{"id":5642,"job_id":11406,"problem_id":1,"lane_id":3,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Job #11406 - Route 373 rescue: exact branch-decomposed antiderivative certificate for the rational-regime 3-fold saving\n\nGeneral mode, lane formalize, route 373 (job 11406). Predecessors: #5631 (published certificate, sha a7371e8c...) and #5639 (refuted box attempt, recipe source). **No proof claim beyond the stated certified bounds; conditional on the audited statement-level inputs of #5321 (review #1855).**\n\n## 1. The obligation and what was delivered\n\nThe named obligation: does a branch-decomposed exact antiderivative reduction of the FULL 3-fold Buchstab saving (t3 >= 1/20, including v3 > 3) certify Delta Phi_3(1/20) >= 0.1120 in 34-digit outward-rounded arithmetic, with Phi_lo(1/C, 1/20) > 0 at C = 3.296 (anatomy) and C = 3.299 (plain)?\n\nDelivered:\n1. **Exact t3-antiderivative certificate on the rational regime** (new): for t3 >= (5/8 - t1a - t2a)/4 both sieve variables are affine in 1/t3 (v2hi = R2hi/t3 - 1, v3lo = A/t3 + (b-1) with A = a - t1b - t2b), so the two t3-integrals evaluate in closed form with directed ln enclosures: (kappa/R2hi)[ln(t/(R2hi-t))] and (kappa/A)[ln(t/(A+(b-1)t))]. Branch-decomposed at every knot of a rigorous piecewise-linear lower envelope of theta_3 (five Lemma 2' pairings; exact per-pairing admissibility bounds T_p from the monotone constraints; concave-tent chords recomputed per knot interval with tent-peak crossings as knots). **Delta Phi_3^rat_lo(1/20) >= +0.0588556865** (7183 boxes at step 1e-3, 2088 active, 21.1 s) - improving #5631's per-cell monotone bound +0.0581303786 and within 20% of the float 0.07386998.\n2. **Outer-layer per-cell bound** (t3 < (5/8-t1a-t2a)/4, where v can exceed 3): **Delta Phi_3^outer_lo >= +0.0017162975** (1232 active vs 42409 vacuous cells at (t1,t2)-step 1e-3, t3-step 1e-2). The vacuity is structural: the v-interval widths (0.05-0.1 from the cell parameter variation) exceed the boundary-layer sliver |v3 - v2| ~ 0.02 - exactly what #5639 section D diagnosed; the outer layer carries 0.0402 of the 0.1141 float value and is the entire remaining gap.\n3. **Combined certified thresholds**: Delta Phi_3_lo(1/20) >= +0.0605719841, E3_lo = 2.0838935157 (published 2.0814519102). With the verbatim Part 4 Q-enclosure: **C = 3.312 anatomy Phi_lo = +0.0028666205 > 0 CERTIFIED** (published margin +0.0004250150, improved 6.7x); C = 3.310 anatomy Phi_lo = -0.0037930 < 0, so the certified anatomy threshold lands between 3.310 and 3.312 (~3.311). The full-domain targets are NOT established: Phi_lo(1/3.296) = -0.0504 and Phi_lo(1/3.299) = -0.0404 (anatomy) at the certified E3_lo, and Delta Phi_3_lo >= 0.1120 is not claimed.\n4. **Quantified obstruction for the outer layer** (the distinct next experiment): a rigorous Psi(v) = int_1^v F evaluation needs absolute enclosure width <= ~1e-9 (per-box budget ~4e-8 over ~2e4 active boxes with coefficients 1/R2 <= 10, 1/A <= 13), but range-enclosure tables accumulate an enclosure gap ~ H*TV(F) >= 6.5e-6 even at step H = 1e-5 - a four-order shortfall. The fix is a Taylor/midpoint-accumulated Psi table (per-cell second-order enclosures of F via the ODE recurrences, prefix gap ~ H^2*TV(F)/24 ~ 3e-12) plus either a graded (t1,t2) grid near the theta_3 = theta_2 boundary (corner loss ~ h, measured 15% at h = 1e-3 and 7.7% at h = 5e-4 by the float experiment) or gradient-corrected cell bounds.\n\n## 2. Baseline control and conventions check\n\nThe float feasibility experiment reproduces the published conventions: midpoint float at (t1,t2)-steps 1e-3/5e-4/2.5e-4 gives 0.114410/0.114193/0.114096, converging to the published 0.11406258; the float rational-regime split reproduces 0.07386998. Max v over the active region is 8.50, fixing the F-table range at [1, 9.5]. No published computation was rerun for credit; #5631's certified E(19/2) in [1.8886166329, 1.8886851233] and Delta Phi_2(1/20) in [0.1347048987, 0.1354309216] are consumed as published inputs (outward-rounded, hence valid certified bounds).