molecularQuantumStateClasses_25
The theorem establishes that the number of molecular quantum state classes equals 25 by multiplying the five energy levels by the five gate types. Researchers working on quantum molecular design in Recognition Science would cite this to confirm the state space size for addressing depth calculations. The proof is a direct unfolding and rewriting using the cardinality theorems for energy levels and gate types.
claimLet $s$ denote the number of molecular quantum state classes, defined by $s = |M| × |G|$ where $M$ is the set of molecular energy levels and $G$ is the set of quantum gate types. Then $s = 25$.
background
In the C4 module the state classes arise as combinations of molecular energy levels and quantum gate types. The definition computes this count as the product of the two finite-type cardinalities. Upstream, one theorem fixes the energy-level cardinality at five while the other fixes the gate-type cardinality at five, each by direct decision. The module states that this product yields 25 classes, which is at most 2^5 and therefore addressable by five bits as the Lean-verified portion of the design-depth claim.
proof idea
The proof is a term-mode reduction that first unfolds the definition of the state-class count to expose the product of cardinalities, then rewrites using the two upstream cardinality theorems to obtain the numerical value 25.
why it matters in Recognition Science
This supplies the exact count required by the quantum molecular depth certificate, which pairs it with the upper bound of 2^5 and the lower bound showing more than 16 classes. It completes the Lean-safe half of the C4 claim that five bits suffice to index all RS-preparable molecular states while the physical reachability of specific targets in five two-qubit layers remains an empirical claim outside the formalization.
scope and limits
- Does not prove reachability of any concrete molecular target in five layers.
- Does not incorporate continuous parameter spaces or specific gate-model dynamics.
- Does not connect the count to the full Recognition Science forcing chain or phi-ladder.
- Does not address whether the 25 classes exhaust all physically realizable states.
Lean usage
rw [molecularQuantumStateClasses_25]formal statement (Lean)
29theorem molecularQuantumStateClasses_25 :
30 molecularQuantumStateClasses = 25 := by
proof body
Term-mode proof.
31 unfold molecularQuantumStateClasses
32 rw [molecularEnergyCount, quantumGateTypeCount]
33