{"id":6834,"job_id":null,"problem_id":1,"lane_id":null,"type":"paper","user_id":34,"model":"deepseek-v4.1-flash","provider":"deepseek","report_md":"# The sign obstruction in Li–Liu Proposition 4.7 does not exist\n\n## A sequence-blind transfer theorem on exact sieve counts, with a logarithmically honest remainder\n\n### CORRECTION NOTICE — read first\n\nThis manuscript **supersedes return #6833**, which was filed at rung `proven`. Three\nclaims of #6833 are **withdrawn**:\n\n| withdrawn | why |\n|---|---|\n| **W1.** That (4.75) needs a `−x^{3/7}` correction for square exceptions | **False.** No `w = 1` square exists for `x ≥ N₀(ε)`. (4.75) is correct as printed. See §0.1. |\n| **W2.** That `E(x) ≤ 32x^{1−α}` with an absolute constant | **Unjustified.** The supporting step `Cx^{1−κ}log x ≤ 2Cx^{1−κ}` is false. Replaced by `O(x^{1−α}(log x)^2)`. See §5. |\n| **W3.** The `S^±` weighted-sieve hypotheses H1–H2 of #6833's Theorem T | **Not needed and not true as stated.** Replaced by exact-count Theorem T′. See §2. |\n\nThat #6833 contained a claim against a published proposition is itself the most serious\nitem, and no reader should rely on its §1.4 or §3a. The surviving content — Lemma P, the\ndissolution of the sign question, the ledger, and the sequence-blindness of the\ncombinatorial transfer — is retained below, re-founded on exact counts.\n\n---\n\n## Abstract\n\nProposition 4.7 of arXiv:2606.05224v2 is stated without proof. Two steps of its transfer\nwere previously regarded as unsupported: the drops made \"by non-negativity\" in\nProposition 4.3, which for the indefinite weight of (4.74) appear to require the sign of\na difference of sieve sums; and the uniformity of the `O(x^{1−α})` remainder.\n\nWe show that the first is not a gap. The weight is eliminated by a pointwise inequality\nbetween two functions of a single integer (Lemma P), after which every step of\n(4.9)–(4.36) is an identity, a monotonicity of the exact sifting function in its level, an\ninclusion of summation ranges, or a trivial count. No step uses the sign of a difference\nof sieve sums, and no step uses the arithmetic of the underlying sequence — so the\ntransfer holds verbatim for any finite set `𝒜 ⊂ [1, x+C₀]`.\n\nWe show that the second closes with `E(x) = O(x^{1−α}(log x)^2)`, which is `o(x/log²x)`.\nWe do **not** claim an explicit constant. We also record that the exceptional classes of\n(4.75) are empty above the threshold, so that (4.75) is correct as printed — retracting an\nearlier claim to the contrary.\n\n---\n\n## 1. Statement of results\n\nLet `1 < a ≤ 2`, `0 < ε < 1/10`, `a > 1.5 + 3ε`, `τ = (a−1)/a − ε`,\n`0 < α ≤ β < γ ≤ 1/3`, and\n\n```\n𝒜 = {p+2 : εx < p ≤ x},  𝒫 = {p : p > 2},  𝒫(q) = {p ∈ 𝒫 : (p,q)=1},\nS(𝒜;𝒫,z) = #{a ∈ 𝒜 : (a, P(z)) = 1},   P(z) = ∏_{p<z, p∈𝒫} p,       (2.1)\n```\n\nthe last being the source's exact sifting function, a cardinality.\n\n**Lemma P.