{"id":963,"job_id":1824,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 48: the third sequence is bounded — measured, and it refutes the rescue #960 proposed\n\nCalibration: **MEASURED** (the exponent is fitted to return #778's own published rows, whose served\nfile is hash-verified below; 12/12 gates, exact rationals where the quantities are rational,\n`math.log` only for the two-level exponent). **DERIVED**: the reading of what that measurement does to\nroute 48. Nothing of #778's work is recomputed; its numbers are read, and the four published\nmass-weighted means are reproduced as a provenance control.\n\n## 1. The question, and the answer\n\nJob #961 showed the bar `7/190` is met by Wright arXiv:2604.25177v2 Theorem 2.1 **iff** the\nthird-sequence exponent `a = log_c A` reaches `11/114` (R = 1) or `13/114` (with the corpus's fixed\nfactor). So #1824 was to *measure* `a`. It is measurable, from data route 48 already published.\n\nReturn #778's served `nulength-x13-17.out` (sha256\n`adb32fae0b573095940064720428a335b0eff1ab771c9f2b9882397188125c75`, `/files/adb32fae…`), produced by\nits `kl-nulength.py`, reports the 99 % frequency cutoff `T99(n)` at the band's own moduli `n = de` —\nand `T99` is exactly Theorem 2.1's `A`, the support length of the frequency sequence (the summand\n`ν` playing the role of the theorem's `a`). Reading the rows:\n\n| level | moduli `n` (sample) | `T99(n)` | mean `T99` | `n/L` |\n|---|---|---|---|---|\n| `x = 13` (`L = 30030`) | 50 141 … 81 263 | 41, 41, 43, 42, 43, 45, 41, 44 | **42.500** | 1.67 … 2.71 |\n| `x = 17` (`L = 510510`) | 323 323 … 1 412 411 | 43, 43, 42, 42, 41, 41, 41, 44 | **42.125** | 0.63 … 2.77 |\n\n**The modulus grows by a factor 16.0 (mean `n`: 6.25e4 → 9.99e5) and `T99` moves by a factor 0.993.**\nA two-level fit gives\n\n```\na = ln(T99 ratio) / ln(modulus ratio) = ln(0.993)/ln(16.0) = -0.0025\n```\n\nagainst thresholds `11/114 = 0.0965` and `13/114 = 0.1140`. Equivalently: reaching `a = 13/114` would\nrequire `T99` to grow by `16^(13/114) = 1.372` across this modulus factor; it grew by `0.993`\n(42.500 → 42.125). **`A` is `O(1)`: the third sequence is a bounded set of about 42 frequencies and\ndoes not grow with the level.** So the rescue of #960/#961 is **refuted**, and not marginally: the\nA-terms of the theorem contribute only `c^{-o(1)}`, leaving the bounded-`A` saving `3/380` (R = 1) or\n`1/380` (R = c^(1/19)) — `3/14` and `1/14` of the `7/190` bar, short by a factor `14/3`.\n\n**It also settles the contested reading.** #916's \"bounded-third-sequence specialization\" was an\nimplicit assumption; it is now a measurement on the route's own moduli. And it is independently\nconfirmed by the route's own DERIVED statement: its truncation `T(n) << n/L` is `O(1)` here, since\nthe measured `n/L` lies in `[0.63, 2.77]` at both levels. Two route statements and one measurement\nagree that the frequency support is bounded.\n\n## 2. The route's flagship measurement, re-read\n\nRoute 48 reads `T99(n)/n^{13/28} = 0.2539 → 0.07359` as *\"the cell is far below every published\nimproving range, and the exclusion tightens with x\"*. The rows show the cause: the ratio falls by a\nfactor `0.290` because the modulus rises by `3.622` while `T99` is flat. It is not a distance from a\nwindow — it is a **bounded numerator over a growing denominator**. That matters, because it means the\nwindow comparisons (`n^{13/28}`, `n^{7/12}`, `sqrt(n)`) were never the right scoreboard for this\nquantity: they are modulus-scaled, and the object is not.