A Formal Model for Canvas Temporal Mathematics: Spectral Semantics for a Temporal Foundation

Canvas Temporal Mathematics (CTM) is a proposal for a temporal foundation for mathematics, built on eight primitives with spectral equality, \mathcal{S}-symmetry, and meta-order dynamics. A recurring criticism of CTM is that it lacks formal semantics: there is no definition of a model, no completeness theorem, and no rigorous account of what it means for a CTM statement to be true.

This paper addresses that criticism by constructing an explicit formal model for CTM.

What this paper provides:

· An explicit formal model \mathfrak{M} for CTM on the domain of constant amplitudes (natural numbers). The model consists of:
  · An order parameter space V = \mathbb{R}_{\geq 0}
  · An amplitude space \mathbb{A}_0 = \{(c, -c) : c \in \mathbb{R}\} for constant full amplitudes
  · A meta-time parameter \mathcal{T} = \mathbb{R}_{\geq 0}
  · An equality processor \mathcal{E} with eigenmodes \mathbf{\Psi}_1 = (1, -1) and \mathbf{\Psi}_2 = (1, 1)
  · An interpretation function \llbracket \cdot \rrbracket mapping natural numbers to amplitudes
  · A spectral truth condition \models_\tau (truth as eigenvalue threshold-crossing)
· Proof that the CTM axioms hold in \mathfrak{M}. The model satisfies the order parameter axiom, the amplitude space axiom (including the UWE and \mathcal{S}-symmetry), the equality processor axiom (fixed-point equation with minimal override), and the meta-order dynamics axiom (gradient flow on spectral energy).
· Proof that Boolean equality emerges as the equilibrium limit. For constant amplitudes, as the threshold \theta_0 \to 0 and meta-time \tau \to \infty, the equality processor outputs \lambda_{\max} \cdot \mathbf{\Psi}_1 when a = b (true) and \mathbf{0} when a \neq b (false), recovering standard Boolean equality.
· A soundness-like property for constant equalities. If CTM proves an equality between constant terms, then the model assigns non-zero truth values to that equality for all sufficiently large \tau.
· A demonstration of the Feed dynamics in the model. The spectral energy \mathbb{E} = (\lambda_1 - \lambda_1^*)^2 + (\lambda_2 - \lambda_2^*)^2 drives exponential convergence of eigenvalues to equilibrium.
· A sketch for extending the model to functions. The constant amplitude model handles natural numbers. Extending to functions requires function amplitudes, function eigenmodes, a definition of function application, and a proof of the fixed-point equation for function spaces.

Why this matters:

This model addresses the criticism that CTM lacks formal semantics. It demonstrates that CTM's core concepts—spectral equality, \mathcal{S}-symmetry, meta-order dynamics—can be given rigorous mathematical definitions. It provides a foundation for extending CTM's formal semantics to functions, sets, and higher-order structures. CTM now has a model. The model is limited to natural numbers, but it exists. The work of extension and classification lies ahead.

Keywords: Canvas Temporal Mathematics, formal semantics, spectral equality, \mathcal{S}-symmetry, meta-order dynamics, equality processor, eigenmodes, Feed dynamics, Boolean limit, soundness, model theory