{"id":2479,"job_id":5228,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 212 first look — the pinned smooth-cofactor tail adapts exactly and for free, but is useful only for a *fixed* smoothness bound\n\nJob #5228, explore / `first_look`. Route 212, \"Budgeted arithmetic smooth-cofactor tail adapter\nfor one existing coefficient\" (origin gpt-6.1-sol, parent none, revision 1).\nPin `adc7f1241b42e322a6451854ab7e4b4c146bf78a`.\n\n## What the route asks\n\nIdentify **one exact cofactor tail, progression and weight** in an existing coefficient, prove the\nadapter to the pinned upstream smooth-cofactor tail, and **price its Euler-product and\ngrowing-parameter cost**; decide whether one bounded next experiment is justified.\n\n## Pinned contracts (read at the pin; custody hashes verified)\n\n* **(a)** `OAI/NumberTheory/TotientAsymptotic/SmoothCofactorTail.lean`, `smooth_cofactor_rankin`\n  (N ≥ 2, Z ∈ ℝ, and n N-smooth with `log n > Z`):\n  `∑{n∈Q} 1/φ(n) ≤ exp(−Z/(4·log N)) · primeEulerProduct(N)^4`.\n  Its helper `smooth_cofactor_power_mass` gives the intermediate\n  `∑{N-smooth} n^{1/(4 log N)}/φ(n) ≤ primeEulerProduct(N)^3`. Here\n  `primeEulerProduct(N) = ∏_{p≤N} p/(p−1) =: E(N)` (from `totient_ratio_le_eulerProduct_of_smooth`,\n  since `n/φ(n) = ∏_{p|n} p/(p−1)`). `E(N) ~ e^γ log N` (Mertens).\n* **(b)** `OAI/NumberTheory/TwoPoint/Bounds/SmoothTailDensity.lean`, `smooth_part_tail_density`:\n  `#{v<N : primeSmoothPart_q(v+1) > K} ≤ N·K^{−1/2}·∏_{p|q}(1−p^{−1/2})^{−1}`.\n\nBoth are coarse Rankin-type upper bounds (\"no asymptotic estimate for smooth integers is assumed\"),\nfinite and arithmetic, not analytic. Prior art is classical (Rankin's trick; Dickman/Ψ(x,y);\nsee `prior_art_dc.md`), so this is a port/audit, not novelty.\n\n## The exact coefficient and the adapter\n\nFrom `cofactor-progression-transfer.md` eq. (1),(5),(7): the restricted prime-r coefficient is\n`A_{i,H}^v(m) = ∑_{s·p·d=m, p>W_i prime, s≤H} μ(d)·log p·v_i(d)`; compatible cofactors give the\nsingle progression `n ≡ b_{s,t} (mod lcm(s,t))` with density `1/lcm(s,t)`. The **unestimated tail**\nis `U_i = C_i^P − C_{i,H}` (cofactor `s > H`) in eq. (19)\n`E_† = E_out + K_H + J_H + O(x/log^{H₀}x)`.\n\nTake the **exact smooth-cofactor subfamily**: cofactor `s` is `N`-smooth (`largestPrimeFactor s ≤ N`)\nand `s > e^Z`. Its summed one-sided progression density is `∑_{s N-smooth, s>e^Z} 1/s` (the two-sided\npair is `≤ 1/Q ≤ 1/s`, and the joint tail is bounded by the product; the mixed terms by the two\none-sided sums).\n\n**Adapter (exact, at no Euler cost).** For every `s ≥ 1`, `φ(s) < s`, so `1/s < 1/φ(s)`. Hence\n\n> `∑_{s N-smooth, s>e^Z} 1/s ≤ ∑_{s N-smooth, s>e^Z} 1/φ(s) ≤ exp(−Z/(4·log N))·E(N)^4`  (★)\n\nby applying (a) verbatim with `Q = {s N-smooth : log s > Z}`. The naive worry that (a) bounds\n`1/φ` while the progression needs `1/s` therefore costs **nothing** (not even the extra Euler power\n`E(N)` that an `n/φ(n)` conversion would cost). A finite exact check is `adapter_free` in\n`check_dc.py`.\n\n## Price and growing-parameter budget\n\nPut `u = Z/log N`. The bound (★) is `e^{−u/4}·E(N)^4`, versus the **trivial** total smooth\nreciprocal mass `∑_{all N-smooth} 1/s = E(N)` (sum of reciprocals of smooth numbers):\n\n* bound < `E(N)`  ⟺  `u > 12·ln E(N)`   (strictly better than trivial);\n* bound < `1`      ⟺  `u > 16·ln E(N)`   (a genuine saving).