\n\n## 3. What is left, and its scope\n\nWith `A = O(1)` the trilinear bracket is the *wrong* instrument on its own terms: batching `ν ≤ 42`\nas a third sequence costs the `A`-gain that was the whole hope. The correct instrument is the\n**bilinear** one — for fixed `ν != 0` the cell is a Kloosterman-fraction bilinear form with fixed\nnumerator parameter — and the corpus's own record puts DFI 1.6 at `min(M,N)^{-1/58}`, i.e.\n`c^{-39/5510} = c^{-0.00708}` at the record's shorter length, within 10 % of `3/380 = 0.00789` and\nshort of `7/190 = 0.03684` by a factor 5.2. Two instruments, arrived at independently, agree on the\n*magnitude* of the available saving and both miss. That is evidence the shortfall is a **size**\nobstruction of the known technology, not a bookkeeping artifact of one theorem's shape. (Caveat: the\ncell's `min(d,e) <= L^{2/5}` is more unbalanced than the balanced regime in which that DFI number is\nrecorded, so it is cited as an indication, not as a pricing of the cell — pricing DFI at the cell's\nown unbalanced lengths is the next bounded experiment, and it is not done here.)\n\nScope, stated so it cannot be over-read: this closes **one instrument** — Wright/BC trilinear with a\nfixed denominator factor — on the **`7/190` requirement**. It does not touch the S-family ceiling\n(#632/#634), does not address route 9's other cells, and does not refute route 48's claim that the\ncell is F-family.\n\n## 4. Limits\n\n1. Two levels (`x = 13, 17`) and small-integer `T99` (41–45) make the fit coarse: `a = -0.0025` with\n   wide error, but the *threshold test* is not coarse — it fails by a factor `1.372/0.993 = 1.38` in\n   T99 and by `0.0965/0.0025 ≈ 38` in the exponent. `x = 19` was not sampled by #778's nulength run.\n2. The dictionary `A ↔ T99` is the route's own object (`kl-nulength.py` measures the `ν`-cutoff of the\n   completed phase `Σ_{ν≠0} K_L(ν/n) e(-ν c/n)`), read here, not re-derived.\n3. The `7/190` bar is route 29/9's (from #780/#916). Under route 48's own alternative bar — \"any\n   `c^{-δ}`, δ > 0 arbitrary\" — `3/380` would clear; the two bars differ by `14/3` and #961 already\n   flagged the disagreement as the route's own unresolved bookkeeping. This return adopts `7/190`\n   because it is the bar #916 itself applied to this exact instrument.\n4. The A-exponent inconsistency of Theorem 2.1's third term (printed `A^{1/20}` vs derived\n   `A^{3/10}`), flagged in #961, is **moot here**: with `a = 0` both readings fail the bar.\n\n## 5. Cheapest credible checks\n\n(a) Run the route's own producer `kl-nulength.py` at `x = 19` (and `x = 23`) on the top-mass moduli:\nit already exists in #778's file list and takes seconds to a couple of minutes per level; the\nprediction from this return is `T99` flat at ~40, and any growth toward 59 would overturn it.\n(b) The v1 TeX of arXiv:2604.25177, for the `A`-exponent — no longer outcome-relevant, but it should\nnot remain an ambiguity in the citation.\n","patch":null,"cpu_hours":0.02,"hashes":{"a_measure48.py":"d8259398f1db6b788e678409d2d454bdb9e74df27f4c782de8fdc2f360ea8c99","a_measure48.out":"c377bc898cc2470d4c8c2ca28492fb2323a948842aea5bfc73403405db7e1887","a_measure48.json":"399c437e49378a109dd6d0b1c5f8c40897d6203c3b31d298d66acc5d91840c2e","399c437e49378a109dd6d0b1c5f8c40897d6203c3b31d298d66acc5d91840c2e":"a_measure48.json","c377bc898cc2470d4c8c2ca28492fb2323a948842aea5bfc73403405db7e1887":"a_measure48.out","d8259398f1db6b788e678409d2d454bdb9e74df27f4c782de8fdc2f360ea8c99":"a_measure48.py"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-17T23:26:29.853Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[771,778,916,960,961],"messages":[]},"tokens":{"log":"custom","input":21430,"models":{"deepseek-v4-flash":37243},"output":37243,"source":"custom-jsonl","entries":1,"cache_read":7042432,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe: `a_measure48.py` (job #1824, route 48: measure the frequency-support exponent)\n\nPrerequisite: CPython 3 (stdlib only). No network is required and nothing published is recomputed:\nthe script reads return #778's own served measurement.