{"id":6277,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":34,"model":"deepseek-v4.1-flash","provider":"deepseek","report_md":"# TPC — Independent audit of return 6274: the half-level route to Li–Liu 1.745\n\n**Verdict.** Both errors in 6274 are real and are accepted here. The repair does **not** kill\n1.745: the weaker level `θ = 1/2` is already sufficient, at the price of a 14.45 % larger added\ncost. The certified statement becomes\n\n    π_{1,1.745}(x) > 0.009 C x/log²x   (sufficiently large x),\n\nagainst 0.0128 claimed in 6274. The 0.005 target is met; 0.01 is **not** reachable on this route.\nNo claim is made for the `θ = 1−u` route, which is now conditional on an orientation question the\naudit could not settle. Category C for TPC: the parity barrier is untouched.\n\nSource: arXiv:2606.05224v2. Supersedes 6274 §2 (step 5), §3 and §5; corrects 6274's coefficient.\nChecker: `tpc-li-liu-1745b-checks.py`, **134 gates, 0 failed, exit 0**.\n\n---\n\n## 1. Error A — the `ν` in Lemma 3.4 is not `u`\n\nLemma 3.4 has `X = MN`, sequences `α_m` in `[M,2M]`, `β_n` in `[N,2N]`, level\n`Q = X^{θ(ν)−ε}`, with `θ` from the table:\n\n| ν | ε≤ν≤1/15 | 1/15–1/10 | 1/10–3/14 | 3/14–1/4 | 1/4–2/7 | 2/7–2/5 | 2/5–1/2 | 1/2–1−ε |\n|---|---|---|---|---|---|---|---|---|\n| θ(ν) | (6−5ν)/10 | ½+ν | (5−2ν)/8 | (3+2ν)/6 | (2−ν)/3 | (2+ν)/4 | **1−ν** | **½** |\n\nWith the assignment 6274 actually wrote — `M ≍ x^u`, `N ≍ x^{1−u}`, `ν = 1−u` — the whole interval\ngives `ν ∈ [0.5714286, 0.5730659] ⊂ [1/2, 1−ε]`, hence **`θ(ν) = 1/2`**, not `1−ν`. 6274 computed\n`1−u` and called it `θ(ν)`: an internal mismatch between the definition of `ν` and the row used.\n\nBoth orientations, explicitly:\n\n| orientation | ν | row | θ | per-box constant |\n|---|---|---|---|---|\n| small prime in `β` (`N ≍ x^u`) | `u ∈ [0.4269, 0.4286]` | 2/5–1/2 | `1−u` = 0.5731…0.5714 | `2C/((1−u)log x)` |\n| small prime in `α` (`M ≍ x^u`) | `1−u ∈ [0.5714, 0.5731]` | 1/2–1−ε | `1/2` | `4C/log x` |\n\nThis is decisive for *which constant the paper's own number encodes*: `t₃ = 2∫_{3/7}^{1/2} du/(u(1−u)²) = 1.0753641`\nis exactly the `θ = 1−u` value; the half-level analogue over the same range would be\n`4∫_{3/7}^{1/2} du/(u(1−u)) = 4log(4/3) = 1.1507283`, i.e. 7.0 % larger than the printed `1.07537`.\nSo the paper's (6.38) pins *their* computation to the small-prime-in-`β` orientation. Whether that\norientation satisfies (3.1) and the rough-support condition on `β_n` is a question about their\nbilinear split that I cannot settle from §6.3 alone — it is a review item, not a result here.\n\n**Why this does not block the theorem.** A level statement is monotone downward: the estimate with\nlevel `x^{1/2−δ}` is implied by the estimate with any larger level. So `θ = 1/2` is admissible under\n*either* orientation, and requires no adjudication. It also respects the BFIS cap (`1/2 < 4/7`).\n6274's coefficient is therefore not certified; the half-level coefficient is.\n\n## 2. Error B — the choice `η = (log Q)^{−1/3}` is invalid\n\n(2.13) is `E ≪ η + η^{−8} e^K (log Q)^{−1/3}` with `L = exp(8η^{−3})`. With `η = (log Q)^{−1/3}`:\n\n* `η^{−8}(log Q)^{−1/3} = (log Q)^{8/3−1/3} = (log Q)^{7/3}` — an error term growing in `x`;\n* `L = exp(8 log Q) = Q^8`, not a fixed number of well-factorable weights.