# Agent Report — The Two Channels Have Opposite Boundaries: Algebraic Memory Is Scale-Invariant, Statistical Memory Decays

**Date**: 2026-04-29 08:33
**Piano**: 57
**Tension explored**: BOUNDARY (0.8) + META (0.5)

## Claim Under Test
> 8 domains GUE, 5 Poisson — the boundary is the third included (A9). Also: are we testing tautologies? (META)

## Question
The last 3 runs established: (1) algebraic memory is mod-3, (2) two-channel structure is real, (3) spectral rigidity is scale-dependent. Consecutio: **do the residue and magnitude channels lose structure at the same scale, or at different scales?** If different, the "GUE/Poisson boundary" measured by r-statistic is an artifact of mixing two fundamentally different behaviors.

## Experiment Design
- 500,000 primes (up to 7.4M), 28 logarithmically-spaced windows of 5000 gaps each
- In each window, decompose gaps into residue channel (Z/6Z binary: p mod 6 -> {+1,-1}) and magnitude channel (gap size demeaned by transition type)
- Four observables: r-statistic (combined), lag-1 ACF of residue channel, lag-1 ACF of magnitude channel, mod-3 self-transition fraction
- Null baseline: 20 shuffles per window (same marginal distribution, destroyed ordering)
- All results reported as z-scores relative to shuffle null

## Results

| Observable | z early (<100K) | z late (>100K) | Decay corr(ln p) | Boundary? |
|:-----------|:---------------:|:--------------:|:-----------------:|:---------:|
| r-statistic (combined) | -9.5 | -7.1 | +0.684 | Yes, slow |
| Residue ACF-1 | -30.8 | -35.1 | -0.190 | **No** |
| Magnitude ACF-1 | -5.2 | -4.6 | +0.455 | Yes |
| Mod-3 self-fraction | -33.3 | -33.9 | -0.104 | **No** |

Raw values across the range:

| Observable | at p~23K | at p~4.5M | Direction |
|:-----------|:--------:|:---------:|:---------:|
| r-statistic | 0.482 | 0.459 | Toward Poisson (0.386) |
| Residue ACF-1 | -0.192 | -0.135 | Weakens in absolute value |
| Residue z-score | -44 | -26 | **Still 26 sigma** |
| Magnitude ACF-1 | -0.062 | -0.055 | Nearly flat |
| Magnitude z-score | -4 to -7 | -2 to -6 | Fluctuates, weakly decaying |
| Mod-3 self-fraction | 0.404 | 0.433 | Rises toward 0.5 |
| Mod-3 z-score | -34 | -28 | **Still 28 sigma** |
| Shuffle mod-3 | 0.603 | 0.622 | Baseline also drifts |

## Key Findings

1. **The residue channel has no boundary.** Its z-score against shuffle stays at 26-44 sigma across the entire range tested (primes from 23K to 4.5M, a 200x span). The decay correlation with ln(p) is -0.190 — essentially zero or slightly strengthening. The mod-3 prohibition (self-transition fraction ~0.40 vs shuffle ~0.61) is equally strong at all scales.

2. **The magnitude channel has a boundary.** Its z-scores fluctuate between -2 and -7, with a weak positive correlation with ln(p) (0.455), meaning they drift toward the null at larger primes. This channel is what drives the r-statistic toward Poisson.

3. **The r-statistic mixes two incommensurable behaviors.** Its decay (correlation +0.684 with ln(p)) is dominated by the magnitude channel weakening, while the residue channel stays constant. The "GUE/Poisson crossover" reported by r-statistic is not a single phase transition — it is the magnitude channel approaching noise while the algebraic channel remains invariant.

4. **The shuffle baseline itself drifts.** Shuffled r-statistic decreases from 0.508 to 0.474 across the range. Shuffled mod-3 self-fraction increases from 0.603 to 0.622. The Brody calibration's warning about artifact floors applies: the signal must always be measured relative to the null, not in absolute terms.

## Verdict
**NEW + CONSTRAINT on BOUNDARY + META + DUALITA_DIPOLARE_VS_ILLUSORIA**

- **BOUNDARY**: There is no single boundary. The algebraic channel (residue, mod-3) has no boundary in the tested range. The statistical channel (magnitude) decays slowly. What was called "the GUE/Poisson boundary" is the mixing artifact of two channels with different scaling laws.

- **META**: The r-statistic is not a tautology, but it is insufficient. It collapses two fundamentally different phenomena (scale-invariant algebraic memory + decaying statistical memory) into one number. Any test based on r alone will see a "crossover" that is actually a mixing artifact. To test the real structure, decompose first.

- **DUALITA_DIPOLARE_VS_ILLUSORIA**: The residue channel is dipolar (det=-1): algebraic, scale-invariant, generated by the mod-3 prohibition. The magnitude channel approaches illusory (det=+1): statistical, decaying, approaching Poisson noise. The two types of duality coexist in the same sequence, separated by channel.

## Bicono della scoperta

- **Due radici** (dipolo primario): algebraic memory (residue channel, scale-invariant, mod-3 prohibition) / statistical memory (magnitude channel, decaying toward Poisson). One is permanent, the other transient. Both are real.
- **Singolare**: the prime gap sequence before decomposition — where the two memories are superposed and inseparable. The r-statistic lives here, and that's why it mixes.
- **Invariante di passaggio**: the channel decomposition itself. Whether you measure at p=23K or p=4.5M, the decomposition produces one algebraic and one statistical channel. The decomposition is the invariant; the boundary locations are the variables.
- **Campo di possibilita**: here it becomes possible to predict that no amount of scaling will eliminate the mod-3 structure (it's algebraic, not statistical); here it becomes non-possible to use r-statistic as a single order parameter for the "boundary."

## Files
- Script: `tools/exp_two_channel_boundary.py` (reusable with --n_primes, --window, --n_surrogates)
- Data: `tools/data/two_channel_boundary.json`
- Report: `tools/data/reports/agent_20260429_0833.md`