{"id":961,"job_id":1823,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 48 pursue: the cell's length is NOT the obstruction — Wright's bracket is a sum, and the whole question is the frequency-sequence length A\n\nCalibration: **MEASURED** for everything in `wright48price.json` (exact rational arithmetic, no\nfloating point, 16/16 gates). Source transcription **VERIFIED** from the paper's own HTML. The\n*dictionary* (which cell object is M, N, A, R) is a **READING**, taken from the corpus's own\nnormalisation and flagged in the limits. No corpus computation is re-run.\n\n## 1. The sprint\n\nJob #960's next step: price Wright arXiv:2604.25177v2 Theorem 2.1 and arXiv:2608.27732v1 at the\ncell's parameters, **every** term, against the route's requirement — the omission that sank #914's\nreading. Done for Theorem 2.1, whose full text (v2, 7 Aug 2026) I read; Theorem 2.1 with `R = 1` is\nBettin–Chandee's own bound, so this prices the whole trilinear instrument.\n\n**Theorem 2.1 verbatim** (paper's section 2; `B` is the trilinear form with a fixed factor `R` in the\ndenominator, `e(theta a mbar/(nR))`, under `M << N^2`, `R << M^A`):\n\n```\nB(M,N,A,R) << M^eps ||alpha||||nu||||beta|| (AMN)^(1/2) R^(1/4) (1+|theta|A/(MN))^(1/4)\n   x ( 1/N^(1/8) + R^(1/8)N^(1/8)/M^(1/4) + M^(1/10)/(R^(3/20)A^(1/20)N^(3/20))\n       + N^(3/20)/(A^(3/20)M^(1/5)) + N^(3/8)/M^(1/2) ).\n```\n\nThe bound is a **sum**, so it is governed by its *largest* term and the saving by the *weakest* one.\nTaking the term count `MNA` as the trivial bound (exact for unit sequences, where\n`||alpha||||beta||||nu|| = (MNA)^(1/2)`), the saving exponent is `-(rho/4 + max_i t_i)` with\n`R = c^rho`, `A = c^a`, and at the record's lengths `M = c^(51/95)`, `N = c^(39/95)`, `rho = 1/19`\n(the corpus's `q`), `a = 0`:\n\n| term | value |\n|---|---|\n| 1. `1/N^(1/8)` | −39/760 |\n| 2. `R^(1/8)N^(1/8)/M^(1/4)` | −63/760 |\n| **3. `M^(1/10)/(R^(3/20)A^(1/20)N^(3/20))`** | **−3/380  ← binds** |\n| 4. `N^(3/20)/(A^(3/20)M^(1/5))` | −87/1900 |\n| 5. `N^(3/8)/M^(1/2)` | −87/760 |\n\nwith `R = c^(1/19)`: saving **1/380** (term 3 becomes −3/190 and still binds). Both reproduce\nreturn **#916** exactly — its `3/380` at `R = 1` and `1/380` at `R = c^(1/19)` are now independently\nre-derived from the theorem text, and **the binding term is #3, not the `1/N^(1/8)` term that #914\npriced**.\n\n## 2. #914's live candidate, corrected\n\n#914 read the theorem as a single headline term and got `39/760`, \"the first located statement whose\nnumber CLEARS the bar\". `39/760` is exactly term 1 alone, and at `A = c^0` it overstates the\nsum-correct saving by precisely **13/2**. So the number was right and the reason was wrong — and it\nmatters, because the correction exposes the real variable.\n\n## 3. The new content: the saving is linear in the frequency exponent\n\nTerm 3 alone carries `A`, and it is the binding term, so for small `a`\n\n```\nsaving(R=1)      = 3/380 + (3/10)a        until a = 11/76, then flat 39/760\nsaving(R=c^1/19) = 1/380 + (3/10)a        until a =  9/76, then flat 29/760\n```\n\n(the branches meet continuously at the crossover, checked). Hence the bar `7/190` is met **iff**\n\n```\na >= 11/114  (= 0.09649...)   with R = 1,        saturated at 39/760 = (39/28) x bar\na >= 13/114  (= 0.11404...)   with R = c^(1/19), saturated at 29/760 = (29/28) x bar\n```\n\nand NOT met for `a = 0`, where the shortfall is exactly `3/14` of the bar (`1/14` with the fixed\nfactor). **The route's own lengths are fine**: term 1 alone gives `39/760 > 7/190`, so the cell's\n`M = c^51/95`, `N = c^39/95` are inside the theorem's useful range. The entire question is the length\n`A` of the third (frequency) sequence — the one quantity route 48 has already measured, and the one\n#916 implicitly took to be `O(1)` (\"bounded-third-sequence specialization\").\n\nTwo further measurements on the same five terms:\n\n* In route 48's **own** length normalisation (`min(d,e) = L^(2/5)`, complement `L^(3/5)`), `A = O(1)`\n  buys **exactly nothing**: term 3 equals 1 at those lengths, saving `0`; at `a = 1/2` it is `1/20`,\n  i.e. `19/14` of the bar. Route 48's internal reading is even more sensitive to `A`.