Abstract — original text

We propose an axiomatic framework, termed Coexisting Time Arrows (CTA), in which spacetime geometry, matter, energy, and quantum phenomena are investi- gated as potential emergent consequences of the local coexistence of two oppositely oriented temporal sectors,τ + andτ −. The framework extends the cosmological motivation of the CPT-symmetric uni- verse proposed by Boyle, Finn, and Turok (Phys. Rev. Lett. 121, 251301, 2018) toward a local field-theoretic description in which temporal degrees of freedom are treated as fundamental, while spatial geometry is regarded as an induced structure. A central mathematical result concerns the construction of a four-dimensional Lorentzian metric from temporal scalar fields. We demonstrate that a bilinear metric constructed from a single pair of scalar-field gradients has rank at most two and is therefore necessarily degenerate in four dimensions. We introduce a multimode generalization and prove that at least three pairs of temporal modes are necessary to obtain Lorentzian signature (−+ ++) within the proposed metric ansatz. An explicit construction with three mode pairs reproduces the Minkowski metric. The resulting geometry admits an interpretation as an induced metric in an auxiliary six-dimensional flat space with signature (3,3). The framework further proposes that the gravitational effects attributed to dark matter may originate from a CPT-symmetric baryonic sector, while cosmic accel- eration may admit an interpretation based on temporal geometry rather than a fundamental dark-energy component. Electric charge is hypothesized to arise from local asymmetry between oppositely oriented temporal sectors, and quantum behavior is explored through an operator description of temporal degrees of freedom. The representation of general curved spacetimes, recovery of Einstein dynamics, derivation of gauge interactions, and quantitative cosmological predictions remain open problems. CTA thus provides a mathematically explicit realization of flat Lorentzian ge- ometry from temporal fields, together with a broader axiomatic research program for investigating the possible emergence of physical structure from coexisting time arrows. The multimode construction establishes that three pairs of temporal scalar fields are the minimum required to realize a four-dimensional Lorentzian metric within the proposed bilinear ansatz.