Bouncing balls in a heptagon · step-5-preview

Let me write the index.html with a 2D physics simulation.

Design:
- Canvas, maybe 800x800 or responsive. Heptagon (7-sided) rotating at 1 rev / 5 s → angular velocity = 2π/5 rad/s ≈ 1.2566 rad/s.
- 20 balls, numbered 1-20, arranged in compact non-overlapping arrangement near center — e.g., a 5x4 grid with small spacing, or hexagonal packing. Radius must fit.

Let me pick heptagon with circumradius R = 300 px, center at canvas center. Balls radius ~ 18px. 20 balls in 5 columns x 4 rows with spacing 2*r*1.15 → width 5*2*18*1.02 ≈ 185 px. Fits well.

Actually to be "compact, non-overlapping", use hexagonal-ish packing to be compact.

Physics: Use substeps for stability. Gravity 900 px/s². Restitution 0.75, damping 0.999 air drag.

Wall collision with moving wall: compute wall point velocity = ω × r (perpendicular). For a rotating polygon, the wall segment endpoints move with velocity ω * (-(y-cy), (x-cx)). Rather than treating wall as a line, do proper segment-vs-circle collision with moving endpoints — simplest: for each edge (p1,p2), find closest point on segment to ball center. If distance < r, resolve: push out along normal, and compute the velocity of the closest point at time of collision: v_wall = ω × (closestPoint - center). Relative velocity vrel = v_ball - v_wall. Reflect normal component: v_n' = -e * v_n. Tangential: apply friction. Then v_ball = v_wall + v_n' + v_t'*(1-friction). Also account for ball spin affecting tangential contact velocity — could add a bit. Let's include ball spin with a simple tangential friction coupling: contact point velocity of ball = v + ω_ball × r_contact. Then relative tangential velocity computed properly. That's nice and gives spin visible effect.

Ball-ball collision: standard impulse with restitution, mass ∝ r² (equal radii so equal mass). Also add tangential friction for spin exchange.

Ball angular: initial random spin, and torque from tangential friction impulses.

Containment: also add a safety clamp — if ball escapes polygon (distance from center > R), reset it to center-ish. Actually with substeps and proper resolution, should be fine. Add a hard clamp: if outside, put back inside.

Let me implement carefully.

Heptagon vertices: angle_i = theta + i*2π/7, i=0..6. Inscribed in circle radius R. To keep it fully visible, ensure canvas fits: canvas 800x800 (or scale to container). Center (400,400). R = 320. Then max extent 400+320=720 < 800 ok.

But wait — a rotating heptagon: the "fully visible" requirement means the polygon (its vertices) stays within canvas. With R=320 and center 400, fine.

Balls: radius 17. Initial arrangement: 5 columns × 4 rows centered. spacing = 2*r + 4 = 38. width = 5*38 = 190, height=4*38=152. Centered at (400, 400). Good — that's compact and non-overlapping. Hmm, but "compact" - a hexagonal packing is more compact. Let's do rows with offset odd rows by half spacing, 5 in even rows, 4 in odd rows... Actually let's do 4 rows of 5 = 20 with hexagonal offset for compactness. Rows: 5,4,5,4 = 18. Not 20. Let's just do 5x4 grid with slight offsets alternating to look compact. Fine.

Actually let's do hex packing: rows of 5 and 4 alternating: 5,4,5,4,2 = 20. Eh. Simpler: 5 columns × 4 rows with dx = 2r+3, dy = (2r+3)*0.87, odd rows offset by dx/2. That's 20 balls in a compact hex arrangement. 

Gravity makes them fall and collide with the bottom walls.

Timestep: use fixed dt = 1/600 with substeps, accumulating. Actually use dt = 1/120 with 4 substeps → effective 1/480. Let's do: frame dt clamped to 1/30 max, then substeps of 1/480 for stability. With ω = 1.2566 rad/s, per substep rotation = 0.0026 rad — fine.

Wait, with high rotation, walls move fast. Wall tip speed at R=320: 320*1.2566 = 402 px/s. That's fine, less than gravity effect.

Spin: balls radius 17, angular velocity up to maybe 15 rad/s. Contact point velocity = 15*17 = 255 px/s. Reasonable.

Let me now write the code.

Collision resolution approach for walls:
For each edge, compute closest point. If dist < r:
- normal n = (ballPos - closest)/dist (points from wall to ball)
- penetration = r - dist
- positional correction: ballPos += n * penetration (maybe with a small slop)
- contact point velocity: v_wall = ω × (closest - center)
- ball contact velocity: r_contact = -n * r ; v_c = v_ball + ω_ball × r_c  (2D: ω × r = ω * (-r.y, r.x))
- v_rel = v_c - v_wall
- vn = dot(v_rel, n)
- if vn < 0:
  - normal impulse: jn = -(1+e) * vn / (1/m + ... ) — for simplicity with infinite mass wall and no rotational normal coupling, jn = -(1+e)*vn* m. In 2D with rotation, normal impulse doesn't couple to rotation for a circle (since r_c parallel to n, cross = 0). So jn = -(1+e)*vn / (1/m).
  - tangential: vt vector = v_rel - vn*n. Define tangent t = perp(n). jt magnitude limited by Coulomb: |jt| <= μ*|jn|. Apply impulse jt*t to ball, and reaction torque.
  
For simplicity and stability, use a friction coefficient for wall ~0.25 and apply damping.

Let me implement impulse with rotational inertia. Ball: m = 1, I = 0.5*m*r² = 0.5*r².

Impulse application:
- v += J/m
- ω += cross(r_c, J)/I

tangential impulse: Jt = -vt_mag * t, clamped to μ*jn magnitude... Standard:
```
let rvc = cross(r_c, t); // = cross(r_c, t)
let denom = 1/m + rvc*rvc/I;
let jt = -vt / denom;  // where vt = dot(v_rel, t)
jt = clamp(jt, -mu*jn, mu*jn);
apply J = t*jt
```
Hmm, need care. Let's just do:
```
const t = perp(n); // (-n.y, n.x)
const vrel_t = vrel.x*t.x + vrel.y*t.y;
const rn = cross(r_c, t);
const denom = 1/m + rn*rn/I;
let jt = -vrel_t/denom;
const maxF = mu*jn; // jn positive scalar
if (jt > maxF) jt = maxF; else if (jt < -maxF) jt = -maxF;
applyImpulse(jt*t.x, jt*t.y)
```
where jn = -(1+e)*vn/(1/m) ... but careful with sign: jn should be the impulse magnitude along n. v_rel·n = vn < 0 means approaching. jn_scalar = -(1+e)*vn/(1/m) = -(1+e)*vn*m. Positive since vn<0. Then apply J = n * jn_scalar. Then friction clamp uses jn_scalar.

That's proper sequential impulse. With a few iterations per substep it's stable.

Actually simpler approach: just do direct velocity reflection with restitution and a tangential friction term, which is easier to get right and stable:
```
vn = dot(vrel, n)
if vn < 0:
  newVn = -e*vn
  // tangential: reduce by friction
  vt_vec = vrel - vn*n
  // apply damping to tangential & couple spin
  vrel_new = vt_vec*(1-mu) + newVn*n
  v_ball_contact = vrel_new + v_wall
  v_ball = v_ball_contact - ω_ball × r_c
```
This is a "velocity-level" resolution that conserves the spin→translation coupling implicitly. It's stable and simple. Since r_c = -r*n, cross(r_c, ω vector) = ω*(-r*n.y, r*n.x)... Let me compute: v_c = v + ω × r_c where ω × r = ω*(-r.y, r.x). So v = v_c - ω × r_c.

This approach: the impulse changes both translation and spin in a way consistent with sticking. Hmm, but if I set v_c directly, then v changes and ω unchanged — that's physically like the friction impulse being applied only to translation. It's approximate but stable and looks fine. Actually better: it makes spin not change from wall friction, which is less interesting.

I'd rather do proper impulses. Let me write a proper impulse solver — it's not that hard.

Let me write the code now.

```js
function cross(ax, ay, bx, by) { return ax*by - ay*bx; }
```

Ball object: {x,y,vx,vy,r,m,invM,I,invI,ang,angVel,num,hue}

Ball-ball: for pairs, dist < r1+r2 → normal, positional correction (split by invMass), impulse with restitution + friction (and torque from tangential impulse). Since all balls same radius but I'll allow varying sizes? Requirement says "reasonable sizes" — let's make all balls same size for simplicity and uniformity. Actually varied sizes look nicer... but numbering 1-20 with same size is cleaner and safer. I'll do uniform radius 17.

Panels: I'll implement velocity iterations: 3 iterations of collision resolution per substep for contacts, using accumulated impulses? Simpler: run contact solving loop: detect & resolve sequentially (Gauss-Seidel) N times, then integrate positions. That's the standard approach: detect contacts, then iterate solving velocities, then integrate.

Simpler robust scheme per substep:
1. Update wall rotation angle (constant).
2. For each ball: v += g*dt; apply air drag.
3. Detect contacts (ball-ball, ball-wall) → list with normal, penetration, contact point.
4. Solve velocity constraints: for iter in 0..K: for each contact: compute relative velocity, apply normal + friction impulse (solve normal once with restitution bias — Baumgarte style: separate position correction after).
5. Integrate positions.
6. Position correction: push apart overlapping pairs/walls with a percentage (e.g. 0.8) and slop 0.01.

This gives stable results without unbounded energy gain.

Let me be careful about the normal restitution handling in velocity solve: use `restitution` applied on first iteration only, or use target velocity = -e*vn_initial. I'll store initial vn and use bias velocity: `let target = -e*vn0; if (target < 0) target = 0;` Then jn = -(vn - target)/(n mass term). Fine.

Let me write it.

For walls (rotating heptagon), the wall has infinite mass. Contact point velocity computed at current wall rotation. Since wall is rotating, the wall velocity matters — we compute v_wall at contact point each iteration (it's constant per substep).

Also need to handle that the rotating wall "pushes" balls: the normal constraint with infinite mass wall and moving wall velocity naturally handles push-out via Baumgarte position correction and the velocity bias. Good enough.

Edge case: ball near a vertex. Closest point on segment handles it.

Additional safety: after integration, if ball center is outside the polygon circumradius + r (escaped), teleport back toward center. Let me implement point-in-polygon test for safety and re-place if outside.

Actually a simpler robust containment: since the polygon is convex and contains the center, a ball is "inside" if for every edge, dot(pos - v_i, inwardNormal) <= 0. If the ball is beyond any edge by more than r*1.2, it's escaped → then place it at the center-ish position with reduced velocity. Let's add that as a failsafe.

Let me now write the HTML.

Canvas: 900x760 maybe. Let's use a responsive canvas with fixed internal resolution 800x800 and CSS width 100% max 700px. Hmm, for the browser tool checks at 1200x800 viewport, let's make canvas 760x760 with center (380,380) and R = 300. Balls r=16. Actually let's use width=800,height=800 and CSS max-height 78vh.

Let's plan layout: title, canvas, controls (Pause/Resume, Reset), and stats (FPS, ball count). Also keyboard shortcuts: Space pause, R reset.

Heptagon drawing: stroke with lineWidth 4, plus glow. Also maybe draw faint trail? Keep simple.

Number rendering: rotate context by ball.ang, draw text. Font bold 14px.

Also draw a marker on the ball so spin is visible beyond the number — the number itself rotates, that's enough, plus a small dot/line.

Let me write the file now.

Also "moving-wall velocity in bounce response" — done via contact point velocity.

Let me write.

Physics numbers:
- R (circumradius) = 300
- gravity g = 1400 px/s²? With balls r=16 and world scale 800 px ≈ maybe 1.6 m tall. Let's use g = 1200.
- restitution ball-ball = 0.6, ball-wall = 0.55
- friction μ = 0.3
- air drag: v *= 0.9995 per substep? Use linear damping factor: v *= (1 - 0.15*dt) etc.
- angular damping: ω *= (1 - 0.5*dt)

Rotation: ω_wall = 2π/5 = 1.2566 rad/s.

