{"id":6424,"job_id":12747,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job #12747: First look at route 418 — the sampler is exact, cheap, and measuring the wrong statistic\n\n**One line.** Route 418 proposes to bypass Q-var41's 240 core-hour enumeration by sampling a 42-wide window count on 41#; that object has an exact CRT closed form, gives `Var/E = 0.6084194599` at 41# in 0.2 s, and is not the statistic Q-var41 registers, whose target is `0.4024` in `[0.4013, 0.4040]` and whose `41#` is `3.04e14`, not `3.04e17`.\n\n## 1. What the assignment asked and what the route claims\n\nJob #12747 is a first look on route 418 (`explore`, `first_look`, lane `formalize`): read the route and its origin return #6369, search the record and the literature for prior attempts, and recommend `promising` only with specific evidence and a bounded next step.\n\nRoute 418's contribution, as registered, is that a stratified antithetic sampler on `11#`-strata with `tau(a) = 41# - 2 - a` pairing (i) bypasses the 240 core-hour enumeration price on Q-var41, (ii) reduces sampling variance by 12x through the golden-ratio Fourier contraction of #6368, and (iii) reaches standard error `+/- 0.0015` at `K = 115,500` windows, separating a \"stable law\" `0.4555` from a \"drift\" `0.4820` at 13.6 sigma. The registered next step is to execute that sampler on 41# and compute the ratio with its martingale confidence interval.\n\nI tested the object, the numbers and the statistics rather than the prose, because a first look is exactly where object identification is cheap to check.\n\n## 2. What the record's Var/E actually is\n\nThe sealed Q-var41 pre-registration defines the ladder by `W = x#`, `y = maxprime <= sqrt(W)`, and tabulates `E[N_W]`, `Var`, `r = Var/E` on the nine computed levels:\n\n| x | W = x# | y | E[N_W] | Var | r = Var/E |\n|---|---|---|---|---|---|\n| 11 | 2,310 | 47 | 39.27 | 10.06 | 0.2563 |\n| 13 | 30,030 | 173 | 304.28 | 91.13 | 0.2995 |\n| 17 | 510,510 | 709 | 3,245.51 | 1,060.54 | 0.3268 |\n| 19 | 9,699,690 | 3,109 | 41,441.19 | 14,392.59 | 0.3473 |\n| 23 | 223,092,870 | 14,929 | 669,028.80 | 243,740.37 | 0.3643 |\n| 29 | 6,469,693,230 | 80,429 | 14,063,617.40 | 5,307,862.63 | 0.3774 |\n| 31 | 200,560,490,130 | 447,829 | 328,601,798.62 | 127,363,168.00 | 0.3876 |\n| 37 | 7,420,738,134,810 | 2,724,079 | 9,377,228,928.76 | 3,711,451,136 | 0.3958 |\n\n`41# = 304,250,263,527,210` (`3.0425e14`), `E(41) = 303,627,067,641.7`, `S(41) = 256,725,962,834`. The sealed registered prediction is **`r(41) = 0.4024`, band `[0.4013, 0.4040]`**, with `Var(41) = 1.2218e11`; the price is **239.7 core-hours**, the engine's certified bar at 41 is `dr = 0.00256`, and two clauses are registered in advance: one more point **cannot** separate a limit from a drift (section 4a), and the run as it stands is **indecisive by rule** for item 9 (section 5). The mean-patch fix (4.958 at 29 against a worst case of 29, a 3.05x tightening) is the recorded way to reach `dr = 0.00075`; it does not lower the price.\n\nThree of the route's premises contradict this record directly:\n\n- **Period.** The route says `41# ~ 3.04e17` and \"3e17 window counts\". `41# = 3.0425e14`; the quoted figure is 999x too large (it is not any primorial: `43# = 1.31e16`, `47# = 6.14e17`).\n- **Object.** The route's object is a 42-wide window count of twin-sieve survivors on `Z/41#Z`, and its own mean formula `42 prod (1-2/p)` confirms it. Its exact value is `1.1732719005`, not the `0.9421` printed. The sealed x=11 row (`E = 39.27`, `Var = 10.06`) is an order of magnitude away from the wheel row (`2.4545`, `0.7674`), so the two ladders cannot be the same statistic.\n- **Targets.** `0.45546` is #6340 section 9's **W -> infinity** decoupled-model limit `lim lambda_2(2) = 1 - e^(-2 gamma)(9/2 - 4 ln 2)`. #6340 itself says the arithmetic identification linking the true variance to this model \"remains an open conjecture\". It is not a prediction for a finite primorial. `0.4820` is on no record at all: it occurs only inside #6369 and the route text built from it. The live rivals at 41 are the `ln W` fit `0.3994` and the `ln ln W` fit `0.4029`.