Character Stripping, Deterministic Square-Function, and the Precise Localization of H′
ZFCρ Paper L

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

ZFCρ Paper L: Character Stripping, Deterministic Square-Function, and the Precise Localization of H'

Han Qin (秦汉)

ORCID: 0009-0009-9583-0018

April 2026

---

Abstract. We precisely localize the remaining difficulty for Hypothesis H' in the ZFCρ framework. The 94% positive bias of η(p) is identified as an arithmetic masquerade from p mod 12 structure, with an a priori derivation from the {2,3}-cost basis of integer complexity, and removable by character stripping. Four candidate proof routes are rigorously excluded. The stripped dyadic block sums B_j decay at rate |B_j| ~ 2^{−j/3} — exponentially faster than the provable bound |B_j| ≤ C/√j — and this gap is formalized as the deterministic square-function hypothesis DSF_α. The Tauberian condition (slowly decreasing) is proved unconditionally from the Guy upper bound and Chebyshev's prime count. The prime-power layer converges absolutely in a neighborhood of Re(s) = 1, making it analytic past the boundary and non-critical for the prime-layer singularity structure. The full closure chain reduces to two conditions: DSF_α (block regularity) and BL (block localization summability). Twenty-four unconditional results at scales N = 10⁷, 10⁹, 10¹⁰ support the localization. Paper 49 identified two open problems for H'; the present paper transforms them from vague targets into precisely stated, numerically verified, formally testable mathematical objects.

---

§1. Introduction and Strategic Context

1.1. The state after Paper 49

Paper 49 reduced the closure of H' to two open problems:

• Gap 1 (prime-layer cancellation): Does the stripped Dirichlet series H(s) = Σ_p h(p) p^{−s} have Hölder-continuous boundary behavior at Re(s) = 1?
• Gap 2 (UBPD): Is |δ_a(p)| = O(1) for all a ≥ 2?

Both were identified as requiring genuinely new mathematics, but neither was formulated as a precise, testable hypothesis. The present paper transforms both gaps: Gap 1 is localized to a formally stated pair of conditions (DSF_α + BL, §6) with a precise polynomial-to-exponential gap (§5) and four excluded routes (§3); Gap 2 is decomposed into its internal structure (§7) and, within the Dirichlet series architecture, shown to contribute only an analytic (non-singular) term at the boundary Re(s) = 1 — so Gap 2 does not obstruct the prime-layer singularity analysis.

A further simplification: the Tauberian condition (slowly decreasing partial sums) is proved unconditionally in §4.4.

1.2. Convention

We work in the Paper 32 convention throughout: ρ(0) = 0, ρ(1) = 0, multiplication cost +2, giving λ ≈ 3.86. This convention has computational data at N = 10⁹ and N = 10¹⁰. All structural results — character stripping, block decay patterns, Gap 1/Gap 2 decoupling, RQSA, CDR stability — are convention-independent.

For reference, the standard integer complexity convention (‖1‖ = 1, multiplication free, λ ≈ 3.04) is used in certain verification experiments. A translation table appears in Appendix A.

1.3. Notation

• η(p) = ρ(p) − λ log p (prime deficiency)
• δ_a(p) = ρ(p^a) − ρ(p) − ρ(p^{a−1}) (sub-additivity defect at prime power p^a)
• h(p) = η(p) − μ_{12}(p) where μ_{12}(p) = E[η | p mod 12] (character-stripped residual)
• B_j = Σ_{2^j < p ≤ 2^{j+1}} h(p)/p (dyadic block sum)
• H(s) = Σ_p h(p) p^{−s} (stripped Dirichlet series)
• D(s) = Σ_p η(p) p^{−s} (full prime-layer Dirichlet series)
• P⁻(n) = smallest prime factor of n
• CDR(p, a) = ρ(Φ_a(p)) / (λ · log Φ_a(p)) (convergence drift rate)

---

§2. Character Stripping: Arithmetic Masquerade and Q = 12

2.1. A priori conductor selection

In the ρ-function, every integer n is expressed as a tree of 1s under addition and multiplication. The two cheapest primes are 2 (cost 2) and 3 (cost 3), with cost-to-log ratios 2/ln 2 ≈ 2.89 and 3/ln 3 ≈ 2.73 respectively — both near λ. All larger primes have higher cost/log. Consequently, n is "cheap" (ρ(n) well below λ log n) precisely when n factors richly over {2, 3}.

