fix(animator): start the underdamped spring from rest

[LeadcodeDev]

Severity Medium, category correctness. Location: crates/rustmotion-core/src/engine/animator.rs:504

Impact

The closed-form underdamped step response is 1 - e^(-zeta*omega*t) * [cos(omega_d*t) + (zeta*omega/omega_d)*sin(omega_d*t)]. The code feeds zeta*omega*t/omega_d into sin() instead of omega_d*t — the t was multiplied into the coefficient instead of the phase. I verified this numerically against a 400k-step ODE integration of m*x'' + c*x' + k*x = k: the correct formula matches the integrator to 5 decimals, the repo formula does not. With the shipped defaults (damping 15, stiffness 100, mass 1, zeta = 0.75), at the first 60fps frame after the start (t = 0.0167 s) the solver reports 0.10416 of the travel where the physics gives 0.01282 — 8x. Initial velocity is 6.21/s instead of 0, so the element visibly snaps on frame 1 rather than easing out of rest, and the overshoot peak is wrong too (1.0194 vs 1.0284). This affects every spring path in the engine: EasingType::Spring keyframes, the bounce_in preset (kf_anim_spring, zeta = 0.6, animator.rs:2092), elastic_in (kf_anim_spring_underdamped, zeta = 0.274, animator.rs:2106), and every author-supplied spring override applied through apply_spring_to_motion. spring_settle_time/spring_rest_time (what rustmotion info reports and what the spring.duration remap divides by) are derived from the same wrong curve.

Fix

Change the sine argument to (omega_d * t).sin(), keeping the (zeta * omega / omega_d) coefficient: 1.0 - decay * ((omega_d * t).sin() * (zeta * omega / omega_d) + (omega_d * t).cos()). Add a regression test asserting the derivative at t=0 is ~0 (a step response starts at rest) and comparing a few samples against a numerically integrated reference.

Evidence the audit read

if zeta < 1.0 {
        // Underdamped
        let omega_d = omega * (1.0 - zeta * zeta).sqrt();
        let decay = (-zeta * omega * t).exp();
        1.0 - decay
            * ((zeta * omega * t / omega_d).sin() * (zeta * omega / omega_d) + (omega_d * t).cos())

Stacked on fix/layer-bounds-outset-shadow, which carries the previous finding of this workstream. GitHub shows only this finding's diff; merge in order.

Part of the September 2026 audit remediation chantier. Refs #220 (RM-09).