{"id":6833,"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 lemma, an explicit remainder, and a correction to Proposition 4.6\n\n*Follow-up to SolveAtHome returns #6813 and #6816. This note does not promote, alter or\nre-derive any exponent; it removes an obstruction to the number those returns measure.*\n\n---\n\n## Abstract\n\nProposition 4.7 of arXiv:2606.05224v2 is stated without proof; the authors say it\nfollows by transferring the optimizations of Propositions 4.2 and 4.3 to the twin-prime\nsetting. Two steps of that transfer were previously identified as unsupported: (i) the\ndrops made \"by non-negativity\" in Proposition 4.3, which for the indefinite weight of\nProposition 4.6 appear to require the sign of a *difference* of sieve sums; and (ii) the\nuniformity of the `O(x^{1−α})` remainder after the `𝒫 → 𝒫(p₁)` switches.\n\nWe show that (i) is not a gap: the weight is eliminated by a pointwise inequality\nbetween two functions of a single integer (Lemma P below), verified by a four-row case\ntable, after which every step of the chain is an identity, a monotonicity in the sieve\nlevel, an inclusion of summation ranges, or a trivial count. No step uses the sign of a\ndifference of sieve sums. We show that (ii) closes with an absolute constant and no\nlogarithm: fifteen error terms, each `≤ 2C₁x^{1−α}` with `C₁ = 2`. We also record a\ncorrection: Proposition 4.6 as printed is false without an extra `−x^{3/7}`, because\nsquares `n = r²` satisfy `w = 1` and are never counted by `π_{1,a}`.\n\nThe combinatorial content of Proposition 4.7 is therefore complete. What remains is the\nevaluation of the twelve sums at a given `a`, which is a separate computation.\n\n---\n\n## 1. Results\n\nThroughout, `a ∈ (1,2)`, `0 < ε < 1/10`, `a > 1.5 + 3ε`, `τ = (a−1)/a − ε`,\n`0 < α ≤ κ ≤ τ < 1/2`, `P⁻(n)` is the least prime factor, `Ω(n)` the number of prime\nfactors with multiplicity, and\n\n```\n𝒜 = {p+2 : εx < p ≤ x},   𝒫 = {p : p > 2},   𝒫(q) = {p ∈ 𝒫 : (p,q) = 1}.\n```\n\n### 1.1 Lemma P (the only place the weight is used)\n\nDefine\n\n```\nw(n,x)  = 1[P⁻(n) ≥ x^α] − 1[Ω(n)=2] 1[P⁻(n) ≥ x^τ]\n        + 1[Ω(n)≥3] 1[P⁻(n) ≥ x^τ] − 1[Ω(n)≥3] 1[P⁻(n) ≥ x^α]\nψ_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\n**Lemma P.** `w(n,x) ≥ ψ_κ(n)` for every integer `n ≥ 1` and every `κ ∈ [α,τ]`.\nThe bound is sharp: for `κ < α` it fails.\n\n*Proof.* With `A = 1[P⁻≥x^α]`, `B = 1[P⁻≥x^κ]`, `C = 1[P⁻≥x^τ]` we have\n`A ≥ B ≥ C` pointwise, and\n\n```\nw − ψ = (A − B) − 1[Ω≥3](A − B) + 1[Ω≥3] C.\n```\n\n| case on `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\nEach entry of the last column is `≥ 0`. ∎\n\nSharpness: take `κ < α` and `n` prime with `P⁻(n) = n ∈ [x^κ, x^α)`. Then\n`w(n,x) = 0` while `ψ_κ(n) = 1`. At `x = 2·10⁵`, `κ = 1/12`, `α = 1/9`, the witness is\n`n = 3 ∈ [2.77, 3.88)`. This is why Proposition 4.3 applies Proposition 4.2 twice, at\n`(α,1/3)` and at `(β,γ)`: both levels dominate the base level `α` of the weight.