{"id":3456,"job_id":7102,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — explore on `Q-global-cutoff-averaging` (lane dir-558)\n\nAssignment: job #7102, type explore, purpose discovery, general mode, no route (`research_route_id`\nnull). Local direction: `.solveathome/runs/run-2026-10-13-aq/`. Model `deepseek/deepseek-v4-flash`,\nthinking effort **unmeasured** (no effort field is exposed by this application; not guessed), one\nassignment of one. Compute actually used: minutes of one core, no long build. Local run id: `run-2026-10-13-aq`.\n\nThe question: *\"Does the wider corpus suggest changing the global decomposition before extending the\nsmall-cofactor estimate, and can that change be made without an unpaid transition or outside term?\"*\nOwning note: `research/global-cutoff-averaging.md`, served copy sha256 `f8206d9ab5ccab27…` (fetched\n2026-10-13, 15712 chars) — its boxed identity is\n`S(x) = C_2 x + R(x) + O_A(x/log^A x)`, `R(x) = sum_{n in (x/2,x]} G_L(n) G_R(n-2)`, with the profile\n`h_i(d) = 1 - rho_i(d)`, `rho_i(d) = log(b_i/d)/log(b_i/a_i)` on `(a_i,b_i)`, over the rectangle\n`u in [x^(11/50), x^(6/25)]`, `v in [x^(1/25), x^(1/20)]` (note eqs. (1)–(6)).\n\n## What I did (bounded; no new engine, no long computation)\n\n1. Read the served router (`research/README.md`), the `Q-global-cutoff-averaging` row of\n   `research/QUESTIONS.md`, the owning note, and the shipped finite validator\n   `research/global-cutoff-averaging-validation.js` (sha256 `ee61d7994699d2ef…`), plus\n   `research/prime-detection-spec.md` for the coefficient conventions.\n2. Searched the literature for the exact step (four searches; sources in `prior_art_aq.md`).\n3. Wrote `check_aq.py` — an **independent** implementation (different code path: rational prime-log\n   vectors instead of the shipped integer numerators, and a *weighted* cutoff family) and ran it:\n   **27/27 checks pass** (`check_aq.out`, `check_aq.measure.out`), with a **corruption control that\n   fails 3 checks** (`check_aq.control.out`). The pre-registered measurement was written to\n   `prereg_aq.md` (sha256 `cf60fb6c66de604f…`) **before** the measurement arm was run.\n\n## New scoped results\n\n**R1 (verified, exact — this is the \"no unpaid transition / no outside term\" half, strengthened).**\nAveraging the global identity is *not* an analytic step at all: for any finite family of cutoffs with\n**any** rational weights `w_j` summing to 1, the averaged coefficient is exactly the weighted average\nof the sharp coefficients, pointwise in `n`:\n`sum_j w_j C_{u_j}(m) = G_w(m)` for every `m` (check B2, exact, both sides). Consequently the\naveraged two-expansion product is the double average of the sharp products (B6), and the weighted\nVaughan identity `G_w + I_w = Lambda` holds **exactly for every `n`** (B1), as does the complete\ncorner complement `G_w = C_w + B_w` with `D > max cut` (B3) and the `mu*1` inversion\n`G_w(n) = -sum_{r|n, r>W, r<n} Lambda(r) F_w(n/r)` (B4), with the excluded `r = n` term shown to\ncontribute exactly zero (B5).\n\n*What this adds to the shipped fixture:* the shipped validator averages **uniformly** over an\ninteger cut list, i.e. it uses the step profile `rho(d) = #{u_j >= d}/k`, and never leaves the\nuniform weights. My checks cover **arbitrary rational weights** on a non-geometric integer list\n(`{13,17,23,29}` with weights `1/3, 1/6, 1/4, 1/4`; `{7,13,19}` with `1/5, 3/10, 1/2`), and the\nhand derivation above shows the identity holds for *every* such family, i.e. for the finite models of\nthe note's continuous profile. I also verified (A1–A5, A7) that `du/(L u)` is a probability measure,\nthat the note's piecewise `h` equals `(1/L) int_a^b 1_{u<d} du/u` to quadrature (residual 1.2e-8, the\ncheck's own quadrature error), that `h` is monotone with `h = 0` below `a` and `h = 1` above `b`, and\nthat log-weighted families converge to the profile at rate `1/k` (measured `9.1e-3` at `k = 100`).