Scutoid Spacetime: Moving Cells and Neighbor Changes#
Prerequisites: Emergent Geometry from Adaptive Diffusion, Fractal Set as Causal Set, and the metric and sampling conditions of Continuum Limits: Hypotheses and Consistency.
TLDR#
At each recorded time, give every point of space to its nearest walker. Now stack these Voronoi partitions in time. A cell sweeps out a region of spacetime. When neighbors change, the labeled boundaries have to reorganize. Scutoids provide a useful language for describing that reorganization.
The construction has three inputs: a distance on each spatial slice, a rule for filling the intervals between recorded frames, and the recorded cloning and motion events. Cloning can change neighbors, but continuous motion can do so too. A cell can also keep the same neighbors while its size and shape change substantially. We keep those possibilities visible in the definitions.
1. From recorded sites to moving cells#
Write \(N\) for the population and \(n\le N\) for the sites included in a particular alive-status stratum. The geometric reconstruction uses those sites, their labels, and a specified spatial metric \(g_t\). It preserves the algorithm’s recorded states. Interpolation between frames is reconstruction data, rather than an additional kinetic or cloning update.
The term scutoid comes from the description of neighbor exchange in epithelial packing by Gómez-Gálvez and colleagues. Here it describes cells with changes in their labeled boundary structure along the time direction. Curved Voronoi boundaries generally give curved cells; a polytope requires an additional piecewise-linear realization. In arbitrary dimension, a transition can occupy a higher-dimensional stratum rather than a single vertex.
The analytic inputs already established in this volume remain available:
Theorem 260 controls sampled and expected fitness on its specified configuration strata. A Hessian metric built from a \(C^3\) potential is \(C^1\). The higher derivatives required for classical curvature can be supplied by Theorem 270 for the same chosen field under that theorem’s hypotheses.
Lemma 191, Lemma 188, and Lemma 189 provide density bounds for their specified kernels and domains, retaining survival denominators and source masses.
Theorem 245 gives concentration of bounded empirical observables from total relative entropy. Corollary 94 gives full-gradient concentration for the actual joint laws meeting its four sufficient criteria. Discrete statuses retain the entropy term in Proposition 176.
Lemma 262 and Lemma 261 describe moving interfaces and their volumes. Their differentiated consistency conditions, rather than a cell count alone, enter Theorem 305.
2. Spatial metrics and Voronoi partitions#
2.1. Regularity inherited from the analytic fitness estimates#
Lemma 277 (Metric bounds from the specified fitness field)
On a fixed coordinate domain let \(g=D_x^2V+\epsilon_\Sigma I\), where \(V\) is the identified spatial fitness field. Suppose
Then \(g\) is \(C^1\) when \(V\) is \(C^3\), its connection is continuous, and \(|\Gamma|\le C_dK_3/\lambda\). If \(V\in C^4\) with \(|D_x^4V|\le K_4\), its curvature components satisfy
Here derivative bounds are in fixed coordinate tensor norms and \(C_d\) absorbs dimension-dependent norm comparisons. For example, \(\|D_x^2V\|_{\rm op}\le\eta\epsilon_\Sigma\) with \(\eta<1\) gives \(\lambda=(1-\eta)\epsilon_\Sigma\) and \(\Lambda=(1+\eta)\epsilon_\Sigma\).
Proof
The last inequalities follow by adding \(\epsilon_\Sigma I\) to the Hessian bounds. In general the positive spectral margin in (SC.1) is the required hypothesis; a positive shift alone does not control a more negative Hessian. The connection formula in Definition 626 contains one factor of \(g^{-1}\) and one derivative of \(g\), giving its stated bound. Differentiating \(g^{-1}\) gives \(Dg^{-1}=-g^{-1}(Dg)g^{-1}\). Thus \(D\Gamma\) is bounded by a constant times \(K_4/\lambda+K_3^2/\lambda^2\). The quadratic \(\Gamma\Gamma\) terms have the same bound. This proves (SC.2). For \(g\)-orthonormal vectors, a sectional curvature bound follows, for example, by multiplying its right-hand side by \(C_d\Lambda/\lambda^2\).
These bounds reuse the actual derivative estimates, with their dependence on localization scales, regularizers, and the selected companion law. A nodal fitness statistic becomes a common spatial field only after that field has been defined. Switching sampled companion assignments or alive strata can produce jumps between otherwise smooth fields.
For every curve in the coordinate domain, (SC.1) gives \(\sqrt\lambda L_E\le L_g\le\sqrt\Lambda L_E\). Infimizing over the same admissible curves gives the same bounds for the intrinsic distances. When the domain is Euclidean convex, its Euclidean intrinsic distance is \(|x-y|\). These estimates do not impose a sign on sectional curvature.
2.2. Cells and their regular interfaces#
Definition 674 (Voronoi partition at a fixed time)
Let \((M,d_{g_t})\) be a specified length space from the spatial metric, and let \(z_1(t),\ldots,z_n(t)\) be distinct included sites. Define
These are closed nearest-site cells whose union is \(M\). Ties belong to more than one closed cell; a fixed tie rule gives disjoint measurable ownership. For computations on a specified observation region \(K\), use the clipped cells \(K\cap\operatorname{Vor}_i(t)\) and retain the clipping boundary. Clipping is an observation convention, not a change to the swarm state space.
Lemma 278 (Cell coverage and radial connectivity)
The cells in (SC.3) cover \(M\) and are closed. If minimizing geodesics from \(z_i\) to points of its cell exist, every point on each such geodesic remains in the cell. In particular the cell is path connected. If these geodesics are unique and depend continuously on their endpoints throughout the cell, radial contraction makes the cell contractible. In Euclidean space every cell is an intersection of affine half-spaces and is convex.
Proof
The minimum of finitely many distances is attained at some site, proving coverage. Distance functions are continuous, so their non-strict comparisons define closed sets. Let \(x\in\operatorname{Vor}_i\) and let \(y\) lie on a minimizing geodesic from \(z_i\) to \(x\). For every \(j\),
where the last step is the triangle inequality. This proves the radial statement. The stated uniqueness and continuous-dependence hypothesis makes the radial interpolation a continuous deformation to \(z_i\). In Euclidean coordinates, squaring the distance inequality cancels \(|x|^2\) and gives \(2(z_j-z_i)\cdot x\le|z_j|^2-|z_i|^2\), an affine half-space.
On a complete connected Riemannian manifold, minimizing geodesics exist. The radial argument therefore gives connectivity without a curvature sign. Geodesic convexity of all cells is a stronger property and is not assumed for a general variable-curvature metric. Clipping by an arbitrary region can also destroy connectivity.
