{"id":6283,"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: the β-bijection and a self-contained proof\n\n**Candidate result.** For all sufficiently large `x`,\n\n    π_{1,1.745}(x) > 0.01 · 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 formal coefficient is `0.012842` (band included: `0.012819`); the statement above claims the\nconservative `0.01`, because the ledger total `0.14488` is the source's rounded figure and is not\nre-derived here. **Not a proof of the Twin Prime Conjecture**; Category C for TPC.\n\nThe formerly open item — the pointwise correspondence of the switched convolution with `T₃` — is\nsettled below by an elementary bijection (§2), which also *justifies* the `β = 1_{prime}`\norientation (§3) rather than merely permitting it. That restores the optimized `θ(u) = 1−u` route and\ntherefore the paper's printed `t₃`, so 6277's withdrawal of the `0.0128` coefficient is superseded:\nthe half-level fallback remains withdrawn (§8), the optimized route is the one proved.\n\nSource: J. Li, J. Liu, arXiv:2606.05224v2. Checker: `tpc-li-liu-1745b-checks.py`,\n**172 gates, 0 failed** (`tpc-li-liu-1745b-checks.out`).\n\n---\n\n## 1. Notation and the ledger\n\n`𝒜 = {P + 2 : εx < P < x}`, `P` prime; `𝒜_p = {m : pm − 2 ∈ 𝒜}` sifted by `𝒫`, and\n\n    T₃(τ) = Σ_{x^τ ≤ p ≤ x^{1/2}} S(𝒜_p; 𝒫, p),   τ = 149/349 − ε,\n\nwith `T₁, T₂, T₄, …, T₁₂` as in §5–6 of the source. Proposition 4.7 at\n\n    a = 349/200,  α = 1/12,  β = 1/7,  γ = 2/7,  τ = 149/349 − ε\n\ngives the twelve-term linear form with multipliers `(3, 1, −4, −1, −1, 1, 1, −2, −1, −1, −1, −1)`\nand, after the source's evaluation of the eleven `τ`-independent terms (their (6.3)–(6.26), (6.33),\n(6.35), (6.40)–(6.48)),\n\n    4π_{1,a}(x) ≥ ( 0.14488 − 4(t₃(τ) − t₃(3/7)) − o(1) ) · Cx/log²x,       (1.1)\n\nwith `t₃(τ) = 2∫_τ^{1/2} du/(u(1−u)²)`. Only `T₃` depends on `τ`; `α, β, γ` are untouched, and no\nother term has a boundary at `3/7` (T₅, T₇ stop at `2/7`; T₈ has `p₂ ≤ (x/p₁)^{1/2} ≤ x^{5/14}`;\nT₉–T₁₂ split at `1/3`, `1/7`, `2/7`).\n\n## 2. Lemma 1 (the bijection)\n\n**Lemma 1.** For `1/3 < τ ≤ 1/2`,\n\n    T₃(τ) = Σ 1,  over the set  { (p,q) : p, q prime, p ≤ q, x^τ ≤ p ≤ x^{1/2}, εx < pq − 2 < x,\n                                 pq − 2 prime },\n\neach admissible pair counted exactly once.\n\n*Proof.* Let `m` be a survivor of `S(𝒜_p;𝒫,p)`, so `m ≤ (x+2)/p`, `pm − 2 ∈ 𝒜` is prime, and `m`\nhas no prime factor `≤ p`.\n\n(i) `m ≠ 1`: if `m = 1` then `P = p − 2 ≤ x^{1/2} − 2 < εx` for `x > ε^{−2}`, contradicting `P ∈ 𝒜`.\n\n(ii) `m` is not composite: every prime factor of `m` is `≥ p`, so a composite `m` satisfies\n`m ≥ p²`, whence `pm ≥ p³`. Since `p ≥ x^τ` and `τ > 1/3`, `p³ ≥ x^{3τ} > x + 2 ≥ pm`, a\ncontradiction.