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A registered result in its source context

Original ledger classification: Identity/Lattice [I]/[L] (the finite rigidity theorem exact)

Recorded statement: THE QGEO RIGIDITY THEOREM (the FINITE half of the Tier-1 hinge GATE.QGEO, hardened to [E]; v167 Tier 1): a genus-0 boundary with a faithful D4 action, one orbit of four parabolic marks, H^1 of rank 3 and the nontrivial mu4-character…

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QGEO.RIGID.01

Original ledger classification: Identity/Lattice [I]/[L] (the finite rigidity theorem exact)

Original claim text

THE QGEO RIGIDITY THEOREM (the FINITE half of the Tier-1 hinge GATE.QGEO, hardened to [E]; v167 Tier 1): a genus-0 boundary with a faithful D4 action, one orbit of four parabolic marks, H^1 of rank 3 and the nontrivial mu4-character decomposition chi_1+chi_2+chi_3 is Moebius-equivalent to P^1 minus mu4 with the (1,2,3) grading. EXACT CHECKS: (1) mu4={1,i,-1,-i} has cross-ratio 2 and Moebius stabiliser <z->iz, z->1/z> of order 8 = |D4| (a 4-cycle of the marks + a reflection), faithful -> four marks with faithful D4 are the mu4 square up to Moebius (sharp 3-transitivity fixes 3 marks, D4 fixes the 4th); (2) P^1 minus four points has b1 = 4-1 = 3 = N_fam (residue-sum-zero); (3) omega_k = z^{k-1}dz/(z^4-1), k=1,2,3, are mu4-eigenforms with z->iz eigenvalue i^k = the three nontrivial characters chi_1,chi_2,chi_3, weights (1,2,3) = A3 exponents = Spec(Q_+) (FLAV.QGEO.02, Q_+ spec {1,2,3}). The cohomological + rigidity content of GATE.QGEO is thus closed exactly; NOT a new structural class -- it hardens the existing Tier-1 hinge

Original classification detail

Identity/Lattice [I]/[L]: QGEO finite rigidity closed (cross-ratio 2, D4 order 8, b1=3, chi_k=A3 exps); the math half of GATE.QGEO

Dependencies as recorded

GATE.QGEO.01; FLAV.QGEO.02; FLAV.QGEO.01; v137_qplus_cohomology.py

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Registered verification modules

  • v168_qgeo_rigidity.pyRegistered verification module · registry
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    the QGEO Rigidity Theorem — the FINITE half of the Tier-1 hinge GATE.QGEO, hardened. EXACT: μ4={1,i,−1,−i} has cross-ratio 2 and Möbius stabiliser ⟨z→iz, z→1/z⟩ of order 8=|D4| (faithful: 4-cycle + reflection) ⇒ four marks with faithful D4 are the μ4 square up to Möbius; H¹(ℙ¹∖μ4) rank 4−1=3=N_fam; the forms ω_k=z^{k−1}dz/(z⁴−1) (k=1,2,3) are μ4-eigenforms with z→iz eigenvalue i^k = the three nontrivial characters χ1,χ2,χ3, weights (1,2,3) = A3 exponents = Spec(Q+). QGEO.RIGID.01 [E] (the math half closed); QGEO.REALIZE.01 [C] (seam-collar = ℙ¹∖μ4 stays the realisation premise). Not a new structural class — it hardens the existing hinge.

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