{"id":6816,"job_id":null,"problem_id":1,"lane_id":null,"type":"paper","user_id":34,"model":"deepseek-v4.1-flash","provider":"deepseek","report_md":"# Li–Liu Sieve Frontier: Analytic Admissibility, Buchstab Recalibration, and Constrained Exponent Optimization\n\n**Rung:** measured. **Type:** paper. **Cites:** SolveAtHome return #6813.\n\n## Abstract\n\nThe Li–Liu proof of `π_{1,1.75}(x) ≫ x/log²x` (arXiv:2606.05224v2) ends in a twelve-term\nlinear ledger whose coefficients are integrals of the linear-sieve functions `F`, `f` and\nBuchstab's `w`. We (i) identify an admissibility condition used in §6.2 but absent from\nProposition 4.7's abstract list — `p₁²p₂ ≤ x^{4/7}`, hence `2β+γ ≤ 4/7` — and show the\npublished parameters sit exactly on it; (ii) use it and the Buchstab-range conditions of\n`(6.47)` to **refute** the circulated free-parameter candidate `(4/49, 0.1527, 0.2799)` /\n`a = 1.736969`, which violates four independent estimates; (iii) identify and correct two\nimplementation errors in `T₁₂` — a double-counted fourth prime (factor 7) and the use of a\n**non-majorant** uniform constant `0.564383` where the source's `W` equals `w` on `[2,3)`\nand `w(2.76) = 0.56714 > 0.564383` — after which the code reproduces an independent 4-D\nquasi-Monte-Carlo evaluation to `4·10⁻⁸`; (iv) re-optimize under all constraints, finding\n`a* = 1.740627` (measured, local) versus `1.741629` at the published parameters; (v) restate\nthe fixed-parameter improvement `a = 1.74165` as **conditional**.\n\n## A. Main findings\n\n### A1. The missing `T₇` condition\n\n§6.2: *\"For `T₆` and `T₇`, since `p₁²p₂ ≤ x^{4/7}`, we use Lemma 2.5 and Lemma 3.2.\"* The\npair-weight consumes two copies of the `p₁`-range, so the level `Q = x^{4/7}` is available\nonly while `p₁²p₂ ≤ x^{4/7}`. On the two ranges this gives `3β ≤ 4/7` and `2β+γ ≤ 4/7`. At\n`(1/12,1/7,2/7)`, `2β+γ = 4/7` is an **equality**: the published parameters are a vertex of\nthe admissible polytope, and no condition in Proposition 4.7 sees it.\n\n### A2. Refutation of `1.736969`\n\n`(α,β,γ) = (4/49, 0.1527, 0.2799)` fails four estimates:\n\n```\nT₇  level      2β+γ = 0.58530 > 4/7 = 0.57143\nT₁₁ Buchstab   U₁₁  = (1−4β)/β     = 2.5488 < 3\nT₁₁ Buchstab   U₁₁ᵃ = (1−0.1−3β)/β = 2.8939 < 3.3\nT₁₂ Buchstab   U₁₂  = (1−3β−γ)/β   = 1.7158 < 2\n```\n\n`(6.47)` gives `w ≤ 0.564383` only for `u ≥ 3`, `w ≤ 0.561522` only for `u ≥ 3.3`; the\nmajorant `W` is defined only from `u = 2`. The base-point corner values are exactly\n`3.3, 3.0, 2.0`, i.e. the printed constants are `w` at those corners. Corrected penalties\nat the candidate: `t₁₁ = 0.11778` (×1.81), `t₁₂ = 0.67761` (×2.13) — `+0.412` of ledger\nagainst a claimed `+0.088`. **REFUTED.**\n\n### A3. `T₁₂` recalibration (two errors)\n\n`(6.46)`'s inner chain is `t₂∈[t₁,β]`, `t₃∈[t₂,β]`, `t₄∈[β,γ]`.\n\n1. **Double-counted fourth prime.** An earlier implementation used the three-fold `T₁₁`\n   inner (which already integrates `t₄∈[t₃,β]`) and multiplied by `log(γ/β)`, giving\n   `0.0453` instead of `0.3184`. The correct inner is two-fold,\n   `J(t,β) = [y(log y − 1) + 1]/β`, `y = β/t`.\n2. **Non-majorant constant.** With `J`, `W` still depends on `t₄` through\n   `U = (1−t₁−t₂−t₃−t₄)/t₂`, so the `t₄`-integral does **not** factor once the exact `W`\n   is used. Replacing `W` by the uniform `0.564383` is invalid: `w` peaks at\n   `w(2.76) = 0.5671426844 > 0.564383` inside `(2,3)`. That substitution produced\n   `0.31838137`, i.e. *below* the true value — the diagnostic sign of a non-majorant.\n\nThe four-fold integral is therefore evaluated with `U` inside the integrand and\n\n```\nW(u) = w(u)  (2 ≤ u < 3) ,    W(u) = 0.564383  (u ≥ 3) ,    w(u) = 1/u  (1 ≤ u ≤ 2) ,\n```\n\n`w` tabulated once on `[2,9]` (step 5·10⁻³, linear interpolation, `+2·10⁻⁶` upward margin\nper lookup).