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[2] ai.viXra.org:2606.0082 [pdf] submitted on 2026-06-30 05:52:42
Authors: Jason Merwin
Comments: 10 Pages. A GitHub repository link is provided
We exhibit a minimal combinatorial model in which a single structural property—second-order self-reference—produces three features usually treated separately: an intrinsic arrow of time; a sharp distinction between a closed ("block") and an open future; and the stabilization of a shared, observer-independent reality. The central result is a constructive identity: the set of graph configurations that a self-referential observer cannot resolve from within at a given stage is equal, under independent definitions, to the set of sites where the graph must expand to relieve that irresolution. Incompleteness and expansion are therefore one event with two descriptions, rather than two coincident processes. Iterating the resolution rule makes time a generative process—completing the incomplete seeds new incompleteness—and introduces a control parameter, the branching ratio of the abstraction, whose critical value separates a sub-critical phase in which paradoxes exhaust and time halts from a super-critical phase in which they self-sustain and the open future is guaranteed; in the generic toy sweep we locate this critical point at βc ≈ 1.08, near the unit replacement threshold expected on heuristic grounds.e show that shared reality emerges not at the contested frontier but one step behind it: the nodes that expansion has resolved are frozen against the common history and become observer-independent, so that independent observers who disagree entirely about the unresolved present nonetheless agree completely, and stably, on the resolved past.Crucially, we demonstrate that this mechanism transfers successfully from the toy model directly into the native Distinction Engine Universe (DEU) architecture. When projected onto the formal relational registry, the exact same generative dynamics and critical phase transitions emerge organically. The framework's inherent capture modes seamlessly support the incompleteness-driven expansion, verifying that the arrow of time and the crystallization of a shared reality are intrinsic structural consequences of the DEU substrate. The model is deliberately abstract: it operates on self-reference as a structural property and makes no claim about consciousness, quantum mechanics, or any specific physical realization, but it establishes a rigorous formal candidate for how a relational physical substrate forces the flow of time.
Category: Combinatorics and Graph Theory
[1] ai.viXra.org:2606.0078 [pdf] submitted on 2026-06-29 19:53:17
Authors: Peilin Wen
Comments: 9 Pages.
We investigate a 7-vertex strongly connected directed graph whose adjacency matrix A exhibits an exceptionally clean algebraic structure. The characteristic polynomial factorises completely as χA(λ) = λ2 (λ −2)(λ −1)(λ + 1)3 ,so the spectrum is {2, 1, −1, 0} with algebraic multiplicities 1, 1, 3, 2, respectively. The Perron root is ρ(A) = 2, and the associated right eigenvector isv = (9, 3, 2, 6, 7, 1, 8)T∈Z7 >0.For every directed edge i →j we define the rational weight wij = vj/(2vi). These weights form a row-stochastic matrix. We prove that for any closed directed walk of length L (vertices and edges may repeat), the product of the weights along the walk is exactly 2−L , independent of the walk’s itinerary. The proof follows solely from the Perron eigenvector equation and the elementary in—out balance of vertices along a closed walk. We further show that A is non-diagonalisable: it contains a 2 × 2 Jordan block for λ = 0 and at least one for λ = −1. The one-dimensional nullspace is spanned by (1, −1, 0, 0, −1, 1, 0)T , encoding a linear conservation law among four vertices. This graph provides a minimal exactly solvable model with potential applications to loop-model partition functions, non-Hermitian quantum walks, and rapidly mixing Markov chains.Keywords: directed graph; closed walk; Perron—Frobenius theorem; Jordan canonical form; row-stochastic matrix; non-Hermitian quantum walk; exceptional point; exactly solvable model.
Category: Combinatorics and Graph Theory