gojo 1;

let $space = carrier();
let $x = carrier();
let $y = carrier(unit_of($x));
let $u = carrier();
let $normalized = span($u,0,1);
let $basis = group_on($space, span($x,0,1), span($y,0,1));
let $zero = group_on($space, at($x,0), at($y,0));
let $p0 = $zero;
let $p1 = group_on($space, at($x,1), at($y,0));
let $p2 = group_on($space, at($x,2), at($y,0));
let $joint = group_on($space, at($x,3), at($y,0));
let $q1 = group_on($space, at($x,4), at($y,0));
let $q2 = group_on($space, at($x,5), at($y,1));
let $q3 = group_on($space, at($x,6), at($y,1));
let $incoming = ports($p0,$p1,$p2,$joint);
let $outgoing = ports($joint,$q1,$q2,$q3);

// Adjacent difference and an explicit threefold calibration construct the
// endpoint derivatives of cubics with respect to the SAME normalized parameter.
let $triple = ports(span($x,0,1) -> span($x,0,3), span($y,0,1) -> span($y,0,3));
let $sixfold = ports(span($x,0,1) -> span($x,0,6), span($y,0,1) -> span($y,0,6));
let $entry_leg = group($p2,$joint,$basis);
let $exit_leg = group($joint,$q1,$basis);
let $entry_tangent = group(?entry_leg,$triple,$basis);
let $exit_tangent = group(?exit_leg,$triple,$basis);
let $endpoint_check = group($joint,$joint,$basis);
let $tangent_check = group(?entry_tangent,?exit_tangent,$basis);
// Desired zero is attached to each request. The observed displacement is a
// conditional witness; mismatch does not identify which participant must change.
root group($incoming,$outgoing,$normalized,$basis,
    $endpoint_check -> $zero, $tangent_check -> $zero,
    frame($entry_leg -> ?entry_leg, $exit_leg -> ?exit_leg,
        $entry_tangent -> ?entry_tangent, $exit_tangent -> ?exit_tangent),
    $triple,$sixfold,?curvature_relationship,?join_history);

// Same forward tangent direction with a different speed is weaker than C1.
let $faster_control = group_on($space, at($x,5), at($y,0));
// A zero first derivative is not evidence of a well-defined tangent direction.
root group(group($joint,$faster_control,$basis), group($joint,$joint,$basis),
    $triple, ports(span($x,0,2) -> span($x,0,1), span($y,0,2) -> span($y,0,1)), $basis);

// Coordinate agreement alone does not give a second point the joint's identity.
let $lookalike = group_on($space, at($x,3), at($y,0));
root group(group($joint,$lookalike,$basis), $joint, $lookalike);

// Second derivatives use the difference of adjacent legs and degree*(degree-1).
// C2 compares these parameter derivatives. Curvature still needs a metric and
// regularity; it is deliberately a separate open relationship in root 0.
root group(frame(
    group($p1,$p2,$basis) -> ?entry_preleg,
    group($q1,$q2,$basis) -> ?exit_nextleg,
    group(?entry_preleg,?entry_leg,$basis) -> ?entry_second,
    group(?exit_leg,?exit_nextleg,$basis) -> ?exit_second,
    group(?entry_second,$sixfold,$basis) -> ?entry_acceleration,
    group(?exit_second,$sixfold,$basis) -> ?exit_acceleration),
    group(?entry_acceleration,?exit_acceleration,$basis) -> $zero,
    $normalized);