The UAP_Ω Theorem, Slow Decrease, Cyclotomic Discount Rates, and the Two Final Gaps for H′
ZFCρ Paper XLIX

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

The UAP_Ω Theorem, Slow Decrease, Cyclotomic Discount Rates, and the Two Final Gaps for H'

ZFCρ Paper XLIX

Han Qin

ORCID: 0009-0009-9583-0018

March 2026

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Abstract

Paper 48 reduced H' to two core open problems (prime-layer cancellation, UBPD) and three weak auxiliary inputs. This paper advances all five directions.

First, UAP_Ω^{SW} (Ω-shell-in-AP equidistribution in the Siegel-Walfisz range) is derived from Gafni-Robles 2025 AP Selberg-Delange expansion. The three auxiliary inputs become theorems. Coprime-shell numerical verification: ratio ≈ 1.00 ± 0.01.

Second, the Slow Decrease Theorem is proved: A(X) = Σ_{p≤e^X} η(p)/p satisfies one-sided bounded decrease (A(X+h)-A(X) ≥ -Ch). This eliminates one of two Tauberian conditions.

Third, the prime-layer gap is precisely identified: not slow decrease (proved), but the boundary pseudofunction behavior of (D(1+s)-C)/s. A natural and sufficient intermediate target is Hölder continuity of D(1+it) at t=0 — D(1+it)-C = O(|t|^α) — which combined with upgrading to very slowly decreasing would close prime-layer. The ℓ²/H² route (proposed by Gemini, reviewed by ChatGPT) is fully assessed and shown insufficient: Hardy space boundary lies at Re(s)=1/2, not Re(s)=0; division by s requires A_loc regularity, not L².

Fourth, the Cyclotomic Discount Rate CDR(p,d) = ρ_E(Φ_d(p))/(λ·log Φ_d(p)) ≈ 0.97 (Zsigmondy explanation) establishes the complete physical mechanism for UBPD. Experiment C verifies: large |δ| occurs only under "simultaneous smoothness coincidence."

Fifth, the Restricted Quasi-Super-Additivity (RQSA) hypothesis is proposed. UBPD's remaining bridges include at least three: RQSA (multiplicative incompressibility of p^a-1), quotient rigidity (ρ_E(R_a(p)) ≥ ρ_E(p^{a-1})-C), and benchmark-split rigidity (min_j split ≥ j=1 split - C). General QSA fails (heuristic family shows O(log n) discount), but p^a-1's special algebraic structure (additive successor is a prime power, not smooth) makes RQSA plausible.

H''s two core open problems now have precise mathematical targets: the direct Tauberian input for prime-layer is (D-C)/s boundary pseudofunction behavior (Hölder continuity is a natural intermediate target); UBPD reduces to RQSA + quotient rigidity + benchmark-split rigidity.

Keywords: integer complexity, UAP_Ω, slow decrease, Hölder continuity, cyclotomic discount rate, Zsigmondy, RQSA, UBPD, H' closure

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§1 Introduction

Paper 48 (DOI: 10.5281/zenodo.19328550) reduced H' to two core open problems and three weak auxiliaries. This paper contributes: UAP_Ω^{SW} theorem (§2), Slow Decrease theorem (§3), prime-layer gap identification (§4), CDR + Zsigmondy mechanism (§5), RQSA hypothesis (§6).

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§2 UAP_Ω^{SW} Theorem

2.1 Theorem

For d ≤ (log x)^b, (c,d)=1, 1 ≤ k ≤ K·log₂x (K<2):

S_k(x;d,c) = (1/φ(d))·S_k^{(d)}(x)·(1 + O((log log d)^A / log x))

2.2 Proof sketch

Gafni-Robles Lemma 3.1 → AP Selberg-Delange expansion → coprime shell summation → leading-term cancellation in difference → Cauchy coefficient extraction.

2.3 Coprime-shell data

2.4 Master Proposition

UAP_Ω^{SW} → Lemma A (q-adic a≥2) + Lemma B (E[ω]=O(log log x)) + Lemma C (E[Ω²]=O((log log x)²), needs UAP^×) → tail harmlessness + weak remainder mean + RT predecessor moment.

