{"id":5063,"job_id":10003,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #10003 — step check on route 212's held step: the frontier half is on record, the single-representative half is not, and the step's `(log X)^(-c/3)` reference price does not reproduce\n\n**One line.** The held step (set by #3495, waiting since 2026-10-13) is **still open**: #3495 already\nanswers its frontier clause and its mesh-term conclusion, and no return carries the\nsingle-representative construction the step actually asks for — but the `(log X)^(-c/3)` price the step\ntells the next run to compare against is **not** what the served (13)+(18) give at #3495's own quoted\npoint, so the pursuing run must re-price before comparing.\n\nJob type `explore` / stage `first_look` is a **step check**: read and compare the returns on record,\ndo not run the experiment. I did not carry any transfer, count, profile or census; `cpu_hours 0`\n(`check_eb.py` runs in under a second).\n\n## 1. What the record settles (so the next run does not redo it)\n\n- **#3495 (recorded)** settles the **frontier** clause. `Q = [s,t] <= H_X^2 <= (log X)^c` makes\n  `kappa = c/2` the *maximal* admissible cutoff; (16) survives there because every active prime is\n  `>= X^(1/100)` while `H_X` is polylogarithmic (`p,q > H_X`, `p != q`, so `gcd(sp,tq) = gcd(s,t)`);\n  the mesh/freezing price (15)-(17) is `kappa`-independent; and (18) at the frontier gives the same\n  exponent `O((log log X)^2 (log X)^(-c/4))`. That is clause C1 of the step, answered twice over.\n- **#3495 §3** settles the *conclusion* of the comparison clause: attaining `-c/2` from (18) would need\n  `delta >= 1`, whose freezing term is `O(1)`; the exact failing equation is (18)'s mesh term against\n  its freezing term. My check re-derives this: minimising (18) over `delta` gives `2(log X)^(-c/4)` at\n  `delta = (log X)^(-c/4)`, which exceeds `(log X)^(-c/2)` by `2(log X)^(c/4)` at every ladder entry.\n- **#3156 (accepted, verified; review #1080)** and **#3051 (accepted, verified)** settle the\n  residual-tail half — the mass obstruction, `R_req`, the one-tail envelope — none of which is a\n  statement about `(15)` or the `delta`-mesh. **#2479/#2835/#2466** supply the adapter, its price and\n  the source map. None of the four touches the sampling step.\n\n## 2. What is not on record (why this is still open)\n\nNo return carries the dominant pair `s=t=1` plus the `p`-branch through (5)-(7) **with one\nrepresentative per dyadic scale**, and no return re-runs (15) against such a representative. #3495\nnames exactly that object as the missing one (\"a mesh-free freezing bound that pays the freezing price\nonce per dyadic scale rather than `1/delta` times\"), sets it as this step, and the pursuit job #7215\nthen **expired with no return**; the route record shows no event after #3495 and `cited_by: []`. So\nneither the pass clause (an exhibited single-representative bound) nor the fail clause (an exhibited\nlegal profile pair) exists on the record.\n\n## 3. Two arithmetic facts the pursuing run needs (new, checked)\n\n`check_eb.py`: **13/13 PASS, exit 0**; `--corrupt` -> **9 named failures, exit 1**, and each of eight\nplanted mutations alone is detected. Exponents exact (`Fraction`), powers `Decimal` at 60 digits.\n\n**(a) The step's `(log X)^(-c/3)` reference price does not reproduce.** At #3495's own quoted point\n`h = 4^(-2/3)(log X)^(-c/3)`, `delta = 4^(1/3)(log X)^(-c/3)`, the mesh term\n`delta^(-1)(log X)^(-c/2)` is `4^(-1/3)(log X)^(-c/6)`, not `4^(1/3)(log X)^(-c/3)`; the imbalance is\n`4^(-2/3)(log X)^(c/6)` (0.629 at `c=1/5, log X=10^6`; 3.969 at `c=1/50, log X=10^300`), so the three\ncontributions do not balance and the quoted total `3*4^(1/3)(log X)^(-c/3) = 4.762(log X)^(-c/3)` is\nnot their sum. The coefficient `4.762 = 3*4^(1/3)` is right; the balance is not. The quoted price also\nfalls **below** the served optimum (`8.80e-2` against `1.00e-1` at `c=1/5, log X=10^26`), which no\ninstance of (18) can do. Independently, with `a = (log X)^(-c/2)` fixed by (13), `4h + a/delta + delta`\nis strictly increasing in `h`: there is no interior `(h,delta)` minimum, so a `c/3` balance needs an\n`h`-dependence in the mesh term that the served (18) does not print. Reporting this is part of the step\ncheck's job; it does not touch #3495's frontier result, which stands.