molecularQuantumStateClasses
The definition computes the total number of molecular quantum state classes as the product of the cardinalities of the sets of molecular energy levels and quantum gate types. Researchers in Recognition Science quantum molecular modeling cite this count when verifying that five bits suffice to address all preparable states. The implementation is a direct multiplication of Fintype.card applied to two inductive types each having five constructors.
claimLet $N$ be the number of molecular quantum state classes. Then $N = |E| × |G|$, where $E$ is the set of molecular energy levels and $G$ is the set of quantum gate types.
background
The C4 module examines quantum molecular design depth in the Recognition Science framework. MolecularEnergyLevel is an inductive type with five constructors (electronic, vibrational, rotational, translational, spin). QuantumGateType is an inductive type with five constructors (pauli, clifford, tGate, cnot, toffoli). The module documentation states that five levels times five gate types yields 25 prepared state classes, which is at most 2^5 and thus addressable by five bits. This supplies the concrete count for the Lean certification of the depth bound, while the stronger claim of reaching specific molecular targets remains outside the formalization.
proof idea
The definition is realized by multiplying Fintype.card MolecularEnergyLevel by Fintype.card QuantumGateType. Both inductive types derive their Fintype instances automatically from their five constructors, so the product evaluates to 25.
why it matters in Recognition Science
This definition provides the state class count referenced by the theorems molecularQuantumStateClasses_25, which proves equality to 25, and molecularQuantumStateClasses_le_2pow5, which shows the inequality with 2^5. It is a component of the QuantumMolecularDepthCert structure that certifies five-bit addressability. Within the Recognition Science framework it completes the formal part of the C4 claim on quantum molecular design depth, leaving empirical questions about specific molecular preparation open.
scope and limits
- Does not prove that any particular molecule reaches a target state in five layers.
- Does not derive the five energy levels or gate types from the Recognition Science forcing chain.
- Does not incorporate decoherence or noise models.
- Does not address the physical implementation of the gates.
Lean usage
theorem depth_cert : molecularQuantumStateClasses ≤ 2 ^ 5 := by rw [molecularQuantumStateClasses_le_2pow5]formal statement (Lean)
26def molecularQuantumStateClasses : ℕ :=
proof body
Definition body.
27 Fintype.card MolecularEnergyLevel * Fintype.card QuantumGateType
28