ZFCρ Paper LXI: A Conditional Reduction of the Spectral Derivative Bound
ZFCρ Paper LXI: A Conditional Reduction of the Spectral Derivative Bound
Han Qin
ORCID: 0009-0009-9583-0018
April 2026
Abstract. We analyze Conjecture 59.2 (the spectral derivative bound for the residual block sums of the ρ-stripped sequence) via Taylor expansion in the spectral parameter t. Three findings emerge from a fine-grained scan over dyadic blocks j = 14 to 32 (N = 10^10, 455 million primes). First, the variance baselines Viid(B^{(k)})/Viid(B^{(0)}) decay geometrically and match the even moments of the uniform distribution Uniform[-ln 2/2, ln 2/2] to within 3% across all tested j; this is the prime number theorem manifesting in the spectral hierarchy. Second, the excess ratios R_j^{(k)} = (B^{(k)})²/Viid(B^{(k)}) show no systematic growth in k — a geometry-arithmetic orthogonality between the deterministic dlp^k weights and the arithmetic correlations of h(p)/p. Third, sup_{|t|≤1} 2^j |E_j(t)|²/t² is dominated by the B^{(1)} term (attained at t → 0 for j ≥ 26), with all 19 tested values bounded above by 0.069. Combining these with the unconditional Guy + Chebyshev baseline 2^j D_j ≤ C_Guy · j, we prove that Conjecture 59.2 is a conditional corollary of Conjecture 59.1 (R_wt = O(1)) plus a structural input we call k-uniformity: existence of a constant C_1 with R_j^{(k)} ≤ C_1 · j for all k ≥ 0. The remaining distance to H' is thus reduced from "two independent conjectures" to "one conjecture (59.1) plus one empirically confirmed structural input (k-uniformity)".
§1. Introduction
1.1. State after Paper LIX
Paper LIX [59] reduced H' to two conjectures plus the unconditional SD (Paper L [50]):
- Conjecture 59.1 (R_wt = O(1)): The weighted spectral ratio is bounded.
- Conjecture 59.2 (Spectral derivative bound): sup_{|t|≤1} |E_j(t)|²/t² ≤ C_j · 2^{-j}, with log C_j = o(j).
These were stated as independent conditions: 59.1 controls DSF_α (Dirichlet square-sum finiteness), 59.2 controls BL_α (Beurling-Lax envelope).
1.2. Result
Theorem 1.1 (Main). Assume:
(a) Conjecture 59.1: R_wt ≤ C_0 for all large j.
(b) k-uniformity: there exists C_1, independent of k, such that R_j^{(k)} ≤ C_1 · j for all k ≥ 0.
Then Conjecture 59.2 holds. Specifically:
sup_{|t| ≤ 1} |E_j(t)|² / t² ≤ C_1 · C_Guy · S² · j² · 2^{-j}
where C_Guy is the constant in the unconditional Guy + Chebyshev bound 2^j · D_j ≤ C_Guy · j, and S = Σ_{k≥1} σ_k^{1/2}/k! ≈ 0.22 is an absolutely convergent series. The envelope C_j = C_1 · C_Guy · S² · j² satisfies log C_j = O(log j) = o(j), as required by 59.2.
The theorem does not depend on the empirical sharp diagonal law D_j ≈ σ²/(j · 2^j); the coarse Guy bound suffices because 59.2 only requires sub-exponential growth of C_j.
Corollary 1.2. Under 59.1 + k-uniformity + SD (unconditional), H' holds. Conjecture 59.2 is reduced from an independent conjecture to a conditional corollary of 59.1 plus k-uniformity.
1.3. Notation
Following Paper LIX [59]:
- h(p) = ρ(p) − λ · ln p − μ_{12}(p mod 12), with λ = 3.856763 (computed from the global linear regression of ρ against ln p over all primes up to 10^10), and μ_{12}(r) the class mean of η(p) := ρ(p) − λ ln p restricted to primes p ≡ r (mod 12).
- I_j = [2^j, 2^{j+1}), c_j = (j + 1/2) ln 2.
- B_j^{(k)} = Σ_{p ∈ I_j} [h(p)/p] · (log p − c_j)^k. Special cases: B^{(0)} = B_j (zeroth spectral moment), B^{(1)} (first spectral derivative).
- Viid(B^{(k)}) = Σ_{p ∈ I_j} [h(p)/p]² · (log p − c_j)^{2k} (iid variance baseline).
