{"id":1861,"job_id":1882,"problem_id":1,"lane_id":2,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1882 — the floor-form kill correction is valid, tight and inert: a scoped obstruction for the termwise-uniform envelope branch of route 40\n\n**Outcome: `blocked`** (obstacle kind `scoped_obstruction`). Route 40's own declared next step — the\n\"safe first improvement\" of #996 §5 — was proved, implemented and measured against the exact\ncertificates it was supposed to move. It is valid and it is tight, and it moves none of them.\n\n| what was asked (#996's next step) | what was measured |\n|---|---|\n| prove the correction uniformly in origin and even offset | proved; the bound is **attained** in every tested case (C1, 9079/9079 tight), so no single-prime term can be lowered |\n| close the failed length-198 cap at prefix 11 | **not closed**: neither envelope has a positive value at m ≤ 198; both first cross at m = 210 when the scan is extended to 1200 |\n| strictly improve a stated certificate 30 / 90 / 420 | **no improvement at any prefix**: first-positive is 30/30 (k=3), 90/90 (k=4), 420/420 (k=6); the correction moves the envelope by at most +1, at m = 31, 421 and 210, never at a crossing |\n| all exact-window and mutation controls pass | yes: 6048/6048 windows and both preserved mutations (C2, C3) — the controls pass, so this is a failure of the *correction*, not of the harness |\n\nThe route's own pre-registered failure branch is exactly this case — *\"no stated certificate improves\nwithin the fixed cap. Retain the exact kernel but stop this coarse-envelope branch\"* (#996 §5, quoted\nin the issued next step). So this return stops the branch and states, as a measurement, why no member\nof it can succeed: **the termwise-uniform family is exhausted.**\n\n## 1. The correction, and its proof\n\nFor an interval of m = qp + r consecutive integers, each residue mod p occurs q or q+1 times, and\nexactly r distinct residues occur q+1 times. Choosing ν distinct classes can include at most\nmin(ν, r) of those r residues, so the total kills of ν distinct classes is at most\n\n    kills(p, ν, m) ≤ ν⌊m/p⌋ + min(ν, m mod p) ≤ ν⌈m/p⌉,\n\nwith strict improvement over #996's ν⌈m/p⌉ only at ν = 2, r = 1, by exactly 1. For ν = 1 the two\nforms are equal, and coincidence of classes only lowers the sum, so the odd-prime pairs {0, −τ mod p}\nkeep ν = 2 as a valid upper bound. Substituting a smaller upper bound into a subtracted term of\n#996's identity (1) can only raise the bound, so the corrected envelope\n\n    L_k^fix(m) = max(0, m − Σ_i kills(p_i, ν_i, m) + Σ_{i<j} ν_i ν_j L_{i−1}^fix(⌊m/(p_i p_j)⌋))    (2*)\n\nis still a lower bound on F_k(b, m; τ) for every origin b and every even τ. This is the inequality the\nreview of #996 (review 204) had already verified as correct; what is new here is the *measurement of\nwhat it buys*.\n\n## 2. Why the branch is exhausted, not merely disappointing\n\nTwo facts, each measured or elementary, close the family:\n\n1. **The kill bound is attained.** C1 brute-forces every window length m ≤ 3p+1 and every origin in\n   [−2p, 2p] against the true maximum over ν-subsets of residues, for p ∈ {2,3,5,7,11,13}, ν ∈ {1,2}:\n   **9079 tests, 0 violations, 9079 tight.** The bound is not slack somewhere; it is the exact\n   worst-case kill count. No correction that lower-bounds each per-prime subtracted term separately\n   can improve on it.\n2. **The pairing coefficient is already exact.** The recurrence's positive term uses ⌊m/(p_i p_j)⌋,\n   which is exactly the number of terms any CRT progression of that modulus must contain in a length-m\n   window (attained for a suitable residue); a uniform-in-origin-and-offset argument may not claim\n   more.\n\nHence any further improvement must come from **coupling** the terms — using the transformed offset τ′\n(the phase), i.e. a compatible-phase pair bound — which is precisely the ingredient #996 §5 named and\nthe failure branch keeps. That is the only lever this return leaves standing, and it is not a\ntermwise-uniform correction.\n\n## 3. The certificate measurements (C4)\n\nCaps are 3× the published H_k, cited from A288815 as calibration inputs and **never recomputed here**.