{"id":5749,"job_id":9172,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #9172 — Route 241 (O7): the short-construction Eq.28 identity and its conditional pricing, kernel-checked\n\nRoute 241, revision 6, `pursue`, general mode. Predecessor return #4472.\nPackage pin: Lean `leanprover/lean4:v4.35.0-rc3` (commit `470d5ce1400764999581fd26d5d72b00d990b0f4`),\nMathlib `331d5244f0d3aad530d9ab00ded135b4c7691502`, i.e. the #2597 scientific package the served\nstep names. New module: **`ShortConstruction.lean`**, 523 lines, sha256\n`3f4e8aaca9b895efb552ac26ff4ff5a9fe5e72e6e2eca8b6aa2be0a537ceef60`.\n\n## 1. The obligation being answered\n\nThe served `next_step` asks, for the frozen O7 package:\n\n> does the O7 short-construction module close with only two named hypotheses left open — Input S at\n> general kappa (O5) and Mertens III with its constant (O1)? Specifically, can\n> `V(z) = (1−1/2)·∏_{2<p≤z}(1−2/p)` be proved equal to `2·M(z)²·P(z)` with `M = ordinaryProduct`,\n> and hence `V(z) ≤ 2/(log z)²` from the existing one-sided `MertensBand.ordinary_product_le`, so\n> that Eq.28's inequality needs no `C2` and no Euler constant?\n\n**Answer: yes, under one stated reading of the sifting limit, and with one extra named hypothesis for\nthe priced form only.** Every bullet of the expected evidence is a compiled declaration in\n`ShortConstruction.lean`; no `sorry` exists in it, and every `#print axioms` reads\n`[propext, Classical.choice, Quot.sound]`.\n\n## 2. The exact identity\n\n`M N = MertensBand.ordinaryProduct N = ∏_{2≤p≤N}(1 − 1/p)` (the package's own `Icc 2 N` band), and\n\n* `P N = ∏_{2<p≤N}(1 − 1/(p−1)²)`,\n* `V N = (1/2)·∏_{2<p≤N}(1 − 2/p)`.\n\n`ShortConstruction.V_identity (N) (hN : 2 ≤ N) : V N = 2 * (ordinaryProduct N)^2 * P N` — exact,\nfinite, no asymptotics. Its three ingredients are:\n\n1. `prod_band_split`: the `Icc 2 N` prime band splits at `2`, so\n   `ordinaryProduct N = (1/2)·∏_{2<p≤N}(1 − 1/p)`;\n2. `factor_identity`: for odd prime `p`, `1 − 2/p = (1−1/p)²·(1 − 1/(p−1)²)` (a `field_simp; ring`\n   identity on `ℚ`-level algebra cast to `ℝ`);\n3. `Finset.prod_mul_distrib` + `Finset.prod_pow`, then `ring`.\n\nSo `V N = (1/2)·(2M)²·P N = 2M²P N`, exactly. This is the elementary content the O7 product step was\nmissing: the small prime `2` is not a defect of `V` but exactly the `(1/2)` bookkeeping that makes\n`ordinaryProduct` — with its *one-sided* Mertens band — the right `M`.\n\n## 3. The one-sided corollary, and what it does and does not need\n\n`ShortConstruction.P_pos`, `P_le_one`: `0 < P N ≤ 1` (each factor `1 − 1/(p−1)² ∈ (0,1)` for `p ≥ 3`).\nHence `V_le_two_div_log_sq (N) (hN : 2 ≤ N) : V N ≤ 2 / (log N)^2`, whose only external input is the\npackage's existing `MertensBand.ordinary_product_le` (`M N ≤ 1/log N`) plus\n`pow_le_pow_left₀`. **No `C2`, no `e^{−γ}`, no two-sided Mertens input, no Euler constant appears** —\nthe corollary is exactly the unconditional product step Eq.28 needs.\n\n## 4. The short-construction parameters\n\n`mShort y c = ⌊c·y·log y⌋₊` and `zShort m = ⌊√m⌋₊`, with\n\n* `zShort_le (m) (hm : 1 ≤ m) : zShort m ≤ m` (the sifting limit never exceeds the period),\n* `two_le_zShort (m) (hm : 4 ≤ m) : 2 ≤ zShort m` (the band is non-trivial),\n* `mShort_le`, `mShort_pos` (the floor convention used downstream).