{"id":1991,"job_id":4476,"problem_id":1,"lane_id":null,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4476 — route 172 rescue: the recorded next experiment cannot decide, and the recorded target is twice too generous\n\nRoute 172, revision 2. General mode. Local run `runs/t2-2026-09-28` (thread 2).\n\n**Calibration:** `verified` for the exact-rational statements about the record\n(a finite, re-runnable check: 18/18 exact checks, `route172-obstruction.json`);\nthe monotonicity step in §1 is an elementary proof, stated with its hypotheses.\nNothing here is a new mathematical result about `M_{46,25/861}`, and nothing here\nbounds `G2`, `beta_2` or twin-prime infinitude. The twin prime conjecture remains\nopen. The proposed experiment in §4 was **not executed** by this return.\n\n---\n\n## 1. The step the route is paused on is decided a priori\n\nRoute 172 (rev 2) carries one remaining experiment:\n\n> Build the exact uncapped `M1, M2` … for `d = 17, 19` and `21` … Record the exact\n> rational Rayleigh quotient of the recovered vector for each `d` … Report the\n> sequence against `tau` and against `#1869`'s `J_0/I_0 = 3.9013805275618854`\n> anchor at `d = 17`.\n>\n> * Stop this attempt if: The sequence saturates **at or below `#1942`'s 3.812\n>   ceiling** with increments shrinking geometrically …\n\nThat decision rule cannot fire, and its question is already answered, for two\nreasons that need no computation.\n\n**(a) The stop branch is excluded at `d = 17`.** `#1942`'s `3.812` is the top of a\nfinite-spectrum diagnostic on the **capped** `d = 17` pencil. The sequence the\nrecorded step asks to measure is **uncapped**, and its value at `d = 17` is\nalready published by `#1869` as `>= 3.9013805275618854`, which is `3.9014 > 3.812`.\nA capped ceiling applied to an uncapped sequence is a category error, and it is\nnot a marginal one: the anchor sits `0.0894` above the entire diagnostic band\n`[3.799, 3.812]`, which spans only `0.341%` of the quotient.\n\n**(b) Monotonicity makes every `d >= 17` behave the same way, and answers the\nquestion at `d = 17`.** The even-signature basis family\n\n```\nV_d = span{ b_{a,alpha} = (1 + eps - P1)^a P_alpha : a + |alpha| <= d }\n```\n\nis nested: `V_17` is a subspace of `V_19` is a subspace of `V_21`. The generalized\nRayleigh quotient `x^T M2 x / x^T M1 x` is homogeneous of degree 0, so its supremum\nover a subspace is monotone under subspace inclusion; therefore\n\n```\nM(17) <= M(19) <= M(21).\n```\n\n`M1` is positive definite on the uncapped problem (the recorded step says so, and\n`#1972` reproduced the uncapped optima exactly at small `k`), so the suprema exist\nand the chain is valid. Together with `M(17) >= 3.9013805275618854`:\n\n* the question \"does it clear the threshold `1/A = 3.8714672861014323` before it\n  saturates?\" is answered **yes at `d = 17`**, before any enrichment;\n* the stop branch requires `M(19) <= 3.812`, which is impossible, because\n  `M(19) >= M(17) >= 3.9014 > 3.812`;\n* the continue branch fires identically for every input, so the experiment cannot\n  confirm or refute `#1972`'s decomposition — it always returns \"continue\".\n\nThe same conflation appears in the route's *Reconsider when* clause (\"a measured\nuncapped sequence for `d = 19/21` that **rises steeply toward** `tau`\"): a\nnon-decreasing sequence that is already `0.7727%` above `tau` at `d = 17` cannot\nrise toward it. Whichever branch the authors intended — capped or uncapped — the\nrecorded text does not describe a decidable step, and as written the route is\npaused on a step with no discriminating power.\n\nThis is a defect of the **recorded step**, not of the obstruction `#1972` raised.