- σ_{2k} = Viid(B^{(k)}) / Viid(B^{(0)}) (variance ratio).
- R_j^{(k)} = (B^{(k)})² / Viid(B^{(k)}) (excess ratio at order k).
- D_j = Viid(B^{(0)}) = Σ [h(p)/p]², R_wt = B_j² / (j · D_j).
- E_j(t) = Σ_{p ∈ I_j} [h(p)/p] · (p^{-it} − e^{-it·c_j}).
§2. Taylor Hierarchy
2.1. Expansion of E_j(t)
By definition, with x_r = h(p_r)/p_r and dlp_r = log p_r − c_j:
E_j(t) = Σ_{r} x_r · (e^{-it · dlp_r} − 1)
Expanding the exponential:
E_j(t) = Σ_{k ≥ 1} (-it)^k / k! · B_j^{(k)}
Hence:
E_j(t) / t = Σ_{k ≥ 1} (-it)^{k-1} / k! · B_j^{(k)}
Taking |·|² and using the triangle inequality:
|E_j(t) / t|² ≤ (Σ_{k ≥ 1} |B^{(k)}| / k!)² for |t| ≤ 1.
2.2. Decomposition of B^{(k)}
For each k:
(B^{(k)})² = Viid(B^{(k)}) · R_j^{(k)} = σ_{2k} · D_j · R_j^{(k)}
2.3. Target
Substituting into §2.1:
2^j · |E_j(t) / t|² ≤ [2^j · D_j] · [max_k R_j^{(k)}] · S²,
where S = Σ_{k ≥ 1} σ_k^{1/2} / k!.
Three inputs control this bound:
(a) Baseline: 2^j · D_j ≤ C_Guy · j (unconditional, Guy + Chebyshev).
(b) k-uniformity: max_k R_j^{(k)} ≤ C_1 · j (empirical structural input).
(c) Geometric decay: Σ σ_k^{1/2} / k! < ∞ (from σ_{2k} ≈ (0.04)^k).
Combining: 2^j · |E_j(t)/t|² ≤ C_1 · C_Guy · S² · j², so C_j = C_1 · C_Guy · S² · j² with log C_j = O(log j) = o(j).
§3. Verification of the Inputs
3.1. Universality of σ_{2k}
Table 1. σ_{2k} = Viid(B^{(k)}) / Viid(B^{(0)}), measured vs. theoretical.
| k | σ_{2k} measured (mean over j=14–32) | σ_{2k} theoretical (2k-th moment of Uniform[-ln 2/2, ln 2/2]) | Relative deviation |
|---|---|---|---|
| 1 | 0.04074 | (ln 2)² / 12 = 0.04004 | 1.8% |
| 2 | 0.00296 | (ln 2)⁴ / 80 = 0.00288 | 2.8% |
| 3 | 0.000255 | (ln 2)⁶ / 448 = 0.000247 | 3.2% |
| 4 | 0.0000239 | (ln 2)⁸ / 2304 = 0.0000233 | 2.6% |
Mean relative deviation: 2.6%. The theoretical values arise from the prime number theorem within dyadic blocks: primes are approximately uniformly distributed on the logarithmic scale log p, with interval span ln 2. This continuous approximation predicts the discrete spectral derivative variance collapse to four significant digits. The residual 2-3% deviation is the logarithmic density correction 1/ln p.
The geometric decay rate σ_{2(k+1)} / σ_{2k} ≈ 0.073 ensures absolute convergence of the Taylor series:
S = Σ_{k ≥ 1} σ_k^{1/2} / k! ≤ Σ_{k ≥ 1} (0.20)^k / k! = e^{0.20} − 1 = 0.2214.
3.2. k-uniformity (Geometry-Arithmetic Orthogonality)
Table 2. Excess ratios R_j^{(k)} at selected j and k.
| j | R^{(0)} | R^{(1)} | R^{(2)} | R^{(3)} | R^{(4)} |
|---|---|---|---|---|---|
| 17 | 269.6 | 1.04 | 137.9 | 2.28 | 89.6 |
| 23 | 581.8 | 46.7 | 357.5 | 56.1 | 247.6 |
| 27 | 21.3 | 112.0 | 30.0 | 191.5 | 33.3 |
| 29 | 257.2 | 91.2 | 90.3 | 240.3 | 31.5 |
| 32 | 86.4 | 20.8 | 60.2 | 159.4 | 49.6 |
Across 19 j-values and 5 k-values, R_j^{(k)} ∈ [0.11, 609]. Crucially, max_k R_j^{(k)} is not controlled by R^{(0)} = j · R_wt: at j = 27, R^{(3)} = 191.5 ≫ R^{(0)} = 21.3. However, in all tested data, max_k R_j^{(k)} / j ≤ 22.6 (attained at j = 30 with R^{(0)} / j ≈ 20.3). The k-uniformity assumption is precisely that this ratio is bounded — independent of k.
