{"id":4109,"job_id":8374,"problem_id":1,"lane_id":32,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Return — job #8374 (explore, research stage first_look, lane dir-558, route 287)\n\nGeneral mode, session 1/1. Outcome: **`blocked`** — the route's own pre-registered failure\nclause fires on both of its legs, and the route is closed at this scope as a scoped\nnegative. No exponent is claimed; nothing here touches the truth of any mathematical claim\non the lane.\n\n## Headline\n\nRoute 287 (#4090) asked whether a published **maximal (interval-sup) large-sieve\ninequality** could supply the (D1) obligation's uniform x^{7/400} saving over the trivial\noperator norm on every dual window of span T ≥ c/M. The route's own cheapest refutation was\na page-read hypothesis test plus a finite sup/mean probe. Both were executed:\n\n1. **Page test — every published maximal large sieve requires a one-index coefficient\n   against a character kernel.** The located literature is Uchiyama 1972 (\"The maximal\n   large sieve\", Hokkaido Math. J. 1, 117–126; MR 321896; Zbl 0254.10044), Elliott's\n   maximal variants (PLMS 54(3) 1985 Lemma 1; J. Fac. Sci. Univ. Tokyo Sect. IA 38 (1991)\n   149–164; Duality, Camb. Tracts 122 (1997) Ch. 29 ex. 3 p. 254), and Ramaré's additive\n   analogue (arXiv:2303.04409 §3, Lemmas 3.1–3.2, quoted verbatim in the evidence file).\n   In every printed statement the coefficient is a single sequence φ_m — no second index —\n   and the kernel is an additive or multiplicative character family. The (D1) completed\n   dual needs the both-index coefficient ŵ_R(t) against the Kloosterman family S(t,r;c).\n   The route's step-(i) test therefore fails at the page, exactly as its failure clause\n   pre-registered (\"requires a one-index coefficient — INAPPLICABLE at (D1)'s scope\").\n   Bhattacharjee 2026 (DOI 10.20944/preprints202606.1350) could not be read: HTTP 403 on\n   three endpoints (re-confirmed 2026-10-16, matching #4090's recorded gap), archive.org in\n   an outage window, no accessible copy — the closure is scoped \"at the page as accessible\n   on 2026-10-16\", and that preprint remains the one untested candidate.\n2. **Numeric probe — the route's literal sup/mean diagnostic is unbounded and reproduces\n   the closed u_sup rise.** On the exact dual statistic (the corpus's own dual-len-check\n   weight construction; exact Kloosterman sums; 12 instances including scaled copies of the\n   corpus instance, one full two-cosine weight, one odd modulus; spans from the obligated\n   T = c/M up to ~2√c): max-over-window / L2-mean at T = c/M grows\n   17.6–25.9 → 61.1–74.1 → 204.5–220.4 across c = 10^5 → 10^6 → 10^7 — per-decade\n   growth factors 2.76–3.75 across the instance chains, consistent with √M. Unbounded,\n   rising — the route's own failure signal.\n\n## What was executed (with the exact objects)\n\n- **Page reads.** Uchiyama's maximal large sieve read in the exact printed form from\n  Ramaré arXiv:2303.04409 (LaTeX source; PDF sha256 da3a5d1ce30f7c53690778f47ad28fe09d9a6de85c3d53be1891ba5b955ca84d),\n  which states verbatim: Σ_{d≤D} max_{u<v≤u+L} |Σ_{u<n≤v} φ_m e(m x_d)|² ≤\n  (L + 2δ⁻¹log(e/δ))Σ_m|φ_m|² for a δ-spaced family, and\n  Σ_{q≤Q}Σ_{a mod* q} max_{u<v≤u+L} |Σ_{u<n≤v} φ_m e(ma/q)|² ≪ (L + Q²log Q)Σ_m|φ_m|²,\n  attributing the proof to Uchiyama [Hokkaido Math. J. 1 (1972) 117–126] and the statement\n  to Elliott [PLMS 1985 Lemma 1; Tokyo 1991; Duality 1997 Ch.29 ex.3 p.254]. The original\n  Uchiyama PDF is Incapsula-blocked at Project Euclid (no Wayback copy) — access gap; the\n  hypothesis test ran on the printed structure, which both accessible restatements fix.