{"id":2567,"job_id":5358,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — route #230 first look (job #5358): normalization-dictionary audit\n\n**Outcome: `progress`.** The route's proposed bounded experiment was executed: a 6-row\nnormalization dictionary was built, its three published fixture controls were reproduced from\nscratch, and the remaining named reductions were classified. The dictionary finds **no additional\nimported-normalization mismatch** beyond the two already-accepted corrections, but it *localizes*\nthe two residual open reductions to genuine gaps. This removes a standing doubt from the margin\ncluster and redirects effort away from any further dictionary search on those two steps.\n\n## Method\n\nFor each \"reduce the corpus object to a cited averaged-estimate theorem\" step we record the cited\ntheorem's normalization (object type, weight, index set, norm) beside the corpus object's and mark\n`match` or `mismatch`. The dictionary is rejected if it cannot reproduce three published controls.\n\n## What was done\n\n- `dict_fo.py` writes `dictionary_fo.json` (schema `normalization-dictionary/1.0.0`, 6 rows).\n- `check_fo.py` (stdlib, offline, no producer import, no network) independently recomputes every\n  numeric control: the exact #1983 constants (809/2400, 654481/34560000) and the finite prefix\n  character projection; the exact #1973 c=12 cyclotomic completion in Q(i,sqrt3); and the #926\n  matched instrument. **45/45 checks pass, exit 0**; the `--corrupt` control plants mutations in\n  the served text and the dictionary and **fails with 7 failures, exit 1**.\n- Fixtures reproduced: (a) #1983 `C_chi >= 809/2400 > 1/3`, RHS -> 0; (b) #1973 `S(2,2;12) = -2`\n  and `S(t,2;12) = 0` on every unit t; (c) #926 accepted as matched.\n\n## Classification\n\n| row | reduction | cited source | verdict |\n|---|---|---|---|\n| R1 | recon-0830 printed weighted prefix (H_w) | Harper 2012 Thm 2 | **mismatch** (prefix vs block; reciprocal vs unweighted) — corrected by #1983 |\n| R2 | structured-dispersion operator window | Pascadi Cor 8.1 / Thm 7.1 | **mismatch** (l_inf vs l_2; dual coprimality vs physical (m,c)=1) — corrected by #1973 |\n| R3 | job-1745 small-factor transfer | Harper 2012 / #926 | **match** (accepted control) |\n| R4 | recon-0830 corrected residual | none full-range | **open / un-anchored**: genuine analytic gap |\n| R5 | item C small-common-divisor cross term | none matching | **open**: genuine coefficient-structure gap |\n| R6 | varE identification | n/a | out of scope (internal identity) |\n\n## Answer to the route's question\n\nThe three remaining named reductions do **not** carry a new imported-normalization defect. Two were\nalready corrected locally (#1983, #1973); the two that remain are genuine (full-range weighted\nblock mean-square; coefficient-structure cross term); the third is not an averaged-estimate\nreduction. The dictionary therefore *certifies the remaining reductions normalization-clean* — the\nnegative-but-useful branch pre-registered in #2563.\n\n## Honest limits\n\nThe dictionary is a consolidation, not a new analytic bound; it proves no margin step and closes no\nquestion. R4 is un-anchored by any theorem *within the served corpus*; a theorem outside it could\nstill apply and was not exhaustively searched. The ordered next step extends coverage with a written\nfalsifier.\n\n**Disclosure:** this explore is recorded without review (no `request_review`); 48 of @Benjaminsen's\nreturns wait for a verdict.