# AI-Lab Report: Excess Correlation Scaling
**Date**: 2026-04-05 09:16
**Piano**: 39
**Tension explored**: BOUNDARY (0.7)

## Claim Under Test
> "I primi hanno eccesso di correlazione (z medio=2.6, max=5.6) che CRESCE con la scala."

## Experiment Design
- **Metric**: Gap ratio statistic `<r> = min(g_i, g_{i+1}) / max(g_i, g_{i+1})`
- **Scope**: 5,761,455 primes up to 10^8
- **Windows**: 30 log-spaced, 5000 primes each (from p~32K to p~100M)
- **Null**: 20 Cramer surrogates per window (exponential gaps, same density)
- **Test**: Linear fit of `delta_r = <r>_primes - <r>_Cramer` vs `log(p_center)`

## Results

| Quantity | Value |
|----------|-------|
| Mean delta_r | 0.0582 |
| Mean z-score | 14.89 |
| Min z-score | 8.39 |
| Slope delta_r vs ln(p) | **-0.00237** (NEGATIVE) |
| Pearson r | -0.890 (p = 4.5e-11) |
| Spearman rho | -0.894 (p = 2.8e-11) |
| <r> small primes (p~32K) | 0.476 |
| <r> large primes (p~100M) | 0.444 |
| <r> Cramer small | 0.407 |
| <r> Cramer large | 0.394 |

## Verdict: PARTIAL FALSIFICATION

The excess correlation is **real** (z >> 2 at every scale), but it does **NOT grow** with scale. It **shrinks logarithmically**:

- delta_r decreases from ~0.070 (small primes) to ~0.049 (large primes)
- The correlation between delta_r and log(p) is very strong: r = -0.89
- Both <r>_primes and <r>_Cramer decrease, but primes decrease faster
- Primes move from near-GUE (0.476) toward intermediate (0.444), never reaching Poisson (0.386)

## Interpretation

The previous claim "il gap CRESCE con n" is falsified. The corrected picture:

1. Primes genuinely have more short-range correlation than Cramer (Hardy-Littlewood effect)
2. This excess is persistent (still z~13 at p=10^8) but slowly decaying
3. The GUE/Poisson classification is an oversimplification: primes sit between them, drifting toward Poisson
4. The structural content is NOT the boundary or its direction, but the **persistence** of the excess across 4 orders of magnitude

## What This Means for D-ND

The prime gap structure is NOT a fixed universality class. It's a slowly evolving system that retains memory of its correlations (Hardy-Littlewood) but asymptotically loses them. This is consistent with:
- F3 (convergence to phi is trivial) — the interesting structure is the *rate* of approach
- The metric tensor claim (METRIC_TENSOR) — g=(p/2)^2 growing implies gaps decorrelate

## Next Steps
- Test whether the decay rate (-0.0024/ln(p)) matches any Hardy-Littlewood prediction
- Verify METRIC_TENSOR claim independently (highest intensity tension, 0.9)

## Files
- Script: `tools/exp_excess_scaling.py`
- Raw data: `tools/data/reports/exp_excess_scaling_20260405.json`