\n\n## 3. Validity notes\n\n- v2hi <= 3 and v3lo <= 3 hold exactly on t3 >= (5/8 - t1a - t2a)/4 because R2, R3 <= 5/8 - t1a - t2a (theta_1 <= 5/8 with tent peak at s = 1/4).\n- v3lo <= v3 pointwise: theta_3 >= a + b*t3 pointwise (the chord of the concave tent lies below it, composed with the pointwise bound s1 >= (alpha_min if the chord increases else alpha_max) + s11*t3).\n- A > 0 on active pieces: v3lo = A/t3 + (b-1) <= v3 with v3 > 1 forces A > (1-b)t3 > 0.\n- The (t1,t2) dependence is corner-bounded: 1/(t1t2) >= 1/(t1b t2b), R2 <= R2hi, theta_3 >= envelope; the area factor (t1b-t1a)(t2b-t2a) is included.\n- Defects found and fixed during the work: a dropped cell-area factor (first run returned 37210.64, caught by the float cross-check); box-wide pairing admissibility (a 0.0372 regression, fixed with exact T_p bounds); wide-range tent chords (up to 0.05 sag at the peak, fixed with per-interval chords).\n\n## 4. Verdict\n\nOutcome progress: the exact-antiderivative reduction is implemented, rigorous, and improves the certified rational-regime bound and the C = 3.312 anatomy margin 6.7x; the full-domain obligation (Delta Phi_3 >= 0.1120, C = 3.296/3.299) remains open with the gap located entirely in the outer layer and the Psi-enclosure precision requirement quantified at four orders.\n- Post-science-review fixes (all six non-blocking findings applied): quoted bound values are now floor-directed 10-decimal quotes of the 34-digit certified Decimals (full precision in analysis_11406.json and the *_full fields of thresholds_11406.json/outer_11406.json); two directed-rounding slips fixed (the v2-term lower-log argument now rounds up with ln_hi, and the corner factor 1/(t1b t2b) now rounds down); the slope statement corrected to any chord slope of the concave tent; the Part 3b fast path is selectable as `certify_11406.py 1e-2`; the dead helper build_F_table removed. The certificate was re-run under the reviewed bwrap isolation (exit 0, 163.8 s; 7183/2088 boxes and 1232/42409 outer cells unchanged; every row sign and the C = 3.312 anatomy margin sign unchanged).","patch":null,"cpu_hours":0.12,"hashes":{"feas_11406.py":"0ffc20324b6b81b0972d20068b834c3857e96ebd3d753dccc22f486be4e3bc51","certify_11406.py":"f49cc935668001bf1063eed58333b65c77a9b01ffad91c46a667df7cebba9a48","outer_11406.json":"55bcf81ac51e19c8d98fcf693b0dc40c02803a9a1a9d243001dae7b30fe92b16","thresholds_11406.json":"ff85f860321e1810ff7154e35c10d4e1facf1ecab398bb24348b9028a66aad35"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-24T03:08:06.897Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5631,5321,5639,4511],"messages":[]},"tokens":{"log":"summary","input":246554,"models":{"glm-5.3-flash":251558},"output":251558,"source":"reported","entries":0,"cache_read":47030548,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Reproduction (deterministic, single-threaded, ~6 min total):\n\n```\npython3 -I certify_11406.py          # default Part 3b t3-step 1e-3 (slow, ~10 min):\n                                     #   Part 3a -> dp3_rat_lo >= 0.0588556865 (21 s),\n                                     #   Part 3b outer layer, Part 4 thresholds, Part 5 float reference\npython3 -I certify_11406.py 1e-2     # frozen-artifact fast path (Part 3b t3-step 1e-2)\npython3 -I feas_11406.py             # float feasibility: midpoint 0.114410/0.114193/0.114096\n                                     #   vs corner 0.097284/0.105434; max v 8.50 (~4 min)\n```\n\nThe certified rows quoted in the report were produced by the fast path `python3 -I certify_11406.py 1e-2` (Part 3b at t3-step 1e-2, dp3_outer_lo >= 0.0017162975, 1232 active / 42409 vacuous cells) plus the Part 4 threshold sweep. The producing run of the fixed post-review bytes (isolated bwrap execution, exit 0 in 163.8 s) wrote analysis_11406.json; every quoted bound below is a floor-directed (Q_hi ceiling-directed) 10-decimal quote of the 34-digit certified Decimals stored there. All arithmetic is 34-digit decimal with ROUND_FLOOR/ROUND_CEILING directed rounding; the only inputs taken from the literature are the published certified enclosures of #5631 (E(19/2), Delta Phi_2), quoted at their published 10-digit outward-rounded values, and kappa = 2*e^gamma enclosed from a 19-digit gamma bracket.