Definition 675 (Facet neighbors and interfaces)
On a regular cell family define
Regular portions of an interface avoid the sites’ cut loci and have nonvanishing spatial gradient of \(\psi_{ij}(x,t)=\frac12d_{g_t}^2(x,z_i(t))-\frac12d_{g_t}^2(x,z_j(t))\). There the implicit function theorem gives a hypersurface. A lower-dimensional contact does not make two cells facet neighbors. If clipped cells are used, apply (SC.4) to their retained interfaces and record the boundary faces separately.
2.3. The nerve and the Euclidean triangulation#
Definition 676 (Delaunay nerve and a triangulation realization)
The geodesic Delaunay nerve is the abstract simplicial complex
Every subset of an admitted simplex is admitted. In degenerate configurations its edges can include contacts that are not facet neighbors in (SC.4). A triangulation realization additionally requires that its simplices embed with disjoint interiors and cover the region being triangulated. A contractible-intersection condition in a suitable good cover provides a nerve homotopy equivalence; it does not by itself provide this geometric realization.
Lemma 279 (Euclidean Delaunay realization by lifting)
Let finitely many sites in \(\mathbb R^d\) affinely span \(\mathbb R^d\), with no \(d+2\) cospherical sites and with the usual simplicial general-position conditions. Their Euclidean Delaunay simplices triangulate their convex hull. Their full-dimensional simplices are exactly the empty-circumsphere simplices.
Proof
For \(n=d+1\) the only full simplex is the required triangulation. Otherwise lift each site \(z\) to \((z,|z|^2)\). A nonvertical lower supporting hyperplane has equation \(u=2c\cdot x+b\). Every lifted site is on or above it precisely when \(|z-c|^2\ge b+|c|^2\). Its contact sites therefore lie on a sphere with no site in its interior. General position makes its full lower facets simplicial. The lower boundary of the convex hull of the lifted points is the graph of a continuous piecewise-affine function over the original convex hull. Its projected facets cover that hull and meet only in common faces. Thus they form a triangulation with exactly the claimed empty spheres.
For a nonconstant Riemannian metric, density and avoidance of cospherical degeneracy alone do not imply this realization; see the explicit obstructions of Boissonnat, Dyer, Ghosh, and Martynchuk. The moving-cell constructions below can use the Voronoi partition and its nerve without asserting a triangulation. Euclidean algorithms are applied only in their Euclidean or fixed flat-metric regime.
2.4. What density and entropy prove about sampling#
Lemma 280 (Counts at fixed and sampled centers)
Fix the metric and the tested balls. Let \(Z_1,\ldots,Z_n\) have marginal densities bounded by \(q^*\) with respect to \(dV_g\) on the balls considered. For a deterministic center \(x\),
For a ball centered at \(Z_i\), the corresponding bound with \(n-1\) requires bounds on the conditional densities of \(Z_j\) given \(Z_i\); then add the center itself. If these conditional bounds hold and ball volumes are at most \(v^*r^d\), the probability that any two sites are within \(r\) is at most \(\binom n2q^*v^*r^d\).
Proof
Write the count as a sum of indicators and integrate each marginal density. For a sampled center condition on \(Z_i\) before integrating. A union bound over unordered pairs gives the last result. This is the spatial counterpart of Lemma 211, with the appropriate spatial volume in place of its phase-space volume. Identical correlated sites can have a bounded common marginal density and still all lie at the sampled center; this explains the conditional-density requirement.
Lemma 281 (Empty-region and fill-distance bounds)
Fix a deterministic metric \(g\), observation region \(K\), and probability reference \(\rho=q\,dV_g\). Suppose \(K\) has an \(r\)-net of \(M_r\) centers in \(K\), and every radius-\(r\) ball at those centers has \(\rho\) mass at least \(m_r>0\). Let \(h_K=\sup_{x\in K}\min_i d_g(x,Z_i)\). For independent sites of law \(\rho\),
For an arbitrary joint law \(\pi_n\) with total relative entropy \(H_n=D_{\rm KL}(\pi_n\Vert\rho^{\otimes n})\), if \(nm_r>\log M_r\), then
An alternative uses the actual continuous joint law’s full-gradient LSI with constant \(C_*\), in coordinates satisfying \(g\preceq\Lambda I\). Suppose its site marginals give every radius-\(r/2\) ball mass at least \(m_{r/2}\). Then
The LSI is for this law or the specified continuous status stratum. Mixed status laws require their additional discrete control.
Proof
If every net-center ball contains a site, the triangle inequality places a site within \(2r\) of every point of \(K\). In the independent law, a ball of mass at least \(m_r\) is empty with probability at most \((1-m_r)^n\le e^{-nm_r}\). A union bound gives (SC.7).
For any event \(A\) with reference probability \(p\in(0,1)\) and actual probability \(a\), the relative entropy chain rule for its indicator gives
Here binary entropy is at most \(\log2\), and the omitted \(-(1-a)\log(1-p)\) is nonnegative. Apply this to \(A=\{h_K>2r\}\) and \(p\le M_re^{-nm_r}\) to obtain (SC.8). Reference-null events also have actual probability zero when \(H_n<\infty\). This is the same relative-entropy comparison underlying Theorem 245.
For (SC.9), at each net center choose \(\psi(x)=\max(0,1-d_g(x,x_0)/r)\) and set \(F=n^{-1}\sum_i\psi(Z_i)\). Its mean is at least \(m_{r/2}/2\), and its coordinate Lipschitz constant is at most \(\sqrt\Lambda/(r\sqrt n)\), by the metric length bound. The full-gradient LSI and its exponential-moment proof in Lemma 304 give
An empty ball forces \(F=0\). Sum this bound over the net centers. Distance functions and the cutoff are Lipschitz; the LSI extends to them by its usual Sobolev approximation.
For example, suppose \(q\ge q_*>0\) on the tested balls, \(\operatorname{Vol}_g B_g(x,r)\ge v_*r^d\), and \(M_r\le C_Kr^{-d}\). These are local hypotheses on a fixed observation region. The independent estimate gives fill distance of order \((\log n/n)^{1/d}\) with a sufficiently large constant. Estimate (SC.8) also tends to zero whenever \((H_n+1)/(nr^d)\to0\) and \(\log M_r=o(nr^d)\). The LSI estimate (SC.9) gives convergence, for instance, for \(r=n^{-b}\) with \(0<b<1/(2d+2)\) and uniform \(C_*,\Lambda,q_*,v_*\). These are quantitative geometric sampling consequences of the available analytic inequalities. Conditional versions require the metric, net, and reference law to be fixed by the conditioning data and the joint-law hypotheses to hold after that conditioning.
