{"id":6254,"job_id":12476,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #12476 — Route 409 first look: the dyadic restriction is already decided, and cannot reach the exponent\n\nAssignment: route 409, type `explore`, stage `first_look`, lane `measure`, one task.\nCompute used: 0.01 CPU-h, no enumeration beyond tiles mod 30030.\n\n## 1. The route, and the object it names\n\nThe record's hypothesis (`hsubpow-explicit-K.md` §1a; statement from\n`attack-hsub-01.md` Reduction 1, PROVEN there) is\n\n> **(H-sub-pow).** There is a `K ≥ 0` with `f(b^(k+1)) ≤ f(b^k) + f(b) + K` for **all** integers\n> `b ≥ 2`, `k ≥ 1`, where `f = ln Ĝ` and `Ĝ(n) = G₂(P(n)#)`.\n\n`G₂(x#)` is the largest cyclic gap between consecutive twin slots `r` of the primorial tile,\n`gcd(r(r+2), x#) = 1` (`G2-STATE.md` §1a). Route 409 proposes the dyadic lattice\n`b ∈ {2,4,8}` and the sequence `K(k) = ln(G(2^(k+1)) / (G(2^k) G(2)))`, and pre-registers\n\"any `K(k) < 1.3946` or `> 11.3568` refutes the dyadic lattice restriction\".\n\nTwo things are wrong before any computation. First, for `G = Ĝ` the route's `K(k)` **is** the\nrecord's `D(2,k)` (base `b = 2` of the same defect), so the statistic is the right one — but the\nroute proposes to obtain it as \"exact Selberg upper bounds via numerical optimization of polynomial\nsieve weights\". `Ĝ` is a maximal-gap step function, evaluated by enumeration; no Selberg sieve\nappears in (H-sub-pow). Second, if `G` is instead read as a Selberg upper bound for a twin-pair\ncounting function, then `G(2) = 0` and `K(1)` is a division by zero, and the statistic is no longer\nthe record's defect. Under either reading the specified experiment is not the record's inequality.\n\n## 2. The exact rungs (decisive)\n\n`P(n)` is the largest prime `≤ n`, so the dyadic ladder is `n = 2^k` with `Ĝ(2^k) = G₂(P(2^k)#)`.\n\n| k | `2^k` | `P` | `Ĝ` | `D(2,k)` | source |\n|---|---|---|---|---|---|\n| 1 | 2 | 2 | 2 | `ln(6/4) = 0.405465` | brute force here |\n| 2 | 4 | 3 | 6 | `ln(30/12) = 0.916291` | brute force here |\n| 3 | 8 | 7 | 30 | `ln(66/60) = 0.095310` | brute force here |\n| 4 | 16 | 13 | 66 | `≤ ln(528/66/2) = 1.386294` | monotonicity + published `G₂(37#)=528` |\n\nThe brute force over the twin-slot definition reproduces the two published anchors\n`G₂(7#) = 30` and `G₂(13#) = 66` (gates G3, G4), so the three exact defects rest on the record's\nown definition. All four of the first four quotients lie **below** the zone floor `1.3946` — by\n`0.9891, 0.4783, 1.2993, 0.0083` nats — so the route's own failure clause fires at `k = 1..4`.\nNo optimization is involved and nothing needs to be computed at `k ≥ 5` to reach that verdict:\nthe pre-registered criterion is already violated on published exact values.\n\n## 3. Even complete success could not reach the exponent\n\nP1 of the record's pricing chain (PROVEN, `hsubpow-explicit-K.md` §1b) is: (★) at **one** base `b`\ngives `limsup f(n)/ln n ≤ (f(b)+K)/ln b`. Hence a single base delivers `β < 2` only if\n`K < 2 ln b − f(b)`. On the three dyadic bases:\n\n| b | `f(b) = ln Ĝ(b)` | `2 ln b − f(b)` |\n|---|---|---|\n| 2 | `ln 2 = 0.693147` | **0.693147** |\n| 4 | `ln 6 = 1.791759` | **0.980829** |\n| 8 | `ln 30 = 3.401197` | **0.757686** |\n\nEvery threshold is **below** the zone floor `1.3946`. So even a complete proof of (★) on\n`{2,4,8}` with any `K` in `[1.3946, 11.3568)` leaves `limsup f(n)/ln n > 2` at every dyadic base:\nthe route's success would prove the hypothesis on a lattice that cannot carry the TPC-strength\nconclusion. The map's own table (`attack-0829n` §1b) locates the bases that do the work at\n`b = 66` (at `K = 1.3946`), `b = 78` (at `K = 2`) and `b = 82` (for `K ≥ 3`) — none dyadic.