The Discrete Population Limit and Propagation of Chaos#
A large swarm has two kinds of randomness. A walker fluctuates around the population distribution, and the population distribution itself can fluctuate from run to run. Exchangeability says that the labels carry no information. Propagation of chaos says more: the second kind of randomness disappears, and any fixed number of walkers become independent in the limit.
We follow one complete programmed update. Measurement companions create sampled fitness marks; accepted donor edges assemble collision components; one rotation acts on each component; and BAOAB, position diffusion, the velocity cap, and boundary classification finish the step. The component can contain several walkers, so its shared randomness must survive in the one-walker population map. Independence emerges between separately tagged components as the swarm grows.
The resulting evolution is \(\mu_{n+1}=\mathcal F_h(\mu_n)\) at the actual fixed timestep. We prove its one-step approximation and then iterate that proof. Stationary existence, stationary concentration, and global attraction are different claims; the last section states precisely which of them follow.
1. Swarm Laws, Empirical Measures, and Tightness#
The correct object to follow is the empirical measure of a whole swarm. Its expectation is a one-walker marginal, but an expectation can hide a mixture. For example, half the runs could put every walker near one point and half near another. Every label then has the same marginal, while the walkers remain strongly correlated.
Definition 584 (Finite-swarm QSDs and marked marginals)
Let \(\mathsf Z=\mathbb R^d\times\overline B(0,V)\times\{0,1\}\) be the complete marked slot state at the end of an update. Its coordinates are position, capped velocity, and alive indicator. Dead coordinates are retained: their position enters weighted revival-donor sampling and their velocity enters the connected-component collision. The status space has its discrete topology. The canonical process has no donor history; configurations with history require a correspondingly enlarged state.
For each \(N\ge2\), let \(Q_N\) be the killed full-swarm kernel on its noncemetery space. For the canonical quadratic-force terminal-box configuration, Theorem 225 establishes its unique QSD \(\nu_N\). For other configurations, use an applicable QSD theorem for their complete transition. Write
For a swarm \(S=(z_1,\ldots,z_N)\) define
Here \(\mu_N\) is a probability on \(\mathsf Z\), while \(\Lambda_N\) is a probability on \(\mathcal P(\mathsf Z)\). Propagation of chaos toward \(\mu_*\) means that, for every fixed \(l\), the first \(l\) coordinate marginal converges weakly to \(\mu_*^{\otimes l}\).
Definition 585 (Alive mass and normalized alive distribution)
For a marked probability \(\mu\), let \(f=\mu|_{\{a=1\}}\) be its alive subprobability and \(m_a=f(\mathsf Z)\). If \(m_a>0\), its normalized alive law is \(\rho=f/m_a\). The finite-swarm counterparts are
The mass-one marked law, alive subprobability, and normalized alive law have different normalization equations. A scalar dead reservoir is a closed model only when the revival rule depends on dead walkers through that scalar alone. Companion selection that depends on their retained coordinates requires their distribution in the state.
For a fixed number of walkers, the final two independent noises provide a particularly direct route to equilibrium under survival. The OU noise spreads velocity, and the final position noise spreads position. Revival first brings every position back to a donor inside the box. Consequently even a walker whose retained dead position is very far away has a controlled distribution at the next update.
The far-away coordinate is still part of the state. The useful observation is that the next donor choice sees it through the bounded distance feature; once the donor is chosen, revival replaces its position before the kinetic motion. We can exploit this exact dependence to prove the finite-swarm QSD theorem.
Theorem 225 (Unique QSD and conditioned convergence for the canonical box gas)
Fix \(N\geq1\) and the canonical terminal absorbing-box update with \(U(x)=|x|^2/2\), positive OU and final position-noise amplitudes, and smooth velocity cap \(V\). For \(h>0\) with \(h\ne2\), this actual killed kernel has a unique QSD \(\nu_N\) and a survival eigenvalue \(0<\alpha_N<1\). There are \(C_N<\infty\) and \(r_N<1\) such that
The supremum is over probability laws on nonextinct, terminally consistent full marked states with capped velocities. Dead physical positions remain unbounded and are retained in this statement. In particular the theorem applies to the canonical \(h=0.04\) configuration. Its constants may depend on \(N\); no assumption of positional contraction by cloning is needed.
Proof
1. Represent exactly the input coordinates used by the transition. Let \(S_R(x)=Rx/(R+|x|)\) be the position feature in the donor distance. For an alive slot retain \(x\in\overline D\); for a dead slot represent its position by \(u=S_R(x)\in\overline{S_R(D^c)}\). Retain the velocity in \(\overline B_V\) and its discrete alive/dead mark in both cases. The finite union of the resulting products over all nonempty alive masks is a compact space \(K_N\).
For \(|u|<R\), the inverse is \(x=Ru/(R-|u|)\), so every finite physical dead coordinate remains recoverable. Points with \(|u|=R\) compactify only the input representation. The donor weights extend continuously to these points. Canonical reward statistics use eligible walkers, whose physical positions are bounded by the box. No other canonical operation requires the raw dead position: mandatory revival copies a frozen eligible donor position before jitter, and the collision uses retained capped velocities. Thus the actual kernel extends to \(K_N\). The status components are disjoint, and the exact single-eligible-donor convention is retained on those components.
Every position before jitter is in \(\overline D\). The frozen component rule bounds each post-collision velocity by \((1+2\alpha)V\), where \(\alpha\) is the restitution coefficient. These are bounds on kinetic input means, not a claim that physical dead output positions have compact support.
2. Verify full phase-space smoothing for the declared schedule. Condition on the finite measurement-companion, cloning-companion, and gate pattern and on its component rotations. The collision coordinates are bounded before independent Gaussian jitter. The remaining quadratic-force BAOAB stages are affine. With \(x_1\) the A1 position, \(v_1\) the B1 velocity, \(a=e^{-\gamma h}\), and actual OU innovation amplitude \(q>0\), write
The independent standard Gaussian innovations \((\xi,\zeta)\) have coefficient matrix, for the output ordered as \((x_3,v_3)\),
It is invertible for \(h\ne2\). After integrating Gaussian cloning jitter, the pre-cap joint phase-space law is therefore a Gaussian with strictly positive density on \(\mathbb R^{2dN}\). Its mean ranges over a compact set as the effective input and rotations vary. Its covariance belongs to a finite positive-definite family indexed by the accepted-recipient pattern.
The final cap \(v\mapsto Vv/(V+|v|)\) is a \(C^1\) diffeomorphism from \(\mathbb R^d\) onto \(B_V\), with positive Jacobian. The resulting density is positive on \(\mathbb R^{dN}\times B_V^N\). Terminal classification assigns its actual marks; removing the all-dead output gives \(Q_N\). Every nonempty relatively open subset of \(K_N\) contains a positive-measure set of physical interior outputs, so \(Q_N(k,\cdot)\) has full support on \(K_N\). Both survival and extinction have positive probability at every input.
3. Prove compactness of the transition operator. At fixed \(N\) there are finitely many companion and gate patterns. Within each fixed alive mask their probabilities are continuous in the effective input: weighted normalizers stay positive, regularized population statistics are continuous, and the positive-part acceptance formula is continuous at ties. For a fixed pattern, component membership is fixed and its Haar rotations range over a compact product of orthogonal groups.
Gaussian laws with fixed positive covariance vary continuously in total variation with their means, uniformly over these compact parameter sets. Finite mixing and integration over the rotations preserve this continuity. The cap, terminal marking, and survival restriction contract total variation. Consequently \(k\mapsto Q_N(k,\cdot)\) is TV-continuous on \(K_N\).
The operator \(Q_N:C(K_N)\to C(K_N)\) maps the unit ball to a bounded uniformly equicontinuous family, since
where the TV norm in this proof is \(\sup_{|f|\leq1}|\mu f|\). Arzelà–Ascoli makes \(Q_N\) compact. Full support gives \(Q_Nf>0\) for every nonzero continuous \(f\geq0\), and compactness of \(K_N\) gives \(\min Q_Nf>0\). Moreover continuity of \(Q_N1\) gives constants \(0<a_N\leq Q_N1\leq b_N<1\).
4. Obtain an eigenfunction and verify a Markov minorization. The positive cone of \(C(K_N)\) is total. The compact positive operator has spectral radius at least \(a_N>0\), because \(Q_N^n1\geq a_N^n1\). The compact-operator Krein–Rutman theorem therefore supplies an eigenfunction \(e_N\geq0\), \(e_N\not\equiv0\), with \(Q_Ne_N=\alpha_Ne_N\) and \(\alpha_N=r(Q_N)\). Strong positivity gives \(0<m_N\leq e_N\leq M_N<\infty\), and \(a_N\leq\alpha_N\leq b_N<1\). These verify the hypotheses of Zhang, Theorem 1.1.
Choose a compact target box strictly inside the all-alive position domain and the open velocity ball, and let \(\theta_N\) be normalized Lebesgue measure there. The conditional Gaussian means are bounded and the covariance family is finite and positive definite. Their densities consequently have a common positive lower bound on the inverse-cap image of this target. The inverse-cap Jacobian has a positive lower bound there as well. Hence, for some \(\epsilon_N>0\),
This bound holds for every accepted graph and rotation and therefore also for their actual mixture. The Doob kernel
is Markov and satisfies
Splitting this common part from \(P_N\) contracts total variation by \(1-\delta_N\). Iteration gives its unique invariant law \(\pi_N\) and uniform geometric mixing.
5. Recover the QSD and conditioned convergence. Define
Invariance of \(\pi_N\) gives \(\nu_NQ_N=\alpha_N\nu_N\). Any other QSD \(\nu\) with eigenvalue \(\beta\) satisfies \(\beta\nu(e_N)=\nu Q_Ne_N=\alpha_N\nu(e_N)\), so \(\beta=\alpha_N\). Its normalized \(e_N\)-weighted law must be invariant for \(P_N\) and hence equals \(\pi_N\). This proves QSD uniqueness.
For every bounded measurable \(f\),
Divide by the same formula for \(f=1\). Uniform mixing of \(P_N\) and the positive bounds on \(e_N\) give the stated estimate with, for example, \(r_N=1-\delta_N\) and \(C_N=4(M_N/m_N)^2\).
6. Lift to the complete physical state. Let \(\iota:S_{\rm phys}\to K_{\rm real}\) retain alive positions, every velocity and every mark, and apply \(S_R\) only to dead position coordinates. It is a measurable bijection, with the inverse given in Step 1. Couple the two coordinate descriptions with identical measurement draws, cloning draws, gates, component rotations, jitters, OU innovations, and final position noises. Stage by stage, the complete updates satisfy
with the same extinction event. Consequently, for all bounded measurable \(f\) on \(K_{\rm real}\) and \(g\) on the physical marked state,
These are exact transition identities. The coordinate map changes neither the mechanism nor its survival probabilities.
