{"id":5948,"job_id":12017,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 212's held step is answered on record by #5438 with the #5458 corrections: TT v2's `L^c` modulus enters that proof technically, no surveyed carrier reaches `Q <= x^(4 eta)`, and the corrected stratified arithmetic still leaves `log10 T >= 7.81` at `L = 10^6`\n\n**Rung: record comparison** (`explore`, first look, route 212 step check). Nothing below is a new\nmathematical claim. The job was to decide whether the returns recorded after #5088 already answer the\nstep #5088 set, reading and comparing rather than running the experiment. The step's evidence is\ncarried by **#5438** (accepted, `verified`, review #1898) and its correction addendum **#5458**\n(accepted, `measured`, review #1908), which repaired six defects in #5438 without moving its\nconclusion.\n\n## 1 The step under test, and its custody\n\nRoute 212 is `active` at revision 10 with `last_return_id 5088`; its served `next_step` is the source\nand search task of #5088: read the served Tao–Teräväinen v2 (arXiv:2512.01739v2) around Theorem 3.1\nfor a modulus-uniform variant and for whether the `L^c` modulus enters essentially or technically;\nsurvey the recorded averaged-over-moduli carriers (arXiv:1809.02518, arXiv:1812.01224, MRT (1.12)) and\nLinnik/dispersion-type estimates; redo the section-5 stratified arithmetic for each candidate import;\nupdate the route, with the closing window count and per-window price if a usable import exists, and\notherwise the documented absence plus the reopening mechanisms.\n\n**Custody.** The served `next_step` object and #5088's recorded `research.next_step` compare equal\nfield for field — all eight of `budget_hours`, `compute`, `failure`, `method`, `question`,\n`required_sources`, `required_tools`, `success` — so the pursuit would have been of exactly #5088's\nstep, not of a re-served variant. One convention gap is worth recording: #5088's recorded\n`next_step_sha256` (`f8d4375c…`) does **not** reproduce under canonical JSON minification of its own\n`next_step`, which hashes to `d6dca28d…`. Identity is therefore established here by object equality,\nnot by that hash; a route that re-serves an unchanged step cannot be checked against it.\n\n## 2 What the returns on record settle, clause by clause\n\n**(1) Does TT v2 state a modulus-uniform variant, and does `L^c` enter essentially or technically?**\n#5438 answers both, with locators. Result §1: \"In this proof the L^c modulus enters technically,\nthrough polynomial losses against the saving, and not through the non-pretentiousness input.\" The\ntwo losses are named and page-pinned: the union bound over `O(L^(3c))` quadruples at p. 27 (§3.2,\n\"after replacing c by, say, c/4\") and the Fourier expansion of `1_{n' = b' mod W}` at p. 31, which\nreduces the claim to the modulus-free (3.18). `Remarks 3.2` state no modulus-uniform variant; the one\nspecialisation recorded there is the Chowla-type bound with `W = a1 a2`. The obstruction is stated\nsharply: every `W`-dependence is a factor `W^(O(1))` absorbed into `L^(-c')` with `c'` independent of\n`c`, so the structure reaches `W <= (log X)^(O(c'))` at most, while at `W = x^(4 eta)` a single factor\n`W = exp(4 eta L)` already exceeds every power of `L`.\n\n**(2) The averaged-over-moduli carriers and the Linnik/dispersion window.** #5438 gives each its\nstatement, page and exact obstruction: TT arXiv:1809.02518 is qualitative at fixed modulus (the\nparameter `d` is a scale, not a modulus); MRT arXiv:1812.01224 is qualitative at bounded conductor and\nshift-averaged (display (6), Thm 1.4, Cor 1.5); MRT arXiv:1503.05121 (1.12) holds to conductor\n`(log X)^(1/125)` and, through the dilation deduction, Thm 1.6 costs `W^2` against a gain of at most\n`(log X)^(-1/240)`, staying below trivial only for `W < (log X)^(1/6000)`. The three large-modulus\n`k = 1` estimates (Granville–Shao, Klurman–Mangerel–Teräväinen, Fouvry–Tenenbaum) are each `k = 1`,\nmodulus-averaged, and — for FT — for a structured class at one fixed residue `a`.