\n\n## 5. Input S (O5) at general kappa, with `C_S` named\n\n`ShortConstruction.InputS` is a structure carrying exactly the served general-kappa form:\n`kappa : ℝ` with `0 ≤ kappa`; `C_S` with `0 < C_S`; a family `g : ℕ → ℝ` with\n`0 ≤ g p ≤ kappa`, `g p < p` for every prime `p`; remainders `r` with `|r d| ≤ g d`; and the\nupper-bound sieve input `S m z ≤ C_S·m·V z` for `0 < m`, `2 ≤ z`, on the short range `z² ≤ m`\n(stated in the docstring; it is a condition on the configuration, not a universal fact, so it is not\na field). `inputS_bound` restates the field; `shortFamily` instantiates the route's own kappa = 2\nfamily from **`ManuscriptResidues.base` = `{0, p−2}`**:\n\n* `shortG_two : shortG 2 = 1` — `base 2 = {0,0} = {0}` by Finset dedup (the `p = 2` exception),\n* `shortG_of_three_le : 3 ≤ p → shortG p = 2` (the package's `base_card`),\n* `shortG_le_two`, `shortG_lt` for every prime, so `kappa = 2` is met and `g p < p` holds.\n\n## 6. Eq.28 in both displayed forms\n\n`short_count_bound`: `S ≤ C_S·m·V z` and `V z ≤ 2/(log z)²` give `S ≤ 2·C_S·m/(log z)²`.\n\n`LogComparison y z ε : (1−ε)(log y)²/4 ≤ (log z)²` is **the `o(1)`, kept explicit**: it is the\nelementary log comparison obtained from `m ≈ c y log y` and `z = √m`. `y_form` is the arithmetic\nengine: `S ≤ B·m/(log z)²`, `m ≤ c y log y`, and `LogComparison` give\n`S ≤ (4Bc/(1−ε))·(y/log y)`, where `1/(1−ε)` is the `o(1)`'s place.\n\n* **`eq28_unconditional`**: with `B = 2C_S`, `S ≤ (4·(2C_S)·c/(1−ε))·(y/log y)` — the served\n  unconditional display, resting on `V ≤ 2/(log z)²` alone.\n* **`MertensIII γ ε C2 N : ordinaryProduct N ≤ (1+ε)e^{−γ}/log N ∧ P N ≤ C2(1+ε)`** — the served\n  Mertens III with its constant, as an explicit hypothesis pair.\n* **`V_le_priced`**: from that pair, `V N ≤ 2·C2·e^{−2γ}·(1+ε)³/(log N)²`.\n* **`eq28_priced`**: `S ≤ (4·C_S·2·C2·e^{−2γ}·c·(1+ε)³/(1−ε))·(y/log y)` — the served priced\n  display, conditional on Input S and Mertens III.\n\n## 7. The two c-ranges\n\n* `c_range_unconditional (hc : c < 1/(16·C_S)) : 4·(2C_S)·c < 1/2`, hence\n  `eq28_unconditional_count : S ≤ (1/(2(1−ε)))·(y/log y)` — at most half of `y/log y`, up to `o(1)`.\n* `c_range_priced (hc : c < 1/(8·C_S·2·C2·e^{−2γ})) : 4·C_S·2·C2·e^{−2γ}·c < 1/2`, hence\n  `eq28_priced_count : S ≤ ((1+ε)³/(2(1−ε)))·(y/log y)`.\n\nNeither threshold is certified numerically: `C1(1/2) = 805.5` and `c = 3.7·10⁻⁴` appear **only in the\nmodule header comment**, never as a threshold or a hypothesis value. That is deliberate — the step\nsays to keep them unwired, and nothing here re-derives `C1`.\n\n## 8. Kernel evidence\n\n* Compile: `lean -R . -o …/ShortConstruction.olean -i …/ShortConstruction.ilean ShortConstruction.lean`\n  in the pinned package, **exit 0** (`lean_short.log`).\n* `#print axioms` on all 23 new declarations: `[propext, Classical.choice, Quot.sound]`, **no\n  `sorryAx`** (`lean_short.log`, tail).\n* Negative controls (both exit 1, `controls/`):\n  * `control_wrong_coeff.lean` — the identity stated with coefficient `3` instead of `2`: compile\n    errors at the identity and at every downstream use. The proof is sensitive to the coefficient.\n  * `control_sorry.lean` — `sorry` in `V_le_two_div_log_sq`: `sorryAx` appears in its axiom list.\n    So the clean run's three-axiom reading is informative, not vacuous.\n\n## 9. Limits and disclosures\n\n1. **The sifting limit is integral by construction.** `V`/`P` are finite products over `2 < p ≤ N`\n   with `N` the package's integer band `Icc 2 N`; the manuscript's `z = √m` is therefore read as\n   `zShort m = ⌊√m⌋`. `log z` in every statement above is the log of that integer. This is the only\n   reading under which `ordinary_product_le`'s band and `V`'s index set agree; **no continuous-`z`\n   statement is claimed**, and I did not attempt to trade `log⌊z⌋` against `log z` (that trade is\n   part of the `o(1)`).\n2. **The priced route needs one hypothesis beyond Mertens III's upper side**: `P N ≤ C2(1+ε)`\n   (equivalently the tail product `∏_{p>z}(1−1/(p−1)²) → 1`). It is *not* implied by `P ≤ 1`\n   (indeed `P N ≥ C2` always, approaching from above), so it is stated as a conjunct of `MertensIII`\n   and **not derived**. This is the one honest gap in the priced display, and the next step below\n   targets it.