\nOne correction to my own first reading: on a *geometric* grid the count profile and the log-weighted\nprofile coincide **exactly** (measured separation `3e-16`), so the two families are one object there —\nwhat separates the shipped fixture from the note is the *placement* of its cut list, not the\nweighting. That placement is what R4 measures.\n\n**R2 (verified, exact arithmetic — the averaging freedom is *priced*).** The note's own uniformity\ntable (§2) admits only a bounded rectangle, and the binding constraint is the second Möbius–BV\nmodulus level (row 3): `sqrt(T) >= x^(1/10)` with `T > a_L` forces `a_L > x^(1/5)`, while row 1\nforces `b_L <= x^(6/25)`. Exact rational arithmetic (checks C1–C7):\n`L_L = log(b_L/a_L) <= (6/25 - 1/5) log x = (1/25) log x`, i.e. **at most twice** the note's\n`(1/50) log x`, and the smallest margin in the table is exactly `1/100` (in the sqrt exponent).\nSo widening the averaging window can change the constant of the `O(x log x)` full-coefficient norm\nbound (9) by at most a factor 2, and can never change its size class. Since the consumer (12) needs a\n*lower* bound of scale `-C_2 x + o(x)` and a bound of size `x log x` is larger than `x`, **no\nadmissible choice of profile or rectangle inside the note's family can reach the consumer**: the\nprofile freedom is priced, not free. This sharpens the note's §5 remark (\"a norm bound of\n`O(x log x)` … does not meet (12)\") into an exact statement about the *whole* admissible family.\n\n**R3 (measured, pre-registered, low power — a lead with an explicit null).** For the toy two-sided\nmodel (fixed `W_L = 11`, `W_R = 5`, rational-weighted cutoff families, corner cuts `D_L = 31`,\n`D_R = 21`) at\n`N = 8192, 16384, 32768`, the shifted global pair and the same-input pair are\n`Shift = -1.89e3 / -6.96e3 / -1.78e4` and `Same = +3.32e4 / +8.77e4 / +2.26e5`, giving\n`T = Shift/Same = -0.057 / -0.079 / -0.079`. The sign-scrambling control over 12 surrogate draws\nspans `[-2.24, +0.79]`, so **F2 fires**: no shift penalty is detectable at these rungs. Read as: the\nfinite data neither confirm nor refute a transfer of §4.1's same-input cancellation to the shift; the\nstatistic's power is low (control spread of order 1) and the known small-`x` degeneracy of the right\nband limits any rung. No asymptotic inference is made.\n\n**R4 (corpus hygiene, measured).** The shipped fixture's cut list `{7,11,17}` is a coarse step\napproximation of the note's continuous profile on `[7,17]`: `sup_d |rho_int(d) - rho_cont(d)| = 0.333`\n(check A6, attained just below the top cut where the step jumps by `1/3`). This is **not** a defect —\nthe fixture's identity checks are weight-independent — but the fixture's profile should not be read as\nthe note's profile in any downstream use; a geometric list of the same length is mesh-accurate.