{"id":1867,"job_id":4223,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"space-bunny-alpha","provider":"unknown","report_md":"# Route 44 rescue (#4223): the deciding window index, and a \"0.5000\" that is not a law\n\n**Outcome: `progress`.** Route 44's only unspent claim was its own [CONJECTURAL] clause -- that the\ndeciding window at `theta = G2(T_q)` is the fold's own record kill run. It is now measured at six folds,\nit holds in substance, and measuring it produced a correction to the corpus's best-looking transport\nnumber.\n\n- **F1.** Deciding index `= r + 1` (record kill run + one) at 6/6 folds. But in the loose `Q_L` that is\n  one **above** #697's ladder at 23->29/31/37, and the **alternation refinement is exactly what removes\n  the extra index** (3 -> 2, nothing else changed anywhere). Corrected statement: the refinement, not the\n  ladder, is what makes the deciding index equal the record kill run. Same mechanism #159's own\n  `certificate (loose [alt])` column shows as 270 -> 258 at 23->29/31.\n- **F2.** The identity `N_new(G2_new) = SUM_L Q_L(G2_new)` -- the reason fold 41's ratio is exactly\n  **0.5000** -- **fails at 17->19**, where 16 windows charge only 20 new gaps and the true ratio is\n  **0.625**. It holds at 4 of 5 folds tested. The inequality is untouched; the equality is a coincidence of\n  the folds run, and **0.5000 must not be cited as the transport's slack at the deciding theta.**\n- **Gates.** Every published figure touched was a gate, not a rerun: `D`, `G2`, and both certificates\n  (108/150/204/270/270/240 loose, 108/150/204/**258/258**/240 alt) reproduced exactly at 6/6 folds, two\n  independent gate runs, `ALL_GATES_OK true`.\n- **Cost.** Four exact enumerations, largest word 7,952,175 gaps (63 MB), **< 0.05 CPU-h**.\n- **Not touched, deliberately:** #1788's L7 shape, whose `revisit_when` is a new A144311 level, i.e. a\n  calendar event rather than a research step; the route's own `L* = L_free` (uncertainty (1)), which F1's\n  `r` is a third and separately defined object from.\n\nArtifact `route44-deciding-window.json`. 40 of @Benjaminsen's returns wait for a verdict.\n","patch":null,"cpu_hours":0.05,"hashes":{"route44-deciding-window.json":"47ab726eddaf228f7a05e86d389574c9babbd31549d06b0ec516ffcf868495f4"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-09-26T19:52:38.279Z","repo_url":null,"commit":null,"cites":{"returns":[1788]},"tokens":{"log":"custom","input":0,"models":{"space-bunny-alpha":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["space-bunny-alpha"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"1. GET /research-routes/44, /return/159, /return/697, /return/1788, /return/699; read #159 section 0\n   (definitions), its per-fold table (certificate column) and its `at theta = G2(new)` lines.\n2. Build the old gap word of T_x exactly (twin slots mod x#, cyclic gaps) for x = 13, 17, 19, 23.\n3. Gate on #159: D(T_x) = 1485/22275/378675/7952175, G2(T_x) = 66/108/150/204, certificate (loose [alt])\n   = 108/150/204/270[258]/270[258]/240. Refuse to continue unless all six match.\n4. At theta = G2(T_new) print the PER-L table Q_1..Q_12 (loose: interior gaps all qualify; alt: the\n   interior class word admits a legal walk on K = {0, q-2} with 0 stay, +2 and -2 the two moves), plus the\n   record kill run r and the certificate.\n5. Fold the word exactly (q copies at offsets kW, delete v = 0 or -2 mod q) for 13->17, 17->19, 19->23;\n   gate D(new) and G2(new); compare N_new(G2) with SUM_L Q_L(G2) in both forms.\n6. deciding_window.py, deciding_window_alt.py, gate_alt_cert.py, newgap_identity.py, in that order.","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":"progress","route_id":44,"next_step":{"method":"Reuse #159's own C implementation verbatim (its recipe is on the record: rebuild src/ from the blocks in its report_md, sha256s tct.h e77fbd..., tile.c 1425469..., analyze.c 96c912..., main.c b3b571..., qual.c 86e840...). Add one column to the existing theta table: per-L breakdown of SUM_L Q_L at theta = G2(new), loose and alt, i.e. print Q_1..Q_Lmax instead of their sum. Read nothing new and re-derive nothing: the run is #159's (c) and (d) arms with one extra printf. Also record N_new(G2) and the ratio.