§1. Introduction

§1.1 Remaining Landscape

Paper 22 conditionally closed Assumption A' ($p_\infty(k) \to 1$). The $D(N) \to 1$ proof chain (Paper 15, Theorem F) now has only Assumption A remaining: for each fixed $k$, $p_k(N) := P(G(n) > 0 \mid \Omega(n) = k, n \leq N)$ converges to $p_\infty(k)$.

Paper 15's original version requires monotone convergence ($p_k(N) \downarrow p_\infty(k)$). ChatGPT (Paper 22 review) suggested replacing monotonicity with quantitative controlled convergence. This paper executes that replacement strategy.

§1.2 Three Contributions

1. K_p distribution constancy (§2): $K_p$'s distribution is approximately constant across $k$ — each SPF insertion yields a stable positive discount of roughly 0.2–0.3. This explains why $E[K_p]$ in Paper 22 is always positive.
2. The 80/3 Law (§3): The reason $D(N)$ does not approach 1 is precisely located: $k = 1$ (primes) $+ k = 2 + k = 3$ contribute 80.5% of $1 - D(N)$. High-$k$ shell contributions are negligible.
3. Quantitative Theorem F (§4): Replacing monotone A with $A_q^\sharp$ (quantitative controlled convergence), combined with Erdős-Kac weight migration, conditionally closes $D(N) \to 1$.

§2. K_p Distribution Constancy (Numerical Finding)

§2.1 K_p Distribution is Approximately k-Independent

Recall: $K_p(n) = G_{\mathrm{spf}}(n) - G_{\mathrm{spf}}(m)$, where $m = n/P^-(n)$, $\Omega(n) = k+1$.

$P(K_p > 0) \approx 0.43$ and $E[K_p \mid {>}0] \approx 1.34$ are nearly $k$-independent. The slow decrease of $E[K_p]$ comes from $P(K_p < 0)$ rising from 0.238 to 0.288 — not from changes in conditional means, but from a slight increase in negative-event probability.

§2.2 Physical Decomposition of K_p

$$K_p = [\rho_E(n-1) - \rho_E(m-1)] - [M_n - M_m]$$

i.e., $K_p$ = predecessor gap − target gap.

§2.3 Cofactor Representativeness Confirmed

$\Omega(m) = k$ holds by definition ($\Omega(n) = k+1$ and $m = n/P^-(n)$ removes one prime factor count). Cofactor bias $= E_{I_k}[G_{\mathrm{spf}}(m)] - \mu_k$:

$P^- = 2$ (95–99% of events) has minimal bias (< 0.05). $P^- = 3$ has large bias (0.5–1.3) but negligible weight (< 5%). Weighted bias is < 0.09 throughout — cofactors are highly representative.

§3. The 80/3 Law

§3.1 Shell Decomposition of 1 − D(N)

$$1 - D(N) = \frac{\pi(N)}{N} + \sum_{k=2}^{\infty} w_k(N) \cdot q_k(N)$$

where $q_k(N) = 1 - p_k(N)$.

Exact decomposition at $N = 10^7$:

§3.2 Physical Interpretation

The entire content of $D(N) \to 1$ is: waiting for the Erdős-Kac distribution center to migrate from $k \approx 3$ (its position at $N = 10^7$) to $k \gg 10$.

On $k \geq 10$ shells, $q_k < 0.005$ — nearly all integers jump. But at $N = 10^7$, these shells carry only 0.9% of the weight. The Erdős-Kac distribution center sits at $\ln\ln N$, and its migration toward higher $k$ is doubly exponentially slow.

The $N$-scale required for $D(N) \to 1$: when $\ln\ln N \approx 10$, the Erdős-Kac center reaches $k \approx 10$, requiring $N \approx e^{e^{10}} \approx 10^{9600}$.

§3.3 Implications for Quantitative Theorem F

The 80/3 Law shows: high-$k$ shells have already "solved their own problem" ($q_k \approx 0$ for $k \geq 7$). Theorem 8 does not need high-$k$ shells to improve — they are already good enough.

For low-$k$ shells ($k = 2, 3, 4$), data shows $p_k(N)$ monotonically decreasing in $N$ (approaching $p_\infty(k)$ from above), so $q_k(N)$ slightly increases with $N$. But this does not obstruct $D(N) \to 1$: the key for low $k$ is not that $q_k(N)$ shrinks, but that the weight $w_k(N) \to 0$ (Sathe-Selberg). Even if $q_k(N)$ slightly worsens, as long as $w_k(N)$ decays fast enough, the product $w_k \cdot q_k$ still tends to zero. This is exactly the logic of Theorem 8: low $k$ relies on vanishing weights, high $k$ relies on tiny $q_k$.

§4. Quantitative Theorem F

§4.1 Replacement Assumption $A_q^\sharp$

§4.2 Theorem 8 (Quantitative Theorem F)

§4.3 Numerical Consistency Check for $A_q^\sharp$

Fit: $p_k(N) \approx p_\infty(k) + c_k / \ln N$ ($N = 10^4$ to $3 \times 10^7$):

$c_k$ decays rapidly. Numerical check of $A_q^\sharp$'s second condition ($N = 10^7$):

$$\eta(N) \sum_k w_k \cdot c_k = \sum_k w_k \cdot c_k / \ln N = 0.038$$

Dominated by $k = 4$ (89% contribution). As $N \to \infty$, $w_4 \to 0$ (Erdős-Kac migration), so the data is consistent with the second condition of $A_q^\sharp$.

§5. Updated Proof Landscape

§6. Thermodynamic Interface

References

1. Qin, H. (2025a). ZFCρ Paper XV. DOI: 10.5281/zenodo.19007312.
2. Qin, H. (2025b). ZFCρ Paper XVI. DOI: 10.5281/zenodo.19013602.
3. Qin, H. (2025c). ZFCρ Paper XVIII. DOI: 10.5281/zenodo.19023418.
4. Qin, H. (2025d). ZFCρ Paper XX. DOI: 10.5281/zenodo.19027893.
5. Qin, H. (2025e). ZFCρ Paper XXI. DOI: 10.5281/zenodo.19037934.
6. Qin, H. (2025f). ZFCρ Paper XXII. DOI: 10.5281/zenodo.19039953.
7. Sathe, L.G. (1953). J. Indian Math. Soc. 17, 63–141.
8. Selberg, A. (1954). J. Indian Math. Soc. 18, 83–87.

A.1 Third Validation of "Data Before Direction"

This round's 8-block exploration (Assumption A investigation + (L1)(L3) deep dive) discovered the 80/3 Law and $K_p$ distribution constancy BEFORE Paper 23 was defined — both became core contributions. Without running data first, we might have pursued Route 1 (direct monotonicity proof), wasting significant time.

A.2 Strategic Convergence of Three AIs

ChatGPT, Gemini, and Grok independently analyzed the working note and all recommended Route 2 (quantitative controlled convergence). ChatGPT further noted: the error term does not need to be as precise as $c_k/\ln N$ — $O((\ln\ln N)^A / \ln N)$ suffices, because the exponential decay of Erdős-Kac weights absorbs $c_k$ growth. This judgment significantly reduced the formalization difficulty of $A_q^\sharp$.

B.1 K_p Distribution (k+1 = 8, N = 10^7)

B.2 Cofactor Bias by P⁻(n) (k+1 = 8)

B.3 Shell Decomposition of 1 − D(N) (N = 10^7)

(See §3.1 table.)