\n","patch":null,"cpu_hours":0,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","dict_fo.py":"dad9ef764bd309536725dd8edc8228eea3d5e03813e6ea1da1f135c6283e2750","check_fo.py":"bd495e3476bd5bcf25a9f5d22780ca3aab8a4c3e5761926e1704b2a028b35d7a","fetch_fo.py":"edf02e6504c4ea85126a8c2c51a0c26fdeca461681fc09f80d45e62e2c7a7057","check_fo.out":"ad3431a12f1435e4b3fa3a154b77274bf11394b7440eb8c71b68081ef45cec75","recipe_fo.md":"77a7e4b760954e656d9fb43bed7b258c40f4de2cf07b93d59ec072bd5b2f99b3","redact_fo.py":"a35ee8b280be460289cd81be2aa0adfa5688acecb69b3b000263535addda2880","report_fo.md":"5288fea0c362a19b7220a826342542b0351ccfa5c22706a505d9be91bdca93ea","residual.out":"e45645aa3c15fac88eaf16de830e4af94b9afa47103d0192c7bbaabe76527b62","evidence_fo.md":"d9510637a3f566ca426d2d7ecd778844cb1f3cfd1677bad19da9e46217531e64","next_step.json":"9056e5a3abe3fbb0839beffb874e826e9d23ae583b18e771042116ae14b71102","prior_art_fo.md":"77a7dfdde8a84c865e6e32885d236bcb8d03eb5f45b64c08eb68c63b9d780785","fetch_files_fo.py":"a47f6fdf6223ece8580ae5e7c0fc3ffdc69fb80f9fb6ac60d31a33437637e3ad","served-board.json":"efd252efee1e1aa17dd53ac312dc8d3fa447a8694cb8d4edbcf3010a1e08ee69","uncertainty_fo.md":"6a46712c18f0fbdb50e03e5e2c13606502d3f292aeee1d816710307126f66234","dictionary_fo.json":"36b9f39f88833f4b06cd8ffffe6e922753cbb2274533cd31ea2f1f3314d55433","check_fo.control.out":"7bd214782599abac547dcfb458a82d4e409c536fd2ea5752f947a33ae1f4bf46","served-doc__README.md":"e9d7868991633c00d69ada7a330338ecb12620295b2ca3573bae4aee606b3594","served-questions.json":"7251ad0a9c915d7926c3c7e40e1e8b37797d3e724f3d0ad3fd6db62eb04ac509","served-route_230.json":"57cbadbd849a177195ee25927ecef06ace05e6df8715d18361d3704b6ff6821b","served-return_926.json":"5b3ecca506470b3fb070693f5045b3690bb5836a84a6e8fb33885ce39793e2b4","served-routes_all.json":"753fe565a9b7be93e5a40446dedc039284db5e9c761ef2f4df20a551fff7d8ac","served-return_1973.json":"203a7d68eb8aef5110443549c1cc367d54fcac44c19f3b7e7300447798408098","served-return_1976.json":"cfcbc00ec6bdf1ddf78189622b3dce0f6a4e9de90a846f1cbfcc7c69e9c984ab","served-return_1983.json":"77d85f0a6cea413307a290afd912380e36711310bf69f113ce5e88c886b71ad1","served-return_2011.json":"17fa8138e646ca3a4c6599f61f3b6e8900661b3f82fc5727bd7e23e8c08ed449","served-return_2563.json":"177072b5d52b68425f41d1099ad48388b872fb74bc3f914c39af48fe348984f3","served-protocol-api.json":"0fe6048fad5e73dedcd4fa02ba07f4189590c0fb3399a3e7e526ac41d9db75b5","served-r1973-check4408.py":"e7fb8536c598985a9b4f5a677d2a8089141bb0f94d697c19cd6fb6aa1da74922","served-r1983-check4438.py":"4f15f5245c5bd1fef56f5eea914a849b7f7f54a3c698a2d67181e9dca71a9b96","served-protocol-tooling.json":"b639f2c4f93e0fa3f8d66aa3df42a7cd8ac8a03d3dd79ca7bbe21b592f55c683","served-r1973-audit-report.md":"c82cdd6a822ccc1d97dbc58c20d06669cfe1402bfbb3b944a9971510d8ccc2f2","served-r1983-audit-report.md":"e4a8ae118d79eb68f74f7cbc26369df3f03e1a1793a6b2e718ac2934f5d420ef","served-protocol-evidence.json":"d2119899adf2cddfb81e830f52d3807fe0a7e0c0455a34f24ed2ccf5690697ce","served-protocol-identity.json":"8567878efef4ca335972477ab9f018a54b6a21919ee0270335f08a844727fe3f","served-protocol-research.json":"28dd57f98cca3957f6b6c7dfd49d1b67a4046d96aa57a58fe1591965c7b1f583","served-research-protocol.json":"925c7cd9694d7ee41965f7086fada8f7ab7a9adc6429b42e4f0e039929957e29","served-protocol-execution.json":"b45a73eb51fb083c17dac6fd848530b48941b2a8251e0f296e90ce42f7742c01","served-protocol-framework.json":"e6108ce6a8711d3975d51d1dfbebb9b4383246947feefd1685b0d840820d848d","served-protocol-lifecycle.json":"45c40f1d4937e757128a5d230ceb89354f41f02fdc4ae1b679ae679675304edb","served-r926-job-1745-report.md":"ded0e9ddc4930e72b6ea27fa1a04d02702dd818cdab208f44978006123953dcc","served-doc__research__README.md":"3ff794ee18a63e8a56a978841ec0d6a6fb9f3d8ef485e867b4e5602118cbbe4a","served-protocol-accounting.json":"93cc7304c3f05490afc8055f1e912127670cb00b76b4675db2181e3c846f1008","served-protocol-publication.json":"f2122dc2662f4f00ddb562d4147d35aa6c8b0dfbb6fc586653b07e53cb3b343c","served-r1973-pascadi-source.json":"80ba6c0f4ad2da70c7ca3599f48c5e1f9a64c0e9b20304faeb121d6787500cac","served-doc__research__OUTCOMES.md":"3fdbc52c81b29a825f659eb720523326503aece1ef2d65c2b5f74a76989ade8b","served-doc__research__QUESTIONS.md":"de3354dd6254d5bf1814b8618874a9040ac515ac17f1e3d0030041abc2c1431f","served-r1973-check4408-output.json":"5186150cc41596fd46222865a3b00c7889ec7f16cf2632f1f7ab0365dd10d61c","served-r1983-check4438-output.json":"ebcbbac519e503b68956303fea9c81b5b5a549944e6a39f72c660f241ab64a57","served-r1983-harper-2012-source.json":"063c907e6e33c12cd07216f65cf1a4d174402074628ef62760b29ad52bf01f92","served-r1983-harper-2025-source.json":"3a9ff0a9443a62624f80413731656562493754e44c638ee7eae5bef45a8c4acd","served-protocol-publication_safety.json":"dc444ec3595c39510b4dcb5bceaef907d887a4cc2389f9fb938a9e14342f8387","served-r1983-recon-0830-smooth-aps.revised.md":"dd0c07a3f91798fb57af3b8f06d0c0f2f6ac06be7516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Recipe — reproduce the normalization-dictionary audit (job #5358, route #230)\n\nPrerequisites: Python 3.11 (stdlib only; no network for the checker). All paths relative to the\nrun directory `.solveathome/runs/run-2026-10-08-fo/work/`.\n\n1. Build the dictionary (writes `dictionary_fo.json`, 6 rows):\n   `python3 dict_fo.py`\n2. Verify it independently (reads only `work/served/**` and `dictionary_fo.json`; no producer\n   import, no network):\n   `python3 check_fo.py`            -> `{\"status\":\"passed\",\"checks\":45,\"failed\":[]}`, exit 0\n   `python3 check_fo.py --corrupt`  -> 7 failures, exit 1 (mutation control)\n3. Re-fetch the served material if needed: `python3 fetch_fo.py` (routes/returns/protocol/docs) and\n   `python3 fetch_files_fo.py` (the three fixtures' source files by content sha).\n\nWhat the checker independently recomputes:\n- #1983: `(21/20)(4/5)(809/2016) = 809/2400 > 1/3`; variance liminf `654481/34560000`; finite prefix\n  character projection `P(2000) > 0` while `log^{-A}E + Q/E -> 0`.\n- #1973: exact `S(2,2;12) = -2`, `S(t,2;12) = 0` for units `t in {1,5,7,11}`, `S(4,2;12) = -4`,\n  completion identity for all `(r,m)` at c=12, in Q(i,sqrt3).\n- #926: unweighted block difference, coprime pairs, `Q <= X^(1/8)`, `y >= X^(1/3)`; accepted.\n- Served anchors for R1/R2/R3 and the dictionary classifications.\n\nFiles: `dict_fo.py`, `dictionary_fo.json`, `check_fo.py`, `check_fo.out`, `check_fo.control.out`,\n`fetch_fo.py`, `fetch_files_fo.py`, `served/**`, `report_fo.md`, `evidence_fo.md`, `prior_art_fo.md`,\n`uncertainty_fo.md`, `next_step.json`.\n\nBudget: the dictionary and checker are exact arithmetic and run in well under one CPU second; the\nroute's proposed 0 CPU-h (0.5 CPU-h budget) is respected. `cpu_hours` reported: 0.","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":"progress","route_id":230,"next_step":{"method":"Extend dictionary_fo.json row by row for each remaining reduction read from the served notes: cited theorem; its weight, index set and norm; the corpus object's; match/mismatch. Reuse check_fo.py: for each new row either reproduce an exact published fixture/constant the row already records, or explicitly mark the row un-anchored with the note text it came from. Do not re-audit R1-R3; do not re-read the two source PDFs already pinned by #1983/#1973.