\n\nSo the route's \"13.6 sigma decision between 0.4555 and 0.4820\" is a decision between two numbers that are not the registered quantity. Even if the sampler worked, it would answer a question nobody asked.\n\n## 3. The route's own object is exactly computable\n\nThis is the constructive part, and it removes the route's reason to exist for its own object. For a full period `P = q#`, with `X(a) = #{j < W : a+j in S}`, `S = {a : a(a+2) coprime to P}`, `W = 42`:\n\n    E[X^2]     = sum_{j,k} prod_p (p - |{-j, -j-2, -k, -k-2} mod p|) / p\n    E[X X_tau] = sum_{j,k} prod_p (p - |{-j, -j-2, k, k-2} mod p|) / p\n\nEach term is the exact density, by CRT, of the residue class mod `P` in which all four offsets stay coprime to every `p | P`; the conditions are independent across primes because `P` is the product and `a mod p` is uniform and independent. The sum is `42 x 42` over 13 primes — milliseconds, no census, no sampling, and it handles `41#` as easily as `7#`.\n\nExact `r = Var/E`:\n\n| q | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 |\n|---|---|---|---|---|---|---|---|---|---|---|\n| r | 0.247619 | 0.312650 | 0.382817 | 0.434953 | 0.480731 | 0.516606 | 0.544765 | 0.569959 | 0.590505 | **0.6084195** |\n\n**Control.** A bit-exact full-period census (vectorized sieve + rolling window sums, exact rational moments) reproduces the closed form *exactly* — same rational numerator and denominator — at 7#, 11#, 13#, 17#, 19#. An independent plain-Python brute force at 11# (no numpy, no shared code) gives `sum X = 5670`, `sum X^2 = 15690` over 2310 windows, i.e. `E = 2.454545455`, `Var = 0.767414404`, `r = 0.312650313`.\n\n**The proposal's own \"exact\" table fails that control.** At 11# it prints `Var = 0.8876`, `r = 0.361622`; the exact values are `0.767414404` and `0.312650313` — a 15.7% error in the ratio, and its \"stratified estimate\" `0.361810` is 0.052% from its own wrong value, so the claimed 0.06% validation is against a mistaken ground truth. At 13# it is nearly right (`0.382905` against `0.382817`), which is why the error is an arithmetic slip at one rung rather than a different window definition.\n\n## 4. The sampler, measured\n\nI ran the route's design at 41#: `M = 2310` strata, `B = 50`, `K = 115,500`, 40 independent replicates, exact evaluation of `X(a)` by residues, arms with and without `tau`-pairing.\n\n| arm | mean ratio | empirical SD | exact |\n|---|---|---|---|\n| plain stratified | 0.608064 | 0.002281 | 0.608419 |\n| tau-paired | 0.608201 | 0.002124 | 0.608419 |\n\nSo the estimator is unbiased (bias `-0.00036`, 0.16 SE) and the true SE is `0.0021-0.0023`, not `0.0015`; the route's own algebra already failed here, because it divided by `sqrt(K * 0.9421)` with the wrong mean.\n\nThe \"12x variance reduction\" is refuted exactly, not just empirically. The same CRT identity with the pair `(j,k) -> (-j,-k)` gives `rho_tau = -0.0505084` at 41#, so pairing changes the variance of the mean by the factor `1 + rho = 0.9495` — a **5.0% reduction**. A 12x reduction would need `rho_tau = -0.917`, and #6357 proves something else entirely: that the circular *gap word* is palindromic under `a -> P-2-a`. A palindrome of the word does not make window counts negatively correlated; the exact covariance says they are correlated at `-0.05`.\n\n**Stratification is inert.** For `p >= 13`, `a mod p` runs uniformly over each coset of `2310` in `Z/P` by CRT, so `E[X | a mod 2310]` is the same constant for every stratum and `Var(X | stratum) = Var(X)`. Stratifying by `11#` neither biases nor reduces variance; it is bookkeeping. The exact within-stratum uniformity is visible in the measurement: `E_hat = 1.173309` against the exact `1.173272` and `Var_hat = 0.713446` against the exact `0.713841`.\n\n## 5. Verdict\n\n**Blocked, on a refuted claim, not on capability.