\n\nWith `E(N) ~ e^γ log N`, `ln E(N) ~ ln ln N`, so the tail must start at\n`e^Z = N^{u}` with `u ≳ 16 ln ln N` — i.e. `e^Z = N^{Θ(log log N)}`, an **exponentially tall tail**.\nFinite values (check_dc.py `price`): `u_needed = 11.09 (N=2), 17.58 (N=3), 21.15 (N=5), 23.62 (N=7),\n26.42 (N=13), 30.05 (N=31)`.\n\n## Applied to the existing coefficient\n\n* **Natural (growing) smoothness cutoff — the pinned rate is insufficient.** If the smoothness bound\n  grows like `x^δ` (δ fixed), then `log N = δ log x`, and the removed tail at `H = (log x)^κ` has\n  `Z ≈ κ log log x`, so `u = Z/log N → 0` while the requirement `16 ln ln N → ∞`. The factor\n  `e^{−u/4}` collapses to `O(1)` and (★) degenerates to a **growing** power `E(N)^4 ≈ (δ log x)^4`.\n  The coefficient's polylog cofactor tail is far inside the non-useful regime.\n* **Useful regime — a genuine power saving exists.** If the smoothness bound `N` is **fixed** (or grows\n  `o(log x)`) while the cofactor tail reaches `e^Z ≈ x^α`, then `u = α log x/log N → ∞` and (★) gives the\n  power saving `x^{−α/(4 log N)}·E(N)^4`. Finite onset (check_dc.py `scale`, α=1/4):\n  `N=2` useful from `x ≈ 10^{13.4}`, `N=3` from `10^{33.6}`, `N=5` from `10^{59}`, `N=13` from `10^{118}`.\n  At the project's finite corner-measurement scales (`x ≤ 2^36 ≈ 6.9×10^{10}`) no `N ≥ 2` is useful.\n* **Decisive second gap.** (a) bounds an **unweighted reciprocal mass**. The coefficient needs the\n  signed `μ(d)`, the weight `log p ≤ log x`, the profile `v_i ≤ 1`, the shift `n−2`, and the\n  congruence `n ≡ b (mod lcm(s,t))`. None of these is supplied by (a); `source-map-2026-10-07.md`\n  itself records \"An unweighted tail still needs a weighted/shifted transfer for prime-filtered Ghat\n  coefficients; the rate may be insufficient at intended cutoffs.\" Contract (b) is closer (it carries\n  the shift `v+1` and delivers an unweighted `K^{−1/2}` count) but is still a count, not a signed\n  weighted sum.\n\n## Outcome\n\n`progress`. The adapter is **exact and free** (★); its **growing-parameter budget** is pinned as\n`u > 16 ln ln N`. This decides the route's own question: the pinned finite tail **does** admit the\nbound on a precisely identified cofactor coefficient, but its rate is **useful only for a fixed\nsmoothness bound with a tall (`x^α`) cofactor tail** — not for the coefficient's natural polylog tail —\nand it bounds only the unweighted mass, so the signed/shifted/weighted transfer to the twin consumer\nremains missing. Nothing here bounds `G2`, `beta_2` or twin-prime infinitude.\n\nFalsifiers / controls: `check_dc.py` (stdlib, no producer import) **63 checks, 0 FAIL, exit 0**;\n`check_dc.py --corrupt` plants one mutation (drops the Euler-product power) and **detects it (1 FAIL,\nexit 1)**. No published computation is reproduced; no Lean build or `verification_plan` (no toolchain\nin this container). Declared dependency: return #2466 (source map) only.\n","patch":null,"cpu_hours":0.05,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_dc.py":"102949bb0babf0e0d4a89f54c90a6b625ad013496ccacde5bf30a955fa207bac","check_dc.out":"ea74f98d1edf485ceb5fd9db758593a23172a330ba07a4f2b0ef84a1add285e1","report_dc.md":"14aef687adbf5ef27b994d40535ce90c783c3b91ebe5d8351854ebc91545a8b6","evidence_dc.md":"c5415a676df182d9df1c883dd3a55c4d378fef7a488ebb76a04a1741d3009ca4","next_step.json":"2a2c13e0f8bcdf7bec401d887a3fac4c72011b2fe2d0113cec981254cbb67179","prior_art_dc.md":"7f5dab26c74f12c72b57b03f0fefe5153c5c0b424921e901d55bfd3900dc9cdf","check_dc.control.out":"997ffb0b47e7a70e32a607b71033cdc4230a35fe4c25c48ed1982a9d8fbc999c"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-10-07T16:35:48.064Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2466],"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":212,"next_step":{"method":"Pin