k = 4
(1, 2, (0, 0, 1)) dim S, rank I = (0, 0)
(2, 1, (0, 1, 1)) dim S, rank I = (2, 2)
(2, 2, (0, 0, 1)) dim S, rank I = (4, 2)
(3, 1, (0, 0, 1)) dim S, rank I = (6, 4)
(4, 0, (0, 1, 1)) dim S, rank I = (6, 6)
(6, 0, (0, 0, 1)) dim S, rank I = (12, 11)
(6, 0, (1, 1, 1)) dim S, rank I = (12, 11)
k = 5
(1, 2, (0, 0, 1)) dim S, rank I = (0, 0)
(2, 1, (0, 1, 1)) dim S, rank I = (3, 3)
(2, 2, (0, 0, 1)) dim S, rank I = (6, 5)
(3, 1, (0, 0, 1)) dim S, rank I = (9, 9)
(4, 0, (0, 1, 1)) dim S, rank I = (9, 9)
(6, 0, (0, 0, 1)) dim S, rank I = (18, 18)
(6, 0, (1, 1, 1)) dim S, rank I = (18, 18)
All 37 oriented edge-star signatures pass exact rational rank checks.
face augmentation 4 ((0, 0, 0), (1, 0, 0)) (2, 4)
face augmentation 4 ((1, 0, 0), (1, 1, 0)) (4, 6)
face augmentation 4 ((0, 0, 0), (1, 1, 1)) (11, 12)
face augmentation 5 ((0, 0, 0), (1, 0, 0)) (5, 6)
paired interior quotient rank: 22 -> 24
All listed face-bubble complement checks pass exactly over Q.
All deficient-edge face-selection checks pass on the complete N=3 template set.
All deficient oriented signatures have an exact triangular face-correction route: {4: 22, 5: 6}
N=4,5,6 have the same seven local edge classes as N=3.
All 117 boundary-decorated N=3 edge types have the stated exact admissible S_e dimensions (k=4,5).
N=4,5,6 have exactly the same 117 boundary-decorated local edge types as N=3.
N=4,5,6 have exactly the same 37 canonical local signatures as N=3.
Quartic gamma fields span every admissible divergence-vertex pattern on all N=3 vertex stars (k=4,5).
