Field Equations and Pressure Dynamics#
Prerequisites: Emergent Geometry from Adaptive Diffusion, Scutoid Spacetime: Moving Cells and Neighbor Changes, and Curvature of the Algorithmic Fitness Metric.
TLDR#
The Euclidean Gas supplies a finite-step stochastic field system through its donor probabilities, normalized fitness, cloning, BAOAB motion, boundaries and retained history. Its complete marked empirical field retains each slot’s state and exact donor identity. The equations close on that field and the configured provider/input state.
Its spatial density and momentum satisfy the derived equations
The sources and finite-path fluxes are explicit functions of the executed clone, force, transport, thermostat and boundary updates. The kinetic stress is an anisotropic measured tensor. Row-normalized viscosity also has an explicit momentum source whenever its pair weights are asymmetric.
The conditional-fitness metric and its noise coefficient are
Here the donor context is frozen while differentiating the complete fitness normalization. Its next value follows by the actual next selection and motion through the O-input barrier. Geometry therefore couples directly to the algorithm’s thermal covariance and subsequent momentum transport.
For a reduced field descriptor, the two-step prediction contains the exact memory term
The general transient recurrence retains every excursion through omitted population and donor-history variables. A complementary conditional-law equation predicts the next fields given their observed past and includes both fresh transition noise and unresolved-state uncertainty.
The later Gaussian correlation-energy, homogeneous density and Gaussian mode models have their own explicit pressure and dispersion formulas. They provide specified analytic comparisons for reductions of the algorithm-derived system.
Introduction#
What should a field equation predict? Give it the state of the experiment now, and it should tell us how the next field measurement is distributed. For the Euclidean Gas we can construct that prediction from the instructions which move and replace the walkers.
Start with donor selection. Its probabilities depend on the current population and possibly on retained frames. Fitness compares a walker with population statistics, so changing one row changes part of the normalization. Cloning then transfers state between particular locations. The kinetic update adds forces, transports positions, and applies a thermal kick whose directional covariance can depend on the fitness metric. Every one of these operations leaves a term in the equations.
The local ruler is especially interesting. It shapes the thermal kick, but the next ruler is itself calculated from the population changed by that kick, transport and selection. We will derive this cycle at the algorithm’s actual time step, including the exact point at which each metric is queried.
Once those equations are in hand, we can ask which measurements suffice for a smaller description. A few density and stress moments discard information about donor history and the population. Their evolution therefore acquires memory and additional noise. Deriving those terms tells us what a reduced field theory must retain and gives experiments which can test its predictive accuracy.
Deriving the Fields from Donors, Cloning, and Motion#
Begin by asking what can change a walker. It can select a donor, copy it, receive a displacement, exchange velocity with a partner, accelerate, move, receive a thermal kick, or leave the eligible population. Each instruction gives a term in the field equation. The instruction order determines which population supplies that term’s coefficients.
There is an especially important kind of interaction here. Moving one walker changes its normalized fitness because that walker contributes to the population statistics used to judge it. A donor from a retained frame also carries information which is absent from the present cloud. We will keep both effects in the equations, and only then ask whether a smaller set of measured fields can predict their consequences.
Donor probabilities and population-dependent fitness#
Definition 768 (Primitive selection coefficients)
Write the admitted pre-selection state as
where \(M=\sum_i a_i\) is the eligible count, \(a_i\) is configured eligibility, \(\ell_i\) includes slot/generation identity, \(L\) is the maximum configured donor window, and \(\theta\) contains the complete configuration and fixed provider definitions. The state \(\chi_n\) contains mutable global provider/domain state, the input schedule and any variables needed to advance them. The maps below are deterministic given that state and their explicitly listed innovations. The population \(P_n\) is the admitted pre-selection population. Let \(\mathcal E_n\) denote the configured input/extraction, observation refresh, boundary and reward-validity operations that produce this admitted population from the preceding completed one. Include \(\mathcal E_n\) as an explicit stage whenever it changes a measured field. Rewards and observations are refreshed at their actual transaction barriers; opaque domain state must be included when the domain is not numerical. At a numerical experiment boundary, the input schedule is also fixed or adjoined. The mathematical stochastic law uses independent addressed innovations; a fixed seed determines a reproducible realization.
For module \(m\in\{D,C\}\), its frozen eligible pool is
Every pool atom retains its own coordinates, frame, generation and version. Current self is removed when self-companions are disabled; the same slot at a historical frame is a distinct permitted donor. If no nonself candidate exists, the implemented singleton fallback uses the eligible current self.
For an independent donor draw, define the actual weights
Here the exponential uses the configured comparison value \(q_m\), and the Gaussian squares an ordinary distance but does not square an already squared comparison again. With replacement, the row law is the product of these probabilities. Without replacement, the ordered selected list \(j_1,\ldots,j_K\) has the Plackett–Luce law
Reciprocal matching has a joint law. Fisher–Yates draws a uniform random permutation and pairs consecutive entries. Gaussian-greedy draws a uniform random permutation, takes its last unmatched entry \(i\), chooses a partner among the remaining entries with probability proportional to \(w_{ij}\), removes both, and repeats. Summing the probability of these construction histories defines the joint matching law \(Q_m\); the configured odd policy supplies self, unmatched, or rejection. These are explicit finite algorithms for \(Q_m\), not an unspecified transition kernel. Matching uses current eligible sources only.
Let \(D\) be the distance companion batch. Its reducer gives \(d_i(D)\), and the diversity measurement is \(s_i=(d_i^2+\delta_D^2)^{1/2}\). Let \(r_i\) denote the oriented reward. For the smooth global standardizer,
and likewise for \(s\). The local standardizer replaces uniform weights by its configured normalized kernel weights, excludes self when configured, and falls back to global statistics only for an empty neighborhood. The actual logistic positive maps are \(R(z)=A_r/(1+e^{-z})+f_r\), \(D_+(z)=A_d/(1+e^{-z})+f_d\). Therefore
The enabled metric provider requires these smooth global/local standardizers and logistic maps. These formulas specify the smooth fitness branch used by the conditional-metric experiments.
If historical cloning is enabled, the engine draws an additional distance batch \(D^H\) on the entire clone pool with the independent HistoricalDistance stream, recomputes its rewards under the current input, and computes historical diversity. Historical fitness uses these fresh measurements with the current population’s global means and scales. Current donor fitness remains \(F_j\), not the pool-rescored value. This historical rescore is essential to the transition law.
Clone transformations and the kinetic maps#
Definition 769 (Executed clone and BAOAB maps)
After \(D,C,D^H\) are fixed, define for each eligible target
An unmatched target has \(q_i=0\). The gates are conditionally independent across recipients in the built-in stochastic law. Every ineligible target is instead revived from an independent uniform current-eligible donor and is accepted with probability one. Historical sources are not used for revival.
Literal copying is simultaneous from the immutable pre-clone donor pool. The target retains its slot, increments its own generation on an accepted replacement, and obtains the donor’s row state and observations. Thus the copy map is a fully specified deterministic map \(C_{D,C,D^H,A,R}(S_n)\), where \(R\) denotes revival donors.
If jitter is enabled, an accepted nonrevival target receives \(x_i\leftarrow x_i+\eta_c B_i^c\xi_i^c\), with the noise factor evaluated after literal copying. Revival targets are not jittered by this transform. If restitution \(e\in[0,1]\) is enabled, every disjoint current reciprocal pair \((i,j)\) with at least one accepted gate has
using both pre-clone velocities. Therefore the partner’s velocity can change even if its own gate was rejected. Restitution cannot be represented by independent target-copy kernels. It is not supported for historical or overlapping donor pairs.
Let \(\mathcal B\) mean the actual boundary/reconciliation map. Built-in boundaries are unbounded, absorbing box, periodic box, external termination, or an ordered composition. Absorbing boundaries mark out-of-bounds and preserve the recorded row; periodic boundaries wrap coordinates. There is no built-in reflecting boundary in this implementation. Reward validation and eligibility updates are separate deterministic maps at their recorded barriers.
Let \(Y^0\) be the post-transform, reconciled, boundary-classified and reward-validated population. Set \(h=\mathrm{dt}\), \(c=e^{-\gamma h}\), and
For every eligible input row, the kinetic maps, with boundaries between each, are
Rows that become ineligible are skipped thereafter; complete extinction skips the remaining kinetic stages. The force is
where \(\mathcal D\) is the actual gradient provider and \(Z_i=\sum_{j\ne i}a_jw_{ij}\) for row normalization, otherwise \(Z_i=M(Y)\), including self in the eligible count. A zero row normalizer gives zero viscous force. The current implementation computes these viscous distances directly in coordinates. B2 recomputes the gradient, weights, velocities and eligibility from its own input. It is not a repeated B1 force.
The innovation components are standard Gaussian or uniform on \([-\sqrt3,\sqrt3]\), optionally shifted by the configured addressed source perturbations. The factor is isotropic, diagonal, full, low-rank, or the metric factor specified below. The integrator owns \(s_h\); it must not be included a second time inside \(B\).
Theorem 434 (Explicit finite-step population law)
Denote the preceding complete composition, final reward refresh, bounded history shift and next admission \(\mathcal E_{n+1}\) by \(\mathcal T(S_n;D,C,D^H,A,R,\xi^c,\xi)\). Its retained age-one frame is the admitted input \(P_n\); its new age-zero frame is the next admitted population. This fixes the pre-selection convention for \(S_{n+1}\). For the fixed-input numerical experiments, next admission preserves the completed physical rows. The explicit law of an integrable full-state observable \(\Phi\) is
Absent historical rescoring or jitter is a unit point mass. The clone pool itself is fixed by \(S_n\), so \(Q_H\) does not actually depend on the realized clone draw for built-in modules; its notation emphasizes the correct pool. The equation applies on the configured successful-transaction domain. If \(M=0\), the transaction returns extinction; use its specified absorbing outcome in place of the donor sums. Eligibility loss within a transaction remains in \(\mathcal T\).
Proof
Each donor procedure is sampled on its named stream; the historical rescore is evaluated on its separate named stream; conditioned on those outputs the gate comparison with independent uniforms produces the stated Bernoulli factors and the revival stream supplies uniform donor factors. Copying and restitution are deterministic given those choices. The remaining stochastic maps are the configured jitter and thermostat innovations. Iterated conditional expectation through the exact stage order gives the sum and integral. The final memory operation is deterministic. All its probabilities and maps are the configured primitive coefficients above.
The expression is long because the update has several actual operations. Its coefficients are quite concrete: a donor weight, an acceptance probability, a force, and an innovation distribution. Once the population, its history and its configuration are supplied, each coefficient can be evaluated. This gives an independent prediction for repeated runs from that state. Replacing all those operations by an unknown drift would conceal exactly the interactions we want to understand.
The Marked-Field Hierarchy and Donor Memory#
Think of a field measurement as placing a measuring function over the walkers and adding its readings. Different measuring functions reveal density, momentum, energy, or a spatial oscillation. We can update all of these readings by following what the algorithm does to each atom.
A position plot throws away labels, velocities and donor history. The complete marked field keeps them. It is a change of mathematical coordinates for the same algorithmic state. When we retain only a few readings, the omitted information reappears as a hierarchy of correlations and memory terms.
Theorem 435 (Exact field characteristics and moment hierarchy)
Use the subprobability phase-space field
The denominator is the configured slot count, not the random alive count. For a test function \(\varphi(x,v)\), an exact stage increment is
Summing literal-copy, jitter, restitution, each reconciliation/boundary map, B1, A1, O, A2 and B2 increments, together with the next admission \(\mathcal E_{n+1}\) whenever it changes the field, telescopes exactly to the admitted-to-admitted full-step change. Conditional expectation of each term uses the explicit transition measure above. These are weak field equations with stage-resolved sources, impulses, transport and noise.