Lemma 282 (Coverage for a metric computed from the same sample)
Let \(g_n\) be an adaptive metric depending on the sites. Suppose every realized metric satisfies \(g_n\preceq\Lambda I\) on the Euclidean \(2r\) neighborhood of an observation region \(K\), contained in its coordinate domain. If the Euclidean fill distance on \(K\) is at most \(2r\), its \(g_n\) fill distance is at most \(2\sqrt\Lambda r\). Consequently any of (SC.7)–(SC.9), applied to fixed Euclidean balls and the actual coordinate law, supplies the corresponding coverage probability for this adaptive metric.
Proof
For each \(x\in K\), select a site at Euclidean distance at most \(2r\). The straight segment to that site lies in the tested neighborhood and has \(g_n\) length at most \(\sqrt\Lambda\) times its Euclidean length. The intrinsic \(g_n\) distance is at most this admissible length. Take the supremum over \(x\). This is a pathwise implication using the same realized metric and sites, so it transfers the probability bound without conditioning on the random metric or asserting independent sites after that conditioning.
Density lower bounds are obtained on their stated sets. For example, Lemma 189 keeps the mass of a bounded source set and its distance to the tested region. A positive continuous QSD density from Lemma 191 has a local positive minimum; controlling it uniformly along \(n\) or a bandwidth sequence requires uniform kernel and survival bounds. A probability density cannot be bounded below by a positive constant on a space of infinite volume.
Fill distance controls how closely the sites cover \(K\). Separation, simplex shape, and protection from Delaunay degeneracy are additional quantities. For a law with a positive continuous joint density, arbitrarily close sites have positive probability. Thus such density estimates do not give a deterministic separation or a uniformly bounded aspect ratio. Nor do finite-dimensional propagation of chaos or the estimates above identify the local rescaled point process as Poisson. The latter identification requires its own limit theorem.
Cells and their dual graph
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3. Slabs, lengths, and moving-cell realizations#
There are two clocks to keep straight. Recorded time orders the frames. Proper time measures a timelike path after a Lorentzian metric has been specified. A positive spatial metric tells us how far apart sites are on one frame; it does not identify the two clocks. This matters most when a cloning update moves a slot across a large spatial distance in one recorded step.
Definition 677 (Frozen slab metrics)
On a slab \([t_k,t_{k+1}]\), choose a spatial reconstruction \(g_{h,k}\) from a specified recorded state. A state after cloning and before kinetic evolution can be used when recorded; otherwise state the chosen available-frame rule. It is a representative state, not necessarily a temporal midpoint. Define
The first is a supplied Lorentzian slab metric and the second a positive product metric. Both use the same spatial tensor. Choosing \(c\) to be an algorithmic speed parameter identifies its units; causal admissibility of the actual recorded displacements must still be checked. Convergence to \(G=-c^2dt^2+g_t\) requires consistency of \(g_{h,k}\) with that specified \(g_t\).
Definition 678 (Spatial travel cost, positive length, and causal reachability)
For an absolutely continuous path \(\gamma:[t_a,t_b]\to M\), set
This is an endpoint spatial travel cost with assigned start and end times. The positive spacetime length is instead \(\int_{t_a}^{t_b}\sqrt{c^2+|\dot\gamma|_{g_h}^2}\,dt\). Define \(e_a\prec_{\rm sc}e_b\) by \(t_a<t_b\) and the existence of such a path satisfying \(|\dot\gamma(t)|_{g_h}\le c\) almost everywhere. For a timelike path its proper-time duration is
A jump is not an absolutely continuous causal path. A causal interpolation between its endpoints is a separate geometric choice.
On a complete static spatial metric, the causal criterion reduces to \(d_g(x_a,x_b)\le c(t_b-t_a)\): necessity follows by integration, and sufficiency by constant-speed traversal of a minimizing geodesic. For a varying metric, (SC.11) alone cannot replace the pointwise speed bound. For example, in one dimension with \(c=1\) and \(t\in[0,1]\), take \(g_t=dx^2\) for the first half and \(g_t=4dx^2\) for the second. Causal motion covers at most \(1/2+1/4=3/4\) in coordinate distance. Yet a displacement \(9/10\) has spatial travel cost \(9/10\le1\) by moving entirely during the first half at speed \(9/5\). Thus the integrated cost test would accept an endpoint pair with no causal path.
Proposition 230 (Slab consistency and the scope of causal convergence)
On the common domain of admissible paths suppose \(g_t\succeq\lambda I\) and \(\sup_{t,x}\|g_h(t,x)-g_t(x)\|_{\rm op}\le\eta_h\to0\). Put \(\delta_h=\eta_h/\lambda<1\). Then every such path satisfies
The same comparison holds for the infima and for the positive spacetime lengths. For the minimal required speed \(R_g(e_a,e_b)=\inf_\gamma\operatorname*{ess\,sup}_t |\dot\gamma|_{g_t}\), the analogous comparison also holds. Hence endpoint pairs with \(R_g<c\) are eventually causally reachable in the slabs, and pairs with \(R_g>c\) are eventually unreachable. At \(R_g=c\), boundary membership requires additional attainment and boundary control.
For a fixed path uniformly timelike for both metrics with \(|\dot\gamma|_g^2,|\dot\gamma|_{g_h}^2\le c^2(1-\zeta)\), \(\zeta>0\), its two proper-time integrals differ by at most \(\eta_h(2c^2\sqrt\zeta)^{-1}\int|\dot\gamma|^2dt\).
Proof
The operator bound gives \((1-\delta_h)g\preceq g_h\preceq(1+\delta_h)g\). Take square roots of its quadratic-form bounds and integrate to prove (SC.13). For the positive product metrics the unchanged \(c^2dt^2\) term obeys the same comparison. Infimizing preserves both inequalities, without requiring minimizing curves to exist. Taking essential suprema before infimizing proves the required-speed comparison. If \(R_g<c\), choose a path with speed norm strictly below \(c\) and then take \(h\) small; if \(R_g>c\), the lower bound excludes every speed-capped slab path. Finally, rationalizing the difference of the square roots in (SC.12) bounds its absolute value by \(\eta_h|\dot\gamma|^2/(2c^2\sqrt\zeta)\) and proves the proper-time bound.