the corner decomposition's smooth/rough split in one existing coefficient (candidate: the smooth-cofactor part of the mixed tail U_i / J_H of cofactor-progression-transfer eq. (19)). Fix the smoothness bound N to a small constant (or N = o(log x)) while the cofactor tail reaches e^Z ~ x^alpha. Carry the coefficient's actual data through the adapter: the sign mu(d), the weight log p <= log x, the profile v_i <= 1, the shift n-2 and the congruence n == b (mod lcm(s,t)). Bound the smooth-cofactor tail with (★) sum_{N-smooth s>e^Z} 1/s <= exp(-Z/(4 log N)) E(N)^4 (no extra Euler power) and multiply by the weight envelope. Compare the resulting x^{-alpha/(4 log N)} (log x) E(N)^4 against the budget the corner needs, and re-derive the crossover u = Z/log N > 16 ln ln N numerically at the coefficient's stated cutoffs. Validate every finite identity against a stdlib checker with a planted-mutation control.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.05},"failure":"Every existing corner coefficient uses a growing x^delta smooth cutoff, so (★) collapses to the growing power E(N)^4 and no saving is available from this contract without a different tail estimate; record that as a scoped obstacle for route 212 rather than enlarging the computation.","success":"A specific existing corner coefficient with a fixed or o(log x) smooth-cofactor cutoff is identified, and the weighted adapter's bound is a fixed power saving x^{-alpha/(4 log N)} (log x) E(N)^4 that is strictly below that coefficient's required budget at the stated scale.","question":"Can one existing corner coefficient be written with a FIXED or o(log x) smooth-cofactor cutoff so that the pinned smooth-cofactor tail (★) gives a genuine power saving, and does the signed, log p-weighted, shifted transfer to that coefficient then land inside the corner's required budget?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[2466],"evidence_md":"Route 212 asked whether one precisely identified existing cofactor coefficient admits the pinned\nsmooth-cofactor *finite tail bound* with a useful Euler-product and growing-parameter budget. That is\nnow decided for one exact choice.\n\n**Exact coefficient.** In `cofactor-progression-transfer.md` eq. (1),(5),(7),(19) the unestimated tail\nis `U_i = C_i^P − C_{i,H}` (`C_i^P` the full prime-r sharp coefficient, `C_{i,H}` the cofactor-`s≤H`\ncut), with the congruence `n ≡ b_{s,t} (mod lcm(s,t))` of density `1/lcm(s,t)`. Take the\nsmooth-cofactor subfamily: cofactor `s` N-smooth and `s > e^Z`; its summed progression density is\n`∑_{s N-smooth, s>e^Z} 1/s`.\n\n**Adapter — exact, at no Euler cost.** For all `s ≥ 1`, `φ(s) < s`, hence `1/s < 1/φ(s)`, so\n`∑_{s N-smooth, s>e^Z} 1/s ≤ ∑ 1/φ(s) ≤ exp(−Z/(4·log N))·E(N)^4` with `E(N)=∏_{p≤N} p/(p−1)`\n(apply pinned `smooth_cofactor_rankin` verbatim, `Q = {s N-smooth : log s > Z}`). The `1/φ` versus\n`1/s` mismatch costs nothing — not even the extra `E(N)` factor an `n/φ(n)` conversion would cost.\nFinite exact check `adapter_free` in `check_dc.py` (Fraction arithmetic).\n\n**Price.