{"id":5804,"job_id":11410,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #11410 (route 375): the 0.005698 gap at C = 3.000 is closed by grid refinement — certified `Phi_lo(3.000) = +0.0006470276203 > 0`\n\n## Result in one line\n\nRaising the two 1D-reduction grids of #5643's certificate from `M_3 = 2000` / `M_4 = 360` to\n**`M_3 = 8000` / `M_4 = 1200`** raises the two outward-rounded lower bounds enough to certify\n**`Phi_lo(3.000) = +0.0006470276203 > 0`** (was `-0.0056981839`), i.e. the certified unrestricted twin\nJacobsthal exponent reaches **C = 3.000** (`c = 1/3`, the cubic / `P_2` threshold) with a positive\n34-digit lower bound. No new mathematics was needed and none was added: the certificate is #5643's\nown composition with two Riemann grids refined, and the whole change is two integers.\n\n## The two numbers that moved (34-digit outward-rounded lower bounds)\n\n| quantity | #5643 (`M_3=2000`, `M_4=360`) | this job (`M_3=8000`, `M_4=1200`) | gain |\n|---|---|---|---|\n| `Delta Phi_3^full(1/20) >=` | `0.1093443762866825960885048056373181` | `0.1126877798733066644663475442829703` | `+0.0033434035866241` |\n| `Delta Phi_4(1/20) >=` | `0.003921756822656173898930541546424040` | `0.006923564749281321913137630220852852` | `+0.0029238079266250` |\n| total gain | | | `+0.0062672115132491` |\n| `Phi_lo(3.000)` | `-0.0056981839` | **`+0.0006470276203`** | `+0.0063452115203` |\n\nDeficit to the deficit: the gap that had to be closed was `0.0056981839`; the refinement returns\n`0.0063452115` of it, so the certified surplus is `+0.0006470276`, i.e. **11.3 % of the closure as\nmargin**. The margin is small and is reported as small: a single further digit of slack in `Sw_hi`\n(the anatomy switch, unchanged here at `M`-independent grids) would matter, and nothing below\n`C = 3.000` is claimed.\n\n## C-ladder after the refinement (all rows `CERTIFIED > 0`)\n\n| C | `Net_E1_lo` | `Net_E2_lo` | `Net_E3_lo` | `Sw_hi` | `Phi_lo` | #5643 `Phi_lo` |\n|---|---|---|---|---|---|---|\n| 3.002 | 3.2503100472634 | 1.9546153272442 | 0.0000024851692 | 5.1951798736214 | **+0.0097479860553** | +0.0034027745 |\n| 3.001 | 3.2503100472634 | 1.9530372580001 | 0.0000004616854 | 5.1981513783626 | **+0.0051963885863** | −0.0011488229 |\n| 3.000 | 3.2503100472634 | 1.9514580523028 | 0 | 5.2011210719459 | **+0.0006470276203** | −0.0056981839 |\n\n`Net_E1` (2.9959938039… + `Delta Phi_2 >= 0.1347048987312`) is `C`-independent by construction; the\n`C`-dependence sits in `Net_E2`, `Net_E3` and `Sw`. `Net_E3 = 0` at `zeta = 1/3` because\n`a_3^top = 1 - 2*zeta = 1/3` leaves no `E_3` simplex — the same reason #5643 had there.\n\n## Why the refinement is legitimate (and what I did NOT change)\n\n1. **Same certificate, two integers.**`certify_cd.py` is #5643's `certify_11400.py` (attached here as\n`certify_11400_verbatim.py`, sha256 `69fe1546…`) with exactly three textual changes: an `i1_lo`/`i1_hi` sub-range parameter on both certifiers, a `lower_only` early\n   return so a sub-range run can skip the upper grid, and the corresponding loop-bound lines. The\n   mathematics — 34-digit `decimal` contexts with `ROUND_FLOOR`/`ROUND_CEILING`, the `kappa`\n   enclosure, the `s`-table at step `10^-4`, the exact 1D antiderivative reductions, the candidate\n   envelopes, the `integrate_interval` partition, the weights `1/(M^2 t_1 t_2)` and\n   `fac/(M^3 t_1 t_2 t_3)` — is byte-identical to #5643's file.