The Determinant Ladder
A human–AI field expedition, August 2026 — down to the determinant as signed volume, and back up rung by rung to the conjecture that fell.
The main paper
One Formula, Many Worlds
A walk down to the determinant and back up to the broken Jacobian conjecture. Thirteen sections, twenty-two figures, and one ladder with no missing rungs.
An expedition that began with a freshly broken conjecture and refused to be satisfied with the headline: down to the most primitive object underneath it — the determinant as signed volume — and back up, rung by rung, watching one small formula reappear in richer and richer worlds.
The conjecture fell in 2026. This is what went down after it: what the determinant is actually measuring, why the schoolroom prohibition on dividing by zero is the same rule as Jacobi’s dependence alarm, and what the counterexample looks like when you stand close enough to see its shape.
Four documents. The main paper is the spine; the other three came out of writing it.
The papers
Bracket and Unique MechanismThe freedom-versus-constraint crossing at d ≈ 2.489, the unique-mechanism programme, and the hidden-hypothesis principle.
Down the NeedleThe fiber of the 2026 counterexample as one geometric object — valleys, mirror, branches, horizons, and the ground floor.Behind them
The instruments and the working papers — what the figures were drawn with, the conventions they were drawn under, and the question the research note was written towards.