{"id":1014,"job_id":1912,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1912 — triage of route 77: the two-tail orbit reading is already published; the repair's citation base is not\n\nRun `run_20260918_164400_X2wqng` · attempt `02b3a24a3a6e2da3cc500fb5c746a50e` · session\n`41760691b4dc804a2779d830` (1 of 1) · department `dept_c326cb5ae203e5d0d94f8db1` · type `explore`,\nstage `triage`, route **77**, general mode · model `deepseek/deepseek-v4-flash`, `X-Effort:\nunmeasured` · tool `sah/14` (`38a08cad…`) · ledger `job1912-ledger.py` / `.log` (19/19 PASS,\n`exit_code 0`, 0.04 s wall, one bounded `exec`).\n\n## Verdict\n\n**`promising`, with a corrected and much cheaper next step than the one route 77 pre-registered.**\nTriage searched first (see `prior_art_md` in `research-1912.json`) and read the served sources the\nroute names before any computation (README gotcha 49). Three findings:\n\n1. **The orbit reading is already published, so nothing here is new.** The served producer\n   `research/natal-cap-13-anchored-calm.js` (embedded run 2026-08-18, node v22.21.0) states it in its\n   own header, before #41 and before return #1013: mirror pairing `stat(t) = stat(W−t)` **as integer\n   phases**, *\"the unique fixed point INSIDE the window is t = W/2; and the anchor's mirror partner is\n   the integer phase W itself … which lies just OUTSIDE the enumerated window\"* (ledger A6). No\n   novelty is claimed for the route's reading; it is a corpus reading.\n2. **The route's pre-registered next experiment is now partly a reproduction, so it must not be run\n   as written.** The route asks to extend the census to @17 and @19 with that same producer. The\n   producer already **enumerates the full ensemble at @17** (`FULL ENSEMBLE (all 510510 rotations)`,\n   anchor VR row `0.553 0.002 10/510510`; also exact at @11 `42/2310` and @13 `1182/30030` — ledger\n   A7/A9), and it already answers the referee's open sub-item at three levels: the `W/2` (δ=0) VR\n   percentile is `100.0 / 99.3 / 100.0` at @11/@13/@17 with `G(W/2,q)` even for all q and `SP(W/2)=0`\n   exactly (ledger A10/A11). @19's full ensemble (9.7 M rotations) is unaffordable *and* not\n   discriminating: the loudness of `W/2` is a per-prime theorem (calm-lemma: *Mirror-Phase Doubling\n   Lemma [PROVEN, every q, every x]*), not an ensemble statistic.\n3. **New, and the reason the route still earns one bounded experiment: the repair's own citation base\n   is wrong on both of its claims, and the one corpus document that carries the slip is a different\n   document.** Return #1013's `prior_art_md` says `research/anchored-windows.md` §4 *\"ALREADY states\n   the correct reading\"*. The served file does not contain the word *mirror* anywhere (0 occurrences,\n   ledger A1) and its §4 is *\"The dip constant is e^{2γ}/4, and the 0.89 reading is a blend\"* (A2) —\n   the repair sentence lives only in the producer's header, which is not a document. The\n   `natal-cap-19-calm-lemma.md` claim is right in location and is *stronger* than reported: its\n   **Lemma 3** is titled *\"the two degenerate phases, concatenation and duplication\"* and states\n   *\"the pair is **self-paired iff 2t ≡ 0 (mod W)**, i.e. exactly t = 0 and t = W/2\"* (A3/A4) — a\n   modulo-W condition, while the ensemble's own definition (the producer again: *\"t and t+W are\n   different phases here, since W mod q ≠ 0\"*) pairs the anchor with the phase **W**. So the corpus\n   contradicts itself on exactly the point route 77 repairs, and the manuscript regressed against\n   its own producer rather than against a document.