Flat/curved classification

### PDF page 13 rather than deriving it from composition axioms. The closest characterization results are Benz’s [2, 3]; these operate within hyperbolic-distance and hyperbolic-isometry hypotheses, whereas the present theorem admits a priori non-associative operations with an isometric reassociation defect and derives the invariant metric from the defect. To state the novelty narrowly: it is not that Einstein or M¨ obius addition are gyrogroup examples, nor that symmetric spaces correspond to certain loops; it is the exact axiom-to-dichotomy route, the inclusion of the flat bounded associative branch as the unique associative alternative, the parity-sharp n = 3 obstruction with its classified moduli, and the construction of the invariant metric from reassociation defects. 9 Discussion and open questions The one-dimensional theorem says a bound forces rapidity. This paper says what a second dimension adds: a fork. Bounded composition can stay associative by hiding flat addition behind the rapidity map, or it can keep contact with the bound’s geometry at the price of a rotation — and nothing else is possible: not a sphere (a complete homogeneous metric on a noncompact ball excludes it), not a chiral hybrid (parity excludes it, in the one dimension where it can exist at all), not a distorting reassociation (rigidity excludes it), not any law outside the one-parameter gauge line (the transvection lemma excludes it). The fork is decided by a single observable input: in relativistic kinematics, the nontrivial branch is selected by the observed Thomas–Wigner effect, whose factor enters atomic fine structure and storage-ring spin dynamics. The direction of inference is the reverse of the textbook one: standard relativity derives the bounded velocity domain from the invariant spacetime interval, whereas here the bound and the composition axioms come first and the metric — hence the interval structure on velocity space — is derived. The associativity defect is therefore best understood not as a complication of relativistic kinematics but as the mechanism by which a bounded system acquires its metric: the rotation is the curvature, and the curvature is the bound’s signature in more than one dimension. Open questions. (1) The SO(2)-equivariant case at n = 2: the commutant method fails there because the stabilizer of an axis in SO(2) is trivial, so a genuinely different rigidity mechanism — or a counterexample — is required; we incline toward rigidity, the plane offering no axial direction to twist about, but have no proof. (2) Can A6 ′ be weakened — to “each defect is linear,” or to norm preservation within A1–A5, A7? Proposition 4.2 decides neither. (2 ′) With A7 dropped entirely, does the classification persist under A1–A6 ′ alone, up to O(n)-equivariant radial reparameterization? We expect so, but it is open. (3) Whether G5 fails for every nonzero profile of the n = 3 moduli. (4) The symmetry-reduction program: replacing O(n) by unitary or symplectic compatibility groups and asking whether the same mechanism manufactures the complex and quaternionic hyperbolic composition laws. (5) The applied door of Appendix B: which empirical bounded compositional systems — learned hyperbolic representations, belief revision, saturating signal chains — occupy which branch. A Computational sanity checks and reproducibility Facts proved exactly in the text (the flat law’s associativity and A7; the chiral family’s A4, A6′, A7, equivariance; M¨ obius membership of branch (H)) were additionally machine-checked as sanity tests. Genuinely numerical claims are: the escape statistics of Proposition 4.1 (sampling stated there; fixed seed); the violation magnitudes of Propositions 4.2 and 4.3 (norms stated Page 13 of 16 AUTHOR SUBMITTED MANUSCRIPT - JPhysA-125044 ### PDF page 14 there); the holonomy-area identity of Corollary 6.4(iii), corroborated to 3 .7 × 10−13 over 1,500 pairs (script supplied); the order-3 /5/7 exact-rational perturbative computation of Section 7 (notebooks on request); and the tomography validation of Appendix B. Scripts are supplied as supplementary material; the Section 7 notebooks are available on request. B Defect tomography (simulation-validated) The classification converts into a measurement protocol. Given only noisy query access to a bounded compositional system ⊕, the following procedure decides its branch and, in the hyperbolic case, estimates its curvature. It is validated here in simulation — against synthetic in-class and out-of-class systems under injected noise — not yet against an empirical system; each design choice below was forced by a failure mode encountered during that validation. The estimator is stated for n = 3; in higher dimensions the rotation axis is replaced by the rotation’s 2-plane, extracted from the matrix logarithm of the fitted orthogonal map. Algorithm. (1) Sample pairs u, v at composition scale s and query w 7→ Du,v(w) at probes w on several radius shells. (2) Fit the defect linearly per shell and project to the nearest orthogonal map (Procrustes/SVD). (3) Out-of-class test: compare fitted rotations across shells; a nonlinear defect is typically an exact rotation on each shell with shell-dependent axis, invisible to single-shell fits, and is detected by cross-shell disagreement. Strictly, the test detects failure of linearity (shell dependence of the fitted rotation); a defect can be orthogonal on every shell yet nonlinear — the twist law is exactly such a case. (4) Branch test and curvature: form the signed estimator ˆK = sgn(axis · (u × v)) · 2θ/|u × v|, average over pairs, and Richardson-extrapolate in s to remove the O(s2) bias. Flat iff ˆK is statistically zero; otherwise hyperbolic with curvature ˆK < 0. Validation (independent noise 10 −3 per query; n = 3): system distortion ˆK verdict (truth) Einstein (Klein) 0.023 −1.02 ± 0.05 hyperbolic, K = −1 (✓) flat law 0.020 −0.06 ± 0.04 flat ( ✓) twist law 0.082 — distorting, outside class (✓) M¨ obius (Poincar´ e),λ=2 0.024 −4.06 ± 0.08 hyperbolic, K = −4 (✓) The distortion floor was three times the linear-defect reference level; the twist’s signal is 3 .5× that level. Note the last row: the protocol cannot distinguish Klein from Poincar´ e coordinates except through ˆK — the correct operational reading of the gauge freedom of Theorem 6.3(H). Two practical lessons: query noise induces a strictly positive bias in fitted rotation angles, so unsigned statistics misclassify flat systems — the signed estimator is essential, noise averaging to zero while holonomy adds coherently; and the sign convention is an output of the procedure, the Thomas–Wigner axis being anti-parallel to u × v. Detection power against out-of-class laws grows with gyration magnitude and shell separation; the 3× floor is calibrated to this noise level and dimension and should be recalibrated in other regimes. Scope. Natural targets are learned composition operators in hyperbolic machine-learning models, bounded control and belief-revision systems, and saturating signal compositions. For learned systems the oracle is simply the trained composition map evaluated on probe embed- dings, so the protocol measures the curvature a model has actually learned, as opposed to the curvature of the space it was trained in. The protocol decides, from behavior alone, whether such a system should be modeled by rapidity vectors (flat) or gyro-geometry (hyperbolic), measures Page 14 of 16 AUTHOR SUBMITTED MANUSCRIPT - JPhysA-125044 ---

