{"id":275,"job_id":643,"problem_id":1,"lane_id":4,"type":"explore","user_id":35,"model":"gpt-6-astra","provider":"openai","report_md":"# Return 151: owned source family, unmatched full band estimate\n\n**Owned at the convenient-divisor theorem; not an unconditional source for (4.9) or the signed Type II margin.** Return #151 is an editorial scope repair, not a new prime-distribution theorem. Its distinction between paying P_band and paying B=T_II^low+P_band survives this source check. No claim of literature absence or novelty is made.\n\n## Positive source match\n\nOwning convention: primes in arithmetic progressions to large moduli, fixed residue class, convenient divisor. James Maynard, arXiv:2006.06572v2 (2021), printed page 3, Theorem 1.1 and Corollaries 1.2–1.3. The Memoirs AMS edition is volume 306, number 1542; the locators verified here are explicitly the arXiv version. The PDF pages were rendered and read, not inferred from an abstract. The theorem averages absolute errors on products with three size restrictions. The corollary restricts moduli by a divisor-size condition; the exceptional-set corollary retains exceptions. PDF: https://arxiv.org/pdf/2006.06572v2 .\n\nThat theorem is already named by the project's fixed-endpoint source matrix and SEARCH-CONVENTIONS row for the band. The contribution is a checked attribution boundary, not a discovery that the paper exists. IMPORT-MAP does not yet have this specific fixed-endpoint/convenient-divisor row; a proposed row is supplied separately.\n\n## Exact local parameter check\n\nAt delta=1/60 and eta=1/1000, the source's divisor interval becomes exponents (103/3000,227/3000); its second upper bound is 547/3000. A prime modulus near x^(31/60) has only divisor exponents 0 and 31/60, so is not covered by that sufficient divisor condition. This is a scope witness, not a claim that the prime modulus actually has a large distribution error.\n\nA product with factor exponents 1/20 and 7/15 has total 31/60 and meets the theorem's three strict inequalities for theorem-epsilon 1/10000: left exponents 59/60,58/15,148/15 are below 99/100,399/100,999/100 respectively. Thus the source supplies a real subclass, rather than failing every band shape. These are exact parameter substitutions in check.py, not experiments on primes.\n\nThe project's (4.9) instead ranges over every odd q up to 2x^(1/2+eps')(log x)^(3L), with tau(q)^3 and a supremum over prefixes. The factor condition alone does not imply that all these moduli are covered. Even a hypothetical proof of (4.9) leaves the separate T_II^low sum in the exact decomposition. That last statement is project algebra (2.5)–(2.9), not an imported analytic impossibility theorem.\n\n## Conditional EH bookkeeping, not a new theorem\n\nHere is the elementary implication behind #151's EH comment, with assumptions explicit. Assume a prefix form of EH at some fixed theta strictly between 1/2+eps' and 1: for every A, sum_{q<=x^theta} D(q) << x/log^A x, where D(q)=sup_{t<=x}|Delta_q(t;-2)| on reduced odd classes. The fixed extra theta margin absorbs the polylogarithmic cutoff. The elementary bound D(q)<