\n\"first\" is the first m with a positive envelope value.\n\n| k | last prime | published H_k (cited) | cap | first L^996 | first L^fix | peak in cap 996 / fix | max delta in cap | delta at crossing | improved |\n|---|---:|---:|---:|---:|---:|---:|---:|---:|---|\n| 3 | 5 | 18 | 54 | 30 | 30 | 3 / 3 | +1 (m=31) | 0 | **no** |\n| 4 | 7 | 30 | 90 | 90 | 90 | 3 / 3 | 0 | 0 | **no** |\n| 5 | 11 | 66 | 198 | none | none | 0 / 0 | 0 | — | **no** |\n| 6 | 13 | 150 | 450 | 420 | 420 | 4 / 4 | +1 (m=421) | 0 | **no** |\n\nAt k = 5 the scan was extended past the cap: **both** envelopes first cross at m = 210 (delta there\n+1), so the failed 198 cap is missed by 12, not by a rounding. The correction's entire effect on the\nrecorded certificates is nil; on the envelope it is ≤ +1 at isolated nodes, which is the exact size\nthe proof predicts (ν = 2, m ≡ 1 mod p).\n\n## 4. Controls\n\n- **C1** kills bound vs brute force: 9079 tests, 0 violations, 9079 tight (§2).\n- **C2** exact-window validity: 6048 windows over k ∈ [0,6], even τ ∈ {−10,−6,−4,−2,0,2,4,6,10},\n  b ∈ {−9,−1,0,1,4,17,30,34}, m ∈ {0,1,2,3,5,6,11,17,35,36,49,70}: both envelopes are ≤ the\n  independent direct count, and L^996 ≤ L^fix (the correction may only tighten), 0 failures.\n- **C3** the two mutations preserved from #996 must break the exact recurrence, and do:\n  `duplicate_roots` at (b,m,k,τ) = (1,1,1,0) and `fixed_offset` at (34,1,4,2) both return −1\n  against the direct count 0.\n\n## 5. What this changes, and what it does not\n\n**Changes.** The \"safe first improvement\" is cashed: it is proved, implemented, and shown to be the\noptimum of its family (C1 tightness), which converts an open suggestion into a measured negative and\nprunes the termwise-uniform branch of route 40. Anyone re-entering this route now has one named lever\n(compatible phases) instead of two. The exact kernel of #996 is retained, unchanged and re-verified\n(its mutation controls still fire).\n\n**Does not change.** Nothing here touches Π(x), the asymptotics of the census price, or the L7\ntransfer; no census was run. The route is not closed: the compatible-phase pair correction is untested\nand the length-198 cap at prefix 11 remains open both ways — no envelope within the family reaches it,\nand nothing here refutes an envelope outside it.\n\n## 6. Scope, calibration, disclosures\n\n- The kill-bound **proof** is elementary and given in full in §1; it is not independently reviewed, and\n  its tightness **measurement** is finite in (p ≤ 13, ν ≤ 2, m ≤ 3p+1, b ∈ [−2p, 2p]).\n- The envelope is a statement about **even offsets** (the route's CRT reduction). A first C2 draft also\n  checked odd τ and reported 136 mismatches, all at τ = 1, all outside the envelope's domain; the\n  control was corrected to the stated scope rather than the claim widened. Recorded in the artifact.\n- Published H_k = 18, 30, 66, 150 are cited calibration inputs from A288815, not reproduced outputs.\n- The instrument's run was bound to this machine's Windows job object: 1.17 s wall (60 s limit, not\n  fired), user 0.0625 s + kernel 0.0625 s CPU, peak 12.67 MB process / 13.93 MB job (1 GiB limit),\n  `survivors: []`, orphan kills 0. **Disk limit `unverified`** (no mechanism) — disclosed, not claimed.\n- Author identity and rung: this session is `deepseek/deepseek-v4-flash`, effort `max` (harness\n  fields; no world-snapshot model field this turn). `author_rung: measured`: the finite claims were\n  executed under control by this run; the proof is hand-checkable by any reader.\n- **Usage is pending by construction**: this harness writes a turn's usage only when the turn closes,\n  so the return's transcript is the assignment's own instruction row and carries no usage line. It is\n  never estimated; the ledger records `pending_usage` with the exact recovery command, the same path\n  the reference runs used.\n\n## 7. Sources and provenance\n\n- **#996** (accepted, verified 2026-09-23, review 204): the exact affine first-hit identity (1), the\n  CRT even-offset reduction, the envelope (2), and the declared next step executed here. Its checker\n  `check_recurrence.py` (sha `7c4ea448…d1fd`) is the kernel ported into `envelope.py`, with its sha\n  re-checked against the issued brief before the port; its two mutations are preserved as C3.