Curvature of the Algorithmic Fitness Metric

Prerequisites: Emergent Geometry from Adaptive Diffusion and Scutoid Spacetime: Moving Cells and Neighbor Changes.

TLDR

Start with the algorithm’s recorded population, eligibility mask, companion sources, and fitness configuration. Move one query through that fixed conditioning data, recomputing the affected measurements, statistics, and local weights. This defines the conditional fitness field \(V_i(z\mid\Xi)\) and its spatial derivatives.

The configured diffusion provider converts its Hessian \(H\) into \(g=H+\epsilon_\Sigma I\) on the strict positive branch, or \(g=H_++\epsilon_\Sigma I\) on the clipped branch. Its normalized covariance shape is \(g^{-1}\). These are identified algorithmic constructions; their curvature follows from differentiating them.

A smooth Hessian metric needs six Hessian and ten third-derivative components in three dimensions. Direct contractions give Ricci, scalar, and sectional curvature without evaluated fourth derivatives or a materialized Riemann tensor. Mixed-sign clipping generally needs fourth fitness derivatives through the first two derivatives of its metric. Clipping thresholds require separate smoothness analysis.

Transport supplies a second measurement: consistent small-loop holonomy recovers the same curvature with a quantified error. Spacetime transport, Raychaudhuri, and geodesic focusing apply only after the additional Lorentzian construction and their stated hypotheses. They impose no evolution equation on the gas. The finite-step metric evolution and mechanical balances of the executed updates are developed in Field Equations and Pressure Dynamics.

Polyhedral Gauss–Bonnet constrains curvature on a fixed surface; subdividing a cell preserves the Euler characteristic.

Introduction

Choose a walker at the stage where the algorithm constructs its adaptive noise. Keep the recorded population and donor sources in view, and move the walker’s query position a little. Its reward and companion distance change. The statistics used to normalize those measurements can change too. Calculating all of those changes gives the derivatives of the conditional fitness actually used by this provider.

The next step is a concrete matrix calculation. Apply the configured Hessian shift or spectral clipping, and obtain the positive metric that sets the shape of the noise. Its inverse tells us which directions the walker explores easily. Spatial derivatives of that same metric give the connection and curvature. We can evaluate these quantities on a snapshot and compare two independent formulas before asking what they do over time.

Only after this calculation do we carry an arrow around a loop. That experiment measures the already identified curvature when the transport rule is sufficiently accurate. Introducing a time coordinate and a Lorentzian metric lets us study additional geometric identities, but their hypotheses must be stated separately. To find how the gas changes its own geometry, we return to its transition law and calculate the metric increment produced by the full update.

From Conditional Fitness to the Metric

Suppose every walker carries a little ruler, with squared length \(g_{ab}\,dx^a dx^b\). Its principal axes tell us which directions the adaptive noise explores more easily. Now put identical rulers everywhere, all stretched by the same amount along the same axes. There is anisotropy, but there is no geometric curvature: a fixed change of coordinates makes every ruler Euclidean.

This distinguishes two uses of the word curvature. The Hessian measures how a scalar fitness changes near a point. Riemann curvature measures how the rulers fit together across neighboring points. Even a positive, large fitness Hessian gives a flat metric when it is constant. Nor does every varying Hessian metric have nonzero Riemann curvature; a separable potential provides another useful flat example.

Three dimensions need not make the local calculation expensive. For a smooth Hessian metric, the cancellation in Lemma 312 leaves only six independent Hessian entries and ten independent third derivatives. We can compute the scalar curvature from those sixteen numbers without building an 81-entry Riemann array. The important preliminary work is to specify which fitness function those derivatives belong to. Moving a query while keeping its sampled donor fixed and moving the whole population are different experiments, and their derivatives answer different questions.

Definition 752 (Conditional fitness field and derivative convention)

Fix a recorded population, its eligibility mask, the sampled companion identities, their immutable source coordinates, the fitness parameters, and a target row \(i\). Denote these conditioning data by \(\Xi\). An identified conditional fitness provider specifies a scalar field \(V_i(z\mid\Xi)\) by replacing the target query by \(z\) and executing its stated measurement and fitness calculation. Its derivatives are spatial derivatives with respect to \(z\) on one smooth stratum of that calculation. The stratum excludes changes of eligibility, donor identity, periodic image, or an active nonsmooth floor.

For the conditional global-statistics construction, the target reward and separation vary with \(z\), and the population means and variances containing those measurements are recomputed and differentiated. Other rows and donor source coordinates remain fixed. Freezing the numerical means and variances instead defines a different comparison field; it must be identified as such. For a neighborhood-dependent construction, the dependence of its weights and affected measurements is also part of the derivative.

Differentiating the complete measurement stage with respect to a source population coordinate follows all uses of that coordinate. Even with companion indices fixed, moving row \(i\) changes the separation of each row that selects \(i\) as its source. For example, on a noncoincident Euclidean stratum, a row \(j\) selecting \(i\) has

\[ \nabla_{x_i}|x_j-x_i|=\frac{x_i-x_j}{|x_i-x_j|}. \]

Those affected measurements also enter the differentiated population statistics. This population derivative and the provider’s query derivative have different data dependencies: the former moves the source coordinate wherever it is used; the latter evaluates a query against the immutable source snapshot supplied to that operator. Holding companion indices fixed specifies the source identities, while holding source coordinates fixed additionally specifies which inputs are constant. Two such calculations can agree in value at a recorded point and have different gradients and Hessians.

In the fixed affine coordinates of the algorithm, define the smooth shifted construction by

\[ \Phi_i(z\mid\Xi)=V_i(z\mid\Xi)+\frac{\epsilon_\Sigma}{2}|z|^2, \qquad g_{ab}=\partial_a\partial_b\Phi_i. \]

Its domain of classical Hessian curvature is an open region where \(\Phi_i\in C^4\) and \(g\succ0\). The configured shift or clipping is the algorithm’s noise rule; the derivative convention follows the inputs of the operator that executes it. If coordinates have units \(L\) and fitness has units \(F\), then \([g]=F/L^2\), \([\epsilon_\Sigma]=F/L^2\), \([C]=F/L^3\), and scalar and sectional curvature have units \(F^{-1}\). All are dimensionless for dimensionless algorithm coordinates and fitness. A nonlinear coordinate change transports \(g\) as a tensor; taking an ordinary Hessian again in the new coordinates generally constructs a different metric.

INTERACTIVE EXPERIMENT · VI-39

Curvature of the executed fitness metric

How does fitness normalization and clipping shape curvature? Reconstruct conditional fitness derivatives using the actual donor and normalization context, then compute the configured metric, connection, and curvature tensors. Independently refine spatial finite differences until successive numerical scalar and Ricci estimates stabilize, and display their discrepancy with the analytic jet.

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From a Metric to a Connection

Once the provider has supplied a smooth positive metric field, we can compare arrows at neighboring query positions. Requiring that comparison to preserve the metric and have zero torsion determines the Levi-Civita connection. Its coefficients follow from the spatial derivatives of the same field; a collection of metric matrices without a spatial reconstruction does not supply those derivatives.

The first calculation uses only the algorithm’s spatial coordinates. A spacetime comparison introduces additional coefficients under Assumption 21.

Definition 753 (Levi-Civita connection)

For a \(C^1\) nondegenerate metric \(q\), the Levi-Civita connection has coefficients

\[ \Gamma^a_{bc}[q] =\frac12q^{ae} (\partial_bq_{ec}+\partial_cq_{eb}-\partial_eq_{bc}). \]

It is metric compatible and torsion free. Here \(q=g_t\) gives the intrinsic spatial connection. A spacetime application uses the separately specified \(q=G\) in Assumption 21. For the smooth shifted Hessian field, spatial coefficients use third derivatives of the fitness potential.

Metric compatibility and zero torsion determine these coefficients uniquely: add \(\partial_bq_{ac}=\Gamma_{abc}+\Gamma_{cba}\) and \(\partial_cq_{ab}=\Gamma_{acb}+\Gamma_{bca}\), subtract \(\partial_aq_{bc}\), and use symmetry in the two lower connection indices to solve for \(\Gamma\).

INTERACTIVE EXPERIMENT · VI-37

Metric transport along executed displacements

Does numerical transport preserve the measured metric norm? Reconstruct the selected conditional fitness metric from its executed donor context. Integrate its Levi-Civita transport along segments joining recorded positions and compare coarse and refined transport and metric-norm residuals.

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Curvature Tensors and Hessian Metrics

A varying metric produces a connection, and a varying connection produces curvature. For a general metric this uses second derivatives of the metric.

A Hessian metric has extra symmetry. All its entries are second derivatives of the same scalar function. When we antisymmetrize to form curvature, the fourth derivatives of that scalar cancel. What remains is a product of third derivatives. This gives a useful direct bound from the fitness regularity estimates.

