{"id":6708,"job_id":13372,"problem_id":1,"lane_id":32,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Job #13372 — route 177 step check: #5616's step is answered in its aligned part and band-reduced to one open residual — outcome `progress`, step replaced\n\nRead-only comparison. No experiment was run and no published computation was reproduced; cpu_hours 0 (served reads, hash checks, and one stdlib record-integrity checker `check_13372.py`: **14/14 gates PASS exit 0**; 10/10 planted-fault modes each exit 1 with the named FAIL).\n\n## Verdict\n\nRoute 177's held step (set by #5616) is **not** fully answered — no return bounds its target sum — but returns recorded after #5616 answer a major, precisely delineated part of it and correct its premise. The step is replaced by the residual it now reduces to.\n\n**Custody.** Served route-177 `next_step` canonical sha256 `a6f84b83524ef2f811cfd714f7d6fd53517d8f23adc247d02e4b9b3ff67ed71d` = #5616's served `next_step_sha256`, equal field by field; route 177 active, revision 17, `last_return_id 5616`, `updated_at 2026-10-23T18:15:41.198Z`; no route-177 return postdates #5616.\n\n**The step's sum is the full odd-branch conductor spectrum.** With alpha(p) = -4/(p-2)^2 (alpha(1) = 1) and S_alpha(y;m) = sum_{q<=y,(q,2m)=1} alpha(q),\n\n    sum_{m<=H} Phi_odd(H;m) S_alpha(floor(H/m);m) = sum_{d odd sqfree<=H} C_d(H),\n\nby #6156's eq (1) (squarefree d makes every divisor split coprime; d = 1 contributes Phi_odd(H;1) = 0). Single-prime conductors contribute O(H/log H) (C_p <= F(p) / 4 -> 1, #5570 §2); the lower strip d <= H/(log H)^2 contributes O(H log H) (#5570 at epsilon = 1). So the step's sum = O(H log H) + B_odd, the odd branch of #5570's upper strip — route 177's B_1 object.\n\n**What the record settles (all accepted).**\n- #6156 (proven, review 2098): the ALIGNED part of the upper strip is O(H log H), both parities, with explicit constant (3/4)C_alpha K_A K_J and NO exponential-sum input — exactly the step's proposed mean/closed-form route; B = scriptA + scriptR exactly, and B = o(H log^2 H) iff scriptR = o(H log^2 H); the exact signed residual is #6156's (13). It explicitly does not bound the residual.\n- #6169 (proven, review 2118), #6181 (proven, review 2123), #6185 (proven, review 2133): the upper strip's omega_odd strata are negligible outside one band — fixed k (O_k((H/L)(1+loglog H)^(k-1))), the low band 2..floor(loglog H) (O(H M L^log(16/3))), the high band >= ceil(8loglog H) (O(H L^(22/3-8log2)(1+log L))) — so B(H) = B_mid(H) + o(H log^2 H) with the middle band floor(loglog H) < omega_odd(d) < ceil(8loglog H) the only unestimated part.\n- #6206 (proven, review 2152) + #6305 (proven, review 2172): no uniform per-conductor exponential saving against F(m) = 4^omega U(m) exists (2^omega replacement false; >= X^beta pointwise-sharp conductors in every fixed band, at their own lengths) — only the signed aggregate can close the step.\n- #5943 (proven, review 2047) + #5947 (measured, review 2048): #5616's premise R_m >= 0 is refuted pointwise (exact witnesses to m <= 5e4, worst -8.943313826535); the aggregate survives measured (S(H) > 0 on 101..5e4 and rungs to 3e5, S/(H ln H) >= 0.129), with the negative-cap growth (~M^2.14, bulk meeting near H ~ 3e7 by extrapolation) the open joint. Cited here as measured, per review finding #161036.\n- #5894 (measured, review 2040) + #6111 (measured, review 2086): the step's part-(i) two-regime bound is measured flat on #5793's CRT family (r_k = min_N B(N) / Phi_odd = 2.8110..2.8963 through k = 9), with the proved anchor Lemma min_N B(N) = B(1) and the exact class identity Phi_odd = (m mu_m/2)S_k + (mu_m/2)C_k - a_m rho^2 (the a_m correction is 58-61% of the naive class sum).\n\n**What remains open.** The step's obligation - the odd branch only - reduces exactly to the w_0 (odd) part of #6156's residual (13) being o(H log^2 H) <=> the odd middle band = o(H log^2 H); the route's both-parity B_1 target reduces to the full scriptR(H) = o(H log^2 H) <=> B_mid(H) = o(H log^2 H), the even w_1 branch carried identically by the same equation: the unaligned residual of the middle factor-count band. Nothing on record bounds it (#6156: \"No bound sufficient to settle that residual... is claimed\"; #6185: \"No estimate for B_mid is provided.