Recovering the primes in a dyadic interval from exact moments: two constructive routes and an obstruction theorem

Companion papers supply two exact moment channels for the primes in a dyadic window (m, 2m]: integer power sums produced from divisor-lattice data with no prime input, and weighted logarithmic moments produced from the primes below the window. This paper closes the circle: the channels determine the window primes individually. Route I uses r+1 integer moments, Newton's identities, and integer root reading — no primality test anywhere in the pipeline. Route II uses a single real moment, Chebyshev's functional θ(2m) − θ(m), delivered to about log2(N) bits where N is the product of the window primes, recovered by exponentiation, rounding, and complete trial division; a Prony decoder interpolates between the two. Against these, an obstruction theorem: any decoder receiving at most D bits that correctly decodes every r-subset of the window needs D at least log2 C(m, r) − 1, which diverges — so no uniform scheme of bounded description decodes the dyadic windows for all m, while a single aggregate of entropy O(log m) admits a log-length certified determination: one pigeonhole, two entropy scales. Machine-verified at m = 500: both routes recover the 73 primes of (500, 1000] exactly, the second from a single 695-bit real number. Developed in collaboration with Claude (Anthropic); a full AI-use disclosure appears in the paper. Verification code: https://github.com/carlgribble-caa/prime-moments