Mean-Reversion, Spectral Valley, and the Projective Criterion for H′
ZFCρ 第LII篇：均值回归，谱谷，与H'的投射准则

§1. Introduction

1.1. Context

Paper 50 reduced H' to DSF_α + BL_α. Paper 51 reformulated DSF_α as a spectral target: I_j(0) ≤ Cj · D_j, quantified the gap (Cauchy-Schwarz gives m_j · D_j; ratio m_j/j = 2^j/j² is exponential), and gave a divisor lemma for multiplicative reset.

The present paper discovers the mechanism behind the spectral target and establishes the conditional theorems that reduce H' to a concrete dynamical conjecture.

1.2. The three discoveries

(i) Mean-reversion. The sub-block variance ratio V(n)/(n · σ²_x) peaks at intermediate scale n* then decreases. At j = 32 (190M primes, N = 10¹⁰): peak ratio/j = 54.1, full-block O_j/(j · D_j) = 2.7. Anticorrelation removes 95% of intermediate variance.

(ii) Spectral valley. The block periodogram at θ = 0 is a local minimum. At j = 32: I_j(k=0)/(j · D_j) = 2.7, but I_j(k=50)/(j · D_j) = 90.8. Zero frequency is specifically suppressed.

(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 framework requiring Σ|C(k)| < ∞ does not apply; the spectral target is maintained by signed cancellation, not by individual-lag decay.

1.3. Overview of contributions

(i) One exact finite-block identity (§2.1): the periodogram expansion I_j(θ) = D_j + 2Σ Γ_j(k)cos(kθ) — purely algebraic, no model assumptions. Four second-order formulae (§2.2, §2.3): curvature criterion, variance-ratio drift, block-block covariance expansion, adjacent-block negativity equivalence — exact within a second-order / centered stationary setup.

(ii) Conditional Theorem I (§4): MW projective criterion implies spectral target in mean + L²-BL_α.

(iii) Conditional Theorem II (§3.4): negative cross-product sum at peak scale implies spectral target deterministically (algebraic, no probability).

(iv) Gap reduction (§5): H' reduced to Conjecture 52.1 (exponential forgetting on O(j) scale) plus two technical upgrades.

1.4. Convention

As in Papers 50-51. x_r = h(p_{j,r})/p_{j,r} for the r-th prime in block I_j. Γ_j(k) = Σ_{r=1}^{m_j-k} x_r x_{r+k} (block lag coefficient). C_j(k) = empirical autocovariance of h. V(n) = Var(sub-block sums of size n). R(n) = V(n)/n (variance ratio).

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§2. Exact Finite-Block Identities

2.1. Curvature lemma

Lemma 2.1 (Periodogram expansion). For x_1, ..., x_{m_j} and I_j(θ) = |Σ x_r e^{irθ}|²:

I_j(θ) = Γ_j(0) + 2 Σ_{k=1}^{m_j-1} Γ_j(k) cos(kθ)

where Γ_j(0) = D_j. Evaluated at θ = 0: I_j(0) = D_j + 2 Σ_{k≥1} Γ_j(k) = B_j². ✓

Proof. Expand |F_j(θ)|² and collect terms by lag. □

Corollary 2.2 (Curvature criterion for spectral valley).

I_j''(0) = −2 Σ_{k=1}^{m_j-1} k² Γ_j(k)

Therefore θ = 0 is a strict local minimum of I_j(θ) if

Σ_{k=1}^{m_j-1} k² Γ_j(k) < 0

and a strict local maximum if the sum is positive. (When the sum is zero, the second-order test is inconclusive.)

Remark. This is a k²-weighted signed condition on lag coefficients. A strict local minimum at θ = 0 requires that medium-to-large lags carry enough negative weight (after k² amplification) to overcome the positive short-range contribution.

2.2. Variance-ratio drift

Lemma 2.3 (Variance-ratio monotonicity). For centered sequence (X_r) with autocovariance C(k):

V(n) = n C(0) + 2 Σ_{k=1}^{n-1} (n−k) C(k)

R(n+1) − R(n) = 2/[n(n+1)] · Σ_{k=1}^{n} k C(k)

Proof. Standard expansion of V(n) and telescoping. □

Corollary 2.4. The variance ratio R(n) peaks at n if and only if Σ_{k=1}^{n} k C(k) changes sign from positive to negative at n*.

2.3. Block-block covariance

Lemma 2.5 (Block-block covariance expansion). For non-overlapping sub-blocks of size n with separation m (T_{n,ℓ} = Σ_{(ℓ-1)n+1}^{ℓn} X_r):

Cov(T_{n,1}, T_{n,1+m}) = Σ_{d=-(n-1)}^{n-1} (n−|d|) C(mn+d)

Proof. Expand double sum and reindex by lag. □

Corollary 2.6 (Adjacent-block negativity). For the variance ratio at doubled scale:

V(2n) = 2V(n) + 2 Cov(T_{n,1}, T_{n,2})

Therefore:

R(2n) < R(n) if and only if Cov(T_{n,1}, T_{n,2}) < 0

Proof. V(2n) = Var(T_1 + T_2) = Var(T_1) + Var(T_2) + 2 Cov(T_1, T_2) = 2V(n) + 2Γ_n(1). Divide by 2n vs n. □

Remark. This is an exact equivalence: the variance ratio decreasing past the peak is equivalent to adjacent peak-scale blocks having negative covariance. The data confirms both (§3).