Every output of \(Q_N\) has finite physical positions and velocity strictly inside the cap. The eigenmeasure identity \(\nu_N=\alpha_N^{-1}\nu_NQ_N\) therefore gives zero QSD mass to artificial compactification points. Applying the inverse feature coordinate to dead slots reconstructs their full physical law. This measurable bijection on actual states preserves the kernel identities and TV bounds, proving the claim for the retained-coordinate algorithm itself.
Remark 179 (Exact velocity collapse at the resonant timestep)
The restriction \(h\ne2\) in the preceding theorem corresponds to an actual change in the canonical dynamics. Take \(N=1\), \(h=2\), and \(U(x)=|x|^2/2\). The sole live walker has no accepted cloning edge and receives no cloning jitter. Writing its entering state as \((x,v)\), the actual stages give
Final position noise and terminal classification leave this pre-cap velocity calculation unchanged. Therefore, on every surviving trajectory,
Any QSD is supported on \(v=0\). Indeed, the \(n\)-step killed output is supported on \(\{|v|\leq V/(n+1)\}\) for every entering state. Its QSD identity \(\nu Q^n=\alpha^n\nu\) with \(\alpha>0\) forces the same support for \(\nu\); intersecting these sets over \(n\) gives \(v=0\). This argument retains the survival weights exactly.
On the invariant set \(v=0\), the physical position update is
It is the actual restricted Gaussian transition, with strictly positive variance \(q^2+2\sigma_x^2\). On the compact box closure its killed kernel has a continuous strictly positive density. The compact-positive-operator and Doob-minorization argument in the preceding proof therefore gives a unique QSD on this invariant set. Since every QSD must be supported there, it is also the unique QSD of the complete one-walker killed process at \(h=2\).
Nevertheless, starting from any \(v_0\ne0\), the velocity at each finite surviving update is the nonzero deterministic vector displayed above. Its conditional law and the QSD have disjoint velocity supports, so their TV norm distance is \(2\) at every finite \(n\). Each such conditioning event has positive probability because the final position noise has positive density. Thus the velocity norm tends to zero, but total-variation convergence to the QSD fails. The loss of the preceding convergence conclusion at \(h=2\) is a property of the programmed BAOAB and cap composition.
Lemma 160 (Exchangeability from kernel symmetry)
If \(Q_N\) commutes with every permutation of walker labels and has a unique QSD, then \(\nu_N\) is exchangeable. Consequently \(\mathbb E_{\nu_N}L_N\varphi=\mu_N\varphi\) for integrable \(\varphi\).
Proof
For a permutation \(\sigma\), the commutation identity implies that \(\sigma_\#\nu_N\) satisfies the same QSD equation as \(\nu_N\). It is a probability, so uniqueness gives \(\sigma_\#\nu_N=\nu_N\). All coordinate marginals therefore coincide, and averaging their expectations proves the last assertion. The required kernel symmetry is proved for the component update in Lemma 167. For the canonical quadratic-force box configuration, Theorem 225 supplies the unique QSD, so both hypotheses are verified for this algorithm.
Theorem 226 (Tightness from a uniform confining moment)
Suppose \(\nu_N\) is exchangeable and there is a nonnegative lower semicontinuous function \(W\) on \(\mathsf Z\) with compact sublevel sets such that
Then the sequence \(\{\mu_N\}\) is tight. The moment hypothesis follows, for example, from integrable swarm Lyapunov functions satisfying
with \(a,b\) fixed, \(\sup_NB_N<\infty\), and \(\inf_N(\alpha_N-r_N)>0\).
Proof
Exchangeability gives \(\mu_NW=\mathbb E_{\nu_N}L_NW\le C_W\). Therefore
The compact set \(\{W\le R\}\) contains at least \(1-C_W/R\) of every marginal. This is tightness; Prokhorov’s theorem then gives weakly convergent subsequences on the Polish space.
For the stated Lyapunov criterion, Theorem 210 gives \(\nu_NV_N\le B_N/(\alpha_N-r_N)\). Substitute this into the domination of \(L_NW\) to obtain the uniform moment bound.
Corollary 74 (Tightness of the random empirical measures)
Under the preceding moment hypothesis, \(\{\Lambda_N\}\) is tight in \(\mathcal P(\mathcal P(\mathsf Z))\).
Proof
For \(R>0\), the set \(\mathcal K_R=\{\mu\in\mathcal P(\mathsf Z):\mu W\le R\}\) is tight by the same compact-sublevel estimate. It is closed under weak convergence by lower semicontinuity, so it is compact by Prokhorov’s theorem. Markov’s inequality now gives \(\Lambda_N(\mathcal K_R^c)\le C_W/R\).
Remark 180 (The confining observable on an open or unbounded domain)
Centred swarm variance alone does not bound the spatial barycentre. On an unbounded domain, \(W\) must use the established confining position envelope. On an open bounded domain, a Euclidean ball intersected with that domain need not be compact in its interior topology; control near the excluded boundary may also be needed. The integrable barrier analysis in The Keystone Principle and the Contractive Nature of Cloning provides the corresponding moment inputs where its hypotheses hold. Tightness permits atomic limits. Absolute continuity or regularity of a limit requires a smoothing or equation argument.
Population-size scaling of collective fluctuations
Rust engine experiment · Seed 7 · Poster after 1 experiment steps. Interactive view starts from the same seed.
2. Empirical Convergence and the Measurement Pipeline#
Once a population law converges, bounded continuous measurements of it converge too. The regularizers in the algorithm keep divisions stable. There are two points to watch: the companion law is weighted by its kernel, and fitness is a nonlinear function of a sampled distance. Replacing that sampled distance by its expectation before evaluating fitness changes the quantity being averaged.
Lemma 161 (Empirical measures, mixtures, and chaos)
For exchangeable \(\nu_N\), if \(\Lambda_N\Rightarrow\Lambda\), then every fixed \(l\)-coordinate marginal converges to
In particular, \(\Lambda_N\Rightarrow\delta_\rho\) implies chaos toward \(\rho\). Conversely, convergence of the first two coordinate marginals to \(\rho\) and \(\rho^{\otimes2}\) implies \(\Lambda_N\Rightarrow\delta_\rho\) whenever the empirical laws are tight. Removing one tagged walker does not change the empirical limit, since for \(\|\varphi\|_\infty\le1\),
For a companion population of \(k_N\) alive walkers, the same estimate is \(2/k_N\) when the excluded tag is alive. A positive alive-fraction bound makes this \(O(1/N)\); the full-population formula does not replace that normalization.
Proof
For \(\|\varphi_j\|_\infty\le1\), the product \(\prod_{j=1}^lL_N\varphi_j\) averages over sampling indices with replacement. Conditioned on all sampled indices being distinct, exchangeability gives the expectation of \(\prod_{j=1}^l\varphi_j(z_j)\). The probability of a repeated index is at most \(l(l-1)/(2N)\), so the two expectations differ by at most \(l(l-1)/N\). The map \(\eta\mapsto\prod_j\eta\varphi_j\) is bounded continuous for bounded continuous \(\varphi_j\). Pass to the limit in its expectation to obtain the mixture formula. Product test functions determine probability measures on finite products; tightness allows the usual approximation on compact sets.
For the converse, exchangeability gives
The assumed first and second marginal limits therefore imply \(\mathbb E|L_N\varphi-\rho\varphi|^2\to0\) for every bounded continuous \(\varphi\). Apply this to a countable convergence-determining family and use tightness to identify every empirical-law limit as \(\delta_\rho\). For the final inequality, subtract the two finite averages and bound the tagged summand and the mean by one. The identical calculation with \(k_N\) proves the alive-population version.
Remark 181 (A symmetric mixture need not become independent)
For distinct points \(a,b\), the law \(\nu_N=\tfrac12\delta_a^{\otimes N}+\tfrac12\delta_b^{\otimes N}\) is exchangeable for every \(N\). Its first marginal is always \(\tfrac12(\delta_a+\delta_b)\), while its empirical law is always \(\tfrac12\delta_{\delta_a}+\tfrac12\delta_{\delta_b}\). The empirical law therefore remains random. No infinite exchangeable extension or law of large numbers can turn this mixture into the product of its first marginal.
Lemma 162 (Continuity of reward moments)
If \(\rho_n\Rightarrow\rho\) and \(R\) is bounded continuous, then
For unbounded \(R\), the same statement holds when the squared rewards are uniformly integrable under \(\rho_n\) and integrable under \(\rho\).
Proof
For bounded \(R\), apply weak convergence to \(R\) and \(R^2\) and then continuity of multiplication. For unbounded \(R\), truncate with continuous cutoffs, pass to the limit for the bounded functions, and let the cutoff grow. Uniform integrability controls the discarded tails uniformly in \(n\); Cauchy-Schwarz also controls the first-moment tails.
Lemma 163 (Continuity of companion-weighted distance moments)
Let \(k(z,z')\) be a bounded continuous companion weight, and define
Suppose the denominators are positive and bounded away from zero on compact sets of \(z\), uniformly along a weakly convergent sequence \(\rho_n\Rightarrow\rho\). Then \(J_{\rho_n}\Rightarrow J_\rho\). Consequently, for bounded continuous algorithmic distance \(d\),
For unbounded \(d\), require uniform integrability of \(d^2\) under the joint companion laws. Uniform companion selection is the special case \(k=1\).
Proof
For a bounded continuous test \(g(z,z')\), the integrals \(\int k(z,z')g(z,z')\rho_n(dz')\) converge uniformly for \(z\) in a fixed compact set. To see this, restrict \(z'\) to a compact set containing all but arbitrarily small mass of the tight family \(\{\rho_n,\rho\}\); uniform continuity on the product compact gives a finite net in \(z\). Weak convergence handles its finitely many points, and boundedness controls the tail. The same argument applies to \(Z_{\rho_n}\). The denominator bound therefore gives uniform convergence of their ratios on each compact set. Integrate the ratio against \(\rho_n\), using tightness again, to obtain convergence of \(J_{\rho_n}g\). Apply this to \(d\) and \(d^2\), or truncate their tails under the stated integrability hypothesis.
Lemma 164 (Explicit Lipschitz bounds for normalized measurements)
Use \(\|\cdot\|_1\) for the full variation norm of signed measures or the \(L^1\) norm of densities. If \(|R|\le M_R\), then
If \(0\le k\le1\) and \(Z_\rho,Z_\eta\ge a>0\) uniformly, then
The corresponding distance mean and variance constants are \(M_D(1+2/a)\) and \(3M_D^2(1+2/a)\) when \(|d|\le M_D\). For alive subprobabilities of mass at least \(m_*>0\),
Proof
The mean estimate is the dual bound for integration. The second moment difference is at most \(M_R^2\|\rho-\eta\|_1\), while the difference of the squared means is at most \(2M_R^2\|\rho-\eta\|_1\). For a fixed \(z\), add and subtract \(k(z,\cdot)\eta/Z_\rho(z)\). The numerator difference contributes at most \(\|\rho-\eta\|_1/Z_\rho(z)\) and the normalization difference contributes at most the same quantity. For \(J\), also change its first marginal, which contributes \(\|\rho-\eta\|_1\). The distance estimates follow by the reward calculation on \(J\). The alive normalization bound follows from the same add-and-subtract argument and \(|\int f-\int g|\le\|f-g\|_1\).
