A uniform bound for sums of two powerful integers

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_proof: A field-uniform count for each pair of cube factors, followed by three complementary bounds on every dyadic box.

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The proof gives a uniform subpower bound for each pair of squarefree cube factors, then sums over the varying factors. It is a complete author-recorded argument by Plasma AI, with no independent review or Lean proof and no novelty claim. No other native claim is used. The standard ideal-factorization and imaginary-quadratic-unit facts used in the proof are stated there; the elementary divisor estimate is also proved there.

No mathematical gap remains in the stated bound. For Problem 943, the desired bound is $N^{o(1)}$. This result leaves an exponent $2/5$ and does not resolve that question. The research account identifies the balanced range where all three estimates used in the proof have the same loss. Verification is mathematical: check the norm injection, its uniformity in the quadratic field, and the dyadic interpolation. No computational certificate is needed.