{
  "target_id": "eq:nmUpToM",
  "proof_status": "defective",
  "summary": "Conditional on lem:kEllD and the intended finite summation range, the fixed-gcd counting argument and final numerical estimate are sound. However, the displayed enlargement to all divisors w of d1 introduces undefined modular inverses. This is a localized indexing defect, not a counterexample to the claimed bound. No external results are imported; although model knowledge may postdate the 2012-05-24 cutoff, it is not used as historical evidence here.",
  "used_dependencies": [
    "lem:kEllD"
  ],
  "missing_context": [
    "Explicit integer domains, positivity of d, and the lower bounds on the m,n summation range.",
    "A valid interpretation or corrected indexing of the enlarged w-sum, ensuring every displayed modular inverse exists.",
    "The proof of lem:kEllD, if an unconditional assessment rather than a conditional local assessment is required."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "\\sum_{w\\mid d_{1}}\n\\sum_{m,n<M}\\bo_{\\{m\\equiv0(\\mod w)\\}}\\bo_{\\{ n\\equiv-\\overline{\\frac \\ell w}\\frac mw k(\\mod \\frac{d_{1}}w)\\}}",
      "explanation": "The inverse of ell/w modulo d1/w is defined when w is the previously fixed value gcd(ell,d1), but not for every divisor w of d1. Some added indices do not even satisfy w|ell; others give a nonunit. For example, k=1, ell=2 and d=d1=6 satisfy gcd(k,ell,d)=1, but the added term w=1 requires the nonexistent inverse of 2 modulo 6. Thus this displayed multiple sum is not well-defined. The remaining obligation is to state a valid indexing or residue-class convention for the enlarged sum."
    },
    {
      "kind": "context",
      "source_quote": "Let $w=(\\ell, d_{1})$; then $mk\\equiv0(w)$ implies $m\\equiv0(w)$ since $(k,\\ell,d)=1$.",
      "explanation": "This step is correct with w fixed as stated. Since w divides both ell and d, the supplied coprimality gives gcd(k,w)=1. Once w|m, division by w leaves ell/w invertible modulo d1/w, so n occupies exactly one residue class. The two resulting counting factors are therefore justified for this fixed gcd."
    },
    {
      "kind": "context",
      "source_quote": "\\left(\n{M\\over w}\n+1\n\\right)\n\\left(\n{wM\\over d_{1}}\n+1\n\\right)",
      "explanation": "The final numerical estimate is valid for positive divisors and M>=1. Expanding gives M^2/d1 + M/w + wM/d1 + 1. Here d|d1^2 implies d1>=sqrt(d), while 1<=w<=d1, giving O(M^2/sqrt(d)+M) per pair. There are at most tau(d)^2 pairs. The elementary bound tau(d)^2 <<_epsilon d^epsilon follows directly from prime factorization: for sufficiently large primes, (a+1)^2<=p^(epsilon*a), and the finitely many smaller primes contribute a bounded factor."
    },
    {
      "kind": "context",
      "source_quote": "\\#\\{m,n<M:Am^{2}+2Bmn+Cn^{2}\\equiv0(d)\\}",
      "explanation": "The estimate requires a finite range such as 0<=m,n<M or 1<=m,n<M, and a positive integer modulus d. These conventions are not explicitly supplied. If m,n range over all integers below M, the count need not be finite; for negative or zero d, the displayed right-hand side is not defined as intended."
    },
    {
      "kind": "unverified",
      "source_quote": "As in Lemma \\ref{lem:kEllD}, $A,B,C$ and $d$ determine   $k,\\ell$",
      "explanation": "The dependency statement supplies exactly the uniform implication and coprimality needed, under the same coefficient and divisibility hypotheses. Its proof is not included, so this assessment uses it only conditionally. The proposed dependency edge is appropriate; the packet does not establish the dependency itself."
    }
  ],
  "dependency_proposals": [
    {
      "label": "lem:kEllD",
      "role": "conditional",
      "source_quote": "Then there are integers  $k,\\ell          \n$ with $(k,\\ell,d)=1$",
      "explanation": "Retain this as the conditional proof input. Together with its stated implication from the quadratic congruence to the square congruence, it supplies the required k,ell independently of m,n. No additional external dependency is needed for the elementary counting steps."
    }
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