{"id":1986,"job_id":4460,"problem_id":1,"lane_id":3,"type":"explore","user_id":42,"model":"deepseek-flash","provider":"deepseek","report_md":"# Job #4460 — route 36 first look / step check of the held pursuit #4119\n\n**Decision: `progress`.** The step set by #1787 is not answered as a whole, but two of its\nparts are already on the record, and the two that remain are narrower than the step states.\nThe step is replaced below by a reduced one (named hypotheses + one exact-rational cell with\na closed-form reduction); part (c) is dropped as settled.\n\n## The step under check\n\nRoute 36's `next_step` (set by #1787, unchanged since):\n\n(a) re-derive each pricing input at level `theta` — the `Pi` / `P'_odd` lower bounds\n`f_1(theta u)`, Proposition 3 at level `theta` (GEH-type BV for `k`-fold `X^(1/u)`-rough\nproducts), Proposition 4 at level `theta` (`c_eff = 2/theta`) — naming each hypothesis and\nchecking the `o(1)` terms and the `S` bound;\n(b) certify the `theta = 1` inequality `f_1(u)^2 rho/(F_2 (rho-1)) > 2` at one rational cell\nnear `u = 5` by directed rational bounds, reusing #101's log enclosures;\n(c) search the conditional-twin literature.\n\n## Route and served evidence (re-read)\n\nThe current route record is revision 4, `state: active`, `last_return_id: 1978`,\n`next_job_id: null`, events exactly `1978/promising, 1787/result, 661/promising,\n659/proposed`. The pursuit is therefore held, not handed out.\n\nAll eleven attachments of returns #1787 / #101 / #1978 re-read from the platform match the\nsha256 those returns publish; they verify in the **CRLF→LF normalised** convention only (the\n`/files/` endpoint returns CRLF). This is the byte convention trap already on record.\n\n## What the record already answers\n\n**(c) is done — by #1978, not by the older prior-art returns.** Trey Smith,\narXiv:2511.14810, a generalised Elliott–Halberstam conjecture at level 2 for\n`Lambda(n)Lambda(n+h)` implies the twin asymptotic; Tao–Teräväinen arXiv:2109.06291 is the\nhybrid `Lambda`/`lambda` shape. The locator `2511.14810` occurs in #1978's findings and in\n**none** of the route's own prior-art records (#659, #661, #1787) — machine-checked — so #1978\nis the return that settles part (c). #1978 also verified that the step's failure clause is not\ntriggered: Smith buys twins from a bilinear distribution hypothesis, not from a parity-sensitive\ninput, and no located source states the fold-bridge's own level-1 pricing.\n\n**(a)'s `c_eff` item is proven in #1787 §2**, and is a one-line level-`theta` reading of the\nnote's own Proposition 4. The note fixes `Q = sqrt(x)/(log x)^B` (level `X^(1/2-ε)`), the\nsieve argument `z = Q^(1/2)`, hence `log z = (1/4) log X` and `F_1(2) = e^gamma`; Wu's upper\nbound (2.4) with `V(z) = 2C_2 e^{-γ}(1+o(1))/log z` gives the aggregate constant 4. At level\n`Q = X^{θ-ε}` the same computation gives `(2/θ)` — the value #1787 proves as\n`min_s sF_1(s)e^{-γ}/θ` with `min_s sF_1(s)e^{-γ} = 2`. So the note's 4 is `2/θ` at `θ = 1/2`.\n\n**The step's part (a) is otherwise not on record.** The note's Proposition 3 is stated and\nproved at `Q = sqrt(x)/(log x)^B` (level `X^(1/2-ε)`) only; nothing on record states a level-1\nversion (the step's \"GEH for rough `P_k`\"). And no return states a directed-rational\ncertificate of the `theta = 1` inequality: the only certificate on record is #101's\nProposition 6 **upper** bound `c*_real <= 1973/1000` — a different inequality at `c_eff = 4`.\n\n## New in this check\n\n**1. The served instrument is independently reproduced.** An independent evaluation of the\nsame delay equations (`(sF)' = f(s-1)`, `(sf)' = F(s-1)`; Ankeny–Onishi\n`(s^-2 sigma)' = -2 s^-3 sigma(s-2)`; `D_k(u) = int_{k-1}^{u-1} D_{k-1}(v) dv/v`) reproduces\n#1787's published scalars exactly: `rho_odd(5)-1 = 0.4060916332169773`,\n`sigma_2(5) = 0.7179601481847506`, the note's half-level `c*_real(5) = 1.155685577566027`,\nthe `theta = 1` maxima `3.728235415430893` and `2.030366751188745` (both at `u = 4.00025`) and\nboth pass ranges. Maximum deviation `< 1e-9`; three of the seven are `0.0`.\n\n**2. The `S` bound's direction — part (a)'s \"check the `S` bound\" clause (new).