A nearest-neighbour update gives an explicit continuum approximation and an exact positivity bound. Its dispersion exposes finite-resolution anisotropy, showing why dimensional counts, metric convergence and propagation must be checked independently.
A continuum is a relation between models
Increasing the number of records is not enough to define a continuum limit. One must specify which physical region is held fixed, how graph lengths and update durations change, and how observables at different resolutions correspond. Without that dictionary, two larger archives may describe different systems rather than better descriptions of one system.
For SMM/DTT, refinement should preserve the meaning of completed records and material comparisons. The six nested roles can organize finer generative work, but their extra operations must be reflected in memory and clock budgets. A limit that silently makes an ever-larger serial construction instantaneous changes the physical assumptions.
A concrete local evolution
Consider a scalar record value f on a cubic lattice of spacing a. A familiar nearest-neighbour diffusion update is obtained from the discrete Laplacian. Write κd for a positive diffusion coefficient to distinguish it from a dimension or a source frequency.
The right side uses only declared neighbours. It is therefore an example of dynamics compatible with a retained local graph. In a full source model this generator would need to follow from completed microscopic operations; here it provides a transparent test of what a continuum approximation actually establishes.
Deriving the leading continuum equation
For a smooth f, Taylor expansion of opposite neighbours cancels the odd powers of a. The second difference becomes ∂j²f plus a²∂j⁴f/12 and higher terms. Summing directions gives diffusion with an explicit leading correction:
The correction shrinks for variations whose length scale is much larger than a. This is a consistency calculation, not yet a complete convergence theorem. Such a theorem also needs controlled initial data, boundary conditions, a stable evolution and estimates over a declared time interval.
Worked example: a stable completed step
Use an explicit time step Δt and define r=κdΔt/a². The new value is (1−6r) times the old value plus r times each of its six neighbours. If 0≤r≤1/6, every coefficient is nonnegative and their sum is one.
Central weight = 1/4; each neighbour weight = 1/8.
The update preserves positivity, preserves total mass on a periodic lattice and cannot increase the maximum absolute value. These are exact stability properties of the discrete rule. If r exceeds 1/6, the simple nonnegative averaging interpretation fails. Reducing spatial spacing therefore requires a corresponding adjustment of time step for this scheme.
Finite resolution leaves directional corrections
Insert a plane-wave mode exp(ik·x) into the discrete diffusion generator. Its decay rate is known exactly:
=κd|k|²−(κda²/12)Σjkj⁴+O(a⁴|k|⁶).
The leading term depends only on |k| and is isotropic. The correction depends on direction: a fixed |k| along an axis has Σkj⁴=|k|⁴, while equal components along a diagonal give |k|⁴/3. This gives a quantitative diagnostic of lattice effects. A cubic dimensional count does not by itself guarantee exact rotational invariance at finite resolution.
Diffusion is not a relativistic signal law
Each discrete averaging step reaches at most one graph edge beyond existing support. The usual diffusion scaling takes Δt proportional to a², making the bound a/Δt diverge as a decreases. The continuum diffusion equation accordingly has nonzero tails arbitrarily far away at any positive time.
This is appropriate for a coarse-grained transport approximation, but it does not yield a finite fundamental causal speed. A relativistic SMM/DTT reconstruction must control a different joint limit of source activity, physical clock and matter propagation. Matching a diffusion exponent or obtaining a smooth equation cannot replace that causal analysis.
Check several observables on the same archive
A finite spectral benchmark can count integer modes n in a box and use the heat trace H(σ)=Σexp(−σ|n|²). Its scale-dependent heat dimension is −2d log H/d log σ. It tends to zero at extremely small σ because the finite mode count saturates, and at very large σ because the zero mode dominates.
An intermediate plateau near three becomes meaningful only when it persists under regulator changes with calibration fixed. On the same archive, calculate distances, neighbourhood defects and transport coefficients. These tests diagnose different properties. A successful continuum claim reports their common window of validity rather than selecting whichever plot resembles the desired dimension.
A robust target for SMM/DTT
The next model should hold a physical region and clock standard fixed while refining the retained archive. It should bound distance errors, control detector conditioning and quantify leakage beyond the intended transport neighbourhood. Small perturbations of source operations should produce controlled changes in these measurements.
That programme turns emergence into a reproducible comparison across models. Its eventual goal is a common limit in which spatial comparisons, material clocks and matter propagation agree because they descend from one source interaction. The finite examples already show which calculations can support that goal and which additional convergence and universality results remain to be established.
Continue with the articles
- Generating a neighborhood and preserving local transport
A generated graph predicts local influence only when later operations respect its edges and retained ports. Guarded transport supplies a finite-update support bound while exposing runtime and controller-memory requirements.
- A finite-update cone
A stated quadratic update bound produces a candidate cone and an integrated speed limit. Its interpretation as physical causality also requires calibrated readings and dynamics that keep every relevant signal inside it.
- A random clock behind dimensional flow
A renewal clock with heavy-tailed waits accumulates inner events sublinearly in observed duration. The explicit event-count model illustrates temporal intermittency while leaving the waiting law's physical origin open.
- Subordination: diffusion on another clock
Subordination averages ordinary evolution over a normalized random operational time. It preserves probability while allowing phase decay and memory in the observed-time description.
- Spectral dimension from return probabilities
A return-probability slope defines a spectral diagnostic only when the return observable is finite. The inverse-stable point-return integral diverges in two or more dimensions, so the broad scaling claim needs a specified finite probe.
- A running temporal exponent
A scale-dependent temporal exponent can describe a crossover without deriving its underlying clock process. A normalized kernel, integrated local slope, and regulator-dependent return calculation are needed to assess the model.
- Why spectral dimension does not determine distance
An operator can have cubic spectral growth while assigning infinite distances between localized evaluation states. The counterexample shows why record comparisons require commutator control beyond mode counting.
- A finite regulator for spectral experiments
Finite eigenvalue and heat-trace sums provide a reproducible test of spectral scaling. Comparing them with a separate graph-distance calculation exposes cutoff effects and prevents a numerical plateau from standing in for complete geometry.
- What Lorentzian reconstruction leaves open
Lorentzian reconstruction provides a conditional account of observer geometry without determining matter, coupling constants or gravitational dynamics. This guide identifies the source-level results still needed to connect clocks, propagation and curvature.