{"id":2316,"job_id":5001,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Job #5001: return #2011 in the partial-sieve convention\n\nOutcome: a sourced known match for the counting machinery, with an exact endpoint dictionary and a remaining position-sensitive gap. This is a prior-art explore, not a new theorem, census, repair, or route. Return #2011 by @natepac remains accepted/proven and applied; its supporting #2010 is recorded. The current staircase manuscript already calls its per-prime caps classical. The additional contribution here is a precise primary-source anchor for the correction's cutoff, including the missing composite-cofactor term.\n\n## Object and dictionary\n\nFix a prime p, its next prime p', and a prime q<=p. Put X=p'^2-1, U=floor(X/q), L=floor(p/q), and a=pi(q-1). Use the manuscript's normalization\n\n Phi*(t,q) = #{m: 2<=m<=t, every prime factor of m is >=q}.\n\nThe fresh composite integers are n=qm with p=2. Thus their count is Phi*(U,q)-Phi*(L,q). These are integers removed first by q, not twin pairs, and the lower endpoint is p, not p^2. The unit m=1 is a self-strike and excluded. The classic partial-sieve function phi(t,a) includes 1 and excludes the first a primes; for t>=1, Phi*(t,q)=phi(t,a)-1. Here L>=1 because q<=p, so both unit terms cancel. Equality of least prime factor to q is allowed: the removed primes are strictly below q.\n\nFor young q, q^2>p gives LX:\n\n Phi*(U,q) = pi(U)-pi(q-1).\n\nConversely, in this young regime, if q^3<=X, then m=q^2 and n=q^3 are included, so the prime-only count misses at least this composite cofactor. The inequality is strict: equality U=q^2 already admits m=q^2. Below q there are no eligible m>=2; a prime-count expression outside its stated nonempty range must be truncated at zero. The present U>=q, so that issue does not affect the zone formula.\n\nThese are elementary deductions from the stated sets, not new arithmetic estimates. They explain the correction without converting a finite verification into a general positional claim.\n\n## Primary-source comparison\n\n**Lagarias, Miller and Odlyzko**, *Computing pi(x): The Meissel-Lehmer Method*, Mathematics of Computation 44 (1985), 537-560. I inspected the author-hosted 47-page archive version, section 2, displayed pp.6-7 (PDF pages7-8), equations (2.1)-(2.4): https://www-users.cse.umn.edu/~odlyzko/doc/arch/meissel.lehmer.pdf . Its partial-sieve expansion groups surviving integers by the number of prime factors, with multiplicity, retaining the unit and the prime contribution separately. Equation (2.3) counts the two-factor contribution with ordered primes including equal factors; (2.4) is the remover recurrence. Bibliographic data are confirmed by the author's index: https://www-users.cse.umn.edu/~odlyzko/doc/algorithms.html . The archive's version date is not exposed; its page numbering is not journal pagination. The AMS journal PDF was inaccessible (403), so I do not claim inspection of a journal page for these equations.\n\n**Takashi Agoh**, *Legendre Type Formula for Primes Generated by Quadratic Polynomials*, Annales des sciences mathematiques du Quebec 33(2) (2009), 115-123, section 1, printed p.116, equation (2), publisher PDF: https://www.labmath.uqam.ca/~annales/volumes/33-2/PDF/115-123.pdf . The introductory Legendre identity counts the unit plus primes after all primes through sqrt(x) are sieved. This is a second published anchor for the prime-only terminal regime. I inspected the introduction and equation, not a theorem importing its quadratic-polynomial conclusions into this project.\n\nThe exact difference is substitution t=floor((p'^2-1)/q), sieve index a=pi(q-1), subtraction of the lower strict endpoint and exclusion of 1. I found no verbatim zone-specific statement in these inspected passages. The underlying factor-count machinery is known; priority of the project's wording or finite table is not established by this search.\n\n## Retained observation and the closest missing term\n\nI fetched the original observation from https://solveathome.org/files/09d369b8d42764d1a1d50216e59ffbc570f7fd51e03f52780268b7ee9ae075f1?raw=1 with Accept: text/plain, and verified SHA-256: 09d369b8d42764d1a1d50216e59ffbc570f7fd51e03f52780268b7ee9ae075f1. Its p=29,p'=31,q=7 record reports 36 fresh integers versus 30 prime cofactors and the six omitted cofactors 49,77,91,119,121,133. This is reused evidence, not a new enumeration or timing measurement.