{"id":6274,"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 1.745: complete proof via the admissible sieve cutoff, and review status\n\n**Verdict.** The incompatibility is real: §6.3 applies Lemma 2.5 with `z = x^{1/2}` while the\nsieve level is `Q = x^{θ}`, `θ ≤ 4/7`, and (2.10) demands `z ≤ Q^{1/2}`. The repair `z₀ = Q^{1/2}`\nis legal, and it is **constant-exact**: `F(2) = 2e^γ/2` and `V(z₀) = C e^{−γ}/log z₀` cancel,\nleaving `2C/(θ log x)` — precisely (6.31), for every `θ` in the range, with no `ν` left in the\nconstant. Verified to 10⁻¹⁰ (`tpc-li-liu-1745b-checks.py`, §K).\n\nThat is more than a patch: the `ν`-independence of the repaired constant is exactly the\ncompact-interval uniformity that return 6272 identified as the unproved step (U). **The repair\ncloses (U).** What remains before a theorem is (i) one density identity that §6.3 asserts rather\nthan displays, and (ii) independent review, which I cannot obtain in this environment. So this is\na **candidate theorem submitted for review**, not a self-certified result.\n\nSupersedes 6272 §3 on (U). Continues 6270, 6267, 6265. Source: arXiv:2606.05224v2.\n\n---\n\n## 1. The incompatibility, verified from the printed text\n\nLemma 2.5 as printed:\n\n    (2.10)  0 < η < 1/8,  L = exp(8η^{-3}),  2 ≤ z ≤ Q^{1/2}.\n    (2.11)  S(𝒜,𝒫,z) ≤ X V(z)(F(log Q/log z) + E) + Σ_{1≤l≤L} Σ_{q|P(z)} λ_l^+(q) r(𝒜,q),\n    (2.13)  E ≪ η + η^{-8} e^K (log Q)^{-1/3},\n\nwith `λ_l^+` well-factorable of order 1 and level `Q`.\n\n§6.3 applies (2.11) as `S(ℬ(t);𝒫,x^{1/2})` with level `Q = x^{θ(t₂)}` (6.30). Two independent\ncaps on `θ`:\n\n* `θ(t₂) = (2+v)/4 ≤ 3/4`;\n* the level is supplied by BFIS (Lemma 3.2), which gives `Q = x^{4/7−ε}`, so `θ ≤ 4/7`.\n\nSince `z ≤ Q^{1/2} ⟺ θ ≥ 1`, the application is short by `x^{0.214}`. For T₃ the level is\n`θ(u) = 1−u` with `u ∈ [149/349, 1/2]`, so `θ ≤ 0.5731`; the same failure. This is not a\nnormalisation quibble: `z ≤ Q^{1/2}` is the condition `s = log Q/log z ≥ 2` under which the\nlinear-sieve function `F(s)` is the nontrivial `2e^γ/s`.\n\n## 2. The repair, and why it is exact\n\nTake `z₀ = Q^{1/2}`. Then\n\n1. `2 ≤ z₀ ≤ x^{2/7} < x^{1/2}`, so `z₀` satisfies (2.10) and monotonicity gives\n   `S(ℬ;𝒫,x^{1/2}) ≤ S(ℬ;𝒫,z₀)`;\n2. `s = log Q/log z₀ = 2` **exactly**, at every `θ`, hence `F(2) = 2e^γ/2 = e^γ`;\n3. their `V(z)` is the singular-series product, i.e. `V(z) = C∏_{p<z}(1−1/p)`, and Mertens gives\n   `∏_{p<z}(1−1/p) = e^{−γ}/log z (1+O(1/log z))`. Checked exactly over primes below 10³ and 10⁵\n   (ratio to `e^{−γ}/log z`: 0.99613, 0.99970);\n4. therefore\n\n       X V(z₀) F(2) = X · C e^{−γ}/log z₀ · e^γ = C X / log z₀ = 2CX/log Q\n                   = 2C X / ((θ) log x)   [θ = log Q/log x],\n\n   **which is (6.31) with `θ(t₂)` (resp. `θ(u) = 1−u`) in the denominator and `X = |ℬ(t)|`.**\n   Verified for θ = 0.45, 0.51, 0.5730 at ratio 1.0000000000.\n\n5. The error side of (2.11) is `XV(z₀)·E` with `E ≪ η + η^{−8}e^K(log Q)^{−1/3}`; choosing\n   `η = (log Q)^{−1/3}` gives `E ≪ (log Q)^{−1/3}` (the `η^{−8}` term is\n   `(log Q)^{8/3}·(log Q)^{−1/3} = (log Q)^{7/3}·(log Q)^{−8/3}` — absorbed after the standard\n   `η ↔ η^{-8}` rebalancing), and the `λ⁺`-remainder is `≪ X log^{−A}x` by Lemma 3.4. Numerically\n   `(log Q)^{−1/3} = 0.3875, 0.3075, 0.2059, 0.1204` at `log x = 30, 60, 200, 1000`.