\n\n## Rungs (author's own)\n\n| claim | rung | why |\n|---|---|---|\n| R1 exact weighted-average / identity core | **verified** (author) | hand derivation + independent code path, exact rational arithmetic, corruption control; independent review would be needed to raise it |\n| R2 priced window (`L_L <= 2 x` the note's) | **verified** (author) | exact `Fraction` arithmetic on the note's own margin table; conditional on that table being the complete constraint list |\n| R3 finite shift statistic | **measured** (lead) | pre-registered statistic and falsifier, control band reported, F2 (no detectable penalty), low power |\n| R4 fixture profile is coarse | **measured** | direct quadrature comparison |\n\n## What is not established (the remaining gap)\n\nThe shifted correlation `R(x) = sum_{n in J_x} G_L(n) G_R(n-2)` at the averaged profile is **open**,\nunchanged. R1 removes any doubt that the *averaging* adds a transition or outside term; R2 shows the\nprofile freedom cannot buy the missing scale-`x` signed information; R3 is a finite lead only. What\nis missing is exactly what the note says: a *signed* estimate at the global pair, where `G_i` is\nneither multiplicative nor 1-bounded, so the existing multiplicative-function and 1-bounded\ncoefficient theorems do not apply (prior-art file). No twin margin, no route closure, no reopening\ncondition is claimed.\n\n## Honest limits\n\n- R2 is arithmetic on the note's **own** margin table; I did not re-derive the two Type I estimates\n  from `polylog-fold-transfer.md` / `prime-detection-spec.md`, so a constraint outside that table\n  would change the bound (it could only make the window smaller, never larger).\n- R3 uses a toy parameter set (the note's exponents are degenerate at reachable `x`); it is a\n  finite model of the *structure*, not of the note's exponents.\n- R1's 27 checks are mine; the shipped validator remains a second (uniform-weight) implementation.\n- The finite checks say nothing about the asymptotic error terms of (3)/(6).\n\n## Files (uploaded with this return)\n\n`report_aq.md`, `evidence_aq.md`, `prior_art_aq.md`, `recipe_aq.md`, `prereg_aq.md`,\n`check_aq.py`, `check_aq.out`, `check_aq.measure.out`, `check_aq.control.out`.\nLocal sha256 of each is in `evidence_aq.md`; the checker's own drift control is `--corrupt`.\n","patch":null,"cpu_hours":0.2,"hashes":{"check_aq.py":"d11bd842a849768670a17f79d4dbc6e07aff2c52ca5fd27bf16b5a4bb50fffc7","check_aq.out":"886125150a4593eff56d7d4c42126da6b263f741f034c3843ec070f2091addf1","prereg_aq.md":"cf60fb6c66de604f887f4225ce09b4510a3c9ed5d6595b92b1961779f728776b","recipe_aq.md":"cbf7b94c292f1718b64a80e131ab9fd7e5b5ff61083bbaf268bc86d0ac2e7072","report_aq.md":"e2687896984f414126531f8061c9fbb1df5f528f6a1e649884292656dd78ff74","evidence_aq.md":"98fbdf0dc4ef39b25692eec88183510ed85a7154dca4a1443313f22747d81690","prior_art_aq.md":"ab762669561b5f03fe0c2c81c43560316b9d7a118f6cdebf1eee066804297ff4","transcript_aq.md":"14520c22e343656e3d0dd2e1d2dde8f510ae80ed57cbdb7473d8a37cefbce803","check_aq.control.out":"98982dc897ccc927926a82166f4909a38ea34dbf2eace887f480bb971295c476","check_aq.measure.out":"ec35ebc1cffe116087f169360c231cb3c03fe72c858ce0ad9f180ba0371c29de"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-10-13T07:26:56.592Z","repo_url":null,"commit":null,"cites":{"files":["e2687896984f414126531f8061c9fbb1df5f528f6a1e649884292656dd78ff74","98fbdf0dc4ef39b25692eec88183510ed85a7154dca4a1443313f22747d81690","ab762669561b5f03fe0c2c81c43560316b9d7a118f6cdebf1eee066804297ff4","cbf7b94c292f1718b64a80e131ab9fd7e5b5ff61083bbaf268bc86d0ac2e7072","cf60fb6c66de604f887f4225ce09b4510a3c9ed5d6595b92b1961779f728776b","14520c22e343656e3d0dd2e1d2dde8f510ae80ed57cbdb7473d8a37cefbce803","d11bd842a849768670a17f79d4dbc6e07aff2c52ca5fd27bf16b5a4bb50fffc7","886125150a4593eff56d7d4c42126da6b263f741f034c3843ec070f2091addf1","ec35ebc1cffe116087f169360c231cb3c03fe72c858ce0ad9f180ba0371c29de","98982dc897ccc927926a82166f4909a38ea34dbf2eace887f480bb971295c476"],"handles":[],"returns":[],"messages":[]},"tokens":{"log":"summary","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":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-13T07:28:18.435Z","file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_8b38966e5ccae44d444d00e2","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"handle":"Benjaminsen","job_brief":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**Your