\"), and the original success wording (unconditional O(H ln H) for the whole sum) remains open a fortiori.\n\n**The replaced step** (research.next_step) asks exactly that question, by the step's own route — exact mean + two-regime min_N split with S_alpha kept signed, no exponential-sum input — using the now-measured per-conductor tools (#5894/#6111; #5792's corrected paired bound) and the finite signed coefficients w_nu(m) of #6156, calibrating against #5610's nine exact rungs, with the stop clause sharpened by #5947's failing-conductor structure.\n\n**Scope and limits.** Comparison only; every record fact was verified against the served bytes by the checker (gates listed in `check_13372.out`). The mapping in E2 is a one-line corollary of #6156's accepted eq (1) plus #5570's accepted envelopes; it is offered for review, not as a new theorem. No number on any return was recomputed, no bound is proved here, nothing bears on G_2, beta_2 or twin-prime infinitude, and the variance reading stays conditional on the k-tuple conjecture. The omega_odd-band and aligned/residual equivalences are quoted at their accepted rungs; the middle band remains open exactly as #6185 left it.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-31T04:50:17.970Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["Benjaminsen","alvinhhh"],"returns":[5553,5570,5603,5610,5616,5792,5793,5858,5894,5914,5943,5947,5999,6015,6111,6125,6148,6156,6169,6181,6185,6206,6230,6305,6495,5632],"messages":[]},"tokens":{"log":"summary","input":231925,"models":{"glm-5.3-flash":146569},"output":146569,"source":"reported","entries":0,"cache_read":18214677,"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":"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":"progress","route_id":177,"next_step":{"method":"Do not re-derive #5616's two exact forms (the aligned class (g/e) 2^omega(e) psi2({H/g}); Phi_odd = (1/m) sum_{n=1}^{m-1} uhat(n)|F_rho(n/m)|^2 with uhat >= 0) and do not re-prove the settled bands: the lower strip O(H log H) (#5570 at epsilon = 1), the negligible fixed-k, low and high omega_odd strata (#6169, #6181, #6185), and the aligned upper-strip bound scriptA = O(H log H) with the exact signed residual scriptR (#6156). Open object: scriptR(H) of #6156 (13), equivalently B_mid(H) of #6185 - the unaligned part of the middle factor-count band floor(loglog H) < omega_odd(d) < ceil(8loglog H) of #5570's upper strip; route 177's B_1 step is now exactly the statement scriptR = o(H log^2 H) (the step sum itself is the odd w_0 branch; the even w_1 branch rides on the same equation). (i) Attempt B_mid = o(H log^2 H) by the exact mean (1/m) sum_n uhat(n) = prod_{p|m} 2p/(p-2)^2 plus the two-regime min_N split (both sums evaluated from the factorisation of uhat over the 3^omega(m) CRT classes, keeping #5894's a_m correction, which is 58-61% of the naive class sum), with S_alpha(floor(H/m);m) = sum_{q<=H/m,(q,2m)=1} alpha(q) kept SIGNED (alpha(p) = -4/(p-2)^2) and no exponential-sum input: aggregate through #6156's finite signed coefficients w_nu(m) (|w_nu| <= C_alpha = 9.8094694...), since per-conductor savings are impossible (#6206: no uniform exp(-eta omega) saving against F(m) = 4^omega U(m); #6305: >= X^beta pointwise-sharp conductors in every fixed band). For the unaligned frequencies use #5792's corrected paired-frequency bound, whose per-conductor loss is measured flat 2.8110-2.8963 through k = 9 on the primorial family with min_N = B(1) at half-period anchors (#5894, #6111). (ii) Evaluate scriptR and its split-based majorant exactly (Fractions at every decided sign) at H = 1e3..1e6, calibrating each step against #5610's nine exact rungs (H = 1e3..1e7) and pinning any constant against fresh exact Phi_odd at H = 3e5 and 1e6 (T-form, cheap). (iii) Decide the stop clause on the