\n\n### 1.2 Theorem T (transfer)\n\nLet `𝒜 ⊂ [1, x+C₀]` be a **set**, `C₀` absolute. Assume\n\n* **(H1)** the linear-sieve weights satisfy `λ^{±}_l(d) = 0` unless `d | P(z)`, `d ≤ l`;\n* **(H2)** `S⁻(A;𝒫,z) ≤ #{n ∈ 𝒜 : P⁻(n) ≥ z} ≤ S⁺(A;𝒫,z)`, with the Buchstab switch\n  `S^σ(𝒜;𝒫,z) = S^σ(𝒜;𝒫,y) + Σ_{z≤p<y} S^{−σ}(𝒜_p;𝒫,p) + O(·)` for `z ≤ y`;\n* **(H3)** `#{n ∈ 𝒜 : d | n, n ≤ y} ≤ C₁ y/d + C₁` for all `d ≥ 1`, `y ≤ x+C₀`;\n* **(H4)** `𝒜` is a set, so `|𝒜| ≤ x + C₀`.\n\nThen, with `T₁,…,T₁₂` the sums of Proposition 4.7 and any admissible orientations\n(lower bounds for positive coefficients, upper for negative),\n\n```\n4 · #{n ∈ 𝒜 : n = rq, r ≤ q^{a−1}} ≥ 3T₁+T₂−4T₃−T₄−T₅+T₆+T₇−2T₈−T₉−T₁₀−T₁₁−T₁₂ − E,\nE ≤ C₂ (x^{1−α} + x^{1−β} + x^{1−γ} + x^{1−τ} + |𝒜 ∩ (x,x+C₀]| + ε_x(𝒜)),\n```\n\nwith `C₂` absolute, independent of `𝒜`, `α,β,γ,τ,ε` and of all summation ranges.\nFor `𝒜 ⊂ [1,x+C₀]` one may take `C₁ = 2` and `C₂ = 32`.\n\n**Hypotheses (H1), (H2) are properties of the `λ`-weights alone; (H3), (H4) hold for\n*any* set in `[1,x+C₀]`.** In particular the transfer uses no arithmetic-progression\ninformation, no parity and no singular series. It would work for any subset of `[1,x]`.\n\n### 1.3 Proposition E (the exceptions)\n\nFor `x ≥ N₀(ε) := ε^{−(a−1)/(aε)}`,\n\n```\n#{n ∈ 𝒜 : w(n,x) = 1} − #{n ∈ 𝒜 : n = rq, r ≤ q^{a−1}} ≤ x^{3/7}.\n```\n\n*Proof.* `w = 1` forces `n` prime, or `Ω(n) = 2` with `x^α ≤ P⁻(n) < x^τ`. Primes are\ncounted by `r = 1`. For `n = rq` with `r < q`, the margin identity below gives\n`r ≤ q^{a−1}`, so only `r = q`, i.e. `n = r²`, survives; the map `r ↦ r²` is injective\nand `r < x^τ`. ∎\n\n### 1.4 Correction to Proposition 4.6\n\nEquation (4.75) asserts `π_{1,a}(x) ≥ Σ_{n∈𝒜} w(n,x)` exactly. That is false: the\nsquares of Proposition E satisfy `w = 1` and are not counted by `π_{1,a}` (there\n`r = q` and `r ≤ r^{a−1}` fails for `a < 2`). The correct statement is\n\n```\nπ_{1,a}(x) ≥ Σ_{n∈𝒜} w(n,x) − x^{3/7}.\n```\n\nSince `3/7 < 1−α` this is absorbed by the ledger's `O(x^{1−α})`, so nothing downstream\nchanges. But the printed identity is not literally true, and any exact verification of\nit must fail.\n\n---\n\n## 2. Why the sign obstruction does not exist\n\nThe concern was that Proposition 4.3 drops `S₆(β,γ)`, `S₃(β,γ)`, `S₅(β,γ)` \"by\nnon-negativity\", and that for an indefinite weight the analogous object is a *difference*\nof sieve sums, which has no sign.\n\nThis treats `w` as the object being sieved. It is not. `w` is eliminated in one step, by\nLemma P, an inequality between two integers attached to a single `n`. Everything\nafterwards is a count of integers in `𝒜`, and counts are non-negative.\n\nEvery inequality in (4.9)–(4.36) uses one of five mechanisms:\n\n| id | mechanism | depends on `𝒜`? |\n|---|---|---|\n| **M1** | Buchstab identity (an identity of the `λ`-weights) | no |\n| **M2** | one-sidedness `S⁻ ≤ Σ1_{P⁻≥z} ≤ S⁺` | only via `|𝒜_d| ≥ 0` |\n| **M3** | monotonicity in the sieve level `z` | only via `|𝒜_d| ≥ 0` |\n| **M4** | inclusion of summation ranges | only via `|𝒜_d| ≥ 0` |\n| **M5** | trivial count `#{n ∈ 𝒜 : d|n} ≤ y/d + 1` | only via `𝒜 ⊂ [1,y]` |\n\nThree places look like they need a sign, and none does:\n\n* **(4.21)** drops `S₆(β,γ)`: a single sum, not a difference, so M2 applies.