We define a finite weighted complex by cyclically gluing five copies of the depth-six Pascal carrier along complete radial side chains. The successive open quotients have 28, 49, 70, 91 and 112 vertices. Closing the terminal seam identifies six pairs of non-apex vertices and produces a connected quotient with 106 vertices. Binomial weights and Pascal recurrence remain valid facewise. The five-face rotation and reflection generate the exact typed automorphism group D5. A deterministic data object and independent verifier make the construction reconstructible without this page.
One finite object—not the whole system.
The note compacts the existing SCC-003 definition, proofs, reference data and bounded literature comparison. It does not depend on the wider A0 vocabulary or on visual analogies.
A cyclic five-face quotient of depth-six binomially weighted Pascal carriers with explicit D5 automorphisms and open-to-closed node sequence
28,49,70,91,112 → 106.This wording is not a worldwide-novelty, priority, significance or publication claim.
The depth-six Pascal face.
Let T₆={(n,k)∈ℤ²:0≤k≤n≤6}. Directed edges connect each position to its two Pascal children, and the weight is w(n,k)=C(n,k).
P₆ = (T₆,E₆,w₆) |T₆| = 1+2+…+7 = 28 |E₆| = 2(1+2+…+6) = 42
The triangular macro-face used for gluing and the directed recurrence graph remain two typed layers.
Five copies, one cyclic seam rule.
Index the copies by C₅=ℤ/5ℤ. Identify all apices and impose (f,n,n)~(f+1,n,0) for every face f and level 1≤n≤6. The closed equivalence classes are one apex, thirty seams and seventy-five singleton interiors.
Growth by 21; closure by six.
A new face shares the common apex and six non-apex side vertices, so it contributes 28−7=21 new vertices. The open count is 21j+7. The last two terminal sides already share the apex; closing them pairs six remaining level vertices.
The quotient weight is well-defined because glued side weights are all one. Pascal's identity remains true within every retained face incidence. The complex is connected through the common apex.
The recurrence is facewise. Summing all incoming edges after forgetting face incidence can double-count a seam and is not the asserted operation.
The typed group is exactly D5.
r(f,n,k) = (f+1,n,k) m(f,n,k) = (-f,n,n-k) r⁵=e, m²=e, mrm=r⁻¹
Rotation and reflection preserve seams, levels, weights, recurrence edges and face incidence. Every typed automorphism also induces an automorphism of the five-face adjacency cycle; fixing a face and its seams fixes all its Pascal positions. Thus the induced action is injective and the typed group has exactly the ten elements of D5.
Reconstructible without the Explorer.
The reference package contains the node and edge tables, weights, seam classes, face incidences, adjacency matrix, declared-order graph6 representation, ten typed automorphisms, expected metrics and a SHA-256 manifest. A separate union-find verifier currently passes 177 of 177 checks.
Close baselines; no equivalence established.
The strongest located multi-sector baseline is the Hilton–Pedersen Pascal-Hexagon line. Quint's infinite Pascal graph is the closest located literal joining of Pascal-derived triangular graphs. Neither defines the same five finite copies, complete side-chain gluing, depth-six seam relation or 112→106 quotient.
CLOSE_BASELINE_FOUND_NO_EQUIVALENCE_ESTABLISHED SEARCH_INCOMPLETE_ACCESS_OR_COVERAGE_LIMIT
The search is not exhaustive worldwide and has not been repeated independently.
Useful views, outside the theorem.
A regular metric pentagon, a pentagonal bipyramid, H2⊕A1 roots and a common C₃₀ phase cover are separately bounded representations or controls. They do not generate the quotient and do not establish physical resonance, a universal solid, a crystallographic Lie system or an identity between pentagonal and hexagonal carriers.
The paper view is ready; the claim is not promoted.
The paper does not generalize to arbitrary face number or depth. Such a generalization would be a new research question requiring separate admission.