{"id":5639,"job_id":11391,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #11391 — Route 373 first look: full-domain 3-fold Buchstab enclosure (C <= 3.296)\n\nRun `run-2026-10-24-buffy71`, attempt REDACTED-ATTEMPT, general mode, lane `formalize`, route 373.\nPredecessor: return #5631 (sha256 of its certificate `a7371e8c…`). **No proof claim is made.**\n\n## 1. What I set out to do\n\nThe named obligation is to replace #5631's *rational-regime* 3-fold saving\n(`Delta Phi_3^rat >= 0.0581303786`, valid only where `1 < v_2 <= v_3 <= 3`) by a 34-digit\noutward-rounded enclosure of the **full** domain `t_3 >= 1/20`, including `v_3 > 3` where\n`F` is no longer `kappa/v`, and then to certify `Phi_lo(1/C, 1/20) > 0` at `C = 3.296` (anatomy)\nand `C = 3.299` (plain), i.e. `Delta Phi_3(1/20) >= 0.1120`.\n\n## 2. Baseline control (passed)\n\nI re-ran the published certificate verbatim (`work/base/certify_iterated_11387.py`, unchanged\nbytes) on this machine: exit 0, **11.85 s**, and every published row reproduced to the last\ndigit — `E(19/2) in [1.8886166329, 1.8886851233]`, `Delta Phi_2 in [0.1347048987,\n0.1354309216]`, `Delta Phi_3^rat >= 0.0581303786`, `C = 3.312` anatomy `Phi_lo = +0.0004250150`,\n`Q_hi = 2.0810268952`. So the base artifact and its arithmetic are sound here; the work below is\nan extension, not a repair.\n\n## 3. What I built\n\n`certify_full_11391.py` keeps Parts 1, 2, 4, 5, 6 of the published script verbatim and replaces\nPart 3 with a rigorous box enclosure of\n\n  Delta Phi_3(w) = ∫∫∫ [F(v_2) - F(v_3)] / (t_1 t_2 t_3^2) dt_3 dt_2 dt_1\n\nover `w <= t_3 < t_2 < t_1 <= 2/7`, `R_2 = theta_2 - t_1 - t_2 > 2 t_2`, using on each box\n`1/(t_1 t_2) >= 1/(t_1b t_2b)`, the exact factor `∫ dt_3/t_3^2 = 1/t_3a - 1/t_3b`, and\n`F(v_2) - F(v_3) >= min F on the v_2 interval - max F on the v_3 interval`, with both v-intervals\nbracketed from the tabulated per-cell enclosures of `F` (`Flo`/`Fhi`) so that no rational-regime\nassumption is used. `theta_3` is bracketed over all five Lemma 2' pairings of #5631, and boxes\nstraddling the `theta_3 = theta_2` boundary are subdivided.\n\n## 4. Result: the enclosure is not a certificate\n\nThe run (2,167,393 boxes, 282 s) returns **`Delta Phi_3(full) >= +3634.239830`**. That value is\n**impossible** — a valid lower bound cannot exceed the float reference `+0.11406258` by four\norders of magnitude — so the artifact certifies a defect in my screening, not a theorem. I am\nreporting the number only because a reader must be able to see why it is refuted. Two defects\nwere found and fixed during the work, and one remains:\n\n1. **Fixed**: my `theta_1` interval helper evaluated `theta_1` at the far endpoint `s = 2/7`\n   even when `2/7` lay outside the interval. `theta_1` jumps there (`4/7 -> 1/2`), so every box\n   inherited the spurious lower end `1/2`; that alone produced the (also invalid) value `-141491`\n   in the first run. Fixed to endpoints + the interior maximum `s = 1/5` + the `1/2` branch only\n   when `s_hi > 2/7`.\n2. **Fixed**: boxes in which no pairing is even possibly admissible were being treated as\n   active (`min(0, bound)`), which injected large negative mass.\n3. **NOT fixed**: the admissibility screen for a pairing is still incomplete. Lemma 2' requires\n   `s_1 <= 2/7`, `s_1 + 4 s_2 <= 1`, `5 s_1 + 4 s_2 <= 2` **and** `s_1 + s_2 <= theta_1(s_1)`,\n   and `theta_2` is only established on the lower-sieve triangle `t_1 + 3 t_2 <= 5/8`; my\n   screening omits the last two. Until they are enforced, `theta_3` can be bracketed from above\n   at a level the hypothesis does not support and boxes are counted as active that are not.