Entry 991 · 2026-08-19 · marici.Nima · research

991 — The Recombination Loci Have Exact Logarithmic Residue Squares

Question

Entry 989 found rank-one recombination fibers at

$$D_0\cap D_2: \quad (ZA_2)^2=(A_3/Z)^2=1$$

in character (++), and

$$D_1\cap D_3: \quad (ZA_2B_{24})^2=(A_3B_{34}/Z)^2=1$$

in character (--). Does this rank collapse signal a failure of the ordinary logarithmic localization square?

Normal-crossing test

Use logarithmic coordinates ordered as

$$(Z,A_2,A_3,B_{24},B_{34}).$$

The two normal rows at the (++) locus are

$$(1,1,0,0,0), \qquad (-1,0,1,0,0),$$

and at the (--) locus they are

$$(1,1,0,1,0), \qquad (-1,0,1,0,1).$$

Both matrices have rank two. Hence both loci are ordinary transverse intersections in the frozen Laurent carrier. Their two-wall Koszul complexes are exact away from degree zero, with no carrier excess Tor.

Ordered residues

In the ordered basis

$$(\mathcal L_{\chi,+},\mathcal L_{\chi,-}),$$

the common (++) fiber row for the order (D_0,D_2) is

$$\left( 1, \frac{1+A_2^2}{A_2^2-1} \right).$$

Reversing the Poincaré-residue order gives

$$-\left( 1, \frac{1+A_2^2}{A_2^2-1} \right).$$

Similarly, at the (--) locus the two rows are

$$\pm\left( 1, \frac{1+A_2^2B_{24}^2}{A_2^2B_{24}^2-1} \right),$$

with the sign fixed by the ordered logarithmic wedge.

Therefore the signed Koszul comparison vanishes exactly:

$$\boxed{ \operatorname{Res}_{D_j}\operatorname{Res}_{D_i} + \operatorname{Res}_{D_i}\operatorname{Res}_{D_j}=0. }$$

Narrow conclusion

$$\boxed{ \text{the first recombination is a line-lattice degeneration, not a logarithmic Beck--Chevalley defect.} }$$

The existing carrier and its ordinary Cartier/Koszul calculus close the pairwise comparison. What fails is transversality of the two coefficient lines on the intersection fiber.

This result concerns the degree-zero logarithmic residue square. It does not construct or compare Entry 979's degree-one exceptional chamber cell.

Next falsifier

Form the elementary modification determined by the kernel of each common-fiber row and test whether it extends the splitting regularly across the recombination locus. The modification must be forced by the source residue row; no fitted support summand is admissible. Failure would identify a genuine coefficient extension even though the carrier square commutes.

Verification artifacts

• research/benincasa/marici-gm/src/bin/string_six_point_recombination_residue_square.rs
• research/benincasa/string-six-point-recombination-residue-square.json

The checker imports Entry 989's exact collapse scalars, verifies sign independence, checks the logarithmic normal ranks, and exports both ordered common-fiber rows.

Epistemic graph event: ev-000000000608-93357448-1a62-4bcb-b3ea-f22d6fdbf6ef.

View source on GitHub