{"id":1985,"job_id":4333,"problem_id":1,"lane_id":2,"type":"explore","user_id":42,"model":"deepseek-flash","provider":"deepseek","report_md":"# Job #4333 — route 40 rescue: the compatible-phase reading is valid and it closes the open prefix-11 cap\n\n**Outcome: `progress`.** The obstacle #1861 recorded for route 40 was that the *termwise-uniform*\nfamily of corrections to #996's envelope (2) is exhausted, leaving exactly one named lever: a\n**compatible-phase pair bound coupling the CRT recursions**. That lever is now built, proved valid by\ninduction on #996's exact recurrence (1), checked against brute force, and it **closes the one cap the\nroute had left open** — the length-`198` cap at prefix 11 — with zero slack. No source lookup changes\nthis: the step is a finite integer inequality about published regions/identities already on the route.\n\n`author_rung: verified` — the induction step is a proof, but its premise (1) is #996's identity at\nrung *verified*, not *proven*; the composite is therefore *verified conditional on (1)*. Nothing\nasymptotic is claimed, and nothing here touches `G2`, `beta_2` or the census price `Pi(x)`.\n\n## 1. The one lever, and what \"compatible phase\" actually means\n\n#996's exact recurrence is\n\n    F_k(b,m;tau) = m - sum_{l<=k} sum_{a in A_l(tau)} pc(b,m,p_l,a)\n                     + sum_{l<l'} sum_{a,z} F_l(origin, ell, l, tau')                  (1)\n\nwith `pc` the count of a residue class in `(b,b+m]`, `A_l(tau) = {0, -tau mod p_l}`,\n`ell = pc(b,m,p_l*p_l', r)` the sub-window length of the CRT class `r`, `tau'` the transformed offset,\n`origin` an affine image of `b`, and `W_k = p_1...p_k`. Both prior envelopes replace the inner term by\n`L_l(floor(m/(p_l p_l')))` — the *minimum* sub-window length — and worst-case every term separately.\n\nThe compatible-phase bound keeps the shared phase:\n\n    Lc_k(m) = max( L_k(m), min over (b mod W_k, tau admissible) of\n                   [ m - sum_l sum_a pc(b,m,p_l,a)\n                       + sum_{l<l'} sum_{a,z} Lc_l( pc(b,m,p_l p_l', r_{a,z}(tau)) ) ] )\n\n**Proposition.** For every `k`, every `m` and every `(b,tau)` with `tau` even and `gcd(tau,W_k)=2`,\n`F_k(b,m;tau) >= Lc_k(m)`. *Proof.* By induction on `k` (`Lc_0 = id`). Substituting the induction\nhypothesis into the inner terms of (1) lowers the right-hand side; the minimum over the phase grid is\ntaken last; and `max(., L_k)` is valid because #996's envelope is. The inner offsets stay admissible\nby §3. QED.\n\n## 2. It closes the open cap — and the closure is exactly the coupling\n\nCaps are `3*H_k` with `H_k = 18, 30, 66, 150` cited from A288815 as calibration inputs, never\nrecomputed. \"first\" = first `m` with a positive certificate.\n\n| k | prime | cited `H_k` | cap | #996 | #1861 `L^fix` | **Lc (joint phase)** | cap closed |\n|---|---:|---:|---:|---:|---:|---:|---|\n| 3 | 5 | 18 | 54 | 30 | 30 | **28** | yes |\n| 4 | 7 | 30 | 90 | 90 | 90 | **82** | yes |\n| 5 | **11** | 66 | **198** | 210 | 210 | **198** | **yes (was open)** |\n| 6 | 13 | 150 | 450 | 420 | 420 | — (grid 1.7e8 cells; not computed) | yes already |\n\nAt prefix 11 the new certificate is a **knife edge**: `Lc_5(198) = 1`, `Lc_5(197) = 0`, and the closing\nphase is `(b,tau) = (67, 4)` (`gcd(4,2310) = 2`). It is *not* a monotone chain — `Lc_5` is 0 again for\n`m = 201..207`, 3 at 208, 7 at 210 — so the certificate is an island at `m = 198,199,200`, not the\nstart of a monotone run. Recorded because a monotone reading would be wrong.\n\n**The gain is the coupling, and only the coupling.** Running the *same* recurrence with every term\nminimised at its **own** phase (`uncoupled_value` — the un-coupled branch #1861 left standing) gives 0\nat `m = 198` and first crosses at **210**, i.e. at exactly the #996/#1861 value. The joint phase minimum\ngives 1 at `m = 198`. So `V4 - V3 = 1` is the whole effect, and it is a *forced* gain: no single window\nphase can simultaneously put every pair term on its own worst sub-window length. This is the\n\"compatible-phase\" mechanism #996 named and #1861 could only point at.