§1 Introduction

1.1 The problem

Paper 46 (DOI: 10.5281/zenodo.19303511) reduced H' to four hypotheses: UBPD, PMH-DS, PCF, RT. This paper advances on three fronts: (1) character refinement and second-order reduction of PMH-DS, (2) Shell-Depth Lemma, (3) parity-mediated PCF architecture.

1.2 A priori principles

(P1) η and residue class are systematically linked. ρ_E(p) = ρ_E(p-1)+1; small factors of p-1 are determined by p mod q.

(P2) The link does not block PMH. β(t) absorbs the principal character; residue bias enters log L(s,χ) for non-principal χ. Same order in t ≠ same order in singularity.

(P3) PMH is an external mathematical input.

(P4) θ-mediated separation. f(n-1) = λ·(log N - θ) + ξ⁻(n). θ's leading-order k-drift is zero (Shell-Depth Lemma); finite-N correction is O(1/log x). ξ⁻ has an O(1) parity-driven shift.

(P5) Variance orthogonal, mean non-orthogonal. Var(ξ⁻) is k-independent; E[ξ⁻] has k-drift.

(P6) θ is k-independent in leading order. Shell-Depth Lemma.

(P7) μ⁻ slope = θ contribution + ξ⁻ contribution. +0.077 = 0.035 + 0.042.

(P8) ξ⁻ positive drift from parity constraint. High Ω(n) → n even → n-1 odd → P⁻(n-1) large → η positive.

(P9) ξ(n) negative drift from small-prime dominance. High Ω = small primes → η negative.

1.3 Notation

As in Paper 46. Additional:

η(p) := ρ_E(p) - λ·log p (centered prime residual).

ξ and ξ⁻ definitions (both centered relative to current n):

ξ(n) := f(n) - λ·log n

ξ⁻(n) := f(n-1) - λ·log n (not f(n-1) - λ·log(n-1))

This ensures f(n) = λ·(log N - θ) + ξ(n) and f(n-1) = λ·(log N - θ) + ξ⁻(n) share the same θ, making the decomposition check an exact identity.

Characters mod 12: χ₀ (principal), χ₃, χ₄, χ₁₂. Theoretical definition of c_χ(t): harmonic-prime Fourier coefficient c_χ(t) ~ lim_{x→∞} [1/log log x]·Σ_{p≤x, p∤Q} e^{tη(p)}·χ̄(p)/p. Numerical approximation uses classwise empirical mean c_χ^{emp}(t).

1.4 Main results

(1) η(p) residue-class non-uniformity. Q=12 between/total = 17.6%.

(2) CR-PMH-DS₂(12). Second-order character-refined prime input. Numerically verified.

(3) Shell-Depth Lemma (theorem-level). log E[e^{-uθ} | Ω=k] = -log(1+u) + O(ε_x + 1/log x).

(4) μ⁻ slope decomposition + parity constraint.

(5) PCF₂ numerically confirmed. μ⁻ = O(k) ✓, Var⁻ = O(1) ✓.

§2 Residue-Class Structure of η(p)

2.1 Data (N = 10⁷)

2.2 Classwise η (Q=12)

Structural explanation (P1): p ≡ 1 mod 3 ⟹ 3|p-1 ⟹ p-1 has small factor ⟹ ρ_E(p-1) smaller ⟹ η more negative.

2.3 Character coefficients

c_χ(0) = 0 for χ ≠ χ₀ (character orthogonality). Residue bias enters log L(s,χ) for non-principal χ. Same order in t ≠ same order in singularity.

§3 CR-PMH-DS₂(12)

3.1 Strategic reduction

Conjecture 2 uses only first- and second-order cumulants. Full PMH-DS far exceeds requirements.

3.2 Correct form

CR-PMH-DS(12):

Σ_{p∤12} e^{t·η(p)} / p^w = β(t)·log(1/(w-1)) + Σ_{χ≠χ₀} c_χ(t)·log L(w,χ) + H₁₂(w,t)

H₁₂ analytic in Re(w) > 1-δ.

Key (P2): c_χ(t) and β(t)-1 are both O(t), but multiply different singularities. β multiplies log(1/(w-1)) (divergent at w=1); c_χ multiplies log L(w,χ) (finite at w=1 for fixed conductor).

3.3 Second-order reduction

Hypothesis CR-PMH-DS₂(12).