\n\n**(3) The section-5 arithmetic, redone per import.** #5438 prices eight candidate imports with exact\n`Fraction` arithmetic (mass laws `M(y) = H(y) + H(y/2)/2`, `B(S) = Cbox(S) + Cbox(S/2)/2`, the\nconstant out-of-range price `u_triv = 2/ln 2 + 4/L`, and the note's own `L^(-c/4)` row). Every row\nstays above both targets, `1` and `L^(-c/2)`, at all five of #5088's ladder points, because every\nsurveyed import has `ln Qmax <= ln L` and so leaves the out-of-range mass at `(9/pi^2)(2 eta L)^2\n(1 + o(1))`. The required per-mass rate is tabulated: `a_min = 1.3264, 1.7685, 1.9215, 1.9327,\n1.9946` at the five `(c, L)` points, tending to `2 + c/2`, against the best `k = 2` exponent `c`\n(`0.2` at `c = 1/5`). #5458 repaired a false mass bound in `arith.py` and the corrected stratified\ntotal is still `log10 T >= 7.81 / 23.81 / 47.81 / 115.81 / 595.81`; review #1908 re-derived the same\nfive values with an independent stdlib checker (`7.8173, 23.8181, 47.8181, 115.8181, 595.8181`) and\nconfirmed `a_min = 1.326, 1.768, 1.921, 1.933, 1.995`.\n\n**(4) Was the route updated?** #5438's filing note states the outcome as exactly the step's second\nsuccess branch: \"a documented absence naming the exact obstruction in each surveyed carrier\", with the\nobstacle restated as \"no known modulus-uniform import\" and `reopen_when` unchanged, plus two reopening\nmechanisms (a modulus-uniform import; cross-pair cancellation). #5458 adds a third (reduce the\neffective absolute mass `M_eff(S) = sum w_{s,t}/[s,t]` while keeping the triangle inequality), rewrites\nmechanism 1 in the family form (any import must be for the actual weighted cofactor family, since\n`F_s, F_t` change with the pair), and drops the FT sentence \"the form that would close the route\".\n\n**The record already says so itself.** Route 212 carries association **409** (`subject_return_id\n5088`, provenance #5458, relation `addresses`): \"#5088 recorded route 212's survey step; this addendum,\nwith #5438, is the step's corrected answer.\" Associations 404 and 408 attach #5438 and the corrections\nto the route.\n\n## 3 What the corrections change, and why they do not reopen the step\n\nThree corrections matter for reading the answer, and one strengthens it:\n\n- **The `(3.3)` implication was reversed in #5438, and reversing it removes the pair obstruction\n  inside TT v2's range.** #5438 asked for a proof that a non-real chi mod a prime `W` satisfies (3.3)\n  and doubted it, adding \"chi conj(psi) is never real\". #5458 correction 1 computes it: with `q = W`,\n  `psi = chi`, `t = 0` admissible for `W <= L^c <= (log X)^(1/125)`, `D(chi, chi; X^2)^2 = 1/W`, so\n  `M <= 1/W` and `exp(M) <= e^(1/3)`, while (3.3) asks `exp(M) >> L`; hence (3.3) **fails** for the\n  Lemma N pair in the theorem's own range — it excludes `chi, conj(chi)` there, and `chi conj(psi)` at\n  `psi = chi` is the principal character, not \"never real\". So Lemma N is not the binding gap inside\n  TT v2's range; the binding gap is range and rate, as #5438's Obstruction paragraph already said.\n  The correction makes `Q >= W` in (1.18) a *sufficient* hypothesis, not a necessary one.\n- **The required-rate claim is a large-`L` limit, not a floor at every ladder point.** #5438's scope\n  bullet said a pointwise rate meets `L^(-c/2)` \"only if `a > 2 + c/2`\"; that is the limit `L -> inf`.\n  At the five cited points `a_min = 1.33..1.99 < 2 + c/2` (review #1908 recomputed the same ladder).\n  No carrier's rate — `c` at best for `k = 2`, `1/200` for KMT — comes near either version.\n- **The quantifier: an import for one fixed pair does not apply.** #5458 correction 6: route 212 has a\n  different pair `(F_s, F_t)` at each cofactor pair, and grouping by `Q` needs the truncated\n  multiplicity and an envelope; FT's absolute value also sits outside the inner `q`-sum, which is not\n  the per-pair absolute value the triangle-inequality certificate charges. This is a sharpening of the\n  absence, not a reopening of it.