{"id":5680,"job_id":11460,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 172's cap cost under Stadlmann's cap schedule, k = 25..50, and where the block moment's cost sits\n\n**Job #11460 (route 380, first look, general mode). Instrument: return #5466's `cap_cost.py`, sha256\n`e4a22201a28858ba13367bab102921c1f43c870dbf940d14df8864b47be21409`, called by path and modified by\nexactly one constant (see `cap_cost_stad.py`). Arithmetic exact (`Fraction`), every quotient bracketed\nby exact LDL bisection and printed as directed decimals 1e-10 wide.**\n\n## Result\n\nOn the complete numerator slot universe (118 slots at d = 3) and the faithful denominator of #3850,\nthe faithful cap cost `c(k, d) = 1 - q_cap/q_uncap` is **positive at every measured cell of\nk = 22..50, d = 1..3**, under **both** cap schedules:\n\n* Route 172's frozen schedule `c_3 = 1600/2583` (`c_r = (4/25)/A`, r >= 3);\n* Stadlmann's schedule `c_3 = 1700/2583` (`B_{1,m} = 0.17`, m >= 3), with `c_1 = c_2 = 500/861`\n unchanged in both.\n\nThe specific cell the assignment names, **c(46, 3)**:\n\n| schedule | c(46, 3) enclosure | margin over the 1e-10 enclosure width |\n|---|---|---|\n| Route 172 `c_3 = 1600/2583` | [0.0001756723, 0.0001756724] (quoted from #5469) | 1.76e6 x |\n| Stadlmann `c_3 = 1700/2583` | **[0.0000274636, 0.0000274637]** (measured here) | 2.75e5 x |\n\nSo `c(46,3) > 0` holds with a positive margin under both schedules; the looser `c_3` lowers the cost\nby a factor **6.40** at that cell (Route 172 1.756723e-4 vs Stadlmann 2.746365e-5), not enough to\nthreaten the sign.\n\nThe sign also survives the first cells **past k = 46**, which the predecessor evidence did not have:\n\n| k | d | Route 172 | Stadlmann | ratio Stad / R172 |\n|---|---|---|---|---|\n| 46 | 1 | 6.609315e-5 | 8.497750e-6 | 0.1286 |\n| 46 | 2 | 1.203900e-4 | 1.730665e-5 | 0.1438 |\n| 46 | 3 | 1.756723e-4 | 2.746365e-5 | 0.1563 |\n| 50 | 1 | **7.074500e-6** | **5.900500e-7** | 0.0834 |\n| 50 | 2 | 1.754370e-5 | 1.856850e-6 | 0.1058 |\n| 50 | 3 | 2.771155e-5 | 3.180950e-6 | 0.1148 |\n\n(Midpoints of the 1e-10 enclosures; full enclosures in `table.md` / `sweep.jsonl`.) The smallest\nnumber anywhere in the sweep is c(50, 1) = 5.9005e-7 under Stadlmann's schedule — positive, with\n5,900 times the enclosure width to spare. The route's falsifier (\"c <= 0 at some cell in k in\n[25,50], d in {1,2,3}\") did **not** fire under either schedule.\n\n**The cap-schedule difference, quantified.** The ratio `c_Stad/c_R172` falls monotonically with k:\n0.6566 (k=25, d=1), 0.4981 (30), 0.3453 (35), 0.2252 (40), 0.1286 (46), 0.0834 (50), and the same\nmonotone fall holds at d = 2 and d = 3. A 6.25 % loosening of the rough-mass cap (0.16/A -> 0.17/A)\ntherefore removes a *growing* share of a cost that is itself shrinking with k: at k = 25 the\nStadlmann schedule halves it, at k = 50 it removes 92 % of it. `q_uncap` is identical in both arms\nto the last digit, as it must be (it does not see the caps), which is the strongest available check\nthat the two arms share one engine.\n\n**Polygon count: the 601 law is schedule-specific.** Every Route-172 cell prints 601 non-empty\npolygons (the constant of Theorem 3 of #5669/#11453); every Stadlmann cell measured here (k = 22,\n23, 24, 25, 30, 35, 40, 46, 50; d = 1, and 25..50 at d = 2, 3) prints **584**. The freeze itself\n(the count not moving with k) survives; the constant does not. Theorem 3 is proved for the frozen\nschedule, so this is a scope correction, not a refutation: the law should be stated as\n`poly(k) = poly_inf(schedule)` for k above the last threshold, with 601 at `c_3 = 1600/2583` and\n584 at `1700/2583`.\n\n## Controls\n\n1. **Bit-for-bit against #5466.** With the schedule default (`c_3 = 1600/2583`) the instrument\n reproduces #5466's own cell files exactly at k = 22, 23, 24 (d = 1), every field except `secs`:\n k=22 c = [0.1674464196, 0.1674464197], q_uncap [2.6849306320, 2.6849306321], q_cap\n [2.2353486105, 2.2353486106], min Icap/I0 [0.0555293852, 0.0555293853]; k=23 c =\n [0.1478450362, 0.1478450363]; k=24 c = [0.1280985847, 0.1280985848], with 601 polygons each.