Substep dt: 1/480 s. Per substep, wall rotation = 1.2566/480 = 0.0026 rad. At R=300, displacement = 0.78 px. Good, less than typical penetration.

Number of substeps per frame: dt_frame clamped to 1/60 max (0.01667). substeps = ceil(dt/dt_max) where dt_max = 1/480 → 8 substeps. That's 8*20 balls*... fine performance wise.

Position correction: percent 0.6, slop 0.05.

Let's also add a max velocity clamp: speed <= 3000 px/s; angular vel clamp ±60 rad/s.

Write code.

Initial arrangement: hex packing:
rows: y offsets. r=16, gap=4 → dx = 2*r+gap = 36 spacing horizontally between centers 36? For hex packing, vertical spacing = dx*sqrt(3)/2 = 31.2. 4 rows → height = 3*31.2 + 2*r = 125.6. Columns: row0: 5 balls, row1: 4 balls offset, row2: 5, row3: 4 → 18. Hmm need 20. Let's do 5 columns × 4 rows: 20. Rows alternate offset by dx/2. So row widths: full 5, then 4 offset... no — with all rows having 5 and offset alternating, the hex look is achieved by offsetting every other row by dx/2 (rows with 5 balls offset → extends beyond). Total width 5*36 = 180 for rows 0,2 and 4*36+18=162 for rows 1,3.

Simplest: all 20 balls in a 5x4 grid, x_i = -2..2 * dx, y_j = -1.5..1.5 * dy with odd j offset by dx/2. Good — compact and non-overlapping.

Place near center: center = (400,400). But gravity pulls down; starting centered is fine.

Actually to make the "initial fall" visible and interesting, place them slightly above center? Requirement says "near the enclosure center", so center is fine. They'll fall ~a bit and settle.

Let me also record the initial state for Reset: array of {x,y} and wall angle 0.

Let me write it.

One more: "Keep it fully visible" — ensure the polygon + stroke fits. With R=300 at center 400 and canvas 800: vertices up to x=100..700. Good, and vertices of a heptagon rotated arbitrarily: max |x-cx| = R = 300 → within [100,700]. 

Now the code.

For the browser verification, I need to run index.html via the browser tool. Let me write and then check.

Let me write the file.

Now let me write the index.html file.

There's a leftover unused `invMassSum` variable in solveContact which is harmless but let me clean it up. Also the `insideHeptagon` check uses `> POLY_R + BALL_R` which is odd — the heptagon is inscribed, so the distance from center to a vertex is POLY_R, and edges are at distance POLY_R*cos(π/7) ≈ 0.9*POLY_R from center. My check `nx*(x-p1)+ny*(y-p1) > POLY_R + BALL_R` is a loose containment check — it will basically never trigger unless the ball is way outside. Let me make it a proper point-in-polygon test instead: for each edge, signed distance to the edge line along inward normal should be >= -(BALL_R + margin).

Let me fix that: for each edge, inward normal points toward center. If dot(pos - p1, inward) < -(BALL_R*1.5), then escaped. Actually if the ball center is more than some margin outside the polygon, reset it. Let's use: dist_to_edge_line (signed, positive inside) < -BALL_R*0.5 means center is outside beyond half a radius → failsafe. Since penetration is corrected each frame, a ball center shouldn't be outside by more than penetration. Hmm, during a fast rotation, penetration can be up to ~2px. Using -BALL_R*0.5 = -8 as threshold is safe from false triggers.

Let me clean up both.

Also note: number font 15px in a 32px diameter ball is fine.

Let me also double-check the degenerate normal handling in buildContacts: for edges, the normal should point from the closest point on the edge toward the ball (inward). I do that when d >= 1e-6. Good. But there's an issue: if the ball's center is exactly on the edge line (d == 0), we use edge normal with inward correction — fine.

Another potential issue: for a ball resting on an edge and penetrating, the normal points inward, and `solveContact` treats `a` = ball, `b` = null (wall). In my formulation, n points from a towards b/wall, i.e., from ball towards the wall = outward! Wait.

Let's check: contacts.push({a: b, b: null, nx, ny, ...}) where nx = ball.x - qx normalized → points from wall toward ball = inward. So in solveContact, c.nx points from the "wall" (surface) toward ball a. Hmm, I set rax = -c.nx * a.r → contact point = ball center - n*r = ball center + outward*r. Wait: if n points from wall toward ball (inward), then the contact point on the ball surface is at ball.center - n*r... no. Let's think: wall surface at distance d below the ball (from the wall toward ball is n). The ball center is at distance r from the contact point along -n from ball center? The closest point on the wall is at q, ball center is at q + n*d. The contact point on the ball is at ball.center - n*r = q + n*d - n*r = q - n*(r-d) — which is on the wall side. Yes: contact point = ball.center - n*r. So rax = -c.nx * a.r is correct for the contact point.

Now relative velocity: rv = v_ballContact - v_wallContact. Normal direction: n points from wall to ball. Approaching means the ball moves toward the wall, i.e., v_rel·n < 0 (ball moving opposite to n, into the wall). Yes: vn < 0 means approaching. Correct.

Normal impulse: jn positive, applied along +n to the ball (pushing it away from the wall, inward). applyImpulse with ix = c.nx*jn → a.vx += n*jn/invM → positive, pushing the ball along +n = away from the wall. Correct.

For ball-ball: nx = (b.x - a.x)/d → points from a to b. Contact point on a is a.pos + n*a.r (toward b). But I wrote `rax = -c.nx * a.r` for both cases! That's wrong for ball-ball.

For ball-ball, contact point relative to a should be +n*a.r, and relative to b it's -n*b.r. Let me fix that.

Also in the ball-ball case, the relative velocity convention: vn = (v_aContact - v_bContact)·n where n points from a to b. Approaching when vn < 0. Impulse: apply +n*jn to a and -n*jn to b. That pushes them apart. Correct.

So I need to fix the contact-point radius signs. Let me restructure: define in the contact object `rax, ray` (a's vector from center to contact) and `rbx, rby`. For wall: n points wall→ball: ra = -n*a.r, rb = 0. For ball-ball: n points a→b: ra = +n*a.r, rb = -n*b.r.

But then the normal direction semantics for ball-ball differ from wall. Let's unify: define n as the "separation direction" from a's surface toward b's surface... Hmm, for wall the wall is like an infinitely heavy b, and n from a to b would be outward (from ball to wall). For ball-ball, n from a to b is from ball a's center toward ball b's center.

So unify: n = direction from a's center to b's center (or toward the wall surface for wall contacts). Then contact point on a = a.center + n*a.r; contact point on b = b.center - n*b.r (for ball-ball, contact point = a.center + n*a.r = b.center - n*b.r ✓). For a wall: b is at infinity-ish, contact point = a.center + n*a.r, where n is the outward normal of the wall (from ball toward wall). Wall contact velocity at that point.

So I should store, for wall contacts, the OUTWARD normal (pointing from the ball toward the wall). But the wall normal direction should be consistent — I should use the wall's own normal (from the edge outward), not the direction toward the closest point q. Actually, for a segment vs circle, the closest-point direction is the natural normal. For the containment case, that direction from q to the ball is inward, so from the ball toward wall is -that = outward. So store nx, ny = outward = (q - ball)/d.

Hmm but careful: if the ball center is exactly on the wall line, using the wall's own normal is better. Let's just use: normal = (closestPoint - ballCenter)/d normalized if d > eps; that points from the ball toward the wall = outward. Good.

Then for the wall: contact point = ball.center + n*ball.r (which lies on the wall side). Good, and the moving-wall velocity at that point.

Let me rewrite solveContact with the unified convention:
- n: unit vector pointing from a's center toward the surface of b (or toward the wall).
- Penetration positive.
- ra = n * a.r (contact point rel to a)
- rb = -n * b.r for ball-ball (rel to b), for wall just 0.
- rv = (v_a + ω_a × ra) - (v_b + ω_b × rb + wallVel)
- vn = rv·n; approaching if vn < 0.
- impulse: a gets +n*jn, b gets -n*jn.

And position correction: a moves -n*corr, b moves +n*corr. Since n points from a toward b/wall, a should move away from b, i.e., along -n. Wait: if n points from a toward b, then to separate them, a moves along -n and b moves along +n. Currently my code does `a.x -= c.nx * im * a.invM` — with the new convention (n from a to b), a.x -= n*... = move away from b ✓, and `b.x += c.nx*im*b.invM` ✓. Good — that part already matches the ball-ball convention, and for the wall (b=null) it just moves the ball away from the wall ✓.

Now the booleans: previously for wall contacts with the inward normal, `a.x -= nx*...` would push the ball INTO the wall. But position correction only runs when pen > slop... it would have made things worse! Let's check: pen = r - d > 0 when overlapping. With inward normal (wall→ball), `a.x -= nx*im` moves the ball along -inward = outward = INTO the wall. That's a bug that would push balls out of the polygon. Good catch. Must fix.

Let me rewrite the contact building and solver cleanly.

Also there's `cross(rax,ray,c.nx,c.ny)` — for a circle, ra ∥ n so cross = 0. Fine; remove the weird invMassSum leftover.

Let me rewrite the whole script section carefully with a clean structure. I'll rewrite the file to keep it coherent.

Also, potential issue: with `impulse` on ball-ball where both are same mass, that's fine.

Let me also reconsider friction clamp: jt = clamp(-vt/denom, -mu*jn, mu*jn). With jn from the first iteration. Fine.

One more thing: in the iteration loop, solving normal impulse repeatedly without accumulated clamping can add energy. Standard approach uses accumulated impulses. But with restitution only on iteration 0 and 4 iterations, energy gain is limited by restitution ≤ 1 in the normal direction... Actually repeated application of `jn = -(vn - target)/invMass` each iteration sets vn to the target exactly each time it's violated; since after iteration 0 vn ≈ target (≥ 0), subsequent iterations see vn ≥ 0 → jn = 0. So no energy gain. 

But careful: after the friction impulse is applied, the tangential change doesn't affect vn (perpendicular). And later contacts (Gauss-Seidel) can re-violate. Still fine.

Now let's rewrite the file.

Also let me double check the initial arrangement: 5 cols x 4 rows, dx = 2*16+5 = 37, dy = 37*0.866 = 32.04. Height = 3*32.04 + 32 = 128. Width = 4*37 + 32 = 180. Row 1 and 3 offset by 18.5 → extends to ±(2*37+18.5+16) = ±108.5. All within R=300. Good.

Wall tip speed at R=300: 377 px/s. Ball fall speed after falling 300px: sqrt(2*1200*300) ≈ 849 px/s. That's fine — the wall moves slower than balls fall. Good, balls stay contained.

Let me now write the corrected version.

Now let's test the physics logic headlessly with node to verify stability before running the browser. I can extract the physics and run it in node. Let me create a quick test harness that loads the script portion... Easier: write a small node script that duplicates the physics by extracting from the HTML between markers. Actually, simplest: use node to read index.html, extract the <script> content, stub out `document` and `window`, and eval.

Let me do that.

The `sim` variable — the `window.__sim` assignment was in the cut-off portion. Let me instead reference the globals directly (balls, wallAngle, verts, etc.) which are declared with `let`/`const` at eval top-level in a script — but eval in module scope... `eval(src)` inside a function scope? It's at module top-level in CommonJS, so `let` declarations inside eval are scoped to the eval and don't leak. I need to use indirect eval or append code to the source.

Simplest: append the test code to src and eval the whole thing together.

Something's wrong: at t=0.5s the balls are frozen (maxSpd=5) and they collapsed to the center, and wallOverlap=272 means all balls are outside the heptagon?? Wait, "wallOverlap" here is signed distance to edges being negative — meaning balls are outside. But `insideHeptagon` returns true apparently since escaped=0 initially... Actually escaped=72120 over 60s.

Wait, maxSpd=5 at t=0.5 and balls collapsed to center at x=399-401. And contacts=190 = huge (should be at most 20*7 + 190 pairs = 320). Hmm 190 is plausible for ball-ball contacts (190 = C(20,2) = 190!). So all pairs overlapping = all balls at the center. And wallOverlap 272 means all balls are outside the polygon...

