Foundational Temporal Theory (FTT): A Mathematical Program for Emergent Quantum Mechanics and Spacetime from Causal Growth

We present a mathematical program for a theory in which time (temporal order)is fundamental, while space, geometry, and quantum mechanics emerge from thestatistical properties of a growing causal network. The framework is built on rigorousdefinitions, explicit axioms, and provable theorems under clearly stated assumptions.Key results include: (1) the existence of an effective constant ℏeff = σ2S/µ fromstationary growth statistics, derived as the ratio of fluctuation intensity to mean action density; (2) a rigorous construction of a Hilbert space and complex amplitudesvia the Gelfand-Naimark-Segal (GNS) method applied to a positive definite correlation kernel between classical histories—crucially, no complex weights are assigned toindividual histories; (3) a causally covariant coarse-graining procedure that yields aquadratic local effective action; and (4) a demonstration that the emergent structurereproduces the Feynman path integral in the continuum limit.All results are stated as theorems with proofs under explicit probabilistic assumptions. The program identifies open problems in hierarchical order and provides afoundation for further development toward continuum geometry, quantum dynamics,and testable cosmological predictions. This revised version (0.3) addresses criticalpoints raised in peer review, including a strengthened proof of positive definiteness,clarification of the logical status of ℏeff, and a detailed discussion of the unitarityproblem.