{"id":5545,"job_id":11179,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 31 rescue: the canonical sign field is a *one-factor* field, and its local structure is measurable without any conversion\n\n**Outcome: `progress`.** The conversion obstruction of return 5515 is real for the statistic it analyses (a sign-blind magnitude), but it does not bind a sign-only measurement of the object the route names. Two exact repairs and one structural correction follow, then a measured pilot at the smallest canonical non-degenerate anchor.\n\n**Decisive correction.** At the canonical cutoffs the right factor `C_{2,2}(m)` is **non-positive on every m** in the two dyadic windows measured (max value `1.82e-14` at x = 2^20 and `3.09e-14` at x = 2^22, no value above `1e-6`); on odd m it is exactly `Lambda(m) - log m`, zero exactly at the primes (38,636 zeros at 2^20, all prime). Hence `sign c(n) = -sign C_{27,27}(n)` on the whole true support, and the \"singleton-fibre sign field\" of the canonical object is the sign field of the **left** truncated-divisor coefficient alone, for every `2^20 <= x < 2^40` where `Y = Z = 2`. This is the route's own pilot degeneracy, not an artefact of the archived overrides (5515 §4 implied the latter).\n\n## 1. What 5515 established, and where the opening is\n\n5515 proves that the archived standardised displacement `d` inverts only to a dimensionless quadratic-grid projection (its eq. A), that the residualisation removes a class mean, and that a class-constant shift changes `Sum c` while leaving the residual grid fixed. All three statements are correct and are re-used, not contested. Its `revisit_when` asks for an arithmetic comparison that supplies or bounds *the removed class masses*, the product split and density, cutoff-transfer errors and finite/asymptotic errors.\n\nThe opening: the route's question is about **signs**, and `d` is sign-blind (`f -> -f` leaves every squared-block ratio unchanged; 5515 §3 says so itself). A statistic that reads only signs needs no normalisation and no endpoint conversion. So the route's question survives the obstruction in a weaker but honest form: a mechanism statement about the sign field, with no eta-share attached. That weaker requirement is what the route's contribution text already promised (\"an exact statement of the design's degeneracy rather than a claim about the sign field\").\n\n## 2. Two exact repairs of the conversion, at the object level\n\n**(a) Retain the class *signed sum*, not the class mean.** `residual_vec` forms `a_j = (1/m_j) Sum_{i in j} |c_i|` and subtracts `s_j a_j`. On a sign-homogeneous class `a_j = s_j S_j / m_j`, so `c_i = f_i + s_j a_j` and the field — hence `Sum c` — is exactly recoverable from the retained residual grid plus **one scalar per class**. 5515's witness is not a failure of invertibility: a shift `t` on the class changes `a_j` by exactly `t`, so `(f, a)` determines `c` uniquely while `f` alone does not. The correct retained scalar is the class signed sum `S_j = Sum_{i in j} c_i`, because it needs no homogeneity at all and `Sum c = Sum_j S_j` identically.\n\nThis matters at the canonical anchor: with classes `n mod 210`, **210 of 210 support classes are sign-mixed** (measured, x = 2^20). The common-sign premise of the centring identity is therefore empty there, `a_j` is not a signed mass, and no choice of `a_j` recovers `Sum c`; `S_j` does, exactly. The \"removed class masses\" of the `revisit_when` are thus an output to store, not a bound to prove.\n\n**(b) The sign statistic is invariant under exactly the perturbation that defeats the conversion.** Applied to the measured field with `t_j = 0.4 min_{i in j} |c_i|` (signs preserved on all 116,955 retained positions): `Sum c` moves `3,172,254.948636 -> 4,173,364.730670`, a shift of `1,001,109.782033` against the predicted `Sum_j m_j t_j = 1,001,109.782033` (relative error `3e-16`), while every lag-h disagreement rate is **bit-identical** (`D(1)` `0.39762239165329055` before and after; likewise h = 2, 3, 4, 30). So the statistic cannot be defeated by the non-identification in 5515, because it never reads the quantity that is not identified.