Original ledger status
QGEO.EMBED.01
QGEO.EMBED.01
The joint embedding -- the compiler pair (U, V) is realized on the mu3-cover (one module, 15 checks, ~4 s; discovery probe joint_embedding_probe.py 15/15, verdict JOINT-EMBEDDING-LANDED). THE PAIR [E]: in the Gamma-even DEGREE-2 sector (the v599 grading functor applied to cusp-Gram products), U* = -2E1 + (G1 Gamma(G1) + Gamma(G1) G1) + 3 and V* = -E3 + (1/2)(G2 Gamma(G2) + Gamma(G2) G2) + 1 are both integral over Z[omega], Gamma-even, with char U* = x^2(x-3), char V* = x(x-1)(x-2), char(U*+V*) = (x-1)(x^2-5x+3) = char Q, and the compiler joint traces tr(U*V*) = 3, tr(U*V*^2) = 6. THE ALGEBRA EMBEDDING [E]: the seven words {1, U*, V*, U*V*, V*U*, V*^2, V*U*V*} are independent (rank 7 = the full 2|1 parabolic) and the simultaneous-conjugation system is EXACTLY solved: an explicit invertible g with g U* g^-1 = U_c and g V* g^-1 = V_c -- the v566 parabolic self-code algebra lives on the cover. THE INTEGRAL FORM [E]: the real structure R as an integer matrix on Z^6 has R^2 = 1; its fixed lattice has the explicit saturated basis (1,2,2,0,0,0), (0,1,1,-1,0,0), (0,0,1,1,-1,1) (SNF certificate (1,1,1)); U*, V* restrict to INTEGER 3x3 matrices with the same chars. THE 81 [E, the hard test]: the Z-order of the seven restricted words has Smith elementary divisors (1,1,1,3,3,3,3) -- saturation index 81 = 3^4, EXACTLY the v566 compiler self-code index with the same divisor pattern. HONEST LIMITS: must-break control passes (the v599 degree-1 pair fails the joint criterion); CANONICITY IS OPEN and typed open -- the degree-2 scan admits many joint pairs, so this is an EXISTENCE claim, not uniqueness; GATE.QGEO does not move until the canonicity round and the two-demand continuum derivation land; no marker changes. [2026-08-01 XVII: the canonicity question is answered -- v601: the J-adapted projector presentation is D4-equivariant by construction (mark choice = winding choice) and realizes the DUAL parabolic exactly; the Klein group lives integrally on the curve; see QGEO.CANON.01.] [2026-08-01 XVIII: the integer core is Lean-certified -- FORM.QGEO.04 in TfptCarrier/CoverEmbedding.lean: W = T*S, Smith certificate P*T = diag(1,1,1,3,3,3,3) with unimodular P, S-primitivity via unimodular minor (determinant-free, kernel decide; lake build green).]
Recorded status: Exact [E] existence realization (integrality, conjugation, saturation index all exact sympy) + typed open canonicity [O]; NO marker moves
Dependencies, status detail and source
Canonical status: JOINT-EMBEDDING-LANDED: (U*, V*) Gamma-even degree-2, Z[omega]-integral, chars + joint traces compiler-exact, word rank 7, conjugation solved, integer on saturated Fix(R), Smith (1,1,1,3,3,3,3) index 81 = v566; canonicity open; 15/15 checks
Dependencies: QGEO.REAL.01; QGEO.DICT.01; QGEO.COVER.01; QGEO.INVOL.01; DIAMOND.PARA.SELFCODE.01; GATE.QGEO.01; v566; v597; v598; v599
Scripts: v600_joint_embedding.py
Location: tfpt_1;tfpt_2;tfpt_3
Original CSV, line 618