Sun's conjecture on p_m(a) + 2p_m(b) + 4p_m(c) + 8p_m(d)

Z.-W. Sun conjectured that for m ∈ {7, 9, 10, 11, 12, 13, 14} every natural number n can be written as p_m(a)+2p_m(b)+4p_m(c)+8p_m(d) with a, b, c, d integers, where p_m(ℓ) = ((m−2)ℓ²−(m−4)ℓ)/2 is the generalized m-gonal number. Bringmann and Kane proved this for n ≥ C_m with C_m between 4.8·10⁸⁰ and 2.1·10⁸⁹, and for m = 12 only when n ≢ 4 (mod 16). We prove the conjecture in full for all seven values of m. The proof is computer-assisted: we decompose the relevant theta series explicitly into Eisenstein series and newforms (exactly for most components, and by a ball-arithmetic certificate for m = 13 and the order-3 part of m = 11), which reduces each case to a finite list of at most 218,591 explicit integers n ≤ 1.6·10⁸, and we check these directly. For m = 12 we show that the excluded class is in fact the easiest one: there the cusp part vanishes identically. The mathematics, code and audits were produced by the AI system Claude (Anthropic) under the author's direction; see the paper's final section. This record contains the paper and all programs, certificates and logs.