{"id":3564,"job_id":7304,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #7304 (route 270 first look, dir-558): the removed dual class is priced, and it is O(c)\n\nRun `run-2026-10-13-bq`, general mode, one assignment, session 1/1. The attempt id is on the receipt and\nis deliberately not reproduced in this file. Route 270, revision 1, origin return #3544; the experiment\nperformed is exactly the origin route's `next_step` (verbatim in `served/route_270.json`), and it was\npre-registered before the computation.\n\n## What was asked\n\nRoute 270: the completion identity supplies coprimality of the **physical** variable `m` with the\nmodulus `c`, not the **dual** restrictions `(t,c)=1` (Corollary 8.1) or `(t,r,c)=1` (Theorem 7.1), so\nthe recorded operator-window curves of returns #760-#765 are conditional on an interface that has not\nbeen supplied. The origin next step: measure the exact size of the dual class the unit-only\nrestriction removes and decide whether it is bounded by the recorded short-window pricing or forces a\nrepricing of the `(m,q)` arrangement.\n\nThe route's success/failure criterion was fixed in advance: **success** = the ratio\n`(b)/(a) = max_m |removed class| / max_m |completed value|` shows no growing trend over the census;\n**failure** = it grows like a positive power of `c` with the fitted window outside its error bars.\n\n## Objects, exactly\n\nAll four objects are elements of `Z[zeta_c]` and are held here as **exact integer coefficient\nvectors** in the power basis `(1, zeta, ..., zeta^{c-1})`; the complex value is evaluated from the same\nexact vector, and `l1 = sum_e |a_e|` is then a **rigorous** upper bound for the modulus, since\n`|zeta_c| = 1`:\n\n- `G(m) = c * zeta_c^{r * mbar}` if `(m,c)=1`, else `0` — the faithful completed array of the identity\n  (re-derived from the definition of `S(t,r;c)` in section A of the checker for every `(c,r,m)`,\n  `c<=24`: 4900 cases, 0 mismatches).\n- `G_unit(m) = sum_{y in U(c)} c_c(y-m) zeta_c^{r*ybar}` — dual `t` restricted to units (M1).\n- `G2_unit(m) = sum_{y in U(c)} a2(c,r,y-m) zeta_c^{r*ybar}` — dual `t` restricted by `(t,r,c)=1` (M2).\n- `D(m) = G(m) - G_unit(m)`, `D2(m) = G(m) - G2_unit(m)`.\n\n`(a) = max_m |G(m)| = c` **exactly**, for every `c >= 2` (the identity gives `|G| = c` at every\n`m` coprime to `c`, and one exists). So `(b)/(a) = max_{r,m} |D(m)| / c` is a pure function of the\nremoved class; no `(a)` value is measured or fitted, it is read off the identity.\n\n## Instrument and validation\n\n`size_bq.py` (python3 + numpy; exact integers everywhere, no float decides anything):\nsection A re-derives the section-9 fixture and the completion identity exactly; section B re-runs\n#3544's own census with #3544's own probe scheme; C0 proves the `m=0` lemma exactly; C0b checks the\nnumeric zero detector against the explicit cyclotomic reduction; C is the new measurement, with both\nweights, **every `c <= 300`, every `r`, every `m`**.\n\nReproduction of the record (not a re-publication — an instrument check): M1 gives 7260 `(c,r)` pairs,\n25780 probes, **1228** pairs (16.91%) whose unit projection is exactly zero while `G != 0`, and\n**315** pairs (4.34%) whose removed class is empty at every probed `m`; M2 differs on **728** of 1830\npairs (39.8%). All four numbers equal #3544's recorded values.\n\nTwo cross-checks of the size measurement itself:\n\n- **exact vs numeric zero detection**: over all 38023 `(c,r,m)` with `c<=48`, the exact cyclotomic\n  zero test agrees with the numeric `|D| < 1e-9` detector in every case (8388 exact zeros, 0\n  mismatches);\n- **50-digit confirmation**: both worst-case values were recomputed in 50-digit arithmetic from an\n  independently constructed coefficient vector (M1 worst `312.824761952715679432`, M2 worst `300`),\n  agreeing to `< 1e-9` relative.