FailureMode
The inductive definition enumerates five canonical institutional failure modes, each corresponding to one of the five democratic institutions whose recognition ratio crosses the J(φ) threshold. Governance and institutional-design researchers cite it when mapping J-cost violations to collapse patterns in democratic systems. It is introduced directly as an inductive type with five constructors and automatic derivations for decidability, equality, and finiteness.
claimThe five canonical failure modes are authoritarianism (executive capture, $r_{exec} ≪ 1$), oligarchy (legislative gridlock, $r_{legis} → ∞$), rule-of-men (judicial politicization, $r_{judic} ≠ 1$), coup risk (military overreach, $r_{milit} > 1$), and information collapse (press censorship, $r_{press} ≪ 1$), where each mode is the transition $J(r_i) > J(φ)$ for the recognition ratio $r_i$ of the corresponding institution.
background
The module models five democratic institutions (executive, legislative, judicial, military, free press) as a configuration space of dimension 5. Each institution carries a recognition ratio $r_i$ (actual competence over mandated competence); healthy states satisfy $J(r_i) ≈ 0$. The module states that each failure mode is exactly the transition from $J(r_i) ≤ J(φ)$ to $J(r_i) > J(φ)$, the same threshold used for disease, ecosystem collapse, and phase transitions. Upstream inductive FailureMode definitions appear in StructuralSafetyFromJCost (yielding, buckling, fatigue, fracture, creep), FatigueFractureMechanicsFromJCost (ductileFracture, brittleFracture, fatigue, creep, stressCorrosion), and OptionAFalsifierRegistry, each deriving DecidableEq, Repr, BEq, Fintype.
proof idea
The declaration is a plain inductive definition that enumerates the five constructors corresponding to the five institutions. No lemmas or tactics are invoked; the type is introduced directly and equipped with the standard deriving clauses for decidability and finiteness.
why it matters in Recognition Science
The definition supplies the failure-mode type required by StructuralSafetyCert (which asserts Fintype.card FailureMode = 5 together with a CanonicalCert) and by ProtocolFalsifiable (which pairs each combination with a dataset, observable, and failure mode). It completes the E7 governance step by embedding the universal J(φ) threshold into institutional analysis, consistent with the five-dimensional configuration space and the forcing-chain landmarks T5–T8. Downstream results use it to define failureMode assignments and bijections in the OptionAFalsifierRegistry.
scope and limits
- Does not assign quantitative J-cost values or explicit r_i intervals to each mode.
- Does not derive the mapping from specific recognition-ratio trajectories to modes.
- Does not include empirical datasets or falsification criteria for the modes.
- Does not extend to non-democratic or hybrid institutional forms.
- Does not prove stability or transition probabilities between modes.
formal statement (Lean)
40inductive FailureMode where
41 | authoritarianism | oligarchy | ruleOfMen | coupRisk | informationCollapse
42 deriving DecidableEq, Repr, BEq, Fintype
43