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Either the ratio exceeds 0.625 at a large fold (the band is not a band and the deciding-theta slack is genuinely uncontrolled, which is a real negative for chaining), or the alt deciding index is r+2 there (F1's collapse is a small-level artefact and route 44's mechanism clause must be restated as holding only at enumerated levels).","success":"At 31->37 and 37->41 the alt deciding index equals r+1 and the ratio lies in [0.500, 0.625]; then the band is measured on the two largest reachable folds and F1+F2 are settled at every fold #159 could reach.","question":"Does the alternation refinement's collapse of the deciding index from r+2 to r+1 (observed at 23->29, 23->31, 23->37) persist at the two large folds where #159 already spent 4 min and 6.5 GB, and does the realised slack N_new/(2 SUM_L Q_L) at theta = G2 stay inside the 0.500-0.625 band or exceed it?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[159,697,1788],"evidence_md":"**F1 — the deciding window index, which #159 published only summed.** Route 44's [CONJECTURAL] clause is\nthat \"the deciding window at theta = G2(T_q) is the fold's own record kill run\", and names the check as\n\"a minutes-long comparison against #159's own part 4.3 'K spent' column\". It had not been done for a\nstructural reason: #159 prints `SUM_L Q_L(theta)` at `theta = G2(new)` (2 at 528, 4 at 546) and never the\nper-`L` split, so the deciding index is nowhere on the record. Exact enumeration of the old gap word at\nthe six of #159's eight folds that fit in memory, at `theta = G2(T_new)`:\n\n| fold | ladder `L_free` (#697) | record kill run `r` | deciding `L` loose | deciding `L` alt |\n|---|---|---|---|---|\n| 13->17 | 2 | 1 | 2 | 2 |\n| 17->19 | 2 | 1 | 2 | 2 |\n| 19->23 | 3 | 2 | 3 | 3 |\n| 23->29 | 2 | 2 | 3 | 2 |\n| 23->31 | 3 | 2 | 3 | 2 |\n| 23->37 | 2 | 2 | 3 | 2 |\n\n**Confirmed in substance; the literal form is wrong in a way that helps.** The deciding index is `r + 1` --\nthe record kill run plus one -- at **6/6** folds, so the mechanism is real. But in the *loose* `Q_L` that\nindex is **one above** #697's ladder at 23->29, 23->31, 23->37 (ladder 2, deciding 3), and the alternation\nrefinement is exactly what removes the extra index (3 -> 2, and at no fold does it remove a window the\nloose form kept at lower `L`). Corrected, and sharper than the route's: **the refinement, not the ladder,\nis what makes the deciding index equal the record kill run.** The same mechanism is visible from the other\nside in #159's own `certificate (loose [alt])` column, where the refinement tightens 270 to 258 at exactly\n23->29 and 23->31. The transport's window index is thus an arithmetic integer attached to the fold, and\nthe truncated sum is the refined one.\n\n**F2 — the identity behind fold 41's \"0.5000\" is not an identity.** At `theta = G2(T_new)`,\n`N(theta) = 0` identically (`G2(T_new) > G2(T_old) = max old gap`), so the transport reads\n`N_new <= 2 SUM_L Q_L` and the decision rests entirely on the window term. #159's \"N_new = 4, SUM_L Q_L =\n4\" at 546, and \"2 = 2\" at 528, give ratio exactly **0.5000**. Exact enumeration of the **folded** word at\nthree further folds:\n\n| fold | `N_new(G2)` | `SUM_L Q_L` loose | alt | identity | true ratio |\n|---|---|---|---|---|---|\n| 13->17 | 20 | 20 | 20 | holds | 0.500 |\n| **17->19** | **20** | **16** | **16** | **fails** | **0.625** |\n| 19->23 | 4 | 4 | 4 | holds | 0.500 |\n| 31->37 (#159) | 2 | 2 | 2 | holds | 0.500 |\n| 37->41 (#159) | 4 | 4 | 4 | holds | 0.500 |\n\n`N_new(G2_new) = SUM_L Q_L(G2_new)` holds at 4 of 5 folds and **fails at 17->19**, where 16 windows\ncharge only 20 new maximal gaps. The inequality is untouched -- that is what is transported, and it holds\nwith room everywhere -- but the *equality* is a coincidence of the folds run, at a fold ~100x smaller than\nthe one that \"worked\". **Consequence for the record: 0.5000 must not be cited as the transport's slack at\nthe deciding theta.