Target structure: 2D torus Z_n × Z_n with added chord edges.
Base operator: Graph Laplacian on the toroidal-chord graph.
Goal: Determine whether Π V Π = 0 holds for all URF-admissible V.
Baseline torus: G0 = Z_n × Z_n ⋊ D4
With chords: Global symmetry reduced to subgroup G ⊂ G0.
Exact G: (TBD — must be computed explicitly)
Laplacian spectrum: λ(k,l) = 4 - 2 cos(2πk/n) - 2 cos(2πl/n) (without chords)
Chords modify: (TBD)
Multiplicity structure: (TBD)
Decompose: E_k ⊕ E_{k+1} as a representation of G.
Question: Does the trivial representation occur in:
Hom_G(E_k ⊕ E_{k+1}, E_k ⊕ E_{k+1}) ?
YES → ΠVΠ ≠ 0 possible → wall collapses
NO → ΠVΠ = 0 forced → rigidity survives
Symmetry group: OPEN
Representation decomposition: OPEN
ΠVΠ verdict: OPEN