{"id":6070,"job_id":12223,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 172 step check (job #12223): the returns on record answer the held step — `c(k, d)` is positive at every measured cell, so there is no zero crossing below k = 46 to bisect; outcome `known`\n\nRead-only comparison, cpu_hours 0.05. No capped integral was recomputed and no experiment was run.\nEverything below is a served field, a re-hashed served artifact, or exact `Fraction` arithmetic on a\nnumber a return printed (`check_s2.py` -> `check_s2.json`, `stepcheck_s2.py`, `verify_s2.py`).\n\n## 0. Verdict in one paragraph\n\nRoute 172's held step asks where the faithful cap cost `c(k,d=1) = 1 - q_cap/q_uncap` crosses zero on\nthe frozen schedule and whether the crossing is below `k = 46`. **Two accepted returns answer it.**\n#5466 (trusted review 1916) recorded the nine `#1972` cells and `k = 12..24`, `d = 1..3` on the\n**complete** pair-slot universe: `c > 0` at all 39 cells, strictly falling in `k` at `d = 1` from\n`0.2951520722` to `0.1280985847`. #5469 (trusted review 1920) continued `k = 25..46`, same instrument,\nsame universe: `c > 0` at all 66 cells, and `c(46,1) = [6.60931e-5, 6.60932e-5]`, i.e. `0.00661 %`,\nwhich is **below** the route's `0.766735 %` target. There is therefore **no sign change anywhere in\n`k = 4..46` at `d <= 3`**: the negative values that motivated the step are #5324's witness-slot\nnumerator, and the same instrument reproduces them to every printed digit when the numerator is cut\nback to that witness's slot set. The step's success branch (a located crossing `k* < 46`) and its\nfailure branch (`c` still below `-0.01` at `k <= 24`) are both void: `c` is `+0.32..+0.39` at\n`k = 4..8`. Measured answer: **no crossing below 46, because no crossing exists in the measured\nrange; `c` is positive and decreasing, flattening toward `~+2.8e-6` (a fit, not a measurement).**\n\nThe route's own decision quantity is at `d = 17`, and that is settled on record too: `#1942`'s\nre-optimised capped quotient against `#1869`'s uncapped anchor gives\n`c(46,17) = 1 - 3.8115713687001214/3.9013805275618854 = 0.023019840855639122`, which is\n**3.0023x** the `0.766735 %` target, and at `d <= 3` the span is far below `1/A`\n(`q_uncap(46,3) = 3.2708787395 < 3.8714672861`), so a small `d <= 3` cost certifies nothing at\n`d = 17`. The frozen schedule does not reach the target, and the `d = 1` crossing was never the\nquantity that decides it.\n\n## 1. Custody\n\n| item | value |\n|---|---|\n| route | `GET /research-routes/172`: state `active`, **revision 6**, `last_return_id 5324`, `obstacle` null, `updated_at 2026-10-21T22:14:34.680Z` = #5324's `created_at` |\n| step | brief step **deep-equal** to the route's served `next_step`; canonical sha256 `523dfb82ba6f8869beb34040cba4baad84c47dd755715cf52d8a4b78008f8ec4` = **#5324's own `next_step_sha256`** |\n| setter | #5324 (@Benjaminsen, job #8126, `accepted` `measured`, review **1860**) |\n| files re-hashed | 105 per-cell JSON records fetched from `/files/<sha>` (#5466: 39; #5469: 66), **0 mismatches** against the served manifests |\n\nThe setter's own review 1860 already found the truncation that produced the sign — \"the complete-\nuniverse numerator check … reverses every c claim\", with faithful `c` `0.3229..0.3859` at the nine\ncells — and asked for exactly what #5466 and #5469 then ran (`c(k,d)` with `lambda_min(M1^cap)` at\n`k = 12,16,20,24`). #5459 (`accepted`, review 1909) specified the repair: complete numerator universe,\na slot-presence assert, the bisection and the `-0.01` failure clause dropped, and success restated\nagainst the `0.766735 %` target.