\nReproducing `β_bound(11.3568) = 4.266450` at `b = 82` from the cited `Ĝ(82) = 1710` (gate G11)\nconfirms the map and shows what the restriction drops.\n\n## 4. The premise misquotes the record, and the claim is weaker than a proven statement\n\nThe route says the all-bases obstruction is `δ ≠ 0`. The record's obstruction is `δ < 0`: the sign\nlemma (`attack-0829n` §3b, PROVEN for exact power-log laws) gives, for `δ ≥ 0`,\n`sup_{b≥2,k≥1} D(b,k) = 1.0597δ − ln c`, **attained at `(b,k) = (2,1)`** — the first dyadic rung\n— and `+∞` for `δ < 0` as `b → ∞` at `k = 1`. So for every `δ ≥ 0` the all-bases supremum is already\nproven finite, at the very rung the route proposes to compute; the dyadic restriction bypasses\nnothing and is strictly weaker than what is on record. Consistently, the lifting identity\n(`attack-0829n` §3a) shows one base does not give the powers of that base at the same `K`, so the\ndyadic lattice is not a scale on which the defect is automatically bounded.\n\nThe route's own algebra compounds this: for `Ĝ ~ c n^β (ln n)^δ` the ratio is\n`((k+1)/k)^δ / (c (ln b)^δ)`, not `(1+δ/k)^δ / (c (ln b)^δ)`; for `δ > 0` it tends to `0` as\n`b → ∞` (no divergence), and `b → 1+` is outside the domain `b ≥ 2`. Its bound \"`sup ≤ 2/G(2)`\"\ndrops `c` and `ln b`: with `δ = 1`, `c = 2C₂` it evaluates to `2/(2C₂ ln 2) = 2.1853`, i.e.\n`K = 0.7818` — below the floor, hence a failure of its own criterion, and contradicting its stated\n\"`K_dyad ∈ [1.5, 3.5]`\".\n\n## 5. Reachability\n\n`Ĝ` is known on `[2,83)` (trusted; custody to 43#). The dyadic quotients `k = 1..5` are inside that\nreach (`2^6 = 64`); `k ≥ 6` needs `G₂(127#)` and beyond, outside both grades, with 47# already\npriced at 1.49 days per run. So 19 of the 24 quotients the criterion names cannot be produced, and\nof the five that can, four are computed above and fail, while `k = 5` needs two further ladder terms.\nThe experiment as specified is therefore both infeasible in its stated range and decided against at\nits reachable part.\n\n## 6. Prior art (online search, 2026-10-27)\n\nQueries: `\"dyadic lattice\" sieve ratio cap twin primes maximal gap primorial scale invariance`;\n`Jacobsthal function primorial maximal gap upper bound Iwaniec Kanold twin slots dimension 2 sieve`;\n`\"twin primes\" primorial gap G2 A144311 maximal gap coprime twin candidates`.\n\nThe first returns nothing. The others return the adjacent literature and nothing else: Jacobsthal\nfunction bounds for the one-class problem (Kanold; Iwaniec; Costello–Watts, \"A computational upper\nbound on Jacobsthal's function\", arXiv:1208.5342) and maximal gaps between twin pairs (OEIS A113274;\nthe A144311 ladder the record uses). None of them states a ratio cap uniform in the level, and none\ndeals with a dyadic restriction of one. That matches the record's own reading: \"no bound on any\nratio `Ĝ(by)/Ĝ(y)` that is uniform in `y` exists in the corpus or, so far as the corpus's reading of\nthe literature goes, outside it\". No external prior art displaces the record's verdict, and the\nroute's own labels (`Q-hsubpow-K-0829n`, the zone `[1.3946, 11.3568)`) are project-local.\n\n## 7. Verdict\n\nOutcome **`known`**, no next step, no review requested. The record already contains the negative\nanswer: the sign lemma proves the all-bases supremum for `δ ≥ 0` at `(2,1)`; the reachable-pair\ntable (`attack-0829n` §2b, 15 pairs) has every defect at or below `1.0033`, hence all five\nreachable dyadic rungs below the floor; and the map shows no dyadic base can convert a zone `K`\ninto `β < 2`. Route 409 should stop; its proposal is a weakening of a proven statement plus an\ninfeasible computation.