The hook turns as hard as the geometry needs, or not at all

A slice hooking onto a body two metres to the side flew a closed loop around
it forever: 285 frames of the same circuit, never arriving, never falling.
A constant turn rate has a constant turn radius (v/w = 2 m), and anything
inside that circle cannot be reached however long the ball steers — while
the hook pins vy at 0, so nothing else ends it either. A crowd is nothing
but close bodies, which made the one place this weapon should shine the one
place it hung.

The rate is no longer a constant. An arc leaving along the current heading
and passing through a body at distance d, bearing a, has radius d/(2 sin a),
so the ball needs 2 v sin(a) / d — solved every frame, because the body is
walking while the ball travels. sliceTurn stays as the FLOOR, which is what
it was always really for: how wide the curl looks when there is room.
Turning faster than needed is safe, the aim being re-solved each frame.

Two things the maths cannot cover, so they are ruled out instead: a body
behind the ball, which no tangent arc reaches at any rate (sliceArc), and a
target that walks out of reach mid-hook (sliceHookT, a hard 0.7 s licence to
steer). An endless orbit is now impossible by construction rather than by
the geometry happening to work out.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>

config.js

@@ -247,8 +247,19 @@ export const TUNE = {
   sliceHops: 2,     // extra bodies one perfect serve hooks to. 2 => TRIPLE KILL is the ceiling
   sliceR: 6.5,      // m the hook reaches for the next body. Wider reads as the game aiming for you;
                     // much narrower and a spread-out crowd never chains at all
-  sliceTurn: 11,    // rad/s the ball swings onto the next one, and THE knob for how it looks: past
-                    // ~30 it snaps like a ricochet, under ~6 it sails wide before finishing the turn
+  sliceTurn: 11,    // rad/s the ball swings onto the next one: a FLOOR, not the rate. It is the knob
+                    // for how the shot LOOKS when there is room — past ~30 it snaps like a ricochet,
+                    // under ~6 it sails wide. The rate actually used is whatever the geometry needs
+                    // (see the hook), because a fixed rate has a fixed turn radius (v/w = 2 m here)
+                    // and ANY body inside that circle is unreachable: the ball flew a closed loop
+                    // around it forever, never arriving, never falling. A crowd is all close bodies,
+                    // so the one place the weapon should shine was the one place it hung.
+  sliceArc: 120,    // deg: the widest bearing off the ball's heading that can be hooked at all. No
+                    // single tangent arc reaches a body directly BEHIND the ball at any turn rate,
+                    // so past this the chain just ends and it plows on
+  sliceHookT: 0.7,  // s a hook may steer before it lets go. The geometry above is solved fresh every
+                    // frame and still cannot promise an arrival when the target WALKS, so this is the
+                    // backstop that makes an endless orbit impossible by construction, not by maths
   // The schedule calls the DUEL, not the tin: weapons are only ever won, but a duel comes round
   // on its own so there is always a chance at one. Fires off the next hand-cleared wave.
   duelFirst: 2,     // game after which the first scheduled sudden-death duel is called