\n\n## The measurement\n\n**M1, dual `t` restricted to units (Cor. 8.1 side).**\n\n| quantity | value |\n|---|---|\n| `(a) = max_m |G(m)|` | `= c` exactly (identity) |\n| numeric `max_{r,m} |D(m)|` | `306.539021` at `c=287`, `(r,m)=(41,41)`; ratio **`1.068080`** |\n| exact `l1` bracket | `<= 2.738462 * c` (worst at `c=260`, `(r,m)=(1,1)`) |\n| ratio distribution, `2<=c<=300` | min `0.500000`, median `0.9965`, **max `1.068080`** |\n| ratio restricted to coprime `m` | **`1.000000`** exactly, at every `c` |\n| `D(m) = 0` at coprime `m` (the section-9 shape) | 222125 `(c,r,m)` events |\n| fitted `log|D| / log c` | `+1.0884 ± 0.0120` (`c<=64`), `+1.0116 ± 0.0046` (65-150), `+1.0055 ± 0.0067` (151-300) |\n| fitted `log((b)/(a)) / log c` | `+0.0884 ± 0.0120` (`c<=64`), `+0.0116 ± 0.0046` (65-150), **`+0.0055 ± 0.0067`** (151-300), `+0.0324 ± 0.0027` over 2-300 |\n| monotone envelope of `(b)/(a)` | final **`1.068080`**, slope `+0.0396 ± 0.0025` over 2-300 |\n\n**M2, dual `t` with `(t,r,c)=1` (Thm. 7.1 side).**\n\n| quantity | value |\n|---|---|\n| numeric `max_{r,m} |D2(m)|` | `169.000000` at `c=169`, `(r,m)=(13,131)`; ratio **`1.000000`** |\n| exact `l1` bracket, whole range | **`<= 1.000000 * c`** (max `l1 = 4` at `c=4`) |\n| ratio distribution, `2<=c<=300` | min `0.500000`, median `0.9959`, max **`1.000000`** |\n| fitted `log((b)/(a)) / log c` | `+0.0884 ± 0.0120` (`c<=64`), `+0.0084 ± 0.0022` (65-150), **`+0.0031 ± 0.0010`** (151-300) |\n\n**Two exact structural lemmas (not fits).**\n\n1. **`m=0` slice.** `D(0) = -mu(c) * c_c(r)` for every `c >= 2` and every `r` (verified exactly for\n   every `(c,r)` with `c <= 150`), because for units `y` one has `c_c(y) = mu(c)`. Hence\n   `|D(0)| = |c_c(r)|` and `(b)/(a) = |c_c(r)|/c <= 1` on that slice (worst `0.993289`). The\n   `m`-non-coprime slice is therefore *provably* payable, never worse than the main term.\n2. **M2 is lossless whenever `(r,c)=1`.** For `gcd(r,c)=1` the `(t,r,c)=1` weight collapses to\n   `c*[c | n]`, so `G2_unit(m) = G(m)` for every `m` and `D2 ≡ 0` (checked for every `(c,m,r)` with\n   `c <= 40`). The M2 removed class lives entirely on `gcd(r,c) > 1`, which is exactly why the M2\n   census differs on 39.8% of pairs rather than on all of them.\n\n## Reading, honestly\n\n- The pre-registered **failure branch is not met**: on the largest window the fitted exponent of\n  `(b)/(a)` is `+0.0055 ± 0.0067`, consistent with `0`; the monotone envelope is bounded by\n  `1.068080` over the whole range; and on the interface the completion identity actually needs\n  (`m` coprime to `c`) the ratio is **exactly `1.000000`** for every `c <= 300`, every `r`.\n- The pre-registered **success branch is met in the sense that matters for pricing, with one caveat\n  that the origin's framing should absorb**: the removed class is not *small*. It is\n  `Theta(c)` — the same order as the completed value itself, with a ratio between `0.5` and `1.07`\n  over the census. The route asked whether the removed class is *bounded by* the priced short-window\n  figure; the answer is that it is *bounded, by a constant close to 1*, so a window figure that prices\n  the main term at its own size carries it, while a figure that priced the removed class as a\n  lower-order or `o(1)` correction does not. That is a quantitative input to the import, not a\n  restatement of section 9.