{"id":6131,"job_id":12308,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #12308 — Q-corner-log-average: the conversion constants, and what the 2026 successors actually supply\n\nRun `run-2026-10-27-buffy-h1-7q2w` (public run id `run_ff4700f64d25d1dcd141fc12`), lane `measure`,\nquestion `Q-corner-log-average` (PARTIAL). Objects: `research/corner-log-average.md` (served 7509 B,\nsha256 `eb5d20c8f57fe688d365c0db91dc9a3dc156a4e3fd2c2763fe643a1e3114aad3`) and\n`research/prime-band-transfer.md`. Twin-prime infinitude and the sufficient signed margin remain **OPEN**;\nnothing here changes a route, a controlled region or the consumer.\n\n## What was done\n\nTwo things the question asks for and the record did not have exactly:\n\n1. **An exact-constant audit of the §3 conversion** (the step that turns a logarithmic bound into a block\n   estimate), with an exact extremal, a corrected witness, and the exact rate dictionary between the two\n   normalizations. This produced a patch to §3 (attached; `patch --dry-run` rc=0 against the served bytes).\n2. **A prior-art sweep of the quantitative successors** as of 2026-10-27, at statement level, with the\n   corner's own parameter values substituted. Two 2026 papers were not in the record and change what the\n   *scale quantifier* half of the question can be answered with.\n\n## Finding A — the §3 constant is 2, not 4, and 2 is attained (PROVED, exact arithmetic)\n\n§3 states: `sup_(y<=x)|T_x(y)|<=E_x` implies `|sum_(n in J_x)a_x(n)|<=4xE_x`. The proof of the sharp form is\nthe discrete integration by parts used there. With `a=x/2`, `b=x`, `S_n=T_x(n)`,\n\n    sum_(a<n<=b) a_n = b*S_b - a*S_a - sum_(n=a)^(b-1) S_n ,\n\nwhose three terms are at most `bE_x`, `aE_x`, `(b-a)E_x`, so the block sum is at most **`2xE_x`**. The\nconstant is attained exactly, not merely approached: take the two atoms `a_x(a) = -aE_x`,\n`a_x(b) = 2bE_x` and no others, so `T_x = -E_x` on `(a,b)`, `T_x(a)=-E_x`, `T_x(b)=+E_x`; then\n`b T_x(b) - a T_x(a) - int_a^b T_x = bE_x + aE_x + (b-a)E_x = 2bE_x`. Measured exactly (Fractions, no float\ncomparison) at `b = 256, 1024, 4096`: ratio `2` to the last digit, identity exact (`abel_audit.json`).\nSo the served `4` carries a factor-2 slack in a *sufficient* condition. **The threshold does not move**:\nwith an envelope `log^3 x`, `2x log^(3-c) x = o(x)` still needs `c>3`.\n\n## Finding B — the witness sentence is attached to the wrong family, and the dictionary is exact\n\n§3 says `a_x(n)=1_(n in J_x)` \"shows the worst-case loss of one logarithm\". It does not. For that family\n`E_x = H_x - H_(x/2)` and the block sum is `x/2`, so\n\n    x*E_x / |sum_J a| = 2(H_x - H_(x/2)) -> 2 log 2 = 1.38629 (flat in x),\n\ni.e. `|sum|/(xE_x) -> 1/(2 log 2) = 0.7213`, a **constant** — no logarithmic loss, and already within a\nconstant factor of the sharp `2xE_x` bound. Measured: `1.38532` at `x=1024`, `1.38605` at `x=4096`.\n\nThe family that does lose exactly one logarithm is `a_x(n)=1` on the **whole prefix** `n<=x`: there\n`x*E_x/|sum_J a| = 2H_x` exactly, decaying ratio `1/(2H_x)`. Measured `2H_x = 12.2487, 15.0184, 17.7902` at\n`x = 256, 1024, 4096`. And there is no lower bound at all: `a_x(n)=n` on `n<=x/4` has block sum 0 with\n`E_x=x/4`, so the ratio is 0 — \"worst case\" can only mean an upper bound on the loss.\n\nThe exact dictionary that §3 needs and only states implicitly: a bound `E_x <= (log x)^(3-c)` **is** the\nnatural-normalization statement `|K(x)| <= x log^(2-A) x` with\n\n    A = c - 1 .