ZFCρ Paper 55: Damped Oscillation, the 70/30 Law, and the Positive-Tail Theorem

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

§1. Introduction

1.1 Position in the H' program

Paper 53 established the Screening Return Theorem (Theorem 4.1, Corollary 4.2): if the normalized cumulative coarse-block covariance G̃_j(M) = G_j(M)/(j·b·σ²) has buildup height F̃_j = sup_M G̃_j(M) and screening return time L_j, and if F̃_j and L_j are both polynomial in j, then F̃_j · L_j / N_j → 0 (since N_j ~ 2^j/j² is exponential), and the mean spectral target E[I_j(0)] ≤ Cj · D_j follows.

Paper 54 proved F̃_j polynomial (Corollary 54.2) via the Layered Innovation Theorem, under conjectures on geometric innovation decay and cross-cell decorrelation. The remaining input for Conjecture 53.1 is that L_j is polynomial.

Paper 55 does not independently close L_j polynomial. Instead, it provides the structural mechanism behind Conjecture 53.1: the post-peak G_j(M) profile is a damped oscillation with universal rebound ratio r ≈ 0.80, connecting the screening return to the fluctuation absorption rate η ≈ 0.2 from the thermodynamic interface paper. The geometric envelope decay (Theorem 55.3) bounds the positive tail mass, providing the first mechanistic explanation for why F̃·L/N < 0.4 at tested blocks j = 20–32.

1.2 The η field

For primes p, define the detrended residual

η(p) = ρ_E(p) − λ · ln p − μ_{p mod 12}, (1.2)

where λ = 3.856763 is the regression slope of ρ_E vs ln p over all primes up to N = 10^10, and μ_r is the class mean of ρ_E − λ ln p by residue r mod 12.

Since ρ_E(p) = ρ_E(p−1) + 1 for all primes (the additive path is always optimal when no nontrivial factorization exists), η(p) captures the fluctuation of ρ_E(p−1) around its trend. The field η has mean zero by construction.

Class means for nontrivial residue classes:

(Classes r = 0, 2, 3, 4, 6, 8, 9, 10 contain at most one prime each.)

1.3 Summary of results

§2. Scale-Invariant Variance

2.1 Dyadic layer decomposition

Each prime p > 2 has p − 1 even. Define the dyadic layer ℓ = v_2(p−1), the 2-adic valuation of p − 1. By Dirichlet's theorem on primes in arithmetic progressions, the fraction of primes with v_2(p−1) = ℓ converges to π_ℓ = 2^−ℓ. The spatial range (mean gap in prime index between consecutive same-layer primes) is L_ℓ = 2^ℓ.

Both relations are verified to high precision:

The structural identity π_ℓ · L_ℓ = 1 holds exactly.

2.2 The variance law

Conjecture 55.1 (Scale-invariant shell variance).

sup_j sup_{ℓ ≤ j} Var_{I_j}(η | v_2 = ℓ) < ∞.

Empirical evidence (Table 1). Across 455,052,511 primes, Var(η | v_2 = ℓ) oscillates between ~0.46 (odd ℓ) and ~0.53 (even ℓ), stable from ℓ = 1 to ℓ = 25:

Odd-layer mean: 0.474. Even-layer mean: 0.521. Overall mean: 0.497 ≈ 1/2.

The odd-even oscillation has an empirical structure

V_ℓ ≈ V_* + Δ·(−1)^ℓ + o(1),

with V_* ≈ 0.50 and Δ ≈ 0.024, plausibly connected to the mod 4 structure of 2^ℓ.

2.3 Mechanism: scale invariance of the DP

The recursion ρ_E(n) = min(ρ_E(n−1) + 1, min_ab=n ρ_E(a) + ρ_E(b) + 2) competes two paths. The multiplicative efficiency — the fraction of ρ_E(n) saved by factoring rather than successor-stepping — depends on the relative factor structure (what share of ρ can be captured multiplicatively), not the absolute scale of n.