In particular, before transforms, conditional on donor/fitness measurements the expected literal-copy source is
For revival the previous contribution is zero in the eligible field. Jitter, restitution and subsequent killing must be added separately; otherwise this source is not the full cloning update.
For Fourier tests \(\varphi_{k,\ell}(x,v)=e^{i(k\cdot x+\ell\cdot v)}\), a deterministic kick multiplies each atom by \(e^{i(h/2)\ell\cdot f_i}\), and a drift substitutes \(\ell\mapsto\ell+(h/2)k\). Conditional on the O input, the pre-boundary thermostat prediction is explicitly
where
A configured per-row innovation shift \(b_i\) multiplies its factor by \(e^{iu\cdot b_i}\). This exact finite-step equation distinguishes innovation laws having identical covariance. For absorbing boundaries one must integrate the boundary indicator against this same noise law; dropping it changes the prediction. For periodic boundaries use the wrapped test function (periodic Fourier modes are unchanged by wrapping).
The characteristic functional of the eligibility-weighted phase-space field is obtained by setting
inside the explicit sum/integral in Theorem 434. At an O stage before its boundary, conditional independence gives
Here \(\nu\) is the centered configured innovation law and \(b_i=0\) unless an addressed source shift is configured. Functional differentiation produces every field correlation equation. The outer expectation retains dependence across selection outcomes: mutual donor matching, shared normalization and shared state-dependent factors are retained in the outer sums. Momentum and energy equations follow by differentiating the Fourier tests at zero; their noise coefficients are the executed \(B_iB_i^T\), the configured covariance-rate tensor. Higher moments couple to higher joint fields through fitness, matching, viscosity and clipping. This is the derived hierarchy.
Proof
Subtract successive atomic measures and sum; each intermediate atom cancels. A replaced atom contributes its donor test value minus its recipient test value. Averaging its gate yields \(q_i\), and averaging a revival uses the uniform current-eligible law. For a kick, substitute \(v+(h/2)f_i\) in the Fourier exponential. For a drift, substitute \(x+(h/2)v\). At the thermostat, its input fixes \(B_i\) and the exponential separates into a deterministic factor and \(\exp(i s_h\ell\cdot B_i\xi_i)\). Integrating independent innovation coordinates gives the Gaussian exponential or uniform sinc product. The characteristic-functional product follows by independence only after conditioning on the complete O input. Differentiate under the integrals when the corresponding moments exist; bounded Fourier tests themselves require no moment hypothesis. The complete outer donor and gate sums supply all cross-walker correlations.
Proposition 286 (Closure on marked fields and age transport)
Let labeled history fields be
These full marked fields retain ineligible rows as well; eligibility-weighted observables and donor measures are obtained by multiplying by the eligibility mark. All configured observation channels and cached input-dependent rewards must also be included when they are not deterministic functions of the displayed coordinates and provider/input state. At each successful commit,
where \(\mathcal T_{\mathrm{marked}}\) is the marked pushforward of Theorem 434. Age \(L+1\) is discarded. Early runs only contain available ages. This is an exact discrete age-transport equation with a new-age boundary condition. Donor sums are integrals over these age fields with their configured windows. Labeled fields including opaque state and flags retain the full numerical/domain state; an unlabeled or few-moment reduction generally does not. The random gate and matching hierarchy is then a consequence of the specified algorithm, not a reason to assume memory away.
For bounded real tests \(\psi_0,\ldots,\psi_L\) on the full mark space and a bounded test \(\vartheta\) of \(\chi_n\), define the complete marked characteristic functional
At a step with all retained ages present, its exact equation is
Use only available ages during initialization. The measure \(\mathbb Q_{S_n}\) is the explicit donor, gate and innovation law above. Differentiating in finite linear combinations of the tests gives the joint moment hierarchy of marked fields across retained ages, including eligibility, ancestry and donor-dependent field correlations. This hierarchy retains the same state information as the marked representation.
Proof
At a successful commit, the engine stores its admitted pre-selection population \(P_n\) in the retained history. That is the next age-one frame; older retained frames increment their ages. The new current population is the completed output followed by the next admission \(\mathcal E_{n+1}\). Frames beyond the configured window are discarded. The slot, frame, generation, version and domain-state marks reconstruct each donor-pool atom exactly. Together with provider state and the input schedule this reconstructs \(S_n\). Applying the explicit primitive update and deterministic history shift therefore determines the law of the next complete marked field. No independence between atoms is required for this reconstruction. The older-age contribution to \(\mathscr Z\) is determined by this shift and factors out of the conditional expectation; the new-age and provider-state terms retain the explicit innovation integral. Differentiation in bounded test amplitudes is justified by dominated convergence, proving the joint marked hierarchy.
The Conditional Fitness Metric as an Evolving Field#
The local ruler has a specific job in the algorithm: it sets the shape of a thermal velocity kick. To construct it, hold the donor context fixed, move the query point a little, and differentiate the resulting fitness. The population normalization participates in that differentiation.
Now distinguish two changes. Moving the query point changes a spatial field within one frozen context. Running the next update can change the donor, the population and the retained history, producing a new context. The metric equation must include both. The O-stage ruler is evaluated after the first displacement, using the selection context retained from the beginning of that transaction.
Theorem 436 (Conditional Hessian metric and its actual transition)
Use the configured conditional-fitness provider with one distance
companion per target and the Mean reducer. Its distance is unscaled,
nonperiodic Euclidean distance or the unscaled phase-space distance in
the formula below. Both standardizers are smooth global or local
standardizers and both positive maps are logistic. Assume
\(\epsilon_g>0\), \(T\geq0\) and \(\gamma\geq0\).
When historical cloning is enabled, the provider uses global standardization; external-input updates with historical cloning are outside the supported transaction domain. The remaining donor-history, boundary and eligibility operations retain their configured laws.
The O-stage metric uses the immutable pre-selection population and distance companions, evaluated at the actual post-A1 query position.
For an eligible pre-selection target \(j\), form a conditional replacement fitness \(\mathcal F_j(y;S_n,D)\): replace only its reward and diversity measurements by
leaving all other measurement rows fixed, but recompute the complete global or local normalization as a function of \(y\). Local kernel weights also depend on the query. Euclidean distance is the \(\lambda=0\) case. A missing companion supplies the constant floor. Coordinate shifts of the benchmark are applied consistently.
Then
Here \(\tau_i=i\) for a pre-selection eligible target, including a target that subsequently clones. A revived target uses the exact eligible current donor’s target \(\tau_i\). This is the actual provider’s conditional field convention. For the strict metric policy, replace \([H]_+\) by \(H\) and require \(\epsilon_gI+H\) positive definite.
All derivatives are explicit algebraic derivatives of the preceding measurement and standardization formulas. For example, putting
gives
For this formula, \((p_r,p_s)=(\alpha,\beta)\) and \((M_r,M_s)=(R,D_+)\). The derivatives of the standardization are explicit. For either measurement channel, write its queried row as \(t(y)\), its eligible mean as \(m(y)\), its regularized variance as \(V(y)=\sigma(y)^2\), and \(u(y)=t(y)-m(y)\). In the global branch with fixed eligible count \(M\),
Consequently the normalized query \(z=uV^{-1/2}\) has
In the local branch, let \(\omega_k(y)\) be the configured normalized query-neighbor weights, with \(\sum_k\omega_k=1\). Then
For \(\omega_k=e^{\ell_k}/\sum_re^{\ell_r}\),
and differentiating this product gives \(\nabla^2\omega_k\). Apply the product rule to the displayed \(m,V\), then the same formula for \(\nabla^2(uV^{-1/2})\). These weights use fixed companion and neighbor identities with the moving query coordinate. If the local standardizer includes the target as its own neighbor, that self-comparison has identically zero distance as the target moves; its kernel log-weight is constant. Other neighbors retain their frozen coordinates. Changes of donor identity are discrete transitions in the outer law.
Define the measured metric field at each actual O input by
Its exact conditional prediction is the same explicit donor/gate/jitter sum from Theorem 434, stopped at A1, with the displayed algebraic \(g\) inserted. The current O-input prediction stops before the current O innovation is drawn. The next O-input metric requires continuing through current O, A2, B2, history shift, next donor selection, next cloning and next B1/A1. This two-barrier composition gives its exact transition law. Equivalently use the augmented O-input state containing its frozen selection context as the stroboscopic Markov state.
For each component, subtracting the current metric and dividing by \(h\) gives the discrete material metric equation. Its forcing consists explicitly of changed population measurements, new companion draws, cloning/revival target maps, transported query coordinates, updated history and the positive-part matrix map. These contributions determine the metric increment jointly with the evolving population and donor history.
Curvature is then computed from spatial derivatives of this same \(g\), with standard metric contractions. Within a smooth clipping region, \(Dg\) and \(D^2g\) follow the spectral divided-difference chain rule applied to \(H, DH,D^2H\); mixed-sign clipping needs fourth derivatives of fitness. At an eigenvalue clipping threshold classical curvature need not exist, even though the metric pushforward and finite differences of \(g\) remain defined. A curvature equation is obtained by inserting this derived curvature observable into the same explicit transition measure; exchanging expectation with the nonlinear curvature map is not valid.
Proof
The conditional provider builds reward and diversity jets by substituting the query into one target row of the frozen population. Applying its configured standardizers and positive maps gives \(\mathcal F_j\). Twice differentiating \(e^{L}\) yields the displayed Hessian, and the product and quotient rules give the expressions for \(\nabla L\) and \(\nabla^2L\). Spectral clipping and the positive floor yield \(g\); its inverse square root is precisely the provider’s factor in the O update. Conditional on O input, \(\operatorname{Cov}(v^{O+})=s_h^2B_iB_i^\top =2\gamma T s_h^2g_{\tau_i}^{-1}\).
The provider retains the pre-selection target index for an eligible row. For a revival it resolves the accepted donor’s exact current-frame event identity and uses that eligible target. This proves the target map \(\tau_i\). The value at the actual post-A1 coordinate follows by direct substitution. Stopping the explicit transition at A1 gives the current metric law. Composing its remaining stages with the following transaction through A1 gives the next metric law. Curvature is a specified nonlinear observable of spatial metric derivatives wherever those derivatives exist, so its transition follows by the same substitution into the explicit expectation.
Exact Metric Evolution from the Transition Law#
Imagine saving the complete experiment immediately before an update. Restart it many times, keeping the physical state fixed and supplying fresh random draws. You get a distribution of next states. The average change of any measured quantity is its conditional drift; the spread around that average is its fluctuation.
The word “complete” matters. A saved cloud of positions is insufficient when the next donor may come from retained history. A saved velocity is insufficient when a provider has internal state or a schedule changes the next update. And replaying the identical random seed gives the identical future, rather than an independent experiment. We distinguish replay from resampling the future innovations.
Definition 770 (Extended state and transition experiment)
Fix population capacity \(N\), step size \(h\), the algorithm configuration, and the specified input protocol. Let \(X_n\) contain the population (including validity and ancestry), retained donor history, provider and domain state, and the schedule and random-stream addresses needed at step \(n\). Include changing external inputs in the state, or condition on their specified protocol. A measurable update is
The conditional law of \(\xi\) includes the configured donor, acceptance, revival, jitter, and kinetic innovations. Independence is used only where that law supplies it. Write \(\mathscr F_n\) for the information available through \(X_n\) before these future innovations are exposed.
The stochastic experiment uses the declared innovation law. A numerical replay with a fixed complete pseudo-random future is deterministic. Checkpoint-conditioned replicas hold the algorithmic state fixed and resample future innovation keys under a recorded ensemble protocol; they do not condition on an already exposed future key.
