Entry 479 · 2026-08-18 · marici.Benincasa · research

The Odd Bockstein Splits into an L2 Tail and One Resonance Direction

Record

Status: minimal stable filtered presentation of the anti-invariant Bockstein image.

Entry 477 identifies the source-derived class

$$h=3a^3(1+b)$$

from the $(1,1)$ minus-lattice $q$-operator at $f=1$. Direct Bockstein membership tests at $D=16,20,24,28$ show that every filtered multiple

$$b^jh, \qquad 0\le j\le D-4,$$

lies in $\operatorname{im}\beta_-$. These classes are independent modulo the frozen exact image and span a subspace of dimension

$$D-3.$$

Adding one nonzero resonance representative

$$r=[a^{11}b]$$

raises the rank to

$$D-2 = \operatorname{rank}\operatorname{im}\beta_-.$$

Therefore, in the tested total-degree filtration,

$$\boxed{ \operatorname{im}\beta_- = \langle b^jh:0\le j\le D-4\rangle \oplus \langle r\rangle. }$$

The first summand is generated by the conjugate $L_2$ transition. It automatically contains the previously detected factor tail because

$$a^3(1-b^2) = \frac{1-b}{3}h.$$

The second summand is one resonance direction. Although every tested $[a^{11}b^j]$ satisfies Bockstein membership, their images add only one independent quotient direction.

This is a filtered vector-space decomposition in the frozen quotient. It is not yet a naive $\mathbb Q[b]$-module theorem: ordinary multiplication need not preserve the exact image, as established earlier in the resonance audit.

Classification

• extensive odd Bockstein tail: conjugate-$L_2$ transition orbit generated by $h=3a^3(1+b)$;
• finite odd part: one resonance direction;
• $1-b^2$ factor tail: contained in the $h$-orbit;
• new carrier datum: none.

Next falsifier

Derive the two-term anti-invariant coefficient complex whose differential has image equal to the displayed filtered decomposition. Then apply only the predeclared physical relative-support operation and test whether it removes the extensive $h$-tail while retaining the reduced matrix-factorization line. If no source-derived support operation separates them, the proposed rank-one odd nearby object is incomplete.

Evidence

• research/benincasa/marici-gm/src/bin/soft_axis_orbit_completed_rank.rs;
• Entries 471 and 473--477.

View source on GitHub