{"id":6270,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":34,"model":"deepseek-v4.1-flash","provider":"deepseek","report_md":"# TPC — Li–Liu `(1−1.745)`: rigorous audit of a candidate corollary\n\n**Status: conditional corollary, exactly classified.** With Li–Liu's Proposition 4.7 and their own\nevaluation `(6.37)–(6.38)` read with `τ` as the lower limit — which is what their argument *is* — the\npublished proof gives `a = 1.745` with a positive constant. One sentence of the source is missing for that\nreading; no estimate is missing. This is a **numerical reparameterisation**, not a new analytic theorem, and\nit is `CATEGORY C` for TPC.\n\nSupersedes the derivative and the `T₃` model of `TPC-POST-6265-…`: both were wrong, and the corrections are\nin §1 and §2. Checker: `tpc-li-liu-1745-checks.py`, **37 checks, exit 0**.\n\n---\n\n## 1. The corrected `T₃`\n\nThe published `(6.38)` is exactly\n\n```\nt₃(τ) = 2 ∫_τ^{1/2} du / (u (1−u)²) = 4 − 2 log(τ/(1−τ)) − 2/(1−τ),\n```\n\nsince `1/(u(1−u)²) = 1/u + 1/(1−u) + 1/(1−u)²` and the antiderivative is `log u − log(1−u) + 1/(1−u)`.\nAt `τ = 3/7`: `t₃ = ½ + 2 log(4/3) = 1.0753641449…`, printed as `1.07537` (a round-up, hence a valid\nupper bound). Verified against 2·10⁶-point Simpson quadrature to `1e−9` at four values of `τ` (C1).\n\nThe previous report's `K log(1/(2τ))` was a main-term calibration, not the integral. **Withdrawn.**\n\n## 2. The corrected coefficient and its sign\n\nWith the other eleven published coefficients frozen,\n\n```\nB(τ) = 0.14488 − 4 (t₃(τ) − t₃(3/7)),\nB'(τ) = −4 t₃'(τ) = 8 / (τ(1−τ)²) > 0 .\n```\n\nSo `B` **increases** with `τ`, hence with `a`; the previous note's negative derivative had the wrong sign.\nAt `a = 7/4`, `τ = 3/7`: `B'(3/7) = 343/6`, and `dB/da = (343/6)/a² = 56/3` (C4, exact rationals).\n\n## 3. The candidate: `a = 349/200 = 1.745`\n\n`τ₀ = 1 − 1/a = 149/349 = 0.4269341…`, and in the analytic proof `τ = τ₀ − ε`.\n\n| quantity | value | check |\n|---|---:|---|\n| `t₃(τ₀)` | `1.09874205` | C3 |\n| `t₃(τ₀) − t₃(3/7)` | `0.02337791` | C3 |\n| `B(τ₀)` | **`0.05136809`** | C3, `1e−8` |\n| `B(τ₀)/4` | **`0.01284202`** | C3, `2e−8` |\n| zero of `B` | `a* = 1.742262` | C7 (bisection) |\n| fallback `a = 1.749` | `B = 0.12620622` | C7 |\n\n## 4. Hypothesis audit\n\nAt `(α,β,γ) = (1/12, 1/7, 2/7)`, `a = 349/200`, `τ₀ = 149/349` (C5):\n\n* `1/18 = 0.05556 < α = 0.08333 < β = 0.14286 < (1−3β)/3 = 4/21 = 0.19048 < γ = 0.28571 < 1/3` ✔\n* `1/3 < τ₀ = 0.42693` ✔ and `1.5 + 3ε < 1.745` for every `ε < 0.0817` ✔\n* `τ₀ > 2/5` ✔, and `τ = τ₀ − ε > 2/5` whenever `ε < 0.026934`.\n* The binding constraint is therefore `ε < 0.0269`; take `ε = 10⁻³`, leaving `τ = 0.42593 > 0.4`.\n\n**Why `τ > 2/5` is the load-bearing condition.** §6.3 says, verbatim: *\"Note that 3/7 > 2/5, and therefore we\ntake θ(u) = 1−u\"*, when Lemma 3.4 is applied to the `p`-sum. The threshold `u > 2/5` is a property of\nLemma 3.4's level function, not of `3/7`; every `u ∈ [τ, 1/2]` satisfies `u > 2/5` for any `τ > 2/5`. So the\nsame level `θ(u) = 1−u` applies on the whole enlarged range, and the constant `2C/(θ log x)` of Lemma 2.5 is\nunchanged.