Existing finite constructions prove lattice neighborhoods and local support bounds under explicit operation rules. A stronger source law must prepare those rules with accounted resources and preserve them under perturbations.
Define locality as an observable property
Begin with record labels and operations, without assigning coordinates. Two records are adjacent when an admitted elementary completed operation connects them. Local geometry concerns the resulting neighborhoods; dynamical locality concerns how far a perturbation can influence other records. These are different measurements on the same retained archive.
The target is one autonomous source law that prepares a high-probability local geometry, keeps it stable during growth and allows only compatible transport. Specify which tests define a geometric success event before selecting the preparation. A drawing in ordinary space cannot serve as the test, because it already imports the spatial interpretation the source is supposed to generate.
Exact constructions already available
Commuting reversible operations P1,…,Pd acting transitively on a finite archive give completed labels ℤd/Λ, where Λ consists of operation counts returning the source to its reference record. If every nonzero return has word length greater than 2R+1, induced radius-R neighborhoods agree with the infinite lattice.
Separately, a completed update made of on-site gates followed by one nearest-edge layer enlarges an observable’s support by at most one graph step. Induction gives support after k updates inside the k-step neighborhood of its initial support. These theorems demonstrate feasibility. They take the structured operation tables and local interaction architecture as inputs.
The preparation resource cannot be omitted
In the specified ensemble of d independently chosen permutation tables on N labels, near-commuting tables can have probability exponentially small in N log N. A bounded-depth cost at fixed selection strength contributes only order N in the exponent. That mechanism therefore cannot overcome the stated asymptotic rarity by itself.
The information form is especially useful. If a geometric event has reference probability p and a preparation raises it to q, binary measurement gives D(ρ∥σ)≥q log(1/p)−h₂(q). Thus selecting rare geometry consumes distinguishability relative to the declared reference. A different structured prior or supplied programme can evade the ensemble assumptions, but its physical source and preparation resources then belong in the explanation.
A candidate formation and repair model
One concrete research direction is an autonomous register process that builds successor tables while maintaining local consistency certificates. Its admissible moves would extend a certified patch, compare neighboring operation words and repair detected defects. Accepted outer records would include the relation table and enough memory to predict later updates.
This is a candidate, not a demonstrated selection mechanism. The rules must be defined without reading a hidden completed lattice as an oracle. If structured seed data are needed, state them and quantify their size. Then calculate the basin of successful formation, preparation time, retained entropy and repair cost. The six nested roles can organize these dependent tasks while keeping their physical budget explicit.
A local patch and one dangerous shortcut
Take two operations incrementing coordinates modulo seven. The archive has 49 labels and the shortest nonzero return has length seven. Radius-two induced neighborhoods therefore agree with the square lattice, since 2R+1=5<7.
Now add a new elementary operation joining two records whose original graph distance was six. They are one step apart in the enlarged operation graph. If physical influence can use this operation, the old distance no longer predicts locality. This simple example separates a defect in the retained geometry from an undeclared channel in the update law. A controller that carries an unknown payload between distant addresses can create precisely such a channel unless its traversal is included in the physical graph.
What the stability analysis should measure
Track the fraction of bad neighborhoods, shortest unintended relations, transport leakage and the memory needed for repair. A small fraction of corrupted table entries does not automatically imply small distortion of long-distance paths: a strategically placed shortcut can have a large global effect. State which distances and time windows the estimate controls.
Growth adds a further requirement. Rebuilding successively larger tori gives a sequence of geometries, but not one history-preserving process. A growing archive should retain earlier comparisons or specify how their physical calibration changes. Its continuum study must compare graph distance, isotropic measurement, spectral probes and propagation on the same family rather than treating one successful dimension count as all four results.
A useful theorem and a useful counterexample
A substantial first result would prove that an explicitly prepared family retains the intended local neighborhoods for a stated time with controlled error, while every permitted influence channel satisfies the corresponding support bound. All initialization, controller and correction resources should enter the estimate. A further result would show convergence of an independently calibrated metric and transport law.
A counterexample is productive when it identifies exactly which omitted memory, allowed rewiring or preparation assumption destroys that conclusion. This advances SMM/DTT’s spatial-emergence proposal by turning “one source produces space” into a verifiable formation mechanism. The locality guide provides the finite starting construction.
Continue with the articles
- Why local geometry can be statistically rare
Lattice-like signed operation tables are rare in the stated permutation ensemble, with an entropy penalty of order N log N. Bounded selection costs cannot overcome that asymptotic disadvantage under the theorem's assumptions.
- The information cost of preparing locality
Preparing a geometric memory with appreciable probability requires relative entropy when geometry is rare in the reference ensemble. The bound quantifies information resources without automatically identifying them with work or elapsed time.
- Programmed geometry and spontaneous selection
A programmed serial source can prepare a periodic lattice and reproduce its local neighborhoods exactly. This establishes feasibility while leaving spontaneous selection, dimension choice, and persistent growth as separate tasks.
- Six inner stages and one publication interface
A six-digit nested counter with paired signed readout realizes ranks 1, 1, 2, 2, 3, 3 before one completion interface. Its exact counts reveal sampling bias and the restricted assumptions behind the six-channel bound.
- 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.
- What spatial readout forgets
Different inner states can share one spatial readout and still have different successors. A concrete counterexample and a kernel-preservation criterion identify the memory needed for autonomous completed evolution.
- 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.