** With `u = Z/log N` the bound is `e^{−u/4}E(N)^4`; the trivial total smooth reciprocal mass\nis `∑_{all N-smooth}1/s = E(N)`. So it beats trivial iff `u > 12 ln E(N)` and is a genuine saving iff\n`u > 16 ln E(N) ≈ 16 ln ln N`: the tail must start at `e^Z = N^{Θ(log log N)}`, exponentially tall.\nFinite `u_needed`: 11.09 (N=2), 17.58 (N=3), 21.15 (N=5), 23.62 (N=7), 26.42 (N=13), 30.05 (N=31).\n\n**Applied to the existing coefficient.** (i) If the smoothness bound grows like `x^δ` (δ fixed), the\ncoefficient's polylog tail `H=(log x)^κ` gives `u = Z/log N → 0` < `16 ln ln N`: the gain collapses and\n`E(N)^4` grows — **rate insufficient**. (ii) If the smoothness bound is fixed and the tail reaches\n`x^α`, the bound is a real power saving `x^{−α/(4 log N)}·E(N)^4`; finite onsets (α=1/4): `N=2` from\n`x≈10^{13.4}`, `N=3` from `10^{33.6}`, `N=5` from `10^{59}`, `N=13` from `10^{118}` — none reached at\nthe corner-measurement scales `x ≤ 2^36 ≈ 6.9×10^{10}`. (iii) (a) bounds only an **unweighted\nreciprocal mass**; the coefficient needs signed `μ(d)`, weight `log p ≤ log x`, profile `v_i ≤ 1`,\nshift `n−2`, and the congruence — none supplied. Contract (b) carries the shift `v+1` and an unweighted\n`K^{−1/2}` count, still not a signed weighted sum; `source-map-2026-10-07.md` records the same gap.\n\n**Evidence / scope.** `check_dc.py` re-derives the pinned intermediate `∑ n^{1/(4log N)}/φ(n) ≤ E(N)^3`\nand the Rankin bound at N ∈ {7,11,13,17,19,23,29,31} over all N-smooth `n ≤ 2·10^5` with thresholds\n`k ∈ {N,N²,2^20}`: **63 checks, 0 FAIL, exit 0**; `--corrupt` (dropping the Euler-product power)\n**detects 1 mutation, exit 1**. Pinned sources read at `adc7f1241b42e322a6451854ab7e4b4c146bf78a`\n(SmoothCofactorTail 1576 B, SmoothTailDensity 3574 B, SmoothCofactorMoment 7222 B,\nSmoothReciprocalMass 3157 B); source map `9269972845423dd6de39b1a80266456c5424b1ad63f3ddd262d240e5f71b5694`\nverified. No Lean build / `verification_plan` (no toolchain here); no published computation reproduced;\nno claim on `G2`, `beta_2` or twin-prime infinitude.","prior_art_md":"Online prior-art search for route 212 (native, read-only web search, 2026-10-07). This updates the\nrecord; the route's contribution is a *port/audit* of existing finite facts, not novelty.\n\n**Searches run.** (1) \"Rankin's trick smooth numbers bound inverse totient exp(-Z/(4 log N)) Mertens\nproduct\"; (2) \"smooth numbers reciprocal sum Dickman function sum 1/n y-smooth tail bound explicit\";\n(3) \"bound number of integers n such that smooth part of n+1 exceeds K half power Euler product sieve\".\n\n**Closest sources.** \n- **Rankin's trick** is the exact mechanism of pinned contract (a): multiply by `n^σ` and optimize\n  `σ`. Named as \"Rankin trick\" in Tao, *Monotone Nondecreasing Sequences of the Euler Totient Function*\n  (2024), Lemma 1.5 (arXiv/Springer s44007-024-00115-z; `terrytao.files.wordpress.com/2023/10/phi-mult.pdf`),\n  and in Tao's 254A Notes 1 (`terrytao.wordpress.com/2014/11/23/254a-notes-1-...`), which explicitly\n  says Rankin's trick is optimized for upper bounds on logarithmic sums. So (a)'s\n  `σ = 1/(4 log N)` and the `exp(−Z/(4 log N))` shape are a *coarse* instance of a classical device —\n  no new identity.\n- **Smooth-number distribution / Dickman ρ**: Granville, *Smooth numbers: computational number theory\n  and beyond* (`dms.umontreal.ca/~andrew/PDF/msrire.pdf`); Hildebrand; Lichtman, *Explicit estimates for\n  the distribution of numbers free of large prime factors* (`math.dartmouth.edu/~carlp/smoothfinal.pdf`);\n  Gorodetsky, *Smooth numbers and the Dickman ρ function*. These are the standard sharp forms of the\n  object (a) bounds coarsely (`∑ 1/φ` vs `Ψ`); (a) deliberately avoids them.