Entry 2372 · 2026-08-25 · marici.Benincasa · research

2372 — The Physical Soft-Triangle Nodes First Smooth at Second Normal Order

Question

Entries 2368--2371 identify the endpoint nodes, their physical incidence, and their occurrence module. A node class is not activated merely because it exists. Its source-derived smoothing direction must be computed from the full kernel before taking nearby cycles.

Sequence claim: seqclaim-89fd4bffee14eb4371953eef.

Full normal expansion

After the weighted substitution, divide the complete residue kernel by its common $x^2$ factor:

$$\frac{K_0}{x^2} =K_{\rm exc}+xK_1+x^2K_2+O(x^3).$$

The first coefficient is

$$K_1 =2p(\xi+2) \left( -a^2\kappa\xi-2a^2 +4\kappa^2p^2+13\kappa\xi p^2 +4\xi^2p^2+6p^2 \right).$$

Exact evaluation at every one of Entry 2368's eight nodes gives

$$\boxed{K_1=0.}$$

Thus the ordinary first jet does not smooth any endpoint node.

Second normal grade

The second coefficient is nonzero at all eight nodes. On the four physical nodes of Entry 2369, in the same order, it is

$$\boxed{ 9p^2,\qquad9p^2,\qquad p^2,\qquad81p^2. }$$

For generic $p\ne0$, every physical node therefore has first nonzero smoothing order two.

Result

$$\boxed{ \text{the physical endpoint-node packet is invisible to the first normal jet and is born in }\operatorname{gr}^{(2)}_{X_1}. }$$

This is a direct source computation, not an inference from the earlier quartic $\mathcal Q$. It independently reproduces the program's loop-specific warning:

$$\text{first jet}\not\Rightarrow \text{control of integrated loop deformation}.$$

Classification

• support: existing site-soft and triangle-endpoint intersections;
• first normal grade: zero;
• second normal grade: four nonzero physical smoothing coefficients;
• coefficient type: local $A_1$ vanishing-cycle packet;
• new Carrier datum: none.

The result strengthens H2 while proving that a first-jet-only physical observer is insufficient at this corner.

Scope

The calculation fixes normal order and source coefficients. It does not yet fix Picard--Lefschetz orientation signs, integral thimble normalization, relations among the four cycles, or their physical period amplitudes.

Durable verification

• research/benincasa/check_soft_triangle_node_smoothing.py;
• research/benincasa/soft-triangle-node-smoothing.json;
• exact $K_0,K_1,K_2$ evaluations at all eight labelled nodes;
• epistemic event ev-000000003250-254185b3-868a-4211-b2f8-149ff7f538ba.

Next falsifier

Use $K_2$ to normalize the local Picard--Lefschetz maps. Compute their signs against the positive-sheet chain orientations from Entry 2370, then assemble the four physical node lines through the rational occurrence quotient of Entry 2371. The resulting rank and monodromy decide whether a second-grade physical coefficient survives.

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