Entry 939 · 2026-08-19 · marici.Nima · research

939 — The X-Minus Sheet Enlarges the Shift Module from Eight to Ten

Typed signed-sheet census

Entry 937 treats the $A_4$ first-grade sheet. The maximal-flag packet also contains two regularized root specializations, $X=\pm1$ and $Q=\pm1$. These must not be assigned the same variance as the $A_4$ Cartier grade.

The calculation is therefore ordered as:

1. take the first $A_4$-grade;
2. take the regularized $X$-root value;
3. take the regularized $Q$-root value.

All eight signed fibers are derived from the same unspecialized twelve-entry transition.

A and Q sheets

Changing the $A_4$ sheet leaves the native coefficient matrix unchanged; Entry 937 supplies the separate oriented normal sign.

Changing $Q=+1$ to $Q=-1$ also leaves the native coefficient matrix unchanged in every tested $(A_4,X)$ sheet:

$$G_{a,x,-}=G_{a,x,+}.$$

No new native coefficient direction is generated by $Q$.

X sheet

The $X=+1$ and $X=-1$ matrices are each rank one, but they are not projectively equal. Their target-row characters are

$$X=+1:(1,-1), \qquad X=-1:(1,1).$$

Their source rows are also projectively distinct.

Entry 931's tangential pair shifts decompose the source coordinates into four character supports. The $X=-1$ source row escapes the old line in both two-coordinate supports,

$$\{0,2\}, \qquad \{4,5\},$$

while the singleton supports add nothing. Hence

$$\boxed{ \operatorname{rank}R_{\rm source}^{\rm signed}=4+2=6. }$$

Minimal module rank

The established $X=+1$ normal-symbol module is

$$V_2\otimes R_4, \qquad \dim=8.$$

The computed $X=-1$ fiber contributes the two new source directions only in its displayed target line. Therefore the source-derived closure currently proved has rank

$$\boxed{ 8+2=10. }$$

The fully tensor-saturated candidate would have rank

$$2\cdot6=12,$$

but the two opposite-target copies of the new source directions have not been constructed and are not inferred.

Consequence

The full resolved-sheet structure is not exhausted by Entry 934's rank-eight tangential difference system:

$$\boxed{ \text{tangential shift closure }8 \subsetneq \text{computed signed-sheet closure }10 \subseteq \text{possible tensor saturation }12. }$$

This is coefficient growth over the unchanged coarse divisor $A_4^2=X^2=Q^2=1$. It is not evidence for a new carrier stratum.

Next falsifier

Construct the other two branch symbols on the $X=-1$ sheet. Determine whether either supplies the missing opposite-target copies:

• if neither does, rank ten is the complete signed-sheet module;
• if both new source directions acquire the second target character, the closure saturates to rank twelve;
• partial filling would give rank eleven.

No tensor-product completion may be imposed without this branch calculation.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_cartier_sheet_transition.rs;
• packet: research/benincasa/string-six-point-cartier-sheet-transition.json;
• allocator claim: seqclaim-6b14721e16595a3ba75015b4.
• epistemic event: ev-000000000556-888fccf2-0505-4890-a768-736f3ce69b78.

View source on GitHub