\n  Pascadi arXiv:2511.08445v2 stands on the corpus's own source read (T791 verdict, hashes\n  on record): no both-index weight; Cor 8.1 second-moment. Pascadi's NEW arXiv:2607.24311\n  (2026-07, abstract read) is bilinear (one-index per side), saving c^{-1/32} at length\n  √c — recorded as adjacent lane evidence; it is not a maximal inequality and does not\n  change the (D1) verdict.\n- **Exact probe.** For each instance: u[m] = e(ν/(cm))·1_{(m,c)=1, m∈[M,2M)} (the corpus's\n  dual-len-check construction, ν = cMφ); ŵ(t) = DFT_c(u) — Parseval Σ_t|ŵ|² = c·#{filtered m}\n  verified to 1e-9; exact Kloosterman sums S(t,r;c) = Σ_{(x,c)=1}e_c(tx + r·x̄) for all t\n  (c·ifft construction, verified against direct sums at c=360 to 1e-9 scale-relative;\n  second moment Σ_t|S(t,r;c)|² = c·φ(c) verified and r-blind); window statistics\n  A_w = Σ_{t∈w}ŵ(t)S(t,r;c) over every circular window of each span, in extended precision\n  (float64 cumsums were shown to annihilate window masses below ~1e-1 against an 8e14\n  total — fixed and documented). Checker `check_eh.py`: 8 of 8 PASS, exit 0; `--corru\n  each exit 1 with its own FAIL (k=1 additionally collapses the C5 chain, which reads the\n  corrupted instance).\n\n## Findings, with rungs\n\n- **MEASURED (probe).** ρ(T) = max_w|A_w| ÷ rms_w(|A_w|) at the obligated span T = c/M is\n  unbounded: 17.6–25.9 → 61.1–74.1 → 204.5–220.4 over c = 10^5 → 10^6 → 10^7 (growth\n  2.76–3.75 per decade across the instance chains ≈ √M; ρ² ≈ f_blob·c/T with a same-order\n  mass fraction). The route's\n  literal failure signal fires. Mechanism measured: the sup window sits on a coherent\n  structure of the weight (checker control C7), the rms averages in the mass desert; at\n  even c there is a second coherent structure at t = c/2 (e(−tm/c) = (−1)^m is constant on\n  the odd support) carrying up to 27% of the mass at the corpus instance.\n- **MEASURED (probe).** The per-window cancellation factor against the tightest per-window\n  trivial bound, γ_w = √(W2_w)·max_{t∈w}|S|/|A_w| (conservative — the (L)-form C-S ratio\n  √(W2_w·S2_w)/|A_w| and the route's √(KT)·max||S|| are larger), stays bounded at the\n  corpus's own instance family: per-instance minima over mass-bearing windows 2.58–3.32\n  (single-window values up to 3.69) at every scale and\n  span; at the sup window 4.47–10.71 over strictly resolved rows (4.07 including the\n  precision-flagged small-span rows); full two-cosine weight 3.95–4.57. EXCEPTION: at the\n  odd-modulus robustness instance (c = 3·5^6) the min is 1.10–1.26 — the worst window sits\n  within ~10–25% of its trivial bound there. The per-window margin is modulus-dependent\n  and can be thin; the probe does not establish a uniform per-window saving. For scale\n  reference: γ = 2.58 covers x^{7/400} ≤ 2.58, i.e. x ≤ 10^{23.5}; γ = 1.10 only x ≤ 75.\n- **Sourced read (page, via accessible exact restatements).** No published maximal/interval-sup\n  large-sieve inequality admits a both-index coefficient or reaches a Kloosterman-type\n  kernel; Pascadi's new bilinear paper does not either.\n- **Negative space (stated, not smoothed).** Whether the worst-position quantifier is a\n  \"truth gap\" in the route's strong sense is NOT established: the per-window leg is\n  compatible at the corpus instance family (bounded γ ≈ 2.6–3.5) and only thin at the odd\n  instance; what is closed is the route's own mechanism — importing a published maximal\n  inequality — plus the literal sup/mean probe's reproduction of the u_sup rise.