Definition 754 (Riemann curvature tensor)

For a \(C^2\) metric and its Levi-Civita connection, use the convention

\[ R^a{}_{bcd} =\partial_c\Gamma^a_{bd}-\partial_d\Gamma^a_{bc} +\Gamma^a_{ce}\Gamma^e_{bd} -\Gamma^a_{de}\Gamma^e_{bc}. \]

Thus \(R(X,Y)V\) has components \(R^a{}_{bcd}V^bX^cY^d\). The metric and connection are both spatial or both spacetime according to the application.

Lemma 312 (Cancellation of fourth derivatives in a Hessian metric)

Let \(\Phi\in C^4\) in the fixed Euclidean coordinates and let \(g_{ab}=\partial_a\partial_b\Phi\) be positive definite. Write \(C_{abc}=\partial_a\partial_b\partial_c\Phi\). Then

\[ \Gamma^a_{bc}=\frac12g^{ae}C_{ebc},\qquad R_{abcd} =\frac14g^{pq} \left(C_{adp}C_{bcq}-C_{acp}C_{bdq}\right). \]

In particular, the smooth construction \(\Phi=V_{\mathrm{fit}}+\epsilon_\Sigma|x|^2/2\) has this formula.

If \(g\succeq mI\) and \(|C(X,Y,\cdot)|\leq K_3|X|\,|Y|\) in Euclidean norm, then the absolute value of every sectional curvature is at most \(K_3^2/(2m^3)\). Classical use of this identity is justified by the stated \(C^4\) regularity. A clipped field has this Hessian identity only where it actually agrees with the smooth Hessian metric.

Proof

Symmetry of \(C\) reduces the connection formula to \(\Gamma^a_{bc}=g^{ae}C_{ebc}/2\). Differentiating the inverse gives

\[ \partial_cg^{ae}=-g^{ap}C_{pqc}g^{qe}. \]

Substitute into the curvature definition. The two terms \(\tfrac12g^{ae}\partial_c C_{ebd}\) and \(-\tfrac12g^{ae}\partial_d C_{ebc}\) cancel because the fourth derivatives are symmetric. The remaining derivative-of-inverse terms combine with the two connection products. Lowering the first index gives

\[ R_{abcd} =\tfrac14g^{pq}C_{adp}C_{bcq} -\tfrac14g^{pq}C_{acp}C_{bdq}. \]

For \(g\)-orthonormal \(X,Y\), Euclidean lengths satisfy \(|X|,|Y|\leq m^{-1/2}\). Each cubic tensor contracted with two of these vectors has Euclidean norm at most \(K_3/m\). Each of the two inverse-metric products is therefore at most \(K_3^2/m^3\). Their coefficients sum to \(1/2\), proving the sectional bound.

Definition 755 (Ricci tensor and scalar curvature)

For the specified metric \(q\),

\[ \operatorname{Ric}_{bd}=R^a{}_{bad},\qquad R=q^{bd}\operatorname{Ric}_{bd}. \]

Geodesic focusing in the timelike direction \(u\) uses \(\operatorname{Ric}(u,u)\). Scalar curvature is a trace over directions; its sign alone does not determine that contraction or the sign of every sectional curvature.

Direct contractions and three-dimensional computation

Lemma 313 (Direct Hessian Ricci and scalar contractions)

Under Lemma 312, fix a point and let \(E\) be an invertible matrix satisfying \(E^{\mathsf T}gE=I\). Its columns form a \(g\)-orthonormal frame. Set

\[ \widehat C_{ijk}=C_{abc}E^a{}_iE^b{}_jE^c{}_k, \qquad (S_i)_{jk}=\widehat C_{ijk}, \qquad t_k=\sum_i\widehat C_{iik}. \]

Then the Ricci tensor in that frame and its scalar trace are

\[ \boxed{\widehat{\operatorname{Ric}}_{ij} =\frac14\left(\langle S_i,S_j\rangle_F -\sum_k t_k\widehat C_{ijk}\right)}, \qquad \boxed{R=\frac14\left(\|\widehat C\|_F^2-|t|^2\right)}. \]

The coordinate Ricci tensor is \(\operatorname{Ric}=E^{-\mathsf T}\widehat{\operatorname{Ric}}E^{-1}\). For linearly independent coordinate vectors \(u,v\), sectional curvature is

\[ K(u,v)= \frac{ C(u,v,\cdot)^{\mathsf T}g^{-1}C(u,v,\cdot) -C(u,u,\cdot)^{\mathsf T}g^{-1}C(v,v,\cdot)} {4\bigl(g(u,u)g(v,v)-g(u,v)^2\bigr)}. \]

These expressions require no evaluated fourth derivatives and no materialized fourth-order curvature tensor. They retain the \(C^4\) hypothesis used to justify the classical cancellation identity.

Proof

Because \(EE^{\mathsf T}=g^{-1}\), changing all four free indices in Lemma 312 to the orthonormal frame gives

\[ \widehat R_{abcd} =\frac14\sum_p (\widehat C_{adp}\widehat C_{bcp} -\widehat C_{acp}\widehat C_{bdp}). \]

The Ricci convention of Definition 755 contracts the first and third indices. Thus

\[ \widehat{\operatorname{Ric}}_{bd} =\frac14\sum_{a,p} (\widehat C_{adp}\widehat C_{bap} -\widehat C_{aap}\widehat C_{bdp}). \]

Symmetry of the cubic tensor identifies the first term as \(\langle S_d,S_b\rangle_F\) and the second as \(\sum_p t_p\widehat C_{bdp}\). Taking the trace sums the first term to \(\|\widehat C\|_F^2\) and the second to \(\sum_p t_p^2\). Since \(\widehat{\operatorname{Ric}}=E^{\mathsf T}\operatorname{Ric}E\), inverting that change of basis gives the coordinate formula. Finally, contract \(R_{abcd}\) with \(u^av^bu^cv^d\) and divide by the squared \(g\)-area of the parallelogram. Positivity of \(g\) and independence of \(u,v\) make this denominator positive.

Corollary 135 (Packed three-dimensional calculation)

In \(d\) dimensions, fully symmetric derivatives of order \(k\) have \(\binom{d+k-1}{k}\) independent components. For \(d=3\), store the Hessian at indices \(a\leq b\) and the cubic tensor at indices \(a\leq b\leq c\). They require six and ten components respectively. The complete scalar jet through order three contains \(1+3+6+10=20\) coefficients when stored as derivatives rather than Taylor coefficients.

Given an orthonormal frame, scalar contraction costs \(O(d^3)\) arithmetic and cubic storage; the direct Ricci contraction costs \(O(d^4)\). Whitening the dense cubic by three successive index contractions costs \(O(d^4)\), and a dense metric factorization costs \(O(d^3)\). These counts exclude evaluation of the fitness derivatives. A full Riemann output additionally has \(d^4\) entries and is unnecessary for scalar or Ricci queries.

Proof

A symmetric component corresponds to a multiset of \(k\) indices selected from \(d\) possibilities, giving the stated binomial count. In three dimensions the cubic entries are \(111,112,113,122,123,133,222,223,233,333\). For each transformed index, multiplication by \(E\) sums \(d\) terms for each of \(d^3\) output entries. The scalar expression then sums cubic entries and the \(d\) traces. Each of \(d^2\) Ricci entries uses an inner product with \(d^2\) terms. Packed storage must account for permutation multiplicities in full contractions: \(\|\widehat C\|_F^2\) counts an all-equal entry once, a two-equal entry three times, and an all-distinct entry six times. These are the numbers of distinct permutations of their indices.

Whitening means choosing units and axes that make the ruler the identity at the point where we are calculating. It does not flatten a neighborhood. The third derivatives still describe how the neighboring rulers change, and their contractions retain the curvature.

Two checks are particularly revealing. If the fitness is quadratic, the cubic tensor vanishes and all geometric curvature vanishes. If the potential is separable, \(\Phi(x)=\sum_a\phi_a(x^a)\) with positive \(\phi_a''\), the metric varies but is still flat: replace each coordinate by \(y^a=\int^{x^a}\sqrt{\phi_a''(s)}\,ds\). The line element becomes \(\sum_a(dy^a)^2\). Both the direct contraction and an independent general metric calculation must reproduce these answers. A nonzero scalar curvature cannot be manufactured merely by measuring a large Hessian.

Lemma 314 (Constant-work global-statistics update at a fixed query)

Suppose precisely one of \(n\geq2\) eligible scalar measurements is a variable \(x(z)\). Let \(\bar x_-\) and \(M_{2,-}\) be the mean and sum of squared centered deviations of the other \(n-1\) measurements. The population mean and population variance, with denominator \(n\), are

\[ \bar x(z)=\bar x_-+\frac{x(z)-\bar x_-}{n}, \qquad \sigma^2(z)=\frac{M_{2,-}}{n} +\frac{n-1}{n^2}\bigl(x(z)-\bar x_-\bigr)^2. \]

After preparing these summaries, every derivative of these statistics through the requested order follows by differentiating these formulas; no new population scan is needed per query. For \(n=1\), the mean is \(x(z)\) and the population variance is zero.