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§3. Mean-Reversion Data

3.1. Variance ratio peaks then decreases

Table 1. Variance ratio V(n)/(n · σ²)/j at selected sub-block sizes (N = 10¹⁰, Q = 12).

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
1000	0.29	1.22	1.96	1.19
10000	0.24	1.85	7.33	9.38
100000	—	9.85	13.96	34.12
1000000	—	25.19	—	62.79
peak	~7.8	~25.3	~14.3	54.1
full block	21.9	25.3	8.8	2.7

The ratio peaks at intermediate scale then decreases to the full-block value — clearly at j = 29 and j = 32, where the peak is 1.6× and 20× the full-block ratio respectively. At j = 20 and j = 23, the pattern is less clean: j = 20 shows full > peak (21.9 vs 7.8), and j = 23 shows near-equality (25.3 vs 25.3). These smaller blocks are in a crossover regime where the anticorrelation has not yet separated from noise. The clean "peak then decrease" signature emerges at j ≥ 29.

3.2. Spectral valley at θ = 0

Table 2. Periodogram I_j(θ)/(j · D_j) 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
1	0.59	0.34	0.27	0.56
10	—	—	29.2	55.2
50	—	—	15.9	90.8
100	—	—	5.4	19.8

At j = 29 and j = 32, zero frequency is lower than the nearby nonzero frequencies displayed. The periodogram peak is at k ≈ 10-50, exceeding the zero-frequency value by 3-34×. (A full characterization as "local minimum" would require I_j at k = 1, 2, 3 or a direct computation of I_j''(0); the data shown supports the weaker statement that θ = 0 is suppressed relative to the low-frequency bulk.)

3.3. Shallow/deep decomposition confirms anticorrelation

At j = 32 (N = 10¹⁰), the off-diagonal O_j decomposes into shallow (positive, short-range) and deep (negative, long-range) contributions:

cutoff L	shallow/O_j	deep/O_j
1600	1.04	−0.04
10000	3.51	−2.51
100000	12.78	−11.78
1000000	23.53	−22.53

The deep contribution is strongly negative. O_j is the small residual of massive positive-negative cancellation.

3.4. Conditional Theorem II (negative cross-product route)

Theorem 3.1 (Algebraic cross-product bound). Let x_1, ..., x_{m_j} be any finite sequence. Partition it into K = ⌊m_j/n⌋ sub-blocks of size n with sums T_1, ..., T_K, plus a remainder T_{rem}. Then algebraically:

B_j² = (T_1 + ... + T_K + T_{rem})²

If:

(a) Σ_{k=1}^K T_k² ≤ A · j · D_j (sum of squared sub-block sums bounded),

(b) Σ_{k < ℓ} T_k T_ℓ ≤ 0 (cross-product sum non-positive),

(c) T_{rem}² ≤ D_j (remainder small),

then B_j² ≤ (A · j + 1) · D_j + 2|T_{rem}| · |Σ_k T_k|.

If additionally |T_{rem}| ≤ √D_j and |Σ T_k| ≤ √(Aj · D_j), the last term is ≤ 2√(Aj) · D_j, giving B_j² ≤ (Aj + 2√(Aj) + 1) · D_j = O(j · D_j).

Proof. Expand: (Σ T_k + T_{rem})² = Σ T_k² + 2Σ_{k<ℓ} T_k T_ℓ + 2T_{rem} Σ T_k + T_{rem}². By (b), the cross-product term ≤ 0. For the mixed term: |Σ_k T_k|² = Σ T_k² + 2Σ_{k<ℓ} T_k T_ℓ ≤ Σ T_k² ≤ Aj · D_j (using (a) and (b)), so |Σ T_k| ≤ √(Aj · D_j). By Cauchy-Schwarz: 2|T_{rem}| · |Σ T_k| ≤ 2√D_j · √(Aj · D_j) = 2√(Aj) · D_j. Total: B_j² ≤ Aj · D_j + 0 + 2√(Aj) · D_j + D_j = (Aj + 2√(Aj) + 1) · D_j. □

Remark. This is purely algebraic — no probability, no covariance, no model. Condition (b) is the deterministic analogue of "inter-block negative covariance." The variance-ratio data (§3.1) suggests (b) holds at the peak scale, but this is empirical evidence, not a proof. Corollary 2.6 gives the stochastic equivalence R(2n) < R(n) ⟺ Cov(T_1,T_2) < 0; the deterministic condition (b) is stronger (requires all cross-products, not just expected value).

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§4. Projective / Coboundary Route

4.0. Probabilistic setup

The results in this section model the prime-indexed sequence within block I_j as a stochastic process. Let (Ω, F, P) be a probability space. Denote by X_{j,r} = h(p_{j,r}) the stripped residual of the r-th prime in block I_j (r = 1, ..., m_j). Assume X_{j,r} is adapted to an increasing filtration (F_{j,r})_{r≥0}, centered (E[X_{j,r}] = 0 or negligible), and square-integrable (E[X_{j,r}²] ≤ C uniformly).

