Shifted-Product Deficit and the Logarithmic Growth of Mloc at the Ω=2 Layer
DOI: 10.5281/zenodo.19164588Through seven progressive numerical experiments, this paper reveals that the local margin \(M_{\mathrm{loc}}(p, t)\) at the Ω=2 layer exhibits a clear logarithmic growth signal when the ratio \(t = q/p\) is held fixed, and identifies the driving mechanism as the shifted-product deficit.
(1) At fixed \(t = q/p \in [3,10]\) with full window (\(p \leq 31{,}623\)), \(M_{\mathrm{loc}}\) has a positive slope of +0.096 against \(\ln(p)\), with \(R^2 = 0.89\). The previously reported \(\bar{M}_N \approx 1.5\) (mixed \(t\), \(R^2 = 0.0001\)) is identified as a composition shift artifact — the same mechanism discovered in Paper XXXVIII for \(G\).
(2) The growth of \(M_{\mathrm{loc}}\) admits a precise decomposition: \(\text{slope} = h_{\tau} + h_q - 2h_{pq}\), where \(h_{\tau}, h_q\) are the ρ/ln growth rates for primes and \(h_{pq}\) is the ρ/ln growth rate for the shifted prime product \(pq-1\). The sum of the three component slopes exactly equals the \(M_{\mathrm{loc}}\) slope (algebraic identity verification).
(3) Multi-scale \(1/\ln(s)\) extrapolation is consistent with \(h_{\tau}(\infty) \approx 3.923\) and \(h_{pq}(\infty) \approx 3.848\), giving \(\Delta(\infty) \approx 0.076 > 0\) (\(R^2 = 0.92\)). The resulting unconstrained estimate for the asymptotic slope is \(\text{slope}(\infty) \approx 0.117\), consistent with the Block 3 measurement of 0.096 at moderate scales.
(4) A parity control experiment decomposes the total deficit \(h_{\mathrm{gen}} - h_{pq}\) (≈ 0.0165) into a parity effect (approximately 54%) and a beyond-parity effect (approximately 46%). The difference \(h_{pq} - h_{\mathrm{even}} = -0.0075 \pm 0.0025\) (>3σ) confirms that the multiplicative shadow effect is real, not merely a parity artifact.
(5) Ω=2 closure is reduced to a precise open lemma — the ratio-mixing lemma: proving that the positive slope of \(M_{\mathrm{loc}}\) is not averaged away under the actual \(q/p\)-mixing.
Keywords: shifted-product deficit, \(M_{\mathrm{loc}}\), composition shift, ratio-mixing lemma, Ω=2 closure
§1 Introduction
1.1 Background
Paper XXXVIII established the ratio-parametrization \(G = f(q/p)\), identified the "B–C erosion crossing zero" as a composition shift artifact, and introduced the three-layer G distinction (\(G_{\mathrm{loc}}, \bar{G}_N, G_X\)). Paper XXXIV established the Harmonic Cesàro Lemma: if \(V(p) = O(1)\) and \(M_p \to \infty\), then \(\bar{c}_h \to 0\) and the Ω=2 layer closes. Since \(V(p) \approx 1.4\) was globally confirmed in Paper XXXIV, Ω=2 closure reduces to the question: does \(M_p \to \infty\)?
1.2 Scope of the Present Paper
This paper answers through precise numerical experiments: What is the true asymptotic behavior of \(M_p\)? What drives its growth? What remains to be established for Ω=2 closure?
§2 Three-Layer M Distinction
Paralleling the three-layer G framework from Paper XXXVIII:
- \(M_{\mathrm{loc}}(p, t)\): The local margin at fixed ratio \(t = q/p\). For each prime \(p\), only primes \(q\) with \(q/p \in [t_{\mathrm{lo}}, t_{\mathrm{hi}}]\) are used to compute \(\mathrm{diff}_{\mathrm{loc}} = E[\rho(pq-1)] - E[\rho(q)]\) (harmonic-weighted), then \(M_{\mathrm{loc}} = \tau_p - \mathrm{diff}_{\mathrm{loc}}\), where \(\tau_p = \rho(p) + 2\).
- \(\bar{M}_N(p)\): The mixed-average margin at fixed cutoff \(N\) and prime \(p\), averaging over all available \(q\). Equals \(\int M_{\mathrm{loc}}(p, t)\, d\mu_{N,p}(t)\) over \(t \in [1, N/p^2]\).
- \(M_X(p)\): The expanding-window limit with \(p\) fixed and \(N \to \infty\). A theoretical object not directly measurable.
§3 Experimental Design and Data
3.1 Data Source
rho_1e10.bin: ρE values up to \(N = 10^{10}\), in int16 format, generated by rho_dp.c (Paper XXXII convention). Sanity checks: ρ(1)=0, ρ(2)=1, ρ(3)=2, ρ(100)=15, ρ(10⁷)=58, ρ(10⁸)=66. All passed.
3.2 Overview of Seven Experiments
| Block | Object Measured | p Range | Core Result |
|---|---|---|---|
| 1 | M̄ (q > p) | p ≤ 10⁵ | M̄ ~ 2.4, R²=0.013 |
| 2 | M̄ (mixed t) | p to 10⁷ | M̄ ≈ 1.5, R²=0.0001 |
| 3 | M_loc (fixed t∈[3,10]) | p ≤ 31,623 | 0.096·ln(p), R²=0.89 |
| 4 | h_prime − h₀ | 10³–10⁹ | ~0.71/ln(n) → 0 |
| 4b | Gemini formula check | p ≤ 10⁶ | Systematic bias ~0.85 |
| 5b | Direct component slopes | p ≤ 31,623 | h_tau+h_q−2h_pq=0.100 |
| 6 | Multi-scale h convergence | p 500–40,000 | Δ(∞)≈0.076 |
| 7 | Parity control | p 500–40,000 | parity 54%, shadow 46% |
§4 Blocks 1–2: Surface Behavior of the Mixed Average M̄
4.1 Block 1: p ≤ √N = 10⁵, q > p
For 9,587 primes \(p \in [3, 10^5]\), we compute \(\bar{M} = \tau_p - (E[\rho(pq-1)] - E[\rho(q)])\), with \(q\) ranging over primes in \((p, N/p)\).
| Band | n_p | ⟨ρ(p)⟩ | ⟨τ_p⟩ | ⟨diff⟩ | ⟨M̄⟩ | std |
|---|---|---|---|---|---|---|
| [3, 10) | 3 | 4.00 | 6.00 | 5.999 | 0.001 | 0.815 |
| [100, 300) | 37 | 19.27 | 21.27 | 19.582 | 1.688 | 1.217 |
| [1000, 3000) | 262 | 28.50 | 30.50 | 28.270 | 2.230 | 0.798 |
| [10000, 30000) | 2016 | 37.51 | 39.51 | 37.133 | 2.381 | 0.828 |
| [30000, 100000) | 6343 | 42.08 | 44.08 | 41.629 | 2.450 | 0.824 |
M̄ rises from ~1.0 to ~2.45. The slope of \(\tau_p\) against \(\ln(p)\) is 3.918; that of diff is 3.834; the difference is only 0.084. A direct fit of M̄ against \(\ln(p)\) yields \(R^2 = 0.013\). \(V(p) \approx 1.40\), globally bounded, consistent with Paper XXXIV.
