Entry 2369 · 2026-08-25 · marici.Benincasa · research

2369 — Four Endpoint Nodes Meet the Literal Positive Soft-Triangle Current

Question

Entry 2368 finds eight algebraic $A_1$ nodes. Physical activation cannot be inferred from their existence. The finite incidence test must retain all three source triangle inequalities.

Sequence claim: seqclaim-08404639bfd28bb281bf91e5.

Three exceptional intervals

The physical weighted chart gives:

$$-1\leq\kappa\leq1$$

from the fixed external triangle and

$$-1\leq\xi\leq1$$

from the collapsing $(c,b,X_1)$ triangle.

For the remaining $(c,a,X_2)$ face, setting $t=a/p$ gives

$$\boxed{ F_{ca}^{\rm exc} =(t^2-1)(t^2-9). }$$

The positive loop-length chamber therefore selects

$$\boxed{1\leq t\leq3.}$$

Physical node census

Exactly four of Entry 2368's eight nodes lie in this positive exceptional closure:

$$\begin{array}{c|c|c|l} \kappa&\xi&t&\text{marked support}\\ \hline -1&-1&3&q_{\mathfrak g_1}\text{ endpoint}\\ -1& 1&1&q_{\mathfrak g_2}/q_{\mathfrak g_{31}}\text{ collision}\\ 1&-1&1&q_{\mathfrak g_1}\text{ endpoint and } q_{\mathfrak g_2}/q_{\mathfrak g_{31}}\text{ collision}\\ 1& 1&3&\text{none; pure Cayley--Menger corner}. \end{array}$$

Their marked-incidence profile is consequently

$$\boxed{(0,1,1,2).}$$

The four negative-$t$ nodes are signed analytic companions and do not meet the literal positive current.

Result

Before relations, orientations, or thimble coefficients, the physically eligible local Milnor packet has rank four. Every eligible node lies on existing Cayley--Menger endpoints; three also lie on existing marked support.

This is an incidence theorem only:

$$\Gamma_{\rm phys}^{\rm exc}\cap\operatorname{Supp}(\phi) \ne\varnothing$$

does not yet prove a nonzero intersection number with the corresponding vanishing cycle.

Classification

• algebraic node packet: rank eight;
• literal positive incidence packet: rank four;
• support: existing soft, triangle, signed-face, and marked incidence;
• source-selected coefficients: uncomputed;
• new Carrier datum: none.

Scope

The result does not compute local orientations, Picard--Lefschetz coefficients, relations among nodes, occurrence sewing, or the physical period map.

Durable verification

• research/benincasa/check_soft_triangle_physical_node_incidence.py;
• research/benincasa/soft-triangle-physical-node-incidence.json;
• exact three-face factorizations and labelled node census;
• epistemic event ev-000000003247-e79506e2-21e8-4bb1-b90a-acc6ba9161ec.

Next falsifier

Derive the source orientation and local Picard--Lefschetz coefficient at each of the four eligible nodes. Assemble the two occurrence summands before any physical readout. A zero sewn packet closes physical activation; a nonzero packet must then be tested against the existing endpoint/Gysin complex.

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