\n\nDeterministic outputs: analysis_11406.json (full run), thresholds_11406.json (threshold rows), outer_11406.json (outer-layer bound at t3-step 1e-2).","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":"f49cc935668001bf1063eed58333b65c77a9b01ffad91c46a667df7cebba9a48","name":"certify_11406.py","notes":["prints what looks like progress or timing to stdout on line 812 (\"print(f\"elapsed {analysis['elapsed_s']}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":373,"next_step":{"method":"Keep Parts 1, 2, 4 of the published certificate and this return's Part 3a rational-regime bound. Build the Psi table on [3, 9.5] with per-cell Taylor enclosures of F, f, F', f', F'', f'' from the ODE recurrences (shifted-cell interval arithmetic), accumulate IF = int_3^s F with midpoint second-order accuracy, and enclose each Psi difference as [IF_lo(j_a) + part_lo] - [IF_hi(j_b) + part_hi] (width ~ 2e-10, independent of the v-distance). On the outer layer t3 < (5/8-t1a-t2a)/4 use the same envelope pieces as this return and evaluate the exact antiderivative (1/A)[Psi(v3lo(ta)) - Psi(v3lo(tb))] per piece with corner-bounded (R2hi, A, 1/(t1b t2b)). Handle the (t1,t2) corner spread (measured 15% loss at h = 1e-3, 7.7% at h = 5e-4, scaling ~ h) with a graded grid near the theta_3 = theta_2 boundary or gradient-corrected cell bounds (K monotone decreasing in R2, increasing in the envelope intercept a and slope b per branch, so corner evaluation is valid once the gradient correction bounds |dK/dt1| + |dK/dt2| per cell). Certify the sliver t3 in [t2a, t2b) either by fine (t2,t3) sub-cells or as a quantified dropped mass.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The Psi-based enclosure still returns Delta Phi_3_lo < 0.1120, or the (t1,t2) spread alone costs more than the 3.8e-4 margin at C = 3.299 plain; report the achieved certified value, the measured corner-loss curve, and the exact remaining gap.","success":"Delta Phi_3_lo(1/20) >= 0.1120 with all admissibility conditions enforced and Phi_lo(1/C, 1/20) > 0 certified at C = 3.296 (anatomy) and C = 3.299 (plain), exit 0 in under 30 s for the production certificate.","question":"Does a Taylor/midpoint-accumulated rigorous Psi table (per-cell second-order enclosures of F via the ODE recurrences d/du(uF) = f(u-1), d/du(uf) = F(u-1), giving a prefix enclosure gap ~ H^2*TV(F)/24 ~ 3e-12 at H = 1e-5) certify Delta Phi_3(1/20) >= 0.1120 over the full domain t3 >= 1/20, with Phi_lo(1/C, 1/20) > 0 at C = 3.296 (anatomy) and C = 3.299 (plain)?","budget_hours":4,"required_tools":[],"required_sources":[]},"depends_on":[5321,5631,5639],"evidence_md":"Exact branch-decomposed t3-antiderivative certificate on the rational regime (t3 >= (5/8-t1a-t2a)/4, where 1 < v2 <= v3 <= 3 and F = kappa/v exactly): Delta Phi_3^rat_lo(1/20) >= +0.0588556865 in 34-digit outward-rounded arithmetic (7183 boxes at step 1e-3, 2088 active; 21.1 s), improving #5631's per-cell monotone bound +0.0581303786 by 0.0007 and lying within 20% of the float 0.07386998. The reduction: on each rigorous piecewise-linear envelope piece theta_3lo = a + b*t3, v2hi = R2hi/t3 - 1 and v3lo = A/t3 + (b-1) are affine in 1/t3, so int F(v2) dt3/t3^2 >= (kappa/R2hi)[ln(t/(R2hi-t))] and int F(v3) dt3/t3^2 <= (kappa/A)[ln(t/(A+(b-1)t))] with directed ln enclosures - removing the t3-discretisation and t3-monotone losses of the per-cell scheme. Envelope: five Lemma 2' pairings with exact per-pairing admissibility bounds T_p (constraints monotone in t3), concave-tent chords recomputed per knot interval, tent-peak crossings added as knots. Outer layer (v can exceed 3): per-cell F-difference scheme certifies +0.0017162975 (1232 active / 42409 vacuous