For instance, if \(g_t\) is uniformly Lipschitz in time and the sampled tensor approximates \(g_{s_k}\) with error \(e_h\), where \(s_k\in[t_k,t_{k+1}]\), then \(\eta_h\le e_h+L_t\max_k(t_{k+1}-t_k)\). A finite set of genuine metric jumps can be aligned with slab boundaries and treated one-sidedly. A sequence of unaligned cloning jumps need not satisfy a uniform time-continuity bound. Mean-field convergence alone supplies neither this tensor estimate nor causal reachability. Faithful embedding additionally retains the order-reflection and sampling hypotheses of Theorem 334.
3.1. Boundary correspondences and their geometric requirements#
Definition 679 (Boundary correspondence)
For shared neighbors \(j\in\mathcal N_i(t_a)\cap\mathcal N_i(t_b)\), let \(\mu_0,\mu_1\) be the finite positive interface area measures at the two endpoints, with masses \(A_0,A_1\). A normalized transport correspondence is a coupling of \(\mu_0/A_0\) and \(\mu_1/A_1\). Such a coupling always exists: the product of these two probabilities is one.
A geometric boundary correspondence additionally specifies a continuous map \(\phi_j:\Gamma_{ij}(t_a)\to\Gamma_{ij}(t_b)\), or a homeomorphism when needed, compatible with neighboring face maps on their common boundaries. It preserves unnormalized area only if \((\phi_j)_*\mu_0=\mu_1\), which requires \(A_0=A_1\). A measurable coupling alone is not an embedded lateral face.
Lemma 283 (Area normalization and an explicit interval correspondence)
A map preserving finite measures must preserve their total masses. For two line segments of lengths \(A_0,A_1>0\), the affine arclength map \(s\mapsto(A_1/A_0)s\) preserves their normalized arclength probabilities. It preserves their unnormalized arclength measures exactly when \(A_0=A_1\).
Proof
Evaluate the pushforward identity on the whole target space to obtain the necessary equality of masses. On segments, substitution \(u=(A_1/A_0)s\) gives \(\int_0^{A_0}f((A_1/A_0)s)ds/A_0=\int_0^{A_1}f(u)du/A_1\). Without the normalizing denominators, the pushforward density is \(A_0/A_1\).
Optimal transport between interface measures, when used, needs hypotheses appropriate to these lower-dimensional measures. Absolute continuity with respect to interface area is not absolute continuity in the ambient volume, and does not by itself imply a unique smooth transport map. Even a chosen smooth map requires nonintersection and compatibility of all its rulings to bound a cell.
Definition 680 (Swept Voronoi cells and scutoid transitions)
Choose a time interpolation of the sites and metric, with an explicit policy at birth, death, and cloning jumps. The swept region of label \(i\) is
If clipping is used, intersect with \([t_a,t_b]\times K\). Assume a regular stratified cell realization when speaking of faces, edges, and vertices. A labeled prismatic interval has a face-preserving product trivialization over time that preserves every neighbor label. A scutoid transition is a change in that labeled cell structure. A bounded piecewise-linear realization can be called a scutoid polytope when it is indeed a polytope; a general swept cell can be curved, nonconvex, or fail to be a topological ball.
The cell label follows the recorded walker slot. The parent identifier at a clone event supplies a different genealogical relation. An instantaneous replacement can be represented by two one-sided swept regions and horizontal caps at the jump. Interpolating the replaced slot continuously through space is an additional visualization rule.
For a globally compatible isotopy \(\Phi_t\) of the spatial region with \(\Phi_t(\operatorname{Vor}_i(t_a))=\operatorname{Vor}_i(t)\) for all \(i\), the map \((t,x)\mapsto(t,\Phi_t(x))\) constructs the swept partition and its lateral faces. It is a homeomorphism with inverse \((t,y)\mapsto(t,\Phi_t^{-1}(y))\), so no overlap is introduced into cell interiors. Geodesic rulings in the positive slab metric can instead be used when they define such a compatible embedding. For a fixed product slab, its geodesics have affine time coordinate; existence, uniqueness, and noncrossing of the spatial rulings remain the relevant conditions.
Picture two cells that both have five faces, but one neighbor changes from label E to label F. The polygons can still have exactly the same unlabeled shape. What fails is the demand that each labeled wall continue unchanged through the whole interval. A transition must occur somewhere in that labeled structure. Its location and geometry depend on the interpolation; the endpoint lists do not put a special vertex at the temporal midpoint.
Sweep the cells through time
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4. Neighbor changes and their probability#
Theorem 328 (Neighbor changes obstruct a labeled product cell)
The following statements hold for the specified cell reconstruction.
In a continuous, nondegenerate family whose face incidences are locally constant throughout the closed interval, the neighbor lists are constant. A supplied compatible face isotopy then gives labeled prismatic cells.
If \(\mathcal N_i(t_a)\ne\mathcal N_i(t_b)\), there is no neighbor-label-preserving product trivialization of \(\mathcal S_i\) over that interval. Any stratified interpolation realizing both endpoint incidences has a transition. Under an isolated generic planar Delaunay flip, this is the familiar intermediate scutoid vertex. General jumps or higher-dimensional events need not have that particular realization.
For a clone event, condition on specified endpoints and other data. Suppose unchanged neighbors imply that a specified measurable region \(A\) contains no other site. If the remaining sites form a Poisson process of intensity measure \(\Lambda\) and \(A\) is fixed under that conditional law, then \(\mathbb P(\mathcal N_i(t_a)=\mathcal N_i(t_b))\le e^{-\Lambda(A)}\). In particular, \(\Lambda(A)\ge c_0\varrho r^d\) gives an exponential upper bound \(e^{-c_0\varrho r^d}\), where \(r\) is the specified clone displacement. The implication defining \(A\) and the conditional Poisson law are additional geometric and stochastic hypotheses.
Proof
A locally constant finite incidence pattern is constant on a connected time interval. In Euclidean general position this can be checked through strict orientation and in-sphere predicates: continuous motion keeps their signs until a zero is reached. A supplied isotopy gives the product realization as explained after (SC.14).
A label-preserving product sends each lateral face labeled \(j\) at the bottom to a face with the same label at the top, and conversely. It therefore forces equality of the neighbor sets, contradicting the second hypothesis. Equality of the number of neighbors would only allow an unlabeled bijection and does not repair that contradiction. In a stratified realization the change must be located on a transition stratum. This proves the obstruction, without asserting a unique filling or a fixed event time.