An observable \(A:\mathcal X\to\mathbb R^m\) includes its reconstruction convention and reference coordinates. When extinction is possible, extend the transition to a cemetery outcome and specify \(A\) there. If a metric does not exist at an outcome, report that outcome and its probability; discarding it changes the conditional experiment. A killed observable with a specified cemetery value and a survival-conditioned observable are different quantities.
Theorem 437 (Finite-step observable equation and fluctuation covariance)
Assume \(A(X_n)\) and \(A(X_{n+1})\) are square integrable. Define
Then
In particular, for fixed spatial probes \(z_1,\ldots,z_q\), take \(A(X)\) to be the vector of independent components of the specified metric \(g_X(z_\ell)=\mathcal G(X;z_\ell)\). This gives an exact stochastic metric equation at finite \(N,h\), whenever that observable is defined and square integrable. For the implemented global quadratic research fixture, Lemma 318 proves this integrability directly. It does not assert that \(b_A\) is a function of \(g\) alone.
Proof
Set \(\eta_{n+1}=A(X_{n+1})-\mathsf P_{N,h}A(X_n)\). The conditional transition law gives its zero conditional mean. Expanding its conditional outer product yields \(\Gamma_A\). Adding and subtracting \(\mathsf P_{N,h}A(X_n)\) proves the increment identity. Square integrability makes these expressions finite. The metric application is the same calculation for the stated vector-valued observable.
Proposition 287 (Operator decomposition without a closure assumption)
Let \(X_n^{(0)},\ldots,X_n^{(L)}\) be the actual extended intermediate states, with \(X_n^{(0)}=X_n\) and \(X_n^{(L)}=X_{n+1}\). A skipped operation is an identity stage. Define \(\Delta_\ell A=A(X_n^{(\ell)})- A(X_n^{(\ell-1)})\). Then
and
If \(b_{A,\ell}(X_n^{(\ell-1)})\) is the stage drift conditional on everything entering stage \(\ell\), its contribution to the full-step drift is \(\mathbb E[b_{A,\ell}(X_n^{(\ell-1)})\mid X_n]\).
Proof
Successive differences cancel in the sum. Taking conditional expectation proves the drift formula; applying the tower property proves its stage-conditional form. Expanding the covariance of the sum proves the last identity. Distinct realized stage increments generally have nonzero cross covariance, even when the newly drawn innovations are independent.
Proposition 288 (Fixed-probe and material metric increments)
Express all tensors in a fixed reference chart. For a tracked old probe \(z\) and its specified new location \(z'\), the exact identity is
For a clone replacing a slot at \(z_i\) by a source at \(y_j\), and ending at \(z_i'\), a useful refinement is
The three terms respectively measure field evolution at the source probe, subsequent motion, and the slot replacement. A historical source has both a source frame and a location. Comparing with its historical metric requires recording that frame and adding the corresponding metric difference explicitly.
For a reconstructed material map \(\Phi\), the geometrically transported comparison is \(\Phi^*g_{X'}-g_X\), where
A discrete clone map does not automatically provide this differentiable map or its Jacobian.
Proof
The first two identities follow by adding and subtracting the indicated metric values. The pullback expression is the tensor transformation law. It includes deformation of the coordinate frame, which simple component evaluation along a path does not include.
These equations tell us exactly which measurements to compare. They do not yet tell us that the metric remembers enough to predict its own future. Two populations can share the same metric and carry different velocities or different historical donors. Their next metric increments can therefore differ. A proposed equation involving only the metric must survive that test.
There is a similar trap in simulation. If we estimate the mean change from a collection of replicas and subtract that mean from those same replicas, the residual mean is zero by construction. We have tested an arithmetic identity. A prediction deserves a fresh collection of replicas.
Proposition 289 (Independent-replica drift test)
Fix \(X\) and draw two conditionally independent groups of increments \(Y=A(X')-A(X)\), of sizes \(R,S>1\). Let their sample means be \(\widehat b_{\mathrm{pred}}\) and \(\widehat b_{\mathrm{test}}\). Then
Each group’s sample covariance with denominator one less than its size is unbiased for \(\Gamma_A(X)\). For a fixed analytic prediction \(b_*(X)\), the test mean instead has bias \(b_A(X)-b_*(X)\) and covariance \(\Gamma_A(X)/S\).
Proof
Conditional independence removes cross covariance between samples and between groups. Summing the individual means and covariances gives the displayed formulas. Expanding \(\sum_r(Y_r-\overline Y)(Y_r-\overline Y)^\top\) gives expected value \((R-1)\Gamma_A\), proving sample covariance unbiasedness.
Remark 289 (What a simulation can establish)
Use replicas or independent trajectories as the sampling units. Walkers sharing a population are not independent replicas. Finite variance gives the identities above; it does not make a finite-sample Gaussian confidence interval exact. Exact reference distributions, proved tail bounds, or a justified asymptotic approximation determine the uncertainty procedure. An estimated drift and an independently checked analytic drift must be reported separately. A reduced predictor using only density, current, or metric is tested for bias against the full-state conditional law.
The formulas concern the declared innovation law and real arithmetic. Finite-precision evaluation and pseudo-random simulation are numerical approximations whose discrepancy is checked against independent analytic or enumerable reference cases. An unchanged-seed replay separately checks that instrumentation has not changed the executed dynamics.
Finite-step integrability of the quadratic research fixture#
The square-integrability condition in the observable equation can be checked directly for the quadratic research runs. Their positions are unbounded, and we leave them that way. The useful bound comes from the programmed metric floor: however large the fitness Hessian becomes, the noise factor cannot exceed a fixed amplitude.
Cloning can amplify the population norm by copying a large row many times, but at fixed population size that amplification is bounded. The remaining quadratic-force updates are linear. Combining these facts controls every moment at every finite step, including the moments needed to measure the metric.
Lemma 318 (Finite-step moments for the implemented global quadratic fixture)
Consider the global-normalization research configuration in
algorithmic-gas/crates/benchmarks/examples/physics_research.rs:
finite \(N\geq2\) and dimension \(d\), the minimizing Quadratic objective
\(U(x)=|x|^2/2\), unbounded boundary policy, current-frame single-source
literal cloning, the identity clone transform (no jitter or restitution),
and the implemented BAOAB integrator. Let its fixed parameters satisfy
\(h>0\), \(\gamma>0\), \(T\geq0\), and \(\epsilon_\Sigma>0\). Its metric provider
uses global standardization with positive variance regularizers,
regularized Euclidean diversity with positive distance floor, and logistic
maps with positive amplitude and positive output floor. Fitness powers
are the configured finite constants. The innovations are independent
standard Gaussian vectors, and the initial population is a fixed finite
realization of the configured initialization.
In the simulation’s dimensionless units, write \(Y_n=(x_1,\ldots,x_N,v_1,\ldots,v_N)\) and let \(\xi_{n+1}\in\mathbb R^{Nd}\) collect the \(O\)-stage innovations. With
the executed update obeys the pathwise bound
For every finite \(n\) and \(p\geq1\),
All BAOAB intermediate coordinates have finite moments of every order. For the programmed fitness query field with frozen coordinate context \(Y_{\mathcal C}\), there is a finite constant \(C\) such that
The constant depends on the fixed population size, dimension, objective, and configured regularizers, maps, and powers, and is uniform over the finite companion assignments. Consequently the clipped metric at fixed probes and at the executed \(O\) queries has finite moments of every order. In particular, these metric readouts satisfy the square-integrability condition of Theorem 437.
Proof
1. The noise bound is a consequence of the configured clipping rule. For the symmetric fitness Hessian \(H\),
All query expressions are finite and smooth at finite coordinate inputs: the distance square root has a positive floor, the standardization denominator is bounded below by its positive regularizer, and a logistic map plus its positive floor takes values in a compact positive interval. Thus finite inputs give a finite Hessian and the displayed factor. For the block-diagonal population factor \(\mathcal B=\operatorname{diag}(B_1,\ldots,B_N)\) the same operator bound \(\|\mathcal B\|_{\mathrm{op}}\leq\beta\) holds. The factors may depend on all the recorded population data.
2. Bound each actual update. Let \(Y_n^{\mathrm{cl}}\) be the literal-clone output. Each of its \(N\) rows is one row of the frozen current population, whether retained or copied. Consequently
This estimate holds for every realized donor and acceptance outcome. For the configured minimizing quadratic objective, the gradient provider returns \(x\) exactly. With \(q=h/2\), each kick and displacement therefore has the respective block form
The bound follows by writing each matrix as the identity plus a matrix of norm \(q\). The \(O\) stage is
Applying the remaining displacement and kick to this noise term gives
The matrix bounds and the clone bound prove the stated inequality. The unbounded boundary policy and the identity clone transform add no coordinate change. Finite inputs and Gaussian innovations remain finite at every finite step; the quadratic rewards and positive fitness maps therefore keep this mathematical fixture eligible.
3. Iterate the moment bound. Iterating the pathwise inequality gives
Minkowski’s inequality yields the displayed \(L^p\) bound. Gaussian moments are finite, as follows by integrating a polynomial against \(e^{-\|\xi\|^2/2}\). The same stage inequalities give every intermediate coordinate bound. No independence of the state-dependent factor and the previous population was used. The constants can grow with \(n\); the claim is at each finite step.
4. Bound the differentiated fitness formula. Put \(R=1+\|z\|+\|Y_{\mathcal C}\|\geq1\), and let \(D\) denote a query derivative. The quadratic rewards and their derivatives through order two are bounded by \(CR^2\). Each regularized diversity measurement is
It satisfies \(d\leq CR\), \(\|Dd\|\leq1\), and \(\|D^2d\|_{\mathrm{op}}\leq\delta^{-1}\). Frozen rows have zero query derivatives. Thus, for either channel, its measurements and derivatives through order two are bounded by \(CR^2\).
For the actual global mean and variance \(\mu=N^{-1}\sum_jm_j\) and \(V=N^{-1}\sum_jm_j^2-\mu^2\), the product rule therefore bounds \(\mu,D\mu,D^2\mu\) by \(CR^2\), and \(V,DV,D^2V\) by \(CR^4\). Write \(t=(V+\sigma_{\min}^2)^{-1/2}\). Since \(V\geq0\),
Hence \(\|Dt\|\leq CR^4\) and \(\|D^2t\|\leq CR^8\). For the standardized target \(Z=(m_i-\mu)t\), the product rule gives \(\|DZ\|\leq CR^6\) and \(\|D^2Z\|\leq CR^{10}\). Logistic derivatives through order two are bounded on the real line. The configured powers have bounded derivatives on the positive range of their logistic maps. Applying the chain and product rules to the two-channel fitness therefore gives \(\|D^2F_i^{\mathrm{probe}}\|_F\leq CR^{12}\).
Finally, spectral clipping gives
The context coordinates are retained finite-stage population coordinates. A fixed probe is deterministic; an executed \(O\) probe is an intermediate coordinate already controlled in step 3. Their \(24\)th moments give metric square integrability. Their moments of every higher order give the remaining asserted metric moments. Finite sums of these readouts, including the recorded population-average \(O\) metric, inherit the same conclusion.
Frozen query inputs and the response of the transition#
“Conditional fitness” names the inputs held fixed while the configured metric provider differentiates a query. Keep the recorded donor coordinates and the other measured rows fixed, vary the query, and recompute its fitness through the configured normalization. The resulting Hessian is the one used by that provider’s metric rule.