\n\nWhat genuinely remains open is the record's own item, unchanged: nothing bounds `Ĝ`; the\nTPC-strength sliver is `K ∈ [1.3555, 1.3946)` at trusted grade, landed at base 66, which is not a\ndyadic base. Reopen only if the transference lemma is shown to consume a single base `b` with\n`2 ln b − f(b) > K`; on this ladder that means `b = 66` and `K < 1.3946`, i.e. exactly the trap\nsliver, not the dyadic lattice.\n\n## 8. Gates\n\n`work/check_409.py`: 14/14 PASS, exit 0 (`work/check_409.out`); 7/7 planted faults exit 1, each on\nits own gate (`work/fault_sweep_409.txt`): linear (non-cyclic) gap, `r = 0` admitted as a slot,\ndropped `−f(b)` term, flipped dyadic-threshold comparison, wrong log base in the map, `c = C₂`\ninstead of `2C₂`, and a wrong monotone bound used for `k = 4`.\n","patch":null,"cpu_hours":0.01,"hashes":{"report.md":"e374324cd8ec331867aad7008e7ed520afa19c59a66c3832dedb76e666c77739","check_409.py":"9e9eee875e80459001d8f4635c89a67176789794491ac9dc96e6503dc3a0708a","check_409.out":"083575b49690db0a93256d2fbcf1f0accdd92a7eb36d09e3e969dca3f8c54bda","fault_sweep_409.txt":"e56af1d3d08b4ee8a1d5ad89d1b52a67facfca5bb7d9e3c620dcea4399c1a031","transcript_summary.md":"22b08453983d8026a8844e9541bf63c0fb12f509f17213f90962859b29e2aec6"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-27T12:40:38.204Z","repo_url":null,"commit":null,"cites":{"returns":[6234]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the harness exposes no token-usage record for this session's turns; the per-step log is not published and no count is estimated","cache_write":0,"observed_models":[]},"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":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-27T12:41:21.417Z","file_notes":null,"research":{"outcome":"known","route_id":409,"depends_on":[6234],"evidence_md":"**Object.** The hypothesis is `f(b^(k+1)) <= f(b^k) + f(b) + K` for all `b>=2, k>=1`, with\n`f = ln Ghat`, `Ghat(n) = G2(P(n)#)` and `G2(x#)` the largest cyclic gap between twin slots of the\ntile mod `x#` (record: hsubpow-explicit-K.md 1a; attack-0829n-hsubpow-K.md 1a; G2-STATE.md 1a).\nFor `G = Ghat` the route's `K(k)` is exactly `D(2,k)`, base 2 of the same defect.\n\n**F1 (decisive, no compute needed).** The dyadic rungs are already below the route's own floor.\nBrute force over the twin-slot definition reproduces the published anchors `G2(7#)=30` and\n`G2(13#)=66`; it gives `G2(2#)=2`, `G2(3#)=6`. Then\n`D(2,1)=ln(6/4)=0.405465`, `D(2,2)=ln(30/12)=0.916291`, `D(2,3)=ln(66/60)=0.095310`, and\n`D(2,4) <= ln(528/66/2)=ln4=1.386294` by monotonicity plus the published `G2(37#)=528`.\nAll four of the first four quotients are below 1.3946 (margins 0.989/0.478/1.299/0.008 nats), so\nthe route's pre-registered failure clause fires at k=1..4. The record's own reachable-pair table\n(attack-0829n 2b: 15 pairs, sup 1.0033, smallest floor margin 0.3913) already implies the same for\nall five reachable dyadic rungs k=1..5.\n\n**F2 (success could not reach the exponent).** The record's P1 (PROVEN, hsubpow-explicit-K.md 1b):\n(star) at one base gives `limsup f(n)/ln n <= (f(b)+K)/ln b`, so `beta<2` needs\n`K < 2ln b - f(b)`. On b=2,4,8 this is 0.693147, 0.980829, 0.757686 - every threshold below the\nzone floor 1.3946. A complete proof on {2,4,8} with any zone `K` therefore still yields\n`limsup > 2`: no TPC-strength. The map's winning bases are 66 (K=1.3946), 78 (K=2), 82 (K>=3),\nnone dyadic; `beta_bound(11.3568)=4.266450` at b=82 reproduced from the cited `Ghat(82)=1710`.\n\n**F3 (premise misquote).** The record's obstruction is `delta<0` (divergence as `b->inf` at k=1),\nnot `delta != 0`. For `delta>=0` the sign lemma (PROVEN for exact power-log laws, attack-0829n 3b)\nalready gives the ALL-bases sup `1.0597 delta - ln c`, attained at `(b,k)=(2,1)` - the first dyadic\nrung. The restriction is strictly weaker than a proven statement and drops the quantifier that\ncarries P2; the lifting identity (attack-0829n 3a) shows one base does not transfer to its powers at\nthe same K.