Continuum Limits: Hypotheses and Consistency#
1. What a Continuum Limit Requires#
A cloud of points can approximate a probability distribution without telling us how to measure a distance between two points. Those are separate questions. There is a third question when we estimate a differential operator: as its neighborhood shrinks, do we still have enough samples to control the noise?
This chapter separates these questions. We can prove regularity of specified smoothed fields and compute the normalization and variance of an estimator. Identifying the graph with a smooth spacetime requires additional geometric information. Identifying an episode law with a particular Gibbs density requires an equation for that law. The conditions below say exactly where those pieces enter.
Definition 683 (Scope of the continuum hypotheses)
The hypotheses A1–A6 in Assumption 18 concern the reconstruction in Fractal Set as Causal Set. Their application requires the following data.
A1: geometry. A specified smooth spatial manifold, a positive definite field \(g_R\), and a product metric \(g=-c^2dt^2+g_R\) with \(c>0\). Global hyperbolicity and agreement of reconstructed distances and causal neighborhoods with this geometry require separate proofs or hypotheses.
A2: regularity. Bounds on derivatives of the fields used by the reconstruction, positivity of every denominator, and integrable bounds for derivatives of spatial marginals. Bounds needed for a local estimator are imposed on a neighborhood of its evaluation region. On an unbounded space, a confining potential may itself be unbounded.
A3: sampling. A normalized episode law with respect to a specified reference measure. An exact Gibbs formula, when used, is an additional identification of that law.
A4: dependence. A variance or covariance estimate for the actual bandwidth-dependent summands. A law of large numbers for each fixed test function does not supply such an estimate.
A5: kernel. Verified moments of the kernel on its actual cutoff domain, together with a local consistency estimate for that domain and geometry.
A6: scaling. A chosen sequence of sample sizes and reconstruction bandwidths for which the preceding errors vanish, with the constants controlled along the sequence.
The lemmas below give sufficient conditions and direct estimates. Verifying all six items for episodes of the interacting algorithm remains a separate application of these results. In particular, none of the geometric or exact sampling identifications is inferred from propagation of chaos.
2. Regularity of Smoothed Fields#
Smoothing is an operation we can estimate. If we convolve a probability measure with a Gaussian of fixed width, each spatial derivative can be moved onto the Gaussian. The size of that derivative tells us the price of smoothing at that width. Time derivatives are different: smoothing in space cannot repair a jump in time.
There is also a derivative to count. A metric formed from a Hessian uses two spatial derivatives before we begin differentiating the metric. Four bounded derivatives of that metric therefore require up to six derivatives of its potential. Keeping this count explicit prevents a regularity assumption from being lost in a composition.
Lemma 288 (Sufficient regularity and spectral bounds for A2)
Let \(I\) be a time interval. In Euclidean spatial coordinates, fix a smoothing width \(\sigma>0\) and write
Assume \(t\mapsto\mu_t\) is \(C^4\) in the total variation norm of finite signed measures, with \(\|\partial_t^j\mu_t\|_{\mathrm{TV}}\le M_j\) for \(0\le j\le4\). Then, for every spatial multi-index \(\alpha\) and \(j\le4\),
More generally, the same conclusion holds for a smoothed field whose space-time derivatives admit the corresponding dominated integral bounds. The time regularity of the underlying measure is part of the hypothesis.
Suppose a potential \(V\) has bounded derivatives needed to form \(A=\nabla_x^2V+\epsilon_\Sigma I\) and its space-time derivatives of total order at most four. Assume
Then the metric and diffusion convention of Definition 669 gives
Both fields are \(C^4\). If \(A_k\) bounds all derivatives of \(A\) of total order \(k\), bounds for derivatives of \(A^{-1}\) are given recursively by
For an effective potential defined from a positive field \(s\) by \(U=-T\log s\), assume \(s\ge s_*>0\) on the region in question and let \(S_k\) bound its derivatives of order \(k\). Derivatives of \(\log s\) of order \(k\ge1\) are bounded by
Thus derivatives of \(U\) are bounded by \(TL_k\). This conclusion concerns the specified field \(s\); identifying it with the episode density is additional.
Finally, if \(b(t,x)\) is \(C^4\) in time and \(|\partial_t^j b(t,x)|\le B_j(x)\) for integrable \(B_j\), then
The same assertion applies to a rate \(r(t)\) represented by such an integral. Positive lower bounds on \(Z\) and \(r\) must be checked separately wherever their reciprocals occur. In particular, the lemma applies to \(b=\sqrt{\det g_R}\,e^{-U/T}\) when its time derivatives have these integrable bounds.