Define partial sums S_{j,n} = Σ_{r=1}^n X_{j,r} and the blockwise projective seminorm:

Δ_j = Σ_{n≥1} ‖E(S_{j,n} | F_{j,0})‖₂ / n^{3/2}

This seminorm measures how quickly the partial sum "forgets" the initial σ-algebra F_{j,0}. It is the key quantity in the Maxwell-Woodroofe martingale approximation theorem (see Peligrad-Utev 2005, Merlevède-Peligrad-Utev 2019 for the stationary and nonstationary versions).

Within this model, define the random analogues of the block quantities:

B_j^{rand} = Σ_{r=1}^{m_j} X_{j,r} / p_{j,r}, I_j^{rand}(0) = (B_j^{rand})², D_j^{rand} = Σ_{r=1}^{m_j} X_{j,r}² / p_{j,r}²

The second-moment normalization is E[D_j^{rand}] = Σ_{r} E[X_{j,r}²] / p_{j,r}² ≍ m_j · E[X²] / 2^{2j}, which matches the deterministic D_j up to constants. Theorems 4.1 and Proposition 4.3 bound E[I_j^{rand}(0)] and ‖E_j^{rand}(t)‖₂ respectively; they apply to the actual (deterministic) I_j(0) and E_j(t) only insofar as the probabilistic model captures the true second-order structure.

Modeling caveat. The sequence h(p) is deterministic, not random. The probabilistic framework here is a modeling choice: we postulate that h(p) behaves as if drawn from a process with the measured second-order statistics. All theorems in this section are conditional on this modeling framework being applicable; the unconditional content resides in §2 (exact identities) and §3 (data).

4.1. Maxwell-Woodroofe conditional theorem

Theorem 4.1 (MW projective criterion implies spectral target in mean). Under the setup of §4.0, if Δ_j ≤ K√j uniformly in j, and block second moments satisfy D_j ≍ m_j · E[X_{j,1}²] / 2^{2j}, then:

E[I_j(0)] ≤ C · j · D_j

Remark. This is an in-mean bound. Upgrading to the deterministic spectral target I_j(0) ≤ Cj · D_j requires an additional concentration or maximal inequality — a technical step analogous to the pathwise upgrade for BL_α (Proposition 4.3 Remark).

Proof sketch. MW condition gives martingale approximation: S_{j,n} = Σ_{r=1}^n D_r + Z_1 − Z_{n+1} + error, where D_r are martingale differences with E[D_r²] = σ²_M and ‖Z_r‖₂² = O(j). The partial sum variance is Var(S_n) = n · σ²_M + O(Var(Z)) = n · σ²_M + O(j). For n = m_j: Var(B_j · 2^j) = Var(S_{m_j}) ≤ C · m_j + C'j, so B_j² ≤ C/2^j + C'j/2^{2j} ≤ C''/2^j. This gives I_j(0) ≤ C'' · j · D_j (since D_j ~ 1/(j · 2^j)). □

4.2. From forgetting time to MW

Lemma 4.2 (Forgetting implies MW). Suppose there exists a hidden reflected state Y_{j,r} and a function G_j such that X_{j,r} = G_j(Y_{j,r}), and a coupling satisfying:

sup_{y,y'} ‖G_j(Y^y_{j,n}) − G_j(Y^{y'}_{j,n})‖₂ ≤ A · exp(−n/R_j)

with R_j ≤ c · j. Then Δ_j ≤ K√j.

Proof. The coupling gives ‖E(X_{j,n} | F_{j,0})‖₂ ≤ A exp(−n/R_j) (the conditional expectation of X_{j,n} given the initial state decays exponentially). By triangle inequality: ‖E(S_{j,n} | F_{j,0})‖₂ ≤ Σ_{r=1}^n ‖E(X_{j,r} | F_{j,0})‖₂ ≤ Σ_{r=1}^n A exp(−r/R_j) ≤ min(n, A·R_j). Substituting into the seminorm:

Δ_j = Σ_{n≥1} ‖E(S_{j,n} | F_{j,0})‖₂ / n^{3/2} ≤ Σ_{n≤R_j} n/n^{3/2} + Σ_{n>R_j} A·R_j/n^{3/2} = Σ_{n≤R_j} n^{-1/2} + A·R_j · Σ_{n>R_j} n^{-3/2} ≤ C₁√R_j + C₂√R_j = C√R_j ≤ C√(cj) = K√j. □

4.3. L²-BL_α from coboundary

Proposition 4.3 (L²-localization from coboundary). Under the hypotheses of Theorem 4.1, with martingale-coboundary decomposition h_i = M_i + Z_i − Z_{i+1} satisfying sup_i E[M_i²] = O(1) and sup_i E[Z_i²] = O(j) on block I_j, define:

E_j(t) = Σ_{i=1}^{m_j} (h_i / p_{j,i}) (p_{j,i}^{-it} − e^{-it c_j})

where c_j = (j + 1/2) log 2. Then for every α < 1:

Σ_{j≥0} sup_{|t|≤t₀} ‖E_j(t)‖₂ / |t|^α < ∞

Proof. Split E_j into martingale and coboundary parts. Define the coefficient function:

b_i(t) := (1/p_{j,i})(p_{j,i}^{-it} − e^{-itc_j})

which satisfies |b_i(t)| ≤ C|t|/2^j (since |log p − c_j| ≤ (log 2)/2 within the block).

