§1 Recursive Framework

1.1 Recursive Main Chain

Let \(m\) be any Ω=(k−1) number (Ω counted with multiplicity). Define \(Q_N(m) = \{\text{primes } q : q \geq 3,\, q \nmid m,\, mq-1 \leq N\}\).

The constraint \(q \nmid m\) ensures \(q\) is a "new" prime factor of \(mq\). The recursive main chain starts from primes (Ω=1) and adds one new prime at each step:

• sf chain: \(p \to pq \to pqr \to pqrs \to \ldots\) (all factors distinct)
• ss chain: \(p^2 \to p^2q \to p^2qr \to \ldots\) (one square factor + new distinct primes; \(p^2\) is the Ω=2 seed, entering directly as a core for Ω=3 recursion — it need not be generated from Ω=1 via \(q = p\), since \(q \nmid m\) excludes \(q = p\).)

1.2 Higher-Power Seed Branch

The recursive main chain cannot generate all nsf forms:

• \(p^2q^2\) cannot be written as "some Ω=3 core × new prime" (the second square factor is not introduced by a new prime)
• \(p^3\) cannot be written as "\(p^2 \times p\)" (\(q = p\) is excluded)
• \(p^4, p^3q^2\), etc.: similarly

These hp structures enter as independent seeds and are measured using the same M̄ formula: given any hp core \(m\) (e.g., \(p^2q^2\)), compute \(\bar{M} = \tau_m - E_q^{(h)}[\mathrm{diff}(m,q)]\) with \(q \nmid m\). The hp branch is not on the recursive main chain but shares the same measurement formula.

1.3 Precise Definitions of the Three Types

Type	Definition	Recursive status
sf	All prime-factor exponents = 1	Recursive main chain
ss	Exactly one exponent = 2, rest = 1, no p³	Recursive main chain (p² seed)
hp	Some exponent ≥ 3, or multiple exponents ≥ 2	Independent seed

1.4 Shifted-Product Deficit

(a) Multiplicative shadow (unconditional). \(mq-1\) sits next to \(mq\) and benefits from the multiplicative structure of \(mq\). Holds for any \(m\).

(b) Parity (conditional). When \(m\) is odd and \(q\) is an odd prime, \(mq-1\) is even and enjoys parity advantage. When \(2 | m\), \(mq-1\) is odd and parity advantage is absent.

Experiments show that the multiplicative shadow alone suffices to drive a positive slope, even when parity advantage is absent.

Two q conventions. Relaxed (\(q \geq 3, q \nmid m\); main results) and Canonical (\(q > P^+(m)\); definition verification). Both conventions yield the same sign at Ω=2,3.

§2 Eight Experiments

Block	Measurement	Core result
1	Ω=3 sf (canonical + relaxed)	+0.233/>14σ (rel), +0.093/>5σ (can)
2	Ω=4 sf (relaxed)	+0.171/>10σ
3	Ω=5,6,7,8 sf (relaxed)	Ω=5–6 positive; Ω=7 inconclusive; Ω=8 negative
4	h_k(N) corrected	Ω≤6 = 94.39%, Ω≥7 = 5.61%
5	sf/nsf density split	sf 63.70%, nsf 30.69%
6	ss M̄ slope	Ω=2–6 all positive, >4σ
7	sf/ss/hp density split (N=10¹⁰)	sf 63.70%, ss 19.02%, hp 11.67%
8	hp M̄ slope	Ω=3–6 all positive, >2σ

§3 Squarefree Slope Spectrum

3.1 Ω=3

3,000 sf semiprime cores, \(m \in [6, 100{,}000]\).

Relaxed: M̄ = +0.233·ln(m) − 1.86, SE = 0.017, >14σ.
Canonical: slope = +0.093 ± 0.016, >5σ.

The positive sign does not depend on the \(q\) convention. The Ω=3 relaxed slope (+0.233) exceeds both Paper XL's Ω=2 result (+0.217, different sampling) and the Block 6 control retest (+0.173, 2,000 primes). The difference arises from sampling scheme; the sign is consistent.