For the dyadic decomposition: p − 1 = 2^ℓ · q with q odd. The trend component λ·ℓ·ln 2 is absorbed by the 2^ℓ factor (this appears in E[η|ℓ], which decreases by ~0.35 per layer). The residual variance comes from ρ(q) − λ ln q, which is the same detrended complexity problem at scale q. Since the DP's competition is scale-invariant, Var(ρ(q) − λ ln q) is independent of the scale of q.

2.4 Spectral budget

Since π_ℓ · L_ℓ = 1 and Var(η|ℓ) = O(1), the per-layer spectral budget is

π_ℓ · Var(η|ℓ) · L_ℓ = Var(η|ℓ) ≈ 0.5 = O(1).

The total budget across ℓ = 1 to log₂ N is O(log N) = O(j). This explains why Paper 53's spectral bound is j-linear (not constant): the total available variance grows with the number of active layers.

2.5 Within-layer autocorrelation

For consecutive same-layer primes (both with v_2(p−1) = ℓ), the autocorrelation of ρ(q) (where q is the odd part) satisfies

C(1)_ρ(q) ≈ 0.96 across all layers.

This means 96% of the ρ(q) value is inherited from the previous same-layer prime, with only 4% independent innovation. The innovation is approximately white noise (Paper 54, Experiment 10).

§3. The 70/30 Law

3.1 Cross-layer decomposition

For dyadic block I_j = [2^j, 2^j+1), partition primes by layer ℓ = v_2(p−1). Define coarse block sums Y_ℓ,t = Σ_{p in block t, v_2(p−1)=ℓ} η(p) and Y_t = Σ_ℓ Y_ℓ,t. The cumulative aggregate covariance decomposes as

G_j(M) = Σ_ℓ,ℓ' G_j,ℓ,ℓ'(M)

where G_j,ℓ,ℓ'(M) = Σ_m=0^M Cov(Y_ℓ,t, Y_ℓ',t+m).

Define same-layer and cross-layer components:

G_same(M) = Σ_ℓ G_j,ℓ,ℓ(M), G_cross(M) = G_j(M) − G_same(M).

3.2 Empirical result

The 70/30 Law. G_same/G_total ≈ 30% and G_cross/G_total ≈ 70%, constant across all M and in both buildup and screening phases.

This falsifies the hypothesis that screening is driven by cross-layer compensation opposing same-layer buildup. Both components share the same sign at every lag and maintain their ratio through the entire profile.

3.3 Theorem 55.2 (Approximate synchronization)

Theorem 55.2. Suppose the layer block sums satisfy the approximate rank-one decomposition

Y_ℓ,t = π_ℓ · Z_t + ε_ℓ,t, (3.1)

where Z_t is the aggregate mode and the error field has bounded relative contribution:

sup_M |G_err(M)| / |G_Z(M)| ≤ ε_j. (3.2)

Then

G_same(M) / G_total(M) = Σ_ℓ π_ℓ² + O(ε_j), (3.3) G_cross(M) / G_total(M) = 1 − Σ_ℓ π_ℓ² + O(ε_j). (3.4)

Proof. Under (3.1), Cov(Y_ℓ,t, Y_ℓ',t+m) ≈ π_ℓ π_ℓ' · Cov(Z_t, Z_t+m). Summing over same-layer pairs: G_same ≈ (Σ π_ℓ²) · G_Z. Summing over all pairs: G_total ≈ (Σ π_ℓ)² · G_Z = G_Z. The ratio is Σ π_ℓ². For π_ℓ = 2^−ℓ:

Σ_ℓ=1^∞ (2^−ℓ)² = Σ 4^−ℓ = 1/3. (3.5)

Therefore G_same/G_total ≈ 1/3 and G_cross/G_total ≈ 2/3. □

The observed 29.3% (vs 33.3%) reflects the O(ε_j) correction from imperfect synchronization (C(1)_rq = 0.96 ≠ 1).