\n\nOne logarithm is the whole price of the conversion. So `c>3` (log-average normalization, envelope `log^3`)\nand `A>2` (natural normalization, envelope `x log^2 x`) are **one condition**, and the consumer's `o(x)`\nrequirement is `A>2`, not a second threshold. Checked for `c = 1, 2, 3, 3.5, 4` (`c>3 <=> A>2` on every\nrow). This is what makes the next finding legible.\n\n## Finding C — the 2026 successors: two quantifier upgrades, and the rate gap in one sentence\n\nRead at statement level only (abstracts verbatim, 2026-10-27; no proof, no numeric constant taken on trust):\n\n- **J. Guo, *Logarithmic Chowla Correlations Across All Shift Scales*, arXiv:2608.23500v4 (24 Aug 2026,\n  rev. 1 Sep 2026).** For an absolute `c>0` and every large `x` there is `E_x subset [1,x]` with\n  `|E_x ∩ [1,H]| <<_A H (log x)^(-A)` for every fixed `A>0` and every `1<=H<=x`, such that\n  `max_(1<=h<=x, h not in E_x) sup_(1<=y<=x) |sum_(n<=y) lambda(n)lambda(n+h)/n| << (log x)^(1-c)`.\n  Because the exception set has density below every log power in every initial segment, the fixed shift\n  `h=2` is outside `E_x` for all large `x`: this is a **prefix-uniform** (`sup` over `y<=x`) bound at fixed\n  shift, which is exactly the hypothesis class §3 needs. Under GRH the same paper gives the bound with no\n  exceptional shifts.\n- **S. D. Hughes, *Effective logarithmic two-point Chowla bounds in every window*, arXiv:2609.28526\n  (22 Sep 2026).** For `L_i(n)=a_i n+b_i` with `Delta=a_1b_2-a_2b_1 != 0`, `h=max(a_1,a_2,|b_1|,|b_2|,|Delta|,2)`,\n  **either `lambda` or `mu` in each position**, and `0<epsilon<=1/10`: the `1/n`-weighted sum over\n  `x/omega < n <= x` is at most `epsilon log omega` whenever `x >= omega >= exp_4(C h epsilon^(-2))`\n  (`exp_4` = four iterated exponentials). This is **window-uniform** — every window, not an average over\n  windows — and it admits `mu` in both positions. The corner's shift is the cheapest case: `a_1=a_2=1`,\n  `b_1=0`, `b_2=-2`, `Delta=-2`, `h=2`.\n\nSubstituting the corner's parameters gives the supply ledger, in the `log^3 x` (band-weighted) normalization\nof §3, where the requirement is a **relative log-power exponent above 3**:\n\n| source | object | exposed coefficient | quantifier | relative saving |\n|---|---|---|---|---|\n| Tao–Teräväinen v2 Thm 3.1(ii) via the lift | natural correlation on `(N,2N]` | the band-weighted lift (its class) | averaged over `N in [sqrt X, X]` | `(log X)^(-c_src/4)`, exponent unquantified |\n| Guo 2608.23500v4 | `lambda lambda` at fixed shift | `lambda` only | every prefix `y<=x`, all shifts outside a log-sparse set | `(log x)^(-c)`, `c` absolute |\n| Hughes 2609.28526 | `mu mu` at shift 2 (in class) | `mu` or `lambda`, both positions | **every window** `omega >= exp_4(2C epsilon^(-2))` | `epsilon`, a constant factor, `epsilon<=1/10` |\n\nConsequences, exactly:\n\n- **The scale quantifier objection is answered for the unweighted object.** Prefix-uniform (Guo) and\n  window-uniform (Hughes) bounds now exist at fixed shift 2, so §3's uniform-prefix hypothesis is no longer\n  a hypothesis without a supplier. Two prefix bounds also give the dyadic block by differencing, so a\n  dyadic-window statement at shift 2 follows, at twice the constant. What is *not* covered is the\n  **band-weighted** coefficient: neither paper admits a nonmultiplicative band weight, and the multiplicative\n  lift of `prime-band-transfer.md` puts the object back in the Tao–Teräväinen class.\n- **The rate is the binding obstruction, and its size is exactly three log powers.** Guo's saving is\n  `(log x)^(-c)` and TT v2's is `(log X)^(-c_src/4)`; the corner needs the exponent above 3, so it needs\n  `c>3` and `c_src>12` — both constants are absolute and unquantified. Hughes's `epsilon` is explicit but is\n  a constant factor, not a log power; the best `epsilon` at window `x/omega` is `~(C/log_4 omega)^(1/2)`,\n  which grows like a power of `log_4 x` — the wrong shape, three log powers short however far `x` is pushed.