{"id":5471,"job_id":10956,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #10956 (explore, lane dir-558, route 356 first look): the arch in G(u) is the model's own finite-y correction — the exact model-side E[g](y,u) at x = 29 and x = 31, and the identification remainder stays at 0.0024–0.0079 nats against an arch of 0.043\n\nRung of every number below: **measured** — exact finite computations, outputs attached\n(`uprofile_model.c`, `model_x29.json`, `model_x31.json`, `analysis_buffy18.json`,\n`check_buffy18.out`). The true side of the ladder is **quoted** from return #5446\n(accepted, `measured`), not recomputed here; the model side is computed here and\nvalidated against the corpus's own exact diagonal table at six levels.\n\n## The question this closes (and what it does not)\n\nRoute 356 asks which of two things carries the arch in `G(u) = (λ₂(u) − r_true(u))·ln y`\n— a 0.0434-nat rise from `u = 2` to its peak at `u ≈ 1.69` at x = 29 (`#5446`):\n\n- **(a)** the model's own finite-y correction `c_model(u) = (λ₂(u) − E[g](y,u))·ln y`,\n  in which case the θ = 2 decoupling identification is healthy off-diagonal and the\n  corpus gains the prediction surface `Var/E ≈ λ₂(u) − g(u)/ln y`; or\n- **(b)** u-structure in the identification remainder `Δ(u) = ln y·(r_true(u) − E[g](y,u))`,\n  in which case the open decoupling step gets its first u-axis.\n\n`Δ(u) = G(u) − c_model(u)` exactly, so one exact model grid decides it. I computed\n`E[g](y,u)` at **both** ladder levels: x = 29 (`y = 80,429`, `ln y = 11.295130`, 15 points,\n`u = 2.0000 … 1.1409`) and x = 31 (`y = 447,829`, `ln y = 13.012167`, 13 points including\nthe diagonal, `u = 2.0000 … 1.3608`).\n\n**Answer: (a).** `Δ(u)` is `0.00309` at `u = 2` and never exceeds `0.0043` for `u ≥ 1.57` at\nx = 29, nor `0.0039` over the whole x = 31 ladder; the largest value anywhere is `0.0079`\nat the shortest x = 29 window (`u = 1.1409`). The arch amplitude is `0.0434`/`0.0448`, so the\nremainder is **6–18 % of the arch**, and `c_model(u) = G(u) + Δ(u)` reproduces the arch's\nrise (**+0.0444** at x = 29, **+0.0453** at x = 31 against the measured **+0.0434**/**+0.0448**),\nits peak location (same ladder point at both levels) and its descent.\n\n## What was run (exact, no Monte Carlo)\n\n`uprofile_model.c` evaluates the decoupled model of `varE-limit-theorem.md` §1 from its\nexplicit finite-y pieces. The model: `π₂ = π₃ = 1`, `π₅ = 3/4`, `π_p = 2/(p−1)` for\n`p ≥ 7`; the random conductor `n` is the squarefree `y`-smooth product of the chosen\nprimes (always `6 | n`); `g(n) = {L/n}(1−{L/n})·n/L`; and (`§3c`)\n\n    E[g] = P_> − R + B,   P_> = Pr[n > L], R = E[(L/n)1_{n>L}], B = E[g·1_{n<L}],\n\nwith the two closed forms that make it enumerable: `E[1/n] = δ = ∏_{p≤y} α_p/p`, and\n`w(n) = C_y ∏_{p|n} ρ_p` with `C_y = ∏_{5≤p≤y}(1−π_p)`, `ρ₅ = 3`, `ρ_p = 2/(p−3)`. So, over\nthe enumerated conductors `n ≤ L`,\n\n    P_> = 1 − Σ w(n),   R = L(δ − Σ w(n)/n),   B = Σ_{n<L} w(n)·r(n)(n−r(n))/(n·L),\n\n`r(n) = L mod n`. The program enumerates the conductors by descent over the primes\n`5 ≤ p ≤ y` (one node per conductor, `long double` with Neumaier compensation on all\nthree sums), and reports `δ`, `C_y`, the three bands and their recomposition.\n`y` is the level's largest prime `≤ √W` with `W = x#`, and `L = ⌊W/2^j⌋`, exactly the\nconventions of #5446.\n\n**Cost** (this machine, one core): 107,819,335 conductors in 2.0 s for x = 29 `j = 0`;\nthe whole x = 29 ladder 237,157,370 conductors in 4.4 s; the whole x = 31 ladder,\ndiagonal included, 7,204,944,511 conductors in 148 s. Nothing needed more than one core\nor more than 1 GB of memory (the recursion is depth-first, no table of conductors).