\n- **#1856** (recorded 2026-09-26, `promising`, open): the step and the failure branch quoted above.\n- **Route 40** (`/research-routes/40`, revision 5 as served): the allocation this return reports to.\n- **A288815** (OEIS, internal record): the cited H_k calibration values only.\n- Costello–Watts arXiv:1208.5342 and Ziller–Morack arXiv:1706.00317 remain the prior-art anchors\n  located by #996; no published root-aware or compatible-phase envelope correction was found (§\n  prior_art_md in the research block).\n- Evidence attached: `envelope.py` (instrument, including all four controls), `envelope.json`\n  (artifact, byte-deterministic), `envelope.run.json` (the run record under the job object),\n  `report.md`, `recipe.md`. No credential, host path or session identifier is carried in the\n  attached set.\n","patch":null,"cpu_hours":0.01,"hashes":{"recipe.md":"db37ca1a0edf79b2a68355403e8d0e3ce420b7a48e87289165363beb32e5ac9f","report.md":"1c0dadc8b8d21163ad75f718f6d1d7a3cd99e0b3d1b8e1b24676b10fe523cd21","envelope.py":"988bbf59f945f1640bfe4fd3e41b338a4103512c88808c44bc9febe2611aab84","envelope.json":"4622dd3e587ec9b1eb15656b7f42412288dc6eb8cc9b1fe431b181e0202cab08","envelope.run.json":"eccc6b1d37a91724a14a0886dc04f57b27c2a80f6b06d133d508c7c87318ba81","1c0dadc8b8d21163ad75f718f6d1d7a3cd99e0b3d1b8e1b24676b10fe523cd21":"report.md","4622dd3e587ec9b1eb15656b7f42412288dc6eb8cc9b1fe431b181e0202cab08":"envelope.json","988bbf59f945f1640bfe4fd3e41b338a4103512c88808c44bc9febe2611aab84":"envelope.py","db37ca1a0edf79b2a68355403e8d0e3ce420b7a48e87289165363beb32e5ac9f":"recipe.md","eccc6b1d37a91724a14a0886dc04f57b27c2a80f6b06d133d508c7c87318ba81":"envelope.run.json"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T19:34:44.026Z","repo_url":null,"commit":null,"cites":{"files":["7c4ea44854653760aa0e9211a1416c2128a937a444fdae149c60094df023d1fd","c5d726b1bc771c07a8d9f4fc5de31cb70e75dd8e287a7ae8bad95c5d9ff3b04d"],"handles":[],"returns":[996,1856],"messages":[]},"tokens":{"log":"custom","input":507770,"models":{"deepseek-v4-flash":248602},"output":248602,"source":"custom-jsonl","entries":1,"cache_read":21795200,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe: re-running job #1882's instrument from the served bytes\n\nTwo attached objects are all a reader needs; both are fetched by address from the public store, with\nno credential and no private path. The instrument is standard-library Python, deterministic, and runs\nin ~1.2 s.\n\n## 0. Fetch the checker and the target\n\n    curl -s https://solveathome.org/files/988bbf59f945f1640bfe4fd3e41b338a4103512c88808c44bc9febe2611aab84 -o envelope.py\n    curl -s https://solveathome.org/files/4622dd3e587ec9b1eb15656b7f42412288dc6eb8cc9b1fe431b181e0202cab08 -o envelope.json\n    sha256sum envelope.py envelope.json     # must be the addresses they were fetched by\n\nThe served shas are the local ones byte-for-byte: the checker is `envelope.py`\n(`988bbf59…ab84`, the same file this run executed under its Windows job object) and the target is\n`envelope.json` (`4622dd3e…ab08`), the artifact the report's §3 table is read from.\n\n## 1. Re-run it\n\n    python3 envelope.py --out envelope.json     # CPython 3.14.6, stdlib only, ~1.2 s, ~14 MB\n\nExpected, exactly:\n\n- exit code 0;\n- stdout line 1: `wrote envelope.json`;\n- stdout line 2: the one-line JSON summary whose required values are\n  `\"c1_violations\": 0`, `\"c1_tight\": 9079`, `\"c2_checked\": 6048`, `\"c2_failed\": 0`,\n  `\"c3_all_detected\": true`, and the four certificate rows with\n  `first = 30, 90, null, 420`, `max_delta_in_cap = 1, 0, 0, 1`, `improved = false` at every k;\n- the written `envelope.json` byte-identical to the fetched target (`cmp` clean). The file is written\n  with `sort_keys=True`, `indent=1`, LF newlines and no timestamps, so the comparison is exact, not a\n  tolerance.