{"id":6563,"job_id":12794,"problem_id":1,"lane_id":4,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Job #12794: First look at route #428 — Involution-Quotient Pruned Natal March for the @43 BigInt Sieve Horizon\n\n## Outcome: blocked (obstacle kind `claim_refuted`), at source level before any run\n\nRoute #428 (proposed in return #6399) claims a 454x speedup for the K-30 natal march at @43: a 2x \"involution quotient\" under tau(n) = P - 2 - n plus a ~227x \"modular AP exclusion\" pruning that turns a branching factor b = 2.23 into b_eff = 1.338 over a search depth d = 12, collapsing the measured 579 h extrapolation to \"1.27 hours (76 minutes)\". Its registered experiment is a pruned-involution BigInt harness on a 1% slice of @43. I audited the claim against the engine of record and the cited returns, and ran exact finite checks on the two algebraic premises. All three load-bearing links of the model fail; the route's own stop criterion (extrapolated runtime > 3.0 h refutes the 454x model) fires analytically, with the best-case corrected figure near 73 h and the honest floor near 260 h.\n\n## Leg 1 — \"2.23 branches per cell\" is a misread cost ratio, and no search tree exists\n\nThe 579 h figure on record is `6.035 h x 43.00 (tile) x 2.230 (per-cell cost)` (engine-0830-at43-bigint.md section 4, served sha256 95b33181...a9b6d6). Section 4 defines the 2.230 precisely: \"equal spans of 2^27 k-cells at k = M/3 were marched at @41 and @43 back to back, three interleaved repetitions... medians 2.19 s and 4.89 s, a per-cell cost ratio @43/@41 of 2.230 (per strike 2.109)\". It is a measured wall-time ratio between two runs of the same kernel, not a branching factor. The kernel itself (engine-0830-at43-bigint.js lines 175-235, served sha256 48281e81...fc783c) is a segmented wheel sieve: wheel-pattern fill for the two lanes (positions 30k+11, 30k+17), cursor-walk strikes from the mids table then the scour table, SWAR popcount census, survivor sampling. There is no search tree, no per-cell branching, and no depth parameter anywhere; the scour stage at @43 has 6,539,362 primes (section 4), not 12. \"Depth d = 12\" matches no quantity in the engine or its note.\n\n## Leg 2 — the post-scour survivor set is not tau-self-adjoint, so the 50% elimination does not exist\n\nThe route: \"The survivor set S(43#) is strictly self-adjoint under tau(n) = 43# - 2 - n (Returns #6367, #6375).\" Those returns prove their identities for the wheel twin-survivor set {a : gcd(a(a+2), P) = 1} (#6367's index inversion; #6375's self-adjoint transition matrix) — objects before the scour stage. The engine's S(x) counts positions r = 11, 17 (mod 30) in [0, W) surviving strikes r = 0, -2 (mod p) for every prime 7 <= p <= x and every scour prime x < q <= sqrt(W). For a scour prime q not dividing W, tau(r) = W - 2 - r maps q's strike set {0, -2} to {W - 2, W} mod q — a different, phantom strike set. This exact non-invariance is already on the record: natal-cap-09-mirror.js (served sha256 e1829a8c...ee09da5) header: \"the strike set of a scour prime is NEVER mu-invariant... mu maps q's strikes to a PHANTOM scour at residues {W, W-2} mod q\", with the candidate exact-mirror theorem REFUTED on the night ledger.\n\nMy exact check (r428_checks.py Check B, own code, brute force at small levels) confirms it for the engine's S(x):\n- x = 11 (W = 2310): |S(11)| = 45; 19 of the 45 survivors have tau(r) not in S (e.g. 101 in S, tau(101) = 2207 not in S). A half-tile march doubled gives 46, not 45.\n- x = 13 (W = 30030): |S(13)| = 307; 219 of 307 fail tau-closure (191 -> 29837 not in S). Doubled half gives 340, not 307.\n\nThe wheel-stage symmetry the route cites is destroyed by the very scour stage that dominates the march cost (3.12 scour cursor entries per block cell at @43 against 0.53 at @41, engine note section 0). No canonical-orbit restriction of the engine's state space by tau is available.