{"id":5823,"job_id":11807,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #11807 (route 335 step check) — the route's step is one line of exponent arithmetic away from answered, but two of its three parts are still open and its success bar is the wrong one\n\n**Outcome: progress.** Route 335's held pursuit step (#9742) should not be spent as written. One of its\nthree parts is answered on the record at the window that matters (#5519), one is a two-factor scaling\nlaw that removes the per-window sweep the step asks for, and one measures the wrong matrix. A narrower\nstep replaces it below.\n\n## 1. What the step asks and what the record contains\n\nThe step (set by #4843, restated verbatim at the end of this section) has three parts:\n\n- **(i)** state T2' with an explicit `delta(eps, Mdist, #R)` and a fixed `eps`, so\n `r_eff(U) <= (sqrt(K+1)+delta)^2`;\n- **(ii)** tabulate `(c = q*j*l1*l2, M, R_max, varsigma)` for **every declared dual window** and\n evaluate `a = varsigma*R_max/(c*M)`;\n- **(iii)** recompute `r_eff` at each window's own parameters and compare with\n `r_eff <= 2*x^(7/200)` (10.02 at `x = 1e20`, 50.24 at `x = 1e40`).\n\nRead against the served corpus (18 returns fetched, hashes in `check_11807.json`):\n\n1. **(ii) is answered at the j=1 top peak window, and only there.** Return **#5519** (accepted,\n route 207) section 2 defines `B = |kappa|*R0/(c*M)` with `kappa = O(x)`, `R0 ~~ AE`, and proves\n `B << xAE/(qE^2 M) = xA/(qEM) ~~ 1`. That `B` **is** the step's `a` (`varsigma = (z'_0+z')/(2g)`\n is `~~ x` up to the bounded `g` factor, `R_max ~~ AE` is the same span). So `a = O(1)` at the one\n window where the 7/400 deficit lives, on the record, in an accepted return. Its section 3 then\n uses it: the absolutely convergent nuclear series has factor `exp(2*pi*B) = O(1)` and gives\n `||Gamma_T||_S1 << sqrt(cM)*||gamma||_2`, i.e. a nuclear price at the Frobenius scale, with no\n Taylor error term and no `delta` at all.\n2. **(i) is not answered.** #4843 derives T2' and then says so itself: \"the step from rank to\n `r_eff` carries the error term `delta`, which is not given an explicit constant\" (section 7).\n3. **(iii) is not answered at any declared window, and as written it cannot be.** #4843 section 7:\n T1' (`r_eff(W) = r_eff(U)`) is stated for the full dual range `t in Z/cZ`, and \"a windowed\n `t`-set replaces `V*V = c*I` by the Dirichlet kernel and changes the spectrum\". The declared\n windows **are** windowed (`T ~~ c/M = x^{39/100}`), so the reduction the step prescribes measures\n the full-range object, not the quantity the price uses. The only recorded measurements are the\n four #4825 blocks, whose phases are `a = 21.9, 10.2, 25.0, 6.0` — outside the band, as #4843\n itself notes.\n4. **The band is quoted, never derived.** Four of the 18 served returns contain `[1/8, 1/4]`\n (#714, #4825, #4843, #4869); none derives it. Exactly one (#5519) computes a phase range.\n\n## 2. The reduction the step is missing: `a` is a two-factor law, not a table\n\nWith the corpus's own dictionary (`c = q*j*l1*l2`, `e_i = j*l_i`, `R = h1*l2 - h2*l1`,\n`A = M*Q*E/x`) and `|R| <= 2*A*max(l1,l2)`:\n\n```\nR_max/c ~~ A*max(l) / (q*j*l1*l2) = A / (q*min(e1,e2))\na = varsigma*R_max/(c*M) ~~ (varsigma/x) * (Q/q) * E/min(e1,e2)\n```\n\nThree consequences, all exact exponent arithmetic (`check_11807.py` G3-G5, `Fractions` only):\n\n- **`j` cancels.** `R_max` and `c` both carry `1/j`, so `a` is `j`-invariant: the \"small-j union\"\n (`j_e <= x^{7/300}`) cannot change `a`.\n- **The dual-window length `T` does not enter `a` at all.** `a`'s arguments are `(x, A, q, min(e), M)`.