{"id":3255,"job_id":5311,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# job #5311 (route 221, pursue): the two-sided bracket is two-signed in every reading, and both sides are now sized — the lower one by an inexplicit constant, the upper one by the record's own slack\n\n**Type** `explore`, stage `pursue`, route **221** rev 3, lane dir-558. **Outcome `inconclusive`** (the\nroute's own `stop_if`/failure branch: the floor is inexplicit and asymptotic-only at the reachable\nladder and the ceiling inputs are too loose). **Compute 0.0 CPU-h** — arithmetic on quoted inputs plus\nsource reads; no enumeration, no new `G2` value, no producer rerun. **Files:** `report.md`,\n`transcript.md`, `bracket.py`, `bracket.out`, `inputs.json` (quoted values and their locators).\n\n## Answer\n\n`delta` is two-signed at the reachable ladder **in every reading of the inputs on record**, so the\nbracket does not force the sign, and `(H-sub-pow)` stays a **proof gap with an unquantified `K`\nobligation** rather than a truth gap. Both sides are loose, and each is now sized:\n\n| ceiling used at the top rung (`n = 82`, `x = 79`) | `beta` | induced `delta` interval |\n|---|---|---|\n| the trusted measured point itself (tightest) | 1.00 | `[-9.28, +14.73]` |\n| the trusted measured point | 1.50 | `[-11.24, +12.77]` |\n| the trusted measured point | 4.26645 | `[-22.07, +1.93]` |\n| proven upper bound `x^{4.2665+eps}` | 1.50 | `[-47.87, +59.39]` |\n| period bound `Ghat(n) <= P(n)#`, `ln = theta(79) = 70.246` | 1.50 | `[-95.22, +274.22]` |\n\n**Lower side: the floor's constant is not a number.** The FGKMT floor is a Vinogradov `>>`, so\n`delta_lo = -infinity`; even at the corpus's own best reading (`C = 1`, all implied constants 1 in the\n[INFERRED] two-class form), `delta_lo >= 0` would need **`C >= 9.29` (`beta = 1`), `>= 14.87`\n(`beta = 1.50`), `>= 200.8` (`beta = 4.26645`)** — and that form is valid only above `y0 = 10^{134.1}`\n(`A = 4.05`), i.e. ~100 decades above the ladder top (`79#`, `theta(79) = 70.246`). So the lower input\nis missing in *magnitude* and in *range*.\n\n**Upper side: a finite number.** `delta_hi <= 0` would need the top-rung ceiling to fall to\n`U(82) <= e^{3.91}` (`beta = 1`), `e^{4.38}` (`beta = 1.50`), `e^{6.98}` (`beta = 4.26645`), while the\nrecord's **trusted measured value is `e^{7.44}`** — short by `~34x` (`beta = 1`) / `~21x`\n(`beta = 1.50`) — and the period bound (`e^{70.25}`, i.e. **62.8 nats of slack**, the step-check's own\nnumber) and the proven power ceiling (`e^{18.64}`) are looser still. The record already marks both of\nthese ceilings as empty tools for this purpose: the owner note's table lists **T5 (the period bound)**\nas `\"legal, empty: diverges in k\"` and **T3 (the two power envelopes)** as `\"CLOSED: gamma - lambda =\n3.26645\"`.\n\n**Sub-finding (new).** A **single** rung excludes nothing about `delta` at all: the floor and the\nceiling are the same product form, so the unknown `Ghat` constant `c` absorbs the inequality for every\n`delta` (printed for `beta` in {1, 1.5, 4.26645} and `delta` in {-20, -5, 0, +5, +20}). Any `delta`\ninterval must therefore come from a **slope across >= 2 rungs** — which is exactly where the inexplicit\n`C` and the loose ceilings destroy it.\n\n**Which input would have to improve (the scoped statement the step asks for).** A floor with an\n**explicit constant `>= ~9` (`beta = 1`; `>= ~15` at `beta = 1.5`) valid at `n <= 82`**, or a ceiling\ntighter than the record's own trusted values. Nothing weaker changes the sign question.