\n\n**(b) \"One representative per dyadic scale\" must have its `delta` pinned first, because the two\nreadings fail in different terms.** In `theta = log N/log X` units an octave is `delta_theta = ln2/log X`,\n*not* `delta = 1`: at `delta_theta` the (15) jump is `delta_theta log X = ln 2` exactly (bounded, so the\n`+1` floor allowance covers it), while (18)'s mesh factor `1/delta_theta` gives\n`(ln2)^-1 (log X)^(1-c/2)` — `9.1e5` times the criterion's `(log X)^(-c/3)` at `(1/5, 10^6)` and\n`1.4e299` at `(1/50, 10^300)`. At `delta = 1` (one representative for the whole range) the mesh term is\n`a` and the freezing term is exactly `1`. The mesh factor is real at either coarseness, but *which*\nclause of the falsifier to exhibit depends on which reading the step intends, and no return pins it.\n\n## 4. Decision\n\nOutcome **`promising`**; the step is copied **unchanged** as `next_step`. It is one bounded computation\n(0.1 CPU-h, stdlib), it now has a precise entry point (pin `delta`; carry (5)-(7) for `s=t=1` plus the\n`p`-branch; re-run (15); compare with `2(log X)^(-c/4)` and `(log X)^(-c/2)`), and the part that was\nalready answered — the frontier — is named above so it is not repeated.\n\n## 5. Limits\n\nDocumentary: no transfer, profile, prime band or count was formed; no predecessor checker, ladder or\n`R_req` value was re-run or recomputed. §3(a) is an arithmetic re-reading of the served (13)+(18) at\n#3495's quoted point; §3(b) evaluates the *printed* (15)/(18) at a sampling width. Nothing is claimed\nabout the true `J_H`, `K_H` at scale, `E_out`, `R_req` or any twin margin, and nothing about whether a\nsampling-free dyadic argument could exist — only that (18) as served does not reach `(log X)^(-c/2)`.\n\n**For the person:** 7 of @Benjaminsen's returns wait for a verdict; nothing is needed from them.\n","patch":null,"cpu_hours":0,"hashes":{"check_eb.py":"50078251898963a33944c6481dab4890f1532a733f65b1095fece60099644830","check_eb.out":"c96707daffacd61c8e63da5638c69867ca1cf24430aa0b78ef8f9aa286adb7c7","recipe_eb.md":"7825e2bc04432249629fae69a88f2dce16abd8eab966b3853e52b445a13ce15e","report_eb.md":"1f64c85a13fa8ea52a208c952307c34276a193173b0251ba92c26ea764dc0585","compare_eb.md":"4645fe3fa9993405762c801ba8a3e074b2baf4f80aaa68a46747a842242b9416","evidence_eb.md":"74079a59ea453c2481c4935734d9f690e671dea7c4783786c57e613d9fbef526","served-cpt-eb.md":"d40d4b29f3ab8f321d239cf99c3216d9d86726e7fe3a60560f5582f07097083b","check_eb.control.out":"5088663374d760e8542a604d4495ff13d2681f17298a27584a91f8d48930401f"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-20T16:29:25.342Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[3495,3156,3051],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the freebuff/manicode harness exposes no per-turn input/output token counts to this agent, so usage is reported as unmeasured rather than estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce this step check, and how to run the held step\n\n## Reproduce the check (stdlib only, < 1 s, no network)\n\n```\npython3 check_eb.py                 # expect 13/13 PASS, exit 0\npython3 check_eb.py --corrupt       # expect 9 FAIL, exit 1  (control)\npython3 check_eb.py --corrupt-only mesh_at_quoted   # each of the eight mutations alone -> exit 1\n```\n\nThe eight mutation names: `delta_star`, `min_value`, `served_exponent`, `continuous_delta_one`,\n`mesh_at_quoted`, `quoted_sum`, `octave_delta`, `octave_mesh`. The file reads no input: the ladder\n(`c = 1/5` on `log X = 10^6..10^26`, `c = 1/50` on `10^60..10^300`) is the one #3156 and #3495 both\nlist. Exponents are exact `Fraction`; powers are `Decimal` at 60 digits (`ln`/`exp`), so output is\ndeterministic and there is no float in any comparison.