4.2 Block 2: Extending p to 10⁷
Key insight: we do not require \(p \leq \sqrt{N}\); we only need \(pq-1 < N\). For \(p = 10^6\), \(q < 10^4\) suffices; for \(p = 10^7\), \(q < 10^3\).
| p Range | ⟨M̄⟩ | std | gap = τ/ln(p) − diff/ln(p) |
|---|---|---|---|
| [5×10⁴, 10⁵) | 1.431 | 0.848 | 0.128 |
| [10⁵, 3.16×10⁵) | 1.528 | 0.786 | 0.125 |
| [3.16×10⁵, 10⁶) | 1.557 | 0.802 | 0.117 |
| [10⁶, 3.16×10⁶) | 1.506 | 0.798 | 0.104 |
| [3.16×10⁶, 10⁷) | 1.468 | 0.798 | 0.094 |
Surface conclusion: \(\bar{M} = \Theta(1)\). Block 3 shows that this flat appearance is a consequence of composition shift and cannot serve as a diagnostic for the asymptotic behavior of \(M_{\mathrm{loc}}\) or \(M_X\) (see §5).
§5 Block 3: Identifying Composition Shift
5.1 Motivation
Paper XXXVIII proved \(G\) depends only on \(t = q/p\), and "B-C erosion through zero" is a composition shift. A separate analysis thread noted: "Paper XXXVIII's composition shift may also affect the \(M_p\) diagnostic — large \(p\) can only access small \(q\), and the decline in gap may partly arise from windowing effects." If \(\bar{M}\) is similarly affected, Block 2's "\(\bar{M} = \mathrm{constant}\)" is an artifact.
5.2 Fixed t ∈ [3,10]: The Decisive Experiment
For each \(p\), only primes \(q\) with \(q/p \in [3,10]\) are used. The full window requires \(10p^2 \leq N\), i.e., \(p \leq \sqrt{N/10} \approx 31{,}623\).
Main fit (first 9 bins, full [3,10] window, p ≤ 31,623):
| p Band | n | ⟨ln(p)⟩ | ⟨M_loc⟩ | τ/lnp | diff/lnp | gap |
|---|---|---|---|---|---|---|
| [500, 1000) | 73 | 6.60 | 2.068 | 4.056 | 3.743 | 0.313 |
| [1000, 2000) | 135 | 7.29 | 2.099 | 4.039 | 3.751 | 0.288 |
| [2000, 3000) | 127 | 7.82 | 2.268 | 4.048 | 3.758 | 0.290 |
| [3000, 5000) | 239 | 8.28 | 2.226 | 4.030 | 3.761 | 0.269 |
| [5000, 7000) | 231 | 8.70 | 2.288 | 4.028 | 3.765 | 0.263 |
| [7000, 10000) | 329 | 9.04 | 2.288 | 4.019 | 3.766 | 0.253 |
| [10000, 15000) | 525 | 9.43 | 2.380 | 4.021 | 3.768 | 0.253 |
| [15000, 20000) | 508 | 9.77 | 2.330 | 4.008 | 3.769 | 0.239 |
| [20000, 30000) | 983 | 10.12 | 2.429 | 4.010 | 3.770 | 0.240 |
\(M_{\mathrm{loc}} = +0.096 \cdot \ln(p) + 1.43\), \(R^2 = 0.89\).
Truncated-window robustness check (p > 31,623; upper end of t=10 inaccessible; not included in main fit):
| p Band | n | ⟨ln(p)⟩ | ⟨M_loc⟩ | τ/lnp | diff/lnp | gap |
|---|---|---|---|---|---|---|
| [30000, 50000) | 1000 | 10.58 | 2.443 | 4.004 | 3.774 | 0.231 |
| [50000, 70000) | 396 | 10.89 | 2.417 | 3.998 | 3.776 | 0.222 |
Broadly compatible with the main-fit extrapolation.
5.3 Multi-t-Window Verification
All four tested t-windows ([1,3], [3,10], [10,30], [30,100]) show \(M_{\mathrm{loc}}\) increasing with \(p\):
| t Window | p = 1k–3k | p = 3k–10k | p = 10k–30k | p = 30k–100k |
|---|---|---|---|---|
| t∈[1,3] | 2.136 | 2.223 | 2.317 | 2.403 |
| t∈[3,10] | 2.181 | 2.270 | 2.374 | 2.431 |
| t∈[10,30] | 2.223 | 2.326 | 2.408 | — |
| t∈[30,100] | 2.270 | 2.365 | 2.401 | — |
5.4 Block 2 vs Block 3
| Block 2 (mixed t) | Block 3 (fixed t, full window) | |
|---|---|---|
| Slope | +0.005 | +0.096 |
| R² | 0.0001 | 0.89 |
| Diagnosis | Composition shift artifact | Growth signal |
\(R^2\) jumps from 0.0001 to 0.89. In the presence of composition shift, the constant appearance of the mixed average is compatible with logarithmic growth of the local margin: Block 2's \(\bar{M} \approx 1.5\) reflects the shifting \(q/p\) composition as \(p\) increases, not the asymptotic behavior of \(M_{\mathrm{loc}}\).
§6 Blocks 4–4b: Failure of the Gemini Formula
6.1 Gemini's Derivation and Block 4 Measurement
Gemini derived: at fixed \(t\), \(M_{\mathrm{loc}} \sim (2h_{\mathrm{prime}} - 2h_0) \cdot \ln(p) + (h_{\mathrm{prime}} - h_0) \cdot \ln(t)\), where \(h_{\mathrm{prime}}\) and \(h_0\) are the ρ/ln growth rates for primes and general integers respectively.
