Done. Summary of results:

**Experiment**: Does mod-3 ordering memory scale with position like Brody beta?

**Answer**: No. They are separate channels.

- **Brody beta** decays at -0.030/ln(p) (R²=0.78) — metric, statistical, weakening
- **Mod-3 self-transition** z-score is flat at -42 (slope ≈ 0, R²=0.0004) — algebraic, exact, eternal
- The non-zero mod-3 self-transition is **algebraically forbidden** (theorem: same non-zero residue would force p_{n+2} ≡ 0 mod 3). Only 1 violation at p=3.
- Markov depth M2/M1 = 0.472 is constant across all scales

**Three-layer decomposition** of prime gap structure:
1. **Magnitude** (PNT, decays) — gap distribution approaches exponential
2. **Statistical ordering** (Hardy-Littlewood/Lemke-Oliver, decays) — pairwise correlations weaken
3. **Algebraic ordering** (Z/3Z, exact) — mod-3 prohibition is a theorem, scale-invariant

The Brody beta flow only sees layers 1-2. Layer 3 is invisible to it. The GUE/Poisson boundary model needs an algebraic floor.

**Consecutio**: Does the algebraic layer exist for mod-5, mod-7? Is the M2/M1 = 0.47 ratio derivable?