{ "target_id": "eq:cI3bnd", "proof_status": "needs_context", "summary": "The displayed estimation is algebraically correct conditional on the supplied lemmas, the ensemble bound |F| \u226a T^\u03b4, uniform coefficient bounds |a| \u226a T, and a modest upper bound on \u03b4. These additional prerequisites are not established in the packet. No local arithmetic defect was found, including at Q = X. This is not a verdict on the dependency proofs. Model knowledge may include later developments; no external result or later development is used here.", "used_dependencies": [ "eq:Mis", "eq:TXNis", "eq:cIQ3def", "eq:fuIs", "eq:modBndaNeqAp", "eq:tilQis" ], "missing_context": [ "The definition and counting estimate for F sufficient to justify |F|^2 \u226a T^(2\u03b4), or an explicitly controlled variant.", "The identification of each coefficient a with the order-T coefficient announced in the ensemble construction, including its sign if the displayed factor aa' is used literally.", "Definitions of S, S_f, and the family of forms, sufficient to check the hypotheses and uniformity of eq:modBndaNeqAp in this application.", "The definition or an upper bound for \u03b4 ensuring (\u03b4\u2212\u03981)\u03b70^2 < 7375/3." ], "findings": [ { "kind": "context", "source_quote": "\\tilde q:=(u^{2},q)", "explanation": "The elementary loss calculation checks out: (q/q0)^2 \u2264 u^4, and the square root of the two gcd factors is at most |aa'|. For a \u2260 a', gcd(a\u2212a',q) \u2264 |a\u2212a'|. The u^4 cancels the weight u^\u22124. Counting O(U) values of u, O(Q) moduli, and O((UQ/X)^4) frequency quadruples yields U^6 X^\u22121 Q^(11/4) T^(2\u03b4+5/4), assuming the ensemble bounds below. Since Q < XT, this is at most U^6 X^(7/4) T^(2\u03b4+4), exactly as claimed." }, { "kind": "unverified", "source_quote": "T^{2\\gd}", "explanation": "The second displayed inequality uses an unstated bound |F|^2 \u226a T^(2\u03b4), together with |a|, |a'| \u226a T and |a\u2212a'| \u226a T. The contextual discussion announces an order-T condition for a_gamma, but the packet does not define F or establish its size and its relation to those coefficients. If positivity of a and a' is not part of that definition, the first displayed upper bound should be read with |aa'|." }, { "kind": "unverified", "source_quote": "With the above notation,", "explanation": "The modular lemma is supplied only with this notation-dependent qualification. The section setup identifies its modular component with S, but the definitions of S, the normalized sums S_f, and the permitted family of forms are absent. Its applicability to the forms and all frequencies in eq:cIQ3def therefore remains conditional, rather than independently verified." }, { "kind": "context", "source_quote": "X^{-1/4}T^{5}=N^{-59/800}", "explanation": "This exponent is correct. Combining the supplied definitions gives U^6 = N^[3(\u03b4\u2212\u03981)\u03b70^2/100000]. Thus the remaining factor is N^[\u221259/800 + 3(\u03b4\u2212\u03981)\u03b70^2/100000]. A positive saving follows provided (\u03b4\u2212\u03981)\u03b70^2 < 7375/3. The packet gives 1 < \u03981 < \u03b4 and 0 < \u03b70 < 1, but no upper bound on \u03b4. The intended geometric range would readily suffice, but its definition or bound must be supplied rather than imported." }, { "kind": "map_correction", "source_quote": "so together with \\eqref{eq:fuIs}, this is clearly a substantial power savings.", "explanation": "The final step also uses eq:Q0is to turn the definition of U into a power of N. That definition appears inside the supplied eq:fuIs context but has no separate incoming edge. The ensemble size and coefficient estimates are additional omitted proof inputs. These are mapping/context limitations, not established proof defects." } ], "dependency_proposals": [ { "label": "eq:Q0is", "role": "proof_input", "source_quote": "Q_{0}= T ^{(\\gd-\\gT_{1})/20}", "explanation": "Needed alongside eq:fuIs and eq:TXNis to calculate the exponent contributed by U^6. Already present in the packet, but absent as a separate frozen-map dependency." }, { "label": "Ensemble cardinality estimate", "role": "proof_input", "source_quote": "\\sum_{\\ff,\\ff'\\in\\fF}", "explanation": "Supply the ensemble definition and the counting bound used to replace this pair sum by O(T^(2\u03b4))." }, { "label": "Uniform coefficient bounds", "role": "proof_input", "source_quote": "the value of $a_{\\g}=\\$ (see \\eqref{eq:vgIs}) is of order $T$.", "explanation": "The announced condition must apply to every form in F; it supplies the losses T^2 and T^(1/4)." }, { "label": "Definition and range of \u03b4", "role": "proof_input", "source_quote": "1<\\gT_{1}<\\gd", "explanation": "This lower comparison alone does not bound the exponent of U. An upper bound on \u03b4, or directly on (\u03b4\u2212\u03981)\u03b70^2, completes the final power-saving check." } ], "_meta": { "pipeline": "claim_review", "model": "gpt-6.1-sol", "reasoning_effort": "high", "provider": "openai", "batch_job_id": "batch_6ad4023e64a48190a34e245075554b92", "custom_id": "claim-5daba88a9dd809133bbb525a", "source_sha256": "f14dd8ff2ed84ccb83179bd2b7e91bfcbd0a91cef90005c8743f976bd703f5f6", "usage": { "input_tokens": 13215, "input_tokens_details": { "cache_write_tokens": 13212, "cached_tokens": 0 }, "output_tokens": 3450, "output_tokens_details": { "reasoning_tokens": 2070 }, "total_tokens": 16665 }, "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 } }