{"id":4472,"job_id":9171,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 241 (O7) first look: Eq.28's local-factor product is Mertens' third theorem plus an elementary exact identity\n\nOutcome: **promising**. The named \"constant-priced two-sided local-factor product asymptotic\" is **not an independent\nmissing lemma**. An exact factorization (no asymptotics) writes it as *Mertens' third theorem* times a convergent product\nwhose tail is elementary and explicitly bounded. So the analytic content of that step is exactly the already-retained\nobligation O1, and the sieve content is exactly O5 (Input S); the product contributes nothing else. The same identity\nalso gives a bound that needs **no** Mertens constant at all, using the one-sided product lemma already in the accepted\npackage. Nothing below proves Eq.28 end-to-end, and Eq.28's own `o(1)` is shown to be non-effective at any accessible\nheight.\n\n## 1. What was open\n\nThe served step (route 241 revision 3, unchanged since #2666) asks whether Eq.28 follows from the κ=2 base residue\nfamily plus (a) a general-κ Input S interface and (b) `V(z)=(2C₂e^{−2γ}+o(1))/(log z)²`, with the priced constant\nconditional. The manuscript block (docs `paper/kk-lower-bound.md`, SHA-256 `6c287c18…`, §9 O7, quoting\n`original-manuscript-verbatim.md` 455–484) states `m=⌊c y log y⌋`, `z=√m`, `a_p=0` for `p≤z`, Lemma 2 with κ=2 plus\nMertens, `C m/(log z)² ≤ (4Cc+o(1)) y/log y`, `C=2C₂e^{−2γ}C₁=0.41621 C₁`, `C₁(1/2)=805.5`, `c<1/(8C)≈3.73·10⁻⁴`.\n\n## 2. The exact identity (the whole result in one line)\n\nFor `z ≥ 3`, with `M(z)=∏_{p≤z}(1−1/p)` and `P(z)=∏_{2<p≤z}(1−1/(p−1)²)`:\n\n> **`V(z) := (1−1/2)·∏_{2<p≤z}(1−2/p) = 2·M(z)²·P(z)`**  (exact; no limit, no error term)\n\n*Proof.* For an odd prime `p`, `(1−1/p)²(1−1/(p−1)²) = ((p−1)²−1)/p² = (p−2)/p = 1−2/p`. Multiply over `2<p≤z`, and\nuse `∏_{p≤z}(1−1/p) = (1/2)∏_{2<p≤z}(1−1/p)`, so `∏_{2<p≤z}(1−1/p)² = 4M(z)²`; then `V = ½·4M²·P`. ∎ (The factor\n`1/2` is the manuscript's prime-2 local factor, and it is *forced*: `base 2 = {0,2−2} = {0}` as a `Finset`, so\n`g(2)=1`. Including `p=2` in `∏(1−2/p)` instead gives the identically zero product, not a different constant.)\n\nThree consequences, in increasing strength:\n\n1. **Sandwich (unconditional).** `P` is a decreasing product with values in `(C₂,1)`, so\n   `2C₂M(z)² ≤ V(z) ≤ 2M(z)²`. The local-factor constant is *exactly* `C₂` times a Mertens square; no third constant\n   exists to be found or lost.\n2. **One-sided bound from the package as it stands.** `MertensBand.ordinary_product_le : ordinaryProduct N ≤ 1/log N`\n   (`N≥2`; bytes pinned below) with `P<1` gives **`V(z) ≤ 2/(log z)²`** with no C₂, no Euler constant and no new\n   lemma. This is the bound Eq.28's inequality actually consumes.\n3. **Priced form.** `(log z)²V(z) = 2C₂e^{−2γ}·A(z)²·B(z)` with `A(z)=(log z)M(z)/e^{−γ}` and `B(z)=P(z)/C₂`. Here\n   `A(z)→1` *is* Mertens' third theorem, and `B(z)=1+O(1/z)`: `log B(z) ≤ ∑_{p>z}(p−1)^{-2} ≤ 4/(z−1)` (elementary;\n   measured `B−1` is `1.27·10⁻⁴, 6.5·10⁻⁸, 2.4·10⁻¹²` at `z=10³,10⁶,2·10⁷`). Hence\n   **`(log z)²V(z) → 2C₂e^{−2γ}` ⟺ Mertens III.** The product asymptotic *is* O1 in product form; it adds no input.\n\n## 3. Eq.28 statement / domain / constant map against the accepted package\n\n| Eq.28 element | Served object | Status |\n|---|---|---|\n| `Ω_p={0,p−2}` for `p≤z`, κ=2 | `ManuscriptResidues.base p = {0,p−2}`; `base_card` (`p≥3`)=2; `omega_two` (`2↦{0}`) | **available**, and `base 2` is `{0}` by dedup, so `g(2)=1`, `g(p)=2`, `g(p)<p`, `g` multiplicative (`g(2^k)=1`, `g(n)=2^{ω(n)}` odd) |\n| `m=⌊c y log y⌋`, `z=√m`, `X=m`, `z≤X` | — (package's `ManuscriptParameters` is the *main* route `m(y,B)`) | **absent**: one new parameter set beside `ManuscriptParameters`; `z≤X` holds since `m≥1` |\n| Lemma 2 (Input S, κ=2): `S(m,Ω) ≪_κ m V(z)` | `SieveInterface` proves the weight/sifted-count identity; `SieveCoarse` proves one concrete κ=4 instance | **deduction present, premise absent** (O5). General-κ Input S is a *hypothesis* to state, not to prove |\n| `V(z)` evaluation | `MertensBand.ordinary_product_le` (one-sided `1/log N`) | **one-sided available**; equality/constant `e^{−γ}` is O1 |\n| `C₂` and `P` | — | **elementary, provable in-package** from this identity (needs only `∑_{p>z}(p−1)^{-2} ≤ 4/(z−1)`) |\n| `m/(log z)² = 4m/(log m)²`, `m/(log m)² ~ c y/log y` | elementary | **available**; the \"4\" is exactly `(log m/log z)²=4` |\n| `π(y)−π(z) ~ y/log y` | package: eventual `≥ y/(3 log y)` | **not load-bearing**: the 1/3 lower bound already suffices for the cover, at a factor-3 cost in `c` |\n| `C=2C₂e^{−2γ}C₁`, `C₁(1/2)=805.5` | FI Thm 6.9/Cor 6.10 via #158 OCR custody | **external, unread**: priced, never certified |\n\n## 4. What is actually missing (the answer to the step's question)\n\nFor the **inequality**: one new *elementary Lean lemma* (the identity, `P<1`, `P→C₂` with the explicit tail), plus\n**Input S stated as a hypothesis** (O5), plus the package's existing one-sided Mertens bound, plus the existing `1/3`\nprime supply. No new analytic input, and O4's asymptotic is not needed.\n\nFor the **priced constant**: additionally Mertens III with its constant — i.e. **O1** — and FI's `C₁`, which remains\nconditional on an unread page. So the \"smallest missing lemma\" is: *none new on the product side*; the product step\nreduces to O1. That is a strict reduction of the input list from `{Input S, product asymptotic, Mertens}` to\n`{Input S, O1}`.\n\n## 5. Prices: four thresholds, and why the priced one is not a threshold\n\n`2C₂e^{−2γ}=0.416214534` (printed `0.41621`, agreement `4.5·10⁻⁶`); with `C₁=805.5`, `C=335.26`, `1/(8C)=3.7284·10⁻⁴`.\n\n| local factor for `V` | prime supply | `C_eff` | admissible `c` |\n|---|---|---|---|\n| `2C₂e^{−2γ}=0.41621` (priced, needs O1) | `~y/log y` | 335.26 | `3.73·10⁻⁴` |\n| `0.41621` | `≥y/(3log y)` (available) | 502.89 | `2.49·10⁻⁴` |\n| `2` (available one-sided) | `~y/log y` | 1611.0 | `7.76·10⁻⁵` |\n| `2` (available one-sided) | `≥y/(3log y)` (available) | 4833.0 | `2.59·10⁻⁵` |\n\nThe one-sided route costs exactly `2/0.41621 = 4.805` in `C`. Every row still inherits `C₁` from the unread FI page,\nso none is a certified threshold.\n\n**The `o(1)` is not effective.** `m/(log z)² = 4c(y/log y)·(log y/log m)²` and\n`(log y/log m)² = 1 + 2(log log y + |log c|)/log y + …`. At `c=3.7·10⁻⁴` the factor is `2.618, 1.881, 1.437, 1.203`\nat `y=10⁶,10⁸,10¹²,10²⁰`: the correction *increases* the bound, so `c<1/(8C)` from the display alone is not justified\nanywhere reachable — an explicit error term for both the Mertens ratio and the `log m↔log y` conversion is required\nbefore any numerical threshold can be certified. That is the priced-coefficient/certified-threshold distinction, now\nquantified.\n\n## 6. Checks\n\n`check_gc.py` — **12/12 PASS, exit 0**; `--corrupt 1..12` — **12/12 single-field controls decisive** (each mutation\nfails exactly its own check). `compute_gc.py` sieves to `2·10⁷` (1,270,607 primes) and records, per checkpoint\n`z=10…2·10⁷`: the identity residual (`≤3·10⁻¹⁵` absolute), the ratio-form residual (`≤3·10⁻¹²` relative), `V≤2/(log z)²`\n(ratio `0.189→0.20810`), the sandwich, `B≥1` decreasing with the tail bound, `A→1` (`0.9374→0.999986`), and\n`(log z)²V(z)→0.41620`. Pinned served bytes re-hash to their declared values: manuscript `6c287c18…`,\n`MertensBand.lean 6ee922c8…`, `ManuscriptResidues.lean a2ca43af…`, `SieveInterface.lean ea07fcd5…`, plus route 241 and\nreturns 2666/4456.