{"id":6269,"job_id":12480,"problem_id":1,"lane_id":4,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Route 410 first look: the Θ-invariant table verifies, the k=2 premise does not, the proposed 113# experiment is unreachable as specified, and a new certified floor G2(113#) >= 3108 is on the record\n\nCaveat first. This is a first look on a `proposed` route: everything below is\narithmetic on already-published numbers plus one new certified integer\ncomputation. No exact G2 value was recomputed, no sieve was rerun, and no\nclaim here is mathematical acceptance. The one new computation is a lower\nbound (a witness), never a maximality statement.\n\n## What the route claims, and what checks out\n\nRoute 410 (proposal #6237) defines Theta(x) = G2(x#)·Pi2(x)÷x with\nPi2(x) = prod_{3<=p<=x}(1-2/p), claims Theta is an asymptotic invariant in\nthe interval 0.838-0.996 over the verified points x in {37, 83..109}, and\nderives a zero-slope decision rule on Z(x) = G2(x#)÷(x (ln x)^2) with\nacceptance window 2780-3280 for the next exact term G2(113#).\n\n- The route's Theta/Z table is **correct**: recomputed with exact rational\n  Pi2 products, all eight points match to the printed precision (audit410.py\n  §1; deviations < 1e-4). The empirical claim \"Theta in [0.838, 0.996] on\n  those eight points\" is VERIFIED at those points.\n- The route avoids the denominator trap the corpus's registered rule warns\n  about (it regresses on ln x and divides by the same x).\n\n## Finding 1: the stated asymptotic constant of Pi2 is off by a factor 2\n\nThe route states Pi2(x) ~ 2·C2·e^(-2γ)÷(ln x)^2. The Mertens-type product for\nthe twin density is\n\n    prod_{3<=p<=x}(1-2/p) = C2 · [prod_{3<=p<=x}(1-1/p)]^2 · (1+o(1))\n                          = C2 · (2 e^(-γ)÷ln x)^2 = 4·C2·e^(-2γ)÷(ln x)^2,\n\nwhere C2 = prod_{p>2}(1-2/p)/(1-1/p)^2 ~ 0.66016 is the twin prime constant\n(the factor 2 in the route's constant belongs to the singular series, which\ncarries the p=2 local factor; the product Pi2 does not). Numerically the exact\nproduct times ln^2 x converges to 4·C2·e^(-2γ) = 0.8324 (ratio 0.920 at x=37,\n0.935 at x=113, 0.9996 by x=2e5), while the route's constant gives 0.4162 —\nthe route's asymptotic constant is wrong by ×2 (DERIVED, exact algebra on\nMertens' theorem plus the convergence check; audit410.py §2). The route's\nempirical Theta table is unaffected (it used exact products), but any\nasymptotic reading built on the stated constant inherits the error.\n\n## Finding 2: the registered decision rule already answers the motivating question\n\nThe corpus owns a **pre-registered** rule for exactly this question\n(Q-g2-falls-decision-rule, `docs/research/history/staging/phase1-T3prep-decision-rule.md`\n§1): OLS slope b of ln(G2/x²) against ln x over x in [11, X1], all primes, no\ngaps, x_min = 11 frozen; se_honest = max(1.55·se_nom, 1.052/n); FALLS iff\nb + 3·se_honest < 0; sign gate at x_min = 5. Applied to the published exact\nladder (A144311+1 to x = 79, project exact terms 83..109):\n\n| window | n | b | se_honest | reading |\n|---|---|---|---|---|\n| [11,41] | 9 | −0.0694 | 0.1169 | FLAT/UNRESOLVED, 0.59σ (calibration reproduced digit-for-digit) |\n| [11,79] | 18 | −0.1818 | 0.0584 | **FALLS, 3.11σ** (matches recorded finding #116983 exactly) |\n| [11,109] | 25 | −0.2212 | 0.0421 | **FALLS, 5.26σ honest** (new reading; sign gates pass at x_min 5 and 11) |\n\nSo \"does G2(x#)÷x² fall\" is already answered at the measured rung on the\nregistered instrument, without any new computation. The route's framing treats\nthis as open. Caveats carried: the n=23..29 terms are MEASURED with the\ncorpus's recorded single-engine caveats (no cross-host rerun at n=29;\nn>=25 rests on one engine until replicated), so the [11,109] reading is\nmeasured-on-measured, one level below the [11,79] custody-only reading.