_r=|\mathcal A_r|$, and $\mathcal A_{\mathrm{stable}}=\mathcal A_1\cap\mathcal A_2$. Set $$ n_c(\mathcal S_1,\mathcal S_2)=\sum_{i=1}^N(s_{1,i}-s_{2,i})^2,\qquad \Delta_{\mathrm{pos,Sasaki}}^2= \sum_{i=1}^N d_{\mathcal Y}^{\mathrm{Sasaki}}(\varphi(w_{1,i}),\varphi(w_{2,i}))^2, $$ and $D_{\mathcal Y}=\operatorname{diam}(\mathcal Y)$. Expected-distance statements below use unregularized comparison distance and the uniform-companion extension. The deterministic scalar-array statements assume their displayed uniform bound $V_{\max}$; physical reward comparisons are restricted to compact sets where the specified $L_R^{\mathrm{Sasaki}}$ is finite.

For the canonical global regularizer, the notation $\sigma_{\min,\mathrm{patch}}$ in these scalar-array inequalities means $\sigma_{\min}$: take $\kappa_{\mathrm{var,min}}=0$ and $\varepsilon_{\mathrm{std}}=\sigma_{\min}$. As a function of variance $t\ge0$, the scale $\sqrt{t+\sigma_{\min}^2}$ has derivative at most $1/(2\sigma_{\min})$. Thus its denominator bounds are available directly. Its application to sampled arrays still requires averaging after, rather than before, the nonlinear pipeline.

Lemma 82 (Single-walker positional error bound in the Sasaki metric)

Let $\mathcal S_1$ and $\mathcal S_2$ be two swarm states. For a given walker $i$ that is alive in swarm $\mathcal S_1$ ($s_{1,i}=1$), let $\mathbb C_i(\mathcal S_1)$ be its companion selection measure.

The absolute error in its expected distance due to the positional displacement of the walkers between the two states, evaluated over the fixed companion set from $\mathcal S_1$, is bounded by the sum of its own displacement and the average displacement of its potential companions.

Referenced by Lemma 84 and Theorem 165.

\[ \left| \mathbb{E}_{c \sim \mathbb{C}_i(\mathcal{S}_1)} \left[ d_{\mathcal Y}^{\mathrm{Sasaki}}(\varphi(w_{1,i}), \varphi(w_{1,c})) \right] - \mathbb{E}_{c \sim \mathbb{C}_i(\mathcal{S}_1)} \left[ d_{\mathcal Y}^{\mathrm{Sasaki}}(\varphi(w_{2,i}), \varphi(w_{2,c})) \right] \right| \le d_{\mathcal Y}^{\mathrm{Sasaki}}(\varphi(w_{1,i}), \varphi(w_{2,i})) + \mathbb{E}_{c \sim \mathbb{C}_i(\mathcal{S}_1)} \left[ d_{\mathcal Y}^{\mathrm{Sasaki}}(\varphi(w_{1,c}), \varphi(w_{2,c})) \right]\]

Lemma 83 (Single-walker structural error bound in the Sasaki metric)

Let $i\in\mathcal A_{\mathrm{stable}}$ and keep the second swarm’s capped positions fixed. Let the initial swarm have at least two alive walkers, $k_1=|\mathcal A(\mathcal S_1)| \ge 2$. The absolute error in the expected distance for walker $i$ due to the change in the companion selection measure is bounded by:

Referenced by Theorem 159.

\[ \left| \mathbb{E}_{c \sim \mathbb{C}_i(\mathcal{S}_1)} \left[ d_{\mathcal Y}^{\mathrm{Sasaki}}(\varphi(w_{2,i}), \varphi(w_{2,c})) \right] - \mathbb{E}_{c \sim \mathbb{C}_i(\mathcal{S}_2)} \left[ d_{\mathcal Y}^{\mathrm{Sasaki}}(\varphi(w_{2,i}), \varphi(w_{2,c})) \right] \right| \le \frac{2 D_{\mathcal Y}}{k_1-1} \cdot n_c(\mathcal S_1, \mathcal S_2)\]

where $D_{\mathcal Y}$ is the diameter of the algorithmic space.

Lemma 84 (Mean-square error on stable walkers (Sasaki))

Let $\mathcal S_1,\mathcal S_2$ be swarms with alive sets $\mathcal A_r$ and let $\mathbf d^{(r)}$ denote the expected raw distance vector produced by the measurement operator on $\mathcal S_r$. Write $\mathcal A_{\mathrm{stable}}:=\mathcal A_1\cap\mathcal A_2$ and $k_{\mathrm{stable}}:=|\mathcal A_{\mathrm{stable}}|$. Then

\[ \sum_{i\in\mathcal A_{\mathrm{stable}}}\big|d^{(1)}_i-d^{(2)}_i\big|^2\le C_{\mathrm{pos}}^{\mathrm{Sasaki}}(k_1,k_{\mathrm{stable}})\,\Delta_{\mathrm{pos,Sasaki}}^2(\mathcal S_1,\mathcal S_2),\]

where $C_{\mathrm{pos}}^{\mathrm{Sasaki}}(k_1,k_{\mathrm{stable}}):=2\Big(1+\frac{k_{\mathrm{stable}}}{\max\{1,k_1-1\}}\Big)$.

Referenced by Theorem 159.

Theorem 159 (Mean-square continuity of the distance measurement (Sasaki))

Let $\mathbf d^{(r)}$ be the expected raw distance vectors of swarms $\mathcal S_r$. With $k_{\min}:=\max\{1,\min(k_1,k_2)\}$, $k_{\mathrm{stable}}:=|\mathcal A_{\mathrm{stable}}|$, and alive-difference count $n_c:=\sum_{i=1}^N(s_{1,i}-s_{2,i})^2$, define

\[ F_{d,ms}^{\mathrm{Sasaki}}(\Delta_{\mathrm{pos}}^2,n_c):=C_{\mathrm{pos}}^{\mathrm{Sasaki}}(k_1,k_{\mathrm{stable}})\,\Delta_{\mathrm{pos}}^2+4k_{\mathrm{stable}}\frac{D_{\mathcal Y}^2}{\max\{1,k_1-1\}^2}\,n_c^2+D_{\mathcal Y}^2 n_c.\]

Referenced by Theorem 165.

Then

\[ \big\|\mathbf d^{(1)}-\mathbf d^{(2)}\big\|_2^2\le F_{d,ms}^{\mathrm{Sasaki}}\big(\Delta_{\mathrm{pos,Sasaki}}^2(\mathcal S_1,\mathcal S_2),n_c(\mathcal S_1,\mathcal S_2)\big).\]

Non-degenerate noise. The positive canonical velocity diffusion and final position diffusion give non-Dirac kinetic noise. Clone jitter has its own scale $\sigma_{\mathrm{clone}}$. Shared component rotations add collision randomness when relative velocities and restitution are nonzero; this randomness can be degenerate when either vanishes.
Sufficient amplification. The weights $\alpha,\beta\ge 0$ satisfy $\alpha+\beta>0$ exactly as in the canonical swarm (Axiom 25).
Aggregator axioms. Let $R_{\max}:=\sup_{x\in\mathcal X}|R_{\mathrm{pos}}(x)|+\lambda_{\mathrm{vel}}V_{\mathrm{alg}}^2$ and recall from Lemma Lemma 78 that the reward satisfies the Lipschitz bound

\[ |R(x_1,v_1)-R(x_2,v_2)|\le L_R^{\mathrm{Sasaki}}\,d_{\mathcal Y}^{\mathrm{Sasaki}}\big((x_1,v_1),(x_2,v_2)\big),\qquad L_R^{\mathrm{Sasaki}}:=L_{\mathrm{pos}}+\frac{2\lambda_{\mathrm{vel}}V_{\mathrm{alg}}}{\sqrt{\lambda_v}}.\]

Whenever aggregators act on reward vectors we use the uniform bound $V_{\mathrm{max}}^{(R)}:=\max\{|R_{\min}|,R_{\max}\}$; for distance vectors we use $V_{\mathrm{max}}^{(d)}:=D_{\mathcal Y}$. For swarms $\mathcal S_r$ write $k_r:=|\mathcal A(\mathcal S_r)|$, define $k_{\min}:=\max\{1,\min(k_1,k_2)\}$, and let $n_c:=\sum_{i=1}^N(s_{1,i}-s_{2,i})^2$ count the status changes.