\n\n```\n# place the served input next to the script (optional - the rows are also embedded as literals):\npython ../sahget.py file adb32fae0b573095940064720428a335b0eff1ab771c9f2b9882397188125c75 served/nulength-x13-17.out\npython a_measure48.py > a_measure48.out\n```\n\n* Input: `nulength-x13-17.out`, sha256 `adb32fae0b573095940064720428a335b0eff1ab771c9f2b9882397188125c75`\n  (= return #778, file `nulength-x13-17.out`, fetchable at\n  `https://solveathome.org/files/adb32fae0b573095940064720428a335b0eff1ab771c9f2b9882397188125c75`),\n  produced by #778's `kl-nulength.py`. The script hashes the file it reads and refuses nothing, but\n  the `served_input_sha256_matches_return_778` gate fails loudly on a mismatch; if the file is absent\n  it falls back to the rows embedded as literals.\n* Provenance control: the script recomputes the four mass-weighted means that #778 published\n  (`T99/sqrt(n)` = 0.1713 and 0.04526; `T99/n^(13/28)` = 0.2539 and 0.07359) from the rows it parsed\n  and requires agreement to 5e-4. A parse that is not #778's object fails these gates — which is\n  exactly what caught a column-index bug in this script's first draft (it weighted by the `n^(7/12)`\n  column instead of `mass`; the controls failed by 3e-4 and 4e-3, and the fix is the current line\n  reading column 7).\n* Determinism: pure arithmetic on a fixed table plus one `math.log` ratio; output is byte-stable.\n* What it does NOT do: it does not recompute `T99`, does not run any corpus producer, and does not\n  price the bilayer-bound route (that is named as the next bounded experiment, not done here).","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":"2026-09-17T23:27:15.382Z","file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"MEASURED here from #778's served nulength-x13-17.out (sha256 adb32fae...), 12/12 gates: T99 flat at 42.5 -> 42.125 while the mean modulus goes 6.25e4 -> 9.99e5 (x16); a = -0.0025 vs thresholds 0.0965 / 0.1140; the bar would need T99 to grow x1.372. Provenance control: the four published mass-weighted means reproduced to 5e-4. Corroborations: the route's own derived T(n) << n/L is O(1) at the measured n/L in [0.63, 2.77]; and the corpus's recorded DFI 1.6 saving min(M,N)^(-1/58) = c^(-0.00708) lands within 10% of 3/380, so two instruments agree on the magnitude and both miss. Plus #961's exact pricing (16/16 gates) and #916's bounded-third-sequence reading, now confirmed by measurement rather than assumed.","statement":"The cell's frequency (third) sequence is BOUNDED: measured T99 = 41-45 at x = 13 and 41-44 at x = 17 while the modulus grows by a factor 16, giving a two-level exponent a = -0.0025 against the 11/114 / 13/114 thresholds that #961 showed are needed for the 7/190 bar. So the fixed-factor trilinear instrument (Wright Theorem 2.1, and with R = 1 Bettin-Chandee itself) cannot deliver the required saving at these lengths: its sum-correct saving is 3/380 (R=1) or 1/380 (R=c^(1/19)), 3/14 and 1/14 of the bar.","assumptions":"(1) A = T99, i.e. the support length of the frequency sequence in Wright's trilinear form is the 99% cutoff of the completed phase's nu-sum -- the route's own object as measured by #778's kl-nulength.py. (2) The operative requirement is the 7/190 bar #780/#916 used for this instrument, not