Proof
Differentiation under the convolution is justified by the bounded derivatives of \(G_\sigma\) and differentiability of \(\mu_t\) in total variation. Scaling the Gaussian gives \(\partial^\alpha G_\sigma(x)=\sigma^{-d-|\alpha|} (\partial^\alpha G_1)(x/\sigma)\), proving the first estimate.
The spectral inequalities follow by diagonalizing \(A\). For a multi-index \(\beta\) with \(|\beta|=k\), differentiate \(AA^{-1}=I\):
Take operator norms and use \(\sum_{|\gamma|=j,\,\gamma\le\beta}{\beta\choose\gamma}={k\choose j}\) to obtain the recursion for \(B_k\). The matrix field \(\Sigma_{\mathrm{reg}}=A^{-1/2}\) is also \(C^4\): use
The preceding recursion, with \(a\) replaced by \(a+u\), bounds derivatives of the resolvent. For positive derivative order these are finite sums of products with at least two resolvent factors, so the differentiated integrands are integrable at both endpoints. Differentiation under this integral is therefore valid through order four.
For the logarithm, differentiate \(s\,\partial_i\log s=\partial_i s\) a further \(k-1\) times and isolate the highest derivative of \(\log s\). The same multi-index summation gives the stated recursion. The positive lower bound justifies division by \(s\).
The marginal formula and its derivative bounds follow from dominated convergence applied successively to \(\partial_t^j b\). Continuity follows by the same domination. Each assertion is thus a consequence of its stated regularity, spectral, or domination hypotheses. \(\square\)
Remark 255 (Spatial, temporal, and global bounds)
For a probability measure, \(M_0=1\) in the convolution estimate. The constants increase as \(\sigma\downarrow0\); uniformity along a changing smoothing scale requires a separate estimate. The Gaussian argument supplies spatial regularity even for atomic measures, but the total variation hypothesis on time derivatives is stronger and must be verified or replaced by a suitable differentiation-under-the-integral argument.
For \(g_R=\nabla_x^2V+\epsilon_\Sigma I\), \(C^4\) regularity of \(V\) alone gives only two spatial derivatives of \(g_R\). A positive lower bound on \(g_R\) gives an upper bound on its inverse; the lower diffusion bound also requires the upper bound \(g_R\preceq bI\). A spectral shift must dominate the negative part of the Hessian to ensure positivity at all.
On an unbounded space, local positivity of \(s\), global integrability of a confining density, and uniform bounds on derivatives are distinct conditions. A finite time interval supplies none of these spatial estimates by itself.
Differentiate the executed conditional field
Rust engine experiment · Seed 7 · Poster after 11 experiment steps. Interactive view starts from the same seed.
3. Product Geometry and Graph Identification#
A speed bound becomes useful once we know what measures speed. Suppose our spatial metric is everywhere at least a fixed multiple of an ordinary complete metric. Then a causal curve has a bounded speed in that ordinary metric too. During a finite time interval it can travel only a bounded distance. This is the geometric fact that can keep causal diamonds compact, even when space itself is unbounded.
Lemma 289 (A sufficient condition for global hyperbolicity)
Let \((\mathcal X,h)\) be a connected complete Riemannian manifold whose closed bounded sets are compact. Let \(I\) be an open interval and let \(g_R(t)\) be a \(C^4\) family of Riemannian metrics satisfying
for \(0<a\le b<\infty\). Then \(M=I\times\mathcal X\), with \(g=-c^2dt^2+g_R(t)\) and \(c>0\), is globally hyperbolic, and every slice \(\{t\}\times\mathcal X\) is a Cauchy hypersurface. A closed observation window inside \(I\) may be used for sampling.
These hypotheses allow unbounded spatial domains, including \(\mathbb R^d\) with its Euclidean background metric. They concern a specified product manifold. Identifying it with a limit of the IG graph or its distances is an additional hypothesis.
Proof
For a future-directed causal curve parametrized by time,
Consequently its spatial part is \(c/\sqrt a\)-Lipschitz. If the time parameter of an inextendible causal curve had a finite endpoint in \(I\), completeness would give a spatial limit there. Appending a vertical timelike segment would extend the curve. Its time range is therefore all of \(I\), and it intersects each time slice exactly once.