Lemma 85 (Value continuity of the empirical moments)

Fix a swarm $\mathcal S$ with alive index set $\mathcal A(\mathcal S)$ of size $k\ge 1$. Let $\mathbf v_1,\mathbf v_2\in\mathbb R^k$ be two scalar value vectors whose components satisfy $|v_{j,i}|\le V_{\max}$. Then the empirical mean and second moment obey

Referenced by Lemma 87 and Lemma 90.

\[ |\mu(\mathcal S,\mathbf v_1)-\mu(\mathcal S,\mathbf v_2)|\le \frac{1}{\sqrt{k}}\,\|\mathbf v_1-\mathbf v_2\|_2,\qquad|m_2(\mathcal S,\mathbf v_1)-m_2(\mathcal S,\mathbf v_2)|\le \frac{2V_{\max}}{\sqrt{k}}\,\|\mathbf v_1-\mathbf v_2\|_2.\]

Lemma 86 (Structural continuity of the empirical moments)

Let $\mathcal S_r=((x_{r,i},v_{r,i},s_{r,i}))_{i=1}^N$ with alive counts $k_r\ge 1$ and let $\mathbf v$ be a scalar vector on the union of alive indices satisfying $|v_i|\le V_{\max}$. Set $k_{\min}:=\max\{1,\min(k_1,k_2)\}$ and $n_c:=\sum_{i=1}^N(s_{1,i}-s_{2,i})^2$. Then

Referenced by Lemma 87 and Lemma 94.

\[ |\mu(\mathcal S_1,\mathbf v)-\mu(\mathcal S_2,\mathbf v)|\le \frac{3V_{\max}}{k_{\min}}\,n_c,\qquad|m_2(\mathcal S_1,\mathbf v)-m_2(\mathcal S_2,\mathbf v)|\le \frac{3V_{\max}^2}{k_{\min}}\,n_c.\]

Lemma 87 (Lipschitz data for the Sasaki empirical aggregators)

For reward vectors take $V_{\max}=V_{\mathrm{max}}^{(R)}$; for distance vectors take $V_{\max}=V_{\mathrm{max}}^{(d)}$. The empirical mean and second moment satisfy the aggregator axioms with

Referenced by Lemma 90 and Theorem 165.

\[ L_{\mu,M}^{\mathrm{Sasaki}}(k)=\frac{1}{\sqrt{k}},\qquad L_{m_2,M}^{\mathrm{Sasaki}}(k)=\frac{2V_{\max}}{\sqrt{k}},\]

\[ L_{\mu,S}^{\mathrm{Sasaki}}(k_{\min})=\frac{3V_{\max}}{k_{\min}},\qquad L_{m_2,S}^{\mathrm{Sasaki}}(k_{\min})=\frac{3V_{\max}^2}{k_{\min}},\]

and growth exponents $p_{\mu,S}=p_{m_2,S}=p_{\mathrm{worst\text{-}case}}=-1$. Consequently $\kappa_{\mathrm{var}}^{\mathrm{Sasaki}}=\kappa_{\mathrm{range}}^{\mathrm{Sasaki}}=1$ as in the canonical framework.

Standardization & rescale continuity. Let $\sigma_{\min,\mathrm{patch}}:=\sqrt{\kappa_{\mathrm{var,min}}+\varepsilon_{\mathrm{std}}^2}$ be the lower bound supplied by the regularized standard deviation operator, and denote by $L_{\sigma'_{\mathrm{patch}}}$ the global derivative bound from Lemma Lemma 33. For notational compactness write

\[ L_{\sigma',M}^{\mathrm{Sasaki}}(k):=L_{\sigma'_{\mathrm{patch}}}\Big(L_{m_2,M}^{\mathrm{Sasaki}}(k)+2V_{\mathrm{max}}^{(R)}L_{\mu,M}^{\mathrm{Sasaki}}(k)\Big).\]

Definition 487 (Standardization constants (Sasaki geometry))

Let $\sigma_{\min,\mathrm{patch}}:=\sqrt{\kappa_{\mathrm{var,min}}+\varepsilon_{\mathrm{std}}^2}$ be the uniform lower bound on the regularized standard deviation, and let $L_{\sigma'_{\mathrm{patch}}}$ be its global Lipschitz constant from Lemma Lemma 33.

The squared coefficients used in the mean-square bounds are defined in Definition 488.

Set $C_R:=L_R^{\mathrm{Sasaki}}\sqrt{N}+R_{\max}\sqrt{\tfrac{N}{\lambda_{\mathrm{status}}}}$ for later use.

These constants verify the continuity axioms for the patched standardization and logistic rescale operators in the Sasaki geometry.

Value Error Coefficients#

The following coefficients bound the error in the standardization operator when the swarm structure is fixed but the raw values change due to positional displacement. They are notably independent of the number of alive walkers, k.

Direct Shift Coefficient ($C_{V,\mathrm{direct}}$): Bounding the error from the direct change in the raw value vector.

\[ C_{V,\mathrm{direct}} := \frac{1}{\sigma_{\min,\mathrm{patch}}}\]
Mean Shift Coefficient ($C_{V,\mathrm{mean}}$): Bounding the error from the resulting change in the empirical mean.

\[ C_{V,\mathrm{mean}} := \frac{1}{\sigma_{\min,\mathrm{patch}}}\]
Denominator Shift Coefficient ($C_{V,\mathrm{denom}}$): Bounding the error from the resulting change in the regularized standard deviation.

\[ C_{V,\mathrm{denom}} := \frac{8\big(V_{\mathrm{max}}^{(R)}\big)^2 L_{\sigma'_{\mathrm{patch}}}}{\sigma_{\min,\mathrm{patch}}^2}\]
Total Value Error Coefficient (Linear Form) ($C_{V,\mathrm{total,lin}}^{\mathrm{Sasaki}}$): The composite coefficient for the full (unsquared) Lipschitz bound on the value error, which aggregates the component-wise effects.

\[ C_{V,\mathrm{total,lin}}^{\mathrm{Sasaki}} := L_R^{\mathrm{Sasaki}} \left( C_{V,\mathrm{direct}} + C_{V,\mathrm{mean}} + C_{V,\mathrm{denom}} \right) = L_R^{\mathrm{Sasaki}} \left( \frac{2}{\sigma_{\min,\mathrm{patch}}} + \frac{8\big(V_{\mathrm{max}}^{(R)}\big)^2 L_{\sigma'_{\mathrm{patch}}}}{\sigma_{\min,\mathrm{patch}}^2} \right)\]