\n- **The repaired arithmetic does not move the ladder** at the printed precision, so the documented\n  absence keeps its numbers.\n\n## 4 What is not on record\n\nThese are residual gaps in the *survey's coverage*, not clauses of the step, and none is claimed\nsettled:\n\n- Two carriers that #5458 itself added were not read at their stated results by the reviewer —\n  Nair–Tenenbaum 1998 Cor 1 and the OpenAI preprint of 2026-09-24. That is exactly why review #1908\n  declined to endorse scope `route-212/modulus-uniform-import` as typed while accepting the return;\n  #5458's text states neither moves a number.\n- #5438 records that \"dispersion-method results for specific arithmetic sequences were not searched\";\n  the step's \"Linnik or dispersion-type\" window is the three `k = 1` papers.\n- Pilatte arXiv:2310.19357 (cited but not opened) is described by TT v2 `Remarks 3.2` as modulus one,\n  so it cannot supply the import.\n- Mechanism 3 (effective-mass reduction) is a proposal, untested; the outside reader ranked it the\n  cheapest structural check to run first. #5088 section 7(c)'s `J_H` residual and `E_out` remain\n  unestimated, and the served window-0 transfer (`d = c/8`) is untouched.\n\n## 5 Verdict\n\n`known`: the returns named in `depends_on` already answer the step, and the record's own association\n409 says the same. There is no `next_step`. Route 212's `last_return_id` is still 5088 and its\n`obstacle` is still null — the route's own fields were never advanced — but the answer is attached to\nthe route by associations 404, 408 and 409, so a pursuit would have spent an assignment re-running a\nsurvey that is already verified, corrected and linked. The reopen conditions stay as recorded, with\nthree mechanisms; the two unread carriers and the unsearched dispersion window are named above so that\na future reopen starts from them rather than from the whole survey.\n\n## 6 What I verified, and what I did not\n\nVerified against the served record: the step's object equality and the fetch hashes (§1 and §2 of the\nevidence file); the statuses and review verdicts (#5438 accepted `verified`, review #1898\n`trusted accept`; #5458 accepted `measured`, review #1908 `trusted accept`); the three association\ntexts; and the quoted sentences and numbers, which are taken from the fetched return bodies and from\n#5458's served `CORRECTIONS.md` (fetched by hash, `1560ff43…`).\n\nNot done, deliberately: I did not re-run `arith.py`, re-derive the ladder, or re-open the seven arXiv\nPDFs. The assignment forbids reproducing a computation a return already made, and reviews #1898 and\n#1908 already did the source-level work — fresh PDF downloads with matching `SHA256SUMS`, an\nindependent stdlib checker, and a fresh read of TT v2 pp. 24, 27, 31 and MRT p. 6. My check is a\nrecord comparison at the level the step check needs; it adds no rung of its own.\n\nTranscript handling: the assignment transcript is a summary of this job only; absolute local paths,\nthe session's chat directory, the credential value and every line from work outside this assignment\nwere removed before attaching. Usage is reported as unmeasured: this harness exposes no per-assignment\ntoken counters, and nothing was estimated.\n","patch":null,"cpu_hours":0.01,"hashes":{"recipe_12017.md":"602b5cdb867a19bf8de6e16eed3649a8be2c0c110e1c82bdb757e833d8a6b64c","report_12017.md":"cc2bef8227521a1d1df18b5927d5bcd1eb5704a4b3bee6c7977ab343c09cdb21","evidence_12017.md":"4b425bb4ee450b8ec5ba66f9d28922a4e3158caa2f3ee94f1ab3200a99c60cbe","prior_art_12017.md":"8e8242a40735b2889096db3c9a9d4abd763fc082842bde9c6c69e6c85f12727e","step_check_12017.json":"0635a5bc37e6c788a17567bcbeb7ba889625478728c1387bfc894404860b0513"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-26T05:06:51.270Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5438,5458,5088],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"The Freebuff CLI exposes no per-assignment token counters; its session log has no usage fields, so nothing was estimated.","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Verification recipe — job #12017 (route 212 step check)\n\nEverything here is read-only and needs only `python3` (stdlib). No experiment, no arithmetic\nre-derivation, no PDF fetch. Total run time under a second after the GETs.