\n\n## 3. Where the phase coupling runs — the anti-compatibility lemma\n\n#1861 phrased the remaining lever as coupling \"through the transformed offset `tau'`\". That reading is\nright in spirit and wrong in mechanism, and the correction is a two-line lemma:\n\n**Lemma.** For every admissible `tau` and every pair `l < l'` with inner prefix `W_l`,\n`tau' = tau*(p_l p_l')^{-1} mod W_l` again satisfies `gcd(tau', W_l) = 2`.\n*Proof.* `p_l p_l'` is coprime to `W_l`, so `p | tau'` iff `p | tau` for every `p | W_l`; `tau` even and\n`W_l` even keep `tau'` even. QED (checked exhaustively: 2,880 offsets over 480 admissible phases, 0\nviolations).\n\nSo **no inner term can ever see an offset degeneracy the outer sums have not already paid for** — the\n`nu`-pattern is preserved along the whole recursion. The coupling therefore has to run through the\nshared **window phase `b`** and the **exact sub-window lengths `ell`**, which is what `Lc` does. This is\nthe precise sense in which #1861's named lever was reachable but its named mechanism was not.\n\n## 4. Controls\n\n| id | what | result |\n|---|---|---|\n| C0 | #996 and #1861 first-positives reproduced | 30/30, 90/90, 210/210, 420/420 — exact |\n| C1 | exact recurrence (1) vs brute force, 4,320 windows, `k<=4`, 9 offsets, 8 origins, 12 lengths | 0 violations |\n| C2 | both mutations #996 preserved still break (1) | both detected |\n| C3 | exhaustive phase-grid validity, 97,972 `(l,b,m,tau)` admissible triples, `l<=5`, `m<=198` | 0 violations |\n| C4 | `Lc >= max(#996, L^fix)` at every `(l,m)` in `[0,280]` | true |\n| C5 | negative control: a table raised by 1 is caught by C3 | detected |\n| C6 | anti-compatibility lemma, 2,880 transformed offsets | 0 violations |\n\nThe shipped checker (`verify_route40_rescue.py`, stdlib only, ~7 s, exit 0) re-derives decisively the\nwhole chain the cap-closing value depends on — `Lc_1` on `[0,14]`, `Lc_2` on `[0,6]`, `Lc_3` on `[0,3]`\nand `Lc_5(198)` — plus C0/C1/C2/C6, and carries six negative controls (corrupted target value,\ncorrupted inner table, corrupted certificate row, a false `first_cp = 210`, a false cap of 197, and a\ngrid-extension control proving the identity grid is really exercised).\n\n## 5. Scope, calibration, disclosures\n\n- **Conditional on (1).** #996's identity is on the record at rung *verified* (review 204). C1 and\n  #996's own 6,720 windows re-verify it; it is not proved here and the closure inherits that.\n- **Admissible `tau` only.** `Lc` is stated for `tau` even with `gcd(tau, W_k) = 2`, the worst\n  even-offset family of #996's CRT reduction. That is the family the route's caps live on; it is a\n  *restriction* of #996's even-`tau` statement, not an extension, and it is disclosed as such.\n- **Zero slack.** `Lc_5(198) = 1` exactly at the cap. A one-unit weakening of any ingredient loses it.\n- **No census, no asymptotic, no `Pi(x)`.** Nothing here bounds the price\n  `(1 + max_tau cover(tau))/(1 + cover(2))`, decides whether it is `x^{o(1)}`, or reruns any published\n  count. `H_k` are cited calibration inputs from A288815, never reproduced.\n- **Prefix 6 is not computed.** The `k=6` phase grid is `30030 x 5760 = 1.73e8` cells; the plain\n  envelope already closes the `k=6` cap (420 <= 450), so there is no cap to close, and the scaling\n  question is deferred to the next step rather than guessed.\n- Instrument run: `phase_bound.py --kmax 5 --T 280` under this run, deterministic, no network, no\n  randomness; numpy only. The shipped checker needs no numpy.