Prime-Replacement Matchings in Dyadic Squarefree Shells: A Poset Hall Criterion for Redheffer Determinants

This paper develops a combinatorial framework for studying Redheffer determinants and the Mertens function through matchings in finite divisibility posets. The Möbius sum defining the Redheffer determinant is first reduced exactly to the odd squarefree integers in a dyadic interval. Removing the largest prime factor partitions this shell into weighted core bags whose multiplicities are given by unsigned prime counts. The nonempty cores form a finite down-set in the odd squarefree divisibility poset and admit a natural rooted-tree structure. Three nested bipartite graphs are constructed from largest-prime deletion, divisibility covers, and admissible prime replacements. Matchings in these graphs pair terms of opposite Möbius sign. The resulting unmatched mass provides an upper bound for the absolute value of the Mertens function. For the rooted-tree graph, this defect has equivalent formulations using maximum weighted independent sets, weighted Hall deficiencies, weighted vertex covers, and integral capacitated flows. The paper derives a positive and constructive sufficient criterion for the Riemann hypothesis: it is enough to prove a square-root-scale upper bound for the unmatched capacity in the rooted core tree. A weaker sufficient criterion is obtained from the full Hasse diagram of the core poset. The construction also controls a distinguished one-point deletion response of a larger Redheffer matrix. An exact bottom-up recurrence computes the rooted-tree defect using only nonnegative prime-count weights. Accompanying exact-arithmetic computations verify the principal identities, test matching saturation for the larger replacement graphs, and evaluate the tree criterion through thirty million. These computations support the proposed matching conjectures but are not claimed to prove the Riemann hypothesis.