Structural Error Coefficients#

The structural error coefficients, which are used in the subsequent theorem for structural continuity, remain as defined:

\[ C_{S,\mathrm{direct}}^{\mathrm{Sasaki}}(k_{\min}):=\frac{V_{\mathrm{max}}^{(R)}}{\sigma_{\min,\mathrm{patch}}}+\frac{2\big(V_{\mathrm{max}}^{(R)}\big)^2}{\sigma_{\min,\mathrm{patch}}^2}, \qquad C_{S,\mathrm{indirect}}^{\mathrm{Sasaki}}(k_{\min}):=\frac{3V_{\mathrm{max}}^{(R)}}{\sigma_{\min,\mathrm{patch}}k_{\min}}+\frac{6\big(V_{\mathrm{max}}^{(R)}\big)^2}{\sigma_{\min,\mathrm{patch}}^2k_{\min}}L_{\sigma',M}^{\mathrm{Sasaki}}(k_{\min}).\]

2.3.4. Theorem: Value Continuity of Patched Standardization (Sasaki)#

Theorem 160 (Value continuity of patched standardization (Sasaki))

Suppose $\mathcal S_1$ and $\mathcal S_2$ share the same alive set $\mathcal A$ of size $k\ge 1$ (so $n_c(\mathcal S_1,\mathcal S_2)=0$). Let $\mathbf r^{(r)}$ denote the raw reward vectors on $\mathcal A$. The N-dimensional standardization operator is Lipschitz continuous with respect to positional changes in the Sasaki metric. The squared L2-norm of the output error is bounded as follows:

\[ \big\|z(\mathcal S_1)-z(\mathcal S_2)\big\|_2^2 \le C_{V,\mathrm{total}}^{\mathrm{Sasaki}}(\mathcal S_1)\cdot\big\|\mathbf r^{(1)}-\mathbf r^{(2)}\big\|_2^2 \le C_{V,\mathrm{total}}^{\mathrm{Sasaki}}(\mathcal S_1)\cdot\left(L_R^{\mathrm{Sasaki}}\right)^2 \Delta_{\mathrm{pos,Sasaki}}^2(\mathcal S_1,\mathcal S_2).\]

where $C_{V,\mathrm{total}}^{\mathrm{Sasaki}}$ is the Total Value Error Coefficient, a deterministic constant defined in Definition 488. The proof is provided in the subsequent sections by decomposing the total error into its constituent parts.

2.3.4.1. Sub-Lemma: Algebraic Decomposition of the Value Error#

Lemma 88 (Decomposition of the Value Error)

Let $\mathcal S$ be a fixed swarm state with alive set $\mathcal A$ of size $k$. Let $\mathbf r_1$ and $\mathbf r_2$ be two raw value vectors for the alive set. Let $(\mu_1, \sigma'_1)$ and $(\mu_2, \sigma'_2)$ be the corresponding statistical properties, and let $\mathbf z_1$ and $\mathbf z_2$ be the corresponding standardized vectors.

The total value error vector, $\Delta\mathbf{z} = \mathbf z_1 - \mathbf z_2$, can be expressed as the sum of three components:

\[ \Delta\mathbf{z} = \Delta_{\text{direct}} + \Delta_{\text{mean}} + \Delta_{\text{denom}}\]

where:

The Direct Shift ($\Delta_{\text{direct}}$): The error from the change in the raw value vector itself, scaled by the initial standard deviation.

\[ \Delta_{\text{direct}} := \frac{\mathbf r_1 - \mathbf r_2}{\sigma'_1}\]
The Mean Shift ($\Delta_{\text{mean}}$): The error from the change in the aggregator’s computed mean, applied uniformly to all walkers.

\[ \Delta_{\text{mean}} := \frac{\mu_2 - \mu_1}{\sigma'_1} \cdot \mathbf{1}\]

where $\mathbf{1}$ is a k-dimensional vector of ones.
The Denominator Shift ($\Delta_{\text{denom}}$): The error from the change in the regularized standard deviation, which rescales the second standardized vector.

\[ \Delta_{\text{denom}} := \mathbf z_2 \cdot \frac{\sigma'_2 - \sigma'_1}{\sigma'_1}\]

Furthermore, the total squared error is bounded by three times the sum of the squared norms of these components:

\[ \|\Delta\mathbf{z}\|_2^2 \le 3\left( \|\Delta_{\text{direct}}\|_2^2 + \|\Delta_{\text{mean}}\|_2^2 + \|\Delta_{\text{denom}}\|_2^2 \right)\]

2.3.4.2. Sub-Lemma: Bounding the Squared Direct Shift Component#

Lemma 89 (Bound on the Squared Direct Shift Component)

Let $\mathcal S$ be a fixed swarm state. Let $\mathbf r_1$ and $\mathbf r_2$ be two raw value vectors for the alive set. The squared Euclidean norm of the direct shift error component, $\Delta_{\text{direct}} = (\mathbf r_1 - \mathbf r_2) / \sigma'_1$, is bounded as follows:

\[ \|\Delta_{\text{direct}}\|_2^2 \le \frac{1}{\sigma_{\min,\mathrm{patch}}^2} \cdot \|\mathbf r_1 - \mathbf r_2\|_2^2\]

where $\sigma_{\min,\mathrm{patch}} := \sqrt{\kappa_{\mathrm{var,min}}+\varepsilon_{\mathrm{std}}^2}$ is the uniform lower bound from the regularized standard deviation.

2.3.4.3. Sub-Lemma: Bounding the Squared Mean Shift Component#

Lemma 90 (Bound on the Squared Mean Shift Component)

Let $\mathcal S$ be a fixed swarm state with alive set $\mathcal A$ of size $k \ge 1$. Let $\mathbf r_1$ and $\mathbf r_2$ be two raw value vectors. The squared Euclidean norm of the mean shift error component, $\Delta_{\text{mean}} = ((\mu_2 - \mu_1) / \sigma'_1) \cdot \mathbf{1}$, is bounded as follows:

\[ \|\Delta_{\text{mean}}\|_2^2 \le \frac{k \cdot (L_{\mu,M}^{\mathrm{Sasaki}}(k))^2}{\sigma_{\min,\mathrm{patch}}^2} \cdot \|\mathbf r_1 - \mathbf r_2\|_2^2\]

where $L_{\mu,M}^{\mathrm{Sasaki}}(k)$ is the axiomatic Value Lipschitz Function for the aggregator’s mean from Lemma 87.

2.3.4.4. Sub-Lemma: Bounding the Squared Denominator Shift Component#

Lemma 91 (Bounding the Squared Denominator Shift Component)

Let $\mathcal S$ be a fixed swarm state with alive set $\mathcal A$ of size $k \ge 1$. Let $\mathbf r_1$ and $\mathbf r_2$ be two raw value vectors with components bounded by $V_{\max}^{(R)}$. The squared Euclidean norm of the denominator shift error component, $\Delta_{\text{denom}} = \mathbf z_2 \cdot ((\sigma'_2 - \sigma'_1) / \sigma'_1)$, is bounded as follows:

\[ \|\Delta_{\text{denom}}\|_2^2 \le k \left( \frac{2V_{\max}^{(R)}}{\sigma_{\min,\mathrm{patch}}} \right)^2 \left( \frac{L_{\sigma',M}^{\mathrm{Sasaki}}(k)}{\sigma_{\min,\mathrm{patch}}} \right)^2 \cdot \|\mathbf r_1 - \mathbf r_2\|_2^2\]

where $L_{\sigma',M}^{\mathrm{Sasaki}}(k)$ is the derived Lipschitz constant for the regularized standard deviation.