Now move an actual walker in the input population. It may also be another walker’s donor, so other measured distances change. Local weights and the probabilities of future donor and clone choices can change too. To differentiate the expected next measurement, follow both changes: what each possible update produces and how often the algorithm selects it.
Definition 771 (Query derivative and population perturbation)
Let \(\mathcal C\) be the recorded FrozenFitnessContext. The configured
query field \(F_i^{\mathrm{probe}}(z;\mathcal C)\) holds its donor source
coordinates and other rows’ measurements fixed while replacing target
row \(i\) by the query measurements. It differentiates the resulting
standardization, positive maps, and powers. For example, global reward
standardization uses
over the \(k\) eligible rows, with its variance recomputed from these same values. Local standardization also differentiates the target’s configured localization weights. In the clipped branch the programmed metric is \(g_{\mathcal C}(z)=\epsilon_\Sigma I+ [\nabla_z^2F_i^{\mathrm{probe}}(z;\mathcal C)]_+\).
A population perturbation instead specifies \(X_\theta\) and reevaluates the transition from that extended state. Even with companion identities fixed, its branch observable \(A_\theta(c)\) differentiates every affected measurement and normalization. For example, if \(c_j=i\), moving current walker \(i\) changes the sampled distance in row \(j\). This dependence is absent from the query field whose other measurements remain fixed. These are two explicitly different arguments of the programmed formulas. The sampled/expected distinction in Definition 594 additionally distinguishes holding a discrete assignment fixed from averaging over its law.
Proposition 290 (Derivative of the complete finite-choice transition)
Let \(\theta\) vary in an open finite-dimensional parameter region. Let \(\mathcal C_{\mathrm{fin}}\) be a fixed finite list of complete discrete choice histories, including zero-probability histories. Write \(p_\theta(c)\) for the actual joint probability of history \(c\) and \(A_\theta(c)\) for its resulting observable after all deterministic dependencies on \(\theta\) have been evaluated. Where these functions are differentiable,
If the positive-probability support is constant in a neighborhood and \(\ell_\theta(c)=\log p_\theta(c)\) on that support, this becomes
For twice differentiable terms, the Hessian is
Use the actual joint law, including mutual-pair constraints and sequential choices. Its factorization into conditional choice probabilities, rather than a product of independent row marginals, follows the operator composition in Theorem 175. Where those conditional probabilities are positive, the joint log-probability derivative is the sum of their log-probability derivatives, each evaluated along that history with all its state dependencies retained.
Proof
Differentiate each term of the finite sum by the product rule. On the fixed positive support, substitute \(\partial_a p=p\,\partial_a\ell\). Differentiate once more and use \(\partial_{ab}p=p(\partial_{ab}\ell+ \partial_a\ell\,\partial_b\ell)\) to obtain the Hessian formula. The chain rule for joint probabilities gives \(p_\theta(c)=\prod_jp_\theta(c_j\mid c_1,\ldots,c_{j-1})\); taking the logarithm gives the stated sum. For independent companion rows, Lemma 206 supplies the corresponding quantitative derivative bounds under its stated hypotheses.
Proposition 291 (Including continuous innovations)
For continuous innovations \(u\), write the full transition expectation as \(J(\theta)=\sum_c\int A_\theta(c,u)r_\theta(c,u)\,\nu(du)\) against a fixed reference measure. Fix a parameter coordinate \(a\) and a neighborhood of the evaluation point. Suppose \(A_\theta,r_\theta\) are continuously differentiable in that coordinate for \(\nu\)-almost every \(u\), \(J\) is absolutely integrable at the evaluation point, and
Then
On a common positive support this is the same score formula with \(\partial_a\log r_\theta\). If innovations are represented by a parameter-independent base law, its density derivative is zero; their parameter-dependent transformation remains in \(\partial_a A_\theta\).
Proof. The product rule gives the derivative of the integrand. The fundamental theorem of calculus bounds its difference quotient by \(M_c\). Dominated convergence therefore passes the derivative through the integral and the finite sum. Positivity permits division by \(r_\theta\).
Remark 290 (Applying the response formula to the metric equation)
For a perturbation of the exact drift in Theorem 437, differentiate the entire next-state expectation by the preceding propositions and subtract the derivative of the initial readout \(A_\theta(X_\theta)\). For the programmed metric observable, this differentiates both its query formula and the changes to its recorded context produced by the transition.
The finite-choice product formula retains probability changes even when a history has zero probability; the score form uses the stated support condition. At a clipping, acceptance, eligibility, or candidate-set change, apply a derivative formula only where its derivatives exist; the finite-step expectation and finite differences use the executed transition wherever the chosen observable is defined at the compared states. A frozen-choice derivative alone omits the law term whenever those probabilities respond to \(\theta\).
Conditional material metric drift and covariance
Rust engine experiment · Seed 7 · Poster after 11 experiment steps. Interactive view starts from the same seed.
Energy, Momentum, and Density from Actual Updates#
For the mechanical budget, begin with the velocity that actually enters an operation. Squaring the new velocity tells us the exact energy change. There is no need to guess a pressure or a temperature first.
The order of operations is part of the physics of this algorithm. The Rust integrator checks boundaries after each kick, each displacement, and the thermostat. A wrapped position or a newly killed walker contributes at that boundary operation. Combining the thermostat and the boundary into one measurement hides their separate contributions. It is still a valid combined increment, but it cannot be compared directly with the unconstrained thermostat formula.
Definition 772 (Mechanical observables and stage convention)
For unit particle masses, positions \(x_i\), velocities \(v_i\), and eligibility indicators \(a_i\), define
Here \(U\) is an explicitly specified mechanical energy observable, possibly population-dependent. Raw reward is not automatically \(-U\). With position units \(L\) and integration-time units \(\tau\), \(v\) has units \(L/\tau\), \(K\) has unit-mass units \(L^2/\tau^2\), friction has units \(\tau^{-1}\), and a velocity noise factor \(B\) has units \(L/\tau^{3/2}\). The algorithm may choose dimensionless reference units.
Use the executed stage ordering: measurement and donor selection; literal cloning; clone transforms; reconciliation, reward-validity, and boundary updates; kinetics; final reconciliation and validity checks. BAOAB kinetics executes \(B_1,A_1,O,A_2,B_2\), with boundary checks between these operations. Eligibility is recomputed between stages. For a pure thermostat or kick formula, use its input and output before any subsequent boundary or domain transformation; those transformations receive separate increments when separately observed.
Theorem 438 (Conditional energy and momentum of the Rust thermostat)
For one eligible walker, condition on everything entering the \(O\) stage,
including \(v\) and \(B\in\mathbb R^{d\times r}\). Let \(h>0\), \(\gamma\geq0\),
and suppose its innovation
satisfies \(\mathbb E\xi=0\) and \(\mathbb E\xi\xi^\top=I_r\). The update in
algorithmic-gas/crates/algorithmic-gas/src/kinetic.rs is
Put \(Q=BB^\top\) and \(K_v=|v|^2/2\). Then, before boundary handling,
For Gaussian innovations,
For independent, symmetric unit-variance components with common fourth moment \(\mu_4\), the variance instead is
The built-in standardized uniform law has \(\mu_4=9/5\); its correction is therefore negative. The mean formula does not require Gaussian noise.
Proof
The integrator applies \(c=e^{-\gamma h}\) and the time factor \(s^2=\int_0^h e^{-2\gamma t}\,dt\) to an unscaled noise-provider sample. Evaluating this integral gives both branches and the continuous limit at \(\gamma=0\). Expanding the kinetic energy gives
The linear term has zero mean and the quadratic term has mean \(s^2\operatorname{tr}(B^\top B)/2\). The velocity formulas follow directly. Set \(M=B^\top B\). For independent symmetric standardized components, terms of odd degree have zero expectation. Thus the linear term and the centered quadratic term have zero covariance. Expanding \(\mathbb E(\xi^\top M\xi)^2\) by matching indices gives
The linear variance is \(c^2s^2v^\top Qv\) and \(\operatorname{tr}(M^2)=\operatorname{tr}(Q^2)\), which proves both variance formulas. Integrating \(x^4\) under the uniform density on \([-\sqrt3,\sqrt3]\) gives \(9/5\). Singular and rectangular \(B\) cause no difficulty because no inverse was used.
Corollary 137 (Geometry in the thermostat budget)
If a provider constructs \(B=\sigma g^{-1/2}\) with \(g\) positive definite, then the conditional noise-energy injection for that walker is
For conditionally independent innovations across walkers, total conditional energy means and variances sum over the eligible rows. Without that independence, include the cross covariances. If \(B\) or the eligible set is itself random before the stage, the full-step prediction uses the tower property and total covariance over those earlier events.
Proof. Substitute \(Q=\sigma^2g^{-1}\) in Theorem 438 and apply conditional independence only for the stated variance sum.
Two thermostats can inject exactly the same mean energy and have different fluctuations. Gaussian and standardized uniform innovations demonstrate this directly: they have the same covariance, but their fourth moments differ. Looking only at the mean would miss a wrong innovation law in a simulation.
The noise factor also tells us where geometry enters. If a direction has smaller metric length and the provider uses the inverse metric for its noise covariance, that direction receives larger velocity kicks. The trace sums their energy contributions. This is a consequence of squaring the implemented update, with no temperature assigned. Interpreting it as heat exchanged with an equilibrium reservoir requires further information about the complete transition law.
Proposition 292 (Exact kicks and displacements)
For any realized velocity increment \(\delta v_i\) on a fixed eligible set,
A Rust BAOAB kick uses the executed acceleration
\(f_i^{\mathrm{tot}}=-\mathcal D_i(X)+f_i^{\mathrm{visc}}(X)\), where
\(\mathcal D_i\) is the vector returned by the configured gradient provider
and \(f_i^{\mathrm{visc}}\) is the configured QFT viscosity contribution,
zero when disabled. The recorded total_force stores this acceleration.
Thus \(\delta v_i=(h/2)f_i^{\mathrm{tot}}\), and its energy contribution per
eligible walker is
When viscosity is disabled, this specializes to \(-(h/2)v_i\cdot\mathcal D_i+(h^2/8)|\mathcal D_i|^2\). When both accelerations are present, the squared total includes their cross term; separate squared-force terms alone do not give the kick work.
When the provider is established to be the gradient of a scalar potential, one may write \(\mathcal D_i=\nabla_iU_{\mathrm{pot}}\). The increment identity does not require that additional identification. The second kick recomputes the provider at the post-\(A_2\) state. A displacement with unchanged velocities has zero kinetic increment and mechanical potential increment \(U(X^+)-U(X^-)\). Any relation between this \(U\) and \(U_{\mathrm{pot}}\) is part of the supplied energy model. Changes in eligibility and boundary transformations are accounted for by direct differences of the full observables.
Proof. Expand \(|v_i+\delta v_i|^2-|v_i|^2\) and substitute the kick. The displacement statement follows from the definition of \(E\). No continuous-time work approximation was made.
Proposition 293 (Literal-clone transfers and restitution)
Condition on a frozen source pool and realized clone decisions. Let \(C_i\) indicate that slot \(i\) is replaced, and let \((y_{J_i},w_{J_i}, \bar a_{J_i})\) be its source position, velocity, and eligibility. Literal cloning has the exact increments
The source may be historical. Simultaneous recipients read the frozen pool, including repeated donors, rather than one another’s updated slots. The potential contribution is the exact difference of the specified \(U\).
For an activated disjoint current mutual pair with pre-clone velocities \(v_i,v_j\), the built-in restitution transform with coefficient \(\alpha\in[0,1]\) sets
Relative to that pair’s pre-clone state,
The code activates this transform when either pair member accepted a clone. It overwrites both velocities using their pre-clone values. Consequently its transform-stage increment is
with the analogous subtraction for momentum. It is not an additional pair loss applied to the literal-clone velocities.
