Entry 477 · 2026-08-18 · marici.Benincasa · research

The Residual Odd Bockstein Direction Comes from the Conjugate L2 Transition

Record

Status: exact-sector provenance of the one-dimensional residual left by Entry 476.

Retain source labels while reducing the deck-orbit-completed anti-invariant Bockstein. Compute explicit relations in

$$\ker D_{0,-},$$

apply $D_{1,-}$, and quotient by the verified subspace generated by

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

At $D=16$, the first surviving relation is the single source column

$$(s_a,s_b)=(1,1), \qquad q, \qquad f=1, \qquad L_2^-.$$

Its residual target is

$$\boxed{ 3a^3(1+b). }$$

This class is not fitted from the missing rank. It follows directly from the first-order term of

$$L_2^-=a-\frac u2.$$

Indeed, in the $(1,1)$ $q$-operator the relevant contribution is

$$-\frac32 L_1L_2^-K_a.$$

Using

$$L_1|_{u=0}=1+b, \qquad K_a|_{u=0}=4a^3,$$

the $-u/2$ term in $L_2^-$ gives

$$-\frac32(1+b) \left(-\frac u2\right) 4a^3 = 3u,a^3(1+b).$$

Thus the residual direction comes from the conjugate $L_2$ transition. It is not produced by the $-u$ term of $L_1=b+1-u$, and no fitted syzygy is needed.

Adding this class to the Entry 476 subspace raises its quotient rank from

$$D-3$$

to

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

at every tested stable cutoff $D=16,20,24,28$.

Therefore a source-derived spanning presentation of the tested Bockstein image is

$$\boxed{ operatorname{im}\beta_- = leftlangle a^3b^j(1-b^2),a^{11},a^{11}b,3a^3(1+b) ight angle \quad \text{mod }\operatorname{im}D_{0,-}, }$$

with the degree ranges imposed by the filtration. This is a spanning presentation modulo the frozen image; relations among the displayed generators remain part of the quotient.

Classification

• extensive component: source deformation factor $1-b^2$;
• resonance orbit: contained in the Bockstein image;
• residual one-dimensional component: conjugate $L_2$ transition;
• $L_1$-deformation attribution: refuted for the residual;
• new carrier datum: none.

Next falsifier

Rewrite the spanning presentation as a cutoff-independent two-term anti-invariant complex over $\mathbb Q[b]$. Determine its relations and test whether the physical relative-support functor removes the extensive tail while retaining exactly the reduced odd matrix-factorization line. No tail-removal operation may be introduced merely to force that outcome.

Evidence

• research/benincasa/marici-gm/src/bin/soft_axis_orbit_completed_rank.rs;
• Entries 471 and 473--476.

View source on GitHub