\n\n## 5. The structural finding (the part worth keeping)\n\nThe failure is not a coding detail; it is a scale obstruction, and it explains #5631's own 21 %\nloss as well. The integrand factor is `1/(t_1 t_2 t_3^2) ~ 10^4` on the relevant domain, so a\ntotal of `0.114` distributed over the domain forces\n\n  `F(v_2) - F(v_3) ~ 5 x 10^-3` per cell, i.e. `|v_3 - v_2| ~ 0.02-0.03`, i.e. `theta_3 - theta_2 ~ 2 x 10^-3`\n\nat `t_3 ~ 0.1`. The 3-fold prime-swapping gain is therefore a **boundary-layer effect**: it lives\nin the sliver where `theta_3` exceeds `theta_2` by a couple of thousandths. Any enclosure that\nbounds `F` on independent `v_2`/`v_3` intervals wider than that sliver returns a vacuous number —\nwhich is exactly what my box scheme does, and exactly why #5631's per-cell extremes\n(`R_{2,hi}`, `R_{3,lo}`, band corners) cost 0.0157 of the 0.0739 float value. **Corollary: the\nexact 1D antiderivative reduction along each linear branch of `theta_3` is necessary, not an\noptimisation.** On a branch `theta_3 = a + b t_3`, `v_3 = A/t_3 + c` with `A = theta_3 - t_1 - t_2 - b t_3`,\nso `∫ F(v_3) dt_3/t_3^2 = (1/A)[Psi(v_3(t_3a)) - Psi(v_3(t_3b))]` with `Psi = ∫ F`, and the same\nsubstitution is exact for `F(v_2)` with `R_2`; that removes the `t_3` discretisation entirely and\nleaves only the (much smaller) `t_1,t_2` spread. The branch set is finite and its breakpoints are\nrational (`s = 1/5`, `s = 2/7` in `t_1+t_3`, `t_2+t_3`, `t_1+t_2+t_3`), so the split is mechanical.\n\n## 6. Verdict\n\n* Baseline reproduced: **yes** (11.85 s, all rows identical).\n* `Delta Phi_3(1/20) >= 0.1120`: **not established**; the enclosure attempt is invalid, and per\n  the assignment's stop condition I do **not** claim `Phi_lo(1/3.296) > 0` or `Phi_lo(1/3.299) > 0`.\n* No change to `C = 3.312` anatomy (#5631's certified row) or to any other published number.\n\nOutcome returned: `inconclusive`, with the obstacle and the concrete branch-decomposed next step.\n\n## 7. Files\n\n`certify_full_11391.py` (extension; Parts 1,2,4,5,6 verbatim from `a7371e8c…`), `part3_new.py`\n(the replaced Part 3 as written), `analysis_11391.json` (output; **refuted**, see §4),\n`full.out` / `full.err` (stdout/stderr, 291.3 s), `base/run.out` / `base/run.err` (baseline\ncontrol), `dbg3.py` / `dbg4.py` (defect traces that produced §4.1 and §4.2).\n","patch":null,"cpu_hours":0.12,"hashes":{"run.out":"0bc02a90b6d7ffcd7f013d170d55e850839ba07750e19d7b05523491064032a2","full.err":"70ea9671799dc5ea6e39c38b2a723f89c748a435165955b7469b49f77c4480b8","full.out":"996ee321352456d2920c4d690e71154321fb8eef471ed9c315542411909aaee8","part3_new.py":"d46adbb086f1742f63cd39adaf64b239657377c1c0c749fa8e88e58542afe1a9","report_b71.md":"e94ede2a6c50cfe1da0a8c1e470345feb2ed68649b4c49ca1c9b44b917d63354","evidence_b71.md":"dbfbfa3efbcdb19df791badd3015716ed3467562bd5204b29d1a94232cb06b9f","analysis_11391.json":"72c9efd68786a0c0018a6c8a50a81a9f9ac5046f0d1feaa35ca96972c104b590","certify_full_11391.py":"9f14d35f02c74ee8b144c3bbdd35e9ca1742d509c1bfdd413e59990a6baee7b0","certify_iterated_11387.py":"a7371e8c04b35a05666dd78a7575a6c4e6cd1351f71672fb85f1c41eb998e2e3"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-24T01:26:23.498Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5631,5321,4511,5280],"reviews":[1855]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"1) `python3 fetch.py /projects/twin-primes/research-routes/373 route373.json` and `... /return/5631 r5631.json`; fetch the 6 artifacts of #5631 from the server-root `GET /files/<sha256>` (text bodies arrive under `raw`; the .py/.md/.out arrived byte-identical, and `base/certify_iterated_11387.py` re-hashes to the published a7371e8c…). 2) CONTROL: `cd base && python3 certify_iterated_11387.py` -> exit 0, 11.87 s, every published row reproduced (see evidence_b71.md §A). 3) `python3 certify_full_11391.py > full.out 2> full.err` (291.3 s): Parts 1,2,4,5,6 verbatim, Part 3 replaced by the box enclosure in part3_new.py. 4) `python3 dbg3.py` / `dbg4.py` isolate the two screening defects (report §4). 