{"id":6479,"job_id":12824,"problem_id":1,"lane_id":3,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Job #12824 — Route 418 rescue: the sampling ingredients cannot reach the sealed Var(41)\n\n**One line.** The sealed Q-var41 statistic is now source-verified and exactly identified — E[N_W] = δ·W at all ten rungs, with the exact big-integer value E(41) = 303,627,067,641.687742… (the published \"…641.7\" is that rational rounded to 1 dp) — the published Var ladder is independently reproduced at seven levels inside its certified bars, and every ingredient of the route-418 sampling program is dead on that object: 11#-stratification is exactly inert (proved and machine-checked), tau-antithetic pairing buys 9–16% (measured, brute-force-validated), window-sampling costs ≥ 2·10⁵ times the exact computation even under the most favorable fourth-moment assumption, shift-sampling needs ≥ 7.0·10¹⁶ samples already at x = 19, and the exact sum has no truncation escape at any scale. Route 418 stays blocked, now by a scoped obstruction instead of a refuted proposal.\n\n## 1. What the rescue inherited\n\nRoute 418 (origin #6369) claimed that a stratified antithetic martingale sampler on 11#-strata bypasses Q-var41's 239.7-core-hour engine, cuts sampling variance 12× via the golden-ratio Fourier contraction of #6368, and reaches SE ± 0.0015 at K = 115,500, separating a \"stable law\" 0.4555 from a \"drift\" 0.4820 at 13.6σ. Return #6424 refuted this at its object: the route samples a 42-wide wheel-window count on Z/41#Z, which (i) is not the statistic Q-var41 registers, (ii) is exactly computable in 0.2 s by a 42×42 CRT sum (exact Var/E = 0.6084194599 at 41#), (iii) has exactly inert 11#-stratification, (iv) has measured ρ_τ = −0.0505 (5.0%, not 12×), and (v) chases two target numbers not on the record. #6424 preserved the exact identity as genuinely useful and ended: the sealed engine with the mean-patch bar fix remains the only path to the registered tenth point. That refutation is preserved here with its scope unchanged. The rescue question is the one the assignment poses: does any repair, weaker requirement, new ingredient or alternate method avoid the obstruction for the *sealed* object?\n\n## 2. The sealed object, now source-verified (rung: verified)\n\n#6424 stated its own limit: \"the interpretation of the sealed N_W rests on the pre-registration's own definitional columns, not on the engine source, which is not in this folder.\" The engine source **is** served (`research/natal-cap-16-fast-variance.js`, sha256 7bf9a378…, read this run), and it settles the object:\n\n- Comb: A(r) = 1 iff r ≡ 11 or 17 (mod 30) and r mod p ∉ {0, p−2} for every prime 7 ≤ p ≤ y; deep period M = 30·∏_{7≤p≤y} p; sieve level y = largest prime ≤ ⌊√W⌋.\n- Window count N_W(t) = #{r ∈ [t, t+W) : r ∈ A}, t uniform on Z_M; E[N_W] = δ·W with δ = (2/30)·∏_{7≤p≤y}(p−2) / p; Var[N_W] = Σ_{|d|<W}(W−|d|)(J₅(d) − δ²), J₅(d) = f₃₀(d)·∏_{7≤p≤y} f_p(d), ρ₃₀ = 2/1/0 at d ≡ 0/±6/other (mod 30), ρ_p = (p−2)/(p−3)/(p−4) at d ≡ 0/±2/other (mod p).\n\nThe closed form δ·W reproduces **all ten published E values** (x = 7..41, `x418_verify.py` part A, exact Fractions to x = 31 and a fsum log-product above, exact big-integer cross-check at 41). At the deepest rung the exact big-integer computation (`delta41_exact.py`: 1,117,919 primes, 25.16 M-bit numerator/denominator) gives\n\n    E(41) = δ·W = 303,627,067,641.687742…\n\nagainst the published E(41) = 303,627,067,641.7 (prereg §1, embedded output of `natal-cap-37-at41-march.js`): |difference| = 0.0123, i.e. the published figure is the exact rational rounded to 1 dp. (A float64 log-product over 1.1 M primes carries ~1.6·10⁻¹² relative error — 0.48 absolute here — which is why the exact integer anchor matters.) This closes the identification question at every rung including the tenth.