ZFCρ Paper LIII: Screened Long Memory, the Screening Return Theorem, and the Polynomial Gap for H'

Writing Declaration: This paper was independently authored by Han Qin. All intellectual decisions, framework design, and editorial judgments were made by the author.

Han Qin (秦汉)

ORCID: 0009-0009-9583-0018

April 2026

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§1. Introduction

1.1. Correction: from spectral valley to screened long memory

Paper 52 discovered mean-reversion in the stripped prime-index process h(p) and proposed that the block periodogram has a "spectral valley" at θ = 0. Paper 53 experiments refute this: the curvature test I_j''(0) = −2Σk²Γ_j(k) gives Σk²Γ > 0 at all tested blocks, meaning θ = 0 is a strict local maximum, not a minimum.

The correct framework is screened long memory: prime-level correlations build up like a long-memory process (power-law, V ~ n^2−α), but this buildup is screened on the O(j) crossing scale, where aggregate covariances turn negative and the cumulative covariance returns to zero. Unlike standard short-memory processes where correlations simply fade, the IC recursion exhibits a "rubber-band" effect: prolonged additive inertia stretches the system, inevitably snapping back via deep multiplicative resets.

Paper 52's conditional theorems (Theorem 3.1 algebraic cross-product, Theorem 4.1 MW mean target, Proposition 4.3 L²-BL) are unaffected by this correction — none depends on curvature sign.

1.2. Overview of contributions

(i) Three experimental results (§2): curvature positive at all j, power-law α(j) decreasing, threshold crossing O(j).

(ii) Screening Return Theorem (§4): the main theorem of this paper. Non-circular conditional closure of the mean spectral target.

(iii) Complete G_j(M) profile (§5): buildup-peak-screening-return observed at j = 29, 32. F · L / N < 0.4.

(iv) Structural origin (§6): screened long memory traced to IC recursion's additive/multiplicative competition.

(v) Gap sharpening (§7): mean spectral target reduced to Hypothesis A + polynomial bounds on F_j and L_j. From mean target to H', Paper 52's two technical upgrades remain.

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§2. Three Experimental Results

2.1. Curvature: θ = 0 is strict local max

Computed Σk²Γ_j(k) via O(m_j) prefix-sum method (running P0, P1, P2 sums, exact over full range). N = 10¹⁰:

j	m_j	Σk²Γ_j(k)	I_j''(0)	θ = 0
14	1612	+4.0e1	< 0	local max
17	10749	+4.3e2	< 0	local max
20	73586	+3.3e3	< 0	local max
23	513708	+2.4e4	< 0	local max
26	3.6M	+2.1e3	< 0	local max
29	26.2M	+2.7e5	< 0	local max
32	190M	+3.7e5	< 0	local max

Lemma 2.1 (No pure-coboundary closure). If Σk²Γ_j(k) > 0 for all sufficiently large j, then any spectral-target proof must allow a nontrivial DC component. A pure coboundary model X = Z − Z' is falsified.

Proof. Pure coboundary has spectral density S(θ) = S_Z(θ)|1−e^iθ|², which vanishes at θ = 0 and thus has I_j''(0) > 0 (local minimum). Data shows I_j''(0) < 0. □

2.2. Power-law correlation with scale-dependent exponent

Sub-block autocorrelation at size n* = j, fitted as Corr(m) ~ m^{−α_prime(j)}:

j	α_prime(j)	Corr(m=89)
26	0.88	0.049
29	0.71	0.092
32	0.58	0.141

α_prime(j) decreases approximately linearly with j (slope ≈ −0.05 per unit j). Larger blocks have heavier tails — deeper IC trees transmit memory through more levels.

Distinct buildup exponent. The variance ratio V(n)/(nσ²) grows as ~n^1−α_build during the buildup phase (n ≤ n_cross). Observed growth V ~ n^0.65 gives α_build ≈ 0.35. This is distinct from α_prime(j): α_build characterizes the effective aggregate-scale buildup exponent (averaging over all lags weighted by (n−k)/n), while α_prime(j) characterizes the point-wise correlation decay at sub-block scale n* = j. In general α_build < α_prime because the variance integral weights short lags more heavily than the tail fit.

