Two-adic towers of exceptions for sums of four numbers x(ax+b), and a conjecture of Sun

Z.-W. Sun conjectured (arXiv:2411.14308, Conjecture 5.1) that for integers a > b > 0 with b even, gcd(a,b) = 1 and (a,b) ≠ (3,2), infinitely many positive multiples of 4 are not of the form w(aw+b)+x(ax+b)+y(ay+b)+z(az+b) with integers w, x, y, z. We give a short elementary proof, valid for every odd a ≥ 5 and every even b prime to a. The reason is 2-adic: a sum of four squares divisible by 8 has only even terms, and each forced halving moves the residue class of the terms modulo a. Combined with two theorems of Sun, the same descent shows that the exceptional set is always a finite union of explicit "towers" plus finitely many values, and we determine the exceptional sets for (a,b) = (5,2) and (5,4) completely, answering Sun's Remark 1.5. We also note that 15 is missing from the list in Sun's Corollary 1.5. The method is standard. The mathematics, programs and text were produced by the AI system Claude (Anthropic) under the author's direction, and checked by a separate audit run of the same AI with its own code, a SageMath re-check, and an adversarial review by ChatGPT (OpenAI), which found no error; no human expert has yet checked the note. This record contains the note, all programs and the audit.