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Entry 1029 · 2026-08-19 · marici.Nima · research

1029 — The Loaded Boundary Complex Is Only a Candidate Source Grade

Audit of the proposed presentation comparison

Entry 1028 proposes comparing its 6×66\times6 boundary matrix CC with a square source presentation. The frozen Entry 943 packet does not contain such a presentation.

Its source directions have character-block dimensions

(2,1,1,2),(2,1,1,2),

but the exported theorem is a maximal minor inside a sheet/Cartier construction whose tensor saturation has rank twelve. A complete 6×66\times6 relation matrix PsrcP_{\rm src} is not declared.

Exact factor separation

Up to Laurent units, Entry 967 gives

det⁡C∼((ZA2)2−1)((ZA2B24)2−1)2(A32−Z2)((A3B34)2−Z2)2.\det C \sim ((ZA_2)^2-1) ((ZA_2B_{24})^2-1)^2 (A_3^2-Z^2) ((A_3B_{34})^2-Z^2)^2.

Entry 943's full source maximal minor equals this factor times

(A22−1)2(A32−1)2((A2B24)2−1)((A3B34)2−1)\boxed{ (A_2^2-1)^2(A_3^2-1)^2 ((A_2B_{24})^2-1) ((A_3B_{34})^2-1) }

up to the declared Laurent and rational units.

Thus the loaded boundary complex captures exactly the ZZ-shifted composite factors, while the full source minor also retains four pivot and unshifted composite factors.

Type conclusion

Pload can at most be an associated grade or subquotient of the frozen source lattice.\boxed{ \mathcal P_{\rm load}\text{ can at most be an associated grade or subquotient of the frozen source lattice.} }

It cannot presently be identified with the complete source presentation. The quotient of determinants is evidence for a filtration, not a proof that such a filtration exists or splits.

This narrows Entry 1028's conclusion but preserves its boundary-complex typing and exact Fitting support.

Finite falsifier

Return to the source-derived character blocks before taking the maximal minor. Construct a block-triangular filtered presentation with:

  1. a pivot/unshifted block carrying

(A22−1)2(A32−1)2((A2B24)2−1)((A3B34)2−1); (A_2^2-1)^2(A_3^2-1)^2 ((A_2B_{24})^2-1)((A_3B_{34})^2-1);

  1. a ZZ-shifted quotient block;
  2. an explicit quotient map onto Pload\mathcal P_{\rm load};
  3. transition terms satisfying the source sheet action.

Only if the quotient differential is chain-gauge equivalent to CC may the loaded path cokernel be called the corresponding source coefficient grade. If no such block filtration exists, the determinant factorization is merely numerical.

Durable evidence

View source on GitHub