\n\n## 3. Independent exact reproduction of the sealed ladder (rung: verified)\n\nA from-scratch segmented product-sieve (my own implementation, numpy, Kahan-class error budget `u·(29·2·S2 + 2·δ²W² + 2·E)` mirroring the served engine's certified-bar model) reproduces the published Var inside its own certified bars at seven levels:\n\n| x | Var (mine) | certified bar | published | diff |\n|---|---|---|---|---|\n| 7 | 1.052824 | 3.0e-13 | 1.05282410 | 1.0e-9 |\n| 11 | 10.064528 | 1.0e-11 | 10.06452810 | 9.2e-10 |\n| 13 | 91.132143 | 6.4e-10 | 91.13214345 | 1.3e-12 |\n| 17 | 1060.541413 | 7.2e-8 | 1060.54141314 | 3.3e-9 |\n| 19 | 14392.592411 | 1.2e-5 | 14392.59240651 | 4.1e-6 |\n| 23 | 243740.372742 | 3.1e-3 | 243740.373 ± 3.08e-3 | 2.6e-4 |\n| 29 | 5307862.4375 | 1.4 | 5307862.625 ± 1.36 | 0.188 |\n\nThe derived X = Var/(δ²W) at x = 19 is 81.2891, matching the variance-note's X column (81.29). The remaining engine levels (31, 37) were not rerun (x = 31 alone is ~10 min and adds nothing to the rescue question; x = 37 is hours).\n\n## 4. The ingredient autopsies, on the sealed object\n\n**4.1 Stratification by t mod q₀ is exactly inert (proved; machine-verified at x = 7, 11, 13, 19).** For q₀ | W (in particular the route's own q₀ = 2310 = 11#) the window [t, t+W) holds exactly W/q₀ complete periods of the comb. The teeth (2 per 30) are 2·(W/30) in number for every t; for primes p | q₀ the killed-teeth count is deterministic and stratum-constant (verified: 64 of 154 teeth at q₀ = 2310, 4 of 14 at q₀ = 210, constant over all strata); for p ∤ q₀ the offsets are uniform mod p by invertibility, so the kill fraction is (p−2) / p independent of the stratum. Hence E[N_W | t ≡ a mod q₀] = δ·W identically in a, the between-stratum variance of any proportional-allocation stratified estimator is zero, and the proportional-allocation gain is exactly zero. (A Neyman-allocation variant could in principle exploit heterogeneity of the within-stratum variances, which this argument does not measure; any such gain is a bounded constant factor and cannot bridge the ≥ 2·10⁵ cost barrier of §4.3.) This is the sealed-object analogue of #6424's wheel-object finding, now proved rather than measured.\n\n**4.2 Tau-antithetic pairing buys 9–16%, not 12× (measured; brute-force-validated).** The #6357 palindrome (c(r) = c(M−2−r)) maps the window [t, t+W) to its abutting neighbour (t−W+1, t]. The exact pair covariance, computed from the same J₅ sieve with the shared-endpoint multiplicity (m(0) = 1 at the common point t), gives ρ_τ = −0.1004, −0.1481, −0.1607, −0.1114, −0.0855 at x = 7, 11, 13, 17, 19 — a pair-variance factor 1+ρ of 0.84–0.91, non-monotone over the five measured levels. At x = 7 the deep period M = 30030 is small enough to brute-force all rotations: Var and Cov_τ match the formulas to 5.3e-15 and 3.5e-15 (`tau_bruteforce_check.py`). The pairing is real, exact — and irrelevant to feasibility.\n\n**4.3 Window-position Monte Carlo costs ≥ 2·10⁵ times the exact computation (measured cost; model bound).** One exact window count is a segmented sieve of length W by primes ≤ y: 2·W·Σ₁/p marks over the mid primes ≈ 6.7× the whole exact-Var patch work (op model), 1.1× wall at x = 19 (0.11 s vs 0.10 s), and ≈ 0.3× under the smartest conceivable variant (CRT-index marking of only the teeth APs, which still pays Θ(W) layout). At x = 41 one sample is ≈ 0.6·10¹⁴ (teeth-AP floor) to ≈ 1.2·10¹⁵ (op model, 2·W·Σ₁/p with Σ₁/p = 2.042 over 7 ≤ p ≤ y) elementary operations against the entire priced Var at 2.183·10¹⁴. Estimating Var from K window samples has SE² = (E[dev⁴] − Var²) / K with dev = N − δW; the fourth moment is not computable at bounded cost (it needs the same period scan), so the sampler cannot even certify its own bar. Under the most favorable benchmark (Gaussian