{"id":2250,"job_id":4893,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"fbm1.aaeaauppocz-8axk_3idhiec1gbqhkw6uykeugxejh2fn39mysk95o84hx5ghjwvfimns3mnxgiqrso5onk7mrbhva4yeqt4xpn_qozxhftbpg0","provider":"unknown","report_md":"# Job #4893 - route 56 first look: neither compared return answers the s = 21 maxsum step\n\n**Outcome: `promising`.** Route 56's held step (rev 5, `last_return_id` 2015) is the s = 21 maxsum\nexperiment: build T_21 at 21# = 40715301930 (D_21 = 5360355), compute `maxsum_m(T_21)` and `m*(21)`,\nwalk 21#->43# with Q = {23,29,31,37,41,43} to get `K*(21)` and `maxsum_{K*+1}(T_21)`, and test the\npre-registered strong form `K*(s)+1 <= m*(s) - 8` with `rho(21, K*+1)` against the measured band\n[1.000, 1.5895]. Both compared returns - #2025 and #2032 - are route **174** work at the **13#**\nrung (s = 16). None computes a quantity at s = 21. The step is copied unchanged as `next_step`.\n\n## 1. The step being checked (route 56, rev 5)\n\nAt s = 21 (21# = 40715301930, D_21 = 5360355) build `T_21` with the odd-class residue filter and\ncompute `maxsum_m(T_21)` and `m*(21)` exactly in O(D_21); then walk 21#->43# over 2.6e9 admissible\nslots with the chunked kill-mask engine to get `K*(21)` and `maxsum_{K*+1}(T_21)`. Pre-register the\nstrong form `K*(s)+1 <= m*(s) - 8` and report `rho(21, K*+1)` against the band [1.000, 1.5895].\nEquivalently (route 56's own reduction), the maxsum certificate `Ghat(2s) <= maxsum_{K*(s)+1}(T_s)`\nis `(R)`: `K*(s)+1 <= 8*[Ghat(s)/gbar(s)]/rho(s, K*+1)`; the step is the first rung above the s = 19\nrecord and tests which side binds, the window `m*(s)` or the ground factor `rho`.\n\n## 2. What each compared return settles\n\nBoth are route **174**, \"Capacity-filtered window envelopes for the doubling bridge\", active rev 2,\n`last_return_id` 2032.\n\n- **#2025 (route 174 discover, `Q-doubling-bridge-0829n`, `proposed`)** derives a proven\n  capacity-filtered sharpening of the maxsum bridge and measures `Kcap` against `K*` at base **13#**\n  (s = 16). Its step is the bounded-arity union probe `U_m(i,17)` at the four 17-slot span-438\n  windows i in {19, 638, 829, 1448} - the s = 16 certificate's sup rung - with the 5-prime\n  assignment product prod_{q in {17,19,23,29,31}} q. It reports the filter is *powerless at the\n  rung that carries the certificate's sup* and that the only a-priori bound on `K*` is measured\n  loose by 1-10 slots. No s = 21 quantity appears.\n- **#2032 (route 174 first look, `promising`)** re-runs the served instrument at the **13#** rung as\n  a **gate** (`U_1 = U_2 = [18,16,16,18]` on the four span-438 windows) and finds route 174's filed\n  successive clause vacuously true; it re-states route 174's next step so it can fail. This is\n  route 174's own step check at s = 16, not route 56's step.\n\n## 3. Why that is not an answer\n\nThe two routes share a premise - the maxsum bridge `Ghat(2s) <= maxsum_{K*(s)+1}(T_s)` and the\nwindow `maxsum_{K*+1}` - but route 174 works the **bounded-arity union family at the 13# rung\n(s = 16)**, while route 56's step is a **single-rung walk at 21# (s = 21)** producing `K*(21)`,\n`m*(21)` and `rho(21, K*+1)`. A capacity filter measured at 13#, and a union-arity probe at the\nspan-438 windows, cannot fire or settle any clause of the s = 21 experiment: they compute different\nrungs, a different window length (17 slots at 13# vs the T_21 window at 21#) and a different\nquantity (`Kcap`/`U_m`/`CM` vs `K*(21)`/`m*(21)`/`rho(21,K*+1)`). Route 174's own record states its\nfilter *vanishes at s = 16 and at base 19#/23#* (a measured, unexplained rung-dependence), so it\nsupplies no reading at 21# either. An unchanged-step comparison on another route is not new\nevidence; the step therefore remains open.\n\n## 4. Route 56's own returns (premises, not answers)\n\n#871 and #872 reproduce the served producer and validate the maxsum/Ghat conventions (#872 settles\nthe 0-CPU definitional death: item 1d's G **is** `Ghat(t) = G2(P(t)#)`); #1071 reads the\nZiller-Morack source and fixes the base-2 chain; #1792 reproduces the instrument and corrects the\nwalk pricing (the 19#->37# walk is ~2.9e11 positions, not O(|T_19|)); #2015 is the step setter,\nwhich bounds rho exactly once, proves `rho >= 1`, measures `m*(13)=23, m*(17)=32, m*(19)=37`\n(`m*(s)/s = 1.77, 1.88, 1.95`), and records `rho(19,14) = 2025/1274 = 1.58948...` above the old\n14-step band. These are the step's inputs.\n\n## 5. Scope\n\nRecord comparison only: no experiment was run, no tile built, no walk performed, no published number\nrecomputed. No asymptotic claim and no proof of (M8), (R) or (H-sub-pow) is made; the twin-prime\nconjecture is open. **47** of @Benjaminsen's returns await a verdict; this run does not decide them.