Monodromy
Definition
Monodromy in the UOR Framework describes how constraint types transform under parallel transport around closed loops in the constraint nerve. For a ConstrainedType over the ring R_n, the monodromy group is a subgroup of the dihedral group D_{2^n}.
A ClosedConstraintPath represents a closed loop in the constraint nerve returning to the same site assignment. The pathLength counts the number of constraint steps and pathConstraints lists the traversed constraints.
Holonomy Group
The HolonomyGroup accumulates DihedralElement generators from all closed paths. Its order is given by holonomyGroupOrder.
A MonodromyClass classifies the type according to its holonomy subgroup structure.
Each Monodromy individual records:
- monodromyLoop — the closed path
- monodromyElement — the dihedral element produced by parallel transport
- isTrivialMonodromy — whether the element is the identity
Flat and Twisted Types
Types are classified by their holonomy:
- FlatType — trivial holonomy (identity subgroup); disjoint with TwistedType.
- TwistedType — non-trivial holonomy (proper subgroup of D_{2^n}); disjoint with FlatType.
The MonodromyResolver computes the HolonomyGroup by enumerating closed paths via monodromyTarget and returns the result via holonomyResult.
Identity Algebra (MN_ series)
| Identity | Statement |
|---|---|
| MN_1 | HolonomyGroup(T) ≤ D_{2^n} for any ConstrainedType over R_n |
| MN_2 | Purely additive constraints give trivial holonomy (FlatType) |
| MN_3 | neg + bnot in closed path gives full dihedral holonomy |
| MN_4 | Non-trivial HolonomyGroup ⟹ β₁ ≥ 1 |
| MN_5 | CompleteType ⟹ β₁=0 ⟹ trivial holonomy ⟹ FlatType |
| MN_6 | monodromy(p₁ · p₂) = monodromy(p₁) · monodromy(p₂) in D_{2^n} |
| MN_7 | TwistedType ⟹ H²(N(C(T')); ℤ/2ℤ) ≠ 0 for any QuantumLift |