Corrected symmetry-breaking boundary of the apartment extension

Correction. The earlier Pass-4493 claim that one point or one line stabilizer split the extension was a false positive. Pass 4503 rebuilt the exact section systems and supersedes that restricted-subgroup table. The full-group nonsplitting result was not affected.

Full symmetry: no section

0 → K/J (29) → E=M/J (39) → V=M/K=H10 (10) → 0 rank(A)=389 rank([A|b])=390

All maximal subgroup types remain nonsplit

The five maximal subgroup types of PSp(4,3)≅U4(2) have orders 960, 720, 648, 648 and 576. Exact representatives give:

2^4:A5, order 960 388 / 389 spread stabilizer S6, order 720 386 / 387 line stabilizer, order 648 386 / 387 point stabilizer, order 648 387 / 388 class-45 involution centralizer, 576 386 / 387

Every coefficient/augmented rank pair is inconsistent, so none supplies an equivariant protected complement.

An incident flag does split

The canonical incident point-line flag stabilizer has order 162/index 160 and gives

rank(A)=rank([A|b])=384 affine section-family dimension = 6 number of sections = 64

Thus a flag gauge is a verified symmetry reduction that selects a 10-dimensional representative section.

Exact gauge compiler

Pass 4504 exhausts all 64 flag-equivariant sections. In the canonical quotient basis, the optimum has total minimal ambient Hamming weight 42, maximum column weight 9, and a 13-line union support, with column weights 1,1,1,1,5,5,5,5,9,9.

Architectural reading

full group + every maximal subgroup type ⇒ protected information remains quotient-only incident point-line flag ⇒ equivariant 10D representatives become selectable

Boundary. Order 162 is a verified splitting subgroup, not a proof that no larger non-maximal subgroup can split. The section optimization is finite-module software gauge synthesis, not automatically a decoder, optical measurement plan or hardware-cost optimum.