IndisputableMonolith.Cosmology.DarkEnergyDensity4FromJCost
Module packaging the claim that a dark-energy density factor of four is forced by the Recognition J-cost on a canonical domain threshold. Cosmologists working in RS units would cite the certificate bundle DEDensity4Cert. The development is mostly definitional: nonnegativity and evaluation lemmas for a domain cost, positivity of a threshold, and an inhabited certificate record.
claimIn RS units, a domain cost built from the J-cost $J(x)=(x+x^{-1})/2-1$ is nonnegative, agrees with its pointwise evaluation, and meets a positive canonical threshold; the package $DEDensity4Cert$ records that the dark-energy density factor four is the corresponding certified consequence of that threshold.
background
Recognition Science measures mismatch by the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced unique by the Recognition Composition Law and the T5 step of the unified forcing chain. The Cost import supplies that functional; Constants supplies the RS time quantum $\tau_0=1$ tick and the golden-ratio ladder constants used elsewhere in cosmology.
This module sits in the Cosmology domain. It introduces a domain-level cost (domainCost) evaluated on a canonical threshold, together with elementary comparison facts: the cost equals its pointwise form, is nonnegative, and the threshold is strictly positive. Those facts are then bundled into a certificate type DEDensity4Cert whose witness is the factor-four dark-energy density claim read off the J-cost geometry.
No external dynamical model of $\Lambda$ is assumed; the factor four is treated as an RS-native dimensionless readout of the cost threshold, not a fit parameter.
proof idea
Definition-and-certificate module rather than a deep derivation. domainCost is introduced from the Cost layer; domainCost_at_eq and domainCost_nonneg are short algebraic or positivity facts about that definition. canonicalThreshold and canonicalThreshold_pos fix a positive scale. DEDensity4Cert is a structure packaging those ingredients; cert and cert_inhabited supply a concrete inhabited instance. There is no multi-step tactic proof of a new identity beyond assembling and checking the certificate fields.
why it matters in Recognition Science
Gives Cosmology a named, checkable handle on the claim that dark-energy density carries a factor four forced by J-cost geometry rather than by an external $\Lambda$ fit. Downstream graph edges are empty in the current mirror, so this module is a leaf certificate: it closes a local scaffolding path (inhabited DEDensity4Cert) without yet feeding a larger proved theorem. In the broader RS picture it sits beside the T5 J-uniqueness landmark and the RS-native constants ($c=1$, $\hbar=\varphi^{-5}$, $G=\varphi^5/\pi$), offering a cosmology-side readout of the same cost that forces $\varphi$ and the eight-tick octave elsewhere. Referees should treat it as a certified packaging step, not as a full derivation of observed $\Omega_\Lambda$.
scope and limits
- Does not derive observed $\Omega_\Lambda$ or match Planck numerical dark-energy density.
- Does not prove uniqueness of the factor four outside the chosen canonical threshold.
- Does not couple to FLRW dynamics, equation-of-state $w$, or structure growth.
- Does not discharge broader Cosmology theorems; used_by is currently empty.
- Does not re-prove J-uniqueness (T5) or the Recognition Composition Law.