Omega_0
plain-language theorem explainer
Omega_0 defines the log-periodic modulation frequency in the RS inflationary spectrum as 2 pi divided by the natural logarithm of pi over phi. Cosmologists modeling primordial power spectrum oscillations cite it for the modulation period of approximately 0.664 in ln(k). The definition is a direct real-number expression built from the optimal recognition ratio X_opt equals phi over pi.
Claim. Define the modulation frequency by the expression $2π / ln(π / φ)$, where $φ$ denotes the golden ratio. Equivalently, $Ω_0 = 2π / ln(1/X_{opt})$ with the optimal recognition ratio $X_{opt} = φ / π$.
background
The Inflation from phi module derives RS inflationary predictions from the phi-ladder and recognition cost functional. It sets the alpha-attractor parameter to phi squared, the spectral index to approximately 1 minus 2 over N, and the tensor-to-scalar ratio to approximately 12 phi squared over N squared. The log-periodic modulation frequency is introduced as Omega_0 equals 2 pi over ln(pi over phi), yielding oscillations with period Delta ln(k) approximately 0.664. The referenced definition X_opt equals phi over pi supplies the ratio at which recognition cost and geometric constraint balance.
proof idea
The declaration is a one-line definition that substitutes the expression for X_opt directly into the logarithm argument and applies the standard real division and multiplication operations on pi and phi.
why it matters
Omega_0 supplies the modulation frequency required by the InflationCert structure that certifies curvature bounds and spectral properties. It fills the log-periodic modulation step stated in the module documentation for the Universe-Origin Paper. The frequency rests on the phi self-similarity fixed point and the eight-tick octave that fix the optimal ratio X_opt.
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