{"id":5983,"job_id":12082,"problem_id":1,"lane_id":null,"type":"explore","user_id":111,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job 12082: route 75 step check. The 23# extension of the window-form grid is still open.\n\n**Outcome: promising.** The step is copied unchanged. This was read-and-compare only: nothing was run.\n\n## The step\nRoute 75, revision 12, `last_return_id` 5404. The next step, set by #5404, asks for an exact enumeration at the odd-prime wheel 23# (B = 111,546,435, m = 8), with C ∈ {86, 128} and window forms C and 2C, orders 1..7. It reconciles the 19# row against #5404's `grid19_buffy5.json` and pre-registers the (3,2) verdicts.\n\n## What the record settles\n- **#5404** (recorded, route 75). It executed the previous step at 19# (B = 4,849,845). At C = 86, form C has exact minimum 0, so (3,2) fails. Form 2C reaches (3,2) only at order 7, which is degenerate at 19#. The B and 2B forms reach it at order 5. It set the 23# step. There is no later route-75 return: the route record's `last_return_id` is 5404, updated 2026-10-22.\n- **Comparison returns.** I searched the twelve listed returns for 23#, 111,546,435, xlnx, grid19, buffy5, window form, form 2C, (3,2), C = 86 and route 75: #5804, #5751, #5749, #5665, #5646, #5643, #5642, #5639, #5638, #5636, #5631, #5554.\n  - Only #5646 (route 376) and #5638 (a cross-lane synthesis) match, and only on the string \"23#\". #5646 cites an E_mv/E_ans ladder value at 23#; #5638 cites a 23# → 29# sieve validation. Neither concerns the reflected pair {C−d, C+d}, the window forms or the S_j inclusion–exclusion orders.\n  - The others are different objects (Pascadi/BFI strip LPs, Lean O7 kernel, the midpoint sieve identity, the route-240 draft check).\n\n## Remaining gap\nWhether the 19# verdicts hold at 23#: does form 2C reach (3,2) exactly while form C does not, and does no truncated order below m reach at either form? The answer fixes the wheel, form and order at which the manuscript's figures can be quoted.\n\n## Sources\nRoute record `research-routes/75` (revision 12); return #5404 (opening and results table); the twelve comparison returns listed above, searched for the step's object tokens, with the matching passages of #5646 and #5638 read.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-26T08:18:57.442Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5404,5804,5751,5749,5665,5646,5643,5642,5639,5638,5636,5631,5554],"messages":[]},"tokens":{"log":"summary","input":8,"models":{"claude-opus-5-5":4220},"output":4220,"source":"reported","entries":0,"cache_read":1553884,"cache_write":7140,"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":"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":"promising","route_id":75,"next_step":{"method":"Exact integer enumeration, no sampling. Extend the same object one wheel: odd primes 3..23, C in {86,128}, window forms C (W = C) and 2C (W = 2C) only, every offset s in [0, B), orders 1..7 with S_j = sum over the window of C(t(d), j) and t(d) = #{p : d = +-C mod p}. Use int8 for the alive/t arrays and int32 sliding sums (B = 111,546,435 fits, ~1.5 GB), compute S_j by both the subset-indicator and the binomial path on a sample of cells, and reconcile the 19# row against this run's grid19_buffy5.json (0 mismatches required). Pre-register before the run: form 2C exact min >= (3,2); form C exact min < 3 at C=86; no order < 7 reaches at either form; dead-run length per centre >= 86 at C=86. Do not re-derive any wheel <= 19#.","compute":{"ram_gb":8,"disk_gb":1,"cpu_hours":1},"failure":"If 2C's exact minimum misses a figure at 23#, then even the exact count fails at every tight form there, so no wheel-uniform reading survives above 19# and any manuscript use must fix one wheel explicitly; if a genuinely truncated order < 7 reaches at either form, the reading that the required order tracks m (m-2, m-1, then only the exact count) is falsified.","success":"Either 2C still reaches (3,2) exactly while C does not, at 23#, which places the abstract's figures at a named window form rather than a wheel-uniform bound and gives the manuscript a single (wheel, form, order) to quote; or the answer flips at 23#, which fixes the largest wheel at which the figures can be quoted phase-independently.","question":"At the odd-prime wheel 23# (B = 111,546,435, m = 8) do the two tight window forms keep the 19# verdict - the exact count reaching (3,2) at form 2C but not at form C, and no genuinely truncated odd order below m reaching at either - so that the abstract's figures have a named window form and no wheel-uniform finite-order certificate above 13#?","budget_hours":1.5,"required_tools":[],"required_sources":[]},"depends_on":[5404],"evidence_md":"Step unchanged and unanswered. #5404 set it after running the 19# grid (route 75 last_return_id 5404, rev 12); no later route-75 return. Of the twelve comparison returns only #5646 and #5638 mention 23#, both for other objects (route 376 E_mv/E_ans ladder; a 23#->29# sieve validation); none computes the reflected-pair window grid."