Lemma 165 (Lipschitz continuity of the actual regularized fitness pipeline)
On a class where the preceding measurement bounds hold, suppose the fixed regularized standard-deviation map \(s\) has lower bound \(s_*>0\) and Lipschitz constant \(L_s\) on the attained variance range. Let the rescale map \(g\) be Lipschitz and have range in \([g_*,g^*]\subset(0,\infty)\). For fixed \(\alpha,\beta\ge0\), the sampled fitness
is bounded and Lipschitz as a function of \(\rho\) in the uniform norm over \((z,z')\). These are the multiplicative fitness operations of Definition 496, with moments taken under the specified companion law. The expected fitness is obtained by integrating this sampled quantity against that law.
Proof
For bounded raw measurement \(u\) and two means and scales,
The preceding lemma bounds the mean and variance differences; Lipschitz continuity of \(s\) bounds the scale difference. Apply the Lipschitz bound for \(g\). The maps \(x\mapsto x^\alpha\) and \(x\mapsto x^\beta\) have finite Lipschitz constants on \([g_*,g^*]\), including the constant map when an exponent is zero. Finally, expand the difference of the two products. Integrating against \(K_\rho\) adds its Lipschitz contribution, bounded by the fitness supremum times \(\|K_\rho-K_\eta\|_1\).
When cloning probability is nonlinear in sampled fitness, its mean must be computed by integrating that probability over the sampled fitness variables. Evaluating it at expected fitness is a different operation, as specified in Remark 118.
Lemma 166 (Uniform integrability and passage to expectations)
If random variables \(Y_n\) converge in distribution to \(Y\), their absolute values are uniformly integrable, and \(Y\) is integrable, then \(\mathbb EY_n\to\mathbb EY\). In particular, this applies to bounded continuous functions of a convergent random empirical measure. A sufficient uniform-integrability condition is \(\sup_n\mathbb E|Y_n|^{1+\epsilon}<\infty\) for some \(\epsilon>0\).
Proof
Clip \(Y_n\) to \([-R,R]\). Expectations of the clipped variables converge by weak convergence. The difference from their original expectations is bounded by \(\mathbb E[|Y_n|1_{\{|Y_n|>R\}}]\), uniformly small as \(R\to\infty\). The same tail vanishes for \(Y\). For the sufficient condition, this tail is at most \(R^{-\epsilon}\mathbb E|Y_n|^{1+\epsilon}\).
3. The Actual One-Step Consistency Estimate#
Sit on a randomly chosen walker and look at all the accepted cloning edges attached to it. Some point toward its donor. Others come from walkers that selected it. Those incoming walkers matter: they change the component’s centre-of-mass velocity, and they share its random rotation.
There is a useful ordering hidden in this graph. An accepted live edge goes strictly uphill in sampled fitness. A dead walker points to a live donor, and no walker points to a dead donor. Thus a component has no cycle. The ordering also bounds long paths, even when acceptance probabilities are large. This is the estimate that lets a local component describe a walker in a very large swarm.
Definition 586 (Canonical fixed-step regime and convergence class)
Use the full update \(\mathcal F_h\) constructed in The Mean-Field Law of the Euclidean Gas: independent current Gaussian-weighted measurement and cloning companions, global regularized fitness statistics, frozen order-one acceptance, mandatory revival through the same weighted current-donor law, jitter for all accepted recipients, and one independent Haar \(O(d)\) rotation per accepted undirected component. Follow this by the specified constant-noise BAOAB step, independent final position diffusion, radial velocity cap, and terminal boundary classification. The canonical configuration has no donor history and no viscosity; the potential has globally Lipschitz gradient with linear growth. The timestep \(h>0\), positive final position-noise amplitude, finite cap \(V\), positive fitness floors, positive standardization regularizers, and donor widths are fixed independently of \(N\).
The squashed algorithmic distances give actual weight bounds \(0<\kappa_D\le w_D\le1\) and \(0<\kappa_C\le w_C\le1\). They do not bound physical positions. Take either the terminal absorbing box, or the unbounded confining configuration with quadratic reward. In the box, alive positions are bounded and all retained velocities obey the cap. In the unbounded case, assume initial empirical convergence and a uniform initial \((4+\delta)\) position moment for some \(\delta>0\); independent initialization with that moment supplies this condition. All slots are alive in the unbounded case.
Write \(\mu_N=L_N(S_N)\) for deterministic input arrays and suppose \(\mu_N\Rightarrow\mu\) with \(m(\mu)>0\). In the unbounded case use the stated moment bound as well. Every sufficiently large \(N\) then has \(m(\mu_N)\ge m_*>0\). The finite-step moment and terminal-survival estimates in The Mean-Field Law of the Euclidean Gas propagate this class over each fixed finite horizon.
Lemma 167 (Permutation equivariance of the complete canonical kernel)
The canonical transition kernel commutes with every permutation of slot labels. Consequently it preserves exchangeability, including the marked alive/dead state and the cemetery event.
Proof
Let \(\pi\) relabel an input array. Transport a donor index \(j\) to \(\pi(j)\), each row innovation to the corresponding relabelled row, and each component rotation to the relabelled vertex set. Distances, eligible donor sets, and normalization sums are unchanged by this operation. The conditional probability of every transported donor choice is therefore unchanged. The empirical reward and diversity statistics are symmetric sums; thus fitness, acceptance probabilities, and transported gate outcomes agree.
The accepted undirected graph is relabelled by \(\pi\). Its components, their sizes, their frozen position sources, and their centre-of-mass velocities are transported exactly. Assigning independent Haar matrices to components has the same joint law after any such component permutation. With those matrices transported, the component update commutes pointwise with \(\pi\). Row jitter, the prescribed force evaluations, BAOAB noises, cap, and terminal boundary tests do likewise. Extinction depends on the number of eligible slots, which is unchanged. Integrating the transported innovations proves the kernel identity. This is a statement about the random kernel: keeping numerical random addresses fixed while relabelling the inputs need not give the same sample path.
Lemma 168 (Concentration of the actual sampled measurement marks)
Let \(\eta_\mu\) be the limiting type law \((z,y_D,F)\) constructed from the weighted measurement companion and the actual sampled fitness in The Mean-Field Law of the Euclidean Gas. Under Definition 586, the finite empirical type law \(\eta_N\) converges in probability to \(\eta_\mu\). Every bounded continuous type test converges in \(L^2\) as well. In the box, the random diversity mean and second moment have conditional variances bounded by constants times \(1/N\).
Proof
Measurement before normalization. Condition on the deterministic input array. A live row’s companion has probabilities
The full empirical-kernel denominator is at least \(\kappa_Dm_*\). Removing the self atom changes the normalized companion law by at most \(2/[\kappa_D(m_*N-1)]\) in full variation, for sufficiently large \(N\). The companion draws of distinct measurement rows are conditionally independent, although the selected indices can coincide. For any bounded test \(g\) of \((z,y_D)\), the conditional variance of its empirical average is at most \(\|g\|_\infty^2/N\). Its conditional expectation tends to the integral against the weighted joint law, by Lemma 163 and the self-exclusion bound. Assign a dummy measurement mark to dead rows; their contribution is a deterministic empirical integral.
The global statistics. Apply the same calculation to the sampled bounded diversity and its square, restricted to live rows, and divide by the deterministic alive fraction. Their conditional variances are bounded by constants times \(1/(m_*^2N)\). Reward statistics are deterministic functions of the input array. In the box they converge by boundedness; for quadratic reward in the unbounded case the \((4+\delta)\) position moment gives uniform integrability of squared reward. Lemma 162 applies.
Fitness remains a mark. Replace only the empirical normalization statistics by their limits. Positive regularizers and positive bounded rescale maps make the fitness map continuous. On bounded reward sets the replacement error tends uniformly to zero; the unbounded case follows by restricting the row type to a compact set and then using tightness. The sampled diversity itself is retained in every row’s fitness. It is not replaced by its mean. Combining this replacement with the preceding empirical joint-law convergence proves convergence of \(\eta_N\). A bounded continuous type test is uniformly bounded, so convergence in probability also gives its \(L^2\) convergence.
Lemma 169 (Uniform collision-component truncation)
Condition on the input swarm and every measurement mark. For sufficiently large \(N\) with at least \(m_*N\) live rows, each accepted recipient-to-donor edge has probability at most \(C/N\), where one may take \(C=2/(\kappa_Cm_*)\). Let \(\mathcal C_N(i)\) be the component of a tagged vertex \(i\). Then
and \(\mathbb P(|\mathcal C_N(i)|>K\mid S,\text{measurements})\le e^{2C}/K\). These bounds require no small-acceptance assumption.
Proof
A live row has at most one accepted outgoing edge, and that edge strictly increases frozen fitness. A dead row has one outgoing revival edge to a live row and cannot be a target. Put dead vertices below live vertices in a total ordering compatible with fitness, breaking ties arbitrarily. Every edge strictly increases this ordering. An undirected cycle would have as many edges as vertices; because each vertex has at most one outgoing edge, all cycle vertices would have exactly one outgoing cycle edge, giving a directed cycle. This contradicts strict ordering. The graph is a forest.
Consider a simple length-\(\ell\) path from \(i\). Along that path arrows cannot point outward in both directions from an internal vertex. They therefore point toward a single sink, with an increasing leg of length \(a\) and a decreasing leg of length \(\ell-a\), for some \(0\le a\le\ell\). There are at most \(N^a/a!\) possible label sequences on the increasing leg: choose its labels, whose order is then determined. There are at most \(N^{\ell-a}/(\ell-a)!\) choices for the other leg. Ignoring overlap and additional constraints at the sink only enlarges this bound.
Each path edge uses a distinct recipient’s outgoing draw. Conditional on all measurement marks these row draws are independent, and each required edge has probability at most \(C/N\). This includes weighted mandatory revival: its denominator is at least \(\kappa_Cm_*N\). Thus the expected number of such paths is at most
A vertex at radius \(r\) supplies a length-\(r\) path. Summing the path bound over \(\ell\ge0\) bounds the number of vertices by \(e^{2C}\). Markov’s inequality gives the size-tail estimate. Averaging over measurement marks preserves all bounds. The constant can be large; the proof asserts finite component control, not a sharp practical estimate of component size.
