This release executes five queued research fronts and three outside-box falsifier/construction probes. Every promoted statement below is backed by an executable witness and a frozen certificate in the repository.
The smallest connected self-paired PSp(4,3) orbital graph on the 810 selected flags is 16-regular with 6480 edges, diameter 5 and first Betti number 5671. Its equivariant lift to the 1620 apartments is a connected regular C2 double cover. The base triangle 0 → 482 → 367 → 0 changes apartment sheet upstairs, proving the deck class nonzero.
The outer PGSp(4,3) action preserves both base and lift. The deck class therefore spans a genuine one-dimensional trivial PGSp submodule of H¹.
The 45 complements of the dual weight-132 shell partition the 135 selected coordinates into 45 triples. The 270 dual minimum words project to 270 triples on those packets. Their pair graph is SRG(45,12,3,3).
Its maximal cliques are exactly 27 copies of K5. Every projected triple lies in exactly one such K5, and each K5 supplies all ten of its three-subsets. Thus the dual shells intrinsically reconstruct the point-line geometry GQ(4,2): 45 points, 27 lines, 3 lines through each point and 5 points on each line. The dual line graph is SRG(27,10,1,5).
The 45×270 triangle incidence has Smith profile 1^44 3; the 45×27 point-line incidence has nonzero Smith profile 1^21.
The full 20-class ordinary character algebra of PSp(4,3) was reconstructed from the finite permutation group and used to decompose both routing-edge permutation characters.
The 405-edge Petersen-fiber kernel is
378 = 2(15_L)+20+24+2(30_C)+2(conj 30_C)+2(60)+64.
The 1620-edge base-fiber kernel is
1485 = 4(15_L)+2(20)+2(24)+30_R+4(30_C)+4(conj 30_C)+40_C+conj 40_C+45_C+conj 45_C+5(60)+3(64)+5(81).
Independent local induced-module decompositions cross-check both results.
The selected135 graph is a connected three-sheet cover of the GQ(4,2) point graph. Each base edge carries a perfect matching of its two three-point packets, producing an S3 connection. In spanning-tree gauge, the 226 cotree voltages consist of 64 identities and 54 copies of each transposition.
Every one of the 270 base triangles has transposition holonomy, split 90/90/90 across the three transpositions. Its unique fixed sheet reconstructs one selected singular line. The 270 reconstructed lines reproduce the original selected incidence and selected270 intersection graph exactly.
The 27 quotient lines are the 27 Petersen fibers. Across each of the 135 adjacent fiber pairs, the 12 robust-base edges are exactly three disjoint K2,2 blocks indexed by the three internal coordinates of the shared packet.
Let rho = C_shortcut / C_base. Symmetry reduces uniform all-pairs multicommodity flow to the exact two-edge-orbit path frontier. The maximum concurrent-flow envelope changes policy only at
63/155, 111/137, 239/105.
| Capacity-ratio region | Optimal mix | Maximum normalized demand |
|---|---|---|
| 0 < rho < 63/155 | P2–P3 | 3(rho+2)/373 |
| 63/155 < rho < 111/137 | P1–P2 | 3(3rho+8)/1429 |
| 111/137 < rho < 239/105 | P0–P1 | 3(rho+4)/659 |
| rho ≥ 239/105 | P0 | 1/35 |
At equal capacities the exact optimum is 13/80 P0 + 67/80 P1, giving equal per-edge load 659/15 and maximum demand 15/659. Queue and erasure extensions are explicitly labeled modelling assumptions rather than measurements.
Petersen from the dual-shell design. Within each reconstructed five-point GQ line, the ten projected triples form a Petersen graph under intersection size one and its six-regular complement under intersection size two. These 27 Petersen graphs are literally the 405 shortcut edges.
Regular S3 closure. The three-cover has a canonical connected six-sheet Galois closure with 270 vertices and 1620 edges. A parity gauge identifies it with the bipartite double cover of selected135. It is therefore not the non-bipartite selected270 base router despite the matching counts.
Petersen coding. Every 15-edge Petersen fiber carries the canonical [15,6,5]₂ cycle code and dual [15,9,3]₂ cut code. The cycle code is invariant under the full local S5, has 12 minimum words corresponding to the twelve Petersen 5-cycles, and corrects every pattern of at most four erased edge symbols. The 27-fiber direct sum is [405,162,5]₂.
The C2 and S3 holonomies are finite graph-cover data, not optical phases or gauge fields. The capacity, queueing and erasure formulas are mathematical network models with stated assumptions, not measured Holonet performance. The Petersen code adds redundancy and does not create free fault-free throughput.