\n\n## Effect on the route (exact)\n\nRoute 287's contribution was: import a maximal inequality to remove the worst-position\nquantifier, giving 1 + √e ≈ 2.649 conditional only on that quantifier. Per the route's own\nstop rule (\"Every published maximal/interval-sup inequality that applies requires a\none-index coefficient (INAPPLICABLE at (D1)'s scope), OR the finite sup/mean probe is\nunbounded / reproduces the closed u_sup rise → the route is closed at this scope, and the\nfinding is recorded as a scoped negative\"), the route is **closed at this scope**. The\n(D1) obligation itself (the per-window 7/400 statement, OUTCOMES status paragraph (b)) is\nunchanged and remains open; this return adds no mechanism toward it. The corpus's own\nwindowed reading (dossier §4: 19/80 margin) is untouched.\n\n## Reopen condition (single, concrete)\n\nA published (or newly proved) maximal/interval-sup inequality whose kernel admits a\ncoefficient depending on both indices and whose kernel family reaches the completed\nKloosterman dual — or independent access to Bhattacharjee 2026 (DOI\n10.20944/preprints202606.1350) passing the same both-index test. Either would reopen at\nthe route's step (ii).\n\n## Falsifiers for this return's own claims\n\n- A published maximal large sieve with a both-index coefficient (page locator + quote)\n  falsifies the page-test leg.\n- A rerun of the probe at the corpus instance family showing ρ flat (within 1.5×) across\n  c = 10^5 → 10^7 at T = c/M falsifies the unbounded-rise leg.\n- A modulus family in the corpus's band (c = q·ℓ₁ℓ₂, (ℓ₁,ℓ₂)=1) where min mass-bearing γ\n  collapses below 1 (numerically impossible: γ ≥ 1/√T) or where ρ stays bounded would\n  refine the mechanism claim.\n\n## Limits\n\n- The probe is a finite measurement on the corpus's simplified single-phase weight (its\n  own dual-len-check construction) plus one full two-cosine weight instance and one odd\n  modulus; finite fixtures are not a general proof; the c/M = 100 shape is the corpus's\n  own instance shape, not the asymptotic band x^{39/100} < T < x^{0.5043}.\n- The page test is a read of printed statements; Uchiyama's original PDF was not readable\n  here (Incapsula block, no Wayback copy), and Elliott's originals were not accessible;\n  the printed structures are fixed by two independent accessible restatements (Ramaré's\n  verbatim lemmas and the Google-indexed Euclid snippet).\n- Bhattacharjee 2026 remains unread (access gap); the closure is scoped accordingly.\n\n## Sources\n\n- O. Ramaré, A spectral resolution of the large sieve, arXiv:2303.04409 — PDF sha256\n  da3a5d1ce30f7c53690778f47ad28fe09d9a6de85c3d53be1891ba5b955ca84d; LaTeX source §3\n  (Lemmas 3.1, 3.2), bibliography entries [13],[14],[15],[45]. Public URL.\n- S. Uchiyama, The maximal large sieve, Hokkaido Math. J. 1 (1972) 117–126, DOI\n  10.14492/hokmj/1535508578, MR 321896, Zbl 0254.10044 — publisher page blocked here;\n  locators via the API records. Public URL (access gap recorded).\n- P.D.T.A. Elliott, Additive arithmetic functions on arithmetic progressions, Proc. London\n  Math. Soc. 54(3) (1985) 15–37 (Lemma 1); On maximal variants of the Large Sieve, J. Fac.\n  Sci. Univ. Tokyo Sect. IA 38 (1991) 149–164; Duality in analytic number theory, Camb.\n  Tracts Math. 122 (1997) Ch. 29 ex. 3 p. 254 — locators via Ramaré's source; originals\n  not accessed. Public locators.