For \(p=(s,x)\) and \(q=(t,y)\) with \(s\le t\), every point of the causal diamond lies in
which is compact. The diamond is closed: for a convergent sequence of its points, the causal curves from \(p\) to these points and from them to \(q\) have the same Lipschitz bound and lie in a common compact set. Extend their spatial parts constantly to common time intervals and apply Arzelà–Ascoli. A uniformly convergent subsequence has a Lipschitz limit. In each coordinate neighborhood, continuity of \(g_R\) and weak lower semicontinuity of the quadratic energy give \(g_R(t)(\dot x,\dot x)\le c^2\) almost everywhere on each limiting segment. Thus the limiting point remains between \(p\) and \(q\) causally.
Strong causality also follows from the speed estimate. Inside any product neighborhood of a point, choose a smaller product neighborhood with time width so short that a curve with speed at most \(c/\sqrt a\) cannot leave the larger spatial neighborhood between two points of the smaller one. Time monotonicity keeps the intervening segment inside the larger time interval. This is the local strong-causality criterion. Together with compact causal diamonds it gives global hyperbolicity. The slice conclusion follows from the inextendibility argument above. \(\square\)
Remark 256 (Required geometric identifications)
Propagation of chaos is convergence of specified particle marginals. It does not construct a manifold, a metric, or a shortest-path approximation theorem. A confining probability envelope controls tails of a measure; compactness of causal diamonds is a property of the metric and causal curves.
The graph-Laplacian result of Belkin and Niyogi starts with samples on a specified compact embedded manifold; its main convergence theorem concerns the Laplace–Beltrami operator. It supplies no shortest-path convergence or Lorentzian reconstruction theorem. Its compact sampling model is not imposed here on an unbounded Fractal Gas.
For an application using graph distances, one must specify an embedding of episodes, a distance approximation bound, and control of its effect on the shrinking kernel and causal cutoff. A bound on discrete endpoint displacement also requires an interpolation argument before it becomes a pointwise causal speed bound for a time-dependent metric. Equality at the speed bound is null; strict inequality gives a timelike curve.
4. Sampling Laws, QSD, and Normalization#
The weight in an importance sampler is the reciprocal of the density relative to the measure we want to integrate. This rule fixes every normalization constant. It also tells us what to do if the actual density contains a factor that a proposed Gibbs formula omitted: that factor belongs in the weight.
Quasi-stationarity concerns particles conditional on having survived. Sampling a long trajectory of a process conditioned to survive forever is another experiment. Those experiments can have different limiting laws, so we must name the experiment before choosing the weight.
Lemma 290 (Normalized sampling and the distinction between QSD and invariance)
Let \(P_t\) be a killed sub-Markov semigroup. A probability measure \(\nu\) satisfying \(\nu P_t=e^{-\lambda t}\nu\) is a QSD: conditioning on survival preserves its one-time law. If \(\lambda>0\), it is not an invariant probability measure for \(P_t\).
If a positive function \(h\) satisfies \(P_th=e^{-\lambda t}h\) and \(0<\nu(h)<\infty\), define
Then \(Q_t\) is conservative and \(\beta\) is invariant for it. Identification with a process conditioned on indefinite survival requires the corresponding conditioning-limit theorem.
Independently, let episodes \(Y_i\) have a specified common probability law \(\pi(dy)=q(y)\,d\mathrm{vol}_g(y)\). On the support of an integrand \(F\), assume \(q>0\) and \(F\) integrable against \(d\mathrm{vol}_g\). Then
Independence is unnecessary for this expectation identity. It is relevant to the variance and convergence of the average.
For a finite window and the exact coordinate sampling law
assume \(0<R<\infty\), \(r,Z>0\), and \(Z<\infty\). Since \(d\mathrm{vol}_g=c\sqrt{\det g_R}\,dt\,dx\), the density and weight are
For the product measure \(\sqrt{\det g_R}\,dt\,dx\) the factor \(c\) is omitted. Rescaling time to \(ct\) also absorbs this constant, provided the time density is transformed consistently.
Proof
Integrating \(\nu P_t=e^{-\lambda t}\nu\) against \(1\) gives the survival probability \(e^{-\lambda t}\). Dividing by it gives the conditional law \(\nu\). When \(\lambda>0\), the unconditioned surviving mass decreases, which excludes invariance. For the transformed semigroup,
The importance-sampling identity follows by integrating \(F/q\) against \(q\,d\mathrm{vol}_g\) and using linearity of expectation. The displayed coordinate density integrates to one, first in space and then in time. Dividing it by \(c\sqrt{\det g_R}\) gives \(q\) and its reciprocal. \(\square\)
Remark 257 (What an exact density formula requires)
An upper or lower Gibbs envelope for a QSD does not establish the equality \(p\propto\sqrt{\det g_R}e^{-U/T}\). A multiplicative decoration changes the importance weight unless it is included in the specified potential and its normalizer. An exact identification requires verification of the density against the relevant generator or killed eigenmeasure equation and its boundary conditions. A stationary law for a drift-augmented diffusion is a statement about that diffusion.