2.3.4.5. Proof of Theorem 2.3.4#

Proof

of Theorem 160

The proof establishes the final bound by assembling the deterministic bounds for each of the three error components derived in the preceding sub-lemmas.

Step 1: Start with the Decomposed Error Bound. From the algebraic decomposition in Lemma 88, the total squared value error is bounded by:

\[ \|z(\mathcal S_1)-z(\mathcal S_2)\|_2^2 \le 3\left( \|\Delta_{\text{direct}}\|_2^2 + \|\Delta_{\text{mean}}\|_2^2 + \|\Delta_{\text{denom}}\|_2^2 \right)\]

Step 2: Substitute the Bounds for Each Component. We substitute the deterministic bounds for the squared norm of each component, which all relate the component error to the squared norm of the raw value difference, $\|\mathbf r_1 - \mathbf r_2\|_2^2$.

From Lemma 89:

\[ \|\Delta_{\text{direct}}\|_2^2 \le C_{V,\mathrm{direct}}^{\mathrm{sq}}(\mathcal S_1) \cdot \|\mathbf r_1 - \mathbf r_2\|_2^2\]
From Lemma 90:

\[ \|\Delta_{\text{mean}}\|_2^2 \le C_{V,\mathrm{mean}}^{\mathrm{sq}}(\mathcal S_1) \cdot \|\mathbf r_1 - \mathbf r_2\|_2^2\]
From Lemma 91:

\[ \|\Delta_{\text{denom}}\|_2^2 \le C_{V,\mathrm{denom}}^{\mathrm{sq}}(\mathcal S_1) \cdot \|\mathbf r_1 - \mathbf r_2\|_2^2\]

Step 3: Combine and Factor. Substituting these into the inequality from Step 1 and factoring out the common term $\|\mathbf r_1 - \mathbf r_2\|_2^2$ gives:

\[ \|z_1 - z_2\|_2^2 \le 3 \left( C_{V,\mathrm{direct}}^{\mathrm{sq}}(\mathcal S_1) + C_{V,\mathrm{mean}}^{\mathrm{sq}}(\mathcal S_1) + C_{V,\mathrm{denom}}^{\mathrm{sq}}(\mathcal S_1) \right) \cdot \|\mathbf r_1 - \mathbf r_2\|_2^2\]

By definition (Definition 488), the term in parentheses is the Total Value Error Coefficient, $C_{V,\mathrm{total}}^{\mathrm{Sasaki}}(\mathcal S_1)$.

Step 4: Relate Raw Value Error to Positional Displacement. The raw reward vector difference is bounded by the positional displacement via the Lipschitz continuity of the reward function (Lemma 78):

\[ \|\mathbf r_1 - \mathbf r_2\|_2^2 = \sum_{i \in \mathcal A} |R(x_{1,i},v_{1,i}) - R(x_{2,i},v_{2,i})|^2 \le \sum_{i \in \mathcal A} \left(L_R^{\mathrm{Sasaki}}\right)^2 d_{\mathcal Y}^{\mathrm{Sasaki}}(\varphi(w_{1,i}), \varphi(w_{2,i}))^2 \le \left(L_R^{\mathrm{Sasaki}}\right)^2 \Delta_{\mathrm{pos,Sasaki}}^2(\mathcal S_1,\mathcal S_2)\]

Step 5: Final Assembly. Substituting the bound from Step 4 into the inequality from Step 3 gives

\[ \|z_1 - z_2\|_2^2 \le C_{V,\mathrm{total}}^{\mathrm{Sasaki}}(\mathcal S_1)\cdot\left(L_R^{\mathrm{Sasaki}}\right)^2 \Delta_{\mathrm{pos,Sasaki}}^2(\mathcal S_1,\mathcal S_2), \]

where $C_{V,\mathrm{total}}^{\mathrm{Sasaki}}$ is defined in Definition 488.

This completes the proof.

Q.E.D.

2.3.5. Definition: Value Error Coefficients (Squared Form)#

Definition 488 (Value Error Coefficients (Squared Form))

Let $\mathcal S$ be a fixed swarm state with alive set $\mathcal A$ of size $k \ge 1$, and let $M$ be the chosen Swarm Aggregation Operator. The coefficients for the bounds on the squared value error are defined as follows:

Referenced by Theorem 160.

The Squared Direct Shift Coefficient ($C_{V,\mathrm{direct}}^{\mathrm{sq}}(\mathcal S)$):

\[ C_{V,\mathrm{direct}}^{\mathrm{sq}}(\mathcal S) := \frac{1}{\sigma_{\min,\mathrm{patch}}^2}\]
The Squared Mean Shift Coefficient ($C_{V,\mathrm{mean}}^{\mathrm{sq}}(\mathcal S)$):

\[ C_{V,\mathrm{mean}}^{\mathrm{sq}}(\mathcal S) := \frac{k \cdot (L_{\mu,M}^{\mathrm{Sasaki}}(k))^2}{\sigma_{\min,\mathrm{patch}}^2}\]
The Squared Denominator Shift Coefficient ($C_{V,\mathrm{denom}}^{\mathrm{sq}}(\mathcal S)$):

\[ C_{V,\mathrm{denom}}^{\mathrm{sq}}(\mathcal S) := k \left( \frac{2V_{\max}^{(R)}}{\sigma_{\min,\mathrm{patch}}} \right)^2 \left( \frac{L_{\sigma',M}^{\mathrm{Sasaki}}(k)}{\sigma_{\min,\mathrm{patch}}} \right)^2\]
The Total Value Error Coefficient ($C_{V,\mathrm{total}}^{\mathrm{Sasaki}}(\mathcal S)$): The composite coefficient that bounds the total squared value error, incorporating the factor of 3 from the error decomposition.

\[ C_{V,\mathrm{total}}^{\mathrm{Sasaki}}(\mathcal S) := 3 \cdot \left( C_{V,\mathrm{direct}}^{\mathrm{sq}}(\mathcal S) + C_{V,\mathrm{mean}}^{\mathrm{sq}}(\mathcal S) + C_{V,\mathrm{denom}}^{\mathrm{sq}}(\mathcal S) \right)\]

When raw values are induced by positions, the positional coefficient in Theorem Theorem 160 is $\left(L_R^{\mathrm{Sasaki}}\right)^2 C_{V,\mathrm{total}}^{\mathrm{Sasaki}}$.

where $L_{\mu,M}^{\mathrm{Sasaki}}(k)$ and $L_{\sigma',M}^{\mathrm{Sasaki}}(k)$ are the value Lipschitz functions for the aggregator’s mean and regularized standard deviation, respectively. For the canonical empirical aggregator, these coefficients simplify, notably making the mean shift coefficient independent of $k$: $C_{V,\mathrm{mean}}^{\mathrm{sq}}(\mathcal S) = 1/\sigma_{\min,\mathrm{patch}}^2$.

2.3.6. Theorem: Structural Continuity of Patched Standardization (Sasaki)#

Theorem 161 (Structural Continuity of Patched Standardization (Sasaki))

For general swarms $\mathcal S_1,\mathcal S_2$ with alive counts $k_r\ge 1$, the squared L2-norm of the output error of the standardization operator is bounded by a function of the number of status changes, $n_c(\mathcal S_1,\mathcal S_2)$.

