A registered result in its source context

QGEO.DUALFORMS.01

Original ledger classification: Exact [E] duality/dictionary/positivity identities + [X/O] quantified negatives and typed opens (exact sympy; two numeric inertia counts of exact matrices) + Lean certificate FORM.QGEO.04

Original claim text

The duality-and-forms round (one module, 17 checks, ~8 s; discovery probe duality_forms_probe.py 17/17, verdict DUALITY-FORMS-LANDED). THE DUALITY MECHANISM [E]: U_1 = 3 GJ_1 is J-self-adjoint; the J-adjoint of V_1 = 1 + GJ_4 - GJ_1 GJ_2 is the exact product reversal 1 + GJ_4 - GJ_2 GJ_1; and (U_1, V_1^J) is EXACTLY conjugate to the compiler line-stabilizer pair while (U_1, V_1) is conjugate to the transposed pair (v601) -- the J-adjoint exchanges the two parabolic types inside ONE equivariant structure: the cover's realization is J-self-dual as a pair of pairs. DICTIONARY COMPLETION [E]: H = 1 + V(V-1)/2 has char (x-1)^2(x-2) -- the corpus anchor formula holds VERBATIM on the curve; Q = U + V has char (x-1)(x^2-5x+3); Q_+ ~ 2 - GJ_1 + GJ_2 GJ_3 has char (x-1)(x-2)(x-3); the binary AnchorLadder ~ 2 - GJ_1 + 2 GJ_2 GJ_3 has char (x-1)(x-2)(x-4); the S_0-grading halves of Q are computed and documented. WEIL POSITIVITY [E, Strategy II at module level]: the Weil operator C from J's spectral split (Hodge numbers (1,2)) gives a POSITIVE DEFINITE twisted form JC with exact eigenvalues {2sqrt2-2, 2, 2+2sqrt2} = |Spec J| -- the Riemann bilinear positivity statement holds on the omega-sheet. THE CANONICAL BILINEAR FORM [E]: B(v,w) = (Rv)^dagger J w has matrix B = M^dagger J, SYMMETRIC and nondegenerate with det B = 8 -- the curve's complex orthogonal structure combining the real structure with the polarization. HONEST NEGATIVES [X/O, quantified]: (i) g^T g is NOT proportional to B -- the corpus transpose structure is not the (R,J)-bilinear form (corpus RP word-Gram inertia (4,3,0) documented); (ii) the naive mark-local cokernel map fails (U_2, U_3 lie outside the seven-word algebra, U_4 inside with integer coefficients) -- the carrier of the mark-local Z/3 mechanism stays open; (iii) the seam-plane normal is explicit but NOT a deck eigenvector -- the seam identification stays open. COMPANION LEAN CERTIFICATE (FORM.QGEO.04): the v600 integer core is certified in TfptCarrier/CoverEmbedding.lean -- compiler relations of (C_U, C_V), the word factorization W = T*S, the Smith certificate P*T = diag(1,1,1,3,3,3,3) with unimodular P (P*Pinv = 1), and S-primitivity via a unimodular minor: determinant-free, kernel decide, lake build green (3375 jobs). GATE.QGEO does not move; no marker changes. [2026-08-01 XIX: the seam question advances -- v603: the seam line is a REAL line of the cover (R-fixed), S_0 reverses it geometrically; the infinity cusp separates in the unreduced module; the first per-mark Z/3 witness lands (E_1, order 3); see QGEO.SEAMMARKS.01.] [2026-08-01 XXII: the corpus-transpose translator question is RESOLVED -- v605: theta_corpus = Jadj o inner(c) exactly, c unique up to scale, 14c integral; c's independent geometric identity is the remaining open; see QGEO.TRANSLATOR.01.]

Original classification detail

DUALITY-FORMS-LANDED: J-adjoint exchanges the parabolic types (exact conjugations both ways); dictionary complete in J-projector language (anchor formula verbatim); Weil-twisted form positive definite (exact); B = M^dag J symmetric, det 8; corpus-transpose / mark-local / seam questions typed open; Lean FORM.QGEO.04 green; 17/17 checks

Dependencies as recorded

QGEO.CANON.01; QGEO.EMBED.01; QGEO.REAL.01; DIAMOND.PARA.SELFCODE.01; v572; v590; v600; v601