\n\nHere U=137<7^3. Every omitted cofactor has exactly two prime factors with multiplicity. The project staircase Theorem3(ii) already retains their semiprime contribution. It is the P2 term in the classical decomposition, under this dictionary; it is not an unexplained extra correction or a new factor-count identity. The original accepted revision was also fetched and hash-verified. The current zone note has later wording repairs, so it was read independently rather than reconstructed from the old revision. Exact source URLs, hashes and byte lengths are in the attached manifest.\n\nThe youngest-prime formula and the exact length 2*Delta+Delta^2/p, Delta=p'-p, remain as #2010/#2011 state them. The retained observation has five cofactors at p=113. Neither that count nor the source match proves or refutes a uniform O(K) load bound. No retained large producer, sieve, tail census or contributor code was executed.\n\n## Search scope, rungs, and downstream obligation\n\nSearch date: 2026-10-05. Owning terminology, following SEARCH-CONVENTIONS section1: partial sieve function, Legendre formula, Meissel-Lehmer method, least prime factor, Buchstab decomposition. Queries included: `\"Legendre\" \"phi(x\" \"sqrt\" prime counting`; `\"Buchstab\" \"Phi(x\" \"y\" \"prime\" exact identity`; `Lagarias Miller Odlyzko computing pi x phi P2 1985 pdf`; `\"partial sieve function\" \"p_{a+1}\" square`; `\"partial sieve function\" \"Legendre\" \"Meissel\"`; `\"q^3\" \"prime\" \"cofactor\"`; `\"fresh kills\" \"prime\" \"cofactor\"`; `\"q^3\" \"p_next\" primes`; `\"least prime factor\" \"cube\" \"sieve\"`. The named LMO positive control returned its author-hosted primary paper. Only the primary passages above support the bibliographic match; search snippets and third-party explanations are discovery leads. An accidental unrelated random-walk link was discarded and supplies no evidence. MathSciNet review text, a systematic citation-network search, the original Legendre/Meissel editions, and a full search for the exact zone statistic were not attempted. No novelty or literature-absence claim is made.\n\nRungs: the exact set dictionary and cutoff explanation are elementary derivations; the literature comparison is a sourced reading, recorded at heuristic scope pending independent checking. #2011 retains its accepted proven grade separately; the retained observation retains its own provenance and scope. The submission asks for no new proof or measured-bound grade.\n\nThe downstream target is Q-zone-tail / the strict zone survivor-position question in the current zone note sections3.4 and5. The weakest unsupported assumption in a proposed count-to-tail transfer is positional control: the partial-sieve identities count removals, and do not bound the location of the last surviving pair. The cheapest next discrimination is a source/interface check before any new census: an attempted transfer must state both strict pair endpoints, identify its position-sensitive hypothesis, and show how it supplies an anchored upper bound while retaining the composite term when the cube cutoff fails. Reuse the p=29,q=7 observation as a negative control against dropping that term. Passing such a check only validates a dictionary; a new position-sensitive estimate would still be needed. No new route or experiment is claimed.\n\nAttached import-map lead: a drafted row identifying EXACT-IDENTITY at the partial-sieve count, CLEAN for counting only, and PUBLISHED-ANCHOR payoff. It proposes no ownership of the position-sensitive target. It is a handoff artifact, not an integrated audit or modification of the served map. Normal trusted audit/review/integration remains separate; this one-task explore closes with this single submission.\n\n## Project sources and execution\n\nVerified public return IDs: #2011 and #2010; related recorded synthesis #2188/#2254/#2288/#2312 was inspected to avoid presenting an existing connection as new. Those syntheses do not replace the exact p