Referenced by Definition 489 and Lemma 95.

\[ \|z(\mathcal S_1)-z(\mathcal S_2)\|_2^2 \le C_{S,\mathrm{direct}}^{\mathrm{sq}}(\mathcal S_1, \mathcal S_2) \cdot n_c(\mathcal S_1, \mathcal S_2) + C_{S,\mathrm{indirect}}^{\mathrm{sq}}(\mathcal S_1, \mathcal S_2) \cdot n_c(\mathcal S_1, \mathcal S_2)^2\]

where $C_{S,\mathrm{direct}}^{\mathrm{sq}}$ and $C_{S,\mathrm{indirect}}^{\mathrm{sq}}$ are the Squared Structural Error Coefficients defined in Definition 489. The proof is provided in the subsequent sections.

2.3.6.1. Sub-Lemma: Decomposition of the Structural Error#

Lemma 92 (Decomposition of the Structural Error)

Let $\mathbf r_2$ be a fixed raw value vector. Let $\mathcal S_1$ and $\mathcal S_2$ be two swarm states. Let $\mathbf z_1 = z(\mathcal S_1, \mathbf r_2)$ and $\mathbf z_2 = z(\mathcal S_2, \mathbf r_2)$ be the corresponding N-dimensional standardized vectors, computed using the fixed raw values from the second swarm but the structure of each respective swarm.

Referenced by Definition 489.

The total structural error vector, $\Delta\mathbf{z} = \mathbf z_1 - \mathbf z_2$, can be expressed as the sum of two orthogonal components:

\[ \Delta\mathbf{z} = \Delta_{\text{direct}} + \Delta_{\text{indirect}}\]

where:

The Direct Error ($\Delta_{\text{direct}}$): The error vector whose non-zero components correspond to walkers whose status changes between $\mathcal S_1$ and $\mathcal S_2$.
The Indirect Error ($\Delta_{\text{indirect}}$): The error vector whose non-zero components correspond to walkers that are alive in both swarms.

Because these two vectors have disjoint support, the squared L2-norm of the total error is the sum of the squared L2-norms of the components:

\[ \|\Delta\mathbf{z}\|_2^2 = \|\Delta_{\text{direct}}\|_2^2 + \|\Delta_{\text{indirect}}\|_2^2\]

2.3.6.2. Sub-Lemma: Bounding the Squared Direct Structural Error#

Lemma 93 (Bound on the Squared Direct Structural Error)

Let $\mathbf r_2$ be a fixed raw value vector with components bounded by $V_{\max}^{(R)}$. The squared Euclidean norm of the direct structural error component, $\|\Delta_{\text{direct}}\|_2^2$, is bounded by a term linear in the number of status changes, $n_c(\mathcal S_1, \mathcal S_2)$.

Referenced by Definition 489.

\[ \|\Delta_{\text{direct}}\|_2^2 \le \left( \frac{2V_{\max}^{(R)}}{\sigma_{\min,\mathrm{patch}}} \right)^2 \cdot n_c(\mathcal S_1, \mathcal S_2)\]

2.3.6.3. Sub-Lemma: Bounding the Squared Indirect Structural Error#

Lemma 94 (Bound on the Squared Indirect Structural Error)

Let $\mathbf r_2$ be a fixed raw value vector. Let $\mathcal S_1$ and $\mathcal S_2$ be two swarm states. The squared Euclidean norm of the indirect structural error component, $\|\Delta_{\text{indirect}}\|_2^2$, is bounded by a term quadratic in the number of status changes, $n_c(\mathcal S_1, \mathcal S_2)$.

Referenced by Definition 489.

\[ \|\Delta_{\text{indirect}}\|_2^2 \le C_{S,\mathrm{indirect}}^{\mathrm{sq}}(\mathcal S_1, \mathcal S_2) \cdot n_c(\mathcal S_1, \mathcal S_2)^2\]

where $C_{S,\mathrm{indirect}}^{\mathrm{sq}}$ is the Squared Indirect Structural Error Coefficient defined in Definition 489.

2.3.6.4. Proof of Theorem 2.3.6#

Proof

of Theorem 161

The proof establishes the final bound by assembling the deterministic bounds for the two orthogonal error components derived in the preceding sub-lemmas.

Step 1: Start with the Orthogonal Decomposition. From Lemma 92, the total squared structural error is the sum of the squared norms of the direct and indirect error components:

\[ \|z(\mathcal S_1)-z(\mathcal S_2)\|_2^2 = \|\Delta_{\text{direct}}\|_2^2 + \|\Delta_{\text{indirect}}\|_2^2\]

Step 2: Substitute the Bounds for Each Component. We substitute the deterministic bounds for the squared norm of each component.

From Lemma 93, the direct error is bounded by a term linear in $n_c$:

\[ \|\Delta_{\text{direct}}\|_2^2 \le C_{S,\mathrm{direct}}^{\mathrm{sq}} \cdot n_c(\mathcal S_1, \mathcal S_2)\]
From Lemma 94, the indirect error is bounded by a term quadratic in $n_c$:

\[ \|\Delta_{\text{indirect}}\|_2^2 \le C_{S,\mathrm{indirect}}^{\mathrm{sq}}(\mathcal S_1, \mathcal S_2) \cdot n_c(\mathcal S_1, \mathcal S_2)^2\]

Step 3: Combine the Bounds. Summing the two bounds from Step 2 directly gives the final inequality as stated in Theorem Theorem 161.

\[ \|z(\mathcal S_1)-z(\mathcal S_2)\|_2^2 \le C_{S,\mathrm{direct}}^{\mathrm{sq}} \cdot n_c(\mathcal S_1, \mathcal S_2) + C_{S,\mathrm{indirect}}^{\mathrm{sq}}(\mathcal S_1, \mathcal S_2) \cdot n_c(\mathcal S_1, \mathcal S_2)^2\]

This completes the proof, establishing a deterministic, worst-case bound on the operator’s output error due to structural changes.

Q.E.D.

2.3.7. Structural Error Coefficients (Squared Form)#

Definition 489 (Structural Error Coefficients (Squared Form))

Let $\mathcal S_1$ and $\mathcal S_2$ be two swarm states with alive sets $\mathcal A_1$ and $\mathcal A_2$, of sizes $k_1:=|\mathcal A_1|$ and $k_2:=|\mathcal A_2|$. Let $k_{\mathrm{stable}}:=|\mathcal A_1\cap\mathcal A_2|$. The coefficients for the bounds on the squared structural error are defined as follows:

Referenced by Lemma 94 and Theorem 161.

The Squared Direct Structural Error Coefficient ($C_{S,\mathrm{direct}}^{\mathrm{sq}}$): The coefficient of the term linear in $n_c$.

\[ C_{S,\mathrm{direct}}^{\mathrm{sq}} := \left( \frac{2V_{\max}^{(R)}}{\sigma_{\min,\mathrm{patch}}} \right)^2\]
The Squared Indirect Structural Error Coefficient ($C_{S,\mathrm{indirect}}^{\mathrm{sq}}(\mathcal S_1, \mathcal S_2)$): The coefficient of the term quadratic in $n_c$, which bounds the error for the stable walkers.

\[ C_{S,\mathrm{indirect}}^{\mathrm{sq}}(\mathcal S_1, \mathcal S_2) := 2 k_{\mathrm{stable}} \frac{(L_{\mu,S}^{\mathrm{Sasaki}})^2}{\sigma_{\min,\mathrm{patch}}^{2}} + 2 k_2 \left(\frac{2V_{\max}^{(R)}}{\sigma_{\min,\mathrm{patch}}}\right)^2 \frac{(L_{\sigma',S}^{\mathrm{Sasaki}})^2}{\sigma_{\min,\mathrm{patch}}^{2}}\]

where $L_{\mu,S}^{\mathrm{Sasaki}}$ and $L_{\sigma',S}^{\mathrm{Sasaki}}$ are the structural continuity functions for the aggregator’s mean and regularized standard deviation, respectively, which depend on the swarm states.