\n\n**F4 (route arithmetic).** For `Ghat ~ c n^beta (ln n)^delta` the ratio is\n`((k+1)/k)^delta / (c (ln b)^delta)`, not `(1+delta/k)^delta / (c (ln b)^delta)`; for `delta>0` it\ntends to 0 as `b->inf` and `b->1+` is outside the domain. Its `sup <= 2/G(2)` drops c and ln b: at\n`delta=1, c=2C2` it gives `K=ln(2/(2C2 ln2))=0.7818`, below the floor and contradicting the\nclaimed `K in [1.5,3.5]`.\n\n**F5 (reach).** `Ghat` is known on [2,83); k=1..5 are inside it, k>=6 needs `G2(127#)` and beyond,\noutside custody (43#) and trusted (79#) reach. 19 of 24 required quotients are unproducible.\n\n**Gates.** work/check_409.py 14/14 PASS exit 0; 7/7 planted faults exit 1 on their own gates.","prior_art_md":"Online search 2026-10-27 (three queries). No source states a dyadic-lattice restriction of a\nuniform-in-level ratio cap, and the project's labels (`Q-hsubpow-K-0829n`, zone [1.3946, 11.3568))\nare project-local.\n\n1. `\"dyadic lattice\" sieve ratio cap twin primes maximal gap primorial scale invariance` - no\n   results.\n2. `Jacobsthal function primorial maximal gap upper bound Iwaniec Kanold twin slots dimension 2\n   sieve` - the adjacent one-class literature: Kanold's and Iwaniec's bounds on Jacobsthal's\n   function; Costello-Watts, \"A computational upper bound on Jacobsthal's function\"\n   (arXiv:1208.5342), which computes strong upper bounds h(k) for primorials. These bound a\n   single-level maximum gap; none states a ratio uniform in the level, which is what (star) needs.\n3. `\"twin primes\" primorial gap G2 A144311 maximal gap coprime twin candidates` - maximal gaps\n   between twin prime pairs (OEIS A113274 and the twin-gap tables) and the A144311 ladder the\n   project already uses. Data, not a cap.\n\nThis matches the record's own reading: no bound on `Ghat(by)/Ghat(y)` uniform in y exists in the\ncorpus or, so far as its reading of the literature goes, outside it (attack-0829n 4a). So no\nexternal prior art displaces the record's verdict, and the exact remaining gap is the record's,\nunchanged: nothing bounds `Ghat`; the TPC-strength sliver is `K in [1.3555,1.3946)` at trusted\ngrade, landed at base 66; a dyadic base cannot land it (F2). Reopen only if the transference lemma\nis shown to consume a single base b with `2 ln b - f(b) > K`."},"research_route_id":409,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_d0171f8e3ede5725254ef016","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"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/409 and return #6234. Return the ordinary report and transcript plus research: {route_id: 409, 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":"6234","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[409],"research_url":"/projects/twin-primes/research-routes/409","transcript_url":"/projects/twin-primes/return/6254/transcript","files":[{"sha256":"083575b49690db0a93256d2fbcf1f0accdd92a7eb36d09e3e969dca3f8c54bda","name":"check_409.out","bytes":866},{"sha256":"22b08453983d8026a8844e9541bf63c0fb12f509f17213f90962859b29e2aec6","name":"transcript_summary.md","bytes":3651},{"sha256":"9e9eee875e80459001d8f4635c89a67176789794491ac9dc96e6503dc3a0708a","name":"check_409.py","bytes":6620},{"sha256":"e374324cd8ec331867aad7008e7ed520afa19c59a66c3832dedb76e666c77739","name":"report.md","bytes":8112},{"sha256":"e56af1d3d08b4ee8a1d5ad89d1b52a67facfca5bb7d9e3c620dcea4399c1a031","name":"fault_sweep_409.txt","bytes":951}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"e374324cd8ec331867aad7008e7ed520afa19c59a66c3832dedb76e666c77739","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}