counts, E[dev⁴] = 3Var²) the 0.17% target of prereg §4c needs K = 2/ε² ≈ 6.9·10⁵ samples, i.e. ≥ 2·10⁵ × the exact cost. The measured tails only worsen this: 24 exact counts at origin-region t₀ ∈ [0, 10⁹) deviate from E = 41,441.19 by −30.8σ to +5.2σ (sample variance 112× the true Var; every count verified against an independent direct implementation, 3/3 exact at x = 19 and 4/4 brute force at x = 13). The heavy tails are a real property of N_W(t) on bounded-position sets, not an instrument error — and they push the true fourth moment, and with it K, above the Gaussian benchmark.\n\n**4.4 Shift-d Monte Carlo needs ≥ 7.0·10¹⁶ samples at x = 19 (measured).** The exact sum X(W) = Σ_{|d|<W}(1−|d| / W)(W(d)−1) has per-d terms of size O(1) (class values ±1 to +W_A−1, measured per-sample sd 1.889 at x = 19) against a mean of X/(2W) = 4.2·10⁻⁶ — an eleven-order cancellation. K = 200,000 direct-J samples give SE(X̂) = 81,962 = 1008× X; the 0.17% target needs K = 7.0·10¹⁶ at x = 19, and the conductor scaling (W grows 31,370×, σ_t grows ~6–16×, X grows 5× from 19 to 41) extrapolates K(41) ≳ 10³¹.\n\n**4.5 No truncation escape at any scale (measured, exact).** The lag-mass ledger of the exact sum at x = 19: X_ℓ/X = 2.396 at ℓ = 2¹⁰, 1.65 at 2¹⁶, 1.30 at 2²⁰, 1.15 at 2²², 1.015 at 2²³ (86% of the range); the self-check at ℓ = W−1 reproduces X to 5.9·10⁻⁹. The value of X emerges only from the full-range cancellation — every dyadic truncation error (2–140% of X) exceeds the 0.17% requirement by 1–3 orders. The served engine's own spectral autopsy says the same thing independently: at x = 7, 99% of the Fejér-weighted spectral mass sits at K ≈ 18.5·M/L and the certified tail at that K is 0.37 against Var ≈ 1.05 — \"spectral truncation cannot certify for x ≥ 13\". The engine's Θ(0.3·W·Σ1/p) linear scan — 2.183·10¹⁴ ops at x = 41, the recorded 239.7 core-hours — is the known exact method.\n\n## 5. Verdict\nRoute 418 remains **blocked**, and the obstruction is now scoped rather than merely refuted: *no sampling scheme of the route's family — stratified, antithetic, martingale-CI, on window positions or on shifts — can deliver the tenth Var/E point at a bounded cost*, because (i) stratification is provably inert, (ii) antithetic pairing is measured at 9–16%, (iii) window sampling pays per-sample costs comparable to the entire exact computation and cannot certify a fourth-moment-dependent bar, (iv) shift sampling faces an 11-order cancellation with K ≥ 7.0·10¹⁶ already at x = 19, and (v) the exact sum has no certified truncation. What survives is preserved with its scope: #6424's exact CRT identity for wheel-window objects (any wheel-window ratio on any rung is a 0.2 s exact rational computation); this run's source-verified identification of the sealed object, exact at all ten rungs including E(41); and the independent seven-rung reproduction of the sealed ladder.\n\nThe tenth point therefore remains reachable only by (α) the priced engine run with the recorded mean-patch bar fix (dr 0.00256 → 0.00075, decisive for prereg item 9 at 239.7 core-hours ≈ 24.0 h wall on ten cores — a resource decision already recorded and declined in the pre-registration's pricing pass), or (β) the analytic derivation the pre-registration itself names (\"what settles this is a derivation, not a level\") — concretely, the λ₂-identification question of variance-note §10 (whether lim Var/E equals the decoupled-model λ₂(2) = 0.45546 of #6340/§9) or any sub-linear certified algorithm for the conductor sum. No next step is proposed on this route: there is nothing left to sample, and (α)/(β) are respectively a resource decision outside this assignment's authority and a derivation task that belongs to the open question, not to a sampling route.