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§3 Slow Decrease Theorem

3.1 Theorem

There exists an absolute constant C > 0 such that for all sufficiently large X and 0 < h ≤ 1:

A(X+h) - A(X) ≥ -Ch

3.2 Proof

3.3 Numerical support

Window minimum ≈ A(start) throughout. Oscillation = O(1/log x).

3.4 Significance

Debruyne-Vindas one-sided Tauberian requires two conditions: (a) slow decrease, (b) boundary pseudofunction behavior. This theorem closes (a).

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§4 Prime-Layer Gap Identification

4.1 ℓ² route assessment

d_p = η(p)/p ∈ ℓ² (|η|=O(log p), Σ(log p)²/p² < ∞).

But ℓ² cannot close prime-layer. Two gaps:

(a) Dirichlet-series H² has natural boundary at Re(s)=1/2 (Hedenmalm), not Re(s)=0. Boundary values of D(1+it) at Re(s)=0 cannot be obtained from H² directly.

(b) Even with L²_loc boundary for D, (D(1+s)-C)/s requires A_loc (local Wiener algebra) regularity (Debruyne thesis Lemma 8.3.2). L² is insufficient.

However, L²_loc ⊂ pseudofunction is correct — the problem is regularity loss under division by s.

4.2 Direct Tauberian target vs intermediate target

Direct target: (D(1+s)-C)/s has local pseudofunction boundary behavior on Re(s)=0. Combined with slow decrease from §3, Debruyne-Vindas gives A(X) → C.

Natural intermediate target: D(1+it) - C = O(|t|^α) at t=0 for some α > 0 (Hölder continuity).

If Hölder holds → (D(1+it)-C)/(it) = O(|t|^{α-1}), locally integrable → L¹_loc ⊂ pseudofunction. But this is local information near t=0. To trigger Debruyne-Vindas from local information requires upgrading §3's slow decrease to very slowly decreasing, or supplementing with full-boundary pseudofunction control.

Two viable paths:

(a) Hölder at t=0 + very slowly decreasing → closure.

(b) (D-C)/s full-boundary pseudofunction + slow decrease → closure.

§3's theorem currently closes the slow decrease half of path (b). Hölder is the intermediate target for path (a).

4.3 Hölder vs CR-PMH-DS

Full CR-PMH-DS → D(w) analytic at w=1 → at least Lipschitz (α=1 Hölder), hence Hölder for all 0 < α ≤ 1. Hölder is an intermediate target between CR-PMH-DS and β'(0)=0. Much weaker than the former, stronger than the latter.

4.4 Possible attack routes

(a) Character structure: D = Σc'_χ·log L + H'₁₂. Non-principal log L is Hölder at t=0. H'₁₂'s Hölder depends on CR-PMH-DS remainder.

(b) Additive structure + second moment control.

(c) Direct partial-sum with uniform-in-t control.

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§5 CDR + Zsigmondy Mechanism

5.1 CDR definition and data

CDR(p,d) = ρ_E(Φ_d(p)) / (λ·log Φ_d(p))

CDR ≈ 0.97. Zsigmondy's theorem: Φ_d(p) contains primitive prime factors → no multiplicative discount.

5.2 The "cyclotomic trap" clarified

False intuition: Σ φ(d)/2 = a/2 → additive cost (a/2)·λ·log p.

Correction: Φ_d(p) is not a perfect square. Primitive prime factors → CDR ≈ 1 → ρ_E(Φ_d) ≈ λ·φ(d)·log p (full price) → Σ ρ_E(Φ_d) ≈ a·λ·log p = multiplicative path.

5.3 Experiment C: |δ| and simultaneous smoothness

p=23, a=6, δ=-4 (largest |δ|): Φ₁=22, Φ₂=24, Φ₃=553(7·79), Φ₆=507(3·13²). All smooth (largest prime factor 79).

p=23, a=7, δ=-2: Φ₇=292,561 (prime). Primitive prime factor restores rigidity.

5.4 Large-a search

No counterexamples.

