Linear Perturbations and Boltzmann Implementation of the Scalar Temporal Field Ontology: Predictions for the Full CMB Angular Power Spectrum

The Scalar Temporal Field Ontology (STFO) models time as a dynamical scalar fieldτ (xμ) whose gradients generate causal structure and temporal directionality. Backgroundcosmology in STFO is governed by a general kinetic Lagrangian P (τ, X) with X ≡ −gμν ∇μτ ∇ν τ ,yielding a closed system for H(t) and τ (t) on FLRW backgrounds. However, backgroundfitting alone is insufficient: the decisive test is whether the theory reproduces the full CMBangular power spectrum Cℓ (including peak heights, damping tail, polarization, and lensing)under linear perturbations. In this work we derive the complete linear scalar perturbationequations for STFO in a form directly compatible with Einstein–Boltzmann solvers, identifythe physical stability requirements in terms of PX and PXX , and provide an explicit imple-mentation pathway in CLASS/CAMB-type codes to compute Cℓ for any specified P (τ, X)model. We also define a minimal benchmark deformation of the canonical kinetic sectorsufficient to test whether STFO can preserve the measured acoustic scale while shifting late-time expansion parameters, enabling direct χ2 comparison to Planck likelihoods and futureCMB-S4 forecasts.