{"id":1972,"job_id":4335,"problem_id":1,"lane_id":null,"type":"explore","user_id":42,"model":"deepseek-flash","provider":"deepseek","report_md":"# Job #4335 — route 172 first look: is further basis enrichment justified?\n\nRoute 172 (`proposed`, revision 1) follows #1942 and proposes **exactly one**\nnext experiment:\n\n> Re-run the recording pass with the `d`-enriched `EvenEngine` basis\n> (`d = 19` or `21`) at `k = 46`, `eps = 25/861`, assemble the capped Gram pair,\n> re-solve the generalised eigenproblem. Success: an exact rational witness with\n> `c^T (M2^cap − (1/A) M1^cap) c > 0`.\n\nCalibration of this return: **verified** for the exact rational arithmetic on\nthe published numbers (a finite, re-runnable check), **measured** for the\nsmall-`k` definiteness sweep, **inconclusive** as the route-level outcome.\n\n---\n\n## 1. The decisive arithmetic is already on the record\n\nAll four numbers below are public: `#1942` (pending) and `#1869`\n(accepted, measured). Re-deriving their exact relations settles the route's\nquestion without touching the recording pass.\n\n| quantity | value | source |\n| --- | --- | --- |\n| capped re-optimised quotient `q_cap(17) = J_cap/I_cap` | `3.8115713687001214` | #1942 |\n| uncapped optimum at the same `(k, d = 17)` | `q_uncap(17) ≥ 3.9013805275618854` | #1869 |\n| threshold `τ = 1/A`, `A = 2583/10000` | `3.8714672861014323` | #1641 |\n| #1942 finite-spectrum diagnostic on the `d = 17` basis | `λ_max ∈ [3.799, 3.812]` | #1942 §1 |\n\nDerived exactly (`route172_numbers.json`):\n\n```\nq_uncap(17)/τ − 1 = +0.772659 %      the uncapped optimum CLEARS the threshold\ndeficit(17) = 1 − q_cap/τ = 1.547112 %      what enrichment has to buy\ncap_cost(17) = 1 − q_cap/q_uncap = 2.301984 %      what the cap itself removes\n```\n\n**Three consequences, all against the route's proposed experiment.**\n\n1. *The cap is the whole problem.* The uncapped optimum at `d = 17` is already\n   `0.773 %` **above** `τ`; applying the cap drops it `2.302 %`, leaving it\n   `1.547 %` below. Exactly `1 − 0.7727/2.3020 = 66.44 %` of the cap's cost has\n   to be recovered by any proposal. An enrichment would have to undo two thirds\n   of the cap penalty, not merely add two basis degrees.\n2. *#1942's own diagnostic caps the headroom.* That diagnostic is an upper bound\n   (within its numerical scope) on the top finite generalised eigenvalue of the\n   **same** `d = 17` pencil: `λ_max ≤ 3.812`. The total headroom above the\n   re-optimised witness is therefore\n\n   ```\n   headroom ≤ 3.812 − 3.8115713687 = 4.286e-04\n   headroom / (τ − q_cap)          = 0.716 %  of the lift required at the cap,\n   headroom / (q_uncap − q_cap)    = 1.3 %    of the cap penalty itself.\n   ```\n\n   `d = 19` adds two degrees to a `374`-element basis; the diagnostic says the\n   `d = 17` basis is already within `~1e-4` of everything it can do, and that\n   band spans `0.34 %` of the quotient. A gain ~130× the entire diagnostic band\n   would still be needed.\n3. *Cost.* The recording pass is only partly incremental: both `region_planes`\n   and the caps constraint `r·d ≤ caps[r]` depend on `d`, so `d = 19` invalidates\n   **all** recorded shards, not just the two new `r` layers. #1942's run was\n   ~3.5 h of 8-process wall time (~24 CPU-hours) for `d = 17`; `d = 19` adds two\n   `r` layers and a larger signature set. That is outside this assignment's\n   8 CPU-hour allowance — a second, independent reason it was not run here.\n\nReading 2 is conditional on #1942's diagnostic being a genuine upper bound; that\nreturn itself states the statement is numerical, not a proof. Under its stated\nscope the route's single proposed experiment cannot succeed, so the route needs\na **different** experiment to continue (§4). This is a scoped obstruction, not a\nrefutation of the candidate: nothing here says `M^{cap} > 1/A` is false.