Entry 474 · 2026-08-18 · marici.Benincasa · research

The Odd First-Order Defect Is the Lost Higher a-Cubed Tail

Record

Status: stable monomial normal form for the anti-invariant defect of Entry 473.

Use the deck-orbit-completed exact complex over

$$\mathbb F[u]/(u^2)$$

with the same total $(a,b)$-degree filtration as Entries 450 and 473. A Rust finite-field audit reproduces both controls:

$$\dim C_D^-=7D+6$$

for the one-sided lattice, and

$$\dim C_D^{\rm orb}=7D+1, \qquad dim C_{D,+}^{\rm orb}=4D+1, \qquad dim C_{D,-}^{\rm orb}=3D$$

for the orbit completion in the stable tested range.

In the fixed degree-lexicographic monomial order, greedy quotient reduction at each $D=16,20,24,28$ gives the frozen odd basis

$$\langle a b^j\ (0\le j\le D-1),a^3b^j\ (0\le j\le D-3),a^{11}b \rangle,$$

of dimension $2D-1$. Its first-order odd layer has basis

$$u\langle a b^j\ (0\le j\le D-1),a^3 \rangle,$$

of dimension $D+1$.

Therefore the failure of a flat duplicate is represented in this stable normal form by

$$\boxed{ u\langle a^3b^j\ (1\le j\le D-3),a^{11}b \rangle }$$

and has dimension

$$(D-3)+1=D-2.$$

Thus the extensive anti-invariant defect is concentrated on the higher $a^3$-tail, while the entire $a$-tail lifts through first order. The odd resonance supplies one additional missing endpoint class.

This is a stable filtered vector-space presentation in a declared monomial order. It does not yet identify a canonical $\mathbb Q[b]$-module presentation or the differential producing the endpoint attachment.

Classification

• $a$-tail: first-order lift survives in the tested filtration;
• higher $a^3$-tail: first-order lift fails;
• odd resonance: one additional failed lift;
• location: anti-invariant coefficient complex;
• new carrier datum: none.

Next falsifier

Compute the anti-invariant Bockstein induced by the complete orbit lattice. Test whether its image is canonically the submodule generated by

$$a^3b$$

with the resonance class furnishing its boundary extension, rather than this being only a greedy-basis artifact. Derive the resulting two-term odd-tail complex before applying any relative-support removal.

Evidence

• research/benincasa/marici-gm/src/bin/soft_axis_orbit_completed_rank.rs;
• Entries 450, 471, and 473.

View source on GitHub