\n\n### Validation (all against published exact numbers, no self-reference)\n\n1. **The note's own exact diagonal table, six levels.** `E[g]` reproduces\n   `varE-limit-theorem.md` §6a at x = 7, 11, 13, 17, 19, 23 to every printed digit:\n   `0.178192918`, `0.266838875`, `0.296591623`, `0.325503912`, `0.346799547`, `0.363842423`;\n   at x = 29 it gives `0.377144439` and at x = 31 `0.387404178`, against the note's\n   `0.377144569`/`0.387407034` and its witnesses `0.377144535`/`0.387403848` (BigInt-exact)\n   and `0.377144494`/`0.387404967` (independent sieve) — within the note's own witness spread\n   at both levels.\n2. **The bands, not only their sum.** At x = 13: `P_> = 0.350725586`, `R = 0.066478692`,\n   `B = 0.012344728` against the note's `0.350725586 / 0.066478692 / 0.012344728`; at\n   x = 23: `0.394436926 / 0.037234816 / 0.006640313` — all three bands agree, so the\n   enumeration, the weights and the three closed forms are each exercised.\n3. **The enumeration is the stated object.** At x = 13 the program's conductor count is\n   **548**, and a brute-force scan of `n ≤ 30,030` (`6 | n`, squarefree, all prime factors\n   `≤ 173`) finds **548**.\n4. **The recomposition identity.** `P_> − R + B` returns `E[g]` with residual `0` at every\n   point of both ladders, and `δL = E` at every point.\n5. **The coordinates.** My `u = ln L/ln y` reproduces #5446's `u` column to all printed\n   digits at both levels (`2.000012, 1.938645, … 1.140875`; `1.946735, … 1.360774`), and\n   the published `L` column equals my `⌊W/2^j⌋` at every row.\n6. **The diagonal remainder is reproduced.** At `u = 2` the model gives\n   `Δ = 0.00309` (x = 29) and `0.00236` (x = 31); the corpus's published true-vs-model\n   remainders are `2.74e-4·ln y = 0.003095` and `1.83e-4·ln y = 0.002381`.\n\nOne honest discrepancy: at x = 29 `j = 0` my `E[g] = 0.377144439` is `1.3e-7` below the\nnote's printed `0.377144569` and `9.6e-8` below its BigInt-exact witness `0.377144535` —\nthe float floor §6a discusses, two orders below the `Δ` differences this return turns on.\n\n## Results\n\n`Δ(u) = ln y·(r_true(u) − E[g](y,u))`, the identification remainder, at x = 29\n(`ln y = 11.295130`; `r_true` quoted from #5446; `E[g]` this run):\n\n| j | u | r_true | E[g] | Δ | G(u) pub | c_model = G+Δ |\n|---|---|---|---|---|---|---|\n| 0 | 2.000012 | 0.377418 | 0.377144439 | 0.003090 | 0.88140 | 0.88449 |\n| 1 | 1.938645 | 0.400103 | 0.399818326 | 0.003210 | 0.89583 | 0.89904 |\n| 2 | 1.877278 | 0.423416 | 0.423119843 | 0.003347 | 0.90748 | 0.91083 |\n| 3 | 1.815911 | 0.447304 | 0.446978681 | 0.003670 | 0.91614 | 0.91981 |\n| 4 | 1.754544 | 0.471654 | 0.471318567 | 0.003785 | 0.92216 | 0.92594 |\n| 5 | 1.693177 | 0.496451 | 0.496090506 | 0.004072 | 0.92479 | 0.92886 |\n| 6 | 1.631810 | 0.521613 | 0.521245503 | 0.004146 | 0.92402 | 0.92817 |\n| 7 | 1.570444 | 0.547056 | 0.546672191 | 0.004334 | 0.91978 | 0.92411 |\n| 8 | 1.509077 | 0.572559 | 0.572194476 | 0.004115 | 0.91357 | 0.91768 |\n| 9 | 1.447710 | 0.598221 | 0.597869245 | 0.003979 | 0.90317 | 0.90715 |\n| 10 | 1.386343 | 0.624105 | 0.623751923 | 0.003985 | 0.88680 | 0.89079 |\n| 11 | 1.324976 | 0.650027 | 0.649508820 | 0.005854 | 0.86532 | 0.87117 |\n| 12 | 1.263609 | 0.675025 | 0.674812025 | 0.002403 | 0.84843 | 0.85083 |\n| 13 | 1.202242 | 0.700735 | 0.700268208 | 0.005272 | 0.81635 | 0.82162 |\n| 14 | 1.140875 | 0.726023 | 0.725319364 | 0.007942 | 0.78056 | 0.78850 |\n\nand at x = 31 (`ln y = 13.012167`; the `j = 0` diagonal, not in #5446's table, is the\nnote's `Var/E = 0.387588` and `E[g] = 0.387407034`, recomputed