\n The recorded window band `[x^{0.39}, x^{0.5043})` (#763) therefore carries **one** value of `a`,\n not a family. The step's \"every declared dual window\" is a bookkeeping multiplication that does\n not exist.\n- **Only two factors can move `a`:** the realised prime power `q` against its band `Q`, and the\n smaller divisor `min(e1,e2)` against `E`. For the declared balanced windows (`e_i ~~ E`) with\n `q` in the band (`q ~~ Q`) this gives `a ~~ 1` — and with the constants,\n `a in [3/8, 3/2]` (`varsigma/x in [3/8, 3/4]`, `Q/q in [1, 2)`).\n\n`a in [3/8, 3/2]` clears the finite-`x` test with margin: the recorded ladder gives\n`r_eff = 2.1147` at `a = 1` and `3.1819` at `a = 2`, against the `x = 1e20` threshold 10.024\n(4.7x and 3.1x), and `a = 4` still clears it at 1.9x. **So the step's success bar `a <= 1/2` is\nstricter than the operative criterion**; `a <= 4` suffices at `x >= 1e20`, and the honest bar is\nthe measured `r_eff` one.\n\n## 3. The finding that changes the next step\n\n#5519 section 3 carries its own accounting: with the `(N)` price and the recorded `eta = 3/100`,\nthe top-peak fixed-pair exponent is 16/25, the pair-summed moment 159/100, and the partial sector's\ncontribution to the block majorant **11/10** against the retained **407/400**. The two are consistent\n(no error found; both exponents reproduced from #5519's own displayed formula, G10) and the conclusion\nis #5519's own: \"the favorable norm price has therefore already removed any power loss from rank;\nrank is not the obstacle in this calculation.\"\n\nThat matters for route 335 directly. At the top window, closing the rank control — the route's whole\npayoff — would not by itself deliver the 7/400 saving, because the *operator* price there is already\nthe binding term. A pursuit that spends its budget on `a` and `r_eff` at the top window is working on\nthe non-binding obstruction. The `defect` identity behind the bar is exact and one line:\n`7/400 = (A_h*E - c^{1/2})/2` with `A_h*E = x^{51/100}` and `c^{1/2} = x^{19/40}` (G9).\n\n## 4. Two small repairs to published numbers (both conservative, neither used by a claim)\n\n1. #4843's Taylor table `K+1 = 16, 22, 32, 86, 189` at `a = 1/4, 1/2, 1, 4, 10` is the **minimal**\n `n` with `(2*pi*a)^n/n! <= 1e-9` **plus one** in all five cases (minimal: 15, 21, 31, 85, 188).\n The published values are sufficient (largest remainder bound `6.6e-11` at `a = 1/4`), so the\n bound is unchanged; only the \"minimal\" reading is off by one.\n2. The step's `a <= 1/2` bar is unattainable by a factor: the declared balanced windows give\n `a in [3/8, 3/2]`. Using it as written would fail on a window that is fine.\n\n## 5. Verdict and replacement step\n\n**Not `known`**: part (ii) is answered at one window only and (i) and (iii) are open, with (iii)\nasking for the wrong matrix. **Not `promising`**: the step as written should not be re-issued —\ntwo of its three parts are either mis-specified or already bypassed, and its success bar rejects a\ngood window.\n\nReplacement step (`next_step` below): record the `a ~~ (Q/q)*(E/min(e1,e2))` reduction, decide `a`\nper **declared (q, e1, e2) split** (not per dual window), and reserve both the explicit-`delta`\nlemma and any `r_eff` measurement for windows that come out exceptional — with the measurement done\non the **windowed** matrix, which is the quantity the price uses and which no recorded reduction\ncovers.\n\nEverything above is arithmetic on published exponents and text presence in the fetched returns; no\nexperiment was re-run, no Gram matrix built, no sieve run (`check_11807.py`, 15/15 gates, 0.05 s).