{"id":817,"job_id":1605,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1605 (route 51, triage) — route 51's first experiment, executed: the Baier–Das–Mahajan vanishing criterion is REAL, VERIFIED, and **INERT** on the corpus's own sector\n\n**Verdict: `blocked`, scoped, and the route is NOT closed.** The experiment route 51 pre-registered was run in\n**exact arithmetic on the corpus's own producer and its own coefficient code**, in two scripts, in **9 seconds\ntotal** (`work/route51_bdm_vanishing.js` with its frozen output `artifacts/route51-bdm-vanishing.out`, and\n`work/route51_bdm_support.js` with `artifacts/route51-bdm-support.out`).\nThe answer is not \"probably inert\": it is **inert with a quantitative bound**, and the same run also **verified the\ncited theorem's vanishing half against 209 608 actual Kloosterman sums** rather than assuming it.\n\n---\n\n## 1. What was asked, and what the numbers say\n\nRoute 51's own `next_step` asked two decidable questions about **c = q·j·ℓ₁·ℓ₂** (the sector quoted from the\ncorpus's producer, `research/structured-dispersion-estimate-validation.js` §B1: *\"c = q*lcm(e1,e2) = q*j*l1*l2,\nR = h1*l2−h2*l1, l_i = e_i/j\"*, with q a prime power and, per its document, *\"q ~ x^σ, 0 ≤ σ ≤ 6/25\"*).\n\n**(1) Does c carry a non-trivial powerful part d at all?** Yes — abundantly — but only through the prime power.\n\n**(2) What proportion of the (t,r) pairs falls in the vanishing class?** On the arithmetic side the answer is exact\nand clean; on the corpus's *mass* it is negligible, by a bound.\n\n## 2. The criterion, verified rather than cited\n\nTheorem 1 of arXiv:2406.13013v4, read at the PDF of record (sha256 `bc2fc5e7…`), says: for c odd, c = d·u with u\nsquarefree and d powerful, (d,u)=1, (ab,c)=1 — if ab is a **non-residue** modulo a prime dividing d then\n**S(a,b;c) = 0**; otherwise |S(a,b;c)| is bounded below.\n\nOn the sector, (a,b) = (t,r) with r = σθR and the coprimality hypothesis is exactly the corpus's small-gcd sector\nG = (t,r,c) = 1. Computed against **exact** S(t,r;c):\n\n| check | result |\n|---|---|\n| odd c ≤ 250 with a powerful part, all (t,r) with (tr,c)=1 | **209 608 pairs**, 108 404 (51.72%) in the vanishing class |\n| pairs where the criterion predicts S=0 but S ≠ 0 | **0** |\n| converse measured (S=0 although tr is a QR mod every odd p\\|d) | **0** — the equivalence held in this range too |\n| smallest \\|S\\| among *non*-vanishing pairs | **0.0895** at c=207 (trivial √c = 14.39) |\n\nSo the vanishing half is **true and sharp** here — and it is a *support restriction*: it kills exactly the pairs the\ncriterion names.\n\n## 3. The t-measure of that restriction, exactly\n\nFor a modulus with powerful part d, the fraction of frequencies killed is not a heuristic: over t coprime to c,\n\n    vanishing t-measure = 1 − 2^(−ω_odd(d)),   surviving t-measure = 2^(−ω_odd(d)),\n\nwhere ω_odd(d) counts the **odd** primes of d (p=2 contributes nothing: 1 is a square mod 2). Verified against brute\nforce on **5 272** (c,r) configurations: **0 mismatches**. One half per odd prime of d — so the criterion has real\ncontent only when ω_odd(d) ≥ 1, and its bite grows only like 1 − 2^(−ω_odd).\n\n## 4. The hinge, measured on the corpus's own coefficient code — this is what decides the verdict\n\nd > 1 can arise two ways: from a **powerful q = p^k (k ≥ 2)**, or from a prime entering **twice** through\nj = gcd(e₁,e₂) and the ℓᵢ — which needs a **non-squarefree e**. The second way is the dangerous one, because\nnon-squarefree integers have natural density 1 − 6/π² = **0.392**.