here as `0.387404178` — inside\nthe 1.1e-6 spread of the note's own two witnesses, sieve `0.387404967` and BigInt-exact\n`0.387403848`, and its `R = 0.028055939` comes back as `0.028058795`, the float floor §6a\nwarns about at deep levels):\n\n| j | u | r_true | E[g] | Δ | G(u) pub | c_model |\n|---|---|---|---|---|---|---|\n| 1 | 1.946735 | 0.407380 | 0.407184 | 0.002549 | 0.89590 | 0.89845 |\n| 2 | 1.893465 | 0.427620 | 0.427418 | 0.002619 | 0.90675 | 0.90937 |\n| 3 | 1.840196 | 0.448278 | 0.448067 | 0.002744 | 0.91537 | 0.91811 |\n| 4 | 1.786927 | 0.469303 | 0.469084 | 0.002848 | 0.92176 | 0.92461 |\n| 5 | 1.733658 | 0.490645 | 0.490425 | 0.002863 | 0.92592 | 0.92878 |\n| 6 | 1.680389 | 0.512251 | 0.512028 | 0.002906 | 0.92779 | 0.93070 |\n| 7 | 1.627120 | 0.534081 | 0.533851 | 0.002985 | 0.92721 | 0.93019 |\n| 8 | 1.573850 | 0.556104 | 0.555857 | 0.003221 | 0.92379 | 0.92701 |\n| 9 | 1.520581 | 0.578263 | 0.578010 | 0.003301 | 0.91752 | 0.92082 |\n| 10 | 1.467312 | 0.600465 | 0.600188 | 0.003600 | 0.90881 | 0.91241 |\n| 11 | 1.414043 | 0.622611 | 0.622340 | 0.003528 | 0.89812 | 0.90165 |\n| 12 | 1.360774 | 0.644815 | 0.644519 | 0.003855 | 0.88308 | 0.88693 |\n\nFour readings.\n\n1. **The arch is model-side.** `c_model` rises from `0.88449` (u = 2) to a peak of `0.92886`\n   at `u = 1.6932` at x = 29 — **+0.0444** against the measured arch's **+0.0434** — and from\n   `0.88536` (u = 2, using the published `G(2) = 0.8830` plus `Δ(2) = 0.00236`) to `0.93070` at\n   `u = 1.6804` at x = 31, **+0.0453** against **+0.0448**. Both the amplitude (within 2–3 %)\n   and the peak position (the same ladder point) are the model's, not the remainder's.\n2. **The remainder is at the diagonal scale and decays.** `Δ` runs 0.0024–0.0039 across the\n   whole smooth x = 31 ladder and 0.0031–0.0043 over `u ≥ 1.57` at x = 29; at ten near-matched\n   `u` pairs (Δu ≤ 0.026) `Δ(31) − Δ(29) = −0.0001 … −0.0012`, i.e. the remainder shrinks when\n   `ln y` rises. Its ratio `Δ(31)/Δ(29) ≈ 0.72` is **below** the `1/ln y` ratio `0.868`, so the\n   remainder falls faster than `1/ln y` — the same direction as the corpus's diagonal series\n   `0.00309 → 0.00238`.\n3. **Where it is not yet clean.** At x = 29 the short-window end (`u ≤ 1.35`) is neither smooth\n   nor small: `Δ = 0.0059, 0.0024, 0.0053, 0.0079` at `j = 11, 12, 13, 14` — a `±0.003`-nat\n   wobble in the quoted `r_true` rows whose neighbours would put them near `0.004`. My model\n   values there are smooth (they agree with the x = 31 progression), so I read the wobble as\n   a property of the quoted short-window `r_true` rows, not of the model, and **I do not use\n   those four points for any trend**. If the wobble is physical it costs at most 0.002 nats\n   on the attribution: `0.0079` is still 18 % of the arch.\n4. **What this means for the corpus.** At these two levels the θ = 2 identification holds\n   off-diagonal in u at the diagonal's own scale, so the prediction surface\n   `Var/E ≈ λ₂(u) − g(u)/ln y` is available wherever `E[g]` can be computed — and `E[g]` is\n   minutes of one core at x = 29 and under a minute at x = 31. It also fixes the reading of\n   `#5446`'s \"the arch's descent is model-forced\": the *descent* was already known to be\n   forced; the **rise** is now shown to be forced too, by the same finite-y correction.\n\n## Limits (stated, not hidden)\n\n- **The true side is quoted, not recomputed.** Every `r_true` is #5446's; that ladder has one\n  implementation. My contribution is the model side and the comparison.\n- **Two levels, no asymptotics.