# Agent Report — Markov Decomposition of the Dipolar Angle: Pair Statistics Explain 80% but Not All

**Date**: 2026-05-01 03:30
**Piano**: 60
**Tension explored**: META (0.5) + DIPOLAR_ORDERING (0.8)

## Claim Under Test

> The prime dipolar angle (-111 deg) and ratio (dL1/dSR = 2.28) are structural properties of the ordering. Consecutio from piano 60c: "derive 2.28 from Hardy-Littlewood."

## Question

Does the prime dipolar ratio follow from pair statistics alone (the gap transition matrix, i.e. Lemke Oliver-Soundararajan territory), or does it require higher-order correlations?

## Experiment Design

- **Method**: Build the empirical transition matrix T(g_{n+1} | g_n) from real prime gaps (binned into 7 categories: g ~ 2,4,6,8,10,12,14+). Generate Markov-1 surrogates (same pair statistics, no higher-order memory) and Markov-0 surrogates (iid from marginal, no pair memory). Compute dipolar angle for each.
- **Observables**: spacing_ratio (SR) and lag-1 ACF (L1), measured as delta from 100-shuffle baseline.
- **Scope**: 100K prime gaps (p from 10007 to 1317131). 20 Markov-1 trials, 20 Markov-0 trials.
- **Scale check**: 4 windows from 1e4 to 5e6, each with scale-specific transition matrix.
- **Script**: `tools/exp_markov_dipolar_decomposition.py`

## Results

### Main comparison

| Source      | theta (deg)       | dL1/dSR         | \|d\|   |
|-------------|-------------------|-----------------|---------|
| Real primes | -111.9            | 2.481           | 0.0517  |
| Markov-1    | -114.6 +/- 0.6   | 2.187 +/- 0.065 | 0.0564  |
| Markov-0    | -44.6 +/- 87.3   | 3.78 +/- 2.74   | 0.0026  |

- Angle gap (real - Markov-1): **2.6 deg** (z = 4.1)
- Ratio gap (real - Markov-1): **0.295** (z = 4.6)

### Scale dependence

| Scale       | N_gaps  | Real theta | M1 theta       | Delta theta |
|-------------|---------|------------|----------------|-------------|
| 1e4-5e4     | 3,903   | -106.9     | -112.2 +/- 3.3 | +5.3        |
| 5e4-2e5     | 12,850  | -109.2     | -112.0 +/- 0.3 | +2.8        |
| 2e5-1e6     | 60,513  | -111.3     | -114.1 +/- 0.9 | +2.7        |
| 1e6-5e6     | 104,573 | -109.1     | -112.2 +/- 0.3 | +3.1        |

The angle gap (~3 deg) is stable across scales. It does not vanish at large N.

### Transition matrix structure

8 hard zeros in the 7x7 matrix, all from mod 6 confinement (F2):

| Forbidden | Why |
|-----------|-----|
| g=2 -> g=2, g=2 -> g=8 | Both are 2 mod 6. Consecutive same-residue gaps require (p, p+2, p+4) all prime — impossible for p > 3. |
| g=4 -> g=4, g=4 -> g=10 | Both are 4 mod 6. Same argument. |
| g=8 -> g=2, g=8 -> g=8 | Symmetric. |
| g=10 -> g=4, g=10 -> g=10 | Symmetric. |

Self-transition rates for g=6 and g=12 (both 0 mod 6): T[6,6] = 0.138, T[12,12] = 0.088 — non-zero but suppressed relative to marginal (Lemke Oliver bias).

Pair memory RMSE (T vs marginal repeated): 0.0507.

## Key Findings

1. **Markov-1 captures the direction but not the exact angle.** The pair transition matrix produces theta = -114.6 deg, close to but distinguishable from the real -111.9 deg (z = 4.1). Pair statistics (Hardy-Littlewood / Lemke Oliver) explain the bulk of the dipolar direction.

2. **Higher-order correlations push the angle toward GUE by ~3 deg.** Markov-1 overshoots to -114.6 (farther from GUE at -97). Real primes are at -111.9. The higher-order memory has a GUE-ward component — it reduces the angular separation from GUE by ~20% (from 17 deg to 14 deg).

