{ "target_id": "lm_generation", "proof_status": "needs_context", "summary": "The local-to-global argument is valid conditional on lm_local and simultaneous CRT lifting for both factor groups \u0393\u2081 and \u0393\u2082. The supplied dependency lm_local alone does not provide that lifting property. The packet omits \u0393\u2082's definition and does not justify the asserted factor-group decompositions. There is also a notation error: the simultaneous lifts must belong to quotients modulo q, not the undefined p^m. No counterexample to the target lemma is established.", "used_dependencies": [ "lm_local" ], "missing_context": [ "Definitions of \u0393\u2082 and of \u0393_j(r), including their embeddings into \u0393\u0304/\u0393\u0304(r).", "A proof or source anchor establishing \u0393_j/\u0393_j(q) \u2245 \u220f_k \u0393_j/\u0393_j(p_k^{m_k}) for j=1,2 and arbitrary prime factorizations of q.", "In particular, a congruence description or congruence-compatible identification of \u0393\u2082 sufficient to verify that CRT property." ], "findings": [ { "kind": "unverified", "source_quote": "Since $\\Ga_1/\\Ga_1(p^m)$ and $\\Ga_2/\\Ga_2(p^m)$ are the direct product\nof local factors", "explanation": "This is the essential additional prerequisite. For each j=1,2, the proof needs surjectivity of \u0393_j/\u0393_j(q) \u2192 \u220f_k \u0393_j/\u0393_j(p_k^{m_k}), with distinct primes p_k and positive exponents m_k. CRT for the ambient coefficient ring does not automatically imply this for an arbitrary subgroup. The packet asserts, but does not justify, the required subgroup property." }, { "kind": "context", "source_quote": "there are $g_1,\\ldots, g_{10^{13}}\\in\\Ga_1/\\Ga_1(q)$ and\n$h_1,\\ldots, h_{10^{13}}\\in\\Ga_2/\\Ga_2(q)$", "explanation": "\u0393\u2082 is not defined in the supplied excerpts, and the congruence kernels \u0393_j(q) are not explicitly defined. An explicit congruence description of \u0393\u2081 appears later in the proposed scope, but no corresponding description or congruence-compatible identification for \u0393\u2082 is supplied. These definitions are needed to check the simultaneous-lifting assertion." }, { "kind": "error", "source_quote": "find elements $g_1,\\ldots, g_{10^{13}}\\in\\Ga_1/\\Ga_1(p^m)$ and\n$h_1,\\ldots, h_{10^{13}}\\in\\Ga_2/\\Ga_2(p^m)$", "explanation": "Here p and m are not defined in the general-modulus proof. The displayed congruences concern all prime-power factors of q, and the subsequent conclusion requires elements modulo q. These quotient labels must therefore be modulo q. This is a localized notation error, not evidence that the lemma itself is false." }, { "kind": "context", "source_quote": "hence all these containments must be equality.", "explanation": "This final inference checks out conditional on the factor-group CRT property and lm_local. Reduction embeds \u0393\u0304/\u0393\u0304(q) into the product of its prime-power images: an element in every prime-power kernel is congruent to the identity modulo their product q. Simultaneous lifting of each alternating factor gives the reverse containment for A_{10^{13}}. Applying lm_local to each prime-power factor then proves the target. For q a prime power, lm_local directly supplies the conclusion." }, { "kind": "map_correction", "source_quote": "Using Lemma \\ref{lm_local}", "explanation": "The supplied lm_local \u2192 lm_generation edge is appropriate but incomplete as a dependency account. The argument additionally uses CRT-surjectivity for both factor groups. The proof of lm_local appearing in the source slice is not certified by this bounded assessment; lm_local is used only as the supplied unverified premise." } ], "dependency_proposals": [ { "label": "Factor-group CRT decomposition", "role": "proof_input", "source_quote": "are the direct product\nof local factors", "explanation": "Add an explicit prerequisite giving simultaneous lifting for \u0393\u2081 and \u0393\u2082. This is separate from prime-power bounded generation and is the exact remaining mathematical obligation in the global argument." }, { "label": "Factor-group definitions and congruence structure", "role": "proof_input", "source_quote": "\\Ga_1=\\left\\{\\mat{a}{b}{c}{d}\\in\\SL(2,\\Z): b\\equiv0\\; \\mod 4\\right\\}.", "explanation": "This supplied description is relevant to checking \u0393\u2081's CRT behavior. Supply the corresponding definition or structural identification for \u0393\u2082 and definitions of its congruence kernels." }, { "label": "Adjacent spectral-gap results and transference", "role": "excluded", "source_quote": "Now \\cite{Selberg1965} and \\cite[Theorem 1]{Brooks2007} provides us with", "explanation": "These concern an application of lm_generation, not its CRT proof. No external theorem is imported in this assessment. The recorded cutoff is 2012-05-24; later model knowledge is not evidence of historical availability and is not used to close any obligation." } ], "_meta": { "pipeline": "claim_review", "model": "gpt-6.1-sol", "reasoning_effort": "high", "provider": "openai", "batch_job_id": "batch_6ad4023e64a48190a34e245075554b92", "custom_id": "claim-912f0453846060eb8bcc1c9e", "source_sha256": "3a686e58dae4eb31f38a07406975061b8d2eaf88e91aec4ae23804f6c560b19f", "usage": { "input_tokens": 15161, "input_tokens_details": { "cache_write_tokens": 15158, "cached_tokens": 0 }, "output_tokens": 2645, "output_tokens_details": { "reasoning_tokens": 1393 }, "total_tokens": 17806 }, "max_output_tokens": 24576, "prompt_profile": "claim_review-v2", "prompt_profile_sha256": "4aa7aa2ef9a18c790385b9e0e0f7ad6c44ff664be253c2d9f790a1469765c3a4", "review_policy": { "schema": 1, "cutoff_date": "2012-05-24", "basis": "journal_received", "evidence": "https://link.springer.com/article/10.1007/s00222-013-0475-y : Received 24 May 2012; publisher.html retained locally", "status": "recorded", "rule": "external-results-available-by-cutoff" }, "chronology_prompted": true } }