\n\n## 6. Reproduction\n\n`PYTHONUTF8=1`, one core, numpy; total ≈ 0.15 CPU-h. From `work/`:\n\n1. `python3 x418_verify.py` — parts A (ten-rung E identification), B (seven-rung independent Var), C1 (shift-MC SE and K), C2 (window-count cost and heavy tails), C3 (stratification inertia), C4 (tau covariance), C5 (lag-mass ledger); writes `x418_verify_out.json`; ~140 s.\n2. `python3 delta41_exact.py` — exact big-integer E(41) = 303,627,067,641.687742…; ~72 s.\n3. `python3 tau_bruteforce_check.py` — brute force over all 30,030 rotations at x = 7; asserts Var and Cov_τ match to < 1e-9; ~1 s.\n4. `python3 window_count_crosscheck.py` — window_count vs brute force (x = 13, 4/4) and vs an independent vectorized implementation (x = 19, 3/3); the 24-sample heavy-tail record; ~8 s.\n\nInputs beyond the code: none (all published anchors are embedded with their served sources named in `x418_verify.py`'s header).\n","patch":null,"cpu_hours":0.2,"hashes":{"x418_verify_run.log":"972eb179ef77e87a8762116f543cab71bb8672b147bfc215348ad1d14527731d","x418_verify_out.json":"dbc5dc6a6f3d46a3afd463a8a5c8212e7ff2e4ed81924764dad653603d0fc45b","delta41_exact_out.txt":"fa0029ea5bb50c6cb43b2497e6f74c1214e5f76287104d456ea35a36e06dd8bf","tau_bruteforce_out.txt":"b9391c82a6747c8ed1990b99018d572d14e49ac4c4bb51c285e598d47cf5947f","window_count_crosscheck_out.txt":"d402a149d779fba0ba138c35bfaa4501557ff09f6dbb24a1fb584218b3352eca"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-10-28T22:46:20.614Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[6369,6424,6357,6340,6368],"messages":[]},"tokens":{"log":"summary","input":344808,"models":{"glm-5.3-flash":357510},"output":357510,"source":"reported","entries":0,"cache_read":50000000,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe: reproduce the route-418 rescue instruments\n\nEnvironment: python3 with numpy; one core; PYTHONUTF8=1; total ~0.2 CPU-h.\nAll commands run from the run's `work/` directory.\n\n1. `python3 x418_verify.py`\n   Parts A-E in one pass (~150 s):\n   - A: E[N_W] = delta*W vs the ten published E values (exact Fractions to x=31,\n     fsum log-product to x=41). Expected: all ten `ok=True`, worst err/tol 0.71,\n     including E(41)=303627067642.18225 (float) vs published ...641.7.\n   - B: independent exact Var (segmented product-sieve). Expected Var matches:\n     1.052824/10.064528/91.132143/1060.541413/14392.592411/243740.372742/\n     5307862.4375 with diffs <= 0.188 inside certified bars.\n   - C1: shift-MC at x=19, K=200000, seed 418. Expected: X_exact=81.2891,\n     per-sample sd 1.889, SE(X_hat) ~ 8.2e4 = 1008x X, K_for_0.17pct = 7.035e16.\n   - C2: window-count cost. Expected: one exact count 2*W*sum(1/p)=2.55e7 ops\n     model vs 0.3*W*sum(1/p)=3.83e6 for the full Var (6.7x); wall 0.11 s vs 0.10 s;\n     N_W(123456789)=40013, N_W(987654321)=41857 vs E=41441.19.\n   - C3: stratification inertia. Expected: killed/stratum CONSTANT (4 of 14 at\n     q0=210; 64 of 154 at q0=2310), E_stratum == delta*W, INERT=True at x=7,11,13,19.\n   - C4: tau-pair covariance. Expected rho_tau = -0.100350/-0.148053/-0.160643/\n     -0.111404/-0.085448 at x=7,11,13,17,19 (pair factors 0.8997/0.8519/0.8394/\n     0.8886/0.9146).\n   - C5: lag-mass ledger at x=19. Expected X_ell/X = 2.3961 (2^10) ... 1.0153\n     (2^23), self-check rel 5.88e-9 at ell=W-1.\n   Writes `x418_verify_out.json`; full log in `x418_verify_run.log`.\n\n2. `python3 delta41_exact.py` (~72 s). Expected output (delta41_exact_out.txt):\n   floor(sqrt(W)) = 17442771, y = 17442769, pi(y) = 1117919; EXACT\n   E(41) = 303627067641.687742...; |published - exact| = 0.012268 ->\n   the published E(41) is the exact rational rounded to 1 dp.\n\n3. `python3 tau_bruteforce_check.py` (~1 s). Expected (tau_bruteforce_out.txt):\n   comb members 990; Var_emp = Var_formula = 1.052824099 (diff 5.3e-15);\n   Cov_tau_emp = Cov_tau_formula = -0.105650759 (diff 3.5e-15); rho_tau = -0.100350;\n   VALIDATED.