Multi-Scale Persistence
Patterns that stay recognizable across declared scales
Formal Definition
Axiom A3 is an interpretive postulate — one of five research hypotheses (dated 2025-12-23), not an empirically established law. As a working definition it treats a pattern x as persistent on a scale interval when its coarse-grained descriptions stay within a declared tolerance ε under specified coarse-graining operators Gs. Read this way, a candidate pattern counts as scale-persistent when it stays recognizable across the declared scales rather than being a scale-dependent artifact. The Axioms of Existence organize conceptual and narrative work; they do not replace the repository’s finite proofs, executable validation, or empirical evidence.
In the Hypercube
When the hypercube is driven by controlled symmetric bit-flip noise, state occupancy converges to uniform over all 512 vertices of F₂⁹, and the Hamming-weight marginal of that uniform occupancy is exactly Binomial(9, ½), C(9,k)/512 (mean 4.5, variance 2.25). Binning states by Hamming weight therefore reproduces the familiar bell-shaped histogram. This binomial envelope is a generic finite-hypercube baseline — it is reproduced by non-ASH controls and is not an emergent or ASH-specific result, and on its own it is not evidence that the ASH transforms cause it. It is used here as a control baseline, not as a demonstration of the axiom. With XOR transforms and no noise the dynamics instead stay confined to a 16-state code orbit on weights {0, 4, 8} and do not converge (TV ≈ 0.73 from the binomial). The Hamming-weight shells provide natural scales of observation.
Micro Scale
Under the controlled-noise limit every vertex is occupied with equal probability (~1/512); no vertex or region is preferred, so there is no micro-scale clustering. ASH transforms with no noise instead stay confined to a single 16-state code orbit on weights 0, 4, and 8 — a control showing the binomial spread is not produced by the transforms alone.
Macro Scale
Aggregating vertices into Hamming-weight shells yields the Binomial(9, ½) distribution, C(9,k)/512, whose two modal shells are weights 4 and 5 (each with 126 states). This is the generic uniform-hypercube baseline, reproduced by non-ASH controls — not an ASH-specific effect.
Scientific status: This page describes interpretive and computational ASH content, not empirically established physics. ASH (v1.1.0) proves specific finite mathematics and specifies deterministic computations; it is not a validated theory of cosmology or consciousness, and the Axioms of Existence are interpretive postulates, not laws.