Block 4 measures \(h_{\mathrm{prime}} - h_0\) at 30 log-spaced scales (10³ to 10⁹):
| Scale | h_prime | h₀ | Δ |
|---|---|---|---|
| 10⁴ | 3.798 | 3.720 | 0.078 |
| 10⁶ | 3.833 | 3.781 | 0.052 |
| 10⁸ | 3.843 | 3.803 | 0.039 |
| 10⁹ | 3.845 | 3.810 | 0.035 |
Fit: \(\Delta = 0.710/\ln(n) + 0.00019\), \(R^2 = 0.996\). The intercept is numerically indistinguishable from zero, implying \(h_{\mathrm{prime}} \to h_0\) asymptotically. If the Gemini formula were correct, this would imply \(M_{\mathrm{loc}} \to \mathrm{constant} \approx 1.42\).
6.2 Block 4b: Cross-Validation
\(\rho(p) = \rho(p-1) + 1\), with 100% confirmation (78,496 primes, no exceptions). The predecessor \(p-1\) is systematically cheaper than a general integer by ~0.3 (\(p-1\) is always even).
Plugging Block 4's h-values into the Gemini formula and comparing with Block 3's measured \(M_{\mathrm{loc}}\):
| p | M_loc (measured) | M_loc (Gemini) | Bias |
|---|---|---|---|
| 750 | 2.068 | 1.165 | −0.903 |
| 8500 | 2.288 | 1.544 | −0.744 |
| 60000 | 2.417 | 1.454 | −0.963 |
The formula is systematically low by ~0.85. The reason: it assumes \(E[\rho(pq-1)] \sim h_0 \cdot \ln(pq)\), but \(pq-1\) is not a general integer — it is a shifted product with its own effective growth rate \(h_{pq} \neq h_0\).
§7 Block 5b: The Shifted-Product Deficit
7.1 Direct Component Slopes
At fixed \(t \in [3,10]\), we directly measure the slopes of all five raw quantities against \(\ln(p)\):
| Component | Slope vs ln(p) | R² |
|---|---|---|
| τ_p | +3.91746 | 0.9488 |
| E[ρ(q)] | +3.90829 | 0.9999+ |
| E[ρ(pq−1)] | +7.72529 | 0.9999+ |
| diff = E[ρ(pq−1)] − E[ρ(q)] | +3.81700 | 0.9999 |
| M_loc = τ_p − diff | +0.10045 | 0.012 |
Verification: τ slope − diff slope = 3.91746 − 3.81700 = 0.10045 = M_loc slope. Exact closure.
7.2 Effective h-Coefficients
Since \(E[\rho(pq-1)]\) has slope ≈ \(2h_{pq}\) (because \(\ln(pq) \sim \ln(t) + 2\ln(p)\) at fixed \(t\)) and \(E[\rho(q)]\) has slope ≈ \(h_q\):
| Quantity | Effective h | Identity |
|---|---|---|
| h_tau | 3.917 | ρ-slope for prime p |
| h_q | 3.908 | ρ-slope for prime q |
| h_pq | 3.863 | ρ-slope for shifted prime product pq−1 |
\(M_{\mathrm{loc}}\) slope \(= h_{\tau} + h_q - 2h_{pq} = 3.917 + 3.908 - 2 \times 3.863 = 0.100\).
7.3 Core Mechanism: Shifted-Product Deficit
The driver of \(M_{\mathrm{loc}}\) growth is \(h_{\tau} + h_q > 2h_{pq}\): the ρ/ln growth rate of the shifted prime product \(pq-1\) is systematically lower than that of primes. The original Gemini formula equates \(pq-1\) with a general integer (\(h_{pq} = h_0\)), but in fact \(h_{pq} < h_0 < h_{\mathrm{prime}}\). The corrected formula:
$$M_{\mathrm{loc}} \sim (h_{\tau} + h_q - 2h_{pq}) \cdot \ln(p) + \text{lower-order terms}$$§8 Block 6: Multi-Scale Convergence
8.1 Extrapolation of Four h-Values
At 20 log-spaced scales (\(p = 500\) to 40,824), we simultaneously measure \(h_{\tau}, h_q, h_{pq}\), and \(h_{\mathrm{gen}}\):
| h | Extrapolated Limit | Fit | R² |
|---|---|---|---|
| h_tau(∞) | 3.923 | −1.109/ln(s) + 3.923 | 0.986 |
| h_q(∞) | 3.890 | −0.651/ln(s) + 3.890 | 0.972 |
| h_pq(∞) | 3.848 | −0.487/ln(s) + 3.848 | 0.928 |
| h_gen(∞) | 3.856 | −0.435/ln(s) + 3.856 | 0.995 |
The data are consistent with the ordering \(h_{\tau} > h_q > h_{\mathrm{gen}} > h_{pq}\).
8.2 Convergence of h_tau − h_pq
Observed values rise from −0.03 (\(p \sim 500\), small-sample noise) to +0.013 (\(p \sim 40{,}000\)). The \(1/\ln(s)\) fit: \(\Delta = -0.622/\ln(s) + 0.0756\), \(R^2 = 0.92\). The data are consistent with \(\Delta(\infty) \approx 0.076 > 0\).
8.3 Unconstrained Estimate of the Asymptotic Slope
\(\text{slope}(\infty) = h_{\tau}(\infty) + h_q(\infty) - 2h_{pq}(\infty) = 3.923 + 3.890 - 2 \times 3.848 = \mathbf{0.117}\).
This is an unconstrained estimate from three independent \(1/\ln(s)\) fits. Block 3's measured slope of 0.096 at \(p \leq 31{,}623\) is below the extrapolated 0.117, consistent with the current window not yet having reached the asymptotic regime.
§9 Block 7: Parity Control Experiment
9.1 Motivation
It was pointed out in review that \(pq-1\) is always even for odd primes \(p, q\). How much of \(h_{pq} < h_{\mathrm{gen}}\) comes from parity, and how much from a "multiplicative shadow"?
9.2 Five Control Groups
At 15 log-spaced scales (\(p = 500\) to 40,000), we simultaneously measure:
| Control Group | Definition | Mean h |
|---|---|---|
| h_pq | E[ρ(pq−1)/ln(pq−1)], p,q prime | 3.787 |
| h_2prod | E[ρ(ab−1)/ln(ab−1)], a,b random integers | 3.820 |
| h_even | Random even integers of same size | 3.795 |
| h_gen | Random integers (mixed parity) | 3.804 |
| h_odd | Random odd integers of same size | 3.812 |
9.3 Decomposition of the Deficit
Total deficit \(h_{\mathrm{gen}} - h_{pq} \approx 0.0165\):
| Effect | Magnitude | Share |
|---|---|---|
| Parity (h_gen − h_even) | 0.0090 | ~54% |
| Beyond-parity (h_even − h_pq) | 0.0075 | ~46% |
\(h_{pq} - h_{\mathrm{even}} = -0.0075 \pm 0.0025\) (>3σ). The multiplicative shadow effect is real, not entirely a parity artifact.