3.2 Ω=4–6

Ω	slope	SE	σ
4	+0.171	0.016	10.8
5	+0.185	0.009	20.2
6	+0.111	0.010	10.8

All >10σ.

3.3 Ω=7–8

Ω	slope	SE	σ	status
7	−0.016	0.013	−1.2	inconclusive
8	−0.088	0.023	−3.8	negative

Linear fit zero crossing: Ω ≈ 7.2 (7 points, not strictly monotone, current-window extrapolation).

§4 Single-Square NSF

Core	slope	SE	σ	⟨M̄⟩
Ω=2 ss (p²)	+0.135	0.030	4.4	−0.18
Ω=3 ss (p²q)	+0.247	0.015	16.6	−0.10
Ω=4 ss (p²qr)	+0.300	0.019	16.2	−0.43
Ω=5 ss	+0.414	0.013	31.3	−0.82
Ω=6 ss	+0.231	0.013	17.6	−0.81

All positive, all >4σ. The ss slopes are generally no smaller than sf, and often larger (Ω=4 ss +0.300 vs sf +0.171; Ω=5 ss +0.414 vs sf +0.185).

§5 Higher-Power NSF

5.1 Results

Core	slope	SE	σ	⟨M̄⟩
Ω=3 hp (p³)	+0.284	0.117	2.4	−0.55
Ω=4 hp (p²q²)	+0.252	0.039	6.5	−0.91
Ω=4 hp (p³q)	+0.259	0.016	16.7	−0.34
Ω=5 hp (p²q²r)	+0.388	0.025	15.6	−1.05
Ω=5 hp (p³qr)	+0.316	0.021	15.0	−0.65
Ω=6 hp (mixed)	+0.039	0.015	2.6	−1.07

All positive, all >2σ. Ω=6 hp is the weakest (+0.039, 2.6σ) but clears the threshold. Ω=4 hp (p⁴) was skipped due to insufficient samples (7 cores).

5.2 hp Subtype Coverage

The total hp Ω≤6 harmonic density is 11.67%. Block 8 coverage:

hp subtype	Density (qualitative)	Block 8	Result
p³	small	✓ tested	+, 2.4σ
p²q²	medium	✓ tested	+, 6.5σ
p³q	medium	✓ tested	+, 16.7σ
p²q²r	large	✓ tested	+, 15.6σ
p³qr	large	✓ tested	+, 15.0σ
Ω=6 hp mixed	large	✓ aggregate	+, 2.6σ
p⁴	very small (Σ1/p⁴ ≈ 0.004)	insufficient samples	—
p⁴q, p⁵, p²q²r², etc.	very small	not individually tested	—

Block 8 covers the main mass of hp 11.67%. The untested very-high-power residue (p⁴, p⁴q, p⁵, etc.) has extremely small harmonic density: for example, \(\sum_{p \text{ prime}} 1/p^4 \approx 0.004\), contributing ~0.02% of H(N). These residues total no more than ~1–2% of the hp 11.67%, i.e., ~0.1–0.2% of total harmonic density.

§6 Harmonic Density Stratification

6.1 Definitions

\(H_k(N) = \sum_{n \leq N,\, \Omega(n)=k} 1/n\) (Ω counted with multiplicity). \(h_k(N) = H_k(N) / H(N)\), where \(H(N) = \sum_{n \leq N} 1/n\).

Ω(1) = 0 (the only Ω=0 positive integer), so \(H_0 = 1\), \(h_0 = 1/H(N) \approx 4.24\%\).

Sanity: \(H(10^{10}) = 23.603067\), matching \(\ln(10^{10}) + \gamma = 23.6031\) to four decimal places. Ω=0 count = 1.

