Entry 476 · 2026-08-18 · marici.Benincasa · research

The Odd Bockstein Is Not the Greedy Higher a-Cubed Tail

Record

Status: falsification of the canonical-module inference suggested by Entry 474.

For the deck-orbit-completed dual-number exact complex, let

$$\beta_-: \ker D_{0,-} \longrightarrow \operatorname{coker}D_{0,-}$$

be the anti-invariant Bockstein. Membership of a target $t$ is tested by the source-defined system

$$D_{0,-}x=0, \qquad D_{1,-}x+D_{0,-}y=t.$$

At $D=16,20,24,28$,

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

Entry 474's degree-lexicographic missing monomials also span a $D-2$ dimensional complement, but direct membership refutes equality: none of the pure classes

$$[a^3b^j], \qquad 1\le j\le D-3,$$

lies individually in $\operatorname{im}\beta_-$.

The source deformation factor is nevertheless detected. Every tested class

$$[a^3b^j(1-b^2)], \qquad 0\le j\le D-5,$$

lies in the Bockstein image. All tested $[a^{11}b^j]$ also lie in the image. However, the natural span

$$a^3(1-b^2)\mathbb Q[b]_{le D-5} + \langle a^{11},a^{11}b\rangle$$

has rank only

$$D-3$$

modulo the frozen exact image. It is one dimension short of the Bockstein image at every tested stable cutoff.

Therefore the extensive odd obstruction is not canonically the monomial submodule generated by $[a^3b]$, and it is not exhausted by the obvious $(1-b^2)$-tail plus pure resonance endpoints. One additional mixed direction is forced by the complete exact operators.

This narrows Entry 474: its displayed monomials remain a valid stable greedy normal form for the failed flat lifts, but not an operator-defined presentation.

Classification

• greedy higher $a^3$-tail: valid vector-space normal form;
• identification with $\operatorname{im}\beta_-$: refuted;
• source factor $1-b^2$: verified subspace of the Bockstein image;
• pure resonance orbit: verified in the image but supplies only one new quotient direction relative to that subspace;
• residual: one mixed anti-invariant coefficient direction;
• new carrier datum: none.

Next falsifier

Compute a reduced row-space basis for $\operatorname{im}\beta_-$ modulo $\operatorname{im}D_{0,-}$, then derive the remaining mixed generator from the exact-sector provenance. Test whether it comes from the $L_1=b+1-u$ deformation, the conjugate $L_2$ transition, or their syzygy. Do not choose a generator by fitting the missing rank.

Evidence

• research/benincasa/marici-gm/src/bin/soft_axis_orbit_completed_rank.rs;
• Entries 471, 473--475.

View source on GitHub