{
  "target_id": "eq:cIQ2bnd",
  "proof_status": "needs_context",
  "summary": "The divisor-sum manipulation and final power saving check conditionally on the supplied premises. The remaining obligation is to verify from the omitted ensemble definitions that the form sums are controlled, with their actual multiplicities, by the gamma-count in eq:spec3, and that 0<|a|\u226aT uniformly. No local arithmetic error is identified. No external result is imported; the recorded cutoff is 2012-05-24, and pretrained knowledge is not intrinsically date-filtered.",
  "used_dependencies": [
    "eq:TXNis",
    "eq:cIQ1def",
    "eq:fuIs",
    "eq:spec3"
  ],
  "missing_context": [
    "The formal definition eq:fFdef and the gamma-to-form construction, including multiplicities, sufficient to show that both total form counts and fixed-a congruence counts are bounded by the corresponding counts in eq:spec3.",
    "The norm and coefficient definitions, including eq:vgIs, establishing uniformly that a is a nonzero integer and |a|\u226aT for every indexed form."
  ],
  "findings": [
    {
      "kind": "context",
      "source_quote": "{(a^{2},q)\\over q}",
      "explanation": "For a'=a, eq:cIQ1def gives this factor exactly, since (0,q)=q. For nonzero a, use (a\u00b2,q)\u2264\u03a3_{d|a\u00b2, d|q}d. Each d\u226aQ divides O(Q/d) integers q\u224dQ, so the q-sum is O(\u03c4(a\u00b2)). Prime factorization gives \u03c4(a\u00b2)\u226a\u03b5 T^\u03b5 when |a|\u226aT, after adjusting \u03b5. Thus the displayed divisor step is valid under the announced size condition; a=0 must be excluded."
    },
    {
      "kind": "context",
      "source_quote": "Recalling that $a=a_{\\g}=\\<e_{1},\\g v_{0}\\>$, replace the condition $a'=a$ with $a'\\equiv a(\\mod \\lfloor Q_{0}\\rfloor)$, and apply \\eqref{eq:spec3}:",
      "explanation": "Equality implies this congruence, so enlarging the nonnegative counting sum is valid. In the section range Q\u2080\u2264Q<X=N^(99/200)<N, the modulus floor(Q\u2080) lies in [1,N) for large N and is comparable to Q\u2080. Equation spec3 then supplies the inner bound T^\u03b4/Q\u2080^\u03b7\u2080, provided its gamma-family controls the form-family with the required multiplicities. The outer bound #F\u226aT^\u03b4 follows from spec3 itself at q=1; no additional counting theorem is needed. The formal ensemble and form definitions needed to check that transfer are absent."
    },
    {
      "kind": "context",
      "source_quote": "We further need the technical condition that in the $T$-ball, the value of $a_{\\g}=\\<e_{1},\\g\\, v_{0}\\>$ (see \\eqref{eq:vgIs}) is of order $T$.",
      "explanation": "This announces precisely the size property needed for the divisor estimate. The excerpt of the spec3 proof also mentions the restriction a>T/100, supporting nonvanishing. However, eq:fFdef, the norm definition, and the formal gamma-to-form construction are not supplied, so uniformity of 0<|a|\u226aT over the actual indexed sums cannot be independently verified here."
    },
    {
      "kind": "map_correction",
      "source_quote": "Q_{0}= T ^{(\\gd-\\gT_{1})/20},",
      "explanation": "Equation Q0is is an essential additional input, although it appears inside the excerpt labeled reference:eq:fuIs rather than among the dependency IDs. Put \u03b1=(\u03b4\u2212\u0398\u2081)/20>0 and \u03ba=\u03b7\u2080\u22126\u03b7\u2080\u00b2/100>0, using 0<\u03b7\u2080<1. The displayed estimate becomes N T^{2(\u03b4\u22121)}T^{\u03b5\u2212\u03b1\u03ba}. Taking \u03b5=\u03b1\u03ba/2 and using T=N^(1/100) gives the claimed bound with \u03b7=\u03b1\u03ba/200>0. The four existing dependency edges are relevant; this is an omitted parameter-definition edge, not a proof defect."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:Q0is",
      "role": "proof_input",
      "source_quote": "Q_{0}= T ^{(\\gd-\\gT_{1})/20},",
      "explanation": "Together with \u0398\u2081<\u03b4, this converts the Q\u2080-saving into a fixed positive power of T and hence N."
    },
    {
      "label": "eq:fFdef and coefficient-size prerequisites",
      "role": "conditional",
      "source_quote": "Dropping the condition $\\<e_{1},\\g_{1}\\,\\g_{2}v_{0}\\>>T/100$ in \\eqref{eq:fFdef}",
      "explanation": "The formal ensemble definition should confirm nonvanishing, the upper size bound, and applicability of spec3 to the same indexed family. The supplied passage identifies the prerequisite but does not provide its full definition."
    },
    {
      "label": "gamma-to-form counting correspondence",
      "role": "conditional",
      "source_quote": "Recalling that $a=a_{\\g}=\\<e_{1},\\g v_{0}\\>$",
      "explanation": "A multiplicity-compatible correspondence is needed to transfer spec3 from gamma counts to the form sums. Injectivity is unnecessary if the relevant form counts are otherwise dominated by those gamma counts."
    }
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