This packet starts from the all-q Levi Hodge and apartment-spectrum theorems and extracts exact reconstruction, stability, determinant, a modular radical corollary, q=5 post-minimum structure, and q=3 decoder consequences.
For the signed flag-by-apartment matrix C,
CCᵀ = N Ecyc, N=(q+1)²(q²+1), C⁺=(1/N)Cᵀ.
Every real Levi cycle flow therefore has a canonical minimum-norm apartment reconstruction. Normalized apartments form a unit-norm tight frame of bound N/8.
The unsigned apartment visibility matrix has full row rank and
κ(BBᵀ)=8q⁴ / ((q−1)²(q²+1)).
The smallest mode is always the terminal q⁴-multiplicity sector. Changing only odd-distance shell signs collapses the full five-sector Gram operator exactly onto the Hodge cycle projector.
The conjugate eigenvalues 2q²(q−1)(q+1±√(2q)) pair into
Qq(x)=x²−4q²(q−1)(q+1)x+4q⁴(q−1)²(q²+1).
Hence the complete characteristic polynomial is integral, and the radical pair cancels from the exact Gram determinant. At q=3, det(BBᵀ)=2⁴⁵⁶·3¹⁶⁰·5¹⁰⁵.
Pass5376 already proves rank₂(F)=g and ker(Fᵀ)=CW for every odd prime power. Combining that theorem with Pass5350 gives
M₀/CW ≅ Rad(im Fᵀ), dim=g−1.
The transient extra point-kernel quotient Xq=ker(Fᵀ)/CW is therefore identically zero. This is distinct from Pass5421's nonzero apartment-to-footprint kernel Dq in 0→Dq→H₁*→CF→0, where dim Dq=q⁴−g.
For q=5, K₀=[73125,560,1000]₂ has no weights 1001–1039; every non-minimum word has weight at least 1040.
The provenance decoder now has a global eventual guaranteed radius of 10. At weight ten, V=1 and V=2 false-center sectors are infeasible by exact MILP. For V=3, direction counting leaves exactly 1,296 completion candidates and every one clears. The V=4 raw census of 20,544,048 cases prunes to 349,568 necessary survivors; every one clears under the actual deterministic decoder.
reventual=10, rmonotone true-only=7.
Radius 11 remains open.
Evidence boundary: Pass5431 is a corollary/reconciliation of the already-proved Pass5376 all-odd rank theorem, not a second proof of it; tomography/spectral statements are characteristic-zero real-linear results; the decoder theorem is computational and specific to the deterministic q=3 provenance decoder.