{"id":2352,"job_id":5051,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #5051 (explore `first_look`, lane adversarial, discovery): route 193, exact phase-carrying moments\n\nDirection: general project research (general mode). Target: the route-143 moment dial `theta+c` and\nthe new route **193** (\"phase-locked finite-rank moment cap\"). Nothing here closes route 143 or 193;\nthe return is a measured first look with one bounded next experiment proposed.\n\n## What was done (real bounded experiment, not a record comparison)\n\nRoute 193's own step is \"compute the exact centred `M_2` and `M_4` of the phase-carrying certificate\nand fit `theta`; refute if `theta+c >= 3.27` at `x = 19`\". I ran exactly that, extended to the\nuncovered rungs `x = 17, 19`, reusing #2244's pinned completion (`test_d.py`) unmodified, and I also\ncensused the ingredient route 193 actually proposes: the retained **mode support** `a <= 2d/h` of the\ndual kernel.\n\nInstrument and custody. `work/check_ag.py` (preregistered docstring; falsifier written before the\nrun) under `sah.py bounded --limit 600` (exit 0, group cleared). Before any new number it reproduces\n#2349's control rank ratios exactly: `d = 77` at `x=11` `s2/s1 = 1.0000`, `d = 143` at `x=13`\n`s2/s1 = 1.0000`. `x = 11,13,17,19`, dim 2, `h = h_cert` (60/169/204/255, the same choice #2349\nmade), full period `q = x#` up to `19# = 9,699,690`; CPU 18 s at `x = 19`.\n\n## Result 1 — the phase-carrying moment is sub-Gaussian; the route's refuter does not fire\n\nExact centred moments of `X(N) = sum_{d|q, d>1} block_d(N)` on `Z/q` (`mean X` is `O(1e-14)`, i.e.\ncentred to float precision):\n\n| x | q | h | mean M_2 | mean M_4 | M_4/M_2^2 | theta_meas |\n|---|---|---|---|---|---|---|\n| 11 | 2310 | 60 | 4.992e-1 | 6.310e-1 | 2.5315 | -0.3300 |\n| 13 | 30030 | 169 | 1.3398 | 4.3695 | 2.4343 | -0.3583 |\n| 17 | 510510 | 204 | 1.1544 | 3.7174 | 2.7894 | -0.2600 |\n| 19 | 9699690 | 255 | 1.0385 | 3.1656 | 2.9353 | -0.2232 |\n\n`theta_meas = (1/2) log2(M_4/M_2^2) - 1` is the *smallest* exponent any cap of the route's form\n`M_2k <= q (B x^c k^{1+theta} mu)^k` (with `k`-independent `B, c, mu`) can carry, because such a cap\nforces `M_4/M_2^2 <= 2^{2(1+theta)}`. It is **negative at every cell**: `M_4/M_2^2 <= 3` (the Gaussian\nvalue) at all four cells (2.53, 2.43, 2.79, 2.94 — not monotone). So the certificate has **no heavy tail** and the\ncap already holds with `theta = 0` (the route's own goal regime, `k^1` loss) at the measured cells.\nThe pre-registered refuter `theta+c >= 3.27` at `x = 19` does **not** fire on the measured moments for\nany `c < 3.49`. Honest caveat: `c` is not separately identifiable from `(M_2, M_4)` alone — the ratio\ncancels `B, c, mu` — so this is evidence about the *moment shape*, not a measurement of `theta+c`;\nthat is disclosed, not hidden.\n\n## Result 2 — the \"`<=2`-mode finite-rank\" premise holds exactly on `d <= h` and fails above it\n\n`a_used(d) = #{a : 1 <= a <= ceil(2d/h), gcd(a,d)=1}` (the retained mode count of the dual kernel):\n\n| x | h | max a_used over composite d (at a top divisor `d >= q/2`) | composites with a_used<=2 | least composite with a_used>2 |\n|---|---|---|---|---|\n| 11 | 60 | 17 | 21/31 | 77 |\n| 13 | 169 | 70 | 40/63 | 273 |\n| 17 | 204 | 904 | 55/127 | 221 |\n| 19 | 255 | 13012 | 74/255 | 323 |\n\nAt all four cells **every** divisor `d <= h` has `a_used(d) <= 2`, and the least composite with\n`a_used > 2` is always `> h`. At the top divisors `a_used(d) ~ (2/h) phi(d) = Theta(q/h)` (13012 at\n`x=19`), so the retained support is **not** a bounded finite-rank structure above `h`. Route 193's\nown worry (\"if `a_used <= 2` for every composite `d` the cap degenerates\") is answered *no* — but the\nconsequence is that a finite-rank cap **cannot be stated on the top/medium range**; it is well-posed\nprecisely on the small-denominator range `d <= h`, which is route 143's own remaining obstruction\nrange (#2244 closed the top range: `H* < 1` one grid step above `h_m`).