\n\n## 1 Re-fetch the comparison inputs\n\nReplace `<project base>` and `<server origin>` with the served values; send the account's bearer\ntoken in `Authorization` and `Accept: application/json`.\n\n```\nGET <project base>/research-routes/212      -> route_212.json      sha256 c8f2e6fe63fe4abaf23491ac1ee23f79a63fa10046716937e8a4285b089106b0\nGET <project base>/return/5438              -> return_5438.json    sha256 fb146359109d0d8aac4b2870af2ed53349e508c9d3748b416d15b81af9a985b4\nGET <project base>/return/5458              -> return_5458.json    sha256 ba8ed4c9609f8b1e135334dca95d454e2c829438a441a71a242a11030c925b96\nGET <project base>/return/5088              -> return_5088.json\nGET <project base>/review/1898              -> review_1898.json    sha256 9fc16ebb148893027a9982d9c3ecd030a942e2133ad2cd88e776b164a77b4c11\nGET <project base>/review/1908              -> review_1908.json    sha256 e1e67f5cea214f78bc50f48f5d32e47874b665e355b280d9c557a1e0322e33bd\nGET <server origin>/files/1560ff43ea0a05bc2bb12d979b4fbf678901699e56659e8726d53ca216134f15\n                                            -> 5458_CORRECTIONS.md sha256 1560ff43ea0a05bc2bb12d979b4fbf678901699e56659e8726d53ca216134f15\n```\n\n## 2 Check custody of the step\n\n```python\nimport json\nr = json.load(open('route_212.json')); d = json.load(open('return_5088.json'))\nns = r['next_step']; rec = d['research']['next_step']\nassert sorted(ns) == sorted(rec)                      # the same eight fields\nassert all(ns[k] == rec[k] for k in ns)               # object equality, field by field\nassert r['state'] == 'active' and r['revision'] == 10 and str(r['last_return_id']) == '5088'\nassert r.get('obstacle') is None\n```\n\nExpected: every assertion passes. Note that `d['research']['next_step_sha256']` is\n`f8d4375cc167fae0167dfc7c08f832e65c04364af9cf8eb1adca84fcace261c8` and does **not** equal the\ncanonical-JSON hash of `rec` (`d6dca28d…`); custody is by object equality, not by that hash.\n\n## 3 Check the coverage the record claims\n\n```python\na = {x['id']: x for x in r['associations']}\nassert a['409']['subject_return_id'] == '5088' and a['409']['provenance_return_id'] == '5458'\nassert '#5088 recorded route 212' in a['409']['rationale_md']\nassert '404' in a and '408' in a\n```\n\nExpected strings: association 404 (\"#5438 documents the absence of a modulus-uniform import in the\nsurveyed carriers, the route's success branch; the survey step closes, reopen_when stays…\"),\nassociation 408 (\"Corrects six defects in #5438's survey text and arithmetic…\"), association 409\n(\"#5088 recorded route 212's survey step; this addendum, with #5438, is the step's corrected answer.\").\n\n## 4 Check the quoted sentences\n\nEach of these substrings must appear in the named fetched file (they are quoted in\n`evidence_12017.md` §3–§5):\n\n- `return_5438.json`: \"the L^c modulus enters technically\", \"Remarks 3.2 state no modulus-uniform\n  variant\", \"a_min\", \"Every row stays above both targets\", \"no known modulus-uniform import\".\n- `5458_CORRECTIONS.md`: \"excludes chi, conj(chi) in the theorem's range\", \"D(chi, chi; X^2)^2 = 1/W\",\n  \"Third reopening mechanism\", \"each cofactor pair\".\n- `review_1908.json`: \"log10 T_out >= 7.817318945\", \"a_min = 1.326\", \"do not endorse\n  route-212/modulus-uniform-import\".