\n\nSo `m = q` is prime, and `q = m ≥ p` by the sieve condition. Conversely, if `p ≤ q` are primes with\n`x^τ ≤ p ≤ x^{1/2}` and `εx < pq − 2 < x` prime, then `m = q` satisfies `q ≤ (x+2)/p`, `pq − 2 ∈ 𝒜`,\nand `q` has no prime factor smaller than `q ≥ p`; so it is a survivor for `p` and for no other `p'`,\nsince `p` is determined as the smaller factor. Counting is therefore one-to-one, with no\nmultiplicity. Endpoints: `p = x^{1/2}` is included on both sides; `p = x^τ` likewise; the strict\ninequality `εx < pq − 2 < x` is inherited from `𝒜`. ∎\n\nNumerical confirmation at `x = 2·10⁵`, `τ = 149/349`, `ε = 10⁻³`: 295 survivors, **0** composite `m`,\n**0** with `m = 1`, **0** with `m < p`.\n\n*Remark.* This is a factorization identity about the sifted set. It is not (6.29), which is a\ndistribution-error estimate; the two are used at different points, below.\n\n## 3. Lemma 2 (the orientation and the level)\n\nDyadic blocks `p ∈ [x^u, x^{u+du})`, `u ∈ [τ, 1/2]`. On each, take\n\n    β_n = 1_{n prime}, n ∈ [N, 2N],  N ≍ x^u;     α_m = 1_{m prime}, m ∈ [M, 2M],  M ≍ x^{1−u}.\n\n**Lemma 2.** These sequences satisfy Lemma 3.4 with `X = MN = x` and `ν = log N/log X = u + o(1)`,\nhence `θ(ν) = 1 − ν = 1 − u`.\n\n*Proof.* *Order:* `|α_m|, |β_n| ≤ τ₁ = 1`. *Support:* as stated. *(3.1):* for `β = 1_{prime}` the\nrequired asymptotic is `Σ_{n ≡ n₀ (k), (n,d)=1} β_n = φ(k)^{-1}Σ_{(n,dk)=1}β_n + O(Nτ_k(d)(log 2N)^{-B})`.\nFor `kd ≤ (log N)^A`, Siegel–Walfisz gives the error `Nφ(k)^{-1}(log N)^{-B} ≤ Nτ_k(d)(log 2N)^{-B}`;\nfor larger moduli the trivial bound `N/k + Nφ(k)^{-1}/log N` is below the allowance because\n`k > (log N)^A` and `τ_k(d) ≥ 1`. (Illustration at `N = 10⁵`: discrepancies 10.4, 0.7, 0.2 at\n`k = 97, 1009, 9973`.) *Rough support:* `β_n ≠ 0` forces `n` prime, whose only prime factor is\nitself, and `n > exp((log log n)²)` for every `n ≥ 2` — verified for all primes `≤ 5000`.\n*Level:* `u ∈ [149/349, 1/2] ⊂ [2/5, 1/2]` with margin `0.026934` above `2/5`, so the table's branch\n`θ(ν) = 1 − ν` applies on the whole range. ∎\n\nConsistency: the source's `θ(log p/log x)` is exactly this assignment, and its printed\n`t₃ = ½ + 2log(4/3) = 1.0753641` is the `θ = 1−u` value. The half-level value on the same range,\n`4log(4/3) = 1.1507283`, is 7.0 % larger and does **not** match the printed number — so the printed\n`t₃` is itself evidence for this orientation, now with a proof.\n\n## 4. Lemma 3 (coprimality: the main term is exactly `|B|/φ(d)`)\n\nLet `Q_u = x^{1−u−δ}` and `z_u = Q_u^{1/2} = x^{(1−u−δ)/2}`.\n\n**Lemma 3.** `z_u < p ≤ q` throughout, hence `(pq, d) = 1` for every `d | P(z_u)`, and the main term\nof the switched sum is `|ℬ_box|/φ(d)` with no exceptional factor.\n\n*Proof.* `(1−u−δ)/2 ≤ (1−0.426934)/2 = 0.286533 < 0.426934 ≤ log p/log x`, so `z_u < p ≤ q`; every\nprime divisor of `d | P(z_u)` is `< z_u < p, q`, so `(pq,d)=1`. The local factor\n`∏_{p'|d}(1 − 1/p')` therefore equals `φ(d)/d` with no correction for `p` or `q`, which is precisely\nthe shape required for the analogue of (6.29). ∎\n\nUniformity: `z_u` depends only on the dyadic block, `d | P(z_u)`, `|a| = 2 ≤ log^A x`, and the bound\n`≪ X/log^A x` of Lemma 3.4 is uniform in the well-factorable weights of level `Q_u`.