\n\n## The exact discriminator the ledger settles (definitions only, no census)\n\nFrom the served definitions — mirror on strike classes `c ↦ (W−2−c) mod q`, rotation `t` holds the\nclass pair `{t, t−2}` — the two zeros of `2t ≡ 0 (mod W)` are **not** the same kind of object:\n\n| phase | class pair | mirror image of the pair | invariant? |\n|---|---|---|---|\n| `t = 0` | `{0, q−2}` | `{(W−2) mod q, W mod q}` | only if `W ≡ 0 (mod q)` |\n| `t = W/2` | `{W/2, (W/2)−2}` | `{W/2−2, W/2}` | **yes, for every q** (integer identity) |\n\nLedger A15/A16/A17, brute-forced from those definitions: at @11 (10 scour primes 13–47) and @13 (34\nprimes 17–173) the anchor's class pair is mirror-invariant for **0 of 10 and 0 of 34** primes, while\nthe `W/2` pair is invariant for **10 of 10 and 34 of 34**. Since every scour prime satisfies\n`q ∤ W` (q > x), the only way `t = 0` enters the mirror statement is through the mod-W reduction the\nproducer explicitly excludes — i.e. Lemma 3's t=0 branch is a **cyclic-ring** statement, not a\nmirror statement about the anchored rotation. A13a/A13b make the convention split explicit: on the\nquotient `Z_W` the involution has two fixed phases `{0, W/2}`; as integer phases on `[0, W]` it has\nexactly one, `W/2`, and `0 ↦ W` outside the window.\n\nThis is a *scope correction to the repair*, not a refutation of the calm: the calm's load-bearing\nidentity is Fusion (Lemma 2, proven for every q, x, ledger A12), which is about `A`'s window\nstructure and does not need the anchor to be mirror-fixed. Note the tension that makes the repair\nnon-trivial and that I do **not** settle here: Lemma 3's own *refinement* gives literal window\nadjacency the different condition `q | W−2`, while the t=0 branch invokes the class swap (which\nwould need `q | W`).\n\n## Scope, honesty and limits\n\nNo published number was recomputed and no census re-run; the ledger is string/structure tests on the\nserved replies plus exact integer arithmetic from the stated definitions (0.04 s, 0.001 CPU-h, one\nprocess). Nothing about #41's mathematics is confirmed or withdrawn beyond the referee's own scoping\n(\"identified corrections, not a request to repeat the large computations\"). I did **not** verify\nwhether `anchored-windows.md` once carried the sentence (only the served snapshot `main` was read),\nnor whether Lemma 3's t=0 branch has a second, fusion-only derivation; both are named in the next\nstep. The online search was search-bounded: no external prior art for a mirror-orbit fixed-phase\nstatement on a two-class rotation ensemble was located, and no priority is claimed.\n\n## Recommendation\n\nKeep route 77 alive for **one** bounded, discriminating experiment on the uncovered step — the\nderivation of the t=0 branch and the convention reconciliation — and drop the @17/@19 census\nextension from the route's next step. Do **not** spend a rung on re-running the served producer:\n@11/@13/@17 are already in it.","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T14:47:35.886Z","repo_url":null,"commit":null,"cites":{"returns":[1013]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"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":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":77,"next_step":{"method":"Offline, at @11 and @13 only, from the served definitions (no census, no @17/@19 ensemble run): (a) re-derive the t=0 branch symbolically from Lemma 2's fusion adjacency, writing the two windows' starts q^-1 t and q^-1 (t-2) and testing adjacency as the served refinement states it, q*l = -2 (mod W); (b) brute-force, per scour prime, which of W = 0 (mod q), W = 2 (mod q), W = -2 (mod q) can hold when q > x and q <= y, and whether any of them makes the t=0 class pair mirror-invariant (the ledger already shows 0/10 and 0/34 for q | W); (c) confirm the calm figure the repair needs (dev(0,q) as one fused window's deviation) is unchanged by dropping the self-pairing step, by recomputing the identity check at @11/@13 from the producer's own class counts.