For a prime p, the dominant access paths are additive: p = (p−1) + 1, with cost ≈ ρ(p−1) + 1. Thus ρ(p) is controlled by the {2, 3}-smooth part of p − 1 (and p + 1, via the shortcut p² − 1 = (p−1)(p+1)). The {2, 3}-valuations of p ± 1 are determined by p mod lcm(4, 3) = 12. The four admissible prime residue classes are:

Q = 12 is the minimal conductor capturing the full {2, 3}-valuation structure of p ± 1. Higher conductors (Q = 24, 60) capture secondary corrections; the between-class variance at Q = 12 already accounts for essentially all of the bias structure.

2.2. Stripping procedure and symmetrization

Define residue class means μ_{12}(r) = lim_{X→∞} (1/π_r(X)) Σ_{p ≤ X, p ≡ r (12)} η(p), and the stripped residual h(p) = η(p) − μ_{12}(p mod 12).

Table 1. Symmetrization at three scales.

The 94% positive bias is entirely explained by low-conductor arithmetic structure. After stripping, h(p) is near-symmetric, mean-zero, with no visible drift.

2.3. Variance decomposition

Total Var(η) = 0.613 decomposes as between-class (Q = 12) = 0.109 (17.8%) and within-class Var(h) = 0.504 (82.2%). This is consistent with Paper 47's 17.6% figure.

2.4. Convergence tests

The critical ℓ² norm Σ|h|²/p² is fully converged at all three scales:

(Different absolute values across conventions; convergence pattern identical.)

2.5. Relationship between D(s) and H(s)

The full prime-layer Dirichlet series decomposes as:

D(s) = Σ_p η(p) p^{−s} = H(s) + Σ_{r ∈ R} μ_{12}(r) · P_r(s)

where R = {1, 5, 7, 11} and P_r(s) = Σ_{p ≡ r (12)} p^{−s}. By the prime number theorem for arithmetic progressions, each P_r(s) has a logarithmic singularity at s = 1:

P_r(s) = (1/φ(12)) · log(1/(s−1)) + analytic

with φ(12) = 4. The combined residue-class contribution is:

Σ_r μ_{12}(r) P_r(s) = (μ̄/4) · log(1/(s−1)) + analytic

where μ̄ = Σ_r μ_{12}(r) ≈ 4.94 (sum of all class means). This is a known singularity — its structure is fully determined by μ_{12} and the Dirichlet L-functions for characters mod 12, all of which are classical.

Therefore: the unknown component of D(s) at s = 1 is entirely H(s). If H(s) has Hölder-continuous boundary behavior at Re(s) = 1, the full prime-layer Dirichlet series has boundary behavior of the form "known log-singularity + Hölder remainder." This is the precise content of Gap 1.

---

§3. Eliminated Routes

Before presenting the successful path, we record four approaches that were investigated and conclusively excluded. This narrows the strategy space and motivates the square-function approach of §4–§6.

3.1. H² boundary shortcut

Claim (excluded): The Dirichlet series (H(1+s) − H(1))/s lies in the Hardy space H² of the half-plane, from which boundary regularity would follow.

Obstruction: The H² norm corresponds to an L² boundary on Re(s) = 1/2 in the original Dirichlet half-plane, not on Re(s) = 1 where the prime-layer singularity lives (Hedenmalm). The quotient (H(1+s) − H(1))/s requires A_loc regularity, not L² regularity (Debruyne, Lemma 8.3.2). H² is the wrong functional space.

3.2. Direction D (absolute Lipschitz)

Claim (excluded): Prove |H(1+it) − H(1)| ≤ C|t| directly from absolute bounds on η(p).

Obstruction: This would require Σ|η(p)| · log p / p < ∞. Data:

Growth rate ≈ 5.5 · log log x. The sum diverges. Direct Lipschitz from absolute bounds is impossible. Hölder requires structural input — cancellation, not size control.

Character stripping does not save Direction D: Σ|h| · log p / p still diverges (8.13 at Q = 12 vs 15.27 raw at N = 10⁷).

Remark. Direction D exclusion is non-load-bearing: the DSF_α proof path (§6) does not depend on it. It is recorded to narrow the strategy space, not as a premise.

3.3. Kolmogorov three-series

Claim (excluded): Apply the Kolmogorov three-series theorem to h(p), treating it as a sequence of "nearly independent" random variables.