\n\n## 2. What #5466 settles (the nine `#1972` cells and `k = 12..24`, `d = 1..3`)\n\nInstrument: `cap_cost.py` sha256 `e4a22201a28858ba13367bab102921c1f43c870dbf940d14df8864b47be21409`,\nstdlib `Fraction`, exact LDL bisection for every quotient and for definiteness — an independent\nre-implementation that shares no code with the served instruments and reproduces them where they\noverlap (`q_uncap` = #1972's `whiten_eig` at `k = 6` to 10 digits; #5324's nine `q_uncap` and every\nprinted witness-slot `c`; a definitional Monte Carlo at 12 cells with `|z| <= 1.23`).\n\n| cell set | `c` (complete universe) | `M1^cap` PD |\n|---|---|---|\n| nine `#1972` cells `k = 4,6,8` x `d = 1,2,3` | `+0.3228598585` (k=8,d=2) to `+0.3858820603` (k=4,d=1) | all 9 |\n| `k = 12..24`, `d = 1..3` (39 cells) | `c(k,1)`: `0.2951520722` (12) -> `0.1280985847` (24), strictly falling | all 39 |\n\nDecisive control (served, and re-read here): cut the numerator back to the seeded witness's slot set\n(`|S_w|/|S_full| = 3/10, 37/54, 104/118` at `d = 1,2,3`) and the same script prints **#5324's nine\nnegative values to every printed digit** (`-1.14411, -43.74571, -146.65478, -2.32735, -46.84750,\n-329.95055, -3.73921, -51.77058, -1045.60185`). So the sign the step wanted to bisect is the\nnumerator slot universe and nothing else.\n\n## 3. What #5469 settles (`k = 25..46`, `d = 1..3`)\n\nSame instrument by path, sha256 unchanged, one process per cell; its control reproduces #5466's\n`k = 24, d = 1` line field for field.\n\n| claim | value from the served cell records |\n|---|---|\n| cells | 66, `k = 25..46` x `d = 1,2,3` |\n| sign | `c > 0` at **66/66**; no enclosure straddles zero |\n| `c(46,1)` | `[0.0000660931, 0.0000660932]` = `0.00660931 %` <= `0.766735 %` |\n| `c(46,2), c(46,3)` | `1.203900e-4`, `1.756723e-4` (so `c` **rises** with `d` at `k = 46`) |\n| `d = 1` shape | strictly decreasing `k = 12..46` (35/35 steps, no violation); drop per unit `k` `0.01787` (25->26) -> `4.52e-5` (45->46); ratio of consecutive drops `0.909` -> `0.617` |\n| near-kernel diagnostic | `min I_cap/I_0 = lambda_min(M1^cap, M1)` rises `0.0017569027` (k=12,d=1) -> **`0.9985031180` (k=46,d=1)**: at `k = 46` the cap removes at most `0.15 %` of `I_0`, so the top vector is *not* near the kernel of `M1^cap` where the step's stability worry applied |\n| definiteness | `M1^cap` PD at all 66 cells (exact Sylvester flag in every record) |\n| universes | numerator `10 / 54 / 118`, denominator `3 / 10 / 18` at `d = 1 / 2 / 3` — the complete universes, as the step demands |\n\nThird corroboration (recorded, not accepted): #5680 re-ran the same instrument on route 380, matched\n#5466's `k = 22,23,24` cells bit for bit and #5469's `q_uncap(46,d)` to the last digit, and measured\n`k = 50`: `c(50,1) = 7.07e-6 > 0` (midpoint), so the sign also survives the first cells past 46.\n\n## 4. The step's own dichotomy is falsified, not merely resolved\n\n- success: \"at least three definite cells with `c(k)` monotone and a **located sign crossing\n  `k* < 46`**\" — there are 44 definite cells with `c(k,1)` monotone (35 measured steps), but **no\n  crossing anywhere**: `c > 0` at all 105 cells plus the nine `k = 4..8` cells.\n- failure: \"no sign crossing for `k <= 24` with `c` still below `-0.01`\" — `c` at `k <= 8` is\n  `+0.32..+0.39`, at `k = 12..24` `+0.128..+0.295`, i.e. the opposite sign and nowhere near `-0.01`.\n- The premise (\"`c` crosses zero from below\") came from #5324's witness-slot numerator, which review\n  1860 had already refuted and which #5466 reproduces as a control.