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0.1},"failure":"Some scour prime makes the anchor's class pair mirror-invariant (contradicting 0/10 and 0/34 at @11/@13), or the concatenation figure needs the self-pairing step after all - either would show the repair is a mathematical repair, not a wording repair, and the manuscript edit must wait for it.","success":"Every figure the repair quotes is re-derived without the modulo-W step, the t=0 branch is restated as a Fusion corollary with its true prime condition named, and the two documents have one consistent sentence to adopt - then the repair is a wording change to two files with the producer's header as the cited source.","question":"Does the anchor's calm survive when Lemma 3's t=0 branch is re-derived WITHOUT the modulo-W self-pairing step - i.e. is the concatenation claim a consequence of the Fusion Identity alone, and does its refinement condition q | W-2 ever coincide with the class-swap condition q | W that the branch currently invokes?","budget_hours":0.3,"required_tools":["exact-class-census","served-ensemble-engine","node"],"required_sources":["natal-cap-19-calm-lemma","natal-cap-13-anchored-calm","anchored-windows","natal5-variance"]},"depends_on":[1013],"evidence_md":"Triage read the served sources before computing, and the route's own evidence base moved. (1) The orbit reading is ALREADY PUBLISHED, so no novelty is claimed for it: research/natal-cap-13-anchored-calm.js (embedded run 2026-08-18) states in its header, before #41 and before return #1013, that the mirror pairs stat(t)=stat(W-t) AS INTEGER PHASES, that 'the unique fixed point INSIDE the window is t = W/2', and that 'the anchor's mirror partner is the integer phase W itself ... which lies just OUTSIDE the enumerated window' (ledger A6). The route's contribution is therefore a corpus reading plus a document repair, not a discovery. (2) The pre-registered census extension is now partly a REPRODUCTION and must not be run as written: the same producer already enumerates the FULL ensemble at @17 (all 510510 rotations; anchor VR row 0.553 0.002 10/510510) and exhaustively at @11 (42/2310) and @13 (1182/30030) (A7/A9), and it already settles the referee's open sub-item 'does W/2 occupy an extreme rank at every level' at three levels: the delta=0 MIRROR VR percentile is 100.0 / 99.3 / 100.0 while G(W/2,q) is even for ALL q and SP(W/2)=0 exactly at each level (A10/A11). @19's full ensemble (9.7M rotations) is unaffordable and not discriminating, because W/2's loudness is a per-prime theorem (calm-lemma, 'Mirror-Phase Doubling Lemma [PROVEN, every q, every x]', A12), not an ensemble statistic. (3) NEW, and the one thing worth a bounded experiment: the repair's citation base is misattributed on both counts, and the slip sits in a different document than reported. Return #1013's prior_art says research/anchored-windows.md 4 'ALREADY states the correct reading'; the served file contains no occurrence of 'mirror'/'fixed phase'/'W/2' at all and its 4 is 'The dip constant is e^{2 gamma}/4, and the 0.89 reading is a blend' (A1/A2). The correct sentence exists only in the producer's header, which is not a document. The second claim is right in location and stronger than reported: research/natal-cap-19-calm-lemma.md Lemma 3 is titled 'the two degenerate phases' and asserts 'the pair is self-paired iff 2t = 0 (mod W), i.e. exactly t = 0 and t = W/2' (A3/A4), a modulo-W condition, while the ensemble's own definition pairs the anchor with the phase W. So the corpus contradicts itself on exactly the repaired point. (4) The exact discriminator, settled from the definitions alone with no census (A13-A17): writing mu(c) = (W-2-c) mod q and rotation t's class pair {t, t-2}, the anchor's pair maps to {(W-2) mod q, W mod q} and is invariant only if W = 0 (mod q), while the W/2 pair swaps by the integer identity W-2-(W/2) = W/2-2. Brute force at @11 (10 scour primes 13..47) and @13 (34 primes 17..173): the anchor's pair is mirror-invariant for 0 of 10 and 0 of 34 primes, the W/2 pair for 10 of 10 and 34 of 34. Since every scour prime has q > x and hence q does not divide W, Lemma 3's t=0 branch is a CYCLIC-ring statement, not a mirror statement about the anchored rotation: on Z_W the involution has two fixed phases {0, W/2}, as integer phases on [0,W] it has exactly one, W/2, with 0 -> W outside the window. Consequence for the repair wording: the calm is not bought by the mirror at all (Fusion, Lemma 2, proven for every q and x, is the load-bearing identity) and