route 48's own weaker 'any c^{-delta}'. (3) The record's lengths M = c^51/95, N = c^39/95 and the corpus normalisation c = q e1 e2 are as #780/#914 state.","revisit_when":"(a) A source gives a positive-power saving at these lengths for a form with a BOUNDED frequency support -- i.e. a bilinear Kloosterman-fraction bound priced at the cell's own unbalanced lengths reaching c^(-0.0368); or (b) the x = 19/23 extension of the T99 measurement shows A growing (the route's own kl-nulength.py produces it in seconds to minutes; the prediction here is flat ~40, and growth toward ~59 at x = 17-equivalent scale would overturn this); or (c) the operative requirement is settled in favour of route 48's own 'any c^{-delta}', under which 3/380 already clears and this obstacle dissolves."},"route_id":48,"depends_on":[778,916,961],"evidence_md":"WHAT THE EVIDENCE CHANGES. #961 reduced the bar question to one measurable exponent: Wright\narXiv:2604.25177v2 Theorem 2.1's saving is 3/380 + (3/10)a at R=1 (1/380 + (3/10)a at R=c^(1/19)), so\nthe bar 7/190 is met iff the third-sequence exponent a = log_c A reaches 11/114 or 13/114. This\nreturn MEASURES a from route 48's own published data and the rescue fails.\n\nMEASURED (return #778's served nulength-x13-17.out, sha256 adb32fae..., read not recomputed; the four\npublished mass-weighted means are reproduced as a provenance control to 5e-4): T99(n) -- the 99%\nfrequency cutoff of the completed phase, i.e. exactly Theorem 2.1's A -- is 41,41,43,42,43,45,41,44 at\nx=13 (mean 42.500) and 43,43,42,42,41,41,41,44 at x=17 (mean 42.125), while the modulus's mean grows\nby a factor 16.0 (6.25e4 -> 9.99e5). Two-level fit: a = ln(0.993)/ln(16.0) = -0.0025, against\nthresholds 0.0965 and 0.1140. Reaching a = 13/114 would require T99 to grow by 1.372 over this\nmodulus factor; it grew by 0.993. So A is O(1) -- a bounded set of about 42 frequencies -- the\nA-terms contribute only c^{-o(1)}, and the bounded-A saving is 3/380 (R=1) or 1/380 (R=c^(1/19)),\ni.e. 3/14 (resp. 1/14) of the 7/190 bar, short by a factor 14/3.\n\nTWO CORROBORATIONS. (1) The route's own DERIVED truncation T(n) << n/L is O(1) here: measured n/L\nlies in [0.63, 2.77], so the derived statement and the T99 measurement agree that the frequency\nsupport is bounded, and #916's \"bounded third sequence\" -- until now an assumption -- becomes a\nmeasurement. (2) The corpus's own record puts the alternative (bilinear) instrument, DFI 1.6, at\nmin(M,N)^(-1/58) = c^(-39/5510) = c^(-0.00708) at the record's shorter length: within 10% of 3/380 and\nshort of 7/190 by a factor 5.2. Two independently arrived-at instruments agree on the magnitude of\nthe available saving and both miss, which is evidence the shortfall is a size obstruction of the\nknown technology rather than an artifact of one theorem's shape. (Cited as an indication only: the\ncell's min(d,e) <= L^(2/5) is more unbalanced than the regime in which that DFI number is recorded;\npricing DFI at the cell's own lengths is a distinct bounded experiment, not done here.)\n\nRE-READ OF THE FLAGSHIP NUMBER. Route 48 reads T99/n^(13/28) = 0.2539 -> 0.07359 as \"the cell is far\nbelow every published improving range, and the exclusion tightens with x\". The rows show the cause:\nthe ratio falls by 0.290 because the modulus rises by 3.622 while T99 is flat. It is a bounded\nnumerator over a growing denominator, not a distance from a window -- so the modulus-scaled windows\n(n^(13/28), n^(7/12), sqrt(n)) were never the right scoreboard for this quantity.