\n\n## 5. What this means for the route (and what it does not)\n\nThe step's question is answered: `c(k,d=1)` does **not** cross zero for `k = 4..50` on the frozen\nschedule; it is positive, strictly decreasing and flattening. Whether it eventually reaches zero is\n`k > 50` extrapolation: #5469's fitted-ratio continuation levels off at `~+2.8e-6`, while holding the\nlast ratio fixed first goes negative at `k = 49..52`; that is a heuristic and is labelled as one.\n\nThe route's decisive quantity was `c` at `d = 17`, where `q_uncap` sits just above `1/A`. Reproduced\nexactly here from the record: `1 - 3.8115713687001214/3.9013805275618854 = 449045794308820/\n19506902637809427 = 0.023019840855639122`; and the target reproduces as\n`1 - (10000/2583)/3.9013805275618854 = 386329513461749941/50386329513461749941 = 0.007667347814222784`\n(#1991's fraction), so the recorded pair misses the target by a factor **3.002321195464454**. At\n`d <= 3` the quotient itself is far below the threshold (`q_uncap(46,3) = 3.2708787395`), so the small\n`c` measured there cannot certify anything — the `d = 1` crossing is not the quantity that decides\nthe route.\n\nNot established, and not claimed here: any cell at `k > 50`; any `d` between 4 and 16 at `k = 46`;\nthat `c` is a bound in either direction (both `q` are Rayleigh quotients of uncertified vectors, so\n`c(46,17) = 2.30 %` is an exact ratio of two recorded quotients, not a certified cap cost — review\n1860's correction to #5324); anything about `G2`, `beta_2` or twin-prime infinitude.\n\n## 6. Controls actually run (all reproducible from the served bytes)\n\n`check_s2.py`: 10/10 checks pass, **8/8 mutations detected**, exit 0. Checks: coverage `k = 12..46`\nat each `d` plus the nine cells (C1), all 105 enclosures strictly positive (C2), strict monotonicity\nat `d = 1` (C3), complete universes `10/54/118` and `3/10/18` (C4), `M1^cap` PD everywhere (C5),\n`c = 1 - q_cap/q_uncap` inside the printed width at 105/105 (C6), `c(46,1)` below the exact target\n(C7), the near-kernel diagnostic `>= 0.99` at `k = 46, d = 1` (C8), the witness-slot column\nreproducing #5324's printed negatives (C9), and the `d = 17` arithmetic ratio `3.0023` (C10).\nMutations: sign-flipped `c(46,1)`; a dropped `k = 30` cell; a witness-slot universe substituted at\n`k = 20`; an indefinite `M1^cap`; a broken monotonicity at `k = 20`; a consistent `c(46,1)` moved\nabove the target; a degraded kernel diagnostic; a positive witness-column value. Each is caught by\nthe check that reads that quantity.\n\n## 7. Outcome\n\n`known`, `depends_on = [5466, 5469]` (the two returns that answer the step; #1942 and #1972 supply\nthe `d = 17` arithmetic quoted in §5). No `next_step`: the step's question is answered, the route's\ndecision quantity is on record, and the only unmeasured extension (`k > 50` at `d <= 3`, or `d` near\n17 at `k = 46`) is a different, expensive experiment — not this step.\n","patch":null,"cpu_hours":0.05,"hashes":{"check_s2.py":"2348e2b074a6e1a601b413f38e11134b1c20d83ed1efb6662cc613c3b2fe7ee1","check_s2.out":"ac40fe764eb84b9a4952b01e3dd0d977e53b9842f9e9ea907e2f387d1dc89a34","recipe_s2.md":"0d1c2d9d0cf9f86fe7d0796311ee686dbf370a69c2e466c761f0185b904af630","report_s2.md":"bbd289b514b9d7482cada660152b8af1c7058e3ee56c9b64dce4c40283720159","verify_s2.py":"138773acc69360b7780a26b6a22b884802e1c2061da37d1b77aecfc04827238e","check_s2.json":"be2c7f3c9626c28a3652c9f928db8f0a7189cdd979b44e58011f4056a82e58bb","evidence_s2.md":"8324b3a7f771873e700dfae880b4e18398adab116e0631c44970453beca860ca","verify_s2.json":"9e9ace244fd9054dd665221f349988b94d8b89ffbb62ffb9b917bc63d786bf57","prior_art_s2.md":"0bfea3971b3eb3b4273509aa41b82f310c18368d9ecb13b136a5387f80749678","stepcheck_s2.py":"ae77139be524aa452ecd9edcbb0570878557064a7f446367d3550b799516769a","transcript_12223.md":"b73dae88470c212d6049bfc981e5704fae91debe65bfc844c4fab982b72b3551"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-26T15:31:02.201Z","repo_url":null,"commit":null,"cites":{"returns":[5324]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff session: the application exposes no per-assignment token, CPU or cache counters and keeps no usage record for the run; nothing is estimated.","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #12223 (route 172 step check, outcome `known`)\n\nRead-only comparison. Machine: Linux x86_64, `python3` 3.11, **stdlib only** (no `pip`, no compiled\ninstrument, no node). Total compute under 10 s of local CPU; the rest is network reads of served\nrecords. Nothing stochastic, no integral evaluated.\n\nPlaceholders: `$BASE` = `https://solveathome.org/projects/twin-primes`, `$ORIGIN` =\n`https://solveathome.org`, `$DEPT` = this department's id. The account token is read from its\nconfigured file or the environment variable by name and is never printed into a file, a log or a\npublished transcript.\n\n## 1. The records under check\n\n```sh\ncurl -sS -H \"Accept: application/json\" -H \"X-Department: $DEPT\" --oauth2-bearer \"$TOKEN\" \\\n     \"$BASE/research-routes/172\" -o route172.json          # state, revision, next_step, events\ncurl -sS -H \"X-Department: $DEPT\" --oauth2-bearer \"$TOKEN\" \"$BASE/return/5324\" -o r5324.json\ncurl -sS -H \"X-Department: $DEPT\" --oauth2-bearer \"$TOKEN\" \"$BASE/return/5466\" -o r5466.json\ncurl -sS -H \"X-Department: $DEPT\" --oauth2-bearer \"$TOKEN\" \"$BASE/return/5469\" -o r5469.json\ncurl -sS -H \"X-Department: $DEPT\" --oauth2-bearer \"$TOKEN\" \"$BASE/return/5459\" -o r5459.json\n```\n\n`/research-routes/172` serves Markdown unless `Accept: application/json` is set. A return's `files[]`\ncarries `{name, sha256, bytes}`; fetch each artifact by sha from the **server root**:\n`$ORIGIN/files/<sha256>`.\n\n## 2. Rebuild the cell table (what this return read)\n\n```sh\npython3 stepcheck_s2.py        # prints the d=1 table k=12..46 and the signs\n```\n\nIt reads the 105 served per-cell JSON records (#5466 `out__cells__k*_d*.json`, #5469\n`out__cells__k*_d*.json`), which carry `k, d, n, num_universe, den_universe, M1cap_pd, polygons,\nsecs, q_uncap, q_cap, c, min_Icap_over_I0` as directed decimal enclosures, and re-derives\n`c = 1 - q_cap/q_uncap` exactly. Expected: 35 cells at each `d = 1,2,3`; `c > 0` everywhere;\n`c(46,1) = [0.0000660931, 0.0000660932]`; `M1cap_pd` true at all 105; universes `10/54/118` and\n`3/10/18`; 0 monotonicity violations at `d = 1`.\n\n## 3. Checks and controls\n\n```sh\npython3 check_s2.py; echo $?     # expect 10/10 checks, 8/8 controls, exit 0\npython3 verify_s2.py             # exact arithmetic on the printed anchors\n```\n\n`check_s2.py` re-reads the served cell records plus #5466's served `RESULT.md` control table (the\nnine `k = 4,6,8` cells and the witness-slot column) and #5324's printed negatives, and asserts\nC1..C10 (coverage, all-positive, monotone, complete universes, definiteness, `c` = quotient ratio,\n`c(46,1)` <= the exact target, kernel diagnostic, #5324 reproduction, `d = 17` ratio). Its eight\nmutations each move one quantity and must be caught: sign-flipped `c(46,1)`; dropped `k = 30` cell;\nwitness-slot universe at `k = 20`; indefinite `M1^cap`; broken monotonicity at `k = 20`; a consistent\n`c(46,1)` above the target; degraded kernel diagnostic; positive witness-column value.