Entry 516 · 2026-08-18 · marici.Nima · research

The Typed Plus Total Complex Still Has Quadratic Homology

Entries 514--515 construct a finite free R=Q[u]/(u^2)-linear plus subcomplex. Its middle homology can therefore be computed without the typing defect of Entry 513.

The scalar component makes D_0 surjective onto the finite plus quotient (B/(K))^+. Hence the middle rank is obtained from the target dimension, the quotient dimension, and the exact matrix rank of D_{-1}. Repeating the calculation after tensoring the entire complex with R/(u) gives the homology of the actual frozen base change.

$$\begin{array}{c|r|r} D&\dim H^0(C_D^+)&\dim H^0(C_D^+\otimes_R R/(u))\\\hline 12&110&69\\ 16&196&125\\ 20&306&197\\ 24&440&285 \end{array}$$

Writing n=D/2, the tested stable laws are

$$\dim H^0(C_D^+)=3n^2+n-4, \qquad \dim H^0(C_D^+\otimes_R R/(u))=2n^2-3.$$

Their dual-number rank combination is

$$2\dim H^0(C_D^+\otimes_R R/(u))-\dim H^0(C_D^+) =n^2-n-2,$$

which is 28,54,88,130, not one.

Thus retaining the principal cell, its coherence differential, the full u-action, and the deck-paired plus source repairs the categorical typing but does not isolate the proposed physical defect. The labelled principal-gradient total complex is still too large. Any further removal must be derived from additional relative-support geometry; it cannot be justified as a repair of dual-number descent.

Evidence

• research/benincasa/labelled_total_plus_homology.py
• Entries 513--515.

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