2.3.8. Theorem: Composite Continuity of the Patched Standardization Operator#

Theorem 162 (Composite Continuity of the Patched Standardization Operator (Sasaki))

The N-dimensional standardization operator $z(\mathcal S)$, when applied to the reward vector, is continuous with respect to the dispersion metric. For any two swarms $\mathcal S_1, \mathcal S_2$ with $k_1=|\mathcal A(\mathcal S_1)|\ge 1$, the squared L2-norm of the output error is bounded by a composite function of the squared dispersion distance:

\[ \|z(\mathcal S_1)-z(\mathcal S_2)\|_2^2 \le L_{z,L}^2(\mathcal S_1,\mathcal S_2) \cdot d_{\mathrm{Disp},\mathcal Y}^{\mathrm{Sasaki}}(\mathcal S_1, \mathcal S_2)^2 + L_{z,H}^2(\mathcal S_1,\mathcal S_2) \cdot d_{\mathrm{Disp},\mathcal Y}^{\mathrm{Sasaki}}(\mathcal S_1, \mathcal S_2)^4\]

where $L_{z,L}^2$ and $L_{z,H}^2$ are state-dependent coefficients representing the Lipschitz and higher-order parts of the bound, respectively. Consequently, the logistic rescale operator $u(\mathcal S) = g_A(z(\mathcal S))$ is also continuous.

Lemma 95 (Lipschitz continuity of patched standardization (Sasaki))

The bounds in Theorem Theorem 162 show that the patched standardization operator $z$ is continuous with respect to the dispersion metric. In particular, $z$ admits the composite Lipschitz–Hölder control

\[ \|z(\mathcal S_1)-z(\mathcal S_2)\|_2^2\le L_{z,L}^2(\mathcal S_1,\mathcal S_2)\,d_{\mathrm{Disp},\mathcal Y}^{\mathrm{Sasaki}}(\mathcal S_1,\mathcal S_2)^2+L_{z,H}^2(\mathcal S_1,\mathcal S_2)\,d_{\mathrm{Disp},\mathcal Y}^{\mathrm{Sasaki}}(\mathcal S_1,\mathcal S_2)^4.\]

4.4 Swarm-level continuity and population evolution#

Remark 110 (Constants and their scope)

The kinetic constants $L_{\mathrm{flow}},s_h,C_x,C_v,C_0$ are given explicitly in Lemma 76 and Lemma 80. The terminal death-probability constant is $L_{\mathrm{flow}}/(\sqrt{2\pi}s_h)$ in physical phase-space distance. On a compact physical set these can be converted to squashed-distance constants using the Lipschitz constant of the inverse projection.

For each fixed $N$, the complete marked kernel is Feller by Theorem 165. Uniform-in-$N$ continuity and concentration require control of the accepted components and sampled global statistics. Their proofs are given for the same canonical transition in The Mean-Field Law of the Euclidean Gas and The Discrete Population Limit and Propagation of Chaos; they do not follow from independent per-walker collision outputs. The shared rotation is part of the population map.

2.6 Axiom for Convergence: Non-Deceptive Landscape#

For the geometric ergodicity proof in Convergence, Survival, and Parameter Dependence to hold, the Euclidean Gas must satisfy an additional environmental axiom that strengthens the Axiom of Environmental Richness. This axiom prevents pathological scenarios where positional diversity and reward signals become decoupled.

Axiom 36 (Axiom of Non-Deceptive Landscapes)

The environment $(X_{\mathrm{valid}}, R_{\mathrm{pos}})$ is non-deceptive if there exist constants $\kappa_{\mathrm{grad}} > 0$ and $L_{\mathrm{grad}} > 0$ such that for any two points $x, y \in X_{\mathrm{valid}}$ with $\|x - y\| \ge L_{\mathrm{grad}}$, the average squared norm of the reward gradient along the line segment connecting them is bounded below:

\[ \frac{1}{\|x-y\|} \int_{0}^{\|x-y\|} \big\|\nabla R_{\mathrm{pos}}\big(x + t\tfrac{y-x}{\|y-x\|}\big)\big\|^2 dt \ge \kappa_{\mathrm{grad}}.\]

Applicability: This inequality is an additional quantitative landscape hypothesis wherever a theorem uses it. Continuity alone does not prove a positive lower bound. It is not required to define the canonical transition or its finite-horizon population limit; any geometric-ergodicity application must verify it for its specified potential and scales.

5. The canonical marked transition#

Theorem 163 (Well-defined canonical Euclidean Gas)

For finite $N\ge1$, positive Gaussian donor widths, positive regularization and fitness floors, finite $h>0$, globally Lipschitz force, and the canonical schedule, Algorithm 3 defines a time-homogeneous Markov kernel on the full marked state. It preserves all-slot momentum during the collision stage and is permutation equivariant. With $\sigma_x>0$ and Lebesgue-null boundary it is Feller.

Proof. With at least one alive row the donor normalizers are positive; self-exclusion and singleton rules specify every draw. Positive regularization denominators make the sampled fitness finite, and all acceptance probabilities lie in $[0,1]$. A finite undirected graph has a unique partition into connected components. The common-rotation formula assigns each row exactly one output. BAOAB, final diffusion, cap, and status classification are measurable, so their composition with the finite draws is a probability kernel. The all-dead branch is absorbing.

For each component, $\sum_{i\in C}(v_i-\bar v_C)=0$, so summing its velocity formula gives $\sum_{i\in C}\widetilde v_i=\sum_{i\in C}v_i$. Relabeling the input, donor indices, innovations, and components relabels the output: weights, empirical statistics, connectedness, and component means are unchanged as unlabeled objects. Haar matrices have the same independent law after this transport. This proves kernel equivariance, without requiring equality of arbitrary fixed-seed trajectories under relabeling. The Feller assertion is proved in Theorem 165. $\square$

Remark 111 (What requires a separate convergence argument)

Kernel existence and symmetry do not imply every axiom of an abstract convergence theorem. In particular, a richness lower bound depends on an explicitly specified reference measure, correlated component outputs cannot satisfy an independent-output assumption, and stationary uniqueness requires an attraction or contraction argument for the actual nonlinear map. The canonical population and finite-horizon chaos statements are established in The Mean-Field Law of the Euclidean Gas and The Discrete Population Limit and Propagation of Chaos; the latter gives the exact stationary identification obligations.

6. Swarm Update Operator Kernel#

We define the one-step kernel $\Psi_{\mathcal F_{\mathrm{EG}}}$ on the ordered swarm space $\Sigma_N=(\mathcal X\times\mathbb R^d\times\{0,1\})^N$. Measurement and donor innovations are independent across recipient rows conditional on the input, while rotations are independent across accepted components. Output walkers within a component share one rotation. When no walkers are alive the process becomes absorbing.

6.1 Stage 1 — Cemetery absorption#

If the alive index set $\mathcal A(\mathcal S_t)$ is empty, the operator returns $\delta_{\mathcal S_t}$ and the run stops. All subsequent stages are skipped.