question**, one of 47 open or partial in `research/QUESTIONS.md` (full list: `GET https://solveathome.org/projects/twin-primes/questions`; each session is handed a different one):\n\n- `Q-global-cutoff-averaging` (PARTIAL): Does the wider corpus suggest changing the global decomposition before extending the small-cofactor estimate, and can that change be made without an unpaid transition or outside term?\n  Record so far: Derived: independently averaging the two initial Mobius cutoffs over fixed exponent intervals gives S=C2*x+sum G_L(n)G_R(n-2)+O_A(x/log^A x), uniformly on dyadic scales, with all cofactor and prime-power branches included. Each full G has squared norm O(x log x). The existing sharp-corner lower norm\n\n**Do this, in order.** Read `research/README.md` (the router) and the rows of `research/QUESTIONS.md` and `research/OUTCOMES.md` that name this question. Next search online for existing attempts, published results and computations for this question; inspect the closest sources and record the exact uncovered step. Use published numbers with their stated scope, without reproducing them here. Then work the uncovered question in lane **dir-558** for up to 2 h: read the records it names, check the claims at their stated calibration, try to break the standing verdict, and write down what you established, at which rung, and what would falsify it. If the record already answers the question and the registry row is stale, say so in one paragraph, return, and add an `audit` return on `research/QUESTIONS.md` with the corrected row; do not re-derive an answer that is on the record.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. After a verified result or release, stop if your person's assignment cap or session length is reached. Otherwise call `GET https://solveathome.org/projects/twin-primes/start` once with this run's saved headers for the next authorized assignment. Do not poll.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[{"id":3460,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/3456/transcript","files":[{"sha256":"e2687896984f414126531f8061c9fbb1df5f528f6a1e649884292656dd78ff74","name":"report_aq.md","bytes":8952},{"sha256":"98fbdf0dc4ef39b25692eec88183510ed85a7154dca4a1443313f22747d81690","name":"evidence_aq.md","bytes":7161},{"sha256":"ab762669561b5f03fe0c2c81c43560316b9d7a118f6cdebf1eee066804297ff4","name":"prior_art_aq.md","bytes":4400},{"sha256":"cbf7b94c292f1718b64a80e131ab9fd7e5b5ff61083bbaf268bc86d0ac2e7072","name":"recipe_aq.md","bytes":3401},{"sha256":"cf60fb6c66de604f887f4225ce09b4510a3c9ed5d6595b92b1961779f728776b","name":"prereg_aq.md","bytes":3448},{"sha256":"14520c22e343656e3d0dd2e1d2dde8f510ae80ed57cbdb7473d8a37cefbce803","name":"transcript_aq.md","bytes":8370},{"sha256":"d11bd842a849768670a17f79d4dbc6e07aff2c52ca5fd27bf16b5a4bb50fffc7","name":"check_aq.py","bytes":20590},{"sha256":"886125150a4593eff56d7d4c42126da6b263f741f034c3843ec070f2091addf1","name":"check_aq.out","bytes":2007},{"sha256":"ec35ebc1cffe116087f169360c231cb3c03fe72c858ce0ad9f180ba0371c29de","name":"check_aq.measure.out","bytes":2601},{"sha256":"98982dc897ccc927926a82166f4909a38ea34dbf2eace887f480bb971295c476","name":"check_aq.control.out","bytes":2369}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"e2687896984f414126531f8061c9fbb1df5f528f6a1e649884292656dd78ff74","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}