Dependencies

At most one row-ordering of a square real matrix is stable, and a stable ordering carries the sign of the determinant.

The arising cone for \(\sigma \): the set of real parameter vectors where \(\sigma \) arises (i.e., \(z_k^*\) is in the upper half-plane at each step).

\(D(s) = \prod _{\sigma \in \Pi \text{-stable}} \mathrm{nonSelProd}_\sigma (s)\), the product of the spectator denominators. It is polynomial in \(s\).

\(N(s) = (2\pi i)^n \sum _\sigma \bigl(\prod _{\sigma ' \neq \sigma } \mathrm{nonSelProd}_{\sigma '}(s)\bigr)\, h(z_\sigma (s))\, \det _\sigma ^{-1}\), obtained by clearing the common denominator from the flag-residue sum. Since the pole point \(z_\sigma (s)\) is affine in \(s\) and \(h\) is holomorphic, \(N\) is holomorphic in \(s\).

The convergence hypothesis: \(\mathrm{Re}(s) {\gt} 0\) and UHP-filtered convergence (the Cauchy residue identity holds at each step of the iterated integral).

The flag residue for a selection \(\sigma \): computed by iterating \(\mathrm{poleIn0}\) through the restricted forms at each step.

The subset \(U \subseteq \Omega \) where, for every selection whose coefficient matrix is invertible, the spectator product is nonzero. It is open, and (when nonempty) dense in \(\Omega \).

A matrix \(J\) is compatible if it is either not stable, or for all \(k\) and \(l {\gt} k\) we have \(q_{k,l}(J) \leq 0\).

\(P\) is \(\Pi \)-stable if its Jacobian square block is stable.

A selection \(\sigma \) is \(\Pi \)-stable.

A matrix \(J\) is soluble if all leading principal minors \(p_1,\ldots ,p_n\) are nonzero.

A matrix \(J\) is stable if all leading principal minors \(p_1,\ldots ,p_n\) are positive and the signed \(r\)-minors \((-1)^{k-j} r_{j,k}\) are non-negative.

The \(k\)-th leading principal minor \(p_k(J)\) is the determinant of the leading principal \(k \times k\) submatrix of \(J\).

The integral \(F(s) = \int _{\mathbb {R}^n} h(x) / \prod _j (f_j(x) - s_j i)\, dx\), viewed as a function of the parameter \(s\).

The domain \(\Omega = \{ s \in \mathbb {C}^R : \mathrm{Re}(s_j) {\gt} 0 \text{ for all } j \} \). It is convex, hence preconnected and open.

The finite set of all \(\Pi \)-stable embeddings \(\mathrm{Fin}\, n \hookrightarrow \mathrm{Fin}\, R\).

A polar data is a collection of \(k\) real linear forms on \(\mathbb {R}^n\).

The minor \(q_{k,l}(J)\) is the determinant of the submatrix with rows \(\{ 0,\ldots ,k{-}1\} \) and columns \(\{ 0,\ldots ,k{-}2, l\} \).

The minor \(r_{j,k}(J)\) is the determinant of the \((k{-}1)\times (k{-}1)\) submatrix with rows \(\{ 0,\ldots ,k{-}1\} \setminus \{ j\} \) and columns \(\{ 0,\ldots ,k{-}2\} \).

An \(n\)-element selection \(\sigma : \mathrm{Fin}\, n \hookrightarrow \mathrm{Fin}\, R\) gives rise to a combined polar data.

The formula \(z_k^* = p_{k+1}^{-1}\bigl((s_k p_k + \sum (-1)^{k-j} s_j\, r_{j,k+1})\, i - \sum q_{k+1,l}\, x_l\bigr)\).

Along the coefficient line of a circuit, the signed contributions of the positive-coordinate and negative-coordinate endpoints cancel: the balanced signed sum vanishes.

If \(f\) and \(g\) are analytic on a preconnected set \(\Omega \) and agree on a nonempty open subset \(U \subseteq \Omega \), then they agree on all of \(\Omega \).

