Original ledger status
QGEO.CANON.01
The canonical equivariant realization (one module, 14 checks, ~12 s; discovery probe equivariant_dual_probe.py 14/14, verdict EQUIVARIANT-DUAL-LANDED). THE REPAIR: the naive cusp-Gram presentation of v600 is not D4-equivariant because the standard hermitian dagger is basis-dependent -- the Burau image preserves J; the canonical building blocks are the J-ADAPTED cusp projectors GJ_k = (1-T_k) J^-1 (1-T_k)^dagger J. THE PROJECTORS [E]: all four are idempotent rank-1 J-self-adjoint projectors (char x^2(x-1)) rotating exactly under D4 (r GJ_k r^-1 = GJ_(k+1)); opposite cusps J-orthogonal (tr(GJ1 GJ3) = 0), neighbors overlap with trace 1. THE EQUIVARIANT PAIR [E]: U_k = 3 GJ_k, V_k = 1 + GJ_(k-1) - GJ_k GJ_(k+1) -- Z[omega]-integral, compiler spectra, compiler joint data (char(U+V) = charQ, tr(UV) = 3, tr(UV^2) = 6), seven-word rank 7, and D4-EQUIVARIANT BY CONSTRUCTION (r^k-conjugation = mark shift, verified for all marks): the mark choice IS the compiler's winding choice. THE DUALITY [E]: the equivariant pair is NOT conjugate to (U_c, V_c) (the intertwiner space is 1-dimensional with identically singular solutions -- must-fail documented) but EXACTLY conjugate to the transposed pair (U_c^T, V_c^T): the equivariant presentation realizes the DUAL (plane-stabilizer) parabolic, the v600 Gamma-even pair the line-stabilizer -- the cover carries BOTH parabolic types, exchanged by duality. THE KLEIN GROUP ON THE CURVE [E]: all eight spectral-sign involutions of V_1 are Z[omega]-integral (v590 rigidity reproduced); the two v590 demands select the explicit integral S_0 (trace -1, commuting with V), and A B = S_0 closes the Klein four-group on the cover. HONEST OPENS [C]: the naive J-word Gram has inertia (3,2,2), not the corpus v572 RP pattern (4,0,3) -- the Hodge/RP comparison needs the Weil-twisted polarization (named); the seam-line geometric identification stays open. GATE.QGEO does not move; no marker changes. [2026-08-01 XVIII: the duality mechanism is identified -- v602: the J-adjoint exchanges the two parabolic types exactly ((U1, V1^J) ~ compiler pair); the dictionary completes in the J-projector language; Weil positivity holds exactly; see QGEO.DUALFORMS.01.]
Recorded status: Exact [E] equivariance/duality/Klein identities + [C] typed opens (exact sympy; one numeric inertia count of an exact hermitian matrix)
Dependencies, status detail and source
Canonical status: EQUIVARIANT-DUAL-LANDED: GJ_k idempotent equivariant projectors; pair family compiler-exact and equivariant by construction; conjugate to the TRANSPOSED compiler pair (dual parabolic); Klein group integral on the curve (8/8, S_0 explicit, A B = S_0); J-word-Gram inertia (3,2,2) typed open vs v572; 14/14 checks
Dependencies: QGEO.EMBED.01; QGEO.REAL.01; QGEO.INVOL.01; DIAMOND.PARA.SELFCODE.01; v590; v597; v599; v600
Scripts: v601_equivariant_dual.py
Location: tfpt_1;tfpt_2;tfpt_3
Original CSV, line 619Original ledger status
QGEO.DUALFORMS.01
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.]
Recorded status: 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
Dependencies, status detail and source
Canonical status: 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: QGEO.CANON.01; QGEO.EMBED.01; QGEO.REAL.01; DIAMOND.PARA.SELFCODE.01; v572; v590; v600; v601
Scripts: v602_duality_forms.py
Location: tfpt_1;tfpt_2;tfpt_3;tfpt_research_contracts
Original CSV, line 620