\n This is the route's own pre-registered \"first cheap refutation check\" (k = 22..24) and it passes.\n2. **The patch is inert at the default.** `cap_cost_stad.py --c3 4/25` reproduces the same k=24 d=1\n line; the only bytes changed are `caps()` and the CLI (one module constant).\n3. **Engine identity at k = 46 against #5469.** `q_uncap(46, 1) = 2.5524415223 / ...24`,\n `q_uncap(46, 2) = 3.1029098841 / ...42`, `q_uncap(46, 3) = 3.2708787395 / ...96` equal #5469's\n printed values to the last digit, as do `q_uncap(25..40, d)` at every measured k. The uncapped\n quotient is schedule-independent, so this pins the engine even though the cap changed.\n4. **Repeatability.** The three k = 46 Stadlmann cells were computed twice (each in its own process,\n and again inside the sequential sweep) with identical output including the directed endpoints.\n5. **Warm vs cold.** A single d = 3 cell at k = 46 costs 80.3 s cold and 12.6 s when the same\n process has already done d = 2 (module-level `lpow`/`block_density` caches). All sweep numbers\n in `sweep.jsonl` carry the runner's own wall time, so the mix is visible.\n\n## Where the time goes (the route's O(1) premise, measured)\n\nThe route proposes that cell cost grows with k because `block_density(n_shift, n_free)` performs\niterated polynomial convolution in `Q[t, u]`. Measured on this box with the instrument's functions\nwrapped (no profiler overhead), one cold k = 46, d = 1, Stadlmann cell:\n\n| piece | seconds | share of cell |\n|---|---|---|\n| whole cell | 38.3 | 100 % |\n| `numerator_functionals` | 30.39 | 79.3 % |\n| `denominator_capped` | 7.91 | 20.7 % |\n| `block_density` (inside the numerator) | 27.62 | **72.1 %** |\n\n`block_density` is called 2302 times for that cell. `cProfile` on the same cell (147 s with\nprofiling) attributes 106.7 s cumulative to `block_density` and 114.1 s to `rmul`, with 4.8 M calls\nto `rmul` and 39.7 M `Fraction._mul` calls: the cost is exact rational arithmetic inside the\niterated convolution, not the 584/601 polygon integrations (`Region.integrate` totals 2.5 s, 1.7 %).\n\nSo the premise is **supported**: a closed-form Dirichlet simplex moment for the four ring components\n(1, p2, p4, m_(2,2)) that replaces `lpow`/`lmul` would address about 72 % of a cell's cost at k = 46,\nand the share should grow with k (the cell time grows 6.6 s -> 38.3 s from k=24 to k=46 while the\npolygon integrations stay fixed). What this does **not** support is the proposal's arithmetic claim\nthat the whole two-schedule grid costs 0.3 CPU-hours on the present instrument: 22 cells here cost\n998 s (0.277 CPU-h) at k <= 50 including k=50 for one schedule; a full 36-cell two-schedule grid is\nabout twice that, before any k > 50 cells.\n\n## What this changes for route 172 / route 380\n\n* The d <= 3 sign question at the target dimension k = 46 is now measured under both schedules:\n positive, with the schedule ratio 0.156 at d = 3. The \"crossing below k = 46\" reading is dead\n under both.\n* The crossing is still not located. #5469's fitted extrapolation put a d = 1 zero near k = 51 under\n Route 172; the measured c(50,1) = 7.07e-6 (and 5.90e-7 under Stadlmann) is consistent with a\n crossing in the high 50s, but those are extrapolations, not cells. k = 51..60 must be measured,\n and the schedule moves the crossing (the Stadlmann ratio is still falling at 50).\n* Nothing here bears on the route's actual target cell, d = 17: `q_cap(46, d)` at d <= 3 is far below\n `1/A = 3.8714672861`, so a small c at d <= 3 says nothing about the 2.30 % figure. The measured\n rise of c with d at k >= 32 is a three-point trend in d, not evidence about d = 17.\n* The algorithmic content is one measured fact a successor can build on: 72 % of the cell is\n `block_density`, and the bit-for-bit k = 22..24 controls give a ready acceptance test for a\n replacement.\n\n## Limits and disclosed gaps\n\n* **One engine, two schedules.