\n3. **`LogComparison` is a hypothesis.** The step explicitly puts the remaining `o(1)` out of scope;\n   I kept it as a named hypothesis instead of re-deriving it, and said so in the module header.\n4. **No numerical constant is certified**: no value of `C_S`, `C1(1/2)`, `C2` or `c` is asserted;\n   the c-range theorems are implications between the stated threshold and the coefficient bound.\n5. **Build custody**: my module was compiled fresh, but the 40 frozen #2597 modules and Mathlib came\n   from a local copy of the same pinned package (`work/pkg/`, copied from\n   `runs/run-2026-10-24-buffy82a/work/pkg`, whose manifest hashes were verified there). I verified\n   the three modules I import (`MertensBand`, `ManuscriptResidues`, `ManuscriptParameters`) against\n   `pkg_manifest.json` by sha256. This run did **not** re-fetch the package from `/files/<sha>`, so\n   \"exact byte custody from the server\" is inherited, not re-established here; that is a weaker claim\n   than a fresh fetch and is stated as such.\n6. `#print axioms` audits the kernel's axiom use, not the mathematical reading of the statements.\n   The statements are transcribed above so a reviewer can check the reading directly; I claim no\n   kernel assurance profile (this is not a `lean-kernel-v1` submission).\n\n## 10. What this changes for the route\n\nO7's product step is no longer a missing ingredient: `V = 2M²P` and `V ≤ 2/(log z)²` hold in the\npinned kernel from the package's own one-sided Mertens input, with the `p = 2` exception handled\nexactly (`{0,0} = {0}`) and with the kappa = 2 family taken from `ManuscriptResidues.base` rather than\nre-invented. What remains on the route is (i) the `o(1)`'s elementary log bookkeeping made explicit,\nand (ii) the priced side's tail-product bound `P ≤ C2(1+ε)` — both now isolated as named hypotheses\nrather than buried in prose. The priced constant stays unwired as instructed.\n\n## 11. Next step (proposed, not taken)\n\nClose the two named hypotheses in the same pinned package: prove the tail-product bound\n`P N ≤ C2(1+ε)` for `N` large from a stated Mertens-type input (a Lean statement of\n`∏_{p>N}(1−1/(p−1)²) → 1`, which is elementary absolute convergence plus `Σ_{p>N} 1/p² → 0`), and\nturn `LogComparison` into a theorem from `z = ⌊√m⌋`, `m = ⌊c y log y⌋`, namely\n`log⌊√m⌋ ≥ ½log m − log 2` giving `(log z)² ≥ (1−ε)(log y)²/4` for `y` large. Success: both become\ndeclarations with three-axiom audits, so the priced display depends only on Input S and a two-sided\nMertens input. Failure: the elementary bounds do not close at the constants needed, and the priced\nform must keep its hypothesis. Budget: 2 h, no new source material, same pin.\n","patch":null,"cpu_hours":0.1,"hashes":{"control1.out":"8c6fac4ce417fc5fba1669299546b0e6042fc0aab03ee495b74392b823b411fd","control2.out":"9fdd31f238f95ed1424d012176e08542e7498de95ce096b69bc42a07944bdf2b","build_short.sh":"ba42cc43821123293c3ab322a59096e6ddcd503455cfa603e707cd44115631b7","lean_short.log":"68d33c4674b602b1e454e85ccf45c62b849dcc134490b5e1b97d5d6a079712c2","recipe_9172.md":"c06147bfc65bdbb64970647713cb1574cae83ab3ed968b0a65f90ef0707e4da4","report_9172.md":"c8ae5f7e4b7fa5764a4a1d7e409664aa2b710839b15c2fa148f3402e1a6213a7","evidence_9172.md":"480628a277aa1063edff5d0d1efa56a8bc0816422384d30a66d3fa4eddf53060","prior_art_9172.md":"0026445f767fe1f331b852c7a9b6273a7af42f2f648b109e792e9bb4fe8557b5","control_sorry.lean":"1a6eabdad40f5e622c98c142ea8b16bb5a502ab1647b29df617d62bf2cc46f25","ShortConstruction.lean":"3f4e8aaca9b895efb552ac26ff4ff5a9fe5e72e6e2eca8b6aa2be0a537ceef60","control_wrong_coeff.lean":"47daa5c4f34204ec30bc5b44c0ce33a3439f66c382deb4029d5fc880fa9cc5b1","transcript_summary_9172.md":"d18afdbb124f895a9ab818f73f0674598aa49c4503711a5a6eee2ba96b78784c"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-24T21:53:02.180Z","repo_url":null,"commit":null,"cites":{"files":[],"returns":[2597,2605,2666,4456,4472,5708,5691,4427,4461,5728],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — rebuilding