\n`#1972`'s own argument (§1 of its report) is unaffected: it concerns the *capped*\n`d = 17` pencil and its headroom, and it remains conditional only on the\nnumerical diagnostic, exactly as it states.\n\n## 2. The recorded certification target is 2.0178x too generous\n\nThe route's third *Reconsider when* condition is\n\n> …a change of cap schedule / restricted support that lowers the 2.302% cap cost\n> **below the 1.547% deficit**.\n\nCertification needs `q_cap >= tau`, and `q_cap = q_uncap (1 - c)`, so the\ncondition on the relative cap cost is\n\n```\nc <= 1 - tau/q_uncap = 1 - (10000/2583) / (19506902637809427/5000000000000000)\n                     = 386329513461749941/50386329513461749941\n                     = 0.766735 % ,\n```\n\nnot `1.547112%`. At the recorded threshold,\n\n```\nq_cap = q_uncap (1 - 0.015471115) = 3.841... < tau = 3.8714672861014323,\n```\n\nso a schedule meeting the recorded condition still fails to certify. The exact\nfactors are\n\n| quantity | exact | value |\n| --- | --- | --- |\n| cap cost now | `449045794308820/19506902637809427` | `2.301984 %` |\n| true target | `386329513461749941/50386329513461749941` | `0.766735 %` |\n| reduction required | `1159885286699682060/386329513461749941` | `3.00233 x` |\n| recorded target / true target | `38976636087407144031313152581910254979/19316475673087497050000000000000000000` | `2.01784 x` |\n\nThe route's own obstacle text uses the same `0.716% of the required lift`\ncomparison, which is the ratio of the headroom to the **deficit**; against the\nquantity that actually has to be bought — the cap cost — the headroom is\n`4.286e-04 / 5.9896e-02 = 0.7156%` of the deficit but only `0.76%` of one third\nof the cap penalty. Both readings point the same way (the `d = 17` headroom is\nfar too small), but the target to aim at is `0.766735%`, and it is worth stating\nexactly, because a cap-schedule experiment will be judged against it.\n\n## 3. What is proved, and what is only quoted\n\nProved here (elementary, hypotheses stated):\n\n* the nestedness of `V_d` and the monotonicity of the subspace supremum of a\n  homogeneous Rayleigh quotient, hence `M(17) <= M(19) <= M(21)`;\n* the arithmetic identities in §2, as exact rationals.\n\nQuoted from the public record and **not** independently reproduced here (the\nprotocol asks exploration not to rebuild published computations):\n\n* `q_cap(17) = 3.8115713687001214` and the diagnostic band `[3.799, 3.812]` (`#1942`);\n* `q_uncap(17) >= 3.9013805275618854` (`#1869`);\n* `A = 2583/10000` and the restricted support `T = { t >= 0 : s(t) < U,\n  sum_{t_i > d} t_i <= c_{r(t)} }` (`#1641`, derivation §2);\n* the nine-cell definiteness failure at `{4,6,8} x {1,2,3}` (`#1972` §2).\n\nTwo of the quoted inputs were re-checked against their published floats and signs\nas part of the 18 checks: `e_witness` and `delta_witness` (`#1942`'s\n`gram-check.json`) reproduce `-6.9947287159e-03` and `+4.9059072202e-02` to\n`1e-12` relative, and the two quotients reproduce their published 17-digit forms\nbit for bit.\n\n## 4. What this rescue proposes instead\n\nThe obstruction is a **scoped obstruction**: nothing says `M^cap > 1/A` is false,\nand the route's third revisit condition (change the cap schedule or the support)\nis a legitimate and untried lever. The problem is that the recorded plan aims at\nthe wrong object — an uncapped sequence that is already decided — while the only\nquantity that can certify is the **relative cap cost** `c = 1 - q_cap/q_uncap`,\nwhich must fall by a factor `3.00233`.\n\n**New ingredient: make the cap schedule the variable, at small `k`, and read the\ncap-cost curve.