9.4 h_pq vs h_2prod: The Role of Primality
\(h_{pq} - h_{2\mathrm{prod}} = -0.033 \pm 0.008\). Shifted prime products are significantly cheaper than shifted random products. However, the \(h_{2\mathrm{prod}}\) control group is not parity-matched, so this comparison still conflates parity composition differences and cannot cleanly isolate a "prime purity effect." A parity-matched control (e.g., \(a, b\) restricted to odd integers) would be required for further confirmation.
9.5 Impact on Core Conclusions
Regardless of how the deficit mechanism decomposes, the positive sign of \(h_{\tau} - h_{pq}\) is unaffected. The shifted-product deficit driving \(M_{\mathrm{loc}}\) growth is robust.
§10 Composition Shift: A Core Methodological Lesson
10.1 Three Occurrences
| Paper | Distorted Quantity | Mechanism |
|---|---|---|
| XXXV–XXXVIII | G (B–C gap) | Large p forced to use q ~ p; mixed average obscures G's t-dependence |
| XXXVIII | G_X(p) vs G_loc(t) | Expanding-window G converges to ≈ −1.9 due to large-t dominance |
| XXXIX | M̄ vs M_loc | Large p forced to use small t; mixed average flattens M_loc's ln(p) growth |
10.2 Methodological Conclusion
Any statistic involving \(q\) must be measured at fixed \(t = q/p\); otherwise, composition shift may produce artifacts. This constraint extends to subsequent Ω ≥ 3 generalizations.
§11 From Mloc to Ω=2 Closure
11.1 What Has Been Established
- At fixed \(t\), the data are consistent with \(M_{\mathrm{loc}} \sim (h_{\tau} + h_q - 2h_{pq}) \cdot \ln(p)\), with an unconstrained extrapolated slope ≈ 0.117.
- \(V(p) \approx 1.4\) is globally bounded (Paper XXXIV).
- The shifted-product deficit is robust (Block 7 parity control, >3σ).
11.2 Ratio-Mixing Lemma (Open Lemma)
Statement. Let \(a(t)\) denote the slope of \(M_{\mathrm{loc}}(p, t)\) with respect to \(\ln(p)\), and let \(\mu_{N,p}(t)\) be the actual \(q/p\) distribution in the Ω=2 defect computation, supported on \(t \in [1, N/p^2]\).
Ratio-Mixing Lemma (to be proved): There exists an admissible growth sequence \(N = N(p)\) (satisfying \(N/p^2 \to \infty\)) and a constant \(c > 0\) such that
$$\liminf_{p \to \infty} \int_1^{N(p)/p^2} a(t)\, d\mu_{N(p),p}(t) \geq c > 0$$The core difficulty is that as \(p \to \infty\), the upper limit of integration \(N/p^2\) contracts (unless \(N\) grows fast enough). Required inputs: (i) the precise structure of \(\mu_{N,p}(t)\); (ii) whether \(a(t)\) has a uniform positive lower bound over relevant \(t\); (iii) the growth rate of \(N(p)\).
11.3 Heuristic Reasons for Optimism
The physical sources of the shifted-product deficit (parity + multiplicative shadow) do not depend on the specific value of \(t\). For any \(t > 0\), \(pq-1 \sim tp^2\) lies near \(p \times (tp)\), benefiting from a similar structural discount. Thus \(a(t) > 0\) may hold for all relevant \(t\). However, this remains a heuristic argument, not a proof.
11.4 Conditional Conclusion
If the ratio-mixing lemma holds: \(\bar{M}_N(p) \to \infty\) → \(\Gamma_p = \bar{M}^2/V \to \infty\) → Cantelli: \(c_p \to 0\) → Harmonic Cesàro Lemma: \(\bar{c}_h \to 0\) → Ω=2 layer closure.
§12 Discussion
12.1 Core Contributions of Paper XXXIX
- (a) Identifying \(\bar{M}_N \approx 1.5\) as a composition shift artifact; \(M_{\mathrm{loc}}\) at fixed \(t\) shows clear \(\ln(p)\) growth.
- (b) Identifying the driver: shifted-product deficit, \(h_{\tau} + h_q > 2h_{pq}\).
- (c) Refuting the Gemini formula — incorrect parameter identification (equating \(pq-1\) with a general integer).
- (d) Multi-scale extrapolation consistent with \(h_{\tau} > h_q > h_{\mathrm{gen}} > h_{pq}\).
- (e) Parity control experiment: approximately 54% parity effect, approximately 46% beyond-parity; multiplicative shadow exceeds 3σ.
- (f) Reducing Ω=2 closure to the ratio-mixing lemma.
12.2 Distance Estimate
Ω=2 closure now reduces to a small number of precise analytical inputs, the core being the ratio-mixing lemma. This is a more precise and more attackable formulation than any prior statement of the problem.
§13 Data Sources and Reproducibility
| Script | Measurement | Block |
|---|---|---|
| p39_mp_precise_v2.py | M̄ full components, 9,587 primes, p ≤ 10⁵ | 1 |
| p39_large_p.py | M̄ extended to p ~ 10⁷, 10,000 primes | 2 |
| p39_fixed_ratio.py | M_loc at fixed t, Experiments 1+2 | 3 |
| p39_hprime.py | h_prime vs h₀ multi-scale, 30 scales | 4 |
| p39_crossval.py | Gemini formula check + ρ(p)=ρ(p−1)+1 + smooth bias | 4b |
| p39_direct_slopes.py | Direct component slopes, h_tau/h_q/h_pq | 5b |
| p39_hpq_multiscale.py | h_tau, h_q, h_pq, h_gen multi-scale convergence | 6 |
| p39_parity_control.py | Parity control: h_pq vs h_even vs h_gen vs h_odd vs h_2prod | 7 |
Data: rho_1e10.bin (int16, N = 10¹⁰, rho_dp.c, Paper XXXII convention).
References
[1] H. Qin. ZFCρ Papers I–XXXVIII. Paper XXXVIII DOI: 10.5281/zenodo.19157939.
[2] H. Qin. ZFCρ Paper XXXIV: Upper-Margin Closure and Bounded-Variance. DOI: 10.5281/zenodo.19140015.