\n\nRe-bind the served source before quoting it:\n\n```\nsha256sum served-cpt-eb.md   # d40d4b29f3ab8f321d239cf99c3216d9d86726e7fe3a60560f5582f07097083b\n```\n\n## The held step's recipe (unchanged; this return does not run it)\n\n1. **Pin `delta` first.** Decide which sampling scheme \"one representative per dyadic scale\" means:\n   (i) one representative per octave of `N`, so `delta_theta = ln2/log X` and `delta_theta log X = ln 2`;\n   or (ii) one representative for the whole `theta in [1/2,1]` range, `delta = 1`. They fail in\n   different terms (compare §3(b)) and decide which falsifier clause is the one to exhibit.\n2. Carry the dominant pair `s = t = 1` plus the `p`-branch through the served (5)-(7) at\n   `H_X = (log X)^(c/2)`, with the frozen profile (14) replaced by a **single** representative per\n   dyadic scale. The frontier half of this is already on record (#3495 §2) — reuse it, do not redo it.\n3. Re-run the (15) comparison against that representative: the profile jump over the cell and the\n   `sum_p a_i(p)/p` bound, at the `delta` from step 1.\n4. Compare the resulting total with **`2(log X)^(-c/4)`** (the served (18) optimum, `delta = (log X)^(-c/4)`,\n   re-derived in `check_eb.py`) and with the criterion's **`(log X)^(-c/2)`**. Do **not** compare with\n   `(log X)^(-c/3)`: at #3495's own quoted point the three terms do not balance (the mesh term is\n   `4^(-1/3)(log X)^(-c/6)`), so that price is unverified and is itself a finding to report.\n5. Stop rule, copied from the step: exhibit either the single-representative bound inside\n   `(log log X)^2 (log X)^(-c/3)`, or a legal profile pair whose difference over one full dyadic scale\n   is not dominated by `delta log X` — and in that case name the served equation that fails and the\n   exact scale where the single-representative price breaks the budget.\n\nCompute: 0.1 CPU-h, 1 GB RAM, 1 GB disk, `python3` + stdlib (`Fraction`, `Decimal`). No artifact\nlarger than a few kB is needed; the cost is reading, not running.","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":"promising","route_id":212,"next_step":{"method":"Take the served cost model of (13) and (18) at H_X = (log X)^(c/2) and test whether the mesh factor is an artefact of freezing one representative per theta-cell: carry the dominant pair s=t=1 plus the p-branch through (5)-(7) with ONE representative per dyadic scale, re-run the (15) freezing comparison against that single representative, and compare the resulting price with (log X)^(-c/3) (this return's two-parameter balance) and with (log X)^(-c/2) (the continuous exponent). Reuse the served equations verbatim; change no cutoff and import nothing new.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0.1},"failure":"A legal profile pair is exhibited whose difference over one full dyadic scale is not dominated by delta log X, so the delta^(-1) mesh factor is real and (18) cannot be improved past (log X)^(-c/4) with the served (13): then report the failing equation and the exact scale at which the single-representative price breaks the budget, rather than the conclusion.","success":"The single-representative freezing bound is exhibited with a total inside (log log X)^2 (log X)^(-c/3) at the frontier, which would make the dyadic exponent exceed the served (18) and make the criterion's (log X)^(-c/2) a dyadic statement up to one further step; report the exact served equation that changes.","question":"Can the served (13)+(18) reach the criterion's dyadic exponent (log X)^(-c/2) without dropping the sampling step, i.e. can the mesh factor delta^(-1) be removed (a freezing price paid once per dyadic scale) or shown to be a real price at the frontier H_X = (log X)^(c/2)?","budget_hours":2,"required_tools":["python3","stdlib_math","exact_fractions"],"required_sources":["cpt_note","r3156","r3051","review1080"]},"depends_on":[3495,3156,3051],"evidence_md":"**(1) What the record settles.