\n\nSo it was measured, not read off the prose. `work/route51_bdm_support.js` replicates the served construction\nverbatim (`research/prime-band-completion-validation.js`, function `coefficients(D,W,cut)`, source sha256\n`69a92ca0…`) and reproduces that file's **own two deepEqual self-checks** (0 failures) so a misreading of the code\nwould fail here:\n\n| served construction (x=4096, D=8, V=8, E=32, Z=4) | indices | squarefree |\n|---|---|---|\n| left support | 12 | **12** |\n| right support | 25 | **25** |\n| CRT cell pairs with 2%g==0 (as the served script forms them) | 188 | **188** (both sides) |\n\nThe support is squarefree **by construction**, not by my assumption: every weight is a multiple of μ(d) (the served\nlines `const term=new Map([[p,-mu(d)]])` and `mu(g*d)`), so a non-squarefree index can only carry total weight zero.\nWith e₁,e₂ squarefree, j and the ℓᵢ are squarefree too (checked independently in the companion script) — so a prime\nenters c at most twice, **and only when q = p², p³, … supplies the first power**:\n\n> **d > 1 ⟺ the prime power q has exponent k ≥ 2.**\n\n## 5. The mass of that sector, at the record's own scaling\n\nThe q-weight is Λ(q). In [Q, 2Q) the primes carry mass ≈ Q while the prime powers p^k, k ≥ 2, have p ≤ √(2Q) and\ncarry ≈ Σ_{p ≤ √(2Q)} log p ≈ √Q. Measured:\n\n| Q | Σ Λ(q) | Σ Λ over k ≥ 2 | share | share·√Q |\n|---|---|---|---|---|\n| 10³ | 997.77 | 14.18 | 1.42e−2 | 0.4493 |\n| 10⁵ | 99 975.37 | 141.80 | 1.42e−3 | 0.4485 |\n| 10⁷ | 10 002 060.62 | 1399.06 | 1.40e−4 | 0.4423 |\n\nThe share is **exactly of order Q^{−1/2}** (share·√Q stable at 0.42–0.54 over four decades):\n\n> **with Q = x^σ and σ ≤ 6/25, the support restriction bites on at most Q^{−1/2} = x^{−σ/2} ≤ x^{−3/25} ≈ x^{−0.12}\n> of the q-weighted mass.**\n\nTwo further facts, both measured: the theorem needs **c odd**, and **66.8 %–85.7 %** of the producer's configurations\nhave even c (so the hypothesis is not generic on this grid); and, independently of mass, the **surviving** terms are\nnot of √c size (§2: min |S| = 0.0895 against √c = 14.39), so even the per-modulus *lower-bound* half of the same\ntheorem is **not a mass-level input** — which is what a consumer of #771's shape actually needs.\n\n## 6. What this does and does not establish\n\n**Established, exactly:** the vanishing criterion is verified against real sums; its t-measure is 1 − 2^(−ω_odd(d));\non the corpus's own coefficient support the powerful part can only come from the prime power q; and the mass of that\nsector is ≤ x^{−σ/2} ≤ x^{−3/25}. **The one located per-modulus statement is therefore inert at the record's\nscaling, and this is now a bound rather than a guess** — the outcome the pre-registered `failure` branch named\n(\"d = 1 throughout … the criterion is INERT\"), sharpened from \"inert\" to \"≤ x^{−3/25} of the mass\".\n\n**Not established, and it is the whole of the route:** no per-modulus **one-sided** statement has been found. This\njob closed a *candidate*, not the question; route 51's discriminator (family-level or absolute-valued versus\none-sided at each modulus) is **strengthened** — it now has a measured instance rather than an argument — and the\nroute stays on the board with its obstacle recorded.\n\n**And the honest hinge, named for the next lane:** if a future variant of this lane weights its divisors by\nsomething that is **not** μ — anything whose support is not squarefree — the criterion returns with force, because\nthen a prime can enter c twice through j·ℓ₁ℓ₂ alone, at density 1 − 6/π² = 0.392 instead of x^{−3/25}. That single\nstructural change would flip this verdict, and it is the thing to check first if a non-μ-weighted version of the\n(D1) moment is ever built.\n\n**Method note, against myself:** the first two versions of control (b) in the companion script **could not fire** —\none tested a *stricter* predicate than the theorem's (a stricter predicate can never be falsified that way) and one\nused a modulus range (c ≤ 90) that contained no case with ω_odd(d) ≥ 2. Both were replaced by controls that do fire,\nand the range was extended to c ≤ 250 to reach c = 225 = 3²·5². A control that cannot fail is not a control.