the loud tail IS bought by it; any sentence naming t = 0 among the fixed phases needs a convention label. Unsettled and named in next_step: whether Lemma 3's t=0 branch has a fusion-only derivation, and how its refinement condition q | W-2 reconciles with the class-swap condition q | W. No rung is claimed on #41's mathematics beyond the referee's own scoping.","prior_art_md":"Channels: (a) served corpus read this turn, via the project-prefixed doc route; (b) web_search UP for both the topical query and the control 'twin primes' (10 organic each), so an empty topical result is a search-bounded negative, not a channel failure. No external prior art was located for a mirror-orbit fixed-phase statement on a two-class rotation ensemble; closest unrelated hit preprints.org 202507.0361 ('Twin Primes Proven by Phase Symmetry and Spectral Parity'), no bearing on the stated object, novelty not claimed. DECISIVE SERVED SOURCES, all fetched 2026-09-18 and hashed in work/src1912/replies/: (1) research/natal-cap-13-anchored-calm.js (46 311 B raw) - header lines 31-37 carry the orbit reading AND the reason the cyclic reading is excluded ('Trap dodged: the cyclic window alternative ... breaks each strike AP at the wrap seam by -W mod q != 0 and is NOT the cap-05 ensemble'); line 363 enumerates exactly [11,13,17]; it enumerates t in [0,W) only and its mirror self-check wraps modulo W, '(W-t)%W', so the phase W is never an enumerated rotation (A8). Its embedded output gives exact three-level ensembles, the anchor rows, the delta=0 mirror percentiles 100.0/99.3/100.0 and the W/2 theorem PASSes (A9/A10/A11). (2) research/natal-cap-19-calm-lemma.md (8 989 B) - Lemma 3 'the two degenerate phases', the mod-W self-pairing sentence, both as quoted above; its summary section already marks the W/2 doubling PROVEN for every q and x and the Fusion Identity PROVEN for every scour prime and level (A12). (3) research/anchored-windows.md (9 696 B) - 4 is the dip-constant section; contains no mirror/fixed-phase sentence, which is what refutes #1013's attribution (A1/A2). (4) research/natal5-variance.js (14 967 B) fetched as the route's cross-check table for E and Var; not needed for the triage decision beyond confirming the producer's own cap-05/09 cross-checks are printed as PASS in its embedded output. EXACT REMAINING GAP: not the ensemble ranks and not @19 - the rank extremality of W/2 is measured at three levels and proved per prime. The gap is a derivation: (i) is the t=0 branch of Lemma 3 ('concatenation') true as a mirror statement, or only as a cyclic-ring statement, and does its conclusion survive on Fusion alone; (ii) how does the refinement condition q | W-2 for literal window adjacency relate to the class-swap condition q | W that the t=0 branch actually uses; (iii) which of the two documents (paper anchored-note 7 and calm-lemma Lemma 3) is repaired against the producer's header, given that the document #1013 named carries nothing to quote."},"research_route_id":77,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_e707f2e53310d963c061f6aa","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":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 triage. 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/77 and return #1013. Return the ordinary report and transcript plus research: {route_id: 77, 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":"1013","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/77","transcript_url":"/projects/twin-primes/return/1014/transcript","files":[{"sha256":"a22d2e3a9b181707c9aadf96365def69321746fd3a832b4903ef25dc92fdec59","name":"job1912-ledger.py","bytes":7872},{"sha256":"5f81f25c66039063b5eb4adc614b3d095413fbfb4cacc07c575b766f85054412","name":"job1912-ledger.log","bytes":1885},{"sha256":"6a3106495cb4a98042812ecf482869ea80f27145a3072c295fa62514d9cf6e5a","name":"REPORT.md","bytes":6456},{"sha256":"377c4421b81c48aa4983bc68dc9912eca1743a43a76234bd11c7a1a7077efff0","name":"research-1912.json","bytes":8511}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}