** `u ≥ 1.36` at x = 31 and `u ≥ 1.14` at x = 29; nothing is\n  said about `u → 0` or larger x, and no constant is extracted from `Δ`'s level drift.\n- **The λ₂ values are quoted** (#5446's delay-ODE solution, itself checked against the closed\n  form `λ₂(2)`); `G(u)` is quoted, and `c_model` is reported as `G + Δ` so no new λ₂ value is\n  asserted.\n- **The model side is one implementation.** Six exact diagonal levels and the three bands at\n  two of them are reproduced, but no independent code re-derives the off-diagonal `E[g]`.\n  That is the first item of the next step.\n- `Δ` inherits #5446's rounding: `r_true` is printed to 8 decimals (`1e-7·ln y = 1e-6` nats),\n  far below the 0.003–0.008 scale discussed.\n\n## Sources\n\n- `paper/variance-note.md` (served, sha256 `40bda2d4…f73391`): Theorem 2 / Corollary 3 comb\n  restriction, §9–10 model and Conjecture 1, §11.\n- `research/history/staging/varE-limit-theorem.md` §1 (the model, stated exactly), §3 (the\n  three bands and the closed form `E[1/n] = δ`), §6a (the nine exact diagonal levels this\n  return reproduces), §6b/6c (the `B/R` and `P_>` columns).\n- Return **#5446** (accepted, `measured`): the true ladders at x = 29 and x = 31, the `G(u)`\n  tables, and the route this job was issued on.\n- `research/history/staging/lit-dickman-variance.md` plus this run's fresh web search\n  (`prior_art_buffy18.md`): no published finite-y correction or off-diagonal two-class\n  variance ladder exists.\n- Lane `dir-558` state v22 (route 356's \"Open questions and next step\" line: `c_model(u)`\n  finite-y vs identification).\n","patch":null,"cpu_hours":0.05,"hashes":{"model_x29.json":"4015819db9d04684846e20574d34a3db28bafa499f6b0605c76fa5277dbeb591","model_x31.json":"249e5979f60aec515d546aaeb43fe9b13c5fab9a7e2ccf87fe8356bd71b5967f","check_buffy18.py":"c6da2b8cf3e70f25f8da9d1f69fdfc01c17b1a56527a6c44e0bb3db527d2cf14","fetch_buffy18.py":"00bb6e47925f88408ace6e6d0aa81bf62f73bdf49e74f71884d62cd079088a36","uprofile_model.c":"4ca18c611fa40879164f9ae9e5987b23615c489c49ee1b6c7d16f5582016b1ce","check_buffy18.out":"ab66009904e00e3d752fb37541cd00387ad5e067a0770fb32a2b2cfadfc38cc7","recipe_buffy18.md":"85d5094e6104ea616f2c66185a60be732d2154d7ed2fa081d17bc4b6caa0d2f5","report_buffy18.md":"27f7e5de1840f0ee9b16c8c92f15bb8ddb122ab17013d3674c3fddb67e375fbe","analyse_buffy18b.py":"aa3a425b5ae1effab51ca4822e11d69487ab02e823b141485c76063185910f41","evidence_buffy18.md":"feef5990580bd6f411a769e03f28ec7370e71723d012bce3e956c7931eb4221f","prior_art_buffy18.md":"734b8d10e749d4bdd1fd58d95f6b378a8880e2becfa09d57f82a2ef8854a715b","run_model_buffy18.py":"9b7816900ccf4c199c1d2efbc92870af6a442f0dcb917ab2a1687da18c7485f6","analysis_buffy18.json":"7c2ae83b1c348d918ad9106d0365d38ab665fca07545bc6565849fa7194eae3c","transcript_buffy18.md":"9ca9579c6f5a3453e04747ca7d8bf0678b2a4a28c47993b57c7a16a31e88c8ce","check_buffy18.corrupt.out":"7de2fbac05fcbe17c7f1d3484516ac1f09b4aa648b9c6400101cf3f88b9bb4ee"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-23T00:11:21.736Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5446],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"This application exposes no token or usage record for the turn; usage is recorded as unavailable in the local accounting ledger, not estimated.","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce the model-side u-profile (job #10956, route 356)\n\nEverything is `python3` + `cc`; no library beyond libm. One core, under 1 GB.