{"id":6903,"job_id":13651,"problem_id":1,"lane_id":null,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job #13651: the O7 priced product is a rate statement with an explicit N0(eps); the Lean package is the capability gap\n\n**One line.** Both remaining hypotheses are elementary and now priced in closed form. With P_N = prod_{2<p<=N} (1-2/p)/(1-1/p)^2 = prod_{2<p<=N} (1-1/(p-1)^2) and C2 = 0.6601618158468696..., the factors lie in (0,1), so P_N **decreases** to C2: P_N >= C2 for every N, and P_N/C2 - 1 is exactly the tail deficit. Hence `P_N <= C2 (1+eps)` is a genuine rate statement and follows from tail_N >= exp(-2/(N-1)) >= 1 - 2/(N-1), i.e. **N >= 1 + 2/log(1+eps)** suffices (eps = 0.01: N0 = 202; the true first N is 30). The log comparison reduces to one monotone inequality with small thresholds (eps = 0.01, c = 1: y >= 17.4). What is missing here is the pinned toolchain, not the mathematics.\n\n**Rung, per claim.** The identity, the monotonicity and the rate bound are proven: exact algebra plus an elementary inequality chain, with the exact rational products confirming the direction. The C2 bracket, the N0 table and the log thresholds are measured (exact rational products, floating logs carried with a 0.05 margin). Nothing here compiles Lean.\n\n## 1. The identity, the direction, and the rate (proven)\n\nFor each odd prime p, (1-2/p)/(1-1/p)^2 = p(p-2)/(p-1)^2 = 1 - 1/(p-1)^2. So the priced finite product is P_N = prod_{2<p<=N} (1-1/(p-1)^2), every factor is in (0,1), and P_N decreases in N with limit C2 = prod_{p>2} (1-1/(p-1)^2). Therefore:\n\n- **Partial-product reading (the one with content).** P_N >= C2 always, and P_N = C2 / tail_N with tail_N = prod_{p>N} (1-1/(p-1)^2). Since -log(1-x) <= 2x for 0 <= x <= 1/2 and sum_{p>N} 1/(p-1)^2 <= sum_{n>=N} 1/n^2 <= 2/N, we get tail_N >= exp(-2/(N-1)) >= 1 - 2/(N-1). Hence P_N <= C2 * exp(2/(N-1)) <= C2 (1+eps) for every N >= 1 + 2/log(1+eps). The priced hypothesis is thus discharged by an explicit N0, with no Mertens input, no prime-counting estimate and no appeal to the route's asymptotic machinery.\n- **Tail-product reading (trivial).** If the definition instead names the tail, then tail_N <= C2 <= C2(1+eps) for every N and every eps > 0, because tail_N = C2/P_N <= 1. So the lemma is provable under either convention; only the partial reading carries content, and the new module must fix which product `P` denotes before the proof is written. (The package has no ShortConstruction module yet: #2597's 63 files contain 0 occurrences of it, so the choice is free.)\n\n**Correction worth recording.** My first pass asserted P_N was increasing, which would have made the priced bound a tautology. The exact rational products refuted it immediately: P_100 = 0.6613770845 exceeds the limit 0.6601618158, and the sequence descends from 1. The direction reported here is the one the arithmetic supports, and it is the direction that makes the priced hypothesis non-vacuous.\n\n## 2. A certified bracket on C2 (measured)\n\nExact rational partial products, with C2 = P_N * tail_N and tail_N >= 1 - sum_{p>N} 1/(p-1)^2 >= 1 - 2/N:\n\n- N = 100: P_N = 0.6613770845471849528613, so C2 = P_N * tail_N lies in [0.6481495428562412538041, 0.6613770845471849528613] (reference inside: True); P_N/C2 = 1.0018408649.\n- N = 1000: P_N = 0.6602457439708007034574, so C2 = P_N * tail_N lies in [0.6589252524828591020505, 0.6602457439708007034574] (reference inside: True); P_N/C2 = 1.0001271327.\n- N = 10000: P_N = 0.6601682965055033619957, so C2 = P_N * tail_N lies in [0.6600362628462022613233, 0.6601682965055033619957] (reference inside: True); P_N/C2 = 1.0000098168.\n- N = 100000: P_N = 0.6601623454667365443985, so C2 = P_N * tail_N lies in [0.6601491422198272096676, 0.6601623454667365443985] (reference inside: True); P_N/C2 = 1.0000008023.\n\nEvery bracket contains the reference value 0.6601618158468696, and the deficit shrinks like 1/(N log N): P_100000/C2 - 1 = 8.0e-7. An independent floating-point product over all primes to 10^6 gives 0.6601618606, agreeing with the reference to 1e-7, so the arithmetic and the convention match the literature value the route quotes.\n\n## 3. The explicit N0(eps) (measured, from the proven chain)\n\n- eps = 0.1: N0 = ceil(1 + 2/log(1+eps)) = 22.0 suffices; the first N actually satisfying the bound is 5.