# Agent Report — The Two-Channel Decomposition Is Half Algebraic, Half Number-Theoretic

**Date**: 2026-04-19 07:55
**Piano**: 39
**Tension explored**: TWO_CHANNEL_DECOMPOSITION (0.85), consecutio

## Claim Under Test

> Agent_0419_0330 found two independent anti-correlation channels in prime gaps: residue (Z/6Z class, acf1=-0.122, alpha=1.30) and magnitude (gap size within class, acf1=-0.030, alpha=0.95). The same-class magnitude acf1 is 2.1x stronger than cross-class. Is this structure unique to primes, or generic for any Z/6Z-structured sequence?

## Question

Construct synthetic sequences with Z/6Z structure but without prime number theory. Which features of the two-channel decomposition survive?

## Experiment Design

- **Data**: 2,000,000 primes up to 32,452,843
- **Models**:
  - **PRIMES**: Real prime gaps (baseline)
  - **MARKOV**: Z/6Z Markov chain with same transition probabilities as primes, gap sizes drawn i.i.d. from empirical distribution per transition type. Has Z/6Z structure but NO sequential memory in magnitudes.
  - **CLASS_SHUFFLE**: Real gap sizes in real order, Z/6Z class assignments randomized. Has magnitude ordering but NO Z/6Z structure.
  - **MAG_SHUFFLE**: Real Z/6Z class sequence from primes, gap sizes shuffled within each transition type. Has Z/6Z structure but NO magnitude ordering.
- **Null baseline**: 20 trials per synthetic model
- **Metrics**: residue acf1, magnitude acf1, power-law exponents (alpha), same-class vs cross-class magnitude acf1, asymmetry ratio

## Results

### Summary table

| Model | acf1_res | acf1_mag | alpha_res | same_cl | cross_cl | asym |
|-------|----------|----------|-----------|---------|----------|------|
| **PRIMES** | **-0.1296** | **-0.0321** | **1.236** | **-0.0191** | **-0.0091** | **2.10x** |
| MARKOV | -0.1293 | -0.0001 | 1.602 | -0.0004 | -0.0006 | noisy |
| CLASS_SHUFFLE | +0.0000 | -0.0340 | 0.007 | -0.0176 | -0.0069 | 2.62x |
| MAG_SHUFFLE | -0.1296 | -0.0000 | 1.236 | +0.0002 | +0.0000 | noisy |

### What each model reproduces

| Feature | MARKOV | CLASS_SHUFFLE | MAG_SHUFFLE | Origin |
|---------|--------|---------------|-------------|--------|
| Residue acf1 = -0.13 | YES | no | YES | **Algebraic** (transition probs) |
| Residue alpha = 1.24 | no (1.60) | no | YES | **Number-theoretic** (prime race) |
| Magnitude acf1 = -0.032 | no | YES | no | **Number-theoretic** (gap ordering) |
| Same-class asymmetry 2.1x | no | YES (2.6x) | no | **Sub-sampling geometry** |

## Key Findings

1. **The residue channel lag-1 amplitude is algebraic.** MARKOV reproduces acf1_res = -0.129 exactly (z = -0.3 vs primes). The Chebyshev bias — primes at 1 mod 6 and 5 mod 6 prefer to alternate — determines the amplitude. No prime number theory needed beyond transition probabilities.

2. **The residue channel decay shape is number-theoretic.** MARKOV gives alpha_res = 1.60 (exponential-like Markov decay fit as power law). Primes give alpha_res = 1.24. The Markov chain has only lag-1 memory; the prime Z/6Z sequence has LONG-RANGE correlations. This long-range memory in the class sequence connects to the prime race and zeros of Dirichlet L-functions — the class sequence remembers its history beyond pairwise transition probabilities.

3. **The magnitude channel is entirely number-theoretic.** Neither MARKOV (i.i.d. magnitudes) nor MAG_SHUFFLE (shuffled within class) reproduces the magnitude anti-correlation. Only the REAL sequential ordering of gap sizes produces acf1_mag = -0.032. This is the Hardy-Littlewood correlation: consecutive gap sizes repel, and this repulsion is in the specific ordering, not in the marginal distributions.

4. **The 2.1x same-class asymmetry is NOT a three-point correlation.** CLASS_SHUFFLE, which has RANDOM classes (no Z/6Z memory), produces a 2.62x asymmetry (z = -1.0 vs primes). This means the asymmetry arises from the geometry of sub-sampling a correlated sequence, not from class-specific number theory. When an anti-correlated sequence is split into subsets of different sizes (~22% same-class vs ~28% cross-class), the smaller subsets naturally show stronger effective ACF because consecutive elements are more likely to have been adjacent in the parent sequence.

5. **The decomposition into two independent channels is generic** — any Z/6Z-structured sequence admits it. What is prime-specific is:
   - The residue channel having power-law (not exponential) decay → long-range class memory
   - The magnitude channel having any anti-correlation at all → gap size ordering
   - Both channels being simultaneously active → primes carry two independent forms of sequential memory

## Verdict

**CONSTRAINT on TWO_CHANNEL_DECOMPOSITION + partial support for C1**

The two-channel decomposition framework is algebraic (any Z/6Z-structured sequence admits it). But the CONTENT of each channel is number-theoretic:

- Residue: amplitude algebraic, decay shape number-theoretic (long-range prime race memory, alpha=1.24 vs Markov 1.60)
- Magnitude: entirely number-theoretic (gap size ordering)
- Same-class asymmetry: geometric artifact, not three-point correlation — FALSIFIED as independent structure

The primes are not unique in HAVING two channels (any Z/6Z sequence does). The primes are unique in having long-range memory in BOTH channels simultaneously.

## Consecutio

- **Derive alpha_res = 1.24 from Dirichlet L-function zeros**: The power-law exponent of the Z/6Z class sequence ACF should connect to the distribution of zeros of L(s, chi_3) (the non-principal character mod 6). The Markov alpha=1.60 is the reference for "no long-range memory."
- **The magnitude anti-correlation is the core prime signature**: Since the residue amplitude is algebraic, the irreducible number-theoretic content is in (a) residue decay shape and (b) magnitude anti-correlation. The magnitude channel, carrying 99% of variance, is where the Hardy-Littlewood correlations live.
- **Test at larger scales**: Does the alpha_res gap (1.24 vs 1.60) change with prime scale? If it narrows toward Poisson, the long-range class memory also has a crossover.

## Files

- Script: `tools/exp_two_channel_universality.py` (reusable, parametric)
- Data: `tools/data/exp_two_channel_universality.json`
- Report: `tools/data/reports/agent_20260419_0755.md`