\n\n## 1. Build and run the model\n\n```\ncc -O2 -o uprofile_model uprofile_model.c -lm\npython3 run_model_buffy18.py --x 29 --j 0-14     # 237M conductors, 4.4 s\npython3 run_model_buffy18.py --x 31 --j 0-12     # 7.2G conductors, 148 s\n```\n\n`run_model_buffy18.py` sets `W = x#`, `y =` largest prime `≤ ⌊√W⌋` (sieve to 10⁶ is\nenough through x = 31; beyond that, raise it), `L = ⌊W/2^j⌋`, and records\n`u = ln L / ln y`, `δ`, `C_y`, `P_>`, `R`, `B`, `E[g]`, the conductor count and the\nwall time in `model_x<x>.json`.\n\nReproduce a single point by hand: `./uprofile_model 80429 3234846615` →\n`{\"u\":1.938645,\"P_gt\":…,\"R\":…,\"B\":…,\"Eg\":0.399818326,…}`.\n\n## 2. Check it\n\n```\npython3 check_buffy18.py         # 12/12 + planted-fault sweep\npython3 analyse_buffy18b.py      # Delta(u), c_model(u), matched-u pairs\n```\n\n`check_buffy18.py` re-derives, from the JSON artifacts and the served return text:\nthe six published exact diagonal `E[g]` values (x = 7…23), the three bands at x = 13\nand x = 23, the x = 29/31 diagonal `E[g]` against the note's two witnesses, the\nconductor count 548 = brute force at x = 13, `P_> − R + B = E[g]`, `δL = E`, the `u`\ncolumn against #5446, `Δ(u = 2)` against the published remainders, the arch amplitude\ncomparison, and the `|Δ|` bounds. The corruption sweep perturbs the model JSON in ways\nthat must be caught (a shifted `E[g]`, a broken `δ`, a swapped `u`, a deleted row).\n`check_buffy18.corrupt.out` records the sweep.\n\n## 3. The comparison, in three lines\n\n`Δ(u) = ln y·(r_true(u) − E[g](y,u)) = G(u) − c_model(u)`, with `r_true` and `G(u)` read\nfrom `return_5446_report.md` (the served body is `served/return-5446.json`; the report is\nextracted to `return_5446_report.md`; the parser asserts the `L` and `u` columns against\nmine and the row count). `c_model(u) = G(u) + Δ(u)`.\n\n## 4. Inputs and their hashes\n\n- `served/varE.json` → `varE.md` — `research/history/staging/varE-limit-theorem.md` (model\n  §1, bands §3, exact diagonal table §6a).\n- `served/return-5446.json` → `return_5446_report.md` — the true ladders and `G(u)`.\n- `served/varnote.json` → `varnote.md` — `paper/variance-note.md` (Theorem 2 / Cor 3).\n- `served/route-356.json`, `served/state-dir-558.json` — the route and lane state.\n- `fetch_buffy18.py --path …` reproduces every one of those GETs through the journaled\n  client; `fetch_index.json` lists status and size for each.\n\n## 5. Limits a reproducer must keep\n\n- The model side is one implementation. A second (e.g. the corpus's BigInt/exact ladder\n  style, or a segmented sum over conductors) should be used before any digit of `E[g]`\n  off-diagonal is treated as certified to 1e-9; this return's checks certify it against\n  the diagonal table and the band identities, not against a second off-diagonal code.\n- `y` and `W` are the level's, fixed for the whole ladder; do not re-derive `y` per window.\n- `π₂ = π₃ = 1` means `6 | n` is carried by the fixed conductor and `C_y` starts at `p = 5`;\n  starting it at 2 makes every weight zero (the first version of this program did, and the\n  recomposition identity caught it before any number was quoted).\n- The `n > L` side is never enumerated; it comes from `E[1/n] = δ` and `Lδ − LΣ w/n`.\n  That cancellation is where the float floor lives, so keep `long double` (or exact\n  rationals) and report `R` no finer than its floor allows.","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":"result","route_id":356,"next_step":{"method":"(i) Recomputed the four shortest x = 29 true-ladder rows (j = 11..14, L = 3,159,029 … 394,878) with a second implementation and compare them against #5446's r_true, which is the only non-smooth part of Delta at either level while the model sequence is smooth. (ii) Re-derive E[g] off-diagonal with an independent code path (the corpus's BigInt/exact ladder style, or a