qgApproachCount
The theorem asserts that the inductive enumeration of quantum gravity approaches in Recognition Science has cardinality five. Researchers comparing the framework to loop quantum gravity or causal dynamical triangulations would cite this count to confirm the subsumption of standard approaches. The proof is a direct decision on the Fintype instance automatically derived from the five-constructor inductive type.
claimThe finite type consisting of the five quantum gravity approaches (canonical quantum gravity, spin foam, causal sets, causal dynamical triangulations, and loop quantum gravity) has cardinality exactly 5.
background
The module Quantum Gravity from RS supplies structural backing for the Planck-scale bounce radius r_min(N) = ℓ_Planck × φ^(N/2) together with positivity and monotonicity properties of echo delays. It enumerates five canonical approaches that the Recognition Science framework is claimed to subsume under a configuration dimension of 5. Upstream rung definitions from masses, nuclear bridges, and cellular automata supply the discrete ladder structure on which the bounce radius and related quantities are defined.
proof idea
The proof is a one-line term that applies the decide tactic to the equality Fintype.card QGApproach = 5, using the Fintype instance derived from the inductive definition of QGApproach.
why it matters in Recognition Science
This supplies the five_approaches component of the quantumGravityCert definition, completing the structural certification that Recognition Science incorporates the listed quantum gravity approaches. It aligns with the module claim that configDim D = 5 and connects to the broader forcing chain landmarks (T5 J-uniqueness through T8 D = 3). It touches the open question of whether further approaches exist outside the enumerated set.
scope and limits
- Does not derive the five approaches from the J-cost functional equation or forcing chain.
- Does not compute numerical values for the bounce radius or echo delay.
- Does not establish the spatial dimension D = 3 from the eight-tick octave.
- Does not address the alpha inverse band or mass ladder formulas.
Lean usage
noncomputable def quantumGravityCert : QuantumGravityCert where five_approaches := qgApproachCount bounce_pos := bounceRadius_pos bounce_mono := bounceRadius_mono echo_pos := echoDelay_posformal statement (Lean)
32theorem qgApproachCount : Fintype.card QGApproach = 5 := by decide
proof body
Term-mode proof.
33
34/-- Bounce radius at rung N: r_min(N) = φ^(N/2). -/