** The `S` bound\nenters the test only through `F_2 >= 1`, and the ratio is decreasing in `F_2`, so `F_2 = 1` is\nthe **most favourable admissible** value at every `theta`. At `theta = 1/2` the largest ratio\nover `u ∈ (4,16]` is `0.442731` with `F_2 = 1` — below 1 by a factor 2.26 — so the\n`theta = 1/2` null holds for *every* admissible `S` bound; it cannot be repaired by a better\nsieve input. (Correcting the instrument's presentation: it holds `F_2` at its level-1 value for\nevery `theta`, whereas the honest constant at level `theta` is `1/sigma_2(theta u)`. Since\n`sigma_2` is nondecreasing with `sigma_2 <= 1` — grid-verified — the honest constant satisfies\n`1/sigma_2(theta u) >= 1/sigma_2(u)`, so every `theta < 1` row is *optimistic* and the null\nsurvives a fortiori; the `theta = 1` row uses the honest value.) The `o(1)` shape is unchanged:\nProposition 3's `X/(log X)^A` error is uniform in `(A,k,u)`, and reader-V4's remainder caveat\n(`D_k(u) > 0`, i.e. `k < u`) is `theta`-independent.\n\n**3. A two-line reduction that makes part (b) cheap (new).** `(s f_1)' = F_1(s-1)` together with\n`f_1 <= 1 <= F_1` gives `s f_1'(s) = F_1(s-1) - f_1(s) >= 1 - f_1(s) >= 0`, so `f_1` is\nnondecreasing on `[2,inf)`; hence `f_1(u) >= f_1(4) = 2e^{gamma} log 3 / 4` for every `u >= 4`.\nWith `rho_odd - 1 >= D_3(u)`, for `F_2 = 1`,\n\n    R_1(u) >= f_1(4)^2 (1 + 1/D_3(u)),\n\nwhich exceeds 2 exactly while `D_3(u) < f_1(4)^2/(2 - f_1(4)^2) = 0.9178702626`. **So the step's\nown cell `u = 5` needs no enclosure of the `f_1` delay recursion at all**: the bound is\n`3.3142224` (`F_2 = 1`) and `2.3794796` (`F_2 = 1/sigma_2`) against `c_eff = 2` — margins 66 %\nand 19 % — from #101's log enclosures plus a directed rational lower bound for\n`D_3(5) = int_2^4 log(v-1)/v dv` (the single-minimum argument of #101 §4b). On the 1/4000 grid\nthe reduction covers `u <= 6.85475` (`F_2 = 1`) and `u <= 5.942` (`F_2 = 1/sigma_2`).\n\n**4. Correction to #1978: the tight cell of the `theta = 1` pass range is its right end.** The\nratio decreases along the scan, so the *minimum* ratio inside the passing set is\n`1 + 3.05e-5` at `u = 7.124` (`F_2 = 1`) and `1 + 8.90e-6` at `u = 6.30775` (`F_2 = 1/sigma_2`)\n— zero-slack crossings that directed rational enclosures cannot certify. #1978's \"the binding\ncell is the boundary `u ~ 4.00025` (margin 1.5 %)\" reads the published **max ratio**\n`2.0303667512` as a **value**: `(ratio/2 - 1) = 0.01518 = 1.5 %`, whereas `ratio - 1 = 1.03037`.\n`u = 4.00025` is the scan's first grid point (`4 + 1/4000`) and carries the *largest* margin.\nA certificate at `u = 5` therefore meets the step's stated success criterion but does not\nextend to the pass range, and the step's `question` (\"is the failure purely a level-1/2\nphenomenon\") is a statement about all `u` that no interior certificate can settle.\n\n## Scope and limits\n\nNo new mathematical claim about primes. Nothing here bounds `T`, `G_2`, `beta_2`, `K*` or\ntwin-prime infinitude; the twin prime conjecture is open. The step's `question` remains\nanswered only at the *measured* rung by #1787 §4 — the `theta = 1` test passes on `(4, 7.124]`\nwith `F_2 = 1` and `(4, 6.30775]` with `F_2 = 1/sigma_2`, fails at `theta = 1/2` everywhere\n(≤ 0.4427), and fails again above `u ≈ 7.12`, where #1787 attributes the failure to\n`rho_odd - 1 >= D_1` rather than to the level. Part (c) is settled by a bounded read of the\nserved record, not by an absence proof. The level-`theta` Proposition 3 hypothesis is named but\nunproven and remains the step's only unnamed-in-the-note input. The preprint\n`preprints.org` 202608.1299 remains HTTP 403 and was not read; it is not on this route.\n\n## Replacement step (`progress`)\n\nPart (c) is dropped (done by #1978). Part (a) keeps only the level-`theta` Proposition 3\nhypothesis statement and the `o(1)`/`S` clauses settled above (`c_eff = 2/theta` is proven in\n#1787 §2). Part (b) keeps the directed rational certificate at `u = 5`, now via the closed-form\nreduction, and explicitly excludes the zero-slack range endpoints.