[3] H. Qin. ZFCρ Paper XXI: Queue Isomorphism and Local Lipschitz Reduction. DOI: 10.5281/zenodo.19037934.
Acknowledgments
Claude (子路) discovered the int16 format bug, designed the Block 2–3–4b–5b–6–7 experiment chain (eight scripts total), and drafted working notes v1–v5. After Block 4 yielded h_prime → h₀, Claude designed Block 4b for cross-validation; after the Gemini formula was found to underpredict by ~0.85, Claude designed Block 5b and discovered the exact decomposition M_loc slope = h_tau + h_q − 2h_pq; Claude then designed Block 6 for multi-scale convergence.
A separate Claude thread (thermodynamic analysis) noted that Block 2 might be affected by composition shift, directly motivating Block 3.
Han Qin (author) asked "Is our data correct?" after Block 4, which led to Block 4b, the exposure of the Gemini formula's flaw, and the entire Block 5b → 6 chain. The author set the methodological direction of "depth first, follow the data."
ChatGPT (公西华) warned at the Paper XXXVIII stage that "the M_p → ∞ derivation has a gap"; in the v2 review proposed the three-layer M distinction, truncated-window correction, and the ratio-mixing lemma; in the v4 review identified the arithmetic inconsistency (0.151 → 0.117), overclaiming language, parity as a competing explanation, and the need for theorem-sized formulation of the mixing lemma — directly motivating Block 7 and comprehensive revisions.
Gemini (子夏) derived the formula M ~ 2(h_prime − h₀)·ln(p); the algebraic framework was correct though the parameter identification was wrong, directly leading to Block 4's measurement and, through its failure, revealing the role of h_pq.
Grok (子贡) provided Lindley parameter estimates and series consistency checks.
The final text was independently completed by the author.
本文通过七组递进式数值实验，揭示了 Ω=2 层局部 margin \(M_{\mathrm{loc}}(p, t)\) 在固定比例 \(t = q/p\) 下具有清晰的对数增长信号，并识别出驱动这一增长的机制——shifted-product deficit。
(1) 在固定 \(t = q/p \in [3,10]\) 且完整窗口（\(p \leq 31623\)）下，\(M_{\mathrm{loc}}\) 对 \(\ln(p)\) 的线性拟合斜率为 +0.096，\(R^2 = 0.89\)。此前报告的 \(\bar{M}_N \approx 1.5\)（混合 t，\(R^2 = 0.0001\)）被识别为 composition shift 伪影——与 Paper XXXVIII 发现的 G 的 composition shift 是同一机制。
(2) \(M_{\mathrm{loc}}\) 的增长可精确分解为 \(\text{slope} = h_\tau + h_q - 2h_{pq}\)，其中 \(h_\tau, h_q\) 是素数的 ρ/ln 增长斜率，\(h_{pq}\) 是 shifted prime product \(pq-1\) 的 ρ/ln 增长斜率。三项分量斜率之和精确等于 \(M_{\mathrm{loc}}\) 斜率（代数恒等式验证）。
(3) 多尺度 \(1/\ln(s)\) 外推与 \(h_\tau(\infty) \approx 3.923\)，\(h_{pq}(\infty) \approx 3.848\) 一致，差值 \(\Delta(\infty) \approx 0.076 > 0\)（\(R^2 = 0.92\)）。据此得到渐近斜率的 unconstrained estimate：\(\text{slope}(\infty) \approx 0.117\)，与 Block 3 在中等尺度的实测（0.096）自洽。
(4) Parity 控制实验将 \(h_{\mathrm{gen}} - h_{pq}\) 的总 deficit（≈ 0.0165）分解为 parity 效应（约 54%）和 beyond-parity 效应（约 46%）。\(h_{pq} - h_{\mathrm{even}} = -0.0075 \pm 0.0025\)（>3σ），确认乘法阴影效应是真实的，不完全是 parity 伪影。
(5) Ω=2 闭合归约为一个精确的开放引理——ratio-mixing lemma：证明 \(M_{\mathrm{loc}}\) 的正斜率在 actual q/p-mixing 下不被平均掉。
关键词：shifted-product deficit，\(M_{\mathrm{loc}}\)，composition shift，ratio-mixing lemma，Ω=2 闭合
§1 引言
1.1 背景
Paper XXXVIII 建立了 \(G = f(q/p)\) 的 ratio-parametrization，发现"B-C 侵蚀穿零"是 composition shift 伪影，并建议三层 G 区分（\(G_{\mathrm{loc}}, \bar{G}_N, G_X\)）。Gemini 和 Grok 据此推测 \(M_p \to \infty\)（"自动闭合"）；公西华（ChatGPT）警告这一推导跳了一步——\(M_p\) 是两个都趋于无穷的量的差，差本身不一定发散。Paper XXXIV 建立了 Harmonic Cesàro Lemma：若 \(V(p) = O(1)\) 且 \(M_p \to \infty\)，则 \(\bar{c}_h \to 0\)，Ω=2 层闭合。因此 Ω=2 闭合归约为：\(M_p\) 是否 → ∞？
1.2 本文的任务
通过精确数值实验回答：\(M_p\) 的真实渐近行为是什么？增长的驱动力是什么？从 \(M_p\) 的增长到 Ω=2 闭合还缺什么？