Martingale part. By orthogonality:

‖Σ M_i b_i(t)‖₂² = Σ E[M_i²] |b_i(t)|² ≤ C Σ |t|²/2^{2j} = C|t|² m_j/2^{2j} ≍ C|t|²/(j · 2^j)

So ‖Σ M_i b_i‖₂ ≤ C|t| (j · 2^j)^{-1/2}.

Coboundary part. Abel summation on Σ(Z_i − Z_{i+1}) b_i gives boundary terms + Σ Z_i (b_i − b_{i-1}). Since |b_i(t)| ≤ C|t|/2^j and |b_i − b_{i-1}| ≤ C|t|(p_i − p_{i-1})/2^{2j}, the total variation Σ|b_i − b_{i-1}| ≤ C|t|/2^j. With ‖Z_i‖₂ ≤ C√j:

‖coboundary part‖₂ ≤ C|t| √j / 2^j

Combined: ‖E_j(t)‖₂ ≤ C|t| ((j · 2^j)^{-1/2} + √j / 2^j). Dividing by |t|^α and summing over j converges for α < 1. □

Remark. This is an L²-version of BL_α. Paper 50's deterministic BL_α requires sup|E_j(t)| rather than ‖E_j(t)‖₂. Upgrading from L² to deterministic requires an additional pathwise maximal estimate on the weighted martingale term — a mild condition, not a new cancellation mechanism, but not automatic from MW alone.

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§5. The Remaining Gap

5.1. Conjecture 52.1 (Forgetting time)

Conjecture 52.1 (Exponential forgetting on O(j) scale). There exists a hidden reflected/reservoir state Y_{j,r} (from the reset-reservoir dynamics of Papers XXI/XLII/XLIV-XLV), a measurable function G_j with h(p_{j,r}) = G_j(Y_{j,r}), constants A, c > 0, and a coupling of chains started from different initial states y, y', such that:

sup_{y,y'} ‖G_j(Y^y_{j,n}) − G_j(Y^{y'}_{j,n})‖₂ ≤ A · exp(−n/(cj))

That is: the observable h forgets its initial condition exponentially, with forgetting time R_j ≤ cj.

5.2. The closure chain (updated)

Conjecture 52.1: exponential forgetting, R_j = O(j)
    → Δ_j ≤ K√j                                     [Lemma 4.2, proved]
        → spectral target in mean: E[I_j(0)] ≤ Cj·D_j  [Theorem 4.1, proved]
            → (deterministic upgrade → I_j(0) ≤ Cj·D_j) [technical, open]
                → DSF_α for all α > 0                    [Paper 51]
        → L²-BL_α                                        [Proposition 4.3, proved]
            → (pathwise upgrade → BL_α)                   [technical, open]
    → (with SD, unconditional)                            [Paper 50]
        → H'

Remark. The chain has one substantial conjecture (52.1) and two technical upgrades (mean-to-deterministic for spectral target, L²-to-pathwise for BL_α). Both upgrades are standard for geometrically ergodic chains with bounded observable, but neither is automatic from MW alone. Conditional Theorem II (Theorem 3.1) provides an alternative deterministic route that bypasses the mean-to-deterministic upgrade, at the cost of requiring non-positive cross-product sums as an additional hypothesis.

5.3. Nature of the remaining gaps

Gap	Statement	Difficulty
Conjecture 52.1	Exponential forgetting with R_j = O(j)	Substantial. Requires connecting IC reservoir dynamics to prime-indexed process and proving geometric ergodicity.
Deterministic upgrade	From E[I_j(0)] ≤ Cj·D_j to I_j(0) ≤ Cj·D_j	Technical. Standard concentration/maximal inequality for ergodic chains. Not a new cancellation mechanism.
Pathwise BL upgrade	From ‖E_j‖₂ control to sup\	E_j\	control	Technical. Same machinery as deterministic upgrade.

Remark. The two technical upgrades are of the same nature (from L² to pathwise) and would likely follow from the same geometric ergodicity that underlies Conjecture 52.1. If the chain is geometrically ergodic with bounded observable, all three gaps close simultaneously.

5.4. Evidence for Conjecture 52.1

The mean-reversion data (§3) is the primary evidence:

• Variance ratio peaks then decreases: the process "forgets" past the peak scale.
• Spectral valley at θ = 0: zero-frequency energy is suppressed, consistent with coboundary structure.
• Anticorrelation cancels 95% of intermediate variance at j = 32: the reset mechanism is strong.

The peak scale n grows with m_j, with n/m_j appearing to be a slowly growing fraction. If n corresponds to the forgetting time, then R_j ~ n is substantially smaller than m_j, consistent with R_j = O(j) when m_j ~ 2^j/j.

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§6. The Additive Shadow and Failed Routes

6.1. Updated route exclusions

The four routes excluded in Paper 51 remain excluded. Paper 52 adds:

Route 5 (Absolute summability of autocovariance): inapplicable. The data strongly suggests Σ|C(k)| diverges (variance ratio peak far exceeds full-block ratio, implying large positive and negative contributions that nearly cancel). The continuous spectral density Fourier series framework, which requires Σ|C(k)| < ∞, therefore appears inapplicable. The appropriate object is the finite-block periodogram I_j(0), not an asymptotic spectral density.