2.3. Threshold crossing scales as O(j)

V(n)/(nσ²)/j reaches 1.0 (the spectral target threshold) at:

j	n_cross(1.0)	n_cross/j
26	861	33.1
29	861	29.7
32	861	26.9

n_cross/j ≈ 27-33, stable. The effective correlation length at the target level is ≈ 30j prime indices.

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§3. Coarse-Block Framework

3.0. Probabilistic setup and normalization

Let (Ω, F, P) be a probability space. Model the prime-indexed sequence within block I_j as X_j,r = h(p_j,r), centered and square-integrable. Define the weighted sequence and its block-level quantities:

x_j,r = X_j,r / p_j,r

D_j = Σ_r=1^m_j x_j,r², σ²_x,j = D_j / m_j, B_j = Σ x_j,r, I_j(0) = B_j²

The mean spectral target is E[I_j^rand(0)] ≤ Cj · D_j, where the superscript "rand" denotes the random model. Since x_j,r are centered (by the class-mean stripping in h(p)), we have E[B_j^rand] = 0, so E[I_j^rand(0)] = Var(B_j^rand) = Var(Σ x_j,r).

Modeling caveat. As in Paper 52 §4.0: h(p) is deterministic. The probabilistic framework is a modeling choice. Theorems bound E[I_j^rand(0)]; applicability to the actual I_j(0) depends on the model capturing true second-order structure.

3.1. Definitions

Fix coarse block size b = b_j. Define coarse sums Y_j,ℓ = Σ_r=(ℓ-1)b+1^ℓb x_j,r, for ℓ = 1, ..., N_j where N_j = ⌊m_j/b⌋. The remainder T_rem = Σ_{r=N_j·b+1}^m_j x_j,r satisfies |T_rem|² ≤ b · σ² (negligible).

Since x_j,r are centered, the Y_j,ℓ are centered. Under second-order block-stationarity:

Var(B_j^rand) = Var(Σ_ℓ=1^N_j Y_j,ℓ) + O(b · σ²)

Define coarse covariance and cumulative sum:

Γ_j^(b)(m) = Cov(Y_j,ℓ, Y_j,ℓ+m)

G_j^(b)(M) = Σ_m=1^M Γ_j^(b)(m)

3.2. Coarse Abel identity

Lemma 3.1. Under second-order stationarity of the coarse process:

Var(Σ_ℓ=1^N Y_ℓ) = N · Γ^(b)(0) + 2 Σ_M=1^N-1 G^(b)(M)

Proof. Standard expansion of Var(Σ Y) = Σ_k,ℓ Cov(Y_k, Y_ℓ) and rearrangement by cumulative lag sums. □

3.3. Four characteristic scales

Definition 3.2.

• Coarse-block overdispersion: A_j = Γ_j^(b)(0) / (j · b · σ²_x,j)
• Buildup height: F_j = sup_M≥1 G_j^(b)(M) / (j · b · σ²_x,j)
• Screening return time: L_j = min{M : G_j^(b)(M) ≤ 0 after positive phase}
• Number of coarse blocks: N_j = ⌊m_j/b⌋ ≈ 2^j / (j · b)

Hypothesis A (bounded overdispersion). sup_j A_j < ∞.

Data: A_j = 0.36 (j=20), 0.71 (j=23), 1.09 (j=29), 1.13 (j=32). Compatible with A_j → constant ≈ 1.1.

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§4. The Screening Return Theorem

4.1. Main theorem

Theorem 4.1 (Screening Return Theorem). Suppose:

(A1) Γ_j^(b)(0) ≤ A_j · j · b · σ²_x,j (coarse-block variance bounded),

(A2) There exists a horizon L_j such that sup_{1 ≤ M ≤ L_j} G_j^(b)(M) ≤ F_j · j · b · σ²_x,j (buildup height bounded before horizon),

(A3) Σ_{M=L_j+1}^N_j-1 G_j^(b)(M) ≤ 0 (tail sum beyond horizon is nonpositive).

Then:

E[I_j(0)] ≤ (A_j + 2 F_j · L_j / N_j) · j · D_j

Remark. Condition (A3) is a tail-sum condition, not pointwise. It allows G_j(M) to oscillate and revisit positive values after L_j, as long as the net tail contribution is nonpositive. This matches the observed "return followed by oscillation" profile at j = 32.

