# AI-Lab Report — 2026-04-05 09:14

## Tension Explored
**BOUNDARY** (0.7): "La transizione GUE->Poisson e' un effetto densita'. Ma i primi hanno eccesso di correlazione che CRESCE con la scala."

## Experiment: Boundary Excess Growth
**Question**: Does the excess `<r>_primes - <r>_Cramer` genuinely grow with scale?

**Method**:
- 5.76M primes up to 10^8
- 25 log-spaced windows of 50K primes each (from p~8K to p~99M)
- 3 baselines: Cramer model (10 trials each), shuffled gaps (10 trials each), and reference constants (GUE=0.5307, Poisson=0.3863)
- Metric: gap ratio `<r> = min(g_i, g_{i+1}) / max(g_i, g_{i+1})`

## Results

### Finding 1: Excess over Cramer is REAL but SHRINKS (PARTIAL FALSIFICATION)
- `<r>_prime - <r>_Cramer > 0` at ALL 25 scales (z-scores: 18 to 50)
- But slope = **-0.0027 per decade** — the excess SHRINKS with scale
- The claim "excess GROWS with n" is **falsified**
- The excess ranges from +0.048 (small scale) to +0.035 (large scale)

### Finding 2: Primes have ANTI-CORRELATED consecutive gaps
- `<r>_prime < <r>_shuffled` at ALL 25 scales (deficit ~0.018)
- Shuffling gaps (destroying order) INCREASES `<r>`
- This means consecutive prime gaps repel: small gap tends to follow large gap
- Lag-1 correlation: corr(g_i, g_{i+1}) ~ -0.03 to -0.06 (always negative)

### Finding 3: Anti-correlation WEAKENS with scale
- Slope of corr vs log10(p): +0.0065 per decade (R²=0.57, p=0.001)
- Primes drift from "weakly GUE-like" toward "Poisson-like" as scale increases
- `<r>` drops from 0.466 (p~8K) to 0.444 (p~99M)
- Both values are between Poisson (0.386) and GUE (0.531), closer to Poisson

### Finding 4: META check passes
- The gap-ratio statistic captures genuine sequential correlation, not just the gap distribution
- Evidence: `<r>_prime != <r>_shuffled` consistently (the order matters)

## Synthesis

The "boundary" between GUE and Poisson is not a static boundary — it's a **trajectory**.
Primes start closer to GUE at small scale and drift toward Poisson at large scale.
The Cramer model is always worse (less correlated) than primes, but primes are slowly approaching it.

The structural content is:
1. **Primes are more correlated than Cramer at every scale** (excess always positive, z >> 2)
2. **But the excess shrinks** — the gap between primes and Cramer is closing
3. **Consecutive gaps anti-correlate** — small-large alternation (Hardy-Littlewood territory)
4. **The anti-correlation weakens** — primes become more "random" at larger scales

## Tension Update
- BOUNDARY claim "excess GROWS with n": **PARTIALLY FALSIFIED**
  - The excess is real and persistent, but it shrinks, not grows
  - New claim: "Prime excess over Cramer is universal (all scales, z>18) but shrinks at -0.003/decade. Consecutive gaps anti-correlate (corr~-0.04) and this weakens at +0.006/decade. The trajectory is toward Poisson."

## What This Means for D-ND
The "third included" at the boundary is not a fixed point — it's the trajectory itself.
The primes don't sit at GUE or Poisson; they trace a path between them.
The rate of approach to Poisson (~0.003/decade in excess, ~0.006/decade in correlation) is itself a structural invariant worth investigating.

Is there a scale where primes actually reach Poisson? Extrapolating linearly: at ~10^20 the excess would be near zero. But linear extrapolation of a logarithmic quantity is unreliable.