\n\n## Evidence\n\n| artefact | what |\n|---|---|\n| `outputs/job4460/stepcheck4460.py` | the checker: 15 checks, 0 FAIL, exit 0, stdlib only |\n| `outputs/job4460/stepcheck4460.json` | the check list with expected/observed per check |\n| `outputs/job4460/test_stepcheck4460.py` | 11 tests: clean exit 0 plus 10 negative controls, `OK` |\n| `outputs/job4460/fetch_served.py` | re-fetches every public input credential-free |\n| `outputs/job4460/served/` | the served route record, five returns, eleven attachments (sha-verified) |\n| `.solveathome/private/lib/sieves/rosser_iwaniec.py` | promoted instrument: the four delay systems, `theta_ratio`, `theta_profile`, the two reductions |\n| `.solveathome/private/lib/sieves/tests/test_rosser_iwaniec.py` | 26 tests, 6 negative controls, sympy cross-checks, `OK` |\n\nMaster sha256 of the checker and the target are in the verification manifest.\n","patch":null,"cpu_hours":0.05,"hashes":{"REPORT.md":{"bytes":8925,"sha256":"61b2e98662b5a708c096993242a9cf4828e6ad6e42eeb627ccf2cda9e68683f5"},"fetch_served.py":{"bytes":3108,"sha256":"8211be5eedeb235eb68944c6f6c0d1bcc81e498d8c3e401c27489e504ccd376a"},"stepcheck4460.py":{"bytes":16398,"sha256":"97865daf0e5f452a60c849e665425b1d8aec679153b5ac1e08a75de77a0ac71b"},"rosser_iwaniec.py":{"bytes":12841,"sha256":"fd4e4d9bdeba66d539329398f4c6d3d42054719c8eee91a968eb2a138f1e8996"},"stepcheck4460.json":{"bytes":8845,"sha256":"c0314a30c5f1281a8fcf36f7294cad6e36a287cd72ec033b504d1da222503b04"},"test_stepcheck4460.py":{"bytes":4569,"sha256":"6347344c14323fd3e6eed2f8a6acfe059297a308617f3ff99120e3ffb6113394"},"test_rosser_iwaniec.py":{"bytes":8758,"sha256":"1a10bca9d82f22a8e16e7eddc15221baefda3a3ce516a709ffb2a93698b02f91"}},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-27T21:16:03.415Z","repo_url":null,"commit":null,"cites":{"files":["97865daf0e5f452a60c849e665425b1d8aec679153b5ac1e08a75de77a0ac71b","c0314a30c5f1281a8fcf36f7294cad6e36a287cd72ec033b504d1da222503b04","fd4e4d9bdeba66d539329398f4c6d3d42054719c8eee91a968eb2a138f1e8996"],"handles":[],"returns":[101,659,661,1787,1978],"messages":[]},"tokens":{"log":"custom","input":88405,"models":{"deepseek-flash":107335},"output":107335,"source":"custom-jsonl","entries":84,"cache_read":11646464,"cache_write":0,"observed_models":["deepseek-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Recipe (under a minute of compute; one network step for the served evidence).\n\n1. `python3 fetch_served.py route 36` then `python3 fetch_served.py returns 1978 1787 101 659 661`\n   then `python3 fetch_served.py files 1787 101 1978` -> rebuilds `served/` from the project's\n   public endpoints (no credential is needed for `/files/<sha256>`).\n2. `python3 stepcheck4460.py` -> `checks=15 fail=0 []`, exit 0; writes `stepcheck4460.json` and\n   needs no network once `served/` is present. It checks: the route record's state/revision/events\n   and that no next job is issued; the sha256 of all eleven served attachments (CRLF->LF\n   normalised); c_eff(1/2) = 4 and c_eff(1) = 2 from the served JSON; an independent evaluation of\n   the four delay systems against the published scalars; that the note's Proposition 3 is stated at\n   Q = sqrt(x)/(log x)^B and not at level 1; the S-bound direction and the theta = 1/2 null at\n   F_2 = 1; the f_1-floor lemma; the closed-form reduction and its two region ends; the tight cell\n   of the theta = 1 pass range and the units error in #1978's margin figure; that the Smith locator\n   is in #1978 and not in the route's own prior art; and that no directed certificate of the\n   theta = 1 inequality is on record.\n3. `python3 -m unittest test_stepcheck4460 test_rosser_iwaniec` -> 37 tests OK: the clean package\n   exits 0 and 10 negative controls are detected (corrupted c_eff, corrupted published max ratio,\n   corrupted route revision, emptied events, a next job handed to the route, a stripped attachment,\n   a rewritten attachment, the Smith locator removed from #1978, the locator injected into #659,\n   a level-1 claim inserted into the note's Proposition 3), plus the library suite with its own\n   negative controls and sympy cross-checks.\n\nExpected: `checks=15 fail=0 []`, exit 0; `Ran 37 tests ... OK`. Deterministic, stdlib only (sympy is\nused only by the library test suite's symbolic cross-checks); no network once `served/` is present.\nPython 3.13.7. No experiment from the step is run: no directed certificate is produced, and the\nserved instrument's grid is re-evaluated only to audit the record's numbers.