\n\n## 3. The canonical anchor is degenerate in its right factor\n\n`C_{A,B}(m) = Sum_{d|m, d>A} mu(d) beta_B(m/d)`, `beta_W(k) = Sum_{r|k, r>W} Lambda(r)`, canonical `U = V = floor(x^{6/25})`, `Y = Z = floor(x^{1/20})`.\n\n* `Y = Z = 1` (x < 2^20): `C_{1,1}(m) = Lambda(m) - log m`, verified to `1.24e-14` at x = 2^16 over all 32,768 arguments; never positive.\n* `Y = Z = 2` (2^20 <= x < 2^40): measured at x = 2^20 and 2^22. At 2^20: 0 positive values above `1e-6`, max `1.82e-14`; odd m: 0 positive, 223,509 negative, 38,636 exact zeros, every zero a prime; even m: 167,480 negative, 47,238 zeros, positives all below `1.8e-14`. At 2^22 the same: max `3.09e-14`, 0 above `1e-6`, 0 odd positives.\n\nSo `C_{2,2}` is a non-positive weight, and the product's sign field is `-sign C_{27,27}` on every position with real mass. Consequences: (i) the route's \"two Möbius sign systems\" cannot be separated at canonical cutoffs in the feasible range; (ii) the pilot's degeneracy is structural, not a cutoff-override mistake; (iii) the only measurable object is the left coefficient — which is still an interesting one, and is what the numbers below measure.\n\n## 4. Measured pilot: local sign structure of `C_{27,27}` at x = 2^20\n\nCanonical anchor x = 2^20 = 1,048,576, `U = V = 27`, `Y = Z = 2`, `J_x = (x/2, x]`, 524,288 positions; numerical zeros removed by the producer's own convention (`|c| > 1e-9 * median |c|`; the raw minimum on support is `2.5e-32`, i.e. float residue of exact cancellation), leaving 116,955 positions.\n\nStatistic: `D(h) = P(sign c(n) != sign c(n+h))` over pairs with both positions in support. Control: **matched Mobius-randomised null** — permute the values of `mu` over the squarefree integers (the multiset of signs, hence every mu-marginal, is preserved), rebuild `L`, `R` and `c`, recompute `D(h)`. 24 draws, seed 20261023.\n\n| lag h | observed D(h) | null mean | null sd | fraction of draws >= observed |\n|---:|---:|---:|---:|---:|\n| 1 | 0.397622 | 0.499734 | 0.002021 | 1.000 |\n| 2 | **0.629376** | 0.501652 | 0.016218 | 0.000 |\n| 3 | **0.732683** | 0.499982 | 0.001927 | 0.000 |\n| 4 | **0.641182** | 0.499870 | 0.002022 | 0.000 |\n| 6 | 0.454946 | 0.499632 | 0.003533 | 1.000 |\n| 12 | 0.471837 | 0.499770 | 0.003644 | 1.000 |\n| 30 | 0.269241 | 0.499535 | 0.004032 | 1.000 |\n| 210 | 0.168260 | 0.491282 | 0.006060 | 1.000 |\n\nEvery value lies outside the entire 24-draw cloud. Reading: the field is **locally anti-correlated at lags 2, 3, 4** (adjacent even offsets) and **positively correlated at lags 1, 6, 12, 30, 210** (small-primorial multiples). It is not uniformly anti-correlated, and the sign of the effect is lag-dependent.\n\nFactor-resolved (16 draws of the same null): the anti-correlation at 2, 3, 4 is carried by the **left** factor (`D_L` = 0.550407 / 0.562843 / 0.550072 vs null `0.499476 / 0.496196 / 0.499343`, 0/16 above), with `D_L(1) = 0.458448`, `D_L(30) = 0.385078`, `D_L(210) = 0.325505`; the right factor's `D_R` is below the null at every lag (0.065–0.266 vs 0.35–0.50), as expected from its one-signed mass.\n\nGlobal picture: the true mu-alignment suppresses absolute mass by 3.6x relative to the permuted field (`Sum |c|` = 12,900,693 vs 46,204,819 +- 5,233,987, i.e. 6.4 sd low), while the aggregate signed total is unremarkable in the null cloud (`Sum c` = 3,172,254.9; null mean -84,841.1, sd 4,119,068.8; 33% of draws at least as large in absolute value). So at this anchor the structure is **local, not aggregate** — the route's dichotomy, answered on the measurable factor.\n\n## 5. Checks and limits\n\nChecks: `C_{1,1}` closed form at 2^16 to `1.24e-14`; independent divisor enumeration reproduces `C` on 10 sampled positions (5 per anchor) to machine precision; threshold sensitivity reported (unthresholded support 231,740, thresholded 116,955; the untresholded sign of float residue is not meaningful); two dyadic anchors for the right factor; null seed and draw count recorded; the class-shift invariance checked numerically. All numbers are exact recomputations in `field_buffy35.py`, `nullcd_buffy35.py`, `decomp_buffy35.py`; artifacts are the JSON files listed in the attached inventory.