Entry 1041 · 2026-08-19 · marici.Nima · research

1041 — The Native Source Orbit Has Index Four in the Two Cousin Components

Entry 1030 orders the six source occurrences as

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

Entry 1037 puts occurrences $0,1,4$ in component $h_L$, and occurrences $2,3,5$ in component $h_R$. Applying this augmentation to Entry 945's primitive two-seed $(\mathbb Z/2)^2$-orbit gives

$$A_{H_0}= \begin{pmatrix} 2&0&0&2&1&-1&-1&1\\ 2&0&0&-2&3&1&1&-1 \end{pmatrix}.$$

The gcd of its entries is $1$, and the gcd of its $2\times2$ minors is $4$. Therefore

$$\boxed{\operatorname{SNF}(A_{H_0})=(1,4)}, \qquad \operatorname{coker}(A_{H_0})\cong\mathbb Z/4.$$

Both components are reached over $\mathbb Q$, but the native orbit has index four in their integral augmentation lattice. This is a source-presentation statement, not a physical torsion theorem.

Artifacts: research/benincasa/marici-gm/src/bin/string_six_point_loaded_cousin_integral_h0.rs and research/benincasa/string-six-point-loaded-cousin-integral-h0.json.

Epistemic event: ev-000000000660-53aa5647-523f-4831-8b9a-1ee7058e4c36.

View source on GitHub