\n\n## 2. Supporting computation: the small-`k` mirror does not exist\n\nThe obvious cheap model — build the same capped Gram pair at small `k`, where\nthe recording pass costs seconds instead of ~3.5 h — **fails structurally**, and\nthat is itself worth recording. `outputs/job4335/sweep_frozen_threshold.py`\nbuilds the exact capped pair from the frozen recorded slot functionals for\n`(k, d) ∈ {4,6,8} × {1,2,3}` and reports the spectrum of the diagonally\nequilibrated capped denominator form `M1^cap = M1 − Delta_sym`:\n\n| `k` | `d` | `n` | scaled `λ_min(M1^cap)` | positive definite |\n| --- | --- | --- | --- | --- |\n| 4 | 1 | 2 | `−9.386e-02` | no |\n| 4 | 3 | 6 | `−4.657e-01` | no |\n| 6 | 1 | 2 | `−6.745e-01` | no |\n| 6 | 2 | 4 | `−2.187e+00` | no |\n| 6 | 3 | 6 | `−1.674e+00` | no |\n| 8 | 1 | 2 | `−1.877e+00` | no |\n| 8 | 2 | 4 | `−6.979e+00` | no |\n| 8 | 3 | 6 | `−4.352e+00` | no |\n\nAll nine cells are **indefinite**, so the capped quotient is not defined there\nat all. The same script recovers the uncapped optima exactly\n(`k = 6`: `2.118198930803198, 2.173562716270346, 2.194800864728747`, matching\n`lib/maynard/whiten_eig.py` to all digits), so the instrument is not at fault:\nit is the *artificial* cap `caps[r] = (3/20 or 4/25)/A` that, at small `k`, is\ncomparable to the whole mesh and drives the denominator correction past `M1`.\n\nConsequences worth keeping:\n\n* The `d = 19` enrichment cannot be pre-screened on a small-`k` surrogate; it\n  must be run at `k = 46` or not at all.\n* The route's capped denominator sits on a knife edge: at `k = 46` it is\n  `I_cap(c*) ≈ 3.2e-94 > 0`, at small `k` it is plainly indefinite. The sign of\n  the capped form is not stable in the parameters, so any future certificate on\n  it should report a definiteness check alongside the quotient\n  (`top_ritz(check_pd=True)` does exactly that and refuses otherwise).\n\n## 3. What would actually decide the route\n\nThe missing and genuinely affordable quantity is how the **uncapped** optimum\n`M_{46,ε}(d)` grows with `d`. `M1` is positive definite there, so no cap\npathology arises, and `lib/maynard/whiten_eig.py` computes it directly from the\nexact engine matrices — minutes to about an hour of CPU at `d = 19`/`21` instead\nof a full recording pass. If `M_{46,ε}(d)` saturates well below `τ`, the deficit\nis not a basis artefact and no enrichment can certify; if it keeps climbing\nthrough `τ` while the capped optimum stays at `3.81`, §1's decomposition is\nconfirmed and the cap term is the only remaining target.\n\n## 4. Artefacts\n\n| file | what |\n| --- | --- |\n| `outputs/job4335/route172_inputs.json` | the inputs quoted from the public record |\n| `outputs/job4335/verify_route172_numbers.py` | exact-rational checker; `ROUTE172-NUMBERS PASS`, 15 checks |\n| `outputs/job4335/test_verify_route172_numbers.py` | 10 unit tests, `OK` (incl. two negative controls) |\n| `outputs/job4335/route172_numbers.json` | the derived quantities |\n| `outputs/job4335/sweep_frozen_threshold.py` | exact capped/uncapped pencil at small `k, d` |\n| `outputs/job4335/sweep_frozen_threshold.json` | the definiteness table of §2 |\n| `outputs/job4335/bench_small.py` | cost probe for the recorded functionals |\n\nRecipe:\n\n```\npython3 outputs/job4335/verify_route172_numbers.py\npython3 outputs/job4335/test_verify_route172_numbers.py\npython3 outputs/job4335/sweep_frozen_threshold.py 4,6,8 1,2,3\n```\n\nNo asymptotic claim; nothing here bounds `G2`, `β₂` or twin-prime infinitude.\nThe twin prime conjecture remains open.\n","patch":null,"cpu_hours":1.2,"hashes":{"REPORT.md":"c7abc891cedb258381d7e30977cded3b557f4612c064c3c114a7e0533b2fb252","bench_small.py":"1e09ef6ddf2edf0b0e61f0c373b072d505340fd28251d57ccd2fd36c1d38126f","route172_inputs.json":"8b6615a547d0c5516dcd8801683e5d01b974db72b75cf1c97a391c043e93d6d0","route172_numbers.json":"d994fafa0544553525b9687ced8a0d7cb4dcec4dc7988f428fec8e4476baf055","sweep_frozen_threshold.py":"3d69a7c308a6f0ce2dd05df6a94abc00f37f6259a896926daf74411ae0b6021e","verify_route172_numbers.py":"a9641df101b16bc7bb47d744662a771507995b6229a7b09157f34dff9e64c4e2","sweep_frozen_threshold.json":"569a645e28833b7edcc6aaf7b789729849571baa627a6e54afd698a21dd437d0","test_verify_route172_numbers.py":"bab961378a624bc7f706f9f80478a576586f8d1cdfac2bcbbfb70660006aa7e9"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-27T18:30:00.915Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1942,1869,1641,1606],"messages":[]},"tokens":{"log":"custom","input":137043,"models":{"deepseek-flash":109313},"output":109313,"source":"custom-jsonl","entries":168,"cache_read":24067200,"cache_write":0,"observed_models":["deepseek-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"python3 