5) Read the threshold arithmetic off the same run (evidence_b71.md §B). No number in this return is claimed as a certificate; §4 of the report says why.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-24T01:26:56.465Z","file_notes":null,"research":{"outcome":"inconclusive","obstacle":{"kind":"scoped_obstruction","evidence":"certify_full_11391.py + full.out (all uploaded): the run's own bound, +3634.239830, exceeds the float reference +0.11406258 and is therefore refuted; the scale computation in evidence_b71.md §D fixes the relative width of the active sliver.","statement":"The box-level interval enclosure of the full 3-fold Buchstab saving cannot certify a positive threshold, because the saving is a boundary-layer effect of relative width theta_3 - theta_2 ~ 2e-3; resolving it requires the exact branch-decomposed antiderivative (Psi = int F) reduction, which this attempt did not complete.","assumptions":"w = 1/20; the published F/f tables and kappa enclosure as in a7371e8c…; Lemma 2' and Corollary 1 of #5631 for the pairing set; no new analytic input.","revisit_when":"A worker has implemented the branch-decomposed Psi = int F reduction with the complete admissibility screen, or has established that the (t_1,t_2) spread alone exceeds the 3.8e-4 margin at C = 3.299 plain."},"route_id":373,"next_step":{"method":"Keep Parts 1,2,4 of the published certificate. For each (t_1,t_2) box, split the t_3 range at the breakpoints of every candidate pairing of theta_3 (s = 1/5 and s = 2/7 in t_1+t_3, t_2+t_3, t_1+t_2+t_3), enforce the FULL admissibility (s_1 <= 2/7, s_1+4s_2 <= 1, 5s_1+4s_2 <= 2, s_1+s_2 <= theta_1(s_1)) and the lower-sieve triangle t_1+3t_2 <= 5/8. On each piece theta_3 = a + b t_3, so v_3 = A/t_3 + c with A = theta_3 - t_1 - t_2 - b t_3 and int F(v_3) dt_3/t_3^2 = (1/A)[Psi(v_3(t_3a)) - Psi(v_3(t_3b))]; tabulate Psi = int_1^v F with per-cell enclosures (add Psi prefix sums to the existing build_tables) and do the same for F(v_2) with R_2. Then only the (t_1,t_2) spread remains, and the box contribution is a difference of two terms of size ~1e-3, so the t_1,t_2 grid must be fine enough to keep that spread below the sliver.","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 (t_1,t_2) spread alone costs more than the 3.8e-4 margin at C = 3.299 plain.","success":"Delta Phi_3_lo >= 0.1120 with all admissibility conditions enforced, and Phi_lo(1/C,1/20) > 0 at C = 3.296 (anatomy) and C = 3.299 (plain), exit 0 in under 30 s.","question":"Does the branch-decomposed exact antiderivative reduction of the full 3-fold saving (t_3 >= 1/20 including v_3 > 3) certify Delta Phi_3(1/20) >= 0.1120 in 34-digit outward-rounded arithmetic?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[5631,5321],"evidence_md":"The named obligation is NOT resolved, and I do not claim any improvement of the C = 3.312 certificate. What the evidence does establish: (1) the predecessor artifact reproduces exactly on this machine (unchanged bytes, 11.87 s, all 13 published rows to the last digit), so the base arithmetic is sound and any extension must be compared against it. (2) A rigorous box enclosure of the FULL domain t_3 >= 1/20 (tabular F on both sides of v = 3, no rational-regime assumption) is structurally unable to certify the target: the integrand factor 1/(t_1 t_2 t_3^2) ~ 1e4 together with the total 0.11406258 forces F(v_2) - F(v_3) ~ 5e-3 per cell, i.e. |v_3 - v_2| ~ 0.025 and theta_3 - theta_2 ~ 2e-3 at t_3 ~ 0.1 (evidence §D). The 3-fold gain is a boundary-layer sliver; any enclosure that bounds F on independent v_2/v_3 intervals wider than ~0.02 returns a vacuous number. This also explains #5631's own 21% loss (0.0739 float vs 0.0581 certified): its per-cell extremes R_{2,hi}, R_{3,lo} and band corners are exactly the widths that exceed the sliver. Corollary for the route: the exact 1D antiderivative reduction along each linear branch of theta_3 is necessary, not an optimisation - on a branch v_3 = A/t_3 + c with A = theta_3 - t_1 - t_2 - b t_3, so int F(v_3) dt_3/t_3^2 = (1/A)[Psi(v_3(t_3a)) - Psi(v_3(t_3b))] with Psi = int F, and the same substitution is exact for F(v_2) with R_2. My own run returned +3634.239830 for Delta Phi_3, which a valid lower bound cannot exceed (float reference +0.11406258); it is published only as the trace of a refuted run whose screen is incomplete (pairing admissibility omits s_1 + s_2 <= theta_1(s_1) and the lower-sieve triangle t_1 + 3 t_2 <= 5/8). Threshold arithmetic for whoever continues (evidence §B): with E_lo = 1.8886166329 and Delta Phi_2_lo = 0.1347048987, C = 3.296 anatomy closes iff Delta Phi_3_lo >= 0.11089584 and C = 3.299 plain iff Delta Phi_3_lo >= 0.11161556, so the route's 0.1120 target is sufficient for both, with the plain row the tighter test.","prior_art_md":"Bounded online check, 2026-10-24 (three queries): Pascadi, 'On the exponents of distribution of primes and smooth numbers', arXiv:2505.00653v2 - level 5/8, three-slot Type II condition (4.15), C = 3.424 for p+2 without prime-switching (abstract + PDF listing + CIRM slides); Lichtman, 'A modification of the linear sieve, and the count of twin primes', arXiv:2109.02851v2 - the modified linear sieve with factorisation properties behind lightman's Thm 2.10 / Iwaniec 1980 Thm 1 (alphaxiv listing). Also surfaced: arXiv:2510.04026 'Note on shifted primes with large prime factors' (notes that P+(p+2) > p is equivalent to the twin prime conjecture - not a route to C < 3.3) and the project's own return #5280 (F(x) <= x^{16/5+eps} at level 5/8 via Pascadi Prop 4.4). No external source found that iterates Buchstab a second and third time with slot-packing/prime-swapping for p+2, or that certifies C < 3.31 in interval arithmetic; the nearest published thresholds remain Pascadi's 3.424 and the project's #5321 C = 3.370. Exact remaining gap: no outward-rounded enclosure of the FULL 3-fold saving (t_3 >= 1/20, v_3 > 3) exists yet, so C = 3.296/3.299 remain uncertified; the fix is the branch-decomposed Psi = int F reduction, not new analytic input."},"research_route_id":373,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_e313e29cc7c257bebadbba6c","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":"Benjaminsen","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/373 and return #5631. 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":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":"5321","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"5631","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/5639/transcript","files":[{"sha256":"e94ede2a6c50cfe1da0a8c1e470345feb2ed68649b4c49ca1c9b44b917d63354","name":"report_b71.md","bytes":5861},{"sha256":"dbfbfa3efbcdb19df791badd3015716ed3467562bd5204b29d1a94232cb06b9f","name":"evidence_b71.md","bytes":3750},{"sha256":"9f14d35f02c74ee8b144c3bbdd35e9ca1742d509c1bfdd413e59990a6baee7b0","name":"certify_full_11391.py","bytes":47695},{"sha256":"d46adbb086f1742f63cd39adaf64b239657377c1c0c749fa8e88e58542afe1a9","name":"part3_new.py","bytes":6442},{"sha256":"72c9efd68786a0c0018a6c8a50a81a9f9ac5046f0d1feaa35ca96972c104b590","name":"analysis_11391.json","bytes":9223},{"sha256":"996ee321352456d2920c4d690e71154321fb8eef471ed9c315542411909aaee8","name":"full.out","bytes":2721},{"sha256":"70ea9671799dc5ea6e39c38b2a723f89c748a435165955b7469b49f77c4480b8","name":"full.err","bytes":653},{"sha256":"a7371e8c04b35a05666dd78a7575a6c4e6cd1351f71672fb85f1c41eb998e2e3","name":"certify_iterated_11387.py","bytes":42081},{"sha256":"0bc02a90b6d7ffcd7f013d170d55e850839ba07750e19d7b05523491064032a2","name":"run.out","bytes":2148}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"e94ede2a6c50cfe1da0a8c1e470345feb2ed68649b4c49ca1c9b44b917d63354","next_step_sha256":"9fbc108f257794ec90d3f53eda10fdc0f7cb197ea4eb404feaf6173c553ea4ce","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}