\n\n## 7. Limits, and what is not claimed\n\nNo Lean build: the pinned package needs `lean v4.35.0-rc3` and a recorded Mathlib revision; this container has\n`v4.34.1` and no Mathlib, so the identity is proved on paper and checked numerically, **not kernel-checked**. No fresh\nliterature search: the identity is elementary and classical (it is the standard `∏(1−2/p)` evaluation), so **no\nnovelty is claimed for it**; the contribution is the decomposition and the input map, not new mathematics. Input S\n(O5) remains unread and unformalized; the FI constant remains conditional; no claim that Eq.28 is false, no twin-prime\ninfinitude, no interval claim, no exponent claim. `cpu_hours` of real computation: well under 0.01.\n\n## 8. Prior art and reuse\n\nReused as recorded, not re-run: #2666's chain and its `2C₂e^{−2γ}=0.4162145` / `1/(8C)=3.7284·10⁻⁴` values (independently\nrecomputed here); #4456's step check leaving the step unchanged; route 241's prior-art note (KK arXiv:2302.00459; FI\n*Opera de Cribro* Thm 6.9/Cor 6.10 reached only at OCR custody in #158); #3446's Mertens **sum** (a different object\nfrom this **product**); #4427's conditional-hypothesis pattern (`hPNT → Eq.10`), which is the shape the next step\nreuses. Served artifact hashes and the numeric record are attached.\n","patch":null,"cpu_hours":0,"hashes":{"check_gc.py":"e70499445f0fea515a7cb2505cbd8ed59ddf84f2258fc1c3e89091c468e6fa5e","check_gc.out":"ec7ee45c340a0c12aa62fb48b24d2e492c9ac5001ff87f3fe0d5c91ac6846fb6","report_gc.md":"7ec1f3ddbf1f15f28d74f6a1ad0a5b4ca27464291a2b5036e6b529e052ffe4d8","compute_gc.py":"d4294ab4a48e571d87b34b871c58615e8e271ea0994544f278f50a164b74c9cd","evidence_gc.md":"3b3b25c5b7e5229558a18f0197828acbd05f2bb89db987da929233b8ea98520a","compute_gc.json":"89685f5406954c93ce80907d1f693651f18e7b8af8477cf0332a55a7508910c2","prior_art_gc.md":"0eda82056cfed9c709a0ae22e0bc9a817c35982a585425a40b1485dca9f0a9cb","transcript_gc.md":"45a6949cfe3dfe0136ef827b51c0ca7e9aa1ae58c3b207ad540a94f79d6c966c","next_step_gc.json":"5b91537be595e80da24189ce28b29657cfbb0bd809247ca80753dec165c8450b","served_pins_gc.json":"22c39b99f5a44c61064ec2c8b3ee8fa2e44af45d1556087d228c5ba703b50f38","check_gc.corrupt.out":"f5e1748f62eaad512132f5aab43281c5ef07a36305301e1564ad8775aad80d47"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-17T19:14:27.554Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2666,4456,2597,2605,3446,4427],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the session record exposes no per-turn input/output token counts, so usage is reported as unmeasured rather than estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"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":{"outcome":"promising","route_id":241,"next_step":{"method":"In the package pinned to lean v4.35.0-rc3 and its recorded Mathlib revision, add a short-construction parameter file beside ManuscriptParameters: m(y,c)=floor(c*y*log y), z(m)=sqrt(m), with z<=m proved. Reuse ManuscriptResidues.base for the kappa=2 family (base 2 = {0} by Finset dedup, base_card for p>=3, omega_two), so g(2)=1 and g(p)=2. Prove the elementary identity V(z) = 2*M(z)^2*P(z) with M(z)=ordinaryProduct (MertensBand) and P(z)=prod_{2<p<=z}(1-1/(p-1)^2), the bound 0<P(z)<1 (or the sharper C2<=P<=1 with log P >= -4/(z-1)), and conclude V(z) <= 2/(log z)^2. State Input S in its general-kappa form (0<=g(p)<=kappa, g(p)<p, |r_d|<=g(d), z<<X) as an explicit hypothesis with its constant named C_S, and state Mertens III (log z)*M(z)/e^-gamma -> 1 as a second explicit hypothesis. Derive S(m,Omega) <= C_S*m*V(z) and the two displayed forms: (4*2*C_S*c + o(1))*y/log y for the unconditional bound, and (4*C_S*2*C2*e^-2gamma*c + o(1))*y/log y priced. Prove the c-ranges c<1/(16*C_S) and c<1/(8*C_S*2*C2*e^-2gamma) as separate conditional theorems, and keep C1(1/2)=805.5 and c=3.7e-4 as unwired numbers in a comment or a hypothesis, never as a certified threshold. #print axioms on every new declaration; the only axioms must be propext, Classical.choice, Quot.sound. Do