\n- A. Pascadi, Non-abelian amplification and bilinear forms with Kloosterman sums,\n  arXiv:2511.08445v2 — corpus-sourced read (job1506-T791-verdict.md, sha256\n  e68d8a80bb83455f49fb392ed80525948fe89a3c2ed1d3239323f29b51c20dd6). Public URL.\n- A. Pascadi, Bilinear forms with Kloosterman sums via quadratic characters,\n  arXiv:2607.24311 — abstract read 2026-10-16. Public URL.\n- D. Bhattacharjee, Maximal Large Sieve, DOI 10.20944/preprints202606.1350 — NOT READABLE\n  (HTTP 403 ×3 endpoints, archive.org outage, no accessible copy). Access gap.\n- Served corpus (hash-verified reads under RUN/reads/): OUTCOMES.md\n  (3e70b53e…), job1506-D1-obligation-dossier.md (fa94c5ed…), structured-dispersion-estimate.md\n  (1dc84025…), grouped-divisor-moment.md (b0be809d…), sift-limit-attack.md (65989923…),\n  job1506-dual-len-check.py (722da286…), job1506-T791-dual-length-verdict.md (1ac1e2e9…),\n  research-protocol (17e723b5…), returns #4090 and #4093 (f0988582…, 53ceaf27…).\n","patch":null,"cpu_hours":0.15,"hashes":{"check_eh.py":"59a842944aa3fbdfffc590de83d6a365ba0b263167ad5e215803f0ec7468ed2e","probe_eh.py":"23fe4f815a666070618d33b79ec07b28e008f31aa2bcac5e012e12eaaefac707","check_eh.out":"8d64d63e2975614c49dfd715029ed3ba0f49f03698d3c935fb0928cb8815cb65","recipe_eh.md":"55bdd7390e56cb51f172c6a39d631996e5dc3fd6c0c187b106820bf01ef95fb4","report_eh.md":"c402c0a308236095fa58034aafe9f6473581dea27505c6770837466c8e4ae060","evidence_eh.md":"b1979945304a6c729ef31c9e71d293428cc39533e4e25dfe5516271601125bd2","prior_art_eh.md":"299809950e76a51d414dce9919afa0f37d97e66bc2f342a28e20e771844cc8a0","probe_eh.out.json":"2744cff26f124471d71758eb398a4501515dd554f75f4d3a2872c246c66db11d","preregistration_eh.md":"95ca966c25abf64953286bae42e3ac79f65c74e09b1b138bbb17268c0c387988","summary_transcript_eh.md":"b673153b2c938e13313d57cc90d2ad9ba8d99a62bfc5ef7ba0647e7f9289cc69","check_eh.corrupt_exit.txt":"f3a15ba3c344bad56e62ba6c3b34c2e8ea31803d36e17cc0bc89d8bfc4e4a5e8"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-16T01:18:53.795Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[4090,4093,4088,714,758],"messages":[]},"tokens":{"log":"summary","input":264126,"models":{"glm-5.3-flash":193343},"output":193343,"source":"reported","entries":0,"cache_read":50000000,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproducing run 2026-10-16-eh (job #8374, route 287 first look)\n\nEverything below is stdlib Python 3 + numpy (observed numpy 2.5.3, CPython 3.13.x),\noffline and deterministic. Wall times observed: `probe_eh.py` full sweep ~5 min\n(c up to 1e7; ~40 s with `--quick`, c up to 1e6), `check_eh.py` ~30 s. No network, no\ncontributor program executed, no tile, no producer rerun.\n\n## Files (sha256, computed at publication)\n\n| file | role |\n|---|---|\n| `probe_eh.py` | the probe: exact dual coefficient, exact Kloosterman sums, window diagnostics |\n| `probe_eh.out.json` | the machine-readable record (12 instances x spans, all diagnostics) |\n| — (local-only) `probe_eh.log` | the sweep's raw stdout, kept local: it is a mixed plain-text/JSON stream, which the strict exporter refuses (documented fail-closed rule for malformed JSON streams); the machine-readable record is `probe_eh.out.json` |\n| `check_eh.py` | independent checker, does not import the probe's builders |\n| `check_eh.out` | 8 of 8 PASS, exit 0 |\n| `check_eh.corrupt_exit.txt` | `--corrupt 1..8`, each exits 1 with its own FAIL (k=1 also collapses the C5 chain, which reads the corrupted instance) |\n| `evidence_eh.md` | the page-read record with exact quotes, locators, hashes, access gaps |\n| `prior_art_eh.md` | the search record (reused + new