inequalityCeilingCert
plain-language theorem explainer
A certificate is supplied asserting that the maximum sustainable Gini coefficient equals φ − 1, lies inside (0.617, 0.623), and carries the same J-cost value as φ. Recognition Science economists would cite it to bound inequality under zero-sigma conditions in the labour-capital ledger. The definition assembles three prior theorems that establish the golden-ratio identity, the numerical band, and the J-cost symmetry.
Claim. Let $g$ denote the Gini ceiling. Then $g = φ - 1$, $0.617 < g < 0.623$, and $J(g) = J(φ)$, where $φ$ is the golden ratio satisfying $φ^2 = φ + 1$ and $J$ is the J-cost function.
background
In the Recognition Science treatment of economics the Gini ceiling is the maximum sustainable Gini coefficient under σ = 0 across the labour-capital ledger. The module states that this ceiling equals 1/φ, which equals φ − 1 by the golden-ratio identity φ² = φ + 1. The J-cost function satisfies J(1/φ) = J(φ) by symmetry at the boundary. Upstream results supply the three supporting facts: the identity theorem states “1/φ = φ − 1 (golden ratio identity: φ² = φ + 1 → φ − 1 = 1/φ)”; the band theorem states “Gini ceiling in (0.617, 0.622)”; and the symmetry theorem states “J(1/φ) = J(φ) (symmetry)”. The local setting is the F3 prediction that values above the ceiling trigger a σ-cascade while values below maintain stable recognition equilibrium.
proof idea
The definition is a one-line wrapper that directly supplies the three fields of the InequalityCeilingCert structure by referencing the theorems that prove the golden-ratio identity, the numerical band, and the J-cost symmetry.
why it matters
This certificate formalizes the Recognition Science prediction that the Gini coefficient cannot sustainably exceed 1/φ without triggering institutional collapse. It draws on the golden-ratio identity and J-cost symmetry proved in the same module. The result sits inside the economics module and supports broader claims about recognition equilibrium under zero-sigma conditions. No open questions or scaffolding are flagged in the supplied material.
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