A Discussion of Density and Passability in Topological Signed Hypergraphs

We state a hypothesis on axes and signs as a set of elementary results. Orthogonal axes combine sign flips only multiplicatively, and a projection between them transfers nothing. A non-zero projection between oblique axes does transfer a sign, but this edge sign depends on orientation. The first orientation-free sign lives on a closed triple. For the edges of a triangle it equals −cos α cos β cos γ, so acute triangles give a negative sign, obtuse triangles a positive one, and right triangles zero. Adding an axis multiplies signs, so stacking is a parity. Right angles stack weakly in a precise sense: every triangle may vanish while a longer loop keeps a non-zero sign. We then cut one right triangle repeatedly by feet of perpendiculars. Each cut is a pair of anticommuting reflections, so path order produces signs, and the feet are dense in the triangle. Next, we let rings with free density parameters (δ, ε) pick convergent subtrees of a Clifford sign tree, and show that a two-sided gap condition orders these groups without cycles. Closing the tree by two reading orders gives a loop sign. It is independent of the hierarchy, but mutual blocking of one-level rings forces it to be positive. We also define an uncertain flip between separation and overlap, without a barrier model. Finally, we distinguish a latent sign from an explicit one, and match separated subtrees in a path space. Reading the nodes of two rings along an original edge gives a word, whose alternating sum is an integer twist. It is invariant under cancelling repeated letters, it needs alternation, and its parity shadow is the path sign. This gives the hypergraph a topological label in a limited sense, not a linking number of curves in space. The last section uses density for the passability of rings: density always allows a pass by size, and we propose a weight for the sign cost of the depth this requires.