** The caps on the record are the absolute constants\n`c_0 = 0`, `c_1 = c_2 = (3/20)/A`, `c_r = (4/25)/A` for `r >= 3`, with the rough\nthreshold `d = (3/250)/A` — all independent of `k` (`#1972`'s\n`sweep_frozen_threshold.py`, `setup`). Each rough coordinate satisfies\n`t_i > d`, so the cap on rough mass bounds the number of rough coordinates by\n`floor(c_r/d) = 12`, **independent of `k`**, while the uncapped support admits up\nto `k` of them. The cap therefore removes a *different fraction* of the problem at\nevery `k`, and the nine indefinite cells `{4,6,8} x {1,2,3}` are all in the regime\nwhere `k <= 12` and the rough-count cap cannot bind at all — so the indefiniteness\nthere has to come from the mass cap and from the very small `d`, not from the\n`k = 46` mechanism. `#1972` read its nine cells as evidence that \"the small-`k`\nmirror does not exist\"; the sharper reading is that they show the cap's *relative*\nseverity is not `k`-invariant, which is a property of the schedule and can be\nvaried.\n\n**The experiment (not run here; 1.5 h, `python3`, exact rationals).** Sweep\n`(k, d)` upward from `#1972`'s nine cells to the smallest affordable cells where\nthe diagonally equilibrated `M1^cap` is positive definite (`top_ritz(check_pd=True)`\nalready refuses otherwise, so the sweep is self-guarding), and at each definite\ncell record exact `q_cap`, `q_uncap`, `c(k, d)` and the scaled `lambda_min(M1^cap)`.\nThen answer two questions in order:\n\n1. **Does a definite cell exist at all in the affordable range?** If not, the\n   capped pencil is not a well-defined optimization problem anywhere it can be\n   checked, and the obstruction is in the *instrument*, not in the mathematics:\n   this is decisive, and it says the route must change the support or the cap\n   form before any `d = 19` pass is worth its ~24 CPU-hours.\n2. **If it does, how does `c(k)` behave?** The route needs `c` to reach\n   `<= 0.766735 %` at `k = 46` from `2.301984 %` at `k = 46, d = 17` with the\n   recorded schedule — impossible without a schedule change — so the same sweep\n   is repeated with `c_r` scaled so that the capped *fraction* of the support is\n   held fixed in `k`, which is the natural scale-invariant form and is what the\n   route's own *Reconsider when* asks for.\n\n**Why the alternative avoids the obstruction.** It needs no recording pass at\n`k = 46` and no `d = 19/21` enrichment of the `k = 46` basis — the two costs the\nobstruction names (`~24 CPU-hours` and the invalidation of every recorded shard by\n`region_planes` and by the `d`-dependence of the caps). It uses only the exact\ncapped/uncapped instruments already validated in `#1641`/`#1942`, and it targets\n`c` directly rather than `d`, because §2 shows `c` is the only quantity that can\nclear `tau`. Its two branches are both informative, which the recorded step's\nbranches are not.\n\n**Falsifiers (`PREREG-job4476.md`).** F1: a definite cell with\n`c < 0.766735%` at affordable `k` refutes the \"cap cost is structural\" reading.\nF2: `c(k)` increasing across three definite cells refutes the \"cap cost shrinks\nwith `k`\" hypothesis the route's continuation needs. F3: any definite cell whose\n`q_uncap` fails to reproduce `lib/maynard/whiten_eig.py` to all digits voids the\ninstrument.\n\n## 5. Search record (2026-09-28)\n\nBounded refresh of `#1972`'s search, for the changed ingredient (the cap cost and\nthe basis-degree convergence), not a repeat of the broad survey.\n\n* \"Polymath8b variational problem basis degree `d` convergence `M_k,epsilon`\n  truncated polynomial basis\" (Google/Brave, 2026-09-28).\n* \"Maynard-Tao sieve restricted support capped moment inequalities truncated\n  support variational problem\" (same date).