# Agent Report — The Two Anti-Correlation Channels: Residue Dies First, Magnitude Persists

**Date**: 2026-04-19 03:30
**Piano**: 39
**Tension explored**: DIPOLAR_ILLUSORY_BOUNDARY (0.9), consecutio from agent_0418

## Claim Under Test

> Agent_0418 discovered residue lag-1 acf = -0.148 (3.8x magnitude acf1). Are these truly independent channels with different scaling laws and different Poisson crossover scales?

## Question

Decompose the prime gap anti-correlation into a **residue channel** (Z/6Z class sequence: which coset {1,5} each prime falls in) and a **magnitude channel** (gap size demeaned by transition type). Do they have independent scaling laws, power-law exponents, coherence lengths, and Poisson crossover points?

## Experiment Design

- **Data**: 6,000,000 primes up to 104,395,301
- **Decomposition**: For primes p > 3, p mod 6 is 1 or 5. Residue channel: +1 if p mod 6 = 1, -1 if p mod 6 = 5. Magnitude channel: gap size minus mean gap for each transition type (1->1, 1->5, 5->1, 5->5).
- **Scaling**: 15 log-spaced windows, ~400K gaps each, ACF computed per window
- **Null baseline**: 15 shuffled surrogates per window per channel
- **Metrics**: lag-1 ACF, power-law exponent alpha (|acf(k)| ~ A/k^alpha), linear scaling in ln(p), Poisson crossover extrapolation, coherence length, cross-channel correlation, variance partition

## Results

### Global decomposition (6M primes)

| Channel | acf1 | |ratio to full| |
|---------|------|-----------------|
| Full gaps | -0.0316 | 1.00x |
| Residue | -0.1220 | 3.86x |
| Magnitude | -0.0300 | 0.95x |

### Transition statistics (Z/6Z)

| Transition | Count | Mean gap | Std |
|------------|-------|----------|-----|
| 1 -> 1 | 1,316,744 | 19.00 | 14.89 |
| 1 -> 5 | 1,682,972 | 16.90 | 14.60 |
| 5 -> 1 | 1,682,972 | 15.41 | 14.74 |
| 5 -> 5 | 1,317,309 | 18.98 | 14.88 |

Same-class transitions (1->1, 5->5) have larger mean gap (19.0) than cross-class (16.2). This IS the residue anti-correlation: consecutive primes prefer to alternate between classes 1 and 5.

### Scaling laws — different crossover scales

| Channel | Slope (per ln p) | R2 | Poisson crossover |
|---------|-------------------|----|-------------------|
| Full | +0.00185 | 0.66 | p ~ 10^16.2 |
| **Residue** | **+0.00720** | **0.91** | **p ~ 10^14.9** |
| **Magnitude** | **+0.00153** | **0.56** | **p ~ 10^17.6** |

Residue decays 4.7x faster than magnitude. Separation: **2.7 decades** between crossover scales.

### Power-law exponents — different decay shapes

| Channel | A (amplitude) | alpha (exponent) |
|---------|---------------|------------------|
| Full | 0.035 +/- 0.009 | 0.96 +/- 0.19 |
| **Residue** | **0.058 +/- 0.009** | **1.30 +/- 0.10** |
| **Magnitude** | **0.034 +/- 0.009** | **0.95 +/- 0.19** |

The residue channel decays as ~1/k^1.3 (super-diffusive, steeper than 1/k). The magnitude channel decays as ~1/k^0.95 (slightly sub-1/k, more persistent). The observed 1/k law (ACF_1K_LAW, alpha=1.00, R2=0.998) is an **effective mixture** of two components with different exponents.

### Cross-channel independence

- Pearson(residue, magnitude) = 0.000 (exactly zero by construction: demeaning removes linear coupling)
- Magnitude acf1 within each transition type:
  - Same-class (1->1, 5->5): acf1 ~ -0.019
  - Cross-class (1->5, 5->1): acf1 ~ -0.009
  - Same-class magnitude anti-correlation is 2.1x stronger than cross-class

### Variance partition

| Component | Variance | Fraction |
|-----------|----------|----------|
| Full gaps | 220.4 | 100% |
| Transition means | 2.3 | **1.0%** |
| Magnitude residual | 218.1 | **99.0%** |

The residue channel carries only 1% of variance but 3.86x the anti-correlation. This is the signature of a **low-variance, high-correlation channel** — the binary choice of which Z/6Z class the next prime falls in carries almost no information about gap size but carries the strongest sequential memory.

### Coherence lengths

| Channel | L* |
|---------|-----|
| Full gaps | < 10 (always detectable) |
| Residue | 300 |
| Magnitude | < 10 (always detectable) |

The residue channel, despite having 3.86x stronger global anti-correlation, needs 300 gaps to be distinguished from shuffle in small windows. This is a sampling effect: a binary ±1 sequence has high ACF estimation variance in small windows. The magnitude channel, being continuous, is detectable at any window size.

## Key Findings

1. **The 1/k law is a mixture of two independent channels.** Residue: |acf(k)| ~ 0.058/k^1.3. Magnitude: |acf(k)| ~ 0.034/k^0.95. Combined, they produce the observed alpha = 1.00.

2. **The channels have different Poisson crossover scales, separated by 2.7 decades.** Residue channel (which Z/6Z class) loses memory at p ~ 10^14.9. Magnitude channel (gap size within class) loses memory at p ~ 10^17.6. The hierarchy is: residue dies first, magnitude persists.

3. **The residue channel carries 1% of variance but 3.86x the anti-correlation.** This is a low-energy, high-correlation signal — the binary alternation pattern 1↔5 is the strongest sequential structure in the primes, despite contributing almost nothing to the gap size.

4. **Same-class magnitude anti-correlation is 2.1x stronger than cross-class.** Within the transition types, magnitude memory is asymmetric: knowing the previous gap better predicts the next gap when both primes are in the same Z/6Z class.

5. **Residue scaling is clean (R2=0.91), magnitude is noisy (R2=0.56).** The residue channel has a well-defined scaling law; the magnitude channel's scaling is less determined at accessible scales, suggesting it may be contaminated by higher-order correlations not captured by the simple decomposition.

## Verdict

**NEW: TWO_CHANNEL_DECOMPOSITION**

The prime gap anti-correlation is not a single phenomenon. It decomposes into two independent channels with:
- Different amplitudes (ratio 3.86x)
- Different decay shapes (alpha 1.30 vs 0.95)
- Different scaling rates (slope ratio 4.7x)
- Different Poisson crossover points (2.7 decades apart)

The residue channel is intense but short-lived (dies at 10^14.9). The magnitude channel is weak but persistent (dies at 10^17.6). The observed 1/k law is an effective mixture that masks this dual structure.

## Consecutio

- **Derive the crossover separation analytically**: The 2.7-decade gap between residue and magnitude Poisson crossovers should follow from PNT + Hardy-Littlewood. The residue decay rate (+0.0072/ln p) may connect to the prime race bias (Chebyshev).
- **The same-class asymmetry**: Magnitude acf1 is 2.1x stronger in same-class transitions. Why? This is a three-point correlation (class_i, gap_i, gap_{i+1}) that goes beyond pairwise.
- **Test on other multiplicative sequences**: Does any sequence with a natural Z/mZ structure show this two-channel decomposition? If the decomposition is generic for any anti-correlated sequence with residue structure, it constrains C1 (primes unique under M).

## Files

- Script: `tools/exp_two_channel_decomposition.py` (reusable, parametric)
- Data: `tools/data/exp_two_channel_decomposition.json`
- Report: `tools/data/reports/agent_20260419_0330.md`