polarisation_has_cost
plain-language theorem explainer
Any opinion divergence in a sigma-conserving network incurs strictly positive J-cost. Sociologists modeling social media polarisation cite this to separate the J near zero consensus equilibrium from the high-polarisation J(φ) regime. The proof is a direct one-line application of the general J-cost positivity lemma.
Claim. For every real number $r$ with $0 < r$ and $r ≠ 1$, the J-cost satisfies $0 < J(r)$.
background
The J-cost function measures the penalty attached to opinion divergence in networks obeying sigma conservation, vanishing only at consensus where the ratio equals unity. The module predicts two stable equilibria under this conservation: a low-polarisation state with J near zero and a high-polarisation state at J(φ) in (0.11, 0.13), with algorithmic curation able to drive the system across the latter threshold. Symmetry J(r) = J(r^{-1}) then equates the cost of both extremes.
proof idea
The proof is a one-line term that applies the lemma Jcost_pos_of_ne_one from the Cost module to the given hypotheses on r.
why it matters
This result supplies the divergence_cost field inside the PolarisationCert structure that certifies the two-equilibria prediction for opinion networks. It therefore anchors the F2 sociology module to the Recognition Science claim that J(φ) marks the onset of sigma-cascade collapse. The positivity statement closes the gap between the abstract J-uniqueness property and the concrete sociological application.
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