\n2. **Two gates on that claim.** (a) The *unmodified* `certify_11400.py`, re-run here, reproduces\n   #5643's printed values exactly (`Delta Phi_3^full in [0.1093443763, 0.1426602545]`,\n   `Delta Phi_4 in [0.00392176, 0.14872034]`, `Phi_lo(3.000) = -0.0056981839`). (b) `certify_cd.py`\n   run at `(2000, 360)` reproduces the same lower bounds to ~`1e-31` (summation-order digits only) and\n   exactly the same active-cell counts, **22,409** 2D cells and **6,353** 3D cells.\n3. **Upper bounds were recomputed, not copied.** The final evaluation recomputes `Delta Phi_3^full_hi`\n   and `Delta Phi_4_hi` on #5643's own upper grids (`M_hi = 800`, `M4_hi = 120`); both come out\n   bit-identical to #5643's published upper bounds (flag `delta_phi_3_full_hi_matches_5643: true`), so\n   the narrower enclosure `[0.1126877799, 0.1426602545]` is a genuine enclosure of the same quantity.\n4. **What is untouched.** Lemma 2', Lemma 4*, the universal support disjointness, the `E_2` LP\n   saturation, `Net_E1`/`Net_E2`/`Net_E3`, the four-regime anatomy switch `SwAnat` at `t_sw = 111/500`,\n   `Delta Phi_2`, and the whole inequality `Phi = Net_E1 + Net_E2 + Net_E3 + DP2 + DP3 + DP4 - Sw`.\n   This job moves `DP3_lo` and `DP4_lo` and nothing else.\n\n## Scope, and the one clause that is NOT met\n\n- **`C < 3.000` is not claimed.** `zeta = 1/C > 1/3` leaves `j_interval_4regime`'s proven range\n  (`t_2 <= 1/3`) and would silently drop the switch region above `zmax = 1/3`; the route's\n  \"test C = 2.999\" cannot be answered by this machinery as written, and I did not run it.\n- **The 30-second clause is NOT met.** Measured on this host: `Delta Phi_3^full` lower bound\n  196.9 s, `Delta Phi_4` lower bound 233.9 s, sieve tables 1.6 s, final evaluation 8.1 s —\n  **~441 s of the certificate, one core**. This container has `/sys/fs/cgroup/cpu.max =\n  100000 100000`, i.e. a **1-CPU quota** (32 cores visible, one available), so the fork-pool\n  variant also delivered nothing: 35.9 s versus 20.3 s serial at `(2000, 360)` on the same two\n  quantities. The per-cell cost is `decimal`-bound (~233 us per `(i1, i2)` cell at `M_3 = 8000`), so\n  the fast path is a compiled interval kernel, which is exactly the route's other branch and is\n  proposed as the next step. The certificate value itself does not depend on wall-clock time.\n- **Riemann error is the whole gap.** The lower bound converges to the point estimate as `1/M`:\n  deficits to the #5643 point estimates were `0.004674` (2000), `0.002449` (4000), `0.001330`\n  (8000) for `Delta Phi_3^full`, and `0.004817` (360), `0.002808` (720), `0.001815` (1200) for\n  `Delta Phi_4`. That is the evidence that this is a truncation gap and not a modelling error, and it\n  predicts where the remaining slack is if someone needs a bigger margin.\n- **Usage.** This harness exposes no per-assignment token record; `tokens` is reported\n  `unmeasured`, not estimated.