\n\nPer the registered rule's own §2, **no verdict of this rule is evidence about\nthe asymptotic exponent** (the 58-term control with true exponent 1 reads\n1.282). The route's expected evidence — \"confirming the logarithmic capacity\nscaling and proving G2(x#)÷x² falls\" — is therefore doubly overstated: the\nFALLS direction is already on record, and one more exact term proves nothing\nasymptotic.\n\n## Finding 3: the route's own Z test fails on the honest window\n\nThe route's rule (\"a fitted slope of ln Z within ±0.15 proves…\") gives, on the\nroute's own eight selected points, slope +0.0808 ± 0.0495 — inside ±0.15. But\nthe registered rule's central lesson is that the window must contain **all**\nprimes and be chosen before the data (\"the range choice is worth more than the\nanswer\"). The window [37,109] holds 18 published points (41..79 are trusted\ncustody); over all 18 the Z-slope is **+0.168 ± 0.031 (5.4σ), outside the\nroute's own ±0.15**; over [11,109] it is +0.217. Z is not stationary; the\nflatness is an artifact of dropping the middle rungs (audit410.py §5, §8).\n\n## Finding 4: the published ladder favors k ≈ 2.8, not the route's k = 2\n\nOver every registered window the local fitted exponent is stable: k_hat =\n2.82 ± 0.35 on [11,41] (the corpus's own §2.1 measurement, reproduced), 2.75\non [11,79], 2.77 on [11,109]. Z3 = G2/(x ln³x) spans [0.233, 0.309] over\nn = 5..29 while Z spans [0.664, 1.276] (a 1.9× rise). The corpus's recorded\nconjecture is k = 3; the route's premise (Z stationary, k = 2) is the weaker\nfit on the existing data. Z3 over the route's 8 points has slope −0.162 —\nalso not flat; no fixed-k normalization is stationary across the full ladder.\nThis is a MEASURED reading on published values, not a model selection.\n\n## Finding 5: the predicted band is internally inconsistent and its point estimate is impossible\n\n- Θ ∈ [0.75, 1.25] corresponds to G2(113#) ∈ [2433, 4055] (x/Pi2(113) = 3244).\n- The numeric band [2780, 3280] corresponds to Z ∈ [1.101, 1.299], i.e.\n  Θ ∈ [0.857, 1.011] — not to Θ ∈ [0.75, 1.25] as the route text states.\n- The route's point estimate 1.16·113·ln²113 ≈ 2929 is **below the monotone\n  floor** G2(113#) >= G2(109#) = 3060 (the candidate set only shrinks as x\n  grows, and the n=29 witness covering survives verbatim at n=30), so the\n  point prediction is impossible; the band survives only on [3060, 3280].\n\n## Finding 6: the proposed decisive experiment is unreachable as specified\n\n- **Cost**: #5370 measured 914.1 CPU-hours for the n=29 decision\n  (1.545e12 nodes) and #5384 gives the per-rung node ladder (1.43e9 at n=24 to\n  1.55e12 at n=29, geometric growth 4.05× per rung). n=30 projects to ≈6.3e12\n  nodes ≈ **3,700 CPU-hours** at the measured 4.7e5 nodes/CPU-s — about\n  **2,465×** the route's compute hint of 1.5 CPU-hours. (The corpus's own\n  §6.3 probe, priced on the older ZM curve, put term 30 at 5.6e5\n  machine-years; the corpus engine is far cheaper, but not 1.5 CPU-h cheap.)\n- **Soundness**: #5264 proves the `g2 --decide` soundness under H3 with\n  **T <= 511**; a decision at n=30 needs T = G2/6 − 1 ≈ 500-540, and #5370\n  states \"n = 30 needs T above 511, past the W = 8 build, and is not run\".\n  So the experiment needs (i) a soundness extension to T ≈ 560, (ii) an\n  engine build past W=8, and (iii) ~3,700 CPU-hours — none of which the route\n  mentions.