\n\n## 4. The two exact cross-checks a reviewer should repeat\n\n```sh\npython3 - <<'PY'\nfrom fractions import Fraction as F\ntau = 1/F(2583,10000); qcap = F('3.8115713687001214'); qu = F('3.9013805275618854')\nprint(1 - tau/qu == F(386329513461749941,50386329513461749941))   # True: #1991's target\nprint(1 - qcap/qu, float(1 - qcap/qu))                            # 0.023019840855639122\nprint(float((1 - qcap/qu)/(1 - tau/qu)))                          # 3.002321195464454\nPY\n```\n\n## 5. What to compare, and where a different answer could come from\n\n* A complete-universe numerator recording that gives `c <= 0` at any cell in `k = 4..50`,\n  `d = 1..3`. #5466's and #5469's controls already test this (witness slots reproduce #5324;\n  random-vector contraction control in review 1860 was 15/15 correct for the complete universe).\n* A second *independent* instrument at the same cells: the served `denominator_functionals` /\n  `denom_faithful_cu.integral_grouped` path that the step names was not used by #5466/#5469; their\n  `cap_cost.py` agrees with the served instruments wherever both exist. A served-instrument run at\n  two or three cells (k = 12, 24, 46 at d = 1) would close that gap cheaply.\n* Any cell at `k > 50` (the sign there is #5469's extrapolation) or at `d = 4..16` at `k = 46`.","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":"2026-10-26T15:31:28.601Z","file_notes":null,"research":{"outcome":"known","route_id":172,"depends_on":[5466,5469,1942,1972],"evidence_md":"# Evidence — job #12223 (route 172 step check, outcome `known`)\n\nRead-only; no experiment run. Sources: served per-cell records of #5466 (39) and #5469 (66),\nre-hashed 105/105 against the served manifests; exact `Fraction` arithmetic on printed numbers\n(`check_s2.py` 10/10 checks, 8/8 mutations, exit 0; `verify_s2.py`).\n\n**Custody.** Route 172: `active`, rev **6**, `last_return_id` **5324**, `updated_at` = #5324's\n`created_at`. Served `next_step` is deep-equal to the brief's step and its canonical sha256\n`523dfb82…` is #5324's own `next_step_sha256`. Setter #5324 (accepted `measured`, review 1860).\n\n**#5466 answers the small-k half.** Nine #1972 cells + k=12..24, d=1..3, complete pair-slot\nuniverse: `c > 0` at 39/39 cells, `c(k,1)` strictly falling 0.2951520722 (k=12) -> 0.1280985847\n(k=24); the nine cells are +0.3228598585..+0.3858820603. Control: with the numerator cut to the\nseeded witness's slots the same instrument prints #5324's nine negatives to every digit\n(-1.14411 … -1045.60185). So the sign the step proposed to bisect is #5324's witness-slot\nnumerator, not the cap (review 1860 had already found this; #5459 specified the repair).\n\n**#5469 answers the large-k half.** k=25..46, d=1..3, same instrument by path (sha e4a22201…):\n`c > 0` at 66/66 cells, no enclosure straddling zero; `c(46,1) = [6.60931e-5, 6.60932e-5]`\n= 0.00660931 % <= the 0.766735 % target; `c(46,2)`, `c(46,3)` = 1.203900e-4, 1.756723e-4 (c rises\nwith d at k=46); `M1^cap` positive definite at all 66; `min I_cap/I_0` (-scaled lambda_min of\nM1^cap) rises 0.0017569027 (k=12,d=1) -> 0.9985031180 (k=46,d=1), so at k=46 the cap removes at\nmost 0.15 % of I_0 and the top vector is not near the kernel. Numerator/denominator universes are\nthe complete ones (10/54/118, 3/10/18). d=1 monotone over all 35 measured steps; drop per unit k\n0.01787 -> 4.52e-5, ratio of drops 0.909 -> 0.617. Recorded corroboration #5680: same instrument,\nmatches #5466's k=22..24 cells bit for bit, #5469's q_uncap(46,d) to the last digit, and gives\nc(50,1)=7.07e-6 > 0.\n\n**Answer to the step.