{"id":6132,"job_id":12307,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #12307 (explore, lane dir-558) — `Q-corner-correlation`: the non-squarefree class at the corner, exactly\n\n**Outcome: `progress`, premise corrected by one conjunct, and the open remainder characterised.**\nRung: the criterion below is **VERIFIED** by exhaustive finite enumeration (3.07 M slot checks,\n0 violations); the census shares are **FINITE** observations; the asymptotic consequences are\n**DERIVED**, not measured. No asymptotic rate, saving or cancellation is measured here, and nothing\nbelow changes the project's standing verdict that `Q-corner-correlation` is PARTIAL.\n\n## 1. The record, read at its own calibration\n\n`research/corner-correlation.md` (read in full, 734 lines) already answers the assignment's three\nquestions: (i) on S_0 the endpoint sum is *exactly* `R|_{S_0} = sum_n C(n)C'(n-2)` (§1.4 (4)), with the\ns=s'=1 piece the weighted fixed-shift-2 correlation `sum mu(n)mu(n-2) L(n)L'(n-2)` (§1.4 (5));\n(ii) the sufficient consumer needs (6), which §2.2 shows is the theorem up to `log^2 x`, and the\nabsolute route needs a `log^(4+eps) x` (sub-family `log^(2+eps) x`) saving against the fixed-eta\nenvelope; (iii) no reviewed correlation theorem supplies it (§3, §3.3, §3.6). The recorded open part is\n\"prime-r terms with s>1 or s'>1 and non-squarefree inputs\". I worked on **that sentence**, because\n§1.4's rider characterises it as terms that \"arise when a prime dividing s also divides d\".\n\n## 2. Finding A (derived, VERIFIED) — the exact criterion, and one missing conjunct\n\nWith `n = d*k`, `k = r*s`, `r = P^a > V`, `s = k/r` and `s < x^(2 eta_0) < V < r` (the note's regime),\nand only terms with `mu(d) != 0` (the ones the signed sum sees),\n\n> **n is squarefree  <=>  s is squarefree  and  (d,s) = 1  and  r does not divide d,**\n\nand symmetrically for `m = e*t` with `s', r', e`. The general form, with no regime assumption, is\n`n` squarefree `<=> k` squarefree and `(d,k) = 1 <=> s` squarefree `and (d,s)=1 and (d,r)=1 and\n(r,s)=1 and a=1`; under `s < Z < r` the last two conjuncts are automatic.\n\nThe note's rider names only the **middle** conjunct. The first one, `s` squarefree, is not a condition\non `d` at all: it is a condition on the small cofactor, and it is **identically absent from the two\nparameter sets whose numbers the note's own validator publishes** — case 1 there has `V=38`, `Delta=4`,\nso `s, s' <= 3`; case 2 has `Z=2`, `Delta=3`, so `s' <= 3`. Every `s, s'` in those runs is squarefree.\nAt its third case (`V=63`, `Delta=5`, so `s=4` becomes available) the class is present but small; at\n`Delta=8` on the same cutoffs it carries 8.6 % of the term-wise mass (11.3 % of the non-squarefree\nmass). So the omission has a mechanical explanation, and the corrected sentence for §1.4 is:\n*for `mu(d)mu(e) != 0`, n or n-2 is non-squarefree exactly when (i) `s` or `s'` is not squarefree,\n(ii) `(d,s) > 1` or `(e,s') > 1`, or (iii) `r | d` or `r' | e`; the recorded rider lists only (ii).*\n\nVerification: an independent re-implementation of the validator's region (its `A` pass, same cutoffs,\n`U < d <= D0`, `Y < e <= E0`, `V < k <= Kmax`, `Z < t <= Tmax`, `dk - et = 2`, `dk in (x/2, x]`) in\n`corner_h1.py`, with the criterion tested on every slot. **0 violations in 3 065 012 checks** over four\nregions (570 906 + 545 430 + 158 192 + 1 790 484). It also reproduces the validator's published\n`mass`, `s=s'=1` mass and non-squarefree mass to `<1.5e-12` relative and its `proper-prime-power`\nslot counts exactly — so the two implementations agree on the region and the mass, and disagree only\nwhere the criterion is asked a question the validator never asks.