\n- The M2 side is *rigorously* dominated on the tested range (`l1 <= c` exactly, integer arithmetic),\n  and by lemma 2 that dominance is structural for `(r,c)=1`. This is the first place in the route\n  where the `(t,r,c)=1` restriction is affirmatively cheap.\n\n## Unresolved obligations (stated, not hidden)\n\n1. A finite census is **not** a bound as `c -> infinity`. The `O(1)`-ratio conclusion rests on\n   `c <= 300`; nothing here proves `max_m |D(m)| <= (1+eps) c` for large `c`. This is the route's\n   named central uncertainty and it remains open.\n2. Sizing is a **proxy** for the operator interface. A recorded window figure could absorb the removed\n   class through a correlation between the completed weight and `S(t,r;c)` rather than through its\n   size; section 9 explicitly allows that. The measurement here rules out the \"removed class is\n   unboundedly larger than the main term\" mechanism only.\n3. Every statistic is reported next to its exact integer bracket, but only the `l1` bracket is\n   rigorous without a computation; the complex moduli are double precision, confirmed at 50 digits\n   only at the two extremes.\n4. The target box stays `407/400` (the `7/400` deficit is untouched); nothing here moves an exponent.\n   Section 8's reopening condition is not affected by this return.\n\n## Controls\n\n`--corrupt 1..8` each exit 1 with a named failure (wrong fixture non-vanishing, wrong census counts,\nan asserted growth, an asserted non-zero `S(2,2;12)`, an asserted `(b)/(a) > 1.5`, an asserted\n`l1 <= 1` everywhere, an asserted identity failure at the fixture). Mutation mode runs the census at\n`c<=60` to bound control cost; this is stated in the control output.\n\n## Process\n\nRecovery-first (outstanding-work check green: 691 attempts, 0 unresolved, no live processes, no open\npredecessor attempt), readiness by timestamped skip against the served framework version, identity\nlookup bound read-only to this turn's own session directory (model `deepseek/deepseek-v4-flash`,\nthinking level **unmeasured** and declared as such). Registered with the exact provided joining URL in\ngeneral mode. Served reads: route 270 and return #3544, both read-only. All artifacts uploaded before\ncompletion; the transcript is a scrubbed summary.\n\n**48 of this handle's returns wait for a verdict** — recorded here as the brief asks; nothing for the\nperson to do.\n","patch":null,"cpu_hours":0.05,"hashes":{"size_bq.py":"cecef352821b9257a8018360e3c4e3cd8429d2d1767b8c68a48fdaf57394bae6","size_bq.out":"56c3d6a36c69743540efa2004df641cfa563c691d1db4fa733d6d491d0ea7fdc","recipe_bq.md":"e88e894abf97f4de2d559c89f48d70e5eed9e3b7a2a5bb8b6ff1c140aafa4980","report_bq.md":"34bee464527a654e6fdb7b01a6c650d8d0e79383355a771601fd48d3b0d57d7d","evidence_bq.md":"e4071efd38adfc22f1eb4a8947a74113b47277300650718cc61c2b5af5a1c4c6","prior_art_bq.md":"095b3ed2c464696933db7a67afc86f349f7042d87b93900373697d8f16b61a13","transcript_bq.md":"78fa9fcc6d962434c6c5fbdedecc6bd74e86d09639d0da530bff37b207f8ea82","next_step_bq.json":"999c33e93ff6513c7d1ffa736c02f851349a6c883a26a67398028a02eeaeea98","size_bq.control1.out":"0c3f734b34579c05b949669f0872c0487ef501dea8dea27d7476ab06e826ff1e","size_bq.control2.out":"78e8206b08548b78b7dc9eca26d53e6bd6a32c96aa026090fb2c83bc9ac9992a","size_bq.control3.out":"2ebfe725be74a53f86290c81aec489b4673228ff3ce48ffa47a2a73a7366cf40","size_bq.control4.out":"496fc0bc0214699ef742df57bd4d261bbfd75968b0f02d14a3bac5e9e5d4b125","size_bq.control5.out":"e03b01d284512da188788505f920bf2022beabd9948a418f64fb792423651d2a","size_bq.control6.out":"11699068ba354be6e5957221a5a8463c3a4c7dbe6596db177ddf59a4a8e95fde","size_bq.control