Lemma 170 (Every collision-component moment is uniformly bounded)
Under the conditioning of Lemma 169, for each integer \(p\geq1\) and every fixed root \(i\),
The estimate also holds when any fixed subset of rows has its outgoing edge removed. It uses neither weak acceptance nor a subcritical branching assumption. In particular, \(M_2(C)=(1+2C)e^{4C}\) and \(M_3(C)=(1+6C+3C^2)e^{8C}\) are admissible constants.
Proof
Freeze all measurement marks, including their global normalization statistics. Order the vertices by fitness, with dead rows below live rows. Every accepted edge increases this order, and different rows choose their outgoing edges independently. Starting from \(i\), follow its outgoing edges to the unique terminal vertex, and let \(A\) denote the number of vertices in this ancestor chain. The increasing-path count gives
Condition now on an exact realized ancestor chain, including the terminal row’s absent outgoing edge. This event only specifies draws of chain rows; all other row draws retain their independent distributions. The entire component is the descendant closure of that chain. Initialize the discovered set with its \(A\) vertices and scan every other row in decreasing order. At the time a row is scanned, all its possible parents have already been scanned or are chain seeds. It joins the component exactly when its outgoing edge hits the currently discovered set. If that set has size \(s\), the conditional probability of joining is at most \(Cs/N\).
For \(s\geq1\), \((s+1)^p-s^p\leq(2^p-1)s^{p-1}\). Hence each scan step satisfies
This inequality remains valid when \(Cs/N>1\), since it is used only as an upper bound on a probability. There are at most \(N\) scan steps, giving \(\mathbb E[|\mathcal C_N(i)|^p\mid\text{chain}] \leq e^{(2^p-1)C}A^p\). Average over the chain. Removing outgoing rows preserves independence, the order, and the edge-probability bound, so the same proof applies. The factorial series converges for every fixed \(p\).
Theorem 227 (Rooted collision convergence and two-root independence)
Under Definition 586, the component around a uniformly sampled root converges to the marked rooted component used to define \(\mathcal J(\mu)\) in The Mean-Field Law of the Euclidean Gas. Two uniformly sampled distinct roots converge jointly to independent copies of that rooted law. Consequently, for every bounded continuous test \(\varphi\) of the post-clone state,
The collision readout uses one shared rotation per component, frozen velocities of every component member including revived rows, and independent jitter for every accepted recipient.
Proof
1. Freeze the type array. By Lemma 168, every subsequence has a further subsequence on which the empirical types converge almost surely to \(\eta_\mu\). It suffices to prove the assertion for each such deterministic convergent type sequence. Write \(t=(z,y_D,F)\) and
Here \(Z_C(\mu;z)=\int1_{\{a_y=1\}}w_C(z,y)\mu(dy)\). The finite edge probabilities are \(N^{-1}\beta\) evaluated using the empirical law, with the explicit self-exclusion correction. The kernels are bounded by \(C\), continuous in their type arguments on each status stratum, and their integrals converge. Fitness ties cause no discontinuity because acceptance vanishes continuously at a tie.
2. Explore a bounded number of vertices. Expose the root’s type and its outgoing edge, if present. A newly exposed target has its own free outgoing draw. To discover incoming edges to a target \(t\), scan the still unexposed rows. Each row independently hits \(t\) with probability at most \(C/N\). For finitely many targets and type-test bins, their incoming counts converge to independent Poisson counts with intensities \(\eta_\mu(du)\beta_\mu(u,t)\).
Here is the rare-event estimate underlying that assertion. A row that hits one of \(k\) specified targets has a categorical law of total probability \(p_j\le Ck/N\). Couple it to independent Poisson counts with the same category means. Expanding \(e^{-p_j}\) shows that the probability of a mismatch is at most \(2p_j^2\) for \(p_j\le1/2\): the errors are absence versus one-hit probabilities and the Poisson probability of two or more hits. Summing over rows bounds the mismatch by \(2C^2k^2/N\). For bounded nonnegative mark tests \(g\), the same limit is read directly from
Taking logarithms changes this product to the exponential of the sum of its linear terms with an error \(O(C^2k^2/N)\). Empirical kernel convergence identifies the limiting marked Poisson intensities. This establishes the point-process statement, rather than just unmarked count convergence.
If a row has already been found not to hit an earlier exposed target, its remaining hit probabilities are divided by \(1-p_j^{\rm old}\), with \(p_j^{\rm old}\le Ck/N\). The resulting total change is \(O(C^2k^2/N)\). Removing the finitely many exposed labels contributes another \(O(Ck^2/N)\). For an exploration stopped after \(K\) vertices, summing these estimates gives an error at most \(A(C,K)/N\), with a finite constant, in addition to the converging empirical kernel integrals. One may use a constant of order \((1+C)^2K^3\); no uniform estimate in unbounded \(K\) is needed here.
3. Respect the direction information. An incoming child has already used its outgoing choice to select its parent. It receives no second free outgoing draw. Its additional incoming children form the corresponding Poisson process. When a known source already points to a newly exposed target, it is excluded from that target’s additional incoming process. These rules are exactly the rooted construction in The Mean-Field Law of the Euclidean Gas. They retain shared donors and all component members.
4. Remove truncation. The chance that a root requires more than \(K\) vertices is at most \(e^{2C}/K\) by Lemma 169. The same bound holds in the limit: apply the finite-exploration convergence to the event of discovering \(K+1\) vertices. Thus the limiting component is finite almost surely. First take \(N\to\infty\) with \(K\) fixed, then \(K\to\infty\). On a finite typed graph, the centre-of-mass formula, shared Haar rotation, copying, and jitter are continuous readouts. Their innovation laws agree in the finite and limiting constructions. Boundedness of \(\varphi\) controls the discarded events and proves the one-root assertion.
5. Explore two roots. Explore the two components together with a \(K\)-vertex cutoff for each. An outgoing draw has probability at most \(2CK/N\) to hit an exposed vertex; the joint incoming Poisson approximation above also controls rows that could connect the two explorations. The chance of a label collision tends to zero. The limiting incoming processes and free outgoing draws are independent between the two explorations. Their component rotations are independent unless the components intersect, an event whose probability vanishes. More directly, for two uniform distinct labels \(I,J\),
The empirical normalization statistics have a deterministic limit by Step 1, so they leave no additional common random variable. Remove both cutoffs to obtain independent rooted limits.
Finally express the second moment of an empirical average as its diagonal \(1/N\) contribution plus the expectation at two uniform distinct roots. The latter converges to \((\mathcal J(\mu)\varphi)^2\), and the first moment converges to \(\mathcal J(\mu)\varphi\). This proves the displayed \(L^2\) consistency without assuming independent collision outputs.
Lemma 171 (Continuity of the canonical population map)
On the convergence class of Definition 586, the actual clone/collision map \(\mathcal J\) and full step \(\mathcal F_h\) are continuous for weak convergence, with the stated moment control in the unbounded case. In particular, their restrictions to any compact subset of this class are uniformly continuous for a metric inducing that convergence.
Proof
For \(\mu_j\to\mu\), positive alive mass bounds all donor denominators away from zero eventually. The companion and reward lemmas give convergence of the type laws and their normalization statistics. The rooted construction truncated at \(K\) involves only finitely many kernel integrations, Poisson intensities, and continuous finite-component readouts. Each is continuous in \(\mu\). The component bound is uniform along this sequence; letting \(K\to\infty\) proves continuity of \(\mathcal J\).
For the canonical kinetic step, the deterministic BAOAB kicks and drifts are continuous because the force is globally Lipschitz. Its O step uses \(c=e^{-\gamma h}\) and the exact integrated constant-noise variance \(s_h^2=(1-e^{-2\gamma h})/(2\gamma)\), with \(s_h^2=h\) at \(\gamma=0\). Independent row noise acts through continuous Markov kernels. The radial velocity cap is continuous. Final position noise has a strictly positive Gaussian variance; its convolution gives zero mass to the box boundary. Thus terminal status classification is continuous almost everywhere under the entering limiting noise law, which suffices for convergence of bounded continuous marked tests. In the unbounded configuration there is no such boundary discontinuity. Finite-horizon moment bounds give tightness and the uniform integrability needed in the unbounded reward passage.
Uniform continuity on a compact subset follows by contradiction: two sequences whose input distance tends to zero have convergent subsequences with the same limit; continuity then makes their output distance tend to zero. This argument supplies a continuity modulus. It does not assert a Lipschitz constant or contraction.
Theorem 228 (Full one-step consistency for the canonical algorithm)
For the deterministic input arrays of Definition 586, let \(L_N'\) be the empirical marked law after the actual full step. On total extinction, assign it any fixed probability measure. For every bounded continuous \(\varphi\),
For a bounded metric \(d_*\) inducing weak convergence,
These are conclusions of the specified donor, fitness, collision, and kinetic rules; they are not assumed consistency hypotheses.
Proof
The rooted theorem proves empirical concentration immediately after cloning. A continuous deterministic row map transports this concentration. For an independent-noise row kernel \(P\), condition on its input population. The empirical test average has conditional variance at most \(\|\varphi\|_\infty^2/N\) and conditional mean \(L_NP\varphi\). The latter converges because \(P\varphi\) is bounded continuous. Iterating this observation over B1, A1, O, A2, B2, final position noise, and cap proves the full-step statement before classification. For the classification step, integrating the final Gaussian position noise gives a continuous kernel even for the alive/dead marked readout: its only spatial discontinuity is a box boundary of Gaussian measure zero. The exact boundary schedule is essential.
The alive-fraction estimate of The Mean-Field Law of the Euclidean Gas makes total-extinction probability tend to zero, uniformly on admitted box inputs. Consequently the arbitrary cemetery extension has no effect on any bounded limit. The moment bounds there provide tightness. Convergence of bounded tests from a countable determining family implies weak convergence in probability of the empirical law; boundedness of \(d_*\) gives convergence of its expected distance to \(\mathcal F_h(\mu)\). The triangle inequality and Lemma 171 replace \(\mu\) by \(\mu_N\).
Shared component collisions and conserved momentum
Rust engine experiment · Seed 7 · Poster after 1 experiment steps. Interactive view starts from the same seed.
Collision forests and tagged-walker dependence
Rust engine experiment · Seed 7 · Poster after 1 experiment steps. Interactive view starts from the same seed.
4. Finite-Horizon Chaos at Fixed Timestep#
We now have the crucial comparison: start a large swarm with a known population distribution, run one complete update, and its empirical output approaches the distribution prescribed by \(\mathcal F_h\). The same reasoning can be applied to the next update. A fixed number of repetitions requires continuity and survival over that horizon. It does not require the evolution to forget its initial state.