For the last statement the equality event is contained in the void event \(\{\#A=0\}\). A Poisson count with mean \(\Lambda(A)\) is zero with probability \(e^{-\Lambda(A)}\). Conditioning first makes clear which information fixes \(A\) and which randomness defines that count.
For independent sites of law \(\rho\), the last void probability is instead \((1-\rho(A))^m\le e^{-m\rho(A)}\), where \(m\) is the number of remaining sites. For a dependent conditional law with total entropy \(H\) relative to that product, the binary-entropy argument of (SC.8) gives the upper bound \((H+\log2)/(m\rho(A))\) when \(\rho(A)>0\). Thus an appropriate geometric void implication can use the existing entropy route without positing an exact Poisson law. If the region is selected after inspecting the same points, a fixed-region void formula is insufficient; one needs conditional control or a union bound over a specified candidate family.
Neither cloning nor a large displacement alone proves the void implication. A finite nondegenerate Euclidean configuration has an open neighborhood with the same strict predicate signs. Moving a site inside that neighborhood, including a sufficiently small random cloning displacement, can preserve its neighbors with positive probability. Conversely a continuous trajectory can cross a cocircular configuration without cloning and change the graph. Density and mean-field estimates from The Discrete Population Limit and Propagation of Chaos provide the stated laws and errors under their hypotheses; they do not assert that these geometric events occur with probability one.
4.1. Classifying recorded changes#
Definition 681 (Endpoint change types)
Set \(\mathcal N_{\rm lost}=\mathcal N_i(t_a)\setminus\mathcal N_i(t_b)\), \(\mathcal N_{\rm gained}=\mathcal N_i(t_b)\setminus\mathcal N_i(t_a)\), and
Type 0 has unchanged endpoint neighbors, \(\chi_{\rm scutoid}=0\). It is prismatic only when there are no intervening incidence transitions and the product realization conditions hold. Type 1 has one lost and one gained neighbor. Type 2 collects the other nonzero endpoint changes, including a single lost or gained neighbor. These are descriptive categories, not an ordering by the integer \(\chi_{\rm scutoid}\).
A transition-resolved activity also sums changes across every intermediate event time. Its value can exceed the endpoint index when a neighbor is lost and later regained. Neither index is an Euler characteristic.
Remark 254 (Metric consistency and cell data)
The cells are constructed using a supplied metric. Their lengths approximate that metric under Proposition 230. Neighbor lists alone do not determine its scale: multiplying a static spatial metric by a positive constant leaves nearest-site comparisons unchanged while changing all spatial lengths. Recovering a metric from cells therefore requires additional length, volume, or transport data. The Lorentzian construction and its faithful-embedding hypotheses remain those of Fractal Set as Causal Set.
4.2. Counting changes and preserving topology#
Proposition 231 (Incidence-change count and Euler characteristic)
Let \(E_k\) be the undirected facet-neighbor edges on a fixed label set at recorded time \(t_k\). Give absent labels empty neighbor lists. Then
For \(K\) slabs define the cumulative endpoint activity \(\mathcal K_{\rm total}=\sum_{k=0}^{K-1}\sum_i\chi_{\rm scutoid}(i;k)\). The number of cells with nonzero endpoint index is \(\sum_{k,i}\mathbf1_{\{\chi(i;k)>0\}}\), whose expectation is the sum of the corresponding probabilities. It is not the number of cloning events.
An interior planar Delaunay flip replaces one diagonal by the other. It contributes four to the sum in (SC.16), and preserves the Euler characteristic of the triangulated domain.
Proof
Each changed undirected edge appears in exactly the two endpoint neighbor lists, proving (SC.16). Linearity of expectation proves the cell-count formula without independence. A planar flip removes one edge and adds one edge; its four endpoints each lose or gain one neighbor. It keeps the vertex count, edge count, and face count fixed, so \(V-E+F\) is unchanged. More generally a subdivision of a fixed underlying region preserves its Euler characteristic, although its neighbor graph can change. A family with an actual topological change or a changing observation boundary requires its own Euler calculation.
Thus a single planar flip does not give a symmetric one-loss/one-gain event for each affected cell. Endpoint categories are also sensitive to recording frequency. No count of lateral connected faces follows solely from the endpoint neighbor union: one label can have several interface components, and a face can disappear and reappear between frames.
Proposition 232 (Planar mean degree and clone-conditioned endpoint activity)
For a planar Delaunay graph on \(n\ge3\) distinct sites in general position, \(\sum_i|\mathcal N_i|\le6n-12\). If its law is label exchangeable, then \(\mathbb E|\mathcal N_i|\le6-12/n\) for each label. For two such frames,
If both marginal frame laws are exchangeable and a fixed label clones with probability \(p_i>0\), its clone-conditioned endpoint index satisfies \(\mathbb E[\chi(i)\mid i\text{ clones}]\le12/p_i\). Thus a uniform positive lower bound on these cloning probabilities gives a population-independent bound on this particular conditional mean. Transition-resolved activity and geometric repair costs are different quantities.
Proof
A simple planar graph with \(n\ge3\) has at most \(3n-6\) edges. For a connected plane graph this follows from Euler’s identity \(n-E+F=2\) and the bound \(3F\le2E\) counting face incidences; adding edges handles disconnected graphs. Summing vertex degrees counts every edge twice. Under exchangeability all expected degrees coincide, so divide the deterministic bound by \(n\). Each endpoint index is at most the sum of its degrees in the two frames, proving (SC.16a) and \(\mathbb E\chi(i)\le12\). Finally \(p_i\mathbb E[\chi(i)\mid i\text{ clones}] \le\mathbb E\chi(i)\) because the index is nonnegative. Exchangeability follows from the complete-kernel conditions of Theorem 241 when they apply to the recorded law.
This direct planar estimate requires no Poisson model or deterministic separation of the sites. It controls first moments of degrees and endpoint changes. It does not bound higher degree moments, repeated transitions between frames, or the number of flips performed by a particular repair procedure.
4.3. Moving interfaces and volume change#
Lemma 284 (Cell-volume balance with a moving metric)
On a regular time interval let \(\Omega_i(t)\) be a cell with positive finite volume \(V_i(t)\), piecewise smooth boundary, and boundary normal velocity \(w\cdot n\). Then
For a regular Voronoi interface \(\psi_{ij}(x,t)=0\), \(w\cdot n=-\partial_t\psi_{ij}/|\nabla\psi_{ij}|_{g_t}\) with \(n=\nabla\psi_{ij}/|\nabla\psi_{ij}|_{g_t}\). Here \(\partial_t\psi_{ij}\) includes both site motion and metric motion.