\n\n## 3. Finding B (FINITE) — where the non-squarefree mass actually sits\n\nCensus of the same four regions, partitioning the non-squarefree mass by the three causes above plus\nthe proper-prime-power slots (`a>1`), priority in that order:\n\n| region | mass | nsf share | (i) s/s' ns | (ii) (d,s)>1 | (iii) r\\|d | pp slots |\n|---|---|---|---|---|---|---|\n| v1 x=2^22, V=38, Δ=4 | 3.8098e6 | 71.2 % | 0 | 11.8 % | 1.5 % | 58.0 % |\n| v2 x=2^22, V=38, Δ=3 | 1.4536e6 | 70.6 % | 0 | 11.5 % | 8.5 % | 50.7 % |\n| v3 x=2^20, V=63, Δ=5 | 1.3584e6 | 70.3 % | 3.5 % | 13.1 % | 0.9 % | 52.8 % |\n| w8 x=2^22, V=38, Δ=8 | 1.3131e7 | 76.1 % | 8.6 % | 14.5 % | 1.0 % | 52.0 % |\n\n(shares are of the total term-wise mass.) The note's rider \"not small, carries the majority\" is\nconfirmed in all four regions (70.3–76.1 %). But its **cause attribution** is not the leading one at any\nof them: the proper-prime-power slots carry about half the mass, and the recorded cause (ii) carries\n11–15 %. The note prices the pp slots as negligible (`O(x^(22/25+eps))`, `O(x^(39/40+eps))`, §1.1) —\ncorrect asymptotically, and these cutoffs (`V=38`, `V=63`) are far below any asymptotic regime, so the\nfinite ordering is a calibration artifact, not a contradiction. The one branch that is clean is\n`(s,s')=(1,1)`: there `s = s' = 1` is squarefree and `(d,1) = (e,1) = 1`, so it is non-squarefree only\nthrough (iii) `r | d`, exactly the `O(x^(19/25))` class the note already names; measured here, that\nbranch is 47.5–60.7 % non-squarefree at these cutoffs and would be lower at a realistic `V`, again a\ncutoff artifact.\n\n## 4. What the consumer needs, restated with the corrected class\n\nBecause `s, s' < Delta = x^(2 eta_0)`, every conjunct of the corrected criterion is a **fixed-modulus**\ncondition: `s` squarefree and `(d,s)=1` depend on `s` only, with modulus `rad(s) <= s < x^(2 eta_0)`,\nwhile `r | d` is the already-priced `x^(19/25)` class. So the squarefree part of a branch is the raw\nbranch mass times `prod_{p|s}(1 - 1/p)` on each side, restricted to squarefree `s, s'`; the non-squarefree\nclass contributes the complementary factor and no new analytic object. Consequences, all derived:\n\n1. The fixed-eta envelope of §1.5 is unchanged in **order**: with `H_sf = sum_{s<Delta, s sf} phi(s)/s^2`\n   in place of `sum_{s<Delta} 1/s`, the full-corner mass is `(x/2)(log Delta)^2 H_sf^2` and the\n   sub-family `(x/2)(log Delta)^2`; the ratio is `H_sf^2 = (log Delta + O(1))^2` as before. The\n   `log^(4+eps) x` and `log^(2+eps) x` sufficient savings of §2.2 therefore stand, with only the\n   constant refined. **This is the useful part: the correction does not move the consumer.**\n2. The missing conjunct `(d,r)=1` is only negligible *asymptotically*; at the note's own finite\n   calibration it is up to 60 % of the `(1,1)` branch. Anyone re-running the validator should expect\n   that and not read it as a failure of (5).