Boundary limits of the matrix geometric mean and the failure of unrestricted GLT closure

Unrefereed research candidate. In one sentence: the matrix geometric mean can jump at singular matrices, every possible jump is described by one rank condition, and the same mechanism refutes a published claim that the symbol of the geometric mean of two GLT matrix-sequences is always the geometric mean of their symbols. The geometric mean A#B of two positive definite matrices is the matrix analogue of √(ab), used to average covariance matrices and structured matrices and to define the Matsumoto fidelity of quantum states. It need not be continuous at singular pairs. Theorem 1 shows that X is a limit of Aj#Bj with positive definite Aj → A, Bj → B exactly when X ≽ 0, the block matrix [[A, X], [X, B]] is positive semidefinite and its rank is at most the dimension N (a dilation description with an auxiliary contraction already follows from recent work of Jones; the explicit elimination is the contribution). At equal inputs this gives the full set of limits and the exact interval of limiting Matsumoto fidelities. For generalized locally Toeplitz (GLT) matrix-sequences, uniformly bounded positive definite inputs with identical scalar symbols can have geometric means with the wrong symbol (4/5 instead of 1 on half the interval) or no eigenvalue distribution at all. This refutes Theorem 4 and Conjecture 1 of the arXiv version of Ilyas, Khan, Loi and Serra-Capizzano and, according to an AI-assisted reading of the published text, Theorem 3 and Conjecture 1 of their article in Linear Algebra Appl. (2025); the failing step of the proof is located. Positive results: closure for every Kubo-Ando mean when the sum of the symbols is positive definite almost everywhere; a one-sided theorem that the regularised symbol is always an upper envelope; and, for bounded Riemann-integrable scalar symbols of the space variable only, a sharp dichotomy (the formula holds for every realisation if and only if the symbols have no common zero set of positive measure or essentially disjoint supports). The classical two-dimensional discontinuity, Ando's maximal characterisation and the envelope X ≼ A#B are earlier work and are credited. The archive contains the manuscript (PDF and Markdown source), two figures with the script that draws them, an exact verifier (verify_exact.py, Python standard library only, 1,030 rational and Gaussian-rational checks of closed forms and finite assemblies, with a mutation self-test) and its retained output, the bibliography with its Crossref/arXiv records, the dated novelty ledger, a revision note, CI and SHA-256 sums. Assurance: written proofs; the verifier checks finite exact identities, not the theorems' limits or quantifiers; internal AI-assisted review only (not human peer review, independent reproduction or formal verification). Creator: Anonymous. Text and figures CC0-1.0; code MIT.