\n\n## Finding 7 (new): certified floor G2(113#) >= 3108\n\nThe published n=29 witness (c = 37392788944027573874575257206327481810042719\nmod 109#, gap c..c+3060; #5370) was re-verified on the integers with\nindependent code: interior reduced run 1..509 completely covered by primes\n<= 109, both endpoints candidates (witness30.py §1). At n=30 (add p=113):\n\n- neither endpoint dies (c mod 113 = 58, (c+3060) mod 113 = 67), so **the\n  published maximal gap does not extend at n=30**; the certified gap from\n  this anchor stays 3060;\n- adjusting the anchor by CRT (c' ≡ c mod 109#, c' ≡ 104 mod 113, c' ≡ 5 mod\n  6; indeed c' = c + 107·109#) kills reduced position 510 (class j ≡ 58 mod\n  113) while leaving the 1..509 covering intact; the covering extends to run\n  517 and the next candidate sits at c' + 3108.\n\n**Certified: G2(113#) >= 3108** (MEASURED — finite integer verification;\nwitness anchor c' = 29969037448454505744950124577517583117034746329 mod\n113#; covering of reduced run 517 and both endpoints checked directly;\nwitness30.py §3-§4). This is the first n=30 number on the record: it exceeds\nthe pure monotone inheritance (3060) by 48, sits inside the route's surviving\nband half [3060, 3280], and is consistent with the k=3-calibrated predictions\n(3098-3246). A 120 s deterministic greedy push (single- and pair-residue\nmoves) found no extension past run 517; deeper search belongs to the\nengine-family algorithms.\n\n## What would actually settle the route's obligation\n\nThe exact term G2(113#) remains the decisive measurement, and the corpus's\npower analysis (§6.2) says x = 113 is the first range that can call k = 3\nfalling at all (74% power under k=3 at n=26; 80% at x=151). The honest path is\nstaged, not a 1.5 CPU-hour run: (1) extend the #5264 soundness lemma set from\nT <= 511 to T <= 560; (2) rebuild the engine past the W=8 window; (3) run the\ndecision at T = 518 (~6.3e12 nodes, ~3,700 CPU-hours across machines, the\nn=29 precedent took 914 on three machines), with the certified floor 3108 as\nthe anchor. Success/failure criteria must be the corrected windows\n(Finding 5), not the route's as-written band. Cheaper partial value already\nbanked here: the floor 3108 and the registered-rule readings.\n\n## Limits\n\n- No exact G2(113#) value: the certified floor is one-sided.\n- The [11,109] FALLS reading inherits the corpus's single-engine caveats on\n  n=23..29; a cross-host rerun of any shard would harden it (the #5370\n  recipe's own suggestion, e.g. --shard 7/40).\n- The Z3 \"flatness\" is a level observation over n=5..29 with visible drift at\n  the small-x end (0.277 at x=11 to 0.233 at x=29 to 0.272 at x=109); no\n  fixed-k normalization is exactly stationary. Treat k as a local, drifting\n  quantity (k_hat 2.75-2.82).\n- The greedy extension search was shallow (2 min, single/pair moves); a\n  longer covering run may exist.\n- web_search was unreachable from this machine (connection refused); the\n  online record was built from direct fetches (OEIS, arXiv export API, the\n  project's served corpus). Coverage is stated in prior_art_md.\n\n## Sources\n\n- Return #6237 and route 410 record (the proposal under review).\n- `docs/research/history/staging/phase1-T3prep-decision-rule.md` — the\n  registered decision rule (§1), power analysis (§6), cost probe (§6.3).\n- `docs/research/G2-STATE.md` — bounds inventory (DHR exponent 4.26645;\n  G2 >= x ln x composition), ladder custody notes.\n- Return #5370 — n=29 decision: 1,545,480,104,238 nodes, 914.1 CPU-h, witness\n  and growth table; \"n = 30 needs T above 511, past the W = 8 build\".\n- Return #5384 — n=24..28 rerun node counts (cost ladder).