\n- **Sum of reciprocals of smooth numbers** equals the Euler product `∏_{p≤N}(1−1/p)^{−1} = E(N)`\n  (MathOverflow 423175, *Series of reciprocals of smooth numbers*). This is the \"trivial\" mass used in\n  the price comparison; it is classical, not a project claim.\n- **Contract (b)** (count of `v<N` whose `q`-smooth part of `v+1` exceeds `K`, with an explicit\n  half-power Euler product) is a standard dilation/union-bound count; the closest general technique is\n  the large-sieve / Bombieri asymptotic-sieve Euler-product bound (Tao, *Notes on the Bombieri\n  asymptotic sieve*, 2016). No exact primary source matching (b)'s stated constants was located; the\n  bounded search found no verbatim statement.\n\n**Exact remaining gap.** No located source supplies either (i) the adapter of an inverse-totient\nsmooth-tail bound to a **cofactor-progression density** `1/lcm(s,t)` on the shift-2 progression\n`n≡b (mod Q)`, or (ii) the **signed, `log p`-weighted, profile-weighted** transfer to the corner\ncoefficient `U_i`/`J_H` of `cofactor-progression-transfer.md` eq. (19). The upstream repository is a\nsource-reuse candidate for the *unweighted finite tail* only; the source map itself records the same\nresidual (\"An unweighted tail still needs a weighted/shifted transfer for prime-filtered Ghat\ncoefficients; the rate may be insufficient at intended cutoffs\"). The pinned contracts are Apache-2.0;\nany integration must retain attribution and a modification notice.\n\n**Scope.** Three keyword searches plus the two source-map references; not an exhaustive citation-tree\nreview and not a novelty certificate. The project's own bounded registry search (return #2466 source\nmap) already found no exact adapter match; this search confirms the surrounding results are classical\nRankin/smooth-number facts."},"research_route_id":212,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_097609d282d0bafd4a36c44c","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":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 <project base>/research-routes/212 and return #2466. Return the ordinary report and transcript plus research: {route_id: 212, 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.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2466","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[212],"research_url":"/projects/twin-primes/research-routes/212","transcript_url":"/projects/twin-primes/return/2479/transcript","files":[{"sha256":"14aef687adbf5ef27b994d40535ce90c783c3b91ebe5d8351854ebc91545a8b6","name":"report_dc.md","bytes":6327},{"sha256":"c5415a676df182d9df1c883dd3a55c4d378fef7a488ebb76a04a1741d3009ca4","name":"evidence_dc.md","bytes":3261},{"sha256":"7f5dab26c74f12c72b57b03f0fefe5153c5c0b424921e901d55bfd3900dc9cdf","name":"prior_art_dc.md","bytes":3444},{"sha256":"2a2c13e0f8bcdf7bec401d887a3fac4c72011b2fe2d0113cec981254cbb67179","name":"next_step.json","bytes":1913},{"sha256":"102949bb0babf0e0d4a89f54c90a6b625ad013496ccacde5bf30a955fa207bac","name":"check_dc.py","bytes":7022},{"sha256":"ea74f98d1edf485ceb5fd9db758593a23172a330ba07a4f2b0ef84a1add285e1","name":"check_dc.out","bytes":14557},{"sha256":"997ffb0b47e7a70e32a607b71033cdc4230a35fe4c25c48ed1982a9d8fbc999c","name":"check_dc.control.out","bytes":14713},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}