Exponential Block Decay, Spectral Reformulation, and the Periodogram Target for H′
ZFCρ Paper LI

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 advance the closure of Hypothesis H′ in the ZFCρ framework with four results. First, the stripped dyadic block sums B_j decay exponentially: the rescaled quantity 2^j · B_j² is bounded (O(1)) across j = 10 to 33 at N = 10¹⁰. This exponential rate is a factor of ~j above pure white noise (which predicts |B_j|² ~ D_j ~ 1/(j · 2^j)); the excess comes from off-diagonal correlations (O_j/D_j ~ j). Paper 50's reported "2^−j/3" is corrected as a comparison artifact. The exponential rate is invariant under conductor stripping Q = 2 to 360, confirming the correlation is IC-intrinsic. Second, we give an exact finite-block spectral reformulation: the condition I_j(0) ≤ Cj · D_j (periodogram at zero bounded by j times its Parseval average) implies |B_j|² ≤ C/2^j, hence DSF_α for all α > 0. This condition decomposes via a window inequality into a low-frequency mass bound and a spectral continuity condition. Third, the gap is quantified: Cauchy-Schwarz gives I_j(0) ≤ m_j · D_j; the target is I_j(0) ≤ Cj · D_j; the ratio is m_j/j = 2^j/j² — an exponential improvement is needed. Fourth, a divisor lemma on factor-subtrees gives the first quantitative bound on multiplicative reset. Thirty unconditional results at scales N = 10⁷ through 10¹⁰ support the framework.

Keywords: integer complexity, exponential block decay, spectral reformulation, periodogram target, dyadic block sums, DSF, divisor lemma, off-diagonal dominance, conductor invariance, ZFCrho, H-prime

Correction of Paper 50

Paper 50 reported the empirical block decay rate as |B_j| ~ 2^−j/3. This paper corrects it: the block sums decay exponentially at rate ~2^−j. The diagnostic 2^j · B_j² is O(1) with no systematic growth (median ≈ 5, range 0.3–10, N = 10¹⁰). Paper 50's "2^−j/3" arose from comparing |B_j| against √D_j without accounting for the O_j/D_j ~ j factor. The ratio |B_j|/√D_j ~ √j was misinterpreted as a sub-WN decay rate. The exponential rate is conductor-invariant (OLS slopes −0.502 to −0.503 across Q = 2, 12, 60, 360), confirming the correlation is IC-intrinsic.

Off-diagonal dominance grows as O_j/D_j ~ j: at j=10, O_j/D_j = 36; at j=20, 439; at j=33, 1351. This is consistent with |B_j|² ~ C/2^j and D_j ~ C′/(j · 2^j) — the ratio |B_j|²/D_j ~ Cj/C′. The linear growth is not anomalous — it is the signature of moderate short-range correlations that boost block variance by a polynomial factor above the independent-summand baseline, without altering the exponential rate.

Spectral Reformulation and the Gap

For x_r = h(p_j,r)/p_j,r, define the block Fourier transform F_j(θ) = Σ_r x_r e^irθ and periodogram I_j(θ) = |F_j(θ)|². Two exact, deterministic, model-free identities hold: I_j(0) = B_j² (definition) and (1/2π)∫I_j(θ)dθ = D_j (Parseval).

Spectral Target: There exists C > 0 such that I_j(0) ≤ C · j · D_j for all j. This is a j-linear spike bound: among the m_j ~ 2^j/j terms in block j, the target requires at least ~2^j/j² to behave as effectively independent. If the target holds, then |B_j|² ≤ C/2^j, and DSF_α = Σ_j (1+j)^1+2α |B_j|² < ∞ for all α > 0 (exponential beats polynomial).

The gap: Cauchy-Schwarz gives I_j(0) ≤ m_j · D_j. The target requires Cj · D_j. The gap factor is m_j/j = 2^j/j² — exponential. A window decomposition (Proposition 4.1) connects the target to the DSF/BL pair of Paper 50: for any δ > 0, I_j(0) ≤ (1/2δ)∫_−δ^δ I_j(θ)dθ + sup_|θ|≤δ|I_j(θ) − I_j(0)|, splitting into low-frequency mass and spectral continuity near zero.

Divisor Lemma and Failed Routes

Proposition 5.1 (Divisor Lemma). If T(n) and T(n+d) share a factor-subtree with root value g, then g | d. For n uniform in [2^j, 2^j+1]: P(∃ shared factor-subtree with root value ≥ M) ≤ 2τ(d)/M. Shallow sharing (small g) is common and explains C_j(1) > 0; deep sharing (large g, strong long-range correlation) decays as 1/M.

Four routes to covariance decay are excluded: Lipschitz → decay (fails; ρ(d) ~ λ log d grows), shared factors → decay (fails; bounded constant), additive path competition (fails; ρ(d) exceeds typical |η|), conductor stripping (fails; slopes identical Q = 2 to 360). All explain the existence of correlation but not decay. Two viable routes remain: regenerative/coboundary methods (reset-reservoir dynamics of Papers XXI–XLV) and direct spectral/divisor combination (divisor lemma + SPF skeleton truncation of Paper XLVI).

Closure Chain (Updated)

The honest state of the chain: Spectral target I_j(0) ≤ Cj · D_j (unproved — the remaining condition) → |B_j|² ≤ C/2^j [§4.4] → DSF_α for all α > 0 [Σ j^1+2α/2^j < ∞] → (with BL_α + SD + Karamata) [Paper 50] → H′. The spectral target implies DSF_α; whether it also implies BL_α is an open question the window decomposition suggests but does not formally resolve. The data is consistent with the spectral target through N = 10¹⁰. What remains is to prove it.