Boundary-Induced Collapse (BIC) in Spectral Language

We propose that in physical systems, structural stability is not an intrinsic property but a consequence of boundary-induced spectral selection. In this work, we formalize the Boundary-Induced Collapse (BIC) by analyzing a quantum system S interacting with a boundary environment B through a total Hamiltonian hatHₓ₄ₗₓₓ₎ₓ = hatHS + hatHB + hatHₒ₁. We demonstrate that the effective dynamics of S is governed by an induced Hermitian operator hatG = textTrB (hatHₒ₁ rhoB hatHₒ₁ᵈagger), which encodes the structural constraints of the boundary. By applying the spectral theorem to hatG, we show that the system’s density matrix necessarily collapses into the spectral basis defined by the operator’s eigenstates (|ᵢ), which we term Ontological Deltas. This process provides a unified framework where quantum decoherence and measurement are seen as specific cases of a more general principle of spectral necessity. Furthermore, we establish that this Hamiltonian-based collapse is the formal quantum analog of the classical AT normalization in linear systems. We conclude that diagonalization is the universal physical mechanism through which nature translates incoherent interactions into stable, observable states of residency.