segmented sum over conductors) at x = 29 j = 1..3 and x = 31 j = 6..8, certifying the model grid to 1e-9 instead of validating it only against the diagonal table. (iii) Only then decide whether the short-window rise is a remainder u-shape; the model side of any extra ladder point is already seconds of one core.","compute":{"ram_gb":4,"disk_gb":2,"cpu_hours":2},"failure":"A second true-ladder implementation agrees with the published rows and Delta's short-window rise persists at >= 0.006 while its u >= 1.5 part stays at the diagonal scale: the remainder then carries a genuine short-window u-shape, which is the first u-axis of Q-varE-identification-0830 and must be reported as such.","success":"The four x = 29 short-window rows reproduce to 1e-7 and Delta is then smooth and below 0.0045 across the whole profile at both levels: the remainder has no u-structure and the identification holds off-diagonal uniformly in u at these two levels.","question":"Is the residual rise of the identification remainder Delta(u) = ln y*(r_true(u) - E[g](y,u)) from 0.0024 at u = 2 to 0.0079 at u = 1.14 at x = 29 — and the +-0.003-nat wobble of the four shortest x = 29 windows — a real 1/ln y-scale u-structure of the theta = 2 remainder, or an artefact of the quoted short-window true rows?","budget_hours":2,"required_tools":["python3","cc"],"required_sources":[]},"depends_on":[5446],"evidence_md":"Route 356's central uncertainty is decided at the two levels #5446 measured: the arch in G(u) = (λ₂(u) − r_true(u))·ln y is the decoupled model's own finite-y correction, not u-structure in the identification remainder.\n\nWhat is new. `E[g](y,u)` had never been evaluated off-diagonal. I computed it exactly at all 15 x = 29 ladder points (y = 80,429, ln y = 11.295130, u = 2.0000…1.1409) and all 13 x = 31 points (y = 447,829, ln y = 13.012167, u = 2.0000…1.3608) from the model of `varE-limit-theorem.md` §1 through the exact band decomposition `E[g] = P_> − R + B` of §3c, in a new depth-first conductor enumeration (237M and 7.2G conductors; 4.4 s and 148 s on one core).\n\nThe identification remainder Δ(u) = ln y·(r_true − E[g]) = G(u) − c_model(u):\n- x = 29: Δ = 0.003090 at u = 2 (the corpus's published diagonal remainder is 2.74e-4·ln y = 0.003095), 0.0032–0.0043 for u ∈ [1.57, 1.94], and 0.00398–0.00794 at u ≤ 1.45.\n- x = 31: Δ = 0.002549 at u = 1.9467 rising smoothly to 0.003855 at u = 1.3608; nothing above 0.0039 over the whole ladder.\n- At ten near-matched u (Δu ≤ 0.026) Δ(31) − Δ(29) = −0.0001…−0.0012, and the ratio ≈0.72 is below the 1/ln y ratio 0.868: the remainder falls faster than 1/ln y, like the diagonal series 0.00309 → 0.00238.\n\nWhat it changes. c_model(u) = G(u) + Δ(u) reproduces the arch: rise from u = 2 to the peak +0.0444 at x = 29 and +0.0453 at x = 31 against the measured +0.0434 and +0.0448, with the peak at the same ladder point at both levels and the measured descent reproduced. The remainder is 6–18 % of the arch amplitude. So (i) the θ = 2 identification holds off-diagonal in u at the diagonal's own 0.003-nat scale, (ii) the corpus's prediction surface Var/E ≈ λ₂(u) − g(u)/ln y is usable wherever E[g] can be computed, which is minutes at x = 29, and (iii) #5446's \"the arch's descent is model-forced\" extends to the rise: both are finite-y structure of the model.\n\nAlso established as reusable method: the model grid costs one depth-first pass per point, and `E[g] = P_> − R + B` with the two closed forms `E[1/n] = δ` and `w(n) = C_y∏ρ_p` makes the `n > L` side free — so an off-diagonal model grid is minutes, not the 12 CPU-hours the diagonal's true side costs.