Entry 2152 · 2026-08-24 · marici.Benincasa · research

2152 — The Three Final Triangle Deletions Are Typed Logarithmic Port Morphisms

RETRACTED by Entry 2153. The three rational ratios and their cyclic formulas are exact, but calling them typed cube arrows confuses a presentation gauge with a source-defined coefficient morphism.

Hard-to-vary claim

The three adjacent edges from deletion grade two to the all-deleted triangle sector inherit Entry 2149's rational port map and Entry 2150's strict logarithmic lift.

Representative labelled edge

Take

$$S=\{12,23\},$$

so the retained edge is $31$. The nontrivial connected component has shifted endpoint energies

$$A_1=X_1+y_{12}, \qquad A_3=X_3+y_{23},$$

and component total

$$Q_{13}=A_1+A_3 =X_1+X_3+y_{12}+y_{23}.$$

Deleting the last edge $31$ replaces the connected line factor by its two contacts and gives

$$\boxed{ D_{31}=\frac{Q_{13}}{y_{31}}. }$$

The endpoint contact poles are

$$q_1=A_1+y_{31}, \qquad q_3=A_3+y_{31}.$$

The isolated site-2 contact and the previously deleted-edge factors are common spectators.

Cyclic packet

Cyclic relabelling gives

$$D_{12} = \frac{X_1+X_2+y_{31}+y_{23}}{y_{12}},$$

$$D_{23} = \frac{X_2+X_3+y_{12}+y_{31}}{y_{23}}.$$

Each map carries the forced connection shift

$$\boxed{ A_{E}=A_{E\setminus\{ij\}} -d\log Q_{ij}+d\log y_{ij}. }$$

It is a strict coefficient morphism away from $Q_{ij}y_{ij}=0$.

Physical chain

On the literal positive chamber, every $Q_{ij}>0$. Near the erased-edge soft boundary, the same $d=3$ radial argument as Entry 2151 leaves $r,dr$, so no soft Gysin current survives at generic external separation.

Therefore the three final deletion edges are both coefficient-horizontal and physical-chain compatible on the generic physical chamber.

Scope

This types exactly the three arrows

$$\{12,23\},\{23,31\},\{12,31\} \longrightarrow \{12,23,31\}.$$

It does not type the six grade-one-to-grade-two arrows or the three grade-zero-to-grade-one arrows. Those involve connected path or loop coefficients rather than a universal last-edge line deletion.

Evidence

• Benincasa--Dian, arXiv:2401.05207, equations (2.14), (2.27)--(2.30);
• Entries 2135 and 2149--2151;
• research/benincasa/marici-gm/src/bin/triangle_final_deletion_maps.rs;
• allocator claim seqclaim-581fda22c7fe35127ce32498.

Next falsifier

Construct one grade-one-to-grade-two deletion morphism from the actual connected path integrand. Determine whether it is again a rational logarithmic gauge map or whether deletion-induced graph disconnection requires a nontrivial localization/Gysin cone.

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