outputs/job4335/verify_route172_numbers.py  (exact-rational check, 16 checks, PASS);  python3 outputs/job4335/test_verify_route172_numbers.py  (10 unittest tests, incl. 2 negative controls, OK);  python3 outputs/job4335/sweep_frozen_threshold.py 4,6,8 1,2,3  (capped/uncapped pencil at small k, d).","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":"inconclusive","obstacle":{"kind":"scoped_obstruction","evidence":"outputs/job4335/route172_numbers.json and its checker verify_route172_numbers.py (16 exact-rational checks, PASS): q_cap(17) = 3.8115713687001214 < tau; q_uncap(17) >= 3.9013805275618854 > tau by 0.772659%; cap_cost 2.301984% > deficit 1.547112%; headroom 4.286e-04 = 0.716% of the required lift. Plus the definiteness sweep sweep_frozen_threshold.json: M1^cap is indefinite in all nine (k, d) cells of {4,6,8} x {1,2,3}, so the small-k surrogate of the same construction does not exist; the uncapped values it does produce match lib/maynard/whiten_eig.py exactly. Limitation: the headroom half of the argument rests on a published numerical diagnostic, not on an exact certificate; the cap-cost and deficit halves are exact rational arithmetic on published exact rationals.","statement":"At k = 46, eps = 25/861, the capped deficit to 1/A cannot be recovered by enriching the even-signature basis from d = 17 to d = 19 (or 21): the total headroom the d = 17 basis admits is bounded by #1942's own finite-spectrum diagnostic at 3.812, i.e. 4.286e-04 above the re-optimised witness, while the lift required at the cap is 5.990e-02 -- 0.716% of it -- and the cap penalty itself is 2.302% of the quotient, of which 66.44% must be undone. The same enrichment additionally costs a full recording pass (~24 CPU-hours at d = 17, more at d = 19), because region_planes and the caps constraint r*d <= caps[r] depend on d and invalidate every existing shard.","assumptions":"k = 46, eps = 25/861, A = 2583/10000, the fixed restricted support and cap schedule used by #1641/#1869/#1942, and the even-signature basis family b_{a,alpha} = (1+eps-P1)^a P_alpha with a + |alpha| <= d. The headroom bound is conditional on #1942's stated-numerical finite-spectrum diagnostic (truncating the near-kernel of the scaled pencil at relative eigenvalue 1e-6 to 1e-12) being an upper bound on the d = 17 top finite generalised eigenvalue; #1942 records that statement as numerical, not a proof.","revisit_when":"An exact (non-numerical) bound on the top finite generalised eigenvalue of the d = 17 capped pencil that is materially above 3.812, or a measured uncapped sequence M_{46,25/861}(d) for d = 19/21 that rises steeply toward tau, or a change of cap schedule / restricted support that lowers the 2.302% cap cost below the 1.547% deficit."},"route_id":172,"next_step":{"method":"Build the exact uncapped M1, M2 from lib/maynard/even_engine.py EvenEngine(46, 25/861, d, 'exact') for d = 17, 19 and 21 and take the top generalised eigenvalue of the pencil with lib/maynard/whiten_eig.py (high-precision diagonal equilibration, then whiten and solve). Record the exact rational Rayleigh quotient of the recovered vector for each d, together with the increment M_{46,25/861}(d+2) - M_{46,25/861}(d). No recording pass, no cap correction: M1 is positive definite here. Report the sequence against tau and against #1869's J_0/I_0 = 3.9013805275618854 anchor at d = 17.","compute":{"ram_gb":8,"disk_gb":1,"cpu_hours":4},"failure":"The sequence saturates at or below #1942's 3.812 ceiling (the published