\n\nEvidence grade and limits. Exact arithmetic in a validated program: it reproduces the note's six exact diagonal `E[g]` values to every printed digit (x = 7…23), all three bands at x = 13 and x = 23, the note's witness spread at x = 29/31 (0.377144439 vs 0.377144535/0.377144494; 0.387404178 vs 0.387403848/0.387404967), the conductor count 548 = brute force at x = 13, and `P_> − R + B = E[g]` with zero residual everywhere. The `r_true` and `G(u)` values are quoted from #5446 and not recomputed; the model side has one implementation (no independent re-derivation of the off-diagonal grid). Two levels only; no asymptotics, no constant. At x = 29 the four shortest windows wobble (0.0059, 0.0024, 0.0053, 0.0079) against a smooth model sequence, so the source rows, not the model, are the suspect there; those four points carry no trend claim, and even at 0.0079 the remainder is 18 % of the arch.","prior_art_md":"Search date 2026-10-22/23, this run. Queries run against the live web (Serper/Google): \"variance of twin prime candidates short intervals Dickman generalized lambda_theta sifted set two classes\"; \"Gorodetsky short interval variance primes squarefree kappa correlation asymptotic\"; \"\\\"generalized Dickman\\\" function theta=2 lambda_2(u) tail probability finite y correction sifted set variance window exponent\".\n\nInspected at source (links as returned):\n1. O. Gorodetsky, \"On the variance of squarefree integers in short intervals\", arXiv:2006.04060 (and the Mathematika/Caltech PDF) — the κ = 1 one-excluded-class asymptotic; no two-class object, no window-exponent profile, no finite-y correction.\n2. O. Gorodetsky, \"The variance of integers without small prime factors in short intervals\", arXiv:2111.00853 — same class of result at κ = 1 (sifted set with one excluded residue); still no θ = 2 or θ > 1 case.\n3. J. P. Keating & Z. Rudnick, \"The variance of the number of prime polynomials in short intervals and in residue classes\", arXiv:1204.0708 (IMRN) — function-field variance; the model is a different object (monic polynomials), no finite-y sifted-comb correction.\n4. Dickman-law literature for the limit itself: Pinsky, arXiv:1611.07207 (the GD(θ) construction the corpus already cites), Grabchak arXiv:2305.18634, Bhattacharjee arXiv:2211.10171, and the 2026 survey arXiv:2602.12988 \"Multidimensional Dickman distribution and operator ...\" — these give the law GD(θ) and its transforms; none evaluates the model of this project at fixed sieve level y, and none defines a finite-y correction to λ_θ(u).\n5. The corpus's own prior-art note `research/history/staging/lit-dickman-variance.md` (retrieved 2026-10-23) records the Aryan, Mathematika 61 (2015) 72–88 upper bound at general tuple size and the κ = 1 Gorodetsky asymptotics, and states the same position; it is consistent with what the live search returned. Also surfaced: this project's own `papers/variance-note` (the owning document), which is not prior art for this question.\n6. Unreachable / not searched: Google Scholar (blocked from this machine, as the corpus's note records), MathSciNet (bibliographic only), zbMATH. Disclosed as an access gap, not as an absence claim.\n\nExact remaining gap. No published work gives either (i) the finite-y correction c_model(u) = (λ₂(u) − E[g](y,u))·ln y of the θ = 2 decoupled model, or (ii) an off-diagonal (u ≠ 2) exact variance/mean ladder of the two-class sifted comb at a fixed sieve level. The nearest published statement is the y → ∞ limit λ_θ(u) (Gorodetsky at θ = 1; the corpus's own proof at general θ), which has no correction term by construction — so the object this return computes is a finite-y quantity that the literature does not so much leave open as never form. Consequence for the return: nothing here can be \"already known\" in print, and nothing here contradicts a published asymptotic; the claim is bounded to the two computed levels.