Intrinsic Diophantine Approximation on Circles and Spheres

The paper proves its main theorems by a clean meta-lemma (Lemma 2.2) and case-by-case verifications of the two key identities ((Φ-i) linear-distance scaling with constant C and (Φ-ii) exact height transport) for all six circle/sphere models; this yields the constants in Theorems 1.1 and 1.3 and their corollaries, with full gcd bookkeeping and correct height normalizations (sections 3–5). In contrast, the candidate solution gets the circle cases essentially right but (i) mis-normalizes the stereographic constants for the sphere cases, concluding L(C,K)=2·L(X,Z) where the paper proves L(C,K)=√2·L(X,Z), and (ii) does not rigorously verify the exact height identity or the gcd conditions (it appeals to parity/gcd ‘bookkeeping’ and suggests a remedial parameter rescaling which violates (Φ-ii)). These issues mean the model’s argument is not correct as stated, while the paper’s is.