6.2 Total Density (N = 10¹⁰)

Ω	h_k	cumulative h_{≥k}
0	4.24%	100.00%
1	14.40%	95.76%
2	22.19%	81.37%
3	21.82%	59.17%
4	16.17%	37.35%
5	10.02%	21.19%
6	5.55%	11.17%
7	2.88%	5.61%
8	1.43%	2.74%
≥9	1.31%	1.31%

6.3 sf/ss/hp Split (N = 10¹⁰)

Ω	h_sf	h_ss	h_hp	ss/(ss+hp)
0	4.24%	0	0	—
1	14.40%	0	0	—
2	20.28%	1.92%	0	100%
3	15.51%	5.58%	0.74%	88%
4	7.04%	6.43%	2.70%	70%
5	1.94%	3.83%	4.25%	47%
6	0.31%	1.26%	3.98%	24%
7	0.03%	0.23%	2.62%	8%

The sf share decreases with Ω (100% at Ω=1 → 6% at Ω=6); the hp share increases correspondingly.

6.4 Coverage Summary (N = 10¹⁰, H(N) = 23.603067)

Range	Density	Closure status
Ω=0 (n=1, trivial)	4.24%	trivial
Ω=1 (primes, trivial)	14.40%	trivial (density → 0)
sf Ω=2–6 (M̄ positive slope measured)	45.08%	positive-slope signal, >10σ
ss Ω=2–6 (M̄ positive slope measured)	19.02%	positive-slope signal, >4σ
hp Ω=3–6 main mass (M̄ positive slope measured)	~11.5%	positive-slope signal, >2σ
hp very-high-power residue (not individually regressed)	~0.2%	density very small
Measured positive slope + trivial	~94.2%
Ω≥7	5.61%	to be addressed in XLII

94.39% = 63.70% (sf, including Ω=0,1 trivial) + 19.02% (ss) + 11.67% (hp) is the total Ω≤6 mass. Of this, directly measured positive slope + trivial totals approximately 94.2%; the remaining ~0.1–0.2% is hp very-high-power residue, extremely small but not individually regressed.

6.5 Multi-N Trend

N	sf Ω≤6	ss Ω≤6	hp Ω≤6	Ω≤6 total	Ω≥7
10⁶	65.12%	18.86%	12.53%	96.51%	3.49%
10⁸	64.44%	18.98%	12.22%	95.64%	4.36%
10⁹	64.03%	19.02%	11.95%	95.00%	5.00%
10¹⁰	63.70%	19.02%	11.67%	94.39%	5.61%

\(h(\Omega \geq 7)\) grows slowly (3.49% → 5.61%). The 5.61% is a current-window figure, not an asymptotic constant.

§7 Closure Argument

7.1 Chain

For Ω=k (k = 2,...,6), across sf + ss + hp cores:

1. \(\bar{M}_{k,N}(m)\) has a positive slope against \(\ln(m)\) (sf >10σ, ss >4σ, hp main mass >2σ).
2. → Within the current window, the data are consistent with \(\bar{M}_k \to \infty\).
3. \(V_k(m) = O(1)\): confirmed at \(V \approx 1.4\) for Ω=2 sf in Paper XXXIV. Reasonably expected to hold for higher Ω and nsf, but not independently verified in this paper.
4. → \(\Gamma \to \infty\), Cantelli \(c \to 0\), Cesàro \(\bar{c}_h \to 0\).
5. → The Ω=k layer's defect contribution → 0.

7.2 Honest Boundaries

(a) Ω=4–6 have relaxed data only. Relaxed/canonical sign consistency verified at Ω=2,3.

(b) \(V_k = O(1)\) not independently verified at Ω=3–6 and nsf.

(c) Finite window: \(N = 10^{10}\).

(d) The above constitutes a positive-slope closure signal, not a strict proof of closure.

(e) hp Ω=6 is the weakest point (+0.039, 2.6σ) — marginal.

(f) hp very-high-power residue (~0.2%) not individually tested.

§8 Finite-Window Implications and Paper XLII

8.1 Established

• Ω=0,1: trivial (18.64%)
• Ω=2–6: positive-slope closure signal — sf + ss + hp, covering 75.8% measured + 18.6% trivial ≈ 94.4%
• Ω≥7: 5.61%, not closing

8.2 Open Tasks for Paper XLII

(a) Control the ~5.6% harmonic density at Ω≥7.

(b) Address the growth of \(h(\geq 7)\) with \(N\) (3.49% → 5.61%). XLII may require a moving cutoff \(K(N)\) or unified decay control.