§2 三层 M 的区分
沿用 Paper XXXVIII 的三层 G 框架，对 M 做平行定义：
- \(M_{\mathrm{loc}}(p, t)\)：固定 ratio \(t = q/p\) 的局部 margin。对每个 \(p\)，只用 \(q/p \in [t_{\mathrm{lo}}, t_{\mathrm{hi}}]\) 的素数 \(q\) 计算 \(\mathrm{diff}_{\mathrm{loc}} = E[\rho(pq-1)] - E[\rho(q)]\)（harmonic 加权），然后 \(M_{\mathrm{loc}} = \tau_p - \mathrm{diff}_{\mathrm{loc}}\)，其中 \(\tau_p = \rho(p) + 2\)。
- \(\bar{M}_N(p)\)：固定截断 \(N\) 和素数 \(p\)，对所有可用 \(q\) 取 harmonic 加权平均。等于 \(\int M_{\mathrm{loc}}(p, t)\, d\mu_{N,p}(t)\)，其中 \(\mu_{N,p}\) 是由 \(N\) 和 \(p\) 决定的 \(q/p\) 分布。
- \(M_X(p)\)：固定 \(p\)，令 \(N \to \infty\) 的极限。理论对象，无法直接测量。
§3 实验设计与数据
3.1 数据来源
rho_1e10.bin：ρ_E 值到 \(N = 10^{10}\)，int16 格式，由 rho_dp.c 生成（Paper XXXII convention）。Sanity checks：ρ(1)=0，ρ(2)=1，ρ(3)=2，ρ(100)=15，ρ(10⁷)=58，ρ(10⁸)=66，全部通过。
3.2 七组实验概览
| Block | 测量对象 | p 范围 | 核心结果 |
|---|---|---|---|
| 1 | M̄（q > p） | p ≤ 10⁵ | M̄ ~ 2.4, R²=0.013 |
| 2 | M̄（混合 t） | p 到 10⁷ | M̄ ≈ 1.5, R²=0.0001 |
| 3 | M_loc（固定 t∈[3,10]） | p ≤ 31623 | 0.096·ln(p), R²=0.89 |
| 4 | h_prime − h₀ | 10³–10⁹ | ~0.71/ln(n) → 0 |
| 4b | Gemini 公式验证 | p ≤ 10⁶ | 系统偏低 ~0.85 |
| 5b | 直接分量斜率 | p ≤ 31623 | h_tau+h_q−2h_pq=0.100 |
| 6 | h 多尺度收敛 | p 500–40000 | Δ(∞)≈0.076 |
| 7 | Parity 控制 | p 500–40000 | parity 54%, shadow 46% |
§4 Block 1–2: 混合 M̄ 的表面行为
4.1 Block 1: p ≤ √N = 10⁵，q > p
对 9587 个素数 \(p \in [3, 10^5]\)，计算 \(\bar{M} = \tau_p - (E[\rho(pq-1)] - E[\rho(q)])\)，\(q\) 遍历 \((p, N/p)\) 中的素数。
| Band | n_p | ⟨ρ(p)⟩ | ⟨τ_p⟩ | ⟨diff⟩ | ⟨M̄⟩ | std |
|---|---|---|---|---|---|---|
| [3, 10) | 3 | 4.00 | 6.00 | 5.999 | 0.001 | 0.815 |
| [100, 300) | 37 | 19.27 | 21.27 | 19.582 | 1.688 | 1.217 |
| [1000, 3000) | 262 | 28.50 | 30.50 | 28.270 | 2.230 | 0.798 |
| [10000, 30000) | 2016 | 37.51 | 39.51 | 37.133 | 2.381 | 0.828 |
| [30000, 100000) | 6343 | 42.08 | 44.08 | 41.629 | 2.450 | 0.824 |
M̄ 从 ~1.0 爬升到 ~2.45。τ_p 对 \(\ln(p)\) 的斜率为 3.918，diff 的斜率为 3.834，差值仅 0.084。M̄ 对 \(\ln(p)\) 的拟合 \(R^2 = 0.013\)，增长信号极弱。\(V(p) \approx 1.40\)，全局有界，与 Paper XXXIV 一致。
4.2 Block 2: p 扩展到 10⁷
| p 范围 | ⟨M̄⟩ | std | gap = τ/ln(p) − diff/ln(p) |
|---|---|---|---|
| [5×10⁴, 10⁵) | 1.431 | 0.848 | 0.128 |
| [10⁵, 3.16×10⁵) | 1.528 | 0.786 | 0.125 |
| [3.16×10⁵, 10⁶) | 1.557 | 0.802 | 0.117 |
| [10⁶, 3.16×10⁶) | 1.506 | 0.798 | 0.104 |
| [3.16×10⁶, 10⁷) | 1.468 | 0.798 | 0.094 |
表面结论：\(\bar{M} = \Theta(1)\)。Block 3 表明此平坦外观是 composition shift 的结果，不能作为 \(M_{\mathrm{loc}}\) 或 \(M_X\) 渐近行为的诊断（见 §5）。
§5 Block 3：Composition Shift 的识别
5.1 动机
Paper XXXVIII 证明 G 只依赖于 \(t = q/p\)，"B-C 侵蚀穿零"是 composition shift。另一 Claude thread 指出："Paper XXXVIII 的 composition shift 可能影响 \(M_p\) 诊断——大 p 只能访问小 q，gap 的下降可能部分来自窗口效应。"如果 \(\bar{M}\) 也受 composition shift 影响，Block 2 的"\(\bar{M} =\) 常数"就是伪影。
5.2 固定 t∈[3,10] 的判决性实验
对每个 \(p\)，只使用 \(q/p \in [3,10]\) 范围内的素数 \(q\) 计算 diff 和 \(M_{\mathrm{loc}}\)。完整窗口要求 \(10p^2 \leq N\)，即 \(p \leq \sqrt{N/10} \approx 31623\)。
主拟合（前 9 bin，完整 [3,10] 窗口，p ≤ 31623）：
| p band | n | ⟨ln(p)⟩ | ⟨M_loc⟩ | τ/lnp | diff/lnp | gap |
|---|---|---|---|---|---|---|
| [500, 1000) | 73 | 6.60 | 2.068 | 4.056 | 3.743 | 0.313 |
| [1000, 2000) | 135 | 7.29 | 2.099 | 4.039 | 3.751 | 0.288 |
| [2000, 3000) | 127 | 7.82 | 2.268 | 4.048 | 3.758 | 0.290 |
| [3000, 5000) | 239 | 8.28 | 2.226 | 4.030 | 3.761 | 0.269 |
| [5000, 7000) | 231 | 8.70 | 2.288 | 4.028 | 3.765 | 0.263 |
| [7000, 10000) | 329 | 9.04 | 2.288 | 4.019 | 3.766 | 0.253 |
| [10000, 15000) | 525 | 9.43 | 2.380 | 4.021 | 3.768 | 0.253 |
| [15000, 20000) | 508 | 9.77 | 2.330 | 4.008 | 3.769 | 0.239 |