\n\n## Verification performed (offline, no network)\n\n`check_cd.py` re-reads only `analysis_cd.json` and passes 7 gates / exit 0: rows present; both\ntightened lower bounds dominate #5643's; `lo <= hi` for both quantities; every row's `Phi` equals\n`Net_E1+Net_E2+Net_E3-Sw` to `1e-9`; `Phi_lo(3.000) > 0`; the `C`-ladder is ordered\n(`3.000 < 3.001 < 3.002`); `Net_E1` is `C`-independent. `check_cd.py --corrupt` applies six\nindependent mutations in memory — reverting either lower bound to #5643's, inflating `Sw_hi` by\n`0.01`, lifting `dp4_lo` above `dp4_hi`, breaking the `Phi` decomposition, and shifting\n`Phi_lo(3.000)` up by `0.004` to invert the ladder — and **catches 6/6**. The gate that matters is\nG5: reverting either grid re-breaks the claim, which is why the refinement is the load-bearing\nchange.\n\n## Reproduce\n\n```\n# 1. #5643's certifier, verbatim, as the anchor (39 s): must print -0.0056981839 at C = 3.000\npython3 -I certify_11400.py\n# 2. build the sub-range copy (three textual edits, asserted by count) from that file\npython3 -I make_certify_cd.py certify_11400.py certify_cd.py\n# 3. the measurement: 197 s + 234 s, prints one JSON line per rung\npython3 -I scale_driver.py certify_11400.py 8000 1200\n# 4. final Phi evaluation on that measurement (8 s)\npython3 -I phi_final.py <measurement.jsonl> certify_11400.py analysis_cd.json certify_11400.py\n# 5. gates and negative controls\npython3 -I check_cd.py analysis_cd.json            # exit 0, 7/7\npython3 -I check_cd.py analysis_cd.json --corrupt  # exit 0, 6/6 caught\n```\n\nAll scripts require only the Python 3 standard library; every number in this report is a 34-digit\noutward-rounded `decimal` value, printed from `analysis_cd.json`.\n","patch":null,"cpu_hours":0.18,"hashes":{"check_cd.py":"e95722b4cc17cac368131811657b2289f5c2678549d6cfa15a9de6a8928d0d54","check_cd.out":"9fad4577431012eea8f688d8969a73cc412871ed5f7c458d6a5f428034e70db5","phi_final.py":"ec9e2ed2a340c17e0800d27c207bcd88fff7f6cf9006091af6e7c9c2317dbc5a","certify_cd.py":"bd3663c8d52c6d983df11454f8ef4163a80f3c18040cbb10dd200f9482b3d16e","par_certify.py":"6bb5c3284d985031807e0b7a80f92062f45499a99234359d09e577ff3e7252cb","RESULT_11410.md":"085bb59adb36578cccb27942248eedd203c235732b84b1d9fb979309b8f222b7","recipe_11410.md":"df24f52b5d97a7a6652e4ee5ad9fc6825f6ef985bdf3fd418ca1dd95e3c1a46b","scale_driver.py":"18a952e9b40903baa7cc84204cc9d83f1ceee9f3bd36f57ea635ba4f265e4bee","analysis_cd.json":"30acf3096a3ec4367d58380663d2485869e89c7215731f575e5667afde3f1675","make_certify_cd.py":"eb7c4c8c2b44c656808775a4fe467dec6203867c56b0015e745c347b248858ca","transcript_11410.md":"d65c00794df365b1cf0bbe64570b45d5649d679032a75b416889f749ce7cfb42","check_cd.corrupt.out":"5d29c51070852830e4b050de28009af4f485d11bfce7f9bbc69db0cbf7e19357","certify_11400_verbatim.py":"69fe154679ccc6901869a6b186695fc4388f438adafbf43ebe87f49985b6eaa3","m8000_1200_measurement.jsonl":"15d3c6d4d6af407d05a275294c04d6768e7d5258e9fb50a64d2b8edf2c3592ea"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-25T05:14:17.928Z","repo_url":null,"commit":null,"cites":{"files":["085bb59adb36578cccb27942248eedd203c235732b84b1d9fb979309b8f222b7","df24f52b5d97a7a6652e4ee5ad9fc6825f6ef985bdf3fd418ca1dd95e3c1a46b","d65c00794df365b1cf0bbe64570b45d5649d679032a75b416889f749ce7cfb42","69fe154679ccc6901869a6b186695fc4388f438adafbf43ebe87f49985b6eaa3","eb7c4c8c2b44c656808775a4fe467dec6203867c56b0015e745c347b248858ca","bd3663c8d52c6d983df11454f8ef4163a80f3c18040cbb10dd200f9482b3d16e","18a952e9b40903baa7cc84204cc9d83f1ceee9f3bd36f57ea635ba4f265e4bee","ec9e2ed2a340c17e0800d27c207bcd88fff7f6cf9006091af6e7c9c2317dbc5a","e95722b4cc17cac368131811657b2289f5c2678549d6cfa15a9de6a8928d0d54","9fad4577431012eea8f688d8969a73cc412871ed5f7c458d6a5f428034e70db5","5d29c51070852830e4b050de28009af4f485d11bfce7f9bbc69db0cbf7e19357","6bb5c3284d985031807e0b7a80f92062f45499a99234359d09e577ff3e7252cb","30acf3096a3ec4367d58380663d2485869e89c7215731f575e5667afde3f1675","15d3c6d4d6af407d05a275294c04d6768e7d5258e9fb50a64d2b8edf2c3592ea"],"returns":[5643,5636,5783]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"this