Wait — I bet the issue is that `substep` is defined but... hold on, `maxSpd=5` at t=0.5 — the gravity integral should give vy = 600 after 0.5s. Unless damping... no.

Hmm, and bbox at the end is 399..401, all balls collapsed into a point. That suggests my `insideHeptagon` failsafe keeps resetting them to the center. And escaped=72120.

So insideHeptagon returns false for balls at the center. Let's check: insideHeptagon computes inward normal from edge... Let me check: for edge from p1 to p2, the inward normal should point toward the center. My formula: nx = ey*il, ny = -ex*il. For edge direction (ex,ey), the normal (-ey, ex) points... depends on winding. Let's test: CX=400, CY=400. verts[0] at angle = -π/2 → (400, 100). verts[1] at angle = -π/2 + 2π/7 ≈ 0.448 → (400+300cos(0.448), 400+300sin(0.448)) = (400+268.4, 400+129.6) = (668.4, 529.6). So edge from (400,100) to (668.4,529.6): ex=268.4, ey=429.6. Normal (nx,ny) = (429.6, -268.4)/508.8 = (0.844, -0.528). Vector from p1 to center: (0, 300). Dot = 0 + (-0.528*300) = -158.4 < 0. So this normal points OUTWARD. So my "inward normal" is actually outward!

That means in `insideHeptagon`, s = outward·(p - p1) which is negative inside the polygon... wait no. If n points outward, then for an interior point, n·(p-p1) < 0. So s < 0 for interior points → `if (s < -BALL_R*0.75) return false` → true for interior points near center → returns false → failsafe triggers constantly!

But wait, then how did maxSpd stay 5 and balls collapse to center? The failsafe resets them near the center with velocity *= 0.2 each frame. Yes! That explains everything. And in buildContacts, the degenerate branch used (nx,ny) = (-ey,ex) with a check `if (nx*(CX-bx)+ny*(CY-by) > 0) negate` — that's also pointing... hmm, in the degenerate case I want n to point from the ball toward the wall (outward). I wrote `nx*(CX-b.x)+...` — if the normal points toward the center, negate. That check itself is fine (it makes n point away from the center). Actually the condition `> 0` means the normal points toward the center → negate → points away from center = outward. Correct, but coincidentally.

So the fix: inward normal = (-ey, ex)*il? Let's check with the above: (-ey,ex)/len = (-429.6, 268.4)/508.8 = (-0.844, 0.528). Dot with (0,300) = 158.4 > 0 → inward ✓.

So swap: nx = -ey*il, ny = ex*il. Wait, but that's exactly the degenerate branch in buildContacts: I used nx = -ey*il, ny = ex*il with the negate-if-toward-center check. Good — so the degenerate branch is right, and insideHeptagon is wrong.

Let me double-check by considering both cases. Edge direction (ex, ey). Rotating by +90° gives (-ey, ex). For a CCW-wound polygon, this is inward. Are my verts CCW? In canvas coords, y is down. verts[0] at angle -π/2 → top. verts[1] at increasing angle → moves clockwise in screen terms... In canvas, increasing angle with y = sin(angle)*R + CY: at angle -π/2, y is minimum (top). As angle increases, y increases → going down. So we go top → right/lower → ... that's clockwise in screen space. In math coords (y up) that's counterclockwise. Anyway, the half-plane test with the correct sign computed from the center is what matters: I verified (-ey, ex) gives positive dot with the vector to the center → inward. But this depends on the winding being consistent for all edges. Since the polygon is convex and all verts are generated by an increasing angle uniformly, the winding is consistent. To be safe, I'll compute the inward normal per-edge as: n = rotate90(e); if n·(center - p1) < 0, negate. That's robust regardless of winding.

Let me fix insideHeptagon to do that robustly. Also double-check buildContacts' normal: it's computed from the closest point to the ball, which is the correct geometric normal (pointing from the ball toward the wall). That's fine.

Let me also verify that the whole thing then works. Let me fix and re-run.

Also, contacts=190 initially because all balls overlapped at the same point due to the failsafe. Should be fine after the fix.

Also: at t=0.5 the initial vertical velocity: g*0.5 = 600 px/s, but maxSpd=5 due to damping by the failsafe each frame (×0.2 per frame ≈ over 30 frames). Yes.

Let me fix.

Progress: no escapes, but `wallOverlap` is huge — my test computes "wallOverlap" using the inward normal sign incorrectly (I used nx=ey*il, ny=-ex*il in the TEST which is the outward one). Let me fix the test's sign too. Also maxBallBallOverlap = 12.177 px which is large — that's concerning. And bbox is 336..447 x 352..464 — quite compact, i.e. all 20 balls in a 111x112 region. With r=16, 20 balls minimum area = 20*π*256 = 16085 px²; the bbox area is 12321 → they're stacked >1 layer which is fine in 2D. But overlap 12px between pairs is bad — that means balls are deeply overlapping.

Hmm wait, but the position correction should fix that. Overlap of 12 px means one ball center is only 20 px from another (rsum=32). That's heavy penetration that never resolves. Something is wrong with the solver.

Let me think about why. Possible cause: BALL_GAP=5 initial → gap 5 between neighbors, so no initial overlap. During the fall, they compress. The solver should separate them.

Hmm, the issue may be the normal impulse denominator for ball-ball: invMass = 1 + 1 = 2, jn = -vn/2. That's correct for equal masses.

Position correction: corr = (pen - slop)*0.65; inv = a.invM + b.invM = 2; a.x -= nx*corr*a.invM/inv = nx*corr/2... wait: a.x -= c.nx * corr * a.invM / inv. With a.invM=1, inv=2 → a moves by corr/2 in -n direction and b by corr/2 in +n. Correct.

So why would 12 px overlap persist? Unless... the contacts list is built BEFORE the position correction, and the pen values used are from the pre-correction positions. But that's a one-frame lag; it should converge over frames.

Hmm, unless the solver is applying impulses wrongly, pushing balls together. Let me check the normal impulse direction for ball-ball again:
- n points from a to b.
- rv = v_aContact - v_bContact.
- vn = rv·n. Approaching when vn < 0 ✓ (a moving toward b).
- impulse J = n*jn applied to a (+), and -n*jn to b. a.vx += jx*a.invM ✓, b.vx -= jx*b.invM ✓.
Good.

Wall: n points from the ball toward the wall (outward). a.vx += n*jn pushes the ball INTO the wall! Because n points outward toward the wall, not away from it!

Wait — let's recheck. In my convention doc: "n = unit vector pointing from body A's centre TOWARD body B's surface (toward the wall for wall contacts)". So for a wall, n points from the ball center toward the wall = outward. Then the impulse should push a AWAY from the wall, i.e. along -n. But my code applies +n*jn. That's backwards!

In the previous version (before the rewrite), the wall normal pointed inward and it was correct. In the rewrite I unified the convention but the wall normal now points outward, so the impulse sign must flip.

Hmm, but wait: is rv·n < 0 for approaching? For a wall with n outward: the ball moving toward the wall means v·n > 0 (moving in the +n direction, i.e., outward toward the wall). So approaching = vn > 0, not < 0. So the sign convention breaks.

Two options: (a) define the wall's n as pointing inward (from the wall toward the ball), keeping approaching = vn<0, but then position correction a.x -= nx*... moves the ball outward into the wall (wrong); (b) keep n outward and flip the signs for the normal impulse and the approaching test for wall contacts only — inconsistent.

Cleanest: make the convention uniform: n points from the wall toward the ball A (i.e., the "separation direction" for A = away from contact, pushing A away from B). Then:
- For wall: n = (ball - closestPoint)/d (inward) ✓ — approaching if vn < 0 where vn = (v_aContact - v_wall)·n... hmm, if the ball moves toward the wall it moves along -n → vn < 0 ✓ approaching.
- Normal impulse pushes A along +n ✓ (a.x += nx*jn).
- Position correction: A moves along +n (away from the wall). For ball-ball, n should then point from B toward A, and B moves along -n.

So use: n = separation direction for A (away from B / into the enclosure). Then:
- ra = -n*a.r (contact point = a.center - n*a.r) ✓ (contact point is toward B).
- For ball-ball with b: n points from b's center to a's center; rb = +n*b.r.
- approaching: vn = (v_aCont - (v_bCont + wv))·n < 0.
- impulse: J = +n*jn to a, -n*jn to b ✓ (a moves along +n away from b, b along -n away from a) ✓.
- position correction: a += n*corr, b -= n*corr ✓.

Let me restructure. I'll store nx,ny = n pointing from B→A. For ball-ball: n = (a - b)/d. For wall: n = (ball - closestPoint)/d.

Let me rewrite the relevant sections:

buildContacts wall:
```
let nx = b.x - qx, ny = b.y - qy;   // wall -> ball (separation dir for the ball)
```
ball-ball:
```
let nx = a.x - b.x, ny = a.y - b.y;  // B -> A
```

contactVelocities: ra = -n*a.r (a's contact point, toward b) ; for b: rb = +n*b.r.

applyImpulse: a gets +J, b gets -J.

position correction: a += n*corr*a.invM/inv, b -= n*corr*b.invM/inv.

Tangent: rotate n by 90°, direction doesn't matter for friction symmetric application; jt sign handles direction. a gets +t*jt, b gets -t*jt.

Let me rewrite these functions. Also fix the physics test's wallOutAt sign.

Let me make the edits.

Now also fix the physics test's wallOutAt sign to use the corrected inward normal. Let me update the test.

Also the initial packing: with the corrected position correction, the initial gap of 5 px means no initial contact. Good.

Let me update the test file's wallOutAt function to compute inward normals robustly.

Improvement: maxBallBallOverlap dropped to 2.2 px. But wallOverlap in the test still reports ~547 (test sign still wrong — I couldn't patch it). Let me fix the test with sed.

Also, "rev in last 60s ~ 0.60" — that's just wallAngle being modulo TAU, so revolutions resets. That's fine, the wall angle per 5s test is separate: at t=0.5 wall=36°, t=3.0 wall=216° → 36°/0.5s = 72°/s = 360°/5s ✓. Good.

Also, hmm, at t=0.5 contacts=0 and maxSpd=549. That means the balls fell 549*... free fall: v = g*t → at t=0.5, v=600*... wait maxSpd = 549 with damping. Distance fallen ≈ 0.5*1200*0.25 = 150px minus damping. Initial center at y=400, so balls now at ~550. Wall bottom edge at y up to 700. OK.

But wait — "maxBallBallOverlap=2.201" after 60s of settled simulation. Hmm, but wait: is the ball pile settled? bbox x=150..405, y=463..656. That's the pile in the lower-left region of the heptagon. Hmm, x from 150 to 405 means the pile is at the left side, but the heptagon center is at 400. With rotating walls, the pile should be roughly centered at the bottom. The center of the pile bbox = (277, 560) — that's off-center to the left. Hmm, that's suspicious. Maybe the balls are piled in one corner. It's plausible because the lowest point of a heptagon isn't necessarily at the center-bottom; the balls roll into a corner where two walls meet.

Actually with 2.2px overlap remaining, that's a bit much. Let's find where it comes from. In a dense pile under gravity with 8 substeps of dt=1/480 and 4 iterations, 2px overlap is plausible for a heavy pile. Let's increase iterations to 6 and see if it drops. Also, the position correction of 0.65 with slop 0.05 — maybe increase to 0.8.

Actually wait. Let me reconsider: is the pile resting on the wall? Overlap 2.2px could be between two balls where one is squeezed against a wall. Let me measure wall penetration separately. Let me fix the test's sign and re-measure.

Let me use sed to patch the test.

Wall overlap = 0 at all sample points, no escapes. 

Now the ball-ball overlap of 2.2 px after 60s. Let me investigate where it occurs and whether it's a persistent stack issue. Let me also check how it evolves — maybe it's a transient at the moment of the last step. Let me measure the max over the entire run and also report the final state's overlaps and speeds.

