gojo 1;

// A local plane: metric pairings are explicit squared-length participation.
let $space = carrier();
let $x = carrier();
let $y = carrier(unit_of($x));
let $squared = carrier(product_units(unit_of($x), unit_of($x)));
let $ux = span($x, 0, 1);
let $uy = span($y, 0, 1);
let $basis = group_on($space, $ux, $uy);
let $metric = group($basis, frame(
    ($ux -> $ux) -> at($squared, 1),
    ($ux -> $uy) -> at($squared, 0),
    ($uy -> $uy) -> at($squared, 1)));

// This later fixture supplies an X/Y representation. It does not generate a
// circle or its landmarks; see circle-construction.gojo for the native stages.
let $origin = point(carrier(), 0);
let $contour = crystal(orthogonal($origin));
let $centre = group_on($space, at($x, 10), at($y, 20));
let $centre_representation = $origin -> $centre;
let $p0 = group_on($space, at($x, 15), at($y, 20));
let $p1 = group_on($space, at($x, 10), at($y, 25));
let $p2 = group_on($space, at($x, 5), at($y, 20));
let $p3 = group_on($space, at($x, 10), at($y, 15));
let $m0 = point($contour, 0);
let $m1 = point($contour, 4);
let $m2 = point($contour, 8);
let $m3 = point($contour, 12);
let $rims = frame($m0 -> ($centre -> $p0), $m1 -> ($centre -> $p1),
    $m2 -> ($centre -> $p2), $m3 -> ($centre -> $p3));
let $cycle = frame($m0 -> $m1, $m1 -> $m2, $m2 -> $m3, $m3 -> $m0);
let $surface = group($centre_representation, $contour, $metric, $rims, $cycle);
root $surface;

// Steps count exposed edges. Marker tick differences are not angular units.
let $steps = carrier();
let $walk = group($surface, $m0, span($steps, 0, 1));
root $walk;

// An attached small frame, a repeated point, and an independent lookalike.
let $a = group_on($space, at($x, 12), at($y, 20));
let $b = group_on($space, at($x, 12), at($y, 21));
let $c = group_on($space, at($x, 13), at($y, 21));
let $d = group_on($space, at($x, 13), at($y, 20));
let $lookalike = group_on($space, at($x, 12), at($y, 20));
let $points = group($a, $b, $c, $d, $a, $lookalike);
root $points;

// Refine one interval with a known equal-reach landmark. Its two resulting
// steps have different geometric sizes: no equal-angle promise is invented.
let $extra = point($contour, 2);
let $extra_rim = group_on($space, at($x, 13), at($y, 24));
let $refined = group($centre_representation, $contour, $metric,
    frame($m0 -> ($centre -> $p0), $extra -> ($centre -> $extra_rim),
        $m1 -> ($centre -> $p1), $m2 -> ($centre -> $p2), $m3 -> ($centre -> $p3)),
    frame($m0 -> $extra, $extra -> $m1, $m1 -> $m2, $m2 -> $m3, $m3 -> $m0));
root $refined;

// Closure remains known when a radial attachment is not yet known.
let $partial = group($centre_representation, $contour, $metric,
    frame($m0 -> ($centre -> $p0), $m1 -> ($centre -> ?missing_rim),
        $m2 -> ($centre -> $p2), $m3 -> ($centre -> $p3)), $cycle);
root $partial;

// The driver binds an actual traversal witness before applying this request.
root group(?traversal, $points);