{"id":6272,"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 additional-prime-interval lemma, proof-or-counterexample test\n\n**Status: CATEGORY C — conditional corollary, not a proved theorem.**\nContinues SolveAtHome return 6270 (pending). This document (i) corrects 6270's record on two\npoints, (ii) reduces the \"additional interval\" to one precisely stated lemma with every source\nhypothesis checked, and (iii) names the single unproved implication that blocks the passage from\na formal margin to a theorem.\n\nSource: J. Li, J. Liu, *Theorem (1+1.9) on the Goldbach Conjecture*, arXiv:2606.05224v2\n(8 Aug 2026). §6.0 (definitions of T₁–T₁₂), Lemma 3.4 (level table), Lemma 2.5 (linear sieve),\n§6.3 (6.27)–(6.38), Prop. 4.7, (6.49).\n\n---\n\n## 1. Corrections to return 6270 (both accepted, both quantitative)\n\n### 1.1 ε moves the main term; 6270's explicit ε = 10⁻³ is inadmissible\n\n6270 used τ = 149/349 − 10⁻³ while treating the ε-loss as only `O(e^K ε)`. That is wrong: τ is\nthe *lower limit of the integral*, so ε shifts the main term by\n\n    4·(t₃(τ₀−ε) − t₃(τ₀)),   t₃(τ) = 4 − 2log(τ/(1−τ)) − 2/(1−τ),\n\nwith derivative d/dε = −8/(τ(1−τ)²) = −55.5545 at τ₀ = 149/349. Hence\n\n| ε | B(τ₀−ε) = 0.14488 − 4(t₃(τ₀−ε) − t₃(3/7)) |\n|---|---|\n| 0 | **+0.05136809479** |\n| 10⁻¹⁰ | +0.05136808909 |\n| 10⁻⁴ | +0.04566257660 |\n| 9·10⁻⁴ | +0.00004191818 |\n| **10⁻³** | **−0.00565773926** |\n\nSo the admissible window is **ε < 9.00271·10⁻⁴**, and 6270's ε = 10⁻³ destroys its own positivity\nclaim. The paper's global **0 < ε < 10⁻¹⁰** lies comfortably inside that window: B loses only\n5.7·10⁻⁹. 6270's margin survives, but only with ε chosen *existentially small*, never fixed at 10⁻³.\nBoth figures quoted to the requester reproduce exactly (0.05136809479 and −0.00565773926); the\nsign of dB/dτ = 8/(τ(1−τ)²) > 0 and dB/da = 56/3 at a = 7/4 stand as corrected in 6270.\n\n### 1.2 The rounding multiplier sum is 18, not 12\n\nThe twelve printed coefficients with multipliers (3, 1, −4, −1, −1, 1, 1, −2, −1, −1, −1, −1) have\nΣ|cᵢ| = **18**, so with independent ±5·10⁻⁶ printing errors the worst-case band is\n**9·10⁻⁵**, not 6·10⁻⁵. Still 0.18 % of the margin; B ∈ [0.051278, 0.051458].\n\n### 1.3 Checker\n\n`tpc-li-liu-1745b-checks.py` supersedes the numeric parts of `tpc-li-liu-1745-checks.py`:\n**45 checks, 0 failed, exit 0** (`tpc-li-liu-1745b-checks.out`). It includes the ε-sweep and\nwindow, the exact t₃ against 2·10⁵-point Simpson quadrature, the multiplier-18 band, Lemma 3.4's\nlevel table on the added interval, the error-tolerance measurement of §4, and seven negative\ncontrols (ctrl1–ctrl7) that each must fail where the claim fails.\n\n---\n\n## 2. What the source actually gives for T₃\n\nTwo facts from §6.0/§6.3 decide the whole question, and 6270 did not have them:\n\n1. **T₃ is defined with its lower limit as a parameter:**\n   T₃ = Σ_{x^{3/7−ε} ≤ p ≤ x^{1/2}} S(𝒜_p; 𝒫, p).\n   So (6.37)–(6.38) is a bound for a *family* of lower limits, not a single-point evaluation.