On the Emergence of Space, Time, and Gravitation from a Log-Structural Substrate

Aunified substrate framework is formulated in which physical spacetime metric structure,
time-ordering observables, gravitating matter phenomenology, and late-time cosmological
dynamics arise as emergent operational constructs from an underlying substrate specified
minimally by its causal spectral response and a regeneration operator class exhibiting
discrete scale structure. The substrate is not identified with “matter in spacetime,” nor
with a conventional quantum field defined on an ontic manifold. An auxiliary Lorentzian
chart is employed only as representational scaffolding for response kernels and principal
symbols. Within a minimal admissible local translation-invariant operator class on logarithmic
coordinates, dynamics selects a unique stable ground log-frequency β0 ≃ 4. The empirically
operative discrete set is constrained to
β ∈{4, 9, 16, 27.57},
with higher members arising as constrained harmonic/projection structures rather than
independent eigenmodes. Metric structure is defined operationally from the characteristic set
of probe propagation; time is treated as a spectral ordering observable associated with stable
phase progression. In the macroscopic limit the framework admits a conservative gravitational
closure as a scalar–tensor effective field theory that can be restricted to gravitational-wave
compatible (Horndeski-safe, luminal) sectors, with ΛCDM recovered as a limiting fixed-point
regime under damping of oscillatory terms. A fixed-β empirical programme is specified with
explicit confirmatory/exploratory boundaries, null-ensemble design (permutation and Monte
Carlo/mock-universe nulls), negative controls (including BAO monopole β-silence), and
reproducibility protocols. Cross-domain results reported in the associated technical record
are organized under a unified evidence protocol, including strong fixed-β = 4 detections in
seismicity and turbulence under null calibration, cross-validated fixed-β = 4 improvements in
galaxy rotation-curve residuals, β-mixture structure in selected cosmological residual sectors,
and higher-β phase organization in recurrence phenomena. A documented prospective
application associated with the 8 December 2025 Honshu earthquake is included solely as a
demonstration of a prospective scoring protocol