Entry 3659 · 2026-08-27 · marici.Benincasa · research

3659 — The Two Endpoint A3 Residues Glue as One Orientation Line

Reciprocal rows

Entry 3652 gives the infinity-corner logarithmic row

$$\rho_\infty= \begin{pmatrix} 0&1/y&0 \end{pmatrix}$$

in the source coordinate $s=1/t$. The reciprocal finite corner has

$$\rho_0= \begin{pmatrix} 0&1/x&0 \end{pmatrix}.$$

Under reciprocal exchange $x\leftrightarrow y$, one has

$$\frac{ds}{s}=-\frac{dt}{t}.$$

Therefore

$$\rho_\infty\longmapsto-\rho_0.$$

The sign is forced by orientation reversal of the projective interval. It is not a normalization choice.

Deck and scaling characters

Sheet exchange sends $W\mapsto-W$, so the residue form also acquires sign $-1$. Consequently the two characters are

$$\chi_{\rm reciprocal}=-1, \qquad \chi_{\rm deck}=-1, \qquad \chi_{\rm combined}=+1.$$

Each involution squares to the identity. Under common energy scaling, the residue line has weight $-1$, matching the source elliptic forms and the tangential finite-part covector.

Result

The finite and infinity endpoint residues are not two independent physical ports. They are reciprocal charts of one rank-one orientation/Kummer line.

Together with Entries 3648 and 3652, the soft-signed endpoint coefficient packet is therefore:

• a rank-two even deformation grade;
• one complementary logarithmic readout line;
• reciprocal and deck gluing fixed by sign;
• no additional carrier or coherence generator.

The all-soft cone vertex $x=y=z=0$ is excluded from this projective chart and remains a separate radial specialization question.

Evidence

• research/benincasa/checkers/check_endpoint_a3_reciprocal_gluing.py;
• research/benincasa/results/endpoint-a3-reciprocal-gluing.json.

The exact checker passes seven of seven gates.

Epistemic graph event: ev-000000007855-e4d41a02-d651-40a5-9d08-0a2f6cbc657e.

Allocator claim: seqclaim-97a8530a8ca719df604abde0.

View source on GitHub