The Fusion Simplex as a Space of Approach Directions

Hoffman, Prakash and Prentner associate to n interacting conscious agents the Markov polytope of n×n row-stochastic matrices, and identify the rank-one stochastic matrices as a fusion polytope, an (n−1)-simplex. This note observes that for n = 2 the asymptotic map sending a kernel to the limit of its powers is undefined at the identity, and depends elsewhere only on the direction along which the identity is approached. The fusion polytope is therefore the space of approach directions to the identity, and carries an involution induced by relabeling the two agents. In the affine coordinate ξ on that space the involution is ξ ↦ 1/ξ. Its fixed point is the unique matrix common to the fusion and Birkhoff polytopes, and the pair it exchanges is the pair of one-dimensional cells carrying the decorated permutations [1,4] and [3,2]. A closing remark identifies the fusion polytope with the moment polytope of complex projective space, under which the Fubini–Study measure pushes forward to the uniform Dirichlet distribution and the coordinates of a fusion are the Born probabilities of the corresponding state. The general-n case is stated as an open question: the fusion polytope has dimension n−1 while the space of approach directions has dimension n(n−1)−1, so the two agree only at n = 2.

All results are verified against the printed statements of Hoffman, Prakash and Prentner, Fusions of Consciousness, Entropy 25 (2023), no. 1, 129.