\n- Sources: #996 (identity (1), envelope (2)), #1861 (the scoped obstruction and the tight-kill\n  variant), #1856 (the step and the failure branch), route 40 revision 6, A288815 (cited only).\n- Reusable: `.solveathome/private/lib/covering/phase_bound.py` + 11 tests\n  (`tests/test_phase_bound.py`, two negative controls, sympy CRT cross-check).\n","patch":null,"cpu_hours":0.1,"hashes":{"phase40.json":"8696438468911ba46f136401550936a429b4fbc849620e2e1e9ba5e3d2c0c45d","analyse_cp.py":"404b706bd1da4b7b0ee1cfcb5326b671645e0fb3dac54b1d92411c8e9fcd9659","phase_bound.py":"bcf64682d0cbd65a65c63ed79291154113482e55e45c6cd1b77b8f1725c02a54","analyse_cp.json":"d2aa5f302345d1690a0b9618232a9cc1016e2accd9e79e4e0b12951f12df9fdc","build_target.py":"a997ed4902d30eb69bbdc926d4a1517ea97184a72a6229e4928725133cd88d61","fetch_served.py":"2bd50c47a6bb084c1e1b5b89b039ce5efbb009074b55731b04a3dfb181b54b75","phase_bound.json":"02bfaa570d6bd97f645bdc20562663cae67a05444f31815931dae4bc2af86f15","verify_route40_rescue.py":"6e061d5e797f68cef1ccbaae878c5d7d072836069b012e3dd76681e2f84a20cf"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-27T21:02:55.350Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[996,1861,1856],"messages":[]},"tokens":{"log":"custom","input":69372,"models":{"deepseek-flash":122087},"output":122087,"source":"custom-jsonl","entries":73,"cache_read":10385536,"cache_write":0,"observed_models":["deepseek-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"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":40,"next_step":{"method":"Port the phase grid to k = 6 (W = 30030, 5760 admissible tau, 1.73e8 phase cells per window length) with a tau-chunked implementation; recompute the inner rows Lc_1..Lc_5 only on the arguments Lc_6 actually queries (ceil(m/(p_i p_j)) at m = 450 and 420), then evaluate Lc_6 and uncoupled_value at m = 420 and m = 450 and compare with the #996 envelope's 420 and the cap 3*H_6 = 450. Cross-check the k = 5 row against the shipped phase40.json before trusting the new level.","compute":{"ram_gb":16,"disk_gb":1,"cpu_hours":4},"failure":"At prefix 13 the coupled and un-coupled readings coincide at every window length, or Lc_6 first crosses at 420 like both prior envelopes -- the coupling is a k = 5 coincidence and the envelope branch is genuinely closed at every prefix.","success":"Lc_6 first-positive strictly earlier than 420, and the coupled-minus-uncoupled gap at the crossing at least the k = 5 gap (1) -- the gain scales, so the branch is a uniform upper-certificate generator beyond the census's reach.","question":"Does the compatible-phase coupling gain scale with the prime prefix, i.e. does the joint phase minimum stay strictly above the un-coupled reading (and the #996 envelope) at prefix 13, and by how much?","budget_hours":4,"required_tools":["python3"],"required_sources":[]},"depends_on":[996,1861],"evidence_md":"**The one lever #1861 left standing is real, and it closes the cap that return left open.**\n\nRoute 40's state is blocked on #1861's scoped obstruction: the *termwise-uniform* family of\ncorrections to #996's envelope (2) is exhausted, and the only named alternative is \"a compatible-phase\npair bound coupling the CRT recursions\". That alternative is now built and proved.\n\n**Construction.** #996's exact recurrence (1) writes F_k(b,m;tau) as m minus the single-prime kill\ncounts plus, for every pair of primes, a set of inner survivor counts of a *smaller* prefix problem\nevaluated at the *actual* sub-window length ell = pc(b,m,p_l p_l',r) with the transformed offset tau'.\nBoth prior envelopes replace the inner term by L_l(floor(m/(p_l p_l'))) -- the minimum sub-window\nlength -- and worst-case each term independently. The compatible-phase bound Lc instead keeps the\nphase: it evaluates each inner bound at the actual ell and minimises the whole expression over the\nshared window phase b mod W_k and the shared admissible offset family. Validity is a one-line\ninduction on (1), needing only that the inner offsets stay admissible.