Obstruction: h(p) is a deterministic sequence — it is a function of p's integer complexity, which is a fixed number. The Kolmogorov theorem requires independent random variables. Despite h(p)'s white-noise-like statistics (near-zero mean, bounded variance, weak autocorrelation), independence is a structural property that deterministic sequences cannot satisfy. At N = 10⁹/10¹⁰, the white-noise prediction breaks down for dyadic blocks j > 22 (actual |B_j| is 5–20× larger than white-noise σ), confirming residual deterministic correlations.

3.4. Montgomery–Vaughan alone

Claim (excluded): The Montgomery–Vaughan inequality gives global L² control sufficient for boundary regularity.

Obstruction: MV gives ∫|H(1+it)|² dt < ∞ — global L² on the boundary. But boundary regularity (Hölder) at a point requires local control. L² and Hölder are independent conditions. MV contributes to the global picture but cannot close the local gap.

---

§4. Dyadic Block Decomposition

4.1. Block sums at three scales

Define the dyadic block sum B_j = Σ_{2^j < p ≤ 2^{j+1}} h(p)/p for stripped residuals h(p).

Table 2. Selected stripped block magnitudes |B_j|.

4.2. Three findings

Finding 1 (Exponential decay, slower than WN). Block sums decay at approximate rate |B_j| ~ 2^{−j/3}, spanning ~3 orders of magnitude from j = 10 to j = 32. This is slower than the white-noise prediction |B_j| ~ 2^{−j/2} but still exponential. The pattern is robust across all three scales.

Finding 2 (Square-function convergence). The cumulative sum Σ|B_j|² converges numerically:

Tail (j > 20) contributes < 0.001% of total. Confirmed across three orders of magnitude.

Finding 3 (A_h convergence). The stripped partial sum A_h(X) = Σ_{p ≤ X} h(p)/p converges numerically at all scales:

(Different values reflect different conventions; convergence behavior identical.)

4.3. Stripped Hölder exponent

Table 3. Effective Hölder exponent α_h at multiple scales.

The effective exponent is scale-dependent (slightly lower at larger N) but remains positive and well above zero at small t across all scales. The target is any α > 0.

4.4. Slowly decreasing — unconditional

Proposition 4.1 (SD). The partial sums A_h(X) = Σ_{p ≤ X} h(p)/p are slowly decreasing: for every ε > 0 there exists δ > 0 such that A_h(Y) − A_h(X) ≥ −ε for all X ≥ 2 and Y ∈ [X, (1+δ)X].

Proof. The Guy upper bound gives ρ(n) ≤ (3/ln 2) log n for all n ≥ 1. With the class means |μ_{12}| bounded, |h(p)| ≤ C log p for a universal constant C. Each term satisfies |h(p)/p| ≤ C log p / p. For Y ∈ [X, (1+δ)X]:

Each summand is at most C log((1+δ)X) / X ≤ 2C log X / X (for δ ≤ 1). The number of primes in (X, (1+δ)X] is at most C'δX / log X (Chebyshev). Therefore:

Given ε > 0, take δ = ε/(2CC'). □

Remark. This is SD, which is strictly stronger than VSD (very slowly decreasing). The proof uses only the Guy upper bound and Chebyshev's prime count — no character stripping, no variance bound, no numerical input. SD is an unconditional, trivially satisfied Tauberian condition for any 1/p-weighted sum with polynomially bounded numerators.

---

§5. The Polynomial–Exponential Gap

This section isolates the precise mathematical gap that the conditional hypotheses must bridge.

5.1. What is provable (polynomial decay)

Proposition 5.1. If Var(h(p)) = O(1) (numerically confirmed: Var(h) = 0.504), then |B_j| ≤ C / √j for all j.

Proof. By Cauchy–Schwarz:

The first factor satisfies Σ_{p ∈ I_j} h(p)²/p² ≤ Var(h) · Σ_{p ∈ I_j} 1/p² ≤ C/2^j (since Σ 1/p² over I_j ~ 1/2^j by partial summation). The second factor is π(2^{j+1}) − π(2^j) ~ 2^j/j (PNT). Therefore:

Note that Σ 1/j diverges, so the Cauchy–Schwarz bound does not imply Σ|B_j|² < ∞. The square-function convergence established numerically in §4.2 — Σ|B_j|² finite, with negligible tail — is a strictly stronger fact than what Proposition 5.1 proves.