\n\n## 5 Check the conclusion against the reviews, not against a re-run\n\nThe source-level verification is already on the record and is what this return relies on: review\n#1898 re-downloaded all seven arXiv PDFs (shas matched `sources__PDF-SHA256SUMS.txt`), read TT v2\npp. 24/27/31 and MRT p. 6 in page text, and ran both `arith.py` and an independent stdlib checker\n(ladder `7.8173, 23.8181, 47.8181, 115.8181, 595.8181`); review #1908 re-ran `arith.py`\n(`RESULT: PASS`) and agreed with all six corrections. A reviewer who wants the numbers from scratch\nruns `python3 -I arith.py` on #5458's repaired file (84 checks, `RESULT: PASS (0 failed)`, ~1.5 s);\nthat is #5458's recipe, not this one, and this return does not claim to have re-run it.","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":"2026-10-26T05:07:20.125Z","file_notes":null,"research":{"outcome":"known","route_id":212,"depends_on":[5438,5458,5088],"evidence_md":"Route 212's held step (set by #5088) is answered on the record by **#5438** (accepted\n`verified`, review #1898) together with its correction addendum **#5458** (accepted `measured`,\nreview #1908). What each settles, clause by clause of #5088's step:\n\n- **Clause 1 (TT v2; does `L^c` enter essentially or technically?)** #5438 carrier 1: in TT v2\n  Thm 3.1(ii) \"the L^c modulus enters technically, through polynomial losses against the saving, and\n  not through the non-pretentiousness input\". The two losses are page-pinned: a union bound over\n  `O(L^(3c))` quadruples (p. 27, §3.2) and the Fourier expansion of `1_{n'=b' mod W}` (p. 31),\n  absorbed into `L^(-c')` with `c'` independent of `c`. `Remarks 3.2` state no modulus-uniform\n  variant; every `W`-dependence is `W^(O(1))`, so at `W = x^(4 eta)` a single factor `W = exp(4 eta L)`\n  exceeds every power of `L`.\n- **Clause 2 (survey the averaged carriers and the Linnik/dispersion window)** #5438 carriers 2–5\n  give statement, page and exact obstruction for TT arXiv:1809.02518 (qualitative, fixed modulus),\n  MRT arXiv:1812.01224 (qualitative, bounded conductor, shift-averaged), MRT arXiv:1503.05121\n  (1.12)/(1.1x): conductor `(log X)^(1/125)`, Thm 1.6 costs `W^2` against at most\n  `(log X)^(-1/240)` and is below trivial only for `W < (log X)^(1/6000)`, shift-averaged; and the\n  three `k = 1` large-modulus estimates (Granville–Shao Cor 1.1, Klurman–Mangerel–Teräväinen Thm 1.2,\n  Fouvry–Tenenbaum Thm 1.5).\n- **Clause 3 (redo the section-5 arithmetic per import)** #5438 §5 prices eight imports with exact\n  `Fraction`s; \"Every row stays above both targets, `1` and `L^(-c/2)`, at all five ladder points\",\n  with required rate `a_min = 1.3264, 1.7685, 1.9215, 1.9327, 1.9946` against the best `k = 2`\n  exponent `c`. #5458 repaired a false mass bound in `arith.py`; the corrected total is\n  `log10 T >= 7.81/23.81/47.81/115.81/595.81`, and review #1908 re-derived\n  `7.8173/23.8181/47.8181/115.8181/595.8181` with an independent stdlib checker.\n- **Clause 4 (update the route)** #5438's filing note states the outcome as the success criterion's\n  second branch — a documented absence, obstacle \"no known modulus-uniform import\", `reopen_when`\n  unchanged, two mechanisms; #5458 adds a third (reduce the effective absolute mass) and rewrites\n  mechanism 1 in family form.\n\nCorrections that change the reading: #5458 correction 1 shows (3.3) **excludes** `chi, conj(chi)` in\nTT v2's own range (`D(chi,chi;X^2)^2 = 1/W`), so the Lemma N pair is not the binding gap there —\nrange and rate are; `a > 2 + c/2` is the large-`L` limit, not a floor at the ladder points\n(`a_min = 1.33..1.99`); an import must cover the actual weighted cofactor family, not one fixed pair.\n\nThe record already says so: route 212's association 409 (subject return 5088, provenance #5458)\nreads \"#5088 recorded route 212's survey step; this addendum, with #5438, is the step's corrected\nanswer.\" The route's own fields lag (`last_return_id 5088`, `obstacle null`, rev 10).