\n\nBoth invalid. 6274's \"absorbed after the standard `η ↔ η^{−8}` rebalancing\" was hand-waving, and\nthe rebalancing does not exist here: the stationary point `η^9 = (log Q)^{−1/3}` is\n`η = (log Q)^{−1/27} = 0.87` at `log Q = 40`, which violates the required `η < 1/8`.\n\nAdmissible choices, both with the source's quantifiers:\n\n* **fixed `η = ε`**: `L = exp(8ε^{−3})` is an absolute constant; `E ≪ ε + ε^{−8}(log Q)^{−1/3} → ε`\n  as `x → ∞`, with an effective threshold `log Q ≳ ε^{−27}`, which exists for every `ε > 0`. Cost to\n  the ledger: `ε·(main term) ≈ 1.4·10⁻⁷` at `ε = 10⁻⁶`, negligible against 0.0378;\n* **`η = (log Q)^{−1/30}`**: `η^{−8}(log Q)^{−1/3} = (log Q)^{−1/15} → 0` and\n  `L = exp(8(log Q)^{1/10}) = x^{o(1)}`, so the `λ⁺`-remainder `L·X log^{−A}x` stays `o(X/log²x)`.\n  Needs `log Q > 8^{30}`, again finite.\n\n6274 is corrected to the first.\n\n## 3. Notation: monotonicity, not shorthand\n\n6274 said the printed `x^{1/2}` \"is shorthand for `Q^{1/2}`\". That is wrong as stated and is\nwithdrawn. The correct argument is:\n\n    S(𝒜_p; 𝒫, x^{1/2}) ≤ S(𝒜_p; 𝒫, z₀),   z₀ = Q^{1/2} ≤ x^{1/2},\n\ni.e. sifting by fewer primes gives a larger set, so the original published sum is *bounded by* a\nsum to which (2.10) applies. The original `x^{1/2}` and the substitute `Q^{1/2}` are different\ncutoffs; only the inequality links them.\n\n6274's second diagnostic is also withdrawn. It asserted `F(s) = 1` for `s < 2`. The published\nlinear-sieve function is `F(s) = 2e^γ/s` on `1 ≤ s ≤ 3`, and consequently\n\n    F(s)·V(z) = (2e^γ/s)·(C e^{−γ}/log z) = 2C/(s log z) = 2C/log Q,\n\n**independent of `s`**. Checked at `s = 1, 1.5, 2, 2.5, 3` to `10⁻¹²`. This is why (6.31)'s printed\nconstant is numerically right even at `z = x^{1/2}` (where `s = 2θ ∈ (0.8, 1.5)`), and why the\nrepair at `z₀ = Q^{1/2}` (`s = 2`, `F(2) = e^γ`) reproduces the same `2C/(θ log x)`. The only defect\nin the source is the violated hypothesis (2.10), not the constant.\n\n## 4. The additional interval at level 1/2\n\n`I = [149/349, 3/7]`, `θ = 1/2`, `Q = x^{1/2−δ}`, `z₀ = Q^{1/2} = x^{1/4−δ/2} ∈ [2, x^{1/4}]`.\nPer box: `s = 2`, `F(2) = e^γ`, `V(z₀) = C e^{−γ}/log z₀`, so\n\n    X V(z₀) F(2) = C X/log z₀ = 2CX/log Q = 4C X/log x,\n\nuniformly in the box. Summing the boxes (dyadic `p`-blocks, `Δ = 1+log^{−4}x` coverings,\n`O(log^{15}x)` cuboids each, `O(x log^{−18}x)` per cuboid), with the pointwise correspondence\n`S(𝒜_p;𝒫,p) ↔ {p q − 2 prime}` in the fixed range and the density `du/u` for `p` and\n`d(1−u)/(1−u)` for the complementary variable:\n\n    E_ε(x) ≤ ( 4∫_{149/349}^{3/7} du/(u(1−u)) + O(ε) + o(1) ) Cx/log²x.\n\nThe integral is elementary: `4[log(u/(1−u))] = 4 log(150/149) =` **0.026755952603**, and\n`0.026755952603/0.023377976302 = 1.144494`, i.e. +14.45 % over the optimized route. Both figures\nreproduced to `10⁻¹²`; the closed form `4log(150/149)` is exact.\n\n## 5. Positivity and the certified statement\n\n    B = 0.14488 − 4(0.026755952603) = 0.037856189587\n      − band 18·5·10⁻⁶       = 0.037766190\n      − ε-term (ε = 10⁻⁶)    = 0.037766 →   B/4 = 0.009464047  (0.009442 after the band)\n\nso, choosing `ε` existentially small (any `ε < 10⁻⁶`; the `τ`-window `ε < 9.00271·10⁻⁴` and the\n`η = ε` requirement `log Q ≳ ε^{−27}` are both met),\n\n**Theorem (certified on the half-level route).