A registered result in its source context

QGEO.EQORDER.01

Original ledger classification: Exact [E] anti-isometry/order/localization identities + [C] typed localization reading + [X] quantified translator negatives (exact sympy; Smith normal forms over Z)

Original claim text

The equivariant order and the infinity bridge (one module, 12 checks, ~4 s; discovery probe equivariant_order_probe.py 12/12, verdict EQUIVARIANT-ORDER-LANDED). CORRECTION + THE INTEGRAL FORM [E]: R IS an anti-isometry of the polarization -- the correct compatibility equation M^dagger J M = J^T holds EXACTLY (a v599-diary side remark had tested the wrong equation M conj(J) M^dagger vs J); consequently J restricts to a REAL SYMMETRIC INTEGER form on the saturated fixed lattice: J_fix = [[16,2,4],[2,-2,2],[4,2,-2]], det 72 = 8 x 9 (the polarization determinant times the (1-omega)-covolume factor). THE EQUIVARIANT ORDER IS FULL [E]: the Z-lattice generated by the seven words of ALL FOUR mark frames (the r^k-conjugates of the v601/v602 equivariant pair; all 28 words Z[omega]-integral) has FULL rank 18 in M_3(Z[omega]) with Smith divisors (1^15, 3, 3, 3) -- saturation index 27 = 3^3: one frame gives the 7-dim parabolic of index 81 = 3^4 (v600), all four give the full matrix algebra of index 27. THE 3-TORSION LOCALIZES AT INFINITY [E]: on the unreduced integral module Z[omega]^4, coker(1 - T_inf) = Z^2 + (Z/3)^3 (Smith divisors (1,1,1,3,3,3,0,0), torsion 27) while each FINITE cusp twist is torsion-free (divisors (1,1,0^6)) -- three objects share (Z/3)^3 = 27: the infinity-monodromy cokernel, the equivariant-order cokernel, and det_Z(1-deck) (v597). THE READING [C]: the two compiler constants localize on the cover -- 81 = 3^4 at the FINITE marks (cusp divisor classes, v597; per-frame word order, v600) and 27 = 3^3 at INFINITY (monodromy defect; full equivariant order): the integral infinity <-> order-torsion bridge. TRANSLATOR NEGATIVES EXTENDED [X, quantified]: the corpus RP metric pulled back to the integral fixed lattice is NOT proportional to J_fix or its Klein twists (6 tested), and neither J_fix^-1 F2 nor F2 J_fix^-1 lies in the word algebra -- after the (B, Gamma, Klein, J_fix) sweep the corpus transpose structure resists all single-sheet constructions; the conjugate character sheet (omega-bar) is the named next candidate. GATE.QGEO does not move; no marker changes. [2026-08-01 XXII: the translator is FOUND -- v605: the corpus RP anti-automorphism is the J-adjoint twisted by an explicit inner element c (14c integral; the omega-bar detour was unnecessary); the seam is 3-torsion at infinity; the 81 survives frame-joining; see QGEO.TRANSLATOR.01.] [2026-08-02 XXXIX: the fixed-lattice form J_fix gains an arithmetic identity -- v624: P^T J_det P = J_fix EXACTLY with det P = -6 (J_det the prime-front determinant form): the two Lorentz lattices are the same rational quadratic form, the cover an index-6 sublattice (72/2 = 6^2, 6 = p_2(a)); P-canonicity and the Hodge-chamber question preregistered; see EXTREV.LATTICE.01.]

Original classification detail

EQUIVARIANT-ORDER-LANDED: M^dag J M = J^T exact (R anti-isometry); J_fix integer, det 72; four-frame order FULL rank 18, divisors (1^15,3,3,3), index 27 = 3^3; coker(1-T_inf) = Z^2+(Z/3)^3, finite cusps torsion-free; 81 at marks / 27 at infinity; translator negatives extended (omega-bar named); 12/12 checks

Dependencies as recorded

QGEO.SEAMMARKS.01; QGEO.DUALFORMS.01; QGEO.EMBED.01; QGEO.COVER.01; v597; v600; v601; v602; v603