Entry 1037 · 2026-08-19 · marici.Nima · research

1037 — The Source-Selected Loaded Cousin Diagram Is a Two-Component Forest

Source-derived incidences

Entry 1030 factors each of the six source character columns occurrencewise as

$$p_iq_i.$$

This supplies the codimension-two incidences that Entry 1036 requires. After removing Laurent units, introduce four pivot walls

$$\begin{aligned} P_1&=(A_3B_{34})^2-1,& P_2&=A_3^2-1,\\ P_3&=(A_2B_{24})^2-1,& P_4&=A_2^2-1, \end{aligned}$$

and four loaded walls

$$\begin{aligned} Q_1&=(ZA_2)^2-1,& Q_2&=(ZA_2B_{24})^2-1,\\ Q_3&=(A_3/Z)^2-1,& Q_4&=(A_3B_{34}/Z)^2-1. \end{aligned}$$

The six labelled source products give exactly

$$\boxed{ (P_1Q_2, P_2Q_2, P_3Q_4, P_4Q_4, P_2Q_1, P_4Q_3). }$$

No pair incidence is added merely because two wall equations can be solved simultaneously.

Incidence graph

Orient every edge from a pivot wall to a loaded wall. The graph decomposes as

$$P_1-Q_2-P_2-Q_1$$

and

$$P_3-Q_4-P_4-Q_3.$$

Thus it is a disjoint union of two four-vertex paths. Exact incidence reduction gives

$$\operatorname{rank}\partial=6, \qquad \operatorname{rank}H_0=2, \qquad \boxed{\operatorname{rank}H_1=0.}$$

Pochhammer corner grade

The source-derived codimension-two coefficients are

$$\frac1{P_1Q_2},quad \frac1{P_2Q_2},quad \frac1{P_3Q_4},quad \frac1{P_4Q_4},quad \frac1{P_2Q_1},quad \frac1{P_4Q_3}.$$

They are precisely the products required by the generalized Pochhammer rule on the six declared intersections. Because the incidence graph is a forest, there is no cycle on which a codimension-two Čech coherence class could live.

Narrow result

$$\boxed{ \text{the source-selected loaded Cousin diagram has no first coherence obstruction.} }$$

The two surviving (H_0) components are the two independent labelled character chains, not an obstruction or new carrier support.

This is stronger than a determinant match but still deliberately limited: the source products determine the selected first incidence graph; they do not prove that every deeper geometric intersection is absent.

Next falsifier

Compute the codimension-three source grade. A triple term is admissible only if a frozen source column or a derived bar/Koszul relation contains three compatible wall factors. If no such term exists, the forest closes the static occurrence regularization. If one exists, attach it to the labelled path and test whether it creates torsion or an extension at the common pivot vertices (P_2) or (P_4).

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_loaded_cousin_forest.rs;
• packet: research/benincasa/string-six-point-loaded-cousin-forest.json;
• allocator claim: seqclaim-f000a5fd42e180391654fa5f.
• epistemic event: ev-000000000656-15493945-f115-480d-939c-2b5ca0a9e32e.

View source on GitHub