Principle of Spacetime Topology Consistency

The standard ΛCDM cosmology relies on the FLRW metric, which admits spatial sections either infinite (flat/open, K=0,−1) or finite (closed, K=+1), while asserting a temporal origin t=0 (Big Bang singularity). This paper argues that the combination of a temporal origin (past timelike geodesic incompleteness) with a non-compact spatial section yields an ∞×0 indeterminate form in the comoving volume measure, indicating that this combination lacks a well-defined volume basis and constitutes an ontological tension to be explained (Proposition A, heuristic). Furthermore, the paper argues that the combination of finite (compact) space with infinite time (especially infinite past) leads to consequences incompatible with observations from two independent angles—causal topology and thermodynamics (Proposition B). Treating spatial geometry, matter, radiation, dark matter, and dark energy as one closed cosmic system, we propose the Principle of Spacetime Topology Consistency (PSTC). Its core formula T≔[t₀,+∞) :Σcompact (PSTC-1) states: the time and space directions of a spacetime manifold should share the same completeness; if time has a beginning (bounded interval), space should be compact (finite-finite coupling); if space is non-compact, time should be unbounded. The connection between PSTC and observations comes from: (1) no matching circles on the CMB sky and no historical superposition traces in acoustic peaks (causal-geometric layer); (2) finite space implies an upper entropy bound while the observed universe is not in heat death, is structurally rich, and has an active arrow of time (thermodynamic layer, based on the Kelvin 1852 heat-death paradox tradition). This paper conducts a computational analysis of the geometric structure of Σ≅S³. All premises are uniformly listed in §2.1; formulas in the main text are tagged with the assumption numbers they depend on.(This record contains both the English and Chinese versions of the manuscript.)