A₁(w) := Σ_{p∤12} [η(p) - Σ_{χ≠χ₀} c'_χ(0)·χ(p)] / p^w

A₂(w) := Σ_{p∤12} [η(p)² - Var(η) - Σ_{χ≠χ₀} c''_χ(0)·χ(p)] / p^w

Require A₁ and A₂ to admit analytic continuation in Re(w) > 1-δ. These are prime sums of fixed additive functions — standard Mertens type.

3.4 Numerical verification

x-flatness: res_raw ≈ -0.504 constant from x = 10⁵ to 10⁷. Scalar PMH-DS is already an excellent approximation. Character correction ~0.017 — formal correction, not numerical necessity.

3.5 Roy Reduction

Roy 2025's LSD for products of L-functions handles the second half: property P → LSD → coefficient extraction → SCF.

The first half (CR-PMH-DS → property P) remains to be proved. The analyticity of H₁₂ is the true gap — requires prime cancellation input.

Roy does not prove PMH-DS, but proves nearly everything except PMH-DS.

3.6 Second-Order Bridge Proposition

Proposition. If CR-PMH-DS₂(12) holds (A₁, A₂ analytic in Re(w)>1-δ), plus UBPD (Local Euler Factor Lemma controlling prime-power tail), then via Selberg-Delange derivative extraction:

(i) μ_{x,k} = λ·log x + k·a₁ + O(1), where a₁ = (log β)'(0).

(ii) Var(f | Ω=k) = k·a₂ + O(1), where a₂ = (log β)''(0).

Remark. Weaker than full SCF — gives only mean and variance k-linearity, not the full log-mgf. But sufficient for Conjecture 2.

§4 Shell-Depth Lemma

4.1 Theorem

Let S_k(y) := #{n ≤ y : Ω(n) = k}, ν_{x,k} := uniform measure on {n ≤ x : Ω(n) = k}, θ_x(n) := log x - log n.

Hypothesis (SS_Ω). There exist K > 0, a locally Lipschitz function Φ_K, and ε_x → 0, such that for all sufficiently large x, uniformly for x^{1/2} ≤ y ≤ x and 1 ≤ k ≤ K·log₂x:

S_k(y) = [y/log y] · [(log₂y)^{k-1}/(k-1)!] · Φ_K((k-1)/log₂y) · (1 + O_K(ε_x))

Remark. This is the standard Ω-Sathe-Selberg asymptotic. Gafni-Robles 2025 and Roy 2025 use equivalent forms.

Shell-Depth Lemma. Under (SS_Ω), for 0 ≤ u ≤ u₀ < 1, uniformly for 1 ≤ k ≤ K·log₂x:

log E_{ν_{x,k}}[e^{-uθ}] = -log(1+u) + O_{K,u₀}(ε_x + 1/log x)

The leading order is independent of k.

Remark. The theorem range is restricted to u ≥ 0 (decreasing weight n^{-u}). Extension to |u| < 1 requires stronger ratio bounds on small-t intervals; this is deferred. Application to Conjecture 2 requires only u ≥ 0.

4.2 Proof

(1) Counting ratio. From (SS_Ω), for x^{-1/2} ≤ t ≤ 1:

S_k(xt)/S_k(x) = t · (1 + O_K(ε_x + (1+|log t|)/log x)) ...... (R)

In the central window k ≤ K·log₂x, the k-dependent corrections are O(1/log x).

(2) Small-t truncation. Split [0,1] into [0, x^{-1/2}] ∪ [x^{-1/2}, 1]. The first segment uses the trivial bound 0 ≤ S_k(xt)/S_k(x) ≤ 1. Since u ≥ 0, ∫₀^{x^{-1/2}} t^{u-1} dt = O(x^{-u₀/2}) → 0.

(3) Main segment. By Abel/partial summation: E[e^{-uθ}] = 1 - u·∫₀¹ t^{u-1}·[S_k(xt)/S_k(x)] dt. Substituting (R) into [x^{-1/2}, 1]:

Main term = 1 - u·∫₀¹ t^u dt = 1/(1+u)

Error controlled by ∫₀¹ t^u·(1+|log t|) dt < ∞.

Hence E[e^{-uθ}] = 1/(1+u) · (1 + O(ε_x + 1/log x)).

(4) Take log. For 0 ≤ u ≤ u₀ < 1, 1/(1+u) is bounded away from 0; taking log yields the conclusion. ■

4.3 Finite-N correction

In the current data window (N = 10⁷, k = 2…14), E[θ|k] has local linear slope = -0.011/k. Multiplied by λ this gives +0.035 — an O(1/log x) correction visible at finite N, not a structural k-linear coefficient. At the theorem level, θ's k-drift = O(1/(log x · log₂x)) → 0 as x → ∞.