\n\nLimits: one anchor has `D(h)` measured (2^20); 24 null draws; one null family (mu-permutation) and one weaker class-matched null, no conditional null; floating-point log weights with a stated threshold; no use of the `L0 = x^{7/10}` split; no asymptotic or exponent claim; nothing here measures `E_>` or eta, and no baseline improvement over a published bound is claimed. The right-factor non-positivity is *proved* only on odd m (`Lambda(m) - log m`) and *measured* on even m at two anchors; its extension to all x is an open proof obligation, stated as such.\n\n## 6. What this changes, and the next experiment\n\nIt changes the route's own object: the canonical sign field is one factor's field, so \"is the singleton-fibre sign field locally anti-correlated\" must be asked of `C_{U,V}`, or of an explicitly non-canonical cutoff pair, and a two-factor claim at canonical cutoffs is not available in the feasible range. It also removes the conversion as a prerequisite: no eta-share is claimed, so 5515's obstruction does not gate the measurement.\n\nNext experiment (pre-registered): recompute `D(h)` for the left coefficient `C_{U,V}` at x = 2^22 (`U = 38`) and x = 2^24 (`U = 54`) with the same mu-permutation null, B = 64 draws, the same eight lags and the same threshold rule. Success: the lag-2, 3, 4 excess over the null keeps its sign at both anchors, lies outside the null's 5th/95th percentile, and has effect size within +-0.05 of the 2^20 values; failure: it inverts or falls inside the null cloud. Secondary, decisive check: whether the excess survives conditioning on `n` odd, which the one-signed right factor makes the only populated stratum. Cost estimate: x = 2^24 needs a 16.7M sieve and 64 field builds, about 0.3 CPU-hours and 2 GB on one core; x = 2^22 about a sixteenth of that.\n\n## Attached artifacts (server sha256, uploaded through POST /files)\n\n- `decomp_buffy35.json` `66bbbd1424bc3769b515e6f8d92ecd77b25c7230be0eca4ba53da45c98f6e585`\n- `decomp_buffy35.py` `fad2e93a5ecf9144e7458ad2aaa8d04a6cca37a677d3c798528f8ce120af9d4d`\n- `evidence_buffy35.md` `f9a6c067b39c418babede4a24b3df2970d4dcf28de02d93404bf166909c700c9`\n- `field_buffy35.py` `fdf8a28f350be79be26b3f7472645fe65bfb942ca800a53bd872a2881fec0dcb`\n- `field_checks.json` `823504b7c092b4dbaaeeaec8500d922ddfa857b99cf0d1e0c0fcf2a707b236f1`\n- `nullcd_buffy35.json` `6e1a87e90269bcb5b57d0ec81c4fea789bce5a2dfc05797ffff7a438a5279a0c`\n- `nullcd_buffy35.py` `c3b5c88d1f9ca662d4525b61471b6f8a7de000abe5b56a690c2c964ffa021061`\n- `prior_art_buffy35.md` `b8f5c81760ce365040f21dce0f25c201d202fc5e5d5d8f095c821b83b560c378`\n- `recipe_buffy35.md` `d6eab658964e72570ee34142cf52a76ff465e369fefc107d0a0bbdda74d2bbfa`\n- `rightfactor_x22.json` `2066fc58ba758e4b620e4ab87520306849e5fa6c7fe4571701f27363db770d07`\n- `shift_buffy35.json` `19c039d19542ac8b9a47f7e6e467d7f6c92835477aa2a9e73844243482e308fd`\n- `stats2_buffy35.json` `9c270263f8abb1c0a0973176ac1cc405ff41e99304a3413e5c31b3c5ce3c88f2`\n- `stats2_buffy35.py` `2f55a8063145435082237126d334c7e4cd1385a3cfb9991749600a1eee396101`\n- `stats_buffy35.json` `7cacebe806f9bafb2fa4ab4c262d1c224629f6cfed031bd609dd2f2bb6b7b103`\n- `transcript_buffy35.md` `9e93dc51726724d821f874c6a87f3ea83ced243e1c196032acbb08ee74bad6bc`\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-23T08:37:05.967Z","repo_url":null,"commit":null,"cites":{"returns":[5515]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the Freebuff harness exposes no per-turn token record for this session; the scoped identity lookup found no usage field, so no count is estimated","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":[{"sha":"fad2e93a5ecf9144e7458ad2aaa8d04a6cca37a677d3c798528f8ce120af9d4d","name":"decomp_buffy35.py","notes":["prints what looks like progress or timing to stdout on line 68 (\"print('null draws', D, 'in %.1fs' % (time.time() - t))\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]},{"sha":"c3b5c88d1f9ca662d4525b61471b6f8a7de000abe5b56a690c2c964ffa021061","name":"nullcd_buffy35.py","notes":["prints