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Admissible Perturbations for URF-SG

This file defines the class of perturbations V allowed in the Spectral Gap Rigidity Wall.

All rigidity results depend only on this definition.

Structural Constraints

An admissible V must satisfy:

  1. Symmetry:

    • Preserves the automorphism group of the base structure.
    • Commutes with all global symmetries of A.

  2. Locality:

    • Acts only on bounded-radius neighborhoods.
    • No long-range couplings.

  3. Uniformity:

    • Same action on all isomorphic local patches.
    • No position-dependent bias.

  4. Boundedness:

    • A-bounded in operator norm.

Forbidden Perturbations

The following are not admissible:

  • Diagonal defects.
  • Local weight injections.
  • Any perturbation that singles out a vertex or cycle.
  • Any perturbation that breaks translational invariance.

Research Problem

Given this class of V, determine whether:

Pi V Pi = 0 holds for all admissible V.

This is the entire URF-SG wall.