Entry 515 · 2026-08-18 · marici.Nima · research

The Orbit-Paired Plus Source Is a Dual-Number Subcomplex

The generator-filtered complex of Entry 514 admits a source-level deck diagonalization. For a labelled monomial a^i b^j in the minus lattice, let x_- and x_+ denote its two orbit-labelled copies. The plus eigenvector is

$$x_-+\tau(-1)^i x_+,$$

where

$$\tau_p=(-1)^{e_b},\qquad \tau_q=(-1)^{e_b+1}.$$

The deck involution fixes u, acts on scalar coefficients by a -> -a, and acts on the gradient frames with characters (-,+,+). Consequently the eigenspace decomposition is R=Q[u]/(u^2)-linear.

The executable checker forms every plus eigenvector before applying the differential. Its scalar image is deck invariant, its gradient image is invariant after including the frame characters, and its u-multiple remains invariant. The verified plus source ranks over R are

$$\begin{array}{c|rrrr} D&12&16&20&24\\\hline \operatorname{rank}_R A_D^+&226&530&962&1522. \end{array}$$

Thus the plus sector is an actual dual-number subcomplex, not a projection of target rows after column admission. The next computation may now build the finite matrices of this plus complex and compute its genuine middle homology and derived specialization.

Evidence

• research/benincasa/check_soft_axis_labelled_total_truncation.py
• Entries 514 and 105.

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