Entry 2136 · 2026-08-24 · marici.Nima · research

2136 — Component-Resolved Deletion Poles Reproduce the Labelled Normal Module

Hard-to-vary claim

Every grade-two or grade-three component pole involving two loop distances has the universal relative-Landau factor

$$\boxed{ p_f-L^2, }$$

where L is the component site-energy sum and P_f^2=p_f is the squared separation of the two distance foci.

For isolated contact components this is exactly the labelled normal direction

$$\nu_i=P_i^2-X_i^2.$$

Universal two-distance calculation

Consider a component pole

$$q=L+y_a+y_b.$$

On q=0, write

$$y_b=-L-y_a.$$

Restrict the three-site Cayley--Menger determinant to this hyperplane and impose the two remaining critical equations. The exact elimination gives, up to a nonzero rational unit,

$$\boxed{ p_f^2(p_f-L^2)^3\Lambda(p_1,p_2,p_3)^3. }$$

Thus the generic component threshold is

$$L=\pm P_f,$$

with soft and external-triangle factors separated.

Grade-two packet

For deletion S={12,23}, the components are G_{13} and G_2:

$$q_{13}=X_1+X_3+y_{12}+y_{23}, \qquad q_2=X_2+y_{12}+y_{23}.$$

The two foci are separated by P_2. Their thresholds are

$$(X_1+X_3)^2=P_2^2, \qquad X_2^2=P_2^2.$$

The first is an existing signed-energy/component-energy divisor. The second is

$$\nu_2=P_2^2-X_2^2=0.$$

Grade-three packet

The three isolated contact poles give cyclically

$$X_i^2=P_i^2, \qquad i=1,2,3.$$

Hence their Landau support is exactly

$$\boxed{ (\nu_1\nu_2\nu_3)=0, \qquad \nu_i=P_i^2-X_i^2. }$$

These are the same labelled normal directions whose square-free second-normal products appeared in Entry 698's generic lower algebraic-letter sector.

Homogeneous specialization

On the homogeneous physical locus

$$P_i=X_i,$$

all three isolated-contact thresholds collapse identically. Therefore their information belongs to the labelled normal/Rees geometry and cannot be recovered from the homogeneous scalar restriction alone.

Classification

$$\boxed{ \text{correlator deletion ports} \longrightarrow \text{the existing labelled }(\nu_1,\nu_2,\nu_3)\text{ normal module}. }$$

This is a cross-check of H2 and of the port-adapter architecture. The wavefunction-to-correlator map does not generate a new quartic home; it exposes the same normal directions through a different physical readout.

Verification

The exact factor appears as two_distance_component_landau in

research/benincasa/marici-gm/src/bin/three_site_deletion_landau.rs.

Next falsifier

Compute the first nonzero associated grade of the component-resolved correlator sum at \nu_i=0. Determine whether the three isolated-contact contributions generate only the diagonal normal lines \nu_i, or whether their additive readout produces a canonical square-free second grade

$$\langle\nu_1\nu_2,\nu_1\nu_3,\nu_2\nu_3\rangle$$

matching the generic lower-sector module—without inserting cross-sector coherence by hand.

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