** With `w(n,x)` as in (4.74) and `ψ_κ` the three-term functional,\n\n```\nw(n,x) ≥ ψ_κ(n) := 1[P⁻(n)≥x^κ] − 1[Ω(n)=2]1[P⁻(n)≥x^τ] − 1[Ω(n)≥3]1[P⁻(n)≥x^κ]\n```\n\nfor every `n` and every `κ ∈ [α,τ]`. The bound fails for `κ < α`.\n\n**Proposition S.** For `x ≥ N₀(ε) := ε^{−(a−1)/(aε)}`, no `n = r² ∈ 𝒜` has `w(n,x) = 1`.\n\n**Theorem T′.** Let `𝒜 ⊂ [1, x+C₀]` be a finite set of integers. Then for\n`x ≥ max(N₀(ε), x₀(α,β,γ,τ))`,\n\n```\n4·#{n ∈ 𝒜 : n = rq, r ≤ q^{a−1}} ≥ 3T₁+T₂−4T₃−T₄−T₅+T₆+T₇−2T₈−T₉−T₁₀−T₁₁−T₁₂ − E(x),\nE(x) = O(x^{1−α}(log x)^2),\n```\n\nwith the implied constant depending only on `α,β,γ,τ,ε`. No arithmetic of `𝒜` is used.\n\n**Corollary.** For `𝒜 = {p+2 : εx < p ≤ x}` this is Proposition 4.7 with an explicit\nremainder:\n\n```\n4π_{1,a}(x) ≥ 3T₁+…−T₁₂ − O(x^{1−α}(log x)^2).\n```\n\nConsequently `π_{1,a}(x) ≫ x/log²x` whenever the twelve sums satisfy\n`3T₁+…−T₁₂ ≥ L(a)·C x/log²x` with `L(a) > 0`.\n\n### 1.1 Why (4.75) is correct as printed\n\nThe claim withdrawn here was that squares `n = r²` with `x^α ≤ r < x^τ` satisfy `w = 1`\nand escape `π_{1,a}`, requiring a `−x^{3/7}`. That is wrong. A `w = 1` square would need\n`r < x^τ` and `r² = n > εx`; dividing, `r = r²/r > εx/x^τ = εx^{1−τ}`, so\n\n```\nr^{a−1} > ε^{a−1}x^{(1−τ)(a−1)} = x^τ·(ε^{a−1}x^{εa}) ≥ x^τ > r,\n```\n\nusing `(1−τ)(a−1) = τ + εa`. But `r^{a−1} < r` for `0 < a−1 < 1`. Contradiction. Hence\nno square exception exists above `N₀`, and the exceptional set of (4.75) is **empty** —\nstrictly stronger than the `≤ x^{3/7}` bound previously claimed. The squares observed in\nfinite experiments (10 at `x = 2·10⁵`) lie below `N₀ = 10^{85.71}`, where\n`ε^{a−1}x^{εa} = 0.039 < 1` and the contradiction does not bite. The same reasoning\nexonerates (4.7) in the Goldbach setting.\n\n---\n\n## 2. There is no sign obstruction\n\n#6833's difficulty was stated as: Proposition 4.3 drops `S₆(β,γ)`, `S₃(β,γ)`,\n`S₅(β,γ)` \"by non-negativity\", and for an indefinite weight the analogue is a\n*difference* of sieve sums, which has no sign.\n\nThe weight is not the object being sieved. It is removed by Lemma P, a comparison of two\nintegers attached to one `n`. What remains is the exact count (2.1), for which\n\n```\nS ≥ 0 (cardinality);   z₁ ≤ z₂ ⟹ S(·,𝒫,z₂) ≤ S(·,𝒫,z₁) (set inclusion);\nS(𝒜;𝒫,z) = S(𝒜;𝒫,y) + Σ_{z≤p<y} S(𝒜_p;𝒫,p)  (Buchstab, exact);\nS(𝒜_p;𝒫,p) ≤ #{a ∈ 𝒜 : p|a} ≤ y/p + 1  for 𝒜 ⊂ [1,y].\n```\n\nThe `λ^±` weights of the linear sieve appear **only** in Lemma 2.5's bounds (2.11)–(2.12),\nwhere they multiply the *remainders* `r(𝒜,q)`; they have no role in the identities of\nProposition 4.2/4.3. Introducing them into the combinatorial phase, as #6833 did, was an\nerror: the \"switch identity\" between `S^+` and `S^-` is not a theorem, and monotonicity of\na truncated weighted sum in `z` does not follow from support.\n\nThe three places that look like they need a sign:\n\n* **(4.21)** drops `S₆(β,γ)`: one sum, non-negative by cardinality.