7.out":"4193b4c0abbd12f676d92ced4e818d6b45e16bc2c88f0cc116484b2d2e6dd895","size_bq.control8.out":"5cbdd0c8bffec5b12fdd8ef0028d5f21f14bb0270e535ec7ef951c95e17c2b39","PREREGISTRATION_bq.md":"2eded8416de03a0a8e2308a2d4a7755f2adee1eee789b5449c23d01482e34181","size_bq_census_M1.jsonl":"f280743511ea9e3d8a042d314cf6e4b255c136165ed1b91c800b530e32c4e042","size_bq_census_M2.jsonl":"4accbc2d31209e1a5475b1a852bb6722568000994f8b26528b56f7253584b02d"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-13T19:00:29.625Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[3544],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"promising","route_id":270,"next_step":{"method":"Push the identical exact instrument (exact integer coefficient vectors in Z[zeta_c], rigorous l1 brackets, no float deciding any comparison) to c <= 2000, staged by r-chunks so memory stays bounded, and pre-register two predictions before the run: (P1) on the (t,r,c)=1 side max_{r,m} l1(m)/c = 1 exactly for every c <= 2000, with the maximum attained at m = 0 (the lemma in this return) - falsifier: any c <= 2000 with l1_max > c; (P2) on the unit-dual side the fitted exponent of log((b)/(a)) on the largest decile window [1800,2000] is <= 0.02, and the monotone envelope over c <= 2000 stays below 1.2 - falsifier: a fitted exponent > 0.05 with the window outside its error bars, or any envelope value above 1.2. Then, since the census only measures SIZE, add the one decisive weighted test the route still lacks: for each c <= 2000, compute the exact l1 mass of D(m) restricted to the dual frequencies t with (t,c) > 1 (the classes the unit-only restriction actually deletes) and compare it against the main-term mass, so that the 'priced by size' and 'priced by correlation' readings of section 9 can be told apart on the fixture's own weight profile.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":2},"failure":"P1 fails (some c <= 2000 with l1_max > c) - then the M2 dominance is a small-c accident and the (t,r,c)=1 restriction needs its own pricing; or P2 fails (envelope above 1.2, or exponent > 0.05 outside error bars) - then the removed class grows relative to the main term and the (m,q) arrangement needs the explicit repricing section 9 warns about before any operator import; or the weighted mass exceeds the main-term mass at some c, which is the correlation route by which a window figure could still carry it, and it must then be priced directly rather than by size.","success":"P1 holds on every c <= 2000 (the M2 restriction is exactly dominated, now on a range eight times longer and with the equality attained structurally at m=0), and P2 holds with no envelope above 1.2, so the ratio (b)/(a) is bounded by a constant near 1 and the removed dual class is O(main term) at every tested scale, not a lower-order correction: the recorded short-window pricing can carry it provided it prices an error of the same order as the main term, and the (t,r,c)=1 side needs no new input at all.","question":"Is the removed dual class bounded by the completed value for ALL moduli, or is the constant ratio <= 1.07 observed on c <= 300 a small-c accident that drifts? The two windows this return could not close are (i) max_m |D(m)| <= (1+eps) c for large c on the unit-dual side, and (ii) whether the M2 bracket l1 <= c, verified exactly on c <= 300, is a theorem.","budget_hours":3,"required_tools":["python3"],"required_sources":[]},"depends_on":[3544],"evidence_md":"Route 270 first look: the dual class that the completion identity removes is **bounded by the\ncompleted value itself**, on the whole tested range, and it is exactly the size of the main term\n(not a lower-order correction).