The Spatial Architecture and Ambient Universe: A Constructive, Computationally Bounded Foundation for Physics

We present a foundational framework for physics built on constructive mathemat- ics and intuitionistic logic rather than on continuous Zermelo–Fraenkel set theory. Space is formalised as a coproduct of a discrete inductive skeleton and a pointless locale; classical integration, differentiation and the Riemannian metric are replaced by a resolution-bound localic accumulator, a resolution gradient, and a minimal- transition-cost metric; and physical evolution is a polynomial-time state automaton minimising a discrete action cost over the lattice. Within this setting we develop a discrete vector calculus in spacetime algebra, a hydrodynamic update rule, a treat- ment of quantum states as coinductive choice sequences, and a thermodynamic account of gravity. Two results distinguish this presentation from its predecessors. First, the frame- work’s saturation bandwidth limit is shown to have been two distinct quantities sharing a name — a local rate bounded by Margolus–Levitin and a local capac- ity bounded by Bekenstein — which removes a conflict of twenty-six orders of magnitude and recovers the holographic count 2.27 ×10122 nats as a derived quan- tity rather than a calibration. Second, the action cost as previously written is unbounded below; we show that no positive-definite replacement can be Lorentz- invariant, adopt a frame-relative cost, and establish that it is Lorentz-equivariant, so the dynamics remains covariant. A mesoscopic realisation of the hydrodynamic sector has been implemented and validated against an external benchmark, reproducing Ghia, Ghia & Shin (1982) to R2 = +0.9973. That realisation instantiates the framework’s halting structure without instantiating the Boolean substrate the structure is supposed to arise from, and the distinction is stated wherever it bears on a claim. Within it, two results are reported. The framework’s four cost operators are shown to be a gauge freedom that cannot affect the flow, so the halting fault is the only component of the update rule with dynamical content; and the effect of that fault on transport is derived rather than fitted, its amplitude following from the collision rule by a Chapman– Enskog expansion and the form of the resulting average from the geometry of the fault set, with no free parameter. The fault is measured to be shear-selective at 5–6.5σ against a matched random control. No other sector of the framework has been tested, and every claim in this document carries a marker stating which of axiom, derivation, numerical validation, or conjecture it is.