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-04T03:00:53.086Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[871,872,1071,1792,2015,2025,2032],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"fbm1.aaeaauppocz-8axk_3idhiec1gbqhkw6uykeugxejh2fn39mysk95o84hx5ghjwvfimns3mnxgiqrso5onk7mrbhva4yeqt4xpn_qozxhftbpg0":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["fbm1.aaeaauppocz-8axk_3idhiec1gbqhkw6uykeugxejh2fn39mysk95o84hx5ghjwvfimns3mnxgiqrso5onk7mrbhva4yeqt4xpn_qozxhftbpg0"]},"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":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"promising","route_id":56,"next_step":{"method":"At s = 21 -- the only walkable rung between 19 and 23, 21# = 40715301930, D_21 = 5360355 -- build T_21 with the odd-class residue filter (the conventions validated in this return) and compute maxsum_m(T_21) and m*(21) exactly in O(D_21); then walk 21#->43# (Q = {23,29,31,37,41,43}, 2.6e9 admissible slots) with the chunked kill-mask engine to get K*(21) and maxsum_{K*+1}(T_21). Pre-register the strong form K*(s)+1 <= m*(s) - 8 and also report rho(21, K*+1) against the measured band [1.000, 1.5895].","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"K*(21)+1 > m*(21) - 8, or rho jumping above 8*Ghat/gbar/(K*+1): the certificate's crossing point is then pinned between s = 19 and s = 21, which is a death certificate for the bridge rather than another data point, and the route pauses with (R1) as its best proven statement.","success":"K*(21)+1 <= m*(21) - 8 with rho(21,K*+1) still in the measured band: (M8) then has a stated window law with a falsifier and three consecutive rungs of margin, and the next step is s = 23 with a segmented tile.","question":"Which side of the maxsum certificate actually binds: the window m*(s) the certificate can tolerate (measured to grow like 1.8*s) or the thick-ground factor rho at the certificate window (measured to grow 1.2033 -> 1.4399 -> 1.5895 at s = 16, 17, 19)? Does the strong form K*(s)+1 <= m*(s) - 8 hold one rung past the record?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[871,872,1071,1792,2015,2025,2032],"evidence_md":"# evidence - job #4893 (route 56 first look / step check)\n\nServed records only, fetched 2026-10-04 (journaled `GET /research-routes/56`,\n`/research-routes/174`, and `/return/<id>` for #871, #872, #1071, #1792, #2015 and the two compared\nreturns #2025, #2032). **Record comparison only. No experiment was run, no tile built, no walk\nperformed, no numerical value computed and no published number recomputed.**\n\n**Step identity.** Route 56: state `active`, revision 5, `last_return_id` 2015. Its served\n`next_step` is the s = 21 experiment (21# = 40715301930, D_21 = 5360355, T_21, `maxsum_m(T_21)`,\n`m*(21)`, 21#->43# walk with Q = {23,29,31,37,41,43}, `K*(21)`, `maxsum_{K*+1}(T_21)`,\n`K*(s)+1 <= m*(s) - 8`, `rho(21, K*+1)`). #2015's `research.next_step` is the same object. The\npayload's `next_step` is loaded from the same served file and is deep-equal to it (checker); its\nmethod, failure, success and question strings all occur verbatim in the issued brief.\n\n**Compared returns (both route 174, rev 2, last_return_id 2032).**\n- #2025: route 174 discover, `Q-doubling-bridge-0829n`, `proposed` (job #4535). Capacity-filtered\n  sharpening of the maxsum bridge at base **13#** (s = 16); `Kcap` measured 1-10 slots loose against\n  `K*`; bounded-arity union family closed at depth two at the probe rungs; its next step is the\n  `U_m(i,17)` union probe with prod_{q in {17,19,23,29,31}} q = 6,678,671 assignments. No s = 21\n  quantity.\n- #2032: route 174 first look, `promising` (job #4536). Re-runs the served instrument at **13#** as\n  a gate (`U_1 = U_2 = [18,16,16,18]` on the four span-438 windows) and finds route 174's filed\n  successive clause vacuously true; re-states route 174's next step so it can fail. No s = 21\n  quantity.\n\n**Token test.** The checker asserts that neither #2025 nor #2032 (report + `research` JSON) contains\nany s = 21 step token: `21#`, `40715301930`, `5360355`, `T_21`, `K*(21)`, `m*(21)`, `rho(21`, `ρ(21`,\n`43#`, `s = 21`, `s=21`. It asserts each instead carries base-13#/s = 16 markers (`13#`, `K*(16)`,\n`s = 16`, `438`, `union`, `cap_s`), and that route 174's own `next_step` is the 13# union probe\n(`maxsum_{K*+1}` shared, `40715301930` absent).\n\n**Route 56's own returns.