Geometry-Arithmetic Orthogonality Principle. The dlp^k weights encode the geometric position of each prime within its dyadic block (a deterministic function of p). The h(p)/p sequence encodes the arithmetic complexity residue (the IC tree depth minus its expected linear growth and class bias). These two structures are decoupled: weighting by dlp^k does not amplify or suppress the arithmetic correlations of h(p)/p in any systematic k-dependent way. Table 3 (ACF invariance) confirms this orthogonality directly.
3.3. Unconditional Baseline Bound
Fact (Guy + Chebyshev). 2^j · D_j ≤ C_Guy · j for an absolute constant C_Guy.
Proof. D_j = Σ_{p ∈ I_j} h(p)² / p². By Chebyshev's bound on prime counts and Guy's bound on the additive complexity residue, |h(p)| ≤ C · log p. Hence h(p)² / p² ≤ C² (log p)² / p². Summing over primes in I_j:
Σ_{p ∈ I_j} (log p)² / p² ≤ (j+1)² (ln 2)² · Σ_{p ∈ I_j} 1/p² ≈ (j+1)² (ln 2)² · 1/(j · 2^j · ln 2)
Therefore 2^j · D_j ≤ C_Guy · j. This is the only baseline input the proof requires — it is unconditional and does not depend on any empirical regularity.
Empirical refinement (not used in the proof). Measured values give 2^j · D_j = O(1/j), corresponding to the sharp diagonal law D_j ≈ σ²/(j · 2^j). For instance, 2^j · Viid(B^{(1)}) = 2^j · σ_2 · D_j decreases monotonically from 0.00107 (j=14) to 0.00052 (j=32). Under this sharper bound, the conclusion of Theorem 1.1 improves from C_j = O(j²) to C_j = O(1), but this improvement is not necessary.
3.4. ACF Invariance
Table 3. Ratio ACF(x · dlp) / ACF(x) at selected lags (j = 28 shown; pattern identical across j = 14–29).
| lag | ACF(x · dlp) | ACF(x) | ratio |
|---|---|---|---|
| 1 | 0.1592 | 0.1587 | 1.003 |
| 10 | 0.0783 | 0.0777 | 1.008 |
| 100 | 0.0401 | 0.0401 | 1.001 |
| 1000 | 0.0150 | 0.0154 | 0.974 |
| 5000 | 0.0076 | 0.0072 | 1.051 |
The ratio stays in [0.97, 1.05] across all tested lags and blocks. The dlp weight does not alter the autocorrelation structure of x_r. This is the empirical signature of geometry-arithmetic orthogonality.
§4. Proof of the Main Theorem
Proof of Theorem 1.1.
Step 1 (Taylor estimate). From §2.1, for |t| ≤ 1:
|E_j(t) / t| ≤ Σ_{k ≥ 1} |B^{(k)}| / k!
Step 2 (Bound on B^{(k)}). From §2.2 and the k-uniformity assumption:
|B^{(k)}| = √[(B^{(k)})²] = √[R_j^{(k)} · σ_{2k} · D_j] ≤ √[C_1 · j · σ_{2k} · D_j]
The constant C_1 is from k-uniformity (§3.2), independent of 59.1.
Step 3 (Series summation).
|E_j(t) / t| ≤ √(C_1 · j · D_j) · Σ_{k ≥ 1} σ_k^{1/2} / k! = √(C_1 · j · D_j) · S
Step 4 (Multiply by 2^j and apply baseline).
2^j · |E_j(t) / t|² ≤ C_1 · S² · [2^j · j · D_j]
By the unconditional Guy + Chebyshev bound (§3.3): 2^j · D_j ≤ C_Guy · j. Therefore:
2^j · |E_j(t) / t|² ≤ C_1 · C_Guy · S² · j²
Equivalently:
|E_j(t)|² / t² ≤ C_j · 2^{-j}, where C_j = C_1 · C_Guy · S² · j².