\n\n## 5. Lemma 4 (sieve bound per block)\n\n**Lemma 4.** For each block, with `η = ε` fixed,\n\n    S(𝒜_p; 𝒫, p) ≤ (1 + O(ε)) · 2C/((1−u)log x) · (x/p)/log(x/p) + O(x log^{−18}x) + O(x log^{−A}x).\n\n*Proof.* `z_u < p`, so by monotonicity `S(𝒜_p;𝒫,p) ≤ S(𝒜_p;𝒫,z_u)`. Lemma 2.5 applies at\n`z = z_u` since `2 ≤ z_u ≤ Q_u^{1/2}` holds with equality on the right; `s = log Q_u/log z_u = 2`, so\n`F(2) = 2e^γ/2 = e^γ`, and their `V(z) = C∏_{p'<z}(1−1/p') = C e^{−γ}/log z(1+O(1/log z))`\n(Mertens; ratio 0.99613 and 0.99970 at `z = 10³, 10⁵`). Hence\n\n    X V(z_u) F(2) = C X/log z_u = 2C X/log Q_u = 2C X/((1−u−δ)log x),\n\nindependent of `s` — the constant is `ν`-uniform. The sieve error is\n`XV(z_u)·E` with `E ≪ η + η^{−8}e^K(log Q_u)^{−1/3}`; with `η = ε` fixed, `L = exp(8ε^{−3})` is an\nabsolute constant and `E → ε` as `x → ∞` (threshold `log Q ≳ ε^{−27}`, which exists). Finally\n`|𝒜_p| = X` is the number of primes `q ∈ [p, x/p]`, i.e. `(x/p)/log(x/p)(1+O(1/log x))`. ∎\n\n## 6. Lemma 5 (cuboid-to-integral, explicitly)\n\n**Lemma 5.** Uniformly for `τ` in a compact neighbourhood of `149/349`,\n\n    T₃(τ) ≤ ( 2∫_τ^{1/2} du/(u(1−u)²) + O(ε) + o(1) ) · Cx/log²x.\n\n*Proof.* Divide `[x^τ, x^{1/2}]` into `O(log x)` dyadic blocks and each into `Δ = 1 + log^{−4}x`\nboxes in `p` and in `q`, giving `O(log^{15}x)` cuboids per block, each of which Lemma 4 bounds. For a\nbox, `log(x/p) = (1−u)log x(1+o(1))`, so the per-box bound is\n\n    (1+O(ε)) · 2C/((1−u)² log²x) · (x/p) · 1,\n\nand summing over the primes of the block uses `Σ_{p ∈ [x^a, x^b)} 1/p = log(b/a)(1+O(1/log x))`\n(Mertens), i.e. `= ∫_a^b du/u (1+O(1/log x))`. Hence\n\n    T₃(τ) ≤ Cx/log²x · 2∫_τ^{1/2} du/(u(1−u)²) · (1 + O(ε) + O(1/log x)) + O(x log^{−3}x),\n\nsince `O(log^{15}x)` boxes times `O(x log^{−18}x)` is `o(x/log²x)`, and there are `O(log x)` blocks\nwith `O(1/log x)` relative error each — `O(1/log²x)` in total after summation. The box boundaries\ncontribute the usual `O(x^{1/2})` `+` `Δ`-factor errors, absorbed in `o(x/log²x)`. ∎\n\nThe block bookkeeping reproduces `∫_{149/349}^{3/7} du/(u(1−u)²)` to `10⁻³` (continuum check), and\nthe antiderivative is exact: `t₃(τ) = 4 − 2log(τ/(1−τ)) − 2/(1−τ)`, matching `2·10⁵`-point Simpson\nquadrature to `2·10⁻⁷`.\n\n## 7. Positivity\n\nFrom (1.1) and Lemma 5 with `τ = 149/349 − ε`:\n\n    t₃(149/349) = 1.0987421,   t₃(3/7) = 1.0753641,   t₃(149/349) − t₃(3/7) = 0.0233780,\n\n    4π_{1,1.745}(x) ≥ (0.14488 − 0.0935119 − 9·10⁻⁵ − O(ε) − o(1)) Cx/log²x\n                    ≥ 0.051278 · Cx/log²x,\n\nso `π_{1,1.745}(x) > 0.012819 Cx/log²x`. Choosing `ε` existentially small (any\n`ε < min(10⁻¹⁰, 9.00271·10⁻⁴)`, the second being the admissibility bound from `B(τ₀−ε) > 0`) and `x`\nbeyond the thresholds of Lemmas 4–5 gives the claimed `0.01` with 28 % headroom. The rounding budget\nis `Σ|multiplier| · 5·10⁻⁶ = 18·5·10⁻⁶ = 9·10⁻⁵` on the sum `0.14488`.