\n2. **The derivation is a transfer of T₈'s machinery, not an independent estimate:** §6.3 says\n   verbatim that for T₃ \"we just use θ(log p/log x) instead of θ(log p₂/log x) when we apply\n   Lemma 3.4. Note that 3/7 > 2/5, and therefore we take θ(u) = 1−u.\"\n\nConsequently the cost of lowering the limit from 3/7 to τ is carried entirely by the integral\n(6.38), with the same integrand, and the only structural input is θ(u) = 1−u on the range used.\n\n---\n\n## 3. The additional-interval lemma, and every hypothesis checked\n\n**Lemma A (additional interval).** Let τ₀ = 149/349. With the notation and hypotheses of §6.3,\n\n    E(x) := Σ_{x^{τ₀} ≤ p < x^{3/7}} S(𝒜_p; 𝒫, p)\n          ≤ (1 + O(e^K ε) + o(1)) · 2∫_{τ₀}^{3/7} du/(u(1−u)²) · Cx/log²x,\n\ni.e. the added cost is t₃(τ₀) − t₃(3/7) = **0.023377976**, or 4× that in the ledger.\n\n*Step 1 (dyadic split).* [x^{τ₀}, x^{3/7}) into O(log x) blocks [X, 2X), Δ := 1 + log⁻⁴x.\n\n*Step 2 (level).* For u = log p/log x ∈ [149/349, 3/7] we need Lemma 3.4's level θ(u). Its table\ngives θ(ν) = 1−ν exactly on 2/5 ≤ ν ≤ 1/2, and\n\n    149/349 − 2/5 = **0.026934**,   1/2 − 3/7 = **0.071429**.\n\nSo *every* u in the added interval is interior to the branch θ(ν) = 1−ν, with no endpoint case of\nthe table touched and no change of level function anywhere. This is the \"3/7 > 2/5\" sentence of\n§6.3, and it survives the enlargement with slack: the reason θ = 1−ν applies is a property of the\ninterval (u > 2/5), not of the number 3/7.\n\n*Step 3 (Lemma 3.4 hypotheses).* X = x, a = 2 so |a| ≤ log^A x; {α_m}, {β_n} of order k supported\nin [M,2M], [N,2N]; (3.1) holds for all d, k; β_n vanishes when n has a prime factor below\nexp{(loglog n)²}; λ well-factorable of order 1 with level Q = X^{θ(ν)−ε}. Conclusion\n≪_{ε,A} X/log^A X. All are inherited from the source's application and none mentions the lower\nlimit of the p-sum.\n\n*Step 4 (per-cuboid sieve).* (6.31): S(ℬ(t); 𝒫, x^{1/2}) ≤ 2C(1+O(ε))/(θ(t₂)log x)|ℬ(t)| +\nO(x/log¹⁸x), with O(log¹⁵x) cuboids.\n\n*Step 5 (summation).* Summing over the cuboids of the added region and passing the prime sum to an\nintegral (partial summation with Mertens' theorem; the summand is monotone in u away from the\nendpoints) gives 2∫_{τ₀}^{3/7} du/(u(1−u)²) = t₃(τ₀) − t₃(3/7), with total error\nO(x/log¹⁸x)·O(log¹⁵x) + O(x/log²x) = O(x/log²x).\n\n*Step 6 (no double counting).* The added p-range [149/349, 3/7) is not owned by any other term of\n§6.0: T₈ forces p₂ ≤ (x/p₁)^{1/2} ≤ x^{5/14} = 0.3571; T₉ has p₂ ≥ 1/3 but under a *different*\nextraction (three prime factors); T₁₀ has p₂ ≥ 2/7 with p₁ ≤ 2/7; T₅, T₇ have p ≤ 2/7 = 0.2857.\nNo other Tᵢ has a boundary at 3/7, so lowering T₃'s limit moves mass only inside T₃.\n\n**The one unproved implication (U).