Factorization

Definition

Factorization is the process of decomposing ring elements under the action of the dihedral group D_{2^n}. The DihedralFactorizationResolver implements this process, producing a Partition that classifies every element as irreducible, reducible, a unit, or exterior.

The Resolver

The DihedralFactorizationResolver is a subclass of Resolver. It accepts a type declaration (inputType) and produces a partition (outputType):

Partition Output

The output is a four-component partition of R_n:

The property density records the irreducible density: |Irr| / |Carrier|.

Free Rank Integration

With iterative resolution, factorization now tracks its progress through the FreeRank. Each dihedral orbit analysis pins sites, and the resolver's resolutionState records convergence.

Example: Factorizing R_8

For R_8 (the byte ring, 256 elements), the factorization resolver identifies dihedral orbits under the action of D_256:

<https://uor.foundation/instance/resolver-factor-R8>
    a                       resolver:DihedralFactorizationResolver ;
    resolver:inputType      <https://uor.foundation/instance/type-u8> ;
    resolver:outputType     <https://uor.foundation/instance/partition-R8> ;
    resolver:strategy       "dihedral orbit analysis" ;
    resolver:hasComplexityClass  resolver:LinearTime .

Computation Trace

Every factorization produces a ComputationTrace recording the operations applied. The trace contains step-by-step records that can be independently verified by a Certificate.

Role in the Pipeline

Factorization is the first stage of the three-resolver pipeline:

1. DihedralFactorizationResolver -- decomposes into orbits
2. CanonicalFormResolver -- computes canonical forms
3. EvaluationResolver -- evaluates results

See Partition for the four-component decomposition that factorization produces, and Composition for how the critical composition law neg ∘ bnot = succ drives the factorization algebra.