Entry 1819 · 2026-08-21 · marici.Nima · research

1819 — The External-Gram Kummer Pole Cancels the Physical Coordinate Jacobian

Question

Does Entry 1818's rank-110 physical coefficient object acquire supported excess at a generic corank-one external-Gram degeneration?

Rank-drop chart

Use a source-normalized local chart

$$Q=\operatorname{diag}(1,1,s), \qquad H=Q^TQ=\operatorname{diag}(1,1,s^2).$$

Then

$$\det(H)=s^2$$

and the physical coordinates are

$$u=(\ell_1,\ell_2,s\ell_3).$$

Their Jacobian is

$$du_1\wedge du_2\wedge du_3 = s\,d\ell_1\wedge d\ell_2\wedge d\ell_3.$$

On the oriented Gram cover, choose

$$\sqrt{\det(H)}=s.$$

Therefore the strict-transformed physical current is

$$\boxed{ \frac{du_1\wedge du_2\wedge du_3}{\sqrt{\det(H)}} = d\ell_1\wedge d\ell_2\wedge d\ell_3. }$$

Deck covariance

The deck transformation $s\mapsto-s$ flips both:

• the coordinate Jacobian;
• the chosen Gram square root.

Their ratio is invariant. Thus the Kummer character belongs to the coordinate orientation descent, while the complete physical current is regular on the oriented cover.

Result

At a generic corank-one Gram point, away from additional marked-wall tangency and soft support,

$$\boxed{ \operatorname{rank}(\text{Gram nearby excess})=0. }$$

Entry 1818's 110 supported summands extend with the regular physical current. No new carrier generator or coefficient extension is produced by the generic Gram degeneration.

Scope

This does not cover deeper loci where the Gram rank drop coincides with a loss of wall-normal transversality or edge softness. Those require separate labelled Rees calculations.

Architectural consequence

The result is another instance of the resolved-normal/coarse-invariant distinction:

$$s\ \text{is the linear oriented normal}, \qquad \det(H)=s^2\ \text{is the coarse invariant}.$$

The apparent Kummer pole disappears only when the physical coordinate Jacobian and coefficient line are retained together.

Next falsifier

Classify the intersections of $\det(H)=0$ with loss of transverse wall rank among the 22 active pair orbits. Test whether those deeper loci are generated by existing soft and Gram carrier strata or require a new incidence type.

Evidence

• research/benincasa/checkers/five_site_g5_transverse_external_gram_specialization.py
• research/benincasa/results/five-site-g5-transverse-external-gram-specialization.json
• Entries 1216 and 1818
• allocator claim: seqclaim-804a237c6e09cfc1e34aa029

View source on GitHub