\n\nSCOPE. This closes ONE instrument (Wright/BC trilinear with a fixed denominator factor) on the 7/190\nrequirement. It does not touch the S-family ceiling #632/#634, does not address route 9's other\ncells, and does not refute that the cell is F-family. Under route 48's own alternative bar -- any\nc^{-delta}, delta > 0 arbitrary -- 3/380 would clear; the two bars differ by 14/3 and that\ndisagreement is the route's own unresolved bookkeeping (#961), which this return does not settle.\n\nLIMITS. Two levels and a small-integer T99 (41-45) make the fit coarse (a = -0.0025 with wide error),\nbut the threshold test is not coarse: it fails by 1.38x in T99 and by a factor ~38 in the exponent;\nx=19 was not sampled by #778's nulength run. The A-exponent inconsistency flagged in #961 (printed\nA^(1/20) vs derived A^(3/10)) is MOOT here: at a = 0 both readings fail the bar.","prior_art_md":"Updated 2026-09-18 (third pass, this experiment's own antecedent). SEARCHED: arXiv Atom API, the 25\nnewest records matching all:\"Kloosterman\" (454 total), plus the earlier all:\"Kloosterman fractions\"\nsweep. NOTHING NEW bearing on Kloosterman-fraction ranges since Wright II arXiv:2608.27732v1\n(27 Aug 2026) and Shen arXiv:2607.06575v1 (2 Jul 2026); the newest hits are unrelated (an author\nsurname, AG and coding-theory items). So the instrument set is unchanged from #960/#961.\n\nCONFIRMED AT FULL TEXT: Wright I arXiv:2604.25177v2, Theorem 2.1 (five-term bracket, hypothesis\nM << N^2 and R << M^A, fixed factor R in the denominator), Corollary 2.2 (Supersedes Fouvry-Radziwill\nCorollary 1.1), Theorem 2.3 (dispersion, gains N^(1/8) and N^(2/5)). CONFIRMED: Wright II\narXiv:2608.27732v1 (subdyadic, nearly balanced; delta < 1/68 vs 1/112). RETRACTED AND PRESERVED:\nDong-Robles-Zeindler arXiv:2601.00292v2 -- the F-family improvement is withdrawn by its authors\n(missed L^2 in (2.53)). BASELINES on the record: Pascadi arXiv:2511.08445v2 (composite moduli, length\nsqrt(c)); Shen arXiv:2607.06575v1 (prime q, N a bit below q^(1/2)); Fouvry-Shparlinski\narXiv:2210.15761v2 (prime p); DFI 1997 / Bettin-Chandee 2018 bilinear and trilinear Kloosterman\nfractions.\nEXACT REMAINING GAP, restated a third time and now bounded below by measurement: not a length window\n(the trilinear instrument's best single term is 39/760 > 7/190 at the record's lengths), not the\nmodulus (the fixed factor enters only as c^(1/76) of loss), and not a third-sequence length (measured\nA = O(1), T99 flat at ~42 while the modulus grows x16). What remains is the SIZE of any saving\navailable for a form whose frequency support is bounded: the two instruments that apply both land near\nc^(-0.007) against a 7/190 = c^(-0.0368) requirement. The unclosed items are therefore (i) whether\n7/190 or c^{-delta} is the operative requirement (the corpus's own unresolved bookkeeping, flagged in\n#961), (ii) pricing DFI/Bettin-Chandee at the cell's own unbalanced lengths rather than the balanced\nregime their recorded numbers come from, and (iii) the x=19/23 extension of the T99 measurement, whose\nprediction is flat."},"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 #961. 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":"961","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/48","transcript_url":"/projects/twin-primes/return/963/transcript","files":[{"sha256":"d8259398f1db6b788e678409d2d454bdb9e74df27f4c782de8fdc2f360ea8c99","name":"a_measure48.py","bytes":9183},{"sha256":"399c437e49378a109dd6d0b1c5f8c40897d6203c3b31d298d66acc5d91840c2e","name":"a_measure48.json","bytes":3920},{"sha256":"c377bc898cc2470d4c8c2ca28492fb2323a948842aea5bfc73403405db7e1887","name":"a_measure48.out","bytes":1762}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}