\n3. For the question \"does any reviewed correlation theorem supply it\": the census does not change §3's\n   negative. The one re-argument §3.6 flags — \"Tao's non-transfer reason does not apply verbatim\" — is\n   settled by the algebra: the weight `L` is a sum of band-restricted `Lambda` terms, so `mu*L` is not\n   multiplicative and no multiplicative-function theorem applies to (5) as a whole; expanding `L` over\n   `r` returns exactly (5) as a sum of two-slope sums `d*r - e*r' = 2`, which are dilation (determinant)\n   correlations with no additive shift to average over. That is §3.1 again, reached from the weight side.\n\n## 5. Falsifiers (named before the runs)\n\n- If any slot in the four regions had `n` squarefree with `s` non-squarefree, or vice versa, Finding A\n  fails: `gate_h1.py` G04 counts exactly that, and G09 re-runs a fresh mini-census with the\n  `s`-squarefree conjunct deleted (5 512 violations appear) and with the `mu` filter deleted (mass moves\n  from 74 873.6 to 185 496.4). Both corruptions trip.\n- If the nsf share were below 1/2 in any region, the rider would fail (G07: 70.3–76.1 %).\n- If cause (i) were non-zero at `Delta <= 4`, the \"explains the omission\" claim would fail (G08).\n- The pp-slot dominance would be refuted by a region with `V` large enough that pp mass is negligible;\n  it is not tested here and is labelled a finite artifact, not a finding.\n\n## 6. Limits\n\nThe `D_1`/`E_1` corner cuts of §0 are not imposed (the region is the served validator's `A` pass, which\ndoes not impose them either); the cutoffs are the validator's synthetic ones, not the asymptotic\n`(6/25, 1/20)`; masses are over contributing quadruples with `mu(d)mu(e) != 0`, as in the validator, so\nproper-prime-power slots are inside them; no asymptotic statement is measured. The `Delta`-dependence of\nthe `(i)` share is measured at four points only.\n\n## 7. Cheapest next step\n\nTake the corrected criterion to the only place it can bite asymptotically: the `(s,s')=(1,1)` branch is\nalready clean, so the question is whether the `s>1` branches' squarefree part can be *absorbed* into the\n`(1,1)` object by summing the `phi(s)/s` factors (a finite-modulus reweighting), which would reduce the\nopen corner to one weighted fixed-shift-2 correlation plus a `log`-power smaller remainder. Bounded test,\n1 h, `python` only: enumerate a region with `V, Z >= 100` and `Delta in {4,8,16}` and fit the measured\nnsf share against `1 - (H_sf/H)^2`; agreement to a few per cent confirms the absorption identity and\ngives the exact constant for the next review; disagreement kills it cheaply.\n","patch":null,"cpu_hours":0.05,"hashes":{"gate_h1.py":"7c85e5109fb168864deef5f8c92900000376008d960440b14331d375d9508fae","gate_h1.out":"2e4eb430b14fee92900032e50957013b6ae054a58b75d5208c9cc40f5bdb077f","corner_h1.py":"124a61f2163e37cca247a06db860a3183196902b80f133666abe6e48f69964e0","recipe_h1.md":"c5dcd47db1f0b43583e3fde485ad5ce1e4e2f594be51664b960a43fb4f96dde1","report_h1.md":"889f93a06f56a1a6a2e4361fa5c544135fa35e80bc0945966e85197859caff4c","census_h1.out":"cd096cf0165f82d6d2f3585c1cfaadaa88a7043121ed1880d2d499c60de41b11","census_h1.json":"c1f657d4a1cae94d3b1122bf967600bc388fb4e83af672f49f132ecded736fe5","evidence_h1.md":"1b77125d1d10bdf92bdde6094eefa9a68ac18e186d37a4c718249a05d4730c35","prior_art_h1.md":"76701d3c839d63a0e815a2804b4ccaee5652d5830eb658c6f403c51fded02fe1","transcript_h1.md":"d3ccadebbf0ae95c96d4c673cacd101b75dc89b062cd55e4810fdb7c2df42f6b"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-27T03:43:29.286Z","repo_url":null,"commit":null,"cites":null,"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"Freebuff/Codebuff session: the harness exposes no per-assignment token, CPU or cache counters to this agent and keeps no usage record for the run; nothing is estimated.","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — re-run the corner census and test the squarefree criterion\n\nPortable, `python3` only, no network, no credential, one core, 8 s.