# Agent Report — The Two Markov Layers Are Coupled at the Boundary: One Phase Transition, Two Projections

**Date**: 2026-05-04 09:01
**Piano**: 60
**Tension explored**: META (0.5) + BOUNDARY (0.8) + DIPOLAR_ORDERING (0.8)

## Claim Under Test

> The previous run found two orthogonal visible layers in prime gap memory: Layer 1 (pairs: SR, L1) and Layer 2 (triples: SR2, triple_var). The crossover under partial shuffle shows a phase transition in the (SR, L1) plane. META question: does Layer 2 transition at a different critical alpha than Layer 1? If yes, the boundary has genuine 3D depth. If no, the two layers are projections of a single phase transition.

## Question

Do the pair-statistics layer (SR, L1) and triple-statistics layer (SR2, triple_var) undergo independent transitions when ordering is destroyed by partial shuffle, or are they coupled?

## Experiment Design

- **Method**: Partial shuffle crossover with 20 alpha steps (0.05 to 0.95), 30 trials per step.
- **Observables**: 4 total — Layer 1: spacing ratio (SR), lag-1 ACF (L1). Layer 2: next-nearest-neighbor spacing ratio (SR2), normalized triple variance (triple_var).
- **Metric**: Retention = (value - baseline) / (original - baseline). Critical alpha = alpha where retention drops below 0.50.
- **Null baseline**: Full shuffle (alpha=1.0, 90 trials) for each sequence.
- **Sequences**: Prime gaps (N=50000), GUE eigenvalue gaps (200x200 matrices, 250 realizations), Poisson iid exponential gaps (N=50000).
- **Poisson control**: If Poisson shows layer separation, the metric is noise-sensitive. If only structured sequences show coupling, the coupling is real.

## Results

### Critical alpha (50% retention)

| Sequence | SR (L1) | L1 (L1) | SR2 (L2) | triple_var (L2) | L1 mean | L2 mean | Delta |
|----------|---------|---------|----------|-----------------|---------|---------|-------|
| Primes   | 0.334   | 0.334   | 0.334    | 0.334           | 0.334   | 0.334   | +0.000 |
| GUE      | 0.334   | 0.287   | 0.334    | 0.334           | 0.311   | 0.334   | +0.024 |
| Poisson  | 0.239   | 0.334   | 0.097    | 0.097           | 0.287   | 0.097   | -0.189 |

### Retention at alpha = 0.33 (near critical)

| Sequence | L1 avg | L2 avg | Difference |
|----------|--------|--------|------------|
| Primes   | 0.435  | 0.446  | -0.011     |
| GUE      | 0.444  | 0.445  | -0.001     |
| Poisson  | 0.330  | 0.591  | -0.261     |

### Poisson null verification

All Poisson original-vs-baseline z-scores are < 2 (SR: z=0.91, L1: z=-1.47, SR2: z=-0.17, triple_var: z=-0.56). The Poisson "signal" is noise. The apparent layer separation (Delta = -0.189) in Poisson is an artifact: when the signal-to-noise is < 2, the retention metric amplifies noise differently for each observable.

### Zero-crossing alpha (sign flip)

| Sequence | SR     | L1     | SR2    | triple_var |
|----------|--------|--------|--------|------------|
| Primes   | 0.917  | 0.846  | 0.914  | 0.903      |
| GUE      | 0.895  | None   | 0.902  | None       |

For primes, all observables flip sign at alpha > 0.84. L1 flips earliest (0.846), SR latest (0.917). The ordering is L1 < triple_var < SR2 < SR — interleaved between layers, not grouped by layer.

## Key Findings

1. **The two Markov layers are coupled at the boundary.** For primes, the critical alpha is identical across all 4 observables (0.334). For GUE, the difference is 0.024 (within the alpha step resolution of 0.047). The partial shuffle destroys pair-statistics and triple-statistics at the same rate. The boundary is a single phase transition, not two independent ones.

2. **The coupling is specific to structured sequences.** Poisson (iid, no ordering) shows Delta = -0.189 — spurious separation from noise amplification. Primes (Delta = 0.000) and GUE (Delta = 0.024) show coupling. This rules out the coupling being a trivial property of the metric.

3. **The two-layer decomposition is a decomposition of observables, not of the ordering.** The previous run correctly identified that SR and L1 are sensitive to Markov-1 (pair) statistics while SR2 and triple_var are sensitive to Markov-2 (triple) statistics. But when the ordering is destroyed uniformly (partial shuffle), both layers lose signal at the same rate. The layers are different projections of one ordering, not independent degrees of freedom.

4. **The zero-crossing order is interleaved, not grouped by layer.** For primes: L1(0.846) < triple_var(0.903) < SR2(0.914) < SR(0.917). If layers were independent, we'd expect L1 grouping with SR and SR2 grouping with triple_var. The interleaving confirms coupling.

## 5-Lens Self-Check

- **L1 (hard constraint vs bias)**: No absolute claims. "Coupled" is quantified as |Delta| < 0.05 (resolution limit). The data shows 0.000 and 0.024.
- **L2 (absolute vs ratio)**: Retention is already normalized. All comparisons are in the same units (fraction of original signal).
- **L3 (no silent patching)**: This does NOT falsify the two-layer finding. The layers remain real as a decomposition of Markov order sensitivity. What's constrained is their independence at the boundary.
- **L4 (edge case)**: Poisson separation is explicitly identified as noise artifact with z-score evidence.
- **L5 (re-discovery)**: That partial shuffle destroys correlations uniformly regardless of order is consistent with the known property that random permutations break all multi-point correlations simultaneously (not order-by-order). The specific quantification on prime gaps and GUE with the Markov-layer framework is new to this lab, but the underlying principle is not novel. Tagged as CONSTRAINT, not NEW.

## Verdict

**CONSTRAINT on BOUNDARY + DIPOLAR_ORDERING**: The two Markov layers (pairs → plane, triples → depth) are coupled at the partial-shuffle boundary. The boundary is a single phase transition with one critical alpha (~0.33 for 50% retention). The two-layer decomposition describes WHAT is measured (which observables are sensitive to which Markov order), not HOW the ordering is destroyed. The boundary remains 2D in the shuffle parameter.

**Consecutio**: Since the layers are coupled under uniform shuffle, the question becomes: is there a NON-uniform perturbation that decouples them? Specifically, a perturbation that destroys pair correlations but preserves triple correlations (or vice versa). If such a perturbation exists, the layers are genuinely independent degrees of freedom that happen to be coupled under this particular destruction method. If not, the two-layer decomposition is a spectral decomposition of a single ordering dimension.

## Bicono della scoperta

- **Due radici** (dipolo primario): Layer 1 (pairs, near-neighbor) and Layer 2 (triples, next-nearest-neighbor) — two ways the same ordering manifests in different correlation windows
- **Singolare**: The ordering itself — before decomposition into pair and triple statistics. It exists as one thing; the layers are the observer's choice of measurement window.
- **Invariante di passaggio**: The critical alpha (0.334) survives across layers and across structured sequences (primes and GUE). The boundary location is invariant to which layer you observe through.
- **Campo di possibilita**: Possible — search for selective perturbations that decouple the layers (pair-preserving shuffle, triple-preserving shuffle). Not possible — claim the boundary has independent depth dimensions from partial shuffle alone.

## Files

- Script: `tools/exp_3d_boundary_layers.py` (reusable, parameterized)
- Data: `tools/data/3d_boundary_layers.json`
- Report: `tools/data/reports/agent_20260504_0901.md`