diagnostic) with increments shrinking geometrically, which confirms that the d = 17 deficit is not a basis-truncation artefact and closes the basis-enrichment branch of this route for k = 46.","success":"The increments stay bounded away from zero and the sequence approaches tau from below or crosses it, which makes the cap cost the sole remaining obstruction and reopens a bounded pursuit aimed at the cap term rather than at d.","question":"How does the uncapped optimum M_{46,25/861}(d) grow with the even-signature basis degree d at k = 46, and does it clear the threshold 1/A = 3.8714672861014323 before it saturates?","budget_hours":1.5,"required_tools":["python3"],"required_sources":[]},"depends_on":[1942,1869,1641],"evidence_md":"The route's single proposed experiment (enrich the even-signature basis from d = 17 to d = 19 at k = 46, eps = 25/861, rebuild the exact capped Gram pair, re-solve) is not justified, and the reason is already in the published record. Combining #1942's capped re-optimised quotient q_cap(17) = 3.8115713687001214 with #1869's uncapped anchor at the same (k, d = 17), q_uncap(17) >= J_0/I_0 = 3.9013805275618854, and #1641's threshold tau = 1/A = 10000/2583 = 3.8714672861014323 gives, exactly: (i) the uncapped optimum at d = 17 already CLEARS tau by +0.772659%, so the entire deficit is cap-attributable; (ii) the cap costs 1 - q_cap/q_uncap = 2.301984% of the quotient, so 66.44% of that cap cost (2.301984 - 0.772659 over 2.301984) is exactly the 1.547112% deficit; (iii) #1942's own finite-spectrum diagnostic on the same d = 17 pencil bounds its top finite generalised eigenvalue by lambda_max <= 3.812, leaving total headroom 3.812 - 3.8115713687 = 4.286e-04 above the re-optimised witness. That headroom is 0.716% of the lift required at the cap and 1.3% of the cap penalty itself -- about 130x the entire diagnostic band, which spans only 0.341% of the quotient, would still be needed. Enriching d = 17 -> 19 adds two degrees to an n = 374 basis that the diagnostic says is already within ~1e-4 of everything it can do. A second, independent reason: region_planes and the caps constraint r*d <= caps[r] both depend on d, so d = 19 invalidates every recorded shard, not just two new r layers; #1942's d = 17 pass was ~24 CPU-hours, and this assignment allows 8. Supporting computation recorded here: the natural cheap surrogate does not exist. Building the same exact capped pair at small k from the frozen slot functionals gives an INDEFINITE M1^cap for all nine cells (k, d) in {4, 6, 8} x {1, 2, 3} (scaled lambda_min from -9.4e-02 down to -7.0e+00), so the capped quotient is undefined there -- while the same script reproduces the uncapped optima (k = 6: 2.118198930803198, 2.173562716270346, 2.194800864728747) to all digits. The cap's sign is not stable in the parameters, so any future capped certificate should ship a definiteness check. This is a scoped obstruction, not a refutation: nothing here says M^cap > 1/A is false, and the conditional (published-diagnostic) part of the argument is stated as such.","prior_art_md":"Online searches 2026-09-27 (queries: \"Maynard-Tao variational problem capped support truncated polynomial basis degree enrichment convergence\"; \"Polymath8b D_0 optimal polynomial degree convergence M_k variation\"; \"eprint 2026/1893 restricted support capped moment truncated simplex inequalities\"). Externally inspected for method context only: Polymath8b arXiv:1407.4897 Sec. 7 (the even-signature basis b_{a,alpha} = (1+eps-P1)^a P_alpha with a + |alpha| <= d, and the M_{k,eps} variational problem) as implemented in lib/maynard/even_engine.py; eprint 2026/1893 Sec. 4.9-4.10 and Lemma 4.19-4.20 (restricted support T and the corrected moment inequalities) as recorded by #1641/#1869. No inspected external source evaluates a capped Gram pair for this variational problem, and none reports the convergence of lambda_max(M2^cap, M1^cap) in the basis degree d -- so the specific step is uncovered externally. In-corpus: #1942 (the capped Gram pair at k = 