The orthogonal absolute-product minimum is attained at one

User3 (MathOverflow user 600125), A nontrivial lower bound for the orthogonally transformed absolute product, question 515549, revision 1, GUID CB719581-BCE3-4573-931E-15327002F861, asks:

Let $P\in\mathbb{R}^{K\times K}$ be nonsingular and define $$M(Q)=|PQ|,|Q^\top P^{-1}|,\qquad Q\in O(K).$$ $|M∣$ means the matrix obtained by taking the absolute value of each entry. Since $$|PQ(Q^\top P^{-1})|=I\le |PQ|,|Q^\top P^{-1}|=M(Q),$$ one obtains the trivial bound $\rho(M(Q))\ge 1$. $\rho(M(Q))$ is the spectral radius of the matrix $M$. The question is whether there exists a nontrivial quantity $\beta(P)$, depending only on $P$, such that $$\rho(M(Q))\ge\beta(P)>1\qquad\text{for all }Q\in O(K),$$ for an appropriate class of nonsingular matrices $P$. Equivalently, can the optimization problem $$\inf_{Q\in O(K)}\rho!\left(|PQ|,|Q^\top P^{-1}|\right)$$ be bounded below, or even characterized explicitly, in terms of intrinsic properties of $P$?

The quotation is attributed to User3, MathOverflow, under CC BY-SA 4.0. Its source identity and license are recorded in the Library note.

The single resolution anchor is the exact optimization characterization: for every natural $K>0$ and every $P\in\mathbb R^{K\times K}$ with $\det P\ne0$,

$$ \left(\forall Q\in O(K),\quad 1\le\rho!\left(|PQ|,|Q^\top P^{-1}|\right)\right) \quad\land\quad \left(\exists Q\in O(K),\quad \rho!\left(|PQ|,|Q^\top P^{-1}|\right)=1\right). $$

Thus the minimum and infimum are both one, with actual attainment. In particular, no $\beta(P)>1$ can bound the objective below for every orthogonal $Q$, for any nonsingular $P$ in positive dimension. This answers the characterization question and negates the proposed strict improvement.

The source distinguishes the entrywise absolute-product spectral radius from a norm or a matrix absolute value. Determining its exact orthogonal minimum decides whether intrinsic features of a nonsingular matrix can force a strict improvement on the elementary bound one.

The source gives the elementary bound but asks whether a stronger bound or an explicit infimum characterization is available. This is a Tier 1 recent externally authored question, preregistered in issue #24893. The retained bounded literature and duplicate screen identified no exact published resolution within its searched scope. That negative search result does not establish exhaustive literature coverage or priority.

For every orthogonal $Q$, the matrices $A=PQ$ and $B=Q^\top P^{-1}$ satisfy $AB=I$. The finite-sum triangle inequality gives $I\le |A||B|$ entrywise; the frozen spectral comparison D5/S1/Digit/Infinite/ResetCodebookOrder.radius_mono supplies the bound one.

Apply ordered Gram-Schmidt to the rows of $P$ and use the resulting orthonormal basis as the columns of $Q$. Then $T=PQ$ is lower triangular, $Q$ satisfies both orthogonality identities, and $T^{-1}=Q^\top P^{-1}$ is lower triangular. The product $|T||T^{-1}|$ is lower triangular with every diagonal entry equal to one. Its complex characteristic polynomial is $(X-1)^K$, so its spectral radius is one for $K>0$. This argument uses the pinned Mathlib Gram-Schmidt, triangular-inverse and characteristic-polynomial results directly.

Replacing entrywise absolute values by a norm or operator absolute value, reversing or changing the matrix products, restricting the orthogonal domain, bounding the dimension, assuming a sign pattern or diagonalizability, or replacing attainment by an approximation would fail to match this problem. The formal result requires $K>0$ and $\det P\ne0$ and explicitly quantifies over every real orthogonal matrix, while separately constructing an attaining one. No classification of all minimizers is asserted.

The result is D5/S1/Eigenstructure/OrthogonalAbsoluteProductMinimum.result in the Lean source. Its complete mathematical exposition is the corresponding Blueprint. Canonical frozen membership is recorded at Golden/Frozen/state/D5/S1/Eigenstructure/OrthogonalAbsoluteProductMinimum.lean.json. The axiom closure uses only Classical.choice, Quot.sound and propext; there is no sorryAx or additional axiom.

The retained Reg source has accepted kernel, sorry and axiom evidence. Its current binding remains incomplete under the linked-issue exception: issue #24912. An extracted declared_validated row is not evidence of completed binding.

• Proved: for every positive natural dimension and every nonsingular real square matrix, all real orthogonal $Q$ have objective at least one and some real orthogonal $Q$ attains one. The resolution kind is proved for this optimization characterization.
• Proved, consequence of the same two clauses: the attained minimum and the infimum over the full orthogonal group equal one. A strict uniform bound $\beta(P)>1$ is impossible for any such $P$, and hence for every nonempty class of them.
• Proved, mechanism of the displayed proof: the chosen orthogonal coordinates make $PQ$ and its inverse lower triangular, and their entrywise absolute product has unit diagonal. The spectral argument does not require diagonalizability.
• Unresolved: a classification of every minimizing $Q$, quantitative stability near the minimum, and variants with a restricted set of orthogonal transformations are outside this theorem. They receive no new theorem or admission from this source resolution.
• Unresolved: exhaustive worldwide literature coverage and historical priority are not established by the bounded source screen.

The source attribution and bounded literature conclusions use the retained primary-source and preregistration evidence. They do not certify present MathOverflow answer status, global novelty or priority. Current Reg binding is explicitly unfinished under issue #24912; successful compilation and CI do not replace the missing binding evidence. The result concerns the stated positive-dimensional nonsingular domain and makes no assertion about singular matrices, dimension zero, or every possible minimizer.