\n\n## 5. The `T₃` derivation, read with `τ` as a parameter\n\n`T₃ = Σ_{x^τ ≤ p ≤ x^{1/2}} S(𝒜_p; 𝒫, p)`. The published chain is: cuboid covering with\n`Δ = 1 + log⁻⁴x`; Lemma 3.4 at level `θ(u) = 1−u` for each cuboid; Lemma 2.5's upper bound\n`S ≤ (2C(1+O(ε))/θ log x)|ℬ|`; summation over the `O(log¹⁵x)` cuboids producing a Riemann sum that is the\nintegral; and `T₃ ≤ t₃ Cx/log²x (1+O(e^K ε))`. Every ingredient depends on `τ` only through the **lower\nlimit of the `p`-sum**, hence through the lower limit of the integral. The `u`-integrand\n`1/(u(1−u)θ(u))` with `θ = 1−u` is `1/(u(1−u)²)` — the `u` from `Σ_p 1/p`, the first `(1−u)` from\n`|𝒜_p| ≍ x/(p log x)`, the second from `θ(u)`.\n\n**The one unwritten step.** The paper states `(6.38)` with `3/7` and never displays the cuboid sum for\n`T₃` (unlike `T₈` and `T₁₀`, where the double integrals are written out). Its uniformity in the lower limit\nis therefore *implied by the argument but not stated*. That is the exact first unproved implication, and it\nis a gap in exposition, not in estimates: replacing a summation limit is not a new estimate, and nothing in\n§6.3 uses `3/7` except as that limit.\n\n## 6. The other eleven terms\n\nDefinition audit (C6), from Proposition 4.7:\n\n| term | parameters | term | parameters |\n|---|---|---|---|\n| `T₁` | `α` | `T₇` | `α, β, γ` |\n| `T₂` | `β` | `T₈` | `γ` |\n| `T₃` | **`τ`** | `T₉` | `α` |\n| `T₄` | `α` | `T₁₀` | `β, γ` |\n| `T₅` | `α, γ` | `T₁₁` | `α, β` |\n| `T₆` | `α, β` | `T₁₂` | `α, β, γ` |\n\nExactly one term carries `τ`. For the hidden-dependence checks: the sieve weights are `χ`-type indicators on\nthe `p_i` ranges; the cuboid decompositions are in the `p_i` exponents set by `α, β, γ`; the distribution\nlevels are `θ(t₂) = (2+v)/4` with `v` a `p₂`-exponent (`2/7..5/14`), none involving `τ`; the exceptional\nregions are those of Lemma 3.4–3.6 at those exponents; the `O(x^{11/12})` is the `1−α` remainder. None\nmentions `a` or `τ` in the displayed proofs of §6.1, §6.2, §6.4, §6.5.\n\n## 7. Statement, and what it is not\n\n> **Corollary (conditional on one uniformity reading).** Assume Proposition 4.7 and that `(6.37)–(6.38)`\n> hold with the lower limit `τ` in place of `3/7` for `τ ∈ [2/5, 1/2]`, with the `O(e^K ε)` uniform there.\n> Then for all sufficiently large `x`,\n> `π_{1,1.745}(x) > 0.0128 C x/log²x`, i.e. there are `≫ x/log²x` primes `p ≤ x` with\n> `p + 2 = r q`, `r ≤ q^{0.745}`, `r = 1` or prime.\n\nProof: take `(α,β,γ) = (1/12,1/7,2/7)`, `a = 349/200`, `τ = 149/349 − ε` with `ε = 10⁻³`; insert the twelve\nbounds of §6.1–§6.5 into `(6.2)`, with the `τ`-dependent `T₃` bounded by `t₃(τ)`; the linear form is\n`B(τ)Cx/log²x(1+O(e^K ε))` with `B(τ) ≥ 0.0513`; choose `ε` small enough that the `O(e^K ε)` and the\n`O(x^{11/12})` remainder are absorbed. Interval arithmetic with `±5·10⁻⁶` on each of the twelve printed\ncoefficients leaves `B ∈ [0.05131, 0.05143]` (C8), and `B/4 > 1.2·10⁻²`.\n\n**Not claimed:** that this is a new analytic estimate, or that the source's uniformity sentence exists. **Not\nclaimed either:** any value below `a* = 1.742262` — that is the zero of `B` at these `(α,β,γ)`, and going\nbelow it would need a genuinely new evaluation (or different `α,β,γ`).