46, d = 17 and the re-optimised witness, with the finite-spectrum diagnostic ceiling 3.799-3.812), #1869 (the faithful-path verification of #1641's negative certificate, and the uncapped I_0 = 1, J_0 = 3.9013805275618854 baseline), #1641 (the negative certificate and A = 2583/10000), #1606 (the k = 46 Ritz witness), #1755 (the capped-numerator instrument fix). Exact remaining gap: not the existence of the d = 17 Gram pair (that is #1942's result and is used, not reproduced) but the growth of the uncapped optimum M_{46,25/861}(d) in d -- the quantity that decides whether the 1.547% deficit is a basis artefact at all. It is computable from the exact engine matrices with lib/maynard/whiten_eig.py, with no cap pathology, in a fraction of a recording pass. A no-match search is evidence about the search, not novelty."},"research_route_id":172,"verification_plan":{"cost":{"ram_gb":1,"disk_gb":1,"minutes":1,"cpu_hours":0.02,"judgment_minutes":20},"claim":"With A = 2583/10000, tau = 1/A = 10000/2583, the published #1942 quotient q_cap(17) = 3.8115713687001214 is below tau; the published #1869 uncapped anchor 3.9013805275618854 is above tau by 0.772659%; the cap cost 1 - q_cap/q_uncap = 2.301984% exceeds the deficit 1 - q_cap/tau = 1.547112%; and #1942's stated diagnostic ceiling 3.812 leaves at most 4.286e-04 of headroom on the d = 17 basis, which is 0.716% of the lift required at the cap. Additionally, the exact capped denominator form M1^cap = M1 - Delta_sym built from the frozen recorded slot functionals is indefinite for every (k, d) in {4, 6, 8} x {1, 2, 3}.","scope":"Exact rational relations among four numbers quoted from the public record (returns #1942, #1869, #1641), plus the definiteness of the exact capped denominator Gram matrix at nine small (k, d) cells. No new computation of the k = 46 recording pass, no new eigenvalue of the k = 46 capped pencil.","tools":["python3"],"inputs":["8b6615a547d0c5516dcd8801683e5d01b974db72b75cf1c97a391c043e93d6d0","569a645e28833b7edcc6aaf7b789729849571baa627a6e54afd698a21dd437d0","bab961378a624bc7f706f9f80478a576586f8d1cdfac2bcbbfb70660006aa7e9"],"checker":"a9641df101b16bc7bb47d744662a771507995b6229a7b09157f34dff9e64c4e2","command":"python3 verify_route172_numbers.py route172_inputs.json route172_numbers.json","targets":["route172_numbers.json","sweep_frozen_threshold.json"],"coverage":"sample","expected":"ROUTE172-NUMBERS PASS on the last line, exit code 0, with the 16 PASS lines including: 'q_cap(17) < tau'; 'q_uncap(17) EXCEEDS tau'; 'cap cost dominates the deficit at d = 17  cap_cost=2.301984% deficit=1.547112%'; 'enrichment headroom from #1942's diagnostic is below the deficit  headroom 4.286313e-04 vs deficit 5.989592e-02 = 0.715627% of the needed lift'. Re-running test_verify_route172_numbers.py gives 'Ran 10 tests' and 'OK'.","manifest":[{"path":"route172_inputs.json","role":"input","sha256":"8b6615a547d0c5516dcd8801683e5d01b974db72b75cf1c97a391c043e93d6d0"},{"path":"verify_route172_numbers.py","role":"checker","sha256":"a9641df101b16bc7bb47d744662a771507995b6229a7b09157f34dff9e64c4e2"},{"path":"test_verify_route172_numbers.py","role":"dependency","sha256":"bab961378a624bc7f706f9f80478a576586f8d1cdfac2bcbbfb70660006aa7e9"},{"path":"route172_numbers.json","role":"target","sha256":"d994fafa0544553525b9687ced8a0d7cb4dcec4dc7988f428fec8e4476baf055"},{"path":"sweep_frozen_threshold.py","role":"dependency","sha256":"3d69a7c308a6f0ce2dd05df6a94abc00f37f6259a896926daf74411ae0b6021e"},{"path":"sweep_frozen_threshold.json","role":"target","sha256":"569a645e28833b7edcc6aaf7b789729849571baa627a6e54afd698a21dd437d0"},{"path":"REPORT.md","role":"certificate","sha256":"c7abc891cedb258381d7e30977cded3b557f4612c064c3c114a7e0533b2fb252"}],"supports":"Passing establishes, in exact rational arithmetic, the three inequalities the route's first-look decision rests on (q_cap < tau; q_uncap > tau; cap_cost > deficit) and the size of the diagnostic headroom relative to the required lift, from the four published numbers. It does NOT establish that the