5.2. What is needed (exponential decay)

For DSF_α with α > 0, we need Σ j^{1+2α} |B_j|² < ∞. With the provable bound |B_j|² ≤ C/j, the summand is j^{1+2α}/j = j^{2α}, which diverges for all α > 0. The Cauchy–Schwarz bound is insufficient for any positive Hölder exponent.

The data shows |B_j| ~ 2^{−j/3}, which is exponential. With this:

j^{1+2α} · |B_j|² ~ j^{1+2α} · 2^{−2j/3}

which converges for all α > 0 (exponential beats any polynomial). The DSF_α series converges numerically:

5.3. The gap

Bridging this gap — explaining why stripped blocks cancel exponentially, not just at the C/√j rate — is the central remaining mathematical problem for Gap 1.

---

§6. Conditional Closure Theorem

6.1. Definitions

Hypothesis DSF_α. For α > 0, the deterministic square-function condition DSF_α is:

Σ_{j ≥ 0} (1 + j)^{1+2α} |B_j|² < ∞

where B_j = Σ_{2^j < p ≤ 2^{j+1}} h(p)/p are stripped dyadic block sums.

Hypothesis BL_α. The block localization condition BL_α requires that the intra-block error is summably small: defining

E_j(t) = Σ_{p ∈ I_j} h(p)/p · (p^{−it} − e^{−it · c_j})

where c_j = (j + ½) ln 2 is the block log-center, we require:

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

for some t₀ > 0.

6.2. Status of BL_α

Proposition 6.1 (Per-Block Bound). |E_j(t)| ≤ (|t| ln 2 / 2) · Σ_{p ∈ I_j} |h(p)|/p for all j.

Proof. For p ∈ I_j, |log p − c_j| ≤ ½ ln 2. Thus |p^{−it} − e^{−it·c_j}| ≤ |t| · |log p − c_j| ≤ |t| · ln 2 / 2. □

This gives |E_j(t)| / |t|^α ≤ C |t|^{1−α} · A_j, where A_j = Σ_{p ∈ I_j} |h(p)|/p. For fixed t, the ratio vanishes as t → 0 for each j individually. However, BL_α requires the sum over j to converge, and Σ_j A_j = Σ_p |h(p)|/p diverges (since E[|h(p)|] > 0). Therefore BL_α does not follow from the per-block bound alone.

Numerical evidence. At t = 0.01, |E_j(t)| / |B_j| < 3.2% for all j ≥ 4 (N = 10⁹ data). The block localization error is empirically negligible relative to the block sum itself. This strongly suggests BL_α holds, but does not constitute a proof.

Assessment. BL_α is a secondary technical condition. The mathematical substance of Gap 1 lies in DSF_α (the block decay rate), not in BL_α (the intra-block frequency spread). A proof of DSF_α would likely carry BL_α as a byproduct, since exponential block decay would make the summability condition easy to satisfy. But we do not claim BL_α is "free" — it is an additional hypothesis, expected to be subordinate to DSF_α but not yet proved from it.

6.3. Main theorem

Theorem 6.2 (Conditional Closure of Gap 1). Suppose DSF_α holds for all α < α (for some α > 0), and BL_α holds for some α ∈ (0, α*). Then:

(i) H(1 + it) − H(1) = O(|t|^{α−ε}) for every ε > 0.

(ii) The stripped Dirichlet series H(s) has Hölder-continuous boundary behavior at Re(s) = 1.

Proof. Write H(1+it) − H(1) = M(t) + E(t), where

M(t) = Σ_j (e^{−it · c_j} − 1) · B_j, E(t) = Σ_j E_j(t).

Main term. |e^{−it·c_j} − 1| ≤ C_α |t · c_j|^α ≤ C'_α |t|^α (1+j)^α. By Cauchy–Schwarz:

Σ_j (1+j)^α |B_j| ≤ (Σ_j (1+j)^{−1−ε})^{1/2} · (Σ_j (1+j)^{1+2α+ε} |B_j|²)^{1/2}

The first factor is finite for ε > 0. The second factor is finite by DSF_{α+ε/2}, which holds for sufficiently small ε since DSF_β holds for all β < α*. Thus |M(t)| = O(|t|^{α−ε}).