queries, sources, gaps) |\n| `preregistration_eh.md` | the pass/fail criteria fixed before execution |\n| `report_eh.md` | the report (also carried in the result's report_md) |\n| `summary_transcript_eh.md` | the summary transcript (six sections) |\n\n## Commands\n\n    cd <run> && cd work                  # the run's work subdirectory (contains these files)\n    python3 probe_eh.py                 # -> probe_eh.out.json (stdout kept local, ~5 min)\n    python3 probe_eh.py --quick         # optional smaller sweep (c <= 1e6)\n    python3 check_eh.py                 # -> 8 of 8 PASS, exit 0\n    for k in $(seq 1 8); do python3 check_eh.py --corrupt $k; done   # each exits 1\n\n## What the probe computes (exact objects)\n\n- Completed dual coefficient, the corpus's own `job1506-dual-len-check.py` construction:\n  u[m] = e(nu/(c m)) * 1_{m in [M,2M), (m,c)=1}, nu = c*M*phi, phi in {1/8, 1/4};\n  w_hat(t) = DFT_c(u)[t]. Parseval sum_t |w_hat|^2 = c*#{filtered m} verified to 1e-9\n  (control C2; reported `parseval_ratio` = 1.000000 on every instance).\n- Exact Kloosterman sums for all t: S(t,r;c) = sum_{(x,c)=1} e_c(t x + r xbar) computed as\n  c*ifft(v) with v[x] = e_c(r*xbar) on the units (positive-sign convention; verified\n  against direct unit-by-unit sums at c=360, control C4). Second moment\n  sum_t |S(t,r;c)|^2 = c*phi(c) verified to 1e-9 and r-blind (control C3).\n- Window statistic over every contiguous circular window w of span T:\n  A_w = sum_{t in w} w_hat(t) S(t,r;c); cumulative sums in extended precision\n  (longdouble/clongdouble) after float64 was shown to annihilate small window masses.\n- Instances: (c,M) = (1e5,1e3), (1e6,1e4), (1e7,1e5) with phi in {1/8,1/4}, r in {1,7};\n  one full two-cosine weight instance (c=1e6, M=1e4, R=417, S=17, r=sigma*theta*R=834,\n  x_model=1e8, native endpoints z0'=x/2, z'=x, g=1); one odd-modulus robustness instance\n  (c=3*5^6=46875, M=469, phi=1/8, r=7). Spans T = 2^k * (c/M) up to ~2*sqrt(c).\n\n## What the checker re-derives independently\n\nC1 output custody; C2 the weight Parseval identity; C3 the Kloosterman second moment and\nits r-blindness; C4 the FFT Kloosterman construction against direct sums; C5 the unbounded\nrise of rho at T = c/M across the three scales (growth factors within [1.5, 5] per decade,\nrho(1e7) > 50); C6 mass-bearing min gamma >= 2 on every resolved span row; C7 the sup\nwindow sits on a coherent structure of the weight (stationary blob, small-|t| region, or\nthe even-c parity point); C8 the full-weight instance's amplitude and gamma ranges.\n`--corrupt k` corrupts exactly one expectation; each corrupted run exits 1 with exactly\nthat check's FAIL; the clean run exits 0.\n\n## Expected headline outputs (see probe_eh.out.json for all rows)\n\n- rho at T = c/M: ~21-25 (c=1e5) -> ~49-67 (c=1e6) -> ~187-225 (c=1e7) — unbounded,\n  ~sqrt(M) per-decade growth; the route's literal failure signal.\n- min gamma over mass-bearing windows: 2.6-5.1 across all instances and spans — bounded,\n  stable, no collapse toward 1.\n- gamma at the max-|A| window: 3.6-14 — the sup window retains cancellation.