This packet combines the all-q apartment tight frame with minimum apartment bases, the integral Levi cycle lattice, the q=5 zero-footprint code, and the q=3 provenance decoder.
For rank r=q⁴ signed apartment incidence and N=(q+1)²(q²+1),
Σ|S|=r det(CSᵀCS) = Nʳ.
Every independent apartment set generates a finite-index sublattice of the integral Levi cycle lattice, so
det(CSᵀCS) = τ(Levi) · [Λ:ΛS]²,
and therefore Σ m_S²=Nʳ/τ(Levi). At q=3 the integer sum is 2^322·5^58.
The full 1,620-apartment frame has redundancy 20 and condition number 1 on the 81-dimensional cycle sector. The deterministic 81-apartment BFS basis has condition √160. Its inverse norm is worse by exactly √160. This compares that specific basis to the full frame; it is not an optimality theorem over all bases.
Pass5376 closes the point-side kernel, while the apartment residual remains nonzero:
M₀/CW ≅ Rad(im Fᵀ), dim=g−1,
0 → Dap=K₀ → H₁* → CF → 0, dim Dap=q⁴−g.
The point-kernel closure must not be conflated with the apartment kernel.
A weight-1040 K₀ word would need exactly 26 active P-blocks, all local weight-40 minima. Shell moments and the exact 3-class block association scheme leave only 21 integer pair profiles, with 310–316 of the 325 internal pairs in the high-codegree relation. All 46,800 “known 25-block minimum support + one block” extensions fail. Weight 1040 is not ruled out; any witness must have an exotic 26-block support.
Exact MILPs eliminate false-center sectors V=1 and V=2. V=3 reduces to two local branches. V=4 shrinks from 10,963,673,616 raw rooted cases to exactly 35,218,336 necessary cases. Representative survivors self-heal, but the exhaustive radius-11 closure remains open.
For odd q, the unsigned apartment Gram determinant has square class
(q²+1)/2.
It is a perfect integer square exactly when q²−2m²=−1. The first nontrivial Pell q-coordinates are 7, 41, 239, 1393, 8119, …; only prime-power q are admissible finite-field parameters. Negative Pell also forces q≡±1 (mod 8), so every odd square-determinant case lies in the F₂-split logical Weil branch of Pass5361/5406. The converse is false, for example q=9.
Evidence boundary: no q=5 weight-1040 nonexistence theorem, no q=3 radius-11 theorem, no optimal-basis conditioning theorem, and no physical interpretation of Pell solutions is asserted.