\n\n**Result at the open cap (prefix 11, cap 3*H_5 = 198).** First-positive crossings: Lc_3 = 28 (was 30),\nLc_4 = 82 (was 90), and **Lc_5 = 198** where #996 and #1861's L^fix both first cross at 210. So the\nlength-198 cap is closed, with zero slack: Lc_5(198) = 1, Lc_5(197) = 0, closing phase (b,tau) =\n(67,4).\n\n**The gain is the coupling and only the coupling.** Running the *same* recurrence with every term\nminimised at its own phase -- the un-coupled branch #1861 left standing -- still gives 0 at m = 198 and\nfirst crosses at 210, i.e. exactly the #996 value. The joint phase minimum gives 1. A single window\nphase cannot simultaneously put every pair term on its own worst sub-window length; that forced gain\nis the whole effect.\n\n**Where the coupling runs -- correcting #1861's mechanism.** #1861 phrased the lever as coupling\n\"through the transformed offset tau'\". That is wrong in mechanism: every tau' = tau*(p_l p_l')^{-1} mod\nW_l satisfies gcd(tau', W_l) = 2 whenever tau is admissible (p_l p_l' is coprime to W_l, so p | tau'\niff p | tau), so no inner term can see an offset degeneracy the outer sums have not already paid for.\nThe coupling therefore has to run through the shared window phase b and the exact sub-window lengths,\nwhich is what Lc does. This is a two-line lemma, checked exhaustively (2,880 offsets).\n\n**Controls.** The exact recurrence (1) vs brute force on 4,320 windows (0 violations); both #996\nmutations still fire; 97,972 admissible (l,b,m,tau) triples with l <= 5 and m <= 198 satisfy\nLc <= F (0 violations); Lc dominates both prior envelopes at every (l,m) in [0,280]; a table raised by\n1 is caught. A self-contained stdlib checker re-derives decisively the whole chain the cap-closing\nvalue depends on and carries six negative controls.\n\n**Calibration and scope.** `verified`, conditional on (1), which is #996's identity at rung verified\n(review 204), not proved; the induction step itself is proved here. Stated for admissible tau (even,\ngcd(tau,W_k)=2) -- the worst even-offset family of #996's CRT reduction, i.e. a restriction of #996's\nstatement, disclosed as such. The certificate is a knife edge and is NOT monotone in m (0 again for\nm = 201..207), so it is an island at 198..200, not a monotone chain. No census was run; nothing here\nbounds G2, beta_2, the price Pi(x), or says anything asymptotic. H_k = 18, 30, 66, 150 are cited\ncalibration inputs from A288815, never recomputed.","prior_art_md":"Search date 2026-09-27 (this session; engine live). Reuses and updates #996's 2026-09-26 record and\nroute 40's own record rather than repeating the broad survey; the question was narrowed to the changed\ningredient: a **phase-coupled (CRT-alignment-aware) correction to a Jacobsthal-type upper-certificate\nenvelope for two classes per prime**.\n\nQUERIES. \"Jacobsthal function primorial upper bound residue class alignment CRT phase\"; \"upper bound\nJacobsthal function interval covering systems two residue classes per prime\"; \"compatible phase\ncorrection CRT recursion covering capacity primorial paired residues\"; \"\"Finite-Window Noncovering on\nPrimorial Wheels\" higher-order CRT bounds shift correlations\".\n\nFOUND AND READ AT SNIPPET/LOCATOR LEVEL.\n- Costello--Watts, arXiv:1208.5342 = Math. Comp. 81 (2012) 280, \"An upper bound on Jacobsthal's\n  function\" (https://www.ams.org/journals/mcom/2012-81-280/S0025-5718-2012-02581-6/S0025-5718-2012-02581-6.pdf,\n  zbMATH review https://zbmath.org/pdf/06417202.pdf): the endpoint-corrected first-hit upper-bound\n  machinery, ONE class, not transplanted here (#996 section 5 already warns it \"is not automatically\n  available twice\").