Mean-Reversion, Spectral Valley, and the Projective Criterion for H′
ZFCρ Paper LII
Writing Declaration: This paper was independently authored by Han Qin. All intellectual decisions, framework design, and editorial judgments were made by the author.
Abstract
We establish the structural mechanism behind the spectral target identified in Paper 51 for Hypothesis H′. Three experiments reveal that the stripped prime-index process h(p) exhibits mean-reversion: the sub-block variance ratio peaks at intermediate scale then decreases (cleanly observed at j ≥ 29); the block periodogram at zero frequency is lower than at nearby nonzero frequencies; and the autocovariance appears not absolutely summable, with the spectral target maintained by massive positive-negative cancellation. One exact finite-block identity is proved (periodogram expansion), along with four second-order formulae (curvature criterion, variance-ratio drift, block-block covariance expansion, adjacent-block negativity equivalence) and an L²-localization bound from coboundary structure.
Two conditional closure theorems are established: (I) a Maxwell-Woodroofe projective criterion Δj ≤ K√j implies the spectral target in mean (E[Ij(0)] ≤ Cj · Dj) and an L²-version of BLα; (II) negative cross-product sum at the peak scale implies the spectral target deterministically. The remaining gap for H′ is reduced to one substantial dynamical conjecture — exponential forgetting of the stripped reset-state on O(j) scale — plus two technical upgrades (from mean to deterministic spectral target, and from L²-BL to deterministic BL). Thirty-six unconditional results at scales N = 10⁷ through 10¹⁰ support the framework.
Keywords: ZFCρ, integer complexity, mean-reversion, spectral valley, projective criterion, Maxwell-Woodroofe, martingale-coboundary, signed cancellation, exponential forgetting, H′, DSFα, BLα
§1 Three Discoveries
Paper 50 reduced H′ to DSFα + BLα. Paper 51 reformulated DSFα as a spectral target: Ij(0) ≤ Cj · Dj, quantified the gap (Cauchy-Schwarz gives mj · Dj; ratio mj/j = 2j/j² is exponential), and gave a divisor lemma for multiplicative reset. The present paper discovers the mechanism behind the spectral target.
(i) Mean-reversion. The sub-block variance ratio V(n)/(n · σ²x) peaks at intermediate scale n* then decreases. At j = 32 (190M primes, N = 10¹⁰): peak ratio/j = 54.1, full-block Oj/(j · Dj) = 2.7. Anticorrelation removes 95% of intermediate variance. The pattern is clean at j ≥ 29; at j ≤ 23 the process is in a crossover regime where anticorrelation has not yet separated from noise.
(ii) Spectral valley. The block periodogram at θ = 0 is a local minimum. At j = 32: Ij(k=0)/(j · Dj) = 2.7, while Ij(k=50)/(j · Dj) = 90.8. Zero frequency is specifically suppressed; the periodogram peak is at k ≈ 10–50, exceeding the zero-frequency value by 3–34×.
(iii) Signed cancellation. The autocovariance is not absolutely summable (Σ|C(k)| diverges), but Σ C(k) = O(j · σ²) through massive positive-negative cancellation. The shallow/deep decomposition at j = 32: at cutoff L = 1,000,000, shallow contribution = 23.53 · Oj, deep contribution = −22.53 · Oj. The framework requiring Σ|C(k)| < ∞ is inapplicable; the spectral target is maintained by signed cancellation, not by individual-lag decay.
§2 Exact Finite-Block Identities
Lemma 2.1 (Periodogram expansion). For x1, …, xmj and Ij(θ) = |Σ xr eirθ|²:
Ij(θ) = Γj(0) + 2 Σk≥1 Γj(k) cos(kθ)
where Γj(0) = Dj. At θ = 0: Ij(0) = Bj². Purely algebraic; no model assumptions.
Corollary 2.2 (Curvature criterion for spectral valley). Ij″(0) = −2 Σ k² Γj(k). Therefore θ = 0 is a strict local minimum of Ij(θ) if Σ k² Γj(k) < 0 — a k²-weighted signed condition on lag coefficients, requiring medium-to-large lags carry enough negative weight to overcome positive short-range contributions.
Lemma 2.3 (Variance-ratio drift). R(n+1) − R(n) = 2/[n(n+1)] · Σk=1n k C(k). The variance ratio R(n) = V(n)/n peaks at n* if and only if Σk=1n* k C(k) changes sign from positive to negative at n*.
Lemma 2.5 + Corollary 2.6 (Adjacent-block negativity equivalence). V(2n) = 2V(n) + 2 Cov(Tn,1, Tn,2). Therefore R(2n) < R(n) if and only if adjacent peak-scale blocks have negative covariance. The data confirms both simultaneously.
§3 Conditional Theorems
Theorem 3.1 (Algebraic cross-product bound — Conditional Theorem II). Partition x1, …, xmj into K sub-blocks of size n* with sums T1, …, TK. If (a) Σ Tk² ≤ A · j · Dj; (b) cross-product sum Σk<ℓ TkTℓ ≤ 0; and (c) remainder Trem² ≤ Dj; then Bj² = O(j · Dj). This is purely algebraic — no probability, no covariance, no model. Condition (b) is the deterministic analogue of "inter-block negative covariance."