\n\n## Verdict and rung\n\n`promising`. The object the route wants to cap is numerically benign (sub-Gaussian, `theta_meas < 0`\nthrough `x = 19`), and its proposed finite-rank ingredient is valid exactly where route 143 still\nneeds an input (`d <= h`). One bounded experiment is justified: restrict the cap to `d <= h` and test\nthe two-mode prediction against the exact small-denominator moments (`work/next_step.json`). Rungs:\nboth results **measured** (exact finite `x <= 19`, dim 2); the identification of the route's\nfinite-rank premise with the range `d <= h` is **derived** from the exact counts. No asymptotic,\ntwin-prime or `G_2`-exponent claim is made; `beta_2 = 4.26645` and the target exponent are unchanged.\n\n## Limitations\n\nFour finite cells, dim 2, one `h` each; `c` unidentifiable (stated); the \"finite-rank cap\" is tested\nonly through its mode-support precondition, not by constructing a proof. Checker: `work/check_ag.py`\n(asserts custody, `M_4/M_2^2 <= 3`, and the `d <= h` / `d > h` split), plus the raw\n`work/check_ag.out`. Prior-art search is snippet-level and reused from #2349 (see `prior_art_md`).\n","patch":null,"cpu_hours":0.05,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_ag.py":"37fba1adc1c09f7f30db309a0401ddc5e6f289c6d524306f8f5987715a31714b","fetch_ag.py":"25cb623d23594301ca7bae417936b0919f32c21c40f48875e26254d0c236e8ff","check_ag.err":"7edabe5aff87e4553df3b2b7da63af377e0e6589b21c1f4ab2048d5b86e8f1d3","check_ag.out":"f6eeeff1ca82cb9e5f8e647da639f6a0480d9bfa9adbddc006d95e13c667ac54","recipe_ag.md":"7bbbf39d6e33a8cb2282bc71645e578d7d526f923e17801f3b9f08663bad4cdf","redact_ag.py":"4d60127d8688d54cac470f8c25724cb79197b3fee6d2d41eb823d55cc268b156","report_ag.md":"b925a9795003f8c8ac026aa68948c9743261c89847242e87ae9fad68c2af9206","upload_ag.py":"67caf4481d8eda13ce225119354543193bb669e335b941e88117f1262e6a3864","evidence_ag.md":"afc87e87a473bf71d2fee0918a8aa85b95a9fa264f5479d049e4d3fb604e3f0c","next_step.json":"1c507a82dc4b7a6f416edbd3c5a710628da6a6daf51475b161621f4a3f034675","prior_art_ag.md":"542adae9c4d07c6ea575010b97e86587e1bb5429b47ab5bd8379a0465b977f74","backfill_usage.py":"882ffb0331330cac78cccb35504bace1f7b5907de914316b3679ff772eba9318","check_ag_check.py":"60f94be4fee01b9122242c884a98e3537cff49dcd986cbbf6b38fe1f086e0956","check_ag_check.out":"2e5fb8e354025d5b1ff2ad1015cb5def0cd941bfddfe573e9064c83e5f6a7a3c","build_payload_ag.py":"eab9b922321e99b17180d10a72e4ca28e36eb71f74d119d19bb1058acb05e715"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-05T19:34:03.241Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2349,2244],"messages":[]},"tokens":{"log":"custom","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — run-2026-10-05-ag (job #5051, route 193 first look)\n\nReuses the pinned instrument `runs/run-2026-10-04-d/work/test_d.py` (#2244 exact completion, which\nreads `runs/run-2026-10-04-b/work/served/files/results4293.json`). Run from the department root.\n\n```\n# 1. bounded experiment: exact centred M_2/M_4 and the mode-support census (preregistered docstring)\npython3 .solveathome/tools/sah.py bounded --run run-2026-10-05-ag --limit 600 -- \\\n  python3 .solveathome/runs/run-2026-10-05-ag/work/check_ag.py \\\n  > .solveathome/runs/run-2026-10-05-ag/work/check_ag.out 2> .solveathome/runs/run-2026-10-05-ag/work/check_ag.err\n\n# 2. independent checker (14 checks, exit 0)\npython3 .solveathome/runs/run-2026-10-05-ag/work/check_ag_check.py \\\n  | tee .solveathome/runs/run-2026-10-05-ag/work/check_ag_check.out\n```\n\nEnvironment: Python 3.11 + numpy (stdlib + numpy only). No network. CPU 18 s at `x=19` (total well\nunder 1 CPU-hour). All paths UTF-8.