Proof

Write \(\delta=x(z)-\bar x_-\). The other measurements have deviations from the new mean equal to their old centered deviations minus \(\delta/n\). Their centered deviations sum to zero, so their new squared deviations sum to \(M_{2,-}+(n-1)\delta^2/n^2\). The target deviation is \((n-1)\delta/n\). Add its square and divide by \(n\) to obtain the variance formula. The mean follows by adding the target to the other measurements’ sum. Differentiation is exact on the stated smooth stratum. The formulas do not apply when changing the query also changes other measurements or query-dependent neighborhood weights.

Remark 284 (Clipped metrics require a different derivative calculation)

For the actual clipped construction, let \(H=\nabla^2 V_i\) and \(g_+=\epsilon_\Sigma I+f(H)\), where \(f(\lambda)=\max(\lambda,0)\). Assume \(V_i\in C^4\) and the spectrum of \(H\) remains separated from zero on an open neighborhood. Matrix spectral calculus then gives

\[ \partial_a g_+=Df(H)[\partial_aH], \qquad \partial_a\partial_b g_+ =Df(H)[\partial_a\partial_bH] +D^2f(H)[\partial_aH,\partial_bH]. \]

In an orthonormal eigenbasis of \(H\), the first derivative has entries \(f[\lambda_i,\lambda_j](\partial_aH)_{ij}\). The symmetric second derivative uses

\[ \bigl(D^2f(H)[A,B]\bigr)_{ij} =\sum_k f[\lambda_i,\lambda_k,\lambda_j] (A_{ik}B_{kj}+B_{ik}A_{kj}). \]

Here brackets denote divided differences, with their continuous derivative limits at repeated eigenvalues of the same sign. Repeated nonzero eigenvalues therefore introduce no mathematical singularity. To verify the formulas, take disjoint complex contours around the positive and negative spectra, and extend \(f\) analytically as \(z\) and \(0\) on their respective interiors. In the contour representation \(f(H)=(2\pi\mathrm i)^{-1}\oint f(z)(zI-H)^{-1}\,dz\), write \(A_z=(zI-H)^{-1}\). Its derivatives are \(DA_z[B]=A_zBA_z\) and \(D^2A_z[B,D]=A_zBA_zDA_z+A_zDA_zBA_z\). Diagonalizing \(H\) inside these integrals gives residues with two and three scalar resolvents, namely the first and second divided differences displayed above. The spectral gap lets one use the same contours throughout a neighborhood. The spatial formulas then follow by the chain rule with \(H(z)=\nabla^2V_i(z)\).

If every eigenvalue is positive, \(g_+\) agrees with the smooth shifted Hessian metric and Lemma 313 applies. If every eigenvalue is negative, \(g_+=\epsilon_\Sigma I\) on the neighborhood and its curvature is zero. In a mixed-sign region, \(g_+\) is generally not a Hessian metric. Compute its Levi-Civita curvature from \(g_+,\partial g_+,\partial^2 g_+\); the latter generally needs fourth fitness derivatives. Substituting clipped eigenvalues in the Hessian curvature identity does not perform this calculation.

At a zero eigenvalue, spectral clipping does not guarantee a \(C^2\) metric. Classical curvature is unavailable without a separate proof of smoothness at that point. A finite-difference estimate across the threshold is a resolution-dependent diagnostic, not a proof of a classical curvature value. A numerical tolerance used to exclude such points must be reported with the result.

Remark 285 (Curvature of a sampled field and of an averaged covariance)

The conditional field above is indexed by its sampled sources and population snapshot. Its curvature is not automatically the curvature of a population-averaged metric. In general,

\[ \mathbb E\bigl[(\epsilon_\Sigma I+H_+)^{-1}\bigr] \ne\bigl(\epsilon_\Sigma I+(\mathbb EH)_+\bigr)^{-1}, \qquad \mathbb E[R(g)]\ne R(\mathbb E[g]). \]

If a provider instead constructs an averaged covariance, its own spatial derivatives define its metric and curvature. Query-dependent sampling probabilities must also be differentiated when differentiating that expectation. Likewise the covariance shape \(g^{-1}\), its noise factor, and a physical increment covariance \(\alpha g^{-1}\) are separate quantities. For constant \(\alpha>0\), interpreting the latter’s inverse as the metric gives \(q=g/\alpha\) and \(R(q)=\alpha R(g)\); a varying \(\alpha\) also contributes metric derivatives. A reported curvature must therefore identify its field, derivative convention, and normalization.

Rust representation and execution

Definition 756 (Computational representation of fitness and metric jets)

The Rust module algorithmic_gas::physics::jet represents multivariate Taylor polynomials in a shared JetSpace::new(dimension, order). An internal coefficient indexed by the multi-index \(\alpha\) equals \(\partial^\alpha V/\alpha!\). Conversion through physics::geometry::FitnessJet::from_jet produces ordinary derivatives: its hessian, third, and optional fourth arrays use lexicographic order over nondecreasing axis tuples. value and gradient complete the scalar jet. Thus a repeated-index derivative must not be copied directly from a normalized Taylor coefficient.

The geometry entry points are:

Function	Mathematical input and result
hessian_curvature(&jet, epsilon, full_riemann)	Strict positive shifted Hessian; uses third derivatives and the direct contractions above
fitness_curvature(&jet, epsilon, policy, threshold, full_riemann)	Strict or clipped construction; the absolute Hessian spectral threshold determines where clipped classical curvature is unavailable
metric_curvature(&metric_jet, full_riemann)	An independently supplied positive metric and its first two spatial derivatives; evaluates the general Levi-Civita formula
ExecutionContext::smooth_curvature_batch(&jets, epsilon, policy, threshold)	Batched smooth Hessian or constant clipped metrics; whitening and curvature contractions use the explicitly selected compute backend

MetricJet uses row-major arrays with layouts \(g_{ij}\), \(\partial_k g_{ij}\), and \(\partial_k\partial_l g_{ij}\): derivative indices precede matrix indices. These entries must be derivatives of the same \(C^2\) metric field. A CurvatureBatch returns its spectrum, coordinate ricci, scalar, einstein, and eigenframe sectional curvatures, with optional covariant row-major riemann. The einstein field is the geometric contraction \(\operatorname{Ric}-Rg/2\); computing it imposes no field equation or identification with stress.

The sectional array enumerates pairs \(i<j\) of orthonormal metric eigenvectors. Within a repeated-eigenvalue eigenspace the eigenbasis is not unique, so these particular sectional planes are frame-dependent. The scalar and the coordinate Ricci tensor do not depend on that choice. The full_riemann=false route avoids allocating a Riemann output. The third-derivative Hessian route additionally avoids fourth fitness derivatives; that saving does not apply to general mixed-sign clipping.

The batch method performs the small eigendecompositions on the host, then uses backend tensor products for whitening, Ricci, scalar, and sectional contractions, with one combined result readback. It retains \(O(Nd^3)\) working storage and \(O(Nd^4)\) contraction work, subject to the execution context’s allocation limits. All queries in a batch must share a dimension and precision. Mixed-sign clipping and clipping thresholds are rejected by this method; there is no automatic backend fallback. Such mixed-sign queries require the separately invoked general metric calculation.

Definition 757 (Computational representation of tessellation geometry)

The Rust module algorithmic_gas::tessellation estimates geometry from the walker positions alone, without fitness derivatives. It is the discrete counterpart of From a Metric to a Connection: the sites are the projections of the eligible walkers onto the tessellated coordinates, and every later quantity is a function of their Delaunay complex.

