This packet executes five queued fronts and three outside-box probes with exact finite graph, module, code, homology, decoder and fractional-flow evidence.
On the 27 quotient lines the binary even permutation module has the nonsplit uniserial chain 0 < W6 < W20 < E26 with dimensions 0,6,20,26 and irreducible successive dimensions 6,14,6. Splitting off the all-one line leaves exactly eight PGSp-invariant outer codes.
| Outer dimension | Outer distance | Physical preimage |
|---|---|---|
| 0 | ∞ | [2025,378,14]₂ |
| 1 | 27 | [2025,379,14]₂ |
| 6 | 12 | [2025,384,14]₂ |
| 7 | 11 | [2025,385,14]₂ |
| 20 | 4 | [2025,398,14]₂ |
| 21 | 3 | [2025,399,14]₂ |
| 26 | 2 | [2025,404,8]₂ |
| 27 | 1 | [2025,405,7]₂ |
Thus 399 is the maximum dimension in this complete invariant line-parity preimage family that retains distance 14.
The six symmetry-breaking cases from Pass 4773 now have exact rational primal/dual certificates:
| Failure | Exact λ |
|---|---|
| one hot Petersen fiber | 67/5952 |
| two adjacent hot fibers | 665/59746 |
| two nonadjacent hot fibers | 133/11946 |
| one ten-vertex fiber removed | 189/16538 |
| two adjacent fibers removed | 1767/153094 |
| two nonadjacent fibers removed | 351/30670 |
Each certificate rechecks commodity conservation, surviving-edge capacities, dual normalization, exact shortest-path separation and primal/dual equality.
The [2025,378,14]₂ code has 1647 independent checks: 1485 local weight-two rows, 135 local weight-four rows and 27 global weight-fifteen line rows. They admit an optimal two-layer schedule under the model that simultaneous checks have disjoint physical-bit support.
Each local [15,3,7]₂ cell has 4096 syndromes and exactly eight representatives per syndrome. Because a nonzero zero-local-syndrome error has weight at least seven, every total error of weight at most six can be decoded by searching only cells with nonzero local syndrome; the raw worst-case candidate count is at most 8⁶=262144, with the 27 global syndrome bits selecting the unique candidate.
The 27 router line-parity checks are exactly pullbacks of the 27 line-vertex star vectors in the binary cut space of the 72-vertex, 135-edge GQ(4,2) Levi graph. They are cut-space checks, not homology checks.
The binary Levi cycle space has dimension 64 and girth 8. Repeating each Levi cycle across the three sheet coordinates and applying the local physical generator map gives a canonical [2025,64,96]₂ homology subcode with physical weight equal to 12 times Levi-cycle weight.
Binary Levi H¹ has neither PSp-fixed vectors nor a trivial quotient: fixed and coinvariant dimensions are both zero. Therefore the four-dimensional trivial socle in the 5671-dimensional flag H¹ cannot map nontrivially and PSp-equivariantly into Levi H¹, and Levi H¹ cannot transfer onto a trivial line. The tempting same-characteristic bridge fails at the module-intertwiner level.
The repository now contains exact producers for the PSp/PGSp actions on the 64-dimensional binary sign-cohomology module and the 225-dimensional twisted-F3 deformation module. It also contains a GAP/CTblLib producer for the exact characteristic-two Brauer decomposition matrix of U4(2)≅PSp(4,3), combined with the frozen ordinary H¹ character. Until those evidence jobs materialize, the complete astronomical S3-moduli orbit census and the full nontrivial Loewy-layer ordering are not promoted. Existing exact module-level data remain: trivial socle dimension four, trivial head dimension one and nonsplitting of the deck line.