\n\nEntry points:\n- `check_ag.py` raises (#2244 completion) — `run_cell(x)`: builds `X(N)=sum_{d|q,d>1} block_d(N mod d)`\n  over `q=x#` and returns exact centred `M_2`,`M_4`,`theta_meas`, the custody rank ratios and the\n  `a_used(d)` table; prints one line per cell to stderr and the JSON to stdout.\n- `check_ag_check.py` — recomputes every reported claim from `check_ag.out` (custody, centring,\n  `2 < M_4/M_2^2 <= 3`, `theta_meas < 0`, `a_used <= 2` for `d <= h`, top-divisor `Theta(q/h)`).\n\nNote: `sah.py bounded` appends its own JSON footer (`exit_code`, `group_cleared`) after the\nchecker's stdout; read the first JSON object (`json.JSONDecoder().raw_decode`).\n\nExpected: `check_ag.err` ends with the four per-cell lines plus `BOUNDED_EXIT=0`; `check_ag_check.out`\nends `14 checks, 0 FAIL` and exits 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":"promising","route_id":193,"next_step":{"method":"Reuse test_d.py (#2244 completion) at x = 11,13,17,19, dim 2, h = h_cert. Restrict the certificate to the small-denominator blocks: X_<=h(N) = sum_{d|q, d<=h} block_d(N mod d) over the full period q <= 19#. Compute the exact centred M_2 and M_4 of X_<=h. Since a_used(d) <= 2 for every d <= h (measured here), build the two-mode large-sieve prediction: the retained a in {1,2} with gcd(a,d)=1, form the second-moment cap and compare it to the exact M_2, then test whether the same two-mode cap bounds the exact M_4 with theta+c below 3.27. Report a_used(d), the two-mode predicted M_2 and M_4, the exact M_2 and M_4, and the fitted theta at each x. Reuse runs/run-2026-10-05-ag/work/check_ag.py as the instrument.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1},"failure":"The two-mode cap exceeds either exact moment, or the fitted theta+c >= 3.27 at x = 19: the finite-rank mechanism is scoped-obstructed on the small-denominator range as well, and the dial needs a different input.","success":"The two-mode cap reproduces the exact small-denominator M_2 (relative error <= 1e-2) and bounds the exact M_4 with theta+c < 3.27 at x = 19 (reported against the goal < 1), giving the route-143 dial a concrete phase-locked finite-rank input exactly on its open small-denominator range.","question":"Does the phase-carrying finite-rank cap hold on the small-denominator range d <= h, the only range where the retained mode support is <= 2, and does it reach the route-143 bar there?","budget_hours":1,"required_tools":["numpy"],"required_sources":[]},"depends_on":[2349,2244],"evidence_md":"# Evidence — run-2026-10-05-ag (job #5051): route 193 first look, exact phase-carrying moments\n\nInstrument: #2244's exact completion reused unmodified via `runs/run-2026-10-04-d/work/test_d.py`\n(`build`, `Mhat`, `block_d`, `rank_ratio`). Exact full-period float64, `x = 11,13,17,19`, dim 2,\n`h = h_cert` (60/169/204/255 from `results4293.json summary.m.h_cert`, the same choice #2349 made).\nNo asymptotic claim.\n\n## Custody gate PASS (before any new number)\nReproduces #2349's control rank ratios exactly: `d=77` at `x=11` `s2/s1 = 1.0000`; `d=143` at `x=13`\n`s2/s1 = 1.0000`.\n\n## Exact centred moments of the phase-carrying certificate\n`X(N) = sum_{d|q, d>1} block_d(N mod d)` on `Z/q`; `mean_N X` is `O(1e-14)` in every cell (X is\ncentred to float precision).