Finite-Rank Quantum Mechanics Axiomatic Foundation, Decoherence Scaling, and the Topological Origin of (π − 3)

We develop a mathematically rigorous framework for quantum mechanics withfinite-dimensional observable subspaces. Starting from four axioms definingfinite observer capacity, we derive: (i) the necessity of Lindblad dissipationfor dynamics in finite-rank subspaces, (ii) a decoherence scaling law Γ ∝ N −1under two explicitly stated additional assumptions.We prove that the spectral defect δ = π − 3 is an inevitable consequence ofdiscrete space structure, following from the Thue–Fejes Tóth–Hales theoremon optimal circle packing.This version (8.0) clarifies the logical structure by explicitly separating the-orems (derived from axioms alone) from predictions (requiring additional as-sumptions). All claims are labeled by their epistemic status.