Reflection and disk derivation

### PDF page 17 budget of the fabrication, not a discrepancy with (4.3). The Pancharatnam-phase formula (4.5) matches interferometric measurements in optical [19, 20, 21, 26] and microwave-acoustic [27] settings; the rapidity forms of VSWR, return loss, and mismatch loss (3.2)–(3.4) are recovered identically from network-analyser measurements on lossless reciprocal cascades. 7. Conclusion The Möbius composition of reflection coefficients and the Poincaré-disk geometry of the Smith chart follow from five compositional axioms — smoothness, identity with U(1)-valued weak associativity, monotonicity, boundary invariance, and U(1) phase covariance — by direct Riemannian construction. The derivation has six ingredients: Aczél's representation theorem on the real diameter (radial rapidity); boundary preservation (rapidity is artanh, properness of left translations); simple connectedness of the disk (left translations are global diffeomorphisms); construction of a Riemannian metric on which the U(1)- valued gyration makes left translations global isometries by a chain-rule reduction to the origin; the maximum-symmetry theorem for Riemannian 2-manifolds (constant Gaussian curvature, via Myers– Steenrod); the Killing–Hopf classification with boundary exclusion of the sphere and the Euclidean plane (the disk is hyperbolic). The orientation-preserving isometry group of the hyperbolic plane is the Möbius group PSU(1,1). The radial rapidity is forced to be η = artanh, the metric is the Poincaré disk metric of curvature K = −1, and the gyration is the Pancharatnam geometric phase. No holomorphic input, no Schwarz–Pick lemma, no symmetric-space classification, no almost-complex structure axiom, no bounded- symmetric-domain framework is required. The engineering corollaries follow from the derivation together with the per-element physical inputs Maxwell's equations supply: the VSWR identity, the Bragg-mirror tanh² reflectance law from rapidity additivity, the Pancharatnam phase as gyration defect equal to hyperbolic area, the σ-duality between open and short circuits. The 2D case fits within a partial family of bounded compositional kinematics in dimensions 1, 2, and 3 sharing an Aczél compositional backbone, with dimension-specific covariance groups (trivial, U(1), SO(3)) supplying the uniqueness mechanism in each case. The framework is kinematic rather than dynamic: it constrains the algebra of bounded composition once the passivity bound is in place, without explaining the bound or the physical impedances. Each axiom supplies one structural ingredient of the derivation. A1 (smoothness) gives the manifold structure; A2 (U(1)-valued weak associativity) gives both the gyrogroup operation and the precise condition — gyration in U(1) — under which the transport-defined metric is globally invariant under left translations; A3 (monotonicity) gives the local diffeomorphism structure that boundary preservation promotes to a global diffeomorphism; A4(a) (boundary preservation) gives properness and completeness; A4(b) (effective boundary action) excludes the Euclidean case in Killing–Hopf; A5 (U(1) covariance) gives the U(1)- invariant inner product at the origin and consistency of the transport construction with the U(1) rotation action. The disk's hyperbolic structure is the unique closure of these five ingredients into a maximally symmetric Riemannian 2-manifold of constant negative curvature. The Smith chart is the consequent, not the assumed. Appendix A. The SU(1,1) action and gyration A.1 The group SU(1,1) and its disk action ### PDF page 18 SU(1,1) is the group of complex 2 × 2 matrices g = [[α, β], [β̄, ᾱ]] with |α|² − |β|² = 1. The Möbius action on ￿ is g · Γ = (αΓ + β)/(β̄Γ + ᾱ), preserving ￿ and ∂￿ by direct calculation: |αΓ + β|² − |β̄Γ + ᾱ|² = (|α|² − |β|²)(|Γ|² − 1) = |Γ|² − 1 < 0 for |Γ| < 1. The stabiliser of the origin is the diagonal U(1) subgroup K = {diag(e^(iφ), e^(−iφ))}, acting on ￿ by rotation Γ ↦ e^(2iφ) Γ. The quotient SU(1,1)/K ≅ ￿ identifies the disk as a homogeneous space with U(1) isotropy. A.2 Polar decomposition and boost parametrisation Every element of SU(1,1) admits a unique polar decomposition g = b(Γ_s) · k(φ), where k(φ) ∈ U(1) is a rotation and b(Γ_s) is a pure boost element parametrised by the image Γ_s := g · 0 of the origin under g. Explicitly, b(Γ_s) = (1/√(1 − |Γ_s|²)) [[1, Γ_s], [Γ̄_s, 1]]. The boost cosets b(Γ_s) · K parametrise ￿ and realise the section ￿ → SU(1,1) underlying Step 6 of Theorem 2.3. A.3 The product of two boosts The product of two boost elements is in general a boost composed with a rotation: b(Γ₁) · b(Γ₂) = b(Γ₁ ⊕ Γ₂) · k(ψ), (A.1) where Γ₁ ⊕ Γ₂ is the Möbius addition (2.2) and ψ = ψ(Γ₁, Γ₂) is the gyration angle. Direct matrix multiplication gives b(Γ₁) b(Γ₂) = [1/√((1−|Γ₁|²)(1−|Γ₂|²))] [[1+Γ₁Γ̄₂, Γ₁+Γ₂], [conj(Γ₁+Γ₂), conj(1+Γ₁Γ̄₂)]], which polar-decomposed against (A.1) yields k(ψ) = (1 + Γ₁ Γ̄₂)/(1 + Γ̄₁ Γ₂) ∈ U(1) (numerator and denominator are complex conjugates, so the ratio is a unit complex number) and Γ₁ ⊕ Γ₂ = (Γ₁ + Γ₂)/(1 + Γ̄₁ Γ₂). Taking arguments, ψ(Γ₁, Γ₂) = arg[(1 + Γ₁ Γ̄₂)/(1 + Γ̄₁ Γ₂)] = −2 arg(1 + Γ̄₁ Γ₂), (A.2) which is (4.5). The gyration vanishes when Γ₁ and Γ₂ are collinear on the real diameter — confirming strict associativity there as required by A2 — and is generically non-zero. The denominator 1 + Γ̄₁ Γ₂ is non-zero for all Γ₁, Γ₂ ∈ ￿ because 1 + Γ̄₁ Γ₂ = 0 would require |Γ̄₁ Γ₂| = 1, but |Γ̄₁ Γ₂| = |Γ₁| |Γ₂| < 1 in the open disk. A.4 Gauss–Bonnet derivation of the area identity We derive (4.6): |ψ(Γ₁, Γ₂)| equals the hyperbolic area of the geodesic triangle △ with vertices 0, Γ₁, Γ₁ ⊕ Γ₂. The Gauss–Bonnet theorem for a geodesic triangle in a Riemannian 2-manifold of constant Gaussian curvature K reads (α + β + γ) − π = K · Area(△), where α, β, γ are the interior angles at the triangle's vertices. For the Poincaré disk metric (3.1), K = −1, so Area(△) = π − (α + β + γ): the area equals the angular defect of the triangle. On the other hand, the gyration gyr[Γ₂, Γ₁] is the holonomy of parallel transport around the closed geodesic loop 0 → Γ₁ → Γ₁ ⊕ Γ₂ → 0 (each arrow being a hyperbolic geodesic segment). The general theorem relating holonomy to curvature in two dimensions [24, Ch. I §13] states that the holonomy angle around a geodesic loop in a Riemannian 2-manifold equals the integral of the curvature over the enclosed region,