3.4 Consequence: reduction to aggregate mode

The 70/30 Law reduces the multi-layer G_j(M) problem to the single aggregate mode G_Z(M). All subsequent analysis operates on G_Z.

§4. Damped Oscillation

4.1 The G_j(M) profile is oscillatory

Contra the monotone-screening narrative of earlier papers, G_j(M) exhibits pronounced oscillation post-peak.

j = 29 profile (30,123 coarse blocks, b = 870):

j = 26 profile (4,674 coarse blocks, b = 780):

4.2 Normalized buildup height

Following Paper 53 Definition 3.2, the normalized buildup height is F̃_j = G_j(peak) / (j · b · σ²), where σ² = Var(η) = 0.576:

F̃_j is polynomial in j (growing from 2.6 to 34.2 over j = 26 to 32). The product F̃·L/N remains below 0.4 at all measured blocks, consistent with Paper 53 result #41. The exponential growth of N_j ~ 2^j/j² absorbs the polynomial growth of F̃_j · L_j.

4.3 Three structural regularities

Regularity 1: Universal rebound ratio r ≈ 0.80.

The rebound ratio is independent of j to the precision of measurement.

Regularity 2: Symmetric half-periods.

At j = 29: half-period (peak → trough) = 394 − 74 = 320. Half-period (trough → peak) = 714 − 394 = 320. Exact symmetry, a signature of harmonic oscillation.

Regularity 3: Period scales as T ~ 2^j.

Ratio: T(j=29)/T(j=26) = 640/80 = 8.000 = 2^3 = 2²⁹⁻²⁶.

This gives T_coarse(j) ~ 2^j / j² and T_prime(j) ~ 2^j.

4.4 Connection to η ≈ 0.2

The rebound ratio r = 0.80 satisfies

1 − r = 0.20 = η,

where η is the fluctuation absorption rate measured independently in the thermodynamic interface paper (DOI: 10.5281/zenodo.19310282) as η ∈ [0.10, 0.31] across 17 Ω-shells.

Microscopic derivation. Paper 54 established C(1)_ρ(q) ≈ 0.96 across all dyadic layers: each layer transmits 96% of correlation and contributes 4% independent innovation. One oscillation cycle spans approximately 5 effective layers (the number of layers with significant population at the oscillation timescale). The compound dissipation is

1 − 0.96^5 ≈ 0.185 ≈ η ≈ 0.20. (4.1)

This closes a circle across the paper series:

4.5 Mechanism: delayed feedback oscillator

The DP recursion creates a delayed feedback loop. The multiplicative path ρ_E(n) = ρ_E(d) + ρ_E(n/d) + 2 references values ρ_E(n/d) that lie outside the current dyadic block (since n/d ≤ n/2 < 2^j for n ∈ [2^j, 2^j+1] and d ≥ 2). These references carry "stale" information from a previous state of the field.

During buildup (G positive, ρ elevated), multiplicative references point to earlier low-ρ regions, eventually triggering screening. During screening (G declining, ρ suppressed), references point to the preceding high-ρ buildup region, triggering rebound. The delay between cause and effect produces oscillation rather than monotone relaxation.

The three elements of classical damped oscillation:

1. Restoring force: multiplicative screening. Elevated ρ makes factorization more efficient, pulling ρ back toward trend.

1. Inertia: the spectral budget O(j). More layers contributing means slower response.

1. Damping: per-layer innovation white noise. Each layer dissipates 4% of coherent energy. Over ~5 layers per cycle, compound dissipation ≈ 20%.

§5. The Positive-Tail Theorem

5.1 Motivation

Paper 53's Corollary 4.2 requires F̃_j and L_j both polynomial in j, so that F̃_j · L_j / N_j → 0 (polynomial / exponential). Paper 54 delivered F̃_j polynomial. The remaining input is L_j polynomial (or more precisely, F̃_j · L_j = o(N_j)).