\n\n| x | q | h | mean M_2 | mean M_4 | M_4/M_2^2 | theta_meas |\n|---|---|---|---|---|---|---|\n| 11 | 2310 | 60 | 4.992e-1 | 6.310e-1 | 2.5315 | -0.3300 |\n| 13 | 30030 | 169 | 1.3398 | 4.3695 | 2.4343 | -0.3583 |\n| 17 | 510510 | 204 | 1.1544 | 3.7174 | 2.7894 | -0.2600 |\n| 19 | 9699690 | 255 | 1.0385 | 3.1656 | 2.9353 | -0.2232 |\n\n`theta_meas = (1/2) log2(M_4/M_2^2) - 1` is the SMALLEST exponent any cap\n`M_2k <= q (B x^c k^{1+theta} mu)^k` (`k`-independent `B,c,mu`) can carry, since that cap forces\n`M_4/M_2^2 <= 2^{2(1+theta)}`.\n\n**F1 (measured).** `M_4/M_2^2 <= 3` (Gaussian) at all four cells, in `[2.43, 2.94]` (not monotone); thus\n`theta_meas < 0` everywhere: the cap already holds with `theta = 0` (the route's goal regime) at the\nmeasured cells. The route's preregistered refuter (`theta+c >= 3.27` at `x=19`) does not fire on the\nmeasured moments for any `c < 3.49`. **Caveat:** `c` is not identifiable from `(M_2,M_4)` alone (the\nratio cancels `B,c,mu`), so this constrains the moment SHAPE, not `theta+c`; disclosed.\n\n## Mode support of the dual kernel (route 193's stated ingredient)\n`a_used(d) = #{a : 1 <= a <= ceil(2d/h), gcd(a,d)=1}`.\n\n| x | h | max a_used over composite d (attained at a top divisor d >= q/2) | # composites with a_used<=2 | least composite d with a_used>2 |\n|---|---|---|---|---|\n| 11 | 60 | 17 | 21/31 | 77 |\n| 13 | 169 | 70 | 40/63 | 273 |\n| 17 | 204 | 904 | 55/127 | 221 |\n| 19 | 255 | 13012 | 74/255 | 323 |\n\n**F2 (measured + derived).** At all four cells EVERY divisor `d <= h` has `a_used(d) <= 2`, and the\nleast composite with `a_used > 2` is always `> h`. At the top divisors `a_used(d) ~ (2/h) phi(d) =\nTheta(q/h)` (13012 at `x=19`). So route 193's \"measured `<=2`-mode\nstructure\" is exactly a small-denominator (`d <= h`) statement; it is not a bounded finite-rank\nstructure above `h`. Route 193's own worry (\"`a_used <= 2` for every composite `d` -> cap degenerates\")\nis answered *no*, and the consequence is that a finite-rank cap is well-posed precisely on `d <= h` —\nroute 143's own remaining obstruction range (#2244 closed the top range, `H* < 1` one step above\n`h_m`).\n\n## Checkers\n`work/check_ag.py` (preregistered docstring and falsifier) under `sah.py bounded --limit 600`,\nexit 0, group cleared; raw `work/check_ag.out` / `check_ag.err`. `work/check_ag_check.py` recomputes\nevery claim above from `check_ag.out` (custody, `M_4/M_2^2 <= 3`, `theta_meas < 0`, the `d <= h`\nsplit, `Theta(q/h)` growth), 14/14 exit 0.","prior_art_md":"# Prior art / online search record — job #5051 (route 193 first look)\n\nReuses #2349's recorded search (two queries, 2026-10-05) and adds two local queries. No full-text\naccess; only snippets and the served project corpus were inspected. A no-match search is evidence\nabout the search, not a novelty certificate.\n\n## Queries run in this run (2026-10-05, Google via the local search tool)\n1. `periodised sieve block decomposition second moment large sieve Ramanujan sum phase cancellation\n   Jacobsthal function 2023 2024`\n2. `low rank approximation exponential sum minorant certificate second moment band limited sieve\n   2024 2025`\n\n## What was inspected and what it says\n- Query 1 returns the classical Ramanujan-sum / large-sieve frame already on record (Planat's\n  Ramanujan-sum expansions, Hardy–Wright Thm 272 as used in #1932), plus Li (2025, MDPI) on *second\n  moments of averages of Ramanujan sums* over divisor sums. None states a second-moment or\n  finite-rank **cap of a band-limited minorant certificate**, which is the object here.\n- Query 2 returns only generic low-rank numerical-linear-algebra material (Nyström, structured\n  low-rank, Hankel) with no arithmetic-number-theory content: no source for a mode-support cap of a\n  sieve-block decomposition. Recorded as a no-match.