radial compositions, each realised by a smooth rapidity. Uniqueness within Aczél is not forced by symmetry — there is no analogue of U(1) or SO(3) on a line segment — and an additional non-symmetric input is required. In the pharmacological dose-response setting, this input is a mass-action rationality hypothesis: the dose-response curve is required to be a rational function of dose, which together with the boundary structure picks out exactly five flows in the PSL(2, ℝ) projective classification (Hill, Loewe, Bliss, inverse- odds, logit). The 1D case is therefore not a symmetry-rigid analogue of the 2D and 3D cases; it is an empirically-anchored projective shadow of the family, with σ-involution σ : e ↔ 1 − e.

6.7 Empirical anchors The bilinear form (2.2) and the bound |Γ| ≤ 1 are built into the standard S-parameter formalism, so the framework's empirical content lies in the kinematic predictions that follow once the per-element physical rapidity is supplied. The closed-form Bragg reflectance R_N = tanh²(N ln(n_H/n_L) + ½ ln(n_s/n_0)) (Appendix B) inverts immediately to a design rule for target reflectance R_target with a symmetric substrate: N ≥ artanh(√R_target) / ln(n_H/n_L). For n_H/n_L = 1.5 and N = 30, the lossless prediction is R₃₀ ≈ 1 − 1.1 × 10⁻¹⁰; the high-N asymptote 1 − R_N ≈ 4 exp(−2N ln(n_H/n_L)) gives the exponential scaling in closed form, with no N-fold matrix multiplication required. Cavity-ringdown measurements on high-reflectance dielectric coatings reach 1 − R ≈ 1.6 × 10⁻⁶ [18], consistent with the lossless ceiling once residual absorption and scatter (outside the lossless axioms) are accounted for; the gap between measured and lossless-ideal reflectance is the loss