polarisationCert
plain-language theorem explainer
The polarisation certificate definition assembles three component results into a record that certifies zero-cost consensus together with positive symmetric costs for any opinion divergence in sigma-conserving networks. Computational sociologists modeling social media dynamics would cite the certificate when confirming the two predicted equilibria under Recognition Science. The construction is a direct record instantiation that assigns the equilibrium theorem to the consensus field, the positivity theorem to the divergence cost field, and theinv
Claim. The polarisation certificate is the structure whose fields assert $J(1)=0$, that $J(r)>0$ for every $r>0$ with $r≠1$, and that $J(r)=J(r^{-1})$ for every $r>0$, where $J$ is the recognition cost function on positive reals.
background
Recognition Science defines the J-cost function on positive reals by $J(r)=(r+r^{-1})/2-1$, which vanishes at unity and is invariant under inversion. The sociology module on sigma-cascade polarisation predicts that any opinion network conserving sigma across nodes admits exactly two stable equilibria: consensus at $J≈0$ and maximal divergence at $J(φ)$ with $φ$ the golden-ratio fixed point. The PolarisationCert structure packages these properties as a record with three fields: consensus requiring $J(1)=0$, divergence cost requiring strict positivity away from one, and symmetry under inversion. Upstream results supply the concrete theorems: consensus equilibrium states $J(1)=0$, polarisation has cost states $0<J(r)$ for $r≠1$, and polarisation symmetric states $J(r)=J(r^{-1})$.
proof idea
The definition is a direct record construction that populates the consensus field with the equilibrium theorem, the divergence cost field with the positivity theorem, and the symmetry field with the inversion invariance theorem.
why it matters
This definition supplies the concrete certificate required by the sociology module to formalize the RS prediction of two stable equilibria in opinion networks. It completes the structural content for low polarisation at consensus and high polarisation at $J(φ)∈(0.11,0.13)$. No downstream uses are recorded, but the certificate supports further development of sigma-cascade models. It touches the framework landmarks of J-uniqueness and the phi fixed point from the forcing chain.
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