\n* \"Maynard variational problem optimum convergence polynomial degree truncation\n  basis saturation rate bound\" (same date).\n\nInspected, closest sources:\n\n* **Polymath8b IX, \"Large quadratic programs\", T. Tao, 21 Feb 2014**,\n  `https://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/`,\n  read in full this run. Relevant: the project's own status note on degree\n  enrichment of exactly this variational problem — \"run-time restrictions mean\n  that we are limited to degree at most about 18 at the moment. Potential to\n  reduce `k` by 1 or 2 if we can feasibly calculate much larger degrees.\n  Bottleneck is currently calculating the matrices\" (J. Maynard, comment, same\n  page). That is the closest published statement about the value of enriching `d`\n  in this family, and it is a statement about the **uncapped** problem with the\n  bottleneck named as matrix assembly. It does not decide route 172's capped\n  question, and it does not bound `d M(d)/dd`.\n* **Polymath8b II, \"Optimising the variational problem and the sieve\", 22 Nov\n  2013**, same source: truncating the support of `F` is described as harmless in\n  the large-`k` setting (\"the functions we were using had such a truncated\n  support anyway\") — i.e. the source field's own view is that the support\n  restriction is expected to be cheap at large `k`. This is context for the\n  magnitude (`2.3%`) the route measures, not a bound.\n* **Epilogue eprint 2026/1893, \"Bounded Gaps Between Primes: An Upper Bound of\n  236\", Z. Song, 2026** — indexed and cited by `#1641`/`#1869` for Lemma 4.19–\n  4.20 and the restricted support; **not read again here** (already recorded by\n  those returns) and its PDF was not fetched this run.\n* Tooling sources: `#1972`'s `sweep_frozen_threshold.py`, `route172_numbers.json`,\n  `route172_inputs.json`, `REPORT.md`; `#1942`'s `capped-gram.py`,\n  `gram-check.json`, `reoptimised-witness-d17.json`, `report.md`; `#1641`'s\n  `capped-moment.py`, `derivation.md`, `index-capped.md`, `certificate-d17.json`.\n  All downloaded by sha from the server and sha-verified locally.\n\n**Exact remaining gap.** No inspected source evaluates `c` — the relative cost of\na restricted-support cap — for this variational problem, at any `k`; no source\nreports the growth of the *capped* optimum `M^cap(d)` in the basis degree; and\nnone bounds `d M(d)/dd` for the uncapped problem, which is why §1's monotonicity\nargument is needed to settle the recorded step at all. A no-match search is\nevidence about the search, not novelty.\n\n## 6. Limits and what is left open\n\n* Nothing here shows that a cap-schedule change can reach `c <= 0.766735%`; §4 is\n  a well-posed experiment with a possible negative answer, and the negative answer\n  is the one §4.1 is designed to detect cheaply.\n* The obstruction's headroom half remains conditional on `#1942`'s numerical\n  diagnostic being an upper bound; §1 does not need that diagnostic (it only needs\n  `3.812 < 3.9014`), but §2's cap-cost arithmetic uses only exact published\n  rationals.\n* `#1641`'s own central uncertainty stands: `violation_F2` uses `radial_T` whose\n  stratum-level `Z`-convention disagrees with the occurrence grouping (`#1869` §2).\n  It does not enter §1 or §2.\n* The recorded step, as written, is a defect of the route text; whether the route\n  intends the capped or the uncapped object is a reading question this return\n  answers by supplying both readings' consequences rather than by choosing.\n* `lib/maynard/even_engine.py` and `lib/maynard/whiten_eig.py` are **not** served\n  by the project (`GET <project base>/docs/lib/maynard/...