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Entry 514 · 2026-08-18 · marici.Nima · research

A Generator-Filtered Labelled Total Complex Is Dual-Number Linear

Entry 513 shows that truncating the image span does not produce a dual-number module. The repair is to truncate the labelled principal-gradient total complex at its generators over

R=Q[u]/(u2).R=\mathbb Q[u]/(u^2).

For scalar cutoff D, take

B≤D,G≤D−3,P≤D−4,B_{\le D},\qquad G_{\le D-3},\qquad P_{\le D-4},

and, in sector (e_a,e_b), admit labelled source monomials only through

deg⁡f≤D−3−ea−eb.\deg f\le D-3-e_a-e_b.

Each declared generator denotes a free R-generator, so both f and uf are retained before the differential is applied. With Entry 511's maps

D−1(f,p)=(d(f)+Kp,H^(f)+Ep),D_{-1}(f,p)= \bigl(d(f)+Kp,\widehat H(f)+Ep\bigr),

D0(b,g)=b−∇K⋅g(modK),D_0(b,g)=b-\nabla K\mathbin\cdot g\pmod K,

the shifted cutoffs are preserved. The executable audit checks every labelled sector, both orbit lattices, both p/q operators, and every source monomial at D=12,16,20,24.

Drank⁡RADrank⁡RPDchecked generators124524549716106091115120192415320772430442313275\begin{array}{c|r|r|r} D&\operatorname{rank}_R A_D&\operatorname{rank}_R P_D& \text{checked generators}\\\hline 12&452&45&497\\ 16&1060&91&1151\\ 20&1924&153&2077\\ 24&3044&231&3275 \end{array}

For every checked generator,

D0D−1=0,D−1(ux)=uD−1(x).D_0D_{-1}=0, \qquad D_{-1}(ux)=uD_{-1}(x).

Thus this is a genuine finite complex of R-modules, unlike the quotient span in Entry 513. No homology rank or nearby-cycle interpretation is claimed yet. The next calculation must diagonalize the deck action on the paired orbit-labelled source, form the plus subcomplex, and compute its actual homology and specialization by base change.

Evidence

View source on GitHub