On sums of four squares with |x+3y−5z| a power of 4

Z.-W. Sun conjectured that every positive integer n can be written as x²+y²+z²+w² with x, y, z, w natural numbers and |x+3y−5z| a power of 4. We verify the conjecture for n ≤ 2·10¹⁰ and show that it holds for all n whose 16-free part is sufficiently large; the bound is ineffective, but it becomes effective (provided certain implied constants in the literature are effective, which we expect but have not checked) with at most one exceptional imaginary quadratic field, which can only affect n ≡ 5 (mod 8). For the integral version of Wu and Sun we prove the conjecture for all n under the Generalized Riemann Hypothesis, unconditionally when the odd part of 70n−2 (resp. 70n−32) is not squarefree, and for all large n with at most two exceptions (effectively, under the same proviso). For the original natural-number version we prove the conjecture for all n under GRH for L(s,χ_D) and for the quadratic twists of E₇₀ and of fifteen explicit newforms of weights 6 and 10; this proof is computer-assisted. We also give a short quaternion proof of an identity of Li–Zhou and Li–Wang, find infinite families of n where x+3y+5z=1 is locally but not globally solvable, and show that the class n ≡ 5 (mod 8) is a Ramanujan x²+y²+10z²-type problem attached to the elliptic curve of conductor 70. 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.