Quantum Logic as Below-Threshold Canvas Temporal Mathematics

Quantum logic is the non-Boolean logic of quantum mechanics, in which propositions correspond to closed subspaces of a Hilbert space, conjunction to intersection, disjunction to the closed span of the union, and negation to the orthogonal complement. The distributive law of classical logic fails, and observables that do not commute cannot simultaneously have definite truth values. For nearly a century, this structure has been taken as a primitive feature of quantum mechanics. This paper shows that quantum logic is precisely Canvas Temporal Mathematics (CTM) in the below-threshold regime. The correspondence is exact and provides a physical explanation for every feature of quantum logic. What this paper provides: · A derivation of quantum propositions as eigenmodes of the Equality Processor. Each eigenmode corresponds to a potential measurement outcome. A proposition is true (at a given instant) if the processor output has a non-zero projection onto that eigenmode.· An explanation of the below-threshold regime. When the combined amplitude |\Phi_i \Phi_j| is below the threshold T_{ij}, the processor operates in the linear regime. The output is a superposition of eigenmodes: \mathbf{\Phi} = \sum_i \lambda_i \mathbf{\Psi}_i, with \lambda_i \in \mathbb{C}. This is the quantum state before measurement. The squared magnitudes |\lambda_i|^2 are the Born-rule probabilities.· A definition of quantum logical connectives in CTM terms: · Conjunction (P \land Q): Simultaneous threshold crossing of both corresponding eigenmodes. This is possible only if the eigenmodes are compatible—that is, if the corresponding observables commute. Non-commuting eigenmodes cannot simultaneously cross threshold, which is the origin of the uncertainty principle. · Disjunction (P \lor Q): The superposition of the corresponding eigenmodes. The processor output projects onto the closed span of the eigenmodes. A disjunction can be true without either disjunct being individually true—the superposition can lie in the span without aligning with either eigenmode. · Negation (\lnot P): The \mathcal{S}-antisymmetric complement of the corresponding eigenmode. The eigenmode \mathbf{\Psi}_{\lnot P} = \mathcal{S}[\mathbf{\Psi}_P] is orthogonal to \mathbf{\Psi}_P and spans the complementary subspace.· An explanation of the failure of distributivity. The distributive law P \land (Q \lor R) = (P \land Q) \lor (P \land R) fails when P, Q, R correspond to non-commuting observables. The left-hand side requires P to cross threshold simultaneously with the superposition of Q and R—a single threshold crossing event. The right-hand side requires either P and Q to cross together, or P and R to cross together—two separate events. When Q and R are non-commuting, the left-hand side can be true even when neither P \land Q nor P \land R individually cross threshold. In the classical limit (commuting observables, zero threshold), the distributive law is recovered.· A formulation of measurement as threshold crossing. Before measurement, the processor operates in the below-threshold regime; the output is a superposition. At the moment of measurement, the combined amplitude of the quantum field and the detector field exceeds the threshold: \sum_i \lambda_i \mathbf{\Psi}_i \rightarrow \mathbf{\Psi}_k (collapse). The probability of outcome k is |\lambda_k|^2, derived from Rice's formula for threshold crossing rates.· A hierarchy of logics within CTM. The Equality Processor contains multiple regimes: below-threshold with non-commuting eigenmodes (quantum logic), below-threshold with commuting eigenmodes (intuitionistic logic), at threshold (measurement collapse), and above-threshold with zero threshold (classical Boolean logic). Classical logic is not fundamental—it is the post-measurement regime. Why this matters: Quantum logic is not a mysterious departure from classical reasoning. It is the natural logic of the below-threshold regime of CTM—the regime before measurement, where propositions are superpositions, where non-commuting observables cannot be simultaneously evaluated, and where the distributive law fails because it presupposes simultaneous definability. Classical logic is the post-measurement regime, where threshold crossings have occurred, superpositions have collapsed, and all relevant observables commute. CTM provides the unified framework that contains both. Keywords: quantum logic, Canvas Temporal Mathematics, below-threshold regime, eigenmodes, equality processor, threshold crossing, non-commuting observables, distributive law, measurement collapse, superpositions, orthomodular lattice, Birkhoff–von Neumann, quantum foundations