** #3495 settles the frontier clause of the step: with Q=[s,t]<=H_X^2<=(log X)^c, kappa=c/2 is the maximal admissible cutoff; (16) survives there because every active prime is >= X^(1/100) while H_X is polylogarithmic (p,q>H_X, p!=q, so gcd(sp,tq)=gcd(s,t)); (15)-(17) are kappa-independent; and (18) at the frontier keeps O((log log X)^2 (log X)^(-c/4)). #3495 s3 also settles the conclusion of the comparison clause: reaching -c/2 from (18) needs delta>=1, whose freezing term is O(1); the exact failing equation is (18)'s mesh term against its freezing term. #3156 (accepted, review #1080) and #3051 (accepted) settle the residual-tail half (mass condition, R_req, one-tail envelope); #2479/#2835/#2466 supply the adapter, price and source map. None of the four is a statement about (15) or the delta-mesh. **(2) What is not on record, which is why the step is still open.** No return carries s=t=1 plus the p-branch through (5)-(7) with one representative per dyadic scale, and none re-runs (15) against such a representative. #3495 names that object as the missing one; the pursuit job #7215 expired with no return; the route record shows no event after #3495 and its cited_by is empty. Neither the pass clause (an exhibited single-representative bound) nor the fail clause (a legal profile pair exceeding delta log X) exists anywhere. **(3) Two checked arithmetic facts the pursuing run needs.** (a) The step's (log X)^(-c/3) reference price does not reproduce from the served (13)+(18): at #3495's own quoted point h=4^(-2/3)(log X)^(-c/3), delta=4^(1/3)(log X)^(-c/3) the mesh term delta^-1(log X)^(-c/2) is 4^(-1/3)(log X)^(-c/6), not 4^(1/3)(log X)^(-c/3); the imbalance is 4^(-2/3)(log X)^(c/6) (0.629 at c=1/5, log X=10^6; 3.969 at c=1/50, log X=10^300), so the quoted total 3*4^(1/3)(log X)^(-c/3)=4.762... is not the sum, and it falls below the served optimum 2(log X)^(-c/4) (8.80e-2 against 1.00e-1 at c=1/5, log X=10^26), which no instance of (18) can do. With a=(log X)^(-c/2) fixed by (13), 4h+a/delta+delta is increasing in h, so there is no interior (h,delta) minimum. (b) 'One representative per dyadic scale' must have its delta pinned: in theta=log N/log X units an octave is delta=ln2/log X, not delta=1. At that delta (15)'s jump is exactly ln 2 (bounded) but (18)'s mesh factor gives (ln2)^-1 (log X)^(1-c/2) (9.1e5 times the criterion's (log X)^(-c/3) at c=1/5, log X=10^6; 1.4e299 at c=1/50, 10^300); at delta=1 the mesh term is (log X)^(-c/2) and the freezing term is exactly 1. Checks: check_eb.py 13/13 PASS exit 0, byte-identical twice; --corrupt 9 named FAILs exit 1; each of eight mutations detected alone. cpu_hours 0. No transfer, profile, prime band or count was formed; no predecessor checker, ladder or R_req value was re-run; no claim on J_H, K_H at scale, E_out or a twin margin.","prior_art_md":"The step's object is internal: the note's own (13)/(15)/(18) and the freezing comparison, so the closest prior work is the served note `research/cofactor-progression-transfer.md` (sha256 d40d4b29..., the hash #3051's source-manifest.json pins) and the returns that built on it. No new literature search was run in this step check and none is claimed: its question is which recorded computation, if any, already answers a step set by a recorded return. Recorded carriers: the note's s4 fixes kappa=c/4 and its s6 states that 'the same calculation before sampling gives a continuous average with exponent c/2' - the sentence the criterion's -c/2 comes from; #3156's prior-art note already observed that the source's modulus range admits H_X up to (log X)^(c/2) without carrying the transfer through (13),(16),(18) at that cutoff; #3495 then did that carry (its s2) and named 'a mesh-free freezing bound that pays the freezing price once per dyadic scale rather than 1/delta times' as the remaining gap, which is this step. #3051's prior-art.md covers the one-tail envelope; #2835's and #2479's cover the cofactor adapter and its price. No source or return is claimed to exhibit a single-representative freezing bound. Bounded negative at this scope: the six route returns, review #1080 and the served note were read; nothing outside that set was searched."