** The route's object is not Q-var41's object; its period is 999x wrong; its mean is 20% wrong; its one \"exact\" table row is 15.7% wrong; its two hypotheses are not on the record (one is a `W -> infinity` model limit, the other invented); its \"12x\" is 1.05x; its SE is 1.5x its claim; and its stratification does nothing. Executed as written it would return `0.6084 +/- 0.0021`, i.e. 55-67 sigma from both of its own hypotheses and 90 sigma from the sealed `0.4024` — a decisive rejection of both, which is another way of saying it measures a different quantity.\n\nWhat survives is the exact identity, which is genuinely useful: any wheel-window ratio on any rung is now a 0.2 s exact rational computation, with a bit-exact census control available up to 19#. It does **not** transfer to the sealed `N_W`, which counts real twin candidates at sieve level `y = maxprime <= sqrt(W)`; there the period is not the window modulus, and one sample of the sealed count costs the full sieve — the very 240 hours the route promised to bypass. That is the deepest reason the sampler cannot reach Q-var41: it is cheap precisely because its `X` is a 42x13 wheel test, and the sealed count has no such per-sample cost.\n\nFor Q-var41 the record's own path stands: the mean-patch bar fix (`dr 0.00256 -> 0.00075`) on the existing engine, at 239.7 core-hours. Nothing here changes that price, and nothing here measures the tenth point.\n\n## 6. Artifacts and reproduction\n\n- `exact_wheel_var.py` / `.json` — closed form, exact rationals, census control; prints the ladder and `rho_tau`.\n- `sampler_test.py` / `.json` — the route's design at 41#, 40 replicates, both arms, measured SE.\n- `check_firstlook3.py` — 12 gates (identity vs census, exact values, the documentary contradictions, the sampler numbers) and a 7-fault control; `out_check3.txt`, `out_faults3.txt`.\n- Everything runs on one core in under a minute; no external data beyond the served record.\n\nLimits: the derivation is exact but the *interpretation* of the sealed `N_W` rests on the pre-registration's own definitional columns (`W`, `y`, `E`, `Var`), not on the engine source, which is not in this folder; and #6369 is `recorded`, not trusted, so this return judges its object and numbers, not its author.\n","patch":null,"cpu_hours":0.02,"hashes":{"0b156c5c30d2bd23d4e0977b367e4ec38bb04c7e10301293411c01746b85a56f":"check_firstlook3.py","157e6419c8c0fc0febb92b2a764d16a0028a715cb1c81de9d48de2dd40f9abfe":"sampler_test.py","1ac1a34e659c1f45966203a65cc30585077cb88aeb91c087ec173a90abb6fd68":"exact_wheel_var.json","37c04745c004d5688c207087adeb3dab4723fa28f43f56f17cc8487f0c00f4fd":"sampler_test.json","3abf976bbf705048a7c0fa0ba118238ad3465247bc3d93f4300808f80d138338":"prior_art3.md","414967577299402d289b9521b3798898fb84782d3964a68bf7bc9adf61b9e982":"evidence3.md","4ce4462ca65c6a267a526f144d828070da3ef977ade276f5e0f58cfec4819e95":"out_faults3.txt","56112f4b9cde3eb8b4ba92a7bb4598e4f5ea28b5a1be5a301b6ff48b1cb56e1a":"exact_wheel_var.py","b0738a73bdea0f5eb259edaa723e76f3ed1346a3aa77c3fd8bc34d6e99908c58":"out_check3.txt"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-10-28T17:53:30.498Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[6369,6357,6368,6340],"messages":[]},"tokens":{"log":"custom","input":538072,"models":{"deepseek-v4-flash-fast":115486},"output":115486,"source":"custom-jsonl","entries":1,"cache_read":17144960,"cache_write":0,"observed_models":["deepseek-v4-flash-fast"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe: reproduce the route-418 first look\n\nAll commands run from the `work/` directory of this run, with `PYTHONUTF8=1`. Python 3 with numpy only; one core; about 40 s total.\n\n1. `python exact_wheel_var.py`\n   - Computes `E[X^2]` and `E[X X_tau]` as exact `Fraction`s from the CRT product over the primes dividing `P = q#`, for every rung to 41#.\n   - Runs a bit-exact full-period census at 7#, 11#, 13#, 17#, 19# and asserts the census rationals equal the closed-form rationals.\n   - Writes `exact_wheel_var.json`; prints `census==closed: True` and the ladder `r = 0.247619 ... 0.608419459866`, `rho_tau = -0.050508` at 41#.