\n\n## Frame of `beta` (settled here, because the step's \"measured exponent\" is undefined on record)\n\nThe served lower bounds and the control all have first-order exponent **1** (`x (ln x)^{2+o(1)}` control;\n`x log x logloglog x/loglog x` floor; `x (ln x)^3 …` [INFERRED] floor), so `beta = 1` is the honest frame,\nand the recorded `1.50 +/- 0.05` is largely an artefact of fitting a *power law* to a *log-power* truth —\nthe corpus measures this itself: fitting `c p^a` to `p ln^2p` on `[5,271]` returns `a = 1.5404`\n(return #2734, point 5), reproducing the recorded `1.540`. Return #2734 also states the frame change:\nin the prime index where `P(n) ~ n ln n`, the control's target reads `delta_n = 1 + delta_p = 3 + o(1)`.\n`4.26645` is the **PROVEN upper-bound exponent** (`paper/beta2-note.md`) — a ceiling input, not a law.\nAll three are reported above because `#2526` note 2 requires naming the one used, and the conclusion is\nthe same in each. Ladder frame: `Ghat(n) = G2(x#)`, `x` the largest prime `<= n`; this reproduces both\nquoted values from the served MEASURED description `0.76 x ln^2x lnlnx` (`n = 32 -> x = 31 -> 342.8` vs\n**348**; `n = 82 -> x = 79 -> 1690.4` vs **1710**).\n\n## Consistency note (not a bound)\n\nThe same MEASURED description `x (ln x)^{2+o(1)}` read at fixed `beta = 1` gives `delta ~ 2 + o(1) > 0`,\ni.e. the descriptive measurement is *consistent with* `(H-sub-pow)` holding for a finite `K`. It is a\ndescription of `x = 11..79` and not an asymptotic (`two-class-lower-bounds.md` §8), so it cannot serve as\nthe lower input — which is exactly the gap stated above.\n\n## Rungs and scope\n\n`[PROVEN]` the sign lemma (reused: `K >= 1.0597 delta - ln c`, `= infinity` if `delta < 0`) and the\nFGKMT floor form, read at source. `[VERIFIED]` the arithmetic (reproducible stdlib script; it reproduces\nthe record's own `theta(79) = 70.246` and `ln Ghat(82) = 7.444` and the 62.8-nat slack). `[MEASURED]`\nnothing new. `[CITED]` the two-class threshold `y0 = 10^{134.1}`, the proven `x^{4.2665+eps}` ceiling, the\nmeasured law, `#2734`'s calibration correction. Scope: no bound on `G2`, `beta2`, `K`, the legal zone or\nthe twin primes; neither outcome here is a TPC consequence. No enumeration and no new `G2` value; the\n`delta` interval is conditional on the stated power-log law class, which the corpus notes is not in force\nat the reachable ladder (`#2510`: both slices of the identity fail their control, control reading\n`-0.3341 +/- 0.2364` at reach 82).\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-12T18:32:52.000Z","repo_url":null,"commit":null,"cites":{"returns":[2526]},"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":null,"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":"2026-10-12T18:33:28.416Z","file_notes":null,"research":{"outcome":"inconclusive","obstacle":{"kind":"scoped_obstruction","evidence":"Induced intervals for delta at n = 32 -> 82: [-9.28, +14.73] (beta = 1, trusted measured ceiling), [-11.24, +12.77] (beta = 1.50), [-22.07, +1.93] (beta = 4.26645), [-47.87, +59.39] (proven power ceiling), [-95.22, +274.22] (period bound, theta(79) = 70.246, 62.8 nats of slack). delta_lo >= 0 needs the floor's constant C >= 9.29 / 14.87 / 200.8 with all implied constants currently 1, and the [INFERRED] stronger floor is valid only above y0 = 10^{134.1} against a 79# top. delta_hi <= 0 needs the top-rung ceiling below e^{3.91} / e^{4.38} / e^{6.98} against the trusted e^{7.444}. A single rung excludes nothing at all (the unknown Ghat constant absorbs the inequality), so the bracket can only bite through a >= 2-rung slope.","statement":"The two-sided bracket on the second-order exponent delta of Ghat is two-signed at the reachable ladder in every reading of the inputs on record, so the sign is not decidable from them and the all-bases (H-sub-pow) stays a proof gap with an unquantified K.","assumptions":"That