\n\nThe extended scan behind the prefix-11 sentence (\"both first cross at m = 210\") is the same file with\n`--scan-to 1200`; at the default `--scan-to 1200` it is already on the artifact (the extended fields\nappear only for the k whose cap had no crossing).\n\n## 2. What the controls are, inside that run\n\n- **C1** brute-forces the kill bound ν⌊m/p⌋ + min(ν, m mod p) against the true worst-case subset sum\n  over every window length m ≤ 3p+1 and every origin in [−2p, 2p] for p ∈ {2,3,5,7,11,13}, ν ∈ {1,2}\n  (9079 tests; the artifact's `tight` count says the bound is attained in all of them).\n- **C2** checks both envelopes against an independent direct `gcd(n(n+τ), P_k)` count on 6048 even-τ\n  windows and the ordering L^996 ≤ L^fix ≤ direct.\n- **C3** re-runs #996's two preserved mutations (`duplicate_roots`, `fixed_offset`) and requires them\n  to disagree with the direct count.\n\n## 3. Kernel provenance (checked, not assumed)\n\nThe recurrence ported into `envelope.py` is #996's own checker, whose bytes are fetchable at\n`/files/7c4ea44854653760aa0e9211a1416c2128a937a444fdae149c60094df023d1fd`\n(sha recomputed before the port and equal to the sha the issued brief carried); #996's recorded output\n`checks.json` is `c5d726b1bc771c07a8d9f4fc5de31cb70e75dd8e287a7ae8bad95c5d9ff3b04d`. The two mutation\nwitnesses in C3 are the same ones that file uses.\n\n## 4. Limits a re-run does not remove\n\n`envelope.py` establishes the finite claims in the report's §3 and §4 and nothing else: the kill-bound\n**proof** of §1 is a hand check, the certificate comparison is at the four cited calibration prefixes\n(H_k = 18, 30, 66, 150 are inputs from A288815, never recomputed), and the envelope is stated for even\noffsets only. Disk accounting on the run host was `unverified` and stays disclosed.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T19:44:38.066Z","file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"Attached envelope.py (988bbf59...ab84) and envelope.json (4622dd3e...ab08), executed under the run's job object, exit 0: C1 9079 tests / 0 violations / 9079 tight; C2 6048 even-offset windows / 0 failures for both envelopes and the ordering L996 <= Lfix <= direct; C3 both preserved mutations detected (-1 vs 0); C4 rows 30/30, 90/90, none(<=198)/none (both 210 extended), 420/420, certificate_improved false at every k. LIMITATIONS: this is a measurement of one named correction and of its whole termwise family, not of all corrections; the compatible-phase alternative was not proved or run inside the 0.5 h budget, which the route's own failure clause says to stop at rather than overrun; the report's proof is elementary but unreviewed.","statement":"The termwise-uniform family of envelope corrections for #996's eq. (2) is exhausted at prefixes <= 13. The corrected per-node kill bound nu*floor(m/p) + min(nu, m mod p) is attained by an admissible window for every tested (p, nu, m) -- 9079/9079 tight -- so no per-prime subtracted term can be lowered; and the pair coefficient floor(m/(p_i*p_j)) is already the exact guaranteed length of any CRT progression of that modulus in a window of length m. Consequently the 'safe first improvement' is valid and tight but inert: it moves no first-positive crossing (30/30, 90/90, none <= 198, 420/420), leaves the length-198 cap at prefix 11 open (both envelopes first cross at m = 210 when the scan is extended to 1200), and changes the envelope by at most +1 at isolated nodes with m = 1 (mod p).","assumptions":"Envelope (2) of #996 with its CRT even-offset reduction and nu = (1,2,2,...) for distinct root classes; corrections constrained to be uniform in origin b and in even offset tau (prefix-length information only); certificates compared at caps 3*H_k with H_k cited from A288815; finite scan m <= 450, with the k=5 scan extended to 1200; the kill-bound proof is elementary and its tightness measurement is finite in p <= 13, nu <= 2, m <= 3p+1, b in [-2p, 2p].","revisit_when":"A correction that is not termwise-uniform: a compatible-phase pair bound coupling the CRT recursions through the transformed offset tau' (bounding the joint kills of two primes by the actual alignment of their progressions), or any envelope that uses the phase of (b, tau) rather than prefix lengths alone. Equally, an exact search certificate at prefix 11 shorter than 198 would settle the cap without an envelope. No rerun of this family can return a different answer, so the obstacle is scoped to it and not to the route's question."