← Exceptional Geometry home

Fixed-point obstruction and the apartment extension cocycle

Pass 4490 — nonsplitting is already visible on fixed points

The exact apartment sequence is

0 → K/J (29) → E=M/J (39) → V=M/K=H10 (10) → 0.

The ambient 40-line permutation module has fixed space exactly J=<1>. After quotienting by J,

E^PSp = 0.

Protected H10 has one fixed line:

V^PSp ≅ F₂.

An equivariant section would have to lift the protected fixed vector to a fixed vector in E. None exists. This conceptually explains the earlier affine rank obstruction 389→390.

Pass 4491 — where the obstruction lives

Choose a lift e of the fixed protected class and define c_g=G_E(g)e−e. The PSp-module generated by those defects is exactly

(K ∩ R⊥)/J, dimension 23.

Its invariant chain is J<U<I<K∩R⊥ with dimensions 1<9<10<24, giving cocycle-support profile

8 | 1 | 14.

The support cannot be gauge-shifted into I/J: the corresponding affine coefficient rank is 30 while the augmented rank is 31. A representative exists in the full 23-space.

Pass 4492 — the 23-space is route hull glued to sentinel code

Let R=ker N, U=R∩R⊥, and let C=ker Nᵀ be the established [40,15,8] sentinel code. Incidence itself gives

0 → U (9) → K∩R⊥ (24) --N→ C (15) → 0.

The image is all of C and the kernel is exactly U. Modding out the line all-ones vector gives

0 → U/J (8) → (K∩R⊥)/J (23) → C (15) → 0.

Pass 176 already owns the route-hull eight-core U/J; Pass 201 owns the sentinel code; Pass 187 owns the sentinel 1|14 filtration. The new result is their exact incidence weld inside the apartment nonsplitting cocycle:

8 | (1 | 14).

Boundary. These are modular representation and cohomology statements. They do not describe a physical error current, energy transfer, or a measured fault path.