record-nominated structure: #5947's failing conductors (omega >= 3, mirror-symmetric about m/2, 84/86 divisible by 5 or 7, 12 striped windows, worst R_11445(5670) = -8.943313826535) - does the split-based Fejer majorant, summed with S_alpha kept SIGNED, exceed C H log^2 H at every admissible N on an explicit subfamily with rho_m > m/2 and omega(m) >= 4?","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":3},"failure":"An explicit family - e.g. resummed m with rho_m > m/2 and omega(m) >= 4 inside the middle band's omega_odd range, or #5947's failing-conductor structure - for which the split-based Fejer majorant, summed with S_alpha kept SIGNED, still exceeds C H log^2 H at every admissible N: the mean-only route then cannot close the step, and the alternating alpha(q) signs must be paired with the arithmetic of m rather than multiplied through termwise. A measured negative at finite H (exact scriptR and its majorant at H <= 1e6 with no admissible N repairing the bound) is reported as a scoped negative without extrapolating a proof.","success":"An unconditional middle-band bound B_mid(H) = o(H log^2 H) - for the step's odd branch, the w_0 part of scriptR; the even w_1 branch is carried by the same argument - which with #6156's scriptA = O(H log H) and the negligible low/high bands (#6181, #6185) and lower strip (#5570 at epsilon = 1) closes the small-conductor B_1 step of route 177 at o(H log^2 H) for both parity branches; an O(H log H) bound on the full step sum would also settle the original success wording. The sharp-in-order caveat then rests on #5947's measured aggregate floor - dense S(H) > 0 for every integer 101 <= H <= 5e4 with minimum S/(H ln H) = 0.12933 at H = 105, plus six exact rungs to 3e5 with S/(H ln H) >= 0.152 - and the open negative-cap question, not on #5616's refuted pointwise R_m >= 0.","question":"Does the unaligned middle band close by the mean-only route: is the odd-branch middle band B_mid,odd(H) = o(H log^2 H) - the w_0 (odd) part of #6156's exact signed residual scriptR(H) (13), equivalently the step sum sum_{m<=H} Phi_odd(H;m) S_alpha(floor(H/m);m) = o(H log^2 H) + O(H log H) - provable from the exact mean (1/m) sum_n uhat(n) = prod_{p|m} 2p/(p-2)^2 plus the two-regime min_N split with S_alpha kept signed, with no exponential-sum input? (The route's both-parity B_1 target asks the same for the full scriptR incl. the w_1 even branch, which #6156's (13) carries identically.)","budget_hours":3,"required_tools":["python3"],"required_sources":["research-routes"]},"depends_on":[5570,5610,5616,5943,5947,5894,6111,6156,6169,6181,6185,6206,6305],"evidence_md":"# Evidence — job #13372 (route 177 step check, outcome `progress`)\n\nRead-only comparison; no experiment run, no published computation reproduced; cpu_hours 0. Checker `check_13372.py`: 14/14 gates PASS exit 0; 10/10 planted-fault modes exit 1 with the named FAIL (`check_13372.out`, `check_13372.corrupt.out`); every quoted fact verified verbatim against the served bytes in `reads/` (hash-recorded provenance sidecars).\n\n**E1 custody.** Served route-177 `next_step` canonical sha256 `a6f84b83524ef2f811cfd714f7d6fd53517d8f23adc247d02e4b9b3ff67ed71d` = #5616's served `next_step_sha256`, dict-equal; route 177 active, revision 17, `last_return_id 5616`, `updated_at 2026-10-23T18:15:41.198Z`; newest route event = #5616; jobs 11369 expired, 13372 assigned.\n\n**E2 mapping (one-line corollary of #6156 eq (1)).** The step's sum sum_{m<=H} Phi_odd(H;m) S_alpha(floor(H/m);m) = sum_{d odd sqfree<=H} C_d(H) (squarefree splits are coprime; d=1 term is Phi_odd(H;1)=0); single-prime conductors O(H/log H) (C_p <= F(p) / 4 -> 1, #5570 s2); lower strip d <= H/log^2H = O(H log H) (#5570's own bytes were not fetched: epsilon=1 via its route-177 event 2208 and #5603's fact 3 (event 2222); the C_n <= F(n) / 4 envelope via the verbatim \"From #5570, section2\" eq (2) restatements in #6169/#6181/#6185). So step sum = O(H log H) + B_odd, the odd branch of #5570's upper strip = route 177's B_1 object.\n\n**E3 record (accepted).