A registered result in its source context

QGEO.INVOL.01

Original ledger classification: Exact [E] + typed relocation [C] (exact sympy; demands declared; continuum derivation open)

Original claim text

The sheet-involution existence question of GATE.QGEO.01, narrowed to a two-demand selection with UNCONDITIONAL existence (one module, 7 checks, <1 s; discovery probe involution_existence_probe.py 7/7, verdict EXISTENCE-UNCONDITIONAL): RIGIDITY -- the integral V-commuting involutions are exactly the spectral sign group (Z/2)^3: all eight sign choices on V's eigenspaces {0,1,2} are integer matrices, genuine involutions, commute with V and stabilize the integral order Z<U,V>, on BOTH twins (nontrivial: generic integral operators do not have integral spectral projectors). THE NARROWING CHAIN: (i) the v574 rank theorem (two-dim-positive involutions cannot carry the binary trace code) kills the trace +1 half, 6 -> 3, all remaining of exact Sigma-signature (1,-1,-1); (ii) the seam-reversal demand S e3 = -e3 narrows 3 -> 2; (iii) the zero-mode-fixing demand (+1 on ker V, flipping exactly the dynamical sheet pair {1,2}) singles out S0 = [[1,-1,0],[0,-1,0],[0,0,-1]] uniquely -- explicit, integral, order-stabilizing: EXISTENCE IS UNCONDITIONAL. HONEST SCOPE: the corpus RP reflection Sigma = diag(1,-1,-1) (an anti-automorphism) does NOT commute with V -- a different structure with a different role; the alt twin admits the same classification (the involution does not separate the twins, consistent with v567/v572). RELOCATION: GATE.QGEO.01's involution-existence input is DERIVED; the residual freedom is the finite two-demand selection, machine-checked; deriving the two demands from the continuum geometry (mu4/D4/H1) is the remaining open [C] of the gate -- named, not claimed. [2026-08-01 V: the grading shortcut is falsified -- NO V-commuting involution implements the corpus mu4 grading S Q S = Sigma Q Sigma (all six fail); Sigma and S0 are genuinely different structures whose product is the infinite-order shear [[1,-1,0],[0,1,0],[0,0,1]]: the two v590 demands stay independent; the continuum derivation remains open.] [2026-08-01 X: a live geometric candidate -- v597: the mu3-cover y^3 = x^4-1 hits four compiler fingerprints (rank-3 Eisenstein, projective D4, (Z/3)^4 = 81 cusp classes, Lorentz (1,2)); see QGEO.COVER.01.] [2026-08-01 XIX: the seam-reversal demand is now GEOMETRIC -- v603: on the mu3-cover the seam line is real (R-fixed) and the two-demand involution reverses it exactly; the zero-mode demand is built into the curve-side classification (v601); the continuum derivation of the demands has its first geometric anchor; see QGEO.SEAMMARKS.01.] [2026-08-01 XXV: BOTH demands now have canonical cover carriers -- v603: the seam-reversal demand is 'reverse the R-real line'; v607: the zero-mode demand is 'fix the only unlabeled mode' (seam = R-real, top = deck-eigen, zero = unlabeled); typed [C] readings, the continuum derivation of the demands from the raw seam stays the gate residual; see QGEO.SELRULE.01.]

Original classification detail

EXISTENCE-UNCONDITIONAL: rigidity (Z/2)^3 on both twins; narrowing 6 -> 3 -> 2 -> 1 via v574 type theorem + seam reversal + zero-mode fixing; S0 explicit with Sigma-signature; the two demands' continuum derivation stays open; 7/7 checks

Dependencies as recorded

QGEO.WINDING.SELF.01; QGEO.QRECON.01; DIAMOND.PARA.SELFCODE.01; v566; v567; v574