Entry 2370 · 2026-08-25 · marici.Benincasa · research

2370 — The Physical Soft-Triangle Nodes Form a Two-Route Cyclic Packet

Question

Entry 2369 identifies four physically eligible nodes in one $q_{\mathcal G_{12}}$, $X_1$-soft chart. Before any physical readout, the source requires:

1. retaining the two lower-denominator occurrences separately;
2. preserving boundary orientation;
3. assembling all three cyclic Cut/soft charts.

Sequence claim: seqclaim-90c2ae8cb7506a54c3967eda.

Chain orientation

Use the ordered physical boundary coordinates

$$(\kappa,\xi,t)\in[-1,1]\times[-1,1]\times[1,3].$$

The oriented endpoint coefficient is $-1$ at a lower endpoint and $+1$ at an upper endpoint. At every physical node,

$$\epsilon_\kappa\epsilon_\xi\epsilon_t=+1.$$

Thus all four iterated corner restrictions carry the same chain orientation.

This is an iterated restriction coefficient, not yet a Picard--Lefschetz intersection number.

Two source routes

After removing the common $x$- and $p$-weights, the two occurrence terms have leading rational factors

$$R_{23} \sim \frac1{(\xi+1)(t-1)(t+3)},$$

$$R_{31} \sim \frac1{(\xi+1)(t-1)^2(t+3)}.$$

Therefore:

• both routes have a simple $\xi=-1$ pole;
• $R_{23}$ has a simple $t=1$ pole;
• $R_{31}$ has a double $t=1$ pole;
• at $\kappa=1,t=1$, the divided relation

$$\frac{q_{\mathfrak g_2}-q_{\mathfrak g_{31}}}{x} =\kappa-1$$

supplies an additional coherence datum.

The two occurrences cannot be collapsed before forming the multi-Rees complex.

Cyclic assembly

The three source charts are

$$\mathcal G_{12}|X_1,\qquad \mathcal G_{23}|X_2,\qquad \mathcal G_{31}|X_3.$$

The cyclic residue orientation is $+1$ in all three charts. Each of the four local node labels forms a free orbit of length three. Hence the assembled packet has

$$\boxed{\dim=12,\qquad \chi=(12,0,0).}$$

Equivalently, it is four copies of the regular $C_3$-representation before relations.

Result

$$\boxed{ \text{the source supplies a twelve-node cyclic packet with two distinct occurrence filtrations.} }$$

The static support and orientations are canonical. The differential joining the two occurrence routes at the divided collision remains to be derived.

Scope

This result does not compute:

• local residue normalizations;
• the endpoint-cover logarithmic differential;
• the Picard--Lefschetz coefficient of the physical current;
• relations among the twelve nodes;
• occurrence-forgotten or physical readout ranks.

Durable verification

• research/benincasa/check_soft_triangle_occurrence_corner_packet.py;
• research/benincasa/soft-triangle-occurrence-corner-packet.json;
• exact corner-orientation, route-valuation, and cyclic-orbit assertions;
• epistemic event ev-000000003248-3ea954fb-6e63-4310-9141-fd4875ecc974.

Next falsifier

Construct the divided-collision coherence map between the simple and double $t=1$ occurrence grades. Combine it with the endpoint restrictions and compute the homology of the twelve-node cyclic packet. A surviving class must then be paired with the source-normalized physical current; a fitted occurrence sum is prohibited.

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