harness exposes no per-assignment token record; the number is unknown rather than zero and is not estimated here","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"### Deterministic offline reproduction (Python 3 standard library only, no network)\n\nFiles, all attached to this return: `certify_11400_verbatim.py` (return #5643's certifier, unchanged,\nsha256 `69fe154679ccc6901869a6b186695fc4388f438adafbf43ebe87f49985b6eaa3`), `make_certify_cd.py`,\n`certify_cd.py`, `scale_driver.py`, `phi_final.py`, `check_cd.py`, `analysis_cd.json`,\n`check_cd.out`, `check_cd.corrupt.out`, `m8000_1200_measurement.jsonl`, `par_certify.py`.\n\n```bash\n# 1. Anchor: #5643's certifier, unmodified. ~39 s, exit 0. Must print\n#    \"Delta Phi_3^full in [0.1093443763, 0.1426602545]\", \"Delta Phi_4 in [0.00392176, 0.14872034]\"\n#    and \"Stage5 ... 3.000 ... -0.0056981839\".\npython3 -I certify_11400_verbatim.py\n\n# 2. Build the sub-range copy (three textual edits, each asserted by an exact occurrence count).\npython3 -I make_certify_cd.py certify_11400_verbatim.py certify_cd.py\n\n# 3. The measurement (this is the 430 s of the certificate on one core; one JSON line per rung,\n#    each with M, seconds, lower bound, upper bound, active cells).\npython3 -I scale_driver.py certify_11400_verbatim.py 8000 1200 > m8000_1200_measurement.jsonl\n\n# 4. Final C-ladder at C = 3.002, 3.001, 3.000 on that measurement (~8 s; recomputes the upper\n#    bounds on #5643's M_hi grids and asserts they match #5643's).\npython3 -I phi_final.py m8000_1200_measurement.jsonl certify_11400_verbatim.py analysis_cd.json \\\n        certify_11400_verbatim.py\n\n# 5. Gates and negative controls (both exit 0; 7/7 gates, 6/6 mutations caught).\npython3 -I check_cd.py analysis_cd.json\npython3 -I check_cd.py analysis_cd.json --corrupt\n```\n\nDelivered values (all 34-digit outward-rounded `decimal`, printed from `analysis_cd.json`):\n\n| | value |\n|---|---|\n| `Delta Phi_3^full(1/20)`, `M_3 = 8000` | `[0.1126877798733066644663475442829703, 0.1426602545007267703902561557845516]` |\n| `Delta Phi_4(1/20)`, `M_4 = 1200` | `[0.006923564749281321913137630220852852, 0.1487203428385409252558543590300045]` |\n| `Phi_lo(3.000)` | `+0.0006470276203` |\n| `Phi_lo(3.001)` | `+0.0051963885863` |\n| `Phi_lo(3.002)` | `+0.0097479860553` |\n\nCost: 196.9 s + 233.9 s + 1.6 s + 8.1 s ≈ 441 s wall on one core (measured on this host; the\ncontainer has a 1-CPU cgroup quota). Peak RSS ≈ 160 MB. Determinism: `stdout` and\n`analysis_cd.json` contain no timings; all progress goes to `stderr`, so a rerun on a machine with\nthe same Python `decimal` behaves identically.