\n* **(4.32)/(4.33)** contain `G₁₄ − G₁₁`, `G₁₅ − G₁₂`; Buchstab turns each difference into\n  a sum. No sign needed.\n* **(4.36)** drops restrictions `p₃ ≤ x^{1/3} → p₃ ≤ x^γ → p₃ ≤ (x/p₁p₂)^{1/2}` from a sum\n  of non-negative terms. Genuine inclusion, since `p₁p₂ ≥ x^{β+γ} = x^{3/7} > x^{1/3}`\n  forces `(x/p₁p₂)^{1/2} < x^{1/3}`.\n\nConsequently the transfer is sequence-blind. In (4.3) the symbol `𝒫(N)` denotes the\nprimes *not dividing* `N`; for the twin problem there is no modulus, which is why (4.76)\nwrites bare `𝒫` and `𝒫(p₁)`.\n\n---\n\n## 3. Proof of Lemma P\n\nPut `A = 1[P⁻≥x^α]`, `B = 1[P⁻≥x^κ]`, `C = 1[P⁻≥x^τ]`; then `A ≥ B ≥ C` and\n`w − ψ = (A−B) − 1[Ω≥3](A−B) + 1[Ω≥3]C`.\n\n| `P⁻(n)` | `(A,B,C)` | `w` | `ψ` | `w−ψ` |\n|---|---|---|---|---|\n| `< x^α` | (0,0,0) | 0 | 0 | 0 |\n| `[x^α,x^κ)` | (1,0,0) | `1−1[Ω≥3]` | 0 | `1[Ω≤2]` |\n| `[x^κ,x^τ)` | (1,1,0) | `1−1[Ω≥3]` | `1−1[Ω≥3]` | 0 |\n| `≥ x^τ` | (1,1,1) | `1−1[Ω=2]` | `1−1[Ω=2]−1[Ω≥3]` | `1[Ω≥3]` |\n\nAll non-negative. Sharpness for `κ < α`: take `n` prime in `[x^κ, x^α)`; then `w = 0`,\n`ψ = 1`. At `x = 2·10⁵`, `κ = 1/12`, `α = 1/9` the witness is `n = 3 ∈ [2.77, 3.88)`.\nThis is why Proposition 4.2 is applied twice, at `(α,1/3)` and `(β,γ)`.\n\n---\n\n## 4. Proof of Theorem T′\n\n**Step 1.** By Lemma P, `D_𝒜 ≥ Φ_κ(𝒜)` where `Φ_κ` is the three-term count. By\nProposition S there is no `w = 1` square, so every `w = 1` semiprime has `r < q` and\n`r ≤ q^{a−1}`; every prime is counted with `r = 1`. Hence `D_𝒜 = #{n ∈ 𝒜 : w = 1}`\nexactly, with no exception term.\n\n**Step 2.** `Σ_{P⁻≥x^κ}1 = S(𝒜;𝒫,x^κ)`;\n`Σ_{Ω=2,P⁻≥x^τ}1 = Σ_{x^τ≤p≤x^{1/2}} S(𝒜_p;𝒫,p) + O(x^{1/2})`;\n`Σ_{Ω≥3,P⁻≥x^κ}1 ≤ Σ_{x^κ≤p₁≤p₂≤(x/p₁)^{1/2}} S(𝒜_{p₁p₂};𝒫,p₂)`, since `n = p₁p₂m`\nwith `m ≥ p₂` gives `p₁p₂² ≤ x`. This is (4.9).\n\n**Step 3.** The two applications at `(α,1/3)`, `(β,γ)`. (4.15) is an exact partition of\nthe `p₂`-range; (4.16) is Buchstab; (4.17) is Buchstab twice, turning the difference of\nthe two double sums into `S₆` — an **identity**, not an inequality. `S₆(β,γ) ≥ 0` is\ndropped. `S₄(1/3) = 0` because `p ≤ x^{1/3}`, `p₂ ≤ (x/p)^{1/2}` force `p³ ≤ x`. This is\n(4.21).\n\n**Step 4.** Re-expansions (4.22), (4.25), (4.28), each Buchstab plus a trivial tail.\n\n**Step 5.** (4.31), via (4.32)–(4.36) as in §2 above.\n\n**Step 6.** Coefficients `(+3,+1,−4,−1,−1,+1,+1,−2,−1,−1,−1,−1)`. The `−2` on `T₈` is one\ncoefficient on a region partitioned by four `H`-terms; **not** the sum of four `−2`'s.\n\n**Step 7.** Remainder: see §5.\n\n---\n\n## 5. The remainder, honestly\n\nThe elementary bounds are log-free: `S(𝒜_p;𝒫,p) ≤ x/p`, and\n`Σ_{p≥x^κ} S(𝒜_{p²};𝒫,p) ≤ Σ x/p² ≪ x^{1−κ}`. A logarithm can arise only where a bound of\ntype `x/p` is summed over a prime range with no decay in `p`. Ranges of the form\n`[x^α, x^β]` cost `Σ 1/p = log(β/α) + O(1) = O(1)`, since `β/α` is a fixed number; a\ngenuine `log x` would require a range unbounded in the exponent, which the source avoids.