IndisputableMonolith.Cosmology.DarkEnergyEquation3_FromJCost
Module packaging a J-cost domain functional and a certificate that the third dark-energy equation-of-state identity holds in RS-native units. Cosmologists citing RS dark-energy claims use the certificate and the nonnegativity/threshold lemmas. Structure is definitional: cost and threshold defs, then an inhabited certificate record.
claimDefine a domain cost $C$ built from the RS $J$-cost on a positive scale variable, a canonical positive threshold $T_\ast$, and a certificate asserting the third dark-energy equation-of-state relation derived from $C$ (with $C\ge 0$ and evaluation identities).
background
Recognition Science fixes the cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$) as the unique symmetric generator of the Recognition Composition Law. Cosmology modules import that cost and the RS constants (including the tick $\tau_0$) to write late-universe observables without free dark-energy parameters.
This module sits in the cosmology layer. It introduces a domain cost obtained by evaluating $J$ on a positive cosmological scale (or ratio), records elementary facts (pointwise evaluation, nonnegativity), and names a canonical positive threshold against which the dark-energy equation-of-state identity is checked.
The certificate type DEoS3Cert packages those ingredients so downstream cosmology proofs can assume a single inhabited record rather than re-proving cost inequalities.
proof idea
Definition-and-certificate module, not a deep derivation. Domain cost is defined from the imported $J$-cost; companion lemmas discharge evaluation and $C\ge 0$. Canonical threshold is a positive constant def with a positivity proof. The main object is an inhabited certificate record bundling those facts for the third DE equation-of-state claim. No long tactic scripts: algebraic identities from Cost plus trivial positivity.
why it matters in Recognition Science
Gives the cosmology stack a single import point for "dark energy EoS form 3 from $J$" instead of ad hoc cost rewrites. Feeds any parent argument that needs a certified nonneg domain cost and threshold when matching RS late-time expansion to $w$-parametrizations. Aligns with the forcing chain's $J$-uniqueness (T5) and RCL: the same $J$ that fixes microphysics is reused as the dark-energy cost. used_by is empty here, so this is a leaf packaging module for certificates rather than a step inside UnifiedForcingChain.
scope and limits
- Does not derive observational $w_0,w_a$ fits or supernova/BAO likelihoods.
- Does not prove uniqueness of the third EoS form among all cosmologies.
- Does not fix $\Lambda$CDM parameters or replace Friedmann dynamics proofs.
- Does not discharge full RS mass-ladder or $\alpha$ claims; only DE cost packaging.
- Does not assert empirical detection of dark energy beyond the formal certificate.