IndisputableMonolith.Unification.RegistryPredictionsProved
Proved registry predictions for the dark-energy density parameter Ω_Λ and related φ-hierarchy bounds used in RS unification. Establishes that the closed-form Ω_Λ expression is positive and strictly less than 11/16, together with elementary lower bounds on α^{-1} and φ-power ladders. Cosmology and constants pipelines cite these inequalities. Arguments are direct real-arithmetic comparisons from the forced φ and gap-weight identities.
claimThe RS registry formula for the dark-energy fraction $\Omega_\Lambda$ is well-defined on the forced golden ratio $\varphi$, satisfies $0 < \Omega_\Lambda < 11/16$, and sits inside a $\varphi$-power hierarchy with explicit lower bounds (including $\varphi^{-6}$ and $\varphi^{-11}$ scales). Companion inequalities include $\alpha^{-1} > 2$. A certificate packages these predictions as a single existence claim.
background
Recognition Science fixes $\varphi$ by self-similarity of a discrete ledger with $J$-cost (PhiForcing): the unique positive solution of the fixed-point relation forced by scale-invariant recognition. Constants are written in RS-native units built from that $\varphi$, with the eight-tick gap weight $w_8$ entering the $\alpha$ pipeline as the parameter-free closed form $f_{\mathrm{gap}} = w_8 \ln\varphi$ (GapWeight).
This module lives in the Unification layer. It does not re-derive $\varphi$ or $w_8$; it consumes those forced constants and turns the registry's predicted dimensionless combinations into Lean inequalities. The headline object is $\Omega_\Lambda$, the dark-energy density parameter whose RS expression is asserted to be calculable and bounded above by $11/16$.
Sibling results also record positivity of $\Omega_\Lambda$, a two-sided bound package, elementary hierarchy comparisons at $\varphi^6$ and $\varphi^{11}$, and the structural claim that the hierarchy is organized by pure powers of $\varphi$. A certificate type bundles the proved predictions for downstream import.
proof idea
The module is a small proved-predictions library, not a single deep theorem. Each inequality is a short real-arithmetic argument from the closed forms of $\varphi$ and the registry expressions: positivity and the strict upper bound $\Omega_\Lambda < 11/16$ are obtained by evaluating or comparing the explicit formula; $\alpha^{-1} > 2$ is an elementary lower estimate; $\varphi$-power hierarchy lemmas compare successive scales by multiplying by $\varphi > 1$. The certificate is assembled by packaging those lemmas into one existence witness. No analytic number theory or measure theory is required beyond Mathlib real facts and the imported constant definitions.
why it matters in Recognition Science
Downstream, CosmologicalConstantDerivation (C-010) imports this module to attack the cosmological-constant problem: QFT vacuum energy is off by $\sim 10^{120}$, while RS claims a finite, registry-fixed $\Lambda$ tied to the same $\varphi$-ladder. The proved bound $\Omega_\Lambda < 11/16$ and positivity supply the calculational spine for that derivation rather than a free fit.
Within Unification, the module converts "registry predictions" from narrative claims into zero-sorry inequalities usable by cosmology and constants consumers. It sits after PhiForcing (T6-style forcing of $\varphi$) and GapWeight (parameter-free $w_8$), so the no-free-parameter stance is inherited rather than re-argued. The certificate existence lemma is the natural import point for any parent that needs "all registry numeric predictions held simultaneously."
scope and limits
- Does not derive the cosmological constant Λ itself; only registry inequalities feeding that derivation.
- Does not re-prove φ-forcing or the closed form of w₈; those are imported assumptions.
- Does not claim observational fits to measured Ω_Λ beyond the analytic 11/16 ceiling.
- Does not address matter or radiation density parameters Ω_m, Ω_r.
- Does not prove uniqueness of the registry formula, only well-definedness and stated bounds.