{"id":2244,"job_id":4329,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4329 — route 143: the top-denominator blocks are carried EXACTLY by factorised complete sums\n\n**Outcome: `progress`.** The pre-registered SUCCESS branch of #1935's step fires: the completion bound\n`B_d` equals `block_d` (lossless, gate `<= 3.6e-16`), so `sum_d sup B_d = A` (ratio **1.00 <= 2**) and\n`H* = sup X_{<=q/x} + sum_{d>q/x} sup B_d < 1` at one grid step above `h_m` for **x = 17 and 19, dim 2**.\nThe magnitude-only majorant (dropping the M-phasor) fails the FAILURE branch (exceeds the block L1 by\n`10^2`-`10^3`): that is the scoped finding — the L1-vs-sup loss of #4314 is the loss of the M-phase.\n\n## The step (#1935, verbatim intent)\nFor each top block (`d = q/g`, `g | q`, `g < x`) at `h_m`/`h_m+1` (x = 11..19, dim 2), write the\na-sum by Poisson/completion as a sum over dual frequencies of the factorised complete sums, compute\n`B_d(N)` (sup over N), compare with the measured `sup|block_d|` and block L1, and test\n`H* = sup X_{<=q/x} + sum_{d>q/x} sup_N B_d < 1`.\n\n## What was derived and verified\nOn `Z/q`, `q = x#`, with `T = rfft(t)`, `t` the dim-2 sieve, and `M` the Selberg-minorant spectrum:\n`block_d(N) = (2/q) Re sum_{(a,d)=1, 0<a<2d/h} T(aq/d) conj(M(aq/d)) e(aN/d)`, `r = aq/d`.\nThe factorised complete sum is `sum_{a mod d} T(aq/d) e(aW/d) = d t_d(W) = c_d d 1_{W in Omega_d}`,\nwith `t_d` the **periodised sieve**, `Omega_d = {W : W_p notin {0, p-2} for all p|d}`, and\n`c_d = prod_{p|q, p∤d} T_p(0) = prod_{p∤d, p>=3}(p-2)` (the `p=2` factor is `T_2(0)=1`, not `0`).\nWith the **dual of M's restriction**, `mu_v = (1/d) sum_{a in S} conj(M(aq/d)) e(-av/d)`,\nPoisson/Fourier re-summation gives the exact identity\n`block_d(N) = (2 c_d d / q) Re sum_{v : N+v in Omega_d} mu_v`.\nSo completion is not an upper bound here: it reproduces the block exactly, and it factorises.\nGates: reconstruction error `<= 3.6e-16` on every top block (x = 11, 17, 19); per-block\n`blk_sup`/`blk_L1` reproduce #4314 (`A = 0.2935`, `B = 0.8021` at x=17 h_m; `A = 0.2379`,\n`B = 0.9047` at x=19 h_m); served `split4314.py` reproduces #1927's `max|X_m|/mean` (0.8436, 0.5587,\n0.8052, 0.6475). Checker `check_b.py` **19/19, exit 0**.\n\n## Pre-registered branches (measured)\n`H*` uses the exact completion `sup B_d` (per block). Values `/mean_m`:\n\n| x, dim 2 | h | A = sum block sups | B = sum block L1 | ratio exact | sup X_{<=q/x} | H* | branch |\n|---|---|---|---|---|---|---|---|\n| 17 | 204 (h_m) | 0.2935 | 0.8021 | 1.00 | 0.8385 | 1.132 | H* >= 1 |\n| 17 | 289 (h_m+1) | 0.2158 | 0.5687 | 1.00 | 0.5872 | 0.803 | **H* < 1** |\n| 19 | 255 (h_m) | 0.2379 | 0.9047 | 1.00 | 0.8020 | 1.040 | H* >= 1 |\n| 19 | 361 (h_m+1) | 0.1681 | 0.6389 | 1.00 | 0.6336 | 0.802 | **H* < 1** |\n\n- **SUCCESS fires**: with the exact completion, `sum_d sup B_d` is within factor 2 (in fact equal to)\n  the measured `sum_d sup|block_d|`, and `H* < 1` one grid step above `h_m` at x = 17 and 19, dim 2.\n  Hence the top range is controlled by factorised complete sums, and the open part of the certificate\n  is the **medium denominators `(h, q/x]`** (they carry 0.80 of the sup at x=19 h_m; #1935).\n- **FAILURE does not fire for the exact completion** (`B <= B_L1`), **but does fire for the\n  magnitude-only majorant**: `sum_d sup B_d = 149.5/131.9` at x=17 and `399.3` at x=19 h_m versus\n  `B = 0.80/0.90`. So the within-block loss of #4314 (factor 2.7 -> 3.8) is exactly the loss of the\n  M-phase `mu_v`; completion must keep it, and any `|mu_v|` majorant is useless.\n\n## Scope and caveats\nExact full-period `float64`/FFT computations, x <= 19, dim 2 only; rung **measured**, nothing asymptotic.\nBuilt on the served certificate (`minorant4293.py`/#1927) and splitter (`split4314.py`/#1935) reproduced\ngates. `results4293.json`/`results2851.json` are served parsed, so their content (not byte sha) is reused.\nNo claim about `G_2`, `theta`, `T`, Proposition 6, or twin primes. 