# Agent Report — Crossover Universality: One Boundary or Many?
**Date**: 2026-04-06 07:14
**Piano**: 39
**Tension explored**: METRIC_TENSOR (0.9) + BRODY_CROSSOVER (0.85) + BOUNDARY (0.75) + GAP_ANTICORR (0.75)

## Claim Under Test
> All four observables (Brody beta, dR_acf1, gap_acf1, <r>) drift toward Poisson with scale.
> Is this ONE universal crossover ("il confine è il terzo incluso") or independent drifts?

## Question
Do beta, dR_acf1, gap_acf1, and <r> share structure beyond the trivial scale trend?
If their residuals (after removing ln(p) dependence) correlate, they measure the same boundary.

## Experiment Design
- 664,579 primes up to 10^7
- 40 log-spaced windows of 10,000 primes each
- For each window: Brody beta, dR_acf1, gap_acf1, <r>
- Linear fits vs ln(p), pairwise partial correlations (removing ln(p) trend)
- Null baseline: 15 Cramer surrogates
- Metric: z-scores for slopes, partial correlation significance

## Results

### Linear fits (all drift toward Poisson)
| Observable | Fit | R² | Direction |
|:-----------|:----|:---|:----------|
| beta | 0.670 - 0.025·ln(p) | 0.965 | toward 0 (Poisson) |
| dR_acf1 | -0.799 + 0.025·ln(p) | 0.850 | toward 0 |
| gap_acf1 | -0.164 + 0.008·ln(p) | 0.710 | toward 0 |
| <r> | 0.515 - 0.004·ln(p) | 0.918 | toward 0.386 (Poisson) |

### Z-scores vs Cramer (which drifts are prime-specific?)
| Observable slope | z-score | Verdict |
|:-----------------|:--------|:--------|
| beta | **-9.3** | PRIME-SPECIFIC (primes decorrelate 2.8x faster) |
| gap_acf1 | **3.9** | PRIME-SPECIFIC |
| r_mean | 1.8 | Marginal |
| dR_acf1 | **-0.5** | TAUTOLOGICAL (Cramer shows same drift: 0.026 vs 0.025) |

### Partial correlations (removing scale trend)
| Pair | raw | partial\|ln(p) | Signal? |
|:-----|:----|:---------------|:--------|
| gap_acf1 vs r_mean | -0.71 | **0.61** | YES — same structure |
| beta vs r_mean | 0.96 | **0.37** | Moderate |
| beta vs gap_acf1 | -0.85 | -0.26 | No |
| dR_acf1 vs gap_acf1 | 0.73 | -0.22 | No |
| beta vs dR_acf1 | -0.92 | -0.15 | No |
| dR_acf1 vs r_mean | -0.89 | -0.07 | No |

### Extrapolation to Poisson (linear in ln p)
| Observable | Poisson at p ~ | ln(p) |
|:-----------|:---------------|:------|
| gap_acf1 | 6 × 10^8 | 20.2 |
| beta | 4 × 10^11 | 26.7 |
| <r> | 5 × 10^13 | 31.6 |
| dR_acf1 | 8 × 10^13 | 32.1 (tautological) |

## Key Findings

1. **dR_acf1 drift is tautological.** Cramer surrogates show the same slope (z=-0.5). The de Sitter autocorrelation drift is an artifact of the g=(p/2)² mapping, not prime structure. This constrains METRIC_TENSOR: the Ricci scalar fluctuation captures scale geometry but NOT prime-specific content.

2. **Two genuine prime-specific drifts survive**: beta slope (z=-9.3) and gap_acf1 slope (z=3.9). Primes decorrelate toward Poisson faster than random (Cramer), and their gap anti-correlation is real.

3. **The crossover is NOT universal — it's multi-scale.** Gap correlations die first (~10^8), then repulsion strength (~10^11), then ratio statistics (~10^13). Three different Poisson horizons, not one.

4. **Two crossover channels exist:**
   - Channel A: gap_acf1 + <r> (partial r = 0.61) — correlation structure
   - Channel B: beta (partial r = 0.37 with <r>) — repulsion strength
   - dR_acf1: not a channel, just geometry

5. **The boundary is not "the third included" — it's a cascade.** Structure peels off in layers as scale increases. The anti-correlation (nearest-neighbor repulsion) dies first; the global repulsion statistics (<r>) persist longest.

## Verdict
**PARTIAL UNIVERSALITY** — confirmed with null baseline.

The prime-to-Poisson crossover is a multi-scale cascade, not a single transition. This falsifies any model (including naive de Sitter) that predicts a single crossover scale. The three Poisson horizons (10^8, 10^11, 10^13) encode a hierarchy of decorrelation — local structure dies before global structure.

New tension: **MULTI_SCALE_CASCADE** — the crossover has at least 2 genuine channels with different Poisson horizons separated by 5 orders of magnitude.

Constraint on METRIC_TENSOR: dR_acf1 is tautological (z=-0.5 vs Cramer). De Sitter geometry captures scale but not prime-specific content. The partial correlation between dR and beta (0.46 from previous exp) was inflated by shared scale dependence; after proper control, it drops to -0.15.

## Files
- Script: `tools/exp_crossover_universality.py`
- Data: `tools/data/reports/exp_crossover_universality.json`
- Report: `tools/data/reports/agent_20260406_0714.md`