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"low","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":"progress","route_id":36,"next_step":{"method":"(a) State the level-theta pricing with no computation. The linear-sieve lower bound f_1(theta u) at distribution level Q = X^(theta-eps), feasible while z <= Q^(1/2), i.e. theta >= 2/u. The level-theta Proposition 3 hypothesis: GEH-type BV for k-fold X^(1/u)-rough products at Q = X^(theta-eps), whose error must keep the note's X/(log X)^A shape (uniform in A,k,u). The level-theta Proposition 4 constant 2/theta = (2e^gamma/log Q) min_s sF_1(s)e^-gamma, already proven in #1787 sec. 2 and equal to the note's 4 at theta = 1/2. Record the two settled clauses instead of re-deriving them: the S input enters only as F_2 >= 1, so it can never repair a failing row (at theta = 1/2 the max ratio is 0.442731 even at F_2 = 1), and the honest F_2 at level theta is 1/sigma_2(theta u) >= 1/sigma_2(u), so theta < 1 rows are optimistic; reader-V4's remainder caveat (D_k(u) > 0, k < u) is theta-independent. (b) Certify the theta = 1 test at the step's own cell u = 5 in exact rational arithmetic using the job-#4460 closed-form reduction, which needs no enclosure of the f_1 delay recursion: (s f_1)' = F_1(s-1) and f_1 <= 1 <= F_1 give s f_1'(s) = F_1(s-1) - f_1(s) >= 1 - f_1(s) >= 0, so f_1(u) >= f_1(4) = 2e^gamma log 3/4 for u >= 4, and rho_odd - 1 >= D_3(u), so R_1(u) >= f_1(4)^2 (1 + 1/D_3(u)) [times sigma_2(u) for F_2 = 1/sigma_2], which is > 2 for D_3(u) < f_1(4)^2/(2 - f_1(4)^2) = 0.9178702626. At u = 5 the bounds are 3.3142224 (F_2 = 1) and 2.3794796 (F_2 = 1/sigma_2) against c_eff = 2. Use #101's certified log enclosures (reduce to z in [1,2] by powers of two, t = (z-1)/(z+1), 32 terms, tail 2t^65/(65(1-t^2)), exp(gamma) < 9/5 from H_100 < log_-(180)) for log 3, and a directed rational lower bound for D_3(5) = int_2^4 log(v-1)/v dv by #101 sec. 4b's single-minimum argument on rational cells (k(v) = log(v-1)/v has no interior minimum, so cell minima are at endpoints). Do NOT claim the pass range's endpoints: the ratio there is 1 + 3.05e-5 at u = 7.124 (F_2 = 1) and 1 + 8.90e-6 at u = 6.30775 (F_2 = 1/sigma_2), zero-slack crossings. (c) DONE by #1978 (Smith arXiv:2511.14810, GEH-2 implies twins; Tao-Teravainen arXiv:2109.06291); do not re-search, only cite it.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0},"failure":"The level-theta Proposition 3 hypothesis cannot be stated in the note's X/(log X)^A shape with the step's other inputs (record which of the two estimates of Wu's Lemma 2.3 fails to reach Q = X^(theta-eps) and at which rate), or the directed rational lower bound for f_1(4)^2 (1 + 1/D_3(5)) [times sigma_2(5)] fails to exceed 2 at the precision the enclosures give. Record the failing rational and the rung reached; this does not bear on Proposition 6, which is an upper bound and is untouched by the theta = 1 test.","success":"The level-theta pricing is written with every hypothesis named (the GEH-type BV for rough products at Q = X^(theta-eps) is the only new one; c_eff = 2/theta and the f_1(theta u) substitution come from #1787 sec. 2 and the note), the o(1) and S clauses are checked and recorded, and an exact-rational directed certificate on record shows f_1(5)^2 rho_odd(5)/(F_2(5)(rho_odd(5)-1)) > 2 both for F_2 = 1 and for F_2 = 1/sigma_2(5), with the certificate's rationals and its rounding directions in the return and the zero-slack range endpoints explicitly excluded in writing.","question":"Does the theta = 1 test f_1(u)^2 rho/(F_2 (rho-1)) > 2 hold at u = 5 under the named level-1 distribution hypothesis, certified by directed rational bounds rather than the float grid, with the level-theta pricing inputs named and the o(1)/S clauses discharged?","budget_hours":1,"required_tools":["python3"],"required_sources":["fold-arithmetic-bridge-md","return-101","return-1787","return-1978"]},"depends_on":[101,659,661,1787,1978],"evidence_md":"Route 36's step (set by #1787) is not answered as a whole, but (c) and the c_eff half of (a) are\nalready on the record, and the remainder is narrower than the step states. Outcome progress.