Error term. |E(t)| ≤ Σ_j |E_j(t)| ≤ |t|^α · Σ_j |E_j(t)| / |t|^α, and BL_α gives Σ_j |E_j(t)| / |t|^α < ∞ for |t| ≤ t₀. So |E(t)| = O(|t|^α).

Combining: |H(1+it) − H(1)| ≤ |M(t)| + |E(t)| = O(|t|^{α−ε}). □

Remark on the hypothesis. The proof uses DSF_{α+ε/2}, not just DSF_α. The condition "DSF_α for all α < α" is the honest hypothesis. Numerically, DSF_α converges for all α tested up to 0.5 (Table in §5.2), with smooth growth in α suggesting α ≥ 0.5.

6.4. From Hölder to convergence of A_h

Proposition 6.3 (Tauberian Closure). If H(s) extends continuously to s = 1 (i.e., H(1) := lim_{σ→1⁺} H(σ) exists and |H(1+it) − H(1)| = O(|t|^β) for some β > 0), then A_h(X) = Σ_{p ≤ X} h(p)/p converges to H(1).

Proof. The argument uses only the real-axis approach to the boundary, avoiding the distributional subtleties of boundary behavior on the imaginary line.

Step 1 (Laplace–Stieltjes representation). By Abel summation, the shifted series admits the representation

H(1 + ε) = ∫₀^∞ e^{−εu} dμ(u), ε > 0,

where μ(u) = A_h(e^u) = Σ_{p ≤ e^u} h(p)/p. This is a Laplace–Stieltjes transform.

Step 2 (Real-axis limit). Hölder continuity at s = 1 implies H extends continuously to Re(s) = 1 near s = 1. In particular, the real-axis limit exists: H(1 + ε) → H(1) as ε → 0⁺.

Step 3 (One-sided Tauberian condition). Proposition 4.1 gives SD: for every δ > 0 sufficiently small, μ(v) − μ(u) ≥ −ε whenever 0 ≤ v − u ≤ δ. This is the one-sided Tauberian condition.

Step 4 (Karamata's Tauberian theorem). The Karamata–Korevaar Tauberian theorem for Laplace–Stieltjes transforms states: if ∫₀^∞ e^{−εu} dμ(u) → L as ε → 0⁺, and μ is slowly decreasing, then μ(u) → L as u → ∞ (Korevaar, Tauberian Theory, Ch. II–III; see also Hardy–Littlewood, 1914, for the classical prototype). Applying this with L = H(1): μ(u) → H(1), i.e., A_h(X) → H(1) as X → ∞. □

Remark. This proof is deliberately elementary: it uses only the real-axis limit H(1+ε) → H(1), not the full boundary behavior of H(1+it) on the imaginary line. The Hölder condition is therefore stronger than what the Tauberian step requires — any condition implying the existence of a finite real-axis limit at s = 1 would suffice. Paper 49 identified two Tauberian routes involving the imaginary-line boundary (Hölder + VSD, or local pseudofunction + SD); the real-axis route via Karamata's theorem is simpler and avoids both complications, since SD (which we prove unconditionally) is exactly the one-sided condition that theorem requires.

6.5. From A_h to the prime-layer boundary behavior

Proposition 6.4 (Recovery). If A_h(X) converges, then the full prime-layer Dirichlet series D(s) = Σ η(p) p^{−s} has boundary behavior at s = 1 of the form

D(s) = (μ̄/4) · log(1/(s−1)) + c + o(1)

as s → 1⁺, where μ̄ = Σ_r μ_{12}(r) ≈ 4.94 and c = A_h(∞) + Σ_r μ_{12}(r) c_r is a computable constant.

Proof sketch. D(s) = H(s) + Σ_r μ_{12}(r) P_r(s) (§2.5). If A_h converges, then H(s) → H(1) = A_h(∞) as s → 1⁺ (Abel's theorem). Each P_r(s) = (1/4) log(1/(s−1)) + c_r + o(1) by PNT in arithmetic progressions. Combining: D(s) = H(1) + (μ̄/4) log(1/(s−1)) + Σ_r μ_{12}(r) c_r + o(1). The singularity is entirely from the known residue-class structure; the unknown component H(1) is a finite constant. □

Remark. The question "Does Σ η(p)/p converge?" has the answer no — η(p) has positive mean ≈ 1.235, so Σ η(p)/p diverges like (5/4) log log X. The correct formulation of Gap 1 is not convergence of Σ η(p)/p, but regularity of H(s) at the boundary, equivalently convergence of Σ h(p)/p. Character stripping separates the known divergent part from the unknown regular part.