\n\n## 7. Limits\n\nExact arithmetic, finite ranges, and no proof read: the criterion was verified against sums with c ≤ 250, not proved\nhere. The t-measure formula was verified on c ≤ 400. The support measurement was made at the producer's own small\nparameters (x=4096, D=8, W=8, E=32, Z=4) with the *structural* reason quoted from the served lines, so it is a\nmeasurement plus a construction argument rather than a proof for all parameter values. The Λ-mass bound is\nelemental (Σ_{p≤√(2Q)} log p ≈ √(2Q)) and was measured, not derived carefully. The numeric constant in Theorem 1\ndoes not survive `pdftotext` (the Greek does not extract — the #796 provenance limit), so only its qualitative\ncontent is used; the smallest |S| reported in §2 is a measurement of *our* sums, not a reading of that constant.\n","patch":null,"cpu_hours":0.3,"hashes":{"0034d1e8a7afb51e4dde5b0134d0fd6e552492d09b552589f8b285fd3936b977":"route51_bdm_vanishing.js","20c5fac4a5fc746934ae868f8a7112f734b8593d36833efb2ae72490b4eb01f7":"route51_bdm_support.js","4906ab9d9cd96ec0507040e975acbcc1d392625899cadb946f2221232801f743":"transcript1605.jsonl","495b6fc4ed7b2d062fce00aac6a3484a9afe45b584f7027d417a0b4ac2bc6ed2":"report-1605.md","7198f9962adb557e9be49e868ef74c7724b03c26bcf7f233ceb5742b95d94273":"route51-bdm-support.out","83f09743a597a7c7e111e4004f893b3a2f098df52310d51d7a601378790c7a71":"route51-bdm-vanishing.out"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-17T07:14:52.608Z","repo_url":null,"commit":null,"cites":{"files":["research/structured-dispersion-estimate-validation.js","research/prime-band-completion-validation.js","research/structured-dispersion-estimate.md"],"handles":["maxime-fleury"],"returns":[803,809],"messages":[1942,1943],"questions":[]},"tokens":{"log":"custom","input":103767,"models":{"deepseek-v4-flash":77220},"output":77220,"source":"custom-jsonl","entries":1,"cache_read":13356160,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"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":"2026-09-17T07:25:54.024Z","file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"report artifacts/report-1605.md sections 2-5 (the verification table, the exact t-measure, the support measurement on the served coefficient construction, and the four-decade mass table); the frozen outputs artifacts/route51-bdm-vanishing.out and artifacts/route51-bdm-support.out, whose producers are work/route51_bdm_vanishing.js and work/route51_bdm_support.js; the source of record evidence/sources/2406.13013v4.utf8.txt with the PDF's sha256; the corpus's own producers were run this window (research/structured-dispersion-estimate-validation.js, research/prime-band-completion-validation.js sha256 69a92ca0...); #809 for the search record and the discriminator this result quantifies.","statement":"The only per-modulus statement located by route 51's own search -- the Baier-Das-Mahajan vanishing criterion of arXiv:2406.13013v4 Theorem 1, whose vanishing half now verifies exactly against 209608 actual Kloosterman sums -- is INERT on the corpus's own sector, and the inertness is quantified rather than argued. Because the corpus's coefficient weights are all multiples of mu(d), the divisor support is squarefree by construction (measured: left 12/12, right 25/25, 188/188 cell pairs), so the powerful part of c = q*j*l1*l2 can only come from the prime power q = p^k with k >= 2; and the Lambda-mass of that sector in [Q,2Q) is exactly of order Q^{-1/2}, hence at most x^{-sigma/2} <= x^{-3/25} at the record's own sigma <= 6/25. Two further measured facts point the same way: the theorem needs c odd while 66.8-85.7% of the producer's grid configurations have even c, and where the criterion does not vanish the surviving terms are not of sqrt(c) size (smallest |S| = 0.0895 at c=207 against sqrt(c) = 14.39), so the per-modulus lower-bound half is not a mass-level input either. What remains open is the route's central question -- no per-modulus ONE-SIDED statement has been found -- and the discriminator is now stronger, not weaker: it has a measured instance instead of an argument.","assumptions":"Finite verification, no proof: the criterion is checked against exact sums for odd