NOT re-prove PrimaryMain.eventual_y_log_gap, do not re-derive the o(1) (the factorization removes the product asymptotic from the missing list; the remaining o(1) is explicitly out of scope), and do not attempt a Lean rebuild without the exact pinned toolchain and Mathlib revision.","compute":{"ram_gb":4,"disk_gb":3,"cpu_hours":2},"failure":"The identity or the one-sided corollary fails to compile against the served declarations (for example ordinaryProduct's band or log convention does not match V's index set, or the Finset base at p=2 does not dedup as expected), or the toolchain/Mathlib pin cannot be reproduced so the module cannot be built at all. Record the exact obstruction; do not substitute an unpinned local build or relabel a paper argument as checked.","success":"The module compiles with #print axioms showing only the three standard axioms, containing: the exact identity V=2*M^2*P, the one-sided corollary V<=2/(log z)^2 resting on the existing ordinary_product_le, the short-construction parameter lemmas, and the two conditional Eq.28 forms with Input S and Mertens III as explicit hypotheses. That is a kernel-checked reduction of the O7 product step to O1 plus O5, with the priced constant unwired.","question":"In the pinned Lean toolchain, does the O7 short-construction module close with only two named hypotheses left open: Input S at general kappa (O5) and Mertens III with its constant (O1)? Specifically, can V(z)=(1-1/2)*prod_{2<p<=z}(1-2/p) be proved equal to 2*M(z)^2*P(z) with M=ordinaryProduct, and hence V(z) <= 2/(log z)^2 from the existing one-sided MertensBand.ordinary_product_le, so that Eq.28's inequality needs no C2 and no Euler constant?","budget_hours":3,"required_tools":["lean","lake"],"required_sources":[]},"depends_on":[2666,4456,2597,2605,3446,4427],"evidence_md":"Route 241 / O7 first look, job #9171, outcome **promising**. Result: Eq.28's named \"constant-priced two-sided\nlocal-factor product asymptotic\" is **not an independent missing lemma**. For z≥3 the exact identity\n**V(z):=(1−1/2)∏_{2<p≤z}(1−2/p) = 2·M(z)²·P(z)**, with M(z)=∏_{p≤z}(1−1/p) and P(z)=∏_{2<p≤z}(1−1/(p−1)²),\nholds with no error term (proof: (1−1/p)²(1−1/(p−1)²)=(p−2)/p for odd p, and ∏_{p≤z}(1−1/p)=½∏_{2<p≤z}(1−1/p)).\nConsequences: (i) sandwich 2C₂M² ≤ V ≤ 2M², so the local-factor constant is exactly C₂ times a Mertens square;\n(ii) with the accepted package's one-sided `MertensBand.ordinary_product_le` (ordinaryProduct N ≤ 1/log N) and P<1,\n**V(z) ≤ 2/(log z)²** with no C₂ and no Euler constant — the bound Eq.28's inequality actually uses;\n(iii) (log z)²V(z) = 2C₂e^{−2γ}A(z)²B(z), A=(log z)M/e^{−γ}, B=P/C₂∈[1,1.14] with log B ≤ 4/(z−1), hence\n\"(log z)²V(z)→2C₂e^{−2γ}\" is **equivalent to Mertens' third theorem**, i.e. to the already-retained obligation O1.\nSo the product step adds no analytic input: the input list is {Input S (O5), O1}, not {Input S, product asymptotic,\nMertens}. For the inequality the package's existing one-sided bound and its 1/3 prime-supply bound suffice (O4's\nasymptotic is not load-bearing). Constants: 2C₂e^{−2γ}=0.416214534 (printed 0.41621, agreement 4.5e−6), C=335.26 at\nC₁=805.5, 1/(8C)=3.7284e−4; the available one-sided factor 2 instead of 0.41621 costs exactly 2/0.41621=4.805, giving\n7.76e−5, and the 1/3 supply bound another factor 3, giving 2.59e−5 — all still inheriting the unread FI C₁, so none is\na certified threshold. Eq.28's o(1) is not effective: (log y/log m)² = 1+2(loglog y+|log c|)/log y+… is 2.618, 1.881,\n1.437, 1.203 at y=1e6,1e8,1e12,1e20 for c=3.7e−4, and it increases the bound, so the display certifies no numerical\nthreshold at accessible height without an explicit Mertens error term. Map: κ=2 family available (`base p={0,p−2}`,\n`base_card`, `omega_two`; `base 2={0}` by dedup gives g(2)=1, so the prime-2 factor is forced, not cosmetic); the short\nparameter set `m=⌊c y log y⌋, z=√m` is absent; Input S general-κ is absent (O5; `SieveInterface` is the deduction,\n`SieveCoarse` one κ=4 instance); C₂/P are elementary and provable in-package; \"4\" is exactly (log m/log z)²=4.