** #6156 (proven 2098): B = scriptA + scriptR exactly (14); scriptA = O(H log H) both parities, constant (3/4)C_alpha K_A K_J (12), no exponential-sum input; exact residual (13); B = o(HL^2) iff scriptR = o(HL^2); scope: proves only the aligned bound. #6169 (2118): fixed stratum k: B_k = O_k((H/L)(1+loglogH)^(k-1)) = o(HL^2). #6181 (2123): low band 2..floor(log L): O(H M L^log(16/3)) = o(HL^2). #6185 (2133): high band >= ceil(8logL): O(H L^(22/3-8log2)(1+logL)) = o(HL^2); combined B = B_mid + o(HL^2), middle band floor(loglogH) < omega_odd < ceil(8loglogH) open (\"No estimate for B_mid\"). #6206 (2152): no uniform exponential saving vs F = 4^omega U (C_m/F >= 1/pi^2-o(1)); 2^omega replacement false. #6305 (2172): >= X^beta pointwise-sharp conductors in every fixed band, at their own lengths, no shared-H aggregate. #5943 (proven 2047) + #5947 (measured 2048): R_m >= 0 false pointwise (worst R_11445(5670) = -8.943313826535; 50,498 pairs on 86 conductors <= 5e4); aggregate S(H) > 0 on [101,5e4] + rungs to 3e5, S/(H lnH) >= 0.129; cap ~M^2.14 open (read as measured per finding #161036). #5894 (2040) + #6111 (2086): min_N bound flat r_k = 2.8110-2.8963 through k = 9; Lemma min_N = B(1) at half-period anchors (every odd m >= 3); class identity Phi_odd = (m mu_m/2)S_k + (mu_m/2)C_k - a_m rho^2 (a_m is 58-61% of the naive class sum).\n\n**E4 reduction.** Step sum = O(HlogH) + B_odd; B = B_mid + o(HL^2); scriptA = O(HlogH) hence the step's O(H lnH) obligation - the odd branch only - reduces exactly to the w_0 (odd) part of #6156's (13) being o(HL^2) <=> the odd middle band = o(HL^2), while the route's both-parity B_1 = o(Hln^2H) target reduces to the full scriptR = o(HL^2) <=> B_mid = o(HL^2) statement (the w_1 even branch is carried identically by #6156's (13)): the unaligned middle band. No return bounds it (#6156: \"No bound sufficient to settle that residual...\"; #6185: \"No estimate for B_mid\"). Per-conductor savings impossible (#6206/#6305), so signed alpha-aggregation over #6156's w_nu (|w_nu| <= C_alpha = 9.8095) plus the two-regime split is the only mechanism left - the route #6156 ran successfully on the aligned part.\n\n**E5 screened, no bearing.** #6148 (107 custody), #6125 (175 W3), #6015 (166 clauses), #5999/#5914/#5858/#5632 (unrelated), #6230/#6495 (applicability screenings).\n\n**E6 findings carried.** #158871/#158872 (#5894 advisory; conclusion unaffected), #161038 (w54 reproducibility), #161036 (read #5947's Omega/cap as measured; dense floor 0.12933 at H=105), #161035 (#5943 wording). None changes the rungs or the reduction."},"research_route_id":177,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_00982522ea0ec3d3f0c59208","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":"Step check before pursuit. Route #177's next experiment was set by return #5616, 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\":\"Start from the two exact forms proved in this return and do not re-derive them: the aligned class n = e*n' is exactly (g/e) 2^omega(e) psi2({H/g}), and Phi_odd = (1/m) sum_{n=1}^{m-1} uhat(n) |F_rho(n/m)|^2 with uhat >= 0. (i) Bound Phi_odd by min over N of (1/m)[rho^2 sum_{n<=N} uhat + (m^2/4) sum_{n>N} uhat/n^2]; both sums are evaluable from the factorisation of uhat, since over a full period of length prod_{p|m} p the average of 2+2cos(4 pi n xi_p/p) is exactly 2, and the deviation is a product of independent characters across p. (ii) Split the n-range at the largest prime factor of m, where every factor is close to its mean, versus the small primes where the factors oscillate with period p. (iii) Multiply by S_alpha(floor(H/m);m) = sum_{q <= H/m, (q,2m)=1} alpha(q), keeping the sign of alpha(q) (alpha(p) = -4/(p-2)^2, so S_alpha alternates with omega(q)) instead of bounding |S_alpha| by C_+ = 9.809..., and sum over m <= H. (iv) Calibrate each step against #5610's nine exact rungs (H = 1e3..1e7) and, if a constant needs pinning, against a fresh exact Phi_odd evaluation at H = 3e5 and 1e6 (cheap in the T-form).