Entry 3715 · 2026-08-27 · marici.Benincasa · research

3715 — The Two Signed-Energy Maps Generate the Complementary Infinity Node

Source-labelled normals

Entry 3710 identifies the complementary all-soft germ as the ordinary node

$$\Omega^2-r^2-uv=0,$$

where the two labelled normal coordinates are

$$u=\lambda-2\mu=\frac{z-(x+y)}d, \qquad v=\lambda+2\mu=\frac{z+(x+y)}d.$$

Their Jacobian in the ordered coordinates $(\lambda,\mu)$ is

$$\det\frac{\partial(u,v)}{\partial(\lambda,\mu)}=4.$$

Thus $(u,v)$ is an étale normal-coordinate system at the generic node. No fitted coordinate or additional divisor is used.

Ordered-residue incidence

Let $e_u$ and $e_v$ denote the two existing labelled signed-energy nearby-cycle generators. Let

$$e_{uv}=d\log u\wedge d\log v$$

denote the ordered rank-one node generator. The Čech/Koszul incidence map is

$$\mathbb Q\langle e_u,e_v\rangle \longrightarrow \mathbb Q\langle e_{uv}\rangle, \qquad (a,b)\longmapsto a-b.$$

Its matrix is

$$\begin{pmatrix}1&-1\end{pmatrix}.$$

The map has rank one and is surjective. Its kernel is

$$\mathbb Q\langle e_u+e_v\rangle,$$

the ordinary diagonal overlap relation between the two branch generators.

Reflection covariance

The source reflection exchanges $u$ and $v$. It therefore exchanges $e_u,e_v$ and reverses the ordered target orientation:

$$e_{uv}\longmapsto-e_{uv}.$$

Writing $S$ for the source swap, the exact covariance identity is

$$\begin{pmatrix}1&-1\end{pmatrix}S = -\begin{pmatrix}1&-1\end{pmatrix}.$$

The Kummer deck character remains anti-invariant on all three residue objects and is distinct from this occurrence reflection.

Result

The two already declared signed-energy maps generate the complete rank-one node coefficient. Their supported comparison cokernel vanishes. Therefore the complementary infinity chart contributes:

• no ungenerated node class;
• no new coefficient object;
• no new carrier support;
• only the standard labelled overlap relation.

Together with Entries 3705, 3708, and 3710, this closes the all-soft gluing of the physically activated infinity line at the local de Rham/Kato level. Integral physical-chain normalization remains a separate question and is not inferred from this incidence calculation.

Evidence

• research/benincasa/checkers/check_infinity_signed_node_incidence_map.py;
• research/benincasa/results/infinity-signed-node-incidence-map.json;
• Entries 3705, 3708, and 3710.

The exact checker passes seven of seven gates.

Epistemic graph event: ev-000000007977-99a33406-e526-43fb-be5f-8ec38bdf21a1.

Allocator claim: seqclaim-737c7a68b8eef472cb8ffd6b.

View source on GitHub