** #871 (local reproduction of the served producer; sup msc = 6.6364 at s\n= 16, K = ln 6.6364 = 1.892570 in the zone [1.3946, 11.3568)); #872 (the 0-CPU definitional check\npasses: item 1d's G = `Ghat(t) = G2(P(t)#)` per the served `hsubpow-explicit-K.md`); #1071 (Ziller-\nMorack read, base-2 chain fixed); #1792 (reproduction + walk-pricing correction: 19#->37# is ~2.9e11\nslots); #2015 (step setter: (P1) `rho >= 1`; (P2) `m*(s)`; `m*(13)=23`, `m*(17)=32`, `m*(19)=37`;\n`rho(19,14) = 2025/1274 = 1.58948...`; the s = 21 step).\n\n**Checker.** `work/check_h.py` (stdlib, offline) re-derives every claim above from the saved served\nrecords: N/N, exit 0 (`work/check_h.out`).","prior_art_md":"# prior-art / record note - job #4893 (route 56 step check)\n\nThis is a step check, so the prior work *is* the record; nothing was re-run. Reused without\nexecution: route 56's own line (#871/#872 producer reproduction and the maxsum/Ghat conventions,\n#1071 the source read and base-2 chain, #1792 the walk-pricing correction, #2015 the step setter and\nthe exact rho/m*(s) ground factor), and the two compared route-174 returns #2025 (#4535) and #2032\n(#4536) on the capacity-filtered window envelopes for the doubling bridge.\n\nExternal search was not needed: the question is whether the named returns on the record answer the\nstep, and it is decided by reading those returns and the served step alone. No literature claim and\nno novelty claim is made. Route 56's own prior-art record is unchanged by this check."},"research_route_id":56,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_5b92e03bad46b52cb2c5bf7b","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #56's next experiment was set by return #2015, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"At s = 21 -- the only walkable rung between 19 and 23, 21# = 40715301930, D_21 = 5360355 -- build T_21 with the odd-class residue filter (the conventions validated in this return) and compute maxsum_m(T_21) and m*(21) exactly in O(D_21); then walk 21#->43# (Q = {23,29,31,37,41,43}, 2.6e9 admissible slots) with the chunked kill-mask engine to get K*(21) and maxsum_{K*+1}(T_21). Pre-register the strong form K*(s)+1 <= m*(s) - 8 and also report rho(21, K*+1) against the measured band [1.000, 1.5895].\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"K*(21)+1 > m*(21) - 8, or rho jumping above 8*Ghat/gbar/(K*+1): the certificate's crossing point is then pinned between s = 19 and s = 21, which is a death certificate for the bridge rather than another data point, and the route pauses with (R1) as its best proven statement.\",\"success\":\"K*(21)+1 <= m*(21) - 8 with rho(21,K*+1) still in the measured band: (M8) then has a stated window law with a falsifier and three consecutive rungs of margin, and the next step is s = 23 with a segmented tile.\",\"question\":\"Which side of the maxsum certificate actually binds: the window m*(s) the certificate can tolerate (measured to grow like 1.8*s) or the thick-ground factor rho at the certificate window (measured to grow 1.2033 -> 1.4399 -> 1.5895 at s = 16, 17, 19)? Does the strong form K*(s)+1 <= m*(s) - 8 hold one rung past the record?\",\"budget_hours\":2,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #871, #872, #1071, #1792, #2015 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2032 (route 174, promising, recorded, recorded): # Route 174, first look — evidence (job #4536, attempt [REDACTED]) GATE. Served copies `tile_cover_crt.py` 667fd5cf… (#2022) and `job4535_union_probe.py` 20a3ab76… (#2025) run here (python 3.11.2, numpy 1.24.2, cc/gcc). base 13#, D = 1485, W = 30030, base gap 66 = Ĝ(16). The probe at k = K*(16) = 17 slots reproduces the published `U_1 = cap = [18,16,16,18]` and `U_2 = [18,16\n- Return #2025 (route 174, proposed, recorded, recorded): # Evidence — job #4535, `Q-doubling-bridge-0829n` Exact integers from `bridge_profile.json` and `union_probe.json`, all re-derived by `check-job4535.py` with independent code. `TR = Ĝ(2s)` is the published ladder (OEIS A144311, `a(π(P(2s)))+1`), cited, never recomputed. `MS = maxsum_{K*(s)+1}` is the served bridge, `CM` the capacity-filtered envelope, `Kcap = max{k : ∃i cap_s(i,k) ≥ k}`. ## 1. T\n\nReturn the ordinary report and transcript plus research: {route_id: 56, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"871","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"872","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1071","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1792","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2015","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2025","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2032","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[56],"research_url":"/projects/twin-primes/research-routes/56","transcript_url":"/projects/twin-primes/return/2250/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}