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
Separate reconstruction of spatial adjacency from the requirement that subsequent dynamics remain local. Explain the guarded transport rules used to preserve the generated metric.
- A finite-update cone
Introduce the bound relating horizontal displacement to temporal projection in the limiting-speed draft. Explain that the finite-update inequality is a stated geometric assumption.
- A random clock behind dimensional flow
Introduce the stochastic-renewal model in which observed time counts irregular inner events. Explain how temporal intermittency changes a diffusion probe without changing the underlying spatial manifold.
- Subordination: diffusion on another clock
Introduce subordination as averaging ordinary diffusion over a random operational time. This gives a concrete bridge between a microscopic event parameter and observed evolution.
- Spectral dimension from return probabilities
Explain how diffusion return probabilities define a scale-dependent spectral dimension. In the draft’s specified regime, temporal scaling changes that diagnostic without directly counting spatial coordinates.
- A running temporal exponent
Explore the model’s use of a scale-dependent clock exponent to reproduce dimensional flow. Distinguish a useful parametrization from an independently derived microscopic running law.
- Why spectral dimension does not determine distance
Use the draft’s counterexample to show that an operator can have a useful eigenvalue count while giving infinite distances between localized states. Dimension and metric require separate checks.
- A finite regulator for spectral experiments
Outline a reproducible finite-model exercise comparing eigenvalue counts, heat traces, and graph distances. Use it to show which diagnostics converge together and which measure different features.
- What Lorentzian reconstruction leaves open
Summarize the boundary of the constraint-relative theorem. Recovering a causal geometry does not itself determine matter dynamics, the numerical limiting speed, or Einstein’s field equations.