PolymerRegime
PolymerRegime enumerates the five canonical scaling regimes for polymer chains under Recognition Science units. Materials physicists deriving persistence lengths and end-to-end distances from the phi-ladder would cite this enumeration when certifying chain statistics. The declaration is a direct inductive type with five constructors that derives Fintype, DecidableEq, and related classes for immediate cardinality use.
claimDefine the inductive type of polymer regimes consisting of the cases rigid rod, worm-like chain, ideal chain, excluded volume, and collapsed.
background
In the Recognition Science treatment of polymers, chains exhibit persistence length Lp proportional to phi to a power in native units, while end-to-end distance scales as R proportional to N to the power nu with nu equal to 1 over phi to the one-third, approximately 0.603 and close to the Flory value 0.588. The module states that five regimes correspond to configDim D equal to 5. The inductive definition supplies the finite set of regimes needed for downstream cardinality and ratio statements.
proof idea
Inductive definition with five constructors, deriving DecidableEq, Repr, BEq, and Fintype in one declaration step.
why it matters in Recognition Science
This supplies the enumeration required by PolymerChainCert to assert Fintype.card PolymerRegime equals 5 together with the phi-ratio property on persistence lengths. It realizes configDim D equals 5 at the materials tier, connecting polymer scaling to the phi-ladder and the forcing chain landmarks. The module records zero-sorry status for the entire construction.
scope and limits
- Does not assign explicit scaling exponents to each regime.
- Does not derive the persistence length formula from the phi-ladder.
- Does not incorporate temperature or bending energy scales.
- Does not model concrete molecular chemistry or solvent quality.
formal statement (Lean)
21inductive PolymerRegime where
22 | rigidRod | wormLikeChain | idealChain | excludedVolume | collapsed
23 deriving DecidableEq, Repr, BEq, Fintype
24