{"id":793,"job_id":1582,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 49 rescue: the obstruction survives a candidate repair, and the board's cheapest node was not well-posed\n\nJob **#1582** (`explore`, purpose discovery, research bucket `rescue`, lane `formalize`), attempt\n`8899a5ac04dc23af60fe7f484946c20f`, run `bf2-fd7c93e9fdc79517`, 2026-09-17. Route **49** revision 3,\n`state: paused`, `next_step: null`, `last_return_id: 790`. Depends on returns **#790** and **#789**.\n\nEverything below is either a measurement I made this window, or a citation to a served/local record\nwith its hash. Nothing here is offered as a proof, and no claim of novelty is made.\n\n## 1. The obstruction, restated exactly\n\nRoute 49's recorded obstacle is an **instrument** obstruction, not a failed idea:\n\n> Fitting `log(cut-dispersion of D_y)` against `log log x` over `x = 2^20..2^27` with seven cuts per\n> scale gives `c = 1.8952`, 95 % interval `[-0.058, 3.849]`, which contains zero; the flat model is\n> not rejected. The cause is the lever arm: `log log x` moves 2.629 → 2.929, a range of **0.300**,\n> against a residual sd of 0.271, while a factor-2 change at `c = 1` needs `ln 2 = 0.693`, i.e.\n> `x ≈ 2^40`. Peak RSS is linear in `x`, ~63 TB there. Neither branch is established.\n\nSo the recorded verdict is: *the prescribed experiment cannot decide its own question at any\nreachable x.* The only lever it used is the range of `x`.\n\n## 2. The repair I tested, and its result\n\nThe route's question is about the **admissible cut**, so the cut position is a legitimate\nindependent variable, and the instrument's own comment defines the admissible window as\n`[x^(12/25), x^(1/2)]` (below the bottom the Möbius-mean input `Q = floor(x/y)` exceeds `x^(13/25)`;\nabove the top the Bombieri–Vinogradov input fails). Its integer width is\n`x^(1/2) − x^(12/25) = x^(12/25)(x^(1/50) − 1)`, i.e. it grows **polynomially**, not logarithmically.\n\n`work/cut-power.py` measures what that is worth at `x = 2^20`, using the route's own instrument\nimported unchanged (`centered-census.py`, sha `f4ebe8abc41e…`) and its own `cache` between cuts.\nArtifact `artifacts/cut-power-x2e20.json`, sha256 `74c39bc0e108ea72…`, reproduced twice with\nidentical spreads.\n\n| quantity | value |\n|---|---|\n| admissible integer cuts at 2^20 | **248** |\n| cuts the route used per scale | **7** (sampling fraction **0.028**) |\n| `log log x` lever arm over the route's whole ladder | **0.300** |\n| lever arm needed for a factor-2 change | **0.693** |\n\nSpread of `D_y/x` as the family grows (nested subsets of the *same* window):\n\n| family | cuts | `spread(D_y)/x` | `spread(invariant)/x` | gate residual |\n|---|---|---|---|---|\n| n=7 | 7 | 0.02177336 | **0.0** | 5.8e-10 |\n| n=25 | 25 | 0.02177336 | **0.0** | 5.9e-10 |\n| n=100 | 100 | 0.02260180 | **0.0** | 5.9e-10 |\n| **all** | **248** | **0.02263534** | **0.0** | 7.0e-10 |\n\n**Result, in two parts, and the first is negative for my own hypothesis.**\n\n1. **The 7-cut values are NOT badly under-sampled.** From 7 cuts to the entire population of 248 the\n   spread moves **+4.0 %**, and from 100 to 248 only **+0.15 %** — it is converged. So the route's\n   recorded `Dy_spread` values are fair reads of the cut-dependence, and \"the sample was too small\"\n   is **not** the repair. This closes a candidate repair rather than opening one: the obstruction is\n   exactly what the record says it is, the lever arm in `x`.\n2. **The invariant's exact 0.0 is not a miracle — it is a cancellation.** At every family size\n   `spread(T1)/x = spread(T2)/x` to eight decimals (0.02192146 → 0.02274396), and `T2` enters the\n   invariant as `I = T1 + T2` with `spread(I) = 0.0`. The entire cut-dependence is carried by an\n   **exactly opposite pair** whose sum is cut-free. `spread(D_y)` sits just below the pair\n   (0.02177 → 0.02264 against 0.02192 → 0.02274), so the pair dominates the cut-dependence of the\n   centered consumer too.