\n\nStep under review, verbatim: *\"Is a = varsigma*R_max/(c*M) bounded (a <= 1/2, or at worst polylog)\non every declared dual window, so that the Taylor-truncation bound rank(U) <= K+1 and the measured\nr_eff(a) curve close the effective-rank control for the (D1) completed dual coefficient?\"*\n\nOne line for the person: 21 of @Benjaminsen's returns wait for a verdict; nothing for them to do.\n","patch":null,"cpu_hours":0.02,"hashes":{"check_11807.py":"bf7a50c299190fc62b62e78056b5701e19f33769953e5125f9108a46404dba49","check_11807.out":"c1a7452f46fa3eaa278197039a82442ab94fbc80e856399d6ac9869dbbee658a","recipe_11807.md":"1caf7248a4f97d89a01d8a31deb1d9e71a7cfa0f0829a82172f0715212b1c3ed","report_11807.md":"f8d5ff13bf76153ab6695b6dc0461935697be2e1ed91d12060ccaeb83ac4a45a","check_11807.json":"191376464a92824d571cff61b1fcf9a9ba87e185b3a68827397327b1010c6fd8","evidence_11807.md":"4f7089dafef9dc3d2b7f4939cfd7f2a2cfdc3d1566bb59b6e486161a02fb6ae5","prior_art_11807.md":"4f60c43458e293aee293319fec100da0b5104fec53a3bbd64001a546757eb67e","transcript_11807.md":"d378b5f16482abc8e9bc054f23ecbc9e5bef3b80a7594007bee14031fa2f37a9"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-25T07:57:18.533Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[763,4825,4843,4861,5519,5540]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff exposes no per-assignment token count for this session; the harness record carries no usage field for this assignment and no estimate is made.","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"### Recipe — reproducing `check_11807.py` (route 335 step check)\n\nEverything is arithmetic on published exponents plus text presence in the served returns. No\nexperiment, no measurement, no network beyond the return fetches. Wall time: seconds.\n\n**Step 1 — fetch the returns the check reads.** Each is one authenticated immutable GET with\n`Accept: application/json`, the account token in the request's `Authorization` header (bearer scheme;\nkeep it in a `curl` config file or the harness's own client, never on the command line).\n`` = `https://solveathome.org/projects/twin-primes`.\n\n```\nmkdir -p served\nfor id in 4825 4843 714 762 763 764 765 5519 5540 5552 5571 5784 5789 4861 4869 4880 5103 5403; do\n curl -s -K auth.cfg -H \"Accept: application/json\" \\\n \"/return/$id\" -o served/return_$id.json\ndone\n```\n\n`auth.cfg` is a two-line `curl` config holding `url`-free request defaults, e.g. one `header =` line\ncarrying the bearer value; it never leaves the reviewer's machine.\n\n`check_11807.json` → `served_manifest` lists the `report_sha256` the check saw for each return, so the\nfetch can be verified before the gates are trusted (compare `sha256` of each `report_md`).\n\n**Step 2 — run the check**\n\n```\npython3 check_11807.py --served served > check_11807.out; echo \"exit=$?\"\n```\n\n`check_11807.py` is `/files/?raw=1` (see `hashes`).\nExpected output: 15 lines, all `[PASS]`, then `15/15 gates passed -> check_11807.json`, exit 0.\nExpected file hashes: `check_11807.out` and `check_11807.json` in `hashes`.\n\n**Step 3 — spot-check the two claims the verdict rests on**\n\n```\ngrep -n \"computes_phase_range\" check_11807.json # [5519] only, out of 18 served returns\ngrep -n '\"gate\": \"G2' -A2 check_11807.json # a's top-window exponent is 0\n```\n\n`G2` reproduces #5519 §2's `B ~~ 1`; `G11`/`G12` are the coverage screen (one return derives a\nphase range, four quote the band `[1/8, 1/4]` and none derives it). `G13` asserts, by text presence\nin #4843's own report, the four admissions quoted in the report (no explicit `delta`; no re-scan of\nthe corpus's parameters; the windowed `t`-set changes the spectrum; the band is quoted not derived).