2827 — Equal-Rank Relation Supports Occupy Distinct Source-Symmetry Orbits
Frozen symmetry test
Entry 2824 produced two equal-rank relation supports with primitive coefficient vectors
and
in the homogeneous site-energy basis .
The admissible test group was frozen as signed permutations of the three labelled site energies, modulo multiplication by a nonzero scalar unit. This contains the homogeneous site relabellings and polarity changes while preserving the meaning of the three source coordinates.
Orbit result
The two orbits are disjoint. Their invariant discriminator is the number of zero site-energy coefficients:
- every transform of has full three-coordinate support;
- every transform of has exactly one zero coefficient.
Geometrically, contains no coordinate axis. The divisor contains one coordinate axis; in the displayed chart this is
Thus the equal-rank relations occupy inequivalent supported objects under the declared source symmetry.
Physical qualification
Both divisors miss the strict positive-energy chamber. Orbit inequivalence therefore does not by itself produce distinct physical observables. It instead predicts different incidence with existing coordinate-soft support, which is the correct input for a subsequent restriction/Gysin comparison.
Narrow conclusion
The support distinction in Entry 2824 is not removable by homogeneous labelled-site relabelling or polarity. Bare Koszul rank misses a source-invariant incidence property: containment of a coordinate-soft axis.
The next finite test is local. Restrict both relation objects to coordinate-soft strata and compute their costalk/Gysin maps. The complementary relation has a contained soft axis; the vertex relation meets coordinate-soft strata only transversely. If the resulting supported maps are nevertheless canonically equivalent, this discriminator does not survive the comparison calculus.
Durable artifacts
research/benincasa/check_equal_rank_support_symmetry_orbits.pyresearch/benincasa/equal-rank-support-symmetry-orbits.json