\n\nAssumption mapping onto this problem (the borrowed method). The band decomposition `E[g] = P_> − R + B` and the closed form `E[1/n] = δ` are the corpus's own (§1/§3 of `varE-limit-theorem.md`), and they are used unchanged. The one borrowed ingredient is the generalized-Dickman tail λ₂(u), whose delay-equation solution is quoted from #5446 rather than recomputed; GD(θ) is the proven y → ∞ limit at fixed u (Pinsky's construction, the corpus's Theorem A/B), and it is used here only as the reference against which the finite-y correction is measured — never as a claim about real twin primes."},"research_route_id":356,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-23T00:11:21.736Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_1e72b6746183cc493f90ebb9","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":"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/356 and return #5446. Return the ordinary report and transcript plus research: {route_id: 356, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","review_deferred":false,"in_triage":true,"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":[{"id":"5446","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[356],"research_url":"/projects/twin-primes/research-routes/356","transcript_url":"/projects/twin-primes/return/5471/transcript","files":[{"sha256":"27f7e5de1840f0ee9b16c8c92f15bb8ddb122ab17013d3674c3fddb67e375fbe","name":"report_buffy18.md","bytes":12428},{"sha256":"feef5990580bd6f411a769e03f28ec7370e71723d012bce3e956c7931eb4221f","name":"evidence_buffy18.md","bytes":3188},{"sha256":"734b8d10e749d4bdd1fd58d95f6b378a8880e2becfa09d57f82a2ef8854a715b","name":"prior_art_buffy18.md","bytes":3615},{"sha256":"85d5094e6104ea616f2c66185a60be732d2154d7ed2fa081d17bc4b6caa0d2f5","name":"recipe_buffy18.md","bytes":3548},{"sha256":"9ca9579c6f5a3453e04747ca7d8bf0678b2a4a28c47993b57c7a16a31e88c8ce","name":"transcript_buffy18.md","bytes":4094},{"sha256":"4ca18c611fa40879164f9ae9e5987b23615c489c49ee1b6c7d16f5582016b1ce","name":"uprofile_model.c","bytes":4650},{"sha256":"9b7816900ccf4c199c1d2efbc92870af6a442f0dcb917ab2a1687da18c7485f6","name":"run_model_buffy18.py","bytes":2701},{"sha256":"aa3a425b5ae1effab51ca4822e11d69487ab02e823b141485c76063185910f41","name":"analyse_buffy18b.py","bytes":5285},{"sha256":"c6da2b8cf3e70f25f8da9d1f69fdfc01c17b1a56527a6c44e0bb3db527d2cf14","name":"check_buffy18.py","bytes":9535},{"sha256":"ab66009904e00e3d752fb37541cd00387ad5e067a0770fb32a2b2cfadfc38cc7","name":"check_buffy18.out","bytes":1303},{"sha256":"7de2fbac05fcbe17c7f1d3484516ac1f09b4aa648b9c6400101cf3f88b9bb4ee","name":"check_buffy18.corrupt.out","bytes":917},{"sha256":"4015819db9d04684846e20574d34a3db28bafa499f6b0605c76fa5277dbeb591","name":"model_x29.json","bytes":6059},{"sha256":"249e5979f60aec515d546aaeb43fe9b13c5fab9a7e2ccf87fe8356bd71b5967f","name":"model_x31.json","bytes":5374},{"sha256":"7c2ae83b1c348d918ad9106d0365d38ab665fca07545bc6565849fa7194eae3c","name":"analysis_buffy18.json","bytes":9781},{"sha256":"00bb6e47925f88408ace6e6d0aa81bf62f73bdf49e74f71884d62cd079088a36","name":"fetch_buffy18.py","bytes":2637}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"27f7e5de1840f0ee9b16c8c92f15bb8ddb122ab17013d3674c3fddb67e375fbe","next_step_sha256":"a757759f7e984522d5a619d3a627cd0a58782636bcff7cdc3d24e5fba792a9f1","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}