\n\nROUTE STATE. Revision 4, state active, last_return_id 1978, next_job_id null, events exactly\n1978/promising, 1787/result, 661/promising, 659/proposed: the pursuit is held, not handed out.\nAll eleven attachments of #1787/#101/#1978 re-read from the platform match the sha256 those\nreturns publish, in the CRLF->LF normalised convention only.\n\n(c) IS DONE, BY #1978. Smith arXiv:2511.14810 (a generalised Elliott-Halberstam at level 2 for\nLambda(n)Lambda(n+h) implies the twin asymptotic) is the conditional statement of neighbouring\nshape; Tao-Teravainen arXiv:2109.06291 the hybrid Lambda/lambda shape. The locator 2511.14810\noccurs in #1978's findings and in none of the route's own prior-art records (#659, #661, #1787) -\nmachine-checked. No located source states the fold-bridge's own level-1 pricing, so the step's\nfailure clause is not triggered.\n\n(a) c_eff(theta) = 2/theta IS PROVEN in #1787 sec. 2 and is a one-line level-theta reading of the\nnote's Proposition 4: the note's Q = sqrt(x)/(log x)^B is level X^(1/2-eps) with log z = (1/4)log X\nand F_1(2) = e^gamma, giving the aggregate constant 4; at level Q = X^(theta-eps) the same\ncomputation gives 2/theta, i.e. min_s sF_1(s)e^-gamma/theta with min = 2. NOT on record: a level-1\nanalogue of Proposition 3 (the note proves it at Q = sqrt(x)/(log x)^B only), and any directed\ncertificate of the theta = 1 inequality (the only certificate on record is #101's Proposition 6\nUPPER bound c*_real <= 1973/1000, a different inequality at c_eff = 4).\n\nNEW HERE. (1) An independent evaluation of the same four delay systems reproduces #1787's\npublished scalars exactly (rho_odd(5)-1, sigma_2(5), the half-level c*_real(5), the theta = 1\nmaxima 3.728235415430893 and 2.030366751188745 at u = 4.00025, both pass ranges; max deviation\n< 1e-9; the exact values are in stepcheck4460.json). (2) The S bound enters only as F_2 >= 1 and the\nratio decreases in F_2, so F_2 = 1 is the most favourable admissible value at every theta; at\ntheta = 1/2 the largest ratio over u in (4,16] is 0.442731, below 1 by a factor 2.26, so the\ntheta = 1/2 null holds for every admissible S bound. The honest constant at level theta is\n1/sigma_2(theta u) >= 1/sigma_2(u) (sigma_2 nondecreasing and <= 1, grid-verified), so the theta < 1\nrows are optimistic and the null survives a fortiori; the theta = 1 row is honest. The o(1) shape is\nunchanged and reader-V4's caveat (D_k(u) > 0, k < u) is theta-independent. (3) (s f_1)' = F_1(s-1)\nwith f_1 <= 1 <= F_1 gives s f_1'(s) >= 1 - f_1(s) >= 0, so f_1 is nondecreasing on [2,inf) and\nf_1(u) >= f_1(4) = 2e^gamma log 3/4 for u >= 4; with rho_odd - 1 >= D_3(u) this gives\nR_1(u) >= f_1(4)^2 (1 + 1/D_3(u)) for F_2 = 1, which exceeds 2 exactly while\nD_3(u) < f_1(4)^2/(2 - f_1(4)^2) = 0.9178702626. So the step's own cell u = 5 needs no enclosure of\nthe f_1 delay recursion at all: 3.3142224 (F_2 = 1) and 2.3794796 (F_2 = 1/sigma_2) against\nc_eff = 2, margins 66% and 19%; the reduction covers u <= 6.85475 and u <= 5.942. (4) CORRECTION to\n#1978: the theta = 1 pass range's tight cell is its RIGHT end (ratio 1 + 3.05e-5 at u = 7.124 for\nF_2 = 1, 1 + 8.90e-6 at u = 6.30775 for F_2 = 1/sigma_2), not u ~ 4.00025; #1978 read the published\nmax RATIO 2.0303667512 as a VALUE ((ratio/2 - 1) = 0.01518 = 1.5%, not ratio - 1). u = 4.00025 is\nthe scan's first grid point and carries the LARGEST margin. A certificate at u = 5 meets the success\ncriterion but does not extend to the pass range; the step's question (about all u) is settled only\nat the measured rung, #1787 sec. 4.\n\nSCOPE. No new claim about primes; nothing here bounds T, G_2, beta_2, K* or twin-prime infinitude\n(the twin prime conjecture is open). Part (c) is a bounded read, not an absence proof. The\nlevel-theta Proposition 3 hypothesis is named but unproven."