IndisputableMonolith.Cosmology.CMB_Power_Spectrum_Peaks_v3
Module packaging RS-native certificates for CMB acoustic-peak positions via a domain cost and a canonical threshold. Cosmologists comparing multipole peaks to the phi-ladder would cite the peak-position certificate and its inhabited proof object. The argument is definitional scaffolding plus nonnegativity and positivity lemmas, not a full Boltzmann-code derivation.
claimDefine a nonnegative domain cost $C$ on the relevant configuration space, a positive canonical threshold $\theta_\ast$, and a certificate asserting that CMB power-spectrum peak positions (v3) lie at the loci fixed by $C$ relative to $\theta_\ast$ in RS-native units ($c=1$, ladder in powers of $\varphi$).
background
Recognition Science cosmology works in RS-native units with $c=1$ and the golden ratio $\varphi$ as the self-similar scale factor forced at T6. Masses and spectral features sit on a $\varphi$-ladder; cost structure comes from the $J$-functional of the Recognition Composition Law, imported here through the Cost module. The Constants import supplies the fundamental tick $\tau_0=1$.
This module sits in the Cosmology domain and introduces a domain cost (with an evaluation identity and a nonnegativity lemma), a canonical threshold (with positivity), and a CMB peak-position certificate together with an inhabited cert object. The intent is to pin acoustic-peak multipoles to threshold crossings of that cost rather than to free-fit $\Lambda$CDM parameters.
No full radiative-transfer or Boltzmann hierarchy is formalized here; the module is the Lean-side certificate layer for peak loci.
proof idea
Definition-heavy module. Domain cost and canonical threshold are introduced as defs; supporting lemmas record evaluation at a point, nonnegativity of the cost, and positivity of the threshold. The peak-position certificate is a Prop-level (or structure-level) bundle; inhabitation is a separate one-liner or constructor application. No deep tactic proof of the acoustic physics itself appears: the module wires Cost/Constants into named cert objects for downstream cosmology claims.
why it matters in Recognition Science
Gives the Recognition framework a named place to hang CMB acoustic-peak positions against the phi-ladder and J-cost structure, parallel to mass-ladder and alpha-band certificates elsewhere in the monolith. Downstream used-by edges are empty in the current graph, so this is a leaf certificate module rather than a proved input to a larger forcing theorem. It touches the cosmology side of RS (peaks as threshold phenomena) without yet closing a full derivation from T0–T8. Referees should treat it as interface-plus-cert scaffolding until linked into a parent spectrum theorem.
scope and limits
- Does not derive the CMB Boltzmann hierarchy or Silk damping from first principles.
- Does not prove numerical multipole values against Planck data inside Lean.
- Does not fix cosmological parameters beyond the RS cost/threshold interface.
- Does not claim a used-by parent theorem in the current dependency graph.
- Does not replace observational peak fitting; it only certifies RS loci.