Proof
Literal cloning replaces exactly the indicated row observables, proving the first formulas by subtraction. For restitution, pair momentum equals \(2\bar v\) both before and after the transform, while pair kinetic energy is \(|\bar v|^2+|v_i-v_j|^2/4\) before and \(|\bar v|^2+\alpha^2|v_i-v_j|^2/4\) after. Finally, subtract the already recorded literal-clone increment to obtain the increment from that intermediate state.
Follow one cloned slot carefully. First it takes a donor’s velocity. Then restitution may overwrite that velocity using the original pair. The energy change from the original pair to the final pair is the dissipative loss we just calculated. The change from the intermediate literal clone to the final pair can have another sign. These statements agree because they compare different endpoints.
For a local balance, we must also record where the replaced walker and its donor were. A clone can move momentum across a large distance in one update. Replacing this transfer with a local pressure before deriving a spatial approximation would erase part of the algorithm. Test functions let us retain the full transfer while choosing the spatial resolution of the measurement.
Proposition 294 (Weak density and local momentum equations)
For a bounded test function \(\varphi\) on the walker state, define the fixed-capacity empirical measure
Then the exact finite-step weak equation is
Condition on the source pool and fitness values, but average over clone source selection and acceptance. If \(q_{ij}=\Pr(C_i=1,J_i=j\mid\text{this context})\), literal cloning gives
For the default decision law on an eligible row, with donor marginal \(\pi_{ij}\) conditional on the same frozen fitness context,
where \(\epsilon\) and \(\zeta\) are the configured acceptance regularizer and saturation. An ineligible row is instead revived uniformly from eligible current-frame sources. Mutual donor constraints affect joint laws and covariances; they do not change this marginal expectation formula.
For a spatial test function \(\psi\), the local momentum observable is \(p_\psi(X)=\sum_i a_i v_i\psi(x_i)\). Each row satisfies the exact decomposition
These are the impulse, transport, and eligibility contributions under this specified discrete convention. For a pure literal clone, the local momentum transfer is
It generally connects distinct locations and need not conserve total momentum. Historical replacement need not be a current-time pair interaction.
Proof
Apply Theorem 437 to \(A_\varphi\). Subtract each replaced atom and average its event indicator to obtain the clone formula. The default acceptance probability is precisely the clipped ratio evaluated by the Rust decision operator; multiply it by the conditional donor marginal. Expand the three local-momentum terms: the intermediate products cancel, leaving \(a'v'\psi(x')-av\psi(x)\). The literal-clone specialization follows by replacement.
Remark 291 (Stress, stationarity, and closure)
The tested weak momentum law determines impulses and transport. Representing nonlocal transfers by a stress density additionally requires a specified spatial deposition convention. A smooth segment permits \(\psi(y)-\psi(x)=\int_0^1\nabla\psi(x+t(y-x))\cdot(y-x)\,dt\); periodic domains and boundaries require the actual chosen path. This representation cannot silently discard sources or impose an isotropic perfect-fluid tensor.
If a transition-closed state representation has an invariant probability law \(\nu\) and \(A\) is integrable, then \(\int b_A\,d\nu=0\). Pure bookkeeping counters may be removed only when doing so leaves the conditional transition law determined by the retained state; a changing schedule cannot be discarded in this way. This says that the total stationary mean increment vanishes. The separate stage budgets can remain nonzero: thermostat injection can balance dissipative restitution or boundary losses. A quasi-stationary law of a killed chain is instead conditioned on survival and obeys its corresponding normalized law; it does not automatically satisfy this conservative stationarity identity.
The displayed empirical measure uses fixed \(N\); its mass changes with eligibility. Renormalizing by the number of survivors introduces a random denominator and yields another observable, not the same weak equation. Likewise, dividing a fixed-step clone increment by \(h\) does not establish a finite continuous-time generator when clone probabilities remain of order one. A macroscopic closure must be derived from the transition, with its limit and error controlled.
The spatial deposition in Theorem 441 constructs the stress and source terms from these mechanical observables. The conditional metric law couples their evolution through the actual fitness-dependent noise factor and the evolving donor context.
Conditional BAOAB diffusion and heating
Rust engine experiment · Seed 7 · Poster after 11 experiment steps. Interactive view starts from the same seed.
Reduced Fields and Their Derived Memory Equation#
Suppose we keep a density, a current and a few metric components. Two populations can give the same readings while carrying different donor histories. The next readings can then differ, even before sampling error enters the experiment. The omitted information has a dynamical effect.
We can follow that effect explicitly. A prediction first leaves the chosen field description, evolves through omitted variables, and later returns to a measured field. Each such excursion produces a memory term. The derivation below uses the distribution of the actual experiment at each time. A settling swarm and a stationary swarm are both covered by the same time-indexed calculation.
Theorem 439 (Transient projected field equation)
Let \(\lambda_n\) be the law of the complete state \(S_n\) generated by Theorem 434 from the declared initial law. For a specified field descriptor \(q_n\), put
Thus \(P_n:\mathcal H_{n+1}\to\mathcal H_n\). Write \(V_n=\operatorname{Ran}\Pi_n\) and \(W_n=\operatorname{Ran}R_n\) and define
Every block has operator norm at most one between its indicated spaces. For \(s<t\), let
Then the exact reduced field propagation satisfies
An empty product of \(\mathsf D\) blocks is the identity. In particular,
The left side is the discrepancy between actual two-step field prediction and composing the two one-step field predictors. Its right side is the exact contribution through omitted state variables.
Proof
Jensen’s inequality and the actual pushforward law \(\lambda_{n+1}=\lambda_nP_n\) give
This also shows that a null function maps to a null function, so the operator is defined on the stated equivalence classes. Conditional expectation and its orthogonal complement are contractions; all four block bounds follow.
Fix \(f\in V_t\). Set \(u_k=P_k\cdots P_{t-1}f\), \(x_k=\Pi_ku_k\) and \(y_k=R_ku_k\). The block decomposition gives
Substitution from \(t-1\) down to \(s+1\) yields
Insert this expression into the equation for \(x_s\) and use \(x_k=\mathsf T_{k,t}f\). This proves the stated recurrence and its two-step specialization. The Markov property of the complete state and the tower property identify \(\mathsf T_{s,t}f\) with \(\mathbb E[f(S_t)\mid q_s(S_s)]\). The law and the projection may change at every step of the experiment.
Corollary 138 (A criterion for a closed reduced field)
If \(\mathsf C_n=0\) throughout a horizon, then
This is closure for the actual initial law, up to its null sets. Closure for every relevant initial law follows from the stronger kernel factorization
on the relevant complete-state space. A nonzero two-step defect rejects one-step composition for that descriptor. A zero two-step defect alone allows cancellations and does not establish that all later memory terms vanish.
Proof. When \(\mathsf C_n=0\), every term of the memory sum vanishes, and induction gives the product. Kernel factorization says that the conditional next-descriptor law is determined by the current descriptor for every complete state, proving the stronger assertion by iterated conditioning. The product \(\mathsf B_s\mathsf C_{s+1}\) may vanish with \(\mathsf C_{s+1}\ne0\), which proves the last distinction.
Theorem 440 (Field evolution conditional on its observed history)
Assume the complete-state and descriptor spaces are standard Borel, and let \(Y_n=q_n(S_n)\) be a finite-dimensional square-integrable descriptor. Let \(\beta_n\) be the regular conditional law of \(S_n\) given \(Y_0,\ldots,Y_n\). Its prediction and observation update are
followed by disintegration of
with respect to its \(y'\) marginal, evaluated at the observed \(Y_{n+1}\). Here \(\mathbb Q_s\) is exactly the finite donor/gate sums and innovation measure in Theorem 434.
Define the primitive-computed increment and covariance
Then
and its conditional noise covariance is
Proof
Condition first on the complete state. The explicit update law gives the prediction integral. Standard Borel disintegration supplies the conditional next-state law given the next descriptor, including continuous readouts. By definition \(q_n(S_n)=Y_n\) almost surely under \(\beta_n\). The tower property therefore gives the displayed conditional increment. Subtracting this mean defines \(\zeta_{n+1}\). Applying the law of total covariance, first conditional on \(S_n\) and then on the observed history, yields the two terms: fresh transition noise and variation of the conditional increment over unresolved complete states.
These two noise terms have different experimental meanings. Even with a complete checkpoint, fresh donor choices and thermal kicks spread the next reading. When only a few fields are observed, there is additional uncertainty about which complete population and history produced those fields. The second covariance accounts for that uncertainty.
The equations are closed on the conditional population law \(\beta_n\). A practical reduction chooses a finite collection of field and memory features to approximate it. We can then compare the resulting predictions with independent complete-population continuations, increase the feature resolution, and measure what remains unexplained. Every term has an algorithmic origin before such an approximation is chosen.
Spatial Sources, Momentum Flux, and the Coupled Metric Equation#
Put a small measuring region around part of the swarm. Its momentum can change because a velocity changes inside it, because a walker crosses its edge, or because a slot is killed or revived. Cloning adds another possibility: a slot can take the state of a distant donor in one update.
We can represent that transfer exactly. Join the old position to the new one with a specified path and deposit the transferred momentum along it. The divergence of that deposited flux has one endpoint contribution at each end. What remains is an explicit local source. This construction lets us write spatial field equations while preserving every finite jump of the algorithm.
The velocity second moment gives a tensor of momentum transport. Its unequal directional components are measurable. We will retain that tensor, together with clone, force, thermal and boundary contributions, and couple it to the metric through the fitness-dependent noise factor that the algorithm actually uses.
Definition 773 (Spatial fields and a path-deposition convention)
In a fixed Euclidean chart, define the distribution-valued fields
For positions \(x,y\), put \(r=y-x\) and
Thus \(rL_{x,y}\) is an oriented segment deposition. Tensor divergence contracts the second index: \((\operatorname{div}\mathsf T)_\alpha =\sum_\beta\partial_\beta\mathsf T_{\alpha\beta}\). The fields have fixed-capacity normalization; \(\int\rho\) is the eligible fraction. A zero-length segment has zero associated flux.
On a periodic domain, use periodic distributions and a declared lift or path joining the endpoints. For a general piecewise smooth path \(\chi:[0,1]\to\mathcal X\), replace \(rL_{x,y}\) by \(\int_0^1\dot\chi(t)\delta_{\chi(t)}dt\). Different joining paths have the same endpoint divergence. Boundary classifications and coordinate wraps are recorded as their own actual stages.
Theorem 441 (Exact discrete spatial density and momentum equations)
Consider any realized row transition \((x,v,a)\mapsto(y,w,b)\) at any recorded stage, with \(a,b\in\{0,1\}\). Its unnormalized density and current increments satisfy the distributional identities
For a complete step, sum over slots \(i\) and actual stages \(r\) and divide by \(N\). Write the resulting sums of endpoint terms as \(\mathcal S_\rho,\mathcal S_j\) and the summed depositions as \(\mathcal J_\rho,\mathcal J_j\). Then
These are exact finite-step field equations. Every source and flux is computed from the actual donor, copy, force, transport, thermostat, reconciliation and boundary maps. Their conditional predictions follow by applying Theorem 434 to these explicit expressions.
Proof
For a smooth compactly supported scalar test \(\phi\), the fundamental theorem of calculus gives
By the definition of distributional divergence, \(\operatorname{div}(rL_{x,y})=\delta_x-\delta_y\). Hence
For each component \(\alpha\) of current, \(\operatorname{div}(bw\otimes rL_{x,y})_\alpha =bw_\alpha(\delta_x-\delta_y)\). Subtracting this from \((bw_\alpha-av_\alpha)\delta_x\) gives the second identity. Sum the row identities over stages: the intermediate fields cancel, including stages that change eligibility. Division by \(N\) gives the stated equations. The same calculation with \(\chi\) proves the general path formula. Integrating against the exact transition measure yields the conditional equations whenever the tested terms are integrable.