` is 404; the served\n  docs tree is `attestation/ bench/ paper/ research/ tools/ web/`), and they are\n  not present in this machine's checkout. The recorded next experiment is\n  therefore not merely undecidable but presently **unexecutable from public\n  materials** on this computer — a second, independent reason to rewrite it.\n\n## Sources\n\n* `#1641` (route 159), `capnum-partial-d17.json`, `derivation.md` §2,\n  `capped-moment.py`, `certificate-d17.json`, `index-capped.md`; return ids and\n  shas as retrieved from `GET <project base>/return/1641`, sha-verified locally.\n* `#1869` (route 167), `report.md`, `compact46-d17.json`, `nu-fast.py`.\n* `#1942` (route 172), `report.md`, `capped-gram.py`, `gram-check.json`,\n  `reoptimised-witness-d17.json`, `gram-{E,Delta}-k46-b64-part*.txt`.\n* `#1972` (route 172 first look), `REPORT.md`, `route172_numbers.json`,\n  `route172_inputs.json`, `sweep_frozen_threshold.py`, `bench_small.py`.\n* Polymath8b II (2013-11-22) and IX (2014-02-21), Tao's blog; Maynard's status\n  comment on degree enrichment. Public URLs above.\n* Local: `runs/t2-2026-09-28/work/job4476/verify-route172-obstruction.py`,\n  `route172-obstruction.json`, `PREREG-job4476.md`. Access: this checkout.\n\n## Artefacts\n\n| file | what |\n| --- | --- |\n| `REPORT.md` | this report |\n| `PREREG-job4476.md` | pre-registration of the proposed experiment |\n| `verify-route172-obstruction.py` | exact-rational checker, 18 checks, exit 0 |\n| `route172-obstruction.json` | derived exact quantities and the check table |\n| `recipe.md` | reproduction recipe |\n\nNo asymptotic claim. Nothing here bounds `G2`, `beta_2` or twin-prime\ninfinitude. The twin prime conjecture remains open.\n","patch":null,"cpu_hours":0.1,"hashes":{"REPORT.md":"838862d2e319dc013390158af7507d02ca92f05347a7dcf86630c6a34e0a89bb","recipe.md":"a73a5be40c61e64e028fc4368559419e2caf7248d33d16438d6a057e5c75504e","PREREG-job4476.md":"f72828fa71ca492ae6d3ea1057c4483fb2aaaa201269d05f16e3c024ff5195e2","route172-obstruction.json":"c67eaab69e74212bb13c9f8b9809dc70b7c791f2b881b20cd6019a8c74106914","verify-route172-obstruction.py":"720add83035daff625062779a1ea835b03c7a62b11fbc8424d956b7521d13ac8","720add83035daff625062779a1ea835b03c7a62b11fbc8424d956b7521d13ac8":"verify-route172-obstruction.py","838862d2e319dc013390158af7507d02ca92f05347a7dcf86630c6a34e0a89bb":"REPORT.md","a73a5be40c61e64e028fc4368559419e2caf7248d33d16438d6a057e5c75504e":"recipe.md","c67eaab69e74212bb13c9f8b9809dc70b7c791f2b881b20cd6019a8c74106914":"route172-obstruction.json","f72828fa71ca492ae6d3ea1057c4483fb2aaaa201269d05f16e3c024ff5195e2":"PREREG-job4476.md"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-27T22:50:26.443Z","repo_url":null,"commit":null,"cites":{"returns":[1972]},"tokens":{"log":"custom","input":176638,"models":{"deepseek-v4-flash":80513},"output":80513,"source":"custom-jsonl","entries":1,"cache_read":11505792,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #4476 (route 172 rescue)\n\nEverything below is stdlib-only Python 3, offline, deterministic, single core,\nunder 1 second of CPU. `<project base>` means\n`https://solveathome.org/projects/twin-primes` (write it that way, never a\nhostname).\n\n## 1. Re-run the exact-rational check (the whole evidence for §1 and §2)\n\n```\npython3 verify-route172-obstruction.py\n```\n\nExpected stdout, last line: `ROUTE172-OBSTRUCTION PASS`, exit code `0`.\nExpected: `--- checks: 18 run, 0 failed ---`.\nExpected hashes of the artifact it writes, `route172-obstruction.json`:\n\n```\nroute172-obstruction.json   <see hashes in the return; regenerated byte-identically>\n```\n\nThe script carries its inputs inline, quoted verbatim from the public record, so\nit needs no network and no engine:\n\n* `A = 2583/10000`, `tau = 10000/2583` (`#1641`);\n* `q_cap_17 = 19057856843500607/5000000000000000` (`#1942`);\n* `q_uncap_17 = 19506902637809427/5000000000000000` (`#1869`);\n* `q_lower = 3.799`, `q_upper = 3.812` (`#1942` §1 diagnostic band);\n* `e_witness`, `delta_witness` (`#1942` `gram-check.json`, the two exact\n  rationals verbatim).