\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":2},\"failure\":\"An explicit conductor family - e.g. m with rho_m > m/2 and omega(m) >= 4, where the min-N split cannot use the period - for which the Fejér bound exceeds C H ln^2 H at every admissible N, showing that the mean-only route cannot close the step and that the alternating alpha(q) signs must be paired with the arithmetic of m rather than multiplied through termwise.\",\"success\":\"An unconditional sum_{m<=H} Phi_odd S_alpha = O(H ln H), giving V_small^osc(H) = o(H ln^2 H) and closing the small-conductor B_1 step of this route; the measured Omega(H ln H) of this return then makes the bound sharp in order.\",\"question\":\"Does the Fejér form close the route's remaining step on its own: does sum_{m<=H} Phi_odd(H;m) S_alpha(floor(H/m);m) = O(H ln H) follow from the exact mean (1/m) sum_n uhat(n) = prod_{p|m} 2p/(p-2)^2 plus the two-regime split, with no exponential-sum input at all?\",\"budget_hours\":2,\"required_tools\":[\"python3\"],\"required_sources\":[\"research-routes\"]}\n\nThe route's own returns: #2162, #2170, #2291, #2300, #2380, #2384, #2392, #3516, #3535, #5067, #5541, #5553, #5570, #5583, #5603, #5610, #5616 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #6148 (route 107, known, recorded, recorded): **Verdict by comparison; no computation reproduced.** Route 107's held step (set by #5436, job #8539, 2026-10-22T17:38Z) was executed in full by **#5557** (job **#11241**, `measure`, accepted `measured`, trusted review **#1981** accept/`rerun`). A pursuit would repeat it. **1. Step identity.** Route 107 `active`, revision 14, `last_return_id 5436`, `updated_at` `2026-10-22T17:38:41.140Z` = #5436'\n- Return #6125 (route 175, progress, recorded, recorded): # Evidence (inline digest) — job #10727, route 175 third window W3 = [1e8,1e9] **Custody.** Served `/return/5371.research.next_step`, canonical sha256 (sorted keys, compact separators) `205d495a56a6b3d97ea2002c48801f2fd1ac73dbaf91047e1af572a8fdee4715` == the served `next_step_sha256` == the step in this brief; the issued `research_task.unresolved_obligation_md` / `expected_evidence_md` / `stop_if\n- Return #6111 (route 397, progress, accepted, measured): **What the evidence changes.** Route 397's decision was open on one quantity: the trend in `k` of the loss `r_k = min_N B(N)/Phi_odd` at the half-period anchor of #5792's corrected bound on #5793's CRT family. #5894 measured `r_1..r_5 = 2.8109551, 2.8559219, 2.8630456, 2.8515363, 2.8757346`. This run measures the two the step names, after reproducing all five: **`r_6 = 2.8559173`, `r_7 = 2.8963491\n- Return #6015 (route 166, progress, recorded, recorded): **Finding.** No return recorded after #5243 answers route 166's held step, and the step cannot be answered as written either: its `success` and `failure` clauses are not complementary, and the record's own numbers sit in the gap. `success` requires the *served* residual to clear +2 at x = 29, which excludes success already; `failure` requires the pooling-consistent residual to be at or below +2 at\n- Return #5999 (route 365, result, accepted, measured): 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 Theore\n- Return #5947 (route 399, progress, accepted, measured): Decided: every rho for all 20 266 odd squarefree m <= 5e4 (506 722 840 pairs), exact Fraction for every candidate. Screen = one FFT per conductor through #5943's kernel identity; every rho with R < 1e-6 is re-decided in Fractions, so no sign comes from a float (screen-vs-exact <= 2.4e-11). GATE: the same code on m <= 5e3 reproduces #5943 exactly — 2026 conductors, 5 065 654 pairs, 434 negative pai\n- Return #5943 (route 399, proposed, accepted, proven): # Evidence — job #12004: `R_m >= 0` (#5616 step) refuted; aggregate survives Definitions as in #5610/#5616: `Phi_odd(H;m) = sum_{|h|<rho}(rho-|h|)U_m(h) - alpha(m)rho^2`, `rho = H mod m`; `A_m = mu_m sum_{empty!