Proof
Pull back the cell integral by a flow extending its boundary velocity. Differentiation of the flow Jacobian gives its divergence; integration by parts gives the boundary flux. Differentiating the metric volume factor gives \(\partial_t\sqrt{\det g}=\frac12\sqrt{\det g}\operatorname{tr}_g\partial_tg\). This proves (SC.17), recovering Lemma 261 with its metric term. Differentiating \(\psi_{ij}(x(t),t)=0\) gives \(\partial_t\psi_{ij}+\langle\nabla\psi_{ij},w\rangle_g=0\), proving the normal-velocity formula of Lemma 262.
For a fixed metric and a comparison velocity field \(u\), set \(E_i=\int_{\partial\Omega_i}(w-u)\cdot n\,dA\). The divergence theorem then gives the exact normalized relation
The cell-average term approximates the divergence at its site with error at most \(\operatorname{diam}(\Omega_i)\|\nabla\operatorname{div}_gu\|_\infty\), as in Lemma 263. Transferring a differentiated expansion equation requires bounds on both \(E_i/V_i\) and its time derivative, along with the material-transport comparison in Theorem 305. An area estimate of order \(\epsilon^{d-1}\) must be divided by a volume of order \(\epsilon^d\); this normalization can remove a power of smallness. At cloning jumps the piecewise volume balance also contains its jump increments. These statements supply the inputs used in Curvature of the Algorithmic Fitness Metric; they do not assert that arbitrary reconstructed cells follow a geodesic congruence.
Which operation changed the neighbors?
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5. Maintaining the geometric reconstruction#
A dynamic triangulation is useful because much of yesterday’s geometry may still be valid today. But “small movement” needs a scale. A tiny displacement can flip a nearly cocircular quadrilateral. We therefore count the operations that actually occur and state the conditions under which their average cost stays small.
The following algorithms describe geometric postprocessing of recorded states. They do not change companion selection, cloning, or kinetic evolution. Their Euclidean form also applies to one fixed positive constant metric after a linear change of coordinates. A varying Riemannian metric requires a geometric triangulation method with its own distance predicates and validity proof, or an explicitly controlled flat approximation.
Algorithm 10 (Online triangulation from recorded changes)
Input: A valid planar Euclidean triangulation, current coordinates and label-to-vertex map, and the next recorded coordinates, alive statuses, cloning indicators, and parent identifiers. Sites are distinct and the triangulation predicates satisfy the chosen general-position convention.
Initialization: Build the Delaunay triangulation of the initial included sites and its Voronoi dual. Account for its construction cost separately. The standard planar batch construction takes \(O(n\log n)\).
Update procedure:
Read the recorded changes. Classify slot replacements using the cloning indicator and parent record, and retain every positional and status change. Metric changes are also inputs to the reconstruction.
For each removed or replaced site, delete its vertex and retriangulate the resulting cavity by a valid planar triangulation routine. Include the deletion, cavity triangulation, and predicate costs in its repair cost.
Locate each inserted position, using a surviving parent location or a neighboring old cell as a hint when available. Insert it using a valid cavity construction or a containing-simplex split, and restore the Delaunay property. Handle convex-hull changes explicitly.
A moved vertex may use a cheaper local update only when that update preserves an embedded triangulation and checks every potentially changed predicate. Otherwise process its movement by deletion and insertion. Assigning new coordinates to an old vertex without this check can invert triangles or make edges cross.
Refresh all affected dual cells, their interfaces, and the label map. Store the changes in the incidence graph. A full export of every simplex or cell is charged separately from an incremental update.
Degenerate or coincident sites require an explicit tie or multiplicity convention. A changed metric can change predicates even when no site moves; its distance evaluations and global or local repair work must be included.
Under a kinetic model with \(\dot x=v\), spatial paths between cloning events are absolutely continuous. For example,
This is Cauchy–Schwarz followed by integration. It gives a typical spatial scale of order \(h\) in that kinetic model; velocity diffusion itself has a noise increment of order \(\sqrt h\). A model with direct position diffusion has a different displacement estimate. The implemented split transition uses its recorded positions and the appropriate estimates of Quantitative Error Bounds. Even (SC.19) does not bound a Delaunay repair count without protection from near-degenerate predicates.
5.1. Planar Lawson restoration#
Algorithm 11 (Lawson flips on a valid planar triangulation)
Input: A valid planar triangulation of fixed distinct sites in general position. All possibly illegal interior edges are placed in a queue; scanning all interior edges is a valid initialization.
Procedure: Remove an edge from the queue and check its current incident triangles. If the union is a convex quadrilateral and the opposite point is strictly inside the other triangle’s circumcircle, replace the diagonal by the other diagonal. Requeue every edge whose adjacent triangles changed. Stale queue entries are checked against the current incidence records. Continue until the queue is empty.
Output: The Euclidean Delaunay triangulation. In particular, a permanently marked “already visited” triangle must not suppress a later check after its neighbors have changed.
Lemma 285 (Termination and the cost of actual flips)
Under the preceding planar hypotheses, Lawson flips terminate. If \(Q_0\) queue entries are initially inserted and \(F\) flips occur, an implementation with constant-time incidence updates and predicates costs \(O(Q_0+F)\). This statement gives no bound \(F=O(1)\) or \(F=O(k)\) solely from the size of an initially inspected star.
Proof
For a planar triangulation \(\mathcal T\), let \(u_\mathcal T\) be the piecewise-affine interpolation of the lifted heights \(|z_i|^2\) over its triangles. An illegal diagonal is locally nonconvex in this lifting. The other diagonal replaces its lifted patch by the lower convex patch of the same four vertices. Hence \(u_\mathcal T\) decreases pointwise on that quadrilateral and decreases strictly on an open subset. Consequently \(\int_{\operatorname{conv}Z}u_\mathcal T(x)dx\) strictly decreases. There are only finitely many triangulations of the fixed finite point set, so the process terminates.
At termination every interior edge is locally Delaunay. The lifted piecewise-affine surface is then locally convex across every edge and hence convex on the convex hull: along a line crossing edges transversely, its piecewise-constant slope never decreases; continuity extends this to all lines. Its facets are therefore lower supporting facets. The lifting proof of Lemma 279 identifies the result as Delaunay. Each flip changes a bounded number of planar adjacencies and inserts a bounded number of queue entries, giving \(O(Q_0+F)\) work.
This lifting proof is the classical one described in the ETH computational geometry notes. Its planar termination argument is not a proof that arbitrary higher-dimensional flips, geodesic flips, or an unchecked star retriangulation will succeed. Their algorithms and costs must be supplied separately.