---

§7. Gap 2: Prime-Power Layer

7.1. Structural decoupling (a corollary)

Proposition 7.1 (Analyticity of Prime-Power Layer). The prime-power contribution

P(s) = Σ_p Σ_{a ≥ 2} δ_a(p) / p^{as}

converges absolutely for Re(s) > 1/2. In particular, P(s) is analytic in the half-plane Re(s) > 1/2, which contains the boundary Re(s) = 1 and a neighborhood. Therefore the singularity structure of the full Dirichlet series at Re(s) = 1 is entirely determined by the prime layer.

Proof. Sub-additivity gives |δ_a(p)| ≤ ρ(p^a) ≤ Ca log p. For Re(s) = σ > 1/2:

The last sum converges for 2σ > 1, i.e. σ > 1/2. □

Remark. This is a one-line consequence of the Euler product architecture. Its significance is strategic: within the Dirichlet series proof path, Gap 2 does not create singularities at the boundary Re(s) = 1. The prime-power layer passes through this boundary analytically. UBPD retains independent interest as a structural property of ρ_E on prime powers, and we develop its internal structure in §7.2–§7.6 for this reason.

7.2. a = 2 versus a ≥ 3: complete separation

Table 5. Defect statistics by power (P32, N = 10¹⁰).

The hard core of Gap 2 is exclusively a = 2. For a ≥ 3, the Zsigmondy mechanism — primitive prime divisors in Φ_a(p) prevent smooth-factorization shortcuts — keeps |δ_a| ≤ 3 across all data.

7.3. Log-UBPD for a = 2

Proposition 7.2 (Unconditional Log-UBPD). |δ₂(p)| ≤ 1.60 · log p + O(1) for all primes p.

Proof. From the established bounds on integer complexity: ρ(n) ≤ (3/ln 2) log n (Guy) and ρ(n) ≥ (3/ln 3) log n (Selfridge). Applied to p²:

δ₂(p) = ρ(p²) − 2ρ(p) ≥ (3/ln 3) log(p²) − 2(3/ln 2) log p = (6/ln 3 − 6/ln 2) log p

The coefficient |6/ln 3 − 6/ln 2| ≈ 1.60. The upper bound is symmetric. □

Empirical envelope: |δ₂| ≤ 0.62 · log p + 1 — much tighter, but the unconditional bound suffices.

7.4. Scissors structure

Proposition 7.3 (Sub-Additivity Bound). ρ(p²) ≤ ρ(p−1) + ρ(p+1) + 1 for all primes p.

Proof. p² − 1 = (p−1)(p+1), so ρ(p² − 1) ≤ ρ(p−1) + ρ(p+1). Then p² = (p² − 1) + 1, giving ρ(p²) ≤ ρ(p² − 1) + 1. □

This gives the unconditional bound δ₂(p) ≤ ρ(p−1) + ρ(p+1) + 1 − 2ρ(p): the defect at p² is controlled by the complexity gap between p + 1 and p − 1 relative to p itself.

Empirical observation (Scissors Formula). In standard IC (N = 10⁷, 445 primes), the sharper inequality δ₂(p) ≤ ρ(p+1) − ρ(p−1) − 1 holds with 0 violations and 232 tight cases. This sharper form would follow from ρ(p) = ρ(p−1) + 1 for all primes — verified for all p ≤ 10⁷ but not yet proved. We record it as a pattern; a formal proof belongs to a future paper.

Interpretation. Worst |δ₂| occurs when the "scissors" opens maximally — p + 1 extremely smooth (cheap) and p − 1 rough (expensive). The δ₂ = −7 cases (p = 26459: p + 1 = 2²·3³·5·7², 7-smooth; p − 1 = 2 × 13229) exhibit exactly this pattern.

7.5. RT moments and CDR stability

The convergence drift rate CDR(p, a) = ρ(Φ_a(p))/(λ · log Φ_a(p)) measures whether cyclotomic factors track λ log faithfully.

Proposition 7.4 (Shell-Side RT Moment). E[(log P⁻(n))² | Ω(n) = k] = O(1), converging to (ln 2)² ≈ 0.48 for k ≥ 4. (Classical.)

Table 6. Predecessor-side RT moment.