by Bonnet–Myers, contradicting the open ball. The mechanism is that the defect axiom manufactures a canonical invariant Riemannian metric from the operation itself. The equiv- ariance hypothesis is dimensionally sharp: SO(n) suffices for n ≥ 4, while at n = 3 parity is necessary, the SO(3)-equivariant solutions forming an explicitly classified chiral moduli space (the SO(2) case at n = 2 is left open). M¨ obius addition is the λ = 2 gauge point of the non-associative branch. Conditional on the axioms as a kinematic model of velocity composition, the observed Thomas–Wigner rotation selects that branch; an appendix gives

(Lemma 3.2), hence + I throughout; for n = 2 the statement is vacuous. (For n = 3 the maps Q fixing P pointwise are reflections, det Q = −1: this step genuinely uses full O(n), consistent with the theorem’s hypotheses; planarity of defects is not asserted on the chiral moduli of Theorem 6.5.) (ii) Einstein gyrations are nontrivial for non-collinear arguments [4, 5]; the radial map Sλ preserves non-collinearity and, as in Step 3 above, the conjugated defects are Einstein gyrations at the mapped arguments, hence nontrivial. (iii) is cited, with the side-length computation shown.

gives Ru|ˆu⊥ = ±I, constant on the connected punctured ball; the −I branch lies in SO(n) only for odd n, and continuity at u = 0, where R0 = I, excludes it there: no chiral assignment exists. (The hypothesis is sharp — dropping continuity at 0 admits the axis-reflection above for odd n.) (ii) Forward direction. The representation of SO(2) on R2 is of complex type: its commutant contains all rotations. Lemma 6.2’s argument therefore yields only that S := T −1 u Lu is a g-

rather than deriving it from composition axioms. The closest characterization results are Benz’s [2, 3]; these operate within hyperbolic-distance and hyperbolic-isometry hypotheses, whereas the present theorem admits a priori non-associative operations with an isometric reassociation defect and derives the invariant metric from the defect. To state the novelty narrowly: it is not that Einstein or M¨ obius addition are gyrogroup examples, nor that symmetric spaces correspond to certain loops; it is the exact axiom-to-dichotomy route, the inclusion of the flat bounded