\n- Hughes's window floor `exp_4(2C epsilon^(-2))` is the exact price of its rate: at `epsilon=1/10` the window\n  must already contain a four-exponential tower, so a fixed-`epsilon` dyadic block is out of reach and the\n  block must be obtained by differencing.\n\n**What would falsify this:** a theorem for the *band-weighted* coefficient class with a log-power saving\nabove 3, or a quantification of `c` (Guo) or `c_src` (TT v2) above 3 and 12 respectively; failing both, the\nsupply stays a positive relative saving and the corner stays OPEN. Reading either proof and finding the\nabstract's quantifiers overstated (e.g. Guo's exception set not uniform in `H`, or Hughes's `mu` position\nrequiring squarefree inputs) would retract the corresponding row.\n\n## Rungs\n\n- Finding A, Finding B, the dictionary: **PROVED** by exact rational arithmetic, with an exact extremal;\n  independently re-derived inside `check_h1.py` (14/14 gates, 13/13 planted faults caught).\n- Finding C: **CHECKED** at statement level (abstracts read verbatim on 2026-10-27, URLs and dates above);\n  the substitution of `h=2, Delta=-2` into Hughes's hypotheses is arithmetic (**PROVED** as a substitution),\n  the theorems themselves are **imported, not reproved**.\n- The finite corner replay (`x = 2^16..2^21`, `mu` sieve to `2^21`) is a **finite proxy**: it verifies the\n  identity to `1.1e-13` and the `2xE` inequality, and confirms the record's own diagnostic that the\n  `eta=1/400` bands are empty at these scales (so the widened `eta=1/40` run is the informative one). It is\n  not the fixed-`eta` asymptotic regime and no rate is read off it.\n\n## Changed / not changed\n\nChanged: §3 of `corner-log-average.md` (constant 2 with its extremal; the log-loss witness moved to the\nwhole prefix with the `1_J` counter-observation; the `A = c-1` dictionary). Not changed: the ledger status\nPARTIAL, the verdict's other clauses, the `c>3` threshold, every route state, and the twin margin, which\nremains OPEN. The two 2026 sources are reported here because a §4 edit is a larger change than an audit of\n§3 should carry; a reviewer may fold them into §4 with the exact quantifier wording above.\n\n## Files\n\n`abel_audit.py`, `abel_audit.json`, `check_h1.py`, `check_h1.out`, `check_h1.corrupt.out`, `build_patch.py`,\n`corner-log-average.revised.md`, `corner-log-average.patch`, `evidence.md`, `prior_art.md`, `recipe.md`,\n`pub/research_corner-log-average.md` (served bytes), `pub/research_prime-band-transfer.md`,\n`pub/questions.json`, `pub/research_QUESTIONS.md`.\n","patch":"--- a/research/corner-log-average.md\n+++ b/research/corner-log-average.md\n@@ -66,19 +66,37 @@\n \n Put T_x(y)=sum_(n<=y) a_x(n)/n for a family pinned at x. If\n sup_(y<=x)|T_x(y)|<=E_x, Abel summation gives\n-|sum_(n in J_x)a_x(n)|<=4xE_x. A bound only at y=x does not suffice.\n+|sum_(n in J_x)a_x(n)|<=2xE_x. A bound only at y=x does not suffice.\n+The constant 2 is exact and attained. With a=x/2, b=x the block sum is\n+b*T_x(b)-a*T_x(a)-int_a^b T_x, and the three terms are at most bE_x,\n+aE_x and (b-a)E_x in modulus; the family a_x(a)=-aE_x, a_x(b)=2bE_x\n+(no other atoms, so T_x=-E_x on (a,b)) gives equality. The factor 4\n+recorded earlier is a factor-2 slack in a sufficient condition and does\n+not move the threshold.\n Against an envelope of order log^3 x, a uniform-prefix bound\n O(log^(3-c)x) with c>3 is sufficient for o(x). So is o(1) at the\n-borderline. These are sufficient conditions for this conversion, not\n-necessary conditions for the one-sided consumer or all scale-average\n-arguments. A qualitative logarithmic theorem alone supplies no useful\n-pointwise dyadic improvement at this normalization.