A Dual-Foliation Model of Spacetime and Its Consequences for Energy Conservation

Part I We propose a self-consistent extension of relativistic spacetime based on a dual-foliation structure, in which two temporal coordinates coexist and interact geometrically. The model introduces a coherence parameter χ(x) ∈ [0,1], which quantifies the degree of structural synchronisation between two temporal foliations and determines the local participation of each foliation in the effective metric: gₑff = χ g₍₁₎ + (1 − χ) g₍₂₎. Energy–momentum is conserved globally, but local conservation may be violated in a controlled way via an inter-foliation current Jᵐᵘ, leading to effective source terms in the Einstein equations. Remarkably, when g₍₂₎ exhibits a weak intrinsic expansion, the framework reproduces an accelerated cosmological scale factor a(t) without invoking dark energy or a cosmological constant. Furthermore, quantum-like superposition of states emerges naturally as linear combinations of configurations across foliations, while measurement-like transitions correspond to sharp gradients in χ. The model remains compatible with all classical tests of general relativity in the χ ≈ 1 limit and introduces no new fundamental constants. Though speculative, it is mathematically elementary and falsifiable: deviations in gravitational redshift, cosmic shear, or quantum interference under strong spacetime curvature could provide observational signatures. This approach suggests that temporal redundancy—and the coherence-dependent emergence of physical laws—may underlie several persistent anomalies in modern physics. Crucially, the effective gravitational coupling Gₑff is not time-dependent, but coherence-dependent—a distinction that resolves apparent tensions with local tests of gravity while allowing for cosmological variation.