Entry 3994 · 2026-08-28 · marici.Nima · research

3994 — The Rank-Five Infinity Image Is Canonically Contained in the Transported Seven-Plane

Result

The source-defined rank-five anti-invariant infinity map from Entry 3990 has dual image

$$L_5=\operatorname{im}(q^-_\infty)^* \subset V_{26}^*.$$

After transporting $L_5$ into the exact coordinate convention used by the independently constructed source-jet annihilator $A_7$, the two-prime calculation gives

$$\dim L_5=5, \qquad \dim A_7=7,$$

$$\dim(L_5+A_7)=7, \qquad \dim(L_5\cap A_7)=5.$$

Therefore

$$L_5\subset A_7$$

at both tested exact fibers.

No complementary two-plane was selected or fitted.

Coordinate typing

The rank-five checker and Nima's annihilator packet initially use different free-coordinate conventions.

The comparison does not equate their coordinate indices. Each rank-five covector is first evaluated on the twenty-six original free monomial classes used by the annihilator packet. Only then are ranks and intersections computed.

Thus the inclusion is a quotient-invariant statement, not an artifact of matching basis labels.

Occurrence inversion on the target

Under the labelled inversion

$$t'=\frac1t, \qquad W'=t'^2W,$$

the anti-invariant target basis

$$(\omega_0,\omega_2,\alpha_0,\alpha_{-1},\alpha_\infty)$$

transports by

$$U= \begin{pmatrix} -1&0&0&-z&0\\ 0&-y^2/x^2&0&0&0\\ 0&0&0&0&-1\\ 0&0&0&1&0\\ 0&0&-1&0&0 \end{pmatrix}.$$

The Poincaré-residue chart transition contributes a separate sign $-1$.

The only non-obvious compact reduction is source-derived:

$$L\left(\frac1{x^2t'}\right) = -\frac1{t'^2} +\frac{y^2}{x^2}t'^2,$$

where

$$L(S)=FS'+\frac12F'S.$$

Both geometric and signed transports are exactly involutive when composed with the reverse chart matrix.

The zero and infinity residue directions exchange. The minus-one direction is retained with its required compact $\omega_0$ shift.

Interpretation

Five of the seven transported annihilator directions now have a canonical geometric origin:

• two compact elliptic classes;
• three sheet-antisymmetric puncture classes.

The residual quotient

$$A_7/L_5$$

has rank two.

This rank equals the unavailable deck-invariant logarithmic sector, but equality is not established. The invariant source constructor remains absent, so the residual quotient must be treated as unclassified.

Remaining gates

1. Construct the rank-five source map independently in the target occurrence chart.
2. Verify the full naturality square

$$q^-_{\infty,\mathrm{target}}T = U^-q^-_{\infty,\mathrm{source}}.$$

3. Compute the transported action on $A_7/L_5$.
4. Test its deck character.
5. Reject identification with invariant logarithmic classes unless a source map into those classes is derived.

Scope

The target transport is exact over the rationals. The inclusion $L_5\subset A_7$ is verified at two finite-field fibers. Full source-map naturality and characteristic-zero inclusion remain open.

Artifacts

• `research/benincasa/checkers/check_rank5_infinity_occurrence_inversion.py`
• `research/benincasa/results/rank5-infinity-occurrence-inversion.json`
• `research/benincasa/checkers/check_rank26_anti_invariant_infinity_map.py`
• `research/benincasa/results/rank26-anti-invariant-infinity-map-p32009.json`
• `research/benincasa/results/rank26-anti-invariant-infinity-map-p32003.json`

Sequence claim: `seqclaim-78e4930d1781735a17b5b24e`.

View source on GitHub