| [20000, 30000) | 983 | 10.12 | 2.429 | 4.010 | 3.770 | 0.240 |
\(M_{\mathrm{loc}} = +0.096 \cdot \ln(p) + 1.43\)，\(R^2 = 0.89\)。
5.3 多 t 窗口验证
| t 窗口 | p = 1k–3k | p = 3k–10k | p = 10k–30k | p = 30k–100k |
|---|---|---|---|---|
| t∈[1,3] | 2.136 | 2.223 | 2.317 | 2.403 |
| t∈[3,10] | 2.181 | 2.270 | 2.374 | 2.431 |
| t∈[10,30] | 2.223 | 2.326 | 2.408 | — |
| t∈[30,100] | 2.270 | 2.365 | 2.401 | — |
5.4 Block 2 vs Block 3
| Block 2（混合 t） | Block 3（固定 t，完整窗口） | |
|---|---|---|
| 斜率 | +0.005 | +0.096 |
| R² | 0.0001 | 0.89 |
| 诊断 | Composition shift 伪影 | 真实增长信号 |
\(R^2\) 从 0.0001 跳到 0.89。Block 2 的 \(\bar{M} \approx 1.5\) 反映的不是 \(M_{\mathrm{loc}}\) 的渐近行为，而是 \(q/p\) 混合比例随 p 变化的伪影。
§6 Block 4–4b：Gemini 公式的失败
6.1 Gemini 的推导与 Block 4 的测量
Gemini 推导：在固定 \(t\) 下，\(M_{\mathrm{loc}} \sim (2h_{\mathrm{prime}} - 2h_0) \cdot \ln(p) + (h_{\mathrm{prime}} - h_0) \cdot \ln(t)\)，其中 \(h_{\mathrm{prime}}\) 是素数的 ρ/ln 斜率，\(h_0\) 是一般整数的 ρ/ln 斜率。
| Scale | h_prime | h₀ | Δ |
|---|---|---|---|
| 10⁴ | 3.798 | 3.720 | 0.078 |
| 10⁶ | 3.833 | 3.781 | 0.052 |
| 10⁸ | 3.843 | 3.803 | 0.039 |
| 10⁹ | 3.845 | 3.810 | 0.035 |
拟合：\(\Delta = 0.710/\ln(n) + 0.00019\)，\(R^2 = 0.996\)。截距在数值上与零不可区分。
6.2 Block 4b：交叉验证
| p | M_loc（实测） | M_loc（Gemini） | 偏差 |
|---|---|---|---|
| 750 | 2.068 | 1.165 | −0.903 |
| 8500 | 2.288 | 1.544 | −0.744 |
| 60000 | 2.417 | 1.454 | −0.963 |
公式系统偏低约 0.85。原因：公式假设 \(E[\rho(pq-1)] \sim h_0 \cdot \ln(pq)\)，但 \(pq-1\) 不是一般整数——它是 shifted product，有自己的有效增长斜率 \(h_{pq} \neq h_0\)。
§7 Block 5b：Shifted-Product Deficit
7.1 直接分量斜率
| 分量 | 斜率 vs ln(p) | R² |
|---|---|---|
| τ_p | +3.91746 | 0.9488 |
| E[ρ(q)] | +3.90829 | 0.9999+ |
| E[ρ(pq−1)] | +7.72529 | 0.9999+ |
| diff = E[ρ(pq−1)] − E[ρ(q)] | +3.81700 | 0.9999 |
| M_loc = τ_p − diff | +0.10045 | 0.012 |
验证：τ slope − diff slope = 3.91746 − 3.81700 = 0.10045 = M_loc slope。精确闭合。
7.2 有效 h 系数
| 量 | 有效 h | 身份 |
|---|---|---|
| h_tau | 3.917 | 素数 p 的 ρ 斜率 |
| h_q | 3.908 | 素数 q 的 ρ 斜率 |
| h_pq | 3.863 | shifted prime product pq−1 的 ρ 斜率 |
\(M_{\mathrm{loc}}\) slope \(= h_\tau + h_q - 2h_{pq} = 3.917 + 3.908 - 2 \times 3.863 = 0.100\)。
7.3 核心机制：Shifted-Product Deficit
\(M_{\mathrm{loc}}\) 增长的驱动力是 \(h_\tau + h_q > 2h_{pq}\)：shifted prime product \(pq-1\) 的 ρ 增长斜率低于素数的 ρ 增长斜率。修正后的公式：
$$M_{\mathrm{loc}} \sim (h_\tau + h_q - 2h_{pq}) \cdot \ln(p) + \text{lower order terms}$$§8 Block 6：多尺度收敛
8.1 四个 h 的外推
| h | 外推极限 | 拟合 | R² |
|---|---|---|---|
| h_tau(∞) | 3.923 | −1.109/ln(s) + 3.923 | 0.986 |
| h_q(∞) | 3.890 | −0.651/ln(s) + 3.890 | 0.972 |
| h_pq(∞) | 3.848 | −0.487/ln(s) + 3.848 | 0.928 |
| h_gen(∞) | 3.856 | −0.435/ln(s) + 3.856 | 0.995 |
数据与排序 \(h_\tau > h_q > h_{\mathrm{gen}} > h_{pq}\) 一致。
8.2 h_tau − h_pq 的收敛
观测值从 −0.03（\(p \sim 500\)，小样本噪声）上升到 +0.013（\(p \sim 40000\)）。\(1/\ln(s)\) 拟合：\(\Delta = -0.622/\ln(s) + 0.0756\)，\(R^2 = 0.92\)。数据与 \(\Delta(\infty) \approx 0.076 > 0\) 一致。
8.3 渐近斜率的 unconstrained estimate
\(\text{slope}(\infty) = h_\tau(\infty) + h_q(\infty) - 2h_{pq}(\infty) = 3.923 + 3.890 - 2 \times 3.848 = \mathbf{0.117}\)。
Block 3 在 \(p \leq 31623\) 的实测斜率 0.096 低于外推值 0.117，与"当前窗口尚未到达渐近区"一致。
§9 Block 7：Parity 控制实验
9.1 动机
公西华（ChatGPT）指出：\(pq-1\) 对奇素数 \(p, q\) 永远是偶数。\(h_{pq} < h_{\mathrm{gen}}\) 中有多少来自 parity，有多少是"乘法阴影"？
9.2 五个控制组
| 控制组 | 定义 | Mean h |
|---|---|---|
| h_pq | E[ρ(pq−1)/ln(pq−1)]，p,q 素数 | 3.787 |
| h_2prod | E[ρ(ab−1)/ln(ab−1)]，a,b 随机整数 | 3.820 |
| h_even | 同尺度随机偶数 | 3.795 |
| h_gen | 同尺度随机整数 | 3.804 |
| h_odd | 同尺度随机奇数 | 3.812 |
9.3 Deficit 的分解
| 效应 | 大小 | 占比 |
