# Regulus programme exhaustion ledger

**Date:** 27 October 2026

**Status:** current corpus-supported branches exhausted

**Programme:** [Regulus closure bridge](README.md)

## Closure criterion

A branch is exhausted here when it has reached one of four states:

1. exact and independently verified;
2. joined to an existing corpus carrier by an explicit intertwiner;
3. rejected by a representation, incidence or orbit obstruction;
4. dependent on new physical data, a new source or a genuinely new theorem.

## Branch ledger

| Branch | Outcome | State |
|---|---|---|
| local regulus closure | three same-ruling lines recover exactly four opposite transversals | exact |
| Spin(3,3) to Spin(2,3) descent | symplectic primitive hyperplane cuts the Klein quadric to the Lagrangian null quadric | exact |
| finite \(Q(4,3)\) carrier | 40 points, 40 pencils, 45 hyperbolic sections | exact |
| cubic lines and tritangents | 27 maximal five-cliques and 45 sections give \(27_5,45_3\) | exact |
| double-sixes | 36 elliptic sections select \(K_{6,6}\setminus6K_2\) | exact |
| ovoid/spread duality | each elliptic section is ten pairwise-disjoint Lagrangian planes partitioning all 80 nonzero vectors of \(\mathbb F_3^4\) | exact |
| two-qutrit measurement frame | every spread gives ten maximal commuting Pauli classes and therefore a complete ten-basis stabilizer MUB | exact standard dictionary; physical realization not inferred |
| frame overlap algebra | 270 syzygetic pairs share four bases and form \(\operatorname{SRG}(36,15,6,6)\); 360 azygetic pairs share one and form its \(\operatorname{SRG}(36,20,10,12)\) complement; the latter retain the unique ternary closure | exact |
| explicit two-qutrit operators | all 40 bases, 360 stabilizer states and 36 complete MUB frames constructed from Weyl projectors | exact |
| invariant noise algebra | all relation-invariant linear channels lie in \(\operatorname{span}\{I,A_4,A_1\}\) and have two measurable nontrivial rates | exact |
| Hecke compression | the 36-frame rank-three algebra is the fiber-constant compression of the rank-26 \(W(E_6)/S_5\) algebra along twelve flags per double-six | exact join |
| MUB clique closure | maximal one-overlap \(K_3\)s are the 120 azygetic closures, maximal \(K_5\)s recover all 216 skew cubic-line pairs, and four-overlap \(K_4\)s recover the 45 tritangents with multiplicity three | exact |
| linear selector | irreducible symmetric noise tends to the uniform frame and cannot select one | rejected |
| nonlinear selector | 15- and 20-dimensional instability thresholds derived; pure relation attraction selects 135 \(K_4\) tritangent-cover clusters or 216 \(K_5\) skew-pair clusters, while a unique frame still needs bias, history or boundary data | exact conditional model |
| implementation selector | weighted Clifford compilation turns calibrated gate costs into an explicit selector; a 4096-profile integer census finds 29 minimizing sets, 17 frames ever minimizing and four uniquely minimizing | exact relative to declared alphabet and weights; continuous fan and device calibration remain |
| equivariant integral lift | quotient exists, but a section would require an impossible order-51,840 finite subgroup of \(GL(4,\mathbb Z)\) | obstructed |
| faithful real four-dimensional lift | finite symmetry would have to embed in \(U(2)\), which \(PSp_4(3)\) cannot do | obstructed |
| device-level Pauli/Plücker map | exact calibration criterion and held-out residual test specified | needs implementation |
| syzygetic/azygetic closure | 270 four-overlap pairs, 360 six-overlap pairs and 120 unique azygetic triples | exact |
| Steiner hierarchy | 120 pairs, 240 trihedra and 40 line-partitioning triads | exact |
| degree-40 carrier | Lagrangian points/tangent sections are equivariantly the Steiner triads; polar lines are the other index-40 action | exact and typed |
| full exceptional symmetry | anti-symplectic similitude extends \(PSp_4(3)\) to order-51,840 \(W(E_6)\) action | exact |
| 432-point Hecke bridge | double-six/line flags are \(W(E_6)/S_5\) and have orbital rank 26 | exact join |
| 240 phase-fiber comparison | both factor as \(40\times3\times2\), but their \(PSp_4(3)\) orbit structures differ | rejected as equivariant identity |
| invariant forty-\(A_2\) root cover | regulus geometry supplies an \(E_6\) incidence object, not the missing \(E_8\) root cover | remains closed by prior no-go |
| real positive lift | Lagrangian boundary, Siegel interior and Riccati transport are separated exactly | kinematic join only |
| physical detector | a closure residual can be defined once line-like observables and noise are supplied | needs experiment-specific input |
| fundamental dynamics | incidence does not select a Hamiltonian, arrow of time or state-dependent feedback | needs new physics |

## What the corpus may now say

The strong defensible statement is:

\[
 \boxed{
 \text{The 40-point finite Spin(2,3) carrier internally generates the full
 classical }E_6\text{ incidence hierarchy and its 432-point Hecke flag
 carrier through regulus closure}.}
\]

The equally important negative statement is:

\[
 \boxed{
 \text{Neither matching 240 factorizations nor matching exceptional group
 names identify the phase fibers with cubic trihedra or }E_8\text{ roots}.}
\]

## What would reopen the programme

The current regulus lane should reopen only if one of the following appears:

1. a canonical integral lift relating the mod-three carrier to a measured real
   or complex response lattice;
2. a source-derived physical observable whose channels form the required
   Plücker line geometry;
3. an explicit intertwiner changing the failed 240-object orbit comparison;
4. a nonstandard \(PSp_4(3)\subset W(E_8)\) embedding with the required
   40-object and 240-object actions;
5. a state-dependent Hamiltonian or open-system law that turns transported
   regulus closure into genuine dynamics;
6. new literature showing an additional classical \(E_6\) configuration not
   generated by the present \(27/36/40/45/72/120/240/432\) hierarchy;
7. a measured device-cost vector, pulse compiler or noise law that tests the
   predicted control-cost chamber and distinguishes one of the 36 complete
   two-qutrit stabilizer-MUB frames.

Until then, further iteration inside the present data would duplicate known
incidences or turn exact proximity into unsupported physical identity.