Component	Mathematical input and result
TessellatorKind	Delaunay complex of the distinct sites: sorted path (\(d=1\)), planar triangulation (\(d=2\)), tetrahedralization with exact orientation and in-sphere predicates (\(d=3\)). Coincident walkers share a site and are mutual neighbors; a swarm confined to an affine subspace is triangulated inside that subspace; no coordinate is perturbed
TessellationDomain	Open space, a clip box realized by mirror-image sites, or a periodic box realized by translated image sites with minimum-image displacements
VoronoiCells	Dual cells from the circumcenters: facet measure \(A_{ij}\) and volume \(V_i=\sum_j A_{ij}\lVert x_j-x_i\rVert/(2d)\)
MetricKind::NeighborCovariance	Emergent metric \(g_i=\bigl(\tfrac{1}{\deg i}\sum_{j\sim i}\Delta x_{ij}\Delta x_{ij}^{\mathsf T}+\varepsilon I\bigr)^{+}\) with clamped spectrum, its determinant, and the diffusion factor \(g_i^{-1/2}\)
VolumeKind	\(\sqrt{\det g_i}\), the Voronoi volume \(V_i\), or their product
WeightMode	Edge weights \(w_{ij}\) from Euclidean or metric edge lengths \(d_g(i,j)^2=\Delta x_{ij}^{\mathsf T}\tfrac12(g_i+g_j)\Delta x_{ij}\), volumes, or facet measures, optionally normalized over each walker’s neighbors
CurvatureKind::ConformalLaplacian	With \(u_i=\log\det g_i/(2d)\), the conformal scalar curvature \(R_i=-2(d-1)\sum_j w_{ij}(u_j-u_i)\)
CurvatureKind::ConformalQuadraticFit	Weighted local quadratic fit of \(u\) giving \(\nabla u\) and \(\nabla^2u\); \(R=-2(d-1)e^{-2u}\bigl(\Delta u+\tfrac{d-2}{2}\lVert\nabla u\rVert^2\bigr)\) and the coordinate Ricci tensor \(R_{ab}=-(d-2)(u_{ab}-u_au_b)-(\Delta u+(d-2)\lVert\nabla u\rVert^2)\delta_{ab}\)
CurvatureKind::ReggeDeficit	Deficit angles \(\delta_h\) of the hinges (vertices for \(d=2\), edges for \(d=3\)) from edge lengths alone, with \(\int R\,dV=2\sum_h\lvert h\rvert\delta_h\) allocated to the sites over their barycentric dual volumes
CurvatureKind::{VolumeDistortion, ShapeDistortion, RaychaudhuriExpansion}	Voronoi-cell indicators: \(1-V_i/\langle V\rangle\), \(1-r_{\mathrm{in}}/r_{\mathrm{circ}}\), and \(-\theta_i\) with \(\theta_i=(V_i-V_i^{\mathrm{prev}})/(\Delta t\,V_i)\) (Expansion from Cell Volumes)
RewardAllocationKind::EinsteinHilbertDensity	The walker’s share \(r_i=\lambda R_i\,\mathrm{vol}_i\) of the Einstein–Hilbert action

The conformal estimators measure the curvature of the conformal class \(e^{2u}\delta\) with \(\det g=e^{2du}\); they are exact for a conformally flat metric and ignore the trace-free part of \(g\). The quadratic fit is exact for quadratic \(u\) up to its ridge, which is absolute and must stay far below the squared neighbor spacing. A Regge deficit angle is \(O(h^2)\) in the neighbor spacing \(h\), the same order as the relative error of a length built from the endpoint metrics; the Regge action converges to the continuum one for exact geodesic edge lengths, and with endpoint-metric lengths it is an indicator with an \(O(1)\) discretization bias. The Voronoi indicators are heuristic. None of these constructions imposes a field equation.

Predicates are evaluated exactly in double precision on the exact images of the run-precision coordinates; numerical geometry uses the run precision. Every per-walker, per-edge and per-cell result is a pure function of the tessellation, so serial and thread-parallel evaluation return identical bits.

Here is a small calculation whose answer can be checked on paper. Take \(V(x,y,z)=(x^2+y^2+z^2)/2+0.1xyz\) and evaluate at the origin, with \(\epsilon_\Sigma=0.2\). The metric there is \(1.2I\). The only nonzero cubic derivatives are the six permutations of \(V_{xyz}=0.1\). Whitening divides each by \(1.2^{3/2}\), while the trace vector \(t\) is zero. Consequently \(R=6(0.1)^2/(4(1.2)^3)\). This example tests a nonzero curvature, all three coordinates, and the multiplicity of a packed mixed derivative at once.

use algorithmic_gas::physics::{
    geometry::{FitnessJet, hessian_curvature},
    jet::JetSpace,
};

fn check_three_dimensional_curvature() -> algorithmic_gas::Result<()> {
    let space = JetSpace::new(3, 3)?;
    let x = space.variable(0.0_f64, 0)?;
    let y = space.variable(0.0_f64, 1)?;
    let z = space.variable(0.0_f64, 2)?;
    let quadratic = x.pow(2.0).add(&y.pow(2.0)).add(&z.pow(2.0));
    let potential = quadratic.scale(0.5).add(&x.mul(&y).mul(&z).scale(0.1));
    let jet = FitnessJet::from_jet(&potential)?;
    let curvature = hessian_curvature(&jet, 0.2, false)?;
    let expected = 6.0 * 0.1_f64.powi(2) / (4.0 * 1.2_f64.powi(3));
    assert!((curvature.scalar - expected).abs() < 1e-12);
    Ok(())
}

Note

The benchmark runner selects the fitness metric provider through RunConfig.physics_metric. The provider supports smooth global and local standardizers with logistic positive maps. Its global cache uses Lemma 314. For local normalization, pipeline_from_measurement_jets and local_log_weights retain the derivatives of both measurements and neighborhood weights, using an \(O(N)\) calculation per query. Supported smooth Euclidean and phase-space kernels retain the recorded eligibility and donor sources; nonsmooth coincident-neighbor distances or periodic seams require their own identified treatment.

When a population-coordinate calculation changes several measurement rows, supply every affected row as a jet to pipeline_from_measurement_jets. The test coupled_population_derivative_matches_executed_fitness_with_fixed_companions checks the resulting gradient and Hessian against finite differences of the production FitnessPipeline::evaluate, with two other rows using the moved walker as their companion. It also checks a matching fitness value and a different gradient for the immutable-source query calculation. This tests the dependency distinction directly.

The provider evaluates conditional jets and its curvature diagnostics on the host in f32 or f64; applying the resulting diffusion factors uses the selected noise backend. The separate smooth_curvature_batch entry point runs its tensor contractions on the selected CPU, WebGPU, or CUDA backend, retaining host eigendecompositions. CPU supports f32 and f64; WebGPU uses f32, and CUDA precision requires the corresponding device capability. A backend build does not establish numerical agreement or a speedup on GPU hardware; those require an actual device run.

At the thermostat stage, the provider records fitness_hessian, fitness_metric, metric_inverse_sqrt, and, when requested, fitness_ricci and fitness_scalar_curvature. Third and, where needed, fourth derivatives are evaluated only when curvature recording is enabled; unrecorded steps retain the second-order metric calculation. Coverage masks distinguish unavailable curvature from a computed zero. In particular, a clipping threshold can make classical curvature unavailable while the positive clipped diffusion metric remains usable. Conditional target-slot and revival-field records identify which reconstructed field was evaluated.

Additional Spacetime Geometry

The conditional fitness calculation supplies a spatial metric. To use that family of metrics in spacetime transport, we must also specify how time enters the line element. The following construction states that extra choice explicitly. Its mixed connection coefficients then follow by substitution; they are not inferred from a spatial curvature map.

Assumption 21 (Spatial and spacetime geometry used in this chapter)

Let \(g_t\) be an identified positive definite spatial metric on a coordinate region \(\mathcal Z\), with the regularity needed by each result below. For spacetime statements choose

\[ M=(t_0,t_1)\times\mathcal Z,\qquad G=-c^2dt^2+g_t,\qquad c>0. \]

This is the specified Lorentzian model, of dimension \(d+1\) when \(\dim\mathcal Z=d\). The metric and its reconstruction are those in Remark 251. For a clipped recorded metric, classical curvature requires the additional smoothness specified in Remark 248.

The continuum cone and normalized-volume hypotheses of Lemma 296 and Corollary 117 can identify an already recovered Lorentzian metric. Applying that identification to the recorded CST requires the corresponding continuum reconstruction and sampling conditions; a finite order alone supplies no smooth metric.

The unit timelike parameter \(\tau\) below satisfies \(G(u,u)=-1\) for \(u=dz/d\tau\). With dimensional \(G=-c^2dt^2+g_t\), \(\tau\) is proper length, equal to \(c\) times physical proper time. Equivalently one may use geometric units \(c=1\).

Lemma 315 (Connection of the time-dependent slab metric)

For \(G=-c^2dt^2+g_t\) with constant \(c\),

\[ \Gamma^k_{ij}[G]=\Gamma^k_{ij}[g_t],\qquad \Gamma^t_{ij}[G]=\frac1{2c^2}\partial_tg_{ij},\qquad \Gamma^i_{tj}[G]=\frac12g^{ik}\partial_tg_{kj}, \]

and \(\Gamma^a_{tt}=\Gamma^t_{ti}=0\). Thus the purely spatial coefficients agree, but ambient parallel transport along a spatial curve need not preserve its tangent space.

Proof

Insert \(G_{tt}=-c^2\), \(G_{ti}=0\), and \(G_{ij}=g_{ij}\) into the connection formula. For a spatial curve and an initially spatial vector,

\[ \frac{dV^t}{ds} =-\frac1{2c^2}\partial_tg_{ij}\,V^i\dot\gamma^j. \]

This component can be nonzero. When \(\partial_tg=0\), the spatial subbundle is preserved and its connection is the intrinsic one.

