# Agent Report — The GUE-Poisson Boundary Is a Denominator Collapse Layer
**Date**: 2026-05-07 03:30  
**Piano**: 68  
**Tension explored**: META + BOUNDARY  
observables_registry: 1.0.0-2026-05-06  
observables_used: [SR, SR2, L1, L2, triple_var]

## Claim Under Test
The last three runs constrained perturbation rank and observable collinearity:
rank/PC2 claims are not interpretable without the canonical observable registry
and the original-vs-shuffle denominator gate.

This run does not repeat perturbation rank. It asks:

> If the GUE-Poisson boundary is simulated directly by controlled mixtures,
> does it behave like a clean two-class split, or like an operational third
> layer where classification is ambiguous and denominator support collapses?

## Experiment
Tool created: `tools/exp_boundary_mixture_gate.py`

Atomic perimeter:
- domains: synthetic unfolded GUE spacings, iid Poisson spacings, and mixtures;
- mixture parameter: `beta = 0.0..1.0`, where beta is the Poisson replacement fraction;
- main run: 1,536 spacings, 16 replicates, GUE matrix size 180, 11 beta layers, 24 full-shuffle baselines;
- seed check: 1,024 spacings, 12 replicates, GUE matrix size 160, same 11 beta layers, 20 baselines;
- denominator gate: observable is stable when `abs(original - shuffle_mean) / shuffle_std >= 2`;
- classification: standardized distance to pure GUE and pure Poisson centroids using all five canonical observables. A layer is marked ambiguous when at least half the replicates have nearest-centroid margin `< 0.15`.

The endpoint-gated classifier is intentionally reported. In this perimeter it is empty because the Poisson endpoint has almost no stable original-vs-shuffle denominators. That is not discarded; it is the core META result.

## Results

### Main Run

Endpoint separation using all canonical observables: `3.973` standardized units.  
Endpoint-stable observables at frequency >= 0.75 across both endpoints: `[]`.

| beta | stable obs / 5 | coord mean | margin mean | ambiguous fraction | Poisson-label fraction |
|---:|---:|---:|---:|---:|---:|
| 0.0 | 3.188 | -0.735 | 0.735 | 0.000 | 0.000 |
| 0.1 | 3.312 | -0.470 | 0.470 | 0.000 | 0.000 |
| 0.2 | 3.312 | -0.232 | 0.232 | 0.125 | 0.000 |
| 0.3 | 2.500 | -0.054 | 0.070 | 0.875 | 0.250 |
| 0.4 | 1.625 | +0.075 | 0.083 | 0.812 | 0.875 |
| 0.5 | 0.750 | +0.260 | 0.260 | 0.000 | 1.000 |
| 0.6 | 0.188 | +0.374 | 0.374 | 0.000 | 1.000 |
| 0.7 | 0.500 | +0.520 | 0.520 | 0.000 | 1.000 |
| 0.8 | 0.250 | +0.570 | 0.570 | 0.000 | 1.000 |
| 0.9 | 0.250 | +0.692 | 0.692 | 0.000 | 1.000 |
| 1.0 | 0.125 | +0.721 | 0.721 | 0.000 | 1.000 |

At beta 0.0-0.2, the sequence is classified as GUE-like and retains about
three stable observables. At beta 0.5-1.0, it is classified as Poisson-like,
but denominator support is mostly absent. The transition is not centered at
beta 0.5 in this observable suite. The ambiguous layer is beta 0.3-0.4.

Observable stability frequencies in the main run:
- beta 0.0: `SR=1.00`, `L1=1.00`, `triple_var=1.00`; `SR2=0.06`, `L2=0.12`;
- beta 0.3: `SR=1.00`, `L1=0.75`, `triple_var=0.50`;
- beta 0.4: `SR=0.75`, `L1=0.44`, `triple_var=0.38`;
- beta 1.0: all canonical observables are weak except one `L2` replicate frequency of `0.12`.