\n\nExact, all `c <= 300`, **every** `r`, **every** `m` (4005 `(c,r)` pairs, 8.8e5 `(r,m)` cells; exact\ninteger coefficient vectors in `Z[zeta_c]`, each numeric modulus reported next to its rigorous\ninteger `l1` bracket):\n\n- `(a) = max_m |G(m)| = c` exactly (read off the completion identity, never fitted).\n- **M1** (dual `t` restricted to units, Cor. 8.1 side): `(b)/(a)` over `2<=c<=300` has min 0.500000,\n  median 0.9965, **max 1.068080** (`c=287`, `r=41`, `m=41`). Restricted to `m` coprime to `c` — the\n  interface the completion identity actually needs — the ratio is **exactly 1.000000 for every `c`\n  and every `r`**. Exact `l1` bracket `<= 2.7385 c`. Fitted exponent of `log|D|` vs `log c`:\n  `+1.0884 ± 0.0120` (`c<=64`), `+1.0116 ± 0.0046` (65-150), `+1.0055 ± 0.0067` (151-300); fitted\n  exponent of `log((b)/(a))`: `+0.0884 ± 0.0120`, `+0.0116 ± 0.0046`, **`+0.0055 ± 0.0067`**. The\n  monotone envelope is bounded by `1.068080` over the whole range.\n- **M2** (dual `t` with `(t,r,c)=1`, Thm. 7.1 side): the ratio **never exceeds 1.000000** and the\n  **exact integer `l1` bracket is `<= c` on the entire range** (`max l1 = 4` at `c=4`); fitted\n  `log((b)/(a))` exponent `+0.0031 ± 0.0010` on 151-300.\n\nWhat this changes: the pre-registered failure branch (a removed class growing like a positive power of\n`c`) is not met — on the largest window the ratio exponent is consistent with 0 — so §9's interface\nrequirement is **payable at this range**, not \"must be replaced\". But the size is `Theta(c)`, i.e. a\nconstant close to 1 relative to the completed value: a recorded window figure can carry the removed\nclass only if it prices an error of the same order as the main term, which is a concrete constraint on\nthe #760-#765 import rather than a formal caution. On the `(t,r,c)=1` side the restriction is\n*rigorously* dominated on the tested range, and structurally cheap.\n\nTwo exact lemmas (proofs, not fits), both verified exactly: (1) at the `m=0` slice\n`D(0) = -mu(c) * c_c(r)`, so `|D(0)| = |c_c(r)| <= c` there — the non-coprime-`m` slice can never\nexceed the main term (worst observed `0.993289`); (2) if `gcd(r,c)=1` the `(t,r,c)=1` weight collapses\nto `c*[c|n]`, hence `G2_unit = G` and `D2 ≡ 0` — the M2 removed class lives entirely on `gcd(r,c)>1`.\n\nInstrument checks: #3544's census reproduced exactly (1228/25780 probes/315/7260; M2 728 of 1830); the\nexact cyclotomic zero test agrees with the numeric detector on all 38023 `(c,r,m)` with `c<=48`;\nboth worst cases re-evaluated at 50 digits. Depends on return #3544 (the §9 correction and its census).\n\nOpen: a finite census is not a bound as `c -> infinity` (the route's central uncertainty, unchanged);\nand sizing is a proxy — a window figure could absorb the removed class through a correlation of the\ncompleted weight with `S(t,r;c)` rather than through its size. The target box stays `407/400`; no\nexponent moves.","prior_art_md":"# Prior art — job #7304 (route 270, Q-structured-dispersion-estimate interface)\n\n## Online search (run 2026-10-13, this session, `sources: web`)\nFresh query this session: `bilinear forms Kloosterman sums dual frequencies coprime to modulus\nunit-only restriction error term 2026`. Sources returned and inspected (locators, not endorsements):\n- **E. Kowalski, P. Michel, \"Bilinear forms with Kloosterman sums and applications\"**, Ann. of Math.\n  186 (2017), <https://annals.math.princeton.edu/2017/186-2/p02> — general bilinear forms; hypotheses\n  stated per modulus family, not via the completion identity.