The q=5 design contains Reye as its zero shell.
The selected q=5 cover, the Reye configuration, the Klein-Latin action, and the twelve antipodal short-root pairs of F4 are now connected by an explicit action map. More sharply, Reye is reconstructed from the q=5 multidesign itself: it is exactly the orbit of triples contained in no design row.
the exact moving-cover action, not an order-576 match
complete triple-containment spectrum
the intrinsic Reye zero shell
every natural PSL(2,7) placement closes fully
The action gate is explicit
The q=5 setwise stabilizer has one fixed cover vertex and one moving twelve-orbit.
Its image has order 576, stabilizer C₂×S₄, subdegrees
1,3,8, and GAP transitive ID 165. The exact cover-to-Latin conjugator is
Pass 5606 already specified this fail-closed computation; this run activates it and its existing composer emits the twelve-row map to F4 short-root pairs.
Third-order incidence finds Reye without naming it
The 312 vertices outside the cover give six-subsets of its thirteen positions.
With multiplicity they form a 2-(13,6,60) design with point replication
144. Among all 286 triples:
| Rows containing a triple | Number of triples | Meaning |
|---|---|---|
| 0 | 16 | the Reye line orbit |
| 16 | 30 | one further group orbit plus the fixed-point sector |
| 24 | 240 | the remaining two moving-triple orbits |
The sixteen zero triples avoid the fixed cover point; every moving point lies in four
of them. Their 28-vertex Levi graph has 48 edges and automorphism group 576, acting as
T12_165 on points and T16_1034 on lines.
Orientation depends on the carrier
The explicit conjugator carries the point-side kernel to the even-Latin group already linked in Pass 5300 to the Hoffman stabilizer quotient. The same abstract group therefore has a nontrivial orientation character on twelve and a trivial one on sixteen.
The natural seven-side opens all of S₁₂
Place the natural degree-seven PSL(2,7) action in twelve symbols by adding
five fixed points. GAP checks every one of its 23,760 conjugates:
| Order-576 action | Relative-placement orbits | Every generated join |
|---|---|---|
| T12_161 | 53 | A₁₂ |
| T12_163 | 62 | A₁₂ |
| T12_165 (Reye) | 58 | S₁₂ |
Parity explains the split. Classes 161 and 163 are even; class 165 is not. The old order-4032 number remains a general divisibility lower bound, but it is not attained on this natural twelve-symbol carrier.
A second twelve, with no canonical identification
The 312 rows collapse to 132 supports:
The twelve multiplicity-six supports carry another T12_165. Their stabilizer
and the moving-point stabilizer have the same abstract type, but are not conjugate inside
the source group. The permutation conjugacy lifts to a source automorphism: its displayed
order-eight representative squares to an inner automorphism, and exactly 48 inner
re-gaugings give outer involutions. The frozen involution exchanges the two stabilizer
classes, but it still supplies no canonical point/support labeling.
The sign twist is not the carrier involution
Exact character enumeration over all 576 source elements separates three operations that an order or parity comparison alone cannot distinguish:
The sign-twisted point permutation module is therefore disjoint from both the heavy and line modules. The point sign twist is neither the explicit source outer involution nor the even line carrier.
Reproduce
gap -q analysis/w33_pass5667_5674_q5_reye_equivariant_orientation.g
python3 -m pytest -q tests/test_w33_pass5667_5674_q5_reye_equivariant_orientation.py7+1⁵ representation. No continuum,
dynamics, particle, gauge-force, or physical-unification claim is made.