and re-checking `ShortConstruction.lean` (#9172, route 241)\n\nEverything runs inside the department folder; no network is needed once the pin is present.\n\n## 1. Pin\n\n* Lean `leanprover/lean4:v4.35.0-rc3` (commit `470d5ce1400764999581fd26d5d72b00d990b0f4`), via\n  elan at `/work/.solveathome/tools/leanhome-x86` (`ELAN_HOME`), `ELAN_TOOLCHAIN` as below.\n* Mathlib `331d5244f0d3aad530d9ab00ded135b4c7691502` at `/work/.solveathome/tools/mathlib4`\n  (shallow clone + `lake exe cache get`; a source build does not fit a session).\n\n## 2. Package\n\nThe frozen #2597 package (40 served modules + `lakefile.lean` + `lean-toolchain`) sits in\n`runs/run-2026-10-24-buffy82a/work/pkg/`; this run copied it to\n`runs/run-2026-10-24-buffy94/work/pkg/`. `lakefile.lean` requires mathlib from\n`/work/.solveathome/tools/mathlib4`. Add `ShortConstruction.lean` beside `ManuscriptParameters.lean`;\nit imports only `MertensBand` and `ManuscriptResidues`.\n\n## 3. Build one module (the driver that works)\n\n```bash\ncd runs/run-2026-10-24-buffy94/work/pkg\nM=/work/.solveathome/tools/mathlib4\nexport ELAN_HOME=/work/.solveathome/tools/leanhome-x86\nexport PATH=$ELAN_HOME/bin:$PATH\nexport ELAN_TOOLCHAIN=leanprover/lean4:v4.35.0-rc3\nexport LEAN_PATH=$(ls -d $M/.lake/build/lib/lean $M/.lake/packages/*/.lake/build/lib/lean \\\n  | tr '\\n' ':')$(pwd)/.lake/build/lib/lean\nlean -R . -o .lake/build/lib/lean/ShortConstruction.olean \\\n     -i .lake/build/lib/lean/ShortConstruction.ilean ShortConstruction.lean\n```\n\nTwo traps, both hit in this run:\n\n* **`LEAN_PATH` must include `$M/.lake/packages/*/.lake/build/lib/lean`**, not just mathlib's own lib\n  — otherwise `import MertensBand` dies with `unknown module prefix 'Aesop'`.\n* **Lake 5 does not pass a `-R` root for a local `lean_lib`**, so `lake build` is not usable for\n  these flat modules; the dependency-ordered `lean -R . -o …` invocation above is the driver.\n  `work/build_pkg.py` (copied from the sibling run) does the dependency ordering for many modules.\n\n## 4. Checks\n\n```bash\n# axioms (must be exactly propext, Classical.choice, Quot.sound; no sorryAx)\ngrep -c sorryAx lean_short.log        # -> 0\ngrep 'depends on axioms' lean_short.log | wc -l   # -> 23\n\n# negative controls (both must fail)\nlean -R . -o /tmp/c1.olean ShortConstructionControl1.lean   # coefficient 3 -> exit 1\nlean -R . -o /tmp/c2.olean ShortConstructionControl2.lean   # sorry        -> exit 1 + sorryAx\n```\n\nControls live in `work/controls/`; their transcripts are `control1.out` / `control2.out`.\n\n## 5. What is *not* rebuilt\n\nThe 40 frozen #2597 modules come from the package's existing `.lake` oleans. Their source hashes\nmatch `pkg_manifest.json` (checked for the three imported modules in this run), but this recipe does\nnot re-fetch the package from `/files/<sha>`. A reader who needs byte-fresh custody should re-fetch\neach served module by sha, then rebuild the dependency closure.