On the Necessity of Fixed–β Log–Periodic Residuals: A Regime–Controlled Substrate Framework and Its Cosmological Falsifiers

A physical framework earns credibility not by scope, but by necessity: a small set of
explicit assumptions must force definite consequences and admit decisive failure. In this
spirit, Unified Substrate Theory (UST) is formulated here not as a microphysical completion,
but as a closed constraints–and–tests structure.
The primitive content is restricted to causal ordering and linear response. All charts,
metrics, and sector observables are treated as operational constructions, introduced only
where justified. To prevent vacuity, domain-independent admissibility rules are fixed a priori,
together with a pre-detection criterion defining a log-stationarity regime in which residual
structure may be inferred without β-selection bias. Coarse-graining is formalised through an
aliasing-excluding kernel class and shown to suppress all nonzero log-Fourier modes under
iteration; hence any robustness-stable nonzero spectral lines cannot arise from coarse-graining
alone and require an explicit regeneration or selection sector.
A conditional universality hypothesis is then declared for this sector: locality and trans
lation invariance in logarithmic scale, weak nonlinearity near threshold, and band-limited
admissible forcing. Within this admissible class, a band-selection normal form implies a
unique ground logarithmic frequency β0. Its value is not derived here and is treated as a
dimensionless constant to be identified empirically under a fixed-β discipline.
When a homogeneous FRW description is operationally meaningful, observability enforces
locality in u = ln(1 + z). Combined with log-stationarity and linear response, this yields a
necessary first-order fixed-β deformation of the late-time expansion history. The framework
therefore terminates in binary outcomes: explicit falsifiers, negative controls, and a decisive
growth no-go bound consistent with luminal gravitational-wave propagation. If the fixed-β
deformation is absent, unstable under admissible robustness tests, or violates its admissibility
envelope, the framework fails in the regime considered