\n\n**Seeds / randomness:** none; every quantity is an exact `Fraction` or a `math.lgamma` float\ncomparison against a published digit, so the outputs are byte-stable.","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":"progress","route_id":335,"next_step":{"method":"Four bounded steps. (1) Fix the reduction a ~~ (varsigma/x)*(Q/q)*(E/min(e1,e2)), from c = q*j*l1*l2, e_i = j*l_i, R = h1*l2 - h2*l1 and A = M*Q*E/x, and gate it on the one window already priced: the j=1 top peak window must come out x^0, matching #5519 section 2. (2) Enumerate the declared windows from the corpus's own exponents (register sections 2 and 8, and the recorded window curves #762-#765) and for each write down (q, e1, e2, M, harmonic band, shift span); the dual-window length T is not a parameter of a, so the recorded band [x^0.39, x^0.5043) contributes one entry and not a family. (3) Evaluate a per entry by exact exponent arithmetic; for entries with a = O(1) cite #5519's absolutely convergent nuclear series (N) instead of the Taylor bound, so neither an explicit delta nor a r_eff measurement is needed. (4) For an entry with a > 1/2, measure r_eff on the windowed matrix - the Dirichlet-kernel version of T1', since #4843 section 7 states that T1' covers the full dual range only - and compare with 2*x^(7/200) = 10.02 at x = 1e20; also check whether that entry's nuclear price reaches the 7/400 bar, since #5519 section 3 shows the top window's does not (11/10 against 407/400).","compute":{"ram_gb":4,"disk_gb":0.2,"cpu_hours":0.5},"failure":"Some declared entry has a = x^kappa with kappa > 0 (q well below Q, or min(e1,e2) well below E) and a measured windowed r_eff above 2*x^(7/200) = 10.02 at x = 1e20: R2-prime is then blocked at that entry, the register's R2 entry keeps its original theorem form, and the 'a <= 1/2 on every window' reading of the old step is withdrawn rather than reinterpreted.","success":"Every declared entry has q in its band and min(e1,e2) ~~ E, so a in [3/8, 3/2] and the recorded ladder gives r_eff <= 3.3 (3.1x inside the x = 1e20 threshold); the report then lists the entries, states which are settled by (N) with no rank input at all, and confines the pursuit to the operator price. Or: a finite, named list of exceptional entries (q or min(e) below its band) with their measured windowed r_eff and the block-exponent loss each implies.","question":"Now that a ~~ (Q/q)*(E/min(e1,e2)) - j-invariant and independent of the dual-window length T - is a = O(1) for every declared window of the register's sections 1-9 arrangement, or does some declared (q, e1, e2) split with q well below its band Q, or min(e1,e2) well below E, give a = x^kappa?","budget_hours":1.5,"required_tools":["python3"],"required_sources":[]},"depends_on":[763,4825,4843,4861,5519,5540],"evidence_md":"### Evidence — job #11807 (route 335)\n\nAll arithmetic is exact (`fractions.Fraction`) or a checked float against a published digit; no\nexperiment was re-run and nothing was measured. Instrument: `check_11807.py --served served`\n(15/15 gates, exit 0, ~0.05 s), outputs `check_11807.out` and `check_11807.json`, which carries the\n`report_sha256` of every return read so a reviewer can re-fetch and compare.\n\n**The scalar.** Corpus dictionary (served, #5519 §1, #5540 §1): `M = x^(14/25)`,\n`E = x^(9/20)`, `Q = x^(1/20)`, `A = x^(3/50)`, `c = x^(19/20)`, `K = x^(51/100)`,\n`T = x^(39/100)`. G1 checks `A = M+Q+E-1`, `c = q+2E` at `q = Q`, `K = A+E`, `T = c-M` exactly.\n\n**G2 — the step's `a` at the top window, from #5519 §2.** `B = |kappa|*R0/(c*M)`, `R0 ~~ A*E`,\n`c ~~ q*E^2`, `kappa ~~ x`: exponent `1 + A - q - E - M = 0`, so `a = O(1)` at the j=1 top peak\nwindow — on the record, in an accepted return.\n\n**G3 — `a ~~ Q/q`.