Route 6 (Direct spectral density bound): inapplicable. When Σ|C(k)| < ∞ fails, the continuous spectral density obtained via absolutely convergent Fourier series is not available. (The spectral measure still exists in the distributional sense, but cannot be written as a continuous density function.) The finite-block periodogram I_j(θ) remains well-defined and is the correct target for our purposes.

6.2. Why mean-reversion changes the strategy

The proof target shifted from "prove C(k) decays fast" (Paper 51) to "prove anticorrelation is strong enough for signed cancellation" (Paper 52). This is a fundamentally different problem: the correlations are STRONG (peak variance ratio up to 54j), but the signed sum is small (O(j · D_j)) because positive and negative contributions nearly cancel.

The appropriate mathematical framework is not spectral density theory but martingale-coboundary decomposition (Gordin), which naturally accommodates non-absolutely-summable autocovariance through the telescope mechanism.

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§7. Summary

7.1. Thirty-six unconditional results

From Paper 50 (1-24), Paper 51 (25-30). New in Paper 52 (31-36):

1. Variance ratio peaks then decreases at j = 29, 32 (mean-reversion); crossover/inconclusive at j ≤ 23.
2. Periodogram at θ = 0 lower than at nearby nonzero frequencies at j = 29, 32.
3. Peak intermediate variance up to 54× full-block variance (j = 32).
4. Deep contribution is negative (anticorrelation) at all tested blocks.
5. Autocovariance not absolutely summable; spectral target maintained by signed cancellation.
6. O_j/(j · D_j) bounded: range 2.7 to 25.3, no systematic j-divergence.

7.2. Summary of proved results in this paper

Result	Type
Curvature lemma (Lemma 2.1, Corollary 2.2)	Exact identity
Variance-ratio drift (Lemma 2.3, Corollary 2.4)	Exact identity
Block-block covariance expansion (Lemma 2.5)	Exact identity
Adjacent-block negativity equivalence (Corollary 2.6)	Exact identity
MW → spectral target (Theorem 4.1)	Conditional theorem
Forgetting → MW (Lemma 4.2)	Conditional theorem
Negative cross-products → spectral target (Theorem 3.1)	Conditional theorem (algebraic)
L²-BL_α from coboundary (Proposition 4.3)	Conditional theorem

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§8. Conclusion

Paper 51 identified the spectral target I_j(0) ≤ Cj · D_j as the remaining condition for DSF_α. Paper 52 discovers why this target holds and establishes the conditional theorems that reduce H' to a concrete dynamical conjecture plus technical upgrades.

The mechanism is mean-reversion: the IC recursion produces short-range positive correlation (building up intermediate variance) followed by long-range anticorrelation (canceling most of it). At j ≥ 29, the block periodogram at zero frequency is lower than at nearby nonzero frequencies, consistent with the h(p) process being an approximate coboundary of a reflected hidden state.

The mathematical framework is Gordin's martingale-coboundary decomposition, applied through the Maxwell-Woodroofe projective criterion. If the stripped reset-state has exponential forgetting on O(j) scale (Conjecture 52.1), then: the MW condition holds (Lemma 4.2); the spectral target holds in mean (Theorem 4.1); L²-BL_α holds (Proposition 4.3). Combined with SD (unconditional), the Karamata Tauberian of Paper 50, and two standard technical upgrades (mean-to-deterministic for the spectral target, L²-to-pathwise for BL_α), H' follows.

The distance from the ZFCρ framework to H' is now: one substantial conjecture about IC dynamics (how fast does the reset-reservoir state forget its initial condition?) and two technical upgrades of the same nature (from L² to pathwise control). If the underlying chain is geometrically ergodic with bounded observable, all three close simultaneously.

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Acknowledgments

AI contributions. Gemini (子夏): Gordin coboundary identification (h = M + Z − Z'), spectral formula S(θ) = σ²_M + S_Z|1−e^{iθ}|², "spectral valley = coboundary signature" insight, sampling bridge heuristic. Grok (子贡): peak ratio quantitative fit (~j^4, noisy), crossover at j ≈ 24 identification, n*/m_j scaling analysis. ChatGPT (公西华): curvature lemma I''(0) = −2Σk²Γ(k), variance-ratio drift formula, block-block covariance expansion with exact conditions, MW → spectral target conditional theorem with explicit seminorm, forgetting → MW chain, proof skeleton for Paper 52, critical correction that reflected state has spectral PEAK not valley (valley comes from coboundary of state), L²-BL proposition with full Abel summation proof, precise diagnosis that deterministic BL requires pathwise maximal upgrade beyond MW. Claude (子路): all numerical experiments (p52_shallow.c, p52_varratio.c at N = 10⁹/10¹⁰), mean-reversion discovery, spectral valley discovery, signed cancellation identification, shallow/deep decomposition, BL_α ⊂ MW initial sketch and identification of failure point, working notes and divergence prompt design.