The geometric envelope provides a mechanism consistent with the observed screening return and a structural bound on the positive tail mass, though it does not by itself prove L_j finite or polynomial. (The oscillation period T_j scales as ~ 2^j, comparable to N_j, so the positive-tail bound Q̃_j ≤ T_j · F̃_j / (1-r) gives Q̃_j / N_j ~ F̃_j, which is polynomial but not → 0.)

Instead, the geometric envelope serves two roles:

1. Mechanism: It provides a mechanism consistent with the observed screening return and a structural bound on the positive tail mass.

1. Positive-tail bound: It provides a tighter estimate of the positive tail mass than the crude L_j × F̃_j rectangle, useful for quantitative prediction.

5.2 Theorem 55.3 (Geometric-envelope positive-tail theorem)

Theorem 55.3. Let M_0 < M_1 < M_2 < ··· be successive local maxima of G̃_j(M) post-peak, with heights P̃_j,k := G̃_j(M_k). Suppose:

(i) Geometric contraction: P̃_j,k+1 ≤ r_j · P̃_j,k for all k, with 0 < r_j < 1.

(ii) Bounded period: M_k+1 − M_k ≤ T_j for all k.

Then the normalized positive tail mass satisfies

Q̃_j = Σ_{M ≥ M_0} G̃_j(M)^+ ≤ T_j · F̃_j / (1 − r_j). (5.1)

Proof. Between consecutive peaks M_k and M_k+1, the positive part of G̃ is bounded by P̃_j,k over an interval of length ≤ T_j. Summing:

Q̃_j ≤ T_j · Σ_k P̃_j,k ≤ T_j · F̃_j · Σ_k r_j^k = T_j · F̃_j / (1 − r_j). □

5.3 What the envelope controls and what it does not

Controls: The positive tail mass Q̃_j. Theorem 55.3 gives Q̃_j ≤ T_j · F̃_j / (1 − r_j). This is a structural bound on the total area under the positive part of G̃_j(M).

Does not control: The first return time L_j. Geometric peak decay does not imply zero-crossing: a profile whose peaks shrink but whose troughs remain positive would never cross zero. To bound L_j from peak contraction alone, one would additionally need a trough-depth or sign-change hypothesis. Paper 55 does not supply this.

Remark. At j = 26 and 29, G_j(M) is observed to cross zero (L_j = 264 and 1,217 respectively), and the troughs are deep (17% and 2.6% of peak). At j = 32, the return is not observed within the measurement window (M_max = 6000), though the post-peak drift is consistent with eventual return. The data is compatible with L_j finite at all j, but this is not proved by Theorem 55.3 alone.

5.4 Connection to Paper 53 closure

Paper 53's Corollary 4.2 requires Conjecture 53.1: F̃_j ≤ j^q and L_j ≤ j^r. The product F̃_j · L_j / N_j then vanishes because N_j ~ 2^j/j² is exponential.

Paper 55 contributes to this program as follows:

What Paper 55 establishes:

• F̃_j is polynomial in j (consistent with Paper 54, verified with normalized data: §4.2)
• The post-peak G̃_j(M) profile is a damped oscillation with universal rebound ratio r ≈ 0.80, connecting the screening mechanism to the thermodynamic absorption rate η ≈ 0.2
• The positive tail mass is bounded by the geometric envelope (Theorem 55.3)
• F̃·L/N < 0.4 at all tested j = 20–32 (empirical support for Conjecture 55.5)

What Paper 55 reduces the gap to:

• Conjecture 55.4 (geometric contraction, r < 1): empirically supported by r ≈ 0.80 at j = 26, 29, and cross-validated by the thermodynamic paper's η ≈ 0.2
• The question of whether L_j is polynomial: not resolved by this paper, but the damped oscillation mechanism and the exponential growth of N_j make F̃·L/N → 0 robust against any sub-exponential growth of L_j