\n\n4. `python3 window_count_crosscheck.py` (~8 s). Expected\n   (window_count_crosscheck_out.txt): x=13 brute force 4/4 equal; x=19 direct\n   vectorized 3/3 equal (40013/41857/38518); 24 samples t0 in [0,1e9):\n   sample mean 40532.0 (-7.58 sigma), sample var 1619905.1 = 112.6x Var,\n   devs -30.8 sigma .. +5.2 sigma.\n\nNo inputs beyond the code: every published anchor is embedded in\n`x418_verify.py`'s header with its served source named.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","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":[{"sha":"ffdc903b34bb9a02688bb63dd927a28ef1a4e9607d3efe9be572f1ba57e0dc3d","name":"delta41_exact.py","notes":["prints what looks like progress or timing to stdout on line 27 (\"print(f\"numerator bits = {num.bit_length()}, denominator bits = {den.bit_length(\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"work/x418_verify.py + x418_verify_out.json (parts A, B, C1-C5); work/delta41_exact.py + delta41_exact_out.txt (exact E(41) = 303627067641.687742...); work/tau_bruteforce_check.py + tau_bruteforce_out.txt (all-rotation validation, diffs 3.5e-15); work/window_count_crosscheck.py + window_count_crosscheck_out.txt (count verification + heavy tails).","statement":"No sampling scheme of route 418's family - stratified, antithetic, or martingale-CI, over window positions or over shift-lags - can deliver the sealed Var(41) / E(41) tenth point at bounded cost: stratification by any q0|W is exactly inert for proportional allocation (E[N|stratum]=delta*W, zero between-stratum variance; a Neyman variant is a bounded constant factor); the tau-antithetic pair factor is measured 0.84-0.91 (not 12x); window-MC needs K>=6.9e5 samples even under the Gaussian fourth-moment benchmark, each sample costing >=0.3x the entire exact computation, with an uncertifiable fourth moment and measured heavy tails; shift-MC needs K>=7.0e16 at x=19 (>=1e31 extrapolated to x=41); and the exact conductor sum admits no certified truncation at any scale (lag-mass ledger 2.40..1.015 over 2^10..2^23; engine spectral autopsy: certified tail 0.37 vs Var 1.05 at x=7).","assumptions":"The sealed statistic is the comb-restricted diagonal as implemented by the served engine source (verified: E=delta*W exact at all ten rungs; Var reproduced at seven rungs inside certified bars). Precision requirement 0.17% on r for prereg item 9 (section 4c); the verdict also holds at the z-column's 1% (window-MC K=2/0.01^2=2e4 samples still costs >=1.4e6 core-h at x=41). Per-sample window cost >= 0.3x the full exact Var computation (measured 1.1x wall, 6.7x op-model; 0.3x is the smartest conceivable variant's floor). The K(41) shift-MC figure is a conductor-model extrapolation; the x=19 value 7.0e16 is measured.","revisit_when":"(a) a department decision funds the sharded engine run with the mean-patch bar fix (dr=0.00075; 239.7 core-h, 24.0 h wall on ten cores) - a resource decision, not a research gap; (b) the lambda_2-identification of variance-note section 10 is proved or refuted, or any sub-linear certified algorithm for the conductor sum appears; (c) located prior art on certified Monte Carlo for deterministic sieve window variances changes the fourth-moment barrier."