FailureMode
plain-language theorem explainer
The inductive definition enumerates five canonical failure modes for democratic institutions when their recognition ratios drive the J-cost above the threshold J(φ). Institutional theorists modeling democratic stability would reference this enumeration when classifying decay patterns as recognition phase transitions. The definition proceeds by direct constructor listing with derived instances for decidability and finiteness, consistent with parallel enumerations in engineering safety modules.
Claim. The inductive type of institutional failure modes consists of the constructors authoritarianism, oligarchy, rule of men, coup risk, and information collapse, each corresponding to a specific deviation in an institution's recognition ratio $r_i$ exceeding the canonical threshold $J(φ)$.
background
Recognition Science models governance through five institutions (executive, legislative, judicial, military, free press) each with a recognition ratio $r_i$ = actual competence over mandate competence. Healthy operation requires $J(r_i) ≈ 0$ for all $i$, where $J$ is the cost function $J(x) = (x + x^{-1})/2 - 1$. Failure occurs when $J(r_i) > J(φ)$, with φ the self-similar fixed point from the forcing chain. This module defines the five failure modes as the canonical patterns for each institution: authoritarianism for executive capture, oligarchy for legislative gridlock, rule of men for judicial politicisation, coup risk for military overreach, and information collapse for press censorship. The definition mirrors the structure of FailureMode in the structural safety module, which enumerates yielding, buckling, fatigue, fracture, and creep. The local setting is the E7 extension of the J-cost framework to governance, with the democratic maintenance condition requiring all five ratios to remain within the canonical band.
proof idea
The declaration is an inductive definition that introduces the five constructors directly. It derives DecidableEq, Repr, BEq, and Fintype automatically, enabling cardinality computations and equality checks without additional lemmas.
why it matters
This definition provides the failure modes used in ProtocolFalsifiable and ProtocolSpec within the empirical protocol module, allowing each combination to be assigned a specific falsifier. It fills the E7 governance step in the Recognition Science chain, extending the J-cost threshold from physical systems to institutional ones and linking to the five-dimensional config space. It supports the GovernanceCert construction and enables downstream counting theorems such as failureModeCount in related modules.
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