§5 μ⁻ Slope Decomposition

5.1 Decomposition formula

μ⁻ = λ·(log N - E[θ|k]) + E[ξ⁻|k] (exact identity, check = 0.000000 ✓)

5.2 Data

5.3 Finite-window local slopes (N = 10⁷, k = 2…14)

Note: These are finite-window local linear fits, not asymptotic coefficients. At the theorem level, θ contributes O(1/log x) → 0, ξ⁻ contributes O(1) (bounded parity shift).

(μ-μ⁻) slope = -0.263. μ decline driven by ξ(n)'s strong negative drift (P9). μ⁻ rise from θ and ξ⁻ contributing roughly equal shares (P7).

§6 Parity Constraint

6.1 Mechanism chain (P8)

High Ω(n)=k → n has many prime factors, more likely divisible by 2 multiple times → n even → n-1 odd → P⁻(n-1) larger → large primes have more positive η(p) → ξ(n-1) positive drift.

6.2 Quantitative verification

Slopes: E[Ω(n-1)] declining -0.143/k, E[log P⁻(n-1)] rising +0.305/k, P(even) declining -0.058/k.

6.3 Even vs odd

Even n-1 has systematically lower ξ (η(2) more negative). At high k, even proportion drops dramatically → E[ξ⁻] pushed toward the more positive odd value.

6.4 Within-stratum drift

Stratified by Ω(n-1), E[ξ] rises with k within each stratum — not only a composition effect but also within-stratum drift (parity's second-order constraint on prime factor sizes).

6.5 Variance orthogonality

Var(ξ⁻) ≈ 1.0, k-independent. Conditioning changes the center but not the spread.

§7 PCF₂

7.1 Target

Prove μ⁻ = O(k), Var⁻ = O(k).

7.2 μ⁻ = O(k)

In the current finite window (N=10⁷), local fit gives μ⁻ ≈ λ·log x + 0.077k + O(1). At the theorem level, only O(1/log x) from θ + O(1) from ξ⁻ is claimed, yielding μ⁻ = λ·log x + O(k).

Sufficient for Conjecture 2's poly(k) target.

7.3 Var⁻ = O(1)

Var⁻(f) = λ²·Var(θ|k) + Var(ξ⁻|k) - 2λ·Cov(θ,ξ⁻|k) ≈ O(1) + O(1) + O(1) = O(1).

Data confirmed: Var(ξ⁻) ≈ 1.0, k-independent (P5).

Note: Var⁻ = O(1) requires two explicit inputs: (a) Var(ξ⁻|Ω=k) = O(1) (shifted-micro variance), (b) Cov(θ,ξ⁻|Ω=k) = O(1) (cross term). Both have strong numerical support but are not yet proved as theorems.

7.4 Strictification route

(a) θ contribution O(k): from Shell-Depth Lemma's O(1/log x) correction.

(b) ξ⁻ mean contribution O(1): high Ω(n) → n even → n-1 odd → E[ξ|odd] ≈ 0.5 (fixed constant). Parity constraint transmits a bounded shift.

(c) Var⁻ = O(1): requires shifted-micro variance input (Var(ξ⁻|k) = O(1)) and cross term input (Cov(θ,ξ⁻|k) = O(1)). Both numerically confirmed.

§8 Complete H' Closure Chain

Proposition 3 (a)(b)(c) (unconditional)
+ UBPD → Local Euler Factor Lemma

+ CR-PMH-DS₂(12) → Shell: μ = λ log x + O(k), Var(f) = O(k)

+ Shell-Depth Lemma → E[θ|k] k-slope = O(1/log x)
+ Parity-mean input → E[ξ⁻|k] = O(1) (bounded shift)
+ Shifted-micro variance input → Var(ξ⁻|k) = O(1), Cov(θ,ξ⁻|k) = O(1)
→ PCF₂: μ⁻ = λ log x + O(k), Var⁻ = O(k)
→ |μ-μ⁻| = O(k)

+ RT → E[(Δr̃)²] ≤ poly(k)

→ E[(Δf)²] = O(k²) → E[(ΔM_spf)²] = O(poly(k)) → Conjecture 2
→ Paper 44 chain → H'

Distance assessment

§9 SAE Interpretation

9.1 Three effects of conditioning

Ω=k conditioning simultaneously does three things:

(a) Compresses within-shell fluctuation (Var(f), Var(ξ) = O(1) — Paper 46 P5)

(b) Separates shell and predecessor means (μ-μ⁻ ≈ -0.26k — Paper 46 P1)

(c) Indirectly distorts predecessor factor structure via parity (ξ⁻ mean drift +0.042/k — this paper P8)

(a) and (b) were understood in Paper 46. (c) is this paper's new discovery. Together they constitute the complete microscopic picture of "conditioning bias."