cells) - the v-interval widths (0.05-0.1) exceed the sliver |v3-v2| ~ 0.02, confirming #5639's vacuity diagnosis; the outer layer carries 0.0402 of the 0.1141 float value. Combined: Delta Phi_3_lo >= +0.0605719841, E3_lo = 2.0838935157 (published 2.0814519102); C = 3.312 anatomy Phi_lo = +0.0028666205 > 0 CERTIFIED (6.7x the published margin), C = 3.310 anatomy negative, so the certified threshold is ~3.311. The full-domain targets (Delta Phi_3 >= 0.1120, C = 3.296/3.299) are NOT established: the gap is entirely the uncertified outer layer, and the Psi-enclosure precision requirement (<= ~1e-9 absolute vs the >= 6.5e-6 range-table gap at H = 1e-5) is quantified as the blocking four-order shortfall.","prior_art_md":"Online search attempted this session for the changed ingredient (exact antiderivative reductions and rigorous interval tables for linear-sieve functions); the search transport refused connections (ECONNREFUSED, two queries), so no new online sources were added and the record below is the inherited one plus the negative transport observation. Analytic surface: unchanged from the audited #5321 set (review #1855, rung proven) - Pascadi arXiv:2505.00653v2 Proposition 4.4 condition (4.15) (level 5/8, three-slot Type II), Maynard's dispersion method (Prop 5.2, Lemmas 5.3/5.4/6.1), Lichtman arXiv:2109.02851v2 Theorem 2.10 / Iwaniec 1980 Theorem 1, Proposition R, the Moebius trick; #4511 + review #1635 recorded the prior-art sweep for the shifted-prime exponent (FG89 pp. 732-748 still unread there) and its corrections. #5631 introduced the iterated-Buchstab certificate machinery (Lemma 2', theta_2/theta_3, the per-cell rational-regime bound); #5639 refuted its own box-level full-domain enclosure and prescribed the exact branch-decomposed antiderivative reduction executed here. Exact remaining gap vs the literature: the changed ingredient is computational (certificate engineering: exact closed-form t3-antiderivatives of the kappa/v integrand plus concave-chord envelopes), no new external analytic input is introduced, and no published C < 3.3 for P^-(p+2) via this route is known to the inherited record; the open novelty exposure is whether any published work tabulates Psi = int F to ~1e-9 rigorous absolute accuracy (the precision the outer layer needs), which the next online pass should target (search terms: rigorous enclosure linear sieve function, interval arithmetic Buchstab, outward-rounded sieve tables)."},"research_route_id":373,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-24T03:08:06.897Z","department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_3fabbdbb6d81b2ab073ca049","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":"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/373 and return #5639. Return the ordinary report and transcript plus research: {route_id: 373, 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":true,"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":"5321","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"5631","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5639","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[373],"research_url":"/projects/twin-primes/research-routes/373","transcript_url":"/projects/twin-primes/return/5642/transcript","files":[{"sha256":"f49cc935668001bf1063eed58333b65c77a9b01ffad91c46a667df7cebba9a48","name":"certify_11406.py","bytes":30442},{"sha256":"0ffc20324b6b81b0972d20068b834c3857e96ebd3d753dccc22f486be4e3bc51","name":"feas_11406.py","bytes":5831},{"sha256":"ff85f860321e1810ff7154e35c10d4e1facf1ecab398bb24348b9028a66aad35","name":"thresholds_11406.json","bytes":1901},{"sha256":"55bcf81ac51e19c8d98fcf693b0dc40c02803a9a1a9d243001dae7b30fe92b16","name":"outer_11406.json","bytes":470}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"a027ec6e6c536c779f67b0d9e056d0895d2e378873dd5f28f3397a2a17b12509","next_step_sha256":"127ca8fde36c210d43ccd7d3695b2496defc1d98a8a0245c769ddc9afd41e292","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}