arcFun_coord1

The module builds high-dimensional linking-vanishing statements from explicit geometry on the circle $S^1\subset\mathbb{R}^2$. Points of the formal sphere object are accessed via ambient coordinates: the helper amb simply reads the underlying Euclidean vector.

The arc map sends $t\in[0,1]$ to the point whose first coordinate is $1-2t$ and whose second is $s\sqrt{1-(1-2t)^2}$, with the sign $s=\pm 1$ selecting the upper or lower semicircle. The point is packaged by a two-coordinate constructor that lands on the unit sphere once the Pythagorean identity is checked.

This coordinate projection is the elementary readout needed before one can identify ranges of the signed arcs or invert the arc-parameter map.

One-line definitional equality. Unfolding the arc map and the ambient-coordinate coercion yields the second component of the underlying pair by construction, so rfl closes the goal.

Feeds three local lemmas: the left-inverse identity that recovers a sphere point from its arc parameter (under a sign-matching hypothesis), and the two range characterizations identifying the images of the upper and lower arc maps with the closed hemispheres ${z_1\ge 0}$ and ${z_1\le 0}$. Those range facts are the geometric engine for showing that certain arc complements are acyclic, which is the high-dimensional half of the linking-vanishing argument in this foundation layer. No forcing-chain landmark (T5–T8) is touched directly; the lemma is pure Euclidean bookkeeping supporting the topological side of Recognition geometry.