That last distinction matters for the long-time problem. A map can be continuous and accurately approximated by large swarms without carrying all initial populations toward the same equilibrium. The finite-time proof and the stationary-attraction problem therefore have different final steps.
Theorem 229 (Finite-horizon propagation of chaos for the actual update)
Let \(S_0^N\) have exchangeable initial laws with \(L_N(S_0^N)\to\mu_0\) in probability, where \(\mu_0\) is deterministic and has positive alive mass. Assume the canonical regime above, including its initial moment condition in the unbounded case. Define
For every fixed integer \(n\ge0\),
The empirical trajectory at any fixed finite list of update indices converges jointly to the corresponding deterministic trajectory. At every fixed horizon, extinction probability tends to zero.
Proof
Pass from deterministic arrays to random inputs. The one-step theorem has the sequential property that every deterministic sequence of admissible arrays with empirical limit \(\mu\) has output empirical limit \(\mathcal F_h(\mu)\). Suppose a random sequence converges in probability to that same \(\mu\). From every subsequence choose a further one with almost-sure empirical convergence. If the conditional expected one-step error did not tend to zero in probability, there would be input realizations converging to \(\mu\) for which that error remains bounded away from zero. This contradicts the deterministic sequential property. Boundedness then turns convergence in probability of the conditional error into convergence of its expectation. For unbounded rewards, restrict first to a compact set in a Wasserstein \(p\) topology with \(4<p<4+\delta\); the uniform \((4+\delta)\) moment bounds make the complement arbitrarily unlikely. The same contradiction argument applies on each such compact set.
Iterate the actual map. Assume the claim at update \(n\). The terminal alive-fraction bound gives a positive limiting alive mass and makes an exceptionally small finite alive count negligible. The moment bounds keep the input sequence in the convergence class. Apply the random-input argument and continuity of \(\mathcal F_h\) to obtain the claim at update \(n+1\). This begins with the stated initialization. The union bound over a fixed number of updates gives the extinction conclusion and joint empirical trajectory convergence.
Convert empirical convergence to particle chaos. Kernel equivariance preserves exchangeability at every update. The sampling-without-replacement argument in Lemma 161 compares the first \(\ell\) coordinates to \(\ell\) independent samples from the empirical law, with error at most \(\ell(\ell-1)/N\) on bounded product tests. The deterministic empirical limit then gives \(\mu_n^{\otimes\ell}\).
Remark 182 (Quantitative errors and the continuity modulus)
For the canonical kernel, Theorem 231 and Theorem 230 give an \(O(N^{-1/2})\) conditional bias and an \(O(N^{-1})\) conditional variance for each bounded observable, relative to \(\mathcal F_h(\mu_N)\) at the entering empirical law. These estimates do not prescribe the rate at which an arbitrary entering sequence \(\mu_N\) approaches a specified law \(\mu\), nor a rate in every metric on probability measures. Iterating an empirical-distance estimate also requires control of that metric and of the map’s continuity modulus. The geometric cluster argument for contraction remains distinct from these one-step fluctuation estimates.
On a compact invariant class let \(\omega\) be the continuity modulus of \(\mathcal F_h\), and let \(e_n\) denote the expected bounded empirical distance to \(\mu_n\). If \(\delta_N\) bounds the expected one-step error on that class, then, for every \(r>0\),
where \(D_*\) bounds the metric diameter. Indeed, on the event that the input distance is at most \(r\), the output-map distance is at most \(\omega(r)\); on its complement use \(D_*\) and Markov’s inequality. Tightness permits a compact restriction with an additional arbitrarily small exceptional probability. Choose \(r\), then \(N\), to iterate convergence over fixed \(n\). This uses the proved continuity, without replacing it by an unproved Lipschitz or contractive estimate.
Remark 183 (Random initial populations)
If the initial empirical law converges to a random directing measure \(M_0\), the same argument, conditional on that limit, gives \(M_n=\mathcal F_h^n(M_0)\). Fixed particle marginals converge to \(\mathbb E[M_n^{\otimes\ell}]\). This preserves the common macroscopic randomness. It reduces to deterministic chaos exactly when the directing measure is almost surely fixed at the observation time.
Remark 184 (Fixed timestep, continuous time, and surviving trajectories)
The theorem concerns the actual map at fixed \(h\). It does not replace simultaneous cloning by a finite-rate jump process. A continuous-time identification requires convergence of the iterates \(\mathcal F_h^{\lfloor t/h\rfloor}\) for the specified parameter family, and finite-population errors controlled over the same growing number of updates. The complete update, including collision, immediate revival, jitter, and repeated cap, must enter that analysis.
The finite-horizon theorem uses an arbitrary extension after total extinction, with vanishing probability of visiting that extension. The law conditioned on survival through the entire fixed horizon has the same limit: conditioning changes any bounded test expectation by at most twice its supremum times the extinction probability. Conditioning each intermediate transition separately is a different path-law operation.
Lemma 172 (The actual boundary and revival contributions converge)
In the canonical box regime, every dead recipient takes a current live donor with the Gaussian weight evaluated from its retained coordinates, receives the prescribed jitter and component collision, and is marked alive before kinetics. The empirical post-revival law and the terminal alive and dead subprobabilities converge to these same stages of \(\mathcal F_h\).
Proof
The dead-row density \(\beta_\mu(t,u)\) in the rooted theorem integrates to one over live targets and contains the actual retained-position donor weight. Thus the rooted output includes every revived recipient and its shared component rotation. There is no unrevived finite-rate reservoir at this stage. The kinetic consistency proof supplies the terminal position law. Its Gaussian final-noise convolution has zero boundary mass, so restriction to the box and its complement converges by the continuity-set criterion. These restrictions give the alive and dead subprobabilities; their sum is the full marked probability.
4.1. An additional variance tool#
Lemma 173 (Independent innovations with controlled influence)
Conditioned on the input swarm, let a proposed full update be a function of independent innovations \(\xi_1,\ldots,\xi_M\). Write \(F=\langle L_N',\varphi\rangle\) and let \(F^{(j)}\) replace innovation \(j\) by an independent copy. Then
If the replacement affects at most \(D_j\) output walker coordinates, then
Thus \(\sum_j\mathbb ED_j^2\le C N\) is a sufficient full-step variance estimate of order \(1/N\), even when the outputs themselves are dependent.
Proof
Reveal the independent innovations in order and form the Doob martingale of \(F\). Its orthogonal differences sum to the conditional variance. For the \(j\)th difference, conditional Jensen’s inequality bounds its second moment by the conditional variance produced by varying innovation \(j\) while holding the others fixed, averaged over the remaining innovations. That variance equals one half of the mean squared difference of two independent copies, giving the first inequality after summation. Changing \(D_j\) bounded summands of the empirical average changes it by at most \(2D_j\|\varphi\|_\infty/N\). Substitute this bound into the first inequality. Any intermediate status checks and common collision updates are part of the function whose influence is being estimated.
Hold the input swarm fixed and repeat one complete update. How much can its empirical average fluctuate? Replace just one measurement draw. Every fitness normalizer must be recomputed, so even distant walkers can acquire different gate probabilities. A changed gate can then join or split collision groups, changing the shared rotation experienced by several walkers. The following proof keeps this entire chain of effects. Its key estimate bounds the expected square of the number of affected output slots independently of \(N\); it does not require a deterministic bound on that number.
There are order \(N\) independent innovation blocks, while each affected slot contributes order \(1/N\) to an empirical average. The squared-influence estimate therefore gives conditional variance of order \(1/N\). The separate bias estimate compares the average output with the actual population map at the same empirical input. That bias is order \(N^{-1/2}\), giving a mean-square one-step error of order \(1/N\) with configuration constants independent of population size.
Collision components serve a specific purpose here: they measure how far fresh randomness can spread during one update. The geometric fitness and error clusters have a different job in the stationary argument: they track the evolution of error already present in the input swarm. The one-step noise estimate supplies that argument’s fluctuation term. Closing the stationary estimate still requires controlling the error flux between those clusters, including its sign for small errors.
Lemma 174 (Squared influence of the actual sampled measurement)
In either canonical regime, fix an input with at least \(m_*N\) alive slots. Couple two updates by replacing one row’s measurement companion and using identical remaining row innovations and identical rotations on every unchanged component. Recompute the global fitness statistics in both updates. Let \(D_r\) be the number of output rows that can differ, before any collapse of the entire marked output to a cemetery state. There is an explicit finite configuration constant \(A_D\), independent of \(N\), such that
Replacing one row’s cloning donor and gate block instead gives the bound \(9M_2(C)\). A local jitter or kinetic block affects at most one final row; replacing the rotation assigned to a component affects only that component.
Proof
1. Quantify every changed acceptance probability. Let \(S_*\) bound the nonnegative squashed separation \(s\), let \(\sigma_s>0\) be its standardization regularizer, and use the fitness parameters of Definition 580. Changing one sampled separation changes the alive empirical mean by at most \(S_*/(m_*N)\), its second moment by at most \(S_*^2/(m_*N)\), and its variance by at most \(3S_*^2/(m_*N)\). Consequently, for every unchanged measurement row, its standardized separation changes by at most \(L_q/N\), where
Reward statistics are unchanged. Define
with \(H_s=0\) when \(p_s=0\). The logistic derivative is at most \(A_s/4\), so \(|F_i-\widetilde F_i|\leq L_0/N\) for \(i\ne r\). This uses the bounded reward rescaling factor, and is valid even when physical rewards or retained dead coordinates are unbounded. Row \(r\) may change by order one. For the actual clipped cloning probability, a Lipschitz constant in the sum of its two fitness arguments is
Retain identical cloning donor draws and gate uniforms. Conditional on both complete measurement arrays, the event that row \(i\)’s accepted outgoing edge changes has probability \(q_i\leq B/N\) for \(i\ne r\), where \(B=C+2L_aL_0\). The \(C/N\) term includes choosing donor \(r\); otherwise both fitness arguments change by at most \(L_0/N\). Mandatory revival is covered: its gate is identically one, and its frozen donor law is unchanged.
2. Expose exceptional rows, not their surrounding components. Put \(r\) in an exceptional set \(E\) regardless of its gate outcome, and include every other row whose accepted outgoing edge changes. These row events are independent under the preceding conditioning. With \(Q=|E|\),
Suppose \(N\geq2B\). Condition on \(E\) and the two outgoing outcomes of every exceptional row. Every remaining row has a common outgoing edge in the two graphs; its conditional probability of a specified edge is at most
These remaining row draws are still independent, since the conditioning factorizes by row. Remove the exceptional outgoing edges. The resulting common graph is an ordered forest, using the first measurement array’s fitness order. This remains true if the two arrays order some fitnesses differently: each common accepted edge respects both orders.