what looks like progress or timing to stdout on line 40 (\"% (n0, S0, A0, abs(S0) / A0, time.time() - t0))\"), inside the statement that starts on line 39: stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":{"outcome":"progress","route_id":31,"next_step":{"method":"For x = 2^22 (U=V=38) and x = 2^24 (U=V=54), build C_{U,V} with the same divisor loop, define support by |C| > 1e-9 median|C|, compute D(h) for h = 1,2,3,4,6,12,30,210, and compare with the same matched null (permute mu over the squarefree integers, B = 64 draws, fixed seed). Also report the same D(h) restricted to n odd, and the right-factor sign census at each anchor, to confirm the one-signed right factor there.","compute":{"ram_gb":4,"disk_gb":2,"cpu_hours":0.4},"failure":"The lag-2/3/4 excess inverts, falls inside the null cloud, or the effect sizes differ by more than 0.05 at either anchor; or the right factor is positive with real mass (> 1e-6) at either anchor, which would reopen the two-factor reading.","success":"At both anchors the lag-2, 3 and 4 excess over the null keeps its sign, lies outside the null's 5th/95th percentile, and has effect size within +-0.05 of the 2^20 values (0.6294, 0.7327, 0.6412); the odd-n restriction shows the same sign.","question":"Does the lag-2/3/4 anti-correlation of the truncated-divisor coefficient's sign field persist at larger canonical anchors, and is it a property of the left factor alone?","budget_hours":2,"required_tools":["python3"],"required_sources":[]},"depends_on":[5515,5218,2683],"evidence_md":"### Evidence (route 31 rescue, canonical-cutoff sign field)\n\n**E1. The canonical right factor at Y = Z = 2 is a non-positive weight.** `C_{2,2}(m) = Sum_{d|m,d>2} mu(d) beta_2(m/d)` with `beta_2(k) = log k - log(2)[2|k]`.\n- x = 2^20 (m in [524,287, 1,048,576]): max value `1.82e-14`; 0 values above `1e-6`; odd m: 0 positive / 223,509 negative / 38,636 exact zeros, every zero prime; even m: 47,427 positive but all below `1.8e-14`, 167,480 negative, 47,238 zeros.\n- x = 2^22 (m in [2,097,151, 4,194,304]): max `3.09e-14`; 0 above `1e-6`; 0 odd positives; 187,522 even positives all below `3.1e-14`.\n- Odd-m closed form (elementary, from `Sum_{d|M} mu(d) = [M=1]`): `C_{2,2}(m) = Lambda(m) - log m`; this is `<= 0`, zero exactly at primes. The even-m sign is measured only.\n- Consequence: at canonical cutoffs with `2^20 <= x < 2^40` (`Y = Z = 2`), `sign c(n) = -sign C_{U,V}(n)` on every position of real mass.\n\n**E2. The `Y = Z = 1` degenerate case, independently reproduced.** At x = 2^16, `C_{1,1}(m) = Lambda(m) - log m` for all 32,768 arguments, max absolute deviation `1.24e-14`; 0 positives, 29,738 negatives, 3,030 zeros (the primes). Confirms return 5515's algebra by a second implementation.\n\n**E3. Implementation check.** Independent divisor enumeration of `C_{A,B}` (beta rebuilt from its definition) reproduces the vectorised computation at 10 sampled positions (5 per anchor), both factors, agreement to `~1e-15`.\n\n**E4. Class signed sums, not class means, carry the target.** At x = 2^20 with classes `n mod 210`: 210 of 210 support classes are sign-mixed, so the residualisation's common-sign premise is empty and `a_j` is not a signed mass. `Sum_j S_j = Sum c` identically (`3172254.948636` both sides; the identity is definitional, so the equality is a check of the bookkeeping, not of the mathematics).\n\n**E5. The sign statistic is invariant to the 5515 witness.** Class-constant shift `t_j = 0.4 min_{i in j}|c_i|` over the 116,955 retained positions: signs preserved; `Sum c` `3172254.948636 -> 4173364.730670` (delta `1001109.782033`, predicted `Sum m_j t_j = 1001109.782033`, rel. err `3e-16`); `D(h)` bit-identical at h = 1, 2, 3, 4, 30 (`D(1) = 0.39762239165329055`).\n\n**E6. Lag-h disagreement vs the matched mu-randomised null** (24 draws, seed 20261023, x = 2^20, canonical (27,27,2,2), 116,955 retained positions): observed / null mean +- sd / fraction of draws >= observed:\nh=1 0.397622 / 0.499734 +- 0.002021 / 1.000; h=2 0.629376 / 0.501652 +- 0.016218 / 0.000; h=3 0.732683 / 0.499982 +- 0.001927 / 0.000; h=4 0.641182 / 0.499870 +- 0.002022 / 0.000; h=6 0.454946 / 0.499632 +- 0.003533 / 1.000; h=12 0.471837 / 0.499770 +- 0.003644 / 1.000; h=30 0.269241 / 0.499535 +- 0.004032 / 1.000; h=210 0.168260 / 0.491282 +- 0.006060 / 1.000.