An Axiomatic Model IV: Axiomatic Identification of the Koide Parameters and the Scale Tier - the Amplitude 2 and the Phase Angle 2/9

Four axioms (orthogonality, atomicity, minimal cost, causality preservation) and one proposition define a discrete computational system. The present paper is the fourth in the Axiomatic Model series and proceeds on the same axiom set as Papers I, II, and III. The complete axiomatic system comprises 15 axioms; the present paper additionally uses its number registry (complete descriptive degrees of freedom), its two reading rules, and the separation of system time and domain time. The paper treats the Koide formula Q = (m_e + m_mu + m_tau) / (sqrt(m_e) + sqrt(m_mu) + sqrt(m_tau))^2 = 2/3. The Z3-symmetric parametrization of this relation was introduced by Koide, the parameter values k_L = 1 (amplitude r^2 = 2) and delta_L = 2/9 (phase angle) were observed by Brannen and Rosen, and the formalism was organized by Zenczykowski. The present paper presents no new values. The paper makes two claims. First, the two parameter values are identified with registered numbers of the axiomatic system: the amplitude r^2 = 2 is the two reversible axes, the phase angle theta = 2/9 is the dimensionless ratio of the reversible-axis 2 to the structural complete descriptive degrees of freedom 9, and since reversible axes draw no arc, no pi conversion can enter. Second, and as the central result, the observed fact that the Koide relation holds only at the pole masses and breaks at high energy scales is shown to be a prediction of the axiomatic system: mass is the juim that CAS Swap records into DATA, hence a domain-time-tier quantity, and the domain-time tier includes thresholds, corresponding to the pole-mass definition. As a candidate for the departure size the paper presents Q(mu) - 2/3 = alpha(mu)/(2 pi) in the threshold-free scheme, checked across eight scales against PDG 2024 anchor values with one-loop threshold running; the sign of the residual flips at the electroweak threshold, fixing the validity range at and below the electroweak scale. Since Paper I assigned the coupling constant sin^2 theta_W to the system-time tier and thereby explained its agreement with the threshold-free scheme, the present paper reinforces the opposite end of the same rule. As an algebraic corollary, k_L^2(M_Z) = 1 + 3 alpha(M_Z)/(2 pi), and the converted departure k_L +0.19% agrees with the deviation Zenczykowski described. The Z3 parametrization itself is the discrete Fourier transform of three real numbers, holds identically for any three positive numbers, and carries no physical input. The registered-number identifications carry postdiction status. The measured Q, propagating the tau-mass uncertainty, is 0.43 sigma from 2/3 and hence indistinguishable from exactly 2/3; the deviation coefficient is indiscriminable in both value and sign at current precision, and current data prefer the no-deviation hypothesis. The derivation yields four unique predictions with explicit falsification conditions: the scale tier itself, the departure size alpha(mu)/(2 pi), the tau mass 1776.985 MeV, and the neutrino mass ratio m_3/m_2 = 5.6634 under the m_1 = 0 premise. Paper I obtained the electromagnetic coupling constant from the signature (5,2), Paper II the metric of gravitational quantization, and Paper III the structural constants of black hole thermodynamics. The present paper shows that the reversible-axis 2 of the same (5,2) appears in both the amplitude and the phase angle of the generation structure, and confirms, as a second case, that the separation of system time and domain time assigns to each physical quantity the scale at which its relation holds.