All affected vertices belong to components of this common forest meeting an exceptional recipient or one of its two donor endpoints. There are at most \(3Q\) such seeds. The seeds are fixed under the conditioning, independently of the remaining row draws. Restoring exceptional edges can only join components already meeting these seeds. Thus, by Cauchy–Schwarz and Lemma 170,
No component-size estimate has been conditioned on the event that a random component is unusually influential; the order of exposure above is what permits the uniform moment bound. Averaging proves the claim with
For the omitted finite sizes \(N<2B\), simply use \(D_r\leq N<2B\).
3. Couple the remaining full mechanism. Give each possible component representative an independent Haar rotation, using its smallest slot label as representative. This realizes exactly one independent Haar matrix per component. If a component is unchanged, its representative and rotation are identical in both updates. Component copying, center-of-mass velocities, and the prescribed correlated rotation can therefore change only rows in the affected components. Jitter, BAOAB, final position noise, capping, and terminal status classification are row-local under identical innovations. They create no additional affected rows.
For replacement of one donor/gate block, remove that row’s outgoing edge from both graphs. The common forest meets at most three seeds (recipient, old donor, new donor); the other row draws are independent with bound \(C/N\). The same squared-sum estimate gives \(9M_2(C)\). The remaining support claims follow from their actual readouts. \(\square\)
Theorem 230 (Quantitative conditional concentration for the full update)
For either canonical regime and every bounded measurable marked-state test \(\varphi\), write \(F=L_N'\varphi\) for the complete physical marked output, retaining its coordinates also on total extinction. Uniformly over fixed inputs with alive fraction at least \(m_*\),
If the output empirical law is instead assigned a fixed probability law on extinction, the bound is \(2A_\varphi/N+8\|\varphi\|_\infty^2\delta_N\), where \(\delta_N\) is the uniform terminal-extinction bound of Corollary 73. It too is \(O(N^{-1})\).
Proof
Represent the full update by four independent innovation blocks per row: measurement companion; cloning donor with gate; addressed Haar rotation; and row-local jitter with both kinetic noises. The force evaluations, cap, and terminal classification are deterministic given these blocks. This is an exact representation of the probabilistic kernel, irrespective of how a particular implementation streams its random numbers.
The measurement and donor/gate blocks have squared influence at most \(A_D\) and \(9M_2(C)\) by the preceding lemma. Changing row \(i\)’s addressed rotation changes no output unless \(i\) represents its component; its squared influence is bounded by \(|\mathcal C_N(i)|^2\), hence in expectation by \(M_2(C)\). A row-local block has influence at most one. Therefore
Substitution into Lemma 173 proves the claim. This calculation retains the actual random global normalization, incoming cloners, shared donors, and correlated component rotations.
For the cemetery extension \(\widehat F\), its difference from \(F\) is at most \(2\|\varphi\|_\infty\) and is supported on extinction. The inequalities \(\operatorname{Var}(X+Y)\leq2\operatorname{Var}X+2\operatorname{Var}Y\) and \(\operatorname{Var}Y\leq\mathbb EY^2\) give the stated correction. Each term of \(\delta_N\) decays exponentially in \(N\); since \(\sup_{x\geq0}xe^{-cx}=1/(ce)\), that correction is bounded by a configuration constant times \(N^{-1}\). \(\square\)
Theorem 231 (Quantitative one-step bias and mean-square consistency)
In either canonical regime, fix an input array \(S\) with alive fraction at least \(m_*\), and let \(\mu_N=L_N(S)\). For every bounded measurable marked observable \(\varphi\), there is a finite constant \(B_*\) depending only on the fixed algorithmic parameters and \(m_*\) such that
Consequently, for the full physical marked output,
Assigning a fixed empirical law on extinction adds the exponentially small correction controlled in Theorem 230. The comparison is to the population map at the actual empirical input; any error between that input and another prescribed initial law is separate.
Proof
1. Integrate the normalization fluctuation. This step uses a coupled integration device to estimate the actual update; it does not redefine its fitness. It suffices to treat \(N\geq N_0\) as defined in Step 3; smaller populations use the elementary bound there. In particular there are at least two alive slots in the calculations below. In that device only, replace the empirical diversity mean and variance by the corresponding donor-weighted moments of \(\mu_N\). Retain every row’s sampled measurement, all reward statistics, the same donor and gate draws, and all subsequent collision and kinetic operations. Denote the device’s empirical output by \(\widetilde L_N'\). Its measurement marks are independent conditional on \(S\), and its deterministic normalizers are exactly those of the target population map.
Use the constants \(S_*,\sigma_s,H_s,L_a\) from Lemma 174, and put
The self-exclusion error in each measurement law is at most \(D_D/N\) in full variation. Conditional independence therefore gives, for the errors \(\Delta_1\) and \(\Delta_2\) of the sampled first and second separation moments relative to their population values,
For the variance error \(\Delta_v\), \(|\Delta_v|\leq|\Delta_2|+2S_*|\Delta_1|\). Thus
Condition on all realized measurement marks. The two acceptance probabilities of every row differ by at most \(L_TT\). On \(L_TT\leq1/2\), expose the set of changed edges and their endpoint labels. Its expected size is at most \(NL_TT\). The common forest after removing these edges has independent row probabilities bounded by \(2C/N\), exactly as in the squared-influence proof. At most three seeds per changed row are needed. Its first component moment gives
On the complementary event use \(D/N\leq1\), and its probability is at most \(4L_T^2A_T/N\). Hence
This term explicitly restores the original random normalization.
2. Compare the marked explorations at the same empirical input. Set \(A=1+C+D_D\). Explore at most \(K\) component vertices in both the integration device and the rooted construction for \(\mu_N\). Also retain the target and its measurement mark of every rejected free cloning proposal. Such an auxiliary target can be revisited later; assigning it a fresh fitness mark would lose the algorithm’s shared fitness information. There are at most \(2K\) exposed component or auxiliary labels. An incoming child already has its outgoing edge fixed to its parent, and receives no free outgoing query.
Here are bounds for this finite exploration, valid when \(K\leq N/(8A)\). All are conditional on matching histories so far.
For an undiscovered row, avoidance of the previous targets has probability at least \(1-2CK/N\). Its joint measurement/edge law is tilted by that avoidance event. Conditioning changes the intensity of a specified new target by at most \(4C^2K/N^2\); summing over rows gives \(4C^2K/N\). The conditioning factorizes over undiscovered rows, preserving their independence. This calculation includes the bias in their measurement marks, instead of assuming that an unrevealed empirical type law is deterministic.
A fresh donor query meets one of the at most \(2K\) exposed labels with probability at most \(4CK/N\) under the same conditioning. A freshly queried mark’s avoidance tilt costs at most another \(4CK/N\) in variation. Removing already exposed labels from an incoming scan changes its intensity by at most \(4CK/N\). Self-exclusion of cloning targets changes the incoming intensity by at most \(2C^2/N\) and a donor-query law by at most \(C/N\).
Replacing each finite row’s self-excluded measurement law by its population companion law costs at most \(D_D/N\) for a directly queried mark, and at most \(2CD_D/N\) in an incoming intensity summed over all rows. These are integrals against the actual atomic input \(\mu_N\); no empirical-law approximation term remains.
In an incoming scan the conditional probability of one row hitting the target is at most \(2C/N\). Coupling its marked Bernoulli event to a Poisson event of the same marked intensity costs at most twice the square of that probability. Summing gives at most \(8C^2/N\). If several targets are scanned together, the categorical bound is at most \(8C^2K^2/N\). Use the latter, larger bound for every scan.
For clarity, the intensity-conditioning estimate in the first item is obtained before dividing: if \(p\) is the probability of the new-target event and \(u\leq2CK/N\) is the probability of any earlier-target event, those outgoing events are disjoint. Their conditioned intensity is \(p/(1-u)\), whose excess is at most \(2up\). This remains an identity of marked measures when the measurement mark is retained in the event. Known incoming children are removed as fixed labels; their outgoing choices are never resampled. The independent marked Poisson processes can be coupled by retaining their common intensity and drawing their two excess intensities separately.
Each of the at most \(K\) vertex-exposure stages uses one incoming scan and at most one free donor query, with at most two new measurement queries. Summing the displayed bounds and enlarging the constant gives total mismatch probability at most
This sum includes failed proposals’ auxiliary labels and all exclusion conditioning. It does not assert independence of two components before excluding their shared finite labels.
3. Remove the finite exploration. Both the integration device and the rooted construction have the component moment bound \(M_3(C)\). For the rooted law it follows by applying finite exploration to repeated copies of the atomic input array and then taking its population limit; the alive fraction and edge bound remain the same throughout. Thus the probability that either component exceeds \(K\) is at most \(2M_3(C)/K^3\). On a matching finite exploration, use identical component rotations, jitter, and row kinetic noises. Its full marked root output is then identical, including the terminal boundary decision. It follows that
Choose \(K=\lfloor N^{1/6}\rfloor\). When \(N\geq N_0=\lceil(8A)^{6/5}\rceil\), the required exploration condition holds. Since \(K\geq N^{1/6}/2\), the last bracket is at most \([64A^2+16M_3(C)]/\sqrt N\). For \(N<N_0\), the elementary coupling bound one is at most \(\sqrt{N_0}/\sqrt N\). Combining all steps proves the theorem with the explicit, population-independent choice
The mean-square identity is conditional variance plus squared conditional bias. Apply Theorem 230 to its first term. \(\square\)
Shared initial randomness and empirical mixtures
Rust engine experiment · Seed 7 · Poster after 1 experiment steps. Interactive view starts from the same seed.
5. Discrete Stationary Identification and Its Remaining Estimate#
At stationarity a finite surviving swarm is governed by a QSD identity. A limiting population is governed by a fixed-point equation. These equations can be connected, but we must retain the survival factor until we have shown that its contribution vanishes.
There is also a second possibility: different stationary swarm runs might converge to different population laws. The object that is then stationary is a probability distribution over population laws. We will identify that object exactly, and state the additional concentration or attraction estimate needed to reduce it to one fixed population.
Remark 185 (Exact stationary balance for a killed discrete kernel)
For a bounded full-swarm test \(H\) extended by zero at the cemetery,
The operator \((Q_N-I)/h\) therefore has balance defect \(-(1-\alpha_N)\nu_NH/h\). This is the exact finite-step difference operator; identifying it with a differential generator requires a further limit. The exponential survival exponent is \(-\log\alpha_N/h\), while the balance coefficient is \((1-\alpha_N)/h\).
Theorem 232 (Vanishing extinction contribution at fixed timestep)
For the canonical terminal-box update, let \(\delta_N\) be the exponentially vanishing uniform probability of failing the positive alive-fraction bound proved in The Mean-Field Law of the Euclidean Gas. For every existing QSD of this same kernel,
Hence its bounded stationary balance defect tends to zero, and \(\mathbb P_{\nu_N}(T_\dagger\le n)\le n\delta_N\) for every fixed \(n\). The unbounded all-alive kernel has no boundary-extinction contribution.
