regimes_exclusive
plain-language theorem explainer
The theorem shows that a discourse network cannot be both non-polarised and polarised under the same Cheeger constant h. Platform algorithm designers and sociologists studying echo chambers cite it to enforce clean separation between cross-exposure and bisection regimes. The proof assumes both predicates, derives h < h via order transitivity, and contradicts the irreflexivity of strict inequality.
Claim. Let τ denote the Cheeger threshold. For any real number h, it is impossible that both τ ≤ h and h < τ hold simultaneously.
background
The Sociology.PolarisationCheegerBound module models social-media discourse as a graph whose Cheeger constant h determines polarisation. IsNonPolarised h holds when τ ≤ h and IsPolarised h holds when h < τ, with τ marking the transition to sub-φ-rational spectral gap and high-conductance bisection. This exclusivity result is one of several parallel theorems across modules that all invoke the same irreflexivity fact from ArithmeticFromLogic.
proof idea
The term-mode proof uses rintro to unpack the conjunction into h ≥ τ and h < τ, applies lt_of_lt_of_le to obtain a strict inequality h < h, and finishes with lt_irrefl.
why it matters
The result supplies the regimes_exclusive field required by PolarisationCheegerCert and its counterparts in LorenzCurveCert and FatigueThresholdCert. It completes the canonical-band separation pattern that runs through the Recognition Science forcing chain and the RCL, ensuring no overlap between the φ-ladder regimes.
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