feat: physics stability, k-NN information metrics & save system

[M1thieu]

Objective

Stabilize N-body simulation, add information-theoretic measurement tools
(k-NN entropy/MI, KL/JS divergence), introduce a production-grade save system
with stable entity identity, and clean up spatial indexing and pairwise staging.

Implementation

utils

• Adaptive neighbor index for spatial queries

forces / energy

• Velocity Verlet integrator (2nd-order symplectic, ~0.01% energy drift over 800+ s)
• Fix missing PreviousAcceleration component (silent Bevy query failure froze all particles)
• Softened Coulomb F = k·q/(r² + ε²)^1.5 — no singularity at r < ε
• All physics to FixedUpdate; GravityForceMode default → Mutual
• egui sliders for gravity/Coulomb/speed multipliers now properly wired

core / pairwise

• Unify pairwise staging; add safety scaffolds for force application ordering

information crate

• k-NN entropy/MI estimators (Kraskov et al. 2004, non-parametric)
• Fix digamma Stirling asymptotic expansion (wrong for all integer inputs)
• KLDivergence: divergence(), jensen_shannon(), cross_entropy(), total_variation()
• Unified discrete + continuous estimator trait API; full test coverage

save_system crate (new)

• PersistentId(u64) + seeded counter — stable cross-session entity identity, no UUID
• Atomic write via std::fs::rename, VecDeque event buffer, chained version migration
• get_saved_entity_components() query by PersistentId

docs

• Standardized crate READMEs; architectural notes across all domains

Testing

• 56 unit tests passing (cargo test --workspace)
• basic_save example: save → reload → entity lookup → migration, end-to-end verified
• N-body tested 830+ seconds at -0.09% energy drift

Visual Changes

// Stable entity identity across sessions
let star_id = id_counter.generate();
save(&game_save, &settings, "sim.json")?;

let loaded = load::<GameSaveData>(&settings, "sim.json")?;
let components = get_saved_entity_components(&loaded, star_id);

Technical Notes

• save_system targets Bevy 0.18
• Entity restore (load → spawn) is TODO — requires Bevy Reflect deserialization
• MPM grid binary serialization is a future addition once the MPM solver is ported