|---|---|---|
| Parity（h_gen − h_even） | 0.0090 | 约 54% |
| Beyond-parity（h_even − h_pq） | 0.0075 | 约 46% |
\(h_{pq} - h_{\mathrm{even}} = -0.0075 \pm 0.0025\)（>3σ）。乘法阴影效应是真实的，不完全是 parity 伪影。
§10 Composition Shift：ZFCρ 的核心方法论教训
| Paper | 被扭曲的量 | 机制 |
|---|---|---|
| XXXV–XXXVIII | G（B-C gap） | 大 p 只能用 q ~ p，混合平均掩盖 G 的 t-dependence |
| XXXVIII | G_X(p) vs G_loc(t) | 扩窗 G 收敛到 ≈ −1.9 是 large-t 主导 |
| XXXIX | M̄ vs M_loc | 大 p 只能用小 t，混合平均抹平 M_loc 的 ln(p) 增长 |
任何涉及 \(q\) 的统计量，都必须在固定 \(t = q/p\) 下测量，否则 composition shift 可能制造伪影。这对后续的 Ω≥3 推广也是核心约束。
§11 从 Mloc 到 Ω=2 闭合
11.1 已建立的链条
- 在固定 \(t\) 下，数据与 \(M_{\mathrm{loc}} \sim (h_\tau + h_q - 2h_{pq}) \cdot \ln(p)\) 一致，unconstrained 外推斜率 ≈ 0.117。
- \(V(p) \approx 1.4\) 全局有界（Paper XXXIV）。
- Shifted-product deficit 是稳健的（Block 7 parity 控制确认，>3σ）。
11.2 Ratio-Mixing Lemma（开放引理）
表述：设 \(a(t)\) 为 \(M_{\mathrm{loc}}(p, t)\) 对 \(\ln(p)\) 的斜率，\(\mu_{N,p}(t)\) 为 Ω=2 defect 计算中 \(q/p\) 的实际分布，其支撑集为 \(t \in [1, N/p^2]\)。
Ratio-Mixing Lemma（待证）：存在 admissible 增长序列 \(N = N(p)\)（满足 \(N/p^2 \to \infty\)）和常数 \(c > 0\)，使得
$$\liminf_{p \to \infty} \int_1^{N(p)/p^2} a(t)\, d\mu_{N(p),p}(t) \geq c > 0$$11.3 条件性结论
若 ratio-mixing lemma 成立：\(\bar{M}_N(p) \to \infty\) → \(\Gamma_p = \bar{M}^2/V \to \infty\) → Cantelli：\(c_p \to 0\) → Harmonic Cesàro Lemma：\(\bar{c}_h \to 0\) → Ω=2 层闭合。
§12 讨论
12.1 Paper XXXIX 的核心贡献
- (a) 发现 \(\bar{M}_N \approx 1.5\) 是 composition shift 伪影；\(M_{\mathrm{loc}}\) 在固定 \(t\) 下有清晰的 \(\ln(p)\) 增长。
- (b) 识别真正的驱动力：shifted-product deficit，\(h_\tau + h_q > 2h_{pq}\)。
- (c) 否决 Gemini 的公式——参数识别错误（把 \(pq-1\) 等同于一般整数）。
- (d) 多尺度外推与 \(h_\tau > h_q > h_{\mathrm{gen}} > h_{pq}\) 一致。
- (e) Parity 控制实验：deficit 中 parity 占约 54%，beyond-parity 占约 46%，乘法阴影效应超过 3σ。
- (f) 将 Ω=2 闭合归约为 ratio-mixing lemma。
12.2 距离估计
Ω=2 闭合现在归约为少数精确的解析输入，核心是 ratio-mixing lemma。这是一个比此前任何表述都更精确、更可攻击的问题。
§13 数据来源与可重复性
| 脚本 | 测量 | Block |
|---|---|---|
| p39_mp_precise_v2.py | M̄ 全分量，9587 primes，p ≤ 10⁵ | 1 |
| p39_large_p.py | M̄ 扩展到 p ~ 10⁷，10000 primes | 2 |
| p39_fixed_ratio.py | M_loc at 固定 t，Exp 1+2 | 3 |
| p39_hprime.py | h_prime vs h₀ 多尺度，30 scales | 4 |
| p39_crossval.py | Gemini 公式验证 + ρ(p)=ρ(p-1)+1 + smooth bias | 4b |
| p39_direct_slopes.py | 直接分量斜率，h_tau/h_q/h_pq | 5b |
| p39_hpq_multiscale.py | h_tau，h_q，h_pq，h_gen 多尺度收敛 | 6 |
| p39_parity_control.py | parity 控制：h_pq vs h_even vs h_gen vs h_odd vs h_2prod | 7 |
数据：rho_1e10.bin（int16，N = 10¹⁰，rho_dp.c，Paper XXXII convention）。
参考文献
[1] H. Qin. ZFCρ Papers I–XXXVIII. Paper XXXVIII DOI: 10.5281/zenodo.19157939.
[2] H. Qin. ZFCρ Paper XXXIV: Upper-Margin Closure and Bounded-Variance. DOI: 10.5281/zenodo.19140015.
[3] H. Qin. ZFCρ Paper XXI: Queue Isomorphism and Local Lipschitz Reduction. DOI: 10.5281/zenodo.19037934.
致谢
Claude（子路）发现 int16 格式 bug，设计了 Block 2–3–4b–5b–6–7 实验链（共八个脚本），起草 working notes v1–v5。在 Block 4 给出 h_prime → h₀ 后设计 Block 4b 交叉验证；在 Gemini 公式偏低 ~0.85 后设计 Block 5b 直接分量分析，发现 M_loc slope = h_tau + h_q − 2h_pq 的精确分解；设计 Block 6 多尺度收敛实验。
热力学 Claude（另一 thread）指出 Block 2 可能受 composition shift 影响，直接催生了 Block 3。
Han Qin（作者）在 Block 4 之后问"我们的数据是对的吗"，催生了 Block 4b，进而揭露 Gemini 公式缺陷，打开了 Block 5b → 6 的整条路径。作者指出"深度优先，先走数据"的方法论方向。
公西华（ChatGPT）在 Paper XXXVIII 阶段警告"M_p → ∞ 的推导跳了一步"；在 v2 审阅中提出三层 M 区分、截断窗口修正和 ratio-mixing lemma；在 v4 审阅中发现算术不一致（0.151 → 0.117）、过度声明语言、parity 作为竞争解释，以及 mixing lemma 需要定理级表述——直接催生了 Block 7 和全面修订。
子夏（Gemini）推导了公式 M ~ 2(h_prime − h₀)·ln(p)；代数框架正确但参数识别有误，直接导向 Block 4 的测量，并通过其失败揭示了 h_pq 的作用。
子贡（Grok）提供了 Lindley 参数估计和系列一致性检验。
最终文本由作者独立完成。