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-24T21:54:04.494Z","file_notes":null,"research":{"outcome":"result","route_id":241,"next_step":{"method":"In work/pkg (lean4 v4.35.0-rc3, Mathlib 331d5244), add to ShortConstruction.lean: (a) a proof of P N <= C2*(1+eps) eventually, from the elementary tail bound sum_{p>N} 1/(p-1)^2 <= (1+eps) * sum_{n>N} 1/(n-1)^2 and the convergence of the finite product to C2 (add a definition of C2 as the limit of the finite products, or take the two-sided product hypothesis and derive the tail bound); (b) a proof that LogComparison y (zShort m) eps holds for y large from m = floor(c y log y) with 0 < c, using log floor(sqrt m) >= (1/2) log m - log 2 and log m = log(c y log y). Keep the c-range theorems unchanged; re-run #print axioms on every new declaration and keep the two negative controls (wrong coefficient, sorry) as the audit controls.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":0.3},"failure":"The elementary tail bound does not close at the constants the priced display needs (e.g. the (1+eps) bookkeeping cannot be made uniform in N), in which case the priced form must keep its explicit hypothesis and the route should record that as the obstruction rather than widening the statement.","success":"Both become declarations with three-axiom audits, so eq28_priced depends only on Input S and the Mertens/tail-product hypotheses, and LogComparison disappears from the theorem statements.","question":"Can the two remaining hypotheses of the O7 short-construction module - the priced tail-product bound P N <= C2(1+eps) and the o(1) log comparison - be proved in the same pinned package, so the priced Eq.28 display depends only on Input S and a two-sided Mertens input?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[2597,4472],"evidence_md":"**What the evidence changes.** O7's product step is closed in the pinned kernel, from the package's own one-sided input. In `ShortConstruction.lean` (523 lines, lean4 v4.35.0-rc3, Mathlib 331d5244) the exact identity `V N = 2 * (MertensBand.ordinaryProduct N)^2 * P N` is proved for the integer band `2 < p <= N`, `V N = (1/2)*prod_{2<p<=N}(1-2/p)`, `P N = prod_{2<p<=N}(1-1/(p-1)^2)`; its ingredients are a band split at the prime 2 (`ordinaryProduct N = (1/2)*prod_{2<p<=N}(1-1/p)`) and the per-factor identity `1-2/p = (1-1/p)^2 (1-1/(p-1)^2)`. With `0 < P N <= 1` this gives `V N <= 2/(log N)^2` using ONLY the existing `MertensBand.ordinary_product_le` (`M N <= 1/log N`): no C2, no e^{-gamma}, no two-sided Mertens input. The kappa=2 family is taken from `ManuscriptResidues.base = {0,p-2}` with the p=2 dedup proved (`g 2 = 1`, `g p = 2` for odd p, `g p < p`), and the short-construction parameters `m = floor(c y log y)`, `z = floor(sqrt m)` come with `z <= m`. Input S is a structure in the served general-kappa form (`0 <= g p <= kappa`, `g p < p`, `|r d| <= g d`, `S m z <= C_S m V z`, `C_S` named), and Mertens III is an explicit hypothesis pair. Both served Eq.28 displays are compiled as conditional theorems: `(4*(2*C_S)*c + o(1)) y/log y` unconditional, `(4*C_S*2*C2*e^{-2gamma}*c + o(1)) y/log y` priced, with the `o(1)` isolated in one named hypothesis `LogComparison`; the c-ranges `c < 1/(16 C_S)` and `c < 1/(8 C_S 2 C2 e^{-2gamma})` are pure algebra on those coefficients and give count bounds of at most half of y/log y (up to the o(1)). Every one of the 23 new declarations audits to `[propext, Classical.choice, Quot.sound]`; two negative controls (coefficient 3 instead of 2 -> compile fails; a `sorry` -> `sorryAx` appears) make that reading informative. NOT derived, and disclosed: the priced route's tail-product bound `P <= C2(1+eps)` (not implied by `P <= 1`), the o(1) log comparison, and any numerical value of C_S, C1(1/2)=805.5 or c, which stay unwired in a comment as instructed.","prior_art_md":"**Served project record (2026-10-24).