Predecessor moment is ~120× larger than shell moment, but stabilizes at O(1) — flat from k = 10 to k = 24, no upward trend. Physical reason: high-Ω(n) numbers are almost all even, so n − 1 is odd, and P⁻(n−1) has the distribution of the smallest odd prime factor of a generic odd number.

Cross-term independence: E[log P⁻(n) · log P⁻(n−1) | Ω = k] / (E[shell] · E[pred]) → 1.00 for k ≥ 17.

Interface with H' proof. RT moments enter through CDR variance control:

E[(CDR − 1)² | Ω = k] ≤ C · E[(log P⁻)² | Ω = k] / (log Φ)²

Since the numerator is O(1) and the denominator grows, CDR fluctuations vanish.

7.6. RQSA

1,877 factor-pair checks on p^a − 1 (p ≤ 47, p^a ≤ 10⁸): zero violations. RQSA remains the strongest single bridge toward a formal UBPD proof.

---

§8. Precise Localization of the Remaining Difficulty

8.1. What Paper 49 left open

Paper 49 stated two open problems: "prime-layer cancellation" and "UBPD." Neither had a formal hypothesis, a precisely identified gap, or a list of excluded approaches. Both were targets, not maps.

8.2. What Paper 50 delivers

For Gap 1 (prime layer):

For Gap 2 (prime powers):

8.3. The closure chain

The full conditional chain to H':

```

DSF_α + BL_α (UNPROVED — the remaining hypotheses)

→ Hölder regularity of H(s) at s = 1 [Thm 6.2, conditional]

→ real-axis limit H(1+ε) → H(1) [continuity, unconditional given Hölder]

+ SD [Prop 4.1, unconditional]

→ A_h(X) converges [Prop 6.3, Karamata Tauberian]

→ D(s) = known log-singularity

+ finite constant [Prop 6.4, unconditional]

+ Prime-power layer analytic [Prop 7.1, unconditional]

→ H' boundary behavior

```

Every link except "DSF_α + BL_α" is either proved unconditionally or is a known theorem from the literature. The remaining difficulty is concentrated in two conditions: DSF_α (block decay rate, the hard mathematical content) and BL_α (block localization summability, a secondary technical condition expected to follow from DSF_α but not yet reduced to it).

8.4. The remaining mathematical problems

DSF_α is the principal remaining problem. BL_α is expected to be subordinate — a proof of exponential block decay would likely carry BL_α as a corollary, since exponentially decaying B_j would make the summability condition easy to satisfy. The Zsigmondy bound and predecessor moment are not H'-critical obstructions within the current proof architecture, but retain independent structural interest for the ZFCρ framework: they characterize the multiplicative regularity of ρ_E at prime powers and the statistical independence of consecutive factorizations.

8.5. What DSF_α requires

The provable bound is |B_j| ≤ C/√j (Proposition 5.1, from bounded variance). The data shows |B_j| ~ 2^{−j/3} (exponential). Bridging this gap requires explaining why stripped dyadic blocks cancel exponentially — why the residual arithmetic structure in h(p), after removing the Q = 12 character component, produces cancellation in dyadic sums far beyond what Cauchy–Schwarz alone guarantees.

Potential approaches:

• Deterministic Littlewood–Paley theory for multiplicative functions restricted to primes
• Almost-orthogonality of character-stripped prime sums across dyadic ranges
• Exponential mixing in the p mod q distribution for q > 12 (higher-conductor depletion)
• Connection to the distribution of smooth numbers adjacent to primes (via the scissors mechanism)

8.6. Twenty-four unconditional results

For reference, the full list of results established without any conditional hypothesis:

Gap 1 (14 items).

1. Character stripping symmetrization: 94% → 52% at three scales.
2. ℓ² summability: Σ|h|²/p² converged (12 decimal places).
3. H² shortcut exclusion.
4. Direction D exclusion.
5. Kolmogorov exclusion.
6. Negative part tame: Σ|η⁻|/p ≈ 0.20, converging.
7. A_h numerical convergence at three scales (precision ±0.001).
8. Dyadic block decay ~ 2^{−j/3} at three scales.
9. Unweighted square-function Σ|B_j|² numerical convergence.
10. Stripped Hölder α_h ≈ 0.7–0.8 at small t.
11. Cauchy–Schwarz bound |B_j| ≤ C/√j (conditional on Var(h) = O(1)).
12. BL numerically negligible (E_j/B_j < 3.2% at t = 0.01).
13. DSF_α numerically converges to α = 0.5.
14. SD proved unconditionally (Proposition 4.1).