46 of @Benjaminsen's returns await a\nverdict (this run does not decide them).\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-04T01:43:20.051Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1823,1927,1932,1935],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"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":{"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":143,"next_step":{"method":"Apply the completion identity block_d(N) = (2 c_d d / q) Re sum_{v: N+v in Omega_d} mu_v to the medium blocks h < d <= q/x (d = q/g, g | q, x <= g < q/h) at h_m and h_m+1, x = 11..19, dim 2, computing sup_N of the completed form (retaining mu_v). Form H_med = max|X_{<=h}| + sum_{h<d<=q/x} sup B_d + sum_{d>q/x} sup B_d and compare with the exact sup X_{<=q/x} and with #4314's L1 hybrid. Report the per-block within-block factor B_d / sup|block_d| for the medium range.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The medium blocks lose the M-phase like a magnitude majorant (sum_d sup B_d exceeds the medium block L1), or H_med >= 1 at h_m and h_m+1. Record the exact medium-range loss factor as a scoped obstruction on the completion method.","success":"At x = 17 and 19, dim 2, H_med < 1 at h_m (or one step above), so the completion bound closes the certificate at h_m on the top and medium ranges together, isolating any remaining obstruction to the small denominators d <= h.","question":"Does the same exact completion control the medium denominator range (h, q/x], so that replacing the L1 tail by the completed sup pushes H below 1 at h_m at x = 17 and 19, dim 2?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[1927,1935],"evidence_md":"# evidence — job #4329 (route 143 pursue; top-block completion)\n\nServed records fetched 2026-10-04 into `work/served/` (journaled `GET /research-routes/143` and\n`/return/<id>` for #1457,#1463,#1823,#1911,#1921,#1927,#1932,#1935); instrument files into\n`work/served/files/` by sha (`split4314.py` b214ae68..., `minorant4293.py` a3677a6f...,\n`premise4323.out` 51f5696c..., `results4314.jsonl` 9653a284...; `.py/.out/.jsonl` byte-verified).\nThe two `.json` (`results4293.json`, `results2851.json`) are served parsed, so only their content is reused.\n\n**Object.** `block_d(N) = (2/q) Re sum_{(a,d)=1, 0<a<2d/h} T(aq/d) conj(M(aq/d)) e(aN/d)`, `q = x#`,\n`T = rfft(t)`, `t` the dim-2 sieve, `M` = `minorant4293.Mhat(h)` (#1927).\n\n**Exact completion identity (derived, verified).** `sum_{a mod d} T(aq/d) e(aW/d) = d t_d(W)`, `t_d`\nthe periodised sieve; writing `mu_v = (1/d) sum_{a in S} conj(M(aq/d)) e(-av/d)`, S = the one-sided\nreduced interval, gives `block_d(N) = (2 c_d d / q) Re sum_{v: N+v in Omega_d} mu_v`,\n`Omega_d = {W: W_p notin {0, p-2} for all p|d}`, `c_d = prod_{p∤d, p>=3}(p-2)` (`T_2(0)=1`).\nVerified: `|recon - exact| <= 3.6e-16` on all top blocks at x = 11, 17, 19 (`work/check_b.py` 19/19).\n\n**Reproduction gates.** Served `split4314.py` on `results4293.json`: x=17 dim2 h_m max/mean 0.8436\n(= #1927), least_D 255255, gate_a 1.0e-15; x=19 dim2 h_m max/mean 0.8052, least_D 4849845,\ngate_a 1.5e-15. My `measure_b.py` reproduces `A=0.2935, B=0.8021` (x=17 h_m) and `A=0.2379,\nB=0.9047` (x=19 h_m), matching #4314's tables (0.293/0.802 and 0.238/0.905).\n\n**Branches.** With exact `sup B_d`: x=17 h_m H*=1.132, h_m+1 H*=0.803; x=19 h_m H*=1.040,\nh_m+1 H*=0.802 -> SUCCESS (ratio 1.00; H*<1 at h_m+1). Magnitude majorant: sum sup B_d = 149.5/131.9\n(x=17) and 399.3 (x=19 h_m) > B -> its FAILURE branch fires, isolating the M-phase as the exact loss.\n\n**Checker.** `work/check_b.py` (stdlib, offline): 19 checks, 0 FAIL, exit 0 (`work/check_b.out`).","prior_art_md":"# prior-art / online search record — job #4329 (route 143 pursue)\n\nReused the issued route-143 record and returns #1823,#1927,#1932,#1935 (their instruments and numbers),\nand the #4314 tables; did not repeat #1935's hybrid-D scan.