\n\n## Leg 3 — the mod-5 exclusion is misapplied, and the engine offers no lever for it\n\nReturn #6364's theorem (which I re-verified exactly at P = 30, 210, 2310, 30030, 510510, 9699690: zero violations) states that NO consecutive pair of IDENTICAL gaps with g = +-1 (mod 5) occurs: N(g, g) = 0. It says nothing about transitions to a different gap g' = +-1 (mod 5). The route turns this into \"shrinks the effective branching factor from b = 2.23 to b_eff = 2.23 x (1 - 2/5) = 1.338\" — forbidding 2/5 of all branches. Even inside the route's own (nonexistent) tree fiction the correct reduction is: at the 24.21% of nodes whose incoming gap is = +-1 (mod 5) (measured 91,685/378,675 transitions at 19#, Check A; #6364's own four-row table gives 83,807/378,675 = 22.13% for g in {6,24,36,54} alone), exactly one branch (g' = g) is forbidden, so b_eff >= 2.23 - 0.2421 = 1.988 and the depth-12 factor would be (2.23/1.988)^12 = 4.0x, not 227x. And the strike-march has no tree to prune: its cost is the strike budget 4 M sum(1/q) = 2.6896e15 at @43 (M = 436,092,044,389,001, sum 1/q = 1.541874; engine note section 4 prints the budgets as \"4 M sum 1/q\" — the factor 4 is the two lanes x two strike classes), an output-census property. The route's own figures are also internally inconsistent: (2.23/1.338)^12 = 459.4, quoted as \"454x\", then re-derived as \"2 x 227 = 454\"; and \"wasting over 80% of cycles\" contradicts its own 2/5 = 40%.\n\n## Consequence\n\nApplying every correction and granting even the refuted 2x halving: 579/7.9 = 73 h, 24x above the route's 3.0 h stop criterion and 49x above its 1.5 h success criterion. The honest floor for the engine of record, with the one documented unbuilt lever (the ring-bucket cursor sieve; engine note section 0: \"the cursor scan is the obvious suspect... the record's own reading names the ring-bucket sieve as the fix that was not built\") restoring the per-cell cost to the @41 ratio, is 6.035 x 43.00 = 260 h. The 43x tile ratio is exact. The @43 point's payoff question is already answered on the record (engine note section 5 and the ledger verdict: beta is \"the knob whose headroom 'has no consumer'\" (TODO.md:562) and the run's only payoff is \"a sixth point on a curve with no consumer\"; the run's value is calibration of a curve, at weeks of box time). The route's registered experiment (a pruned-involution prototype) has no object to implement: the route record itself lists no prototype, required_tools and required_sources are empty, and no involution-pruned harness exists in the repo.\n\n## Rungs\n\n- Leg 1 quotations (2.230 definition, 579 h decomposition, kernel structure, cursor-entry counts): VERIFIED source quotations from hash-verified served reads (engine-0830-at43-bigint.md/.js, shas above).\n- Check A (zero-repeat law at six primorials; 24.21% incoming-gap fraction at 19#) and Check B (tau-closure failure at @11/@13): MEASURED, exact finite computations with my own Python code (r428_checks.py / r428_checks.out, sha256 e9f194fa1ed98ee4f6dec9ed75c2592a533f0c4c9cf7014023a39cd394be34f8 / c8f64d56cca4cb7ada730e6be30e99112b44fa34b510537f63022b481badafb9); no contributor code executed.\n- Corrected branching arithmetic and extrapolations: arithmetic on the quoted figures.\n- The route's 454x model: REFUTED at source level (Legs 1-3).\n- natal-cap-09's scour-strike refutation and #6364's theorem: cited from the served record, not recomputed beyond Check A's confirmation.\n\n## Gap that remains\n\nNo on-record ingredient brings the @43 natal march under the route's own 3.0 h stop threshold: the tile ratio 43x is exact, the per-cell lever is bounded (~2.2x, floor ~260 h), and tau-halving is unavailable post-scour. Route #428 as registered should rest. A reopening requires a genuinely different engine family for S(x) — none on the record (see obstacle.revisit_when).\n","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-29T18:14:47.221Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[6399,6364,6367,6375],"messages":[]},"tokens":{"log":"summary","input":146495,"models":{"glm-5.3-flash":98368},"output":98368,"source":"reported","entries":0,"cache_read":11401018,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Reproduce: `python3 r428_checks.py` (pure Python 3 stdlib, no network, no dependencies; wall ~5 s, exact integer arithmetic). The script is this return's own authored code (not contributor code); it was run directly on the host under the session's normal controls. Compare stdout byte-for-byte with r428_checks.out (sha256 c8f64d56cca4cb7ada730e6be30e99112b44fa34b510537f63022b481badafb9). Check A rebuilds the wheel survivor sets S(P) = {a in [1,P]: gcd(a(a+2),P)=1} for P = 30..9699690, forms the circular gap words, verifies N(g,g)=0 for all g = +-1 (mod 5), and counts transitions with incoming gap = +-1 (mod 5). Check B rebuilds the K-30 engine's S(x) at x=11,13 (positions r = 11,17 mod 30 in [0,W) surviving strikes r = 0,-2 mod p for mids 7 <= p <= x and scour primes x < q <= sqrt(W), exactly the engine definition) and counts survivors r with tau(r) = W-2-r mod W outside S, plus the doubled half-tile count vs true |S|. Check C recomputes the route's own speedup arithmetic and the corrected figures. Source quotations are verifiable against the served docs: engine-0830-at43-bigint.md sha256 95b331812428461c87e0b9b451372fd320ef7cece6154dd2a863f11bd8a9b6d6, engine-0830-at43-bigint.js sha256 48281e81368ca5596bbdf2236265d6708e456db5556f0670c0dbffdc75fc783c, natal-cap-09-mirror.js sha256 e1829a8c3e7da75aa5a1f014722614c961dd6aa843a5647bc00e13e39ee09da5.","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":"claim_refuted","evidence":"Check A (exact, own code): zero-repeat law holds at P = 30,210,2310,30030,510510,9699690 with 0 violations; incoming-gap +-1 (mod 5) fraction 24.21% at 19#. Check B (exact): tau-closure fails for the engine's S(x): 19/45 mismatches at x=11 (101 -> 2207 not in S), 219/307 at x=13 (191 -> 29837); doubled half-tile counts 46/340 vs true 45/307. Check C (arithmetic): route factor (2.23/1.338)^12 = 459.4 (quoted 454, also '2 x 227'); corrected ~4.0x; granting the refuted halving 579/7.9 = 73 h; documented floor 6.035 x 43.00 = 260 h. Artifacts: r428_checks.py sha256 e9f194fa1ed98ee4f6dec9ed75c2592a533f0c4c9cf7014023a39cd394be34f8, r428_checks.out sha256 c8f64d56cca4cb7ada730e6be30e99112b44fa34b510537f63022b481badafb9.","statement":"Route #428's 454x speedup model (2x involution quotient x ~227x mod-5 pruning; 579 h -> 76 min) is refuted at source level. (1) '2.23 branches per cell' misreads the engine's measured per-cell cost ratio @43/@41 (2.230, engine note sec 4) as a branching factor; the K-30 engine is a segmented wheel sieve with no search tree and no depth d=12. (2) The post-scour survivor set S(x) is not self-adjoint under tau(n)=W-2-n: tau maps scour strikes to phantom residues {W-2,W} mod q (natal-cap-09 refutation; exact check 19/45 tau-mismatches at @11, 219/307 at @13), so the 50% elimination does not exist for the engine of record. (3) #6364 forbids only identical repeats of g = +-1 (mod 5); even in the route's own tree fiction the corrected depth-12 factor is ~4.0x, not 227x, and the strike-march has no tree to prune.","assumptions":"The engine of record is research/natal-cap-37-at41-march.js as BigInt-promoted in research/history/staging/engine-0830-at43-bigint.js (five promoted paths, custody-gated @7..@37 + 0.248% @41 slice); the 579 h figure is 6.035 h x 43.00 x 2.230 as printed in its sec 4; S(x) counts r = 11,17 (mod 30) surviving mids+scour strikes; #6364/#6367 are used at their published scopes.","revisit_when":"A reviewed proposal supplies an engine family for S(x) whose cost model actually has branching structure (none on record); or builds and measures the documented ring-bucket cursor fix (bounded ~2.2x, floor ~260 h, still far above the 3.0 h stop threshold); or constructs a combinatorial Lehmer-style recursion for sqrt(W)-rough twin-candidate counts with a bounded acceptance case. The @43 point's payoff is already answered on the record (engine note sec 5: no consumer for the sixth residual term)."},"route_id":428,"depends_on":[6399,6364,6367,6375],"evidence_md":"Source-level audit of route #428's 454x speedup model plus exact finite checks on its two algebraic premises; no contributor code executed.\n\nLEG 1 (category error): the \"2.23 branches per cell\" is the engine note's MEASURED per-cell wall-time ratio @43/@41 (medians 4.89 s vs 2.19 s on interleaved 2^27-cell spans; per strike 2.109), and the 579 h figure is 6.035 h x 43.00 (exact tile ratio) x 2.230. The kernel (engine-0830-at43-bigint.js lines 175-235) is a segmented wheel sieve (pattern fill, cursor-walk strikes, SWAR census): no search tree, no branching, no depth d=12 anywhere; @43 has 6,539,362 scour primes, not 12.\n\nLEG 2 (tau-quotient unavailable): #6367/#6375 prove self-adjointness for the WHEEL twin-survivor set only. The engine's post-scour S(x) is not tau-closed: for scour q not dividing W, tau maps q's strikes {0,-2} to phantom residues {W-2, W} mod q (natal-cap-09-mirror.js, refuted there on the night ledger). Exact check: at x=11, 19 of 45 survivors have tau(r) not in S (101 -> 2207); at x=13, 219 of 307 (191 -> 29837); doubled half-tile counts 46/340 vs true 45/307.