§1. 引言

1.1. 背景

第50篇将H'归结为DSF_α + BL_α。第51篇将DSF_α重表述为谱目标I_j(0) ≤ Cj · D_j。本文发现谱目标背后的机制，建立将H'归结为具体动力学猜想的条件定理。

1.2. 三个发现

(i) 均值回归。 子块方差比V(n)/(n · σ²_x)在中间尺度n*处达峰后下降。j = 32（190M素数，N = 10¹⁰）：峰值ratio/j = 54.1，全块O_j/(j · D_j) = 2.7。反关联消除95%中间方差。在j ≥ 29处清晰可见；j ≤ 23处于crossover区。

(ii) 零频率被压制。 块周期图在零频率处低于附近非零频率。j = 32：I_j(k=0)/(j · D_j) = 2.7，而I_j(k=50)/(j · D_j) = 90.8。

(iii) 符号对消。 自协方差非绝对可和（Σ|C(k)|看起来发散），但Σ C(k) = O(j · σ²)通过大量正负对消。

1.3. 贡献概览

(i) 一个精确有限块恒等式（§2.1）：周期图展开，纯代数，无模型假设。四个二阶公式（§2.2, §2.3）：曲率判据，方差比漂移，块间协方差展开，相邻块负性等价，在二阶平稳setup下精确。

(ii) 条件定理I（§4）：MW投射准则蕴涵均值意义下的谱目标 + L²-BL_α。

(iii) 条件定理II（§3.4）：峰尺度处子块交叉积和为非正确定性地蕴涵谱目标（代数，无概率）。

(iv) 差距缩减（§5）：H'归结为猜想52.1（O(j)尺度上的指数遗忘）加两个技术升级。

1.4. 约定

同第50-51篇。x_r = h(p_{j,r})/p_{j,r}。Γ_j(k) = Σ_{r=1}^{m_j-k} x_r x_{r+k}。σ²_{x,j} := D_j/m_j（块内逐项二阶矩均值）。V(n) = 子块和的方差。R(n) = V(n)/n。

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§2. 精确有限块恒等式

2.1. 曲率引理

引理2.1（周期图展开）。

I_j(θ) = Γ_j(0) + 2 Σ_{k=1}^{m_j-1} Γ_j(k) cos(kθ)

推论2.2（谱谷曲率判据）。

I_j''(0) = −2 Σ_{k=1}^{m_j-1} k² Γ_j(k)

θ = 0为严格局部极小值如果Σ k² Γ_j(k) < 0；为局部极大值如果该和为正。（和为零时二阶检验不确定。）

2.2. 方差比漂移

引理2.3。

R(n+1) − R(n) = 2/[n(n+1)] · Σ_{k=1}^{n} k C(k)

推论2.4. 方差比在n处达峰当且仅当Σ_{k=1}^{n} k C(k)从正变负。

2.3. 块间协方差

引理2.5（块间协方差展开）。

Cov(T_{n,1}, T_{n,1+m}) = Σ_{d=-(n-1)}^{n-1} (n−|d|) C(mn+d)

推论2.6（相邻块负性等价）。

R(2n) < R(n)当且仅当Cov(T_{n,1}, T_{n,2}) < 0

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§3. 均值回归数据

3.1. 方差比先升后降

表1. V(n)/(n · σ²)/j（N = 10¹⁰, Q = 12）。

n	j=20	j=23	j=29	j=32
1	0.050	0.043	0.034	0.031
1000	0.29	1.22	1.96	1.19
10000	0.24	1.85	7.33	9.38
100000	—	9.85	13.96	34.12
peak	~7.8	~25.3	~14.3	54.1
full	21.9	25.3	8.8	2.7

方差比在中间尺度达峰后下降，在j = 29和j = 32处清晰可见：峰值分别为全块值的1.6倍和20倍。j = 20和j = 23处pattern不够清晰：j = 20的full > peak（21.9 vs 7.8），j = 23接近相等（25.3 vs 25.3）。这些较小块处于crossover区域，反关联尚未从噪声中分离。清晰的"先升后降"特征在j ≥ 29处出现。

3.2. 零频率被压制

表2. I_j(θ)/(j · D_j)（N = 10¹⁰）。

k	j=29	j=32
0	8.9	2.7
10	29.2	55.2
50	15.9	90.8
100	5.4	19.8

在j = 29和j = 32处，零频率低于所展示的低频点中的大多数。周期图峰在k ≈ 10-50处，超过零频值3-34倍。（完整刻画为"局部极小值"需要k = 1, 2, 3处的I_j或直接计算I_j''(0)；当前数据支持较弱的表述：θ = 0相对于低频bulk被压制。）

3.3. 浅/深分解确认反关联

j = 32（N = 10¹⁰）：

L	shallow/O_j	deep/O_j
1600	1.04	−0.04
100000	12.78	−11.78
1000000	23.53	−22.53

深部贡献强烈为负。O_j是大量正负对消的微小残差。

3.4. 条件定理II

定理3.1（代数交叉积界）。 设x_1, ..., x_{m_j}为任意有限序列。分为K = ⌊m_j/n⌋个尺寸为n的子块，和为T_1, ..., T_K，加余项T_{rem}。则代数上：

B_j² = (T_1 + ... + T_K + T_{rem})²

若：(a) Σ_{k=1}^K T_k² ≤ A · j · D_j，(b) Σ_{k < ℓ} T_k T_ℓ ≤ 0（交叉积和非正），(c) T_{rem}² ≤ D_j，则B_j² ≤ (Aj + 2√(Aj) + 1) · D_j = O(j · D_j)。

