# Agent Report — Transfer Matrix det(M) Drifts Toward Zero: Dipolar Structure Is Marginal, Not Dynamic
**Date**: 2026-04-12 03:30
**Piano**: 39
**Tension explored**: DUALITA_DIPOLARE_VS_ILLUSORIA (0.9)

## Claim Under Test
> Two types of duality: (1) dipolar — generative, det=-1, (2) illusory — dispersive, det=+1. The drift toward Poisson should correspond to loss of det=-1 structure.

## Question
If we fit a 2x2 transfer matrix M to prime gap dynamics (state-space embedding: (g_i, g_{i-1}) -> (g_{i+1}, g_i)), does det(M) drift from -1 toward +1 as primes grow?

## Experiment Design
- **Metric**: det(M) where M is the least-squares 2x2 transfer matrix mapping consecutive gap pairs
- **Scope**: 7 scale windows from p~100 to p~50M, 50K gaps each
- **Null baseline**: 20 shuffled-gap surrogates per window (same marginal distribution, destroyed order)
- **Fit**: det(M) vs ln(p), linear regression

## Results

| Scale (p~) | ln(p) | det(M)_real | det(M)_shuffled | z-score |
|------------|-------|-------------|-----------------|---------|
| 100 | 12.57 | -0.3880 | -0.3792 +/- 0.003 | -2.8 |
| 1K | 12.58 | -0.3882 | -0.3789 +/- 0.004 | -2.4 |
| 10K | 12.62 | -0.3876 | -0.3791 +/- 0.005 | -1.9 |
| 100K | 12.92 | -0.3877 | -0.3781 +/- 0.003 | -3.0 |
| 1M | 14.11 | -0.3809 | -0.3718 +/- 0.004 | -2.5 |
| 10M | 16.16 | -0.3810 | -0.3720 +/- 0.004 | -2.2 |
| 50M | 17.74 | -0.3707 | -0.3678 +/- 0.003 | -0.9 |

Linear fit: det(M) = -0.4255 + 0.00298 * ln(p), R^2 = 0.90

## Key Findings

1. **det(M) ~ -0.38, not -1.** The transfer matrix of prime gaps has negative determinant (dipolar) but far from unit magnitude. The eigenvalues are ~(+0.82, -0.47) — one attracting, one repelling with orientation flip.

2. **Shuffled gaps ALSO have det ~ -0.38.** The negative determinant comes almost entirely from the marginal gap distribution (the shape of the histogram), NOT from the sequential ordering. The dipolar signature is structural (built into the gap statistics), not dynamic (not from gap-to-gap correlations).

3. **The ordering adds a small extra negativity (delta ~ -0.009) that SHRINKS with scale.** At p~100K the excess is -0.010 (z=-3.0). At p~50M it's -0.003 (z=-0.9). The sequential anti-correlation that makes det(M) more negative than the marginal prediction is decaying — consistent with POISSON_CONVERGENCE and ACF_AMPLITUDE_SCALING.

4. **det(M) drifts toward zero (less negative), slope = +0.003/ln(p), R^2=0.90.** Both real and shuffled drift together. The marginal distribution itself is changing — gaps become more Poisson-like (wider, more symmetric) at larger primes.

5. **The dipolar-vs-illusory distinction maps to real-vs-shuffled, not to det sign.** Both real and shuffled primes have det < 0 (from marginals). The "dipolar excess" — the part that comes from ordering — is the small delta that decays. This is the generative structure the operator describes. It's real but small and shrinking.

## Verdict
**CONSTRAINT + NEW**

- **CONSTRAINT on DUALITA_DIPOLARE_VS_ILLUSORIA**: det(M) does not cleanly separate dipolar (det=-1) from illusory (det=+1). The real det is -0.38, not -1. The sign comes from gap marginals, not dynamics.
- **NEW**: The ordering-induced excess negativity (delta_det ~ -0.009) decays with scale (z from -3.0 to -0.9). This is a new observable that quantifies the "dipolar excess" — the gap between structure and randomness — and it converges to zero. Consistent with the 4-stage Poisson convergence hierarchy.
- **Consecutio**: The eigenvalue ratio |lambda_2/lambda_1| ~ 0.56 is close to 1/phi^2 = 0.382. Coincidence (C2) or structure? Measure on other anti-correlated sequences (Fibonacci mod, logistic map, etc.) to test universality.

## Files
- Script: `tools/exp_det_drift.py` (reusable with any gap sequence)
- Data: `tools/data/exp_det_drift.json`
- Report: `tools/data/reports/agent_20260412_0330.md`