Fixed-Centre Congruences and Complete Finite-Gap Shifted-Triple Exclusion for Nörlund–Stirling Quotients

The paper continues our Nörlund–Stirling arithmetic program by investigating when three equally spaced odd-index quotient polynomials can share a root in a specified p-adic neighborhood. At the fixed centre -3, it develops an exact prime-power congruence criterion and proves that simultaneous vanishing to depth three forces the prime to be Wieferich to both bases 2 and 3. This excludes such fixed-centre triples for all primes up to ten million by a complete finite check. The paper further derives negative-centre interpolation obstructions and a necessary-and-sufficient compatibility test for the first shifted Hensel digit. Its strongest shifted-root result excludes all common triple roots modulo p^2 in the -3 residue disk for every even index gap from 2 through 120, without imposing any upper bound on the prime. Its main novelty is an exact combination of prime-adic structure, local root-shift compatibility, and a complete finite-gap certificate that yields an infinite-prime impossibility theorem. The result does not extend automatically to arbitrary gaps or residue disks.