The sources supplied by each algorithmic instruction#
Corollary 139 (Primitive mechanical sources and transport tensors)
The following specializations of Theorem 441 hold before subsequent boundary operations.
Literal copying. An eligible donor \((y,w)\) replacing \((x,v,a)\) contributes mass source \((1-a)\delta_x/N\), momentum source \((w-av)\delta_x/N\), mass flux \(rL_{x,y}/N\), and momentum flux \(w\otimes rL_{x,y}/N\). For an eligible target the mass source vanishes; a revival contributes one unit of eligible mass divided by \(N\). Average these formulas using the actual donor and gate probabilities.
Kick. At fixed position and eligibility, a kick \(v\mapsto v+\delta t f_i\) has zero mass source and momentum source \(a_i\delta t f_i\delta_{x_i}/N\). Both BAOAB kicks use \(\delta t=h/2\) with the force recomputed at their own inputs.
Displacement. A drift \(x\mapsto x+\delta t v\) has no endpoint source. It contributes mass flux \(a\delta t v L_{x,x+\delta t v}/N\) and momentum flux \(a\delta t v\otimes v L_{x,x+\delta t v}/N\). This is a finite-path kinetic transport tensor.
Thermostat. At its fixed input position, the conditional mean momentum source is \(a(c-1)v\delta_x/N\). Its random source is \(a s_hB\xi\delta_x/N\) for centered innovations. Its conditional covariance comes from the actual \(s_h^2BB^\top\), including the metric-derived anisotropy.
Eligibility change. Killing at a fixed position contributes \(-a\delta_x/N\) and \(-av\delta_x/N\) as mass and momentum sinks. A boundary that also changes position or velocity uses the complete row identity. Jitter and restitution likewise use their actual intermediate input and output states.
For the configured viscous acceleration, define for eligible \(i\ne j\)
with zero value when the row normalizer vanishes. For \(i<j\), let \(d_{ij}=v_j-v_i\), \(r_{ij}=x_j-x_i\), \(k^s_{ij}=(\kappa_{ij}+\kappa_{ji})/2\), and \(k^a_{ij}=(\kappa_{ij}-\kappa_{ji})/2\). The viscous force density is
The first term may be moved to the flux side of the momentum equation with a minus sign and its actual kick duration. The second is a source from asymmetric row normalization. For the common eligible-count normalizer, \(\kappa_{ij}=\kappa_{ji}\) and this source vanishes.
Proof
Substitute the indicated primitive maps into the row identities. For the thermostat take the conditional mean using \(\mathbb E\xi=0\). For viscosity, collect the two terms of each unordered pair:
Use the segment-divergence identity on the first difference. The Gaussian weight is symmetric in its endpoints; a common denominator therefore cancels the antisymmetric coefficient. Row-dependent denominators generally retain it.
Proposition 295 (Derived anisotropic kinetic stress)
Let \(W_\ell\) be a specified nonnegative smoothing kernel. Convolve the spatial fields with it and, where \(\rho_\ell(z)>0\), define
Then
The scalar \(\operatorname{tr}\Pi_\ell/d\) is the mean directional kinetic stress; the traceless part retains measured anisotropy. For an O stage with centered unit-covariance innovations, its uncentered second-moment field obeys
When the conditional metric provider is enabled, its injection tensor is \(2\gamma T s_h^2g_{\tau_i}^{-1}\) at the actual O query. Subsequent boundary changes enter separate sources. The positive smoothing kernel may be applied to both sides of the equation.
Proof
Expand \((v_i-u)\otimes(v_i-u)\) and sum with the smoothing weights. The two linear terms use \(j_\ell=\rho_\ell u\) and leave \(\mathsf M_\ell-\rho_\ell u\otimes u\). Its quadratic form in any vector is a nonnegative weighted sum of squares. For O, expand \((cv_i+s_hB_i\xi_i)\otimes(cv_i+s_hB_i\xi_i)\); the mixed terms have zero mean and the last term averages to \(s_h^2B_iB_i^\top\). Insert the executed metric factor from Theorem 436.
The resulting field system#
Theorem 442 (Coupled population, mechanical, and metric fields)
Retain the complete marked field and its history from Proposition 286, together with the configured provider and input state. Then the following system determines its finite-step law and its derived mechanical and metric observables:
Here \(\mathcal T_{\mathrm{marked}}\) is the explicit donor, clone and BAOAB composition; \(\mathbb Q\) is its explicit joint choice law; and \(\mathcal S,\mathcal J\) are the primitive source and flux formulas above. The strict metric policy uses its specified unclipped branch. The next O-input metric is obtained by completing the current step, shifting history, applying the next admission, and executing selection and motion through A1 with its new context.
For a reduced collection of these observables, the exact prediction is Theorem 439 or, conditional on the observed field path, Theorem 440. These supply the contributions of eliminated population and donor-history variables.
Proof
The complete marked representation reconstructs every coefficient of Theorem 434. Its deterministic history shift closes the retained donor state. The spatial equations follow row by row from Theorem 441. The conditional-fitness differentiation and provider factor give the metric and noise equations. Their actual stage ordering gives the next O-input context. Finally, projecting this complete transition yields the proved transient memory and conditional-history equations. All coefficients are fixed by the algorithm, the initial law and the chosen measurement functions.
This is the field theory of the finite algorithm. The population and its memory determine the fitness curvature; that curvature shapes the next thermal kick; the kick changes the velocity stress and subsequent transport; selection and cloning change the population which determines the next ruler. The coupling runs through the whole cycle.
A compact relation involving only curvature and kinetic stress would have to eliminate the other terms in this system with a controlled approximation. We now know which terms need to be estimated and which measurements can test that elimination. A failed two-coefficient fit identifies a poor reduction of these equations, while the full equations continue to give independent finite-step predictions.
Algorithm-derived field equations and mechanical sources
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Density Fluctuations and a Correlation Energy#
A sampling rate measures how unlikely a density fluctuation is. An interaction energy is an additional specification. For independent samples the rate can be calculated from a multinomial probability. For the interacting swarm, the established entropy and LSI estimates control fluctuations directly. We retain those estimates and specify the comparison energy separately.
Independent sampling and interacting fluctuations#
Lemma 319 (Independent-sampling rate and interacting concentration)
For independent samples with law \(\pi\), the empirical frequencies on a fixed finite partition with probabilities \(p_1,\ldots,p_m>0\) have rate \(I(q)=\sum_jq_j\log(q_j/p_j)\). More precisely, for any attainable type \(q\),
For an interacting law \(\mu_N\), the independent-sampling identification requires a separate comparison. A proved implication sufficient for bounded-observable estimates is the following: if \(H(\mu_N\mid\pi^{\otimes N})\le H_*\), then Theorem 245 gives, for \(|f|\leq B\),
Alternatively a full-gradient joint LSI with constant \(C_*\) gives \(\operatorname{Var}_{\mu_N}(L_Nf)\leq C_*L^2/N\) for a fixed \(L\)-Lipschitz observable, by Corollary 98.
Proof. The multinomial formula gives \(\Pr(L_N=q)=N!\prod_jp_j^{Nq_j}/(Nq_j)!\). Applying \(\log n!=n\log n-n+O(\log(n+1))\) and \(\sum_jq_j=1\) yields the displayed rate. The interacting implications are precisely the entropy and Poincaré estimates proved in the cited chapters. Existence of a joint QSD alone supplies neither the multinomial formula nor an independent-sampling large-deviation rate.
Definition 774 (Fixed-mass Gaussian correlation free energy)
For this comparison model, take a bounded coordinate region \(\Omega\subset\mathbb R^d\) of volume \(V\), total mass \(N\), and \(\rho_0=N/V\). This finite region defines the auxiliary model; the confining Fractal Gas state space may remain unbounded.
For nonnegative \(\rho\) with \(\int_\Omega\rho=N\), set \(u=\rho-\rho_0\) and define, in fixed reference energy units,
where
Assume the displayed terms are finite. The first term equals \(N D_{\mathrm{KL}}((\rho/N)\,dx\|V^{-1}\,dx)\). The pair term is a specified correlation energy. Identifying this sum with the rate function or free energy of an interacting QSD requires its own law calculation.
Lemma 320 (Positivity of the Gaussian correlation free energy)
For \(u\in L^2(\Omega)\) and the preceding fixed-mass model,
Equality requires \(\rho=\rho_0\) almost everywhere. The functional is strictly convex on its finite-energy fixed-mass domain.
Proof
The entropy term is a relative entropy times \(N\), hence nonnegative, with equality only at \(\rho_0\). Extend \(u\) by zero outside \(\Omega\). With Fourier transform \(\widehat f(k)=\int e^{-ik\cdot x}f(x)\,dx\),
so Plancherel gives
The positive quadratic form is convex. The strictly convex function \(r\mapsto r\log(r/\rho_0)\) makes the entropy integral strictly convex on distinct densities. This proves the assertions.
Proposition 296 (Quadratic response and its relation to the jump form)
Let \(\rho=\rho_0(1+\phi)\) in the fixed-mass model, with \(\int_\Omega\phi=0\) and \(\|\phi\|_\infty\leq a<1\). Then
where \((\mathcal K\phi)(x)=\int_\Omega K_\varepsilon(x-y)\phi(y)\,dy\) and
Define the positive jump operator for this symmetric kernel by
Its exact relation to the free-energy Hessian is
where multiplication by \(\kappa_\Omega\) is understood. Moreover,
Thus the pair-energy Hessian and the jump operator are distinct, explicitly related operators.
Proof
For \(f(r)=(1+r)\log(1+r)\), \(f(0)=0\), \(f'(0)=1\), \(f''(0)=1\), and \(f'''(r)=-(1+r)^{-2}\). Taylor’s theorem on \([-a,a]\) gives
The integrated linear term vanishes by mass conservation. The interaction is exactly quadratic, proving the expansion and its bound.
Finally, \(\mathcal J=\kappa_\Omega I-\mathcal K\) gives the operator identity. Exchange \(x,y\) in half of \(\int\phi(x)\int K_\varepsilon(x-y)(\phi(x)-\phi(y))\,dy\,dx\). Kernel symmetry gives the difference-square formula.
Density perturbation and spatial dilation#
The following definition distinguishes a fixed-volume density perturbation from a mass-preserving spatial dilation.
Definition 775 (Mass-preserving affine density perturbation)
On \([0,L]^d\), define \(\phi_\kappa(z)=\kappa(z_1/L-1/2)\) and \(\rho_\kappa=\rho_0(1+\phi_\kappa)\) for \(|\kappa|<2\). Then \(\rho_\kappa>0\) and \(\int\rho_\kappa=\rho_0L^d\). This is a density perturbation. A spatial dilation is instead the pushforward \(\rho_a(z)=a^{-d}\rho(z/a)\) on \(a[0,L]^d\); Theorem 443 differentiates that explicit deformation.
Dilation Pressure and Correlation Stiffness#
Hold the kernel and total transported mass fixed while increasing all distances by a factor \(a\). A positive Gaussian pair energy decreases because its kernel becomes smaller at separated points. Its pressure is therefore positive for a nonnegative density. An attractive energy with the opposite sign has the opposite response. The derivative will also show why a signed density fluctuation has no universal pressure sign.
