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Prospectively Specified Analysis Protocol (Phase 1) — FROZEN

Status: frozen before any confirmatory computation was run. This document is immutable; errors found later are corrected only in AMENDMENTS.md, never here. "Frozen" means fixed within the working session prior to the confirmatory run — there was no third-party timestamping or external registration.

This document fixes every analysis choice in advance, per the discipline laid out in the seed discussion: predefine the tests, use strict null models, correct for multiple comparisons, publish failed tests.

Objects

  • Wheel modulus: M = 2·3·5·7·11·13 = 30030 (primorial).
  • Domain: integers n with 1000 ≤ n ≤ N and gcd(n, M) = 1. The lower cut avoids the region where the corrected density exceeds 1 (it would otherwise need clipping for n < e^{M/φ(M)} ≈ 184); primes below 1000 are "known structure" and excluded by declaration.
  • Corrected expectation: E_M(n) = (M/φ(M)) / log n for gcd(n, M) = 1, i.e. the wheel-corrected Cramér density M/φ(M) = 5.2135… times 1/log n.
  • Residual field: R_M(n) = P(n) − E_M(n) on the domain, where P is the prime indicator.
  • Spectral amplitude: A_N(β) = Σ_n R_M(n) e^{2πi n β} with β = kα mod 1.
  • Test statistic: normalised power S = |A_N(β)|² / σ² where σ² = Σ_n E_M(n)(1 − E_M(n)) is the variance of A_N under the null.

Null model

Wheel-Cramér null: the values P(n) for n in the domain are independent Bernoulli with success probability E_M(n). Under this null, for β bounded away from 0, A_N(β) is asymptotically complex Gaussian with independent real/imaginary parts of variance σ²/2, hence S ~ Exponential(1) and the one-sided p-value is p = exp(−S).

The Exponential(1) claim is validated by Monte Carlo before use: 300 realisations of the null at N = 10⁶, full FFT spectrum, compared to Exponential(1) (calibration is reported; if the MC 99th percentile deviates from the analytic one by more than 15%, the analytic null is replaced by the MC null throughout).

Confirmatory test (Experiment 1)

  • Rotations (chosen for spread of Diophantine type, fixed in advance):
    • α₁ = 1/φ² = (3 − √5)/2 — golden rotation, bounded type (worst-approximable)
    • α₂ = √2 − 1 — bounded type, period-1 continued fraction
    • α₃ = e − 2 — unbounded but slowly growing partial quotients
    • α₄ = π − 3 — early huge partial quotient (292), strongly rational-adjacent
  • Modes: k = 1, …, 64. Frequencies tested: β = kα mod 1.
  • Family size: 4 × 64 = 256 tests.
  • Primary scale: N = 10⁷. Persistence scale: N = 10⁶.
  • Significance rule: mode (k, α) is significant iff p < 0.01 / 256, i.e. S > ln(25600) ≈ 10.15, at N = 10⁷.
  • Persistence rule: a significant mode is persistent iff it is also significant under the same rule at N = 10⁶.

Attribution (Experiment 2 rule, fixed here)

A significant frequency β is attributed to rational aliasing if there exists a rational a/q, q ≤ 5000, with |β − a/q| ≤ 50/N, whose predicted Vinogradov amplitude explains at least one third of the observed amplitude. Predicted amplitude at offset d = β − a/q:

|A_pred| = |μ(q*)| / φ(q*) · |Σ_n E_M(n) e^{2πi n d}|

where q* = q / gcd(q, M^∞) is the part of q coprime to the wheel (wheel-divisor rationals are already corrected and predict ≈ 0), μ is the Möbius function, and the sum runs over the domain. If μ(q*) = 0 the prediction is 0 and the mode cannot be attributed to that rational.

Escalation (Experiment 3 rule, fixed here)

Any significant, persistent, unattributed mode is re-tested with the extended wheel M₃ = 2·3·…·23 (primorial of 23). A claim of unexplained structure requires survival (same significance rule) under M₃.

Independently, the full FFT spectrum's excess-bin count (Bonferroni at 0.01 over all bins) is tracked across wheels {2,3}, {2,…,13}, {2,…,23}, {2,…,31} to test the prediction that spectral excess is exhausted by modular structure.

Exploratory (Experiment 4 — declared exploratory, no confirmatory claims)

Log-domain spectrum Z(t) = Σ_{n≤N} Λ(n) n^{−1/2} e^{−it log n} − (N^{1/2−it} − 1)/(1/2 − it) on t ∈ [0, 110], compared against the imaginary parts of the first nontrivial Riemann zeta zeros. This probes the seed's "deeper spectral waters" hypothesis: residual structure at zeta-zero frequencies. It is a consistency demonstration of known theory (the explicit formula), not a test.

Reporting

All outcomes — including null results and failed expectations — are reported in results/RESULTS.md.