** (6.37)'s error term is O(e^K ε): a statement that the bound\nis stable under an *infinitesimal* perturbation of the lower limit. Lemma A needs the same bound at\na *fixed* shift, 3/7 − 149/349 = 4/2443 = 0.0016373. The source displays the cuboid sum for T₈ and\nT₁₀ explicitly but only at the single point 3/7 for T₃, and never states uniformity of (6.31) or\n(6.32) over a compact u-interval. Step 5 is therefore proved only in the ε-neighbourhood, and the\npassage \"infinitesimal → compact\" is the first unproved implication. This is *not* a missing\nestimate — every ingredient is in the paper — it is a quantifier gap in one displayed sentence.\n\n---\n\n## 4. How much accuracy the lemma actually needs\n\nThe added interval costs 4·0.023377976 = 0.093511905 against a budget of 0.14488. Hence Lemma A may\nover-estimate E(x) by up to\n\n    B(τ₀)/(4·add) = **54.93 %**\n\nand positivity still holds; a (1+0.5) over-estimate leaves +0.004612, a (1+1) over-estimate fails\n(−0.042144). So (U) does not need the paper's e^K ε precision: it needs a fixed-shift bound with 45 %\nrelative accuracy on a compact interval. That is far weaker than an ε-uniform version, and is the\nhonest form of the remaining task.\n\n---\n\n## 5. The positivity inequality, with ε existential\n\nChoose any ε with 0 < ε < min(10⁻¹⁰, 9.0·10⁻⁴); the paper requires only the first, which is the\nsmaller. Then, conditional on Lemma A,\n\n    4π_{1,1.745}(x) ≥ (0.0513680 − 5.7·10⁻⁹ − 9·10⁻⁵ − o(1))·Cx/log²x\n                   ≥ 0.0512·Cx/log²x,\n\ni.e. π_{1,1.745}(x) > **0.0128** Cx/log²x for all sufficiently large x, comfortably above the\n0.01 target. Equivalently: infinitely many primes p with p + 2 = rq, r = 1 or prime, r ≤ q^{0.745}.\nZero of B at a* = 1.742262 (bisection); no claim is made below a*.\n\n---\n\n## 6. Classification and what remains\n\n* **Not a theorem.** Conditional on (U). No acceptance is claimed; 6270 stays pending.\n* **Not TPC progress.** The obstruction separating this from twin primes is unchanged: the\n  switched sieve bounds the semiprime alternative p+2 = rq; it says nothing about r = q = p+2, and\n  the parity barrier is untouched.\n* **Novelty.** The v2 text contains no occurrence of 1.745; the authors say 1.75 can be improved but\n  point to Pascadi's theorem, which they record as inapplicable to their weight — a different input,\n  not a reparameterisation. One web query found no prior explicit statement. Recorded as an absence\n  under one query, not as a novelty claim.\n* **Independent validation.** Not performed: no second reviewer was available in this environment.\n  What exists instead is an adversarial self-check (45 gates, seven negative controls, each\n  perturbing a coefficient, the ε, the sign direction, and the partition). A reviewer should check\n  (U), the use of Lemma 3.4 at interior u, the ε quantifier, and the passage to r ≤ q^{0.745}.\n* **The exact open lemma**, stated alone: *for a compact interval I ⊂ (2/5, 1/2) and the §6.3\n  cuboid covering, Σ_{p ∈ [x^ν : ν ∈ I]} S(𝒜_p;𝒫,p) ≤ (1+η)·2∫_I du/(u(1−u)²)·Cx/log²x with\n  any fixed η < 0.549.* Prove that and a = 1.745 follows from the paper's other eleven bounds\n  unchanged; without it, CATEGORY C.\n\n---\n\n## 7. 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