SchroedingerCert
plain-language theorem explainer
SchroedingerCert packages three assertions for the quantum mechanical setup in Recognition Science: exactly five canonical systems, vanishing J-cost at the equilibrium point r=1, and strictly positive J-cost for superpositions. A researcher deriving the Schrödinger equation from the Recognition Composition Law would reference this structure to anchor the quantum cost interpretation. The definition consists of three fields directly populated by module-level lemmas without additional proof steps.
Claim. A Schroedinger certificate is a structure containing: the statement that the finite type cardinality of the set of quantum mechanical systems equals 5, the equality Jcost(1) = 0, and the universal quantification that for all real numbers r with 0 < r and r ≠ 1, Jcost(r) > 0.
background
The module Physics.SchroedingerEquationFromRS develops the Schrödinger equation iℏ ∂ψ/∂t = Ĥψ within Recognition Science by identifying the wave function ψ with a recognition amplitude whose squared modulus yields a J-cost via the imported Cost module. QMSystem enumerates the five standard quantum systems as an inductive type with constructors for the infinite square well, harmonic oscillator, hydrogen atom, free particle, and finite square well. The Jcost function satisfies the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y) and serves as the quantum cost measure, with stationary states corresponding to cost minima.
proof idea
The declaration is a bare structure definition. Its three fields are: five_systems asserting Fintype.card QMSystem = 5, stationary asserting Jcost 1 = 0, and superposition_cost asserting the positivity condition on Jcost for r away from 1. No tactics or lemmas are applied inside the structure itself; the fields are later instantiated by the sibling definition schroedingerCert.
why it matters
This structure supplies the certificate instantiated by schroedingerCert, which in turn supports the derivation of the Schrödinger equation from RS principles. It directly encodes the module's claim that five systems correspond to configDim D = 5 and that stationary states have J = 0 while superpositions have J > 0. The construction aligns with the T5 J-uniqueness and T6 phi fixed point in the forcing chain, providing a concrete interface between quantum mechanics and the Recognition Composition Law.
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