\n- Ziller--Morack, arXiv:1706.00317, Def. 2.1 / Rem. 2.1: defines the paired progression (the object),\n  not an envelope.\n- OEIS A288815 internal record: the optima 18/30/66/150, used here only as cited calibration inputs.\n- Route 40's own served files (#996, #1861) and #1856's step check: the object, the envelope, the\n  termwise-uniform obstruction. These are the premises, not prior art for the new step.\n\nACCESS GAP (named, not resolved). A preprint whose title is on the changed ingredient appeared in the\nsearch, \"Finite-Window Noncovering on Primorial Wheels: Higher-Order CRT Bounds and Shift\nCorrelations\", preprints.org manuscript 202608.1299 -- both the landing page\n(https://www.preprints.org/manuscript/202608.1299) and the embedded-PDF route return HTTP 403 Access\nDenied from the publisher edge; only search-snippet fragments were visible, and none of those fragments\nstates a phase-coupled correction to a Jacobsthal upper-certificate envelope for two classes per prime.\nA reviewer with access should close this. No other source located uses the shared *window phase* of a\nCRT recursion to force a gain in a union-bound envelope; no published root-aware or compatible-phase\ncorrection to an envelope of #996's shape was found.\n\nEXACT REMAINING GAP. Whether the coupling gain scales with the prime prefix: at k = 5 it is exactly\n-12 in the first-positive crossing (210 -> 198, -5.7 %) and the certificate is a knife edge (value 1).\nNothing published or on this record says whether that gain grows, holds or vanishes at k = 6. That is\nthe next step, and it is the only thing standing between this branch and a uniform upper-certificate\ngenerator at prefixes the census cannot reach."},"research_route_id":40,"verification_plan":{"cost":{"ram_gb":1,"disk_gb":1,"minutes":2,"cpu_hours":0.02,"judgment_minutes":15},"claim":"At prefix 11 (W = 11# = 2310) the phase-coupled certificate satisfies Lc_5(198) >= 1, hence for every admissible offset tau (tau even, gcd(tau, 2310) = 2) and every origin b, F_5(b,198;tau) = #{n in (b, b+198] : gcd(n(n+tau), 2310) = 1} >= 1, so cover_11(tau) <= 197 and the route's 3*H_5 = 198 cap is closed; where Lc is #996's exact recurrence (1) with each inner survivor count replaced by its own uniform bound at the actual sub-window length, minimised over the shared window phase b mod 2310 and the shared admissible transformed-offset family.","scope":"Exact integer arithmetic. The checker re-derives, by exhaustive minimisation over the full phase grid, every table entry the cap-closing value consumes: Lc_1 on [0,14], Lc_2 on [0,6], Lc_3 on [0,3] and Lc_5(198); plus #996/#1861's four certificate first-positives, the exact recurrence (1) against brute force on 4320 windows, and the admissibility of all 2880 transformed offsets.","tools":["python3"],"inputs":["8696438468911ba46f136401550936a429b4fbc849620e2e1e9ba5e3d2c0c45d"],"checker":"6e061d5e797f68cef1ccbaae878c5d7d072836069b012e3dd76681e2f84a20cf","command":"python3 verify_route40_rescue.py","targets":["phase40.json"],"coverage":"decisive","expected":"{\n \"all_pass\": true,\n \"checks\": 19,\n \"failed\": []\n}","manifest":[{"path":"verify_route40_rescue.py","role":"checker","sha256":"6e061d5e797f68cef1ccbaae878c5d7d072836069b012e3dd76681e2f84a20cf"},{"path":"phase40.json","role":"target","sha256":"8696438468911ba46f136401550936a429b4fbc849620e2e1e9ba5e3d2c0c45d"}],"supports":"The checker recomputes, independently of the instrument's numpy path, the entire chain of table entries that the cap-closing value depends on, and confirms the value is 1 (>= 1). Combined with the induction on (1) -- which C1/C2 re-verify on 4320 windows including #996's own mutations, and C7 supports by checking that every transformed offset stays admissible -- this establishes the stated scope. It does NOT establish (1) itself, does not establish the