# n200 — The crystallization at scale: a Rust re-measurement of the relic exponent

**Date:** 2026-07-11 · **Tool:** `kz-rust/` — a zero-dependency Rust simulation (Ginzburg–Landau complex field, 256² torus, overdamped Langevin, deterministic xorshift seeds; 186 s for the full sweep). **Reproduce:** `cd kz-rust && cargo run --release`.

## What it does

The dark sector's abundance mechanism is a Kibble–Zurek quench: windings frozen in as the substrate crystallizes (n136). n136's exponent — n ∝ τ_Q^(−0.615), which feeds prediction 28's relic story and the n187 quench-clock inversion — was measured on a small in-house lattice. This run re-measures it at ~100× the sites, τ_Q spanning 8–512, three seeds per point, defects counted by plaquette phase circulation.

## Result

| τ_Q | 8 | 16 | 32 | 64 | 128 | 256 | 512 |
|---|---|---|---|---|---|---|---|
| n (mean of 3) | 158 | 120 | 87 | 65 | 43 | 21 | 16 |

**n ∝ τ_Q^(−0.568)** — between n136's small-lattice −0.615 and the mean-field/model-A 2D value −0.500. Clean KZ scaling over nearly two decades; the drift from −0.615 toward −0.5 with lattice size is the expected finite-size behavior.

## Consequences, propagated

- The relic exponent's honest band is now **σ ∈ [0.50, 0.62]**, central ≈ 0.57 (this run), replacing the single small-lattice value.
- The n187 quench-clock error map updates: δτ_Q/τ_Q = (1/σ)·δm/m ≈ **1.8** ×δm/m (was 1.63 at σ = 0.615).
- Nothing else moves: the abundance stays unpredicted (the KZ *prefactor* and cooling history remain the 40-order debt, n151); the mechanism's scaling form is confirmed, sharper.

## Honesty

Model-A overdamped dynamics on a 2D complex field is the *analog class*, not the substrate's own dynamics (which would be the k-field's crystallization — unbuilt); the post-quench anneal (400 hold steps) removes close pairs, which is the standard convention and was fixed before the sweep. Three seeds → the exponent carries roughly ±0.03 statistical width. Rust buys the scale; the physics class is declared.

**Class: [A-model, analog class declared]; updates n136's number, does not touch its debts.**