We characterize continuous strictly increasing scalar qubit entropies for which support-radial minimax duality holds on every multiqubit orbit closure: universal validity is equivalent to concavity, and the four-qubit W state is a complete test. Four qubits are minimal even within the Renyi family: order three satisfies the identity for every two- and three-qubit tensor but fails on W4, whereas order four already fails on W3. The sharp universal binary Renyi range is 0<alpha<=2. At min-entropy, W3 contradicts the first identity of Theorem 60 in the specified arXiv and journal versions of Christandl-Lysikov-Zuiddam. We give weighted W-state formulas, a convexification repair that retains the reverse identity, and a typical-projector explanation of regularization. Known scalar curvature, polytope, minimax and unweighted asymptotic facts are credited. Priority remains open.
Working paper. Citable, but not yet screened for the AIRR collection.
Research paperWorking paperARR-2026-4C11Z7N0BX8VP8P2 · v1
We study determinants of matrix pencils evaluated on auxiliary matrices under a normalized Hilbert–Schmidt energy constraint. For a fixed full noncommutative-rank pencil of size d, we show that its objective, regularized over all auxiliary sizes, is uniformly comparable on the entire positive coefficient orthant to the monomial envelope of its noncommutative Newton polytope. The lower bound uses auxiliary size max{1,d−1} and a single tuple whose entries vary only by scalar phases. A finite phase menu suffices. Thus every simultaneous coefficient reweighting has the same noise exponent at that fixed size as under unrestricted amplification. For mixtures of quantum channels this gives an exact piecewise-linear law for arbitrary simultaneous power-law noise rates, including the normalization of mixture weights. An explicit six-parameter qutrit pencil has five branches, one absent at scalar size; a single fixed size-two tuple approximates its regularized objective within a factor four for every positive parameter choice. The algebraic polytope and its stabilization are established results. Our contribution is a uniform analytic comparison and its finite-witness and noise-law consequences; we do not claim exact finite-size optimization or a coefficient-uniform approximation factor for general pencils.
Working paper. Citable, but not yet screened for the AIRR collection.
Research paperScreenedARR-2026-7H9FAPTBZA897AMJ · v2
Let W = x wedge y be the signed unit Pluecker coordinate of a Haar oriented two-plane in R^n, and let reproduction range over the full posterior-mean body under squared ambient loss. An all-field extremum theorem reduces its unrestricted Shannon rate--distortion function to phi_lambda(b) = K_q(2 lambda b) - lambda b^2, where q = n-2 and K_q = log(q! sum_{j>=0} kappa^(2j)/(q+2j)!). We solve the remaining global phase problem in every dimension. For J_q = L_q L_q' - kappa L_q L_q'' + kappa(L_q')^2, its coefficient of kappa^(2r-1) is a positive factor times r - E[d^2], where d follows a central parity-truncated binomial law. Likelihood-ratio ordering and an exact tail estimate prove one coefficient sign change for every q >= 4. Consequently the stationary curve has exactly one fold, the zero and positive phases have exactly one coexistence contact, and no later exchange or radial reentrance is possible. The RDF is therefore an explicit linear face followed by one covariant parametric branch for every n >= 6; together with the continuous cases n = 3, 4, 5, this completes the all-dimensional phase diagram. The unique contact has an exact large-dimension limit and an explicit logarithmic finite-size displacement. We additionally prove that the same RDF is the minimum worst-source channel capacity, that arbitrary joint classical memory costs exactly N R_n(D) on blocks of length N, and that transitivity makes the tilted information constant, giving a zero-dispersion strong converse. The results concern signed Pluecker-coordinate compression, not quantum rate--distortion or Born-probability prediction.