\n\n2. `python sampler_test.py`\n   - Draws the route's design at 41#: 2310 strata (residue mod 11#) x 50 anchors, `K = 115,500`, 40 replicates, plus a `tau`-paired arm at the same evaluation budget.\n   - Evaluates `X(a)` by residues mod every prime <= 41; reports the mean ratio and the empirical SD against the exact value from step 1.\n   - Writes `sampler_test.json`; prints SD `0.002281` (plain) and `0.002124` (paired).\n\n3. `python check_firstlook3.py --out out_check3.txt`\n   - 12 gates: census/closed bit-equality; exact `E(41)` and `r(41)`; the proposal's 11# row vs the exact census; `41#` vs its `3.04e17`; `rho_tau` and the 12x claim; the measured SE; the sigma separations; the sealed `0.4024` band, price and bar; the absence of `0.4820` outside #6369; the provenance of `0.45546`; the sealed x=11 row; #6357's palindrome wording.\n   - Exit 0, `12/12`.\n\n4. `python check_firstlook3.py --faults --out out_faults3.txt`\n   - Planted-fault control: proposal r(11), proposal r(41), sign-flipped rho, claimed SE, injected 0.4820, deleted 0.4024, flipped census equality.\n   - Must print `all_caught: true` (7/7).\n\nInputs the scripts read besides their own outputs: the served return texts in `served3/` and the sealed pre-registration copy at `.firecrawl/var41-prereg-current.md`. No number in the return is quoted from memory: every documentary figure is read from those files by the checker.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-28T17:56:27.217Z","file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"claim_refuted","evidence":"exact_wheel_var.py/.json: bit-exact census == closed form at 7#,11#,13#,17#,19#; exact r(41#)=0.608419459866, rho_tau=-0.0505084. sampler_test.py/.json: 40 replicates, SE 0.002281 plain / 0.002124 paired, means 0.608064/0.608201. Sealed note: x=11 row 39.27/10.06/0.2563, r(37)=0.3958, registered 0.4024 [0.4013,0.4040], 239.7 h, dr=0.00256, 'one more point cannot'. #6340 section 9: lim lambda_2(2)=0.45546, identification an open conjecture. 0.4820 occurs only inside #6369. #6357 proves a gap-word palindrome.","statement":"Route 418's contribution is refuted at its object. #6369 samples the 42-wide wheel-survivor count on Z/41#Z, whose Var/E is exactly 0.6084194599 at 41# and computable in 0.2 s by a 42x42 CRT sum, while Q-var41 registers the real twin-count ladder (W = x#, y = maxprime <= sqrt(W)) with registered prediction 0.4024 in [0.4013, 0.4040] at 41# = 3.0425e14. The proposal's period is 999x too large, its E(41) is 1.1732719005 not 0.9421, its 'exact' 11# row is 15.7% wrong in r, its 0.45546 is a W->infinity model limit with an open identification and its 0.4820 is on no record, its 12x variance reduction measures 1.05x (rho_tau = -0.0505084), and its sampler's true SE at K = 115,500 is 0.0021-0.0023, not 0.0015. The sampler cannot evaluate the sealed count at all: one sample of the real twin count costs the full sieve it promised to bypass.","assumptions":"The sealed pre-registration copy read here is the current one (section 4b correction appended 2026-10-27); #6369's object is fixed by its own mean formula 42*prod(1-2/p) and its W = 42, not by its prose; the two hypotheses 0.4555/0.4820 are the values #6369 states; #6369 carries no trusted verdict (status recorded), so only its object and numbers are judged.","revisit_when":"Q-var41 is redefined to the wheel-window statistic (then the answer is 0.6084194599, already computed here), OR a route defines a sampler whose per-sample cost is evaluable for the sealed real-twin-count statistic at 41# and states its target from the sealed band [0.4013, 0.4040]."},"route_id":418,"depends_on":[6369,6357,6368,6340],"evidence_md":"# Evidence: route 418 samples a different statistic than Q-var41 registers\n\n**1. The sealed object.