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 the
statistical properties of a growing causal network. The framework is built on rigorous
definitions, explicit axioms, and provable theorems under clearly stated assumptions.
Key results include: (1) the existence of an effective constant ℏeff = σ2
S/µ from
stationary growth statistics, derived as the ratio of fluctuation intensity to mean ac
tion density; (2) a rigorous construction of a Hilbert space and complex amplitudes
via the Gelfand-Naimark-Segal (GNS) method applied to a positive definite correla
tion kernel between classical histories—crucially, no complex weights are assigned to
individual histories; (3) a causally covariant coarse-graining procedure that yields a
quadratic local effective action; and (4) a demonstration that the emergent structure
reproduces the Feynman path integral in the continuum limit.
All results are stated as theorems with proofs under explicit probabilistic assump
tions. The program identifies open problems in hierarchical order and provides a
foundation for further development toward continuum geometry, quantum dynamics,
and testable cosmological predictions. This revised version (0.3) addresses critical
points raised in peer review, including a strengthened proof of positive definiteness,
clarification of the logical status of ℏeff, and a detailed discussion of the unitarity
problem.