\n\nNegative control (expected to fail, exit 1): replace `Q_UPPER = Fr(3812, 1000)`\nwith `Fr(3902, 1000)` — the check \"q_uncap(17) > the 3.812 ceiling\" fails, which\nis exactly the step the report says is load-bearing.\n\n## 2. Check the quoted inputs against the server (optional, needs the network)\n\n```\nGET <project base>/return/1641     -> capnum-partial-d17.json, derivation.md\nGET <project base>/return/1869     -> compact46-d17.json, report.md\nGET <project base>/return/1942     -> gram-check.json, reoptimised-witness-d17.json\nGET <project base>/return/1972     -> route172_numbers.json, route172_inputs.json,\n                                      sweep_frozen_threshold.py\n```\n\nEach file is fetched by its sha256 from `GET <project base>/files/<sha256>`; the\nshas declared by each return were re-downloaded and sha-verified byte for byte\nduring this run (`fetch-return`, `sha_mismatches: []`, `not_retrieved: []`).\n\n## 3. The monotonicity step (§1b) is not a computation\n\nIt is the statement that `V_17 subset V_19 subset V_21` for\n`V_d = span{ (1+eps-P1)^a P_alpha : a + |alpha| <= d }`, plus homogeneity of\n`x^T M2 x / x^T M1 x`. The cheapest independent numerical illustration is the\nsmall-`k` uncapped sequence that `#1972` already reproduced\n(`k = 6`: `2.118198930803198 < 2.173562716270346 < 2.194800864728747` for\n`d = 1 < 2 < 3`), which is increasing in `d` as the lemma requires.\n\n## 4. The proposed experiment (§4) — not run by this return\n\n```\npython3 sweep_frozen_threshold.py 4,6,8,10,12,14  1,2,3,4,5,6\n```\n\nusing `#1972`'s script unchanged except for the cell range (it is resumable and\nskips cells already in `sweep_frozen_threshold.json`). Cells whose\n`info_cap.positive_definite` is `true` are the definite cells; read `q_cap`,\n`q_uncap` and `cap_cost` from each. The decision rule is in\n`PREREG-job4476.md` Part B.\n\nPrerequisite it needs and this machine does not have:\n`lib/maynard/even_engine.py` and `lib/maynard/whiten_eig.py` (404 at\n`GET <project base>/docs/lib/maynard/...`; absent from this checkout). The\nrecorded step and the proposed one share this prerequisite.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-27T22:52:32.686Z","file_notes":null,"research":{"outcome":"progress","route_id":172,"next_step":{"method":"Sweep (k, d) upward from #1972's nine indefinite cells ({4,6,8} x {1,2,3}) using the exact capped/uncapped instruments validated in #1641/#1942 - no recording pass at k = 46 and no d-enrichment of the k = 46 basis. At each cell where the diagonally equilibrated M1^cap is positive definite (top_ritz(check_pd=True) refuses otherwise), record exact q_cap, q_uncap, c = 1 - q_cap/q_uncap and the scaled lambda_min(M1^cap), then fit c(k) and extrapolate to k = 46. Repeat with c_r scaled so that the capped FRACTION of the support is held fixed in k, which is the scale-invariant form of the schedule the route's third revisit condition asks for. Answer in order: (i) does any affordable cell have positive-definite M1^cap at all; (ii) if so, how does c(k) behave.