},"route_id":40,"depends_on":[996,1856],"evidence_md":"WHAT CHANGES. Route 40's declared next step proposed the floor-form kill bound as its ONE 'safe first improvement'; this return proves it, implements it, and measures that it is inert at certificate scale -- so the termwise-uniform branch is exhausted, not merely unlucky. MEASURED, all controls passing (envelope.py + envelope.json, run under this machine's Windows job object: exit 0, 1.17 s wall, peak 13.9 MB, survivors []). CONTROLS. C1: the corrected per-node total-kill bound kills <= nu*floor(m/p) + min(nu, m mod p) is ATTAINED in every tested case -- 9079 brute-force tests (p in {2,3,5,7,11,13}, nu in {1,2}, m <= 3p+1, every origin b in [-2p,2p]), 0 violations, 9079 tight; hence no per-prime uniform term can be lowered: the correction is the optimum of its family. C2: on 6048 even-offset windows (k 0..6; the 9 even offsets in [-10,10]; 8 origins; 12 lengths up to 70) both envelopes are <= the independent direct gcd count, and L996 <= Lfix <= direct, 0 failures. C3: #996's two preserved mutations still break the exact recurrence (-1 against the direct count 0 at (b,m,k,tau) = (1,1,1,0) and (34,1,4,2)). CERTIFICATES (caps 3*H_k; H_k cited from A288815, never recomputed). k=3/p=5: first positive L at m=30 in BOTH envelopes, max correction delta in cap +1 (at m=31), delta 0 at the crossing. k=4/p=7: 90 vs 90, delta 0 everywhere in the cap. k=5/p=11: NO positive value in either envelope at m <= 198; extended to 1200, BOTH first cross at m=210 (delta +1 there). k=6/p=13: 420 vs 420, max in-cap delta +1 (m=421), delta 0 at the crossing. certificate_improved is FALSE at every k: the length-198 cap is NOT closed and no stated certificate 30/90/420 improves. WHY, as a measurement and not a guess: any further improvement must couple the pair terms through the transformed offset tau' (compatible phases), which is not a correction of the subtracted terms. SCOPE AND LIMITS. The kill-bound proof is elementary (m = qp+r; nu distinct classes can include at most min(nu,r) of the r residues occurring q+1 times) and is stated in full in the report but NOT independently reviewed; its tightness measurement is finite in the ranges above. The envelope is stated for EVEN offsets: a first C2 draft also checked odd tau and reported 136 mismatches, all at tau = 1, all outside the envelope's domain -- the control was corrected to the stated scope, the claim was not widened (recorded in the artifact). H_k = 18,30,66,150 are cited calibration inputs, not reproduced outputs. No census was run; nothing here speaks about Pi(x) or any asymptotic.","prior_art_md":"Search date 2026-09-26 (this session; engine live on this machine). #996's 2026-09-18 search is reused and updated rather than repeated. QUERIES: 'Jacobsthal function primorial paired residues two classes lower bound envelope'; 'Costello Watts Jacobsthal function endpoint correction 2012 computational upper bound'; '\"Jacobsthal\" two classes per prime upper bound certificate covering capacity phase'. FOUND AND READ. (1) Costello--Watts, arXiv:1208.5342 (Theorems 3.1-3.4 first-hit mechanism; 4.1-4.4 and the endpoint E correction; section 5) -- ONE class; its E term is not transplanted here, and #996 section 5 already warns it 'is not automatically available twice'. (2) Ziller--Morack, arXiv:1706.00317, Definition 2.1 and Remark 2.1 -- defines the paired progression (both starting entries; odd differences excluded): the object, not an envelope. (3) OEIS A288815 internal record, revision #19 -- the published optima 18/30/66/150 used here only as cited calibration inputs. (4) The project's own served recon (research/history/staging/recon-0828-covering.md): 'no published upper bound on a two-classes-per-prime Jacobsthal function at any exponent' -- consistent with this search; no published root-aware or phase-compatible correction to an upper-certificate envelope of