Theorem 4.1 (MW projective criterion — Conditional Theorem I). Under a probabilistic model for the prime-indexed process, if the Maxwell-Woodroofe projective seminorm Δj = Σn≥1 ‖E(Sj,n | Fj,0)‖₂ / n3/2 satisfies Δj ≤ K√j uniformly in j, then E[Ij(0)] ≤ C · j · Dj (spectral target in mean).
Lemma 4.2 (Forgetting implies MW). If the hidden reset-state Yj,r has exponential forgetting with time Rj ≤ cj — i.e., supy,y′ ‖Gj(Yyj,n) − Gj(Yy′j,n)‖₂ ≤ A exp(−n/Rj) — then Δj ≤ K√j.
Proof. ‖E(Sj,n | Fj,0)‖₂ ≤ min(n, A·Rj). Summing over n: Δj ≤ C√Rj ≤ K√j. □
Proposition 4.3 (L²-BLα from coboundary). Under the MW hypotheses, with martingale-coboundary decomposition hi = Mi + Zi − Zi+1 satisfying sup E[Mi²] = O(1) and sup E[Zi²] = O(j) on block Ij: Σj≥0 sup|t|≤t₀ ‖Ej(t)‖₂ / |t|α < ∞ for every α < 1. This is an L²-version of BLα; upgrading to deterministic BLα requires a pathwise maximal estimate.
§4 The Remaining Gap
Conjecture 52.1 (Exponential forgetting on O(j) scale). There exists a hidden reflected state Yj,r, a measurable function Gj with h(pj,r) = Gj(Yj,r), and a coupling satisfying:
supy,y′ ‖Gj(Yyj,n) − Gj(Yy′j,n)‖₂ ≤ A · exp(−n/(cj))
with forgetting time Rj ≤ cj. The closure chain:
Conjecture 52.1 → Δj ≤ K√j [Lemma 4.2] → E[Ij(0)] ≤ Cj·Dj [Theorem 4.1] → (deterministic upgrade) → DSFα [Paper 51]
→ L²-BLα [Prop. 4.3] → (pathwise upgrade) → BLα
→ (with SD, unconditional [Paper 50]) → H′
The remaining gap has one substantial conjecture (52.1: connecting IC reservoir dynamics to prime-indexed process and proving geometric ergodicity) and two technical upgrades (mean-to-deterministic spectral target; L²-to-pathwise BL). If the underlying chain is geometrically ergodic with bounded observable, all three close simultaneously.
The excluded routes are extended: Route 5 (absolute summability of autocovariance) and Route 6 (direct spectral density bound) are both inapplicable, as the data strongly suggests Σ|C(k)| diverges. The appropriate framework is martingale-coboundary decomposition (Gordin), not spectral density theory.
§5 Data Tables
Table 1: Variance ratio V(n)/(n · σ²)/j at selected sub-block sizes (N = 10¹⁰).
| n | j=20 | j=23 | j=29 | j=32 |
|---|---|---|---|---|
| 1 | 0.050 | 0.043 | 0.034 | 0.031 |
| j² | 0.13 | 0.13 | 0.12 | 0.11 |
| 10,000 | 0.24 | 1.85 | 7.33 | 9.38 |
| 100,000 | — | 9.85 | 13.96 | 34.12 |
| peak | ~7.8 | ~25.3 | ~14.3 | 54.1 |
| full block | 21.9 | 25.3 | 8.8 | 2.7 |
Table 2: Periodogram Ij(θ)/(j · Dj) at low frequencies (N = 10¹⁰).
| freq index k | j=20 | j=26 | j=29 | j=32 |
|---|---|---|---|---|
| 0 | 22.0 | 4.2 | 8.9 | 2.7 |
| 10 | — | — | 29.2 | 55.2 |
| 50 | — | — | 15.9 | 90.8 |
| 100 | — | — | 5.4 | 19.8 |
Oj/(j · Dj) range across all tested j: 2.7 to 25.3, no systematic j-divergence (unconditional result 36).
§6 Summary of Results
New in Paper 52 (results 31–36): (31) variance ratio peaks then decreases at j = 29, 32; crossover/inconclusive at j ≤ 23; (32) periodogram at θ = 0 lower than at nearby nonzero frequencies at j = 29, 32; (33) peak intermediate variance up to 54× full-block variance at j = 32; (34) deep contribution is negative (anticorrelation) at all tested blocks; (35) autocovariance not absolutely summable, spectral target maintained by signed cancellation; (36) Oj/(j · Dj) bounded, range 2.7 to 25.3, no systematic j-divergence. Total: 36 unconditional results (Papers 50–52).
Proved results in this paper: curvature lemma; variance-ratio drift formula; block-block covariance expansion; adjacent-block negativity equivalence; MW projective criterion → spectral target in mean (Theorem 4.1); forgetting → MW (Lemma 4.2); negative cross-products → spectral target algebraically (Theorem 3.1); L²-BLα from coboundary (Proposition 4.3).
The proof strategy has shifted from "prove autocovariance decays fast" (Paper 51) to "prove anticorrelation is strong enough for signed cancellation" (Paper 52). The appropriate mathematical framework is Gordin's martingale-coboundary decomposition, which naturally accommodates non-absolutely-summable autocovariance through the telescope mechanism. The distance from ZFCρ to H′ is now: one substantial conjecture (IC dynamics forgetting time) and two technical upgrades (L² to pathwise) — all three likely closing simultaneously under geometric ergodicity.