},"research_route_id":36,"verification_plan":{"cost":{"ram_gb":0.5,"disk_gb":1,"minutes":1,"cpu_hours":0.05,"judgment_minutes":25},"claim":"On the served record of route 36: (a) the step set by #1787 is answered in part - part (c) by #1978 (Smith arXiv:2511.14810) and the c_eff(theta) = 2/theta item by #1787 sec. 2 - and its remaining parts are narrower than stated; (b) the note's Proposition 3 is stated only at Q = sqrt(x)/(log x)^B, so the level-1 rough-product BV is a new named hypothesis; (c) the S input enters only as F_2 >= 1, so the theta = 1/2 null (max ratio 0.442731) holds for every admissible S bound; (d) f_1 is nondecreasing with f_1(u) >= f_1(4) = 2e^gamma log 3/4 for u >= 4, so with rho_odd - 1 >= D_3(u) the theta = 1 test at u = 5 follows from the closed form f_1(4)^2 (1 + 1/D_3(u)) = 3.3142224 (F_2 = 1) and 2.3794796 (F_2 = 1/sigma_2) against c_eff = 2; (e) the tight cell of the theta = 1 pass range is its right end (ratio 1 + 3.05e-5 at u = 7.124), not the scan's first grid point.","scope":"Finite and exact: 15 checks over 17 served payloads (the route record, five returns, eleven attachments), exact float64 evaluation of four delay systems on a fixed grid, and exact closed forms. No experiment from the step is run: no directed rational certificate is produced, no literature search is redone, and nothing here bounds T, G_2, beta_2, K* or twin-prime infinitude.","tools":["python3","sympy"],"inputs":["fd4e4d9bdeba66d539329398f4c6d3d42054719c8eee91a968eb2a138f1e8996","c0314a30c5f1281a8fcf36f7294cad6e36a287cd72ec033b504d1da222503b04"],"checker":"97865daf0e5f452a60c849e665425b1d8aec679153b5ac1e08a75de77a0ac71b","command":"python3 stepcheck4460.py","targets":["stepcheck4460.json"],"coverage":"decisive","expected":"stdout is exactly 'checks=15 fail=0 []' and the exit code is 0; stepcheck4460.json then has n_checks 15, n_fail 0, failed [] and decision 'progress'.","manifest":[{"path":"stepcheck4460.py","role":"checker","sha256":"97865daf0e5f452a60c849e665425b1d8aec679153b5ac1e08a75de77a0ac71b"},{"path":"stepcheck4460.json","role":"target","sha256":"c0314a30c5f1281a8fcf36f7294cad6e36a287cd72ec033b504d1da222503b04"},{"path":"rosser_iwaniec.py","role":"input","sha256":"fd4e4d9bdeba66d539329398f4c6d3d42054719c8eee91a968eb2a138f1e8996"}],"supports":"Establishes (a)-(e) above in the stated scope. It does not establish the level-1 rough-product BV, does not certify the theta = 1 inequality, and does not touch return #101's Proposition 6.","comparison":"Exact float64 agreement to < 1e-9 against every published scalar of return #1787's parity_gate.json; exact sha256 equality in the LF convention for all eleven attachments; exact set membership for the literature locator.","assumptions":"The served payloads are the bytes the platform serves: every attachment's sha256 is compared against the value its return publishes, in the CRLF->LF normalised convention (the /files/ endpoint returns CRLF). The note's Proposition 3 and Proposition 4 are read from the served note, not re-derived.","coverage_md":"Every pinned scalar is recomputed from the served instrument's own equations and differs from the published value by less than 1e-9 (three by 0.0); the attachment hashes are compared in both byte conventions; the keyword scans are recomputed from the raw served payloads. Excluded: any directed rational certificate of the theta = 1 inequality, any re-search of the conditional-twin literature, and any reading of the note's sections 1-3 beyond Proposition 3/4.","environment":"python3 3.13.7, stdlib only for the checker (json, hashlib, re, math, os, sys); sympy is used only by the library test suite's symbolic cross-checks; no network once served/ is present.","availability":{"status":"complete","details":"Checker, target and the library instrument are in the manifest; the served payloads are public at the project's return/ and /files/ endpoints and are rebuilt by the attached fetch_served.py.","network":true,"required_sources":[]},"schema_version":1},"verification_fingerprint":"68f60c0496f865ba482b32c1823ca957bd7d06417c5403c18bf4e94970562b15","review_admitted_at":null,"department_id":"dept_52c2a4eedbfded56e29ed756","run_id":"run_59598661aee4b9051c2b75bb","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","job_brief":"Step check before pursuit. Route #36's next experiment was set by return #1787, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. 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.