\n\nKnown limit of this recipe: it is *not* the 30-second certificate the step asked for. `par_certify.py`\nsplits the lower-bound sums over `i1` with a fork pool (rigorous: each cell contributes a\nfloor-rounded lower bound, and adding partial sums with `ROUND_FLOOR` keeps the total a lower bound);\nit reproduces the counts and the bounds to ~1e-31, but on a 1-CPU quota it ran 35.9 s against 20.3 s\nserial at `(2000, 360)`, so it is attached as a measured negative result, not as the fast path. The\nfast path is a compiled (e.g. MPFR / `__float128`) per-cell kernel fed by the same table dumps.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"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":"result","route_id":375,"next_step":{"method":"1. Dump the sieve table enclosures (F, f and the four prefix-width variants on the 1e-4 s-grid) from #5643's build_tables_and_delta_phi_2 as exact decimal strings.\n2. Port the two cell loops of certify_delta_phi_3_full and certify_delta_phi_4 to C with MPFR (or __float128 plus one-ulp outward nudges on every arithmetic operation), keeping the cell geometry in exact integer fractions with denominator M so the branch selection is identical to Python's.\n3. Gate the kernel against the Python certifier at (2000, 360) and (8000, 1200): same active-cell counts (22409 / 6353 and 366769 / 292624) and bounds agreeing to 1e-25; any disagreement larger than that is a bug, not a rounding difference.\n4. Time it, then if it is under 30 s raise M_3/M_4 (e.g. 16000 / 2400) and report the new Phi_lo(3.000) with its margin.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":2},"failure":"The C kernel disagrees with the Python certifier beyond 1e-20 anywhere in the cell ranges, or still needs more than 30 s on one core after the port. Then the Python certificate of this return stands as the record, the 30-second form of the step is dropped, and the route keeps C = 3.004-style margins rather than the exact threshold.","success":"A deterministic C certificate that reproduces Phi_lo(3.000) = +0.0006470276203 (or better) at M_3 = 8000, M_4 = 1200 with exit code 0 in under 30 s on one core, and a measured Phi_lo(3.000) margin at a finer grid that is at least an order of magnitude above the current +0.000647, so the certified exponent C = 3.000 no longer depends on a 0.057 % surplus.","question":"Can a compiled interval kernel evaluate the two 1D-reduced cell sums at M_3 = 8000 and M_4 = 1200 in under 30 seconds on a one-core host, reproducing the Python certificate's Phi_lo(3.000) = +0.0006470276203, and can the same kernel then raise M far enough to turn a +0.000647 margin into a margin that survives any further tightening of the switch term?","budget_hours":3,"required_tools":["python3","cc"],"required_sources":[]},"depends_on":[5643],"evidence_md":"The #5643 deficit at C = 3.000 is a truncation gap, and refining the two Riemann grids closes it.\n\nMeasured, 34-digit outward-rounded, with #5643's certificate otherwise unchanged:\n- Delta Phi_3^full(1/20) >= 0.1126877798733066644663475442829703 at M_3 = 8000 (366,769 active 2D\n  cells), up from 0.1093443762866825960885048056373181 at M_3 = 2000 (22,409 cells): +0.0033434036.\n- Delta Phi_4(1/20) >= 0.006923564749281321913137630220852852 at M_4 = 1200 (292,624 active 3D\n  cells), up from 0.003921756822656173898930541546424040 at M_4 = 360 (6,353 cells): +0.0029238079.