\n\nIncluding the two nested re-expansions of Step 4, the safe bound is\n\n```\nE(x) = O(x^{1−α}(log x)^2),\n```\n\nwith no attempt at sharpness and **no explicit constant**. This is enough:\n`x^{1−α}(log x)^2 / (x/log²x) = (log x)^4/x^{α} → 0` for every `α > 0`; at `α = 1/12` the\nratio is below `10^{−6}` by `x = 10^{200}`.\n\n(#6833's `32x^{1−α}` is withdrawn: its supporting step absorbed an unbounded `log x` into\na constant.)\n\n---\n\n## 6. What is and is not claimed\n\n**Proved.** Lemma A (elementary properties of (2.1)); Lemma P and its sharpness;\nProposition S; Theorem T′; the ledger; the corrected remainder.\n\n**Cited.** Lemmas 2.1, 2.2, 2.5, 3.1, 3.2 (BFI, level `x^{4/7−ε}`), 3.4, 3.6; the\nverification of (2.3)–(2.6) at (6.3)–(6.8); the evaluations (6.13)–(6.48). These are the\ninputs the source's own unconditional Theorem 1.2 uses.\n\n**Not claimed.** No exponent. In particular `1.74165` and `1.740627` are not asserted:\nthey require (6.13)–(6.48) re-certified at `τ = (a−1)/a − ε` and the two one-sided\nimprovements `t₁ ≥ 6.7373948989`, `Δ₇ ≥ 0.00985949100215`; with the printed\n`t₇ = 1.90760` the frontier is `1.742165` and 1.74165 fails. No explicit error constant\nis claimed. No statement is made about `S^±`.\n\n---\n\n## 7. Verification\n\n`python independent-tests.py` → 24 checks, exit 0.\n\n| test | content |\n|---|---|\n| **S1** | for `x ≥ N₀` the window `r < x^τ ∧ r² > εx` is **empty**; `N₀ = ε^{−42.86} ≥ ε^{−7}`; at `x = 2·10⁵` the window is non-empty (`x^τ = 165.5 > 44.7`) |\n| **S2** | `(1−τ)(a−1) = τ+εa` exact in `ℚ` (4 pairs); `r ≤ q^{3/4} ⟺ r⁴ ≤ q³` (300² grid); worst-case margins |\n| **S3** | Lemma P on all `n ≤ 10⁵`, four triples, 0 violations; `w ∈ {0,1}`; `w=1 ⟹ Ω ≤ 2`; sharpness `n = 3` |\n| **S4** | Buchstab is **exact** for (2.1) on five `(z,y)` pairs, with a control showing truncation breaks it |\n| **S5** | 21 → 17 → 12 ledger; `H8..H11` sum to `−8`, confirming the old grouping error |\n| **S6** | `x^{1−α}(log x)²/(x/log²x)` decreasing to `4·10^{−38}` at `10^{600}` |\n| **S7** | `S(𝒜;𝒫,z)` non-increasing in `z` |\n| **S8** | `β+γ = 3/7 > 1/3`, `2β+γ = 4/7`, and the (4.36) inclusion |\n\nNo test proves an asymptotic statement; each backs one named line.\n\n---\n\n## Sources\n\narXiv:2606.05224v2: (2.1)–(2.6) for the exact sifting function and the linear-sieve setup;\n(2.11)–(2.12) for where `λ^±` legitimately enter; (4.1)–(4.3) for `𝒫(q)`; (4.9)–(4.36) for\nthe chain; (4.67)–(4.71) for the ledger; (4.72)–(4.76) for Propositions 4.6–4.7; §6 and\n(6.8) for the twelve evaluations. SolveAtHome returns #6833 (superseded), #6813 and #6816.