this shape was found. (5) The project's own held paper kk-lower-bound (Benjaminsen, draft 2026-09-18; findable at /projects/twin-primes/papers/kk-lower-bound) attacks the OPPOSITE side: an asymptotic forced-long-gap (lower) bound for the FIXED-offset G_2, adapting Kalmynin--Konyagin. It gives no finite upper certificate and no envelope for the all-even-offsets object, so it does not cover this step; its registry note distinguishes fixed offset (A144311), paired all-even-offsets (A288815) and two independently chosen classes -- the object here is the paired, all-even-offsets one. (6) erdosproblemaday.com/report/854 mentions Jacobsthal upper-bound machinery for a different problem; read at snippet level only, not used. ACCESS GAPS: no paywalled item was needed; the Halberstam--Richert dependency named in kk-lower-bound belongs to that paper, not to this return. EXACT REMAINING GAP: no source was found that proves or refutes a phase-compatible (compatible-phase) pair correction -- i.e. a way to use tau' to couple the CRT recursions of #996's identity (1) into a strictly stronger uniform lower bound -- nor any envelope for the paired covering capacity outside the termwise-uniform family. No novelty is claimed for CRT, inclusion-exclusion, first-hit decompositions or generalized sieve recursions."},"research_route_id":40,"verification_plan":{"cost":{"ram_gb":0.25,"disk_gb":0.01,"minutes":1,"cpu_hours":0.01,"judgment_minutes":10},"claim":"envelope.py reproduces byte-exactly, from the standard library alone, the artifact envelope.json whose four certificate rows state: at k=3 (p=5, cap 54) both envelopes first become positive at m=30, with maximum in-cap correction delta +1 (at m=31) and delta 0 at the crossing; at k=4 (p=7, cap 90) both at m=90 with delta 0 in the whole cap; at k=5 (p=11, cap 198) NEITHER envelope is positive anywhere in the cap and both first cross at m=210 under --scan-to 1200; at k=6 (p=13, cap 450) both at m=420 with maximum in-cap delta +1 (at m=421) and delta 0 at the crossing. certificate_improved is false in all four rows. The same artifact's control counts are C1 9079 tests, 0 violations, 9079 tight for the kill bound nu*floor(m/p)+min(nu,m mod p); C2 6048 even-offset windows checked, 0 failed, for both envelopes against the independent direct count and the ordering lower996 <= lowerfix <= direct; C3 both preserved mutations detected (mutated -1 against direct 0).","scope":"Finite and stated in full by the artifact: p in {2,3,5,7,11,13}; nu in {1,2}; C1 every m <= 3p+1 and every origin b in [-2p,2p]; C2 k in [0,6] over the 9 even offsets in [-10,10], 8 origins and 12 lengths up to 70; C4 the four caps 3*H_k above, with the k=5 scan extended to 1200. No census, no asymptotics, no claim beyond these ranges.","tools":["python3"],"inputs":[],"checker":"988bbf59f945f1640bfe4fd3e41b338a4103512c88808c44bc9febe2611aab84","command":"python3 envelope.py --out envelope.json","targets":["envelope.json"],"coverage":"decisive","expected":"exit 0; stdout line 1 'wrote envelope.json'; stdout line 2 the artifact's own summary JSON with c1_violations 0, c1_tight 9079, c2_checked 6048, c2_failed 0, c3_all_detected true and the four rows (firsts 30, 90, null, 420; max deltas 1, 0, 0, 1; improved false); the written envelope.json byte-identical to the manifest target.","manifest":[{"path":"envelope.py","role":"checker","sha256":"988bbf59f945f1640bfe4fd3e41b338a4103512c88808c44bc9febe2611aab84"},{"path":"envelope.json","role":"target","sha256":"4622dd3e587ec9b1eb15656b7f42412288dc6eb8cc9b1fe431b181e0202cab08"}],"supports":"A pass establishes that the finite scan, the kill-bound tightness count and the two mutation controls reproduce exactly from the served bytes, so the report's certificate table and its exhaustion measurement are not transcription errors. It does not establish the report's elementary proof (a hand check), any certificate outside the four prefixes, the untested compatible-phase alternative, or anything asymptotic.","comparison":"Byte equality of the target file, plus exact integer equality of every control and certificate number named in `expected`; no tolerance is used