Deformed kinematics

PDF page 10 but specific reparametrisations and projections do yield hyperbolic geometries on subsets of state space. A clean operational connection to the present framework would require identifying an associative covariant composition law on quantum state space whose bound is the boundary of the state set; we are not aware that such a structure has been worked out in detail, and we record it as an open question. We have intentionally omitted several other proposed applications (a bounded-time / de Sitter connection, a hyperbolic dose-response curve in toxicology) that appeared in earlier drafts of this work; on closer examination, these do not satisfy the axioms of Section 2 as stated, and they require either substantial reformulation or, in some cases, abandonment. Open questions 1.Higher dimensions and choice of composition. Theorem 3 establishes that Einstein addition is the unique SO(3)-covariant gyro-associative law on Bℓ⊂ R3 satisfying the axioms. Möbius addition on the Poincaré ball is also SO(3)-covariant and gyro- associative but corresponds to a different identification of the ball with H3 (conformal rather than Beltrami–Klein). The choice between these is fixed by the identification of ⊕ with the projection of the SO(3,1) group product onto boost cosets; under Möbius, the group structure is different (composition of orientation- preserving isometries of H3). A precise characterisation of which gyrogroup structures arise from which physical compositional inputs would clarify the relation between the two cases . 2.Dynamics. The framework constrains the kinematic algebra but says nothing about the Hamiltonian. What additional axioms or empirical inputs fix the dynamical content—the dispersion relation, the form of the Hamiltonian in rapidity variables—is a separate question, addressed in the DSR literature . 3.Relation between bounds. If both velocity and momentum satisfy the axioms of Section 2, the bounds c and 1/κ are a priori independent. Whether they should be related—for instance by κ ∼ 1/(Mc) for some mass scale M, as in some DSR models—is not determined by the framework and requires additional physical input. 4.Thermodynamic consequences. The modified phase-space measure (6) alters the density of states and hence the partition functions for relativistic statistical mechanics. The consequences for blackbody radiation, early-universe cosmology, and black-hole thermodynamics merit detailed analysis. Conclusion We have shown that the kinematic structure of doubly special relativity and Snyder non- commutative geometry can be derived from compositional axioms alone, without recourse to a postulated modified dispersion relation or deformed Lorentz algebra. The derivation rests on Aczél’s representation theorem, supplemented by boundary invariance and PDF page 11 rotational covariance conditions, and naturally incorporates the gyrogroup structure required for the 3D extension. The 1D rapidity is ψ(p) = κ−1artanh(κp), the 3D composition is the Einstein gyrovector addition, and quantisation with respect to the rapidity yields the hyperbolic CCR, modified phase-space measure, and Snyder algebra. The construction situates the hyperbolic structure of bounded-momentum theories as a consequence of compositional axioms rather than a postulate. Empirical bounds on κ from ultra-high- energy cosmic rays place κ−1c ≳ 1011 GeV. The framework is conditional: it tells us what the kinematics must look like given a bound and a compositional structure, but does not predict the existence or value of the bound. This is a constraint on theory space, not a theory, and its principal value is in clarifying which features of DSR-type kinematics are robust against changes in dynamical assumptions and which are model-dependent. Author and disclosure. The author is an independent researcher without institutional affiliation. The work received no external funding. Generative AI tools were used as a writing aid for compilation, formatting, and algebraic verification; all theoretical content, derivations, and conclusions are the author’s, and all algebraic claims have been verified directly. Three-dimensional composition and the Snyder algebra We collect here the explicit calculations underlying Section 3. A.1Einstein addition on the ball The Einstein gyrovector addition (2) arises from the action of SO(3,1) on the unit hyperboloid in Minkowski space, restricted to the Beltrami–Klein model of H3. Concretely, parametrise SO(3,1) near the identity by a rotation R ∈ SO(3) and a boost Λ(a) with rapidity vector ϕa= artanh(|a|/ℓ) a (where a = a/|a| and ℓ is the curvature scale). Every group element admits a unique polar decomposition g = Λ(a) R. The product of two boosts is in general not a pure boost but a boost composed with a rotation: Λ(a) Λ(b) = Λ(a ⊕ b) R(a,b), where the rotation R(a,b) ∈ SO(3) is the Wigner rotation (Thomas precession) and a ⊕ b is the Einstein sum (2). The gyration gyr[a,b] ≡ R(a,b) is precisely the source of the deviation from strict associativity; see for the full computation. Key properties of ⊕ : • Identity: 0 ⊕ a = a ⊕ 0 = a. • Non-commutativity: a ⊕ b ≠ b ⊕ a in general, with the discrepancy encoded in gyr [a,b].

changes as a covariant associative (or gyro-associative) composition will exhibit the same kinematic deformations. Conversely, the framework cannot explain why momentum is bounded in the first place, nor predict the scale κ; those remain empirical inputs. The paper is organized as follows. Section 2 states and proves the bounded composition theorem in the form needed. Section 3 applies it to momentum, deriving the hyperbolic canonical commutation relation and connecting to the Snyder algebra. Section 4 discusses experimental constraints and the relation to DSR phenomenology. Section 5 considers what

C1 function ψ:I→R, unique up to a positive multiplicative constant, with ψ(0) = 0 and a ⊕ b = ψ−1(ψ(a) + ψ(b)). Proof. This is Aczél’s theorem on associative operations on a real interval . The hypotheses yield a connected one-parameter Lie group whose unique additive representation up to scale is given by (1). ◻ Lemma 2 (ψ is a bijection onto R). Under Axioms 1–4 in 1D, the map ψ of Lemma 1 satisfies ψ(I) = R.

+ b for |a|,|b| ≪ ℓ. With this choice, ψ has the same dimensions as its argument, and the bound ℓ appears explicitly in the rapidity. This choice is conventional and does not affect any physical predictions, which depend only on dimensionless ratios. Application: hyperbolic quantum kinematics We now apply Corollary 4 to momentum. Postulates •Boundedness. The momentum of a single particle is bounded: |p| < 1/κ for some

momentum and treats sequential changes as a covariant compositional structure will exhibit the kinematic deformations derived in Section 3. The relation is analogous to that between Noether’s theorem and conserved quantities: Noether’s theorem does not predict which symmetries a theory has, but constrains the form of any conservation law given the symmetries. Here, the bounded composition theorem does not predict which observables are bounded, but constrains the kinematics given that an observable is bounded. Velocity composition as the cleanest analogue Velocity in special relativity is the cleanest instance of bounded compositional kinematics. The velocity v is bounded (|v| < c), composes gyro-associatively via Einstein addition, and For a fixed-s representation, the spin projection Sz is bounded by ± sℏ. If sequential projective spin measurements (or some operational analogue) compose in the appropriate sense, the framework would predict a hyperbolic deformation of the spin algebra, with a deformation parameter related to 1/sℏ. This is reminiscent of the SUq(2) quantum group structure with real deformation parameter q . Whether spin measurement composition actually satisfies the axioms of Section 2 is an open operational question; we flag the connection as suggestive rather than established.

energy cosmic rays place κ−1c ≳ 1011 GeV. The framework is conditional: it tells us what the kinematics must look like given a bound and a compositional structure, but does not predict the existence or value of the bound. This is a constraint on theory space, not a theory, and its principal value is in clarifying which features of DSR-type kinematics are robust against changes in dynamical assumptions and which are model-dependent. Author and disclosure.