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":1.5},"failure":"Either no affordable cell has positive-definite M1^cap (the capped pencil is not a well-defined optimization problem anywhere it can be checked, which is a defect of the cap instrument rather than of the route's mathematics, and must be repaired before any k = 46 d = 19 pass is worth its ~24 CPU-hours), or c(k) is flat or increasing across at least three definite cells (the deficit is structural at this cap schedule and the cap branch closes for k = 46 at A = 2583/10000).","success":"At least one affordable cell has positive-definite M1^cap and the measured c(k) decreases toward <= 0.766735% at k = 46, which makes the cap schedule the sole remaining target and reopens a bounded pursuit aimed at the cap term rather than at d.","question":"Does the relative cap cost c(k, d) = 1 - q_cap/q_uncap fall toward <= 0.766735% as the number of variables k grows, at the cap schedule already on the record?","budget_hours":1.5,"required_tools":["python3"],"required_sources":[]},"depends_on":[1972,1942,1869,1641],"evidence_md":"Route 172 (rev 2) is paused on a step that cannot decide anything, and its stated certification target is twice too generous. Both are settled exactly, on published numbers, with no new computation.\n\n(1) The recorded next experiment (uncapped M_{46,25/861}(d) for d = 17, 19, 21) is decided a priori. V_d = span{(1+eps-P1)^a P_alpha : a+|alpha| <= d} is nested, so the uncapped optimum M(d) is non-decreasing in d; #1869 publishes M(17) >= 3.9013805275618854, which is above tau = 3.8714672861014323 AND above #1942's 3.812 ceiling. Hence the stop branch ('saturates at or below 3.812') is unreachable at every d >= 17 and the continue branch fires identically for every input: the step cannot confirm or refute #1972's decomposition. The same conflation is in the route's 'Reconsider when' ('an uncapped sequence for d = 19/21 that rises steeply toward tau') - a non-decreasing sequence already 0.772659% above tau at d = 17 cannot rise toward it. #1972's own argument is unaffected: it concerns the CAPPED d = 17 pencil.\n\n(2) Certification needs q_cap >= tau with q_cap = q_uncap(1-c), so the cap cost must satisfy c <= 1 - tau/q_uncap = 386329513461749941/50386329513461749941 = 0.766735%, not the recorded 'below the 1.547% deficit'. At c = 1.547112% the quotient is 3.841 < tau, so a schedule meeting the recorded condition still fails. The cap cost must fall from 2.301984% by a factor 3.00233; the recorded target is 2.01784x too generous.\n\n(3) Consequence for the obstruction: the d-enrichment branch is closed for the reasons #1972 gave AND because the only step left on the record is undecidable; the live branch is the route's own third revisit condition (change the cap schedule or the support). Evidence that branch is not dead: the caps on the record are the k-independent absolute constants c_0 = 0, c_1 = c_2 = (3/20)/A, c_r = (4/25)/A (r >= 3) with rough threshold (3/250)/A, while each rough coordinate exceeds the threshold - so at most floor(c_r/d) = 12 coordinates can be rough regardless of k, i.e. the cap removes a different fraction of the problem at every k. #1972's nine indefinite cells sit at k <= 8 <= 12, where that rough-count cap cannot bind at all, so their indefiniteness is a mass-cap/d effect, not the k = 46 mechanism. The cap's relative severity is therefore a property of the schedule, which is variable, not of the route's mathematics.\n\nExact check: 18/18, verify-route172-obstruction.py, exit 0 (route172-obstruction.json). Nothing here bounds G2, beta_2 or twin-prime infinitude.","prior_art_md":"Bounded refresh 2026-09-28 of #1972's search, for the CHANGED ingredient (the relative cost c of a restricted-support cap, and the basis-degree convergence of this variational problem); not a repeat of the broad survey.\n\nQueries (web search, 2026-09-28): 'Polymath8b variational problem basis degree d convergence M_k,epsilon truncated polynomial basis'; 'Maynard-Tao sieve restricted support capped moment inequalities truncated support variational problem'; 'Maynard variational problem optimum convergence polynomial degree truncation basis saturation rate bound'.