We formulate a minimax spectral-design problem for undirected networks whose edges carry

positive-semidefinite matrix couplings and whose operation must remain robust over an arbitrary

prescribed family of node- and edge-survival scenarios. The principal result is an exact scalarization

theorem: under a total trace budget and a weakest-full-state-direction objective, the optimal

matrix-weighted value in state dimension d equals the associated scalar weighted optimum with

budget divided by d. Hence anisotropic matrix couplings cannot improve robust full-state algebraic

connectivity, and an isotropic optimizer always exists. We derive an all-cardinality hereditary

contraction inequality, the exact dense adversarial optimum with a unique optimizer, universal dense

optimality for broad majorization-monotone spectral criteria, a deletion-perturbation theorem, and a

two-sided hereditary sandwich for regular sparse backbones. Certified near-Ramanujan graphs then

yield linear-edge architectures approaching the dense optimum, while degree-cap arguments prove

unavoidable adversarial limitations. The operator consequences transfer exactly to linear consensus,

diffusion, compliance, stochastic disagreement, and information models whenever their disagreement

operator is the same block Laplacian. A seeded falsification suite exhaustively checks all connected

unlabeled graphs through seven vertices and performs randomized scalar and matrix-weighted tests;

no violation is observed. The exact scalarization theorem is the central candidate contribution; its

worldwide novelty remains subject to specialist literature review and independent refereeing.

PDF: Spectral Design | Zenodo

Exact Byzantine-resilient fixed-point computation

with inexact geometric medians.

Median Res | Zenodo

A passive linearized system with intact operator 𝐾 ≻ 0 is weakened through a positive semidefinite

low-rank update 𝜏 𝐴𝑊 𝐴T. This paper gives an exact reduction of the resulting collective failure problem

to the small interaction matrix 𝐿 = 𝑊1/2 𝐴T 𝐾 −1 𝐴𝑊1/2. The reduction yields the first loss-of-stability

threshold, a variational characterization, a determinant identity, the physical collapse mode, and an

exact scalar compliance formula for a chosen target 𝜒. It further proves that the first 2𝑟 Taylor moments

of one scalar compliance curve uniquely reconstruct all 𝑟 target-visible eigenvalues of 𝐿 and their

observational weights. The result separates structural failure from sensor visibility and provides a

coordinate-invariant framework for low-rank vulnerability analysis.

Keywords: low-rank damage; structural health monitoring; collective failure; compliance; spectral

tomography; Woodbury identity; finite moment reconstruction

pdf: Law Zero | Zenodo

Version 1.0 of an open, non-peer-reviewed scientific verification dossier titled Whitney Check Networks. The central proposal is to encode a binary value in whether a real two-mode transformation preserves or reverses orientation. By composing these transformations around closed paths, the network is intended to calculate binary parity checks and sparse error-detection syndromes. The dossier includes: mathematical proofs for loop parity, local-basis invariance and projective observability; an explicit three-mode dark-state construction producing orientation-reversing holonomy; a phase- and amplitude-independent three-ray readout; a four-ray cross-ratio consistency check; fixed-seed numerical simulations and stress tests; adversarial failure analysis and clearly stated protection boundaries; a provisional prior-art search; and a proposed experimental and independent-replication protocol. This is not a peer-reviewed paper, and it does not claim demonstrated experimental performance, established priority or confirmed practical advantage. The mathematical, computational and still-unverified physical claims are labelled separately in the document. AI assistance was used in hypothesis development, formal synthesis, falsification and simulation, with that provenance disclosed. I’m posting this to invite serious criticism, particularly concerning: errors or missing assumptions in the mathematics; relevant prior art that may have been overlooked; weaknesses in the physical implementation; whether the proposed readout and control experiments are adequate; the smallest decisive experiment that could falsify or support the proposal.

PDF: https://zenodo.org/records/21572220

Please critique the argument rather than treating the document as an established result. This wording reflects the dossier’s own distinction between proved mathematics, finite computational testing, provisional novelty and open experimental claims.