PolarisationCert
plain-language theorem explainer
PolarisationCert bundles the three minimal properties of the J-cost function required to certify equilibria in opinion networks under sigma conservation. Researchers modeling social media polarisation via Recognition Science would cite it to separate consensus (J=0) from the J(phi) threshold band. The declaration is a bare structure definition that packages the properties without further derivation.
Claim. A structure certifying that the J-cost function satisfies $J(1)=0$, $J(r)>0$ for all $r>0$ with $r≠1$, and $J(r)=J(r^{-1})$ for all $r>0$, where $J$ is the cost measuring opinion divergence.
background
The module Sociology.PolarisationFromSigmaCascade studies opinion networks that conserve a sigma quantity across nodes. The module documentation states that such networks admit two stable equilibria: low polarisation with J≈0 (consensus) and high polarisation with J≈J(φ) (maximal stable divergence), where algorithmic curation drives the system across the J(φ) band in (0.11,0.13). Jcost is the J-cost function imported from IndisputableMonolith.Cost; it is the unique function obeying the Recognition Composition Law with J(1)=0 and strict positivity away from unity. PolarisationCert collects exactly the three properties needed to certify either equilibrium.
proof idea
PolarisationCert is a structure definition. It directly encodes the three fields consensus, divergence_cost, and symmetric with no lemmas, tactics, or reduction steps.
why it matters
The structure supplies the type for the downstream polarisationCert definition in the same module, which constructs an instance from consensus_equilibrium, polarisation_has_cost, and polarisation_symmetric. It operationalises the Recognition Science prediction that J(φ) marks the polarisation threshold and that J is symmetric under inversion, linking directly to the J-uniqueness result (T5) and the self-similar fixed point (T6) in the forcing chain. No open scaffolding is closed here.
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