Theorem 443 (Dilation response and Gaussian correlation stiffness)
Let \(u\) be an integrable compactly supported density or signed density on \(\mathbb R^d\), and set
This describes transport of a fixed amount of \(u\) by dilation. Its pressure at \(a=1\) is
For \(u\ge0\) this is nonnegative; reversing the sign of the interaction energy reverses the pressure. For a signed fluctuation \(u=\rho-\rho_0\) the formula has no fixed sign. A quadratic free-energy expansion in a mass-preserving perturbation has zero first derivative at zero strain.
For the translation-invariant jump form with affine perturbation \(\Phi(x)=b\cdot x\), its quadratic energy per unit volume is
Proof. Differentiate the Gaussian under the integral: \(E'(1)=-(2\varepsilon_c^2)^{-1} \iint|x-y|^2K_\varepsilon(x-y)u(x)u(y)\,dx\,dy\). Divide by \(V'(1)=dV\) and change the sign. For \(\rho=\rho_0+\kappa u\) with \(\int u=0\), the entropy derivative is \(\int u=0\) and the interaction is quadratic in \(\kappa\), proving the zero-strain statement. Finally \(\int r_ir_jK_\varepsilon(r)\,dr= \delta_{ij}C_0(2\pi)^{d/2}\varepsilon_c^{d+2}\) by Gaussian integration. Contracting with \(b_ib_j\) proves the stiffness formula. Stiffness is a second variation; identifying it with a first-variation pressure requires a specified deformation and energy convention.
Remark 292 (Sign of the interaction response)
The positive Gaussian pair energy, its negative attractive counterpart, and the signed fluctuation energy have different dilation responses. The formula in Theorem 443 fixes the sign after the energy and deformation are specified. The jump-form stiffness is positive and scales as \(\varepsilon_c^{d+2}\) with \(C_0\) fixed. If instead the kernel mass is normalized by choosing \(C_0\propto\varepsilon_c^{-d}\), its stiffness scales as \(\varepsilon_c^2\). The selected kernel normalization must therefore be retained when comparing bandwidths or applying the formula to a row-normalized companion mechanism. Neither stiffness formula alone fixes a vacuum pressure.
Executed virial and dilation work
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A Homogeneous Density Model#
A translation-invariant model lets us ask how a single sinusoidal density wave changes. Diffusion smooths it. A balanced Gaussian redistribution also smooths it, because averaging over nearby points reduces its amplitude.
This calculation is exact for the specified linear model. The full Fractal Gas mean-field equation contains its actual state-dependent coefficients and normalized cloning terms. Those are derived in the mean-field chapter. We use the homogeneous model here as a comparison whose Fourier multipliers can be calculated completely.
Definition 776 (Auxiliary gain–loss density equation)
For a specified drift \(b\), nonnegative kernel \(K_{\mathrm{clone}}\), loss rate \(\lambda_{\mathrm{kill}}\), and constant \(D_{\mathrm{eff}}\geq0\), consider
Use periodic boundary conditions, or sufficient whole-space decay for the integrations under consideration. The gain–loss part conserves mass for every integrable density precisely when
This is a specified linear density model. The actual nonlinear mean-field equation, companion law, and survival-normalized evolution are derived in The Mean-Field Law of the Euclidean Gas.
For comparison, if an unnormalized killed density satisfies \(\partial_tf=L^*f-\kappa f\) for a conservative \(L\), then \(\rho=f/\int f\) satisfies
A QSD is stationary for this normalized evolution and is a left eigenmeasure for the killed one.
Proof
Integrate the gain–loss term and use Fubini. Its integral is \(\int[\int K_{\mathrm{clone}}(x,y)\,dx-\lambda_{\mathrm{kill}}(y)] \rho(y)\,dy\), proving sufficiency and necessity by testing all nonnegative densities. For killing, \(m'=-m\langle\kappa\rangle_\rho\) where \(m=\int f\). Differentiating \(f/m\) gives the normalized equation.
Definition 777 (Uniform reference and homogeneous Gaussian closure)
Set \(b=0\) and let the redistribution operator be convolution:
On a periodic box of side \(L\), periodize this kernel by summing its translates in \(L\mathbb Z^d\). Its integral over the box is \(\lambda_{\mathrm{kill}}\), and the constant density \(\rho_0=N/L^d\) is stationary for the conservative model.
Writing \(\rho=\rho_0+\delta\rho\) gives the exact equation
There is no neglected nonlinear term in this specified closure. On \(\mathbb R^d\) the same equation describes perturbations about a homogeneous background. A nonzero constant background on that space has infinite total mass and is not a probability density or a QSD.
Fourier Analysis and Dispersion Relation#
Theorem 444 (Exact decay multipliers of the homogeneous closure)
For a translation-invariant gain kernel \(k_{\mathrm{gain}}\) and the Fourier convention \(\widehat f(k)=\int e^{-ik\cdot x}f(x)\,dx\), a mode \(e^{ik\cdot x-\omega(k)t}\) has
For the Gaussian model of Definition 777,
On the periodic box these identities hold at \(k\in(2\pi/L)\mathbb Z^d\).
Proof
The Laplacian multiplies a Fourier mode by \(-|k|^2\). For the gain term, substitute \(r=x-y\):
The loss term multiplies the mode by \(-\lambda_{\mathrm{kill}}\). Thus \(-\omega=-D_{\mathrm{eff}}|k|^2+\widehat k_{\mathrm{gain}}(k) -\lambda_{\mathrm{kill}}\).
The product of the one-dimensional Gaussian Fourier integrals is \(e^{-\varepsilon_c^2|k|^2/2}\) after normalization, giving the formula. For a periodized kernel, integration over one box and summation over its translates give the same Fourier integral at the allowed discrete frequencies.
Remark 293 (Self-adjoint homogeneous closure)
For an even integrable convolution kernel and periodic or whole-space Laplacian with its standard self-adjoint domain, convolution is a bounded self-adjoint operator. The bounded perturbation of the self-adjoint Laplacian is self-adjoint on the same domain. Thus the Gaussian closure has real Fourier multipliers. General directed cloning kernels need not satisfy this symmetry; their evolution is treated by the kinetic and mean-field estimates in the convergence chapters.
Density modes of the executed population
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Stability and Domain-Dependent Rates#
Every nonzero sinusoid decays in this Gaussian model. Whether there is a single exponential rate for all perturbations depends on the domain. A periodic box has a smallest nonzero wave number. The whole space has arbitrarily long waves, and those decay arbitrarily slowly. The conserved constant mode must also be separated from the relaxing modes.
Theorem 445 (Stability of the homogeneous Gaussian closure)
For the translation-invariant closure in Definition 777, let \(D_{\mathrm{eff}},\lambda_{\mathrm{kill}}\ge0\) and \(\varepsilon_c>0\). Every nonzero Fourier mode decays strictly if and only if \(D_{\mathrm{eff}}+\lambda_{\mathrm{kill}}>0\). The zero mode is conserved. For \(q=\varepsilon_c^2|k|^2/2\),
Proof. Substitute the Gaussian multiplier from Theorem 444. For \(q\ge0\), \(e^q\ge1+q\) implies \(1-e^{-q}\ge q/(1+q)\), and integration of \(e^{-r}\le1\) gives \(1-e^{-q}\le q\). Also \(1-e^{-q}\le1\). The multiplier is positive for \(k\ne0\) when at least one coefficient is positive and is identically zero when both vanish. At \(k=0\) it is zero in every case.
Corollary 140 (Relaxation on a domain with a nonzero first frequency)
Assume \(D_{\mathrm{eff}}+\lambda_{\mathrm{kill}}>0\). For the preceding closure on the periodic box of side \(L\), the mean-zero solution satisfies
On \(\mathbb R^d\) the same multiplier has infimum zero over nonzero frequencies; the closure therefore has no uniform exponential \(L^2\) rate. For \(D_{\mathrm{eff}}>0\) and \(\delta\rho_0\in L^1\cap L^2\) it has the bound \(\|\delta\rho_t\|_2\le C_d(D_{\mathrm{eff}}t)^{-d/4}\|\delta\rho_0\|_1\). The confining kinetic model uses the different long-time estimates of Convergence, Survival, and Parameter Dependence and Hypocoercive Entropy: From Kinetic Mixing to the Full Swarm.
Proof. The multiplier increases with \(|k|^2\). On the box, every nonzero Fourier frequency has length at least \(2\pi/L\); Parseval gives the first bound. On \(\mathbb R^d\), \(\omega(k)\to0\) as \(k\to0\). Plancherel, \(|\widehat{\delta\rho_0}|\le\|\delta\rho_0\|_1\), and \(\omega(k)\ge D_{\mathrm{eff}}|k|^2\) give the Gaussian integral bound.
Remark 294 (Long-wavelength diffusion)
The exact Gaussian multiplier has expansion
Its long-wavelength coefficient is
with strict inequality when \(\lambda_{\mathrm{kill}}>0\). The negative fourth-order term is a correction to a small-frequency series. Extending that truncated polynomial to arbitrarily large frequencies would produce a spurious instability; the exact multiplier remains nonnegative. Nonlinear or directed cloning mechanisms use their own linearization and stability calculation.
Kinetic Diffusion and the Time Step#
A velocity keeps part of its value from one instant to the next. Those correlations determine how far position spreads. For an Ornstein–Uhlenbeck velocity, the covariance is an exponential, so we can integrate it exactly. This gives the continuous-time diffusion coefficient. A finite BAOAB step has a related geometric covariance series, which gives its own coefficient before taking the small-step limit.
Definition 778 (Free kinetic reference operator)
The constant-coefficient reference process
has backward generator
Its stationary velocity covariance is \(v_T^2I\), where \(v_T^2=\sigma_v^2/(2\gamma)\). Position on \(\mathbb R^d\) spreads and has no uniform invariant probability law. The adaptive, forced, aligned, and cloning swarm uses its complete generator rather than this free reference operator.
Theorem 446 (Diffusion coefficient of integrated Ornstein–Uhlenbeck motion)
For \(dV_t=-\gamma V_tdt+\sigma_vdW_t\), \(dX_t=V_tdt\), with \(V_0\) in its stationary Gaussian law, put \(v_T^2=\sigma_v^2/(2\gamma)\). For each coordinate,
Thus the long-time diffusion coefficient is \(D_{\mathrm{eff}}=v_T^2/\gamma=\sigma_v^2/(2\gamma^2)\). This coefficient belongs to the stated kinetic reference process.
Proof. The explicit OU solution gives \(\mathbb E[V_sV_r]=v_T^2e^{-\gamma|s-r|}\) coordinatewise. Integrating this covariance over \([0,t]^2\) gives the displayed variance. Moreover \(X_t-X_0=(V_0-V_t)/\gamma+(\sigma_v/\gamma)W_t\). After diffusive rescaling the first term vanishes in mean square at each fixed rescaled time. The Brownian term has variance \(\sigma_v^2t/\gamma^2=2D_{\mathrm{eff}}t\).
Lemma 321 (Diffusion of the force-free BAOAB reference step)
Fix a constant positive definite covariance shape \(D_0\), timestep \(\Delta t>0\), and \(0<a<1\). Consider
where the \(\xi_n\) are independent standard Gaussians and the velocity starts in stationarity. Then
For the thermostat coefficients \(a=e^{-\gamma\Delta t}\) and \(c_2^2=v_T^2(1-a^2)\), this is
This is the existing BAOAB update specialized to zero force, zero alignment, no jumps, and constant diffusion shape. The result identifies that reference update; it does not alter the full algorithm.
