Two Proofs of the PACK Theorem: Topological Necessity of Zero Acceleration in the Pointwise Continuous Infinite Plenum

The Pressure-Acceleration Continuity Kernel (PACK) has previously been utilized as a foundational axiom in the formalization of the Constant-Pressure Navier-Stokes (CPNS) solution framework and its subsequent proof via Route A. Notably, the utility of PACK has recently been extended to provide a solution path for the Yang-Mills existence and mass gap problem (CMI-YM). While these applications demonstrated the utility of PACK in resolving field ambiguities in the infinite plenum, its status as an axiom necessitated further justification from first principles. We are here to elevate PACK to a theorem, establishing its topological necessity within a Branch 2 configuration. This elevation serves a dual purpose, providing a unified theoretical foundation that simultaneously resolves the field stasis requirements for both fluid and gauge-theoretic manifolds. We provide two independent proofs: an inductive chain argument derived from geometric flux constraints, and an acoustic proof grounded in heritage canonical PDE results regarding the infinite speed of sound in an idealized infinite plenum. Both proofs demonstrate that in an idealized infinite plenum, non-zero acceleration necessitates a divergent mass-acceleration integral over the half-space. This result is incompatible with the requirement for finite total force, thereby confirming PACK as a necessary consequence of the topology of the stage rather than a mere postulate.