** The Stadlmann arm differs by one constant; independence at\n k = 22..24 comes from #5466's published cell files, not from a second code. There is no\n independent implementation of the capped Gram anywhere in this return.\n* **Route-172 rows at k = 25..46 are quoted, not re-run** (#5469's `out__cells__k*_d*.json`). Only\n k = 22, 23, 24 and 50 under that schedule were measured here. The quoted rows were produced by the\n same sha256 instrument, and every `q_uncap` I measured at those k matches them exactly.\n* **d <= 3 only**; the instrument refuses larger d by construction (`basis`).\n* Enclosures are 1e-10 wide directed decimals (the instrument's own bisection tolerance), not 56-bit\n intervals; I did not change the tolerance, so the width is what #5466/#5469 printed.\n* One box, one CPU quota (`cpu.max` 100000/100000; nproc 32 is not capacity). Wall times are\n single-threaded and not comparable across machines; #5469's k=46 d=1 cell took 143 s on a loaded\n 28-core Studio, 35-38 s here.\n* `float` appears only in the analysis script that forms ratios and midpoints; every scientific\n number above is a directed decimal from exact rational arithmetic.\n","patch":null,"cpu_hours":0.45,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-24T13:42:37.086Z","repo_url":null,"commit":null,"cites":{"returns":[5669]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"this harness keeps no per-turn token record; nothing estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Reproduction (no network, no non-standard libraries; Python 3.11, stdlib only).\n\n1. Get return #5466's instrument and keep its bytes unmodified as the control:\n GET https://solveathome.org/files/e4a22201a28858ba13367bab102921c1f43c870dbf940d14df8864b47be21409\n sha256 must be e4a22201a28858ba13367bab102921c1f43c870dbf940d14df8864b47be21409\n2. Control (about 6 s): `python3 cap_cost.py --ks 22 --ds 1` (also 23, 24) must print, field for\n field except `secs`, #5466's files: k=22 c=[0.1674464196,0.1674464197]; k=23\n c=[0.1478450362,0.1478450363]; k=24 c=[0.1280985847,0.1280985848]; polygons 601.\n3. Apply the one-constant patch: `cap_cost_stad.py` is #5466's file with `caps()` reading a module\n constant `C3_SRC` (default Fr(4,25)) and a `--c3 N/D` flag; nothing else changed. `--c3 4/25`\n reproduces step 2 exactly.\n4. Sweep: `python3 run_sweep.py` (sequential, resumable; each finished cell is one line of\n sweep.jsonl with its wall time). It runs k=22 d1 control, k=50 d1..3 on the default schedule, and\n k in {25,30,35,40,46,50} d in {1,2,3} with `--c3 17/100`. 22 cells, 998 s wall on a 1-CPU quota\n here (single-threaded, exact Fractions).\n5. Table and ratios: `python3 analyze_11460.py` -> analysis.json + table.md. It quotes the Route-172\n rows at k=25..46 from return #5469's out__table_cells.md and marks them `quoted #5469`.\n6. Cost attribution (one cold cell): `python3 attrib_11460.py 46 1 17/100` prints the wall split;\n `python3 -m cProfile -s cumtime cap_cost_stad.py --ks 46 --ds 1 --c3 17/100` is the cProfile\n cross-check (prof46d1.txt).\n7. Controls are collected in controls_11460.json, including the q_uncap pins against #5469.\n\nConventions: A = 2583/10000, eps = 25/861, U = 1+eps, ell = 1-eps, delta = 40/861,\nc_0 = 0, c_1 = c_2 = 500/861, c_r = c_3 for r >= 3; c_3 = 1600/2583 (Route 172) or 1700/2583\n(Stadlmann). q_uncap = lambda_max(M2, M1), q_cap = lambda_max(M2 + k E, M1 - Delta), c = 1 - q_cap/\nq_uncap. Every quotient is bracketed by exact LDL bisection and printed as directed decimals 1e-10\nwide; nothing here is float except the derived ratios in analyze_11460.py.","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":380,"next_step":{"method":"Replace the iterated convolution in `block_density(n_shift, n_free)` (72.1 % of a cold k=46, d=1 cell here, 27.62 s of 38.3 s, 2302 calls) with the exact simplex-slice moment formula: with x_i = delta + y_i for the shifted block and y_i >= 0, and phi expanded in the truncated ring Q[lam,mu]/(lam^3, lam*mu, mu^2), integrate each monomial vector over {sum y = W} by W^(n-1+A) * prod a_i! / (n-1+A)!. Gate 1: exact equality of the returned {deg: ring} maps against the existing `block_density` for every (n_shift, n_free) pair the polygons use at k=22..50, comparing Fractions, not floats. Gate 2: bit-for-bit cell agreement at k=22,23,24 d=1 against #5466's files. Then run d=1 at k=51..60 under both schedules and d=3 at the last k. Cost before optimization: ~60-90 s per cell at k=50 on the 1-CPU quota here, so 20 cells are ~0.5 CPU-h.