Also I want to check: is the pile in the corner? With the rotating heptagon, after 60 s, wallAngle is 0.60 rev = 216°... hmm, 0.60 of TAU = 216°. The bbox spans x 150..405, y 463..656. Center of the heptagon is (400,400), inradius = 300*cos(π/7) = 273. So y up to 400+273=673 max. So the pile is at the bottom-left. The balls are piled up in the lower-left area. Makes sense if the "lowest" vertex is at the bottom... Actually, at angle 216°... the vertices are at angles θ + i*51.43° - 90°. Hmm, whatever. But 150 to 405 in x and the pile only occupies 255x193 → that's a 2-layer pile. 20 balls × 32px diameter... 20 balls need ≥ ~4 layers if packed in a 255x193 region... it's fine.

Actually, wait. I should double-check that the balls actually get distributed and tumbled by the rotating walls, and the numbers spin. Let me check the spin distribution after 60s and speed distribution.

But more importantly, let me reduce the ball-ball overlap. Options:
1. More solver iterations (4 → 8).
2. More substeps.
3. Increase POS_PCT.

Let me try iterations=8 and see.

Actually, let me first instrument to find who is overlapping and their velocities.

Overlap around 2.5-3.6 px persistently. That's noticeable (about 10% of the ball diameter). Let me improve: increase the solver iterations from 4 to 10 and see if the overlap drops. Also consider a warm-start / better approach. Actually the simplest big win: more iterations of velocity solve don't reduce static penetration — penetration is resolved by the position correction only. The position correction is limited by POS_PCT=0.65 and it only corrects once per substep, but it does it 8 times per frame at dt=1/480, so it should be plenty (8*0.65 = strong). Unless the contact positions get re-detected and the correction... hmm.

Wait — the issue is that the position correction is applied AFTER the velocity solve, but on the next substep's penetration detection, positions have advanced by v*dt. Actually with 8 substeps/frame, we get 8 corrections per frame, each moving 65% of the penetration. So penetration should converge to near zero within a frame. Unless the balls are being squeezed by gravity faster than the correction can push them apart — which is exactly what happens in a dense pile: gravity keeps pushing them together.

Actually, hold on: 9.7 px max overlap at some point — that's a big spike. This might be at the initial drop when they all fall together. Let me look more carefully — maybe there's a deeper issue.

Hmm, one possible cause of large overlap: the initial arrangement with BALL_GAP=5 is fine, but when the balls fall and land, they compress. Overlap of 3px in a piled state is not terrible but let's do better.

Options to improve:
1. Increase iterations (helps impulse quality → less squeeze).
2. Apply the position correction multiple times (relax) before/after the velocity solve.
3. Increase POS_PCT to 0.9.

Let me experiment with a parameterized version in node to find good values. Let me create a variant script that patches the source string with different parameters and reports.

Also I could implement a proper "block solver" for stacking but that's complex. Let's just tune.

Let me write a tuning harness.

My run function was flawed (I passed both base and patch). Let me rewrite the tuner cleanly using eval with a patched source string.

More iterations don't help; overlap is stuck at ~9.75 max. This suggests a specific event, not a static condition. Note that maxOO=9.75 in all variants — deterministic and unchanged by the solver. That suggests a discrete event, likely at the very start: the initial drop.

Wait — at t=0 the balls have zero velocity and gravity accelerates them. With dt=1/480 and v up to ~700 px/s, per-substep movement = 1.46 px. That's less than the gap (5px). So no tunneling.

Hmm, what about the moment of first impact? They fall as a group and hit the wall at the same time. Impact speed ~ sqrt(2*1200*150) ≈ 600 px/s. With restitution 0.55, the impulse is applied... The velocity solve handles vn<0, so the impact should stop them. Then position correction pushes them apart. Since all 20 balls land simultaneously, the pile compresses. Hmm.

But why would overlap reach 9.75px = 30% of the diameter? Let me find when the max overlap occurs and which balls.

Actually — wait. Maybe the issue is that my new "hedge" for the ball-ball degenerate case (d < 1e-6) is being hit... no.

Let me instrument: find the time and pair of the max overlap.

Actually, I wonder about something else: the FIRST contact built in the list. Hmm.

Let me check whether the issue is the position correction being applied at a moment when contacts were built with positions that then get corrected but then the NEXT substep's contact is built from the corrected positions, so it should shrink geometrically.

Unless... the velocity solver adds energy. Let me measure the total kinetic energy over time to see if it grows. Let me instrument.

Very interesting! The overlap is small (< 0.3 px) here — much better than the 9.75 px reported earlier. The difference: in this diag, I call substep 8 times per frame like the real loop, and the max overlap is measured at frame boundaries. In the earlier test, I called `step(1/60,1)` = one substep per frame, i.e., dt = 1/60 instead of 1/480! That's why overlap blew up — the test was wrong, not the code. dt=1/60 with high speeds = 12px per step → deep penetration.

Wait, in phys_test2 I had `step(dt,n)` and called `step(1/60,1)` → 1 substep of 1/60 s. So yes, the test was simulating with a huge timestep. The real app uses 8 substeps of 1/480.

Great, so the actual overlap is < 0.3 px. Excellent.

Also note: ke peaks at 3.2M at t=0.52 during the initial free fall (before first contact) — that's just gravitational PE conversion, fine. KE values stay bounded around 200-500k after landing.

Let me now redo the 40s tests with the correct stepping (8 substeps per frame).

Excellent. The physics is stable:
- max ball-ball overlap 0.49 px over 90 s
- wall penetration 0.000 (never negative)
- max speed 681 px/s (bounded)
- max spin 24.7 rad/s
- no escapes over 90 s
- revolutions: 18 over 90 s = exactly 1 per 5 s ✓

Let me verify the wall rotation rate precisely and the ball-ball/wall collision response with a targeted test (bounce test): drop a single ball onto a static-ish wall and check the normal velocity reverses with restitution ~0.55, and check that with wall rotation the ball gets tangential velocity.

Also let's verify:
1. Rotation rate: dθ/dt = 2π/5 exactly.
2. Wall velocity included: a ball hit by a rising wall gains tangential momentum.
3. Numbers rotate with balls: b.ang changes → the rendered text rotates. Already true (angVel != 0, mean spin 7.7 rad/s).
4. Reset restores positions and wall angle.

Let me do a focused wall-collision test: reset the simulation, place a ball near the bottom wall, run a few frames, and check the tangential velocity from the moving wall.

Let me write that test.

`reset` is defined in the Controls section, which is cut off. Let me add a reset implementation to the test code (or move the reset function into the physics part). Better: I'll include a `reset` in the test code that replicates it, using `initial`.

Also `balls0` is used before its `var` declaration — hoisted var is fine, but it's assigned at the end. Let me restructure: define balls0 from `initial` at the top of the test.

Also park off-screen balls: setting x=1e6 means they're far away and `insideHeptagon` returns false → the failsafe would teleport them to the center. That will disturb the test! Instead, let me set them out of the way... Actually a simpler approach: create a separate mini-scenario by directly removing them from the array? `balls` is a `let` global in the eval scope, so I can't reassign it from inside... actually yes I can, since the test code is inside the same eval scope. `balls = [balls[0]]` — but `balls` is declared with `let` in the same scope, so assignment works.

But then the pause failsafe in insideHeptagon won't trigger for removed balls. Let me just reassign balls to a single-ball array for tests 2 and 3.

Let me rewrite the test.

Mostly good:
1. Rotation rate exact ✓
2. Restitution ratio -1.252 — my measurement is flawed: I measured the normal velocity 2 frames AFTER the impact, during which gravity has re-accelerated the ball downward, so the measured "separation" includes gravity. Also the ratio > 1 suggests I measured one frame too late (already re-contacting). Let me improve the measurement by capturing the velocity right at the first contact resolution step.

Actually a cleaner approach: use a controlled test. Place a ball just above the floor with a given downward velocity and measure pre/post. Better: measure using the internal solve: instrument by capturing vn before and right after the first substep where a wall contact with vn<0 appears.

Let me write a targeted test: run substeps one at a time; before each substep record v; run one substep; check contacts for a wall contact on that ball; then compute restitution = (-v_after·n_in)/ (v_before·n_out)... Let me define: inward normal m = -n (points into the enclosure, away from the wall). Approach speed along m: v_a = v·m before. After: v_sep = v·m after. Restitution = v_sep/v_a.

Wait, the measured -1.252 might be because my sampling of d (distance to the edge midpoint's line) uses floor.qx/qy as a point on the line but the closest point on the segment matters. Since the ball is near the middle of the bottom edge, that's fine.

The big issue: at 2 frames after the first detected contact, the ball may have already bounced, hit another wall, or been re-captured by gravity. Let's measure on a per-substep basis with high precision.

3. Moving wall: at 0.75s the ball had vy=-45, vx=-219, angVel=7.5 — it's being carried around by the rotating wall (vx changes a lot as the wall rotates). Good, the moving wall imparts tangential motion. Let me make a cleaner test: put a ball resting on a wall with no gravity effect... Actually a cleaner demonstration of moving-wall velocity: temporarily disable gravity, place a ball resting on the rotating wall, and check it gets dragged along at the wall's velocity. Let me do: gravity=0? GRAVITY is a const. Hmm, I could add balls with vy offset... Instead, let's just verify the coupling directly by unit-testing the solver: create a contact with the wall and a ball, and check that the ball gains velocity.

Simplest direct check: put a ball at rest JUST touching the wall (no overlap, so no contact) and let the wall rotate into it. Then the wall's velocity should push it. With gravity present it's muddled.

Alternative: verify the term by checking that in `solveContact` the wall velocity is subtracted properly — I can test the impulse response analytically. Let me do a unit test of the solver: build a synthetic contact state.

Actually, test 3 already shows the wall carries the ball (vx=-219 while the ball started at 0,0 — gravity alone would give vy only, never vx). That's solid evidence the moving-wall velocity is accounted for. Let me additionally show it more cleanly by comparing with gravity off: I can simulate with GRAVITY set... it's a `const`. I'll instead place the ball exactly at the contact and set vy=0, then measure the push from the wall in the normal direction. Hmm.

Let me do the clean unit test of solveContact: 
- Set up a ball at rest at distance r from a wall whose normal n is known.
- Rotate the wall a tiny bit so a contact forms with the wall moving at velocity w.
- Then check the ball's velocity after the solve.

Concretely: create a synthetic scenario where I call substep with a single ball placed so it overlaps a wall, and measure. Let me write it: find the wall midpoint q and inward normal m (from wall toward ball). Place the ball at q + m*(r - 1) (penetrating 1px). Set the ball's velocity to minus the wall's velocity at the contact point (so it "co-rotates" with the wall) — then after the solve, the ball should keep exactly that velocity (no bounce). And if the ball is at rest (v=0), it should gain velocity along m away from the wall with magnitude ~the wall's approach speed. That's a clean unit test of the moving-wall term.

Let me write it. Also improve the restitution measurement with per-substep sampling.

Let me also record the ball's max |v| in test 2 to see the ratio.

Let me write test 2 as: single ball, placed 5px above the floor with vy = -400. Then step substeps one at a time until a wall contact appears with the ball approaching (vn<0), record vn_before, then after the substep record vn_after. But careful: the solve happens after integration, so the recorded position/velocity change within a substep matters. Let me just measure: before substep: v0; after substep: v1. If a wall contact existed during the solve with approaching velocity, the restitution should be ~0.55 (plus gravity's contribution of g*dt = 1200/480 = 2.5 px/s, negligible).

The issue with my earlier measurement: I sampled at frame boundaries (8 substeps), by which time the ball had already bounced and was moving away. Let me sample per substep.

Let me redo tests 2 and 3.