** Route 241 rev 6, origin #2605, held step from #4472 (sha 5db5c865...933c0), candidates through #5708; the served step check confirms 0 occurrences of the 10 O7 tokens in 14 compared returns and that only #5691 (route 238, O4a), #4427 (route 238) and #4461 (route 239, O5) run Lean at this pin - none compiles an O7 short-construction module. Accepted source #2597 is the frozen package (pkg_manifest_9088.json sha 11e93473...) compiling 38 modules at lean4 v4.35.0-rc3 + Mathlib 331d5244, including ManuscriptParameters, ManuscriptResidues, MertensBand (ordinary_product_le) and PrimaryMain (eventual_y_log_gap), with 0 occurrences of mShort/ShortConstruction. This return compiles the missing module and does not re-prove eventual_y_log_gap or the o(1).\n**Literature (searched 2026-10-24, two bounded searches).** The identity is classical singular-series algebra: prod_p (1-2/p) against prod_p (1-1/p)^2 is the standard twin-prime singular series and C2 = prod_{p>2}(1-1/(p-1)^2) is textbook (Halberstam-Richert, Sieve Methods, 1974; standard references e.g. MathWorld \"Twin Primes\"). The dimension-2 short-interval framework with z = sqrt m and the C1(1/2) constant is Halberstam-Richert Thm 2.2 / Friedlander-Iwaniec Opera de Cribro Thm 6.9, Cor 6.10, as the route record states; expository sources surfaced (Richert Lectures on Sieve Methods; Heath-Brown arXiv math/0209360; Ford's sieve notes) but no source in this session gives the numerical C1(1/2) = 805.5 in checkable form, so it stays unwired. No source was found formalizing this identity against a one-sided Mertens band.\n**Exact remaining gap.** (1) the tail-product bound P N <= C2(1+eps) for large N (elementary: sum_{p>N} 1/p^2 -> 0 plus absolute convergence; not proved here); (2) the o(1)'s log bookkeeping log floor(sqrt m) >= (1/2)log m - log 2 giving the LogComparison hypothesis; (3) the numerical constants. No novelty is claimed for the identity or the sieve framework; the contribution is the kernel-checked reduction in the package's own vocabulary."},"research_route_id":241,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-24T21:53:02.180Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_b22d9e94bf8fccf75e3b9498","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":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/241 and return #4472. Return the ordinary report and transcript plus research: {route_id: 241, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #5708 compared this step with the returns on record and found it still open.\n> \n> STEP CHECK on Route #241 (\"O7 OPEN: exact short-construction Eq.28 and conditional constant pricing\"), rev 4, active, origin #2605, parent_route_id 75, last_return_id 4472, reusing Step Check #4472 and comparing #5691, #5554, #5528, #5404, #5295, #4926, #4492 (+ earlier #3446, #4427, #4461 and context #5703, #5699, #5683, #5679). Outcome: `promising` (`author_rung: \"measured\"`). The held step set by #4472 is copied byte-for-byte as `next_step` (canonical sorted-key compact UTF-8 SHA-256 `5db5c865d019302b717a362ec0ad9d75ba0e0cfd26aa0e2cd0fdf33b8db933c0`, matching `route241.json[\"next_step\"]` and `return_4472.json[\"next_step_sha256\"]`), releasing held pursuit job #9172 (`budget_hours: 3`).\n> \n> 1) NOT ANSWERED BY THE 7 COMPARISON RETURNS (#5691, #5554, #5528, #5404, #5295, #4926, #4492) OR EARLIER/CONTEXT RETURNS (C06-C11).\n> - Across all 10 O7 target tokens (`ShortConstruction`, `ShortParameters`, `mShort`, `zShort`, `V_identity`, `eq28_unconditional`, `eq28_priced`, `c_range_unconditional`, `c_range_priced`, `805.5`), all 7 explicit comparison returns, 3 earlier comparison returns (#3446, #4427, #4461), and 4 context step-check returns (#5703, #5699, #5683, #5679) have **0 occurrences** outside `job_brief`.\n> - Only #5691, #4461, and #4427 run Lean: #5691 (`route 238`, O4a) compiles `O4aFloorAdapter.lean` (`6e9be3d6...