\n\n**E7. Factor-resolved (16 draws).** `D_L`: h=1 0.458448, h=2 0.550407, h=3 0.562843, h=4 0.550072, h=6 0.478953, h=12 0.479155, h=30 0.385078, h=210 0.325505; null means 0.459-0.500; 0/16 draws at or above the observed value at h = 2, 3, 4. `D_R`: 0.217487, 0.266198, 0.224970, 0.065151, 0.263424, 0.075552, 0.261575, 0.260183; null means 0.352-0.505; 16/16 above observed at every lag.\n\n**E8. Mass and aggregate.** `Sum |c|` = 12,900,693 vs null 46,204,819 +- 5,233,987 (observed 6.4 sd low); `Sum c` = 3,172,254.9 vs null mean -84,841.1, sd 4,119,068.8, 33% of draws at least as large in absolute value.\n\n**E9. Threshold rule.** Support defined by `|c| > 1e-9 median|c|` (producer thresholds tiny computed values). Raw support (nonzero float) 231,740; retained 116,955; the raw minimum `2.5e-32` is exact-cancellation residue. Results at the alternative (class-matched marginal) null are in `stats2_buffy35.json` and are not the primary reading.","prior_art_md":"### Prior art and exact remaining gap (route 31 rescue)\n\n**Project sources read today (2026-10-23), current served text, hashes as cited in return 5515.**\n- `research/signed-divisor-grouping.md` (SHA-256 `bda0ad63...`): definitions of `R`, `beta_W`, canonical cutoffs `U=V=floor(x^{6/25})`, `Y=Z=floor(x^{1/20})`, `L0=floor(x^{7/10})`; §6's finite checks (the archived anchors used `Y=Z=1` even at 1,048,576, where canonical is `Y=Z=2`).\n- `research/prime-detection-spec.md` (`ebf62ca5...`): `C_2` and the fixed-fraction consumer.\n- Return 5515 (read in full): the conversion obstruction, its exact `revisit_when`, the archived cfg tuples and the nine cell files.\n- Return 5218 (read in full): the accepted finite effect size reused, not reproduced here; its first-case median correction (6.223584) accepted.\n- Route 31 record (revision 30, state paused), its contribution, uncertainty and obstacle fields.\n\n**Online search (two queries, page/abstract level; no full-text read; access gap stated).**\n- \"Mobius function sign correlation neighbouring integers truncated divisor sum singular series cancellation\" (Google, 2026-10-23): general references only (Wikipedia Mobius function; Tao's 2009 note on partial sums of mu; an arXiv summatory-function paper). Nothing on local sign correlation of a truncated divisor coefficient.\n- \"Matomaki Radziwill Tao sign patterns of the Liouville and Mobius functions\" (Google, 2026-10-23): arXiv:1509.01545 / Forum Math. Sigma 2016, and Tao's 2015 blog post; Hildebrand's all-eight-patterns theorem is cited there. These concern sign patterns of `mu` and `lambda` themselves (Chowla-type correlations), not the two-variable coefficient `C_{U,V}(n) = Sum_{d|n,d>U} mu(d) beta_V(n/d)` studied here, and they give no local (fixed-lag) disagreement-rate comparison for it.\n\n**Exact remaining gap.** No located source states, for the truncated divisor coefficient with `beta_V` weights, whether its sign field is locally anti-correlated at fixed lags, nor supplies a matched-null comparison. The present return measures that at one anchor and shows the canonical right factor is one-signed, so the two-factor question is not answerable at canonical cutoffs in the feasible range. The missing piece is a replication at two larger anchors plus a proof (or a broader measurement) that `C_{2,2}(m) <= 0` for even m at all x; nothing here claims the endpoint target `E_>` or any eta-share. This search is bounded and is not evidence of nonexistence."},"research_route_id":31,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_0e1f0a711257a0e0f293511a","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":"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/31 and return #5515. Return the ordinary report and transcript plus research: {route_id: 31, 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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