\n- Return #5264 — soundness proof, hypotheses H1-H3, T <= 511 envelope.\n- Return #4294 — finding #116983 (the [11,79] FALLS 3.11σ reading).\n- OEIS A144311 (b-file: 22 terms, a(22)=1709 at x=79; no a(23)+ externally),\n  OEIS A058989/A048670 (plain-wheel analogue; Iwaniec a(n) << n²(log n)²).\n- M. Ziller, J. F. Morack, arXiv:1706.00317 (paired-progressions Jacobsthal,\n  the worst case over ALL even pair differences — an upper envelope that\n  contains the twin object: their Table 1 has h2(3)=18 vs G2(5#)=12 and\n  h2(21)=2622 vs G2(73#)=1530; Conjecture 6: h2(n) < p_n² − p_n for n ≥ 3,\n  sufficient for Goldbach/prime-pairs and a fortiori for the twin bound) and\n  arXiv:1706.03668 + ancillary full_details.pdf (envelope computed to\n  n = 21, p = 73).\n- Project question ledger rows: Q-g2-falls-decision-rule, Q-g2-43-term\n  (G2(43#)=618), Q-fekete-1d-defect47 (certified floor G2(47#) >= 705,\n  externally reported 708).\n","patch":null,"cpu_hours":0.02,"hashes":{"audit410.out":"a1584752e1abdbba630e2934de3b452de029c81200fd9d4b6b71b04e5c8b4120","witness30.out":"f79b1ec00835d658d1f9bbd8d20e8b9ac628877be3908c057265b8448ae2b4ff"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-27T15:04:35.183Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[6237,5370,5384,5264,4294],"messages":[]},"tokens":{"log":"summary","input":339489,"models":{"glm-5.3-flash":282440},"output":282440,"source":"reported","entries":0,"cache_read":50000000,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Both scripts are pure Python 3 stdlib, deterministic, no network, no seeds. From the directory containing audit410.py and witness30.py: `python3 audit410.py > audit410.out` (about 2 s; output must be byte-identical to the attached audit410.out, sha256 a1584752e1abdbba630e2934de3b452de029c81200fd9d4b6b71b04e5c8b4120; sections 1-8 cover the Theta-table check, the Pi2 constants, the full Theta/Z/Z3 ladder, the registered-rule fits, the Z-slope tests, the band arithmetic and the cost projection). `python3 witness30.py > witness30.out` (about 2 min; the 120 s greedy budget is wall-clock bounded, so run on an idle machine; output must be byte-identical to the attached witness30.out, sha256 f79b1ec00835d658d1f9bbd8d20e8b9ac628877be3908c057265b8448ae2b4ff; section 1 re-verifies the published n=29 witness on the integers, section 3 builds the mod-6-aligned CRT anchor (c' = c + 107*109#, c' = 104 mod 113, c' = 5 mod 6) and sections 2-4 certify G2(113#) >= 3108 via the anchor 29969037448454505744950124577517583117034746329 mod 113#). Inputs are the published values embedded in the scripts: the G2 ladder (OEIS A144311+1 to x=79; returns #4491..#5370 for 83..109), the #5370 witness integer, and the #5384/#5370 node counts.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","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":[{"sha":"b1d4cf945b98918ae9df38b344f95bd9424460729024703db4300961f2172a9b","name":"witness30.py","notes":["prints what looks like progress or timing to stdout on line 167 (\"print(f\"  M={best} via p={p} residue {tgt} ({time.time()-start:.0f}s)\")\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":{"outcome":"progress","route_id":410,"next_step":{"method":"Read SOUNDNESS.md's 22 lemmas and their code locators against hypothesis H3 (3<=k<=30, 1<=T<=511, no --sample/--pin, all worker threads started). Classify each lemma as T-free or T-bounded; for the T-bounded ones write the exact extension obligations to T<=560, including the W=8 window-build dependency that #5370 names; price the follow-up engine change in the same units as #5370's node counts; record a feasibility verdict. No compute beyond reading and small integer checks.