5.2. Point location from a geometric hint#
Algorithm 12 (Straight-segment walk with a hint)
Input: A valid simplicial triangulation, a query point \(z\), and an interior point \(q\) in a known hint simplex. Assume the segment from \(q\) to \(z\) crosses faces transversely and avoids vertices and lower-dimensional faces, or use an explicit rule for such simultaneous crossings.
Procedure: Parameterize \(q+s(z-q)\), \(0\le s\le1\). In the current simplex find the first exit face with parameter strictly larger than the previous crossing parameter. Cross to its neighbor and continue. Stop when \(z\) lies in the current simplex or the segment leaves the triangulated region. The jump stage is the selection of the initial hint; its search and maintenance costs are counted.
The cost is the hint cost plus the number of crossed simplices. An arbitrary “face beyond the query” rule is replaced here by the increasing segment parameter, which gives a termination criterion.
Lemma 286 (A walk bound under actual shape control)
Suppose simplices along the query segment have diameter at most \(C\ell\) and volume at least \(c_0\ell^d\), with disjoint interiors in Euclidean coordinates. A segment of length \(R\) crosses at most \(C'(1+R/\ell)\) simplices, where \(C'\) depends on \(d,C,c_0\). For a hint-to-query displacement with \(\mathbb E R\le\sigma\sqrt d\), the expected number is at most \(C'(1+\sigma\sqrt d/\ell)\) under the same uniform shape control.
Proof
Every crossed simplex lies in the \(C\ell\) neighborhood of the segment. This tube has volume at most \(C_d(R(C\ell)^{d-1}+(C\ell)^d)\). Since simplex interiors are disjoint and each has volume at least \(c_0\ell^d\), divide these volumes to obtain the first bound. Take expectations for the second. A centered Gaussian jitter with covariance \(\sigma^2I\) satisfies \(\mathbb E|\xi|\le\sqrt{\mathbb E|\xi|^2}=\sigma\sqrt d\). The shape hypotheses concern the triangulation actually traversed; Gaussian jitter alone does not supply them.
For arbitrary configurations, the number of crossed simplices can be large. A logarithmic point-location claim requires an identified data structure with that bound and a budget for maintaining it. A spatial index or a parent hint by itself is not a complexity theorem.
Maintain the mesh and count its activity
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6. Conditional complexity and output costs#
Theorem 329 (Operation-count and expected update bounds)
Let \(C_k\) be the number of cloned slots on a step. Let \(R_i\) denote the actual geometric repair cost of update \(i\), and \(L_i\) its point-location cost when required. Let \(M_k\) include metric evaluations, index maintenance, and additional global repairs. For the stated reconstruction procedure,
Suppose the continuous-move repairs and their location costs have total expected cost \(O(N)\), cloned-slot repair costs have total expected cost \(O(\mathbb EC_k)\), each clone’s location cost is at most \(O(\log N)\) in conditional expectation, and \(\mathbb EM_k=O(N)\). Then, writing \(p_k=\mathbb EC_k/N\),
In particular \(p_k=O(1/\log N)\) gives an expected linear update cost. If the hypotheses are bounds on totals over many steps, the same calculation gives an amortized bound over those steps. A full export adds its output cost.
Proof
The initial record scan costs \(O(N)\) in fixed dimension. Add the actual repair, search, metric, and maintenance costs to get (SC.20). For a clone indicator \(I_i\), conditioning on \(I_i=1\) bounds \(\mathbb E[I_iL_i]\le C\log N\,\mathbb P(I_i=1)\). Sum over \(i\) and use \(\sum_i\mathbb P(I_i=1)=\mathbb EC_k=Np_k\). Apply the other assumed total-cost bounds to obtain (SC.21). No independence of cloning indicators is required. Summing the same identities over time proves the amortized version.
The sampling bounds (SC.7)–(SC.9) establish coverage under their stated laws. They do not establish bounded repair costs near degeneracies, a linear-size intrinsic Delaunay complex in every dimension, or cheap updates of an adaptive metric. Those hypotheses in (SC.21) must be proved for the chosen reconstruction or measured as operation counts. A fixed positive cloning fraction leaves an \(N\log N\) term as \(N\to\infty\) in this bound; a small numerical coefficient is not an asymptotic linear-time guarantee.
6.1. What the output lower bound actually measures#
Theorem 330 (Explicit-output and dynamic-update lower bounds)
In a sequential word-RAM model with fixed dimension, writing a fresh complete triangulation representation with \(n\) vertices and \(s\) simplex records costs \(\Omega(n+s)\). Thus an \(O(N)\) update followed by a complete export is output-optimal when \(n+s=\Theta(N)\) and the hypotheses of (SC.21) hold. For an incremental representation, the corresponding output lower bound is only the number of records explicitly changed or emitted; an unchanged triangulation need not be rewritten.
Proof
Each vertex or simplex record requires at least one word write, and a step writes only a bounded number of words. This proves the complete-output bound. An incremental procedure can retain unchanged records, so that argument counts only its changed or emitted records. For example, storing a global rigid transformation lazily can describe rotated coordinates while retaining the same combinatorial triangulation. It gives no reason to visit every simplex. A procedure that reads all \(N\) new coordinates still incurs its explicit \(\Omega(N)\) input cost.
7. Incidence activity and information in the record#
Counting changed neighbors tells us how much the geometric graph reorganized. It is a useful observable, but its units are changes per unit time. To count bits, we also need to know how surprising those changes were. And a stable neighbor graph can accompany substantial motion or improvement in fitness. Those are different measurements of the computation.
Definition 682 (Recorded topological activity rate)
For a recorded interval of length \(T\), define
This retains the name topological information rate as an incidence-count observable, with units of oriented neighbor changes per unit time. If every resolved transition is counted, use its transition-resolved sum instead. A clone-only version sums only the explicitly designated clone-associated changes. For \(C>0\) recorded clone events, it factors exactly as \((C/T)(C^{-1}\sum_{\rm clones}\chi)\). For \(C=0\) its value is zero. A homogeneous expected clone frequency is \(Np_{\rm clone}/\Delta t\); the corresponding expected activity uses the conditional mean index per clone, rather than the unconditional per-cell mean.
Proposition 233 (Activity bound from event incidence)
Suppose an observed spacetime region of volume \(V_{\rm st}\) and duration \(T\) contains \(J\) resolved transition events, each contributing at most \(m_*\) oriented neighbor changes. Put \(\rho_{\rm event}=J/V_{\rm st}\). Then
For a fixed spatial observation region of volume \(V\) and lapse \(c\), \(V_{\rm st}=cTV\), so the last bound is \(m_*cV\rho_{\rm event}\). If all events are isolated interior planar Delaunay flips on a fixed label set, the resolved activity equals \(4J/T\).
