IndisputableMonolith.Cosmology.Spectral_Index3_FromJCost
Packages the link from the J-cost functional to a spectral-index-3 certificate in RS cosmology. Cosmologists cite the nS3 certificate when tying power-spectrum slope or inflationary tilt to cost geometry and D=3. The module defines a domain cost, a positive canonical threshold, non-negativity facts, and an inhabited certificate object rather than a long derivation.
claimThe module defines a domain cost built from the J-cost $J(x)=(x+x^{-1})/2-1$, proves that cost is nonnegative, fixes a positive canonical threshold, and supplies an inhabited certificate that the spectral index satisfies the three-dimensional relation forced by that cost geometry.
background
Recognition Science forces a unique nonnegative cost $J(x)=(x+x^{-1})/2-1$ (T5) and three spatial dimensions (T8). Cosmology modules translate that cost geometry into observables. This module lives in the Cosmology domain and imports RS constants (fundamental tick $\tau_0$) together with the Cost library that supplies $J$ and related functionals.
Sibling names indicate a domain-restricted cost, equality and nonnegativity lemmas, a positive canonical threshold, and an nS3 certificate object with an inhabited wrapper. The eight-tick octave (T7) and $D=3$ supply the dimensional setting in which a spectral index labeled "3" is the natural RS-native target rather than a free fit parameter.
proof idea
Definition-and-certificate module, not a multi-step tactic development. It introduces domainCost from the imported J-cost, records an evaluation equality and nonnegativity, fixes canonicalThreshold with a positivity lemma, then exposes nS3Cert and an inhabited cert wrapper. Argument structure is packaging: the spectral-index-3 claim is reified as a certificate rather than proved from a long chain inside this file.
why it matters in Recognition Science
Anchors the spectral-index-from-J-cost claim inside the RS cosmology layer. No downstream edges are listed yet; consumers would apply the nS3 certificate when assembling CMB or inflationary predictions from cost geometry. The construction sits downstream of J-uniqueness (T5) and $D=3$ (T8): once the cost and spatial dimension are forced, an index-3 certificate is the expected RS-native slot rather than phenomenological tuning. It closes a named certificate interface for spectral index instead of leaving the link informal.
scope and limits
- Does not fit a Planck-scale numerical $n_s$ beyond the certificate packaging.
- Does not prove uniqueness of the index outside the stated domain cost and threshold.
- Does not derive transfer functions, Boltzmann hierarchies, or survey likelihoods.
- Does not discharge scaffolding in other cosmology modules.
- Does not claim observational closure without external data comparison.