fisher_rao/PHASE2.md

Fisher-Rao Phase 2 — coupling & profile invariants

Extends Phase 1 with rotation- and scale-invariant losses on string physics parameters.

Components

Invariant	Module	Symmetry
Coupling skew spectrum	coupling_invariant_loss	O(k) rotation on mode index
Coupling strength	log-ratio in coupling_invariant_loss	positive scale
Speed profile	speed_profile_invariant_loss	global scale quotient + FR distance
Inharmonicity profile	inharm_profile_invariant_loss	scale quotient + FR distance

Phase 1 (damping double-coset) remains under fr_invariant_weight × fr_invariant_damping.

Config

Key	Default
fr_invariant_coupling	0.5
fr_invariant_coupling_strength_weight	0.1
fr_invariant_speed	0.25
fr_invariant_inharm	0.25

Synthetic references

Trainer and smoke test pass ground-truth references when available:

• target_coupling_skew — from synthetic generator or prior_targets["coupling_skew"]
• target_speed_scalars — uniform VELOCITY_SCALE_BASE per mode
• target_inharm_b — TRUE_INHARM_B or estimated from audio

Run

cd physics_audio
python run_fr_smoke_test.py --steps 1200

Trainer

run_single_seed(
    seed=42,
    max_steps=5000,
    fr_invariant_weight=0.3,
    fr_invariant_coupling=0.5,
    fr_invariant_speed=0.25,
    fr_invariant_inharm=0.25,
)

Real audio CLI

python run_real_audio.py --audio note.wav \
  --fr_invariant_coupling 0.5 \
  --fr_invariant_speed 0.25 \
  --fr_invariant_inharm 0.25

build_prior_targets supplies speed_scalars (uniform at f0-derived scale) and inharm_b (estimated B per harmonic) as references.

Next: Phase 3 modal amp pair invariant — fisher_rao/PHASE3.md.

Derivation: ~/Projects/Fisher_Rao/proof.pdf (Thm 2.3, Prop 3.3 — singular values as maximal invariants).