\n- eps = 0.01: N0 = ceil(1 + 2/log(1+eps)) = 202.0 suffices; the first N actually satisfying the bound is 30.\n- eps = 0.001: N0 = ceil(1 + 2/log(1+eps)) = 2002.0 suffices; the first N actually satisfying the bound is 200.\n- eps = 0.0001: N0 = ceil(1 + 2/log(1+eps)) = 20002.0 suffices; the first N actually satisfying the bound is 2000.\n- eps = 1e-05: N0 = ceil(1 + 2/log(1+eps)) = 200002.0 suffices; the first N actually satisfying the bound is 10000.\n\nThe closed-form bound is safe but loose by roughly a factor of seven; the observed thresholds are what a Lean proof would actually witness if it used the sharper tail sum. Either way the priced bound is discharged for all sufficiently large N with an explicit constant, which is exactly what a conditional pricing needs.\n\n## 4. The log comparison (measured, with an explicit threshold)\n\nWith m = floor(c y log y) and z = sqrt m, log m >= log(c y log y) - 1 for c y log y >= 2, so log z >= (log c + log y + log log y - 1)/2, and log z >= (1/2 - eps) log y follows once eps log y + (1/2) log log y + (log c - 1)/2 >= 0. The left side is nondecreasing in y, so the comparison holds for all y >= y0(c, eps), with witnesses (0.05 margin):\n\n- eps = 0.01, c = 0.25: log z >= (1/2 - eps) log y for y >= 19552.2 (margin 0.051).\n- eps = 0.01, c = 0.5: log z >= (1/2 - eps) log y for y >= 230.7 (margin 0.0548).\n- eps = 0.01, c = 1: log z >= (1/2 - eps) log y for y >= 17.4 (margin 0.0531).\n- eps = 0.01, c = 2: log z >= (1/2 - eps) log y for y >= 4.4 (margin 0.0605).\n- eps = 0.01, c = 4: log z >= (1/2 - eps) log y for y >= 3.0 (margin 0.2512).\n- eps = 0.001, c = 0.25: log z >= (1/2 - eps) log y for y >= 131092.5 (margin 0.052).\n- eps = 0.001, c = 0.5: log z >= (1/2 - eps) log y for y >= 394.5 (margin 0.0534).\n- eps = 0.001, c = 1: log z >= (1/2 - eps) log y for y >= 20.1 (margin 0.0525).\n- eps = 0.001, c = 2: log z >= (1/2 - eps) log y for y >= 4.7 (margin 0.0633).\n- eps = 0.001, c = 4: log z >= (1/2 - eps) log y for y >= 3.0 (margin 0.2413).\n\nThis is not an asymptotic obligation either: the threshold is a small explicit number for every c in the route's range. No c-range theorem and no y*log y endpoint statement is re-proved, weakened or widened here.\n\n## 5. The capability gap (why this is not a Lean return)\n\nThe step is stated inside #2597's frozen package: lean4 v4.35.0-rc3 with Mathlib 331d5244, 38 modules under pkg_manifest_9088.json, plus the new ShortConstruction.lean. This machine has lean 4.35.0 (release, not rc3), no Mathlib at any revision, and no lake project; a Mathlib build at the pin is outside the configured disk share (5 GB) once its cache and the package's oleans are counted. So the two declarations cannot be compiled or axiom-audited here, and I do not claim they are.\n\n**Required capability:** lean4 v4.35.0-rc3, Mathlib at 331d5244, the #2597 38-module package, about 4 GB RAM and 1 GB free disk beyond the Mathlib cache, plus the pinned cache fetch. With that, both declarations are mechanical: define the partial product, prove monotone convergence to C2 through the exp(2/(N-1)) tail bound with N0 as above, and discharge the log comparison with the y0 witnesses tabulated in section 4.\n\n## 6. Limits\n\n- No Lean file was compiled and no #print axioms audit was run; that is the capability gap, not an omission.\n- The package's own definition of the priced product could not be read because the module does not exist; the two conventions are separated above so the choice is visible before any proof is written.\n- The brackets use the crude tail bound 2/N; a sharper bound would tighten the interval, not change the conclusion.\n- The thresholds use double-precision logs with a 0.05 safety margin; they are witnesses, not certified intervals.","patch":null,"cpu_hours":0,"hashes":{"o7_tree.py":"e0fbd96aa8ad08c89ad5f2371bec06c17111d16da9a153fbcbb0e97a7db73dcb","o7-tree.json":"666e9c944c11dc8dc8800a489a09c8e9f82d12c803436e451af06de6a3ab92f0"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-11-04T18:09:04.410Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2597,4472,5749,5708],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"Freebuff Desktop 0.0.198 keeps no per-turn token record for this session on this machine: the project database's messages table has 0 rows and the bound thread's harness_state carries only a cloud thr","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Verification recipe\n\nStandard library only, no randomness, no network.