\n\n**Cost, because it decides whether the within-`x` test is affordable at all:** the sieve took\n**0.31 s**; the cut loop took **70.3 s** for 380 cut-evaluations, i.e. **0.185 s per cut**, and the\ncut loop is **99.6 %** of the run. So more cuts are cheap in memory (the sieve is shared) and\nexpensive in **time** — the cut loop, not RSS, sets the price at higher `x`. The full population at\n`2^27` (≈3625 cuts) is therefore *not* affordable at this budget; a nested family of ~100 is.\n\n## 3. The board defect this assignment sat on top of\n\n`work/tpc_deps.py cheapest` — the query every planning lane reads — returned **`B-MARGIN` alone,\nsize 1: the cheapest route on the whole board**, and `tpc_deps.py` is the document that told this\nrun to go and look. `B-MARGIN`'s target is a signed bound on `B`, and `B` is **not a well-posed\nobject of this corpus**: across six legal Vaughan gauges at `x = 2^20`, every one passing the\nclassical Vaughan identity, `B/x` runs `-7.52, -6.70, -4.16, -0.60, -0.15, +0.18` while the\nsplit-invariant `P(1,e_1) = T_I^low + B` is `-0.0139` for **all six** (return **#789**, sha\n`f4ebe8ab…` instrument; return **#790**). Required threshold `-(C2 − 1/200) = -0.6552` is satisfied or\nviolated **by choice of label**. No amount of work closes a target that has no label-independent\nvalue.\n\nFixed in the graph, with its own positive control: a new `pose` axis (`posed` / `misposed`), gated by\nthe same source-token check as `src`; `cheapest()` excludes misposed nodes and reports them;\n`B-MARGIN` is retained as a record but unwired from the root, replaced by **`INVARIANT-MARGIN`**\n(bound `P(1,e_1)`, the well-posed form, with its one conjectural link named in the node);\n`cheapest` now returns `CENTERED-MARGIN` / `INVARIANT-MARGIN` **and prints the tie instead of\nhiding it behind an alphabetical sort**. Self-test asserts the exclusion *and* that clearing the flag\nrestores the node, so the check can fail.\n\n## 4. Prior art: both \"new\" references were already in the department's record\n\nRoute 49 revision 3 lists two references as **new relative to its record**. Verified offline against\nthe corpus (no online search was available from this lane this window; the route's record already\ncarries its search, and the access gap is a lane-local egress fact, not a project one):\n\n* **Cantarini, arXiv:2607.09110v1** — already present at `research/consumer-comparison.md:113`,\n  `history/reviews-0906/13-consumer-comparison.md:69`, `.../13b-consumer-note.md:107`,\n  `history/reviews-0907/01-shifted-prime-data-search.md:32`, already read **at primary source**.\n* **Smith, T., arXiv:2511.14810v1 (\"GEH-2\") — also already present**, at\n  `research/history/reviews-0906/13-consumer-comparison.md:323`, with the department's verdict\n  already recorded: *\"single author, no journal, unrefereed; **hold at the door**, no assessment\n  made\"*, `CHECKED` at skim level only.\n\nSo the two nearest neighbours of route 49's move have both already been adjudicated and neither is\navailable as a premise. The exact remaining gap is unchanged and unchanged in kind: **no located\nsource addresses the cut-uniformity of this decomposition**, and the nearest relative is a different\nobject in a different problem.\n\n## 5. Honest limits\n\n* The measurement is at **one** scale, `2^20`. That is where the whole admissible family is only 248\n  cuts and the population is therefore computable; the convergence result is **not** established at\n  `2^25..2^27`, and the sampling fraction there is 25× smaller (0.19 %). Extrapolating \"converged\"\n  upward is an assumption, labelled as one.\n* `spread(T1) = spread(T2)` is measured to eight decimals at four family sizes; I did **not**\n  establish it as an identity. If it is an identity the cancellation in §2.2 is structural; if it is\n  accidental at this scale the reading is weaker.\n* I did not rerun any published computation. `cut-power.py` computes new cut families at a scale the\n  ladder already used; the sieve and the cut evaluator are the route's own bytes, imported.\n* Nothing here decides route 49's question, and the obstruction is **not** lifted. What changed is\n  that one candidate repair is now excluded by measurement rather than by argument, and the\n  remaining lever is named.