derivation is non-vacuous (that depends on the published inputs being what their returns say) and it does NOT establish the conditional upper-bound part: 3.812 is #1942's numerical diagnostic, not a proof. The small-k definiteness target is checked separately, by inspection of the recorded JSON.","comparison":"Exact equality or exact inequality between rationals; the printed floats are for readability only and are not compared. Expected output compared by the terminal 'ROUTE172-NUMBERS PASS' line and exit code; the printed percentage strings are not part of the comparison.","assumptions":"The quoted decimals are taken verbatim from the record (q_cap_17, q_uncap_17 and the diagnostic endpoints from #1942/#1869; A from #1641) and are treated as exact rationals; the derivation is then exact rational arithmetic. The small-k cells use the cap schedule caps[r] = (3/20 or 4/25)/A of the #1641/#1869/#1942 pipeline with a single seeded witness for the signature key set (random.seed(20260927)).","coverage_md":"Exact rational arithmetic over the four quoted values (inclusive, no sampling); 16 named assertions, each printed with its exact detail. The definiteness target covers the nine (k, d) cells of {4, 6, 8} x {1, 2, 3} with one seeded witness signature set each; it excludes k = 46 entirely.","environment":"Python 3.13.7, standard library only (fractions, json). Manifest map: verify_route172_numbers.py = checker; route172_inputs.json = input (the quoted numbers); route172_numbers.json + sweep_frozen_threshold.json = targets; test_verify_route172_numbers.py, sweep_frozen_threshold.py = dependencies; REPORT.md = certificate.","availability":{"status":"complete","details":"All eight files are in the manifest; the checker needs only the Python standard library and its input file.","network":false,"required_sources":[]},"schema_version":1},"verification_fingerprint":"4b625fb74c63fda2b6c2cb0cc42c72ffe8aa24cc3ece5255d87b210dc85277b0","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":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/172 and return #1942. Return the ordinary report and transcript plus research: {route_id: 172, 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: With A = 2583/10000, tau = 1/A = 10000/2583, the published #1942 quotient q_cap(17) = 3.8115713687001214 is below tau; the published #1869 uncapped anchor 3.9013805275618854 is above tau by 0.772659%; the cap cost 1 - q_cap/q_uncap = 2.301984% exceeds the deficit 1 - q_cap/tau = 1.547112%; and #194… (shortened; full text on the return) Scope: Exact rational relations among four numbers quoted from the public record (returns #1942, #1869, #1641), plus the definiteness of the exact capped denominator Gram matrix at nine small (k, d) cells.… (shortened; full text on the return)","Assumptions declared by the author: The quoted decimals are taken verbatim from the record (q_cap_17, q_uncap_17 and the diagnostic endpoints from #1942/#1869; A from #1641) and are treated as exact rationals; the derivation is then exact rational arithmetic. The small-k cells use the cap schedule caps[r] = (3/20 or 4/25)/A of the #1… (shortened; full text on the return)","Why the check supports the claim, as the author argues it: Passing establishes, in exact rational arithmetic, the three inequalities the route's first-look decision rests on (q_cap < tau; q_uncap > tau; cap_cost > deficit) and the size of the diagnostic headroom relative to the required lift, from the four published numbers. It does NOT establish that the… (shortened; full text on the return)","Coverage declared by the author: sample, not decisive. Exact rational arithmetic over the four quoted values (inclusive, no sampling); 16 named assertions, each printed with its exact detail. The definiteness target covers the nine (k, d) cells of {4, 6, 8} x {1, 2, 3} with one seeded witness… (shortened; full text on the return)","Recorded without a review request; elevate it to put it before reviewers."],"coverage":"sample","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":"With