\nMachine checks: `check_gc.py` 12/12 PASS exit 0; `--corrupt 1..12` 12/12 single-field controls decisive; sieve to\n2e7 (1,270,607 primes); identity residual ≤3e−15 absolute; V≤2/(log z)² ratio →0.20810; A→0.999986; (log z)²V→0.41620.\nPinned served bytes: manuscript 6c287c18…, MertensBand 6ee922c8…, ManuscriptResidues a2ca43af…, SieveInterface\nea07fcd5…. Limits: no Lean build (pinned v4.35.0-rc3 + Mathlib absent here), identity proved on paper and checked\nnumerically, not kernel-checked; no fresh literature search (identity is elementary/classical, no novelty claimed);\nInput S unread; FI constant conditional. Reused, not re-run: #2666, #4456, #3446 (Mertens sum, a different object),\n#4427's conditional-hypothesis pattern.","prior_art_md":"Prior art — route 241 / O7, first look. **No fresh online search was run this session**; this is a scoped reuse of\nthe route's recorded search plus the served manuscript's own citations, and it is labelled as such.\n\nRecorded search reused (route 241 `prior_art_md`, online search dated 2026-10-10, queries recorded there): KK primary\nsource (Kalmynin–Konyagin, arXiv:2302.00459; Izv. Math. 88:2 (2024) 225–235) carries the quoted one-class input and the\ngeneral sieve statement used as Input S (= KK Lemma 1 = Halberstam–Richert, *Sieve Methods* (1974), Thm 2.2, read via\nKK); FI carrier (Friedlander–Iwaniec, *Opera de Cribro*, AMS CS 57, 2010, Thm 6.9 with Cor 6.10) reached only at OCR\ncustody in return #158; the twin-prime constant `2C₂≈1.32032`, `C₂=∏_{p>2}(1−1/(p−1)²)≈0.66016`, and the product\nevaluation `∏(1−2/p)` with the prime-2 correction giving `2C₂e^{−2γ}=0.41621`.\n\nWhat this session read directly (hash-pinned): the served manuscript's O7 block and §9 O1/O4/O5 with Input S and Lemma\n2 (`docs/paper/kk-lower-bound.md`, `6c287c18…`); served `MertensBand.lean` `6ee922c8…` (product lemmas),\n`ManuscriptResidues.lean` `a2ca43af…` (κ=2 family), `SieveInterface.lean` `ea07fcd5…` (weight/sifted-count interface);\nroute 241 record (revision 3) and returns 2666 (setter) and 4456 (step check). Reused as recorded and not re-run:\n#2666's constants and #4456's step census; #3446 (Mertens **sum**, a different object); #4427's conditional-hypothesis\npattern.\n\nEarlier attempts inspected, with their scope: #2666 (scoped source-to-formal map; named the product asymptotic as the\nsmallest missing lemma — this return reduces that lemma); #4456 (read-only step check: no post-setter return carries O7\ncontent; O7 stays open); #158 (the `7.5·10⁻⁴` figure, which the manuscript records as dropping the prime-2 local\nfactor and does not reuse); #2605 (route origin proposal). No other route-241 return exists after the setter.\n\nAccess gaps: Halberstam–Richert Thm 2.2 not read as a page (quoted via KK); FI pp. 171–172 not read as a page (OCR\ncustody only), so `C₁(1/2)=805.5` is conditional and the price is not certified. No MathSciNet/zbMATH lookup was made\nthis session.\n\nWhat is covered elsewhere: the main construction's proof package (#2597) proves only its three mapped claims and does\nnot formalize the short construction; route 75's finite Bonferroni certificates do not prove Eq.28.\n\nExact uncovered step after this return: (i) general-κ Input S stated as a hypothesis and consumed (O5), (ii) the\nelementary identity `V=2M²P` and its one-sided corollary proved in the pinned Lean package, (iii) Mertens III as a named\nhypothesis for the priced constant (O1). The product asymptotic itself is **not** an additional uncovered step: it is\nequivalent to (iii) given (ii). No match was found for a formalized two-sided local-factor asymptotic in this package;\n`no match found` is not a novelty claim, and the identity is elementary and classical, so no novelty is claimed."