Original ledger status
GATE.QGEOM.02
GATE.QGEOM.02
Integer-lift of Q - residual sharpened to ONE named lattice invariant det Q=N_fam (honest reduction, not closure) [I]/[L]. Continuing v69: (1) DERIVED - in the mu4-puncture homology basis {g1,g2,g3} (g4=-(g1+g2+g3)) the Z4=mu4 rotation is the integer matrix R=[[0,0,-1],[1,0,-1],[0,1,-1]], char (t+1)(t^2+1), eigenvalues {-1,i,-i}, R^4=I, R UNIMODULAR (det=-1, SNF=I). (2) the documented transport Q (rows (4,5,6)=(|mu4|,g_car,|R+A3|), cols (9,5,1)=(N_fam^2,g_car,1)) has det Q=3=N_fam, SNF=diag(1,1,3) - the transport DETERMINANT equals the family count. (3) so Q is NOT a unimodular change of basis of R (det -1 vs det 3): the integer lift is precisely the parabolic-degree-0 embedding carrying the extra det=N_fam (family multiplicity) that D4 alone (unimodular) does not supply. RESIDUAL of gate Q-geometry is thus one named invariant det Q=N_fam; spectra+Sigma-split already derived (v69). EHRLICH: D4 alone does NOT close the integer lift - it needs this one lattice datum
Recorded status: Identity/Lie [I]/[L]
Dependencies, status detail and source
Canonical status: Identity/Lie [I]/[L]
Dependencies: GATE.QGEOM.01; v69; v50
Scripts: v70_q_integer_lift.py
Location: tfpt_2
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