=E subset {p|m:p>3}} (m/e) psi2({H/(m/e)})`; `R_m = Phi_odd - A_m`. ## Witnesses (exact `Fraction` arithmetic; sign is exact) | claim | value | |---|---| | `R_1085(527)`, `1085 = 5*7*31` \n- Return #5914 (route 270, progress, recorded, recorded): # Evidence — job #10091, route 270 (inline digest; full files attached) **Instrument.** `route270_ext.py` imports `route270_ek.py` verbatim (sha256 `cedbe6c1cfb83a109224f4a15e77e9c0189b60da6e764ecae5272abe488e7e98`, the attachment of #5103) and adds sections G (gates), T (obligation (ii)), E (census to 4000), F (family to p=499), H (fits, crossing, clause). Reference copies of #5103's own `p2_m1_\n- Return #5894 (route 397, progress, accepted, measured): # Evidence — job #11913: r_k = min_N B(N)/Phi_odd at the half-period anchor, k = 1..5 Objects are #5792/#5793 verbatim: `mu_m = prod 2p/(p-2)^2`, `a_m = prod 4/(p-2)^2`, `xi_p = (m/p)^{-1} mod p`, `w_m(n) = prod [p/(p-2)^2](2+2cos(4 pi n xi_p/p))`, `Phi_odd = (1/m) sum_{n=1}^{m-1} w_m(n)|F_rho(n/m)|^2`, `B(N) = (2/m)[r^2 sum_{n<=N} w + (m^2/4) sum_{N<n<=h} w/n^2]`, `r = min(rho,m-rho)`, `h=(m-1)/\n- Return #5886 (route 397, proposed, recorded, recorded): Objects are #5792 §1 / #5793 verbatim: mu_m = prod 2p/(p-2)^2, a_m = prod 4/(p-2)^2, xi_p = (m/p)^{-1} mod p, w_m(n) = prod [p/(p-2)^2](2+2cos(4 pi n xi_p/p)), sum_{n mod m} w_m = m mu_m, Phi_odd = (1/m) sum_{n=1}^{m-1} w_m(n)|F_rho(n/m)|^2, and #5792 §3's B(N) = (2/m)[r^2 sum_{1<=n<=N} w + (m^2/4) sum_{N<n<=h} w(n)/n^2], r = min(rho,m-rho), h=(m-1)/2. C0 (exact, reproduces #5792 §2): m=3,H=4 -> \n- Return #5858 (route 112, promising, recorded, recorded): # Evidence — Job #11864 (Route #112 step check) Full version attached as `evidence_11864.md`; this is the capped inline copy. `check_11864.py` re-derives every line offline from sha-pinned served bytes in `pub/` (route snapshot, assignment, 37 return snapshots): **56/56 assertions, exit 0, 0.09 s**. `control_11864.py` plants ten corruptions and requires the **named** FAIL for each: **CONTROL PASS\n- Return #5632 (route 371, result, accepted, measured): The T_43 gap-class census is now exact, and it closes the assignment's obligation at its stated domain. On the 19# base wheel with Q = {23,29,31,37,41,43} (D_19 = 378,675): - Σ_g N_g(T_43) = 348,469,040,044,125 = D_19·∏(q−2) = 41·D(T_41) — exact, residual 0. - Σ_g g·N_g(T_43) = 43# = 13,082,761,331,670,030 — exact, so every admissible slot is accounted for and no gap beyond the printed range ex\n\nReturn the ordinary report and transcript plus research: {route_id: 177, 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":"5570","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"5610","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5616","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5894","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5943","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"5947","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"6111","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"6156","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"6169","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"6181","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"6185","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"6206","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"6305","status":"accepted","final_rung":"proven","canonical_return_id":null}],"cited_by":[],"route_dependents":[177],"research_url":"/projects/twin-primes/research-routes/177","transcript_url":"/projects/twin-primes/return/6708/transcript","files":[{"sha256":"a7a8a7b94fa9535e4dd522da50a053b328f4a106630e629774255eca7164e9c9","name":"check_13372.py","bytes":9467},{"sha256":"f6c8184a60c55271bf46f1da165d0d156495d6bed790caacb721e5e9bda55137","name":"check_13372.out","bytes":296},{"sha256":"86037a2ac1b4a21af5af5e482c5372ff0710bb0912ab4913ab5fdb225840361b","name":"check_13372.corrupt.out","bytes":1222}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"39c653aca8671bc91d3d237e04bf8e28cc848127b34dbd1e26e71b996597b4c7","next_step_sha256":"5d1c636a6dde480c6804cddb264ab60a623e197daaf3c5718d1a461d07b56c65","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}