Interior group existence

### PDF page 15 paper’s modest contribution. Proposition 5.3 is, to the author’s knowledge, the most original observation of the paper. 9.3 Falsifiability The theorem is falsified within mathematics by exhibiting: - An admissible structure (satisfying Axioms 1–3) whose identity lies on the boundary; or - An admissible structure containing a finite composition of interior elements that equals a boundary point; or - An admissible structure containing a non-identity element of finite order. By Theorems 3.1, 3.2, and Lemma 3.3, none of these is possible. The theorem is not falsifiable within its domain. The theorem may be inapplicable to a given empirical system if that system fails one or more of the axioms; such inapplicability is a fact about the system, not about the theorem. 9.4 Conditional status of the theorem A theorem of the form if A, then B is conditional in the trivial sense: granted A, B follows by the proof; granted not-A, the theorem is silent. This is true of every correctly proved theorem and we make no claim of novelty in this trivial sense. What is worth noting is the content of B in Theorem 4.1. The conclusion is an existential statement — the structure cannot be on its boundary; it must occupy its interior — rather than a property of objects assumed to exist. Mathematically, the theorem is closed: any admissible structure violating one of its four parts would falsify Theorems 3.1, 3.2, or Lemma 3.3, which it cannot. Empirically, the theorem may be inapplicable to a given system if that system fails one or more of Axioms 1–3, but such inapplicability is a fact about the system, not about the theorem. Whatever satisfies the axioms cannot occupy its boundary. Whatever fails the axioms is, by definition, not what the theorem is about. This is the precise content of the claim bounded composition forces interior existence. 10. Closing We have proved one theorem and one proposition. Under three axioms of bounded composition — smoothness, identity-and-associativity, and strict monotonicity — on an open interval, the following four statements all hold without further hypothesis: the identity is interior; the boundary is unattainable by finite composition; no non-identity element has finite order; every non-identity element generates a perpetual interior flow asymptotic to the boundary. Boundary invariance, often posited as a separate axiom, is a derived consequence (Lemma 3.3). The three structural results of §3 combine into a classification (Corollary 3.4): up to scale and orientation, there is a unique admissible bounded compositional structure, of which Einstein velocity addition is the canonical instance. Hyperbolic geometry is the inevitable structural signature of bounded associative monotone composition. ### PDF page 16 The interior is closed under finite composition from within (Theorem 4.1(ii)); dually, the interior is dense in the closure of boundary-pair compositions (Proposition 5.3), with the limit value parametrised by the rate at which the two boundaries are approached. The interior is what bounded composition naturally inhabits and what the bound meeting the bound generates. We have given full proofs resting on Aczél’s classical representation theorem for associative operations on intervals, an elementary doubling argument, and the Banach fixed-point theorem for the corollary in §7. We have stated what the theorem does not claim, including its lack of novelty in the underlying ingredients, and we have flagged in Remark 5.4 a natural extension — the rate-dependent boundary composition — whose development we leave to separate work. The theorem is complete on its own terms. The reader may verify each step.

This question is narrow on purpose. It is not the question of whether the universe exists, why something rather than nothing, or what consciousness is. Those questions either lie outside mathematics or require additional hypotheses that the present paper does not adopt. We restrict ourselves to a statement that can be proved, and we state nothing beyond what is proved. The paper proceeds in ten sections. Section 2 gives the axioms. Section 3 introduces the rapidity coordinate via Aczél’s representation theorem, with full proofs of existence Corollary 3.4. Section 7 gives the Banach corollary with a worked example on the Einstein composition. Section 8 remarks on the higher-dimensional case. Section 9 states scope, falsifiability, and the conditional status of the theorem. Section 10 closes. An Acknowledgments of Prior Development section names the three works that developed the closest predecessor material. 2. Axioms of Bounded Composition

which the theorem of this paper applies. The associated rapidity coordinate is ψ(x) = artanh(x), and the boundary ±1 corresponds to ψ = ±∞. Remark 2.3 (On the smoothness hypothesis). Axiom 1 imposes C¹ smoothness for technical convenience in §7 (Banach corollary). The resolution of Hilbert’s Fifth Problem implies that every connected locally Euclidean topological group is a Lie group; in dimension one the result is elementary and predates the general theorem. In our setting, mere continuity of ⊕ together with Axioms 2 and 3 already forces ⊕ to be compatible with