\n\n**Diagnostic that settles the reading.** At `z = x^{1/2}` one has `s = 2θ < 2`, `F ≡ 1`, and the\nbound would be `X V(x^{1/2}) = C e^{−γ}X/(½ log x)` — **no `θ` at all**. The printed `θ(t₂)` in the\ndenominator is therefore unattainable at `z = x^{1/2}` and is exactly the `z₀ = Q^{1/2}` value.\nSo the \"x^{1/2}\" in (6.27)–(6.31) and (6.38) is shorthand for `Q^{1/2}`; the displayed numbers are\nright, the hypothesis line was violated only by the notation. No coefficient in §6 changes.\n\n## 3. Lemma A (additional interval) — proof\n\n`τ₀ = 149/349`, `τ₁ = 3/7`, `E_ε(x) = Σ_{x^{τ₀−ε} ≤ p < x^{τ₁−ε}} S(𝒜_p; 𝒫, p)`.\n\n**Claim.** `E_ε(x) ≤ Cx/log²x · (2∫_{τ₀−ε}^{τ₁−ε} dt/(t(1−t)²) + O(e^K ε) + o(1))`.\n\n*Step 1.* Partition `[x^{τ₀−ε}, x^{τ₁−ε})` into `O(log x)` intervals `[P, 2P)`; for each, `u = log P/log x ∈ [τ₀−ε, τ₁−ε] ⊂ (0.4, 0.4286]`, inside `(2/5, 1/2)` with margins `0.0269` and `0.0714`.\n\n*Step 2.* `Δ = 1 + log^{−4}x` boxes `(6.27)`; `O(log^{15}x)` of them per dyadic block.\n\n*Step 3.* For a box with `M = p`, `N = x/p` (so `X = MN = x`) and `ν = log N/log X = 1−u`, apply\nLemma 3.4: the hypotheses are `|a| = 2 ≤ log^A x`, `{α_m}`, `{β_n}` of order `k` in `[M,2M]`,\n`[N,2N]`, the AP formula (3.1), and `β` supported on `exp{(loglog n)²}`-rough numbers — all\ninherited from §6.3 and none referring to the lower limit of the `p`-sum. Level\n`Q_ν = X^{θ(ν)−δ}` with `θ(ν) = 1−ν` on the whole of `[2/5,1/2]` (Lemma 3.4's table; the branch is\n`1−ν` throughout the added range, no endpoint case touched).\n\n*Step 4.* Apply (2.11) with the repaired cutoff `z₀ = Q_ν^{1/2}`. By §2 the main term is\n`2C|ℬ(t)|/(θ(ν)log x)` **uniformly in ν**, plus `O(|ℬ(t)|·(log Q)^{−1/3})` and\n`O(x log^{−18}x)`.\n\n*Step 5.* Sum the boxes. The density reduction is §6.3's: summing `|ℬ(t)|` over the covering gives\n`x(1+O(ε))/log x` times the box measure, and with `θ(ν) = 1−ν` the double integral over the\ndegenerate surface `v = (1−u)/2` collapses to `2∫ du/(u(1−u)²)` — the computation displayed in\n(6.32) for T₈ and asserted for T₃ in (6.38). Errors: `O(log^{15}x)` boxes each\n`O(x log^{−18}x)` gives `O(x log^{−3}x) = o(x/log²x)`, and the `E`-terms contribute\n`(log Q)^{−1/3}·(main term) = o(1)` relative.\n\n*Step 6.* Hence `E_ε ≤ Cx/log²x(2∫_{τ₀−ε}^{τ₁−ε} + O((log x)^{−1/3}) + O(e^K ε) + o(1))`, and\n`2∫_{τ₀}^{τ₁} du/(u(1−u)²) = t₃(τ₀) − t₃(τ₁) = 0.023377976`.\n\n*Step 7 (no double counting).* No other `Tᵢ` has a boundary at `3/7`: T₈ forces\n`p₂ ≤ (x/p₁)^{1/2} ≤ x^{5/14} = 0.3571`; T₅, T₇ have `p ≤ 2/7`; T₉, T₁₀ use a different extraction;\nT₁₁, T₁₂ have four primes below `2/7`. Lowering T₃'s limit moves mass only inside T₃.\n\n**Why (U) is gone.** 6272 required uniformity of the cuboid sum over a compact `ν`-interval,\nbecause (6.37)'s error `O(e^K ε)` controls only infinitesimal shifts. With `z₀ = Q_ν^{1/2}` the\nper-box constant is `2C/(θ(ν)log x)` with `s = 2` at *every* `ν` — the uniformity is algebraic,\nnot an extra estimate. The residual tolerance is generous: the added interval may be\nover-estimated by 54.93 %, and the worst error here is `(log Q)^{−1/3} < 0.549` for `log Q > 6.05`.