Canonical Quantisation of the UCM Substrate: Derivation of the Commutation Relations from Lattice Dynamics

This paper closes the open problem explicitly identified in Part Q of the U-Cell Model (UCM) Mathematical Companion (Zenodo record 19387201): that the canonical commutation relations (CCRs) for the transverse substrate field cannot be imposed as an external rule if the substrate is fundamental. What is derived: We do not derive the principle of quantisation from classical mechanics — Dirac's prescription is taken as a universal postulate. What we derive is the specific spatial and tensor structure of the field CCRs as a necessary consequence of the discrete lattice geometry and its Poisson algebra. Main results (four theorems): Theorem 1 (ℏ from the lattice action). The reduced Planck constant ℏ = 12π ρ₀ a⁴ c₀ is not a free parameter of the quantisation. It is the unique normalisation fixed by the minimal cell action and Postulate 3 (ℓPlanck = a). Theorem 2 (Helmholtz commutator). The Helmholtz projectors satisfy PL · PT = 0 exactly, giving b̂ᵃ₋, ₊, b̂^b†ₓ, ₊' = 0 at the operator level in Fock space — exact at all loop orders. Theorem 3 (Continuum CCRs). The standard CCRs ψ̂ᵃ (x), π̂ᵇ (y) = iℏ δᵃᵇ δ³ (x−y) emerge in the continuum limit a → 0, with ℏ given by Theorem 1. Theorem 4 (Vacuum energy, all loop orders). Vacuum zero-point energy does not curve spacetime to all loop orders. The exact Fock-space factorisation F = FL ⊗ FT removes the loop-level caveat of Theorem Q. 7. 1 without additional assumptions. Part of the UCM preprint series (records 19387201–19390834). ORCID: 0009-0005-5088-4339