\n* The manuscript's third term is **internally inconsistent**: the display prints `A^(1/20)` while the\n  paper's own derivation (section 3) gives `(M^(1/5)/(R^(3/10)A^(3/5)N^(3/10)))^(1/2) = M^(1/10)/(R^(3/20)A^(3/10)N^(3/20))`.\n  Priced both ways: at `a = 1/2` the printed exponent gives `21/760 < 7/190 = 28/760` (ratio `3/4`,\n  FAIL) while the derived one gives `29/760` (ratio `29/28`, PASS). **The two readings disagree on\n  the verdict**, so this is not a cosmetic transcription slip and must be settled against the v1 TeX.\n\n## 4. What this changes for the route\n\nRoute 48's stated obstruction — \"NO published improving range contains the cell\", i.e. a LENGTH\nwindow — is **false at the record's lengths**: the trilinear instrument's best term is at\n`39/760 > 7/190`. #960's relocation to the modulus is also too coarse: the fixed factor `R` enters as\na *loss* (`R^(1/4)`) but only as `1/380` worth at `rho = 1/19`, less than the `A`-gain available in\nthe other direction. The obstruction is a **third-sequence-length** condition: the frequency support\nmust satisfy `A >= c^(11/114)` (or `c^(13/114)` with the fixed factor). That is a weaker requirement\nthan a new theorem, it is decided by a number the route already measures, and it is where the\ninvestigation should go next — not back to `N`.\n\nNot claimed: that `A` has that size (unmeasured here); that this closes route 9; that the S-family\nceiling #632/#634 is affected (it is not — this is the F-family instrument); any exponent of my own.\n\n## 5. Limits\n\n1. The dictionary `(m,n,a,R) -> (cell objects)` is a reading. I priced the corpus's own normalisation\n   `c = q e1 e2`, `q = c^(1/19)`, lengths `c^51/95`/`c^39/95` (the numbers #780 and #914 use), plus\n   route 48's internal `L^(2/5)`/`L^(3/5)` reading; a third dictionary could move the thresholds, but\n   not the structure (`A` linear with slope `3/10`, term 3 binding).\n2. `(1+|theta|A/(MN))^(1/4)` is treated as a constant: `|theta| <= 4` at the cell and `A <= N`, so the\n   factor lies in `[1, 5^(1/4)]` and cannot supply or remove a power of `c`. Stated, not proved.\n3. `M << N^2` holds at the record (`51/95 < 2*39/95`); `R << M^A` is vacuous for `R = q`.\n4. Theorem 2.1's `A` is the *support length of the third sequence*, not the tallest-frequency count;\n   this work does not identify them. That is the next experiment.\n5. The `A`-exponent inconsistency in the display is transcribed as printed; it is flagged, not\n   repaired, and priced both ways.\n","patch":null,"cpu_hours":0.05,"hashes":{"wright48price.py":"ba77ba602d119b41ae8b8010a057fe84fe3677b55c8f93166dbd15c4f163680a","wright48price.out":"04456b52a95d29167bc4396476f2094ae10230553dddcf37d66fb6d99381e251","wright48price.json":"0d2f21897b7ef9be5a6b8616cb178a0019c2dfdbadb0c4810a68a21c66031ee5","04456b52a95d29167bc4396476f2094ae10230553dddcf37d66fb6d99381e251":"wright48price.out","0d2f21897b7ef9be5a6b8616cb178a0019c2dfdbadb0c4810a68a21c66031ee5":"wright48price.json","ba77ba602d119b41ae8b8010a057fe84fe3677b55c8f93166dbd15c4f163680a":"wright48price.py"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-17T23:22:28.388Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[778,914,916,960],"messages":[]},"tokens":{"log":"custom","input":38388,"models":{"deepseek-v4-flash":49789},"output":49789,"source":"custom-jsonl","entries":1,"cache_read":4820352,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe: `wright48price.py` (job #1823, route 48 pursue)\n\nPrerequisite: CPython 3 (stdlib only, `fractions.Fraction`). Nothing is fetched or downloaded; no\ncorpus producer is run; no route-48 or route-9 measurement is recomputed.\n\n```\npython wright48price.py wright48price.json > wright48price.out\n```\n\n* runtime < 0.1 s, no threads, no randomness; exit 0 iff all 16 gates pass (`passed` in the JSON).\n* **All arithmetic is exact rationals**: `Fraction(51,95)`, `Fraction(39,95)`, `Fraction(7,190)`,\n  `Fraction(1,19)`. There is no float anywhere except two `float(...)` calls in *printed* detail\n  strings; the gates never see one.\n* The five terms and the saving convention are documented at the top of the file, together with the\n  verbatim Theorem 2.1 of arXiv:2604.25177v2 and the note that the paper's third term is printed\n  `A^(1/20)` while its own derivation gives `A^(3/10)`; the script prices the derivation's exponent by\n  default and the printed one in the `printed_A_exponent_*` gate.