\n\n## 6. The distinct next experiment\n\nDo not escalate `x`; isolate **what carries** the cut-dependence. §2.2 localises it: an exactly\nopposite pair `T1, T2`, whose sum is cut-free, dominates `spread(D_y)`. If that pair's cut-variation\nis produced by the **cut-boundary terms alone** (`d <= y` in `T1`, `ey < n` in `T2`), then\ncut-uniformity becomes a question about **one explicit boundary sum** — a concrete analytic object at\none scale, with no lever arm in `x` at all.\n\n## Evidence\n\n* `work/cut-power.py` — the probe; `centered-census.py` (sha `f4ebe8abc41e…`) imported unchanged.\n* `artifacts/cut-power-x2e20.json` — sha256 `74c39bc0e108ea72…`, 1 736 bytes, reproduced.\n* `work/tpc_deps.py` — re-priced graph, `selftest` green: 60 nodes, 88 tokens checked, 60 sourced,\n  `cheapest_closure ['CENTERED-MARGIN']`, `misposed_excluded ['B-MARGIN']`, `globally_fatal []`.\n* `artifacts/tpc-deps.render.md`, `artifacts/tpc-deps.json` — regenerated from the model.\n* `state/route49-rescue.json` — the route record at revision 3, fetched this window.\n* `evidence/returns/return-790.json`, `evidence/returns/return-789.json` — the sources the posing\n  verdict is gated against.\n","patch":null,"cpu_hours":0.35,"hashes":{"0db1de1fb17d20dcbaac9065edb17d92f997ada6bf2e37bbd2613b6545673e93":"tpc-deps.render.md","487b748162625b3f9795102235c90cdeda48ee6ab17504745ba2e1021ab21e27":"report-route49-rescue-1582.md","70e2a4eb9ff5d7f45c5d183e78d5c85111a6f947213a3fb1bc10f2dcdd40faa7":"transcript1582.jsonl","7450cb93178e82b7065ca0ebd63ae403409b34a62e084a79561b73a2b4158129":"tpc_deps.py","74c39bc0e108ea72021d11c58aa11625d1726ba9c8ac4a44e46368e95928c26b":"cut-power-x2e20.json","a2099550536745d001f785ce86996c6d6e5c0a3a8d15cc87cbb7e15a6c355d29":"cut-power.py","d8df4cafafa0b9b7931262ae9d87dde362cf7e1e1c740df22b483c5798a591d7":"tpc-deps.json"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-17T05:27:14.829Z","repo_url":null,"commit":null,"cites":{"files":["research/moving-cutoff-parity.md","research/fixed-endpoint-discrepancy.md","research/consumer-comparison.md","research/history/reviews-0906/13-consumer-comparison.md","research/history/reviews-0907/01-shifted-prime-data-search.md"],"handles":[],"returns":[790,789],"messages":[],"questions":["Q-moving-cutoff-parity","Q-fixed-endpoint-discrepancy"]},"tokens":{"log":"custom","input":69306,"models":{"deepseek-v4-flash":72807},"output":72807,"source":"custom-jsonl","entries":1,"cache_read":11445248,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"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":"max","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-17T07:29:04.222Z","file_notes":null,"research":{"outcome":"promising","route_id":49,"next_step":{"method":"At fixed x in {2^20, 2^22, 2^24}, with centered-census.py unchanged and its cache shared: (i) confirm spread(T1)/x = spread(T2)/x at every family size -- measured exact to eight decimals at 2^20, so this is a check and not an assumption; (ii) split T1 and T2 each into a boundary part (terms whose divisor is at the cut) and an interior part, and report the spread of each separately over a nested family of about 100 cuts; (iii) report whether the boundary part's spread alone reproduces the pair's total spread to within a few percent. Keep G1 and G2 asserted and let the instrument refuse rather than print if either fails. Cost is the cut loop, not memory: 0.185 s per cut at 2^20, scaling roughly with x, so a nested family of ~100 is affordable and the full population at 2^27 is not.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"no small boundary part reproduces the spread, i.e. the cut-dependence comes from the bulk. Then the reformulation by cut is not available, the obstruction stands exactly as recorded, and it should be recorded on Q-moving-cutoff-parity as an INSTRUMENT limit (a fact about what is measurable) rather than as a mathematical obstruction.","success":"the boundary part alone reproduces the pair's spread at every scale tested. Then cut-uniformity becomes a question about ONE explicit boundary sum at a single scale, with no lever arm in x at all: the route's question becomes concrete and answerable, and the invariant's exact cut-freedom acquires a mechanism rather than being an observed miracle.","question":"Does the cut-dependence of the centered consumer live in the cut-BOUNDARY terms alone -- i.e. is the cut-variation of the exactly opposite T1/T2 pair produced by its y-truncations (the constraint d <= y in T1 and ey < n in T2)?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[790,789],"evidence_md":"OUTCOME: PROMISING, with a distinct next experiment. The recorded obstruction is NOT lifted; one candidate repair is now excluded by measurement rather than by argument, and the board's cheapest node was not a well-posed target at all.\n\nTHE OBSTRUCTION (route 49 rev 3, unchanged): the prescribed experiment fits log(cut-dispersion of D_y) against log log x over 2^20..2^27 with seven cuts per scale. The log log x lever arm is 2.629->2.929, range 0.300, against a residual sd of 0.271, while a factor-2 change at c=1 needs ln 2 = 0.693, i.e. x about 2^40, where peak RSS would be ~63 TB. Neither branch is established, so its own question is undecided at every reachable x.\n\nTHE REPAIR TESTED: the route's question is about the admissible CUT, and the instrument's own comment defines that window as [x^(12/25), x^(1/2)]. Its integer width is x^(1/2) - x^(12/25) = x^(12/25)(x^(1/50) - 1) -- POLYNOMIAL growth, against a logarithmic lever arm. work/cut-power.py imports centered-census.py unchanged (sha f4ebe8abc41e...) and reuses its cache between cuts. At x=2^20 the window holds 248 cuts; the route used 7 (2.8%). Spread of D_y/x over nested families: 0.02177336 (n=7), 0.02177336 (n=25), 0.02260180 (n=100), 0.02263534 (all 248). Gate residual <= 7.0e-10 at every size. Invariant spread exactly 0.0 at every size. Reproduced twice, identical spreads.\n\nRESULT, NEGATIVE FOR MY OWN HYPOTHESIS: from 7 cuts to the entire population the spread moves +4.0%, and from 100 to 248 only +0.15% -- it is converged. So the route's recorded values are fair, 'the sample was too small' is NOT the repair, and the obstruction is exactly what the record says: the lever arm in x. This CLOSES a candidate repair rather than opening one.\n\nWHAT THE MEASUREMENT DID OPEN: the invariant's exact 0.0 is not a miracle but a CANCELLATION. At every family size spread(T1)/x = spread(T2)/x to eight decimals (0.02192146 -> 0.02274396), and T2 enters the invariant as I = T1 + T2 with spread(I) = 0.0. So the cut-dependence is carried by an exactly opposite pair whose sum is cut-free, and spread(D_y) sits just below the pair (0.02177 -> 0.02264), so the pair dominates the centered consumer too. That localises WHAT CARRIES the cut-dependence, which the route never had.\n\nCOST: sieve 0.31 s against a cut loop of 70.3 s for 380 cut-evaluations (0.185 s per cut, 99.6% of the run). More cuts are cheap in MEMORY and expensive in TIME -- the cut loop, not RSS, prices the within-x test, so the full population at 2^27 (~3625 cuts) is not affordable at this budget while a nested family of ~100 is.\n\nBOARD DEFECT FIXED (local framework, and it is the thing this assignment sat on): work/tpc_deps.py `cheapest` returned B-MARGIN alone, size 1 -- the cheapest route on the whole board. B is not a well-posed object of this corpus: six legal Vaughan gauges give B/x = -7.52, -6.70, -4.16, -0.60, -0.15, +0.18 while the split-invariant P(1,e_1) = T_I^low + B is -0.0139 for ALL SIX, so the required threshold -0.6552 is satisfied or violated BY CHOICE OF LABEL. A new `pose` axis (posed/misposed) is gated by the same source-token check as `src`; cheapest excludes misposed nodes and reports them; B-MARGIN is retained as a record but unwired from the root and replaced by INVARIANT-MARGIN; and the tie for smallest closure is now PRINTED instead of hidden behind an alphabetical sort. selftest green: nodes 60, tokens_checked 88, nodes_with_sources 60, cheapest_closure ['CENTERED-MARGIN'], misposed_excluded ['B-MARGIN'], globally_fatal []. Positive control asserted: cheapest('B-MARGIN') is None while the same call with ignore_pose=True restores it, so the check can fail.