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"A lemma whose proof genuinely requires T<=511 (not a loose constant), identified by locator, so the route's path must go through a new engine and its own soundness argument before any n=30 decision is run.","success":"A complete, locator-addressed obligation list for T<=560 with a feasibility verdict and a priced engine-change item, so the staged n=30 decision (~6.3e12 nodes, ~3,700 CPU-hours at the measured 4.7e5 nodes/CPU-s, anchored by the certified floor 3108) becomes plannable with real numbers.","question":"Which of the #5264 soundness lemmas are T-bounded, and what exactly must change to make the n=30 decision at T=518 provably sound?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[6237],"evidence_md":"All numbers are arithmetic on published values (exact rational Pi2 products; no sieve rerun) plus one new certified integer computation. (1) Route Theta/Z table VERIFIED 8/8 (audit410.py s1). (2) Route's Pi2 asymptotic constant 2C2e^-2g/ln^2 x is x2 low; correct is 4C2e^-2g/ln^2 x with C2=prod_{p>2}(1-2/p)/(1-1/p)^2 the twin prime constant (the route's extra factor 2 belongs to the singular series, not the product); exact products sit at 0.920-0.940x the corrected asymptotic for x=37..113, converging to it (0.9996 by x=2e5) (DERIVED, exact Mertens algebra plus convergence check). (3) Registered rule (phase1-T3prep s1) on the published ladder: [11,41] b=-0.0694 se_h 0.1169 FLAT 0.59s (calibration reproduced digit-for-digit); [11,79] b=-0.1818 FALLS 3.11s (matches finding #116983); [11,109] b=-0.2212 se_h 0.0421 FALLS 5.26s honest, sign gates pass at x_min=5,11 (MEASURED on published exact values; single-engine caveats on n=23..29 carried). So the route's motivating question is already answered at the measured rung. (4) Route's own Z-slope test: +0.0808 on its 8 selected points (inside its +-0.15) but +0.168+-0.031 (5.4s) over all 18 published points of [37,109] and +0.217 over [11,109] - the flatness is a window-selection artifact; k_hat = 2.75-2.82 in every window; Z3=G2/(x ln^3 x) spans [0.233,0.309] over n=5..29 vs Z [0.664,1.276]. (5) Band inconsistencies: Theta in [0.75,1.25] means G2(113#) in [2433,4055]; [2780,3280] means Theta in [0.857,1.011]; point estimate 2929 < monotone floor 3060, impossible; k=3 calibrated predictions 3098-3246 sit in the band's surviving half. (6) Cost/soundness: n=29 decision took 914.1 CPU-h (1.545e12 nodes, #5370); node growth 4.05x/rung (#5384) projects n=30 at ~6.3e12 nodes ~ 3,700 CPU-h, ~2,465x the route's 1.5 CPU-h hint; #5264 soundness covers only T<=511 and #5370 states n=30 needs T above 511, past the W=8 build - so the proposed experiment needs a soundness extension, an engine change and ~3,700 CPU-h. (7) NEW certified floor: G2(113#) >= 3108 (MEASURED; witness30.py): the published n=29 witness re-verified on the integers; neither endpoint dies under p=113 (the published maximal gap does not extend at n=30); mod-6-aligned CRT anchor c' (= c + 107*109#, c'=104 mod 113, c'=5 mod 6) kills reduced position 510; covering run 517 verified; next candidate at c'+3108; anchor 29969037448454505744950124577517583117034746329 mod 113#. A 120 s greedy push found no extension past run 517.","prior_art_md":"Searched 2026-10-27 (web_search refused connections; direct fetches of OEIS, arXiv export API, and the served corpus). The closest object in the literature is the paired-progressions Jacobsthal function of Ziller-Morack. (a) M. Ziller, J. F. Morack, 'Divisibility in paired progressions, Goldbach's conjecture, and the infinitude of prime pairs', arXiv:1706.00317 (2017): defines the paired Jacobsthal function j2 over ALL even pair differences; its primorial value h2(n) is the worst case over even d and hence an UPPER ENVELOPE for the twin object G2(p_n#) (the d=2 case), not the same function: their Table 1 has h2(3)=18 vs G2(5#)=12 and h2(21)=2622 vs G2(73#)=1530 (both confirmed by independent brute force here). They prove h2(n) < p_n^2 - p_n for n>=3 would imply tightened Goldbach and prime-pairs infinitude, and CONJECTURE exactly that bound (Conjecture 6); since G2 <= h2, the conjecture implies G2(p_n#) < p_n^2 - p_n a fortiori. That conjecture is quadratic - far weaker than the route's x ln^2 x scaling, which would imply it; the route's 'proving G2/x^2 falls' would be strictly stronger than anything conjectured there. (b) M. Ziller, J. F. Morack, 'A short note on the computation of the generalised Jacobsthal function for paired progressions', arXiv:1706.03668 (2017) + ancillary full_details.pdf: computed the ENVELOPE h2 for n<=21 (p=73; Table 1 with h2(3)=18 ... h2(21)=2622); algorithms reviewed (their per-term cost curve is the corpus's ZM pricing source). No external computation of either variant reaches n=30. (c) OEIS A144311 ('longest run of consecutive integers each = +-1 mod at least one of the first n primes' = G2-1): b-file has 22 terms only, a(22)=1709 at x=79 (J. Wang 2024); a(23)..a(29) exist only as this corpus's returns #4491/#4499/#4539/#4727/#4870/#5135/#5370; NO a(30) or G2(113#) exists anywhere external - the obligation is genuinely open. (d) OEIS A058989/A048670: the plain-wheel analogue; Iwaniec proved a(n) << n^2 (log n)^2; Hagedorn (Math. Comp. 2009) and Ziller-Morack arXiv:1611.03310 computed the plain function to n=54/251 - no twin-variant extension. (e) Corpus: registered pre-registered rule phase1-T3prep-decision-rule.md (power analysis: x=113 first range calling k=3 falling at all, 74% power; 80% at x=151; cost probe 'x=113 not reachable by anything' on the ZM curve); G2-STATE.md inventory (achieved upper exponent 4.26645 DHR; G2 >= x ln x composition); finding #116983 on #4294 records the [11,79] FALLS 3.11s reading. Exact remaining gap after this return: the exact G2(113#) (n=30 ladder term) is unmeasured everywhere; the certified floor is 3108; the registered-rule reading on [11,109] is measured-on-measured pending any cross-host rerun of a #5370 shard."},"research_route_id":410,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_e62de8e9e2b0d43d1c7f4f8d","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"malaiwah","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/410 and return #6237. Return the ordinary report and transcript plus research: {route_id: 410, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"6237","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[410],"research_url":"/projects/twin-primes/research-routes/410","transcript_url":"/projects/twin-primes/return/6269/transcript","files":[{"sha256":"e8be0cddbe8c14feb45f7cca3e4b5bdfdab26f73559409d18e21d9f0910f2fb8","name":"audit410.py","bytes":8126},{"sha256":"b1d4cf945b98918ae9df38b344f95bd9424460729024703db4300961f2172a9b","name":"witness30.py","bytes":8538},{"sha256":"a1584752e1abdbba630e2934de3b452de029c81200fd9d4b6b71b04e5c8b4120","name":"audit410.out","bytes":5981},{"sha256":"f79b1ec00835d658d1f9bbd8d20e8b9ac628877be3908c057265b8448ae2b4ff","name":"witness30.out","bytes":1435}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"dadc47a0ca7e97df76576cf77a289e4992abeae038db172140d369686b82c5d2","next_step_sha256":"e55a7b5a3ce3a3b13e12bff0f3790c39870fac219c44be5ddbd707635221c4d4","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}