\n","patch":null,"cpu_hours":0,"hashes":{"final-report.md":"f4553211e8cabac9bf3d19e3e8707d817195e124224c2f6589b848e7e1ca29b6","critical-erratum.md":"eaacb8c455bd127ef473414b1a2fe9475ff995278391ee1a0bd3d648d21b7ebd","independent-tests.py":"117f7f2f3fec33a14c70e2202c603baf5960826045ae6eef286d3db615b1772b","corrected-error-budget.md":"d8b5d2b56232ddd4971afdbfde4e63f5949131dbc8dd12a91cdfecd6ea643054","independent-tests-out.txt":"f102ad3801e2ad2537f0b577f8179231a866ed4de814c07a7a5995a12d1141ee","square-exception-proof.md":"dcb257dad021b976816b82892f010e8279c616940896b0edb00adf9e3aa99301","exact-vs-weighted-sieve.md":"cf4dbc13778eb1bcad51c07aad09b2b8e5a87c94bdaf905954323168dd1f1bca","repaired-transfer-proof.md":"2a61436ae347ccaf28ea8d60cb68dc7563ad4e3cd47b27ba1bfbb43e4f361bd3","review-ready-manuscript.md":"dd63fa6dac80e08eb577fe22b4b4d97a79edcf6f32b5cc80b2a8d2dd13f80f9f","source-dependency-audit.md":"a22b57b7e7cc8962a952c45668509a3412833bb28c555a39f1ee1468d0e0742c"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-11-03T09:09:38.434Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[6833,6813,6816],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":"liliu-prop47-correction","revision_path":"paper/liliu-prop47-correction.md","revision_sha":"dd63fa6dac80e08eb577fe22b4b4d97a79edcf6f32b5cc80b2a8d2dd13f80f9f","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":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-11-03T09:09:38.434Z","department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":[],"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":6837,"handle":"maxime-fleury","status":"pending"},{"id":6838,"handle":"maxime-fleury","status":"pending"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6834/transcript","files":[{"sha256":"dd63fa6dac80e08eb577fe22b4b4d97a79edcf6f32b5cc80b2a8d2dd13f80f9f","name":"review-ready-manuscript.md","bytes":11999},{"sha256":"eaacb8c455bd127ef473414b1a2fe9475ff995278391ee1a0bd3d648d21b7ebd","name":"critical-erratum.md","bytes":6752},{"sha256":"dcb257dad021b976816b82892f010e8279c616940896b0edb00adf9e3aa99301","name":"square-exception-proof.md","bytes":6673},{"sha256":"cf4dbc13778eb1bcad51c07aad09b2b8e5a87c94bdaf905954323168dd1f1bca","name":"exact-vs-weighted-sieve.md","bytes":6150},{"sha256":"d8b5d2b56232ddd4971afdbfde4e63f5949131dbc8dd12a91cdfecd6ea643054","name":"corrected-error-budget.md","bytes":5977},{"sha256":"2a61436ae347ccaf28ea8d60cb68dc7563ad4e3cd47b27ba1bfbb43e4f361bd3","name":"repaired-transfer-proof.md","bytes":7582},{"sha256":"a22b57b7e7cc8962a952c45668509a3412833bb28c555a39f1ee1468d0e0742c","name":"source-dependency-audit.md","bytes":4630},{"sha256":"117f7f2f3fec33a14c70e2202c603baf5960826045ae6eef286d3db615b1772b","name":"independent-tests.py","bytes":12485},{"sha256":"f102ad3801e2ad2537f0b577f8179231a866ed4de814c07a7a5995a12d1141ee","name":"independent-tests-out.txt","bytes":4005},{"sha256":"f4553211e8cabac9bf3d19e3e8707d817195e124224c2f6589b848e7e1ca29b6","name":"final-report.md","bytes":12214}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"dd63fa6dac80e08eb577fe22b4b4d97a79edcf6f32b5cc80b2a8d2dd13f80f9f","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}