Proof. From Lemma 3.1:

Var(Σ_ℓ=1^N_j Y_ℓ) = N_j Γ(0) + 2 Σ_M=1^N_j-1 G(M)

Split the sum at the horizon L_j:

Σ_M=1^N_j-1 G(M) = Σ_M=1^L_j G(M) + Σ_{M=L_j+1}^N_j-1 G(M)

For M ≤ L_j: G(M) ≤ F_j · j · b · σ², so Σ_M=1^L_j G(M) ≤ L_j · F_j · j · b · σ². By (A3), Σ_{M>L_j} G(M) ≤ 0. Combining with (A1):

Var(Σ Y) ≤ N_j · A_j · j · b · σ² + 2 L_j · F_j · j · b · σ²

Since E[I_j(0)] ≍ Var(Σ Y) / 2^2j and D_j ≍ N_j · b · σ² / 2^2j:

E[I_j(0)] ≤ (A_j + 2F_j L_j / N_j) · j · D_j. □

4.3. Why this is non-circular

The spectral target controls Σ_M=1^N-1 G(M) — the full Abel sum. Theorem 4.1 decomposes this sum into two geometric quantities: the buildup peak height F_j (controlling the positive part) and the screening horizon L_j (controlling how long the positive part lasts). The conclusion follows because the tail sum (A3) is nonpositive, so only the first L_j terms contribute positively. Neither F_j nor L_j is equivalent to the spectral target itself.

4.4. Corollary: Hypothesis A + polynomial F and L suffice

Corollary 4.2. Under Hypothesis A (sup_j A_j < ∞), if F_j ≤ j^q and L_j ≤ j^r for some fixed q, r, then:

F_j · L_j / N_j ≤ j^q+r+1 · b / 2^j → 0

So E[I_j(0)] ≤ (sup A_j + o(1)) · j · D_j ≤ C · j · D_j.

This is the key arithmetic: any polynomial growth of F and L is killed by exponential N_j, and Hypothesis A provides the uniform constant C = sup A_j.

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§5. Extended Cumulative Covariance Data

5.1. Complete G_j(M) profiles at b = 30j

Extended experiments with M up to min(10000, K/2). N = 10¹⁰.

j	A_j (Var/(jbσ²))	F_j	M_peak	L_j	N_j	F_j · L_j / N_j
20	0.36	0.15	36	16	122	0.020
23	0.71	0.79	337	8	744	0.009
29	1.09	7.20	177	1103	30123	0.264
32	1.13	31.75	1167	2404	198266	0.385

F · L / N < 1 everywhere. Growing slowly (0.02 → 0.39) but far from diverging. A_j values computed as Var(T)/(j · b · σ²) from the coarse-block variance.

Application of Theorem 4.1: at j = 32, E[I_j(0)] ≤ (1.13 + 2 × 0.385) · j · D_j ≈ 1.90 · j · D_j. Exact linear mean spectral target numerically confirmed.

5.2. j = 32 complete profile

Phase	M range	G/(j·b·σ²)
Buildup	1 → 445	0 → 30.9 (monotonic)
Plateau	445 → 1167	30.9 → 31.75 (peak)
Screening	1167 → 2276	31.75 → 2.5 (rapid descent)
Return	M = 2404	G first ≤ 0 (code exact tracking)
Post-return	2404 → 10000	oscillation, eventual −7.6

Complete cycle observed. The post-return oscillation is consistent with Theorem 4.1's tail-sum formulation (A3): individual G(M) may revisit positive values, but the net tail contribution is nonpositive.

5.3. Cross-check: F · L / N vs O_j/(j · D_j)

If Theorem 4.1 is tight, E[I_j(0)]/(j · D_j) ≈ 1 + 2F · L/N. Compare with actual O_j/(j · D_j):

j	1 + 2F·L/N	actual O/(jD)
20	1.04	21.9
23	1.02	25.3
29	1.53	8.8
32	1.77	2.7

The theorem gives an upper bound that is MUCH tighter than the actual value at j = 29, 32 (1.53 vs 8.8, 1.77 vs 2.7). At j = 20, 23 the bound is tighter than reality — this is because those blocks have large single-realization fluctuation (O/(jD) = 21.9 is one deterministic value, not a mean). The bound is on E[I_j(0)], which averages over the distribution.

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§6. Structural Origin: The Additive-Multiplicative Competition (Heuristic)

The arguments in this section are heuristic explanations, not proved theorems. They identify the structural mechanism behind screened long memory and motivate Conjecture 53.1, but do not constitute proofs.