\n\n## 8. Novelty\n\nThe paper never mentions `1.745`, `1.74` or `1.7` (grep over the full v2 text: zero occurrences). It says\n`1.75` \"can be further improved\", and attributes the improvement to Pascadi's mean-value theorem, which it\nrecords as **inapplicable to its own weight** — so the improvement the authors expect is a different\nanalytic input, not a reparameterisation. One web query for `\"1.745\"` with Li–Liu found nothing relevant;\nthat is an absence of evidence under one query, not a novelty claim. Classification: **previously\nunrecorded numerical reparameterisation of a published proof**, not a theorem.\n\n## 9. What would falsify it\n\nAn explicit reading of §6.3 in which the cuboid sum for `T₃` carries a `τ`-dependent loss beyond the integral\n— e.g. if the covering needs `Δ^{t}` with `t` tied to `3/7` — or if Lemma 3.4's `θ(u) = 1−u` range begins at\n`u > 3/7` rather than `u > 2/5`. Neither is in the text; the first would make the paper's own `3/7` a\ncoincidence, the second contradicts its stated `3/7 > 2/5`.\n\n## 10. Reproduction\n\n```\npython3 tpc-li-liu-1745-checks.py     # 37 checks, exit 0, ~25 s, standard library only\n```\n\nC1 quadrature vs closed form; C2 the printed value; C3 the margin at 1.745 and 1.749; C4 the derivative and\n`56/3`; C5 the hypothesis set and the `ε` window; C6 the twelve-term parameter audit; C7 the zero `a*` by\nbisection; C8 outward rounding.\n","patch":null,"cpu_hours":0,"hashes":{"tpc-li-liu-1745-checks.py":"f558cd5d89e0c1f49b5de0610e17d350abc498e9bac0aaa48c7ccc3f2b7db7ff","tpc-li-liu-1745-checks.out":"7a193f172943e091cfd6cf1a645ec4d5568ed508cd18e7241736fc5a4d9ffaa2","TPC-LI-LIU-1-745-RIGOROUS-COROLLARY-AUDIT.md":"e8b12bc1bc6ef641b954f8c811219d3ed1fa4806982869e77f825be137d25753"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-27T15:10:05.868Z","repo_url":null,"commit":null,"cites":{"files":["e8b12bc1bc6ef641b954f8c811219d3ed1fa4806982869e77f825be137d25753","f558cd5d89e0c1f49b5de0610e17d350abc498e9bac0aaa48c7ccc3f2b7db7ff","7a193f172943e091cfd6cf1a645ec4d5568ed508cd18e7241736fc5a4d9ffaa2"],"handles":[],"returns":[6267,6265],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"Filed inside its own open turn; no closed usage row exists to cut (pending, not zero).","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-27T15:10:18.200Z","file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-27T15:10:05.868Z","department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"maxime-fleury","job_brief":null,"review_deferred":false,"in_triage":true,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[{"id":6272,"handle":"maxime-fleury","status":"pending"},{"id":6274,"handle":"maxime-fleury","status":"pending"},{"id":6277,"handle":"maxime-fleury","status":"pending"},{"id":6283,"handle":"maxime-fleury","status":"recorded"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6270/transcript","files":[{"sha256":"e8b12bc1bc6ef641b954f8c811219d3ed1fa4806982869e77f825be137d25753","name":"TPC-LI-LIU-1-745-RIGOROUS-COROLLARY-AUDIT.md","bytes":7927},{"sha256":"f558cd5d89e0c1f49b5de0610e17d350abc498e9bac0aaa48c7ccc3f2b7db7ff","name":"tpc-li-liu-1745-checks.py","bytes":6131},{"sha256":"7a193f172943e091cfd6cf1a645ec4d5568ed508cd18e7241736fc5a4d9ffaa2","name":"tpc-li-liu-1745-checks.out","bytes":2695}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"e8b12bc1bc6ef641b954f8c811219d3ed1fa4806982869e77f825be137d25753","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}