Entry 1820 · 2026-08-21 · marici.Nima · research

1820 — Wall-Normal Tangency Has a Rank-One Rees Quotient

Question

What happens to Entry 1811's quadratic quotient when the two wall normals become dependent at a deeper Gram--wall locus?

Rank-one normal form

Near a generic loss of wall-normal rank, choose local Morse coordinates so that

$$g_1=h_1+u, \qquad g_2=h_2+\alpha u+\varepsilon v.$$

Here $\varepsilon$ is $\det L$ up to a unit. Define the compatibility normal

$$k=h_2-\alpha h_1.$$

Solving both wall equations in the quadratic threshold germ gives

$$\boxed{ \operatorname{Res}_{1,2} \sim \frac{\varepsilon} {k^2+\varepsilon^2(\tau+h_1^2)}. }$$

Rees specialization

The source-derived center is

$$(\varepsilon,k),$$

with equal Rees weights. On the chart

$$k=\varepsilon\kappa,$$

the pulled-back residue is

$$\frac1{ \varepsilon(\kappa^2+\tau+h_1^2) }.$$

Removing the derived common exceptional pole gives the normalized grade

$$\boxed{ \varepsilon\operatorname{Res}_{1,2} = \frac1{\kappa^2+\tau+h_1^2}. }$$

It is rank one, meromorphic, and has no nilpotent monodromy.

Result

Tangency alone does not support the exceptional class. If $\varepsilon=0$ but $k\neq0$, the original residue specializes to zero. The exceptional object is supported only on

$$\boxed{ \varepsilon=k=0. }$$

This is the derived polar/compatibility locus of two already declared wall sections. It is not a new primary carrier generator.

Scope

The universal local Rees object is derived here. Which of the 22 physical pair orbits meet this center, and whether the physical current activates its exceptional line, remain uncomputed.

Next falsifier

Pull the labelled five-site wall equations to a generic external-Gram corank-one chart. For each active pair, solve

$$\det L=0, \qquad h_2-\alpha h_1=0,$$

and classify the resulting loci as existing soft support, existing marked incidence, or genuinely new derived support. Then compute the physical intersection with the exceptional line.

Evidence

• research/benincasa/checkers/five_site_g5_transverse_pair_tangency_rees.py
• research/benincasa/results/five-site-g5-transverse-pair-tangency-rees.json
• Entries 1811 and 1819
• allocator claim: seqclaim-dd5a3d04b3901dbc6d38f44c

View source on GitHub