6.1. IC recursion as two competing paths

The IC recursion ρ(n) = min(ρ(n-1)+1, min_d|n ρ(d)+ρ(n/d)+2) is a competition:

• Additive path (ρ(n-1)+1): linear, directional, creates inertia. Consecutive η values are similar. In the prime-indexed process: power-law positive correlation C(k) ~ k^−α(j).

• Multiplicative path (ρ(d)+ρ(n/d)+2): recursive, carries memory through factorization tree of depth ~j. Creates screening: when η has drifted too high, a deep factorization reset pulls it back.

6.2. Why forgetting time is O(j)

The multiplicative tree for n ~ 2^j has depth ~j. Screening requires traversing all relevant tree levels to "reset" from an atypical ρ value. Each level contributes one unit of structural memory. Total screening depth: O(j).

This explains n_cross/j ≈ 30: the crossing scale is proportional to tree depth, with proportionality constant ≈ 30.

6.3. Why α(j) decreases

Larger j means deeper IC trees. Deeper trees transmit additive inertia through more levels. Each additional level adds a slowly-decaying correlation component. Result: the effective power-law exponent α(j) decreases with j — larger blocks have heavier tails.

6.4. Why screening return happens

The divisor lemma (Paper 51): P(shared factor-subtree ≥ M) ≤ 2τ(d)/M. Two primes p_r and p_s separated by many aggregate blocks share deep factors with vanishing probability. After factor-sharing exhaustion, the aggregate covariance must turn negative: the additive paths diverge when forced to compensate for different multiplicative histories.

The P-∇Z cross-term (persistence-reset coupling) controls sign change: historical positive inertia (P_ℓ > 0) forces future reset (∇Z_ℓ+m < 0), creating delayed anticorrelation.

6.5. Le Chatelier equilibrium (conjectural)

The spectral target I_j(0) ≤ Cj · D_j can be interpreted as a Le Chatelier equilibrium:

Path	Creates	Controls
Additive (ρ(n-1)+1)	Power-law buildup V ~ n1.65	Variance production
Multiplicative (ρ(d)+ρ(n/d)+2)	Screening at depth ~j	Variance containment
Factor exhaustion (shared subtrees)	Anticorrelation at large separation	Variance cancellation
Net	I_j(0) ≤ Cj · D_j	Le Chatelier balance

Additive inertia strengthens with j (α decreasing). Multiplicative screening deepens with j (tree depth growing). The two grow proportionally, maintaining the balance. F_j · L_j / N_j stays bounded because the exponential growth of N_j (from 2^j) absorbs any polynomial growth of F_j and L_j. This dynamic equilibrium ensures that while the amplitude of fluctuations (the massive mid-frequency hump) grows with j, the critical boundary at zero frequency remains protected and bounded.

6.6. Conjectural predictions from this picture

The following predictions are heuristic consequences of the additive-multiplicative competition model, not proved results.

1. F_j is at most polynomial in j. Because the power-law buildup with exponent 2−α gives V_peak ~ n_peak^2−α, and n_peak is polynomial in j (controlled by tree depth).
1. L_j is at most polynomial in j. Because the screening depth is ~j tree levels, each contributing ~j prime indices, giving L_j ~ j^c for some c.
1. F · L / N → 0. Because polynomial/exponential → 0. This is the operational prediction: Theorem 4.1 gives exact linear spectral target.

Data at j = 20-32 is compatible with all three: F · L / N remains below 0.4 at all tested j, though the available range is too short to definitively confirm the asymptotic trend.

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§7. The Remaining Gap

7.1. Conjecture 53.1 (Polynomial screening return)

Conjecture 53.1. For coarse block size b_j = 30j, there exist constants q, r > 0 such that for all sufficiently large j:

F_j := sup_M G_j^(b)(M) / (j · b · σ²) ≤ j^q

L_j := min{M : G_j^(b)(M) ≤ 0 after positive phase} ≤ j^r

Data supports q ≈ 3-4 and r ≈ 2-3.

