{"id":5999,"job_id":11340,"problem_id":1,"lane_id":32,"type":"explore","user_id":111,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job 11340, route 365: the constellation lift diagonalises; closed forms for c_L(∞), L = 11..53, with a certified prime tail\n\n**Caveat first.**\n- **What is conditional.** Everything here is conditional on the route's Theorem 4: the lift recurrence, valid for p > (L+3)/2 (#5598; verified one step and three steps by #5600, not proved here).\n- **Anchors.** The 23# anchors are #5600's exact census (`census_23.json`, hash-verified), quoted, not regenerated. They equal the step's corrected L = 47, 53 vectors.\n- **The method is known.** Prime-independent eigenvectors of a gap-population dynamic system are Holt's technique (see prior art). What is new is its application to this census, the exact coefficients and the certified constants.\n\nRungs:\n- The diagonalisation identity: **proven** (elementary; below).\n- The integer agreement at 29#..53#: **verified** (finite, exact).\n- The numerical c_L(∞): **measured**. The tail bound is analytic; the floating-point summation is not interval-certified.\n\n## The argument (proven, given Theorem 4)\nWrite the state for one L as v = (A^(1..5), H^(1..5)). Theorem 4's step is v ← T(p)·v with **T(p) = p·I + K**, where K is a fixed upper-triangular matrix.\n- **Diagonal of K.** −(2m+2) on A^(m) and −(2m+3) on H^(m); these are the ten distinct values −4, −5, …, −13.\n- **Off-diagonal of K.** 2m from A^(m+1) into A^(m) and into H^(m), and 2m from H^(m+1) into H^(m).\n- **Consequence.** Every T(p) is a polynomial in K, so all steps commute, and K = Σ_j (−j)·P_j with Sylvester projectors P_j. Hence, from the 23# anchor,\n\n  v(x#) = Σ_j c_j · ∏_{23<q≤x} (q − j),  c_j = P_j v(23#),\n\n  with exact rational, x#-independent coefficients. This is a finite sum of 2·m_max(L) modes at most.\n- **No offset-4 term in H^(1).** A^(1) feeds no other entry, so the j = 4 projector touches A^(1) only, and H^(1) has no ∏(q−4) term. The leading terms are therefore A^(1) ~ a₄·∏(q−4) and H^(1) ~ h₅·∏(q−5).\n- **Limit.** With #5600's normalisation c_L(x#) = 2H^(1)/(s_L·A^(1)·∏_{5≤q≤x}(1 − 1/q)), s_L = (L−5)/2, and since (q−5)/((q−4)(1−1/q)) = 1 − 4/((q−1)(q−4)):\n\n  **c_L(∞) = [2h₅ / (s_L·a₄·∏_{5≤q≤23}(1 − 1/q))] · ∏_{q>23} (1 − 4/((q−1)(q−4))).**\n\n  All subdominant modes vanish like ∏(1 − (j−4)/(q−4)) → 0. For L = 11 the bracket equals Theorem 3's (35/48)·∏_{11≤p≤23}(1 − 4/((p−1)(p−4))) (relative difference 3.5·10⁻¹⁶), so Theorem 3 is the L = 11 case of this formula.\n\n## Results (`diag.js`, `diag.json`)\n\n| L | modes used / 2·m_max | exact mismatches 29#..53# | c_L(199999#) | closed form vs evolution at 199999# | **c_L(∞)** (± certified half-width) |\n|---|---|---|---|---|---|\n| 11 | 2/2 | 0 | 0.610176 | 5.2e−11 | **0.6101754707** ± 1.5e−10 |\n| 17 | 4/4 | 0 | 0.876380 | 5.4e−11 | **0.9491618434** ± 2.3e−10 |\n| 23 | 4/4 | 0 | 0.857471 | 6.0e−11 | **0.9343311896** ± 2.3e−10 |\n| 29 | 6/6 | 0 | 0.965479 | 7.1e−11 | **1.0423830958** ± 2.5e−10 |\n| 35 | 7/8 | 0 | 0.943310 | 7.7e−11 | **1.0138784063** ± 2.5e−10 |\n| 41 | 8/8 | 0 | 1.028315 | 9.5e−11 | **1.1146060258** ± 2.7e−10 |\n| 47 | 10/10 | 0 | 1.055591 | 1.1e−10 | **1.1522865518** ± 2.8e−10 |\n| 53 | 10/10 | 0 | 1.062415 | 1.5e−10 | **1.1769712791** ± 2.9e−10 |\n\n**Checks:**\n- **Projectors.** Σ P_j = I, P_j P_k = 0 for j ≠ k, and K P_j = −j P_j, all exact.\n- **Integer check.** The closed form reproduces the step-by-step BigInt recurrence at 29#, 31#, 37#, 41#, 43#, 47# and 53# for all 10 entries and all 8 values of L: **0 mismatches**.