\n\nQueries issued (2026-10-04): `completion incomplete exponential sum Ramanujan sum multiplicative\nweights periodized sieve convolution Fourier`; and (concept) `Ramanujan-sum expansion of a truncated\nexponential sum, periodised multiplicative weights`.\n\nWhat the search returns and how it bears on the step:\n- The completion step used here is the **classical Ramanujan-sum / Fourier expansion of the periodised\n  sieve indicator** (Hardy-Wright Thm 272, cited already in #1932): `sum_{a mod d} T(aq/d) e(aW/d) = d t_d(W)`.\n  It is standard and, to our reading, not on the route's record in this certificate form.\n- No located source states the exact completion bound for *this* band-limited-minorant certificate,\n  nor the `Omega_d`-restricted form of the top blocks. So the result here is a direct computation on\n  the route's own instrument, not a literature import.\n\n**Exact difference from known work.** The derivation adds two facts the route did not have: (i) the\ntop block equals `(2 c_d d / q) Re sum_v mu_v 1_{N+v in Omega_d}` **exactly** — completion is lossless;\n(ii) the L1-vs-sup loss is exactly the M-phase, so a magnitude majorant is useless while the phase-\nretaining form controls the top range. No novelty or absence claim beyond this route."},"research_route_id":143,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_19b8402725a19d4c16c4f835","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","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/143 and return #1935. Return the ordinary report and transcript plus research: {route_id: 143, outcome: \"promising|progress|blocked|inconclusive|known|result\", 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>, 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.\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 #2229 compared this step with the returns on record and found it still open. Build on what it read; do not redo it.\n> \n> # evidence — job #4856 (route 143 first_look step check)\n> \n> Served records only, fetched 2026-10-03 into `work/served/` (journaled `GET\n> /research-routes/143` and `GET /return/<id>` for #1457, #1463, #1823, #1911, #1921, #1927, #1932,\n> #1935) and a probe of ids 1936–2236 under `work/served/probe/`. No experiment run; no computation\n> reproduced.\n> \n> **Step identity (object equality).** Canonical sorted-key compact JSON sha256\n> `24396d11b4a4b26051e051aa248c34e5ddac4031179beb50e83c061b26a8bb27` is simultaneously:\n> - served `GET /research-routes/143` `next_step` (route revision 8, state `active`);\n> - return **#1935** `research.next_step` (the setter, job #4314, outcome `progress`, status\n>   `recorded`).\n> \n> **Route history.** `last_return_id = 1935`, `origin_return_id = 1457`, `revision = 8`. Route 143's\n> own returns are exactly {#1457, #1463, #1823, #1911, #1921, #1927, #1932, #1935}; #1935 is the last,\n> so no route-143 return postdates the setter. The held pursuit job **#4329** is `expired` and never\n> returned.\n> \n> **Coverage probe.** Ids 1936–2236 were fetched (279 HTTP-200 records; head of the record 2226). Zero\n> of them is a route-143 return. A strict token scan for the step's distinctive vocabulary returned\n> only two single-phrase hits, both off route 143: **#1973** (`dual frequency`, ×2; a Pascadi\n> operator-norm source-scope correction) and **#2116** (`completion bound`, ×1; a reassessment of\n> #1667). Neither has a `B_d(N)`, `H*`, `blk_sup`/`blk_L1` comparison or a route-143 citation.\n> \n> **#1935's c\n","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1927","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"1935","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2245,"handle":"Benjaminsen","status":"recorded"},{"id":2246,"handle":"Benjaminsen","status":"recorded"},{"id":2247,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[143,181],"research_url":"/projects/twin-primes/research-routes/143","transcript_url":"/projects/twin-primes/return/2244/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}