7 The cancellation argument
The reduction from the intermediate residue formula (a sum over arising selections) to the main theorem (a sum over \(\Pi \)-stable selections) is carried out by a direct algebraic cancellation, replacing the analytic wall-crossing / identity-theorem argument of the original manuscript. The core is an oriented-matroid circuit identity, reduced to a one-dimensional “coefficient-line” balance lemma.
Along the coefficient line of a circuit, the signed contributions of the positive-coordinate and negative-coordinate endpoints cancel: the balanced signed sum vanishes.
A one-variable accounting argument: pairing each positive endpoint with the matching negative endpoint, the signs are opposite and the magnitudes equal.
The stable signed sum admits a recursion (the “stable cone” statement): it decomposes along the leading pivot into a stable-on-one-side contribution and a remainder that is handled by induction.
Well-founded induction on dimension, using the Schur/leading-minor structure to pass to the restricted cone.
For a circuit with mixed signs, the associated alternating signed sum of selection contributions vanishes. This is the combinatorial heart of the cancellation.
Reduce the circuit identity to the coefficient-line balance lemma along the circuit direction, combined with the stable cone recursion.
At most one row-ordering of a square real matrix is stable, and a stable ordering carries the sign of the determinant.
Every stable ordering contributes the sign of the determinant to the stable signed sum, which the stable cone identity bounds by one in absolute value.
The signed sum \(\chi \) over all selections that arise equals the signed sum over the \(\Pi \)-stable selections. (Degenerate determinant cases vanish on both sides; the generic case uses the circuit lemma via a \(\chi \)/stable-sum recursion trichotomy.)
Induction reducing to the circuit mixed-sign vanishing lemma.