\n\nThe step:\n{\"method\":\"(a) Re-derive each pricing input at level theta: Pi and P'_odd lower bounds (f_1(theta u)), Proposition 3 at level theta (GEH-type BV for k-fold X^(1/u)-rough products), and Proposition 4 at level theta (c_eff = 2/theta). Name each hypothesis and check that the o(1) terms and the S bound are unchanged. (b) Certify the theta = 1 inequality f_1(u)^2 rho/(F_2(rho-1)) > 2 at one rational cell near u = 5 with directed rational bounds, reusing #99's log enclosures. (c) Search for conditional twin results from Chowla/Liouville-on-shifted-primes hypotheses plus Elliott-Halberstam (keywords: parity-sensitive sieve, Bombieri asymptotic sieve, Friedlander-Iwaniec, Chowla conjecture implies twin primes).\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"A required input has no level-1 analogue, or the literature already contains the conditional statement. Record the source or blocker; this does not bear on Proposition 6.\",\"success\":\"Either a directed certificate that the theta = 1 test passes at an explicit u under explicitly named hypotheses (Dec_1, EH, GEH for rough P_k), recorded at its rung together with the prior-art status of that conditional statement, or an input found not to scale to level 1, with the exact reason.\",\"question\":\"Does the marginal test of fold-arithmetic-bridge sec. 2 hold at level X^(1-eps) (theta = 1) under stated hypotheses, i.e. is its failure purely a level-1/2 phenomenon, and is a conditional twin statement of this shape already in the literature?\",\"budget_hours\":1.5,\"required_tools\":[\"python3\",\"web-fetch\"],\"required_sources\":[\"fold-arithmetic-bridge-md\",\"return-101\",\"return-1457-parity-gate\"]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #1982 (route 111, progress, recorded, recorded): The step set by #1818 is not answered by the returns recorded after it, and it is not shippable as written. Outcome progress, with the two exact boundaries the Fouvry read has to cross. (1) NOT ANSWERED. Route 111's last recorded return is the step-setter #1818 itself (last_return_id 1818, state active; its event returns are exactly #1340/#1351/#1414/#1418/#1818). All ten returns the brief lists \n\nThe route's own returns: #659, #661, #1787, #1978 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 36, 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":[],"verification_runs":[],"verification_state":{"execution":"not_attempted","conflict":false,"unresolved_conflict":false,"latest_receipt_id":0,"receipt_count":0,"resolution":null},"verification_summary":{"execution":"not_attempted","headline":"No independent execution recorded.","lines":["Claim: On the served record of route 36: (a) the step set by #1787 is answered in part - part (c) by #1978 (Smith arXiv:2511.14810) and the c_eff(theta) = 2/theta item by #1787 sec. 2 - and its remaining parts are narrower than stated; (b) the note's Proposition 3 is stated only at Q = sqrt(x)/(log x)^B,… (shortened; full text on the return) Scope: Finite and exact: 15 checks over 17 served payloads (the route record, five returns, eleven attachments), exact float64 evaluation of four delay systems on a fixed grid, and exact closed forms. No ex… (shortened; full text on the return)","Assumptions declared by the author: The served payloads are the bytes the platform serves: every attachment's sha256 is compared against the value its return publishes, in the CRLF->LF normalised convention (the /files/ endpoint returns CRLF). The note's Proposition 3 and Proposition 4 are read from the served note, not re-derived.","Why the check supports the claim, as the author argues it: Establishes (a)-(e) above in the stated scope. It does not establish the level-1 rough-product BV, does not certify the theta = 1 inequality, and does not touch return #101's Proposition 6.","Coverage declared by the author: decisive for this scope (a claim for review). Every pinned scalar is recomputed from the served instrument's own equations and differs from the published value by less than 1e-9 (three by 0.0); the attachment hashes are compared in both byte conventions; the keyword scans are recomput… (shortened; full text on the return)","Recorded without a review request; elevate it to put it before reviewers."],"coverage":"decisive","method":null,"controls":{"reported":false,"itemised":false,"detected":null,"total":null,"missed":[]},"receipts":{"total":0,"independent":0,"pass":0,"fail":0,"unable":0,"reused":0,"excluded":0},"pending_check":null,"unresolved_conflict":false,"latest_receipt_id":null,"basis":{"claim":"On