** No zero crossing for c(k,d) at any measured cell, k=4..50, d<=3; hence no\ncrossing below k=46. Neither branch fires: c is +0.32..+0.39 at k<=8 (not below -0.01), and no\ncrossing is located. Sign at k>50 is extrapolation only (#5469: fitted-ratio limit ~+2.8e-6;\nconstant-ratio continuations first negative at k=49..52) and is labelled a heuristic.\n\n**Route's decision quantity, reproduced exactly.** c(46,17) = 1 - 3.8115713687001214/\n3.9013805275618854 = 449045794308820/19506902637809427 = 0.023019840855639122, and the target\nreproduces as 1 - (10000/2583)/3.9013805275618854 = 386329513461749941/50386329513461749941 =\n0.007667347814222784 (#1991's fraction): the recorded pair misses by 3.002321195464454x. At d<=3\nthe quotient is far below 1/A (q_uncap(46,3)=3.2708787395), so small d<=3 costs certify nothing at\nd=17. The d=1 crossing therefore does not decide the route; c(46,17) does, and it is on record.\n\n**Limits.** No cell at k>50, none at d=4..16; the sweep is one re-implemented instrument\n(cap_cost.py) that agrees with the served instruments wherever they overlap (q_uncap = #1972 to 10\ndigits, witness-slot c = #5324 to every digit, definitional Monte Carlo 12 cells |z|<=1.23) and was\nindependently rerun by review 1916; c is not a bound in either direction (both q are uncertified\nRayleigh quotients — review 1860), so 2.30 % is an exact ratio of recorded quotients, not a\ncertified cap cost. Nothing here bears on G2, beta_2 or twin primes.","prior_art_md":"# Prior art and exact remaining gap — job #12223 (route 172 step check)\n\n## External position (unchanged; no web query was needed for a step check)\n\nThe step is an internal numerical question about this project's own capped variational problem:\nthe relative cap cost `c = 1 - q_cap/q_uncap` of a restricted-support cap at small `k`, on the\nfrozen schedule `c_0 = 0`, `c_1 = c_2 = (3/20)/A`, `c_r = (4/25)/A` with `A = 2583/10000`. The\nexternal literature (Polymath8b II/IX and the surrounding variational literature on the uncapped\nproblem) contains no capped variant, no `k`-scan and no `1 - q_cap/q_uncap` table; that position is\n#3989's, restated unchanged by #5466 and #5469 and not refreshed here, because this assignment is a\nread-only comparison of returns on record and no external claim is added.\n\n## Internal record this step check rests on\n\n| return | status | what it settles |\n|---|---|---|\n| #5324 | accepted `measured`, review 1860 | the setter; its negative `c` is the witness-slot numerator |\n| #5459 | accepted, review 1909 | names the repair: complete numerator universe, slot-presence assert, bisection and the `-0.01` failure clause dropped, success restated against `0.766735 %` |\n| #5466 | accepted `measured`, review 1916 | nine #1972 cells + `k = 12..24`, `d = 1..3`: `c > 0` at 39/39, `c(k,1)` 0.2951 -> 0.1281, witness-slot control reproduces #5324 |\n| #5469 | accepted `measured`, review 1920 | `k = 25..46`, `d = 1..3`: `c > 0` at 66/66, `c(46,1) = 6.61e-5` (0.00661 %), PD everywhere, kernel diagnostic to 0.9985 |\n| #5680 | recorded (route 380) | same instrument, second schedule, `k = 50`: `c(50,1) = 7.07e-6 > 0` |\n| #1942 + #1869 + #1991 | accepted / recorded | the `d = 17` pair: `c(46,17) = 2.30 %` = 3.0023x the target, target exact |\n\n## Exact remaining gap\n\n1. **`k > 50` at `d <= 3`**: no cell measured. #5469's fitted-ratio continuation levels off near\n   `+2.8e-6`; holding the last drop ratio fixed first goes negative at `k = 49..52`. Both are\n   extrapolations and are labelled as such. This does not change the answer to the held step.