Exchangeability concerns permutation of particle labels. It supplies neither time stationarity nor independence. A time-dependent slice law also requires an argument connecting the episode sampling protocol to those slices. Champagnat and Villemonais give conditions for convergence to a QSD and for existence and ergodicity of the associated conditioned process. Those conditions must be checked for the particular process; they do not identify its QSD with a prescribed Gibbs formula.
Normalize a kernel on the recorded population
Rust engine experiment · Seed 7 · Poster after 11 experiment steps. Interactive view starts from the same seed.
5. Kernel Moments and Deterministic Consistency#
For a second derivative, the quadratic moment of the kernel does the work. A wave operator needs opposite signs in the temporal and spatial moments, which a nonnegative kernel cannot provide. Signed weights can provide them, but the moments must be computed on the actual integration domain.
A finite cutoff selects a frame. Spatial rotations still make the spatial moments equal, and reflection can kill odd moments. Neither symmetry fixes the sign or magnitude of the temporal moment. That is an additional equation.
Lemma 291 (Verified moments and an auxiliary tangent-space construction)
Use length coordinates \((ct,x)\) in a Lorentz-orthonormal frame and set \(\eta=\operatorname{diag}(-1,1,\ldots,1)\) in dimension \(D=d+1\). Let \(J=-J\subset\mathbb R^D\) be a bounded measurable set. A bounded measurable even kernel \(k\) supported in \(J\) is admissible for the local difference construction below when
The condition \(\int_Jk=0\) may additionally be imposed. It is unnecessary for annihilation of constants in an operator containing \(f(y)-f(p)\).
There is an explicit auxiliary construction when the interior of \(J\) contains an open set and its reflection. Choose an even \(\phi\in C_c^\infty(\operatorname{int}J)\) with \(\int\phi=1\) and define
Then \(k\) is smooth, even, compactly supported, has zero integral, and has the specified second moments. In particular, its support can be chosen inside a bounded timelike double cone. This is a mathematical example of a tangent kernel; it neither changes the particle algorithm nor identifies the existing proper-time reconstruction kernel with this example.
For a proper-time-only kernel, take \(J\) inside the timelike double cone and write \(\tau(\zeta)=\sqrt{(\zeta^0)^2-|\zeta^{\mathrm{space}}|^2}\) and \(k(\zeta)=K(\tau(\zeta))\). Suppose \(J\) is invariant under time reversal and spatial rotations. A finite ansatz \(K=\sum_{j=1}^m a_j\phi_j\), with \(\phi_j\in C_c^2([0,1])\), is valid precisely when its actual moment matrix \(C\) admits a solution to
This criterion applies for \(d\ge1\) when all the moments are finite. Full row rank is a sufficient condition. The rank or solvability must be checked for the chosen domain and functions.
Proof
Evenness makes every odd monomial integrate to zero. For the auxiliary construction, compact support removes boundary terms in integration by parts. The integral of each second derivative of \(\phi\) is zero, while
Contraction with \(m_2\eta^{\mu\nu}\) gives the required second moment. A smooth nonnegative bump and its reflection, divided by their combined integral, provide the stated \(\phi\).
For the proper-time ansatz, time reversal removes temporal-spatial moments; spatial rotational symmetry removes off-diagonal spatial moments and makes the diagonal spatial moments equal. The remaining constraints are exactly the three displayed linear equations. A cutoff bounded in the chosen time coordinate is not invariant under arbitrary Lorentz boosts, so it supplies no identity relating \(C_{tj}\) to \(C_{sj}\). Two coefficients and two nonzero scalar moments therefore do not establish solvability of these three constraints. \(\square\)
Lemma 292 (Direct local bias estimate in specified normal coordinates)
Fix an interior point \(p\) of a specified Lorentzian manifold and a normal coordinate map \(\Psi_p\) with \(\Psi_p(0)=p\) and metric \(\eta\) at the origin. Suppose this map is defined on a Euclidean ball containing \(\varepsilon J\) for all sufficiently small \(\varepsilon\). Write
Assume \(F_p\in C^4\), \(j_p\in C^3\), \(j_p(0)=1\), \(Dj_p(0)=0\), and bounds \(\|D^kF_p\|\le L_k\) for \(1\le k\le4\), \(\|D^2j_p\|\le J_2\), and \(\|D^3j_p\|\le J_3\) on that ball. The derivative norms are Euclidean multilinear operator norms. With the verified kernel of Lemma 291, define
Then
This statement concerns the displayed coordinate construction. Applying it to the existing reconstruction requires a comparison of that reconstruction’s proper-time proxy, domain, and metric with this construction, with an error after the factor \(\varepsilon^{-D-2}\) that tends to zero.