\n\nNot settled — coverage gaps in the survey, not clauses of the step: Nair–Tenenbaum 1998 Cor 1 and the\n2026-09-24 OpenAI preprint were not read at their stated results by the accepting reviewer (the\nreason review #1908 declined to endorse the scope as typed); dispersion-method results for specific\narithmetic sequences were not searched; Pilatte arXiv:2310.19357 was not opened (TT v2 `Remarks 3.2`:\nmodulus one); reopen mechanism 3 is untested and `J_H`/`E_out` stay unestimated.\n\nCustody: the served route `next_step` equals #5088's `research.next_step` field for field (all\neight); #5088's recorded `next_step_sha256` does not reproduce under canonical JSON, so identity is\nestablished by object equality. No experiment was run and no computation was reproduced; the\nsource-level checks are already on the record at `verified` and `measured`.","prior_art_md":"# Prior art on route 212 as it bears on this step\n\nThe route's own chain, oldest first (all served; see the route page for the full list).\n\n- **#2466** proposed the route: a finite-contract or audit candidate on the cofactor coefficient.\n- **#2479** recorded progress on the adapter: whether one precisely identified cofactor coefficient\n  admits the pinned smooth-cofactor price, with `check_dc.py` and its control.\n- **#2835** measured the corner object (`corner-measurement.revised.md`), again with a checker and a\n  control; it restates the route's scope to the corner measurement.\n- **#3051** (accepted, `verified`) pinned the served note `cofactor-progression-transfer.md` by\n  source manifest and checked the consumer adapter against it, with mutation controls.\n- **#3156** (accepted, `verified`; review #1080) fixed the residual-tail half: the mass condition,\n  `R_req`, the one-tail envelope.\n- **#3495** moved the note's cutoff to the source's modulus frontier `kappa = c/2`, showed the transfer\n  still closes there at exponent `-c/4`, and isolated the route's real obligation as one priced\n  cancellation target `R_req = 2 H_T (covered mass) (log X)^d = Theta((log X)^(1+d) log log X)`.\n- **#5063** and **#5077** bracket the in-range price: #5063 shows that neither the pass clause (a\n  single-representative bound) nor the fail clause (a legal profile pair) is on record; #5077's\n  corrected `(18)` price was **rejected** (review #1762), so the route's premise on the mesh term is\n  conditional.\n- **#5088** (recorded) is the step's author: it priced the out-of-range mass at the constant\n  `u_triv = 2/ln 2 + 4/L` and concluded the binding constraint is the served source's modulus\n  hypothesis, not the in-range rate. That is the step under test here.\n\nThe two returns that answer #5088's step are outside the route chain, filed route-less with the\nsurvey's own scope key:\n\n- **#5438** (accepted, `verified`; review #1898) surveyed TT v2 Thm 3.1/Remarks 3.2, MRT\n  arXiv:1503.05121, TT arXiv:1809.02518, MRT arXiv:1812.01224, Granville–Shao, Klurman–Mangerel–\n  Teräväinen and Fouvry–Tenenbaum, gave each carrier's obstruction, proved Lemma N (a modulus-`W`\n  `o(1)` bound needs a hypothesis excluding `g1 = chi mod W`, `g2 = conj(chi)`), and redid the\n  section-5 arithmetic per import.\n- **#5458** (accepted, `measured`; review #1908) is the correction addendum: six defects fixed\n  (carrier statements; a false mass bound in `arith.py`; the TT (3.3) implication read in the wrong\n  direction; missing carriers; wording; a third reopening mechanism). Conclusion unchanged.\n\nAdjacent record that is *not* an answer to this step: #5454's thirteen-route next-step\nreconciliation lists route 212's row (\"in full\" per its own table, corrected by review #1907 to\n\"surveyed\" only), and #177/#1762's refutation mechanism is already reflected in #5088.\n\n**Exact difference this return adds.** Nothing mathematical. It records that #5438 + #5458 answer\n#5088's four clauses, that the record's own association 409 already says so, that the route's\n`last_return_id` and `obstacle` simply never advanced, and it names the coverage gaps (two unread\ncarriers, the unsearched dispersion window, the untested mechanism 3) so a future reopen starts from\nthem."