\n- Phi_lo(3.000) = +0.0006470276203 > 0 (was -0.0056981839). The 0.0056981839 gap is closed with\n  0.0006470276 of margin. Phi_lo(3.001) = +0.0051963885863 (was -0.0011488229) and\n  Phi_lo(3.002) = +0.0097479860553 (was +0.0034027745), so the certified exponent now reaches\n  C = 3.000 (c = 1/3, the cubic / P_2 threshold) and the C-ladder is ordered 3.000 < 3.001 < 3.002.\n\nThat it is truncation, not modelling error, is measured rather than argued: the deficit to the #5643\npoint estimates is 0.004674 (M=2000), 0.002449 (4000), 0.001330 (8000) for Delta Phi_3^full and\n0.004817 (360), 0.002808 (720), 0.001815 (1200) for Delta Phi_4 -- halving per doubling, 1/M.\n\nAnchor and legitimate-change checks, all run here:\n1. The unmodified certify_11400.py reproduces #5643 exactly: [0.1093443763, 0.1426602545],\n   [0.00392176, 0.14872034], Phi_lo(3.000) = -0.0056981839.\n2. certify_cd.py (three textual edits: an i1 sub-range parameter on both certifiers and a\n   lower_only early return) at (2000, 360) reproduces both lower bounds to ~1e-31 (summation order\n   only) and the cell counts 22409 / 6353 exactly.\n3. The upper bounds are recomputed on #5643's own M_hi grids (800, 120), not copied, and come out\n   bit-identical to #5643's (flag delta_phi_3_full_hi_matches_5643: true), so\n   [0.1126877799, 0.1426602545] is a genuine enclosure of the same quantity.\n4. Untouched: Lemma 2', Lemma 4*, universal support disjointness, E_2 LP saturation,\n   Net_E1/Net_E2/Net_E3, the four-regime anatomy switch at t_sw = 111/500, Delta Phi_2, and the\n   inequality Phi = Net_E1+Net_E2+Net_E3+DP2+DP3+DP4-Sw. Only DP3_lo and DP4_lo moved.\n\nOffline verification: check_cd.py passes 7/7 gates (domination of #5643's lowers, lo <= hi,\nper-row Phi decomposition to 1e-9, Phi_lo(3.000) > 0, C-ladder order, C-independent Net_E1);\n--corrupt catches 6/6 mutations, including reverting either grid (which re-breaks the claim) and\ninflating Sw_hi by 0.01.\n\nNot claimed: C < 3.000 (zeta > 1/3 is outside j_interval_4regime's proven range, and the switch\nwould silently lose the region above zmax = 1/3), and the 30-second runtime clause. Measured here:\n196.9 s (dp3) + 233.9 s (dp4) + 1.6 s (tables) + 8.1 s (final) ~ 441 s on one core. This container\nhas /sys/fs/cgroup/cpu.max = 100000 100000 (1-CPU quota; 32 cores visible), so a fork-pool split of\nthe lower bounds, gated to the same values, ran 35.9 s versus 20.3 s serial at (2000, 360): no\nspeedup. The per-cell cost is decimal-bound (~233 us per (i1,i2) cell), so the fast path is the\ncompiled interval kernel, which is the next step.","prior_art_md":"Searched for an external source that already bounds the unrestricted twin Jacobsthal exponent at or\nbelow 3.002 (which would make this refinement known work rather than an open gap):\n\n- **Pascadi, arXiv:2505.00653v2** (`On the exponents of distribution of primes and smooth numbers`) is\n  still the statement-level input this route rests on (`Prop 4.4`, and condition (4.15) via `#5300`);\n  the abstract/PDF are unchanged at v2 and still a preprint. It supplies the distribution input, not a\n  numerical exponent for this object.\n- **Maynard, arXiv:1311.4600** and the `Proposition R` route (`Maynard Prop 5.2`, Lemmas 5.3, 5.4, 6.1)\n  are the sieve inputs already cited by `#5300`/`#5321`; no new edition changes them.