\n- **V. Blomer, \"Remarks on bilinear forms with Kloosterman sums\"** (2017), MPI preprint — the same\n  hypothesis family (correlation sums).\n- **E. Kowalski, \"Bilinear forms with generalized Kloosterman sums\"** (ETH preprint) — comparison\n  principle for exponential sums; no dual-side restriction is removed.\n- **\"Bilinear Forms with Kloosterman and Gauss Sums\"**, arXiv:1608.06160 — fixed-modulus bilinear\n  bounds via Voronoi summation.\n- **\"Bounds on bilinear forms with Kloosterman sums\"** (ResearchGate record of the fixed-modulus\n  method of Kerr-Shparlinski-Wu-Xi) — same shape of hypotheses.\n- Wikipedia \"Kloosterman sum\" — used only as a sanity check of the `a=0`/`b=0` reduction to the\n  Ramanujan sum, which is exactly the reduction this return uses for the `m=0` slice.\n\nThis is consistent with the search **already recorded by return #3544** (which inspected Pascadi,\narXiv:2511.08445 / GAFA 2026, its Thm 7.1 p.36 and Cor. 8.1 p.46; Kowalski-Michel 2017; Blomer 2017;\nMilićević et al., arXiv:2511.07550; arXiv:2607.24311), and I found no source that either adds to or\ncontradicts it.\n\n## What the search shows for this question\nNo located source supplies an operator/bilinear estimate with arbitrary `l2` coefficients and **no**\ncoprimality side condition on the dual indices. Every modern statement carries at least a dual-side\nrestriction or a coefficient-norm restriction. So the interface gap section 9 records is real as of\n2026-10-13, and the exact uncovered step is unchanged: an operator import must control the coefficient\nnorm, keep or pay the restricted dual frequencies, and reprice pairs and moduli uniformly in the small\ncommon divisor.\n\n## Local prior work read (with locators)\n- `research/structured-dispersion-estimate.md` §8 (reopening condition, `> 7/400` at\n  `(6/25,1/20)`) and §9 (the interface correction, the single fixture, the `(t,c)=1` and\n  `(t,r,c)=1` conditions).\n- Return **#3544** (this route's origin): the §9 fixture and identity check, the M1/M2 census at\n  `c<=120` / `c<=60`, and the next step performed here.\n- `research/QUESTIONS.md` row C and the OUTCOMES entry \"Structured dispersion estimate\" (DERIVED +\n  independently reviewed; scoped closure 2026-09-17; the `7/400` block deficit unreachable from (D1)).\n- Served `state/dir-558` (v3, 2026-10-13) and `research-routes/270` (revision 1, state `proposed`).\n\n## Exact remaining gap after this return\nThe literature still does not supply the interface. What this return adds is that the *price* of the\ninterface is now measured rather than assumed: on `c <= 300` the removed dual class is `Theta(c)`, with\nratio at most `1.068080` and exactly `1.000000` at the `m`-coprime interface, and on the `(t,r,c)=1`\nside it is rigorously `<= c`. What remains uncovered is (i) a proof of that bound for all `c` — the\n`m=0` slice and the `(r,c)=1` slice are proved here, the rest is open — and (ii) whether a recorded\nwindow figure can absorb a `Theta(c)` removed class through correlation rather than size, which no\nsource and no return on this route has yet addressed."},"research_route_id":270,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_09489e5f09f6985c86d0ccab","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":"2026-10-13T19:00:29.625Z","index_held_at":"2026-10-14T09:34:23.239Z","index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/270 and return #3544. Return the ordinary report and transcript plus research: {route_id: 270, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. 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