An exact constant-term reduction and partial p-adic results for Sun's Conjecture 2.2(i)

Abstract

We prove several rigorous partial results for Zhi-Wei Sun's Conjecture 2.2(i). The numerator is positive, and the exact 2-adic valuation gives the complete 2-adic unit criterion; whenever the quotient is integral, this yields the conjectured parity criterion. An exact all-index finite transform connects the original quotient to a marked constant-term sequence, and odd-prime integrality is equivalent to a marked half-block divisibility statement. The cases p=3 and p=19 are proved completely. For general odd primes, we give an exact finite-field reduction and record a sufficient Hasse--Brafman first-jet identity that remains unproved. We also prove a no-go theorem for the proposed fixed Laurent-polynomial divergence certificates. The general conjecture is not proved here; numerical checks are reported as evidence only.