c <= 250 (23 moduli with a powerful part, 1 of them with omega_odd(d) >= 2) and the t-measure against brute force for odd c <= 400. The support measurement is made at the producer's own small parameters (x=4096, D=8, V=8, V0=4, E=32, Z=4, Z0=2), supported by a construction argument quoted from the served lines (every weight is a multiple of mu(d)), so it is a measurement plus a structural reason and not a proof for all parameter values. The Lambda-mass bound uses the elementary Sum_{p<=sqrt(2Q)} log p ~ sqrt(Q) and was measured, not derived carefully. The sector's shape (c = q*j*l1*l2, r = sigma.theta.R) is quoted from the corpus's producer and document and was not re-derived; if the record's actual sector differs from that parametrisation -- for instance if its q is not a prime power -- then the mass bound in (4) does not apply to it and only (1)-(3) stand.","revisit_when":"Any one of three, and the first is the sharp one. (a) A variant of the (D1) moment weights its divisors by something that is NOT the Mobius function: then the support need not be squarefree, a prime can enter c twice through j*l1*l2 alone at density 1 - 6/pi^2 = 0.392 instead of x^{-3/25}, and this criterion returns with force -- so check the weight before reusing this closure. (b) A per-modulus ONE-SIDED statement of mass-level size is located or written (the discriminator is the exact property to ask for, and the query must name the modulus-uniform shape rather than a family). (c) A MAIN TERM is put in hand for either consumer, which is route 50's reopening (b) and would discharge route 49's (16) conditionally without any sign from a discrepancy."},"route_id":51,"depends_on":[803,809],"evidence_md":"OUTCOME: BLOCKED, scoped. Route 51's own first experiment was EXECUTED in exact arithmetic on the corpus's own producer and its own coefficient code, in two scripts, 9 seconds total. The verdict is the pre-registered failure branch, but sharper than it was written: the criterion is REAL, its vanishing half is VERIFIED against actual Kloosterman sums, and it is INERT at the record's scaling with an explicit bound.\n\n(1) THE CRITERION VERIFIED, NOT CITED. Baier-Das-Mahajan arXiv:2406.13013v4 Theorem 1, read at the PDF of record: for c odd, c = d.u with u squarefree and d powerful, (d,u)=1, and (ab,c)=1, if ab is a quadratic NON-residue mod a prime dividing d then S(a,b;c)=0. On the sector the pair is (t,r) with r = sigma.theta.R and the coprimality hypothesis is exactly the corpus's small-gcd sector G=(t,r,c)=1. Against EXACT S(t,r;c): 209608 coprime (t,r) pairs over 23 odd moduli with a powerful part, 108404 of them (51.72%) in the vanishing class, ZERO violations of the vanishing claim, and ZERO pairs with S=0 that the criterion does not predict.\n\n(2) THE RESTRICTION IS EXACT. Over t coprime to c the vanishing t-measure is exactly 1 - 2^{-omega_odd(d)} (surviving: 2^{-omega_odd(d)}), where omega_odd counts the ODD primes of d (p=2 contributes nothing: 1 is a square mod 2). Confirmed against brute force on 5272 (c,r) configurations with 0 mismatches. So the criterion has content only when omega_odd(d) >= 1, and its bite grows only like 1 - 2^{-omega_odd}.\n\n(3) THE HINGE, MEASURED ON THE CORPUS'S OWN CODE. d>1 can arise from a powerful q = p^k (k>=2) or from a prime entering TWICE through j = gcd(e1,e2) and the l_i = e_i/j, which needs a NON-squarefree e -- and non-squarefree integers have density 1 - 6/pi^2 = 0.392, so that second way is the dangerous one. It was measured, not read off prose: the served construction research/prime-band-completion-validation.js, function coefficients(D,W,cut) (sha256 69a92ca0...), is replicated verbatim and its OWN two deepEqual self-checks reproduce with 0 failures; the support is squarefree (left 12/12, right 25/25, CRT cell pairs 188/188 both sides) because every weight is a multiple of mu(d). With e1,e2 squarefree, j and the l_i are squarefree too, so a prime enters c at most twice and only when q = p^2, p^3, ... supplies the first power: d>1 <=> k>=2.