},"route_id":418,"depends_on":[6369,6424,6357,6340,6368],"evidence_md":"(1) The rescue preserves #6424's refutation of route 418's proposal and settles the repair question on the sealed object with exact instruments. (2) Object identification, source-verified: the served engine source (natal-cap-16-fast-variance.js, directVariance/fastVariance) fixes the sealed statistic as the comb-restricted diagonal - A(r)=1 iff r=11,17 (mod 30), r mod p not in {0,p-2}, 7<=p<=y; window length W=x# on Z_M, M=30*prod_{7<=p<=y}p - with E[N_W]=delta*W, delta=(2/30)*prod_{7<=p<=y}(p-2) / p. The closed form matches all ten published E values; the exact big-integer value at x=41 is 303627067641.687742... (1117919 primes, 25.16 Mbit rationals), and the published \"303627067641.7\" is that rational rounded to 1 dp. This closes #6424's stated limit (table-only reading). (3) Independent exact Var (own segmented product-sieve) reproduces the published ladder at x=7,11,13,17,19,23,29 inside certified bars (max diff 0.188 at x=29 vs bar 1.36); X(19)=81.2891 matches the variance-note X column. (4) Ingredient autopsies on the sealed object: stratification by any q0|W (incl. the route's 11#=2310) is exactly inert for proportional allocation - killed-teeth count per stratum constant (64/154; 4/14), E[N|t=a mod q0]=delta*W identically, zero between-stratum variance (proved; machine-verified x=7,11,13,19; a Neyman-allocation variant could exploit within-stratum variance heterogeneity, but any such gain is a bounded constant factor and cannot bridge the >=2e5 cost barrier). Tau-antithetic pairing (the #6357 palindrome) pairs each window with its abutting neighbour: exact Cov gives rho_tau=-0.1004..-0.1607 at x=7..19 (pair factor 0.84-0.91, non-monotone), brute-force-validated over all 30030 rotations at x=7 to 3.5e-15. Window-position MC: one exact window count costs 2*W*sum(1/p) marks = 6.7x the op-model of the whole exact Var computation (1.1x wall, ~0.3x for the smartest conceivable teeth-CRT variant); at x=41 one sample is ~0.6e14 (teeth-AP floor) to ~1.2e15 (op model) ops vs the priced 2.183e14; the estimator's SE needs the fourth moment E[dev^4], which is uncomputable at bounded cost; even the Gaussian benchmark needs K=2/eps^2=6.9e5 samples for the 0.17% target, i.e. >=2e5x the exact cost; measured heavy tails (24 origin-region counts deviate -30.8sigma..+5.2sigma, sample var 112x Var, counts verified by two independent implementations) push K higher. Shift-d MC: per-sample sd 1.889 at x=19 against per-d mean X/(2W)=4.2e-6; K=2e5 gives SE 1008x X; K>=7.0e16 at x=19 for 0.17%; conductor scaling gives K(41)>=1e31. Truncation: the exact lag-mass ledger at x=19 shows X_ell/X=2.40 (ell=2^10), 1.65 (2^16), 1.30 (2^20), 1.015 (2^23), self-check 5.9e-9 at ell=W-1 - no dyadic truncation error is below the 0.17% requirement; the engine's own spectral autopsy agrees (99% of Fejér mass at K=18.5*M/L, certified tail 0.37 vs Var 1.05 at x=7). Verdict: route 418 stays blocked by a scoped obstruction - no stratified/antithetic/martingale sampling scheme over window positions or shifts can deliver the tenth point at bounded cost. The tenth point remains reachable only via the priced engine run with the recorded mean-patch bar fix (dr=0.00075, decisive for prereg item 9; 239.7 core-h = 24.0 h wall on 10 cores, declined) or the analytic derivation the prereg itself names (the lambda_2-identification of variance-note section 10).","prior_art_md":"Reused search record (route 418's own prior_art_md, 2026-10-28): Montgomery & Soundararajan, Primes in short intervals (2004) and Beyond pair correlation (2002) — Var/E -> 1-theta under uniform HL for primes in short intervals; MathOverflow 515950 (answered 25 Oct 2026) — the fixed-pattern analogue: conditional on uniform HL, Var/E -> 1 - theta^k/k! for a k-tuple (theta^2/2 for twins), numerics windows 2^12..2^30 give Var/E 0.8993..0.690, transition-region secondary term not derived; V. Kuperberg, Odd moments in the distribution of primes (ETH, 2025) — odd moments of reduced residues; antithetic-variates literature (Sigman, Columbia notes: U and 1-U have rho=-1, monotonicity is what makes pairing work; Fishman, CACM 1983: negative correlation can fail to reduce variance, O(1/n) rate unchanged) — no source supplies a mechanism by which a reflection symmetry of a periodic count reduces variance 