§6. Updated H' Gap Structure

6.1 Current status

``` Conj 54.A + 54.B → Thm 54.1: C<sup>int</sup>(r) ≤ C·r<sup>-α</sup> → Cor 54.2: F̃_j polynomial

Paper 55 structural analysis: Conj 55.1: Var(η|ℓ) = O(1) (scale invariance) Thm 55.2: synchronization → 70/30 law Thm 55.3: geometric envelope → positive-tail bound Conj 55.4: r < 1 (universal damping, η ≈ 0.2) → structural mechanism supporting Conj 53.1

Paper 53 closure: Cor 54.2 (F̃_j poly) + Conj 53.1 (L_j poly) → F̃_j · L_j / N_j → 0 [polynomial / exponential] → E[I_j(0)] ≤ Cj · D_j (mean spectral target) + Paper 52 upgrade 1 (mean → deterministic) + Paper 52 upgrade 2 (L² → pathwise) → H' ```

6.2 Paper 55's role

Paper 55 is a mechanism paper, not an independent closing paper. It identifies the dynamical system (damped oscillation with delayed feedback) that produces the screening return, connects it to the thermodynamic constant η ≈ 0.2, and reduces the remaining gap to precise conjectural inputs.

The mean spectral target still flows through Paper 53's Corollary 4.2. Paper 55 strengthens this by providing:

1. Structural mechanism consistent with the observed screening return: the damped oscillation with r ≈ 0.80 provides a structural bound on positive tail mass via Theorem 55.3. The connection to the thermodynamic absorption rate η ≈ 0.2 cross-validates the contraction.

1. Empirical consistency: F̃·L/N < 0.4 at tested blocks j = 20–32. At j = 26 and 29, L_j is observed to be approximately 2T_j (about two oscillation periods); at j = 32, the return is not yet resolved within the measurement window.

1. Cross-validation: rebound ratio r ≈ 0.80 = 1 − η independently supported by thermodynamic paper.

6.3 Conjectural inputs

The following are empirical laws with mechanism explanations:

1. Conjecture 55.1 (Scale-invariant shell variance): sup_j sup<sub>ℓ≤j</sub> Var<sub>I_j</sub>(η | v_2 = ℓ) < ∞. Supporting structural law for the spectral budget; not directly on the closing chain but explains why per-layer contribution is O(1).

1. Conjecture 55.4 (Geometric contraction): Successive post-peak local maxima of G̃_j satisfy P̃<sub>j,k+1</sub> ≤ r_j · P̃<sub>j,k</sub> with sup_j r_j < 1. (Empirical: r ≈ 0.80 at j = 26, 29.)

1. Conjecture 55.5 (Screening return): F̃_j · L_j / N_j → 0 as j → ∞. (Empirical: F̃·L/N < 0.4 at j = 20–32. Note F̃·L/N increases from 0.146 at j=26 to 0.343 at j=29 within this range; however the exponential growth of N_j is expected to dominate at larger j. The available range is too narrow to confirm the asymptotic trend.)

6.4 The η chain

The fluctuation absorption rate η ≈ 0.20 appears at every level of the H' program:

• Microscopic: 4% innovation per dyadic layer (C(1)_rq ≈ 0.96).
• Mesoscopic: ~20% dissipation per oscillation cycle (compound of ~5 layers).
• Macroscopic: Rebound ratio r = 1 − η ≈ 0.80, controlling positive tail mass.
• Thermodynamic: η ∈ [0.10, 0.31] as fluctuation absorption rate in Ω-shell Lindley queue.

This suggests η is a fundamental constant of the ρ_E system, connecting the integer-level DP dynamics to the spectral-level return behavior.

§7. AI Contributions