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§6 RQSA, Quotient Rigidity, and the Remaining Bridges for UBPD

6.1 General QSA fails (heuristic family)

For large X with X+1 = 2^k: ρ_E(X(X+2)) ≤ 2k+1 ≪ 2λ·log X. Discount O(log n).

Remark. This family depends on primes near 2^k — heuristic from prime distribution, not a constructive counterexample. But it strongly indicates: uniform quasi-super-additivity cannot be expected for general integers. Even without X being prime, X having large prime factors (probability 1) suffices.

6.2 RQSA for p^a-1

Hypothesis (Restricted QSA). There exists an absolute constant C_R such that for all p, a≥2, any factorization UV = p^a-1:

ρ_E(p^a-1) ≥ ρ_E(U) + ρ_E(V) - C_R

6.3 Why RQSA should hold for p^a-1

General QSA failure requires AB+1 (or AB-1) extremely smooth.

But (p^a-1)+1 = p^a is a prime power — not smooth at all. p^a has no "extra" factors providing multiplicative shortcuts. DP cannot exploit a smooth neighbor of p^a-1 because the nearest candidate (p^a) is one of the least smooth structures possible.

Prime-power rigidity of p^a blocks unlimited discounts for p^a-1.

6.4 Successor branch: RQSA + quotient rigidity

UBPD requires ρ_E(p^a) ≥ ρ_E(p)+ρ_E(p^{a-1})-C.

ρ_E(p^a) = min(successor, best multiplicative split). Two branches.

Successor branch: ρ_E(p^a) = ρ_E(p^a-1)+1.

By RQSA (U=p-1, V=R_a(p)): ρ_E(p^a-1) ≥ ρ_E(p-1)+ρ_E(R_a(p))-C_R = ρ_E(p)-1+ρ_E(R_a(p))-C_R.

With quotient rigidity (ρ_E(R_a(p)) ≥ ρ_E(p^{a-1})-C_Q):

ρ_E(p^a) = ρ_E(p^a-1)+1 ≥ ρ_E(p)+ρ_E(p^{a-1})-C_R-C_Q. UBPD holds on successor branch.

6.5 Multiplicative branch: benchmark-split rigidity

If optimal split is j≠1: need min_j[ρ_E(p^j)+ρ_E(p^{a-j})] ≥ ρ_E(p)+ρ_E(p^{a-1})-C_M.

Equivalent to concavity of ρ_E on prime powers — decreasing marginal cost ρ_E(p^a)/a implies j=1 is optimal ("one large + one small" cheaper than "two medium").

Numerically supported: ρ_E(p^a)/a strictly decreasing for p=2,3,5,7,11. But unconditional proof not in hand.

6.6 Summary

Full UBPD requires at least three bridges:

(i) RQSA: ρ_E(p^a-1) ≥ ρ_E(U)+ρ_E(V)-C_R for all factorizations.

(ii) Quotient rigidity: ρ_E(R_a(p)) ≥ ρ_E(p^{a-1})-C_Q.

(iii) Benchmark-split rigidity: min_j split ≥ j=1 split - C_M.

This is a heuristic reduction, not a theorem. All three bridges are hypotheses.

6.7 RQSA status

Hypothesis with complete physical mechanism + numerical support. Formalization requires proving that p^a's prime-power rigidity prevents p^a-1's DP tree from obtaining unbounded discounts. This is a Diophantine / additive combinatorics problem.

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§7 Twisted Sum Correction

Ση·χ/p diverges (c'_χ ≠ 0, expected). Step 2 should be Σa·χ/p converges.

