An Axiomatic Model III: Axiomatic Derivation of the Structural Constants of Black Hole Thermodynamics - the 4 in S = A/4, the Evaporation Coefficient 5120, and the Imprint of Absent Time

Four axioms (orthogonality, atomicity, minimal cost, causality preservation) and one proposition define a discrete computational system. Because no free parameters enter the deduction, fitting is excluded; because the axioms close the search space, numerological tuning is blocked. The present paper is the third in the Axiomatic Model series and proceeds on the same axiom set as Paper I and Paper II. The complete axiomatic system comprises 15 axioms; the present paper additionally uses its number registry (complete-description degrees of freedom), its recovery rule, and its juim-recording rule. The paper treats three structural constants of black hole thermodynamics. First, the coefficient 4 of the Bekenstein-Hawking entropy S = A/(4 l_P^2) is the structural-bracket domain count 4, while the 4 of the quasinormal-mode (QNM) asymptotic damping spacing M dw_I = 1/4 is the cost-bracket ball value 4 (= 1 time + 3 space). The two 4s are different quantities registered in different brackets, and coincide numerically only in three spatial dimensions (Theorem 1). A discrimination criterion follows. The ball value varies as 1+d with spatial dimension d, whereas the entropy 4 is dimension-independent, so higher-dimensional holography separates the two readings. Second, in the bit reading (observer, superposition, time, space) = (0,1,0,1) of Paper II Theorem 1, time = 0, so the ball value 4 cannot be procured on that slice. Obtaining the entropy coefficient from the Wheeler-DeWitt equation therefore forces a discrete scale to be injected from outside the equation (Theorem 2). The methodological constraint that Vaz's canonical quantization of the Schwarzschild black hole had to introduce a discrete mass spectrum as an external assumption is a post-hoc concordance of this theorem. Third, the temperature-lifetime identity T_H^3 tau_BH = (10/pi^2) T_P^3 t_P holds, with the mass cancelling exactly as M^-3 x M^3 (an identity by definition), and the evaporation coefficient decomposes as 5120 = 10 x 2^9, where 10 = dim SO(5) is the dimension of the maximal compact subgroup SO(5) of the same signature (5,2) from which Paper I obtained the fine-structure constant, and the 9 of 2^9 is the structural complete-description degrees of freedom 9 = 7+2 (Theorem 3). The event-horizon radius is reproduced as r_s = 2GM/c^2 by stacking N contraction units of 2 l_P, one per juim from its two cost-bearing stages (an identity by definition, Theorem 4), wrapped by the picture of a pairwise-cumulative-cost saturation surface (class III). As auxiliary results, taking the results of Paper II as input, the paper organizes the Barbero-Immirzi gamma = sqrt(3) ln 3 / 8 (a definite deviation of 0.136% from the transcendental-equation root of the literature), the LQG area gap, and the QNM real part, and shows that the well-known mismatch between the LQG area quantum and Hod's area quantum reduces, after cancellation of ln 3, to the single geometric factor 3 pi / 8. The derivation yields 4 unique predictions with explicit falsification conditions. Paper I obtained the electromagnetic coupling constant from (5,2), and Paper II obtained the metric of gravitational quantization from the same (5,2). The present result is a third physical domain of the same sign structure. That the electromagnetic coupling constant, the metric of gravitational quantization, and black hole thermodynamics descend from the same root is the structural conclusion of this series.