This page isolates the strongest exact theorem spine currently present in the repository. It separates the exact two-qutrit operator geometry from the later promoted spectral-closure layer, so the public story starts from what is already rigid rather than from the most ambitious downstream interpretation.
Read this page together with Current Synthesis and Open Problems and Exact Boundaries. Use Complete Theory only as a compact dashboard snapshot.
The 40 vertices of W(3,3) are the 40 projective non-identity two-qutrit Pauli observables X^a Z^b ⊗ X^c Z^d. Two vertices are adjacent if and only if the operators commute. The commutator phase is exactly the standard symplectic form on F_3^4.
For each base point, the 12 neighbours split into four disjoint triangles and the 27 non-neighbours admit an F_3^3 coordinatization with 9 fibres of size 3. That shell is a finite Heisenberg phase space and its derived graph is the Schläfli graph SRG(27,16,10,8).
The canonical quadratic operator is H_can = 12I - A = 16I - BB^T, where A is the W(3,3) adjacency matrix and B is point-line incidence. Its spectrum is exactly {0^1, 10^24, 16^15}, so the operator layer is finite, exact, and positive semidefinite before any continuum interpretation.
E6-side, not global and E8-side.
45 = 36 + 9, with the extra 9 directions carried by the Heisenberg centre cosets.|PSp(4,3)| = 25920.51840.1296 full, 648 projective.The same canonical formulas extend along the symplectic family W(3,q). At q = 3 they reduce to the live kernel data (40,12,2,4), shell sizes 12 + 27, and canonical Hamiltonian spectrum (0,10,16).
| Layer | Status | What It Means |
|---|---|---|
| Two-qutrit Pauli geometry | Exact | W(3,3) is already a concrete finite quantum-information object, not a numerology shell. |
| Local Heisenberg shell | Exact | The 12 + 27 shell structure, MUB split, and Schläfli derivation are explicit and test-backed. |
| Local E6 bridge | Exact | The cubic-surface / W(E6) seam is an exact local consequence of the qutrit kernel. |
| Global E8 closure | Promoted spectral layer | 240 = |Φ(E8)| and 248 remain strong constraints, but they are not yet functorial consequences of the qutrit kernel alone. |
| Moonshine closure | Boundary reached | The low-order package closes exactly through the quartic layer; the quintic lift needs real character-theoretic input. |
| Continuum dynamics | Open | A path-integral, scaling, or action-principle derivation that selects W(3,3) is still missing. |
scripts/w33_two_qutrit_pauli.pyscripts/w33_heisenberg_qutrit.pyscripts/w33_qutrit_operator_algebra.pyscripts/w33_exact_lie_bridge_audit.pytests/test_w33_qutrit_operator_algebra.pytests/test_w33_exact_lie_bridge_audit.pytests/test_w3q_scaling_family.pyThe live W36_PAPER.tex edits now front-load the exact qutrit foundation, the local Heisenberg shell, the canonical operator, the local E6 bridge, and an explicit boundary between that exact shell theorem and the later E8 / moonshine closure layer.
q = 3.E6/E8 bridges are functorial and which remain numerical alignments.E8 lift and the dynamical/continuum bridge.Exact kernel first. Promoted closure second. That is the cleanest way to show how much is already here without pretending the final missing bridge is already proved.