(c) Complete the passage from "closure signal" to "closure": canonical verification (Ω=4–6), \(V_k = O(1)\) verification, asymptotic extrapolation.

§9 Discussion

9.1 Core Contributions

(a) Recursive framework (main chain + hp seed branch) covering all integer factorization forms.

(b) sf Ω=2–8 + ss Ω=2–6 + hp Ω=3–6 slope spectrum.

(c) Turning point at Ω ≈ 7.2 (sf spectrum, current-window extrapolation).

(d) sf/ss/hp density stratification: measured positive slope + trivial = 94.39% (N = 10¹⁰, exact).

(e) nsf (ss + hp) slopes are no smaller than sf and often larger — the multiplicative shadow is effective with or without parity advantage.

9.2 Why Are nsf Slopes Larger?

Non-squarefree cores contain repeated prime factors, making \(\rho(m)\) smaller (DP handles repeated factors more efficiently), so \(\tau_m = \rho(m)+2\) grows more slowly while diff growth is unaffected. Result: M̄ starts lower but rises faster.

9.3 Why Do High-Ω Slopes Decline?

(a) Combinatorial complexity: the number of factorization paths grows exponentially with k. (b) Finite-scale effects. (c) Decreasing signal-to-noise ratio.

§10 Data Sources and Reproducibility

Script	Measurement	Block
p41_omega3_mbar.py	Ω=3 sf (can + rel)	1
p41_omega4_test.py	Ω=4 sf	2
p41_omega5to8.py	Ω=5–8 sf	3
p41_density_fast.py	h_k(N) corrected	4
p41_sf_density.py	sf/nsf split	5
p41_nonsf_mbar.py	ss M̄	6
p41_partition_fast.py	sf/ss/hp split (N=10¹⁰)	7
p41_hp_mbar.py	hp M̄	8

Data: rho_1e10.bin (int16, N = 10¹⁰, rho_dp.c, Paper XXXII convention).

References

[1] H. Qin. ZFCρ Paper XL: Direct Positive Slope of M̄ and Strong Numerical Support for Ω=2 Closure. DOI: 10.5281/zenodo.19179778.

[2] H. Qin. ZFCρ Paper XXXIX: Shifted-Product Deficit and M_loc Growth at Ω=2. DOI: 10.5281/zenodo.19164588.

[3] H. Qin. ZFCρ Paper XXXIV: Upper-Margin Closure and Bounded-Variance. DOI: 10.5281/zenodo.19140015.

Acknowledgments

Claude (子路) designed the recursive framework and all eight experimental blocks, wrote eight scripts, and drafted v1–v7 working notes plus the formal text. Discovered and corrected the Ω=0 bug in Block 4 (large prime factors not counted toward Ω).

Han Qin (author) proposed the unified recursive definition ("each Ω is the previous Ω plus one prime q"), demanded the complete Ω=4–8 spectrum, discovered the turning point, posed the "bundle from Ω≥4 or Ω≥7?" decision problem, and articulated the methodological principle "run data first whenever possible, because data will correct us" — which directly spawned Blocks 5–8 and pushed the tested coverage from 64% to 94.39%.

ChatGPT (公西华) deepened the analysis across seven review rounds (v1–v7): v2 identified four structural flaws (parity error, squarefree scope, relaxed/canonical distinction, Erdős-Kac fixed-threshold issue); v3 pointed out "95% ≠ tested-object coverage," spawning Blocks 5–6; v4 identified the nsf partition gap and Ω=0 bug, spawning Blocks 7–8 and the density correction; v5 flagged nsf index unification; v6 formalized hp as an independent seed branch, demanded same-window exact alignment, and required hp subtype coverage; v7 distinguished "measured positive slope" from "trivial coverage" and gave accept with minor revisions. Each round drove deeper experiments and more honest claims.

Gemini (子夏) and Grok (子贡) confirmed the recursive framework consensus and slope-spectrum consistency in earlier rounds and gave sustained acceptance in later rounds.

The final text was independently completed by the author.