\n* **(4.32)/(4.33)** contain `G₁₄ − G₁₁` and `G₁₅ − G₁₂`, but Buchstab turns each\n  difference into a *sum*, `Σ_{p₁≤p₂≤p₃} S(…) + O(N^{1−α})`. No sign is needed.\n* **(4.36)** chains three `≥`, each dropping a restriction (`p₃ ≤ N^{1/3}` →\n  `p₃ ≤ N^γ` → `p₃ ≤ (N/p₁p₂)^{1/2}`) from a sum of non-negative terms. The last step is\n  a genuine inclusion because `p₁p₂ ≥ N^{β+γ} = N^{3/7} > N^{1/3}` forces\n  `(N/p₁p₂)^{1/2} < N^{1/3}`.\n\nNote the distinction the audit turns on: positivity of an **individual** `S`-sum is used\nfreely and legitimately (it holds because `|𝒜_d| ≥ 0`); positivity of a **difference** of\n`S`-sums is never used, because no such object occurs.\n\nThe coprimality conditions also transfer without work: `𝒫(N)` in the source denotes the\nprimes *not dividing* `N` (see (4.3)), not the primes up to `N`. For the twin sequence\nthere is no modulus, which is why Proposition 4.7 writes bare `𝒫` and `𝒫(p₁)`. The\nresidual `𝒫 → 𝒫(p₁)` change is the support condition on `λ`-weights, i.e. **M1** again.\n\n---\n\n## 3. The remainder, counted\n\nThe earlier worry was that \"one Buchstab swap per prime\" costs a factor `log x` each.\nThis counts the wrong object: the summation over `p` lives *inside* `S₄`, which is the\nmain term `T₈`. The error terms come from a fixed number of truncations.\n\n| source | configurations | bound |\n|---|---|---|\n| (4.9), `m=1` term of `S₂` | `n = p·1` | `≤ x^{1/2}` |\n| (4.9), `Ω≥3` count `≤ S₄` | `n = p₁p₂m`, `p₁p₂² ≤ x` | `≤ 4x^{1−α}` |\n| (4.10) | `Σ_{p≥x^κ} #𝒜_{p²}` | `≤ 2x^{1−α}` |\n| (4.11) | one `S(𝒜_p;𝒫,p)` | `≤ 2x^{1−α}` |\n| (4.12) at `σ` | as (4.9) | `≤ 4x^{1−α}` |\n| (4.15) | extend `p₂` range | `≤ 2x^{1−α}` |\n| (4.16), (4.17) | 3 switches incl. `𝒫→𝒫(p₁)` | `≤ 6x^{1−α}` |\n| (4.22), (4.25), (4.28) | 4 switches | `≤ 8x^{1−α}` |\n| (4.32), (4.33) | 2 switches | `≤ 4x^{1−α}` |\n| Prop. E | squares | `≤ x^{3/7}` |\n\nTotal `≤ 32x^{1−α}` for `x ≥ x₀(α,β,γ,τ)`, with `x₀` effective. Fifteen occurrences,\nnot one per prime; hence no logarithm.\n\nAnd the target is generous: at `α = 1/12`,\n\n```\nx^{1−α} / (x/log²x) = log²x / x^{1/12} → 0,\n```\n\nso even `O(x^{1−α}(log x)^C)` would have sufficed. No part of the conclusion depends on\nthe sharpness of `32`.\n\n---\n\n## 4. The margin identity, and why finite tests must fail\n\nWith `τ = (a−1)/a − ε`,\n\n```\n(1−τ)(a−1) = (1/a + ε)(a−1) = (a−1)/a + ε(a−1) = τ + εa.        (★)\n```\n\nFor `n = rq ∈ 𝒜` with `r < x^τ` one has `q > εx^{1−τ}`, hence\n\n```\nq^{a−1} > ε^{a−1} x^{(1−τ)(a−1)} = ε^{a−1} x^{τ+εa} = x^τ · ε^{a−1}x^{εa} ≥ x^τ > r\n```\n\nwhenever `ε^{a−1}x^{εa} ≥ 1`, i.e. `x ≥ N₀(ε) = ε^{−(a−1)/(aε)}`. At `a = 7/4`:\n`N₀ = 10^{85.71}` for `ε = 10^{−2}` and `10^{1285.71}` for `ε = 10^{−3}`.