Proof
Total extinction is included in failure of the positive alive-fraction event. Integrate the uniform one-step bound against the QSD. The QSD survival law then gives \(1-\alpha_N^n\le n(1-\alpha_N)\le n\delta_N\). For bounded \(H\), the exact balance defect is bounded by \(\delta_N\|H\|_\infty\). These estimates do not require independent unconditional walker deaths; independence of terminal position innovations is used inside the conditional survival proof in The Mean-Field Law of the Euclidean Gas.
Remark 186 (Population stability and the order of limits)
The estimate is for fixed \(h\) and fixed observation horizons. At fixed \(N\) it permits eventual extinction. For a joint limit with \(h=h_N\to0\), even survival alone needs \(\delta_{N,h_N}/h_N\to0\); the constants in the terminal-noise bound depend on \(h\). No uniform small-timestep conclusion follows by suppressing that dependence.
Theorem 233 (Exact stationary variance budget for the complete update)
Let \(\nu_NQ_N=\alpha_N\nu_N\) be a QSD of the canonical terminal-box kernel, and let \(H(S)=L_N(S)\varphi\) for a bounded measurable marked test \(\varphi\), with \(M=\|\varphi\|_\infty\). For a nonextinct input define
These are respectively the one-step survival probability, the conditional mean and variance of the empirical output given survival, and the input law reweighted by that survival probability. Then
Let \(G_N\) be the positive alive-fraction input set from Corollary 73, and use its uniform error \(\delta_N\) also as an upper bound for one-step extinction. With the constant \(A_\varphi\) of Theorem 230,
In particular, the same stationary variance has the quantitative balance
For a bounded continuous \(\varphi\), put \(g_N(S)=\mathcal F_h(L_N(S))\varphi\) and \(b_N=\nu_N|r_N-g_N|\). The proved stationary moment bounds and actual one-step consistency give \(b_N\to0\). Consequently
Proof
The QSD identities for \(H\) and \(H^2\) imply
Subtracting the square of the first identity proves the boxed formula. It conditions on survival of the whole swarm; all incoming cloners, component rotations, and alive/dead correlations remain inside \(Q_N\).
Let \(F=L_N'\varphi\) denote the physical marked empirical output, including its retained coordinates on extinction, and let \(E\) be the survival event. For each input \(S\),
The first inequality follows because the conditional mean on \(E\) minimizes the conditional squared error. On \(G_N\) the conditional variance theorem bounds the right side by \(A_\varphi/N\); on its complement it is at most \(M^2\). The QSD equation and the uniform output good-set bound give \(\nu_N(G_N^c)\leq\delta_N/\alpha_N\). Integrating the preceding inequality and dividing by \(\alpha_N\) proves the bound on \(\widetilde\nu_Ns_N\).
Since \(q_N\in[1-\delta_N,1]\) and \(\alpha_N=\nu_Nq_N\), one has
This proves the variation bound for the tilted input law. For a function \(|f|\leq M\) and probability laws \(\rho,\eta\),
Indeed, the second moments differ by at most \(M^2\|\rho-\eta\|_1\), and the squared means differ by at most twice that amount. Apply this to \(f=r_N\) and use the exact budget.
For the last estimate, the stationary output moment bounds, the QSD identity, and the positive alive-fraction estimate place the input family in the compact localization class of the one-step theorem. Its uniform localized conditional-mean consistency gives \(b_N\to0\); passage from an unconditioned mean to \(r_N\) adds at most \(2M\delta_N\). Finally, for \(|f|,|g|\leq M\) under the same probability law,
Use this with \(f=r_N\) and \(g=g_N\). This completes each asserted estimate.
Theorem 234 (Deterministic QSD empirical limits are fixed points)
Suppose \(\nu_N\) are exchangeable QSDs of the canonical kernel and \(L_N\to\mu_*\) in probability under \(\nu_N\). Assume the stationary input family lies in the tightness and moment class needed for the actual one-step theorem. Then
In the terminal-box configuration the required positive alive-fraction, capped-velocity, and output moment bounds follow from the canonical one-step estimates and the QSD equation; they do not require a reservoir model.
Proof
Apply the QSD identity to \(H(S)=L_N(S)\varphi\) for bounded continuous \(\varphi\). The one-step theorem and its random-input extension give \(\nu_NQ_NH\to\mathcal F_h(\mu_*)\varphi\); the cemetery convention changes this by at most the vanishing extinction error. Meanwhile \(\alpha_N\nu_NH\to\mu_*\varphi\). Equality for a determining family gives the fixed-point identity.
For the stated box inputs, if \(G_N\) is the good alive-fraction set, then \(\alpha_N\nu_N(G_N^c)=\nu_NQ_N1_{G_N^c}\le\delta_N\). For an output position moment \(W\) uniformly bounded in conditional expectation by \(B_W\), the same identity gives \(\nu_NL_NW\le B_W/\alpha_N\). The cap holds on every admitted output. Since \(\alpha_N\to1\), these estimates supply the required stationary input control. They use the full marked law, including dead positions.
Theorem 235 (A stationary empirical mixture is invariant under the actual map)
Under the corresponding tightness and moment conditions, if \(\Lambda_N\Rightarrow\Lambda\) for canonical QSDs, then
In particular, for every bounded continuous \(\varphi\),
The invariant-measure assertion is stronger than this averaged weak balance. Neither assertion alone makes the barycentre a fixed point of a nonlinear map.
Proof
For a bounded Lipschitz \(H\) on the space of population measures, the one-step theorem and compact localization from the finite-horizon proof show that replacing \(H(L_N')\) by \(H(\mathcal F_h(L_N))\) changes its expectation by \(o(1)\). The QSD identity changes the former expectation to \(\int H\,d\Lambda_N\) with error at most \(2\|H\|_\infty(1-\alpha_N)\). Continuity of \(\mathcal F_h\) and the stationary tightness/moment bounds permit passage to the limit, giving \(\int H\circ\mathcal F_h\,d\Lambda=\int H\,d\Lambda\). This identifies the pushforward measure. Apply it also to \(H(\mu)=\mu\varphi\) to obtain the averaged balance.
Remark 187 (Stationary existence and the precise attraction problem)
The compact-convex argument in The Mean-Field Law of the Euclidean Gas proves existence of a fixed point for the canonical terminal-box map. The consistency and continuity proofs above do not prove uniqueness or global attraction. To prove attraction it would suffice to establish, for this same map on its invariant class \(\mathcal K\), an integer \(r\ge1\), a complete metric \(d\), and \(q<1\) such that
No such inequality is established here for the canonical parameter values. Its missing terms are concrete: changing the input law changes sampled fitness normalization, accepted-edge probabilities, component membership, and the shared rotation acting on the resulting centre of mass. The finite-component estimate controls truncation, but its constant is not a contraction coefficient. Independent final noise proves continuity and survival; it does not by itself dominate these nonlinear changes.
An alternative completion is a population-uniform concentration estimate for the actual marked QSDs, together with identification of their possible fixed-point limits. The next section retains these two exact routes. Their remaining hypotheses are not certified for the canonical kernel by a continuous kinetic mixing theorem or by the existence of its fixed point.
6. Stationary Chaos and Macroscopic Convergence#
One way to make the stationary population deterministic is to show that every empirical measurement has vanishing variance. Another is to show that every population law is drawn toward the same fixed point under repeated actual updates. The first is a concentration statement about the stationary swarm; the second is a dynamical statement about \(\mathcal F_h\).
Both arguments are useful, and neither should be confused with the finite-horizon result already proved. That result starts with a concentrated population. A stationary swarm still needs a reason to be concentrated.
6.1. Concentration of empirical measurements#
Lemma 175 (A variance criterion for deterministic empirical limits)
Suppose the empirical laws are tight, \(\mu_N\Rightarrow\mu_*\), and a countable convergence-determining family \(\{\varphi_j\}\subset C_b(\mathsf Z)\) satisfies
Then \(\Lambda_N\Rightarrow\delta_{\mu_*}\) and the sequence is \(\mu_*\)-chaotic. A sufficient source of the variance estimate is a joint-law Poincaré inequality
with \(C_P\) uniform in \(N\). The joint LSI of Corollary 94, for the law and form to which it applies, implies such a Poincaré inequality. When status varies, the form must also control observables that distinguish status strata.
An alternative input is the total relative-entropy estimate Theorem 245 from Exchangeability, Empirical Laws, and Functional Inequalities. For a product reference \(\rho^{\otimes N}\) and \(H_N=H(\nu_N\mid\rho^{\otimes N})\), it gives, for bounded \(\varphi\),
Thus \(H_N=o(N)\) yields a deterministic empirical limit \(\rho\); bounded total entropy gives the displayed \(O(1/N)\) estimate. The reference and entropy are those of the actual joint law, including its status convention.
Proof
Exchangeability gives \(\mathbb E L_N\varphi_j=\mu_N\varphi_j\). Therefore
Every subsequential empirical-law limit is thus concentrated on measures whose integrals of all \(\varphi_j\) equal those of \(\mu_*\). The determining property makes that measure unique. Tightness then proves convergence of the whole empirical-law sequence, and Lemma 161 gives chaos. The Poincaré conditions bound each displayed variance by \(C_PC_j/N\).
For an LSI written as \(\operatorname{Ent}_{\nu_N}(f^2)\le2\rho^{-1}\mathcal E_N(f,f)\), substitute \(f=1+\epsilon F\) for bounded centred \(F\) and let \(\epsilon\to0\). Expanding the entropy to second order gives \(2\epsilon^2\operatorname{Var}(F)+o(\epsilon^2)\); the form is \(\epsilon^2\mathcal E_N(F,F)\). Thus \(C_P=1/\rho\), with extension to the form domain by the defining approximation.
Corollary 75 (Stationary identification using actual QSD concentration)
Suppose the canonical QSD family has the tightness and moment control above, the concentration criterion holds along every convergent subsequence, and \(\mathcal F_h\) has exactly one fixed point \(\mu_*\) in the class of possible limits. Then \(\Lambda_N\Rightarrow\delta_{\mu_*}\) and \(\nu_N^{(\ell)}\Rightarrow\mu_*^{\otimes\ell}\) for every fixed \(\ell\).
The canonical one-step consistency and extinction contributions have been proved above. Population-uniform QSD concentration and uniqueness of the actual fixed point remain additional, explicitly unverified applications of this corollary.
Proof
Take any subsequence. Tightness gives a further subsequence of first marginals converging to \(\bar\mu\). The concentration criterion makes its empirical-law limit \(\delta_{\bar\mu}\). The discrete stationary-identification theorem then gives \(\mathcal F_h(\bar\mu)=\bar\mu\), hence \(\bar\mu=\mu_*\) by the assumed uniqueness. All subsequences have the same limit. Apply Lemma 161 for fixed marginal convergence.