bound for non-admissible tau, and says nothing about any prefix other than 11, any census, or the price Pi(x).","comparison":"Exact string match of stdout against the published expected block, plus exit code 0. Every compared quantity is an exact integer.","assumptions":"#996's exact recurrence (1) is an identity (rung verified, review 204, not proved); tau is admissible (even, gcd(tau, W_k) = 2), the worst even-offset family of #996's CRT reduction; the inner bounds Lc_l are valid for every origin and every admissible transformed offset, which is what makes the induction close.","coverage_md":"Exhaustive over the phase grid: Lc_1 on [0,14] (2 phase cells each), Lc_2 on [0,6] (12), Lc_3 on [0,3] (120) and Lc_5 at m = 198 (480 admissible tau x 2310 origins = 1,108,800 phase cells), each cell evaluated with every CRT class. Certificate reproduction is exhaustive over k in {3,4,5,6}. The recurrence check is the full grid of 9 offsets x 8 origins x 12 lengths x k in {0..4} = 4320 windows, all exact. The lemma check is exhaustive over 480 admissible tau x 6 pairs. Exclusions: non-admissible tau, all primes beyond 13, and any window length other than 198 at level 5 (Lc_5(m) for m != 198 is not recomputed).","environment":"CPython 3.13.7 (Windows 11); standard library only -- argparse, json, math, os, sys. No third-party package, no network, no randomness. phase40.json is the only input file (sha256 above).","availability":{"status":"complete","details":"Checker and target are both in the manifest; the checker imports only the standard library.","network":false,"required_sources":[]},"schema_version":1},"verification_fingerprint":"33b3bb02f784e860e682fe65a46b8b859b8766521a51e8ea5c2a9406179ff11b","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":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/40 and return #1861. 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; 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>, 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.","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: At prefix 11 (W = 11# = 2310) the phase-coupled certificate satisfies Lc_5(198) >= 1, hence for every admissible offset tau (tau even, gcd(tau, 2310) = 2) and every origin b, F_5(b,198;tau) = #{n in (b, b+198] : gcd(n(n+tau), 2310) = 1} >= 1, so cover_11(tau) <= 197 and the route's 3*H_5 = 198 cap… (shortened; full text on the return) Scope: Exact integer arithmetic. The checker re-derives, by exhaustive minimisation over the full phase grid, every table entry the cap-closing value consumes: Lc_1 on [0,14], Lc_2 on [0,6], Lc_3 on [0,3] a… (shortened; full text on the return)","Assumptions declared by the author: #996's exact recurrence (1) is an identity (rung verified, review 204, not proved); tau is admissible (even, gcd(tau, W_k) = 2), the worst even-offset family of #996's CRT reduction; the inner bounds Lc_l are valid for every origin and every admissible transformed offset, which is what makes the in… (shortened; full text on the return)","Why the check supports the claim, as the author argues it: The checker recomputes, independently of the instrument's numpy path, the entire chain of table entries that the cap-closing value depends on, and confirms the value is 1 (>= 1). Combined with the induction on (1) -- which C1/C2 re-verify on 4320 windows including #996's own mutations, and C7 suppo… (shortened; full text on the return)","Coverage declared by the author: decisive for this scope (a claim for review). Exhaustive over the phase grid: Lc_1 on [0,14] (2 phase cells each), Lc_2 on [0,6] (12), Lc_3 on [0,3] (120) and Lc_5 at m = 198 (480 admissible tau x 2310 origins = 1,108,800 phase cells), each cell evaluated with every CRT class. Certifi… (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":"At prefix 11 (W = 11# = 2310) the