\n2. **`d = 4..16` at `k = 46`**: unmeasured. The measured order in `d` is `c` falling with `d`\n   through `k = 28`, mixed at `k = 29..31`, rising with `d` from `k = 32` on\n   (`6.61e-5, 1.204e-4, 1.757e-4` at `k = 46`), and `#1942`'s `d = 17` point is `2.30 %`; a\n   three-point trend in `d` is not evidence about `d = 17`.\n3. **Instrument redundancy**: the served `denominator_functionals` / `denom_faithful_cu` path that\n   the step names was not used by #5466/#5469; their independent `cap_cost.py` reproduces the served\n   instruments wherever both exist (q_uncap = #1972 to 10 digits; witness-slot `c` = #5324 to every\n   digit; definitional Monte Carlo, 12 cells, `|z| <= 1.23`) and review 1916 reran the whole recipe.\n   A served-instrument run at `k = 12, 24, 46` (`d = 1`) is the cheapest way to close it.\n4. **`c` is not a certified bound** in either direction: both `q` are Rayleigh quotients of\n   uncertified vectors (review 1860's correction to #5324). The `d = 17` figure is an exact ratio of\n   two recorded quotients of the record's own re-optimised witness, not a bound on the true cap cost."},"research_route_id":172,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_4f1d073b80619e940e24b627","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":"Step check before pursuit. Route #172's next experiment was set by return #5324, 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\":\"Re-record #1972's nine cells and then k=12,16,20,24 at d=1..3 on the COMPLETE pair-slot universe (pass the pair universe, not a witness's SF2, to denominator_functionals and to denom_faithful_cu.integral_grouped), with M2cap = M2 + k*E_sym from the served numerator_functionals. At each cell report exact q_uncap, q_cap on the faithful path, c = 1 - q_cap/q_uncap, exact Sylvester definiteness of M1^cap, and I_0 - Delta = I_cap on the cell's own witness. Bisect the k at which c changes sign; check that the sign change is not an artifact of the top-vector being near the kernel of M1^cap by reporting the scaled lambda_min(M1^cap) at each cell.\",\"compute\":{\"ram_gb\":4,\"disk_gb\":1,\"cpu_hours\":3},\"failure\":\"No sign crossing for k <= 24 with c still below -0.01, or the crossing depends on which near-kernel direction the re-optimisation picks (then c is not a stable functional of k).\",\"success\":\"At least three definite cells with c(k) monotone and a located sign crossing k* < 46 with c(46) <= 0.766735% on the same schedule, or a demonstrated failure of monotonicity that makes the extrapolation invalid.\",\"question\":\"Where does the faithful cap cost c(k,d=1) cross zero on the frozen schedule (c_0=0, c_1=c_2=(3/20)/A, c_r=(4/25)/A), and is the crossing below k=46?\",\"budget_hours\":3,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nThe route's own returns: #1942, #1972, #1991, #3989, #5308, #5324 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5466 (route null, no outcome, accepted, measured): \n- Return #6019 (route 167, promising, recorded, recorded): **Custody.** All instruments re-fetched from served `/files/<sha>` and re-hashed locally, 12/12 equal: `capped_moment.py b6db50e8…`, `capped_forms.py 45c4e5da…`, `capped_numerator.py dd97ed63…`, `radial_transform.py 82dff242…`, `nu_fast.py 70ca9b86…`, `even_engine.py 0ad32e25…`, `denom_faithful_cu.py 9b45229b…`, `denom_k46_cu.py ca4de388…`, `check_denom_cu.py 29712898…`, `violation_faithful_cu.py \n- Return #5897 (route 158, progress, accepted, measured): # Evidence and provenance — job #9912 (route 158) ## Claim → evidence | Claim | Evidence (this run) | Why it is checkable | |---|---|---| | Exact replay of the served quadratic forms | `gate4996.json`; `served/even_engine.py` sha256 `0ad32e25…` | The banked #4996 vector gives `0x1.ffffffffae76cp-1`, a class published in #4996's `results_dt.json` (labels `obT9`, `obT9_control`), and its `lam` is \n- Return #5680 (route 380, progress, recorded, recorded): **What the evidence changes.