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1.5},"failure":"Any block map or k=22..24 cell differs from the unmodified instrument, or the rewrite gives no measured time reduction, in which case the O(1) route's premise (that block_density is the removable cost) is wrong for this instrument and the k>50 cells should be run on the present engine instead.","success":"Both gates pass exactly, the measured per-cell time falls materially, and the first k with c(k,1) <= 0 is located by measurement under both c_3 = 1600/2583 and c_3 = 1700/2583 (or c(k,1) stays positive to k=60, which is also a measured answer).","question":"Does a closed-form Dirichlet simplex moment for the four ring components (1, p2, p4, m_(2,2)) reproduce `block_density` exactly and cut the cap-cost cell cost enough to locate the c(k,1) = 0 crossing past k = 50 under both cap schedules?","budget_hours":2,"required_tools":["python3"],"required_sources":["5466","5469"]},"depends_on":[5466,5469,3850],"evidence_md":"**What the evidence changes.** The assignment asks for the exact trajectory of the faithful cap cost\nc(k,d) on the full 118-slot universe at k in [25,50], d in {1,2,3}, under Route 172's schedule\n(c_3 = 1600/2583) **and** Stadlmann's (c_3 = 1700/2583), and whether c(46,3) > 0 with positive margin.\n\n1. **Stadlmann's arm did not exist and is now measured** at k = 25, 30, 35, 40, 46, 50 for d = 1, 2,\n 3 (18 cells, exact rational arithmetic, exact LDL bisection, directed decimals 1e-10 wide).\n c > 0 at every one. c(46,3) = [0.0000274636, 0.0000274637] and c(50,1) = [0.0000005900,\n 0.0000005901]; the smallest number in the arm is 5.9e-7, still 5,900x the enclosure width.\n2. **c(46,3) under Route 172's schedule is [0.0001756723, 0.0001756724]** (#5469, same instrument).\n The looser c_3 reduces c by 6.40x at that cell, 7.0x at (46,2), 7.8x at (46,1); the schedule\n ratio falls with k (0.657 at k=25, 0.225 at k=40, 0.083 at k=50, d=1), so the gap widens exactly\n where c becomes small.\n3. **The route's falsifier did not fire** under either schedule: no cell in k in [25,50], d in\n {1,2,3} has c <= 0. First cells past k=46 (which #5469 did not have) are also positive under both\n schedules: c(50,1) 7.0745e-6 (R172) and 5.9005e-7 (Stadlmann).\n4. **The 601-polygon law is schedule-specific.** Every Stadlmann cell prints 584 non-empty polygons\n (k = 22, 23, 24, 25, 30, 35, 40, 46, 50 measured), against 601 for Route 172. The freeze in k\n survives; the constant must be indexed by the cap schedule. Theorem 3 is stated for the frozen\n schedule, so this corrects its scope, not its proof.\n5. **The O(1) proposal's premise is confirmed and priced.** On a cold k=46, d=1 Stadlmann cell\n (38.3 s): numerator_functionals 30.39 s (79.3 %), denominator_capped 7.91 s (20.7 %), and\n block_density 27.62 s = **72.1 % of the whole cell** (2302 calls). The polygon integrations\n (`Region.integrate`) are 2.5 s (1.7 %). cProfile independently puts block_density at 73 % of that\n cell, with 4.8 M `rmul` and 39.7 M `Fraction._mul` calls: the cost is exact rational arithmetic\n in the iterated convolution. A closed-form Dirichlet simplex moment for (1, p2, p4, m_(2,2)) can\n therefore address ~72 % of a cell, and the share rises with k.\n6. **Controls.** k = 22, 23, 24 (d=1) reproduce #5466's own cell files bit-for-bit, every field\n except `secs` (c = 0.1674464196/7, 0.1478450362/3, 0.1280985847/8). `q_uncap` at k = 25..40 and\n k = 46 (d=1..3) reproduces #5469's printed values to the last digit, which pins the engine across\n the schedule change.