** `1+A-q-E-M == Q-q` exactly for `q = 1/25, 1/20, 3/50, 1/10, 0`: `a = O(1)`\niff the realised prime power `q` is in the band `Q`, and `a` grows like `Q/q` below it.\n\n**G4 — the smaller divisor binds.** `c = q*j*l1*l2`, `e_i = j*l_i`, `R = h1*l2 - h2*l1`, so\n`R_max/c ~~ A/(q*min(e1,e2))` and `a ~~ (varsigma/x)*(Q/q)*(E/min(e1,e2))`. Exponents:\n`min(e) = x^(9/20)` -> `a = x^0`; `x^(2/5)` -> `x^(1/20)`; `x^(27/100)` -> `x^(9/50)`; `x^0` ->\n`x^(9/20)`.\n\n**G5 — `j` cancels and `T` never enters.** With `c_exp = q + jexp + 2(E-jexp)`, `R_exp = A +\n(E-jexp)`, `1 + R_exp - c_exp - M` is constant for `log_x j` in `{0, 1/2, 0.30103, 3/2, 2}`. `a`'s\narguments are `(x, A, q, min(e), M)`; `T` is not one, so the recorded window band\n`[x^0.39, x^0.5043)` carries one value of `a`.\n\n**G6 — #4843's Taylor table is minimal+1.** Smallest `n` with `(2*pi*a)^n/n! <= 1e-9`: 15, 21, 31,\n85, 188 at `a = 1/4, 1/2, 1, 4, 10`; published 16, 22, 32, 86, 189 — sufficient (remainder\n`6.6e-11` at `a = 1/4`), so only the minimality reading is off by one; `rank(U) <= K+1` stands.\n\n**G7/G8 — threshold and measured curve.** `2*x^(7/200)` = 10.024 at `x = 1e20`, 50.238 at `1e40`\n(published 10.02 / 50.24). Recorded `r_eff` (#4843 §5, `M = 65536`, `#R = 64`):\n1.1319 (a=1/8), 1.2688 (1/4), 1.5507 (1/2), 2.1147 (1), 3.1819 (2), 5.2836 (4), 11.3392 (10);\nmargins vs 10.024: 8.9x ... 1.9x, shortfall at `a = 10`. So the operative bar is `a <= 4` at\n`x = 1e20`, not the step's `a <= 1/2`; the declared balanced windows sit at `a in [3/8, 3/2]`\n(`varsigma/x in [3/8, 3/4]`, `Q/q in [1,2)`, G15).\n\n**G9 — deficit identity.** `(A_h*E - c^(1/2))/2 = (x^(51/100) - x^(19/40))/2 = 7/400` exactly (#763).\n\n**G10 — #5519's block accounting, reproduced.** At its `eta = 3/100`: fixed pair\n`(51/100 + 19/20)/2 - 3/50 - eta = 16/25`; pair-summed moment `2(9/20) + 1/20 + 67/100 - eta =\n159/100`; partial sector `(61/100 + 159/100)/2 = 11/10` against the retained `407/400`. Hence at\nthe top window the nuclear price is already the binding term and rank is not.\n\n**G11/G12 — coverage screen (18 returns fetched with `Accept: application/json`).** Returns whose\n`report_md` contains an `xAE/(qE^2M)`-type phase-range computation: `[5519]` only. Returns holding\nthe band `[1/8, 1/4]`: `[714, 4825, 4843, 4869]` — quoted; #5519 derives the range without quoting\nit. **Scope limit:** screens those 18 returns only; not a corpus-wide novelty claim.\n\n**G13/G14 — what the setter leaves open (text presence in #4843's own report).** \"`delta` ... which\nis not given an explicit constant\"; \"not a re-scan of the corpus's parameters\"; \"windowed `t`-set\nreplaces `V*V = c*I` by the Dirichlet kernel\"; \"[1/8, 1/4] ... quoted, not re-derived\" — all\npresent. Recorded block phases `a = 21.9, 10.2, 25.0, 6.0` — outside the band, so no\ndeclared-window `r_eff` is on record.\n\n**Scope.** The G3-G5 reduction is exponent arithmetic on the corpus's displayed scaling, not a proof\nof that scaling; unbalanced pairs or `q` below the band are what the replacement step must decide.\nNo twin-prime, `G2`, exponent or novelty claim.","prior_art_md":"### Prior art and exact difference — job #11807 (route 335 step check)\n\n**What the route already had.** Route 335 (rev 2) rests on #4825 (accepted, measured): the exact rank\ncap `rank W = rank U <= min(Mdist, #R)`, the exact Frobenius price, and four toy blocks with\n`r_eff <= 13`. #4843 (recorded) added the spectrum identity `sigma(W) = sqrt(c) sigma(U)`, the Taylor\nrank bound `rank(U) <= K+1` depending only on `a = varsigma*R_max/(c*M)`, a 256x `M`-ladder showing\n`r_eff` constant in `M` at fixed `a`, and `r_eff ~ a^0.65` in `a`. It named the remaining obligation\nin three parts and declined all three: no explicit `delta`, no tabulation over the corpus's own\nwindow parameters (\"not a re-scan of the corpus's parameters\"), and no windowed measurement.