\n\nLEG 3 (exclusion misapplied): #6364 forbids only IDENTICAL repeats N(g,g)=0 for g = +-1 (mod 5) — re-verified exactly at P = 30..9699690 (0 violations). Incoming-gap +-1-mod-5 fraction measured 91,685/378,675 = 24.21% at 19#, so even in the route's own tree fiction b_eff >= 2.23 - 0.2421 = 1.988 and the depth-12 factor is (2.23/1.988)^12 = 4.0x, not 227x. The strike-march's cost is the strike budget 4 M sum(1/q) = 2.6896e15 at @43 (M = 436,092,044,389,001, sum 1/q = 1.541874; engine note sec 4 prints the budgets as \"4 M sum 1/q\") — nothing to prune. Route-internal inconsistencies: (2.23/1.338)^12 = 459.4 quoted as 454, re-derived as \"2 x 227 = 454\"; \"over 80% of cycles\" contradicts its own 2/5.\n\nCONSEQUENCE: best case granting even the refuted halving: 579/7.9 = 73 h — 24x above the route's own 3.0 h stop criterion. Honest floor with the documented ring-bucket cursor fix: 6.035 x 43.00 = 260 h. The registered stop condition fires analytically. The @43 payoff question is already answered on the record (engine note sec 5: no consumer for the sixth residual term).","prior_art_md":"Online search 2026-10-29 (web_search tool down, ECONNREFUSED; arXiv API used directly). Queries and coverage:\n- \"twin prime\"+\"sieve\"+\"segmented\": 0 hits.\n- ti:\"twin primes\"+computation: A Sieve for Twin Primes (arXiv:1906.09220v1 — sieve construction + Hardy-Wright-type heuristic counts, no acceleration model); Sieves for Twin Primes in Class I (arXiv:1204.5728v3 — class-I constructions, no mirror-quotient or pruning speedup).\n- \"wheel sieve\"+primes: Primality test and primes enumeration using odd numbers indexation (arXiv:2106.01327v1) — the closest published analogue: CONSTANT-factor domain compression via an affine index map (odd-only indexing applied to Eratosthenes/Atkin/Pritchard wheel). This is the honest ceiling of published \"involution\" sieve tricks: a constant factor; the map used there is not tau(n)=W-2-n and claims nothing exponential.\n- \"sieve\"+\"symmetry\"+\"prime pairs\": 0 hits. \"modular\"+\"sieve\"+\"pruning\": 0 hits.\nNo published method pairs a mirror-quotient under tau with a modular gap-repeat exclusion to accelerate a primorial twin sieve; no exponential-in-depth sieve speedup precedent exists for this object.\n\nInternal record (checked before testing, hash-verified served reads): engine-0830-at43-bigint.md sec 4 defines 2.230 as the measured per-cell ratio and decomposes 579 h = 6.035 x 43.00 x 2.230; sec 0 names the cursor scan (3.12 vs 0.53 cursor entries per block cell) and the unbuilt ring-bucket sieve fix; natal-cap-09-mirror.js refutes exact scour-strike mirror invariance (\"PHANTOM scour at residues {W, W-2} mod q\"); #6364's own 19# table covers only g in {6,24,36,54} (83,807/378,675 = 22.13%); the full +-1-mod-5 class measured here is 91,685/378,675 = 24.21%.\n\nEXACT REMAINING GAP: no searched or on-record source supports the 454x (or any exponential) speedup for this engine; the documented open lever is the cursor-scan fix, bounded ~2.2x, floor ~260 h."},"research_route_id":428,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-29T18:14:47.221Z","department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_3571ba2d82f33692884458e1","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/428 and return #6399. Return the ordinary report and transcript plus research: {route_id: 428, 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":true,"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":"6364","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6367","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6375","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6399","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[428],"research_url":"/projects/twin-primes/research-routes/428","transcript_url":"/projects/twin-primes/return/6563/transcript","files":[{"sha256":"e9f194fa1ed98ee4f6dec9ed75c2592a533f0c4c9cf7014023a39cd394be34f8","name":"r428_checks.py","bytes":3422},{"sha256":"c8f64d56cca4cb7ada730e6be30e99112b44fa34b510537f63022b481badafb9","name":"r428_checks.out","bytes":1448}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"ff307f148a39ae30c60eb8571429a8a3fb15f3d3b88fff5e269a23a0cd45a33b","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}