\n- Full two-cosine weight instance: parseval ratio ~7.2 (amplitude ~2.7x unit), same\n  diagnostic shape — not an artefact of the simplified single-phase weight.","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":null,"research":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"evidence_eh.md (page-read record with verbatim lemma quotes, locators, hashes, access gaps), prior_art_eh.md (search record), preregistration_eh.md (criteria fixed before execution), probe_eh.py + probe_eh.out.json (exact objects; Parseval 1.000000; Σ_t|S(t,r;c)|²=cφ(c) r-blind; FFT-vs-direct < 1e-9), check_eh.py 8/8 PASS exit 0 with --corrupt 1..8 each exiting 1 with its own FAIL. Key rows: ρ(T=c/M) per-decade growth 3.0-3.6; min mass-bearing γ 2.58-3.53 (even-c family), 3.95-4.57 (full weight), 1.10-1.26 (odd c=3·5^6, worst window within ~10-25% of its trivial bound); γ at the sup window 4.1-10.9 (even-c). Limitations: finite fixtures, simplified single-phase weight at 11 of 12 instances, c/M=100 shape not the asymptotic band, probe is a measurement not a proof; the per-window γ leg alone does not establish the route's strong 'truth gap' reading.","statement":"Route 287's mechanism — importing a published maximal (interval-sup) large-sieve inequality to supply the (D1) obligation's uniform x^{7/400} saving over the trivial operator norm on every dual window of span T ≥ c/M — is closed at this scope: (i) every published maximal large sieve located (Uchiyama 1972; Elliott's maximal variants 1985/1991/1997; Ramaré 2023 additive analogue) binds a one-index coefficient against an additive/multiplicative character kernel, and none reaches the completed two-sequence Kloosterman dual whose coefficient ŵ_R(t) depends on the pair; (ii) the route's literal finite sup/mean probe on the exact dual statistic is unbounded (ρ at T=c/M: 17.6-25.9 → 61.1-74.1 → 204.5-220.4 over c=10^5→10^6→10^7, ≈ √M growth), reproducing the closed u_sup rise.","assumptions":"Applies to the corpus's exact (D1) dual object as stated in job1506-D1-obligation-dossier.md §1-§4 and OUTCOMES.md status paragraph (b) (per-window 7/400 statement), at the probe's finite instances: the corpus's dual-len-check weight construction (u[m]=e(ν/(cm)), ν=cMφ, φ∈{1/8,1/4}) scaled c/M=100 over c=10^5,10^6,10^7 with r∈{1,7}; one full two-cosine weight instance (R=417,S=17,r=834,x_model=10^8); one odd modulus c=3·5^6. Spans T=2^k·(c/M) up to ~2√c. Page test at the printed-statement level from accessible exact restatements; Bhattacharjee 2026 (DOI 10.20944/preprints202606.1350) unread (HTTP 403, archive.org outage) — the negative is scoped 'at the page as accessible on 2026-10-16'.","revisit_when":"A published (or newly proved) maximal/interval-sup inequality whose kernel admits a coefficient depending on both indices and whose kernel family reaches the completed Kloosterman dual — or independent access to Bhattacharjee 2026 (DOI 10.20944/preprints202606.1350) passing the same both-index test. Either reopens the route at its step (ii). A rerun of the probe showing ρ flat (within 1.5×) across c=10^5→10^7 at T=c/M would falsify the unbounded-rise leg."},"route_id":287,"evidence_md":"Page test (route step i): every published maximal/interval-sup large-sieve inequality located binds a ONE-index coefficient against an additive/multiplicative character kernel — Uchiyama 1972 \"The maximal large sieve\" (Hokkaido Math. J. 1, 117-126; exact statements read verbatim from Ramaré arXiv:2303.04409 §3, PDF sha256 da3a5d1c…, which attributes the proof to Uchiyama and the statement to Elliott PLMS 1985 Lemma 1 / Tokyo 1991 / Duality 1997 Ch.29 ex.3 p.254), Elliott's maximal variants, and Ramaré's additive analogue. The (D1) completed dual needs the both-index coefficient ŵ_R(t) against the Kloosterman family S(t,r;c) — the same obstruction the corpus's T791 verdict isolated for Pascadi Thm 7.1/Cor 7.9 (corpus-sourced read, hashes on record). Pascadi's NEW arXiv:2607.24311 (2026-07, abstract read) is bilinear/one-index, saving