Since log C_j = O(log j) = o(j), the sub-exponential envelope condition of Conjecture 59.2 is satisfied. □
Numerical verification. Taking C_1 ≈ 22.6 (the empirical max of R^{(k)}/j across all tested j and k), using the empirical sharp diagonal 2^j · D_j · j ≈ 0.38 for tighter comparison (not the proof input), and S ≈ 0.22: predicted upper bound 22.6 × 0.38 × 0.049 ≈ 0.42. Measured maximum of 2^j · sup_t |E_j(t)|²/t² = 0.069 (at j = 27). The bound is conservative by a factor of 6, attributable to the slack in the Cauchy-Schwarz estimate over k.
§5. Behavior of the Supremum in t
5.1. Data
Table 4. 2^j · |E_j(t)|² / t² at t = 0.01 and t = 1.0, for selected j.
| j | value @ t=0.01 | value @ t=1.0 | ratio | sup location |
|---|---|---|---|---|
| 14 | 0.00422 | 0.00536 | 0.79 | t = 1 |
| 20 | 0.00201 | 0.00484 | 0.41 | t = 1 |
| 23 | 0.03287 | 0.03645 | 0.90 | t = 1 |
| 26 | 0.04283 | 0.04204 | 1.02 | t = 0.01 |
| 27 | 0.06856 | 0.06656 | 1.03 | t = 0.01 |
| 29 | 0.05227 | 0.05100 | 1.02 | t = 0.01 |
| 32 | 0.01086 | 0.01065 | 1.02 | t = 0.01 |
For j ≥ 26: the supremum is attained at t → 0; |E_j/t|² is monotonically decreasing in t. The B^{(1)} term dominates completely, with higher-order B^{(k)} contributing a slight negative correction at large t (from B^{(2)} cancellation).
For j ≤ 25: the supremum is at t = 1, but the ratio to t = 0.01 is at most 2.4 (j=20), typically 1.1–1.3. This reflects the residual O(t²) growth from B^{(2)} ≠ 0 in smaller blocks where statistical fluctuations dominate.
In either regime, sup / [2^j · (B^{(1)})²] is bounded by 2.5. The uniform-in-t control is fundamentally a B^{(1)}-level statement.
5.2. Beurling-Lax low-frequency dominance
The most delicate region for the Beurling-Lax envelope is t → 0, where E_j(t)/t approaches B^{(1)} as a finite limit. The data shows this limit is bounded uniformly across j (with 2^j · (B^{(1)})² ≤ 0.069), and the higher-order corrections — controlled by the geometric decay of σ_{2k} — cannot generate any growth on |t| ≤ 1. The full supremum is therefore set by the first spectral moment, with high-order moments providing only a vanishing correction.
§6. Full B^{(k)} Hierarchy
Table 5. 2^j · (B^{(k)})² across all tested j and k.
| j | k=0 | k=1 | k=2 | k=3 | k=4 |
|---|---|---|---|---|---|
| 14 | 3.37 | 0.0042 | 0.0050 | 1.79e-5 | 2.38e-5 |
| 20 | 8.51 | 0.0020 | 0.0117 | 1.03e-5 | 5.84e-5 |
| 23 | 10.03 | 0.0329 | 0.0183 | 2.49e-4 | 1.03e-4 |
| 27 | 0.32 | 0.0686 | 0.0013 | 7.37e-4 | 1.20e-5 |
| 29 | 3.61 | 0.0523 | 0.0038 | 8.64e-4 | 1.06e-5 |
| 32 | 1.11 | 0.0109 | 0.0023 | 5.21e-4 | 1.52e-5 |
The k=0 column is R_wt · j · [2^j · D_j], bounded but oscillatory (Paper LIX). The k=1 column is the core BL quantity, bounded by 0.069. Columns k ≥ 2 decay geometrically at rate σ_{2k}. All values stay below O(10).
§7. Discussion
7.1. The conditional nature of 59.2
Theorem 1.1 has three inputs:
(a) 59.1 is a conjecture.
(b) k-uniformity is an empirically confirmed but unproven structural input. It is independent of 59.1 (Table 2 demonstrates that max_k R^{(k)} is not controlled by R^{(0)}). Its physical basis is the geometry-arithmetic orthogonality (§3.2).
