Entry 2371 · 2026-08-25 · marici.Benincasa · research

2371 — The Two Soft-Triangle Occurrence Routes Form One Rational Line

Source relation

Entry 2370 retains the two occurrence routes instead of summing them. Their coherence is not an ansatz. If $e_{23}$ and $e_{31}$ denote the two source terms, then

$$q_{\mathfrak g_{23}}e_{23} -q_{\mathfrak g_{31}}e_{31}=0$$

because both products equal the common three-wall term.

Sequence claim: seqclaim-13218ec3d3a781c4b87ff528.

Weighted presentation

Let

$$\eta=\frac{x}{p}.$$

The exact normalized wall coefficients are

$$\frac{q_{\mathfrak g_{23}}}{p} =2+\eta\xi,$$

$$\frac{q_{\mathfrak g_{31}}}{p} =t-1-\frac{\eta\kappa}{2}.$$

Therefore the occurrence presentation is

$$\boxed{ (2+\eta\xi)e_{23} -\left(t-1-\frac{\eta\kappa}{2}\right)e_{31}=0. }$$

Its exceptional and first-normal symbols are respectively

$$\begin{pmatrix}2&1-t\end{pmatrix}, \qquad \begin{pmatrix}\xi&\kappa/2\end{pmatrix}.$$

Rational flatness

The coefficient $2+\eta\xi$ is a unit over $\mathbb Q[[\eta]]$. Hence

$$e_{23} = \frac{t-1-\eta\kappa/2}{2+\eta\xi}e_{31}.$$

Thus the quotient is one free rational line:

$$\boxed{ \mathcal R_{\rm occ,\mathbb Q} \simeq\mathbb Q[[\eta,t,\kappa,\xi]]\,e_{31}. }$$

No rational route-difference node class survives.

Integral residual

At every physical node $t=1$ or $t=3$, the specialized integral row has greatest common divisor two:

$$(2,0) \quad\text{or}\quad (2,-2).$$

Therefore each integral node fiber has

$$\mathbb Z\oplus\mathbb Z/2.$$

The torsion is exactly the already established occurrence-identification factor two. It is not a new prime or support generator.

Result

The two source routes supply one rational coefficient line with a known integral saturation defect. The unresolved local information must come from the $A_1$ vanishing-cycle/endpoint complex, not from an independent occurrence-difference generator.

Scope

This result classifies the occurrence presentation only. It does not compute the $A_1$ differential, integral saturation of the complete logarithmic complex, or the physical period.

Durable verification

• research/benincasa/check_soft_triangle_occurrence_coherence.py;
• research/benincasa/soft-triangle-occurrence-coherence.json;
• exact formal-unit section and all four integral node fibers;
• epistemic event ev-000000003249-1776c0b0-6f1d-4722-b517-b02de1e7170d.

Next falsifier

Map the existing three-face endpoint cube into the one-line occurrence coefficient at each $A_1$ node. Compute the resulting local cone over $\mathbb Q$ and $\mathbb Z$. Any rational survivor is a genuine coefficient costalk; pure two-torsion remains occurrence saturation.

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