6.2 Stage 2 — Single-shot measurement and frozen potentials#

Definition 490 (Finite sampled measurement and fitness law)

Let $\mathcal A=\{i:a_i=1\}$. For $M\ge2$, the measurement and cloning kernels for an alive recipient exclude its own label. A dead recipient’s clone kernel uses all of $\mathcal A$. For either role $b\in\{D,C\}$, $$ P_b^N(i,j)=\frac{\mathbf1_{j\in\mathcal A\setminus\{i\}}\exp[-d_{\mathrm{alg}}(i,j)^2/(2\epsilon_b^2)]} {\sum_{k\in\mathcal A\setminus\{i\}}\exp[-d_{\mathrm{alg}}(i,k)^2/(2\epsilon_b^2)]}. $$ An alive singleton uses the zero-distance exception and cannot clone from itself. Dead rows still select the sole alive donor.

Fix the separation floor $\delta_D>0$. Independently for each alive $i$, sample $J_i^D$ and retain $$ r_i=R(x_i,v_i),\qquad d_i=\sqrt{d_{\mathrm{alg}}(i,J_i^D)^2+\delta_D^2}. $$ In the singleton exception the measured raw distance is zero and the separation is $\delta_D$. Set unused dead measurement entries to zero. For $y=r,d$, compute $$ \bar y=\frac1M\sum_{i\in\mathcal A}y_i,\qquad \widehat\sigma_y=\sqrt{\frac1M\sum_{i\in\mathcal A}(y_i-\bar y)^2+\sigma_{\min,y}^2},\qquad z_i^y=(y_i-\bar y)/\widehat\sigma_y. $$ For an objective minimized by the engine, $R$ denotes its negative, so larger $z_i^r$ is better. Define $$ V_{\mathrm{fit},i}= \left(\frac{A_r}{1+e^{-z_i^r}}+\eta_r\right)^\alpha \left(\frac{A_d}{1+e^{-z_i^d}}+\eta_d\right)^\beta \quad(i\in\mathcal A), $$ and set unused dead fitness to zero. Freeze the entire realized fitness vector for acceptance. In particular, its donor entry contains the donor’s own sampled measurement, and common empirical means and variances are retained.

6.3 Stage 3 — Accepted edges and component collisions#

Definition 491 (Frozen component cloning transformation)

Draw cloning donors $J_i^C$ from the kernels in Definition 490 and independent $U_i\sim\operatorname{Unif}[0,1]$. For alive rows with a distinct donor, set $$ p_i=\min\left\{1,\frac{[V_{\mathrm{fit},J_i^C}-V_{\mathrm{fit},i}]_+}{p_{\max}(V_{\mathrm{fit},i}+\varepsilon_{\mathrm{clone}})}\right\}, \qquad A_i=\mathbf1_{\{U_i<p_i\}}. $$ For dead rows set $p_i=A_i=1$; for an alive singleton set $p_i=A_i=0$.

Let $G$ have undirected edges $\{i,J_i^C\}$ whenever $A_i=1$, and let $\mathcal C(G)$ be its nontrivial connected components. From frozen input coordinates set $$ \widetilde x_i=\begin{cases}x_{J_i^C}+\sigma_{\mathrm{clone}}\zeta_i,&A_i=1,\\x_i,&A_i=0,\end{cases} \qquad \zeta_i\overset{\mathrm{iid}}\sim\mathcal N(0,I_d). $$ For each $C\in\mathcal C(G)$, draw one independent Haar orthogonal matrix $R_C$ and set $$ \bar v_C=|C|^{-1}\sum_{i\in C}v_i,\qquad \widetilde v_i=\bar v_C+\alpha_{\mathrm{restitution}}R_C(v_i-\bar v_C). $$ For vertices outside these components set $\widetilde v_i=v_i$. All intermediate marks equal one. A persisting donor can therefore change velocity even though it does not copy a position.

Theorem 164 (Component conservation, restitution, and shared covariance)

For each component, with $u_i=v_i-\bar v_C$, $$ \sum_{i\in C}\widetilde v_i=\sum_{i\in C}v_i,\qquad \sum_{i\in C}\|\widetilde v_i-\bar v_C\|^2= \alpha_{\mathrm{restitution}}^2\sum_{i\in C}\|u_i\|^2. $$ Conditional on the accepted graph and all frozen velocities, $$ \mathbb E\widetilde v_i=\bar v_C,\qquad \operatorname{Cov}(\widetilde v_i,\widetilde v_j)= \frac{\alpha_{\mathrm{restitution}}^2}{d}(u_i\cdot u_j)I_d, \quad i,j\in C. $$ At $\alpha_{\mathrm{restitution}}=0$ all component velocities become their mean. At $\alpha_{\mathrm{restitution}}=1$ total component kinetic energy is conserved.

Proof. The sum of the $u_i$ is zero, and a common linear map preserves that identity. Orthogonality preserves every squared relative norm. Haar invariance under $R\mapsto-R$ gives $\mathbb E R=0$, and left orthogonal invariance implies $\mathbb E[(Ru_i)(Ru_j)^\top]$ is a scalar multiple of $I_d$. Taking its trace gives that scalar as $(u_i\cdot u_j)/d$. Summing the relative-energy identity and the unchanged center-of-mass energy proves the last assertion. $\square$

These momentum sums include dead input slots. Before kinetics, all slots have become alive, so the change in alive-only momentum relative to the input is exactly the sum of the retained dead input velocities. The full-slot collision momentum change is zero. Terminal killing then removes its own measured momentum from the alive sum.

Take three slots with velocities $-1,0,1$ connected in a chain. Their center-of-mass velocity is zero. In one dimension the shared Haar matrix is a single random sign, so the output is either $(-\alpha,0,\alpha)$ or $(\alpha,0,-\alpha)$. Momentum is zero in both cases. Assigning separate signs to the two nonzero relative velocities would sometimes make them point the same way and would fail the conservation identity.

The graph also tells us which errors can travel together. A measurement affects a gate; a changed gate can merge two components; the merged component uses one mean velocity and one rotation. This is why the population proof explores accepted neighborhoods instead of treating collision outputs as independent row updates.

6.4 Stage 4 — Kinetic perturbation and status update#

Apply Definition 485 independently to the rows of $(\widetilde x,\widetilde v)$, conditional on this correlated intermediate population. Use its B1–A1–O–A2–B2 sequence, final position diffusion, smooth cap, and terminal classification in exactly that order. The next swarm is $S^+=((x_i^+,v_i^+,a_i^+))_{i=1}^N$; every coordinate is retained even when $a_i^+=0$.

6.5 Kernel representation#

Definition 492 (Innovation representation of the kernel)

For each input $S$, let $\nu_S$ be the law obtained by: drawing independent measurement and cloning companions with their input-dependent categorical weights; drawing the independent gate uniforms; forming $G$; drawing one Haar matrix per component of $G$; and drawing independent clone and kinetic Gaussian innovations. If $\Phi_h$ executes the specified stages, then $$ \Psi_{\mathcal F_{\mathrm{EG}}}(S,B)=\int\mathbf1_B(\Phi_h(S;\omega))\nu_S(d\omega). $$ The equivalent representation with input-independent uniforms realizes each categorical draw by inverse cumulative probabilities. Component matrices can be preassigned independently to every nonempty vertex subset and the matrix indexed by each realized component selected. Only the selected matrices are used. This supplies a fixed product innovation space without assuming independent collision outputs.