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§8 Complete H' Closure Chain

```

Theorems (Papers 42-49):

Proposition 3, Shell-Depth, Parity, Raw↔Char,

Odd-Pred Reduction, 44+42 chain ✅

UAP_Ω^{SW} (§2) ✅ NEW

→ Lemma A/B/C (SW range) ✅ NEW

Slow Decrease Theorem (§3) ✅ NEW

Heuristic reductions:

RQSA + quotient rigidity + benchmark-split three bridges for UBPD (§6)

rigidity → UBPD

ℓ² route assessment:

d_p ∈ ℓ² ✅

H² boundary at Re=1/2 not Re=0 gap

(D-C)/s needs A_loc not L² gap

Core open (precise mathematical targets):

Prime-layer: (D(1+s)-C)/s boundary pseudofunction

Intermediate: D(1+it) Hölder at t=0 sufficient (with very slow decrease)

UBPD: RQSA + quotient rigidity + benchmark-split three structural hypotheses

Numerical support:

CDR ≈ 0.97, |δ| ≤ 4 ✅

Dyadic blocks → 0, A(X) slow decrease ✅

Coprime-shell ratio ≈ 1.00 ✅

→ Conjecture 2 → H'

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§9 SAE Interpretation

9.1 Two gaps in SAE terms

Hölder continuity = remainder regularity in prime frequency space. β'(0)=0 is zeroth-order information (no principal singularity). Hölder is first-order (bounded rate of change). SAE's "remainder conservation" in the frequency domain.

RQSA = multiplicative incompressibility of DP networks. Prime-power rigidity of p^a blocks additive shortcuts. SAE's "remainder must develop" on the DP tree — no free lunch.

9.2 From SAE axioms to H'

Remainder must develop (Axiom 1) → ρ_E(n) grows → prime-layer fluctuates but conserves (β'(0)=0)

Remainder conserves (Axiom 2) → step-level conservation (U=-D(m-1)) → prime-level regularity (Hölder?)

RQSA is Axiom 1 on multiplicative structure: no unlimited discounts through algebraic identities.

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§10 Open Questions

1. (D(1+s)-C)/s boundary pseudofunction behavior. Hölder continuity of D(1+it) as intermediate target.

1. RQSA for p^a-1. Prime-power rigidity → restricted quasi-super-additivity.

1. Quotient rigidity. ρ_E(R_a(p)) ≥ ρ_E(p^{a-1})-C.

1. Benchmark-split rigidity. Concavity of ρ_E on prime powers.

1. CDR theoretical lower bound. Zsigmondy gives existence; quantitative control needed.

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Data Sources

Scripts: uap_check.c, paper49_exp.c, cdr_exp.c, tauberian_exp.c, zsigmondy_exp.c (C, gcc -O2). N = 10⁷.

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References

[1] H. Qin. Paper XLVIII. DOI: 10.5281/zenodo.19328550.

[2] A. Gafni, N. Robles (2025). arXiv:2502.05298.

[3] A. Roy (2025). arXiv:2511.15928.

[4] K. Zsigmondy (1892). Monatsh. Math. Phys. 3, 265-284.

[5] G. Debruyne, J. Vindas. Complex Tauberian theorems for Laplace transforms.

[6] H. Hedenmalm. Dirichlet series and functional analysis.

[7] P.D.T.A. Elliott (1994). Shifted Primes.

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Acknowledgments

ChatGPT (Gongxihua): UAP_Ω^{SW} complete proof sketch. Slow decrease theorem and "not the bottleneck" judgment. ℓ² route step-by-step review and denial (H² boundary at wrong location + A_loc requirement). Hölder target identification. (D-C)/s requires Debruyne Lemma 8.3.2's A_loc. General QSA counterexample confirmation. RQSA reduction review and correction — identified multiplicative branch as third bridge.

Gemini (Zixia): Zsigmondy/cyclotomic explanation and CDR concept. "Cyclotomic trap" clarification. ℓ²/H² → boundary values route (corrected by ChatGPT). RQSA hypothesis — general QSA fails but p^a-1's prime-power rigidity makes RQSA plausible. "p^a is one of the least smooth structures" as core argument.

Grok (Zigong): Cyclotomic + Lindley initial route (inspired subsequent directions). Concavity data (ρ_E(p^a)/a decreasing). QSA random sample data.

Claude (Zilu): All numerical experiments (coprime-shell UAP, UBPD large-a, CDR, twisted sums, dyadic blocks, oscillation, Experiment C). Text drafting, working notes v1-v5.

Han Qin (author): Experiment design. "What prior is missing?" repeated questioning drove all route discoveries. Final reduction architecture.

Final text independently completed by the author.