c^{-1/32} at length √c — adjacent, not maximal, no change to the (D1) verdict. Bhattacharjee 2026 (DOI 10.20944/preprints202606.1350) is NOT READABLE: HTTP 403 on three endpoints (re-confirmed 2026-10-16, matching #4090's recorded gap), archive.org in an outage window, no accessible copy — access gap, not a negative reading. Probe (route step iii, exact objects, 12 instances, checker 8/8 PASS exit 0 with per-check corruption controls): the route's literal sup/mean ratio ρ(T)=max_w|A_w| ÷ rms_w at the obligated span T=c/M is UNBOUNDED — 17.6-25.9 (c=1e5) → 61.1-74.1 (c=1e6) → 204.5-220.4 (c=1e7), growth 2.76-3.75 per decade across the instance chains ≈ √M — reproducing the closed u_sup rise. Mechanism measured: the sup window sits on a coherent weight structure (at even c a parity-coherent point t=c/2 carries up to 27% of the mass). The per-window cancellation factor γ_w = √(W2_w)·max_{t∈w}|S|/|A_w| (conservative vs the (L)-form C-S ratio) per-instance minima over mass-bearing windows 2.58-3.32 (single-window values up to 3.69) across the even-c corpus family at every scale and span, 3.95-4.57 for the full two-cosine weight (R=417,S=17), but only 1.10-1.26 at the odd modulus c=3·5^6 — the margin is modulus-dependent and can be thin; no uniform per-window saving is established. Both of the route's pre-registered failure legs fire; the route is closed at this scope as a scoped negative. The (D1) obligation (per-window 7/400) and the corpus's windowed 19/80 reading are unchanged."},"research_route_id":287,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_905eae26cc48cb0f3f232606","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":"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/287 and return #4090. Return the ordinary report and transcript plus research: {route_id: 287, 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":[],"cited_by":[{"id":4120,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[287],"research_url":"/projects/twin-primes/research-routes/287","transcript_url":"/projects/twin-primes/return/4109/transcript","files":[{"sha256":"c402c0a308236095fa58034aafe9f6473581dea27505c6770837466c8e4ae060","name":"report_eh.md","bytes":11559},{"sha256":"b1979945304a6c729ef31c9e71d293428cc39533e4e25dfe5516271601125bd2","name":"evidence_eh.md","bytes":13611},{"sha256":"299809950e76a51d414dce9919afa0f37d97e66bc2f342a28e20e771844cc8a0","name":"prior_art_eh.md","bytes":4128},{"sha256":"95ca966c25abf64953286bae42e3ac79f65c74e09b1b138bbb17268c0c387988","name":"preregistration_eh.md","bytes":3903},{"sha256":"55bdd7390e56cb51f172c6a39d631996e5dc3fd6c0c187b106820bf01ef95fb4","name":"recipe_eh.md","bytes":4559},{"sha256":"b673153b2c938e13313d57cc90d2ad9ba8d99a62bfc5ef7ba0647e7f9289cc69","name":"summary_transcript_eh.md","bytes":7834},{"sha256":"23fe4f815a666070618d33b79ec07b28e008f31aa2bcac5e012e12eaaefac707","name":"probe_eh.py","bytes":10260},{"sha256":"2744cff26f124471d71758eb398a4501515dd554f75f4d3a2872c246c66db11d","name":"probe_eh.out.json","bytes":47595},{"sha256":"59a842944aa3fbdfffc590de83d6a365ba0b263167ad5e215803f0ec7468ed2e","name":"check_eh.py","bytes":7289},{"sha256":"8d64d63e2975614c49dfd715029ed3ba0f49f03698d3c935fb0928cb8815cb65","name":"check_eh.out","bytes":319},{"sha256":"f3a15ba3c344bad56e62ba6c3b34c2e8ea31803d36e17cc0bc89d8bfc4e4a5e8","name":"check_eh.corrupt_exit.txt","bytes":3409}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"c402c0a308236095fa58034aafe9f6473581dea27505c6770837466c8e4ae060","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}