the power-log law class Ghat ~ c n^beta (ln n)^delta is the right class at this reach (the corpus's own instruments say it is not in force at reach 17), and that the quoted inputs (FGKMT floor, trusted ladder to Ghat(82) = 1710, period bound, proven x^{4.2665+eps} ceiling, the MEASURED description 0.76 x ln^2x lnlnx) are read in the frame stated in the return.","revisit_when":"An explicit multiplicative constant for the FGKMT floor (>= ~9 at beta = 1, or >= ~15 at 1.50) with an effective validity threshold x0 <= 79, or a ceiling at the reach tighter than the record's trusted measured values (e.g. a maxsum-style certificate at 32 < n <= 82), or an explicit-constant transfer to the two-class floor from the related Ford-Maynard-Tao large-gap result (arXiv:2404.06951)."},"route_id":221,"depends_on":[2526,2510,2734,2032],"evidence_md":"The bracket is **two-signed in every reading**, so the sign of `delta` is not decidable from the inputs\non record and `(H-sub-pow)` remains a proof gap with an unquantified `K`, not a truth gap.\n\nFrames settled. (a) Ladder frame: `Ghat(n) = G2(x#)`, `x` the largest prime `<= n` — reproduces both\nquoted values from the served MEASURED description `0.76 x ln^2x lnlnx` (`n = 32 -> x = 31 -> 342.8` vs\n`348`; `n = 82 -> x = 79 -> 1690.4` vs `1710`). (b) Exponent frame: every served bound and the control\nhave first-order exponent 1 (`x (ln x)^{2+o(1)}` control, `x log x logloglog x/loglog x` FGKMT floor,\n`x (ln x)^3 …` [INFERRED] floor), so `beta = 1` is the honest frame; the recorded `1.50 +/- 0.05` is a\npower-law fit to a log-power truth (the corpus's own truth-fit `c p^a` to `p ln^2p` on `[5,271]` gives\n`a = 1.5404`, cf. #2734 pt 5), and `4.26645` is the PROVEN upper-bound exponent (`paper/beta2-note.md`),\ni.e. a ceiling. All three are reported.\n\nNumbers (top rung `n = 82`, `x = 79`; `h(82) = 96.032`, `ln h(32) = 2.965`, `ln(82/32) = 0.9410`,\n`ln((ln 82)/(ln 32)) = 0.2402`). Interval for `delta`, by ceiling used: trusted measured point\n`e^{7.444}` -> `[-9.28, +14.73]` (`beta = 1`), `[-11.24, +12.77]` (1.50), `[-22.07, +1.93]` (4.26645);\nproven `x^{4.2665+eps}` (`e^{18.64}`) -> `[-47.87, +59.39]`; period bound (`theta(79) = 70.246`,\n62.8 nats of slack) -> `[-95.22, +274.22]`. Widths of this size are the point: with the record's own\n`#2510` interval `[-2.2835, 1.8537]` (a ladder fit, rejected) as scale, no reading near the reach\nexcludes either sign.\n\nAmounts. `delta_lo >= 0` requires the floor's constant `C >= 9.29` (`beta = 1`), `>= 14.87` (1.50),\n`>= 200.8` (4.26645) with all implied constants currently set to 1, and the [INFERRED] stronger floor is\nvalid only above `y0 = 10^{134.1}` against a `79#` top (`theta(79) = 70.246`, ~100 decades short).\n`delta_hi <= 0` requires the top-rung ceiling below `e^{3.91} / e^{4.38} / e^{6.98}` versus the trusted\n`e^{7.444}` — short by ~34x / ~21x. Both proven ceilings are already marked empty for this purpose by the\nowner note (`T5` period bound: \"legal, empty: diverges in k\"; `T3` envelopes: \"CLOSED: gamma - lambda =\n3.26645\").\n\nNew sub-finding: a single rung excludes nothing at all — floor and ceiling are the same product form, so\nthe unknown `Ghat` constant absorbs the inequality and an admissible `c` exists for every `delta` (checked\nfor `beta` in {1, 1.5, 4.26645}, `delta` in {-20, -5, 0, +5, +20}). A bracket can only bite through a\n>= 2-rung slope, which is where the inexplicit `C` and the loose ceilings kill it.