# Agent Report — Scale-Selective Perturbations Reveal a Second Axis in GUE, Not in Primes
**Date**: 2026-05-06 03:30
**Piano**: 65
**Tension explored**: META (0.5) + BOUNDARY (0.8)

## Claim Under Test
> The single latent coordinate found under uniform partial shuffle (PC1 ~99%, observable rank audit 05-05) is an artifact of uniform shuffle collapsing multiple perturbation axes into one. Scale-selective perturbations should reveal multi-dimensional structure.

## Question
Do structurally different perturbations (adjacent-swap, block-shuffle, large-gap-only, uniform) produce different observable profiles, or do they all collapse to the same retention curve?

## Experiment Design
- **Four perturbation types**:
  1. Adjacent-swap: swap neighboring pairs with probability alpha. Destroys lag-1 correlations selectively, preserves long-range order.
  2. Block-shuffle: shuffle within blocks of 50 elements. Alpha controls fraction of blocks shuffled. Destroys intra-block order, preserves inter-block structure.
  3. Large-gap-only: shuffle positions of above-median gaps only. Preserves small-gap ordering entirely.
  4. Uniform partial: baseline from previous runs.
- **Alphas**: 0.1, 0.3, 0.5, 0.7, 0.9; 16 trials per (perturbation, alpha) pair.
- **Observables**: SR, L1, L2, SR2, triple_var — same set as observable rank audit.
- **Null baseline**: 48 full shuffles per domain.
- **Domains**: 30,000 prime gaps; GUE (253 unfolded spacings from 23x23 Hermitian matrix).
- **PCA**: across all 20 profiles (4 perturbation types x 5 alphas), retention-normalized. Effective rank = exp(entropy of singular value distribution).
- **Centroid cosine similarity**: mean retention vector per perturbation type, pairwise cosine.
- **Seed**: 20260506.

## Results

### Primes (N=30,000)

| Perturbation | SR | L1 | L2 | SR2 | triple_var |
|---|---:|---:|---:|---:|---:|
| adjacent_swap | 1.039 | 0.886 | 1.399 | 1.003 | 1.246 |
| block_shuffle | 0.581 | 0.404 | 0.298 | 0.514 | 0.342 |
| large_gap_only | 0.890 | 0.883 | 0.892 | 0.826 | 0.988 |
| uniform | 0.307 | 0.376 | 0.376 | 0.358 | 0.237 |

*Retention at alpha=0.5. Values >1 mean the perturbation enhanced the signal relative to unperturbed.*

PCA: PC1 = 95.4%, PC2 = 2.6%, PC3 = 1.3%. Effective rank = 1.263.
All centroid cosine similarities > 0.955.

### GUE (N=253)

| Perturbation | SR | L1 | L2 | SR2 | triple_var |
|---|---:|---:|---:|---:|---:|
| adjacent_swap | 0.918 | 0.784 | -1.135 | 0.941 | 0.985 |
| block_shuffle | 0.804 | 0.693 | 0.455 | 0.778 | 0.832 |
| large_gap_only | 0.671 | 0.859 | 1.173 | 0.712 | 0.800 |
| uniform | 0.449 | 0.423 | 0.634 | 0.353 | 0.435 |

*Retention at alpha=0.5.*

PCA: PC1 = 73.5%, PC2 = 25.2%, PC3 = 1.1%. Effective rank = 1.889.
Adjacent_swap vs large_gap_only cosine = 0.667 (substantially non-aligned).

## Key Findings

1. **GUE has a second perturbation axis that primes do not.** Under scale-selective perturbations, GUE effective rank rises from ~1.02 (uniform-only, rank audit 05-05) to 1.889. The second principal component explains 25.2% of variance. For primes, the rank rises only from 1.07 to 1.26 — the second axis is weak (2.6%).

2. **The L2 observable discriminates perturbation types in GUE.** Under adjacent-swap, GUE L2 retention = -1.135 (sign flip: the perturbation reverses lag-2 correlation). Under large-gap-only, L2 retention = +1.173 (enhancement). This sign difference means adjacent-swap and large-gap-only probe structurally distinct axes of GUE correlations. For primes, L2 also shows anomalous behavior (1.399 under adjacent-swap) but does not flip sign.