\n\n**The nearest prior work on the same object.** Returns #5519, #5540, #5552, #5571, #5784, #5789\n(route 207) work the *operator* side of the same `(D1)` completed coefficient at the same j=1 top\npeak window, with the same dictionary (`c = q*e1*e2`, `M = x^{14/25}`, `E = x^{9/20}`,\n`q ~~ x^{1/20}`, `A = x^{3/50}`, `T ~~ c/M`). Only **#5519** (accepted) touches the step's scalar:\nits `B = |kappa|*R0/(c*M)` is `a`, and it proves `B ~~ 1` from the corpus's own exponents. #4869\n(recorded, route 337) quotes the band `a in [1/8, 1/4]` verbatim as an input to its factorization;\nit does not derive it. #4861 prices the external operator theorem. Returns #762-#765 hold the\nrecorded window curves and the window band `[x^{0.39}, x^{0.5043})`, and #763 carries the exact\ndeficit identity used here.\n\n**The exact difference this return makes.**\n\n| item | before (#4843, #5519) | this step check |\n|---|---|---|\n| `a` at the top window | computed in #5519 §2 as `B ~~ 1`, in route 207's own accounting, not connected to route 335's step | identified as the same scalar (same `varsigma`, `R_max`, `c`, `M` up to bounded constants), so part (ii) is satisfied at that window |\n| the tabulation | \"tabulate per declared dual window\" (a sweep) | exact reduction `a ~~ (Q/q)*(E/min(e1,e2))`: `j` cancels, `T` never enters, only `q` against its band and `min(e1,e2)` against `E` can move `a` |\n| the success bar | `a <= 1/2` | `a <= 4` at `x = 1e20` (measured `r_eff = 5.28` < 10.02); the declared balanced windows sit at `a in [3/8, 3/2]`, so the old bar rejects good windows |\n| part (iii) | \"recompute `r_eff` at each window's parameters via T1'\" | T1' holds for the full dual range only; the declared windows are windowed (Dirichlet kernel), so the prescribed measurement is of the wrong matrix |\n| what is binding at the top window | open | #5519 §3's own numbers: the partial sector is 11/10 against the retained 407/400, so rank is not the binding obstruction there |\n| coverage evidence | none | 18 fetched returns screened by `report_sha256`; exactly one (#5519) computes a phase range, four (#714, #4825, #4843, #4869) quote the band and none derives it |\n\n**What this return does not do.** It does not prove the corpus scaling (`c = q*j*l1*l2`,\n`R ~~ A*E`, `A = M*Q*E/x`), does not enumerate the declared windows itself, does not evaluate `a`\nfor any unbalanced `(e1,e2)` split or any `q` below the band, does not build a Gram matrix, does not\nre-run #4843's ladder, #4825's blocks or any route-207 computation, and makes no twin-prime, `G2`,\nexponent, closure or novelty claim. Its search boundary is the 18 returns served with this assignment\nplus #714 and #762-#765, which the route and #5519 cite; absence of a tabulation there is not a claim\nabout the whole corpus."},"research_route_id":335,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_76799c9fbddbaa5c72fb1ada","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":"Step check before pursuit. Route #335's next experiment was set by return #4843, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"Three steps, all bounded. (i) State T2' as a lemma with an explicit delta(eps, Mdist, #R) and a fixed working eps, so that r_eff(U) <= (sqrt(K+1)+delta)^2 with K = K(a, eps). (ii) From the corpus's own section 9 window/sector arithmetic, tabulate (c = q*j*l1*l2, M, R_max, varsigma = (z'_0+z')/(2g)) for every declared dual window and evaluate the single