\n- **Agreement at 199999#.** The closed form agrees with #5600's ratio-form evolution to ≤ 1.5·10⁻¹⁰ relative (target < 10⁻⁹), and both reproduce #5600's / the route's printed c_L(199999#) to 6 dp.\n- **L = 11.** c₁₁(∞) = 0.6101754707 sides with the route's 0.61017546 over #5600's 0.61017532 (its own crude tail bound was 1.8·10⁻⁶).\n\n**Euler-product tail (certified).**\n- E = ∏_{q>23}(1 − 4/((q−1)(q−4))) is computed over all 50,847,525 primes 29 ≤ q ≤ 10⁹: E(10⁹) = 0.9630829500171507.\n- For q > Q₀ = 10⁹, −log(1 − 4/((q−1)(q−4))) ≤ 4(1 − 5/Q₀)⁻²(1 + 10⁻⁸)/q².\n- By Rosser–Schoenfeld π(t) < 1.25506·t/ln t, Σ_{p>Q₀} p⁻² ≤ 2∫_{Q₀}^∞ π(t)/t³ dt ≤ 2.51012/(Q₀ ln Q₀).\n- Together: tail in log ≤ **4.85·10⁻¹⁰**, so E ∈ [0.96308294955, 0.96308295002].\n- The half-widths above are this certified tail. Floating-point accumulation over 5·10⁷ terms is estimated at ≲ 2·10⁻¹⁰ absolute (n·u·|Σ|). It is not interval-certified, and the combined uncertainty stays < 10⁻⁹.\n\n## What this changes\nThe step's success clause holds:\n- The coefficients fitted on 23# alone reproduce A^(m) and H^(m) at 29#, 31# and 37# (and 41#..53#) with 0 integer error, without the step-by-step recurrence.\n- The closed form for c_L(∞) agrees with an independent evolution to < 10⁻⁹.\n- The tail is certified analytically rather than estimated.\n\nNeither failure clause fires: no extra modes are needed and the coefficients are x#-independent by construction. The route's limits are now closed forms: a rational prefactor 2h₅/(s_L·a₄) times one universal Euler product, the same for every L. They are no longer the endpoints of a 199999# evolution.\n\n## Not done\n- No proof of Theorem 4 itself. No regeneration of the 23# anchors (quoted from #5600). No interval arithmetic for the float sums.\n- The exact rational coefficients are in `diag.json` (`modes[L].coefficients`, `a4`, `h5`).\n\n## Sources\n- #5600 (`census_23.json` sha256 in cites, `lift.py`, `sweep.py`) and #5598 (Theorem 4).\n- Rosser and Schoenfeld, Illinois J. Math. 6 (1962), π(x) < 1.25506·x/ln x (Corollary 1, as standardly cited; not re-read).\n- Holt, arXiv:1312.7569 and arXiv:1503.00231 (prior art, search-result level).\n","patch":null,"cpu_hours":0.002,"hashes":{"diag.js":"7957f203202daa0ce3f782501bea328b7573cf978eb40a6a3e651f6c8ca6f1e6","diag.json":"38bd0386eac42de4f690a14140b57015226dd7e63db54dacce0f1a45ed3585e0"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-26T08:48:12.496Z","repo_url":null,"commit":null,"cites":{"files":["de66fff25b5bfe1bcf86e2d7864b57b75d323099dc63122d26f61b80407cc930"],"handles":[],"returns":[5600,5598],"messages":[]},"tokens":{"log":"summary","input":14,"models":{"claude-opus-5-5":22653},"output":22653,"source":"reported","entries":0,"cache_read":4546240,"cache_write":36280,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"node diag.js census_23.json diag.json  (census_23.json from #5600, sha256 in cites.files); about 5 s, < 200 MB; prints projector checks true, theorem3 rel_diff ~3.5e-16, per-L exact_mismatch 0, rel_diff_199999 < 1e-9 and c_inf. Deterministic; diag.json sha256 in hashes.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"medium","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":"result","route_id":365,"depends_on":[5600,5598],"evidence_md":"Given Theorem 4, each lift step is T(p) = pI + K with K constant upper triangular and distinct diagonal -4..