Deformation Analysis

The deformation analysis guide explains how the ModuliResolver computes the local geometry of the moduli space at a given CompleteType.

ModuliResolver

The ModuliResolver takes a CompleteType as input via moduliTarget and produces a DeformationComplex via moduliDeformation.

The resolver:

1. Constructs the deformation complex of T
2. Computes tangent and obstruction dimensions
3. Determines the holonomy stratum containing T
4. Records the results in a StratificationRecord

Stratification Record

The StratificationRecord captures the holonomy stratification of M_n at a given Witt level:

• stratificationLevel — the Witt level
• stratificationStratum — links to each HolonomyStratum in the decomposition

Worked Example

Consider a CompleteType T in M_2 (Witt level Q2, R_4 = Z/16Z) with basis size 4 and trivial holonomy (FlatType):

Step 1 — Deformation complex. Construct Def(T):

• H^0 = Aut(T) = {id} (T has no non-trivial automorphisms in D_4)
• H^1 has dimension 3 (three independent first-order deformation directions)
• H^2 = 0 (no obstructions)

The tangentDimension is 3 and the obstructionDimension is 0.

Step 2 — Holonomy stratum. Since T is a FlatType, it lies in the flat stratum with codimension 0 (MD_5). The stratumCodimension is 0.

Step 3 — Versal deformation. By MD_7, T admits a versal deformation of dimension 3 (matching H^1). Since H^2 = 0, any deformation family through T preserves completeness (MD_8).

Step 4 — Tower map site. The site of M_2 -> M_3 over T has dimension 1 (MD_9) since the obstruction to lifting vanishes.

Complexity

The ModuliResolver runs in O(n x basisSize^2) time (MR_3), dominated by the deformation complex computation. Identity MR_1 ensures that the resolver's output agrees with the MorphospaceBoundary when restricted to the achievability boundary.