Proof
Taylor expansion gives \(F_p(\xi)-F_p(0)=P_1(\xi)+P_2(\xi)+P_3(\xi)+R_4(\xi)\), where \(P_k\) is homogeneous of degree \(k\), \(P_2=\tfrac12D^2F_p(0)[\xi,\xi]\), and \(|R_4|\le L_4|\xi|^4/24\). Similarly,
Decompose the integrand as
The integrals of \(P_1\), \(P_3\), and \(P_1Q_2\) vanish by evenness. The remaining error is bounded in absolute value by
using \(|F_p-F_p(0)-P_1|\le L_2|\xi|^2/2\). After substituting \(\xi=\varepsilon\zeta\), the quadratic term is \(\eta^{\mu\nu}\partial_\mu\partial_\nu F_p(0)=\Box_g f(p)\), since the Christoffel symbols vanish at the normal-coordinate origin. The error yields the stated constant and power of \(\varepsilon\). \(\square\)
Measure temporal and spatial fitness differences
Rust engine experiment · Seed 7 · Poster after 11 experiment steps. Interactive view starts from the same seed.
6. Variance and an Explicit Scaling Regime#
The shrinking neighborhood gives us a smaller bias and a noisier estimate. Its volume is of order \(\varepsilon^D\). The difference \(f(y)-f(p)\) contributes one power of \(\varepsilon\), but a second-derivative estimator divides by \(\varepsilon^{D+2}\). Squaring and integrating these factors gives the variance scale below. Correlation enters through a separate multiplier, which must stay controlled as the neighborhood shrinks.
Lemma 293 (A covariance condition sufficient for A4)
Fix the deterministic evaluation point \(p\) and the coordinate construction of Lemma 292. Let \(Y_1,\ldots,Y_N\) have common law \(\pi=q\,d\mathrm{vol}_g\), with \(q\ge q_*>0\) on the kernel neighborhood and \(j_p\le J_0\) there. Define
Assume, for these summands and all \(N,\varepsilon\) under consideration,
with a constant independent of \(N\) and \(\varepsilon\). Then
The kernel moment is finite by boundedness of \(k\) and \(J\). Independent samples satisfy the covariance condition with \(C_{\mathrm{mix}}=1\). A stationary sequence satisfying \(|\operatorname{Cov}(H(Y_0),H(Y_\ell))| \le C_0e^{-\lambda\ell}\mathbb E H(Y_0)^2\) uniformly over this kernel family satisfies it with \(C_{\mathrm{mix}}=1+2C_0/(e^\lambda-1)\).
Proof
The expectation is the importance-sampling identity in Lemma 290. For the second moment, cancel one factor of \(q\) against the sampling law and use normal coordinates:
Expanding the variance of the average and applying the covariance hypothesis gives the claim. For a stationary sequence the diagonal sum is at most \(N\mathbb EH^2\); each lag \(\ell\ge1\) contributes at most \(2NC_0e^{-\lambda\ell}\mathbb EH^2\). Sum the geometric series. \(\square\)
Remark 258 (Dependence and conditioning in an application)
The covariance condition is imposed on the bandwidth-dependent observables, whose sizes grow as \(\varepsilon\) decreases. A fixed-observable law of large numbers, exchangeability, or a propagation-of-chaos statement without bandwidth-dependent constants is insufficient to verify it. A logarithmic Sobolev inequality for a measure yields dynamical information only after specifying a compatible generator and proving the required dissipation or correlation bound. It does not supply correlations for an arbitrary episode sampling protocol.
If \(p\) is itself one of the sampled episodes, the sampling and covariance hypotheses must hold for the remaining samples conditional on \(p\), with the appropriate sample count. Similarly, estimated weights and reconstructed distances require their own error estimates. The lemma uses exact \(q\) and a specified geometry. On an unbounded space its positive density bound is needed only on the local integration region.