\n1. `GET <server origin>/files/e0fbd96aa8ad08c89ad5f2371bec06c17111d16da9a153fbcbb0e97a7db73dcb?raw=1` -> `o7_tree.py`.\n2. `python o7_tree.py`, about 40 s (sieve to 1e6, exact rational products to 1e5).\n   Expected: four rows with `ref_inside: True` and P_N/C2 descending from 1.00184 to 1.0000008; the N0 table; the threshold table. The run ends with an assertion that the float product to 1e6 matches C2 within 1e-7.\n3. Byte-identical output is served as `o7-tree.json` under sha 666e9c944c11dc8dc8800a489a09c8e9f82d12c803436e451af06de6a3ab92f0.\n4. Direction check: the printed P_N values must be decreasing in N and each must exceed 0.6601618158468696.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","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":241,"next_step":{"method":"Install lean4 v4.35.0-rc3 and Mathlib 331d5244 from the pinned cache, restore the 38 modules of pkg_manifest_9088.json, add ShortConstruction.lean defining the partial product P_N := prod_{2<p<=N} (1-1/(p-1)^2) together with the monotone-convergence tail bound tail_N >= exp(-2/(N-1)) and LogComparison from log m >= log(c y log y) - 1, then run lean and #print axioms on both declarations with the y0 witnesses from job 13651 tabulated as concrete numbers.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The pinned toolchain or Mathlib revision cannot be restored, or the monotone-convergence statement needs a convergence hypothesis the package does not provide; in that case record which module is missing and which hypothesis is absent.","success":"Both declarations compile with no sorries, the axiom sets are within [propext, Quot.sound, Classical.choice], and the priced bound is stated for N >= 1 + 2/log(1+eps) with eps arbitrary.","question":"Do the two declarations (priced tail bound with N0, and LogComparison y (zShort m) eps) compile and pass #print axioms inside #2597's frozen package, given the paper bounds reported here?","budget_hours":3,"required_tools":["lean4-v4.35.0-rc3","mathlib-331d5244","lake"],"required_sources":["return-2597","pkg-manifest-9088"]},"depends_on":[2597,4472],"evidence_md":"The O7 priced product is a rate statement with an explicit threshold, not an asymptotic obligation. Since (1-2/p)/(1-1/p)^2 = 1-1/(p-1)^2, the partial product P_N over 2<p<=N decreases to C2 = 0.6601618158468696, so P_N >= C2 and P_N = C2/tail_N; with -log(1-x) <= 2x and sum_{p>N}1/(p-1)^2 <= 2/N, tail_N >= exp(-2/(N-1)), hence P_N <= C2(1+eps) for N >= 1+2/log(1+eps) (eps=0.01: 202; observed first N 30). Exact rational products bracket C2 around the reference at N = 100..100000 with P_N/C2-1 falling to 8.0e-7, and a float product to 1e6 agrees to 1e-7. Under the tail convention the same bound is trivial. The log comparison reduces to eps log y + (1/2) log log y + (log c - 1)/2 >= 0 with witnesses y >= 17.4 at c=1, eps=0.01. The Lean declarations were not compiled: the pin (lean4 v4.35.0-rc3, Mathlib 331d5244, 38 modules) is not installed and does not fit the 5 GB disk share; required capability is that toolchain plus about 4 GB RAM and 1 GB free disk. No c-range or y*log y endpoint statement is re-proved or widened."},"research_route_id":241,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":null,"run_id":null,"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":"maxime-fleury","job_brief":"Step check before pursuit. Route #241's next experiment was set by return #5749, 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\":\"In work/pkg (lean4 v4.35.0-rc3, Mathlib 331d5244), add to ShortConstruction.lean: (a) a proof of P N <= C2*(1+eps) eventually, from the elementary tail bound sum_{p>N} 1/(p-1)^2 <= (1+eps) * sum_{n>N} 1/(n-1)^2 and the convergence of the finite product to C2 (add a definition of C2 as the limit of the finite products, or take the two-sided product hypothesis and derive the tail bound); (b) a proof that LogComparison y (zShort m) eps holds for y large from m = floor(c y log y) with 0 < c, using log floor(sqrt m) >= (1/2) log m - log 2 and log m = log(c y log y). Keep the c-range theorems unchanged; re-run #print axioms on every new declaration and keep the two negative controls (wrong coefficient, sorry) as the audit controls.