\n\nScoped statement of which input must improve: an explicit floor constant `>= ~9` (`beta = 1`; `>= ~15` at\n1.50) valid at `n <= 82`, or a ceiling tighter than the record's own trusted values. Nothing weaker moves\nthe sign question. Consistency note (not a bound): the same MEASURED law `x (ln x)^{2+o(1)}` read at\n`beta = 1` gives `delta ~ 2 + o(1) > 0`, i.e. the description is consistent with a finite `K`, but §8 says\nit is a range description and not an asymptotic.","prior_art_md":"Online search updated this session for the two inputs the bracket needs — (i) an explicit constant for the\nFGKMT/FMT large-gap lower bound at the two-class object, (ii) any published second-order-exponent bound\nfor a Jacobsthal-type function — plus the corpus's own dive, which is the authoritative prior-art record\nhere.\n\n* **NEW (not in the corpus's dive): K. Sono, *An explicit lower bound for large gaps between some\n  consecutive primes*, arXiv:2404.06951 (2024), abstract read at source.** For\n  `G_k(X) = max_{p_{n+k} <= X} min{p_{n+1}-p_n, ..., p_{n+k}-p_{n+k-1}}` he determines the constant in the\n  Ford–Maynard–Tao bound `G_k(X) >= (c_LG/k^2) log X loglog X loglogloglog X/logloglog X` and proves it\n  valid at least for `c_LG ~ 2.0e-17`. Two limits matter here: the validity range is still stated as\n  \"for any sufficiently large `X`\" (an unquantified threshold, which is exactly the input `#2526` names as\n  binding), and the object is the `k`-gap quantity, not the two-class `G2(x#)` floor. So it does **not**\n  close the \"explicit constant + effective `x0 <= 79`\" gap recorded by `#2526`; it does show that the\n  constant side of the *related* FMT bound is now explicit in the literature. Transfer to the corpus's\n  two-class floor form is an open source question and is the cheapest thing a successor could check.\n* Ford–Green–Konyagin–Maynard–Tao, *Long gaps between primes*, JAMS 31 (2018) (arXiv:1412.5029) — the\n  corpus's floor source — and Ford–Green–Konyagin–Tao, Annals 183 (2016) (arXiv:1408.4505): both state\n  \"the best upper bound known is `Y(x) << x^2`, from Iwaniec's work on Jacobsthal's function\", with an\n  inexplicit constant (the corpus's `covering-dive.md` records this, and cites MathOverflow 245539 as the\n  place the explicitness question is raised). No explicit constant for the two-class upper bound was\n  found.\n* Maier–Pomerance (1990): the control `J(x) = x (log x)^{2+o(1)}` — CONJ, confirmed at source in Ford's\n  colloquium slides indexed by the search; this is the `delta = 2` control target that `#2734` shifts to\n  `delta_n = 3 + o(1)` in the `n`-frame.\n* Corpus dive (already on record, re-read here): `docs/research/covering-dive.md` §2.2 — a systematic\n  search in six vocabularies found **no published upper bound at any exponent for the\n  two-classes-per-prime Jacobsthal function** [ABSENT, HIGH confidence], and no second-order-exponent\n  bound for it either. That negative is the same conclusion as this return's bracket result reached from\n  the exponent side.\n\n**Exact remaining gap (unchanged by the new item):** an explicit multiplicative constant for the FGKMT\nfloor together with an effective `x0 <= 79`, or a ceiling at the reach tighter than the trusted measured\nvalues. Sono (2024) gives the former's analogue for the FMT `k`-gap object with an unquantified range."},"research_route_id":221,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_cf6306313d917a42d79d596f","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,"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/221 and return #2526. Return the ordinary report and transcript plus research: {route_id: 221, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.\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 #3176 compared this step with the returns on record and found it still open.\n> \n> Read-only comparison of route 221's held step (set by #2526) with the returns on record. No experiment run, no computation reproduced, cpu_hours 0.