Regulus closure bridge

Opened: 26 October 2026

Status: exact finite incidence pass; physical interpretation open

Result

A regulus is not merely a visual analogy for the corpus's two-face grammar. It is an exact local closure object connecting four structures already present in the corpus:

The executable calculation over \(\mathbb F_3\) finds:

• 40 Lagrangian points and 40 four-point pencil lines in \(Q(4,3)\);
• 45 hyperbolic 16-point sections;
• two opposite four-line reguli in every hyperbolic section;
• exact three-line closure onto the opposite four-line regulus;
• a 45-vertex section graph with parameters \(\operatorname{SRG}(45,12,3,3)\);
• exactly 27 maximal five-cliques, reconstructing the incidence counts of the 27 lines and 45 tritangent planes on a cubic surface;
• 36 elliptic sections whose stabilizers split the 27 lines as \(12+15\), with each 12 forming \(K_{6,6}\) minus a perfect matching: the 36 classical double-sixes;
• the same 36 elliptic sections as ten-plane symplectic spreads, hence as 36 labelled complete stabilizer-MUB frames for the two-qutrit carrier;
• an exact measurement-frame overlap algebra: 270 pairs share four bases and 360 pairs share one, matching the syzygetic/azygetic double-six split;
• an explicit construction of all 360 rank-one two-qutrit stabilizer projectors and all 36 complete MUB frames;
• a three-dimensional invariant frame-noise algebra with spectral sectors \(1+15+20\), plus an exact proof that symmetric linear noise mixes rather than selects a frame;
• a weighted Clifford compiler in which calibrated control costs select cheapest frames in hardware-dependent chambers without making any frame intrinsically preferred;
• maximal-clique recovery of the 120 azygetic closures, all 216 skew cubic-line pairs and all 45 tritangents from measurement-frame overlaps;
• a no-section theorem: the finite carrier is a quotient of the integral/real symplectic carrier, not a canonically embedded finite copy;
• 120 Steiner trihedral pairs and 40 Steiner triads, equivariantly identified with the 40 Lagrangian points;
• an anti-symplectic outer involution completing the finite action to the full order-51,840 \(W(E_6)\) symmetry;
• a 432-point double-six/line flag carrier exactly equal to \(W(E_6)/S_5\), with commutant dimension 26;
• transitive \(PSp_4(3)\) actions on the reconstructed 27 lines, 36 double-sixes and 45 tritangent planes.

This supplies a new finite-geometric route from the corpus's 40 qutrit polarizations to its even-\(E_6\) symmetry. It does not supply dynamics, a physical carrier or an \(E_8\) root identification.

Executable artifacts

• Analysis
• Machine evidence
• Independent verifier
• Two-qutrit operator and selector analysis
• Operator/selector machine evidence
• Operator/selector verifier
• Frame-channel fitter

Run:

node scripts/analyze-regulus-spin23-bridge.mjs --write
node verification/verify/verify_regulus_spin23_bridge.mjs
node scripts/analyze-regulus-mub-selector-dynamics.mjs --write
node verification/verify/verify_regulus_mub_selector_dynamics.mjs

Detailed finding

See Regulus closure, the symplectic slice and the finite E₆ incidence layer.

Continue through:

• Full E₆ completion, the 432-point Hecke carrier and the 240-object boundary
• Real positive-domain, dynamics and detector boundary
• Regulus programme exhaustion ledger
• Double-sixes, symplectic spreads and complete two-qutrit measurements
• Two-qutrit operator, noise and selector completion
• Integral, real and experimental resolution

Authority boundary

The finite incidence structure is exact. Its identification with the abstract Schläfli \(27\)-line/\(36\)-double-six/\(45\)-tritangent configuration is an incidence and group- action result, not the construction of a particular cubic surface in physical space. No incidence count establishes a field equation, interaction, spacetime ontology or exceptional unification.