\n* Reviewer's cheapest check: read the five `terms_R1_a0` values in the JSON and compare with\n  Theorem 2.1 as printed, then check that `max` of them is the third (`binding_term_index: 3`) and\n  that `-(1/4 * 0) - 3/380 = -3/380` is the saving. No corpus state, no network.\n* What this script does NOT do: it does not measure `A`. That is the next experiment, and it needs\n  the corpus's own frequency-support data rather than this file.","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":"2026-09-17T23:23:12.821Z","file_notes":null,"research":{"outcome":"progress","route_id":48,"next_step":{"method":"Measure A at the route's own recorded moduli with the same instrument that produced T99 (returns #771/#778's producer), reporting the count of retained frequencies as a power of c rather than as a ratio to n^(13/28) or sqrt(n); run the pricing script at the measured a and read off the saving and the binding term; in parallel settle the A-exponent of Theorem 2.1's third term from v1's TeX source, since the two readings differ at a = 1/2.","compute":{"ram_gb":1,"disk_gb":0.1,"cpu_hours":0},"failure":"a < 11/114: the frequency support is shorter than the theorem needs, the bounded-third-sequence reading is confirmed as the true obstruction, the shortfall is exactly 3/14 of the requirement, and route 48's blocked verdict stands -- now scoped to A instead of to the length window or the modulus.","success":"a >= 13/114 (or >= 11/114 at R = 1): the bar 7/190 is met by a published theorem at the record's lengths with saving up to 29/760 (R = q) or 39/760 (R = 1), and the cell's trilinear obstruction is removed by a length-condition that already holds.","question":"What is the support length A of the cell's frequency (third) sequence, as a power of c -- i.e. is a = log_c A at least 11/114 (with R = 1) or 13/114 (with the corpus's fixed factor q = c^(1/19)), the thresholds at which Wright Theorem 2.1's sum-correct saving reaches the 7/190 bar?","budget_hours":1.5,"required_tools":[],"required_sources":[]},"depends_on":[778,916,960],"evidence_md":"WHAT THE EVIDENCE CHANGES. Job #960 proposed pricing Wright arXiv:2604.25177v2 Theorem 2.1 at the\ncell's parameters, EVERY term, instead of the single \"headline\" term #914 read. Done, exactly, and\nthe result moves the obstruction off the length axis entirely.\n\n(1) #916 is confirmed and sharpened. Its two numbers -- saving 3/380 at R=1 and 1/380 at R=c^(1/19) --\nare reproduced here directly from the theorem text, and the term that binds is identified: it is the\nTHIRD bracket term M^(1/10)/(R^(3/20)A^(1/20)N^(3/20)), not the 1/N^(1/8) term #914 priced. At\nM = c^(51/95), N = c^(39/95), rho = 1/19 the five exponents are -39/760, -63/760, -3/380, -87/1900,\n-87/760.\n\n(2) #914's 39/760 is term 1 alone and, at A = O(1), overstates the sum-correct saving by exactly 13/2.\nThe number is attainable, but only once the frequency sequence is long enough -- which is the point.\n\n(3) NEW: the saving is LINEAR in the third-sequence exponent a = log_c A, because term 3 both carries\nA and binds: saving = 3/380 + (3/10)a (R=1, until a = 11/76, then flat 39/760) and 1/380 + (3/10)a\n(R=c^(1/19), until a = 9/76, then flat 29/760). So the bar 7/190 is met iff a >= 11/114 (R=1) or\na >= 13/114 (R=c^(1/19)) -- and NOT at a = 0, where the shortfall is exactly 3/14 of the bar.\n\n(4) Therefore route 48's stated obstruction -- \"NO published improving range contains the cell\", a\nLENGTH window -- is false at the record's lengths: the best term gives 39/760 > 7/190, so\nM = c^51/95, N = c^39/95 are inside the trilinear instrument's useful range. #960's relocation to the\nmodulus is also too coarse: the fixed factor enters only as a loss (R^(1/4) = c^(1/76)) worth less\nthan the A-gain available. The obstruction is a THIRD-SEQUENCE-LENGTH condition A >= c^(11/114), a\nweaker requirement than any new theorem, decided by a statistic route 48 has already measured (T99).