Passes 4324–4334 — chamber Hecke machine and audited corrections

On the 160 W(3,3) chambers, P changes the line at fixed point and L changes the point on a fixed line. Native GAP proves:

The module decomposes as 1χ(3,3)+15χ(3,−1)+15χ(−1,3)+81χ(−1,−1)+24V₂, giving dimensions 1+15+15+81+48=160.

Hashimoto factorization and chirality

In the 320 oriented-edge basis, B=[[0,L],[P,0]] and B²=diag(LP,PL). The two products are disjoint 9-out halves of the chamber distance-two relation. For Ω=LP−PL:

The 24+24 channel is literally point plus line

Let Q_p and Q_ℓ be the chamber lifts of the eigenvalue-2 projectors of the point graph and dual line graph. Both have rank 24, their images meet only in zero, and their sum is exactly im Π₄₈. They are uniformly isoclinic:

All 24 principal angles therefore have cosine √6/4. If Q=Q_p+Q_ℓ, then the orthogonal span projector is rational:

Folded cubic normal form

F=−68Π₄₈−31X−(21/2)Ω+(2/3)XΩ;

    (F+68Π₄₈)²=−689Π₄₈.

This is an exact finite operator identity. It does not identify a continuum propagator, particle, mass or coupling.

Corrections established in the same release

The Pass 4253 Z₂₇₃₁ lift has a zero-voltage eight-cycle and girth exactly 8; its former girth-at-least-16 certificate is retracted.
The Kotani–Sunada irregular non-real-pole annulus uses square roots; the former band-filling metric did not.
p/f-swap invariance for A/B/C/D is false/true/false/true. Machine D removes both named asymmetries; its remaining localization is symmetric.
Affine translation acts on 81 frames but descends to neither projective 40-set.
Pass 4331's linear-opcode mismatch incidence census detects 1656/1920 differential substitutions and 0/960 shared-control substitutions. The broader Pass 4367/4374 intrinsic flag-register comparator detects 36/39 arbitrary one-register substitutions at q=3.
The universal-set census is 360; the shipped ISA is tied 7th–12th, not strictly 12th.

The degree-33 bridge

An irreducible degree-33 factor has exactly one root in (5.74,5.75) and exactly one in (3.349,3.350). The global Hashimoto growth root and the reported slow real root are therefore algebraic conjugates. The rational primary projector remains to be built.

Reproduction and boundary

The three native GAP witnesses replay 31/31, 28/28, and 17/17 exact checks. Their panel switches are three-way relations, not a deterministic selector or synthesized B/D datapath.