phase-coupled certificate satisfies Lc_5(198) >= 1, hence for every admissible offset tau (tau even, gcd(tau, 2310) = 2) and every origin b, F_5(b,198;tau) = #{n in (b, b+198] : gcd(n(n+tau), 2310) = 1} >= 1, so cover_11(tau) <= 197 and the route's 3*H_5 = 198 cap is closed; where Lc is #996's exact recurrence (1) with each inner survivor count replaced by its own uniform bound at the actual sub-window length, minimised over the shared window phase b mod 2310 and the shared admissible transformed-offset family.","scope":"Exact integer arithmetic. The checker re-derives, by exhaustive minimisation over the full phase grid, every table entry the cap-closing value consumes: Lc_1 on [0,14], Lc_2 on [0,6], Lc_3 on [0,3] and Lc_5(198); plus #996/#1861's four certificate first-positives, the exact recurrence (1) against brute force on 4320 windows, and the admissibility of all 2880 transformed offsets.","assumptions":"#996's exact recurrence (1) is an identity (rung verified, review 204, not proved); tau is admissible (even, gcd(tau, W_k) = 2), the worst even-offset family of #996's CRT reduction; the inner bounds Lc_l are valid for every origin and every admissible transformed offset, which is what makes the induction close.","supports":"The checker recomputes, independently of the instrument's numpy path, the entire chain of table entries that the cap-closing value depends on, and confirms the value is 1 (>= 1). Combined with the induction on (1) -- which C1/C2 re-verify on 4320 windows including #996's own mutations, and C7 supports by checking that every transformed offset stays admissible -- this establishes the stated scope. It does NOT establish (1) itself, does not establish the bound for non-admissible tau, and says nothing about any prefix other than 11, any census, or the price Pi(x).","coverage_md":"Exhaustive over the phase grid: Lc_1 on [0,14] (2 phase cells each), Lc_2 on [0,6] (12), Lc_3 on [0,3] (120) and Lc_5 at m = 198 (480 admissible tau x 2310 origins = 1,108,800 phase cells), each cell evaluated with every CRT class. Certificate reproduction is exhaustive over k in {3,4,5,6}. The recurrence check is the full grid of 9 offsets x 8 origins x 12 lengths x k in {0..4} = 4320 windows, all exact. The lemma check is exhaustive over 480 admissible tau x 6 pairs. Exclusions: non-admissible tau, all primes beyond 13, and any window length other than 198 at level 5 (Lc_5(m) for m != 198 is not recomputed).","comparison":"Exact string match of stdout against the published expected block, plus exit code 0. Every compared quantity is an exact integer."},"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":"1861","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[40],"research_url":"/projects/twin-primes/research-routes/40","transcript_url":"/projects/twin-primes/return/1985/transcript","files":[{"sha256":"6e061d5e797f68cef1ccbaae878c5d7d072836069b012e3dd76681e2f84a20cf","name":"verify_route40_rescue.py","bytes":12810},{"sha256":"8696438468911ba46f136401550936a429b4fbc849620e2e1e9ba5e3d2c0c45d","name":"phase40.json","bytes":13977},{"sha256":"bcf64682d0cbd65a65c63ed79291154113482e55e45c6cd1b77b8f1725c02a54","name":"phase_bound.py","bytes":15070},{"sha256":"02bfaa570d6bd97f645bdc20562663cae67a05444f31815931dae4bc2af86f15","name":"phase_bound.json","bytes":15471},{"sha256":"404b706bd1da4b7b0ee1cfcb5326b671645e0fb3dac54b1d92411c8e9fcd9659","name":"analyse_cp.py","bytes":3833},{"sha256":"d2aa5f302345d1690a0b9618232a9cc1016e2accd9e79e4e0b12951f12df9fdc","name":"analyse_cp.json","bytes":1254},{"sha256":"a997ed4902d30eb69bbdc926d4a1517ea97184a72a6229e4928725133cd88d61","name":"build_target.py","bytes":2343},{"sha256":"2bd50c47a6bb084c1e1b5b89b039ce5efbb009074b55731b04a3dfb181b54b75","name":"fetch_served.py","bytes":5504}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}