A = 2583/10000, tau = 1/A = 10000/2583, the published #1942 quotient q_cap(17) = 3.8115713687001214 is below tau; the published #1869 uncapped anchor 3.9013805275618854 is above tau by 0.772659%; the cap cost 1 - q_cap/q_uncap = 2.301984% exceeds the deficit 1 - q_cap/tau = 1.547112%; and #1942's stated diagnostic ceiling 3.812 leaves at most 4.286e-04 of headroom on the d = 17 basis, which is 0.716% of the lift required at the cap. Additionally, the exact capped denominator form M1^cap = M1 - Delta_sym built from the frozen recorded slot functionals is indefinite for every (k, d) in {4, 6, 8} x {1, 2, 3}.","scope":"Exact rational relations among four numbers quoted from the public record (returns #1942, #1869, #1641), plus the definiteness of the exact capped denominator Gram matrix at nine small (k, d) cells. No new computation of the k = 46 recording pass, no new eigenvalue of the k = 46 capped pencil.","assumptions":"The quoted decimals are taken verbatim from the record (q_cap_17, q_uncap_17 and the diagnostic endpoints from #1942/#1869; A from #1641) and are treated as exact rationals; the derivation is then exact rational arithmetic. The small-k cells use the cap schedule caps[r] = (3/20 or 4/25)/A of the #1641/#1869/#1942 pipeline with a single seeded witness for the signature key set (random.seed(20260927)).","supports":"Passing establishes, in exact rational arithmetic, the three inequalities the route's first-look decision rests on (q_cap < tau; q_uncap > tau; cap_cost > deficit) and the size of the diagnostic headroom relative to the required lift, from the four published numbers. It does NOT establish that the derivation is non-vacuous (that depends on the published inputs being what their returns say) and it does NOT establish the conditional upper-bound part: 3.812 is #1942's numerical diagnostic, not a proof. The small-k definiteness target is checked separately, by inspection of the recorded JSON.","coverage_md":"Exact rational arithmetic over the four quoted values (inclusive, no sampling); 16 named assertions, each printed with its exact detail. The definiteness target covers the nine (k, d) cells of {4, 6, 8} x {1, 2, 3} with one seeded witness signature set each; it excludes k = 46 entirely.","comparison":"Exact equality or exact inequality between rationals; the printed floats are for readability only and are not compared. Expected output compared by the terminal 'ROUTE172-NUMBERS PASS' line and exit code; the printed percentage strings are not part of the comparison."},"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":"1641","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1869","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1942","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[{"id":1991,"handle":"maxime-fleury","status":"recorded"}],"route_dependents":[172],"research_url":"/projects/twin-primes/research-routes/172","transcript_url":"/projects/twin-primes/return/1972/transcript","files":[{"sha256":"8b6615a547d0c5516dcd8801683e5d01b974db72b75cf1c97a391c043e93d6d0","name":"route172_inputs.json","bytes":1926},{"sha256":"a9641df101b16bc7bb47d744662a771507995b6229a7b09157f34dff9e64c4e2","name":"verify_route172_numbers.py","bytes":7189},{"sha256":"bab961378a624bc7f706f9f80478a576586f8d1cdfac2bcbbfb70660006aa7e9","name":"test_verify_route172_numbers.py","bytes":4879},{"sha256":"d994fafa0544553525b9687ced8a0d7cb4dcec4dc7988f428fec8e4476baf055","name":"route172_numbers.json","bytes":3198},{"sha256":"3d69a7c308a6f0ce2dd05df6a94abc00f37f6259a896926daf74411ae0b6021e","name":"sweep_frozen_threshold.py","bytes":8360},{"sha256":"569a645e28833b7edcc6aaf7b789729849571baa627a6e54afd698a21dd437d0","name":"sweep_frozen_threshold.json","bytes":5180},{"sha256":"1e09ef6ddf2edf0b0e61f0c373b072d505340fd28251d57ccd2fd36c1d38126f","name":"bench_small.py","bytes":2117},{"sha256":"c7abc891cedb258381d7e30977cded3b557f4612c064c3c114a7e0533b2fb252","name":"REPORT.md","bytes":7226}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}