This packet closes the code-interaction, decoder-scheduling, binary-cohomology and binary–ternary minimum-shell lane with explicit finite certificates.
On the identical 2025 physical coordinates, C_Levi ⊂ C_378 ⊂ C_399 with dimensions 64, 378 and 399. The Levi code is the cold-only [2025,64,96]₂ repetition-cycle code. Its inclusion in the zero-outer-syndrome kernel is objectwise: every binary Levi cycle has even degree at both packet and quotient-line vertices. Deleting all 405 hot coordinates is injective on code399.
The old eight-layer schedule was not optimal. In the canonical sparse local-basis model, the 1620 local checks occupy two layers of 945 and 675 checks. A six-word W6 basis with weights 12,12,16,12,12,12 has maximum outer-coordinate multiplicity four. Using the four equivalent physical representatives inside every cold repetition class makes all six global checks mutually disjoint in one third layer. Two layers are impossible in this stated sparse model, so the optimum is exactly three.
The 64-dimensional sign cohomology of the triangle-filled GQ(4,2) point graph and the 64-dimensional binary homology of the GQ Levi graph are isomorphic as both PSp(4,3) and PGSp(4,3) modules. The common-generator intertwiner space has dimension one and its unique nonzero element has rank 64.
Burnside enumeration on the faithful sign-module matrix image gives 711,679,993,497,112 PSp orbits and 355,840,805,040,988 PGSp orbits on the affine sign sectors. The linear fixed space is zero, so the canonical all-odd signing is the unique fully symmetric affine sign sector.
The 1080 minimum words of the punctured [1620,64,96]₂ Levi code are the 1080 Levi 8-cycles, equivalently all four-cycles of SRG(27,10,1,5). They form one PSp orbit with stabilizer 24 and one PGSp orbit with stabilizer 48.
These binary minima are linked objectwise to the 360 canonical ternary induced-K3,3 homology witnesses: every binary four-cycle lies in exactly three induced K3,3s and every K3,3 contains exactly nine binary four-cycles. The incidence is therefore 1080_3 – 360_9, connected, with incidence-matrix ranks 324, 359, 360 and 360 over F2, F3, F5 and F7.
The 405 size-four repetition classes, 135 size-three classes and 135 minimum quotient relations form 135 disconnected typed cells. Their exact class-level automorphism group is S3^135:S135, enormously larger than PGSp(4,3). Hence the low shells alone cannot recover the 45 packets, 27 GQ lines or Petersen-fiber layout; higher-shell/full-code data are required.
The 5671-dimensional invariant-flag H1 has exact characteristic-two Brauer composition multiplicities 71·1 + 70·4a + 70·4b + 154·6 + 70·14 + 56·20a + 56·20b + 14·64. Together with the exact trivial socle dimension four and trivial head dimension one, at least 66 trivial factors are forced into internal nonsplit layers, so Loewy length is at least three. The full nontrivial Loewy ordering remains open.
All six failed-router fractional throughput functions are now exact piecewise-rational functions of the surviving hot/cold capacity ratio. Every failure moves the intact-router phase boundaries, and every value at ratio one reproduces its independently frozen equal-capacity optimum.
All promoted claims are finite graph, code, module, cohomology or fractional-network statements. The scheduling theorem uses a specified sparse disjoint-support check model. The flow theorem is splittable fractional routing. No physical gauge field, particle assignment, measured hardware speed or fault-tolerance threshold follows from these finite identifications.