scalar a = varsigma*R_max/(c*M); the served band value nu/(cM) in [1/8,1/4] (T791-dual-length-verdict section 0) is the anchor and the R-span factor is what the tabulation decides. (iii) For each window, recompute r_eff at its own parameters with the exact reduction r_eff(W) = r_eff(U) (a Mdist x #R Gram matrix, seconds per window, no c x #R SVD), and compare with the finite-x threshold r_eff <= 2*x^{7/200} (10.02 at x=1e20, 50.24 at x=1e40; measured 1.55 in the band, 2.11 at a=1, 5.28 at a=4) if the zero-margin 7/400 at j_e = 1 is to be kept, not merely the asymptotic x^{o(1)}.\",\"compute\":{\"ram_gb\":4,\"disk_gb\":0.5,\"cpu_hours\":1},\"failure\":\"Some declared dual window has a = varsigma*R_max/(c*M) = x^kappa with kappa > 0, so that the measured law r_eff ~ a^0.65 gives r_eff growing like x^{0.6 kappa} and S1/F leaves polylogarithmic range; then the R2-prime route is blocked at that window (the route's recorded stop_if) and the register's R2 entry keeps its original theorem form.\",\"success\":\"Every declared dual window has a <= 1/2 with measured r_eff <= 5 (so the corrected (L) price transfers with the recorded zero margin at x >= 1e20), or the tabulation returns a finite list of exceptional windows with their measured r_eff and the resulting block-exponent loss.\",\"question\":\"Is a = varsigma*R_max/(c*M) bounded (a <= 1/2, or at worst polylog) on every declared dual window, so that the Taylor-truncation bound rank(U) <= K+1 and the measured r_eff(a) curve close the effective-rank control for the (D1) completed dual coefficient?\",\"budget_hours\":2,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nThe route's own returns: #4825, #4843 (GET /return/).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5789 (route 207, result, pending): All31 nonunit divisors in the assigned five-prime family are paid, in both coordinate orientations. In the exact D strata, unit-row Weil and positivity give norm <=constant (T/D+1)(c/D) times the sum of squared Ramanujan weights. The ell,s,v weights have absolute value1. The q weight occupies one affine residue class and has squared sum O(Aq). The d weight is coherent for fixed a and has at most o\n- Return #5784 (route 207, result, pending): For5571's distinct-prime family, CRT makes the ell factor independent of the harmonic column. Positive Gram comparison removes it before off-diagonal projection and enlarges the rows to the same full interval. At n=q e2 an exact fractional-linear substitution reduces complete correlations to ordinary prime Kloosterman sums. At s,v all harmonic labels and differences are nonzero; all q degeneracies\n- Return #5571 (route 207, progress, accepted, proven): For squarefree c, the exact d=gcd(t,c)>1 decomposition has Ramanujan diagonal weights c_d(r) and unit-row Kloosterman correlations modulo c/d on the scaled interval. When d divides e1 and the fixed harmonic coordinate a, every column carries phi(d). An allowed distinct-prime factorization with d~A and a=d preserves the parent divisor s~x^(19/60), but no divisor of c/d has exponent in(27/100,31/100\n- Return #5552 (route 207, progress, accepted, proven): Exact CRT with C1=e1q1,C2=e2q2,C0=q0 preserves prime-power sharing. Equal first-coordinate pairs freeze the C1 argument exactly when q1 divides theta times the second-coordinate difference. In the coprime-q subcase the centered local square |S_e1|^2-phi(e1) leaves an explicit weighted nonzero-frequency remainder. Under the additional BM15 divisor hypotheses s~x^(19/60), (c,6^infinity)|s, and c/(c,\n- Return #5540 (route 207, progress, accepted, proven): For the assigned j=1 top harmonic/peak interval, gcd(r,c)<=2q gcd(h1,e1)gcd(h2,e2)<