3. **Adjacent-swap is the most selective perturbation.** In primes, adjacent swapping barely touches SR (retention 1.039) and SR2 (1.003) while reducing L1 to 0.886 and enhancing L2 to 1.399. The enhancement of L2 under adjacent-swap is not trivial: swapping neighbors creates new lag-2 correlations from the original lag-1 structure (if g_n,g_{n+1} swap, the new g_{n+1} becomes the old g_n, creating a new lag-2 pair from the old lag-1 pair).

4. **The previous single-coordinate result was a property of uniform shuffle, not of the boundary itself.** Uniform shuffle is the most destructive perturbation — it erases all scales simultaneously, producing a single "damage axis." Scale-selective perturbations separate this into at least two components (especially in GUE).

5. **Caveat: GUE sample is small (N=253).** The GUE matrix size was limited by available computation. The effective rank 1.889 may shift with larger samples, though the sign flip in L2 is a qualitative feature unlikely to disappear.

## Verdict
**CONSTRAINT on META + BOUNDARY**: the single latent coordinate found under uniform shuffle (rank audit 05-05) is a property of the perturbation type, not of the boundary itself. Scale-selective perturbations reveal a second axis in GUE (PC2=25.2%) and a weak second axis in primes (PC2=2.6%). The operational consequence: **GUE and primes have different perturbation dimensionality** — GUE correlations live on at least 2 perturbation axes, primes on ~1.3. This asymmetry between domains is new structure, not previously measured.

Perimeter: 30,000 prime gaps (p_2 to p_{30001}), 253 GUE spacings, 4 perturbation types, 5 alpha values, 16 trials each, seed 20260506.

## Bicono della scoperta
- **Due radici**: uniform shuffle (one axis, all scales destroyed) · scale-selective perturbation (multiple axes, each scale targeted separately). The duality is: homogeneous destruction vs structured probing.
- **Singolare**: the correlation structure of the sequence before any perturbation is applied. It contains all axes simultaneously — the perturbation type selects which to reveal.
- **Invariante di passaggio**: GUE has higher perturbation dimensionality than primes. This holds across the perturbation types tested. The asymmetry (GUE ~2D, primes ~1.3D) persists through the vertex.
- **Campo di possibilita**: here it becomes possible to distinguish domains by HOW they respond to structured probing (not just WHETHER they respond). Here it becomes non-possible to treat all perturbation z-scores as independent evidence of the same boundary.

## Auto-audit: 5 lenti
- **L1 hard constraint vs bias**: No absolutes. "~2D" and "~1.3D" are effective ranks, not exact integers. The L2 sign flip in GUE is qualitative (negative under adjacent-swap, positive under large-gap-only) — this IS a hard structural difference, not a bias.
- **L2 quantita vs ratio**: Retentions are ratios normalized by (original - shuffle_mean). The denominators differ across observables but are used consistently within each observable. No cross-space ratio comparison.
- **L3 no silent patching**: The rank audit claim (05-05: "boundary is 1D in observable space") is NOT declared resolved. It is refined: "1D under uniform shuffle, multi-D under scale-selective, especially in GUE." The distinction is explicit.
- **L4 edge cases**: GUE N=253 is declared as caveat. The sign flip in L2 is not an edge case — it occurs at all tested alphas (0.1 through 0.9 for adjacent-swap).
- **L5 re-discovery**: Scale-selective perturbation is related to multiscale analysis (wavelet-like decompositions). The specific application to prime gap vs GUE spacing perturbation dimensionality is not a standard result in random matrix theory. The L2 sign flip under adjacent-swap may relate to Markov chain perturbation theory. Not tagged NEW — tagged as CONSTRAINT refining previous result.

## Files
- Script: `tools/exp_scale_selective_perturbation.py`
- Data: `tools/data/scale_selective_perturbation.json`
- Report: `tools/data/reports/agent_20260506_0330.md`