Proof of the Feller Property for the Euclidean Gas Kernel#

Theorem 165 (Feller continuity of $\Psi_{\mathcal F_{\mathrm{EG}}}$)

For fixed $N$, continuous reward, globally Lipschitz force, positive donor and standardization denominators, $\sigma_x>0$, and Lebesgue-null $\partial D$, the canonical marked kernel maps bounded continuous functions to bounded continuous functions on $$ \Sigma_N=(\mathbb R^d\times\mathbb R^d\times\{0,1\})^N, $$ with the discrete topology on the marks. Restricting to terminally consistent marked states preserves this assertion. The physical and squashed coordinate metrics induce the same topology on finite states.

Proof. Suppose $S_k\to S$. The finite vector of discrete marks is eventually constant. If all marks are zero, the transition is the identity and the result follows. Otherwise the alive set and the singleton convention are eventually fixed.

There are finitely many possible measurement donor arrays, clone donor arrays, and acceptance arrays. Their probabilities depend continuously on $S$: Gaussian weights have positive denominators, measured features and rewards are continuous, and regularized scales and positive fitness denominators do not vanish. Fix one such discrete array. Its accepted graph and connected components are fixed. Each component mean is a continuous linear function of the frozen velocities, and the rotation and position-copy maps are continuous for each fixed set of Haar and Gaussian innovations. BAOAB and the cap are likewise continuous.

The terminal status indicator can be discontinuous only when one output position lies on $\partial D$. Conditional on all earlier innovations, independent final Gaussian position noise gives this event probability zero. Hence the marked outputs converge almost surely under the common innovations for the fixed discrete array. A bounded continuous test function then converges in expectation by dominated convergence. Sum over the finitely many discrete arrays with their continuous weights. This proves the assertion. No independence of output rows is used. $\square$

References (selected)#

Federer, H. Geometric Measure Theory. Springer, 1969. Standard tubular-neighbourhood volume estimates and Weyl’s tube formula.
Ethier, S. N., and Kurtz, T. G. Markov Processes: Characterization and Convergence. Wiley, 1986. Composition properties of Feller kernels.

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@book{MeynTweedie2009,
  author    = {Sean P. Meyn and Richard L. Tweedie},
  title     = {Markov Chains and Stochastic Stability},
  edition   = {2},
  publisher = {Cambridge University Press},
  address   = {Cambridge},
  year      = {2009},
  doi       = {10.1017/CBO9780511626630},
  isbn      = {9780521731829}
}

(Cambridge University Press & Assessment)

@book{BoucheronLugosiMassart2013,
  author    = {St{\'e}phane Boucheron and G{\'a}bor Lugosi and Pascal Massart},
  title     = {Concentration Inequalities: A Nonasymptotic Theory of Independence},
  publisher = {Oxford University Press},
  address   = {Oxford},
  year      = {2013},
  isbn      = {9780198767657}
}

(Oxford Academic)

@book{Federer1969,
  author    = {Herbert Federer},
  title     = {Geometric Measure Theory},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {153},
  publisher = {Springer},
  address   = {Berlin Heidelberg},
  year      = {1969},
  isbn      = {9783540045052}
}

(SpringerLink)

@book{Santambrogio2015,
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  series    = {Progress in Nonlinear Differential Equations and Their Applications},
  volume    = {87},
  publisher = {Birkh{\"a}user},
  address   = {Cham},
  year      = {2015},
  doi       = {10.1007/978-3-319-20828-2},
  isbn      = {9783319208275}
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(SpringerLink)

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  author    = {Alexander S. Kechris},
  title     = {Classical Descriptive Set Theory},
  series    = {Graduate Texts in Mathematics},
  volume    = {156},
  publisher = {Springer},
  address   = {New York},
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  doi       = {10.1007/978-1-4612-4190-4},
  isbn      = {9780387943749}
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(SpringerLink)

@article{FritschCarlson1980,
  author  = {F. N. Fritsch and R. E. Carlson},
  title   = {Monotone Piecewise Cubic Interpolation},
  journal = {SIAM Journal on Numerical Analysis},
  year    = {1980},
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  pages   = {238--246},
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  author  = {James M. Hyman},
  title   = {Accurate Monotonicity Preserving Cubic Interpolation},
  journal = {SIAM Journal on Scientific and Statistical Computing},
  year    = {1983},
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  pages   = {645--654},
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(SIAM E-Books)

@incollection{McDiarmid1989,
  author    = {Colin McDiarmid},
  title     = {On the Method of Bounded Differences},
  booktitle = {Surveys in Combinatorics, 1989},
  editor    = {J. Siemons},
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  pages     = {148--188}
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(Cambridge University Press & Assessment)

@book{AmbrosioFuscoPallara2000,
  author    = {Luigi Ambrosio and Nicola Fusco and Diego Pallara},
  title     = {Functions of Bounded Variation and Free Discontinuity Problems},
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}

(Oxford University Press)

Euclidean Gas: Canonical Transition and Operator Estimates

Contents

Euclidean Gas: Canonical Transition and Operator Estimates#

0. TLDR#

1. Introduction#

2. Physical states and comparison geometry#

3. Definition: the Euclidean Gas#

3.1 Euclidean Gas algorithm (canonical pipeline)#

3.2 Compiled configuration and experiment contract#

3.3 Position–velocity foundations and projection (Sasaki metric)#

3.4 Swarm distance and canonical operators#

3.5 Kinetic Langevin perturbations with velocity capping#

4. Axiom-by-axiom validation (Sasaki formulation)#

4.1 Revival and terminal survival#

4.2 Environmental axioms#

4.3 Algorithmic & operator axioms#

2.3.3 Continuity of the Expected Raw Distance Vector (\(k \ge 2\) Regime)#

Value Error Coefficients#

Structural Error Coefficients#

2.3.4. Theorem: Value Continuity of Patched Standardization (Sasaki)#

2.3.4.1. Sub-Lemma: Algebraic Decomposition of the Value Error#

2.3.4.2. Sub-Lemma: Bounding the Squared Direct Shift Component#

2.3.4.3. Sub-Lemma: Bounding the Squared Mean Shift Component#

2.3.4.4. Sub-Lemma: Bounding the Squared Denominator Shift Component#

2.3.4.5. Proof of Theorem 2.3.4#

2.3.5. Definition: Value Error Coefficients (Squared Form)#

2.3.6. Theorem: Structural Continuity of Patched Standardization (Sasaki)#

2.3.6.1. Sub-Lemma: Decomposition of the Structural Error#

2.3.6.2. Sub-Lemma: Bounding the Squared Direct Structural Error#

2.3.6.3. Sub-Lemma: Bounding the Squared Indirect Structural Error#

2.3.6.4. Proof of Theorem 2.3.6#

2.3.7. Structural Error Coefficients (Squared Form)#

2.3.8. Theorem: Composite Continuity of the Patched Standardization Operator#

4.4 Swarm-level continuity and population evolution#

2.6 Axiom for Convergence: Non-Deceptive Landscape#

5. The canonical marked transition#

6. Swarm Update Operator Kernel#

6.1 Stage 1 — Cemetery absorption#

6.2 Stage 2 — Single-shot measurement and frozen potentials#

6.3 Stage 3 — Accepted edges and component collisions#

6.4 Stage 4 — Kinetic perturbation and status update#

6.5 Kernel representation#

Proof of the Feller Property for the Euclidean Gas Kernel#

References (selected)#