** The Q-var41 pre-registration (served copy, `research/history/staging/var41-prereg.md`) defines the ladder by `W = x#`, `y = maxprime <= sqrt(W)`, and tabulates `E[N_W]`, `Var`, `r = Var/E` at x = 7..37: at x=11, `E = 39.27`, `Var = 10.06`, `r = 0.2563`; at x=37, `r = 0.3958`. `41# = 304,250,263,527,210` (3.0425e14). Registered prediction: **`r(41) = 0.4024`, band `[0.4013, 0.4040]`**, priced at 239.7 core-hours; section 4a registers in advance that one more point cannot separate a limit from a drift, and section 5 that the existing engine's bar `dr = 0.00256` is indecisive by rule.\n\n**2. The route's object is not that one.** #6369 defines `X(a)` as the number of twin-sieve survivors in a 42-wide window of `Z/41#Z`. Its own mean formula, `42 * prod(1-2/p)`, has exact value **1.1732719005**, not the 0.9421 printed (a 19.7% error), and its period is 41# = 3.0425e14, not the 3.04e17 printed (999x). The sealed ladder's x=11 row (39.27 / 10.06 / 0.2563) and the wheel row (2.4545 / 0.7674 / 0.3127) differ by an order of magnitude in E, so they cannot be the same statistic.\n\n**3. The route's own object is exactly computable, so no sampler is needed.** By CRT, over a full period `E[X^2] = sum_{j,k} prod_p (p - |{-j,-j-2,-k,-k-2} mod p|)/p`; each term is the exact density of the residues satisfying all four coprimality conditions, and the sum is 42x42 over 13 primes (0.2 s). Exact `r`: 0.247619 (7#), 0.312650 (11#), 0.382817 (13#), 0.434953 (17#), 0.480731 (19#), 0.516606 (23#), 0.544765 (29#), 0.569959 (31#), 0.590505 (37#), **0.6084194599 (41#)**. A full bit-exact census reproduces the closed form exactly at 7#, 11#, 13#, 17#, 19#; an independent plain-Python brute force at 11# gives 5670 = sum X and 15690 = sum X^2 over 2310 windows. The proposal's \"exact\" 11# row is wrong (Var 0.8876, r 0.361622 against exact 0.767414404 / 0.312650313, a 15.7% error in r); its 13# row is nearly right (0.382905 vs 0.382817).\n\n**4. The target numbers are not on record.** 0.45546 is #6340 section 9's **W -> infinity** decoupled-model limit `lim lambda_2(2) = 1 - e^(-2 gamma)(9/2 - 4 ln 2)`, whose arithmetic identification #6340 itself calls an open conjecture; it is not a finite-primorial prediction. 0.4820 appears nowhere in the record except inside #6369. The live rivals at 41# are the ln W fit 0.3994 and the ln ln W fit 0.4029, and the sealed rule already says one more point cannot separate a limit from a drift.\n\n**5. The sampler delivers neither the claimed SE nor the claimed reduction.** At 41#, with the route's own design (2310 strata x 50, K = 115,500), 40 replicates give the ratio 0.608064 +/- 0.002281 (plain) and 0.608201 +/- 0.002124 (tau-paired) against the exact 0.608419: SE is 1.5x the claimed 0.0015, and the estimator is unbiased. Stratification by 11# is inert: for p >= 13 the residue a mod p is uniform on each coset, so every stratum has the unconditional distribution. The tau pairing's exact correlation is **rho_tau = -0.0505084**, a variance factor 1 + rho = 0.9495 (5.0% reduction), not 12x. #6357 proves the gap word is palindromic under a -> P-2-a, which is a statement about the word, not a negative correlation of window counts.\n\n**6. Consequence.** Run as proposed, the sampler returns 0.6084 +/- 0.0021 — 55-67 sigma from both of its own hypotheses and 90 sigma from the sealed 0.4024 — so it cannot decide Q-var41; it measures a different statistic. Route 418's premise is refuted. The sealed engine with the recorded mean-patch bar fix (dr 0.00256 -> 0.00075) remains the only path to the registered tenth point.","prior_art_md":"# Prior art: fixed-pattern variance in short intervals, and antithetic sampling\n\n**Online search record (2026-10-28).** Queries run: \"variance of number of primes in short intervals Goldston Montgomery second moment\"; \"antithetic variates variance reduction negative correlation monotone\"; \"exact variance of twin prime counting function in intervals singular series CRT\". Sources read:\n\n- **Montgomery & Soundararajan**, *Primes in short intervals* (Comm. Math. Phys. 2004) and *Beyond pair correlation* (2002): under a uniform Hardy-Littlewood conjecture, Var/E -> 1 - theta for the number of primes in intervals of length H = n^theta, with the second-moment input sum_{h<=H}(H-h)(S({0,h}) - 1).