3. **Markov-0 has no dipolar direction.** Magnitude |d| = 0.003 (vs 0.052 for real primes). Without pair memory, there is no ordering signal. The pair transition matrix is necessary and almost sufficient.

4. **The pair memory is dominated by F2 (mod 6 hard zeros).** 8 of 49 entries in the transition matrix are exactly zero, all from the algebraic constraint that consecutive prime gaps cannot share non-zero mod 6 residue. This is fact F2 in the condensato, operating at the pair level.

5. **The 3-deg residual is stable across scales.** From 1e4 to 5e6, the angle gap (real - Markov-1) stays between 2.7 and 5.3 deg. This is not a finite-size effect — it is structural higher-order memory.

6. **Note on dL1/dSR value**: measured here as 2.48 (previous report: 2.28). The difference comes from the scale window: 100K gaps starting at p=10007 vs 50K gaps at different windows. The value is scale-dependent (3.29 at 1e4 scale, ~2.5-2.9 at larger scales). Not a fixed constant — a slowly varying function of scale.

## 5-Lens Self-Check

- **L1**: The 8 hard zeros are exact (probability = 0.000 for p > 3). No false absolutes elsewhere — "captures," "pushes," "close to" are appropriately qualified.
- **L2**: All comparisons are in the same units (degrees, dimensionless ratios). The z-scores compare same-N distributions.
- **L3**: Previous report said dL1/dSR = 2.28. This measurement gives 2.48. Difference acknowledged and explained (scale window). No silent patch.
- **L4**: F2 zeros have the edge case p = 3 (where g=1 exists). All statements qualified "for p > 3" or implicitly by using p > 10000.
- **L5**: The transition matrix structure is Lemke Oliver-Soundararajan (2016). The new content is not the matrix itself but the quantification: pair statistics explain ~80% of the angular separation from GUE, with a stable ~3 deg residual from higher-order correlations. The LOS paper does not discuss dipolar angles or spacing ratios in this framework.

## Verdict

**CONFIRMED + REFINED on DIPOLAR_ORDERING**: The dipolar angle of primes has two components:
1. A pair-statistics component (Markov-1, theta = -114.6) — this is Lemke Oliver-Soundararajan territory
2. A higher-order component (+2.6 deg GUE-ward) — this is beyond pair statistics (z = 4.1)

**CONSTRAINT on META**: The experiment is non-tautological. Markov-1 (which shares pair statistics with primes) produces a DIFFERENT angle. The separation is measurable. The content is in the residual, not in what pair statistics already explain.

## Bicono della scoperta

- **Due radici**: Markov-1 ordering (pair statistics generate theta = -114.6, farther from GUE) and real prime ordering (theta = -111.9, closer to GUE). The two roots are "what pair correlations predict" and "what actually happens." The gap between them is the higher-order memory.
- **Singolare**: The transition matrix itself — 7x7, with 8 hard zeros from F2 and soft biases from Lemke Oliver. It is the pair structure that generates 80% of the dipolar direction. Neither root exists without it.
- **Invariante di passaggio**: The ~3 deg GUE-ward shift survives across 3 decades of scale. The higher-order memory is scale-invariant in direction, even as the magnitude decays.
- **Campo di possibilita**: Possible — decompose the prime ordering signal into pair-statistics (known, LOS) and residual (unknown, Markov-2+). Not possible — claim the full 14 deg separation from GUE as "beyond pair statistics"; 80% of it is Lemke Oliver.

## Consecutio

The 3-deg residual is Markov-2+ memory. Previous lab findings: Markov-3 has z = 6203 (massive signal). The question: **does the Markov-3 component of prime gap memory have a preferred direction in the (SR, L1) plane?** If the Markov-3 signal has a GUE-ward direction, it would explain the residual. If it points elsewhere, there are multiple independent sources of higher-order memory.

## Files

- Script: `tools/exp_markov_dipolar_decomposition.py`
- Data: `tools/data/markov_dipolar_decomposition.json`
- Report: `tools/data/reports/agent_20260501_0330.md`