Parallel Transport and Holonomy

Definition 758 (Parallel transport)

For a piecewise smooth curve \(\gamma\) and a specified connection, parallel transport solves

\[ \frac{dV^a}{ds} +\Gamma^a_{bc}(\gamma(s))V^b\dot\gamma^c=0. \]

The linear map from the initial tangent space to the final one is denoted \(P_\gamma\). Intrinsic slice transport uses \(\Gamma[g_t]\) throughout. Spacetime transport uses \(\Gamma[G]\), including its mixed components.

Definition 759 (Holonomy)

For a closed loop \(\gamma\) based at \(p\), \(\operatorname{Hol}_\gamma=P_\gamma:T_pM\to T_pM\).

Metric compatibility makes spacetime holonomy an element of \(O(1,d)\). On an oriented time-oriented spacetime, transport continuously connected to the identity lies in \(SO^+(1,d)\). Intrinsic holonomy of a \(d\)-dimensional Riemannian slice lies in \(O(d)\).

The restricted holonomy uses loops homotopic to the constant loop. Vanishing curvature is equivalent to trivial restricted holonomy on each connected component. Global flat connections may still have holonomy around noncontractible loops.

Theorem 428 (Ambrose–Singer theorem)

For a connected smooth manifold with the specified metric connection, the Lie algebra of restricted holonomy at \(p\) is

\[ \mathfrak{hol}_p =\operatorname{span} \{P_\gamma^{-1}R_q(X,Y)P_\gamma: \gamma:p\to q,\ X,Y\in T_qM\}. \]

The transported curvature endomorphisms in this formula refer to that same connection.

Proof

This is the classical holonomy theorem, stated with its bundle argument in Theorem 303. Curvature is the vertical component of commutators of horizontal lifts. Infinitesimal horizontal loops therefore generate the displayed transported curvature directions. Conversely, their span is preserved by horizontal transport and contains the infinitesimal vertical displacement of such loops. These two inclusions identify the holonomy algebra. The full bundle theorem is given in [Ambrose and Singer, 1953, Kobayashi and Nomizu, 1963].

Lemma 316 (Holonomy of shape-controlled small loops)

On a fixed normal neighborhood, let the metric be \(C^3\). Consider an oriented coordinate rectangle based at \(p\), with orthonormal initial directions \(X,Y\), side lengths \(r,s\), and \(\ell=\max(r,s)\), where \(\min(r,s)\geq a\ell\) for a fixed \(a>0\). For a Lorentzian plane use a nondegenerate pseudo-orthonormal pair and norms in a fixed auxiliary positive definite frame norm.

With \(A=rs\asymp\ell^2\), \(K=\sup|R|\), and \(K_1=\sup|\nabla R|\), choose the loop orientation so that

\[ \operatorname{Hol}_\gamma V =V+A R_p(X,Y)V+E,\qquad |E|\leq C(K_1A^{3/2}+K^2A^2)|V|. \]

The constants include the neighborhood and frame bounds.

Proof

Use the radial frame and transport integral equation in Lemma 260. In that frame the connection one-form is \(O(K\ell)\). Its curvature differs from transported curvature at \(p\) by \(O(K_1\ell+K^2\ell^2)\). The leading curvature integral is \(rsR_p(X,Y)\); its variation contributes \(O(K_1\ell^3+K^2\ell^4)\). Iterated connection integrals contribute \(O(K^2\ell^4)\). The bounded aspect ratio converts these powers to \(A^{3/2}\) and \(A^2\). The same local matrix estimates apply in the fixed auxiliary norm for the Lorentzian connection.

Curvature from Consistent Plaquette Transport

The small-loop formula measures curvature per unit area. This sets the accuracy required of a discrete rule. If the loop has diameter \(h\) and area comparable to \(h^2\), an \(O(h^3)\) transport error becomes an \(O(h)\) curvature error after division.

A well-shaped mesh supplies useful lengths and areas. The transport error still has to be checked for the face maps or edge maps that are actually used.

Definition 760 (Scutoid plaquette)

A plaquette is a specified closed quadrilateral: a bottom edge from \(z_i(t)\) to \(z_j(t)\), the forward trajectory of \(j\), a top edge from \(z_j(t+\Delta t)\) to \(z_i(t+\Delta t)\), and the reversed trajectory of \(i\). Its spanning surface and connection are specified separately.

For a nondegenerate tangent two-plane with spanning vectors \(X,Y\), the unsigned metric area factor is

\[ \sqrt{|G(X,X)G(Y,Y)-G(X,Y)^2|}. \]

The estimate \(A_\Pi\asymp\ell\,c\Delta t\) requires uniform control of this factor and of the surface shape. A null or nearly null plaquette needs separate treatment.

Definition 761 (Piecewise-flat comparison holonomy)

For a nondegenerate simplicial metric, choose orthonormal frames in adjacent simplices and their metric-preserving face-identification maps \(T_e\). The discrete holonomy around a closed oriented dual path is their ordered product. In spacetime dimension \(d+1\) and signature \((-,+,\ldots,+)\), orientation-preserving maps belong to \(SO(1,d)\).

For a Riemannian hinge, rotation is described by the deficit \(2\pi-\sum_j\theta_j\). Around a Lorentzian spacelike hinge, the normal plane has Lorentzian signature; its parameter is the oriented boost rapidity determined by the face maps. The ordered product defines comparison transport in both cases.

Definition 762 (Shape-controlled scutoid refinement)

Let \(\ell\) be a representative spatial edge length and \(h=\max(\ell,c\Delta t)\to0\), with

\[ 0<\kappa_{\min}\leq\frac{c\Delta t}{\ell}\leq\kappa_{\max}<\infty. \]

In addition, require nondegenerate tangent planes and uniformly controlled plaquette shapes as in Definition 760. Then \(A_\Pi\asymp h^2\). These geometric conditions specify the refining family to which a small-loop estimate is applied.

Lemma 317 (Error accumulation for comparison holonomy)

Suppose a plaquette has \(m\leq m_0\) comparison transports \(T_e\) and smooth transports \(P_e\), expressed in compatible frames, with

\[ \|T_e\|,\|P_e\|\leq M,\qquad \|T_e-P_e\|\leq C_{\mathrm{conn}}h^3,\qquad M\geq1. \]

Then

\[ \|T_m\cdots T_1-P_m\cdots P_1\| \leq m_0M^{m_0-1}C_{\mathrm{conn}}h^3. \]

When \(A_\Pi\asymp h^2\), this is \(O(A_\Pi^{3/2})\). The connection consistency estimate is an input for the chosen maps.

Proof

The exact telescoping identity is

\[ T_m\cdots T_1-P_m\cdots P_1 =\sum_{k=1}^m T_m\cdots T_{k+1}(T_k-P_k)P_{k-1}\cdots P_1. \]

Bound the one error factor and the at most \(m_0-1\) other factors in each summand. Point-set shape regularity and weak convergence of a curvature measure do not by themselves supply this edgewise estimate.

Theorem 429 (Curvature recovery from consistent plaquette transport)

Assume Lemma 316 and Lemma 317 for a shape-controlled family based at \(z\), with limiting oriented orthonormal or pseudo-orthonormal directions \(X,Y\). Then

\[ \lim_{A_\Pi\to0}\frac{(\mathcal H[\Pi]-I)V}{A_\Pi} =R_z(X,Y)V. \]

For fixed directions the error is bounded by

\[ C\bigl((C_{\mathrm{conn}}+K_1)A_\Pi^{1/2} +K^2A_\Pi\bigr)|V|. \]

The coordinate contraction is \(R^a{}_{bcd}V^bX^cY^d\). Using the same vector in both antisymmetric slots gives zero.

Proof

Insert smooth holonomy between \(\mathcal H[\Pi]\) and \(I\). Divide the two remainder estimates by \(A_\Pi\) and let the area tend to zero. Continuity supplies the additional error when directions converge rather than remain fixed.

For intrinsic slice transport, the result is intrinsic spatial curvature. Spacetime curvature uses the full connection of \(G\); the relation between the two also involves the slice’s second fundamental form.

INTERACTIVE EXPERIMENT · VI-38

Holonomy along a recorded-point loop

What happens when metric transport returns to its starting point? Construct a closed path from the selected recorded displacement and a neighboring recorded point. Integrate conditional-metric transport at two resolutions and measure loop departure from the identity and norm preservation.

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Discrete Connection Recovery

Watch an edge between two moving points. It changes because the points have different velocities, and because comparing vectors at different base points requires transport. These effects occur in the same measurement.

There is also a counting issue. If all transport experiments follow one velocity direction, they measure the connection contracted with that direction. Recovering every connection coefficient requires enough independent directions. The singular values of the measurement matrix make this requirement quantitative.