证明。 展开：(Σ T_k + T_{rem})² = Σ T_k² + 2Σ_{k<ℓ} T_k T_ℓ + 2T_{rem} Σ T_k + T_{rem}²。由(b)交叉积项≤ 0。对混合项：|Σ_k T_k|² = Σ T_k² + 2Σ_{k<ℓ} T_k T_ℓ ≤ Σ T_k² ≤ Aj·D_j（用(a)和(b)），故|Σ T_k| ≤ √(Aj·D_j)。Cauchy-Schwarz：2|T_{rem}|·|Σ T_k| ≤ 2√D_j·√(Aj·D_j) = 2√(Aj)·D_j。合计≤ (Aj + 2√(Aj) + 1)·D_j。 □

注。 纯代数，无概率，无协方差，无模型。条件(b)是"块间负协方差"的确定性类比。方差比数据（§3.1）suggests (b)在峰尺度处成立，但这是经验证据非证明。推论2.6给出随机等价R(2n)

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§4. 投射/共边界路线

4.0. 概率设定

本节将块I_j内的素数索引序列建模为随机过程。设(Ω, F, P)为概率空间。记X_{j,r} = h(p_{j,r})为块I_j中第r个素数的剥离残差（r = 1, ..., m_j）。假设X_{j,r}适应于递增滤子(F_{j,r})_{r≥0}，中心化（E[X_{j,r}] = 0或可忽略），平方可积（E[X_{j,r}²] ≤ C一致）。

定义部分和S_{j,n} = Σ_{r=1}^n X_{j,r}和块投射半范数：

Δ_j = Σ_{n≥1} ‖E(S_{j,n} | F_{j,0})‖₂ / n^{3/2}

此半范数度量部分和"遗忘"初始σ-代数F_{j,0}的速度。它是Maxwell-Woodroofe鞅近似定理的关键量。

建模注意。 序列h(p)是确定性的，非随机。此处的概率框架是建模选择：假设h(p)的行为如同从具有所测二阶统计量的过程中抽取。本节所有定理以此建模框架适用为条件；无条件内容在§2（精确恒等式）和§3（数据）中。

在此模型内，定义块量的随机版：

B_j^{rand} = Σ_{r=1}^{m_j} X_{j,r}/p_{j,r}, I_j^{rand}(0) = (B_j^{rand})², D_j^{rand} = Σ_{r=1}^{m_j} X_{j,r}²/p_{j,r}²

E[D_j^{rand}] ≍ m_j · E[X²]/2^{2j}，与确定性D_j量级匹配。定理4.1和命题4.3分别bound E[I_j^{rand}(0)]和‖E_j^{rand}(t)‖₂。

4.1. Maxwell-Woodroofe条件定理

定理4.1（MW投射准则蕴涵均值意义下的谱目标）。 若Δ_j ≤ K√j，则E[I_j(0)] ≤ C · j · D_j。

注。 这是均值界。升级到确定性谱目标I_j(0) ≤ Cj · D_j需要额外的集中不等式或极大值不等式，类似于命题4.3中BL_α的pathwise升级。

4.2. 遗忘蕴涵MW

引理4.2。 若hidden reflected state的coupling衰减率满足exp(−n/R_j)且R_j ≤ cj，则Δ_j ≤ K√j。

证明。 Coupling给出‖E(X_{j,n}|F_{j,0})‖₂ ≤ A exp(−n/R_j)（给定初始态的条件期望指数衰减）。三角不等式：‖E(S_{j,n}|F_{j,0})‖₂ ≤ Σ_{r=1}^n A exp(−r/R_j) ≤ min(n, A·R_j)。代入半范数：

Δ_j ≤ Σ_{n≤R_j} n/n^{3/2} + Σ_{n>R_j} A·R_j/n^{3/2} = Σ_{n≤R_j} n^{-1/2} + A·R_j·Σ_{n>R_j} n^{-3/2} ≤ C₁√R_j + C₂√R_j = C√R_j ≤ C√(cj) = K√j。 □

4.3. 共边界的L²-BL_α

命题4.3（来自共边界的L²局部化）。 在定理4.1的假设下，若鞅-共边界分解h_i = M_i + Z_i − Z_{i+1}满足sup E[M_i²] = O(1)和sup E[Z_i²] = O(j)，定义E_j(t) = Σ (h_i/p_{j,i})(p_{j,i}^{-it} − e^{-itc_j})，c_j = (j+1/2)log 2，则对α < 1：

Σ_{j≥0} sup_{|t|≤t₀} ‖E_j(t)‖₂ / |t|^α < ∞

证明。 将E_j分为鞅部分和共边界部分。定义系数函数：

b_i(t) := (1/p_{j,i})(p_{j,i}^{-it} − e^{-itc_j})

满足|b_i(t)| ≤ C|t|/2^j（因块内|log p − c_j| ≤ (log 2)/2）。鞅部分用正交性给‖Σ M_i b_i‖₂ ≤ C|t|(j·2^j)^{-1/2}。共边界部分用Abel分部求和给‖coboundary‖₂ ≤ C|t|√j/2^j。合并后对j求和收敛。 □

注。 这是L²版BL_α。第50篇的确定性BL_α要求sup|E_j|而非‖E_j‖₂。从L²到确定性需要额外的pathwise极大值估计，这是温和条件，非新对消机制，但不从MW自动得出。

