This packet executes five queued fronts plus three outside-box falsifier/construction probes. The residue realization from Passes 4737–4744 is used as an independent cross-check of the 270-router.
The invariant 810-flag graph has 6480 edges and rational first cohomology dimension 5671. Its complete ordinary PSp(4,3) character has no trivial constituent. The self-paired edge stabilizer has order 4 and two endpoint reversers. Yet the apartment deck class is a nonzero PGSp-fixed element of H¹ over F₂. Therefore this fixed mod-2 line has no PGSp-invariant rational or integral lift.
The explicit PGSp action on the 45 GQ(4,2) packets has order 51840, equal to the full reconstructed base automorphism group, and every generator lifts to the selected three-cover. The monodromy is generated by two triangular fundamental cycles with presentation <a,b | a²=b²=(ab)³=1>. However the rule that all 270 base triangles have transposition holonomy is not sufficient: changing one cotree voltage on edge (1,12) gives a distinct connected S₃ cover while all 270 triangle holonomies remain transpositions.
The cold 270-vertex permutation module is 1+6+15b+2(20)+24+30b+30c+60+64. The two irrational 20-dimensional eigenspaces are the same degree-20 irreducible. Cold adjacency acts on its two-dimensional multiplicity space with trace 2 and determinant −12, hence polynomial x²−2x−12 and eigenvalues 1±√13. The full characteristic factor is (x−12)(x−8)^15(x−2)^84(x+1)^64(x+4)^60(x+6)^6(x²−2x−12)^20.
The 3K₂,₂ connection law partitions all 2025 router edges into 135 disjoint fifteen-edge cells. Each cell contains three hot edges and three four-edge cold blocks. Exhaustive enumeration of all 2825 binary subspaces of F₂⁶ finds exactly 25 invariant under the local triangle S₃. Their weighted Pareto frontier is (1,15),(2,10),(3,7),(4,3),(5,2),(6,1), producing global PGSp-invariant families including [2025,405,7]₂. The latter improves the explicit K·d comparison metric over the uncoupled [2025,567,4]₂ baseline, while lowering rate and increasing distance.
Normalize cold capacity to 1 and hot capacity to ρ>0. Cold edge-connectivity is 12, each Petersen fiber has edge-connectivity 3, the 27-vertex quotient has edge-connectivity 10, and every quotient edge lifts to 12 cold physical edges. Thus the exact global min-cut is min(12+3ρ,120), with breakpoint ρ=36. At equal capacities every unordered vertex pair has min-cut 15. Removing all 15 shortcut edges in one Petersen fiber lowers the global min-cut to exactly 12 for every positive ρ.
The 540 weight-three words of the residue [270,240,3]₂ code are the 540 cold triangles partitioning all 1620 cold edges. Their edge boundaries give an injective map from a free 540-generator circuit module, yielding F₂^540 → F₂^1620 → F₂^270 with ranks 540 and 269 and H₁ dimension 811. But this map cannot descend from the dependency code itself: its 540 minimum words have 300 independent relations while the 540 edge-disjoint triangle boundaries are independent.
The three-sheet permutation representation splits as trivial plus the two-dimensional standard representation. In the integral sum-zero lattice the 90×90 standard voltage block is self-adjoint for Gram matrix [[2,−1],[−1,2]] and satisfies exactly S(S²−36I)=0, giving spectrum 6^15,0^60,(−6)^15. This reproduces the selected135 and regular S₃-closure spectra. It does not produce x²−2x−12, so the router radical sector is a distinct multiplicity-space mechanism. An initial source/target transpose mistake was caught by independent execution and corrected before freezing.
The apartment deck cocycle descends through the canonical 810→270 flag-to-line projection after an exact F₂ gauge solve. The projected base is a distinct 16-regular, 2160-edge orbital relation, disjoint from the hot and cold router relations. Its descended double cover has 540 vertices, 4320 edges, degree 16 and diameter 4. The induced PSp action is transitive of order 25920, with point stabilizer order 48. That stabilizer is literally the Pass4738 subgroup K=CPSp(h) inside the residue stabilizer H of order 96. Hence the cover is exactly PSp/K → PSp/H.
These are exact finite graph/group/cohomology/representation/code statements and an undirected finite-network capacity theorem. “Fourier” denotes finite-group representation decomposition. No optical phase, continuum gauge field, particle assignment, measured hardware performance or fault-tolerance threshold is inferred.