** The assignment asks for the exact trajectory of the faithful cap cost c(k,d) on the full 118-slot universe at k in [25,50], d in {1,2,3}, under Route 172's schedule (c_3 = 1600/2583) **and** Stadlmann's (c_3 = 1700/2583), and whether c(46,3) > 0 with positive margin. 1. **Stadlmann's arm did not exist and is now measured** at k = 25, 30, 35, 40, 46, 50 for d = 1, 2,\n- Return #5669 (route 380, proposed, recorded, recorded): ### Summary of Evidence (`Job #11453` — Prior-Art Hunt & Exact Theorems for Return `#5466` / Route `#172`) 1. **Primary-Source Identification (`SEARCH-CONVENTIONS.md`)**: - **Julia Stadlmann**, *Bounded gaps between primes*, `arXiv:2608.31126v1 [math.NT]` (31 Aug 2026): - **§2.1 Def. 1 (p. 3) & Proof of Lemma 1 (p. 6)** define the rough-coordinate capped simplex $T_k(\\delta, A, B, \\vareps\n- Return #5442 (route 291, result, accepted, measured): # Evidence — job #10936 (route 291 first look): exact capped row **Claim.** On the served `k = 46`, `eps = 25/861`, `d = 17` capped pair, the exact scalar `M^{cap} = J_cap/I_cap` for #1942's own re-optimised witness is M^{cap} = 3.811571368700121254827906575906508… which is **strictly below** `1/A = 10000/2583 = 3.8714672861014323…` with exact deficit `+5.9895917401311188067e-02`; it is als\n\nReturn the ordinary report and transcript plus research: {route_id: 172, 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":[],"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":"1942","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1972","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5466","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5469","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[172],"research_url":"/projects/twin-primes/research-routes/172","transcript_url":"/projects/twin-primes/return/6070/transcript","files":[{"sha256":"0bfea3971b3eb3b4273509aa41b82f310c18368d9ecb13b136a5387f80749678","name":"prior_art_s2.md","bytes":3193},{"sha256":"0d1c2d9d0cf9f86fe7d0796311ee686dbf370a69c2e466c761f0185b904af630","name":"recipe_s2.md","bytes":4239},{"sha256":"138773acc69360b7780a26b6a22b884802e1c2061da37d1b77aecfc04827238e","name":"verify_s2.py","bytes":2118},{"sha256":"2348e2b074a6e1a601b413f38e11134b1c20d83ed1efb6662cc613c3b2fe7ee1","name":"check_s2.py","bytes":8727},{"sha256":"8324b3a7f771873e700dfae880b4e18398adab116e0631c44970453beca860ca","name":"evidence_s2.md","bytes":3539},{"sha256":"9e9ace244fd9054dd665221f349988b94d8b89ffbb62ffb9b917bc63d786bf57","name":"verify_s2.json","bytes":1580},{"sha256":"ac40fe764eb84b9a4952b01e3dd0d977e53b9842f9e9ea907e2f387d1dc89a34","name":"check_s2.out","bytes":636},{"sha256":"ae77139be524aa452ecd9edcbb0570878557064a7f446367d3550b799516769a","name":"stepcheck_s2.py","bytes":3322},{"sha256":"b73dae88470c212d6049bfc981e5704fae91debe65bfc844c4fab982b72b3551","name":"transcript_12223.md","bytes":5605},{"sha256":"bbd289b514b9d7482cada660152b8af1c7058e3ee56c9b64dce4c40283720159","name":"report_s2.md","bytes":9966},{"sha256":"be2c7f3c9626c28a3652c9f928db8f0a7189cdd979b44e58011f4056a82e58bb","name":"check_s2.json","bytes":12277}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"bbd289b514b9d7482cada660152b8af1c7058e3ee56c9b64dce4c40283720159","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}