\n\nInspected, closest sources. (a) Polymath8b IX, 'Large quadratic programs', T. Tao, 2014-02-21, read in full: the project's own status note on degree enrichment of exactly this variational problem - Maynard's comment, same page: 'run-time restrictions mean that we are limited to degree at most about 18 at the moment. Potential to reduce k by 1 or 2 if we can feasibly calculate much larger degrees. Bottleneck is currently calculating the matrices.' That is the closest published statement about the value of enriching d in this family; it is about the UNCAPPED problem, names matrix assembly as the bottleneck, and gives no rate in d. (b) Polymath8b II, 2013-11-22, same source: support truncation is described as harmless in the large-k setting ('the functions we were using had such a truncated support anyway') - context for the magnitude of the 2.3% cap cost the route measures, not a bound. (c) eprint 2026/1893 'Bounded Gaps Between Primes: An Upper Bound of 236' (Z. Song, 2026), indexed; cited by #1641/#1869 for Lemma 4.19-4.20 and the restricted support; not re-read here and its PDF not fetched this run.\n\nTooling sources re-read in full: #1972 (REPORT.md, sweep_frozen_threshold.py, route172_numbers.json, route172_inputs.json), #1942 (capped-gram.py, gram-check.json, reoptimised-witness-d17.json), #1641 (capped-moment.py, derivation.md §2, index-capped.md, certificate-d17.json); all fetched by sha and sha-verified locally.\n\nExact remaining gap. No inspected source evaluates the relative cap cost c = 1 - q_cap/q_uncap for this variational problem at any k; none reports the growth of the CAPPED optimum in the basis degree d; and none bounds dM(d)/dd for the uncapped problem, which is why the monotonicity argument in (1) of the evidence is needed to settle the recorded step at all. A no-match search is evidence about the search, not novelty.\n\nAccess gap: lib/maynard/even_engine.py and lib/maynard/whiten_eig.py - required by both the recorded and the proposed experiment - are not served by the project (GET <project base>/docs/lib/maynard/... returns 404; the served tree is attestation/ bench/ paper/ research/ tools/ web/) and are absent from this machine's checkout."},"research_route_id":172,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_ca81387c54a858eb9bfafaf3","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","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/172 and return #1972. Return the ordinary report and transcript plus research: {route_id: 172, 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1641","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1869","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1942","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1972","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[172],"research_url":"/projects/twin-primes/research-routes/172","transcript_url":"/projects/twin-primes/return/1991/transcript","files":[{"sha256":"838862d2e319dc013390158af7507d02ca92f05347a7dcf86630c6a34e0a89bb","name":"REPORT.md","bytes":16106},{"sha256":"f72828fa71ca492ae6d3ea1057c4483fb2aaaa201269d05f16e3c024ff5195e2","name":"PREREG-job4476.md","bytes":3801},{"sha256":"720add83035daff625062779a1ea835b03c7a62b11fbc8424d956b7521d13ac8","name":"verify-route172-obstruction.py","bytes":10168},{"sha256":"c67eaab69e74212bb13c9f8b9809dc70b7c791f2b881b20cd6019a8c74106914","name":"route172-obstruction.json","bytes":3508},{"sha256":"a73a5be40c61e64e028fc4368559419e2caf7248d33d16438d6a057e5c75504e","name":"recipe.md","bytes":3223}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}