Entry 512 · 2026-08-18 · marici.Nima · research

The Frozen Koszul Filtration Does Not Descend to QD Homology

The authoritative finite census of commit 56fa6a9 was extended without changing its target basis, source basis, orbit completion, plus projection, or whole-column cutoff. Let

$$I_D=\langle\widehat H(A_D),E(P_D)\rangle\subset G_D, \qquad Q_D=G_D/I_D,$$

and use the frozen gradient

$$(K_a,K_b,K_u)=(4a^3,0,a^2(1-b^2)).$$

For the elementary Koszul columns

$$s_{ab}=(-K_b,K_a,0),\quad s_{bu}=(0,-K_u,K_b),\quad s_{au}=(-K_u,0,K_a),$$

define S0 from the first two and S1=S0+<s_au>, with all monomial multiples admitted by the census convention. The necessary cycle condition for their images to define a filtration on H(Q_D,u) is

$$uS_i\subset I_D.$$

It fails by the following exact rank increments:

$$\begin{array}{c|cc} D&\operatorname{rk}(I_D+uS_0)-\operatorname{rk}I_D& \operatorname{rk}(I_D+uS_1)-\operatorname{rk}I_D\\\hline 12&15&22\\ 16&21&32\\ 20&27&42\\ 24&33&52 \end{array}$$

The laws on the tested stable range are 3D/2-3 and 5D/2-8. Consequently the proposed F0 subset F1 is not typed on dual-number homology.

As a secondary diagnostic only, ignoring the cycle failure gives formal rank triples

$$(5,20,6),\ (7,42,8),\ (9,72,10),\ (11,110,12),$$

for (F0,F1/F0,H/F1). In particular the untyped residual is D/2, not one. Thus neither the actual homological construction nor its formal shadow supports the proposed two-triangles-plus-one removal.

The next admissible step is not to quotient these elementary syzygies. One must derive corrected u-closed Koszul representatives from the labelled principal-gradient total complex, or prove that no such correction exists.

View source on GitHub