Gap 2 (10 items).

1. Log-UBPD for a = 2 unconditional.
2. a = 2 / a ≥ 3 complete separation.
3. RQSA zero violations (1,877 checks).
4. Shell-side RT moment O(1).
5. Predecessor-side RT moment O(1) (confirmed 10⁹).
6. CDR stable (mean 0.93, no downward trend).
7. Scissors formula perfect in standard IC (0/445).
8. Log-UBPD suffices for prime-side local factor.
9. Predecessor independence confirmed (cross-term → 1.0 for k ≥ 17).
10. Standard IC concavity cleaner (28 violations vs 3,087 in P32).

---

§9. Conclusion

Paper 49 identified two open problems for H'. Paper 50 does not claim to solve them — it transforms them. Each gap now has a precise formal statement, a list of excluded approaches, numerical verification at multiple scales, and a clear assessment of its role in the proof architecture.

The principal localization: Gap 1's remaining difficulty is the exponential decay of stripped dyadic block sums, formalized as DSF_α, together with the block localization condition BL_α. The Tauberian condition (SD, Proposition 4.1) is proved unconditionally. The correct formulation of Gap 1 is boundary regularity of the stripped Dirichlet series H(s), not convergence of Σ η(p)/p — the latter diverges due to the positive mean of η, a fact that character stripping makes transparent by separating the known divergent part from the unknown regular part.

The secondary localization: Gap 2's internal structure is now fully mapped (a = 2 core isolation, scissors mechanism, Log-UBPD, RT moments, RQSA), and its relationship to H' is clarified — the prime-power layer is analytic at Re(s) = 1 (Proposition 7.1), contributing no singularity to the boundary behavior.

The hypotheses DSF_α and BL_α are not conjectures pulled from thin air. DSF_α is supported by exponential block decay |B_j| ~ 2^{−j/3} confirmed across three orders of magnitude; BL_α by the empirical negligibility of intra-block errors. Both are motivated by the character-stripping framework that provides their a priori context, and by the elimination of every alternative path that might have circumvented them. What remains is to explain the cancellation mechanism that produces exponential block decay — and to prove it.

---

Appendix A. Convention Translation

Structural results (symmetrization ratio, block decay rate, CDR mean, RQSA violations) are convention-independent.

---

Appendix B. Thermodynamic Reading

The thermodynamic perspective developed in ZFCρ Paper 48 provides an interpretive layer for the results of the present paper, though the mathematical content is independent of it.

Q = 12 character stripping (§2) corresponds to the projection of the dissipation mechanism onto the prime layer: 2 and 3 are the "most efficient dissipation factors" in the IC cost function, and their modular accessibility for p ± 1 determines the residue-class structure of η(p). The between-class variance (17.8%) measures the fraction of η-variation attributable to this leading dissipation channel.

DSF_α (§6) is the frequency-domain counterpart of the step-absorption rate η ∈ [0.10, 0.31] established in Paper 48. The step-absorption rate measures how strongly the min/max recursion suppresses inter-step fluctuations; DSF_α measures how rapidly these fluctuations decay when decomposed into dyadic frequency bands. Both express the same phenomenon — the recursion's tendency to absorb perturbations — in complementary domains.

The analyticity of the prime-power layer at Re(s) = 1 (§7.1) corresponds to the automatic arithmetic suppression of higher-order multiplicative effects: the 1/p^{2a} denominator is the Dirichlet-series manifestation of the recursion's rapid damping at higher multiplicative depths.

---

Acknowledgments

AI contributions. Gemini (子夏): character stripping framework, scissors formula, "Log-UBPD harmless," Zsigmondy a = 2 escapee mechanism. Grok (子贡): negative part tame, δ = −7 factor structure, α = 1/2 heuristic. ChatGPT (公西华): DSF_α + BL formulation, three impossibilities (H², A_loc, Kolmogorov), Log-UBPD chain analysis, "one big + one small" Gap 2 verdict, paper structure recommendation; post-draft theorem-level review identifying six corrections incorporated in the present version. Claude (子路): all numerical computations (15 programs at scales 10⁷–10¹⁰), independent Cauchy–Schwarz route to gap identification, RT moment O(1) discovery on both shell and predecessor sides, SD unconditional proof.