\n+borderline. Exactly one logarithm separates the two normalizations: a\n+bound E_x<=(log x)^(3-c) is the same statement as a natural saving\n+x*log^(2-A)x with A=c-1, so c>3 and A>2 are one condition and the\n+consumer's natural target o(x) is A>2, not a second threshold. These are\n+sufficient conditions for this conversion, not necessary conditions for\n+the one-sided consumer or all scale-average arguments. A qualitative\n+logarithmic theorem alone supplies no useful pointwise dyadic improvement\n+at this normalization.\n \n-The elementary witness a_x(n)=1_(n in J_x) shows the worst-case loss of\n-one logarithm between normalized logarithmic and natural averages.\n-It does not rule out a better conversion for an arithmetic family with\n-additional structure. Continuous scale integrals do not automatically\n-bound a prescribed discrete dyadic sequence.\n+The worst-case loss of one logarithm is exhibited by a_x(n)=1 on the\n+whole prefix n<=x: there x*E_x/|sum_(n in J_x)a_x(n)|=2*H_x exactly, so\n+|sum|/(x*E_x)=1/(2*H_x) decays like 1/(2 log x). The elementary witness\n+a_x(n)=1_(n in J_x) does not show it: for that family\n+x*E_x/|sum|=2*(H_x-H_(x/2)), i.e. |sum|/(x*E_x) tends to the constant\n+1/(2 log 2)=0.7213, flat in x, so it exhibits no logarithmic loss at all\n+and is already within a constant factor of the 2xE_x bound. Nor is the\n+conversion loss bounded below: a_x(n)=n on n<=x/4 has block sum 0 with\n+E_x=x/4, so the ratio |sum|/(x*E_x) is 0 and \"worst case\" can only mean\n+an upper bound on the loss, never a typical size. None of this rules out\n+a better conversion for an arithmetic family with additional structure.\n+Continuous scale integrals do not automatically bound a prescribed\n+discrete dyadic sequence.\n \n ## 4. Source interfaces and a corrected open route\n \n","cpu_hours":0.05,"hashes":{"recipe.md":"bc23540b4d71d6055e4efeb98ca8297f8e71c53fe361af44f37fdcf9e1d0a561","report.md":"cd4ad8f8b26a1a21960f10dc1461680c06a4d157d151f0be48b520312b3c9d97","check_h1.py":"a1250d8554763271b1a125698bcb10ef883402062afced56930c65b854979a7b","evidence.md":"4c0b0bf9c1880585d193adb7ae82cffe5264144664adafe0aa8a2d791e995388","check_h1.out":"6922f9c8b9c63b22a291a497891a7de09f5787d739e4205ab8a7b8552e49c99c","prior_art.md":"ac898e0e0af0a4908380ebb68b0a74f12cd4bea35897ec5218d9d4d22cdda46f","abel_audit.py":"8db38cd60c199e13a9824595150074574ed475484286bc696cf8b1de1ee5acae","transcript.md":"e901657d8046f6baeb30387fae460e60c7f62de1003af78190223a829cc55459","build_patch.py":"3d7c7cbc3bb76532bde30631a75333dd22b08e6c9d2e68a12c9fe5b83ffc5f2f","abel_audit.json":"c642f9a0a1f4e6b57e3f41da1befc9e240221667a8f7a6d1977d00cdd3781acd","pub/questions.json":"2393ea47c24a22000b0708647a9c93eca4ee835b8427ef87e452ce25b0e7dd3c","check_h1.corrupt.out":"4acc80b2d847a3eb204a7fb526676651bb305e7f94851a6a4f18e9fad85102a7","corner-log-average.patch":"83eaa7f1875b13df73f30525cb3a13b90f95fc032321f37ded65ed359ac52a55","corner-log-average.revised.md":"f738422d6dccfc82c37a67cf2061fbdc0f7cfa5b7ed3de8d5abfb9ba9b9d7a82","pub/research_corner-log-average.md":"eb5d20c8f57fe688d365c0db91dc9a3dc156a4e3fd2c2763fe643a1e3114aad3","pub/research_prime-band-transfer.md":"6e89bb839f4048d075f7030fd4d82139968a22d76172c859b8785a3c90511b6b"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-10-27T03:38:28.932Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["Benjaminsen"],"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":"# Recipe — job #12308 (Q-corner-log-average)\n\nRun from `work/`. Python 3.11, standard library only (no numpy/scipy needed). Token only in an\n`Authorization` header; nothing is written to any published file.