# AI-Lab Report: Brody Crossover Law for Prime Gaps
**Date**: 2026-04-05 09:19
**Piano**: 39
**Tensions explored**: BOUNDARY (0.75) + METRIC_TENSOR (0.9)

## Claim Under Test
> Direction: "Esplorare il confine: 8 domini GUE, 5 Poisson — il confine è il terzo incluso"
> METRIC_TENSOR: "Il tensore metrico dei primi è g=(p/2)². Nel tempo ln(p), è de Sitter 1+1D."

## Question
Previous experiment showed primes drift from GUE toward Poisson with scale. **What is the functional form of this crossover?** The Brody distribution P(s) = (1+beta)*alpha*s^beta*exp(-alpha*s^{1+beta}) interpolates between Poisson (beta=0) and GOE (beta=1). What is beta(p) for primes?

## Experiment Design
- **Metric**: Brody parameter beta fitted via MLE to normalized gap spacings
- **Cross-validation**: gap ratio <r> computed in parallel
- **Scope**: 5,761,455 primes up to 10^8
- **Windows**: 30 log-spaced, 5000 primes each (from p~49K to p~100M)
- **Null**: 20 Cramer surrogates per window (exponential gaps)
- **Test**: Linear fit of beta_prime vs ln(p)

## Results

| Quantity | Value |
|----------|-------|
| beta_prime range | 0.227 to 0.416 |
| beta_Cramer range | 0.003 to 0.009 (≈ 0, as expected) |
| **Scaling law** | **beta = 0.606 - 0.0203 * ln(p)** |
| R² | 0.901 |
| F-statistic | 255.8 (p = 1.3e-15) |
| Pearson r | -0.949 |
| Spearman rho | -0.959 |
| z-score range | 30 to 102 |
| Mean z-score | 57.3 |
| All delta_beta > 0? | Yes (primes > Cramer at every scale) |
| Residual autocorrelation | 0.314 |
| Extrapolated Poisson (beta=0) | **p ~ 10^13** |

## Key Findings

1. **The crossover is logarithmic**: beta(p) = 0.606 - 0.020 * ln(p). Primes lose their "repulsion" at a rate of ~0.020 per e-fold in p. This is an extremely slow decorrelation.

2. **Primes are always above Cramer**: beta_Cramer ≈ 0 at all scales (pure Poisson, as expected). Primes have beta ≈ 0.23-0.42, solidly intermediate between Poisson and GOE.

3. **The crossover is NOT a phase transition**: there is no sharp boundary between GUE and Poisson regimes. The Brody beta decays smoothly and linearly in ln(p). The "boundary" is the entire range.

4. **Falsifiable prediction**: beta → 0 (Poisson) at p ~ 10^13. This is testable with segmented sieves.

5. **Intercept 0.606 is close to 1/φ = 0.618** (diff = 0.012, ~2%). Per C2 (coincidences are never proof), this is noted but NOT claimed. Could be finite-size effect. Would need data to 10^12 to constrain the intercept better.

6. **Slope -0.020 ≈ -1/50**: no obvious connection to known constants found.

## What This Means for METRIC_TENSOR

The METRIC_TENSOR claim says g=(p/2)² gives de Sitter in ln(p) time. If true, the "correlation decay" should follow 1/g^{1/2} ~ 2/p, which decays as e^{-τ} in τ = ln(p). But we measured beta decaying linearly in ln(p), not exponentially. **The Brody crossover is much slower than de Sitter would predict.** This is a constraint on METRIC_TENSOR: the decorrelation is logarithmic, not exponential.

## What This Means for D-ND

The "third included" at the GUE/Poisson boundary is not a point — it's a **crossover function**. The primes' level repulsion parameter decays as 0.606 - 0.020*ln(p), bridging two universality classes without belonging to either. This is structurally consistent with:
- The D-ND framework: the boundary between two poles (GUE/Poisson) carries its own structure
- F3: convergence is trivial; the content is in the rate
- The crossover function itself is the "third" — neither GUE nor Poisson, but a specific interpolation

## Verdict: NEW CONSTRAINT + PARTIAL FALSIFICATION

- **NEW**: Linear Brody crossover law beta = 0.61 - 0.020*ln(p) with R²=0.90
- **CONSTRAINT on METRIC_TENSOR**: decorrelation is logarithmic, not exponential (inconsistent with naive de Sitter prediction)
- **PREDICTION**: primes reach Poisson at p ~ 10^13

## Files
- Script: `tools/exp_brody_crossover.py`
- Raw data: `tools/data/reports/exp_brody_crossover_20260405.json`