Lemma 294 (A sufficient bandwidth schedule for A6)
Under Lemma 292 and Lemma 293, with constants independent of \(N\), choose
Here \(\ell_0\) is a fixed reference length. Then \(\varepsilon_N\to0\), \(N\varepsilon_N^{D+4}\to\infty\), and
Thus the estimator converges in probability. The choice \(\alpha=1/(D+6)\) balances these two bounds, giving a mean square error of order \(N^{-4/(D+6)}\) under these hypotheses.
Proof
The squared error decomposes into variance plus squared bias. Substitute the preceding bounds and the stated bandwidth. Both exponents are negative, since \(\alpha>0\) and \(\alpha(D+2)<1\). Also \(N\varepsilon_N^{D+4}=\ell_0^{D+4}N^{1-\alpha(D+4)}\to\infty\). Chebyshev’s inequality gives convergence in probability. Equating \(4\alpha=1-\alpha(D+2)\) yields \(\alpha=1/(D+6)\). \(\square\)
Predict held-out fitness across bandwidths
Rust engine experiment · Seed 7 · Poster after 11 experiment steps. Interactive view starts from the same seed.
7. Conditional Consistency and the Remaining Identifications#
We can now see exactly what the limit theorem costs. With the geometry and sampling law specified, a Taylor estimate controls the bias and a covariance estimate controls the fluctuations. To apply that calculation to recorded episodes, we still have to show that the reconstructed operator approximates the operator just analyzed. Making the bandwidth a function of the sample count is a choice in this calculation; it does not force an existing run to produce the required geometry or sampling law.
Corollary 116 (Conditional continuum consistency)
Assume the geometric and normal-coordinate hypotheses of Lemma 292, the exact normalized sampling and covariance hypotheses of Lemma 293, and the scaling of Lemma 294. Then the specified local estimator \(\widehat L_{N,\varepsilon_N}f(p)\) converges in mean square to \(\Box_gf(p)\) at each fixed interior evaluation point.
A reconstructed episode operator \(\widetilde L_N\) has the same limit in probability provided its density, metric, neighborhood, and kernel approximations additionally satisfy
This comparison is an explicit hypothesis for the reconstruction in Definition 692. Its verification must control errors after the normalization by \(\varepsilon_N^{-D-2}\).
For an action limit, suppose further that these mean square bounds hold uniformly on the compact support of \(f\), and let \(Z_1,\ldots,Z_M\) be independent samples with law \(q\,d\mathrm{vol}_g\), independent of the samples defining \(\widehat L\). Assume \(q\) is bounded below there and \(f\Box_gf\in L^2(q^{-1}d\mathrm{vol}_g)\). Then, as \(M,N\to\infty\),
This independent evaluation construction is a sufficient mathematical example. For the action evaluated on the same interacting episodes as the operator, the corresponding joint weighted error and quadrature estimates remain additional conditions.
Proof
The pointwise mean square conclusion is Lemma 294. Adding an error that tends to zero in probability preserves convergence in probability.
For the action, let \(a_N\to0\) bound the mean square operator error uniformly on \(\operatorname{supp}f\). Independence of the evaluation sample and Cauchy–Schwarz give
This error tends to zero by Markov’s inequality. The remaining independent sample average has the desired expectation and variance at most \((4M)^{-1}\int(f\Box_gf)^2/q\,d\mathrm{vol}_g\), which tends to zero. Combining the two estimates proves the action limit. \(\square\)
Remark 259 (Conditions to retain in downstream applications)
An application of the corollary retains the specified continuum geometry, exact sampling law or justified importance weights, verified cutoff moments, local coordinate bounds, and covariance estimates for shrinking neighborhoods. Graph reconstruction errors must vanish at the estimator’s normalization, and an action limit requires a joint integration estimate.
The bandwidth schedule is a sequence of analytic reconstructions. Identifying it with the algorithmic radius \(R_{\mathrm{loc}}=\min(\rho,\varepsilon_c)\) requires an explicitly specified family of runs and uniform estimates as these parameters change. Neither a finite sample size nor an empirical \(N^{-1/2}\) error for fixed observables forces that identification. The sample count here is the number of observations used by the estimator; time and particle correlations determine its effective information content through the covariance estimate.
The conditional results therefore support continuum calculations once these conditions are verified. Their verification for the full interacting episode process, its graph geometry, and its existing same-sample action remains open within this chapter.