WaveFunctionCollapseCert
This structure bundles three properties that certify wave function collapse as minimization of the J-cost function in Recognition Science. A physicist modeling quantum measurement via recognition costs would cite it to record that superpositions carry positive cost while definite outcomes sit at zero cost. The definition is a plain structure that packages the five-basis count, the positivity condition on J-cost, and the equilibrium condition without any internal proof steps.
claimA structure certifying wave function collapse consists of three fields: the measurement basis set has cardinality 5, the J-cost satisfies $J(r) > 0$ for every real $r > 0$ with $r ≠ 1$, and $J(1) = 0$.
background
The module models quantum measurement as the system settling to the J-cost minimum, where the wave function is treated as a recognition cost distribution. The MeasurementBasis inductive type enumerates the five bases (position, momentum, spin, energy, angular momentum) and derives a Fintype instance so that its cardinality can be stated directly. The local setting asserts that superpositions (r ≠ 1) carry positive cost while a definite outcome (r = 1) carries zero cost, with the Born rule arising as probability weights proportional to those costs.
proof idea
This is a structure definition whose three fields directly encode the required properties. The cardinality field uses the Fintype instance on MeasurementBasis; the positivity and equilibrium fields reference the Jcost function imported from the Cost module. No lemmas are applied and no tactics are invoked inside the declaration itself.
why it matters in Recognition Science
The structure is instantiated by the downstream definition waveFunctionCollapseCert, which supplies the concrete values measurementBasisCount, superposition_has_cost, and measurement_outcome_equilibrium. It supplies the formal record for the RS claim that collapse corresponds to J-cost minimization and that five bases match configDim D = 5. It touches the open question of deriving explicit Born-rule probabilities from the J-cost distribution.
scope and limits
- Does not prove existence or uniqueness of the J-cost function.
- Does not derive explicit Born-rule probabilities from the cost weights.
- Does not connect the five bases to the spatial dimension D = 3 of the forcing chain.
- Does not address time evolution or unitary dynamics.
formal statement (Lean)
38structure WaveFunctionCollapseCert where
39 five_bases : Fintype.card MeasurementBasis = 5
40 superposition_cost : ∀ {r : ℝ}, 0 < r → r ≠ 1 → 0 < Jcost r
41 measurement_equilibrium : Jcost 1 = 0
42