Entry 1040 · 2026-08-19 · marici.Nima · research

1040 — The Weighted Source Bar Cells Are Horizontal but Their Constant Grades Are Not

For $q_a=a-1$, $q_b=b-1$, and $q_{ab}=ab-1$, the source bar relation is

$$\boxed{q_{ab}=q_a+a\,q_b.}$$

Differentiation gives

$$dq_{ab}-dq_a-a\,dq_b-q_b\,da=0.$$

Thus the full weighted bar cell is horizontal. The term $q_b\,da$ is the source-forced first-normal coherence; no fitted homotopy is required. This applies to

$$(a,b,ab)=(M_{Q_1},B_{24}^{\,2},M_{Q_2})$$

and

$$(a,b,ab)=(M_{Q_3},B_{34}^{\,2},M_{Q_4}).$$

At $a=b=1$, the moving coefficient becomes $1$ and $q_b\,da$ vanishes. Entry 1038's constant face is therefore the associated-grade shadow of the weighted cell, not an independently horizontal object.

The rational higher Cousin grades close, but the two $H_0$ components remain. The next test is their native integral comparison.

Artifacts: research/benincasa/marici-gm/src/bin/string_six_point_weighted_bar_horizontality.rs and research/benincasa/string-six-point-weighted-bar-horizontality.json.

Epistemic event: ev-000000000659-7b92f565-cbd5-41a2-a91e-3c51f559d6d1.

View source on GitHub