BU05 Mathematical Alignment under Dynamic Boundary Fixation

Abstract This paper formalizes Mathematical Alignment under Dynamic Boundary Fixation as a minimal structural account of alignment under a single-universe boundary framework. Its central claim is that alignment should not be treated first as a semantic, ethical, or behavioral label, but as a standing directly induced by dynamic-complete boundary fixation. Let D(M) denote the real-time dynamically complete interface on the mother-structure M, let B_0 denote the unique fixed boundary set, and let \Pi_{B_0} denote the boundary-projection operator. The paper defines boundary drift as the deviation functional \Delta_B(\Phi)=\|\Phi-\Pi_{B_0}(\Phi)\|, and defines alignment standing by the condition \Delta_B(\Phi)=0. On this basis, the paper derives a minimal alignment chain: dynamic completeness induces boundary projection; boundary projection eliminates drift at the admissible level; drift elimination yields alignment standing. The result is a sharp compression of the alignment problem. Ontological-scale misalignment is no longer treated as an open-ended drift of being, but as a failure of real-time boundary matching. This reclassifies much of the alignment burden from existence-level ambiguity to implementation-level approximation, latency, and matching failure. The paper therefore proposes a clean boundary-grounded foundation for later work on low-entropy drive, minimal-action differentials, information-update cost, and application-specific technical alignment.