consensus_equilibrium
plain-language theorem explainer
Recognition cost vanishes at consensus: the J-cost function satisfies J(1) = 0. Researchers modeling opinion networks under sigma conservation would cite this to anchor the low-polarisation fixed point. The proof is a direct term application of the unit-cost lemma from the Cost module.
Claim. $J(1) = 0$, where the recognition cost is $J(x) = (x-1)^2/(2x)$.
background
The Sociology.PolarisationFromSigmaCascade module develops a recognition-science account of social-media polarisation driven by sigma conservation across opinion nodes. Two equilibria are identified: consensus with vanishing cost J ≈ 0 and high polarisation at J(φ) ∈ (0.11, 0.13). The J-cost function, defined as the squared ratio J(x) = (x-1)^2 / (2x), quantifies divergence from unity; it is symmetric under inversion and vanishes only at x = 1. Upstream results include the base lemma Jcost_unit0 that establishes the zero at unity by direct simplification of the cost definition.
proof idea
The declaration is a one-line term proof that applies the Jcost_unit0 lemma from the Cost module.
why it matters
It populates the consensus field of the PolarisationCert definition, which bundles the three structural properties required to certify the sigma-cascade equilibria. The module documentation states that J = 0 corresponds to consensus while J(φ) marks the polarisation threshold crossed by algorithmic curation. Within the Recognition Science chain this supplies the J-uniqueness fixed point at unity, consistent with the T5 landmark.
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