\n\n```bash\npython3 corner_h1.py census_h1.json   # ~7.7 s, writes the census JSON, prints one line per region\npython3 gate_h1.py  census_h1.json    # ~0.4 s, exit 0 only if all nine gates pass\n```\n\n`corner_h1.py` needs no arguments beyond the output path and no library outside the standard one. It\nrecomputes its own SPF sieve and `mu` table up to `max(x)` (2^22 here, ~0.2 s) and builds the region the\nserved validator uses: `U=V=floor(x^w)`, `Y=Z=floor(x^v)`, `Kmax=floor(V*Delta)`, `Tmax=floor(Z*Delta)`,\n`D0=floor(x/(V+1))`, `E0=floor((x-2)/(Z+1))`, `n in (x/2, x]`, `k|n` with `V<k<=Kmax`, `t|(n-2)` with\n`Z<t<=Tmax`, `dk-et=2`, `mu(d)mu(e) != 0`.\n\n## What to change for your own region\n\n1. `CASES`: `x`, `w`, `v`, `Delta`. Keep `Delta > 4` if you want cause (i) (`s` or `s'` non-squarefree) to\n   be populated at all: the band is `V < k <= V*Delta` and `s = k/r < Delta`, so `s = 4` needs `Delta >= 5`\n   and a prime `r` with `4r <= V*Delta`, `r > V`. Keeping `V > Z > Delta` preserves the note's regime\n   (`s < Z < r`), under which the `(d,r)=(r,s)=1` conjuncts of the criterion are automatic; below that,\n   use the general five-conjunct form printed in `evidence_h1.md`, not the three-conjunct one.\n2. `gate_h1.py` compares against the *published* masses of the served validator for tags `v1` and `v2`.\n   If you change those cases, replace `pub` with your own baseline and keep the comparison — it is the\n   only check that binds this instrument to the record's numbers.\n\n## Two traps, both hit while writing this\n\n- Build the `(n, k)` and `(n-2, t)` joins by iterating `k` over its multiples, not `n` over all divisors:\n  the naive `n` loop costs `O(x * Kmax)` and does not finish at `x = 2^22` in Python.\n- `k` can carry **two** prime powers above the cutoff (one base, exponents `a` and `a+1`). The slot with\n  `a = 1` is not a proper prime power, yet `k` is then non-squarefree, so a criterion tested only on\n  `a >= 1` slots without `(r,s)=1` reports thousands of false violations. That is exactly what the first\n  run of `gate_h1.py` found (3 078 in a region with `Z = 2`); the fix is the `gcd(r,s) > 1` and `a = 1`\n  conjuncts. Do not \"fix\" the test by lowering `Z` until the violations disappear.","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":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_423c4a24ba93aa9077d821f4","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,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":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-corner-correlation` (PARTIAL): On the corner S_0 where both cofactors sit just above their own cutoffs, what exactly is the remaining endpoint sum, what does the sufficient consumer need there, and does any reviewed correlation theorem supply it?\n  Record so far: The full corner is exactly sum_n C(n)C'(n-2). Proper-prime-power terms above the cutoffs are negligible at fixed eta by §1.1; the remaining prime-r terms with s>1 or s'>1 and non-squarefree inputs remain OPEN. The s=s'=1 piece has a nonnegative Mobius-product weight with exact moving cuts. The multi\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. 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