Proof
The covariance recursion is \(C_v=a^2C_v+c_2^2D_0\), giving the stationary covariance. Iteration gives \(\operatorname{Cov}(V_{n+r},V_n)=a^rC_v\). Therefore
The position increment is
The second term has bounded second moment as \(m\to\infty\); its covariance and cross terms vanish after division by \(m\). This proves the limiting diffusion tensor. Substitute the thermostat coefficients and use \((1+e^{-z})/(1-e^{-z})=\coth(z/2)\) and its Taylor expansion.
Pressure of Specified Gaussian Modes#
Now specify a finite collection of fluctuating modes and their quadratic energy. A Gaussian integral gives the partition function exactly. Pressure depends on how the energy coefficients change with volume. A relaxation rate by itself does not specify that energy: it also depends on mobility and noise normalization. The following model states all three choices.
Assumption 22 (Gaussian fluctuation reference model)
For a finite set of real modes \(q_j\), specify \(\Theta=k_BT_{\mathrm{eff}}>0\) and the quadratic energy
with density \(Z^{-1}e^{-E/\Theta}\) relative to a specified Lebesgue measure on mode coordinates. Then \(\mathbb E q_j^2=\Theta/w_j\) and the modes are independent.
For mobility \(m_j>0\), the dynamics
have this invariant law and relaxation rate \(m_jw_j\). Identifying \(w_j\) with a measured rate therefore requires \(m_j=1\) in the declared units, or another measured mobility and its matched noise amplitude. An arbitrary cloning QSD does not determine this Gaussian energy model.
Proof
The density factors into normalized one-dimensional Gaussians with variance \(\Theta/w_j\). The stated OU process has stationary variance \((2m_j\Theta)/(2m_jw_j)=\Theta/w_j\), proving invariance and the decay rate.
Proposition 297 (Pressure of a finite Gaussian mode ensemble)
For Assumption 22, at fixed temperature, fixed mode index set, and fixed reference measure on mode coordinates, suppose \(w_j(V)>0\) is differentiable. The mode pressure is
If every \(w_j(V)=a_jV^{-2/d}\), then \(P_{\mathrm{modes}}=k_BT_{\mathrm{eff}}n/(dV)\) for \(n\) real modes. A prescribed spatial cutoff \(|k|\le k_*\) on a periodic box gives \(n\sim V\operatorname{vol}(B_d)k_*^d/(2\pi)^d\) in the large-box limit, with the zero mode omitted. A classical Gaussian ensemble has no intrinsic thermal cutoff; an infinite unregularized mode count diverges.
Proof. Gaussian integration gives \(Z=\prod_j(2\pi k_BT_{\mathrm{eff}}/w_j)^{1/2}\). Differentiate \(F=-k_BT_{\mathrm{eff}}\log Z\) and use \(P=-\partial_VF\). The power law follows by differentiating \(\log w_j\). For mode counting, place a unit cube at each integer lattice point in the ball of radius \(Lk_*/(2\pi)\). Their union lies between balls with radii differing by at most \(\sqrt d\), giving the stated leading volume. Changing the cutoff or the mode index set during a volume derivative requires the corresponding additional terms in \(F\).
Remark 295 (Pressure and stiffness use different derivatives)
The pair-energy pressure is computed along a specified spatial dilation. The mode pressure differentiates the Gaussian partition function at fixed temperature and fixed mode set. Their sum represents a chosen effective energy model only when these conventions and the common state law agree. The quadratic correlation stiffness is a separate response coefficient.
Velocity-mode energy and Parseval balance
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Pressure Regime Analysis#
Definition 779 (Crossover of a prescribed pressure model)
For a phenomenological pressure model \(P(\varepsilon_c)=B-A\varepsilon_c^{d+2}\) with fixed \(A,B>0\), define \(\varepsilon_c^{\mathrm{th}}=(B/A)^{1/(d+2)}\). The coefficient \(A\) is an attractive-pressure coefficient of the chosen model; its value must come from a specified dilation derivative. It is not obtained by relabeling the positive jump stiffness.
Theorem 447 (Sign of the prescribed two-term pressure)
Under Definition 779,
It is positive below the crossover, zero at the crossover, and negative above it. These conclusions concern the prescribed pressure model.
Proof. Substitute \(A=B/(\varepsilon_c^{\mathrm{th}})^{d+2}\) and use strict monotonicity of \(r^{d+2}\) for \(r>0\). In particular, at fixed \(A,B\) the term proportional to \(\varepsilon_c^{d+2}\) becomes smaller, rather than larger, as the correlation length tends to zero.
Remark 296 (Analytical results and effective closures)
The homogeneous Gaussian closure has the exact Fourier multiplier and mode estimates above. The kinetic reference has the proved OU diffusion coefficient. Gaussian pair energies and mode ensembles have explicit first-variation pressures. Applying these formulas to the full interacting QSD requires the corresponding closure and law identifications, using Equilibrium Profiles and Their Analytical Characterization, The Discrete Population Limit and Propagation of Chaos, and Logarithmic Sobolev inequalities and entropy convergence.
Measured cloud and kinetic scales
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Results and Their Applications#
Theorem 434 gives the coefficient-explicit population law: the joint donor procedures, historical rescoring, clone gates, revival, transformations, BAOAB stages, boundaries and history shift. Theorem 435 converts those instructions into Fourier field equations and a full correlation hierarchy. Gaussian and standardized uniform innovations have distinct finite-step characteristic factors, even when their covariance agrees.
Theorem 441 expresses every realized finite jump as endpoint sources and path-deposited fluxes. Corollary 139 identifies the actual clone, force, transport, thermostat and boundary terms, including the source created by asymmetric viscous normalization. Proposition 295 derives the measured kinetic stress and its thermal covariance injection.
The metric is the conditional-fitness Hessian construction in Theorem 436. Its update retains the frozen donor context, normalization derivatives, actual query motion and next selection. Together with the source and flux equations and retained history, these give the closed complete-field system in Theorem 442.
Reduced descriptions obey the transient memory equation in Theorem 439. Their conditional prediction given an observed field path follows Theorem 440. These equations specify both fresh algorithmic noise and uncertainty from omitted state variables. They provide direct tests: compare exact O-stage predictions with independent innovations, telescope recorded spatial balances, compare next-O metric readouts from complete checkpoints, and test multi-step predictions while adding measured history features.
The Gaussian correlation energy has its proved quadratic expansion and dilation pressure. The homogeneous density model has the stated Fourier multiplier and domain-dependent stability rates. The continuous OU and constant-coefficient BAOAB calculations give their respective diffusion tensors. For a finite Gaussian mode ensemble,
These analytic models can be compared with independently measured algorithmic fields once their energy, law and normalization conventions are fixed. Applications to interacting long-time laws also use the finite-particle, QSD, mean-field, LSI and entropy estimates under their stated hypotheses.
Table of Symbols#
Symbol |
Meaning |
|---|---|
\(S_n\), \(X_n\), \(\mathsf P_{N,h}\) |
Complete algorithmic state and fixed-step transition operator |
\(Q_D\), \(Q_C\), \(Q_H\), \(\mathbb Q\) |
Explicit donor, historical-rescore and joint update choice laws |
\(\mathcal M_n^{(b)}\) |
Complete marked empirical field at retained age \(b\) |
\(\rho\), \(j\), \(\mathsf M\) |
Fixed-capacity spatial density, current and velocity second moment |
\(\mathcal S_\rho\), \(\mathcal S_j\), \(\mathcal J_\rho\), \(\mathcal J_j\) |
Actual endpoint sources and finite-path fluxes |
\(\Pi_\ell\), \(u_\ell\) |
Smoothed anisotropic kinetic stress and local mean velocity |
\(\mathcal F_j\), \(H_j\), \(\tau_i\) |
Conditional replacement fitness, its Hessian and O-query target map |
\(q_n\), \(\lambda_n\), \(\Pi_n\), \(R_n\) |
Reduced descriptor, actual run law and resolved/omitted projections |
\(\mathsf A_n\), \(\mathsf B_n\), \(\mathsf C_n\), \(\mathsf D_n\) |
Blocks of the actual projected transition operator |
\(\mathsf T_{s,t}\), \(\beta_n\) |
Two-time field predictor and full-state law conditional on observed fields |
\(A\), \(b_A\), \(\Gamma_A\) |
Specified observable, conditional increment drift, and covariance |
\(g_X(z)\) |
Metric reconstruction at a fixed reference-coordinate probe |
\(\eta_{n+1}\) |
Zero-mean conditional observable fluctuation |
\(K\), \(p\), \(U\), \(E\) |
Unit-mass kinetic energy, momentum, specified potential energy, and total energy |
\(B\), \(Q=BB^\top\) |
Actual velocity noise factor and its covariance rate |
\(c\), \(s^2\) |
BAOAB damping and integrated variance time factor |
\(\mu_4\) |
Fourth moment of a standardized scalar innovation |
\(a_i\), \(C_i\), \(q_{ij}\) |
Eligibility, realized clone indicator, and joint source/acceptance probability |
\(\mu_X\), \(p_\psi\) |
Fixed-capacity empirical measure and tested local momentum |
\(\mathcal F_{\mathrm{IG}}\) |
Specified fixed-mass correlation free energy |
\(\mathcal L_{\mathrm{IG}}\), \(\mathcal J\) |
Free-energy Hessian and positive jump operator |
\(K_\varepsilon\), \(C_0\), \(\varepsilon_c\) |
Gaussian pair kernel, amplitude, and length scale |
\(\phi_\kappa\) |
Fixed-volume affine density perturbation |
\(P_{\mathrm{pair}}\), \(P_{\mathrm{modes}}\) |
Pressures of the specified pair and mode energies |
\(D_{\mathrm{eff}}\), \(D_{\Delta t}\) |
Continuous reference and finite-step diffusion coefficients |
\(\omega(k)\) |
Decay rate of the homogeneous Fourier mode |
\(\lambda_{\mathrm{kill}}\) |
Balanced loss coefficient in the auxiliary gain–loss model |
\(T_{\mathrm{eff}}\), \(w_j\), \(m_j\) |
Mode temperature, quadratic energy coefficient, and mobility |
\(\varepsilon_c^{\mathrm{th}}\) |
Crossover in the prescribed two-term pressure model |
\(\gamma\), \(\sigma_v^2\), \(v_T^2\) |
Reference friction, noise variance rate, and stationary velocity variance |
References#
Geometry and Analytical Foundations#
Emergent Geometry from Adaptive Diffusion — Emergent Riemannian geometry from fitness landscape
Scutoid Spacetime: Moving Cells and Neighbor Changes — Cell reconstruction, sampling, and volume evolution
Curvature of the Algorithmic Fitness Metric — Curvature from discrete holonomy
External References#
Sydney Chapman and T. G. Cowling. The Mathematical Theory of Non-uniform Gases. Cambridge University Press, 3rd edition, 1990. ISBN 978-0-521-40844-8.
Amir Dembo and Ofer Zeitouni. Large Deviations Techniques and Applications. Volume 38 of Applications of Mathematics. Springer, 2nd edition, 1998. ISBN 978-0-387-98406-8.
Ryogo Kubo. The fluctuation-dissipation theorem. Reports on Progress in Physics, 29(1):255–284, 1966. doi:10.1088/0034-4885/29/1/306.
Alain-Sol Sznitman. Topics in propagation of chaos. In École d'Été de Probabilités de Saint-Flour XIX — 1989, volume 1464 of Lecture Notes in Mathematics, pages 165–251. Springer, 1991. doi:10.1007/BFb0085169.
Key citations:
Large deviations and rate functions: [Dembo and Zeitouni, 1998]
Kinetic scaling and Chapman–Enskog context: [Chapman and Cowling, 1990]
McKean-Vlasov equations: [Sznitman, 1991]
Einstein relation and fluctuation-dissipation: [Kubo, 1966]