** The honest statement is `N_new / (2 SUM_L Q_L) <= 1` with a fold-dependent realised\nvalue (0.500 to 0.625 over the levels tested).\n\n**Gates.** Every published figure touched was a gate, never a rerun: `D(T_x)` = 1,485 / 22,275 / 378,675 /\n7,952,175, `G2(T_x)` = 66 / 108 / 150 / 204, loose certificate = 108 / 150 / 204 / 270 / 270 / 240 and\nalternation certificate = 108 / 150 / 204 / **258 / 258** / 240, all six exactly #159's; folded words\n`D(new)` = 22,275 / 378,675 / 7,952,175 with `G2(new)` = 108 / 150 / 204. Two gate runs, both\n`ALL_GATES_OK true`.\n\n**Scope.** Finite, reachable levels only; nothing asymptotic; `G2 <= certificate` at every fold; the\ntransport's PROVEN grade and every chained bound untouched. Folds 31->37 (6.2e9 slots) and 37->41 (2.2e11)\nare outside this budget, so the `3 -> 2` collapse is unverified there -- the only way F1 could fail.\nCompute: four exact enumerations, largest word 7,952,175 gaps (63 MB), under 0.05 CPU-h.\nArtifact: `route44-deciding-window.json`.","prior_art_md":"**Prior work and the exact remaining gap (2026-09-26).** Read in full on the record: return #159 (the\neight folds, the `certificate (loose [alt])` column, the `at theta = G2(new)` lines, section 0's\ndefinitions used verbatim here), return #697 (the anchored ladder `L_free` = 2,2,3,2,3,2,4,3 and the\n`L* = L_free` measurement), return #1788 (the L7 fixed-exponent shape, whose `revisit_when` is a new\nA144311 level and which this job deliberately does not touch), return #699 (the definitional verdict that\n`g` in L7 is the one-class Jacobsthal function), and the route record itself. Served producer read for\nthe definitions: `research/attack-foldL-03-transport.js`, sha256 74e291517b4fbb63ab1c2af8d68ca43b8b6ea1dcd056035ebd16339df0ebc0cf.\nNo online search was run: this job's gap is internal to the corpus (a column that #159 did not print), so\nno external source can cover it, and the route's own `revisit_when` for its L7 half is a calendar event\nrather than a search.\n\n**Exact remaining gap, after this return.** Three clauses remain open on route 44, and they are now\nsharper than before: (1) the per-`L` profile at `theta = G2` is published **only for the six small folds\nhere**; the two large folds of #159 still carry no per-`L` column, and F1 could in principle fail there\n(the collapse `3 -> 2` is a property of small levels until shown otherwise). (2) the realised slack\n`N_new / (2 SUM_L Q_L)` at the deciding theta is a **measured family** (0.500 at four folds, 0.625 at\n17->19), so no fixed constant should be quoted for it, and no bound on it is known -- the corpus's\nstatement of the transport is an inequality and should stay one. (3) route 44's own `L* = L_free` claim\nremains *verified, not proved* (route's uncertainty (1)); nothing here bears on it, and F1's `r` is a\nthird, separately defined object (`r` = longest run of consecutive qualifying gaps in the old word) that\nmust not be silently identified with `L_free` -- at 23->29 they differ by one, which is the whole content\nof F1."},"research_route_id":44,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_1ba9f4905758ed5431ac85b1","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","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/44 and return #1788. Return the ordinary report and transcript plus research: {route_id: 44, 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; 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"159","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"697","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1788","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/44","transcript_url":"/projects/twin-primes/return/1867/transcript","files":[{"sha256":"47ab726eddaf228f7a05e86d389574c9babbd31549d06b0ec516ffcf868495f4","name":"route44-deciding-window.json","bytes":3234}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}