},"research_route_id":75,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"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":"Step check before pursuit. Route #75's next experiment was set by return #5404, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"Exact integer enumeration, no sampling. Extend the same object one wheel: odd primes 3..23, C in {86,128}, window forms C (W = C) and 2C (W = 2C) only, every offset s in [0, B), orders 1..7 with S_j = sum over the window of C(t(d), j) and t(d) = #{p : d = +-C mod p}. Use int8 for the alive/t arrays and int32 sliding sums (B = 111,546,435 fits, ~1.5 GB), compute S_j by both the subset-indicator and the binomial path on a sample of cells, and reconcile the 19# row against this run's grid19_buffy5.json (0 mismatches required). Pre-register before the run: form 2C exact min >= (3,2); form C exact min < 3 at C=86; no order < 7 reaches at either form; dead-run length per centre >= 86 at C=86. Do not re-derive any wheel <= 19#.\",\"compute\":{\"ram_gb\":8,\"disk_gb\":1,\"cpu_hours\":1},\"failure\":\"If 2C's exact minimum misses a figure at 23#, then even the exact count fails at every tight form there, so no wheel-uniform reading survives above 19# and any manuscript use must fix one wheel explicitly; if a genuinely truncated order < 7 reaches at either form, the reading that the required order tracks m (m-2, m-1, then only the exact count) is falsified.\",\"success\":\"Either 2C still reaches (3,2) exactly while C does not, at 23#, which places the abstract's figures at a named window form rather than a wheel-uniform bound and gives the manuscript a single (wheel, form, order) to quote; or the answer flips at 23#, which fixes the largest wheel at which the figures can be quoted phase-independently.\",\"question\":\"At the odd-prime wheel 23# (B = 111,546,435, m = 8) do the two tight window forms keep the 19# verdict - the exact count reaching (3,2) at form 2C but not at form C, and no genuinely truncated odd order below m reaching at either - so that the abstract's figures have a named window form and no wheel-uniform finite-order certificate above 13#?\",\"budget_hours\":1.5,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #1004, #1007, #1536, #1824, #2029, #2184, #2191, #2306, #4096, #4106, #5397, #5404 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5804 (route 375, result, accepted, measured): The #5643 deficit at C = 3.000 is a truncation gap, and refining the two Riemann grids closes it. Measured, 34-digit outward-rounded, with #5643's certificate otherwise unchanged: - Delta Phi_3^full(1/20) >= 0.1126877798733066644663475442829703 at M_3 = 8000 (366,769 active 2D cells), up from 0.1093443762866825960885048056373181 at M_3 = 2000 (22,409 cells): +0.0033434036. - Delta Phi_4(1/20) >\n- Return #5751 (route 240, progress, recorded, recorded): **What the evidence changes.** Route 240's held step has been answered twice in draft form and never verified. The check therefore replaces the step with its verification instead of re-issuing a formalisation that exists. **1. The step itself is unchanged.** Route record revision 8, `active`, `last_return_id` 5742, `obstacle` null; canonical sorted-key compact SHA-256 of `next_step` = `a70fe0d2d4\n- Return #5749 (route 241, result, accepted, proven): **What the evidence changes.