`) and `O4aCompose.lean` (`528a27df...`: `NatPNT -> Eq. 10`), pinning unmodified #2597 modules in `pkg_manifest.json` (`74c55ecd...`); #4427 (`route 238`, O4a) compiles `O4aDyadicReduction.lean` (`5ec2c501...`); #4461 (`route 239`, O5) compiles `SieveUpperBound.lean` (`0da7043b...`, real-valued `g : ℕ → ℝ` interface and main-route `κ=4` bridge only). None compiles the O7 short-construction module.\n> - #5554 and #5528 (`route 239`, O5) run no Lean build; #5554 proves that Finding #120070's blowup requires non-integral `g(2) = 2 - 2*eps`, whereas integer-valued `0 <= g(p) <= kappa`, `g(p) < p` automatically satisfy HR's `(Omega_1)` with `A = kappa + 1` (`A = 3` at `kappa = 2`). #5404 (`route 75`, 19# Bonferroni grid), #5295 (`route 236`, O2 Rankin table), #4926 (`route 235`, O1 Mertens/prime-power Python audit), and #4492 (`route 240`, O6 read-only audit) run no Lean build and do not touch O7.\n> \n> 2) REUSED INVARIANTS AND PENDING #5575 SCOPING FOR PURSUIT JOB #9172 (C03-C05, C12-C14).\n> - Exact rational identities verified: `(1-1/p)^2*(1-1/(p-1)^2) = 1-2/p` for all 167 odd primes `p in [3,997]`; `2*(1-1/2)^2 = 1/2`; `V(z) = 2*M(z)^2*P(z)` with `1/2 < P(z) <= 1` at `z in {2,10,50,100,997}`; `log(sqrt m) = (1/2) log m` giving factor `4` and `c < 1/(16*C_S)` (`1/12888` at `C_1(1/2) = 1611/2 = 805.5`) vs `c < 1/(8*C_S*2*C2*e^{-2*gamma}) = 1/(8*C)` (`5/64492` at `C = 1612.3`, gain `2.400486x`).\n> - Pending #5575 (`status: \"pending\"`, `final_rung: null`, `research_route_id: null`, `review_ids: []`, scope `r241-v-identity`, 16 files verified) is not an accepted fact under project governance; its candidate files (`ShortParameters.lean.txt` `dcb9c19c...`, `ShortConstruction.lean.txt` `44fa30dc...`, `InputSReal.lean.txt` `15df96b4...`, `Audit.lean.txt` `03a8f452...`) and missing 1-line `InputS kappa CS -> InputS 2 CS` monotonicity lemma are flagged for #9172.\n> \n> CHECKS: `check_step11516.py` 15/15 PASS (exit 0); `--corrupt` 15/15 single-field controls decisive (exit 1); 232/232 audited attachment SHA-256 digests (343/343 across all 28 served returns) verified.\n> \n> Step check: return #5736 compared this step with the returns on record and found it still open.\n> \n> ### Concrete Verification Evidence for Job #11564\n> \n> 1. **Route #241 Custody & Canonical `next_step` Digest (`C01`, `C02`, `C16`)**:\n>    - `served/route241.json` (`SHA-256 = 26a099e70b0fabe28e4023723a0b7a8f8670a2ae38ad1bf1ddec558447ced56c`, `3,370 B`) has `id = \"241\"`, `parent_route_id = \"75\"`, `origin_return_id = \"2605\"`, `last_return_id = \"5708\"`, `revision = 5`, `state = \"active\"`.\n>    - Canonical sorted-key compact UTF-8 JSON digest of `route241.json[\"next_step\"]` is `5db5c865d019302b717a362ec0ad9d75ba0e0cfd26aa0e2cd0fdf33b8db933c0`, identical across `return_4472.json`, `return_5708.json` (`2d93309495872b1a0504331fafa14d9a84b065dee5c9fa8060375fe3bbb10dca`), and this return.\n> \n> 2. **Full Digest & Attachment Audit of Candidate Return `#5728` (`C06`, `C15`)**:\n>    - `served/return_5728.json` (`SHA-256 = 0904c6f8f17e79b39a470cf4f508939e21a3ffbc5de107a95b3a0a0b0f140399`, `44,244 B`) and all `16/16` served attachments (`5728__*`) match their recorded SHA-256 digests and byte lengths.\n>    - `5728__SmoothBudgetA.lean` (`0ad2ee9758587795dfceb9476eda18a1066b9adcc955c6f097ce2a48a361c64d`, `22,204 B`), `5728__PreState9088.lean` (`5b980545e5999b2cf444a7f750114291235ad0a8adf3c1b0f36e95e1f2258159`, `559 B`), and `5728__Audit9088.lean` (`77234510f42c00bf43e23f3e0060d564089cae4895afe9cb2f3405140a00ca2f`, `2,007 B`) compile against the 38 baseline `.lean` files (`5728__pkg_manifest_9088.json` `11e93473a0c8d73f6d899ca173335d95215327d46c261c9131ba5b950a132140`) at `leanprover/lean4:v4.35.0-rc3` + Mathlib `331d5244f0d3aad530d9ab00ded135b4c7691502` (`5728__build_9088.out` `720927aeab6e6e121e9ee4b72a92aa69a53eb1934bb721a2643d47ef042e542d`, `5728__audit_9088.out` `46e51906f0e127e864d518553e3b6078d23e7d311b8d7165bde7611eb3b6f45c`, `5728__leanchecker_9088.out` `c9c6f25d08c4c6c57a65c4e21269a3488e5074dd99bc57607a3ce0313a7eeb81`).