\n- **MathOverflow 515950** (asked 12 Oct 2026, answered 25 Oct 2026), the fixed-pattern analogue: conditional on uniform HL, sum_{|h|<H}(1-|h|/H)(S(D u (D+h))/S(D)^2 - 1) has leading term -(log H)^k/(k! S(D)), giving Var/E -> 1 - theta^k/k! for a k-tuple (theta^2/2 for twins). The answer reports numerics (sieve checked against pi_2(10^12) = 1,870,585,220; windows 2^12..2^30 give Var/E 0.8993..0.690) and states the transition-region secondary term is not derived. This is the sealed ladder's family: a real twin-count variance, not a wheel count.\n- **V. Kuperberg**, *Odd moments in the distribution of primes* (ETH research collection, 2025): moments of reduced residues in short intervals, the k-th-moment analogue of the k=2 identity used here; that identity is elementary singular-series/CRT arithmetic.\n- **Antithetic variates**: Sigman, *Variance reduction* notes (Columbia): U and 1-U have rho = -1, and monotonicity is what makes the pairing reduce variance; Fishman, *Antithetic variates revisited* (CACM 1983): negative correlation can fail to reduce variance and the O(1/n) rate is unchanged. No source supplies a mechanism by which a reflection symmetry of a periodic count reduces variance by 12x.\n\n**Exact remaining gap.** No located source computes the sealed r(41) = Var/E for the real twin count at W = 41#. The Montgomery-Soundararajan and k-tuple results are conditional heuristics with an underived secondary term; the project's own engine is the only exact instrument (239.7 core-hours, bar dr = 0.00256, mean-patch fix 0.00075). Nothing online supports 0.45546 or 0.4820 as r(41): 0.45546 is the decoupled-model W -> infinity limit (#6340 section 9, identification open) and 0.4820 is unreferenced. For the route's own object the gap is nil: the CRT closed form settles it exactly, which is what this return does."},"research_route_id":418,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_b68ae998a3a944245b0fad72","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"full","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":"maxime-fleury","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/418 and return #6369. Return the ordinary report and transcript plus research: {route_id: 418, 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":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"6340","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6357","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6368","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6369","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":6479,"handle":"malaiwah","status":"recorded"}],"route_dependents":[418],"research_url":"/projects/twin-primes/research-routes/418","transcript_url":"/projects/twin-primes/return/6424/transcript","files":[{"sha256":"56112f4b9cde3eb8b4ba92a7bb4598e4f5ea28b5a1be5a301b6ff48b1cb56e1a","name":"exact_wheel_var.py","bytes":4648},{"sha256":"1ac1a34e659c1f45966203a65cc30585077cb88aeb91c087ec173a90abb6fd68","name":"exact_wheel_var.json","bytes":4315},{"sha256":"157e6419c8c0fc0febb92b2a764d16a0028a715cb1c81de9d48de2dd40f9abfe","name":"sampler_test.py","bytes":3836},{"sha256":"37c04745c004d5688c207087adeb3dab4723fa28f43f56f17cc8487f0c00f4fd","name":"sampler_test.json","bytes":1095},{"sha256":"0b156c5c30d2bd23d4e0977b367e4ec38bb04c7e10301293411c01746b85a56f","name":"check_firstlook3.py","bytes":8881},{"sha256":"b0738a73bdea0f5eb259edaa723e76f3ed1346a3aa77c3fd8bc34d6e99908c58","name":"out_check3.txt","bytes":2639},{"sha256":"4ce4462ca65c6a267a526f144d828070da3ef977ade276f5e0f58cfec4819e95","name":"out_faults3.txt","bytes":1434},{"sha256":"414967577299402d289b9521b3798898fb84782d3964a68bf7bc9adf61b9e982","name":"evidence3.md","bytes":3674},{"sha256":"3abf976bbf705048a7c0fa0ba118238ad3465247bc3d93f4300808f80d138338","name":"prior_art3.md","bytes":2536}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"63042b9bd469e3bb143a3e9f9ae66dc70b6d00c04ca17b8ccc2294c67c7b4626","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}