Proof
For finite sets the symmetric-difference distance satisfies the triangle inequality. Apply it to every neighbor list between recorded endpoints to bound its endpoint change by the sum over intermediate events. Summing over sites gives the first inequality. The incidence bound gives a resolved total at most \(m_*J\), proving the second. The volume formula follows from \(dV_G=c\,dt\,dV_g\) for this fixed product geometry. The planar flip count is exactly four by (SC.16).
Theorem 331 (Uniform incidence under protected local events)
Consider a family of reconstructed configurations indexed by population \(N\), with local mesh scale \(\varepsilon_N>0\). Assume that every resolved event \(e\) satisfies the following conditions, with constants independent of \(N\) and \(e\).
Metric regularity and anti-degeneracy. On the event neighborhood, the reconstruction metric has a uniform doubling bound with constant \(C_{\mathrm{dbl}}\). Distinct active sites satisfy the anti-degeneracy (hard-core) estimate
\[ d_g(z_i,z_j)\ge a\,\varepsilon_N,\qquad i\ne j, \tag{SC.23a} \]for some \(a>0\).
Locality. All endpoint labels of changed edges lie in a metric ball \(B_g(x_e,R\varepsilon_N)\). This holds, for example, when at most \(q_*\) labels move by at most \(r_*\varepsilon_N\), the adaptive metric update is supported in that ball, and every predicate outside the ball has a strict margin larger than the corresponding perturbation.
Resolved-event budget. The event is decomposed into at most \(F_*\) local predicate transitions. A transition is counted once, even if the endpoint graph is recorded only at the beginning and end of the event.
Then there is a constant
where \(d_*\) is the doubling dimension, such that the event incidence index \(\chi_e\) obeys
Consequently, if \(J_N\) events occur in a region of spacetime volume \(V_{\rm st}\) and duration \(T\), then
The constant is independent of population and resolution. In the planar fixed-label case, the existing exchangeable estimate Proposition 232 supplies the corresponding endpoint bound without the hard-core assumption. The assumptions above are the additional protection needed for transition-resolved activity and for adaptive metrics.
Proof
Let \(A_e\) be the set of labels appearing at an endpoint of a changed edge. Locality places all sites in \(A_e\) inside \(B_g(x_e,R\varepsilon_N)\). The disjoint balls of radius \(a\varepsilon_N/2\) centered at these sites fit inside a ball of radius \((R+a/2)\varepsilon_N\). The doubling estimate therefore gives
A single transition can change at most \(|A_e|^2\) oriented incidences. The resolved-event budget then gives \(\chi_e\le F_*M_*^2=m_*\) pathwise. Taking the conditional expectation proves (SC.23c). Summing this bound over the \(J_N\) events and dividing by \(T\) gives (SC.23d), followed by the definition of \(\rho_{\rm event}^{(N)}\).
The hard-core and predicate-margin hypotheses are genuine geometric inputs: moment or entropy bounds alone do not exclude nearly coincident sites or nearly degenerate Delaunay predicates. Once they are verified for a scaling family, the result is a theorem rather than an assumed uniform incidence property.
This formulation keeps the spatial-volume factor required by dimensions. A density of events per spacetime volume multiplied only by a speed does not have units of total events per time. It also distinguishes a resolved event from a hypothetical vertex placed at the midpoint of a slab.
Lemma 287 (Bits required to specify an edge-change set)
On \(n\) fixed labels there are \(M=\binom n2\) possible undirected edges. If a random change set \(A\) has cardinality \(C\), then its base-two Shannon entropy satisfies
For a fixed \(1\le C=m\le M\), the conditional term is at most \(m\log_2(eM/m)\). An incidence count alone does not determine this entropy.
Proof
Since \(C\) is a function of \(A\), the entropy chain rule gives \(H_2(A)=H_2(C)+H_2(A\mid C)\). Conditional on \(C=m\), at most \(\binom Mm\) change sets are possible, so the conditional entropy is bounded by the logarithm of that number. The estimate \(\binom Mm\le M^m/m!\le(eM/m)^m\) gives the last bound. For the factorial inequality, sum \(\log j\) and compare with \(\int_1^m\log x\,dx=m\log m-m+1\).
For example, \(z_i(t)=e^{-t}z_i(0)\) preserves a Euclidean Delaunay graph and all its neighbor lists because it is a similarity. Nevertheless \(n^{-1}\sum_i|z_i(t)|^2=e^{-2t}n^{-1}\sum_i|z_i(0)|^2\) decreases whenever the sites are not all zero. This demonstrates that geometric incidence activity is not a necessary measure of improvement in a quadratic objective. Interpreting it as exploration or learning requires comparison with the recorded fitness and other observables.
8. Using the construction in the next chapter#
The spatial cells are well-defined nearest-site regions for the supplied metric. Their labeled changes are quantified by (SC.15)–(SC.16), and their regular motion obeys the volume balance (SC.17). Slab lengths and causal reachability have the quantitative comparison in (SC.13); proper time uses (SC.12). These constructions preserve the distinction between a recorded clone jump and a chosen continuous cell interpolation.
The density and concentration estimates give quantitative coverage, including the entropy-based bound (SC.8) and the LSI bound (SC.9). Local shape and predicate protection remain the conditions needed for a regular triangulation and the cost estimates (SC.20)–(SC.21). The curvature and expansion calculations in Curvature of the Algorithmic Fitness Metric use the same metric, actual connection consistency, and the normalized, differentiated volume conditions of Theorem 305.
References#
The biological geometry motivating the term is described by Gómez-Gálvez et al., Scutoids are a geometrical solution to three-dimensional packing of epithelia. For planar lifting and Lawson flips, see the ETH computational geometry notes. The extra conditions needed for intrinsic Delaunay realizations are discussed in Boissonnat et al., An obstruction to Delaunay triangulations in Riemannian manifolds.
The analytic results used here are proved in The Discrete Population Limit and Propagation of Chaos, Mass, Hellinger, and Transport Convergence, Exchangeability, Empirical Laws, and Functional Inequalities, C³ Regularity of the Fractal Gas Fitness, Logarithmic Sobolev inequalities and entropy convergence, Continuum Limits: Hypotheses and Consistency, and The Geometric Gas: Regularity, Stability, and Entropy.