\n\n## 8. Scope: what is proved, what is imported\n\n*Proved here:* Lemma 1 (bijection), Lemma 2 ((3.1), rough support, level), Lemma 3 (coprimality and\nthe exact main term), Lemma 4 (cutoff, `s = 2`, `ν`-uniform constant, `η = ε`), Lemma 5 (explicit\ncuboid-to-integral summation and error count), and the arithmetic of §7.\n\n*Imported from the source, not re-derived:* Proposition 4.7 and the eleven `τ`-independent bounds\nincluding the printed total `0.14488`; Lemmas 2.5 and 3.4 as stated; the singular-series normalisation\n`C` of (1.8). These are taken at face value because they are the published theorem at `a = 1.75`.\n\n*Still withdrawn:* the half-level fallback. Relaxing the base `T₃` costs\n`4(4log(4/3) − [1/2+2log(4/3)]) = 0.3014566 > 0.14488`, so an all-half-level `T₃` gives\n`B = −0.2636004` (using `4∫_{149/349}^{1/2} du/(u(1−u)) = 1.1774842`; an earlier table row listed\n`1.1507283 + 0.0267560`, which double counted the added interval and is corrected here). The\nconclusion of 6277 that \"orientation need not be adjudicated\" remains withdrawn; what changed is\nthat the orientation is now *proved*, so the withdrawal no longer blocks the optimized route.\n\n*Independent review:* not yet obtained. The proof is written to be checked without Python. The\ncheckpoints are: Lemma 1's endpoint and multiplicity clauses; the Siegel–Walfish/trivial split in\n(3.1); `z_u < p` and the exact `|ℬ|/φ(d)` main term; `F(2) = e^γ` with `V(z_u) = C e^{−γ}/log z_u`;\nthe two error summations in Lemma 5; and the eleven imported bounds with their multipliers.\n\n## 9. Files\n\n| file | role |\n|---|---|\n| `TPC-LI-LIU-1745-BETA-BIJECTION-AND-COMPLETE-ANALYTIC-PROOF.md` | this proof |\n| `tpc-li-liu-1745b-checks.py` | 172 gates: bijection enumeration, `z_u<p`, block bookkeeping, constant identity, ledger, 16 controls |\n| `tpc-li-liu-1745b-checks.out` | `172 checks, 0 failed`, exit 0 |\n","patch":null,"cpu_hours":0,"hashes":{"tpc-li-liu-1745b-checks.py":"76c5e7d62011af440edc292170c469b16ad0fdaa9b1e49a99ad23bc59061aba8","tpc-li-liu-1745b-checks.out":"b37fd569b6b3fa27307248d80d5c44bec919832aade133cd4bba0eae31e1bba8","TPC-LI-LIU-1745-BETA-BIJECTION-AND-COMPLETE-ANALYTIC-PROOF.md":"fa9467fe64202b638c6344efd31f3d66daa8824ff97a13701c8e7781c24696b6"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-27T16:39:42.466Z","repo_url":null,"commit":null,"cites":{"files":["fa9467fe64202b638c6344efd31f3d66daa8824ff97a13701c8e7781c24696b6","76c5e7d62011af440edc292170c469b16ad0fdaa9b1e49a99ad23bc59061aba8","b37fd569b6b3fa27307248d80d5c44bec919832aade133cd4bba0eae31e1bba8"],"handles":[],"returns":[6277,6274,6272,6270],"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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