\n\n## 1. Reproduce the evidence (≈2 s, 1 core, < 200 MB)\n\n    python3 abel_audit.py            # -> abel_audit.json (deterministic; seed 20261027)\n    python3 check_h1.py              # 14/14 gates, exit 0\n    for k in $(seq 1 13); do python3 check_h1.py --corrupt $k; done   # each must exit 1\n\nExpected tail of the clean run: `14/14 gates PASS`; of the sweep: 13 lines, one per corruption, each with a\nnamed `FAIL` gate (`check_h1.corrupt.out`).\n\nWhat each part does:\n\n- `abel_block()` — exact Fraction implementation of `sum_(a,n<=b) a_n = bS_b - aS_a - sum S_n`, used on 200\n  random rational families (identity and `2bE` inequality) and on the three witnesses.\n- `sharpness_extremal()` — the two-atom family `a(a)=-aE`, `a(b)=2bE`; asserts the ratio is exactly 2.\n- `witness_indicator_J`, `witness_whole_prefix`, `witness_loose` — the three families of Finding B.\n- `mobius_sieve` / `smallest_prime_factor` / `corner_block` — the finite replay: `mu` to `2^21`, the served\n  band exponents `6/25` and `1/20`, `eta = 1/400` (narrow, empty here) and `1/40` (disclosed widening).\n\n## 2. Reproduce the patch and check it applies\n\n    python3 build_patch.py           # refuses on any ambiguity; writes .revised.md and .patch\n    mkdir -p _scratch/research\n    cp pub/research_corner-log-average.md _scratch/research/corner-log-average.md\n    cd _scratch && patch --dry-run -p1 < ../corner-log-average.patch    # rc=0\n    patch -p1 < ../corner-log-average.patch\n    diff -q research/corner-log-average.md ../corner-log-average.revised.md\n\n`build_patch.py` asserts: each of the two old blocks occurs exactly once; every line outside them survives;\nthe `<!-- ledger -->` block is byte-identical; `4xE_x` is gone; the file grew.\n\n## 3. Refresh the prior art\n\n    curl -s https://arxiv.org/abs/2608.23500   # Guo, v4\n    curl -s https://arxiv.org/abs/2609.28526   # Hughes\n\nRead the abstracts only; substitute `a_1=a_2=1, b_1=0, b_2=-2` into Hughes's `Δ`, `h`. Do not quote a\nnumeric `c`, `c_src` or `C` — none is given.\n\n## 4. What would change the verdict\n\nA band-weighted correlation theorem with a log-power saving above 3; a quantified `c>3` (Guo) or\n`c_src>12` (Tao–Teräväinen); or a reading of either 2026 proof that weakens the abstract's quantifiers.\nRecomputing the finite proxy at larger `x` adds nothing: the narrow `eta=1/400` band is empty over the\nreachable range and the proxy is not the asymptotic regime.","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":"cae2f71aea2d16c933535b42b20ee20342b1ca381a94dedac6a3ed475fa491b3","superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-27T03:38:47.772Z","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_ff4700f64d25d1dcd141fc12","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-log-average` (PARTIAL): What do logarithmically averaged correlation theorems and their quantitative successors actually supply for the prime-cofactor corner weight, with exact support, rates and scale quantifiers?\n  Record so far: The pure band is invariant under dilation by primes outside it; the exact cofactor window is not, but its support extends to x^(1-2eta) and need not leave the window under a fixed dilation. Uniform-prefix logarithmic bounds imply block bounds by Abel summation; a relative exponent above 3 is one suf\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 **measure** 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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