A registered result in its source context

GATE.QGEOM.01

Original ledger classification: Lie/lattice [L]

Original claim text

D4-equivariant derivation of the Q-geometry (Alessandro residual gate #2) - advances Theorem Q from [P] (algebraically forced) to [L] (geometrically derived). The 4 punctures mu4={1,i,-1,-i} are square vertices with symmetry D4=Z4 semidirect Z2 (the 90-deg rotation z->iz IS mu4; the reflection is the sheet parity Sigma). (1) the Z4 4-cycle on the punctures acts on H_1(P^1\\mu4)=Z^4/(sum=0)=Z^3 (3 families) with eigenvalues {i,-1,-i} (non-trivial 4th roots); (2) the D4 permutation rep on square vertices = A1+B1+E (1+1+2), so H_1(sum=0)=B1(+)E; (3) Q_+ (Sigma-even) = 3*(parabolic cusp weights {0,1/3,2/3})+1 = diag(1,2,3) => char (t-1)(t-2)(t-3), Spec{1,2,3} (matches v50); (4) Q_- (Sigma-odd) = E-block coupling sqrt(3)=sqrt(N_fam) with B1 in kernel => char t(t^2-3), Q_-^2 eigenvalues {0,3,3} (matches v50); (5) the Sigma-split (Q_+/Q_-) IS the D4=Z4(mu4)semidirect Z2(reflection=Sigma) structure. So the Q SPECTRA and the Sigma-split are DERIVED from the D4-equivariant geometry of P^1\\mu4 [L]; residual: the cusp weights {0,1/3,2/3} are the established monodromy exponents (v61), and the explicit integer matrix Q still needs the lattice/SNF datum

Original classification detail

Lie/lattice [L]

Dependencies as recorded

QGEOM (was GATE); v50; v61; Theorem Q