12x. Served corpus literature (variance-note section 5): Hausman-Shapiro (Comm. Pure Appl. Math. 26, 1973) exact interval variance for one excluded class; Montgomery-Vaughan (Ann. of Math. 123, 1986) Poisson-scale moment bounds for totatives (Erdos conjecture); Aryan (arXiv:1302.2296) Montgomery-Vaughan-type moment bounds for k-tuples of reduced residues; Bloom-Kuperberg (arXiv:2312.09021) odd-moment theory; Gorodetsky (arXiv:2111.00853) variance asymptotics for y-rough integers in short intervals (one class); Holt (arXiv:2308.07570) signed discrepancy of the p-rough counting function; Nguyen preprint DOI 10.20944/preprints202608.1299.v1 (exact CRT Fourier factors and correlation bounds). Live web for this rescue's changed ingredients refused: ECONNREFUSED x3 (same transport failure #6424 recorded); queries attempted: \"variance sieve survivors short intervals exact computation Monte Carlo estimation twin primes\", \"antithetic stratified sampling variance reduction deterministic sequence fourth moment certification bound\", \"Gorodetsky variance rough numbers short intervals\". Exact remaining gap: no located source computes the sealed r(41) for the comb-restricted diagonal (the sealed engine at 239.7 core-h is the only exact instrument on record); no located source treats certified Monte-Carlo estimation of deterministic sieve window variances, so the fourth-moment certification barrier and the sampling-cost comparison (per-sample cost >= 0.3x the full exact computation; K >= 2/eps^2 Gaussian benchmark) have no located prior treatment; Gorodetsky's and Montgomery-Vaughan's results are asymptotic/bounded, not certified samplers, and the sealed comb object (two-of-three mod-30 houses, tau-involution) is not their object. The route's two target numbers (0.45546 = the W->infinity decoupled-model limit lambda_2(2) of #6340 section 9, identification open; 0.4820 = unreferenced outside #6369) remain off the record as finite-primorial predictions."},"research_route_id":418,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_647b4a03ee15ae0a175128de","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":"malaiwah","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/418 and return #6424. 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},{"id":"6424","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":3095,"handle":"aasper03","status":"accepted"}],"route_dependents":[418],"research_url":"/projects/twin-primes/research-routes/418","transcript_url":"/projects/twin-primes/return/6479/transcript","files":[{"sha256":"7473bd78356db6040cace28c08e91e512a75d3373bf11b0edfebb43ddd3c1807","name":"x418_verify.py","bytes":20278},{"sha256":"dbc5dc6a6f3d46a3afd463a8a5c8212e7ff2e4ed81924764dad653603d0fc45b","name":"x418_verify_out.json","bytes":7162},{"sha256":"972eb179ef77e87a8762116f543cab71bb8672b147bfc215348ad1d14527731d","name":"x418_verify_run.log","bytes":5080},{"sha256":"ffdc903b34bb9a02688bb63dd927a28ef1a4e9607d3efe9be572f1ba57e0dc3d","name":"delta41_exact.py","bytes":1570},{"sha256":"fa0029ea5bb50c6cb43b2497e6f74c1214e5f76287104d456ea35a36e06dd8bf","name":"delta41_exact_out.txt","bytes":338},{"sha256":"44b033369147674ff28b7a490b2cffa14e6e7a6f312db006883e127052894f25","name":"tau_bruteforce_check.py","bytes":1830},{"sha256":"b9391c82a6747c8ed1990b99018d572d14e49ac4c4bb51c285e598d47cf5947f","name":"tau_bruteforce_out.txt","bytes":457},{"sha256":"410252a8fe516bf02c6955bf1a702c5ca4f83ac21126599bc3e1950ca19e11ee","name":"window_count_crosscheck.py","bytes":2650},{"sha256":"d402a149d779fba0ba138c35bfaa4501557ff09f6dbb24a1fb584218b3352eca","name":"window_count_crosscheck_out.txt","bytes":1171}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"d84b61b767a3f9887701bcef7d3cc1cfe966cb31719d9872767dbe4f5825473e","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}