Claude (子路, primary author's experimental partner): All computational experiments. Designed and executed paper55v2 (η-field analysis, 3-pass), paper55_crosslayer (70/30 decomposition), paper55_windrift (window drift verification). Identified the field definition error (h(p) ≡ 0 in ρ_E convention). Proposed delayed feedback mechanism for oscillation. Extracted precise oscillation parameters (r = 0.803/0.806, T ratio = 8.000). Connected rebound ratio to η ≈ 0.2. Working notes v1–v3.

ChatGPT (公西华, structural gatekeeper): Four rounds of review. R1: identified that Var(η|ℓ) = O(1) controls budget, not return time. Proposed cross-layer compensation (later falsified). R2: accepted 70/30 law and revised architecture. Proposed window-averaged drift theorem (later falsified by j=32 rebounds). R3: critical upgrade to positive-tail mass formulation (Theorem 55.3). Identified that Paper 53 uses Q_j, not L_j, as primitive. R4: caught normalization bug (raw vs normalized G_j) and algebra error in §5.3 (Q̃_j/N_j is polynomial, not → 0; must go through Paper 53's Corollary 4.2 with exponential N_j). Positioned Paper 55 as mechanism paper, not independent closing paper. Verdict: "direction is right, needs the four fixes."

Gemini (子夏, physical intuition): R1: "群岛效应" (Archipelago Effect) for persistent positive excess (partially confirmed, then revised to coherent wave). Scale invariance as RG fixed point. R2: CAPM single-factor analogy for synchronization. R3: delayed feedback (DDEs produce complex eigenvalues) as mechanism for underdamped oscillation. "粗糙数荒漠" as reflecting boundary.

Grok (子贡, numerical analysis): R1: α = 0.286 global fit, class r=3 artifact identification. R2: F/L ratio ≈ 105 (drift constant). R3: Corrected naive L prediction (multi-oscillation envelope vs single-cycle formula). Trough depth decay rate.

Thermodynamic Claude (热力学审查): Connected Var(η|ℓ) ≈ 0.5 to Var(B) ≈ 1/4 (Paper 19). Confirmed η chain: 4%/layer → 20%/cycle → 0.80 rebound → L_j polynomial. Identified 1 − 0.96^5 ≈ 0.185 as microscopic derivation of η. Proposed paper section structure.

§8. Data and Methods

All experiments used the rho_1e10.bin dataset (ρ_E values for n = 0 to 10^10, Paper 32 convention, int16 binary format).

paper55v2.c: 3-pass analysis. Pass 1: λ regression. Pass 1b: class means. Pass 2: per-layer statistics (Var, C(1)_rq, innovation variance), conditional screening profile, global autocorrelation, Hurst exponent. 455,052,511 primes processed.

paper55_crosslayer.c: Per-block analysis at j = 20, 23, 26, 29, 32. Computes G_same(M) and G_total(M); G_cross = G_total − G_same. Optimized: O(ℓ × n_blocks × M_max) per block.

paper55_windrift.c: Window drift verification at j = 26, 29, 32. Tests ChatGPT's condition (55.5) for window sizes W = 5 to 2000. Reports worst-case window average, fraction of negative windows, predicted L from (55.6).

References

• [P19] Qin, H. ZFCρ Paper 19: Splitting cost. DOI: (series)
• [P32] Qin, H. ZFCρ Paper 32: ρ_E DP convention. DOI: (series)
• [P52] Qin, H. ZFCρ Paper 52: Spectral valley and two upgrades. DOI: (series)
• [P53] Qin, H. ZFCρ Paper 53 v2: Screened long memory and the Screening Return Theorem. DOI: 10.5281/zenodo.19415440
• [P54] Qin, H. ZFCρ Paper 54: Recursive ancestor inheritance and power-law covariance. DOI: 10.5281/zenodo.19426094
• [Thermo] Qin, H. SAE Thermodynamic interface paper. DOI: 10.5281/zenodo.19310282