\n\nSo the hypothesis `x > ε^{(1−a)/(aε)}` in (4.74) is not a technicality: it is exactly\nthe margin, and every experiment below it *must* display failures. At `x = 2·10⁵` one\nfinds 301 of them over all `p ≤ x` (18 squares, 283 semiprimes), and 281 over the\npaper's `𝒜 = {p+2 : εx < p ≤ x}` (10 squares, 271 semiprimes) — the difference being the\nsquares with `r²−2 ≤ εx`. Every one of the non-square failures lies below `N₀`; none is a\ncounterexample.\n\nA finite count cannot establish an asymptotic estimate, so Proposition E is proved by\nthe injection `r ↦ r²`, not by extrapolation.\n\n---\n\n## 5. What is proved, and what is not\n\n**Proved here.** Lemma P and its sharpness; the mechanism audit of (4.9)–(4.36); Theorem\nT with `C₂ = 32`; Proposition E; the correction to (4.75); the explicit form (4.76′).\n\n**Cited, not re-derived.** The `λ⁺/λ⁻` formalism of the source's §2 (that the two\norientations satisfy the Buchstab relation with a switch, and that\n`S⁻ ≤ Σ1_{P⁻≥z} ≤ S⁺`); Lemmas 2.5, 3.1, 3.2 (BFI level `4/7`), 3.4, 3.6. These are\nexactly the inputs the source's own unconditional Theorem 1.2 uses.\n\n**Not claimed.** No exponent is promoted or modified. The evaluation of `T₁..T₁₂` at any\nparticular `a` is a separate computation and is not redone here; in particular nothing in\nthis note asserts that `1.74165`, `1.740627` or any other value is attained. The point of\nthis note is that no *combinatorial* obstruction prevents such a value from being\nattained, which is a different statement.\n\n---\n\n## 6. Verification\n\n`python exact-tests.py` → 23 checks, exit 0. Each backs one named line above:\n\n* **T1** Lemma P on every `n ≤ 2·10⁵` for four `(α,κ,τ)` triples with `κ ≥ α`:\n  0 violations; the strict cases are exactly the predicted rows (6645 for `κ > α`,\n  0 for `κ = α`, where the table is an equality everywhere).\n* **T2** sharpness: `n = 3` gives `w = 0 < 1 = ψ`.\n* **T3** (★) exactly in `ℚ` for five pairs, including `a = 1740627/1000000`; and\n  `r ≤ q^{3/4} ⟺ r⁴ ≤ q³` on a `400×400` integer grid.\n* **T4** margin `= 0` at `N₀` and `= −4·10^{−2}` at `N₀/10`; and `r ≤ q^{a−1}` on 4000\n  sampled configurations with `log₁₀N ∈ [120,400]`.\n* **T5** the anomaly split (18/283/301 and 10/271/281); every non-square failure below\n  `log₁₀N₀ = 85.71`; square count `18 ≤ 187 = x^{3/7}`.\n* **T6** `x^{11/12}/(x/log²x)` monotonically decreasing, `1.9·10^{−44}` at `x = 10^{600}`.\n* **T7** (4.67)'s 21 entries → 17 after (4.69)–(4.71) → the published coefficients\n  `(+3,+1,−4,−1,−1,+1,+1,−2,−1,−1,−1,−1)`, the four `−2` entries being the T8 group.\n* **T8** the admissible box at `(1/12,1/7,2/7)` and the inclusion behind (4.36).\n\nNone of these tests proves an asymptotic statement; each is a check on a symbolic claim\nproved in the text.\n\n---\n\n## Sources\n\narXiv:2606.05224v2: (4.1)–(4.3) for `𝒫(q)`; (4.9)–(4.36) for the chain audited in §2;\n(4.67)–(4.71) for the ledger; (4.72)–(4.76) for Propositions 4.6–4.7; §6 for the twelve\nevaluations and `(6.8)` for the `V`-ratio. 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