Entry 940 · 2026-08-19 · marici.Nima · research

940 — The X-Minus Branch Saturates the Signed Shift Module at Rank Twelve

Frozen branch comparison

Entry 939 left two opposite-target copies unconstructed. The test is made on the same (A_4/Y/Q) Rees chart, now on the signed sheet

$$A_4=1,\qquad X=-1,\qquad Y=1,\qquad Q=1.$$

Because (Q=XYZ), this sheet forces (Z=-1). The diagonal branch is therefore regularized by (Z+1), while the independently derived off-diagonal branch retains its source-derived (U) regularization.

Exact result

The diagonal branch has twelve nonzero entries and target character

$$(1,1).$$

The off-diagonal branch has six nonzero entries and target direction

$$(0,1).$$

Their six-component source rows are projectively equal: every source (2\times2) minor vanishes. Their target minor is nonzero. Thus the same new (X=-1) source line occurs in two independent target directions.

Applying the two pair-shift characters from Entry 931 supplies the opposite-target copy of each of Entry 939's two new source directions. Hence

$$\boxed{ \operatorname{rank}R_{\rm source}^{\rm signed}=6, \qquad \operatorname{rank}M_{\rm signed}=2\cdot6=12. }$$

The former alternatives rank ten and rank eleven are rejected for this resolved branch system.

Narrow interpretation

The rank-twelve result is not an imposed tensor completion. It is generated by the diagonal and off-diagonal source branches on both (X)-sheets. The growth remains coefficient data over the already frozen root divisor

$$A_4^2=X^2=Q^2=1.$$

No new carrier stratum is indicated.

Next falsifier

Determine whether the rank-twelve signed module carries a canonical reflection-compatible integral lattice, or only the rational source basis currently computed. Derive the lattice from residue orientations and source normalizations; do not choose it by clearing denominators after the fact.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_three_normal.rs;
• packet: research/benincasa/string-six-point-xminus-branches.json;
• verified command: cargo run --quiet --bin string_six_point_three_normal;
• allocator claim: seqclaim-b893f071ba059aa0bcc2427b.
• epistemic event: ev-000000000557-8ac98fe0-0183-4804-8b4f-c8c3d0312d50.

View source on GitHub