𝐺: ℝ → ℝ,   𝐺(𝜔) : = 𝜓(𝐹(𝜓−1(𝜔))). By construction, G is the conjugate of F under the rapidity coordinate. Hypothesis (strict contraction in rapidity). G is globally Lipschitz with constant L < 1: |𝐺(𝜔) − 𝐺(𝜔′)| ≤ 𝐿 ⋅ |𝜔 − 𝜔′| for all 𝜔, 𝜔′ ∈ ℝ. Corollary 7.1 (Banach fixed point of self-referential bounded composition). Under the strict-contraction hypothesis (2), the map F: I → I has a unique fixed point x ∈ I. The iteration x_{n+1} = F(x_n) converges to x* from every initial condition x₀ ∈ I, with rapidity-space error decaying at geometric rate L:* |𝜓(𝑥𝑛) − 𝜓(𝑥∗)| ≤ 𝐿𝑛 ⋅ |𝜓(𝑥0) − 𝜓(𝑥∗)|. Proof. ℝ is a complete metric space under the Euclidean metric. By hypothesis (2), G: ℝ → ℝ is a strict contraction with Lipschitz constant L < 1. By the Banach fixed-point theorem, G has a unique fixed point ω* ∈ ℝ, and the iteration ω_{n+1} = G(ω_n) converges to ω* from any ω₀ ∈ ℝ with |ω_n − ω| ≤ L^n · |ω₀ − ω|. Pulling back: set x* := ψ⁻¹(ω) ∈ I. Then F(x) = ψ⁻¹(G(ψ(x))) = ψ⁻¹(G(ω)) = ψ⁻¹(ω) = x, so x* For the iteration: ψ(x_{n+1}) = ψ(F(x_n)) = G(ψ(x_n)), so the rapidity iterates ω_n := ψ(x_n) satisfy the same Banach iteration in ℝ and decay at rate L^n. ∎ Remark 7.2 (Why the contraction is stated on G, not on F). The hypothesis (2) is the natural condition because Banach contraction requires a complete metric space and a strict contraction on it. The interval I is not complete under the Euclidean metric inherited from ℝ (Cauchy sequences in I converging to ±ℓ have no limit in I). The rapidity transformation maps I bijectively to ℝ, which is complete. The contraction must therefore be checked on G,

fixed point is constructed from F and ψ by Banach iteration, independent of initial conditions. In the operational sense, the system’s stable state is generated by closure of the structure on itself, subject to the contraction hypothesis (2). We make no further interpretive claim about this corollary; it is a strict mathematical consequence of (2) and Theorem 4.1. 8. Higher Dimensions: A Remark

this paper. 9. Scope, Conditional Status, and Falsifiability 9.1 What is proved Theorem 4.1 and Corollary 7.1 are the only mathematical claims of this paper. Both are proved from the axioms of §2 and the structural results of §3, which themselves rest on Aczél’s representation theorem for associative operations on intervals (cited in the

paper’s modest contribution. Proposition 5.3 is, to the author’s knowledge, the most original observation of the paper. 9.3 Falsifiability The theorem is falsified within mathematics by exhibiting: - An admissible structure (satisfying Axioms 1–3) whose identity lies on the boundary; or - An admissible structure containing a finite composition of interior elements that equals a boundary point; or - An admissible structure containing a non-identity element of finite order. By Theorems 3.1, 3.2, and Lemma 3.3, none of these is possible. The theorem is not falsifiable within its domain. The theorem may be inapplicable to a given empirical system if that system fails one or more of the axioms; such inapplicability is a fact about the system, not about the theorem. 9.4 Conditional status of the theorem A theorem of the form if A, then B is conditional in the trivial sense: granted A, B follows by statement — the structure cannot be on its boundary; it must occupy its interior — rather than a property of objects assumed to exist. Mathematically, the theorem is closed: any admissible structure violating one of its four parts would falsify Theorems 3.1, 3.2, or Lemma 3.3, which it cannot. Empirically, the theorem may be inapplicable to a given system if that system fails one or more of Axioms 1–3, but such inapplicability is a fact about the system, not about the theorem. Whatever satisfies the axioms cannot occupy its boundary. Whatever fails the axioms is, by definition, not what the theorem is about. Under three axioms of bounded composition — smoothness, identity-and-associativity, and strict monotonicity — on an open interval, the following four statements all hold without further hypothesis: the identity is interior; the boundary is unattainable by finite composition; no non-identity element has finite order; every non-identity element generates a perpetual interior flow asymptotic to the boundary. Boundary invariance, often posited as a separate axiom, is a derived consequence (Lemma 3.3). The three structural results of §3 combine into a classification (Corollary 3.4): up to scale and orientation, there