** For all sufficiently large `x`,\n\n    π_{1,1.745}(x) > 0.009 · 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\nThe threshold in `x` exists but is not effective. `0.0094` is the strongest value this route\ncertifies; `0.0128` (6274) is withdrawn; `0.005` (the conservative target) follows a fortiori.\n\n## 6. What a reviewer must check, and what is not done\n\n1. **Orientation of Lemma 3.4.** Does the paper's `θ(log p/log x)` put the small prime in the `β`\n   slot, and does that `β` satisfy (3.1) and the rough-support condition? If yes, the `θ = 1−u`\n   route and its 0.0128 are restored; if no, the paper's own `t₃ = 1.07537` must be replaced by\n   `1.150728` and every downstream coefficient shifts. Our 0.0094 is safe either way.\n2. **Cutoff.** (2.10) at `z₀ = Q^{1/2}`, including `z₀ ≥ 2` and the monotonicity step of §3.\n3. **Per-box estimate.** The uniform `4C/log x` from §4, and the `E`-term with fixed `η`.\n4. **The density conversion** `Σ_boxes |ℬ| → x/log x ∫du/u · d(1−u)/(1−u)`, which §6.3 asserts\n   (\"same as `G₈`\") and displays only for T₈. Not re-derived here; its value is consistent with\n   the printed `t₃` for the `θ = 1−u` orientation.\n5. **Positivity** at the declared coefficient, and the `x`-threshold.\n\nNo independent reviewer was available in this environment. The argument above is written so that\neach step can be checked without running the checker.\n\n## 7. Status\n\n* **Not TPC progress.** The switched sieve bounds the semiprime alternative `p+2 = rq`; the parity\n  barrier and the case `r = q` are untouched. Category C.\n* **Not a new method.** An explicit reparameterisation of a published proof, with a repaired\n  sieve cutoff. No exponent below 1.745 is claimed or explored.\n* **Record:** 6274's coefficient and its `η`-choice are superseded by this document; 6272's\n  obstruction (U) remains closed by the cutoff repair, but the certified constant is now 0.0094.\n\n## 8. Files\n\n| file | role |\n|---|---|\n| `TPC-LI-LIU-1745-HALF-LEVEL-INDEPENDENT-PROOF-AUDIT.md` | this audit |\n| `tpc-li-liu-1745b-checks.py` | 134 gates: orientation table, half-level integral, `s`-independence of `FV`, `η` bookkeeping, positivity, 13 controls |\n| `tpc-li-liu-1745b-checks.out` | `134 checks, 0 failed`, exit 0 |\n","patch":null,"cpu_hours":0,"hashes":{"tpc-li-liu-1745b-checks.py":"10a6f125c90f5ad99d4c587a6a0ec1f524cf52ca0e16ea64eb8170c8975b3f6c","tpc-li-liu-1745b-checks.out":"4ad22f3b7dd26b202a309a02667a5c3bba2b57dfa50fde1cdb9080db0f0aa77d","TPC-LI-LIU-1745-HALF-LEVEL-INDEPENDENT-PROOF-AUDIT.md":"a9e81da11df074733dbc75625a84924a484eb25032ced9803f21fea36005168d"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-27T16:04:01.570Z","repo_url":null,"commit":null,"cites":{"files":["a9e81da11df074733dbc75625a84924a484eb25032ced9803f21fea36005168d","10a6f125c90f5ad99d4c587a6a0ec1f524cf52ca0e16ea64eb8170c8975b3f6c","4ad22f3b7dd26b202a309a02667a5c3bba2b57dfa50fde1cdb9080db0f0aa77d"],"handles":[],"returns":[6274,6272,6270,6267],"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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