Test A is problematic: restitution measured as -1.003 at impact speed 743. The impact speed 743 px/s means the ball traveled far... wait, the ball starts at CY-10 with vy=-500 and accelerates. Hmm, but my `w=wallInfo()` picks the edge with the most negative normal (ny<-0.9), which is the flat bottom edge. At wallAngle=0, the heptagon has a vertex at the top (angle -90°) so the bottom is... vertices at -90 + i*51.43 → -90, -38.57, 12.86, 64.29, 115.7, 167.1, 218.6. In canvas coords (y down), the vertex at -90° is at the top (400,100). The vertex at 64.29° → (400+300cos(64.29), 400+300sin(64.29)) = (400+130, 400+270.6) = (530, 670.6) — bottom right. And 115.7° → (400-130, 670.6) bottom left. The bottom-most edges are between 64.29° and 115.7°... The edge connecting the vertices at 64.29° and 115.71°: both at y=670.6, so this edge is horizontal at the bottom, spanning x from 530 down to 270. So the bottom edge is horizontal ✓, with midpoint (400, 670.6), and the inward normal ny = -1 ✓.

So the ball starts at (400, 390) with vy=-500 (upward!). Wait: `balls[0].vy=-500` means moving UP in canvas coords (negative y = up). And restitution measured: I computed `vnApproach = -(w.nx*vx + w.ny*vy)` = -(0*vx + (-1)*vy) = vy. So vnApproach = vy = -500 → negative, not > 0. Hmm, so my check `vnApproach > 0` fails... but the loop continued and the ball fell (gravity) and eventually approached the bottom wall with vy>0 → vnApproach = vy > 0 ✓. OK, so the ball went up, came down, and hit the bottom wall at 743 px/s. Makes sense (it fell from the top to the bottom, ~280px, gaining speed).

Then `sep = w.nx*vx + w.ny*vy = -vy`. If the ball bounces up (vy < 0), sep > 0. res = sep/vin = -vy/vy_before. Measured -1.003, meaning sep ≈ -vin → the ball kept going DOWN after the substep, i.e., no bounce was detected/registered at that substep.

Why? Because the order of operations in my measurement: I detect the contact from the position BEFORE the substep, then apply one substep (which includes integration + solve + correction). If at the beginning of the substep the ball was overlapping the wall and approaching, the solver should reverse the velocity. Unless... the ball wasn't yet overlapping the wall at the start of that substep (my `d < b.r` check uses d = distance from the ball center to the closest point on the segment... wait, `d = w.nx*(b.x-q.x)+w.ny*(b.y-q.y)` = the signed distance from the closest wall point to the ball center, which is the perpendicular distance to the infinite line of that edge (since q is the closest point). So d < r means overlap. Good.

Hmm, but the impact speed 743 px/s → in one substep of dt=1/480, the ball moves 1.55 px. So detection and resolution should be within a substep.

Wait... unless the issue is that the ball hit the bottom edge's ENDPOINT (the vertex) rather than the middle, in which case q is clamped to the endpoint and d is measured differently. Not likely for a ball at x=400.

Hmm, wait. Let's re-read my measurement:
```
const sep = (w.nx*balls[0].vx + w.ny*balls[0].vy);
res = sep/vin;
```
I compute sep AFTER the substep. The substep: (1) integrate gravity, (2) position update, (3) detect+solve, (4) position correction. So the velocity after the substep should be the bounced velocity. Unless the detection order matters: at the start of the substep the ball was at distance d < r... 

Hmm, hold on! There's a subtlety: `vin = vnApproach` is computed BEFORE the substep, but gravity accelerates the ball during the substep (vy += g*dt → +2.5). And more importantly — wait, the measured ratio is -1.003, meaning vy_after ≈ -743. So the ball's velocity became strongly negative. That means the solve pushed it UP, by an amount roughly equal to -2*vin instead of -1.55*vin... Hmm, that suggests the impulse was applied with restitution reversed... Let's see: the change in vy = vy_after - vy_before = -743 - 743 = -1486. That's a change of -2*vin. If restitution e were 1.0, the change would be -(1+e)*vin = -2*vin ✓ with e=1. So restitution appears to be 1.0, not 0.55!

Hmm wait, but then `sep/vin = -1.0` would be with e=1 → sep = -vin → vy_after = -743 ✓ consistent with e=1.

But wait, gravity also added +2.5 to vy before the solve, so the effective impulse is slightly less. The measured -1.003 ≈ -1. So the effective restitution is ≈1, not 0.55.

Hmm! Why? Let me check `solveContact`. For the wall contact: c.e = REST_BW = 0.55. useRestitution is true for it===0. target = -c.e*vn. jn = -(vn - target)/invMass...

Hold on: vn = rv·n where n is the SEPARATION direction for A = pointing from the wall toward the ball (inward). rv = v_aContact - v_wallContact. If the ball moves toward the wall (downward, vy>0), then rv·n where n=(0,-1) (inward, pointing up) = -vy < 0 ✓ approaching.

target = -e*vn = -0.55*(-743) = +408.7. So target is the desired rebound: vn_after should be +408.7 (separating at 408). jn = -(vn - target)/invMass = -(-743 - 408.7) = 1151.7. Applying J = n*jn = (0,-1)*1151.7 → the ball's velocity change = (0, -1151.7). vy goes from 743 to 743-1151.7 = -408.7 → separation speed 408.7 = 0.55*743 ✓.

But measured vy_after = -743. Hmm, that means the applied impulse was ~2x too large: the change was -1486 ≈ -(1+1)*743. That corresponds to e=1: jn = -(vn - (-vn))/1 = 2*743 = 1486.

Hmm, so maybe `useRestitution` is true but `c.e` is 1? No...

Wait, maybe the contact list contains MULTIPLE wall contacts for this ball? At the bottom edge... no.

OR: maybe the ball contacted a DIFFERENT edge, not the bottom one I selected. The `w` I selected is the bottom edge (ny=-1). But the actual contact might be with the vertex region... no, x=400 is in the middle.

Hmm, wait. Let me reconsider: maybe `d < balls[0].r` triggered on an earlier substep than I think, and the reported `vin` is not what I think. `vin = vnApproach` = vy at that moment (if nx=0, ny=-1) — with gravity, vy grows each substep. So vin is correct at the detection moment.

Hmm, let me just instrument directly: log the ball's velocity around the bounce.

Actually! I bet the issue is the ITERATION: `for (let it = 0; it < 4; it++) for (const c of contacts) solveContact(c, it === 0);`

On iteration 0, restitution is applied: vn becomes target = +408. Then the friction impulse is applied (tangential). On iterations 1-3, `useRestitution` is false, so target = 0. jn = -(vn - 0)/invMass = -(408)/1 = -408 → NEGATIVE! I check `if (jn < 0) jn = 0;` → clamped to 0. OK so that's safe...

Unless vn is slightly negative due to gravity re-applied? No, gravity is applied only once at the start of the substep.

Hmm, so where does the extra impulse come from? Let me check `c.jn` storage: `c.jn = jn` before applying. Then the friction clamp uses `c.mu * c.jn`. If c.jn is from iteration 0 (1151), then maxF = 0.3*1151 = 345 — the tangential impulse could be large. But the tangential direction is perpendicular to n (horizontal), so it would affect vx, not vy. Since vx=0 initially, and the wall is rotating, the tangential impulse could be nonzero, but it wouldn't change vy.

Hmm... unless the contact is with a VERTEX-adjacent edge where n has an x-component.

Let me just directly instrument the bounce. Let me log vy over substeps near the impact.

Actually, hold on. Let me reconsider `sep`. After the substep, I recompute... no I don't; I use w (the edge info computed once at the start). Fine.

Hmm, one thing: after the bounce the ball might immediately hit another edge... no.

Let me just instrument.

Actually — WAIT. I bet I know. The initial condition: `balls[0].y = CY-10 = 390`, and `vy=-500` → moving up. It goes up, hits the TOP wall first! The ball flies up at 500 px/s from y=390, traveling ~104px to reach the top wall region (the top vertex is at y=100, so the wall near the top...). The ball at x=400 is directly below the top vertex at (400,100). The two edges adjacent to the top vertex are slanted. The ball at x=400 would contact the top vertex region... So the ball bounces off the top, comes back down, and hits the bottom wall at 743. That should still be a clean bottom-edge contact.

Unless the ball's x-position drifted. Let me instrument. Let me print the ball's position and velocity every substep near the end.

Interesting. At s=581 the ball is at y=664.4 with n = (-0.27, -0.96) — that's the slanted bottom-right edge, not the flat bottom edge! So the ball drifted in x. vy0 = -498.4 (moving UP!) and after the substep vy = -495.5. Hmm, that means the ball is moving up and there's a wall contact... wait, if the ball is moving up (vy<0) it's separating, so vn should be > 0 (separating) — but the printed vn = 590.2 which is c.nx*b.vx + c.ny*b.vy computed AFTER the substep = -0.27*vx - 0.96*vy. If vy=-495.5, then -0.96*(-495.5) = +475. Plus -0.27*vx... if vx is negative, e.g. vx = -400: -0.27*(-400) = +108 → total 583. So the ball is moving up-left, and n=(-0.27,-0.96) points up-left → the ball IS separating. OK.

So my earlier test A detected the ball moving upward against that slanted edge? No — the test's `vin = -(nx*vx+ny*vy)` with the flat bottom w=(0,-1) → vin = vy. The ball was moving UP (vy<0) so vin < 0... so the condition `vnApproach > 0` wouldn't trigger. Hmm, but the test reported an impact speed of 743, meaning at some point vy=743>0 (moving down) and d < r on the flat bottom edge.

Hmm, so it did hit the flat bottom edge while moving down at 743. Then after the substep, vy = -743 → the ball bounced at full restitution (e=1)?

Hmm, wait. Actually, maybe the contact was detected on the flat edge but the ball's closest point q was at the ENDPOINT (clamped) since the flat bottom edge spans x from 270 to 530, and the ball's x might be outside that range. If the ball is outside the span, then d is measured from the endpoint, and the normal I used (0,-1) doesn't match the actual contact normal, so my "sep" measurement is garbage. Given n=(-0.27,-0.96) in the trace, the ball is in the right corner region.

So my measurement was just wrong because the ball drifted to the corner and hit a slanted edge, while my "w" referred to the flat edge. Also, at s=581, the ball is at y=664.4 — near the bottom.

Hmm, but the ball started at x=400 moving straight up with vy=-500 and vx=0. How did it drift in x? From the top-wall collision: the ball hit the top vertex region and was deflected with a tangential (friction) impulse and the rotating wall → it gained vx. Yes, the top vertex between the two slanted edges at (400,100) — the ball at x=400 would hit the vertex itself, and the wall's rotation moves the vertex sideways, giving it vx. Fine.

So my test A was just measuring against the wrong edge. Let me redo test A properly: use the actual contact normal from buildContacts for the measurement. Let me instrument using the contacts list: find the wall contact involving the ball, and if it's approaching (vn<0 with respect to the contact's own n), measure the pre/post velocities along n.

Let me rewrite test A that way.

Also test B result (dv=2.88 px/s for a co-moving ball) — that's excellent, ~0 (well, gravity + dt gives 2.5, which is exactly what we see). So the wall velocity is correctly accounted for ✓.

Test C: ball at rest on a rotating wall gets pushed out normally (143 px/s) and dragged tangentially (-42.9 vs wall tangential -93) ✓.

Now, test D: peak KE 4M vs 57.6M if all at the speed clamp — bounded ✓.

Let me redo the restitution measurement correctly.

Hmm, A and B returned "none" — no bounce was detected. Why? In A, the ball starts at (CX, CY-30) at rest — but wait, `balls=[balls[0]]` and then `resetSim()` resets it from `initial[0]` — which sets balls[0].x to initial. But `balls[0]` is the ball object of the original array... `balls=[balls[0]]` creates a new array with the same ball object. Then I set balls[0].x=CX etc. — fine.

But `contacts.find(...)` — the `contacts` array is a module-level `let`... it's declared with `const contacts = []`. Fine.

Why no contact? The ball starts at (400, 370), falls, hits the bottom edge at y≈670 → 300px fall. It should hit within ~0.7s = 42 frames = 336 substeps. I looped 900 substeps. Hmm, so it should hit.

OH WAIT. The problem: after `balls=[balls[0]]`, `initial[0]` still refers to the original indexed snapshot... resetSim loops `for i<balls.length` = 1, so it resets balls[0] to initial[0] = (400, 400-ish). Then I set x=CX,y=CY-30. Fine.

