polarizationCert
plain-language theorem explainer
The definition constructs a concrete instance of the PolarizationCert structure by assigning the driver cardinality field directly to the established count of five. Sociologists formalizing recognition-cost models of opinion divergence would cite it to anchor the five-driver account within the J-cost framework. The construction is a direct field assignment that relies on the upstream decision proof of driver cardinality together with a supplied threshold certificate.
Claim. Let PolarizationCert be the structure whose fields assert that the cardinality of the set of polarization drivers equals five and that a canonical recognition threshold holds. The definition supplies an element of this structure by setting the driver count to the proven value five and the threshold to the given certificate.
background
In the Recognition Science sociology module, polarization arises when opinion ratios r depart from 1 such that the J-cost J(r) exceeds J(φ), where φ is the self-similar fixed point. Group cohesion fails once this threshold is crossed, producing in-group/out-group dynamics. The module enumerates five canonical drivers (economic inequality, identity threat, media fragmentation, political sorting, institutional mistrust) and packages them with a recognition threshold into the PolarizationCert structure. The upstream theorem polarizationDriverCount establishes the cardinality by decision, while the structure PolarizationCert itself declares the two fields five_drivers and threshold.
proof idea
The definition is a direct construction that populates the five_drivers field with the result of polarizationDriverCount and the threshold field with the supplied cert. No tactics or reductions are applied beyond the structure literal syntax.
why it matters
This definition supplies the formal certificate for the five-driver polarization model inside the Recognition Science sociology tier. It instantiates the configDim D = 5 for this domain and links the J-cost crossing to the observed growth of affective distance (approximating φ²). Although no downstream theorems currently reference it, the declaration closes the interface between the driver enumeration and the CanonicalCert threshold, preparing the ground for quantitative predictions of polarization growth from recognition deficit.
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