Tovacian Geometric Dimensional Analysis II (Path B): Quaternion Products of Dimensions, Balance by Transmutation, and the Physics of Order

Path B of Tovacian Geometric Dimensional Analysis lets the quaternion coordinates of physical quantities multiply as quaternions, so that order matters. This paper reports three positive results. (1) The minus sign of spacetime comes out of the algebra. Built by Baker's products, the velocity of light is a pure quadrantal versor, c = −k, so c² = −1 for any choice of length and time axes; the interval x² − c²t² acquires its sign without Minkowski's hand-inserted ict. (2) Maxwell's system closes under transmutation. Faraday's law and the equality of mixed partial derivatives hold exactly (up to the ±q identification), the vacuum relation με = c⁻² holds as a versor identity, and the two remaining loops, Ampère–Maxwell and continuity, share one angle, arccos ⅓ = 109.47°, the tetrahedral angle. The author's reading of imbalance as transmutation is made exact through the transmutation quotient τ = YX⁻¹, whose tensor and angle are unchanged by any rewriting of a law. (3) Order has a physical home in the protein backbone. The ordered product of backbone rotations, with bond angles from Tovacian stroke orthogonality (109.47°, 120°), gives 3.60 residues per turn for the α-helix (measured 3.6), 2.99 for the 3₁₀ helix (3.0), 2.04 for the β-strand (2.0) and 2.90 for polyproline II (3.0); the β-strand operator is a pure mean reversion. With the author's hydrogen-bond rule (default N–H···O distance 2.90 Å) and a linear i → i + 4 hydrogen bond, the α-helix angles themselves follow, φ = −55°, ψ = −50° (measured −57°, −47°), and the ordered product gives 3.65 residues per turn. Energy, being path-independent (Hess's law), lives in the order-free parts of the algebra; orientation and handedness live in the vector part.