Hmm, but the ball falls... 300 px at g=1200 takes sqrt(2*300/1200) = 0.707s = 340 substeps. So within 900 it should hit.

Unless... the ball is not hitting a wall but escaping? Or... hmm, "impact speed 0" and `res===null` means the loop ended without finding vn < -50 in a wall contact. But contact detection `c.type===1 && c.a===balls[0]`.

Hmm, could `contacts.find` be failing because `c.a===balls[0]` — after `balls=[balls[0]]` the reference is the same object ✓.

Wait, maybe the issue is that D) reset restores state exactly: false — that suggests `before !== after`. `before` was captured after settle(9)?? No — I captured `before` at the point after the 60s run in C (balls had moved), then `settle(9)` then `resetSim()` then compare `after` with `before`. That comparison SHOULD be false! I made a mistake: I should capture `before` BEFORE settle. So D's test logic was wrong, not the code.

Similarly, in A: after C's 60s run in the previous test... no, A comes first. Hmm, but in this script, A comes right after resetSim() with `balls=[balls[0]]`.

Oh! I see it: `balls[0].y = CY - 30` — but WAIT: `resetSim()` is called AFTER `balls=[balls[0]]`, and resetSim sets b.x = s.x, b.y = s.y — fine.

Hmm, let me debug. Maybe the ball never hits the wall because the wall's flat bottom edge... Let me check: at wallAngle=0, the vertices are at -90 + i*51.43 degrees. The bottom edge between 64.29° and 115.71° vertices is at y = 400 + 300*sin(64.29°) = 400+270.6 = 670.6, spanning x from 400+300cos(64.29)=530 to 400+300cos(115.71)=270. Yes, a horizontal bottom edge from (270,670.6) to (530,670.6).

A ball falling at x=400 will land on it. But the wall rotates, so during the fall (0.7s = 50° of rotation) the bottom edge moves away! The heptagon rotates 360°/5s = 72°/s. In 0.707s, it rotates 51° — half a revolution! So the "bottom" edge is long gone by the time the ball falls to that depth, and the ball might land on a slanted edge... or even miss... no, it can't escape; the enclosure always contains it. But my detection of "the wall contact with the actual normal" should still find SOMETHING.

Unless the ball ends up in a corner. Hmm, but the loop should find a contact whenever the ball touches any wall.

Unless the ball keeps missing the walls entirely... impossible in an enclosure.

Hmm, wait: the issue might be that `vn < -50` never happens with a threshold of 50 px/s, because... no, free fall gives 800+ px/s.

OH WAIT. I see it now! In `buildContacts()`, I push `contacts.push({a: b, ...})` for wall contacts where `b` is the ball. So `c.a` IS the ball ✓. And type=1 for walls ✓.

Hmm, let me just debug directly with prints.

Actually hold on, maybe the bug: after `balls = [balls[0]]`, `resetSim()` resets only balls[0]. Then `balls[0].x = CX` — but I wrote:
```
balls=[balls[0]]; resetSim();
balls[0].x=CX; balls[0].y=CY-30; balls[0].vx=0; balls[0].vy=0;
```
Yes that's what I wrote. Let me debug with a print of the ball's position over time.

Hmm... unless `contacts` is emptied by something. `buildContacts()` sets contacts.length=0 then pushes. Fine.

Let me debug.

The ball is falling very slowly: after 180 substeps (0.375s), vy=422 but the position only moved from 370 to 451 — 81px. Free fall should give 0.5*1200*0.375² = 84px. OK that matches! Good, so the simulation is fine; it just needs more time to fall 300px. 300px needs sqrt(2*300/1200)=0.707s = 340 substeps. My loop was 200, and the earlier test was 900 — should have been enough...

But wait, in the earlier test A, the ball was at CY-30 = 370, needs to fall ~300px → 340 substeps < 900. So the loop should have hit. But it printed "none".

OH! I see the bug in my test: `if (vn < -50){...} else substep(...)`. Inside the `if (vn<-50)` branch I call `substep` too... no wait, that's fine.

Hmm, but look: in the branch, I compute `wallApproachSpeed = Math.abs(vn)` and then `substep()`. So if a contact is found and approaching, res is set. Unless the contact's vn is never < -50 because... the contact is detected only when overlapping (pen > 0). When the ball is approaching at 800 px/s and the detection happens after the position update within a substep... hmm, the ball approaches the wall, and at the substep where it penetrates, the solver resolves it and it bounces. So the contact exists for exactly 1 substep with vn < -50 (the approach speed). Then the next substep it's separating.

But my detection loop calls buildContacts() BEFORE substep. So at the substep before the penetration, there's no contact (d > r). At the substep where penetration occurs, my pre-substep buildContacts... wait, I call buildContacts() at the START of each iteration, then if a contact is found approaching, I do the substep and measure. If no contact, I substep.

The sequence: iteration i: buildContacts (from the position after iteration i-1's substep). If pen>0 → contact found. Penetration occurs during a substep (after the position update). So the contact appears at iteration i+1, AFTER the substep that created it — but the solver already resolved the velocity during that substep! So by the time my loop sees the contact, the ball is already separating (vn > 0) → my condition `vn < -50` fails.

So I need to detect the approach differently: check the position/velocity before the substep and detect penetration after. That's what I did in the very first measurement, which also failed due to the wrong edge normal.

Let me redo: at each iteration, before the substep, record v. Then substep, then check whether the ball's velocity changed such that a bounce occurred. Simpler: check whether the ball's normal velocity (against the wall it's approaching) flipped sign.

Alternative approach that's cleaner: measure over a slightly longer window. Capture v and position before the substep, then substep, then re-run buildContacts and check if there's a wall contact and whether the velocity's normal component flipped.

Let me do this: for each iteration: v0 = ball.vy (or the full vector); substep; then buildContacts and find the wall contact; if a wall contact exists with pen > 0 and the pre-substep velocity had a large normal component toward the wall (using the CURRENT contact normal), compute the restitution = (v_after·n) / (-(v_before·n)).

Actually simplest: use the contact normal from the AFTER state (it barely changes in one substep), and compare the pre/post normal velocities. Let me do that.

Actually even simpler and robust: the standard test — drop a ball onto a static (non-rotating) horizontal-ish surface and check the peak rebound height. But rotation makes it hard.

Let me use the "flip detection" approach with the after-state contact normal.

Let me rewrite test A and B that way.

Now the measured values match the constants exactly:
- Wall restitution 0.560 ≈ 0.55 (small extra from gravity applied in the same substep) ✓
- Ball-ball restitution 0.600 = 0.60 exactly ✓
- Friction induces spin ✓

Now let me run the browser checks. Let me start the browser tool on index.html.

First let me also double check the visual: heptagon fully visible, balls inside, numbers visible.

Let me run the browser.

The initial view looks good: heptagon fully visible with 7 sides, 20 numbered balls in a compact staggered arrangement near the center, all inside.

Now let me sample moments during the initial fall and collisions. Let me take screenshots at various times, and check the console. Note the console has a warning about an iframe sandbox — not from my code.

Let me capture screenshots at intervals. I'll take a screenshot after ~1.5s and ~4s.

I can't easily sleep... I could use the browser tool's scroll/click actions. Let me just take screenshots spaced apart; time passes between calls naturally.

Let me first sample the initial fall: take a screenshot now (already done, t=0.22s). Let me evaluate some state checks via inspect to get the DOM/console. Actually the `inspect` action gives DOM and console. Let me use inspect to get the current state.

Let me take a screenshot for the fall, and check ball positions via console evaluation... The browser tool doesn't offer arbitrary JS eval except through inspect (which shows DOM). Hmm. Let me check if there's an eval capability — the tool has `inspect` returning DOM + console. I can't run JS directly.

Well, I can still verify via screenshots. Let me take a few screenshots at intervals.

At t=5.93s, the heptagon has rotated 67° and the balls have tumbled into a pile along the lower-left walls, with 13 contacts, peak speed 685 (bounded), max spin 27.7 rad/s.

Interesting: the numbers appear rotated (some are sideways, e.g. "2" appears horizontal in some balls; hard to tell from the small resolution). Let me zoom in via a screenshot crop? I can't crop. But I can verify spin programmatically in the browser... I can't eval JS. Hmm.

Actually, the `inspect` action returns DOM text — maybe the canvas content isn't in the DOM. I could add a hidden element to the DOM with the ball angles, so that inspect shows them. That's a nice way to verify from the browser. Let me add a small debug element or use the existing HUD. Actually, let me add an "aria-live"-style hidden snapshot element listing ball angles, which also lets me verify spin from the DOM.

Alternatively, I can verify the numbers' rotation visually by taking a screenshot; the image resolution in the report is decent (the balls are ~30px). In the second screenshot, I can see balls like "2" near the top of the pile that looks rotated. Hmm, it's hard to be sure.

Let me add a hidden DOM element updated each frame with per-ball data (angle, x, y) — this makes verification much easier and is a legit debug affordance. I'll add `<span id="dbg">` inside a hidden div.

Actually, a cleaner approach that's still useful: add a compact "spin readout" per-ball? Too much. I'll add a hidden debug node with JSON, updated every frame, and mention nothing in the UI. Actually, hidden debug data in production code is fine and useful for the harness.

Let me add it. Then reload and inspect.

Also I want to check:
1. Pause works (state shows paused, sim time frozen).
2. Reset restores the arrangement (compare ball positions/angles with initial).

Let me add the debug node.

I'll add to the HTML body: `<div id="dbg" style="display:none"></div>` and in updateHUD set `dbg.textContent = JSON.stringify(...)`. But the DOM text won't show if display:none? The `inspect` tool returns DOM text — a hidden element's text may not be included. Let me instead use an attribute or keep it visible but offscreen. Safer: set an attribute on the body, e.g. `document.body.setAttribute('data-sim', json)`. Attributes are in the DOM.

Let me do: `document.body.dataset.sim = JSON.stringify({t, wallAngle, balls:[{n,x,y,a,w}]})`. Then inspect will show it. But it could be long (20 balls). It's fine.

Actually, does the inspect tool dump attributes? Uncertain. Let me try. Alternatively I can add a visible tiny element in a corner with something. Let me try the data attribute approach and see.

Let me implement: in updateHUD, throttle to 0.1s (already), and set the attribute.

The data attribute isn't visible in the inspect DOM output. The inspect tool seems to dump only the visible text content, not attributes.

Alternative: add a visible debug element that is offscreen (position:absolute; left:-9999px) so it's still in the DOM text. Hidden elements are still in the DOM. Let me try a `<pre id="dbg">` positioned absolutely offscreen.

Let me change the approach: add `<pre id="dbg" style="position:absolute;left:-99999px;top:0;height:1px;overflow:hidden"></pre>` and set textContent. The inspect tool's DOM dump likely includes textContent of all elements (it dumped side panel text). Offscreen should still be included since it's not `display:none`.

Let me do that.

Also, I should double check whether the spin is visible. I'll grab the debug JSON and check that ang values differ from 0 after some time and change between samples.

Let me implement.

Wait, the third edit was a no-op (I replaced `</script>\n</body>` with the same). I need to add the debug element before `</body>`. Let me fix that.

Also the first edit was a no-op. Let me redo properly: add the `<pre id="dbg">` element right before `</body>`.

The debug JSON works. Initial state at t=0.117:
- 20 balls, 5 columns × 4 rows, staggered, centered around (400, 408), non-overlapping (min distance 37 > 32 ✓).
- wall angle 0.14657 rad ≈ 8.4°, matching the HUD.
- contacts 0.

Now let me sample the fall and collisions. Let me take a screenshot after some delay to capture the mid-fall and pile, and inspect to read the JSON.

Let me wait a bit by doing some scroll or just repeated screenshots. Each tool call takes some real time. Let me take screenshots and check the DOM each time.

Let me now test:
1. Mid-fall state.
2. Contacts > 0.
3. Spin nonzero.
4. Pause.
5. Reset.

Let me click Pause and inspect. First, let me sample the running state at a later time.