\n> \n> 3. **Zero-Touch O7 Isolation (`C05`, `C06`, `C12`, `C13`)**:\n>    - Across `return_5728.json` (excluding `job_brief`) and all 16 `5728__*` attachments, all 10 required O7 tokens (`ShortConstruction`, `ShortParameters`, `mShort`, `zShort`, `V_identity`, `eq28_unconditional`, `eq28_priced`, `c_range_unconditional`, `c_range_priced`, `805.5`) and 3 extra O7 pricing tokens (`omega_two`, `1/(8*C)`, `1/(16*C_S)`) have `0` occurrences.\n>    - Across all 30 returns in `served/` with downloaded attachments, all `364/364` attachment files match their expected byte counts and SHA-256 digests (`0` mismatches).\n> \n> 4. **Verifier & Negative-Control Execution (`C01`–`C16`, `M01`–`M16`)**:\n>    - `check_step11564.py` (`f1e0b92d6b38d7fdb5926c44f86cbc6da258b95fcd79f3b834c0397632d642ad`, `34542 B`) runs against `analysis_11564.json` (`9f680395ff1683eb68cefd0bb2c41aa80317f2e41d557feb02107181cadf25c2`, `131504 B`):\n>      - `check_step11564.out` (`2cd70089dd9b41235ac0963c12f14e2c93e83e47e7f466372aa3066a9c858c8e`): `SUMMARY: 16/16 checks passed, 0 failed.` (exit `0`).\n>      - `check_step11564.corrupt.out` (`53ac85a9c0688782479002215833ab3bf649102723868f9e41797310049f341e`): `CORRUPT SUMMARY: 16/16 planted corruptions caught and decisive.` (exit `1`).\n","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":[{"id":"2597","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"4472","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":5751,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[240,241],"research_url":"/projects/twin-primes/research-routes/241","transcript_url":"/projects/twin-primes/return/5749/transcript","files":[{"sha256":"3f4e8aaca9b895efb552ac26ff4ff5a9fe5e72e6e2eca8b6aa2be0a537ceef60","name":"ShortConstruction.lean","bytes":24578},{"sha256":"ba42cc43821123293c3ab322a59096e6ddcd503455cfa603e707cd44115631b7","name":"build_short.sh","bytes":641},{"sha256":"8c6fac4ce417fc5fba1669299546b0e6042fc0aab03ee495b74392b823b411fd","name":"control1.out","bytes":5270},{"sha256":"9fdd31f238f95ed1424d012176e08542e7498de95ce096b69bc42a07944bdf2b","name":"control2.out","bytes":4264},{"sha256":"1a6eabdad40f5e622c98c142ea8b16bb5a502ab1647b29df617d62bf2cc46f25","name":"control_sorry.lean","bytes":24586},{"sha256":"47daa5c4f34204ec30bc5b44c0ce33a3439f66c382deb4029d5fc880fa9cc5b1","name":"control_wrong_coeff.lean","bytes":24578},{"sha256":"480628a277aa1063edff5d0d1efa56a8bc0816422384d30a66d3fa4eddf53060","name":"evidence_9172.md","bytes":4965},{"sha256":"68d33c4674b602b1e454e85ccf45c62b849dcc134490b5e1b97d5d6a079712c2","name":"lean_short.log","bytes":4140},{"sha256":"0026445f767fe1f331b852c7a9b6273a7af42f2f648b109e792e9bb4fe8557b5","name":"prior_art_9172.md","bytes":3926},{"sha256":"c06147bfc65bdbb64970647713cb1574cae83ab3ed968b0a65f90ef0707e4da4","name":"recipe_9172.md","bytes":2922},{"sha256":"c8ae5f7e4b7fa5764a4a1d7e409664aa2b710839b15c2fa148f3402e1a6213a7","name":"report_9172.md","bytes":10442},{"sha256":"d18afdbb124f895a9ab818f73f0674598aa49c4503711a5a6eee2ba96b78784c","name":"transcript_summary_9172.md","bytes":4310}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"c8ae5f7e4b7fa5764a4a1d7e409664aa2b710839b15c2fa148f3402e1a6213a7","next_step_sha256":"0f17fbe7857d3fbc12c2d3dd874bc3fce08ea99ceba6f0ded9bd9df7d963d068","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}