Projective selector and five-flow scope

### PDF page 25 axiom (left-translations as automorphisms of the natural Shilov-boundary compactification). The boundary-identity enrichment of Theorem B is specific to the discrete-boundary structure in n = 1; whether higher-rank analogues exist is open. (Q2) Alternative covariance in n ≥ 2. Continuous rotational covariance forbids boundary- identity structures in n ≥ 2 (Proposition 5.1). Weaker covariance — for example, axial covariance (commutation with rotations about a fixed axis) — might admit boundary- identity structures in n = 3 with identity at axis poles. The existence and physical significance of such structures, if any, is open. (Q3) Lossy semigroup extension. The present framework treats lossless operations (isometries of the rank-one hyperbolic geometry). Lossy reciprocal networks act on the disk by holomorphic self-maps that contract the hyperbolic metric — the Schur class of complex analysis and H^∞ control theory [22], with the Schwarz–Pick lemma quantifying the contraction. The lossy generalisation embeds the framework into a semigroup rather than a group structure; the analogous lossy structures in dimensions 1 and 3 are open. (Q4) Non-gyrocommutative bounded composition. Theorem A produces gyrocommutative gyrogroups. Non-gyrocommutative bounded composition (where the magnitude depends on order) is outside A2 but may be appropriate for systems with broken parity or chirality. (Q5) Independence of A6 from A1–A5. Stage 1 of the proof (A1–A5 → radial Aczél backbone) and Stage 2 (A1–A6 → standard formula) are presented as separate layers. Remark 3.1 shows A6 is strictly stronger than A1–A5 (the flat counterexample). Is any part of A5 derivable from A6 + A4? Under A6, left-translations lie in the projective stabilizer, which contains the rotation group as isotropy; some content of A5 may be redundant given A6. 10. Conclusion Bounded compositional kinematics on a rank-one bounded region is organised by two theorems with one structural bridge, all governed by a single first-principles axiom. The unifying axiom is A6 (canonical projective-chart compatibility): each ⊕-left-translation extends to a projective automorphism of the natural projectivization of the bounded region. Across dimensions, the natural projectivization is real ℝℙ¹ for n = 1, complex ℂℙ¹ for n = 2, real ℝℙ³ for n = 3; the projective stabilizers are the rank-one Lorentz/Möbius groups SO⁺(1, 1), PSU(1, 1), SO⁺(3, 1). Theorem A states that under axioms A1–A6, the operation is uniquely the boost composition in the projective stabilizer of X_n: Logit (n = 1), Möbius addition (n = 2), Einstein gyrovector addition (n = 3). The radial rapidity is uniformly ψ(x) = L · artanh(x/L), and the bounded region inherits the rank-one hyperbolic metric of constant negative sectional curvature −1/L². A6 is the decisive axiom: A1–A5 alone admit a flat-pullback counterexample (Remark 3.1) with Euclidean (not hyperbolic) intrinsic geometry, so the ### PDF page 26 hyperbolic uniqueness genuinely follows from A6 rather than from the regularity axioms A1–A5. Theorem B states that in dimension 1, the boundary-identity Möbius semigroup flows on (0, 1) with marked-triple support {0, 1, ∞} are exactly four: Loewe, Inverse-odds, Bliss, Multiplicative. These flows have no analogue in n ≥ 2 because continuous rotational covariance forces interior identity (Proposition 5.1). The five pharmacological flows decompose as 1 interior-identity flow (Logit, the bridge between Theorems A and B) + 4 boundary-identity flows. The σ-involution σ : e ↦ 1 − e fixes Logit, pairs Loewe with Inverse-odds, and pairs Bliss with Multiplicative. The structural slogan summarising the framework is: Aczél linearises the radial coordinate; canonical projective-chart compatibility selects the boost composition in the projective stabilizer; projective embedding into ℝℙ¹ with marked-triple support enriches the one-dimensional case with four boundary-identity Möbius semigroups. The radial Aczél backbone is uniform across dimensions. The interior-identity hyperbolic geometry is the same in every dimension up to choice of homogeneous Lie group. The boundary-identity enrichment is exclusively one-dimensional. Logit is the bridge: simultaneously the n = 1 case of Theorem A under A6, the radial trace of the n ≥ 2 Theorem A operations (Proposition 4.1), and the σ-fixed point of Theorem B’s classification. Under affine reparametrisation x = 2e − 1, Logit becomes one-dimensional relativistic velocity addition, completing the structural unification of pharmacology, microwave engineering, and relativistic kinematics under a single axiomatic framework.

the negative-curvature hyperbolic case. Theorem A is genuinely a theorem about canonically charted bounded compositional structures, and A6 is the substantive axiom. Remark 3.2 (How A6 forces the hyperbolic case). Under A6, each L_𝐚 is by hypothesis an element of the projective stabilizer of X_n (SO⁺(1,1), PSU(1,1), or SO⁺(3,1)). These groups are exactly the orientation-preserving isometry groups of ℍⁿ in the standard projective coordinate models. The family {L_𝐚 : 𝐚 ∈ X_n} acts transitively on X_n (since L_𝐚(0) = 𝐚), so X_n is identified with the homogeneous space of the projective stabilizer modulo the

transitivity. So the entire boundary sphere consists of identity elements, contradicting uniqueness already established. Therefore e_0 ∉ ∂X_n, so e_0 ∈ X_n. ∎ The proposition is structurally significant: in n ≥ 2, the interior-identity hypothesis of A2 is derivable from A1, A3, A4, A5 together with the bare existence of a two-sided identity. The axiomatic count is effectively one lower for n ≥ 2 than for n = 1. In n = 1 the boundary {−L, L} (or {0, 1} in pharmacological coordinates) is discrete and the rotation group is trivial; the proof obstruction does not arise. Boundary identity is

the golden-ratio reciprocal. The maximum gap is g(e) ≈ 0.090, a fixed dimensionless number. The matched diagonal at e ≈ 0.618 maximises the discrepancy between Bliss and Loewe predictions; whether this point is also experimentally optimal for discriminating* between the two models depends on noise and measurement-error structure not addressed by the algebraic classification. The clean closed-form result — the golden-ratio reciprocal as the maximum-gap matched effect — illustrates that the orbit classification has concrete numerical content; the experimental design question is developed in [7].