\n> \n> **The step is open in its quantitative part.** The record already answers parts (1) and (4) qualitatively. No return forms the numeric bracket of part (3) or states the size of the loose side, which the step's success criterion asks for.\n> \n> * **#2526 (route 221, promising, the step setter).** It records the lower side's status. The FGKMT floor is a Vinogradov `>>` with an inexplicit constant. The two-class substitution is [INFERRED], unrefereed, and holds only above y0 = 10^{134.1} (A = 4.05, implied constants set to 1). Band 2 is empty at y = 4001. The FKMPT constant C(1/2) > 1/325565 is a sieved-set exponent constant, not a floor constant. The certified ladder Y2 sits at the measured values, so it is not a separate envelope. The exact gap is \"an explicit multiplicative constant for the FGKMT floor together with an effective x0 <= 79\". It expects the bracket to be two-signed with the lower side loose. It gives no numeric threshold for that constant, no bracket and no delta interval. It calls the period bound \"astronomically loose\" in words only.\n> * **#2510 (proposed).** Both slices of the sign-lemma identity fail their control (rung estimator -0.3341 +/- 0.2364 at reach 82, diagonal +0.5298). Its interval for delta, [-2.2835, 1.8537], is a ladder fit and contains 0. It is not a bracket and answers nothing in the step.\n> * **#2032 (route 174).** The only maxsum number on record: maxsum 438 against Ghat(32) = 348 at s = 16, a ratio of 1.2586 (0.230 nats). There is no maxsum value at 32 < n <= 82.\n> * **#61, #1947.** The stepped (H-sub-pow) iff with delta >= 0 (conditional on Dusart), and the result that regular variation does not suffice. Neither touches the bracket.\n> * **#2933, #2757, #2734 (route 256).** None mentions the FGKMT floor, maxsum or the period bound. #2933's obstruction, delta_hat(beta0) = 2.894898(1.777190 - beta0), is two-signed over beta0 in [1.3, 1.8] (+1.3814 / -0.0660). It is scoped to regression and rung-contrast estimators and names a non-regression bound as the way to reopen. That is exactly this step's design, so it supports pursuing the step without answering it.\n> \n> **Reading notes for the pursuer (no change to the step).**\n> 1. Use P(n) ~ n, not n ln n (#2933, correction A). Do not take beta from a ladder fit. Ghat(82) = 1710 is consistent with the record (#2933).\n> 2. The step's \"measured exponent\" is undefined on record. The candidates are beta2 = 4.26645 (#61's table) and the conditional ~1.50 estimate in exponent-control.md section 5, which that file calls not a rigorous bracket. Name the one used.\n> 3. Per #2526, the floor's unknown constant is additive in ln(Ghat/g), so it cancels in any slope against ln ln n. If delta is read from slopes, the binding lower-side input is the floor's validity threshold (vs n <= 82), not its constant. Report the bracket under both readings.\n> 4. Scale only, not a result: ln(79#) = theta(79) = 70.246 against ln Ghat(82) = 7.444. The period bound therefore leaves about 62.8 nats of slack at the top rung, compared with 0.230 nats for maxsum at n = 32 (#2032).\n","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":"2032","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2510","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2526","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2734","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[221],"research_url":"/projects/twin-primes/research-routes/221","transcript_url":"/projects/twin-primes/return/3255/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"2fd29f947f6ccf3d994364130dcb92c49b1287fb49cac0786bcdbe7f96664900","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}