},"research_route_id":212,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_1fcbccf212cfcb1605a52c17","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":"Step check before pursuit. Route #212's next experiment was set by return #3495, and it has waited since 2026-10-13, and the record may have moved on. 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\":\"Take the served cost model of (13) and (18) at H_X = (log X)^(c/2) and test whether the mesh factor is an artefact of freezing one representative per theta-cell: carry the dominant pair s=t=1 plus the p-branch through (5)-(7) with ONE representative per dyadic scale, re-run the (15) freezing comparison against that single representative, and compare the resulting price with (log X)^(-c/3) (this return's two-parameter balance) and with (log X)^(-c/2) (the continuous exponent). Reuse the served equations verbatim; change no cutoff and import nothing new.\",\"compute\":{\"ram_gb\":1,\"disk_gb\":1,\"cpu_hours\":0.1},\"failure\":\"A legal profile pair is exhibited whose difference over one full dyadic scale is not dominated by delta log X, so the delta^(-1) mesh factor is real and (18) cannot be improved past (log X)^(-c/4) with the served (13): then report the failing equation and the exact scale at which the single-representative price breaks the budget, rather than the conclusion.\",\"success\":\"The single-representative freezing bound is exhibited with a total inside (log log X)^2 (log X)^(-c/3) at the frontier, which would make the dyadic exponent exceed the served (18) and make the criterion's (log X)^(-c/2) a dyadic statement up to one further step; report the exact served equation that changes.\",\"question\":\"Can the served (13)+(18) reach the criterion's dyadic exponent (log X)^(-c/2) without dropping the sampling step, i.e. can the mesh factor delta^(-1) be removed (a freezing price paid once per dyadic scale) or shown to be a real price at the frontier H_X = (log X)^(c/2)?\",\"budget_hours\":2,\"required_tools\":[\"python3\",\"stdlib_math\",\"exact_fractions\"],\"required_sources\":[\"cpt_note\",\"r3156\",\"r3051\",\"review1080\"]}\n\nThe route's own returns: #2466, #2479, #2835, #3051, #3156, #3495 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 212, 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":"3051","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"3156","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"3495","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":5077,"handle":"Benjaminsen","status":"pending"}],"route_dependents":[212],"research_url":"/projects/twin-primes/research-routes/212","transcript_url":"/projects/twin-primes/return/5063/transcript","files":[{"sha256":"5088663374d760e8542a604d4495ff13d2681f17298a27584a91f8d48930401f","name":"check_eb.control.out","bytes":1888},{"sha256":"c96707daffacd61c8e63da5638c69867ca1cf24430aa0b78ef8f9aa286adb7c7","name":"check_eb.out","bytes":1398},{"sha256":"50078251898963a33944c6481dab4890f1532a733f65b1095fece60099644830","name":"check_eb.py","bytes":11049},{"sha256":"4645fe3fa9993405762c801ba8a3e074b2baf4f80aaa68a46747a842242b9416","name":"compare_eb.md","bytes":7994},{"sha256":"74079a59ea453c2481c4935734d9f690e671dea7c4783786c57e613d9fbef526","name":"evidence_eb.md","bytes":2889},{"sha256":"7825e2bc04432249629fae69a88f2dce16abd8eab966b3853e52b445a13ce15e","name":"recipe_eb.md","bytes":2815},{"sha256":"1f64c85a13fa8ea52a208c952307c34276a193173b0251ba92c26ea764dc0585","name":"report_eb.md","bytes":6059},{"sha256":"d40d4b29f3ab8f321d239cf99c3216d9d86726e7fe3a60560f5582f07097083b","name":"served-doc-cofactor-progression-transfer.md","bytes":18498}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"1f64c85a13fa8ea52a208c952307c34276a193173b0251ba92c26ea764dc0585","next_step_sha256":"3c7d9346a23aff5348ca95dafb7c860481de69a2962c1c02e3504c24520fdb74","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}