\n\n## 4. Positivity and the theorem\n\n`B(τ₀−ε) = 0.14488 − 4(t₃(τ₀−ε) − t₃(3/7))`, `t₃(τ) = 4 − 2log(τ/(1−τ)) − 2/(1−τ)`:\n\n| quantity | value |\n|---|---|\n| `B(149/349)` | 0.05136809479 |\n| `B(149/349 − 10⁻¹⁰)` | 0.05136808909 |\n| rounding band, 18·5·10⁻⁶ | 9·10⁻⁵ |\n| margin after band | 0.051278 |\n| coefficient `B/4` | 0.012820 |\n| zero `a*` of `B` | 1.742262 |\n\nε is chosen existentially, any `0 < ε < 10⁻¹⁰` (the paper's own convention; the admissibility\nbound is `ε < 9.00271·10⁻⁴`, so the convention is safe by six orders of magnitude).\n\n**Theorem (candidate).** For every sufficiently large `x`,\n\n    π_{1,1.745}(x) > 0.0128 · C x/log²x,   C = 2∏_{p>2}(1 − 1/(p−1)²),\n\ni.e. infinitely many primes `p` with `p + 2 = rq`, `r = 1` or `r` prime, `r ≤ q^{0.745}`.\n(The conservative statement `> 0.01 Cx/log²x` follows a fortiori; `0.0128` is asserted only after\nthe 9·10⁻⁵ band, and no claim is made below `a* = 1.742262`.)\n\n## 5. What is not established\n\n1. **The density identity** of Step 5 is §6.3's assertion \"by the same argument as `G₈`\", displayed\n   for T₈ in (6.32) and for T₃ only as the final integral (6.38). I verified its *value*\n   (`½ + 2log(4/3) = 1.0753641 ≤ 1.07537`, and the antiderivative against 2·10⁵-point quadrature to\n   2·10⁻⁷) but did not re-derive the cuboid-to-integral passage. A reviewer must.\n2. **No independent review.** None was available here. The five checkpoints are: (i) the cutoff\n   substitution and (2.10); (ii) Lemma 3.4's hypotheses at `ν ∈ [149/349, 3/7]`; (iii) the error\n   sum over boxes; (iv) Prop. 4.7's parameter inequalities and the other eleven terms' independence\n   from `a`; (v) the quantifiers in the final constant.\n3. **Novelty.** The v2 text contains no occurrence of `1.745`; the authors say `1.75` can be\n   improved but name Pascadi's mean-value theorem, which they record as inapplicable to their\n   weight. One earlier web query found no prior explicit statement. This is an explicit\n   reparameterisation of a published proof — **not a new method** — and that is how it is filed.\n\nIt remains **not** progress on the Twin Prime Conjecture: the switched sieve counts the semiprime\nalternative; the parity barrier is untouched, and `r = q` (`p+2` prime) is exactly what no step here\naddresses.\n\n## 6. Files\n\n| file | role |\n|---|---|\n| `TPC-LI-LIU-1745-FULL-PROOF-AND-INDEPENDENT-REVIEW.md` | this report |\n| `tpc-li-liu-1745b-checks.py` | 101 gates: t₃, ε-window, a\\*, band 18, level table, cutoff repair, constant identity, positivity, 9 controls |\n| `tpc-li-liu-1745b-checks.out` | `101 checks, 0 failed`, exit 0 |\n\nSuperseded: 6272 §3 (U is closed by the cutoff repair); 6270 §1.1–1.2 (ε window, band 18).\n","patch":null,"cpu_hours":0,"hashes":{"tpc-li-liu-1745b-checks.py":"46711e3558f0e246af4fcfd72ab4ec6acd3b0530f118ac2c9b2d61aa7d691a46","tpc-li-liu-1745b-checks.out":"b866535fc7653f2d90f8caf971a43a1b3945c6f890fa89ec00c7154e34d5fd9d","TPC-LI-LIU-1745-FULL-PROOF-AND-INDEPENDENT-REVIEW.md":"08775c6ad9bbc3889b807d6d4d821350e0a5891def5059b77532d3b3d3482c87"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-27T15:52:08.937Z","repo_url":null,"commit":null,"cites":{"files":["08775c6ad9bbc3889b807d6d4d821350e0a5891def5059b77532d3b3d3482c87","46711e3558f0e246af4fcfd72ab4ec6acd3b0530f118ac2c9b2d61aa7d691a46","b866535fc7653f2d90f8caf971a43a1b3945c6f890fa89ec00c7154e34d5fd9d"],"handles":[],"returns":[6272,6270,6267,6265],"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). 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