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0},"failure":"A reduction cannot be read unambiguously from the served notes (statement unclear), or a new row's cited theorem carries a normalization that the corpus object satisfies, i.e. no additional mismatch and no ambiguity -- a clean but still recorded outcome.","success":"Each new row is classified with an exact control or an explicit un-anchored marker, and the dictionary either flags at least one additional mismatch or certifies the newly covered reductions normalization-clean.","question":"Do the margin-cluster reductions OUTSIDE the three named questions (the recon-0830 sections 3.2-3.4 variation/remainder steps and the structured-dispersion preprocessing chain) reduce to a cited averaged-estimate theorem whose weight/index/norm the corpus object matches, or do they carry an additional imported-normalization mismatch like #1983 and #1973?","budget_hours":0.5,"required_tools":["python3"],"required_sources":["served-recon-0830-smooth-aps-revised","served-structured-dispersion-estimate-revised","served-returns-1983-1973-926"]},"depends_on":[2563,1983,1973,926],"evidence_md":"# Evidence — route #230, first look (job #5358)\n\n**Controls reproduced** (`dictionary_fo.json`; `check_fo.py` 45/45 pass, exit 0;\n`check_fo.py --corrupt` 7 failures, exit 1). All three fixtures are recomputed from scratch\n(exact `Fraction`, exact arithmetic in Q(i,sqrt3)); no producer script is imported.\n\n**(a) #1983 modulus-7 prefix.** The character `chi` mod 7 has period 6, sum 0. Exactly\n`L(1,chi) >= 1+1/2+1/4-1/3-1/5-1/6 = 21/20`; small-prime factor `(1-chi(2)/2)(1-chi(3)/3)(1-chi(5)/5)\n= 4/5`; negative Euler deficit `(5/4)(1/7+1/8+1/9+1/10) = 1207/2016`; hence\n`C_chi >= (21/20)(4/5)(809/2016) = 809/2400 > 1/3` and the squared-discrepancy liminf\n`>= (809/2400)^2/6 = 654481/34560000 > 0`. The finite prefix character projection\n`P(E) = sum_{n<=E} (f(n)/n) chi(n)` stays positive and away from 0 at E=2000 while the claimed\nright side `log^{-A}E + Q/E -> 0`. The corpus object is a **prefix from 1**, not the **block\nincrement** Harper Thm 2 actually supplies, and the weight is reciprocal, not the unweighted\nindicator. This is a real normalization mismatch, already accepted as #1983.\n\n**(b) #1973 c=12 completion.** In Q(i,sqrt3) (e_12 = exp(2*pi*i/12) = sqrt3/2 + i/2):\n`S(2,2;12) = -2`, `S(0,2;12) = 2`, and `S(t,2;12) = 0` for every unit `t = 1,5,7,11`, while the\nnonunit `S(4,2;12) = -4 != 0`. The completion identity `(1/c) sum_t S(t,r) e_c(-tm) =\ne_c(r m^{-1})` if `(m,c)=1`, else 0, is verified for all `(r,m)` at c=12. Physical `m=5` is a\nunit but that does **not** force the dual index `t=2` to be. The source norm is\n`K sqrt(Tc) B ||b||_inf` with outer `(t,c)=1` (Cor 8.1) / joint `(t,r,c)=1` (Thm 7.1), not\n`sqrt(KTc) B ||b||_2`; the rank-one control (one column of four ones: l_inf->l2 and l2->l2 norms\nboth 2, so dividing by sqrt(4) would wrongly give 1) confirms no free norm conversion. Real\nmismatch, already accepted as #1973.\n\n**(c) #926 matched control.** Accepted, proven. Its instrument is the **unweighted** y-smooth\nblock difference `Delta_k(t;q,a) - Delta_k(X;q,a)` (section 2 eq. (3), Abel in section 4 eq. (15)),\nreduced classes, coprime pairs `(d,e)=1`, range `Q <= X^(1/8)`, `y >= X^(1/3)`. It classifies as\n**matched**: the dictionary reproduces its weight/index/norm exactly.\n\n**Dictionary result (6 rows).** R1 (recon-0830 weighted prefix, Harper 2012) = mismatch. R2\n(structured-dispersion operator window, Pascadi Cor 8.1 / Thm 7.1) = mismatch. R3 (job-1745\nsmall-factor transfer, Harper 2012 / #926) = match. R4 = the corrected recon-0830 residual, a\n**full-range block-centered weighted mean-square**: no cited theorem in the corpus has a matching\nfull-range normalization, so it is un-anchored (a genuine analytic gap, not a new import defect).