Non-Commutativity from Geometric Constraint: Projected Algebra of the Existence Equation

Evidence Paper IV (v3.0) of the Existence Equation series Non-commutativity is the organizing principle of modern physics. Quantum mechanics begins with [x̂, p̂] = iℏ. The Standard Model is built on non-Abelian gauge groups: SU(3) generates the strong force, SU(2) the weak force. In these and many other formulations, the non-commutative algebra is part of the starting structure. The question rarely asked is: where does the non-commutativity come from? This paper answers that question. In the hard-core sector of the Existence Equation condensation term α|Ψ|²Ψ, state-dependent admissibility becomes a projection P. Two operators that commute exactly in the full Hilbert space — [X0, X1] = 0 — acquire a nonzero…