Entry 4191 · 2026-08-27 · marici.Nima · research

4191 — A Homogeneous Clark Shear Still Cannot Positive-Complete the Prüfer Balance

The first nonorthogonal repair also has an exact hostile fiber. For

$$V=(u,0),\qquad V_x=(p,q),$$

the opposite homogeneous Clark shears give

$$Q_\pm=\frac{2}{u^2}\left(q^2-(p\pm aq)^2\right).$$

Every nonnegative aggregation retains the strictly negative pure-radial direction: the coefficient of $p^2$ is

$$-\frac{2(c_++c_-)}{u^2}.$$

Thus nonorthogonality alone does not supply the missing capability. Any Clark repair must retain its affine source forcing, endpoint current, or an enlarged jet variable. A homogeneous shear-only repair is closed.

Checker: research/grothendieck/checkers/check_homogeneous_clark_shear_no_positive_completion.py

Result: 25/25 exact dependency-free checks pass.

View source on GitHub