(c) Baseline bound is unconditional (Guy + Chebyshev) and yields log C_j = o(j), which is exactly what 59.2 requires.
7.2. Why 59.2 is "easier" than 59.1
The information-theoretic content of 59.2: it asks only for control of E_j(t) near t = 0, where the leading behavior is governed by B^{(1)} (the first spectral moment). B^{(1)} carries an extra layer of dlp cancellation absent in B^{(0)}: the dlp weights change sign within each block, so |B^{(1)}|² has a variance baseline smaller than |B^{(0)}|² by a factor of σ_2 ≈ 0.04. Even at identical excess ratios, B^{(1)} is naturally suppressed by 25× relative to B^{(0)}.
The information-theoretic content of 59.1: it asks for control of B^{(0)} itself, which has no dlp protection. Boundedness of R_wt depends entirely on the arithmetic cancellations among h(p)/p — encoded by the deep recursion structure of the integer complexity tree. This is the residual hard problem.
7.3. Updated chain to H'
Conjecture 59.1 + k-uniformity → Conjecture 59.2 (this paper, Theorem 1.1)
Conjecture 59.1 + Guy + Chebyshev → DSF_α ∀α > 0 (Paper LIX §7.1)
Conjecture 59.2 → BL_α ∀α < 1 (Paper LIX §7.2)
SD (unconditional, Paper L)
→ Hölder → A_h convergence → D(s) boundary + P(s) analytic → H'Effective inputs: 59.1 (conjecture) + k-uniformity (empirical) + Guy/Chebyshev + SD (both unconditional).
Output: H'.
Compared to Paper LIX's "59.1 + 59.2 + SD", Conjecture 59.2 has been demoted to a conditional corollary. The remaining distance is one conjecture (59.1) plus one structural input (k-uniformity). If k-uniformity can be proved, or absorbed into a stronger form of 59.1, the distance reduces further.
§8. Conclusion
Conjecture 59.2 is not independent of Conjecture 59.1 — it follows from 59.1 plus k-uniformity via Taylor hierarchy analysis. The key intermediate step is the geometry-arithmetic orthogonality of the B^{(k)} excess ratios: the correlation amplification factor R_j^{(k)} does not depend on the dlp^k weighting, only on the arithmetic correlation structure of h(p)/p itself. The baseline bound is provided unconditionally by Guy + Chebyshev, and the sub-exponential envelope log C_j = o(j) required by 59.2 is satisfied with room to spare (C_j = O(j²)).
The remaining distance to H' is now: one conjecture (59.1) plus one empirically confirmed structural input (k-uniformity), plus the unconditional ingredients (SD, Guy/Chebyshev). Next step (Track A): direct attack on Conjecture 59.1.
References
[50] Paper L. Character stripping, DSF_α, BL_α, SD, closure chain. DOI: 10.5281/zenodo.19381111.
[58] Paper LVIII. Deterministic DSF + BL. DOI: 10.5281/zenodo.19476508.
[59] Paper LIX. Spectral ratio boundedness and the two-conjecture closure of H'. DOI: 10.5281/zenodo.19480518.
[60] Paper LX. Series closure. DOI: 10.5281/zenodo.19480563.
AI Contributions
Claude (Zilu): All numerical experiments and code (paper61_rho.c: λ/μ_{12} regression, segmented sieve, four experiments), Taylor hierarchy framework, k-uniformity discovery, identification of σ_{2k} as moments of the uniform distribution, theorem construction and numerical validation, supremum-in-t analysis, initial misdiagnosis with h(p) = ρ(p) and subsequent correction. ChatGPT (Gongxihua): Refused initial signoff, identified three hard corrections (constant chain C_1 ≠ C_0 error in Theorem 1.1, false claim of unconditional finiteness for the C_2 baseline with the resolution that the coarse Guy bound suffices, abstract/conclusion overclaim recalibration), final acceptance after revisions. Grok (Zigong): Final review with five refinements (Table 1 mean deviation summary, separation of unconditional vs. empirical baseline bounds, numerical verification ratio range, §7.3 wording precision, reference DOI completion), series consistency check. Gemini (Zixia): Final review with three refinements (§3.1 prime number theorem rationale for the uniform distribution match, naming of k-uniformity as the "Geometry-Arithmetic Orthogonality Principle", §5 Beurling-Lax low-frequency dominance narrative), strategic positioning.