The original integrand with real linear forms equals the generalized integrand constructed via \(\mathrm{ofRealForm}\).

\(J\) is stable if and only if it is stable at every step.

The recursive \(\mathrm{poleIn0}\)-based arising predicate is equivalent to the \(z^*\)-based arising predicate. Proved using the Schur complement identities.

\(p_k(\ell \cdot J) = \det (\ell _k) \cdot p_k(J)\) for lower-triangular \(\ell \).

\(q_{k,l}(\ell ' \cdot J) = \det (\ell '_k) \cdot q_{k,l}(J)\).

If \(J\) is stable, then \(J\) is soluble.

The imaginary part of \(z_k^*\) is \(\mathrm{Im}(z_k^*) = \bigl(\mathrm{Re}(s_k)\, p_k + \sum (-1)^{k-j}\mathrm{Re}(s_j)\, r_{j,k+1} - \sum \mathrm{Im}(x_l)\, q_{k+1,l}\bigr) / p_{k+1}\).

For all \(s\) with \(\mathrm{Re}(s) {\gt} 0\) and all real \(x\), \(\mathrm{Im}(z_k^*) {\gt} 0\) if and only if \(J\) is stable at step \(k\).

Let \(J = LU\) with \(L\) unit lower triangular and \(U\) upper triangular, and suppose \(J\) is stable. Then \(J\) is compatible if and only if every strictly-above-diagonal entry of \(U\) is non-positive. This rests on the identity \(q_{k,l}(LU) = \bigl(\prod _{j{\lt}k-1} u_{jj}\bigr)\, u_{k-1,l}\).

The open Bruhat cell equals the set of matrices with nonvanishing leading principal minors. Equivalently, the values \(z_1^*,\ldots ,z_r^*\) are soluble if and only if all leading principal minors are nonzero.

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.)

For a circuit with mixed signs, the associated alternating signed sum of selection contributions vanishes. This is the combinatorial heart of the cancellation.

The imaginary numerator satisfies \(\mathrm{imNum}(J, k{+}1, t) = J_{0,0} \cdot \mathrm{imNum}(\mathrm{Schur}_{00}(J), k, t')\) where \(t'\) is the Schur-restricted parameter.

\(\Pi \)-compatibility is invariant under left-multiplication by a positive diagonal matrix.

\(\Pi \)-stability is invariant under left-multiplication by a positive diagonal matrix.

\(J\) is stable if and only if \(\mathrm{Im}(z_k^*) {\gt} 0\) at every step \(k\), for all \(s\) with \(\mathrm{Re}(s) {\gt} 0\) and all real \(x\).

Under UHP-filtered convergence, the integral equals \((2\pi i)^n\) times the sum over UHP-arising selections only.

\(p_{k+1}(J) = J_{0,0} \cdot p_k(\mathrm{Schur}_{00}(J))\).

Under convergence hypotheses, the integral equals \((2\pi i)^n\) times the sum of flag residues over arising selections.

Let \(s_j\) have positive real part. Under the convergence hypothesis and on the generic locus (for every selection whose coefficient matrix is invertible, the spectator product is nonzero),

The sum runs only over \(\Pi \)-stable selections.

Assume \(h\) holomorphic, the convergence hypothesis (for all \(s\) with positive real part), and that \(F\) is analytic in \(s\) on \(\Omega \). Then, with no genericity hypothesis,

Where \(D(s) \neq 0\) this recovers the closed-form stable-residue formula; where \(D(s) = 0\) it expresses that \(N/D\) has a removable singularity equal to the integral.

\(r_{j+1,k+2}(J) = J_{0,0} \cdot r_{j,k+1}(\mathrm{Schur}_{00}(J))\).

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.

On the generic locus (for every selection whose coefficient matrix is invertible, the spectator product is nonzero), the sum of flag residues over arising selections equals the sum over \(\Pi \)-stable selections. Selections with singular coefficient matrix are inert: no ordering of them arises or is \(\Pi \)-stable.