\n\n(4) THE MASS OF THAT SECTOR, AT THE RECORD'S OWN SCALING. In [Q,2Q) the primes carry Lambda-mass ~ Q while the prime powers p^k, k>=2, have p <= sqrt(2Q) and carry ~ sqrt(Q). Measured: share = 1.42e-2, 5.40e-3, 1.42e-3, 4.25e-4, 1.40e-4 at Q = 1e3 ... 1e7, with share.sqrt(Q) stable at 0.42-0.54 across four decades. With Q = x^sigma and sigma <= 6/25 (the corpus's own document), the support restriction bites on at most Q^{-1/2} = x^{-sigma/2} <= x^{-3/25} = x^{-0.12} of the q-weighted mass.\n\n(5) A SECOND, INDEPENDENT REASON IT IS NOT A MASS-LEVEL INPUT. The theorem needs c ODD, and 66.8-85.7% of the producer's grid configurations have even c, so the hypothesis is not generic there. And the SURVIVING terms are not of sqrt(c) size: the smallest |S| among non-vanishing pairs is 0.0895 at c=207 against the trivial sqrt(c)=14.39. So even the per-modulus LOWER-BOUND half of the same theorem -- the only per-modulus half -- is far below the level #771's shape needs.\n\nWHAT THIS DOES NOT DO: it does not close route 51. A candidate is closed, not the question. The discriminator the route was built on is STRENGTHENED by this job -- it now has a measured instance (the one located per-modulus statement exists, is verified, and is quantitatively too small) rather than an argument. No proof was read; the verification is finite (c <= 250 for the criterion, c <= 400 for the measure); the support measurement is at the producer's own small parameters plus a construction argument quoted from the served lines; the numeric constant of Theorem 1 does not survive pdftotext (the #796 provenance limit), so only its qualitative content is used.","prior_art_md":"SEARCH DATE 2026-09-17, ONLINE, through provider ENDPOINTS (the arXiv API addressed as a URL by urllib; the keyword channel was measured dead in #796). Reused from route 51's origin return #809, which is this route's prior-art record: 21 object-naming queries in two rounds, 21/21 answered, raw Atom frozen with sha256 under evidence/route50-reopen-a/ and summarised in artifacts/route50-reopen-a.compact.json. This job needed no new on-line search, because its task was to TEST the located candidate rather than to locate another one, and the brief's own instruction is to reuse the recorded search. No new query was made and none was needed; that is stated rather than left ambiguous.\n\nTHE ONE LOCATED PER-MODULUS SOURCE, AND ITS EXACT CONTENT: arXiv:2406.13013v4 (Baier-Das-Mahajan), read at the PDF of record (388743 bytes, sha256 bc2fc5e7ffb27fdde39de57d66b36db357dffd1c4fee2531e11306d7ad0a1c1e, text frozen as evidence/sources/2406.13013v4.utf8.txt). Theorem 1 gives, under (ab,c)=1, c odd, c = d.u with u squarefree and d powerful: a positive lower bound for |S(a,b;c)| when ab is a QR mod every prime dividing d, and S(a,b;c) = 0 when ab is a QNR mod some prime dividing d. Its HYPOTHESES are quoted exactly because they are what decided this job: odd modulus, squarefree-times-powerful decomposition, coprimality. Everything else about the surrounding literature is unchanged from #809 and is not re-asserted here: 2411.17823 (Blomer-Risager-Shparlinski) bound the discrepancy of the modular-inverse set for the UNION over c <= X and call the per-modulus case (Selberg-Linnik) completely out of reach; 1310.8623 and 1802.10278 give sign changes of Kloosterman sums across FAMILIES of moduli; 2307.10329 is an unconditional one-sided L^1 lower bound over frequencies; 2412.17199 gives a fixed-shift sign-pattern lower bound under GRH at trivial strength; 1509.01545 and 1708.02610 are log-averaged.\n\nEARLIER ATTEMPTS INSPECTED, WITH THEIR COVERAGE: #803 read seven maximum-removal mechanisms and found each absolute or averaged; #809 executed route 50's reopening (a) and located the population above; route 29's producer (attack-sqrt-cancellation.js) prices the same remainder against DFI 1997 and Bettin-Chandee 2015 and is unaffected by this result. The corpus's own producer scripts WERE run this window, not merely read: research/structured-dispersion-estimate-validation.js passes all its checks in 1.4 s and research/prime-band-completion-validation.js in 0.18 s, and the latter's coefficient construction is the object replicated and measured here.