Definition 763 (Edge deformation in a declared chart)

For neighboring sites, represent the coordinate displacement by a vector \(\ell_{ij}(t)\) at the base point \(z_i(t)\). Alternatively, use a specified logarithmic displacement in a normal neighborhood. For a specified transport \(P_{\gamma_i}\), define

\[ \Delta\ell_{ij} =\ell_{ij}(t+\Delta t)-P_{\gamma_i}\ell_{ij}(t). \]

For coordinate displacements and smooth trajectories,

\[ \Delta\ell_{ij} =\left[v_j-v_i+\Gamma(v_i)\ell_{ij}\right]\Delta t +O(\Delta t^2), \]

with the remainder retaining the corresponding trajectory and edge derivative bounds. Thus the deformation includes the relative velocity. It also uses the prescribed transport, which must be supplied independently if the expression is used as measurement data.

Proof

Taylor expand the edge and solve the transport equation through first order:

\[ \ell_{ij}(t+\Delta t) =\ell_{ij}(t)+(v_j-v_i)\Delta t+O(\Delta t^2),\qquad P_{\gamma_i}=I-\Gamma(v_i)\Delta t+O(\Delta t^2). \]

Subtract the two expansions.

Proposition 284 (Connection recovery with an identified least-squares design)

Fix a chart and a point. For each output component \(a\), parameterize the torsion-free coefficients \(\Gamma^a_{bc}\) in an orthonormal basis \(E_\alpha\) of symmetric \(d\times d\) matrices, with coefficient vector \(\gamma^a\). Suppose \(m\) transport measurements satisfy

\[ y_r^a=\Delta t_r\,\Gamma^a_{bc}w_r^bv_r^c+\rho_r^a,\qquad X_{r\alpha}=\Delta t_r\,w_r^TE_\alpha v_r, \]

where \(|w_r|=O(h)\), \(\Delta t_r=O(h)\), \(|v_r|=O(1)\), and \(|\rho_r^a|\leq C h^3\). If

\[ \sigma_{\min}(X)\geq\kappa\sqrt m\,h^2,\qquad \kappa>0, \]

then the least-squares estimator obeys

\[ |\widehat\gamma^a-\gamma^a|\leq(C/\kappa)h. \]

One valid transport datum is \(y_r=w_r-P_rw_r\), for an independently specified transport with the stated expansion and error. Raw edge deformation must first account for the relative-velocity term in Definition 763.

For \(d>1\), measurements with a single fixed \(v_r=v\) cannot recover all symmetric coefficients: they identify at most the matrix contraction \(\Gamma(v)\) and do not satisfy the stated full-rank condition.

Proof

The data equation is \(y^a=X\gamma^a+\rho^a\). Full column rank gives

\[ \widehat\gamma^a-\gamma^a=X^\dagger\rho^a,\qquad |\widehat\gamma^a-\gamma^a| \leq\frac{|\rho^a|}{\sigma_{\min}(X)} \leq\frac{\sqrt m\,Ch^3}{\kappa\sqrt m\,h^2}. \]

For a smooth connection and uniformly controlled transport paths, \(P_r=I-\Gamma(v_r)\Delta t_r+O(h^2)\); multiplication by \(w_r=O(h)\) gives the stipulated \(O(h^3)\) measurement remainder. Any discrete transport error must be included in that remainder.

With fixed \(v\), the map from a symmetric coefficient matrix \(A\) to these measurements factors through \(Av\in\mathbb R^d\). Its rank is at most \(d\), whereas the coefficient space has dimension \(d(d+1)/2>d\). This proves the final assertion.

INTERACTIVE EXPERIMENT · VI-40

Connection of a measured conditional metric

Are the connection coefficients numerically resolved? Compute centered metric derivatives at two finite-difference scales in the executed conditional context. Form Christoffel coefficients and compare metric-compatibility and derivative-refinement residuals.

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The Raychaudhuri Equation

Take a small ball of neighboring observers moving along timelike curves. It can grow, stretch into an ellipsoid, or rotate. Expansion, shear, and vorticity measure these three changes.

If the observers fall freely, their acceleration vanishes and the curvature term completes the evolution equation. Driven trajectories have an extra acceleration term. The distinction matters for walkers subject to forces, friction, and random velocity kicks.

Definition 764 (Kinematic decomposition)

Let \(u\) be a smooth unit timelike field, \(G(u,u)=-1\). Set

\[ h_{ab}=G_{ab}+u_au_b,\qquad a_a=u^b\nabla_bu_a,\qquad B_{ab}=h_a{}^ch_b{}^d\nabla_du_c. \]

Define \(\theta=\nabla_au^a\), the symmetric trace-free part \(\sigma_{ab}\), and the antisymmetric part \(\omega_{ab}\) by

\[ B_{ab}=\frac{\theta}{d}h_{ab}+\sigma_{ab}+\omega_{ab}. \]

The complete decomposition is

\[ \nabla_bu_a=B_{ab}-a_a u_b. \]

For a geodesic congruence \(a=0\). The contractions \(\sigma^2=\sigma_{ab}\sigma^{ab}\) and \(\omega^2=\omega_{ab}\omega^{ab}\) are nonnegative, because these tensors act on the positive definite rest space orthogonal to \(u\).

Theorem 430 (Raychaudhuri equation)

For a smooth unit timelike geodesic congruence in dimension \(d+1\),

\[ D_u\theta =-\frac{\theta^2}{d}-\sigma^2+\omega^2-\operatorname{Ric}(u,u). \]

For a smooth accelerated congruence, the right-hand side additionally contains \(\nabla_a a^a\).

Proof

Commute derivatives in \(D_u(\nabla_au^a)\), using the curvature convention above and \(a=\nabla_u u\). The result is

\[ D_u\theta =\nabla_aa^a-(\nabla_au^b)(\nabla_bu^a) -\operatorname{Ric}(u,u). \]

Orthogonality of \(B\) and \(a\) to \(u\) makes the quadratic contraction equal to \(B_{ab}B^{ba}\). Trace-freeness and symmetry remove its cross terms, leaving

\[ B_{ab}B^{ba} =\frac{\theta^2}{d}+\sigma^2-\omega^2. \]

Substitute and set \(a=0\) for the geodesic case. This is the direct calculation of Theorem 304.

INTERACTIVE EXPERIMENT · VI-41

Finite-step cloud expansion balance

Which executed stages change the cloud volume? Define volume by the square root of the determinant of regularized eligible-position covariance. Compare its endpoint log-volume increment with the sum of recorded stage increments.

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Expansion from Cell Volumes

A material cell follows the same particles at all times. A Voronoi cell is redrawn as its neighboring sites move. Their boundaries need not have the same normal velocity.

To compare their expansions, divide the difference in boundary flux by cell volume. Since a small cell has much more boundary area per unit volume, a small error on each face can still produce a large expansion error. This is why the normalized flux and its derivative appear explicitly below.

Definition 765 (Regularity and cell scales)

On the fixed spacetime window, assume a sufficiently smooth metric and unit timelike congruence for the displayed differentiated identities, with uniform bounds on the geometric and flow derivatives they use. For transverse spatial cells let \(\epsilon_N\to0\) be a chosen resolution scale, with

\[ \operatorname{diam}(C_i)\leq C_1\epsilon_N,\qquad \operatorname{Vol}(C_i)\geq C_2\epsilon_N^d, \]

and bounded shape ratios in the specified normal neighborhoods. Cell injectivity and regularity concern the transverse Riemannian geometry. A Lorentzian metric does not supply a positive distance for these diameter bounds.

The scale \(\epsilon_N\) is determined by the actual reconstruction and coverage estimate. The independent, joint-entropy, LSI, and same-sample coverage bounds in Scutoid Spacetime: Moving Cells and Neighbor Changes give distinct sufficient sampling estimates. A uniform maximal cell diameter \(O(N^{-1/d})\) is an additional geometric property, not a consequence of particle count alone.

Theorem 431 (Discrete Raychaudhuri correspondence)

Let the sites follow a smooth unit timelike geodesic congruence in dimension \(d+1\). In addition to Definition 765, suppose the transverse cell volumes \(V_i>0\) satisfy

\[ \theta_i=\frac{\dot V_i}{V_i}=\theta(z_i)+r_i,\qquad |r_i|+|\dot r_i|\leq C\epsilon_N. \]

On a window with bounded expansion, shear, vorticity, and curvature,

\[ \dot\theta_i =-\frac{\theta_i^2}{d}-\sigma^2(z_i)+\omega^2(z_i) -\operatorname{Ric}(u,u)(z_i)+O(\epsilon_N). \]

Material cells satisfy the consistency condition under the spatial and material derivative bounds of Theorem 305. Reconstructed Voronoi cells additionally require their normalized boundary-flux defect and its time derivative to be \(O(\epsilon_N)\). The statement applies on intervals of differentiable positive cell volume; topology events require the corresponding jump terms.

Proof

Differentiate \(\theta_i=\theta(z_i)+r_i\). The Raychaudhuri identity gives

\[ \dot\theta_i =-\frac{(\theta_i-r_i)^2}{d}-\sigma^2(z_i)+\omega^2(z_i) -\operatorname{Ric}(u,u)(z_i)+\dot r_i. \]

Bounded expansion and the two bounds on \(r_i\) give the result.