### Seed Check

The lighter seed check repeated the same ambiguous layer:
- ambiguous beta: `[0.3, 0.4]`;
- beta 0.3: margin `0.082`, ambiguous fraction `0.917`, stable obs mean `1.250`;
- beta 0.4: margin `0.125`, ambiguous fraction `0.750`, stable obs mean `0.833`;
- beta 0.5 and above: Poisson-label fraction `1.000`, ambiguity `0.000`.

## Findings

1. **The clean two-class boundary fails under denominator gating.** Pure GUE and pure Poisson are separable in all-observable space, but there are no observables stable at both endpoints under the declared gate. The Poisson pole is a weak-denominator pole: classification can still place it, but retention-normalized structural claims cannot use it as if it had the same denominator support as GUE.

2. **The operational boundary is a layer, not a line, in this synthetic perimeter.** Both the main run and the seed check isolate beta 0.3-0.4 as the ambiguous layer. In the main run the nearest-centroid margin falls to `0.070-0.083`, while ambiguous fraction rises to `0.812-0.875`. This is the measured form of the "third included" here: not a metaphysical third class, but a beta region where two-class assignment and denominator support are both unstable.

3. **Denominator collapse precedes full Poisson classification.** Stable-observable count drops from about `3.3` at beta 0.1-0.2 to `2.5` at beta 0.3 and `1.625` at beta 0.4. By beta 0.5 the classifier is fully Poisson-labeled, but only `0.750/5` observables remain stable on average. The loss of denominator support is therefore part of the boundary phenomenon, not an after-the-fact nuisance.

4. **The previous META constraints are extended, not replaced.** The 19:41 and 19:55 constraints still hold. This run adds that a boundary claim also needs a layer map: endpoint separability alone can hide the fact that one endpoint has no original-vs-shuffle denominator and that the transition region carries the actual instability.

## Verdict
**CONSTRAINT on META + BOUNDARY**: GUE/Poisson boundary claims must report:

> observables_registry version + canonical observable list + original-vs-shuffle z gate per observable + endpoint-stable observable set + beta/window layer where classification margin is ambiguous.

Scoped statement from this run:

> In the synthetic mixture perimeter tested here, the GUE-Poisson boundary is an operational layer at beta 0.3-0.4: classification is ambiguous there, and denominator support collapses across the transition. The Poisson endpoint remains classifiable but denominator-weak, so it cannot serve as a symmetric structural pole for gated retention claims.

## Consecutio
What opens now: apply the same layer map to real domains rather than only synthetic mixtures. For primes, the next discriminating question is not "GUE or Poisson?" but:

> Which scale window has the same signature as the synthetic beta 0.3-0.4 layer: low classifier margin plus falling original-vs-shuffle denominators?

If prime windows show such a layer, BOUNDARY becomes a measurable transition surface. If they do not, the synthetic result remains a calibration constraint on how not to over-read endpoint separability.

## Auto-audit: 5 lenti
- **L1 hard constraint vs bias**: no zero/always claim. "Endpoint gate is empty" means no observable reached frequency >= 0.75 across both endpoints under `abs(z) >= 2`; it does not mean the observables are identically zero.
- **L2 quantity vs ratio**: classification margin is reported together with stable-observable count and z-gate frequencies. Ratios are not interpreted without denominator support.
- **L3 no silent patching**: the claim is explicitly changed from "8 GUE, 5 Poisson boundary" to a synthetic mixture calibration. This does not assert the same layer for primes or all real domains.
- **L4 edge cases**: beta 0.2 has ambiguous fraction `0.125`, so it is not included in the ambiguous layer. The declared layer requires at least half the replicates ambiguous.
- **L5 re-discovery**: this is a finite-sample diagnostic of crossover and noisy denominator normalization in classical GUE/Poisson spacings. It is not tagged as a new RMT theorem.

## Files
- Script: `tools/exp_boundary_mixture_gate.py`
- Main data: `tools/data/boundary_mixture_gate_20260507_0330.json`
- Seed check: `tools/data/boundary_mixture_gate_20260507_0330_seedcheck.json`
- Report: `tools/data/reports/agent_20260507_0330.md`