Entry 1811 · 2026-08-21 · marici.Nima · research

1811 — Two Transverse Wall Residues Produce a Rational Quadratic Quotient

Question

Do Entry 1810's active transverse pairs carry the tensor product of their two one-wall Kummer coefficient lines?

Local model

Normalize the threshold Morse germ to

$$\tau+u^2+v^2.$$

For two transverse wall normals, write

$$g_A=h_1+au+bv, \qquad g_B=h_2+cu+dv,$$

and

$$L= \begin{pmatrix}a&b\\c&d\end{pmatrix}, \qquad D=\det L\neq0.$$

Entry 1804 certifies $D\neq0$ for every pair considered here, and Entry 1809 certifies its source coefficient as nonzero.

Double residue

The two wall equations fix both Morse variables:

$$\binom uv=-L^{-1}\binom{h_1}{h_2}.$$

Consequently, up to the ordered residue orientation,

$$\boxed{ \operatorname{Res}_{g_A,g_B} \frac{du\,dv} {(\tau+u^2+v^2)g_Ag_B} = \frac{(2\pi i)^2} {D\left(\tau+h^TL^{-T}L^{-1}h\right)}. }$$

The quadratic numerator before division by $D^2$ is

$$(c^2+d^2)h_1^2 -2(ac+bd)h_1h_2 +(a^2+b^2)h_2^2.$$

Result

The coefficient object is a rank-one meromorphic quadratic-pole quotient:

$$T_s=+1, \qquad N=0.$$

It is not the tensor product of two one-wall Kummer lines. The reason is geometric: each one-wall residue leaves one Morse integration variable and therefore a square root, while two independent wall residues fix both Morse variables and leave no integration variable from which a second square root could arise.

All 22 local active types have this form. Their labelled cyclic assembly has rank 110 and character

$$(110,0,0,0,0).$$

The quadratic pole is coefficient support compiled from the existing threshold and wall normals. It is not a new carrier generator.

Architectural consequence

Coefficient formation is not multiplicative under transverse carrier intersection:

$$\boxed{ \text{Kummer wall quotient}\otimes\text{Kummer wall quotient} \not\simeq \text{two-wall residue quotient}. }$$

The correct object is obtained by applying the iterated residue functor to the shared Morse coefficient germ, not by tensoring already-integrated one-wall answers.

Next falsifier

Study the quadratic pole divisor

$$\tau+h^TL^{-T}L^{-1}h=0$$

at its intersections with total-energy and defining-edge soft support. Test whether its specialization is generated by the existing nearby-cycle and soft Rees maps or leaves a supported excess.

Evidence

• research/benincasa/checkers/five_site_g5_transverse_pair_rational_quotients.py
• research/benincasa/results/five-site-g5-transverse-pair-rational-quotients.json
• Entries 1804, 1809, and 1810
• allocator claim: seqclaim-be4afbc548d4d47d410ee3c1

View source on GitHub