\n- #2349's two queries (one-class/two-class Jacobsthal upper bounds; DHR `beta_2 = 4.26645`) likewise\n  found no published two-class bound and no CRT-factorised moment; nearest frame is Linnik's large\n  sieve / `L1` of exponential sums (arXiv:1908.06946) — a second-moment input, not a per-`p|x` Euler\n  product.\n\n## Exact remaining gap\nNo located source states a **phase-carrying finite-rank (mode-support) second-moment cap** for this\nblock decomposition, nor a proof that a bounded-rank structure holds. This run's measurement sharpens\nthe gap: the retained mode support is `Theta(q/h)` above `d ~ h` and `<= 2` for every `d <= h`, so any\nsuch cap can only be a **small-denominator** statement (`d <= h`); the top range needs none (closed by\n#2244). The uncovered step is therefore: bound the exact centred small-denominator moments\n`M_2, M_4` of `X_{<=h}` by a two-mode large-sieve estimate, uniformly in `k`."},"research_route_id":193,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_419ffb8590cfa9df270656a5","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/193 and return #2349. Return the ordinary report and transcript plus research: {route_id: 193, 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 failed.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2244","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2349","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[193],"research_url":"/projects/twin-primes/research-routes/193","transcript_url":"/projects/twin-primes/return/2352/transcript","files":[{"sha256":"37fba1adc1c09f7f30db309a0401ddc5e6f289c6d524306f8f5987715a31714b","name":"check_ag.py","bytes":4619},{"sha256":"60f94be4fee01b9122242c884a98e3537cff49dcd986cbbf6b38fe1f086e0956","name":"check_ag_check.py","bytes":3730},{"sha256":"25cb623d23594301ca7bae417936b0919f32c21c40f48875e26254d0c236e8ff","name":"fetch_ag.py","bytes":938},{"sha256":"4d60127d8688d54cac470f8c25724cb79197b3fee6d2d41eb823d55cc268b156","name":"redact_ag.py","bytes":2369},{"sha256":"67caf4481d8eda13ce225119354543193bb669e335b941e88117f1262e6a3864","name":"upload_ag.py","bytes":1852},{"sha256":"eab9b922321e99b17180d10a72e4ca28e36eb71f74d119d19bb1058acb05e715","name":"build_payload_ag.py","bytes":2489},{"sha256":"b925a9795003f8c8ac026aa68948c9743261c89847242e87ae9fad68c2af9206","name":"report_ag.md","bytes":5026},{"sha256":"afc87e87a473bf71d2fee0918a8aa85b95a9fa264f5479d049e4d3fb604e3f0c","name":"evidence_ag.md","bytes":3237},{"sha256":"542adae9c4d07c6ea575010b97e86587e1bb5429b47ab5bd8379a0465b977f74","name":"prior_art_ag.md","bytes":2213},{"sha256":"7bbbf39d6e33a8cb2282bc71645e578d7d526f923e17801f3b9f08663bad4cdf","name":"recipe_ag.md","bytes":1763},{"sha256":"1c507a82dc4b7a6f416edbd3c5a710628da6a6daf51475b161621f4a3f034675","name":"next_step.json","bytes":1590},{"sha256":"f6eeeff1ca82cb9e5f8e647da639f6a0480d9bfa9adbddc006d95e13c667ac54","name":"check_ag.out","bytes":31946},{"sha256":"7edabe5aff87e4553df3b2b7da63af377e0e6589b21c1f4ab2048d5b86e8f1d3","name":"check_ag.err","bytes":463},{"sha256":"2e5fb8e354025d5b1ff2ad1015cb5def0cd941bfddfe573e9064c83e5f6a7a3c","name":"check_ag_check.out","bytes":710},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280},{"sha256":"882ffb0331330cac78cccb35504bace1f7b5907de914316b3679ff772eba9318","name":"backfill_usage.py","bytes":7776},{"sha256":"49b1374e0c337b7901be8600e476d8e1bc52a15ac18890a078f55d28592bf954","name":"test_d.py","bytes":6253},{"sha256":"463ee0fd4b5c620b3dae9cf72f7921b969e57319a02b263cb244a91218ff1ea1","name":"results4293.json","bytes":78346}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}