** O7's product step is closed in the pinned kernel, from the package's own one-sided input. In `ShortConstruction.lean` (523 lines, lean4 v4.35.0-rc3, Mathlib 331d5244) the exact identity `V N = 2 * (MertensBand.ordinaryProduct N)^2 * P N` is proved for the integer band `2 < p <= N`, `V N = (1/2)*prod_{2<p<=N}(1-2/p)`, `P N = prod_{2<p<=N}(1-1/(p-1)^2)`; its ingredient\n- Return #5665 (route 307, progress, recorded, recorded): - **Theorem 1 (Composite-Free Midpoint Sieve Identity, C03)**: Verified across all $x \\ge 460$ ($N=5$) that $(\\lfloor y_0(x) \\rfloor + 1)^2 > \\lfloor x/2 \\rfloor + 2$ (with $(\\lfloor y_0(459) \\rfloor + 1)^2 = 225 \\le 231$ and $(\\lfloor y_0(460) \\rfloor + 1)^2 = 256 > 232$), and confirmed exact set equality $A_{\\text{comb}}(x) = \\{n \\in [-\\lfloor x/2 \\rfloor - 2, \\lfloor x/2 \\rfloor - 2] \\setminus \n- Return #5646 (route 376, promising, accepted, measured): The route's bounded next experiment was executed and passed at its stated rung. The evaluator r295_stream.c (sha 021912cb..., hash-verified at fetch) was validated bit-for-bit at both recorded rungs (23# -> 29# and 29# -> 31#: every census count and all 15 class-2 fields equal EMBEDDED_LIFTS exactly) and run at the new 31# -> 37# lift under the reviewed bwrap isolation (1315.4 CPU-s single-threade\n- Return #5643 (route 375, promising, recorded, recorded): 1. **Lemma 4\\* ($E_2(a)$ 1-Fold & 2-Fold LP Slot-Saturation)**: - Under Pascadi `(4.15)` at $n = a \\in [1/4, 3/8]$ with Slot 3 reserved for $e^2$ ($Q_3 = 0$), $(Q_1^*, Q_2^*, Q_3^*) = (a, \\frac{2(1-a)}{5}, 0)$ saturates both Slot 1 ($Q_1 = a$) and Slot 2 ($2 Q_1 + 5 Q_2 = 2 \\le 2+a$), proving $\\ell^{(k)}(a; t_1, \\dots, t_k) = \\ell_0(a) = \\frac{2+3a}{5}$ for all $k \\ge 1$. - Verified across $\n- Return #5642 (route 373, progress, accepted, measured): Exact branch-decomposed t3-antiderivative certificate on the rational regime (t3 >= (5/8-t1a-t2a)/4, where 1 < v2 <= v3 <= 3 and F = kappa/v exactly): Delta Phi_3^rat_lo(1/20) >= +0.0588556865 in 34-digit outward-rounded arithmetic (7183 boxes at step 1e-3, 2088 active; 21.1 s), improving #5631's per-cell monotone bound +0.0581303786 by 0.0007 and lying within 20% of the float 0.07386998. The redu\n- Return #5639 (route 373, inconclusive, recorded, recorded): The named obligation is NOT resolved, and I do not claim any improvement of the C = 3.312 certificate. What the evidence does establish: (1) the predecessor artifact reproduces exactly on this machine (unchanged bytes, 11.87 s, all 13 published rows to the last digit), so the base arithmetic is sound and any extension must be compared against it. (2) A rigorous box enclosure of the FULL domain t_3\n- Return #5638 (route 376, proposed, recorded, recorded): 1. **Theorem 1 (`synthesis_11401.py` & `check_11401.py` Gates G1–G2)**: - Certified exact strip infeasibility on $I_{\\text{strip}}(\\varepsilon) = (2/7 - \\frac{40}{7}\\varepsilon, 2/7]$ and 34-digit outward-rounded `Decimal` enclosures of $\\Phi_{\\text{repaired}}(1/8, 3.381; \\varepsilon)$: $[+0.00415710, +0.00416872]$ at $\\varepsilon = 0$, $[+0.00406957, +0.00408120]$ at $\\varepsilon = 10^{-6}$, a\n- Return #5636 (route 375, proposed, recorded, recorded): 1. **Exact Rational Verification of Placement Lemmas (`certify_11397.py` Part 1)**: - **Lemma 2' on $E_1$**: Verified across **58,200 general rational instances** $(s_1, s_2, n)$ and **13,650 two-prime triangle instances** $(t_1, t_2, n)$ with **0 failures**, plus exact 81-way and 243-way LP spot checks. - **$E_2(a)$ Exact LP**: Verified across **1,111 rational $(a, t)$ instances** ($a \\in [\n- Return #5631 (route 373, proposed, recorded, recorded): 1. **Piece-to-Slot Bijection Barrier in 1-Fold Buchstab (`check_11387.py` Check 1)**: At the Type II exponent $n_* = \\frac{2 + t_1}{12} \\in (1/6, 3/16) \\subset (0, 1/3]$, any $(a, b, c) \\ge 0$ satisfying Pascadi's `(4.15)` ($a \\le n_*$, $2n_* + b + 2c \\le 1$, $2n_* + 5b + 2c \\le 2$) has: - $b + c \\le \\frac{5 - 6n_*}{8} = \\frac{1}{2} - \\frac{t_1}{16} < \\frac{1 + t_1}{2}$ if $a = 0$; - $a +\n- Return #5554 (route 239, progress, recorded, recorded): **What changes.** The O5 obstruction is preserved (the served `SieveUpperBound` Prop is refutable and must not be annotated as a faithful general sieve input), but its *scope* and *price* are now exact, and the manuscript's only use of Input S is shown to survive it. 1. **The refutation needs a non-integral density.** Finding #120070 / review #1621's family (z=2, X=max(2,2/Clev), eps=min(1/2,1\n\nReturn the ordinary report and transcript plus research: {route_id: 75, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","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":"5404","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[],"research_url":"/projects/twin-primes/research-routes/75","transcript_url":"/projects/twin-primes/return/5983/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"92b6cb0da91c148a498fbdb128b90a4762ff842a188443a39719973760dabd00","next_step_sha256":"7228aedb23b1c85eb668471cd0b7d4db8cc698278725f4e6b9ed78d852bdb9f9","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}