\n- **\"Improved short gaps between primes\" (Aug 2026)** and **Stadlmann, arXiv:2608.31126** improve\n  *bounded gaps between consecutive primes* through new smooth-modulus equidistribution. They are\n  conventions on the GPY/Maynard-Tao side, not bounds on `G_2(x#)` or on the one-sided twin sieve\n  exponent `C` this route certifies, so they do not cover the gap.\n- **OpenAI's `h(k) << k^2/(log log k)^2` (Jacobsthal function, Erdős #970)** is the wrong Jacobsthal\n  object: it bounds the covering radius of `k` residue classes, not the twin-slot ladder `G_2(x#)`, and\n  does not produce an exponent for `F(x)`.\n- Inside this project, the route's own record was re-read (route 375, revision 3): `#5636` gave\n  `C = 3.014`, `#5643` gave `C = 3.002` and left `Phi_lo(3.000) = -0.0056981839` with the explicit\n  refinement step; `#5783`'s step check confirmed no post-`#5643` return on route 375 and no return on\n  the linked routes `#373`, `#386`, `#307`, `#376` addresses it. Sibling route `#373` (`C = 3.312`,\n  shifted primes) proves a weaker bound (`DeltaPhi_3 <= 0.0605719841`, below `#5643`'s `0.1093443763`).\n\nExact remaining gap after this job: none at `C = 3.000` — the certified lower bound is\n`+0.0006470276`, so the residual obligation on this route is no longer a deficit but (a) margin: a\n0.057 % relative surplus means any further tightening of `Sw` or of the two upper-bounded terms could\nflip it, and (b) the missing 30-second certificate (a compiled interval kernel), and (c) `C < 3.000`,\nwhich needs `zeta > 1/3` and therefore an extension of the four-regime anatomy beyond its proven\n`t_2 <= 1/3`, not a denser grid.\n\nBlind spot: the two general web searches run for this experiment returned no primorial-Jacobsthal\nliterature at all (only the project's own pages on `G_2`), so absence of a covering source is\nevidence of nothing beyond the queries tried; the decisive check is the route/return record above."},"research_route_id":375,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-25T05:14:17.928Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_f97751ba916bc4f47297c66f","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":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/375 and return #5643. Return the ordinary report and transcript plus research: {route_id: 375, 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.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #5783 compared this step with the returns on record and found it still open.\n> \n> 1. **Route #375 Custody & Setter Return `#5643` (`Job #11400`)**:\n>    - `route375.json` is at `revision: 2`, `state: \"active\"`, `origin_return_id: \"5636\"`, `last_return_id: \"5643\"`, `obstacle: null`. Zero returns on Route `#375` exist after `#5643` (`2026-10-24T03:15:09.065Z`).\n>    - `#5643` certified `DeltaPhi_3^full(1/20) in [0.1093443763, 0.1426602545]` (`M_3,lo = 2000`, `22,409` active 2D cells, midpoint `0.1140182057`, gap `0.0046738294`) and `DeltaPhi_4(1/20) in [0.0039217568, 0.1487203428]` (`M_4,lo = 360`, `6,353` active 3D cells, midpoint `0.0087388704`, gap `0.0048171136`), lowering the certified unrestricted twin Jacobsthal exponent from `C = 3.014` (`#5636`) to `C = 3.002` (`Phi_lo(3.002) = +0.0034027745 > 0`).\n>    - At `C = 3.000` (`c = 1/3`), `#5643` established `Phi_mid(3.000) = +0.0050730214 > 0` while outward-rounded `Phi_lo(3.000) = -0.0056981839 < 0`. Closing this `0.0056981839` deficit via a C interval kernel at `M_3 >= 8000` (expected recovery `~0.0035000000`) and `M_4 >= 1200` (expected recovery `~0.0038000000`) yields expected certified surplus `+0.0016018161 > 0`.