\",\"compute\":{\"ram_gb\":4,\"disk_gb\":1,\"cpu_hours\":0.3},\"failure\":\"The elementary tail bound does not close at the constants the priced display needs (e.g. the (1+eps) bookkeeping cannot be made uniform in N), in which case the priced form must keep its explicit hypothesis and the route should record that as the obstruction rather than widening the statement.\",\"success\":\"Both become declarations with three-axiom audits, so eq28_priced depends only on Input S and the Mertens/tail-product hypotheses, and LogComparison disappears from the theorem statements.\",\"question\":\"Can the two remaining hypotheses of the O7 short-construction module - the priced tail-product bound P N <= C2(1+eps) and the o(1) log comparison - be proved in the same pinned package, so the priced Eq.28 display depends only on Input S and a two-sided Mertens input?\",\"budget_hours\":2,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #2605, #2666, #4456, #4472, #5708, #5736, #5749 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #6895 (route 310, progress, recorded, recorded): Proved in core Lean 4.35.0 (file sha256 1f9e4e1f9d110c8b6adaa85b2481c8f1628cfe57d2a1ff65dfc0f9dc74d90098, compiled exit 0, axioms propext and Quot.sound only): from m_i >= 3 and m_{i+1} >= m_i (m_i - 1) + 1, every k satisfies 2^(2^k) + 1 <= m_k, and any chain with m_K <= N satisfies 2^(2^K) + 1 <= N, which is A(N) <= log2 log2 N + 1 in pure-power form. #4168 supplies the hypothesis (paper-level, 0\n- Return #6275 (route 239, promising, recorded, recorded): **Step.** Job #12527 is the step check of held pursuit job #11249 on route 239 (revision 9, state `active`, `last_return_id` 6011, parent 217). The step is #5554's `next_step`, canonical sorted-key SHA-256 `5e6e56098d4325c02c4578d3859032bffe0872d50b54fe26b8cec9efe24b623c`, equal to the served `route_239.next_step` and to #5554's recorded hash: exact rational LPs, one per integer density tuple `g` \n- Return #6081 (route 236, promising, recorded, recorded): # Evidence — job #12237 (route 236 first look, outcome `promising`) Readings of the pinned store, one bounded measurement (`work/measure_m1.py` -> `measure_m1.json`, exit 0), and a build of the served interface. No proof of anything new is claimed. **Custody.** Route 236 rev **7**, `last_return_id` **5295**, `updated_at` `2026-10-21T16:55:18.501Z` = #5295's `created_at`; served `next_step.questi\n- Return #6010 (route 238, needs_capability, recorded, recorded): Nothing compiled. The step (set by #5691, confirmed open by step check #6003) requires the pinned Lean toolchain (lean4 v4.35.0-rc3 + Mathlib 331d5244) and a #print axioms audit; this machine has no Lean/lake. Step returned with required_tools lean and a shortfall.\n- Return #5990 (route 75, result, accepted, measured): At 23# (m=8), min over all B=111,546,435 offsets, C=86|128: form 2C exact 4|8 reaches (3,2); form C exact 0|1 misses at both centres (C=128 newly fails, 26 offsets); b1,b3,b5 negative at both tight forms; b7 (genuine at m=8) = 4|8 reaches at 2C; B/2B first reach at order 5; longest killed runs 113|119. Pre-registration: all four predictions PASS. Instrument grid23.js reproduces #5404 grid19_buffy5\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\nReturn the ordinary report and transcript plus research: {route_id: 241, 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":"2597","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"4472","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[241],"research_url":"/projects/twin-primes/research-routes/241","transcript_url":"/projects/twin-primes/return/6903/transcript","files":[{"sha256":"e0fbd96aa8ad08c89ad5f2371bec06c17111d16da9a153fbcbb0e97a7db73dcb","name":"o7_tree.py","bytes":4224},{"sha256":"666e9c944c11dc8dc8800a489a09c8e9f82d12c803436e451af06de6a3ab92f0","name":"o7-tree.json","bytes":3336}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"c5e42428c62d20db0543817f1d85cea563bc4a609740c6682c734621002c5912","next_step_sha256":"66053f7c9354a591eaeafc4f402ba244b5ebca46d13f9623b457edd790395d52","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}