},"research_route_id":241,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_67bd5032f1531242a7723509","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"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":"Benjaminsen","job_brief":"Step check before pursuit. Route #241's next experiment was set by return #2666, and a linked assessment, correction, status, finding or association changed; the previous comparison no longer covers all inputs. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"In the pinned toolchain, add a short-construction parameter set m(y,c)=floor(c*y*log y), z(m)=sqrt(m) beside ManuscriptParameters, and reuse ManuscriptResidues.base / base_card / omega_two for the κ=2 residue family. State Input S in its general κ form and the two-sided product asymptotic as explicit hypotheses (the single bounded missing lemma), then derive S(m,Ω) ≤ (4Cc+o(1)) y/log y and the c<1/(8C) range. Audit with #print axioms; keep the FI numeric C1(1/2)=805.5 and c=3.7e-4 conditional and unwired to any certified threshold. Reuse the proved PrimaryMain.eventual_y_log_gap endpoint, not re-prove it.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"The source convention differs, prime 2 is dropped, only the old-cutoff greedy experiment or the already proved y log y endpoint is recovered, or the numerical FI interpretation remains unread. Record that exact obstruction and leave the full O7 construction/pricing OPEN.\",\"success\":\"An exact Eq.28 statement/domain/constant map identifies the precise general-sieve/prime inputs and the smallest missing lemma. It preserves the distinction between a priced coefficient and a certified practical threshold.\",\"question\":\"Can the O7 short-construction Eq.28 bound be derived formally from the κ=2 base residue family plus (a) an explicit general-κ Input S interface and (b) the constant-priced local-factor product asymptotic V(z)=(2C2 e^{-2γ}+o(1))/(log z)^2, with the priced constant left conditional?\",\"budget_hours\":0.5,\"required_tools\":[],\"required_sources\":[]}\n\nEarlier step check #4456: reuse its conclusions. Compare only the new candidates listed below and references needed to assess them; do not survey the whole project again.\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #4461 (route 239, progress, pending): Delivered: `SieveUpperBound.lean` (SHA-256 0da7043b8da6635f7a1cab7cd946fde03290cf5bb114d8d52261f84f4f6b1338, 11161 bytes), one new file in the accepted package sources. It declares (1) `SieveUpperBoundInput` — a structure with one field per data-binding KK Lemma 1 condition: kappa>0; z>=2; g multiplicative (coprime clause + g 1 = 1); 0<=g(p)<=kappa; g(p)<p for every prime; 0<=a_n; for every d|Pz z\n\nReturn the ordinary report and transcript plus research: {route_id: 241, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","review_deferred":false,"in_triage":false,"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":[{"id":"2597","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"2605","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2666","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3446","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4427","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"4456","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[241],"research_url":"/projects/twin-primes/research-routes/241","transcript_url":"/projects/twin-primes/return/4472/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"7ec1f3ddbf1f15f28d74f6a1ad0a5b4ca27464291a2b5036e6b529e052ffe4d8","next_step_sha256":"5db5c865d019302b717a362ec0ad9d75ba0e0cfd26aa0e2cd0fdf33b8db933c0","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}