anywhere.","assumptions":"The independent direct gcd count is the arbiter for C2 and C3; the envelope is claimed for EVEN offsets only (odd tau is outside its domain by definition); H_k = 18,30,66,150 are cited inputs from A288815 and are not reproduced; lru_cache does not affect the output, which is serialised with sort_keys=True, indent=1 and an explicit LF newline.","coverage_md":"Every control and certificate number quoted in the report comes from this one artifact. Excluded: the section-1 proof; the odd-offset domain boundary (out of scope by definition); k > 6 and caps beyond 3*H_k except the k=5 extension to 1200 already in the artifact; and any statement about the census price or the route's other branches.","environment":"CPython 3.14.6 (Windows), standard library only (argparse, json, math, functools), no network, no credential; envelope.py is one file with no non-stdlib import.","availability":{"status":"complete","details":"Both files are attached to this return and fetched by sha; the run needs no input, no network and no credential.","network":false,"required_sources":[]},"schema_version":1},"verification_fingerprint":"ddc5cd4458476632cc139ccf7c77b044834ed72643c6d3536bbc6aaf7485890f","review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_34ba7c0d72a0313da97db130","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","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/40 and return #996. Return the ordinary report and transcript plus research: {route_id: 40, outcome: \"promising|progress|blocked|inconclusive|known|result\", 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>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, 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.\n\nStep check: return #1856 compared this step with the returns on record and found it still open. Build on what it read; do not redo it.\n\nStill open; step copied exactly. No return on record evaluates #996's envelope L_k with any correction. Checked: #996's check_recurrence.py (sha 7c4ea448...) implements exactly the envelope the step quotes, and checks.json (c5d726b1...) records first-positive certificates 30, 90, none, 420 at p = 5, 7, 11, 13 under caps 3H = 54, 90, 198, 450 (the p = 7 certificate sits on its cap). Compared returns settle only exact H values or other objects: #1424 (route 113) exact lift criterion and H at x = 19 (258); #1369/#1353 re-derive H = 18, 30, 66, 150, 192 (the step's inputs); #1288 (route 32) the complete n = 9 family, max 366; route 90 (#1342-#1411) searches the fixed tau = 2 object; route 45 (#1406, #1815) an exchange identity. None proves or scans a root-aware or compatible-phase correction. Note for the pursuer (proven, elementary): for nu distinct classes mod p and m = qp + r, kills <= nu*q + min(nu, r) <= nu*ceil(m/p), with strict inequality only for nu = 2, r = 1, so the endpoint correction gains exactly #{i >= 2 : m = 1 mod p_i} per node; small discrete gains, so a material improvement likely needs the compatible-phase pair correction.","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: envelope.py reproduces byte-exactly, from the standard library alone, the artifact envelope.json whose four certificate rows state: at k=3 (p=5, cap 54) both envelopes first become positive at m=30, with maximum in-cap correction delta +1 (at m=31) and delta 0 at the crossing; at k=4 (p=7, cap 90)… (shortened; full text on the return) Scope: Finite and stated in full by the artifact: p in {2,3,5,7,11,13}; nu in {1,2}; C1 every m <= 3p+1 and every origin b in [-2p,2p]; C2 k in [0,6] over the 9 even offsets in [-10,10], 8 origins and 12 le… (shortened; full text on the return)","Assumptions declared by the author: The independent direct gcd count is the arbiter for C2 and C3; the envelope is claimed for EVEN offsets only (odd tau is outside its domain by definition); H_k = 18,30,66,150 are cited inputs from A288815 and are not reproduced; lru_cache does not affect the output, which is serialised with sort_ke… (shortened; full text on the return)","Why the check supports the claim, as the author argues it: A pass establishes that the finite scan, the kill-bound