For material cells use the transverse flux volume form \(\iota_u d\operatorname{vol}_G\), pulled back to a transported transverse section, and write \(\langle f\rangle_i=V_i^{-1}\int_{C_i}f\). This agrees with induced rest-space volume for sections orthogonal to \(u\). Since \(\mathcal L_u(\iota_u d\operatorname{vol}_G) =\theta\,\iota_u d\operatorname{vol}_G\), the material Jacobian satisfies \(\dot J=\theta J\), so

\[ \frac{\dot V_i}{V_i}=\langle\theta\rangle_i,\qquad \frac d{d\tau}\langle\theta\rangle_i =\langle D_u\theta\rangle_i +\langle\theta^2\rangle_i-\langle\theta\rangle_i^2. \]

Spatial Lipschitz bounds on \(\theta\) and \(D_u\theta\), together with cell diameter \(O(\epsilon_N)\), give the differentiated consistency estimate. In particular the variance term is \(O(\epsilon_N^2)\).

For a reconstructed boundary with normal velocity \(w\cdot n\), the additional normalized flux is

\[ b_i=\frac1{V_i}\int_{\partial C_i}(w-u)\cdot n\,dA. \]

Reynolds transport gives this term exactly, with the appropriate time-dependent volume derivative for the chosen transverse chart. The conditions \(|b_i|+|\dot b_i|=O(\epsilon_N)\) retain it and its derivative. These are the flux conditions proved and specified in Theorem 305. A face error is multiplied by the surface-to-volume ratio when estimating \(b_i\).

Corollary 136 (Relaxation under positive expansion damping)

Suppose \(\dot\theta_i=-\theta_i^2/d+F_i\) and \(\dot{\bar\theta}=-\bar\theta^2/d+\bar F\), with

\[ \theta_i+\bar\theta\geq d\lambda>0,\qquad |F_i-\bar F|\leq C\epsilon_N. \]

Then

\[ |\theta_i(t)-\bar\theta(t)| \leq e^{-\lambda t}|\theta_i(0)-\bar\theta(0)| +\frac{C\epsilon_N}{\lambda}(1-e^{-\lambda t}). \]

Proof

The difference \(\delta=\theta_i-\bar\theta\) satisfies

\[ \dot\delta =-\frac{\theta_i+\bar\theta}{d}\delta+(F_i-\bar F). \]

Use its integrating factor and the lower damping bound to estimate the initial term and the forcing integral. Curvature boundedness alone does not provide this positive damping coefficient.

INTERACTIVE EXPERIMENT · VI-42

Population volume and its stage budget

Does a change in population spread come from transport or copying? Plot the regularized covariance length scale together with direct log-volume increments and their executed-stage reconstruction. Keep eligible counts in the budget.

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Geodesic Focusing

Definition 766 (Timelike convergence condition)

The geometric timelike convergence condition is

\[ \operatorname{Ric}(u,u)\geq0 \]

for every timelike \(u\). In relativity it is related to a strong energy condition through specified field equations and their cosmological term. Here it is a condition on the chosen metric. The historical label def-sec refers to this geometric condition.

Theorem 432 (Focusing of an initially contracting congruence)

Let a unit timelike geodesic congruence be vorticity free, satisfy the timelike convergence condition, and have \(\theta(0)=\theta_0<0\) on the geodesic under consideration. If that geodesic extends to

\[ \tau_*=\frac d{|\theta_0|}, \]

then the congruence cannot remain regular with nonzero transverse Jacobian throughout \([0,\tau_*]\). Focusing occurs no later than \(\tau_*\); along a regular branch approaching the focusing time, the expansion becomes unbounded below.

Proof

Raychaudhuri and \(\sigma^2\geq0\) give \(\dot\theta\leq-\theta^2/d\). The initially negative expansion stays negative and decreases. Therefore

\[ \frac d{d\tau}\frac1\theta =-\frac{\dot\theta}{\theta^2}\geq\frac1d, \qquad \theta(\tau)\leq\frac{\theta_0}{1+\theta_0\tau/d} \quad(0\leq\tau<\tau_*). \]

If \(J\) is the positive transverse Jacobian, then \(\dot J/J=\theta\), so integration yields

\[ \frac{J(\tau)}{J(0)} \leq(1+\theta_0\tau/d)^d. \]

The right-hand side tends to zero at \(\tau_*\). Thus regular nonvanishing transverse volume cannot persist to that time. Since \(\theta\) is decreasing, a finite endpoint for a regular branch leading to focusing has unbounded negative expansion.

INTERACTIVE EXPERIMENT · VI-43

Measured cloud focusing

Does a proposed focusing balance describe observed expansion? Measure theta as the finite-step log-volume increment divided by timestep, then display its finite difference plus theta squared divided by dimension.

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Curvature and Changes of Tessellation

Draw another point inside a triangulated disk and redraw the triangles. The numbers of vertices, edges, and faces change, but the disk is still a disk. Euler’s alternating sum keeps track of that fact.

Curvature has a related accounting identity in two dimensions. On a closed surface, the total angle deficit is fixed by topology. On a surface with boundary, turning at the boundary shares that total. Inserting a site can redistribute the bookkeeping; it does not create a new handle or hole by itself.

Definition 767 (Riemannian Regge curvature)

For a nondegenerate piecewise Euclidean manifold of spatial dimension \(d\), a hinge \(h\) has dimension \(d-2\). At an interior hinge define the Riemannian rotation deficit

\[ \varepsilon_h=2\pi-\sum_{\text{incident simplices}}\theta_h. \]

The associated scalar-curvature measure assigns integrated weight

\[ \mathcal R_{\mathrm{Regge}} =2\sum_h\varepsilon_h\,\operatorname{Vol}_{d-2}(h). \]

For \(d=2\), hinges are vertices and the Gaussian-curvature weight is \(\varepsilon_h\), half the scalar-curvature weight. Boundary terms are specified separately.

A smooth-limit comparison uses a chosen metric approximation and its curvature-measure consistency theorem. Lorentzian spacetime hinges have dimension \((d+1)-2=d-1\) and use the face-map transport of Definition 761, with the appropriate rotation or boost parameter.

Theorem 433 (Euler constraints for cell refinements)

Let a finite regular cell decomposition of a fixed space have \(F_k\) cells of dimension \(k\). Then

\[ \chi=\sum_{k=0}^d(-1)^kF_k. \]

Two such decompositions of the same underlying space have the same Euler characteristic. Consequently a tessellation change on that space satisfies

\[ \sum_{k=0}^d(-1)^k\Delta F_k=0. \]

In two dimensions, \(\Delta V-\Delta E+\Delta F=0\). When site insertion or removal only changes the decomposition, this relation applies to all affected face counts together.

Proof

For the finite cellular chain complex, write \(Z_k\) and \(B_k\) for cycles and boundaries. Rank-nullity gives

\[ \dim C_k=\dim Z_k+\dim B_{k-1},\qquad \dim Z_k=\dim H_k+\dim B_k. \]

Alternating sums cancel the boundary dimensions, yielding \(\sum_k(-1)^kF_k=\sum_k(-1)^k\dim H_k\). The latter depends only on the underlying space. Subtract the identity for the two decompositions.

Proposition 285 (Curvature balance under a two-dimensional cell change)

For a finite piecewise Euclidean surface with polygonal boundary, let

\[ \varepsilon_v=2\pi-\sum_{T\ni v}\alpha_{T,v} \quad\text{at interior vertices},\qquad \beta_v=\pi-\sum_{T\ni v}\alpha_{T,v} \quad\text{at boundary vertices}. \]

Then

\[ \sum_{v\ \mathrm{interior}}\varepsilon_v +\sum_{v\ \mathrm{boundary}}\beta_v=2\pi\chi. \]

A change of tessellation on a fixed closed surface therefore has \(\Delta\sum_v\varepsilon_v=0\). For a fixed surface with boundary,

\[ \Delta\sum_{\mathrm{interior}}\varepsilon_v =-\Delta\sum_{\mathrm{boundary}}\beta_v. \]

More generally, a true change of topology contributes \(2\pi\Delta\chi\). Adding a site while retaining the same underlying domain has \(\Delta\chi=0\).

For a generic planar Voronoi modification away from an unchanged boundary, if all affected vertices remain trivalent, then

\[ \Delta V=2\Delta F,\qquad \Delta E=3\Delta F. \]

These counting identities require no curvature creation. For an ordinary Euclidean tessellation every interior deficit is zero because its incident angles fill the full angle \(2\pi\).

If at most \(m_0\) vertices change and each has at most \(q_0\) incident triangles, their local absolute deficit variation is bounded by

\[ \sum_{\mathrm{changed}\ v} |\varepsilon_v^{\mathrm{new}}-\varepsilon_v^{\mathrm{old}}| \leq2m_0(2\pi+q_0\pi), \]

where missing vertices have zero weight. This is an explicit sufficient condition for a local \(O(1)\) bound.