At t=4.15s, wall angle = 5.21483 rad = 299°. Wait, the HUD says 292° and the JSON says 5.21483 rad. 5.21483 rad = 298.8°. Hmm, the HUD's `sAngle` = 292°. Discrepancy!

HUD: `const deg = ((wallAngle * 180 / Math.PI) % 360 + 360) % 360;` — with wallAngle = 5.21483 (which is > 2π=6.283? No! 5.21483 < 6.2832, so no wrap). 5.21483 rad = 298.8°. So the HUD should read 299°, not 292°.

Hmm, the HUD and the JSON snapshot are updated at different times (HUD first, then JSON, and there can be an extra frame in between, or the snapshot is written after a frame). Within updateHUD, wallAngle is the same value used for both. Unless... the frames differ: `updateHUD` is throttled at 0.1s — the snapshot is inside the same function, so they're identical. But the screenshot and the DOM dump come from different times? Both come from the same action... The DOM dump is taken at a moment, the screenshot at another. Actually the tool reports both in one response; the ordering shouldn't matter much since they're within milliseconds.

Hmm, wallAngle at 292° → 5.096 rad; at 299° → 5.215 rad. Difference = 0.118 rad = 0.094s of rotation. That's about 5-6 frames at 60fps. So the HUD and the JSON snapshot are ~6 frames apart. That means the DOM text was captured at a different render than the JSON.

Ah I see — the throttle: `if (hudClock < 0.1) return;` — this returns BEFORE both the HUD text and the JSON are updated. So they're updated together. But then why the discrepancy?

Unless the tool's DOM snapshot and the screenshot are captured at different times (screenshot first, DOM after, or the DOM is captured after a delay). The response shows "dom" and "images" — the images might be captured earlier. The wall rotates 0.118 rad between the two captures = ~94ms. Plausible if there's a delay between the DOM dump and the screenshot.

Either way, it's a measurement artifact, not a bug. The important check: both show ~292-299°, and revolutions 0.81-0.83. Let me verify the rotation rate from consecutive JSON snapshots: t=0.117 → wall=0.14657; t=4.15 → wall=5.21483. Δwall = 5.0683 rad over Δt = 4.033s → 1.2566 rad/s ✓ exactly 2π/5. 

Also contacts = 13-18 in the pile. Balls contained. Spin values are large (ang from -13 to +41 rad, angVel up to 18 rad/s). 

The pile: ball positions from (157,504) to (423,669). The heptagon at wall=299°... the pile is on the left side. Hmm, wait: with 20 balls, the pile occupies the lower-left. Let me check the pile is contained: the balls are all inside the heptagon (verified in node).

Hmm, one thing to consider: the balls all pile into one corner and stay there, which is a bit static/boring. But the rotating walls do tumble them (contacts 13-18 continuously, spin present). Good enough. Actually, could the balls get "stuck" in a corner and stop moving? In the earlier node test at 90s, meanSpd=126, meanSpin=7.7, so they're still moving. Good.

Now let me test Pause and Reset. Click Pause, then inspect to compare. Then Reset and compare with the initial snapshot.

Let me capture a "before reset" snapshot: pause first so the state is frozen, then read the JSON, then click Reset, then read the JSON again and compare positions/angles to the initial values.

Let me click Pause.

The click at (90,365) didn't hit the Pause button (state still "running"). The canvas is at the left; the Pause button is in the right panel. The viewport is 1200x800, and the layout is centered. The canvas is 820px wide (800 + padding) plus the side panel 252px → total ~1100px, centered → the side panel starts around x = (1200-1100)/2 + 830 = 50+830 = 880. The Pause button is around y=365 (Controls card). Let me click at around (940, 365)? Hmm, my click at (90,365) hit the canvas.

Wait — the state says "running" and wall=107° / 1.86313 rad = 106.7° ✓ consistent. So the click did nothing (or it hit the canvas, doing nothing).

Hmm, actually the "Pause" button text is at the top of the side panel. In the screenshot the Pause button appears at roughly y=365 in image coordinates... The screenshot is scaled to the viewport, so the image is 1200x800-ish. Looking at the screenshot: the side panel "CONTROLS" is at the top of the right column, around y=170-200 in the image. The Pause button is at around y=205-235 in image coordinates.

Hmm, but earlier I estimated the layout. Let me look at the first screenshot: the canvas occupies the left ~65% and the side panel the right ~30%. The "CONTROLS" header appears at about y=0.25*800 = 200 px. The Pause button is right below at y≈230.

Wait, but the image shown might be scaled. Let me just click at (960, 230) for Pause. Actually, let me compute from the first screenshot: the right column starts at about x = 0.72*1200 = 864 and the Pause button spans roughly x 880-1000. So clicking (930, 230) should hit Pause.

Hmm, my earlier click at (90,365) definitely hit the canvas. Let me click at (940, 235).

Actually, let me look more carefully at the screenshot proportions. In the attached image (1200x800 presumably), the "CONTROLS" text starts at around y=170/800 → in a 1200x800 image that's y≈170. The Pause button is at y≈205-235.

Let me click (950, 220).

Still "running". My clicks aren't hitting the button. Note the sim time keeps advancing (25.53s), so the clicks are landing on the canvas.

Hmm, the coordinates must be off. Let me figure out the actual layout. The canvas is 800x800 with 10px padding → 820px wide. The side panel is 252px. Total content width = 820 + 16 gap + 252 = 1088. Plus body padding 14px each side → 1116. Centered in 1200 → left margin ~42px. So:
- Canvas: x from 42+10=52 to 852.
- Side panel: x from 884 to 1136.

The Controls card: the body has padding-top 18, h1 ~26px, sub ~40px, wrap gap 14... Roughly the card top is at y ≈ 18+26+40+14+10 ≈ 130. The "CONTROLS" h2 at y≈140, the button row at y≈172-200.

So Pause button center ≈ (960, 190)? But the button is at the left of the .side (which starts at 884), the button row has two buttons each ~113px wide: Pause from 897 to 1010, center ≈ 953. y center ≈ 190.

My click at (950,222) was y=222 — should still be within the button (if the button spans 172-200, then 222 is below it, in the kbd text). Hmm.

But maybe the browser viewport is not 1200x800 for the click coordinates... The tool says "Coordinates use a 1200×800 viewport". So the screenshot should be 1200x800. Looking at the attached screenshot, the image appears to be about 1200x800 (typical aspect).

Let me estimate the button position in the screenshot image directly. In the last screenshot, the right column starts around x=0.73*1200 ≈ 875. The "CONTROLS" label is at y ≈ 0.21*800 = 170. The Pause button (teal) spans y ≈ 0.24*800=195 to 0.27*800=215, x from 0.74*1200=888 to ~0.83*1200=1000.

Hmm, my click at (950, 222) should be very close. But the state didn't change. Maybe the click did land but... hmm, could my click have been intercepted? Or maybe the click DID pause and then a subsequent click... no, only one click.

Wait, look at the responses: the first click (90,365) → still running. Second click (950,222) → still running. The sim time advanced a lot between them (16.48 → 25.53), suggesting significant real time passed (~9s of sim).

Maybe the screenshot's DOM is captured before the click is processed? No, the clicks are separate tool calls with results after.

Hmm, maybe the issue: the browser tool's clicks are dispatched at coordinates, but the page is scrolled. The canvas is 800x800 tall plus the header (~120px) → total page height ~1000px, which exceeds the 800px viewport, so the page is scrolled down? No — the body is a flex column with the header at the top. If the viewport is 800 tall and the content is ~960 tall, the page scrolls. The screenshot shows the full canvas and the header, so the page isn't scrolled (or is scrolled to the top).

Hmm, wait. Maybe the screenshot is scaled to fit, e.g., the image is rendered at a smaller size. If the page is 1116x960 and the viewport 1200x800, the page would need scrolling to show the whole canvas (the canvas bottom would be cut off). The screenshot shows the entire canvas plus the "HOW IT WORKS" card at the bottom, so the page must fit within 800px. That means the canvas is being scaled down by CSS (max-height?) — no, I didn't set max-height. Hmm, but the stage has `padding:10px` and the canvas is 800x800 → the total height would be 18(pad)+26(h1)+40(sub)+14(gap)+820(stage) = 918 > 800. So the page should be scrollable and the screenshot would be clipped.

But the screenshot shows the entire canvas AND the bottom card. So either the screenshot is scaled down, or the viewport height is larger than 800 for the screenshot.

The image dimensions: the attached image looks like it could be scaled. The tool says "Coordinates use a 1200×800 viewport". So the screenshot might be a scaled-down rendering of the full page, meaning my click coordinates (in the 1200x800 space) don't map linearly to the screenshot.

If the screenshot is a full-page screenshot scaled to fit some size, then the y-coordinates in the image are compressed. My (950,222) click in the 1200x800 viewport would map to a different point in the page.

Let me instead try to find the Pause button by scrolling. Or better: use keyboard! Press Space to pause (the window keydown handler). That tests pause behavior. And press R for reset. But I also want to verify the buttons work via clicking.

Let me first try pressing Space to verify the pause functionality, and then use the `scroll` action to see the page structure. Actually, let's reason: if the page is 1116 wide and 918 tall, and the viewport is 1200x800... the body has `padding: 18px 14px 28px`. Content height: h1 (19px font, line-height 1.45 → ~28px) + p.sub (13px font, ~2 lines → ~38px) + gap 14 + wrap (max of canvas 820 and side panel ~600) = 820 → total = 18+28+38+14+820+28 = 946.

So the page is 946 tall > 800 → scrollable. The screenshot showing everything means it's a full-page capture. In that case, the pixel coordinates in the screenshot differ from the viewport coordinates.

Let me just use keyboard shortcuts (Space/R), which I've implemented, and also verify the buttons by clicking after scrolling into view. Let me first try the keyboard.

Actually, a more robust approach for clicking: use the `inspect`/screenshot to identify coordinates in the *viewport*, then click. But the screenshot may be full-page, so I can't rely on it.

Let me try pressing Space first.

Pause works: the button text changed to "Resume", the overlay shows "PAUSED", the state shows "paused" (the DOM dump text shows "running" in the state line — wait, it says `State running` in the text but `"paused":true` in the JSON and the button says Resume. Hmm, "State" shows "running" in the text dump? Let me look: "Max spin\n27.6 rad/s\nSim time\n43.36 s\nState\nrunning". But paused is true.

Hmm! The UI state text says "running" but paused=true. Why? Because `updateHUD` sets `ui.state.textContent = paused ? "paused" : "running"` — and it's inside the throttled section, which runs every frame regardless of paused. So it should show "paused"...

Wait, the order in updateHUD: `hudClock += dt; if (hudClock < 0.1) return; hudClock = 0;` then the updates. The frame loop still calls updateHUD every frame even when paused (it's outside the `if (!paused)` block). So it should update.

Unless... the DOM dump text is from the "Sim time 43.36 s" which is a stale snapshot from before the pause? The t=43.398 in the JSON matches sim time 43.36s. Hmm, so both are from the same frame-ish.

Hmm, wait. Maybe the text dump I'm reading is the *initial* render of the DOM (from page load) plus the JSON? No — "Sim time 43.36 s" is clearly live.

Let me re-read the DOM text: 
```
State
running
```
and `"paused":true`. Contradiction.

OH WAIT. I see it. In `updateHUD`, the `ui.state` line is set BEFORE... let me re-check my edit. I replaced:
```
  ui.time.textContent = ...;
  ui.state.textContent = paused ? "paused" : "running";

  // machine-readable snapshot...
```
So both are set. Hmm.

Unless... `ui.state` is not the element I think. `const ui = { ... state: el("sState"), ... }` ✓.

Hmm, is it possible that the DOM text dump the tool gives is a *serialized* DOM where the text nodes are in a different order and my reading is off? Let me look at the exact ordering again in the last response:

"Peak speed\n685 px/s\nMax spin\n27.6 rad/s\nSim time\n43.36 s\nState\nrunning\nPHYSICS CONSTANTS"

Hmm, so State = running. But wall angle 242°, revolutions 8.67 — and paused:true.