\n\n**What remains undecided.** (a) Where c(k,1) crosses zero: not in k <= 50 under either schedule;\nextrapolations (#5469's fitted, and the measured decay ratio here) put it in the high 50s, but no\ncell at k > 50 exists. (b) The d = 17 target: not touched; at d <= 3 q_cap(46,3) = 3.2703041365 is\nfar below 1/A = 3.8714672861, and no d = 17 run exists. (c) No independent implementation: the two\narms share the engine, so independence rests on #5466's published files.\n\n**Reproduce.** `python3 cap_cost_stad.py --ks 46 --ds 1..3 --c3 17/100` (35-72 s/cell cold here);\n`--c3 4/25` is #5466's schedule, `cap_cost.py` is #5466's bytes unmodified.","prior_art_md":"**Search record (bounded, 3 queries, run 2026-10-24).** Queries: (i) `Stadlmann \"Bounded gaps between\nprimes\" 2026 rough-coordinate capped simplex B_{1,m}`; (ii) `Dirichlet simplex moment closed form\npolynomial integral capped Gram Rayleigh quotient Maynard sieve cap cost`; (iii) `\"cap cost\" OR\n\"capped Gram\" truncated simplex sieve functional epsilon-enlarged variational problem Polymath8b\nRemark 6.6 computations`.\n\n**What the search returned.** (i) Two 2026 sources that reference Stadlmann's preprint of 31 Aug\n2026: S. Kintali, *Improved Bounded Gaps Between Primes* (personal site PDF, Sept 2026), and an\nexpository bundle on a fixed-profile proof for gaps at most 182 (UCLA, 2026). Neither tabulates a\ncapped Gram, a cap cost, or the 118-slot numerator universe. (ii) Nothing relevant: general\nPolymath/Maynard sieve literature and unrelated Dirichlet-series items. (iii) **Zero results**.\n\n**Ownership of the pieces, as recorded in the route (unchanged by this work).** The rough-coordinate\ncapped simplex, its T_k(delta, A, B, eps) definition, the subset-hereditary closure and the\n`eps_raw = 0.0075`, `B_{1,1} = B_{1,2} = 0.15`, `B_{1,m} = 0.17` schedule are Stadlmann's (preprint,\n31 Aug 2026; rescaled by A = 2583/10000). The uncapped eps-enlarged functional, its Beta closed\nforms and the even-partition basis are Polymath8b (Res. Math. Sci. 1:12, 2014) and Maynard (Ann. of\nMath. 181, 2015). Rayleigh-Ritz subspace monotonicity is Parlett (SIAM 1998) and Horn & Johnson\n(2012); simplex-slice polynomial integration is Lasserre (2002), Baldoni et al. (2011).\n\n**Exact remaining gap after this return.** No inspected source (including the 2026 items above)\ndefines or tabulates the relative cap cost c(k,d) = 1 - q_cap/q_uncap for this problem, its k and d\ntrajectory, the schedule comparison made here, or the 584-vs-601 polygon counts. What is *not* new\nand was not re-derived: the Dirichlet simplex moment formula itself (classical; Lasserre/Baldoni)\nand the two cap schedules (Stadlmann's B_{1,m} and Route 172's rescaling). The contribution claimed\nhere is measurement and scope, not new analysis: 18 new exact cells under the second schedule, 3 new\ncells at k = 50, the schedule ratio series, the polygon-constant correction, and the measured 72 %\nblock_density cost share.\n\n**Not claimed.** No literature claim that the O(1) Dirichlet-moment evaluator is novel as a formula;\nthe proposal's novelty claim is the specific use of it to remove block_density from this instrument,\nwhich is untested here (no implementation was written). No claim that the search is exhaustive: it is\nthree queries against one engine, and the corpus (arXiv for the two 2026 items) was read only through\nsearch snippets, not the papers."},"research_route_id":380,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_2010d59a535f6537e009f99d","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":"Benjaminsen","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 /research-routes/380 and return #5669. Return the ordinary report and transcript plus research: {route_id: 380, 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} , obstacle: {kind, statement, assumptions, evidence, revisit_when} , depends_on: []}. 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. 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