Proof

Triangulate each polygon without changing the metric. Let \(V_i,V_b\) be the numbers of interior and boundary vertices, and let \(E_i,E_b,F\) count interior edges, boundary edges, and triangles. The sum of all triangle angles is \(\pi F\), so the left side of the first identity is

\[ 2\pi V_i+\pi V_b-\pi F. \]

For a surface with closed polygonal boundary components, \(E_b=V_b\) and \(3F=2E_i+E_b\). Substitution gives

\[ 2\pi V_i+\pi V_b-\pi F =2\pi(V_i+V_b-E_i-E_b+F)=2\pi\chi. \]

Subtract the identities before and after the change. On a closed surface the boundary sum is absent.

For the trivalent planar modification, unchanged boundary contributions cancel in degree counting, leaving \(3\Delta V=2\Delta E\). Combine this with \(\Delta V-\Delta E+\Delta F=0\) to obtain the displayed face-count changes.

Each triangle angle lies in \([0,\pi]\), so \(|\varepsilon_v|\leq2\pi+q_0\pi\). Applying this bound before and after the modification proves the local variation estimate. In a Euclidean realization the exact full-angle sum proves zero interior deficit, including at irregular vertices.

Remark 286 (Higher-dimensional curvature and scaling)

In spatial dimension \(d=3\), Riemannian hinges are edges; in \(d=4\) they are two-dimensional faces. Their deficits and hinge volumes both enter the Regge curvature measure. Face counts alone do not determine these weights.

For a direct obstruction to a universal curvature-per-cell constant, scale all metric lengths by \(s>0\) in a piecewise Euclidean \(d\)-manifold. Its combinatorics and dihedral angles stay fixed, while each hinge volume scales by \(s^{d-2}\). Hence

\[ \mathcal R_{\mathrm{Regge}}(s) =s^{d-2}\mathcal R_{\mathrm{Regge}}(1). \]

For \(d>2\) and nonzero initial curvature measure, identical cell counts can therefore have different total scalar curvature. A statistical relation for a specified random metric would require its metric law and normalization in addition to an incidence model. The smooth convergence results of [Cheeger et al., 1984] apply with their metric and approximation hypotheses.

INTERACTIVE EXPERIMENT · VI-44

Topology of the recorded Fractal Set

Do the recorded event and edge counts satisfy their graph identity? Build the Fractal Set from executed events and edges, count connected components and graph cycle rank, and compare V minus E with components minus cycle rank. Report resolved edges and interaction triangles.

Load experiment Open full view ↗

Rust engine experiment · Seed 7 · Poster after 11 experiment steps. Interactive view starts from the same seed.

Geometric and Dynamical Interpretations

The equations now tell us exactly what a curvature measurement means. The configured diffusion rule identifies a metric field. Differentiating that field computes its curvature, and consistent discrete transport provides an independent recovery. A specified smooth flow additionally allows us to compare geometric expansion with measured volume changes.

Optimization asks an additional question: does the actual stochastic process approach its target law or improve its objective? The force, noise, and cloning analyses answer that question. A geometric caustic is a loss of regularity of a family of geodesics, so it cannot be substituted for a mixing or optimization theorem.

Remark 287 (The geometric correspondence and its hypotheses)

Geometric quantity	Construction used here	Required identification
Fitness derivatives	Differentiated conditional fitness pipeline	Recorded conditioning data, active smooth stratum, and stated derivative convention
Spatial metric	Inverse adaptive covariance shape	Correct diffusion branch, prefactor, and spatial reconstruction
Spacetime metric	\(G=-c^2dt^2+g_t\)	The specified Lorentzian model and causal consistency
Parallel transport	Levi-Civita or stated face maps	Same connection and compatible frames
Curvature	Metric derivative formula or consistent small-loop holonomy	A \(C^2\) metric; transport recovery also needs nondegenerate loops and \(o(A)\) transport error
Hessian-metric curvature	Quadratic expression in third derivatives	A smooth positive Hessian metric
Expansion	\(\dot V/V\)	Transverse volume and differentiated consistency
Geodesic focusing	Raychaudhuri inequality	Geodesic flow, convergence condition, and zero vorticity

The next dynamical calculation uses the full transition law to derive metric increments and mechanical balances, as in Field Equations and Pressure Dynamics. An Einstein-type equation would require identifying a controlled limit of those independently derived laws, including its stress tensor and any unresolved terms. Computing curvature alone does not determine that evolution or its coefficients.

Remark 288 (Focusing and optimization use different evolution estimates)

The focusing theorem controls a smooth timelike geodesic congruence. Its conclusion is a caustic or vanishing transverse Jacobian. It does not identify an objective minimizer or a stationary probability law.

The actual kinetic walkers have forces, friction, velocity noise, and cloning events. A smooth accelerated comparison has the additional \(\nabla_a a^a\) term in Raychaudhuri; stochastic trajectories require their own stochastic analysis. Approximation by a geodesic congruence requires a stated approximation estimate.

The established analytical routes are Convergence, Survival, and Parameter Dependence for the specified finite-particle transition kernel, Exchangeability, Empirical Laws, and Functional Inequalities for joint-law fluctuations, and Logarithmic Sobolev inequalities and entropy convergence together with The Geometric Gas: Regularity, Stability, and Entropy for the proved joint-law LSI criteria and complete entropy evolution. Their hypotheses determine the dynamical convergence statement.

Table of Symbols

Symbol	Meaning
\(d\)	Spatial dimension; spacetime dimension is \(d+1\)
\(g_t\), \(G\)	Intrinsic spatial and Lorentzian spacetime metrics
\(\Gamma[q]\)	Levi-Civita connection of the specified metric \(q\)
\(R^a{}_{bcd}\)	Curvature convention in Definition 754
\(\operatorname{Ric}\), \(R\)	Ricci contraction and scalar trace
\(\theta\), \(\sigma\), \(\omega\)	Expansion, shear, and vorticity
\(a=\nabla_u u\)	Acceleration of the chosen smooth flow
\(\mathcal H[\Pi]\)	Ordered discrete transport around the plaquette
\(A_\Pi\), \(h\)	Oriented-plane area magnitude and mesh resolution
\(\epsilon_N\)	Cell scale established by the chosen geometric estimate
\(\varepsilon_h\)	Riemannian hinge deficit
\(\chi\)	Euler characteristic of the underlying space

Results Carried Forward

The conditioned algorithm data and differentiated fitness pipeline specify the metric-producing calculation. Smooth Hessian metrics admit direct third-derivative curvature contractions, including the packed three-dimensional calculation and its explicit curvature bound. Mixed-sign clipping uses the full metric derivative rule under a spectral gap; the required smoothness remains part of each result.

The small-loop and product-error proofs give independent curvature recovery for a consistent discrete connection. Least-squares recovery has the stated design-rank and measurement-error requirements.

Raychaudhuri’s identity and the material-volume calculation transfer to reconstructed cells when their normalized flux and differentiated errors are controlled. Positive expansion damping yields the quantified relaxation estimate. Timelike convergence with zero vorticity yields geodesic focusing.

Euler and Gauss–Bonnet constrain changes of a tessellation on its underlying space. In two dimensions, total curvature on a fixed closed surface is preserved. These facts specify the geometric content used by the following field-equation chapter.

References

The local transport and volume arguments also appear in The Geometric Gas: Regularity, Stability, and Entropy. The classical sources used here are the holonomy theorem [Ambrose and Singer, 1953, Kobayashi and Nomizu, 1963], Raychaudhuri’s identity and focusing theory [Hawking and Ellis, 1973, Raychaudhuri, 1955, Wald, 1984], and piecewise-flat curvature [Cheeger et al., 1984, Regge, 1961].

[AS53] (1,2)

W. Ambrose and I. M. Singer. A theorem on holonomy. Transactions of the American Mathematical Society, 75(3):428–443, 1953. doi:10.2307/1990687.

[CMS84] (1,2)

Jeff Cheeger, Werner Muller, and Robert Schrader. On the curvature of piecewise flat spaces. Communications in Mathematical Physics, 92(3):405–454, 1984. doi:10.1007/BF01218616.

[HE73]

Stephen W. Hawking and George F. R. Ellis. The Large Scale Structure of Space-Time. Cambridge University Press, 1973. ISBN 978-0-521-09906-6.

[KN63] (1,2)

Shoshichi Kobayashi and Katsumi Nomizu. Foundations of Differential Geometry, Volume 1. Volume 1. Interscience Publishers, 1963.

[Ray55]

Amal Kumar Raychaudhuri. Relativistic cosmology. I. Physical Review, 98(4):1123–1126, 1955. doi:10.1103/PhysRev.98.1123.

[Reg61]

Tullio Regge. General relativity without coordinates. Il Nuovo Cimento, 19(3):558–571, 1961. doi:10.1007/BF02733251.

[Wal84]

Robert M. Wald. General Relativity. University of Chicago Press, Chicago, 1984.