test(poincare): gyrogroup property tests for Möbius addition

[HomunculusLabs]

What

Adds 11 deterministic tests covering G2–G6 of the gyrocommutative gyrogroup structure of Poincaré-ball Möbius addition—G1 was already covered by mobius_add_zero_is_identity_on_either_side—plus selected derived laws, closure, nonassociativity, and curvature-scaling regressions.

References: Ungar (2008), Analytic Hyperbolic Geometry and Albert Einstein's Special Theory of Relativity, Ch. 2, for the gyrogroup laws; Ganea et al. (2018), §2.3, for the Poincaré-ball Möbius-addition convention.

Why

The existing Poincaré tests cover identity, distance, exp/log round-trips, and conformal Dirichlet penalty — but do not test the algebraic structure of Möbius addition itself. Möbius addition is NOT a group operation (it is non-associative); it is a gyrocommutative gyrogroup. These laws are load-bearing for arbitrary-base Poincaré exp/log maps and other consumers of Möbius translation.

Tests added

Test	Axiom/Law	Notes
mobius_add_inverse_cancels_to_origin	G2 (left inverse) + derived right inverse	Tests both (-a)⊕a=0 and a⊕(-a)=0
mobius_add_satisfies_left_gyroassociative_law	G3 (left gyroassociativity)	a⊕(b⊕c) = (a⊕b)⊕gyra,b
gyration_is_automorphism_of_addition	G4 (gyration is automorphism)	gyr(x⊕y) = gyr(x)⊕gyr(y)
gyration_left_loop_property	G5 (left loop)	gyr[a⊕b,b] = gyr[a,b]
mobius_add_is_gyrocommutative	G6 (gyrocommutativity)	a⊕b = gyra,b
mobius_add_left_cancellation_law	Derived	(-a)⊕(a⊕b) = b
gyration_inverse_is_reversed_gyration	Derived	gyr[b,a]∘gyr[a,b] = id
gyration_is_identity_when_either_generator_is_zero	Derived	gyr[a,0] = gyr[0,a] = id (Ungar Thm 2.34)
mobius_add_is_not_associative	Counterexample	(a⊕b)⊕c ≠ a⊕(b⊕c)
mobius_add_stays_inside_ball_for_interior_points	Closure	‖a⊕b‖ < 1
mobius_add_curvature_scaling_relation	Regression	Homothety + flat limit

The gyration operator is gyr[u,v]w = -(u⊕v) ⊕ (u ⊕ (v ⊕ w)), matching the convention in ConstantCurvature::gyration (cross-checked).

Review

Three review cycles before ready:

1. Oracle review — caught missing automorphism + loop property tests, vague curvature claim
2. Codex engineer review (gpt-5.6-sol) — caught missing left-inverse, isometry→homothety correction, axiom attribution fix. Convention cross-checked against constant_curvature.rs.
3. GPT Pro review — confirmed all 11 assertions mathematically correct. Found citation errors (Ganea §3.1→§2.3, Ungar title incomplete, Theorem 2.34 on wrong test), inaccurate SAE-decoder rationale, and taxonomy wording. All fixed in commit 94cfd6e.

Verification

cargo test -p gam-geometry --lib manifolds::poincare
test result: ok. 34 passed; 0 failed

cargo test -p gam-geometry --lib
test result: ok. 317 passed; 0 failed

Not established

• Gyrogroup identities tested at c = -1 only (curvature scaling test covers K=4 and near-flat limit separately).
• Fixed deterministic examples, not property-based generated inputs.
• The gyr helper is test-local; a production gyr might belong in the public API if consumers need it.