9.2 Variance orthogonal, mean non-orthogonal

Conditioning changes ξ⁻'s center (mean drift) but not its spread (variance k-independent). This is the precise realization of SAE's "fluctuation is the intrinsic quantity" on the predecessor side: the magnitude of fluctuation is determined by the system's internal structure; conditioning can only shift the observation window's center.

9.3 Parity as arithmetic constraint propagation

n and n-1's factor structures are coupled through two channels: (1) shared θ (positional coupling), (2) parity constraint (even-odd coupling). (1) accounts for 89% of Corr(f,f⁻). (2) accounts for ξ⁻'s mean drift. Both are propagation of arithmetic constraints — structural, not random.

§10 Open Questions

1. Proof of CR-PMH-DS₂. Analytic continuation of A₁, A₂ prime sums. Possibly via Bombieri-Vinogradov or large sieve.

2. Strictification of parity constraint. High Ω(n) → n even → n-1 odd → E[ξ|odd] bounded. Requires sieve bound.

3. Global proof of UBPD. sup|δ_a(p)| < ∞.

4. Strict proof of RT. E[(Δr̃)²] ≤ poly(k). Data says O(1) and declining.

5. Can E[ξ(n) | Ω=k]'s -0.221/k be derived from η's prime-size distribution?

Data Sources

Scripts: pmh_uniformity.c, pmh_character.c, theta_latent.c, mu_slope_source.c, xi_drift_breakdown.c (C, gcc -O2). Data: ρ_E via DP min for n ≤ 10⁷+1.

References

[1] H. Qin. ZFCρ Paper XLVI. DOI: 10.5281/zenodo.19303511.

[2] H. Qin. ZFCρ Paper XLIV. DOI: 10.5281/zenodo.19247859.

[3] H. Qin. ZFCρ Paper XLII. DOI: 10.5281/zenodo.19226607.

[4] M. Verwee (2026). arXiv:2603.17682.

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

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

[7] É. Goudout (2020). HAL:hal-02985866.

[8] R. Benatar (2024). arXiv:2408.16576.

[9] A. Mangerel (2016). Bivariate Erdős-Kac.

Acknowledgments

ChatGPT (Gongxihua): Complete proof of the Shell-Depth Lemma (Sathe-Selberg counting ratio + partial summation). Correct form of CR-PMH-DS(12) and character coefficient correction (c_χ(0)=0 by orthogonality). Roy reduction (property P's correct object is the (z,t)-family). Strategic reduction proposal (full PMH → CR-PMH-DS₂). Precise judgment that PMH-DS ≠ SCF ("Roy proves nearly everything except PMH-DS"). Fourier-Sub-PCF₂ proposal. Local Euler Factor Lemma sketch. Demotion of UL-strong to bonus (PMH is SCF's correct prime input).

Claude (Zilu): All numerical experiments (η residue-class uniformity, character-refined PMH verification, μ⁻ slope decomposition, ξ⁻ drift breakdown, parity verification). Text drafting, working notes v1–v5.

Claude (Thermodynamic thread): η–Dirichlet character independence judgment (partially corrected by data but core judgment "does not block PMH" confirmed). PMH-DS three-step attack route proposal.

Gemini (Zixia): First proposal of θ-mediated PCF architecture (Sub-PCF 1 + Sub-PCF 2 macroscopic/microscopic separation). Judgment that moment method cannot give O(1) log-mgf error. Bivariate Selberg-Delange as direct route suggestion.

Grok (Zigong): Identification of two gaps in Goudout + bridge route (ω→Ω transfer + shifted weighted additive moment). Priority judgment that θ-mediated dominates Goudout route. Series consistency confirmation.

Han Qin (author): A priori discovery of P8 (parity constraint) — identifying from data (P(n-1 even) dropping from 82% to 0.1%) that parity drives ξ⁻ drift. A priori audit methodology (incremental construction of P1–P9). Acceptance of strategic reduction. Proposal of μ⁻ slope decomposition.

Final text independently completed by the author.