A registered result in its source context

QGEO.SEAMMARKS.01

Original ledger classification: Exact [E] seam/infinity/witness identities + [X] quantified translator negatives + [C] typed per-mark framing (exact sympy)

Original claim text

The seam-and-marks round (one module, 15 checks, ~7 s; discovery probe seam_marks_probe.py 15/15, verdict SEAM-REAL-LANDED). THE SEAM LINE IS REAL [E]: the line-stabilizer pair (U_1, V_1^J) has a UNIQUE common eigenline v with U v = 0 and V v = v -- the compiler seam span(e3), transported; explicitly v = (0, (sqrt3-i)/(sqrt3+i), 1). The real structure FIXES this line (R v ~ v): the seam is a real line of the cover. The two-demand involution REVERSES it (S_0 v = -v): the v590 seam-reversal demand holds GEOMETRICALLY on the curve. Honest scope: S_0 does not commute with R (the sheet involution is not Gamma-even), and the seam's identification with the infinity cusp stays open. THE INFINITY CUSP SEPARATED [E]: the unreduced Burau of B_4 at t = omega (braid relations verified) has invariant column (1,1,1,1) and invariant row (1, omega, conj omega, 1); the full twist r^4 acts with eigenvalues {1 (x1), omega (x3)} -- the infinity direction canonically separated from the omega-sheet. THE FIRST PER-MARK Z/3 WITNESS [E]: E_1 = G_1 + Gamma(G_1) restricts to an INTEGER matrix on the saturated fixed lattice and lies in the seven-word algebra with coefficient-denominator lcm EXACTLY 3 -- an ORDER-THREE class in the word-order cokernel P/O: the v566/v597 per-mark mechanism gets its first exact witness. Honest structure: E_4's class is trivial (integer coefficients), E_2/E_3 lie OUTSIDE the mark-1-anchored algebra -- the naive global map fails; the equivariant per-mark framing (each mark's class in its own rotated frame) is the typed reading. TRANSLATOR NEGATIVES [X, quantified]: the pullback form F_2 = g^T Sigma g is symmetric but not proportional to any of 7 canonical menu forms, and neither B^-1 F_2 nor F_2 B^-1 lies in the word algebra -- the corpus-transpose translator needs a new canonical object, named open. GATE.QGEO does not move; no marker changes. [2026-08-01 XXI: the integral infinity bridge lands -- v604: the 3-torsion localizes at infinity (coker(1-T_inf) = Z^2 + (Z/3)^3, finite cusps torsion-free), the four frames generate a FULL order of index 27 = 3^3, and R is an anti-isometry (J_fix integer, det 72); see QGEO.EQORDER.01.] [2026-08-01 XXII: the seam-infinity identification lands INTEGRALLY -- v605: the seam lift is a NONZERO pure 3-torsion class in coker(1-T_inf) (free coords zero, torsion (2,2,0)); see QGEO.TRANSLATOR.01.] [2026-08-01 XXIV: the zero-mode selection gains its carrier -- v606: the infinity torsion separates all three modes (pure torsion, distinct classes); the canonical '+1' rule is the remaining step; see QGEO.MODESEP.01.]

Original classification detail

SEAM-REAL-LANDED: seam line unique, real (R-fixed), S_0-reversed (v590 demand geometric); infinity cusp separated (unreduced Burau, full twist {1, omega x3}); E_1 order-3 cokernel class (first per-mark witness); E_4 trivial, E_2/E_3 outside; translator negatives quantified; 15/15 checks

Dependencies as recorded

QGEO.DUALFORMS.01; QGEO.CANON.01; QGEO.EMBED.01; QGEO.INVOL.01; v590; v600; v601; v602