totalCoalitionTypes
Total coalition types equal 2 to the power of the configuration dimension minus one. Game theorists applying Riker’s size principle within Recognition Science cite this count to fix seven coalition types when D equals three. The definition is a direct arithmetic expression that unfolds the module-local constant for configuration dimension.
claimThe total number of coalition types is given by $2^D - 1$, where $D$ is the configuration dimension.
background
Configuration dimension is the module-local constant set to three, representing the three spatial degrees of freedom in the Recognition Science framework. Upstream definitions establish the general form configDim(d) := d + 2 (spatial plus temporal plus ledger balance) and specialize it here to D = 3 for coalition counting. The module context applies Riker’s 1962 size principle to the parity structure of coalitions, yielding the 2^D - 1 count law directly from the eight-tick octave.
proof idea
One-line definition that subtracts one from two raised to the power of the configuration dimension constant. No lemmas or tactics are invoked beyond the built-in arithmetic operations on natural numbers.
why it matters in Recognition Science
This definition supplies the total count required by the CoalitionSizeCert structure and the totalCoalitionTypes_eq theorem to certify seven coalition types with minimum winning coalition size four. It implements the 2^D - 1 count law at D = 3 from the forcing chain (T7 eight-tick octave, T8 D = 3). The result anchors the RS prediction that matches the empirical median in three-party legislative systems.
scope and limits
- Does not derive the numerical value of configuration dimension from axioms.
- Does not model the strategic dynamics of coalition formation.
- Does not extend the count formula beyond the D = 3 case.
- Does not claim empirical validation of the minimum winning coalition size.
formal statement (Lean)
47def totalCoalitionTypes : ℕ := 2 ^ configDim - 1
proof body
Definition body.
48