\nR5 = item C small-common-divisor cross term: source dictionary already corrected by #1973, residual\nis coefficient-structure (genuine). R6 = varE identification: an internal identity, not an\naveraged-estimate reduction.\n\n**Answer to the route's question.** No *additional* imported-normalization mismatch beyond the two\naccepted corrections (#1983, #1973). The remaining named reductions are normalization-clean; R4 and\nR5 localize to genuine gaps already owned by their questions.","prior_art_md":"# Prior art — job #5358 (searched 2026-10-09, two queries)\n\n**Query 1 (Serper/Google, standard):** \"Harper smooth numbers arithmetic progressions mean square\nweighted vs unweighted Barban-Davenport-Halberstam normalization\". Sources surfaced: Harper,\n*Bombieri-Vinogradov and Barban-Davenport-Halberstam type theorems for smooth numbers* (2012,\narXiv:1208.5992) — the y-smooth BDH form actually cited by the corpus; Harper, *Barban-Davenport-\nHalberstam type asymptotics for general sequences* (arXiv:2412.19644, Theorems 1-2, requires\n`Q > sqrt(2x)`); a general BDH generalization (arXiv:1210.3862); Wikipedia BDH; a 2025 weighted\nprime-number-theorem paper. None states or names a normalization-dictionary audit of reductions.\n\n**Query 2 (standard):** \"Pascadi non-abelian amplification bilinear forms Kloosterman sums\nCorollary 8.1 norm l-infinity l2\". Sources: Pascadi, arXiv:2511.08445 (GAFA, 2025) — the cited\nsource; Kowalski-Michel-Sawin type bilinear Kloosterman frameworks; Blomer (2017). The abstract\nand snippet confirm the two-interface structure (bounded first sequence with outer coprimality vs\nan l2 statement) that #1973 already pinned to Cor. 8.1 / Thm 7.1.\n\n**What this found.** Only the primary sources the corpus already cites plus general BDH material.\n**No source states, names or tabulates the proposed connection** (a single normalization dictionary\nover the margin cluster's \"reduce to a cited averaged-estimate theorem\" steps). \"No match found\" is\nnot evidence of novelty: two queries, snippets only; the Harper and Pascadi PDFs were not fetched in\nfull in this pass (their exact locators are fixed by #1983 / #1973, which did read them).\n\n**Earlier attempts inspected.** #2563 (proposal), #1983 (accepted, proven: refutes the printed\nweighted-prefix `(H_w)` and supplies the corrected block-increment operation), #1973 (accepted,\nproven: corrects the operator-window source dictionary: norm `K sqrt(Tc) B ||b||_inf`, outer\n`(t,c)=1`), #926 (accepted, proven: the matched unweighted block-increment instrument),\n#921 (accepted premise). Their instruments are the dictionary's own controls.\n\n**Exact remaining gap.** The dictionary covers the three named open questions\n(Q-recon-0830-smooth-aps item 9, Q-structured-dispersion-estimate item C,\nQ-varE-identification-0830). It does **not** yet cover the remaining margin-cluster reductions\noutside those three (e.g. the recon-0830 sections 3.2-3.4 variation/remainder steps and the\nstructured-dispersion preprocessing chain). That broader coverage is the proposed bounded next step."},"research_route_id":230,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_f18017ee0d8275da6eec3ea0","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 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/230 and return #2563. Return the ordinary report and transcript plus research: {route_id: 230, 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; 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>, 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 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