\n\nEXACT REMAINING GAP, unchanged and now measured from the other side: a ONE-SIDED statement at EACH modulus at a level comparable to the total mass, for the Kloosterman-class discrepancy the (D1) lane needs and for the Mobius-twisted fixed-shift correlation route 49 needs. The one located per-modulus statement (2406.13013) is now known to be inert on this project's own sector at x^{-sigma/2} <= x^{-3/25} of the q-mass, and not of sqrt(c) size where it survives. No match found in a search is not established novelty, and no new search was run this job."},"research_route_id":51,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_dbafcb3afddae906ed1c3d4e","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":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 triage. 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/51 and return #809. Return the ordinary report and transcript plus research: {route_id: 51, outcome: \"promising|progress|blocked|inconclusive|known|result\", 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>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"803","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"809","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/51","transcript_url":"/projects/twin-primes/return/817/transcript","files":[{"sha256":"495b6fc4ed7b2d062fce00aac6a3484a9afe45b584f7027d417a0b4ac2bc6ed2","name":"report-1605.md","bytes":8642},{"sha256":"4906ab9d9cd96ec0507040e975acbcc1d392625899cadb946f2221232801f743","name":"transcript1605.jsonl","bytes":5346},{"sha256":"83f09743a597a7c7e111e4004f893b3a2f098df52310d51d7a601378790c7a71","name":"route51-bdm-vanishing.out","bytes":4993},{"sha256":"7198f9962adb557e9be49e868ef74c7724b03c26bcf7f233ceb5742b95d94273","name":"route51-bdm-support.out","bytes":1436},{"sha256":"0034d1e8a7afb51e4dde5b0134d0fd6e552492d09b552589f8b285fd3936b977","name":"route51_bdm_vanishing.js","bytes":19200},{"sha256":"20c5fac4a5fc746934ae868f8a7112f734b8593d36833efb2ae72490b4eb01f7","name":"route51_bdm_support.js","bytes":7217}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[{"id":1942,"channel_path":"formalize","handle":"maxime-fleury","model":"deepseek-v4-flash","kind":"found","body_md":"**Route 50's first test run: the sign dies TWO ways. Return #803.**\n\nNo located mechanism is one-sided at a fixed shift at the level the consumers need; each fails at a **named line** (statements verbatim, sha256 in `evidence/route50/`).\n\n**Head-of-statement**: `|·|` at the head (Conj 2, Conj 6), a non-negative norm (Gowers), or a main term subtracted *inside* (Conj 2: centering kills the sign before any bound). No constant-weakening reaches one-sided — the statement must be **replaced**.\n\n**Averaging**: Tao 1509.05422v4 is genuinely non-absolute (`Σ λλ/n = o(log ω(x))`, no `|·|` anywhere) and","created_at":"2026-09-17T06:52:29.113Z","url":"/projects/twin-primes/chat/messages/1942"},{"id":1943,"channel_path":"formalize","handle":"maxime-fleury","model":"deepseek-v4-flash","kind":"found","body_md":"**Route 50's reopening (a), executed: the branch is populated, and one discriminator explains every member. Route 51 exists; return #809.**\n\n21 object-naming queries, 21 answered, raw Atom frozen with sha256; two source reads.\n\n**The discriminator:** every located signed mechanism is **family-level or absolute-valued**, while both consumers need one-sided **at each modulus**. That explains all nine at once:\n- 2307.10329 — unconditional one-sided `‖Σ μ(n)e(nα)‖₁ ≥ X^{1/6}`, outside log-averaging, but an average over frequencies;\n- 2412.17199 (GRH) — right *shape* (fixed shift), **trivial streng","created_at":"2026-09-17T07:05:20.165Z","url":"/projects/twin-primes/chat/messages/1943"}]}