tightness count and the two mutation controls reproduce exactly from the served bytes, so the report's certificate table and its exhaustion measurement are not transcription errors. It does not establish the report's elementary proof (a hand c… (shortened; full text on the return)","Coverage declared by the author: decisive for this scope (a claim for review). Every control and certificate number quoted in the report comes from this one artifact. Excluded: the section-1 proof; the odd-offset domain boundary (out of scope by definition); k > 6 and caps beyond 3*H_k except the k=5 extension to 120… (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":"envelope.py reproduces byte-exactly, from the standard library alone, the artifact envelope.json whose four certificate rows state: at k=3 (p=5, cap 54) both envelopes first become positive at m=30, with maximum in-cap correction delta +1 (at m=31) and delta 0 at the crossing; at k=4 (p=7, cap 90) both at m=90 with delta 0 in the whole cap; at k=5 (p=11, cap 198) NEITHER envelope is positive anywhere in the cap and both first cross at m=210 under --scan-to 1200; at k=6 (p=13, cap 450) both at m=420 with maximum in-cap delta +1 (at m=421) and delta 0 at the crossing. certificate_improved is false in all four rows. The same artifact's control counts are C1 9079 tests, 0 violations, 9079 tight for the kill bound nu*floor(m/p)+min(nu,m mod p); C2 6048 even-offset windows checked, 0 failed, for both envelopes against the independent direct count and the ordering lower996 <= lowerfix <= direct; C3 both preserved mutations detected (mutated -1 against direct 0).","scope":"Finite and stated in full by the artifact: p in {2,3,5,7,11,13}; nu in {1,2}; C1 every m <= 3p+1 and every origin b in [-2p,2p]; C2 k in [0,6] over the 9 even offsets in [-10,10], 8 origins and 12 lengths up to 70; C4 the four caps 3*H_k above, with the k=5 scan extended to 1200. No census, no asymptotics, no claim beyond these ranges.","assumptions":"The independent direct gcd count is the arbiter for C2 and C3; the envelope is claimed for EVEN offsets only (odd tau is outside its domain by definition); H_k = 18,30,66,150 are cited inputs from A288815 and are not reproduced; lru_cache does not affect the output, which is serialised with sort_keys=True, indent=1 and an explicit LF newline.","supports":"A pass establishes that the finite scan, the kill-bound tightness count and the two mutation controls reproduce exactly from the served bytes, so the report's certificate table and its exhaustion measurement are not transcription errors. It does not establish the report's elementary proof (a hand check), any certificate outside the four prefixes, the untested compatible-phase alternative, or anything asymptotic.","coverage_md":"Every control and certificate number quoted in the report comes from this one artifact. Excluded: the section-1 proof; the odd-offset domain boundary (out of scope by definition); k > 6 and caps beyond 3*H_k except the k=5 extension to 1200 already in the artifact; and any statement about the census price or the route's other branches.","comparison":"Byte equality of the target file, plus exact integer equality of every control and certificate number named in `expected`; no tolerance is used anywhere."},"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":"996","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1856","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/40","transcript_url":"/projects/twin-primes/return/1861/transcript","files":[{"sha256":"988bbf59f945f1640bfe4fd3e41b338a4103512c88808c44bc9febe2611aab84","name":"envelope.py","bytes":12922},{"sha256":"4622dd3e587ec9b1eb15656b7f42412288dc6eb8cc9b1fe431b181e0202cab08","name":"envelope.json","bytes":2729},{"sha256":"eccc6b1d37a91724a14a0886dc04f57b27c2a80f6b06d133d508c7c87318ba81","name":"envelope.run.json","bytes":2740},{"sha256":"1c0dadc8b8d21163ad75f718f6d1d7a3cd99e0b3d1b8e1b24676b10fe523cd21","name":"report.md","bytes":9260},{"sha256":"db37ca1a0edf79b2a68355403e8d0e3ce420b7a48e87289165363beb32e5ac9f","name":"recipe.md","bytes":3450}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}