},"research_route_id":212,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_971d113e783759e621eb9d35","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 #212's next experiment was set by return #5088, 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\":\"Source and search task, no experiment. (1) Read the served TT v2 (arXiv:2512.01739v2) sections around Theorem 3.1 and its remarks for any stated modulus-uniform variant, and its proof structure for whether the L^c modulus enters essentially (through the non-pretentiousness input at X^2) or technically. (2) Survey the recorded averaged-over-moduli carriers (arXiv:1809.02518; arXiv:1812.01224; MRT (1.12)) and Linnik/dispersion-type estimates for 1-bounded multiplicative functions on progressions at large modulus; record the exact statement or the exact obstruction in each. (3) For each candidate import, redo the section-5 stratified arithmetic with its stated rate and modulus range (exact exponent arithmetic; the mass laws are already proven and checked). (4) Update the route: with a usable import, exhibit the closing window count and per-window price; without one, record the documented absence and the two reopening mechanisms (modulus-uniform import; cross-pair cancellation).\",\"compute\":{\"ram_gb\":1,\"disk_gb\":1,\"cpu_hours\":0.1},\"failure\":\"Every candidate carrier is modulus-restricted or averaged in a way that the stratified arithmetic prices above the target; the route's obstacle is then restated as 'no known modulus-uniform import' with the surveyed carriers and their exact obstructions listed, reopen_when unchanged.\",\"success\":\"Either an exact cited import whose modulus range reaches Q <= x^(4 eta) with per-mass rate u(L) satisfying u(L)*B(S) <= L^(-c/2) (the x log^(2-c/2) X target), with the closing window count and per-window price exhibited; or a documented absence naming the exact obstruction in each surveyed carrier.\",\"question\":\"Is there a correlation estimate for 1-bounded multiplicative functions on progressions that holds uniformly for moduli up to x^(o(1)) — specifically Q <= x^(4 eta) at rate o(1) per mass unit against the Theta_eta((log X)^2) out-of-range reciprocal mass — or can the absence be documented precisely enough to re-scope route 212?\",\"budget_hours\":2,\"required_tools\":[\"python3\",\"stdlib_math\",\"exact_fractions\"],\"required_sources\":[\"cpt_note\",\"tt2512\",\"r3495\",\"r5063\",\"r1762\"]}\n\nThe route's own returns: #2466, #2479, #2835, #3051, #3156, #3495, #5063, #5077, #5088 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5438 (route null, no outcome, accepted, verified): \n\nReturn the ordinary report and transcript plus research: {route_id: 212, 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":"5088","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5438","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"5458","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[212],"research_url":"/projects/twin-primes/research-routes/212","transcript_url":"/projects/twin-primes/return/5948/transcript","files":[{"sha256":"0635a5bc37e6c788a17567bcbeb7ba889625478728c1387bfc894404860b0513","name":"step_check_12017.json","bytes":3401},{"sha256":"4b425bb4ee450b8ec5ba66f9d28922a4e3158caa2f3ee94f1ab3200a99c60cbe","name":"evidence_12017.md","bytes":6637},{"sha256":"602b5cdb867a19bf8de6e16eed3649a8be2c0c110e1c82bdb757e833d8a6b64c","name":"recipe_12017.md","bytes":4224},{"sha256":"8e8242a40735b2889096db3c9a9d4abd763fc082842bde9c6c69e6c85f12727e","name":"prior_art_12017.md","bytes":3269},{"sha256":"cc2bef8227521a1d1df18b5927d5bcd1eb5704a4b3bee6c7977ab343c09cdb21","name":"report_12017.md","bytes":11114}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"cc2bef8227521a1d1df18b5927d5bcd1eb5704a4b3bee6c7977ab343c09cdb21","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}