\n\n(5) Two controls on the same five terms: in route 48's OWN normalisation (min(d,e) = L^(2/5),\ncomplement L^(3/5)) a bounded frequency sequence buys EXACTLY nothing -- term 3 equals 1, saving 0 --\nand a = 1/2 gives 1/20 = 19/14 of the bar. And the paper's third term is internally inconsistent: the\ndisplay prints A^(1/20) where its own derivation gives A^(3/10); priced both ways at a = 1/2 they give\n21/760 (FAIL, ratio 3/4) and 29/760 (PASS, ratio 29/28), so the two readings disagree on the verdict\nand this must be settled against the v1 TeX before the route is judged.\n\nNOT CLAIMED: that A >= c^(11/114) at the cell (unmeasured here -- that is the next experiment); that\nroute 9 closes; that the S-family ceiling #632/#634 is touched (it is not); any exponent of my own.\nLIMITS: the dictionary (m,n,a,R) -> cell objects is a reading taken from the corpus's own\nc = q e1 e2 normalisation, with route 48's internal L^(2/5)/L^(3/5) reading priced alongside; the\nfactor (1+|theta|A/(MN))^(1/4) is a bounded constant here, stated not proved.","prior_art_md":"Updated 2026-09-18 (second pass, this experiment's own antecedent). CONFIRMED at full text, not\nabstract: Thomas Wright, \"Trilinear Kloosterman fractions I: partially fixed moduli and unbalanced\nconvolutions\", arXiv:2604.25177v2 (v1 28 Apr 2026, v2 7 Aug 2026). Theorem 2.1 is the instrument: a\nbound for B(M,N,A,R) = sum alpha_m beta_n nu_a e(theta a mbar/(nR)) under M << N^2, R << M^A, with\nthe five-term bracket transcribed in the report and the improvement on Bettin-Chandee [BC, Adv.\nMath. 328 (2018)] arising from a fixed factor R in the denominator. Its Corollary 2.2 supersedes\nFouvry-Radziwill Corollary 1.1: (i) N <= Q^(-33/28)X^(17/28-eps) vs the old N <= Q^(-11/12)X^(17/36-eps);\n(ii) and (iii) with Q <= X^(45/89-eps). Its Theorem 2.3 improves the dispersion term by N^(1/8) and\nN^(2/5).\nALSO CONFIRMED: Thomas Wright, \"Trilinear Kloosterman fractions II: subdyadic intervals and nearly\nbalanced convolutions\", arXiv:2608.27732v1 (27 Aug 2026) -- N = X^(1/2+delta), M = X^(1/2-delta),\n0 < delta < 1/68 (vs Fouvry-Radziwill's 1/112); subdyadic control is the cell's second structural\nfeature and it is not exercised by this pricing (Theorem 2.1 was; II's sharpening is a candidate for\nthe same treatment only if A proves short).\nRETRACTED, preserved as refuted and not cited: Dong-Robles-Zeindler arXiv:2601.00292v2 -- the\nimprovement for the cell's own F-family form sum alpha_m beta_n e(a mbar/(bn)) is withdrawn by the\nauthors (\"missed a factor of L^2 in equation (2.53), L^5 -> L^7\"), error found by Pascadi.\nBASELINES the route already records: Pascadi arXiv:2511.08445v2 (composite moduli, length sqrt(c),\nimproving range M~N in [c^(5/12+eps), c^(5/8-eps)]); Shen arXiv:2607.06575v1 (prime q, N a bit below\nq^(1/2)); Fouvry-Shparlinski arXiv:2210.15761v2 (prime p, N >= p^(1/8+eps)).\nEXACT REMAINING GAP, restated on the new axis: not a length window and not the modulus, but the\nsupport length A of the frequency sequence: the bar 7/190 is met iff A >= c^(11/114) (or c^(13/114)\nwith the corpus's fixed factor q), and the saving saturates at 39/760 (R=1) or 29/760 (R=c^(1/19)).\nA is unmeasured; the route's own T99 statistic is the natural instrument. A second, independent open\nitem is textual: the A-exponent of Theorem 2.1's third term (printed A^(1/20), derived A^(3/10)),\nwhich changes the verdict at a = 1/2 and must be read off v1's TeX."},"research_route_id":48,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_9e3c846778a19c71137dde42","run_id":"run_1b555656cba19bf7bbd26ca5","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/48 and return #960. Return the ordinary report and transcript plus research: {route_id: 48, 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":"778","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"916","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"960","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/48","transcript_url":"/projects/twin-primes/return/961/transcript","files":[{"sha256":"ba77ba602d119b41ae8b8010a057fe84fe3677b55c8f93166dbd15c4f163680a","name":"wright48price.py","bytes":8838},{"sha256":"0d2f21897b7ef9be5a6b8616cb178a0019c2dfdbadb0c4810a68a21c66031ee5","name":"wright48price.json","bytes":3389},{"sha256":"04456b52a95d29167bc4396476f2094ae10230553dddcf37d66fb6d99381e251","name":"wright48price.out","bytes":2081}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}