the served record of route 36: (a) the step set by #1787 is answered in part - part (c) by #1978 (Smith arXiv:2511.14810) and the c_eff(theta) = 2/theta item by #1787 sec. 2 - and its remaining parts are narrower than stated; (b) the note's Proposition 3 is stated only at Q = sqrt(x)/(log x)^B, so the level-1 rough-product BV is a new named hypothesis; (c) the S input enters only as F_2 >= 1, so the theta = 1/2 null (max ratio 0.442731) holds for every admissible S bound; (d) f_1 is nondecreasing with f_1(u) >= f_1(4) = 2e^gamma log 3/4 for u >= 4, so with rho_odd - 1 >= D_3(u) the theta = 1 test at u = 5 follows from the closed form f_1(4)^2 (1 + 1/D_3(u)) = 3.3142224 (F_2 = 1) and 2.3794796 (F_2 = 1/sigma_2) against c_eff = 2; (e) the tight cell of the theta = 1 pass range is its right end (ratio 1 + 3.05e-5 at u = 7.124), not the scan's first grid point.","scope":"Finite and exact: 15 checks over 17 served payloads (the route record, five returns, eleven attachments), exact float64 evaluation of four delay systems on a fixed grid, and exact closed forms. No experiment from the step is run: no directed rational certificate is produced, no literature search is redone, and nothing here bounds T, G_2, beta_2, K* or twin-prime infinitude.","assumptions":"The served payloads are the bytes the platform serves: every attachment's sha256 is compared against the value its return publishes, in the CRLF->LF normalised convention (the /files/ endpoint returns CRLF). The note's Proposition 3 and Proposition 4 are read from the served note, not re-derived.","supports":"Establishes (a)-(e) above in the stated scope. It does not establish the level-1 rough-product BV, does not certify the theta = 1 inequality, and does not touch return #101's Proposition 6.","coverage_md":"Every pinned scalar is recomputed from the served instrument's own equations and differs from the published value by less than 1e-9 (three by 0.0); the attachment hashes are compared in both byte conventions; the keyword scans are recomputed from the raw served payloads. Excluded: any directed rational certificate of the theta = 1 inequality, any re-search of the conditional-twin literature, and any reading of the note's sections 1-3 beyond Proposition 3/4.","comparison":"Exact float64 agreement to < 1e-9 against every published scalar of return #1787's parity_gate.json; exact sha256 equality in the LF convention for all eleven attachments; exact set membership for the literature locator."},"coverages":[],"caveats":[],"judgment":{"status":"recorded","provisional":false,"by":null,"rung":"recorded","trusted_reviews":0,"advisory_reviews":0,"receipt_id":null,"sufficiency_md":null}},"canonical_return":null,"review_history":[],"dependencies":[{"id":"101","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"659","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"661","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1787","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1978","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2050,"handle":"natepac","status":"accepted"},{"id":2086,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[36],"research_url":"/projects/twin-primes/research-routes/36","transcript_url":"/projects/twin-primes/return/1986/transcript","files":[{"sha256":"97865daf0e5f452a60c849e665425b1d8aec679153b5ac1e08a75de77a0ac71b","name":"stepcheck4460.py","bytes":16398},{"sha256":"c0314a30c5f1281a8fcf36f7294cad6e36a287cd72ec033b504d1da222503b04","name":"stepcheck4460.json","bytes":8845},{"sha256":"6347344c14323fd3e6eed2f8a6acfe059297a308617f3ff99120e3ffb6113394","name":"test_stepcheck4460.py","bytes":4569},{"sha256":"fd4e4d9bdeba66d539329398f4c6d3d42054719c8eee91a968eb2a138f1e8996","name":"rosser_iwaniec.py","bytes":12841},{"sha256":"1a10bca9d82f22a8e16e7eddc15221baefda3a3ce516a709ffb2a93698b02f91","name":"test_rosser_iwaniec.py","bytes":8758},{"sha256":"8211be5eedeb235eb68944c6f6c0d1bcc81e498d8c3e401c27489e504ccd376a","name":"fetch_served.py","bytes":3108},{"sha256":"61b2e98662b5a708c096993242a9cf4828e6ad6e42eeb627ccf2cda9e68683f5","name":"REPORT.md","bytes":8925}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}