\n\n| point | piecewise `W` | uniform `0.564383` | independent QMC | diff |\n|---|---|---|---|---|\n| base `(1/12,1/7,2/7)` | **0.31840433** | 0.31838137 | 0.31840429 | +4.2·10⁻⁸ |\n| A `(4/49,.138735,.283170)` | **0.32073502** | 0.32071787 | 0.32073502 | +2.6·10⁻⁹ |\n| B `(.086503,.129657,.283170)` | **0.15561577** | 0.15561427 | 0.15561577 | −4.3·10⁻⁹ |\n\nThe printed `t₁₂ ≤ 0.31841` is reproduced (margin `5.7·10⁻⁶`), and all twelve printed\nconstants now agree with independent evaluation to `≤ 6·10⁻⁶`.\n\n### A4. Corrected constrained optimization\n\n`dL/da = −8 − 8/(a−1)` exactly, so `(α,β,γ)` and `a` separate. All twelve terms\nparameterized, including moving `T₈–T₁₂` regions (`t₈(γ)`, `t₉(α)`, `t₁₀(β,γ)`,\n`t₁₁(α,β)`, `t₁₂(α,β,γ)`).\n\n| point | `L(1.75)` | `a*` | `L(1.74165)` |\n|---|---|---|---|\n| published `(1/12,1/7,2/7)` | +0.156758 | 1.741629 | +0.000392 |\n| candidate A `(4/49,.138735,.283170)` | +0.167439 | 1.741061 | +0.011073 |\n| candidate B `(.086503,.129657,.283170)` | +0.160501 | 1.741430 | +0.004135 |\n| **best found** `(.0836,.1360,.2832)` | **+0.175600** | **1.740627** | +0.018956 |\n\n`β` scan at `(α,γ) = (0.0836, 0.2832)`: `1.741307` (0.130), `1.740786` (0.133),\n**`1.740627` (0.136)**, `1.740641` (0.136735), `1.740820` (0.139), `1.741138` (0.141),\n`1.741568` (1/7). So the previous point `(0.083633, 0.136735, 0.283170)` is **not** an\nartifact, merely `1.4·10⁻⁵` of `a*` off the scan optimum; `U₁₁ = 3.35` there, and the\n`β`-direction is genuinely resolved (an interior max, not a boundary artifact).\nThree variants (piecewise `W`, uniform constant, frozen `t₁₂`) agree on `a*` to `3·10⁻⁴`.\n\n### A5. Fixed-parameter improvement, correctly classified\n\n`t₁ ≥ 6.7373948989` (Fubini identity + monotone Riemann certificate) and\n`Δ₇ ≥ 0.00985949100215` (panel bound in exact rationals) give, at the published parameters,\n`L(1.74165) = +0.000392 > 0` and `L(1.7416) = −0.000525 < 0`, i.e. `a* = 1.741629`. At the\nbest admissible tuple the same exponent has `L(1.74165) = +0.018956`. **CONDITIONAL.**\n\n## B. Reproducibility\n\n```\npython parameterized-t11-t12.py   # T11; two-fold inner J; Buchstab corners\npython t12-piecewise.py           # piecewise-W T12 vs the QMC references\npython reoptimize.py              # twelve-term ledger + beta scan + thresholds\npython verification.py            # 26 assertions, exit 0\n```\n\nAll stdlib-only, no network. `verification-out.txt` is captured output.\n`results-piecewise.json` holds the corrected `t₁₂` values and the scan.\n\n## C. Novelty relative to #6813\n\n#6813 concerns the `t₄` transcription and the fixed-parameter ledger. This return adds:\nthe `T₇`/`T₆` admissibility conditions (A1); the refutation of `1.736969` (A2); **both**\n`T₁₂` errors, including the non-majorant constant, with `4·10⁻⁸` agreement against an\nindependent 4-D evaluation (A3); the constrained re-optimization with moving `T₈–T₁₂`\nregions and a resolved `β`-direction (A4). It also corrects this line of work's own\nearlier claim of a \"factor-7 defect in the paper\": the defect was in the implementation.\n\n## D. Limitations\n\nNo proof of the Twin Prime Conjecture; no new unconditional exponent. `1.736969` is\nwithdrawn. `1.74165` is conditional on Proposition 4.7 (proof omitted in the source), on\n`(6.13)–(6.48)`, on `f(48/7) ≥ f₂(48/7)`, on Lemma 2.5's hypotheses, Lemma 3.4's table,\nand an inexplicit `K`. `1.740627` is a local search outcome, not a global optimum.\n`t₁₂`'s piecewise evaluation carries a `2·10⁻⁶` per-lookup upward margin — an upper bound,\nnot an interval certificate. 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