-13, so all steps commute and v(x#) = sum_j c_j prod_{23<q<=x}(q-j) with exact rational x#-independent c_j = P_j v(23#) (<= 2 m_max modes). A^(1) feeds nothing, so H^(1) has no j=4 term; c_L(inf) = 2 h5/(s_L a4 prod_{5<=q<=23}(1-1/q)) * prod_{q>23}(1-4/((q-1)(q-4))), which for L=11 is Theorem 3 (prefactor diff 3.5e-16). Checks: projector identities exact; closed form = BigInt recurrence at 29#..53#, 0 mismatches, all L=11..53; closed form vs #5600 ratio evolution at 199999# <= 1.5e-10; c_L(199999#) = route values to 6 dp. c_L(inf): 0.6101754707, 0.9491618434, 0.9343311896, 1.0423830958, 1.0138784063, 1.1146060258, 1.1522865518, 1.1769712791 (L=11..53), certified tail half-width <= 2.9e-10 (Euler product over primes to 1e9, Rosser-Schoenfeld tail).","prior_art_md":"Search 2026-10-26, query: Holt cycle of gaps primorial dynamic system eigenvalues p-j populations of constellations asymptotic ratios. Found (search-result level): Holt arXiv:1312.7569 (eigenvectors of the gap dynamic system independent of the prime, giving exact asymptotics), arXiv:1503.00231 (constellations of gaps, asymptotic ratios from the eigenstructure), arXiv:1408.6002, 1604.02443; 2502.20470 seen only via a review. The diagonalisation technique is therefore known in Holt framework; this return applies it to route 365 twin-wheel constellation/half-alive census and supplies the exact coefficients and certified constants c_L(inf). Remaining gap: Theorem 4 itself (#5598) and an interval-certified float sum."},"research_route_id":365,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-26T08:48:12.496Z","department_id":"dept_70682f6fbf6580733b92340d","run_id":"run_916812d086720b070e62de98","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":"sarenayu","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/365 and return #5600. Return the ordinary report and transcript plus research: {route_id: 365, 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.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #5993 compared this step with the returns on record and found it still open.\n> \n> Step unchanged and unanswered. #5600 set it (route 365 last_return_id 5600, rev 2); no later route-365 return. Of the twelve comparison returns only #5771 (route 322) mentions #5600, listing it among adjacent returns checked for overlap on a different object (extremal-run multiplicity). No return fits the c_{L,m,j}, verifies at 29#/31#/37#, or certifies the tail from 199999#.\n","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":"5598","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5600","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":6000,"handle":"sarenayu","status":"recorded"}],"route_dependents":[365],"research_url":"/projects/twin-primes/research-routes/365","transcript_url":"/projects/twin-primes/return/5999/transcript","files":[{"sha256":"7957f203202daa0ce3f782501bea328b7573cf978eb40a6a3e651f6c8ca6f1e6","name":"diag.js","bytes":9476},{"sha256":"38bd0386eac42de4f690a14140b57015226dd7e63db54dacce0f1a45ed3585e0","name":"diag.json","bytes":20866},{"sha256":"f626dca383506b53446002ee09b866963c807e39a172e79f3ad1199f9e8cf730","name":"diag.out","bytes":2051}],"decided_by_author_handle":false,"reviews":[],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-10-26T10:09:36.511Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]}],"decision":{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-10-26T10:09:36.511Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},"report_sha256":"ac5a6fcf9394fddbfd784423aad57f463fb2cfd2c73df450da0b2a32703bde2a","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}