Hmm! Maybe the tool's DOM dump was captured BEFORE the key press took effect for the text but AFTER for the JSON? No...

Actually wait. Maybe the DOM text dump is a cached snapshot from an earlier poll and the JSON is fresh. The tool returns "dom" and "images". Both should be from the same state. But note the earlier discrepancy between the HUD wall angle (292°) and the JSON (298.8°) — they were ~6 frames apart. So the DOM dump and the JSON are captured at different times! The DOM text is captured first, then the JSON? Or the JSON is read from a later poll.

Given the time between captures (~90ms ≈ 6 frames), and the pause happened during that window: the DOM text was captured while still "running" (before the keypress), and the JSON after. That's consistent: the pause happened between the two captures.

OK so that's a timing artifact of the harness. Fine.

Now, the paused state shows: contacts=20 (all balls in contact), the pile is at the bottom-left. Balls all inside. 

Now let me test Reset. Press R. Then compare the restored positions with the initial layout (from the first snapshot):
Initial: [1,326,360],[2,363,360],[3,400,360],[4,437,360],[5,474,360],[6,345,392],[7,382,392],[8,419,392],[9,456,392],[10,493,392],[11,326,424],[12,363,424],[13,400,424],[14,437,424],[15,474,424],[16,345,456],[17,382,456],[18,419,456],[19,456,456],[20,493,456], wall=0.14657 (at t=0.117, i.e., the wall had already rotated a bit from 0).

After Reset, wall should be exactly 0 and positions exactly the initial ones (y=360,392,424,456 and x=326..493).

Let me press R and inspect.

Reset restored the exact initial arrangement:
- All x values identical to the initial snapshot (326, 363, 400, 437, 474 / 345, 382, ...).
- y values 352, 384, 416, 448 vs the initial snapshot's 360, 392, 424, 456 — an 8px offset because the first snapshot was taken at t=0.117s after the balls had already fallen ~8px under gravity. So the reset state is exactly the initial arrangement ✓.
- Wall angle exactly 0 ✓, revolutions 0.00, sim time 0.00, peak speed 0, max spin 0 ✓.
- Still paused, and the screenshot shows the initial arrangement with wall angle 0.

Now let me test the Reset button click and the Pause button click. Let me resume first (Space), then click the buttons. The click coordinates: since the page is taller than the viewport (946 > 800), the screenshot may be a full-page shot and my click coordinates were wrong.

Actually, let's reconsider. In the full-page screenshot, the aspect looks like 1200x800 but shows the entire page (946 tall). If the screenshot is the full page scaled to 1200x800 fitting... Let's estimate: in the screenshot, the heptagon appears to span from y≈95 to y≈705 (in image pixels of height 800), and the canvas is 800x800 CSS px. So the image height 800 maps to ~946 CSS px → a scale of 0.846. And the width: the canvas spans x≈52 to 852 in CSS (800 px) → in the image, x≈42*0.846... Hmm, hard to be precise.

The earlier failed click at (950,222): in CSS coordinates, x=950 is within the side panel (884-1136) ✓, y=222 is within the button (if the button is at CSS y≈172-200, then 222 is below). So I need y≈185 in CSS, which maps to image y ≈ 185*0.846 = 157... 

Hmm, but the browser tool's click coordinates use the 1200x800 *viewport*, not the page. If the page is scrolled to the top, the viewport shows CSS y from 0 to 800, and the Controls card at CSS y=130-250 is visible at viewport y=130-250. My click at y=222 should be in the button region then...

Unless the page IS scrolled. When the page loads, the scroll is at 0. So viewport y = CSS y for y<800. The Pause button at CSS y ~172-200 → viewport y ~172-200. My click at (950,222) should have hit it or just below it (the kbd hint text area). Hmm, y=222 could be below the button (button height ~38px starting at ~172 → ends at 210). So 222 would be in the gap/kbd area. That's why it missed!

And the first click at (90,365) hit the canvas.

So let me click at (950, 190).

Let me first resume with Space, then click Pause at (950,190).

The Pause button click now works: the button shows "Resume", the PAUSED overlay appears, and the sim time froze (the JSON t=7.366 with wall=2.9735 rad = 170.4° matching the HUD 170°) ✓.

Now let me click the Reset button. It's to the right of Pause: the side panel starts at ~884 CSS, the button row has Pause (897-1010) and Reset (1014-1127) roughly. So Reset center x ≈ 1070, y ≈ 190.

Let me click Reset.

Interesting: after clicking Reset, the button text changed back to "Pause" (so the click hit the Pause/Resume button region, not Reset? No wait — the Reset button's handler calls `reset(); setPaused(false);` which would set the button to "Pause" AND reset the sim. But the state still shows t=7.366, wall=170°, contacts=17 — NOT reset!

Hmm. So the click at (1070,190) toggled the pause state to... no, the button text is "Pause" (meaning paused=false), and state=paused... wait, the HUD says State "paused" and the button says "Pause". That's inconsistent again — this is the timing artifact between the DOM text and JSON captures.

Wait, let's think. The state before the click: paused=true, button "Resume". After clicking at (1070,190): the button text is now "Pause" and the JSON says paused:true, t=7.366 (unchanged).

If the click had hit the Resume button, it would toggle paused to false → the button would say "Pause" and the sim would continue (t would advance). But t is unchanged (7.366). Hmm, but the JSON and DOM are captured at different times...

Alternatively, the click hit the Reset button → reset() + setPaused(false) → button "Pause", t=0, wall=0. But the JSON shows t=7.366, wall=2.9735 — NOT reset. So the click hit neither?!

Hmm, but the button text changed from "Resume" to "Pause". That means setPaused(false) was called. So the click DID trigger something... but the sim wasn't reset.

Hold on. Maybe the click landed on the Reset button, and the handler ran `reset(); setPaused(false);`. Then reset() sets wallAngle=0, simTime=0, and the balls back to the initial positions. But the JSON shows the pile unchanged with t=7.366.

Unless the JSON is stale (captured before the reset). The DOM text and JSON are from different moments. The DOM text says: Wall angle 170°, Revolutions 1.47, Contacts 17, Sim time 7.37 s, State paused. That's also NOT reset. So the DOM text also shows the un-reset state. Hmm, but the button says "Pause" in the DOM text (which means setPaused(false) happened).

So the DOM text is a mix: the HUD text (from before the click?) and the button text (from after). That means the DOM text dump itself is a composite captured at different times?! That seems odd. More likely: the tool captures the DOM text via a serialization that happens... hmm.

Wait. Actually, maybe the issue is the throttle in updateHUD: it returns early if hudClock < 0.1. After the click, only ~1-2 frames elapsed, so the HUD text wasn't updated yet. But the button text is updated immediately by setPaused. So:
- Before the click: paused=true (from the previous Pause click), HUD showing paused state, button "Resume".
- The Reset click fires: reset() → balls back to initial, wallAngle=0, simTime=0. setPaused(false) → button text "Pause".
- Then the next frame: updateHUD is throttled and hasn't updated yet (hudClock < 0.1).

Hmm, but the JSON snapshot is also inside updateHUD (after the throttle check), so the JSON would also be stale.

But the screenshot shows the initial arrangement! Look at the screenshot: the balls are back in the 5x4 grid in the center! And the heptagon is at 0° (vertex at top, flat bottom edge). So the reset DID work.

So the DOM text dump is stale relative to the screenshot. Confirmed: the DOM/JSON text is captured at a different time than the image. The JSON I see is from before the reset.

Wait, but the JSON has "paused":true and the button (in the DOM text) says "Pause". The JSOn was captured before the click, and the DOM text after (or the DOM text is captured earlier and includes the button...). Confusing, but the screenshot clearly shows the reset worked.

Hmm, but note: the button says "Pause" and the screenshot shows the balls reset and NOT paused (no PAUSED overlay). Consistent with the Reset handler: reset + setPaused(false) ✓.

So the Reset button works. The DOM text is just stale/lagging in this harness.

Let me confirm with a fresh inspect to let the HUD catch up.

Let me do a screenshot now (after some delay) to confirm the state has settled to the initial arrangement and is running.

The Reset button worked (reset + resumed). After 17s of running, the state is running with contacts=9, all 20 balls inside.

Note: the reset was confirmed in the previous screenshot.

Now let me do a final visual check of the number rotation. Looking at the screenshots, I can see numbers like "3", "7", "13" which appear rotated. Let me verify spin more rigorously — from the JSON snapshots, `ang` values are large and varied (e.g., 229.45, 155.72, 129.23, 150.93...) and `angVel` is nonzero for most balls. Since the number is drawn inside `ctx.rotate(b.ang)`, the numbers rotate with the balls ✓.

Let me do one final check: capture a close-up screenshot where the numbers' rotation is visible. I could add a zoom... not needed. Instead let me verify visually: in the last screenshot, ball 13 (green, top-left) shows "13" rotated ~90°, and ball 3 (blue) shows "3" rotated. Ball 1 (red) shows "1" rotated. Yes, rotation is clearly visible.

One more consideration: the balls are quite small (r=16) and the numbers are 15px monospace bold. Two-digit numbers like "13", "20" fit within 32px width. Good.

Now let me double check a few remaining concerns:

1. Is the pile getting stuck? At t=17s, contacts=9 and the pile is in the bottom-left corner. The balls continue to move (spin 27.6 max). Fine.

2. Performance: 8 substeps × (20 balls × 7 edges + 190 pairs) × 4 iterations. That's 8 × 330 contacts × 4 iterations ≈ 10k contact solves per frame plus pair checks (190 per substep × 8 = 1520). Fine.

3. Any console errors? Only the sandbox warning. No JS errors ✓.

4. One issue: the `insideHeptagon` failsafe — verified rarely triggers (no escapes in 90s of headless sim).

Let me also double check the "keep it fully visible" requirement — the heptagon with circumradius 300 centered in an 800x800 canvas is fully visible with margin ✓ (verified in screenshots).

Let me now clean up: the debug `<pre id="dbg">` element is a bit hacky. I'll keep it since it's harmless and useful, but let me reconsider... It's inside the page but offscreen. It's fine, but for cleanliness maybe I should keep it — it doesn't affect the visuals. Actually, it could be seen as clutter. But it's genuinely useful for automated checks. I'll keep it (offscreen, 10x10, overflow hidden).

Hmm, one more improvement I should consider: the task says "Initially place them in a compact, non-overlapping arrangement near the enclosure center and let them fall under gravity." ✓ Done.

"Rotate the heptagon at a constant rate of one revolution every five seconds. Keep it fully visible." ✓

"Implement ball-ball and ball-wall collision detection yourself... Account for moving-wall velocity in bounce response, and include damping or friction." ✓

"Give each ball angular motion and rotate its number with the ball so spin is visible." ✓ — but initially, the balls have angVel=0. They gain angular motion from collisions (friction). That satisfies "give each ball angular motion".

"Choose reasonable sizes and physical constants that keep the simulation stable" ✓

"Provide Pause/Resume and Reset." ✓

Now let me do a final sanity check of the file (valid HTML, no syntax errors) and run a final headless regression to be sure the edits didn't break anything. Let me re-run the node tests against the final file.

Also, I want to double check one edge case: if the user resizes the window such that the canvas is bigger... The canvas has a fixed internal resolution of 800x800 with CSS width unset (so it renders at 800px). On a narrow screen it might overflow. The .stage has no overflow handling. Let me add `max-width:100%` and `height:auto` to the canvas CSS so it scales down on small screens. But note: the physics uses fixed 800x800 coordinates, so CSS scaling would just scale the rendering. Let me add that for robustness.

Let me also verify the final file loads without errors after that change.

Let me make the CSS tweak and re-run the headless tests + a browser reload.

The final file still passes all headless checks (C) in behaviour3 is the important one: 0.469px max overlap, 0 wall penetration, 0 escapes, max speed 681 px/s. And behaviour4's A/B (restitution 0.56/0.60) and C (friction spins the ball).

Let me do a final browser reload to confirm no errors after the CSS tweak, then finish.