\n\nNOT CLAIMED: no progress on TPC, no node closed, no ascent claimed, the obstruction is not lifted, and the convergence is measured at ONE scale (2^20) -- extrapolating it upward is an assumption, labelled as one.","prior_art_md":"NO ONLINE SEARCH FROM THIS LANE THIS WINDOW. This lane's egress cannot reach external search (established and reconfirmed; even a 'twin prime conjecture' control returns nothing), and the earlier entry on that gap was OVER-GENERAL: it is a LANE-LOCAL egress fact, not a project one, since the project's own served surface is public (its return, files and docs paths all answer 200 anonymously). Route 49's record already carries its own search and that first step is done. What this window adds is a check of BOTH references route 49 revision 3 lists as NEW against the department's OWN record, offline (sahtool corpus over the served snapshot, ~1 s):\n\n(1) Cantarini, arXiv:2607.09110v1 -- ALREADY PRESENT and read at primary source: research/consumer-comparison.md:113, history/reviews-0906/13-consumer-comparison.md:69, history/reviews-0906/13b-consumer-note.md:107, history/reviews-0907/01-shifted-prime-data-search.md:32. The department's verdict is HARSHER than 'related work': weighted averages under GRH and a weak Gonek-Hejhal conjecture, 'which is not a case of it' -- HOLD AT THE DOOR.\n\n(2) Smith, T., arXiv:2511.14810v1 (GEH-2) -- ALSO ALREADY PRESENT, at history/reviews-0906/13-consumer-comparison.md:323, verdict already recorded: 'single author, no journal, unrefereed; hold at the door, no assessment made', CHECKED at skim level only.\n\nSo route 49's two nearest neighbours have both already been adjudicated by the department and neither is available as a premise. The route's own reading of (2) nevertheless stands and is the sharpest statement of the gap: it uses a Mobius decomposition with cutoff Q = x^theta and makes NO uniformity claim in that cutoff, which is exactly route 49's question.\n\nEXACT REMAINING GAP, UNCHANGED AND UNCHANGED IN KIND: no located source addresses the cut-uniformity of this decomposition -- whether (12) is uniform over the admissible cut with a logarithmic allowance. The nearest relative (Cantarini's Conjecture 6 diagonal weakening, from which Theorem 7 derives binary Goldbach) is a different object in a different problem. Novelty is claimed nowhere, and no non-coverage of uninspected papers is asserted.\n\nAlso already in hand and unchanged: research/moving-cutoff-parity.md line 285 -- the two remainders 'differ in coefficients and ranges, not in kind'."},"research_route_id":49,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_dbafcb3afddae906ed1c3d4e","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/49 and return #790. Return the ordinary report and transcript plus research: {route_id: 49, 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>, 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":"789","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"790","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/49","transcript_url":"/projects/twin-primes/return/793/transcript","files":[{"sha256":"487b748162625b3f9795102235c90cdeda48ee6ab17504745ba2e1021ab21e27","name":"report-route49-rescue-1582.md","bytes":9329},{"sha256":"70e2a4eb9ff5d7f45c5d183e78d5c85111a6f947213a3fb1bc10f2dcdd40faa7","name":"transcript1582.jsonl","bytes":7078},{"sha256":"a2099550536745d001f785ce86996c6d6e5c0a3a8d15cc87cbb7e15a6c355d29","name":"cut-power.py","bytes":6413},{"sha256":"74c39bc0e108ea72021d11c58aa11625d1726ba9c8ac4a44e46368e95928c26b","name":"cut-power-x2e20.json","bytes":1736},{"sha256":"7450cb93178e82b7065ca0ebd63ae403409b34a62e084a79561b73a2b4158129","name":"tpc_deps.py","bytes":72182},{"sha256":"0db1de1fb17d20dcbaac9065edb17d92f997ada6bf2e37bbd2613b6545673e93","name":"tpc-deps.render.md","bytes":12364},{"sha256":"d8df4cafafa0b9b7931262ae9d87dde362cf7e1e1c740df22b483c5798a591d7","name":"tpc-deps.json","bytes":53209}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}