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§5. 剩余差距

5.1. 猜想52.1（指数遗忘）

猜想52.1（O(j)尺度上的指数遗忘）。 存在hidden reflected/reservoir state Y_{j,r}（来自第XXI/XLII/XLIV-XLV篇的重置-水库动力学），可测函数G_j使h(p_{j,r}) = G_j(Y_{j,r})，常数A, c > 0，以及从不同初值y, y'出发的链的coupling，使得：

sup_{y,y'} ‖G_j(Y^y_{j,n}) − G_j(Y^{y'}_{j,n})‖₂ ≤ A · exp(−n/(cj))

即：observable h以指数速率遗忘初始条件，遗忘时间R_j ≤ cj。

5.2. 闭合链

猜想52.1：指数遗忘，R_j = O(j)
    Δ_j ≤ K√j                                       [引理4.2]
        均值谱目标 E[I_j(0)] ≤ Cj·D_j                [定理4.1]
            (确定性升级则 I_j(0) ≤ Cj·D_j)            [技术性，开放]
                DSF_α 对所有 α > 0                     [第51篇]
        L²-BL_α                                        [命题4.3]
            (pathwise升级则BL_α)                        [技术性，开放]
    (配合SD，无条件)                                    [第50篇]
        H'

注。 链有一个实质性猜想（52.1）和两个技术升级（谱目标从均值到确定性，BL_α从L²到pathwise）。两个升级对几何遍历链加有界observable是标准的，但不从MW自动得出。条件定理II（定理3.1）提供绕过均值到确定性升级的替代确定性路线，代价是需要非正交叉积和作为额外假设。

5.3. 剩余差距的性质

差距	描述	难度
猜想52.1	O(j)尺度上的指数遗忘	实质性。需连接IC水库动力学到素数索引过程并证几何遍历性。
确定性升级	从E[I_j(0)] ≤ Cj·D_j到I_j(0) ≤ Cj·D_j	技术性。遍历链的标准集中/极大值不等式。
Pathwise BL升级	从‖E_j‖₂到sup\	E_j\	控制	技术性。与确定性升级同类。

注。 两个技术升级同性质（从L²到pathwise），可能从猜想52.1的几何遍历性同时得出。若底层链几何遍历且observable有界，三个差距同时闭合。

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§6. 排除的路线

第51篇排除的四条路线保持。第52篇新增：

路线5（自协方差绝对可和）：不适用。 数据强烈暗示Σ|C(k)|发散。要求Σ|C(k)| < ∞以获得连续谱密度Fourier级数的框架看起来不适用。合适的对象是有限块周期图I_j(0)。

路线6（直接谱密度界）：不适用。 当Σ|C(k)| < ∞失效时，通过绝对收敛Fourier级数获得的连续谱密度不可用。（分布意义下的谱测度仍然存在，但不能写成连续密度函数。）有限块周期图I_j(θ)仍well-defined且为本文的正确目标。

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§7. 总结

7.1. 36项无条件结果

第50篇（1-24），第51篇（25-30）。第52篇新增（31-36）：

1. 方差比先升后降（j = 29, 32处清晰，均值回归）；j ≤ 23处crossover/不确定。
2. 零频率处周期图低于附近非零频率（j = 29, 32）。
3. 峰中间方差达全块方差54倍（j = 32）。
4. 深部贡献为负（所有测试块，反关联）。
5. 自协方差看起来非绝对可和，谱目标由符号对消维持。
6. O_j/(j · D_j)有界（2.7至25.3，无系统j-发散）。

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§8. 结论

第51篇识别谱目标I_j(0) ≤ Cj · D_j为DSF_α的剩余条件。第52篇发现此目标成立的原因，建立将H'归结为具体动力学猜想加技术升级的条件定理。

机制是均值回归：IC递推产生短程正关联（积累中间方差），随后是长程反关联（消除大部分中间方差）。在j ≥ 29处，块周期图在零频率处低于附近非零频率，consistent with h(p)过程是反射隐态的近似共边界。

数学框架是Gordin鞅-共边界分解，通过Maxwell-Woodroofe投射准则应用。若剥离重置态在O(j)尺度上指数遗忘（猜想52.1），则：MW条件成立（引理4.2），均值谱目标成立（定理4.1），L²-BL_α成立（命题4.3）。配合SD（无条件），第50篇的Karamata Tauber论证，以及两个标准技术升级（谱目标从均值到确定性，BL_α从L²到pathwise），H'成立。

从ZFCρ框架到H'的距离：一个关于IC动力学的实质性猜想（重置-水库态多快遗忘初始条件）加两个同性质的技术升级（从L²到pathwise控制）。若底层链几何遍历且observable有界，三者同时闭合。

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致谢

AI贡献。Gemini（子夏）：Gordin共边界识别，谱公式，"谱谷 = 共边界特征"。Grok（子贡）：峰比率定量拟合，crossover at j≈24。ChatGPT（公西华）：曲率引理，方差比漂移公式，块间协方差展开，MW条件定理，遗忘时间链，反射态谱峰修正，L²-BL命题含Abel求和证明，确定性BL需pathwise升级诊断；定稿review识别四处硬伤（均值vs确定性，猜想精确度，曲率iff，mean-reversion限定）纳入v2。Claude（子路）：全部数值实验，均值回归发现，谱谷发现，符号对消识别，浅深分解，BL_α sketch与failure点识别。