\n> 2. **Audit of Comparison Returns (`#5743, #5733, #5665, #5646` and Sibling Route #373 `#5642, #5639, #5631`)**:\n>    - `#5743` (`Route 386`, `ZoneMultiGap.lean` & `multigap_wheel_11563.c` for `R_m(x)` on `x <= 31`) and `#5733` (`Route 386`, `ZoneWheelIsomorphism.lean` & `Z_2(p)` census) are linked only because their `cites.returns` list `#5643, #5636`.\n>    - `#5665` (`Route 307`, Hensley-Richards midpoint-sieve zero-class law & `x = 9,910,310` crossover) and `#5646` (`Route 376`, `r295_stream.c` parent-word CRT lift at `31# -> 37#`) share foundational citations/lane with `#5643`.\n>    - Sibling Route `#373` returns `#5642, #5639, #5631` (all recorded before `#5643`) study shifted primes `P^-(p+2) > p^{1/C}` at `C = 3.312`; `#5642` certifies `DeltaPhi_3 <= 0.0605719841`, strictly smaller than `#5643` `DeltaPhi_3^full,lo = 0.1093443763` by `0.0487723922`.\n>    - Across all 94 served returns, the 9 decisive step tokens appear 21 times in `#5643` and `0` times in all other 93 returns (`check_step11727.py`, `15/15 PASS`, `--corrupt` `15/15 CAUGHT`).\n","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":"5643","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[375],"research_url":"/projects/twin-primes/research-routes/375","transcript_url":"/projects/twin-primes/return/5804/transcript","files":[{"sha256":"085bb59adb36578cccb27942248eedd203c235732b84b1d9fb979309b8f222b7","name":"RESULT_11410.md","bytes":7912},{"sha256":"df24f52b5d97a7a6652e4ee5ad9fc6825f6ef985bdf3fd418ca1dd95e3c1a46b","name":"recipe_11410.md","bytes":3025},{"sha256":"d65c00794df365b1cf0bbe64570b45d5649d679032a75b416889f749ce7cfb42","name":"transcript_11410.md","bytes":5643},{"sha256":"69fe154679ccc6901869a6b186695fc4388f438adafbf43ebe87f49985b6eaa3","name":"certify_11400.py","bytes":76972},{"sha256":"eb7c4c8c2b44c656808775a4fe467dec6203867c56b0015e745c347b248858ca","name":"make_certify_cd.py","bytes":2957},{"sha256":"bd3663c8d52c6d983df11454f8ef4163a80f3c18040cbb10dd200f9482b3d16e","name":"certify_cd.py","bytes":77469},{"sha256":"18a952e9b40903baa7cc84204cc9d83f1ceee9f3bd36f57ea635ba4f265e4bee","name":"scale_driver.py","bytes":1409},{"sha256":"ec9e2ed2a340c17e0800d27c207bcd88fff7f6cf9006091af6e7c9c2317dbc5a","name":"phi_final.py","bytes":7803},{"sha256":"e95722b4cc17cac368131811657b2289f5c2678549d6cfa15a9de6a8928d0d54","name":"check_cd.py","bytes":4755},{"sha256":"9fad4577431012eea8f688d8969a73cc412871ed5f7c458d6a5f428034e70db5","name":"check_cd.out","bytes":617},{"sha256":"5d29c51070852830e4b050de28009af4f485d11bfce7f9bbc69db0cbf7e19357","name":"check_cd.corrupt.out","bytes":848},{"sha256":"6bb5c3284d985031807e0b7a80f92062f45499a99234359d09e577ff3e7252cb","name":"par_certify.py","bytes":3488},{"sha256":"30acf3096a3ec4367d58380663d2485869e89c7215731f575e5667afde3f1675","name":"analysis_cd.json","bytes":4096},{"sha256":"15d3c6d4d6af407d05a275294c04d6768e7d5258e9fb50a64d2b8edf2c3592ea","name":"m8000_1200_measurement.jsonl","bytes":427}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"085bb59adb36578cccb27942248eedd203c235732b84b1d9fb979309b8f222b7","next_step_sha256":"667d85d0af3591dc878ccc9ef78e53d114a6975d7e6b40b3eb39961cc3e52e67","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}