6.2. Identification by the full discrete evolution#
Lemma 176 (Explicit one-step stability of the full marked population law)
Consider two input laws of the canonical terminal-box algorithm,
with \(m,n\geq m_*>0\), alive positions in the declared box, and all velocities capped. The conditional dead laws retain physical positions. When a dead mass is zero, choose any conditional dead law for its zero-weight term. Use full variation norms and put
The following constants are determined by the actual parameters. Let \(\kappa_D,\kappa_C\) be the Gaussian measurement and donor weight lower bounds, \(R\) the oscillation of reward on the box, and \(D\) a bound on the oscillation of the raw diversity measurement. For the canonical logistic maps \(g_j(u)=\eta_j+A_j/(1+e^{-u})\), standardization regularizers \(\sigma_j>0\), and fitness exponents \(p_j\geq0\), \(j\in\{R,D\}\), define
Set \(T_j=0\) when \(p_j=0\). If \(\epsilon_a>0\) and \(s_a>0\) are the actual acceptance denominator and saturation parameters, put
Then the complete actual population map satisfies
For the tagged frozen source position \(X\) and jitter gate \(J\) immediately before recipient jitter, the stronger bound
is independent of \(m_*\). Both constants are independent of \(N\); \(L_{\mathrm{step}}\) is a stability bound and need not be smaller than one.
Proof
First couple physical input types and their actual measurement companions. Normalization of a weighted donor law with weights in \([\kappa_D,1]\) has variation bound \(2/\kappa_D\). Also \(\|\mu-\eta\|_1\leq2\Delta\). Combining the alive and dead branches gives a coupling whose probability of differing physical or measurement types is at most \(L_M\Delta\). This couples physical samples, rather than treating fitness values computed with two different normalizers as identical marks.
Raw reward may be shifted to \([0,R]\) without changing standardization. Its mean and variance differences are bounded by \(R\Delta\) and \(3R^2\Delta\). The derivative of \(u\mapsto\sqrt{u+\sigma_R^2}\) is at most \(1/(2\sigma_R)\). On matched physical samples these bounds give a standardized reward difference at most \(Q_R\Delta\). Apply the same calculation to the measurement-pair law to get the diversity bound \(Q_D\Delta\). The derivative of \(g_j^{p_j}\) is bounded by \(T_j\). Expanding the two fitness factors therefore gives a difference at most \(L_F\Delta\) on matched physical and measurement types.
The actual clipped acceptance function is
Clipping is nonexpansive. Its two unclipped derivatives on \([F_*,F^*]^2\) have absolute value at most \(L_a\), so matched live recipient–donor pairs have acceptance differences at most \(2L_aL_F\Delta\). Matched revival pairs have equal acceptance one. The donor normalization integral changes by at most \(2\Delta\) and is at least \(\kappa_Cm_*\). Thus the accepted edge densities, expressed against their full marked type laws, are bounded by \(C\) and differ by at most \(L_\beta\Delta\) on matched source and target types.
Use the same coupling of base types when exploring the two actual limiting marked collision components. For a matched vertex, couple the outgoing accepted subprobability measure, including its no-edge outcome. Couple incoming Poisson point measures by their common intensity; these point measures describe incoming labels at a fixed update, not a different time clock. In either direction the unmatched intensity is at most
The first term accounts for unmatched base types on the two sides, and the second for the difference of edge densities on matched types. The probability of an incoming mismatch is at most its unmatched intensity. The same bound applies to the outgoing subprobability, with its remaining mass coupled at the no-edge outcome.
Stop at the first mismatch. The expected number of vertices exposed before stopping is bounded by the expected component size in either complete first exploration, hence by \(e^{2C}\) from the actual fitness-ordered component estimate. A union bound over the two edge directions gives
On the complementary event all physical states, measurement companions, accepted edges, and component membership agree. Use the same single Haar matrix for the matching components and the same root jitter, OU innovation, and position innovation. Frozen-source copying, shared component velocity update, both force evaluations, velocity cap, and terminal classification then agree exactly. Full variation is at most twice the coupling failure probability, proving the first assertion.
The pair \((X,J)\) requires only the root type, its cloning donor and that donor’s sampled measurement, and the root gate. Its donor law is normalized against the conditional alive law, so the factor \(m\) cancels. A root mismatch contributes at most \(L_M\Delta\), an augmented donor mismatch at most \((2/\kappa_C)L_M\Delta\), and a matched gate disagreement at most \(2L_aL_F\Delta\). Incoming component members do not change a recipient’s frozen copied position. Converting this coupling bound to full variation gives \(L_{\mathrm{pos}}\). Every estimate retains the actual sampled fitness and the complete component collision transformation.
Lemma 177 (An exact kinetic observable whose noise excludes the OU amplitude)
For the quadratic-force BAOAB step, let \((X,V)\) be a walker’s input after cloning and its specified recipient jitter. Put \(c=h/2\), \(k=1-c^2\), and \(s=\sigma_x\sqrt h\). Let \((x^+,v^+)\) be its final physical coordinates including the smooth cap, before any cemetery replacement. The inverse cap on \(|v|<V_{\max}\) is
The bounded marked-state observable
has, for every \(t\in\mathbb R^d\), the exact conditional expectation
It is independent of the OU noise amplitude. In particular, increasing velocity noise alone does not make the complete one-step physical output independent of its input. This assertion concerns an exact observable of the declared update and does not assert a failure of multistep attraction.
Proof
Write the BAOAB intermediate variables as
Eliminating \(v_2\) gives
The final operations are \(x^+=x_2+s\zeta\) and \(v^+=C_{V_{\max}}(v_3)\), with \(\zeta\) a standard Gaussian independent of the input and OU innovation. Therefore
Taking its Gaussian characteristic function and the real part proves the formula. Terminal alive/dead classification does not change physical coordinates, and the observable uses both status strata. The inverse cap is finite on every physical output of the smooth cap; its values outside that open ball may be assigned arbitrarily. No cancellation of cloning or collision terms is used: their full outcome is the conditioned pair \((X,V)\).
Theorem 236 (Stationary chaos from actual-map attraction)
Suppose the canonical QSD empirical laws are tight with the stationary moment control above, and every measure in their limiting class is attracted to the same probability \(\mu_*\) under the actual map:
Then \(\mu_*\) is a fixed point and
Finite-horizon consistency and vanishing extinction for this kernel are proved in Theorem 228 and Theorem 232. The stated global attraction remains the unresolved hypothesis for the canonical algorithm; no finite-rate substitute is used to assert it.
Proof
Take a subsequence with \(\Lambda_N\Rightarrow\Lambda\). The preceding invariant-mixture theorem gives \((\mathcal F_h)_\#\Lambda=\Lambda\) and therefore \((\mathcal F_h^n)_\#\Lambda=\Lambda\) for every integer \(n\). For bounded continuous \(H\) on the space of population laws,
The assumed attraction and bounded convergence make the left side tend to \(H(\mu_*)\). Thus \(\Lambda=\delta_{\mu_*}\). Every subsequential limit has this value; tightness gives convergence of the whole sequence. Exchangeability and the empirical-to-chaos lemma give the marginal conclusion. Finally continuity and the iteration identity imply
Remark 188 (Why uniqueness of a fixed point is insufficient)
A nonlinear discrete map may have a unique fixed point and also a periodic orbit. The uniform measure on a finite periodic orbit is an invariant probability on the space of population laws. Fixed-point uniqueness alone does not eliminate that invariant mixture. The attraction theorem excludes it dynamically; the concentration theorem excludes it by vanishing empirical variance. This observation identifies a logical requirement, not an asserted periodic orbit of the canonical gas.
6.3. Observables and Wasserstein convergence#
Theorem 237 (Convergence of macroscopic observables)
Under either stationary-chaos theorem above, for every bounded continuous \(\varphi\),
and
If the limiting alive mass is positive, the same convergence holds for bounded continuous observables averaged over the alive population, with limit under the normalized alive law.
Proof
Convergence of the empirical law to a deterministic measure gives convergence in probability of each bounded continuous test integral. Since these integrals are uniformly bounded, their powers are uniformly integrable, so convergence holds in every finite \(L^p\) and in expectation. For the alive average, both its numerator and denominator converge; the limiting denominator is positive. Apply continuity of their ratio, using boundedness of the normalized observable. On the marked state space the alive indicator is a bounded continuous status function.
Corollary 76 (Wasserstein-2 convergence with control of second-moment tails)
Let \(\mu_N\Rightarrow\mu_*\) be the phase-space marginals obtained above. Suppose their squared distances from a reference point are uniformly integrable:
Then \(W_2(\mu_N,\mu_*)\to0\). A uniform \((2+\epsilon)\)-moment bound for some \(\epsilon>0\), or a proved uniform confining exponential moment, is sufficient. For a bounded phase-space metric the tail condition is automatic.
Proof
The tail condition and weak convergence imply that \(\mu_*\) has a finite second moment. By the representation theorem for weak convergence on a Polish space, choose a coupling \(Z_N\to Z\) almost surely with laws \(\mu_N\) and \(\mu_*\). The variables \(|Z_N-Z|^2\) are uniformly integrable, since they are bounded by \(2|Z_N|^2+2|Z|^2\) and the first family has the stated uniform-integrability property. Almost-sure convergence and truncation thus give \(\mathbb E|Z_N-Z|^2\to0\). This coupling bounds \(W_2^2\) from above. A uniform higher moment bounds the squared tail by \(R^{-\epsilon}\sup_N\mu_N|z|^{2+\epsilon}\).
A bounded second moment alone is insufficient: on the line, \(\mu_N=(1-1/N)\delta_0+(1/N)\delta_{\sqrt N}\) converges weakly to \(\delta_0\) and has second moment one, but \(W_2^2(\mu_N,\delta_0)=1\) for every \(N\).
Remark 189 (What has been established for the actual algorithm)
The canonical fixed-step algorithm has an explicit rooted-component population map, proved normalization and component-size bounds, one-step empirical consistency, continuity, and finite-horizon propagation of chaos. The terminal-box configuration also has at least one nonlinear stationary fixed point. None of these conclusions requires cloning probabilities to vanish with the timestep.
Stationary QSD chaos additionally requires the concentration or attraction step stated above. Continuous-time limits require control over a growing number of the same updates, with the actual collision, cap, revival, and boundary operations retained. History-dependent donors, adaptive diffusion, local fitness normalization, and alternative boundary schedules require analysis of their own specified transition; the canonical theorem does not silently identify those extensions with its kernel. These distinctions carry into Hypocoercive Entropy: From Kinetic Mixing to the Full Swarm, Exchangeability, Empirical Laws, and Functional Inequalities, and Continuum Limits: Hypotheses and Consistency.