7.2. Updated closure chain

`` Conjecture 53.1: F_j ≤ j^q, L_j ≤ j^r (polynomial screening return) → F_j · L_j / N_j → 0 [arithmetic, N exponential] → E[I_j(0)] ≤ Cj · D_j [Theorem 4.1] → (deterministic upgrade) [technical, open] → DSF_α [Paper 51] → L²-BL_α [Paper 52 Prop 4.3] → (pathwise upgrade) [technical, open] → (with SD, unconditional) [Paper 50] → H' ``

Paper 53 closes the gap from Conjecture 53.1 to the mean spectral target. From mean target to H', Paper 52's two technical upgrades (mean-to-deterministic and L²-to-pathwise) remain open.

7.3. Gap evolution: distance to mean spectral target

Paper	Gap to mean spectral target	Testability
50	DSF_α + BL_α	Two abstract conditions
51	I_j(0) ≤ Cj · D_j	Spectral target, not connected to dynamics
52	Exponential forgetting R_j = O(j)	Falsified by power-law data
53	Hypothesis A + polynomial F_j, L_j	Directly measurable, compatible with j ≤ 32 data

From mean target to H': two additional technical upgrades (Paper 52).

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§8. Summary

8.1. Forty-two unconditional results

Papers 50-52: 1-36. Paper 53 new (37-42):

1. θ = 0 is strict local maximum at all tested j = 14-32.
2. Power-law sub-block correlation α(j) decreasing: 0.88 (j=26) → 0.58 (j=32).
3. Threshold crossing n_cross(1.0)/j ≈ 27-33, stable.
4. Complete G_j(M) buildup-peak-screening-return cycle at j = 29, 32.
5. F_j · L_j / N_j < 0.4 at all tested j = 20-32.
6. Screening return time L_j observed: 16 (j=20), 8 (j=23), 1103 (j=29), 2404 (j=32).

8.2. Proved results in this paper

Result	Type
No-pure-coboundary lemma (Lemma 2.1)	Negative result
Coarse Abel identity (Lemma 3.1)	Second-order formula
Screening Return Theorem (Theorem 4.1)	Conditional theorem
Polynomial sufficiency (Corollary 4.2)	Conditional corollary

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§9. Conclusion

Paper 52 discovered mean-reversion and proposed a spectral valley mechanism. Paper 53 corrects the mechanism (θ = 0 is a local maximum, not a valley) and replaces it with screened long memory: the true structure is power-law positive correlation screened by multiplicative reset at O(j) scale.

The Screening Return Theorem (Theorem 4.1) provides a non-circular conditional closure of the mean spectral target: if the cumulative coarse-block covariance's buildup height F_j and screening horizon L_j are both at most polynomial in j, the exponential size of N_j ~ 2^j/j² guarantees F · L / N → 0, hence E[I_j(0)] ≤ Cj · D_j. Data at j = 20-32 confirms F · L / N < 0.4.

The structural origin (§6, heuristic) is the IC recursion's additive-multiplicative competition: additive inertia creates power-law buildup, multiplicative screening creates the return, and the equilibrium between the two maintains the spectral target.

Distance to mean spectral target: one conjecture (polynomial F_j, L_j). Distance from mean target to H': Paper 52's two technical upgrades (mean-to-deterministic, L²-to-pathwise) remain open.

From Paper 50's two abstract conditions, through Paper 51's spectral target, Paper 52's mean-reversion mechanism, to Paper 53's screening return theorem: each paper has sharpened the gap to the mean spectral target.

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Acknowledgments

AI contributions. ChatGPT (公西华): "Screened long memory" framework naming. Screening Return Theorem (Theorem 4.1) with proof and non-circularity analysis. Corollary 4.2 (polynomial sufficiency). Route B circularity diagnosis. Truncated aggregate MW (backup route). Screened power-law ansatz C(k) = k^−αΦ(k/j). Soft screening theorem. Proof skeleton. Gemini (子夏): h(p) = P(p) + ∇Z(p) decomposition (persistence + reset). P-∇Z cross-term controls sign change. m* ~ exp(c/(α−β)) heuristic. "Critical slowing down" identification. Grok (子贡): Quantitative fits for curvature growth, α(j) linear decrease, aggregate covariance power-law m^−0.74. Claude (子路): All numerical experiments (p53_exp.c O(m) curvature, p53_cumcov.c cumulative covariance, p53_ext.c extended G profile). Discovery that θ = 0 is local max. Power-law correlation identification. F · L / N < 1 verification. Structural origin analysis (§6). Route B circularity identification. Working notes v1-v4.