Hyperbolic Fixed Points, a Curvature Invariant, and the β=1 Universality Class in Unified Constraint Dynamics

We prove a structural identity within the Unified Constraint Dynamics (UCD) framework: the derivative of the UCD β-function at its critical point κ_g equals a curvature invariant K(κ) of the associated statistical manifold. Since β_UCD(κ_g)=0 and β'_UCD(κ*_g)≠0, a first-order Taylor expansion gives β=1 scaling for the UCD flow. Both results follow from exact identities of the exponential family (Stein identity and cumulant recursion); no approximations are made. For the Jacobson thermodynamic framework, we introduce a horizon structural invariant K_J and derive closed-form values for standard metrics: K_J=−(p+1)/p for attractive-type and K_J=−(q−1)/q for repulsive-type, with common asymptotic limit K_J→−1. We formulate as a conjecture that the shared exponent β=1 across UCD, Paper I (Ricci eigenvalue flow), and the Jacobson horizon reflects hyperbolicity of fixed points on statistical manifolds with non-vanishing curvature. The proof of this conjecture remains an open problem. Epistemic clarification of Verlinde connection: added Conjecture VJ (Verlinde–Jacobson bridge); revised Section 4 entropic dual structure paragraph to distinguish Verlinde's original claims from UCD-derived predictions v4.0: Paper I slope/invariant estimate revised from previous numerical value (0.1026) to 0.037016 after further derivation. Curvature invariant K_I(DV*) now identified with rational-function representation in Fisher-metric components (Remark 7). Geometric characterisation of DV* added (Remark 8). Open Problem 1 restated as analytic proof of F'(DV*) = K_I(DV*). Internal consistency of Paper I status corrected throughout. v5.0: Added Proposition 3 (F'(DV*) = K_I(DV*)): analytic proof of the Paper I slope via block structure of the Ricci endomorphism. Added Open Problem 6 (analytic sign of even-block eigenvalues). v6.0: Fixed Introduction theorem reference (Thm 1+2 combination). Removed internal session label from Open Problem 1. Softened Paper I status in comparison table to "partial". Adjusted Section 4.1 wording for consistency with Open Problem 6. v7.0: Adds a 2D Cayley-Hamilton identity for the Fisher-Rao tensor (exact), numerical lemmas establishing K_even(DV*) = -0.265714 < 0 and K_even -> -1/2 as DV -> infinity, and resolves all epistemic inconsistencies identified in cross-validation (MAGI). Title updated to reflect that the beta=1 universality class is conjectured. v8.0: Claim A and Claim B analytically proved. Lemma (Claim A) upgraded to Proposition: all four asymptotic limits and their ratio derived from the Laplace expansion of the partition function. Remark (Claim B) upgraded to Proposition: kappa_3/G_22 -> 6 proved via Jacobi's formula kappa_3 = d/d(theta^3) log det G_even combined with det G_even = 1/(4|theta^3|^3) + O(1/DV). Convergence rate O(1/DV) established analytically in both cases. Open Problems 6(b) and 6(c) resolved; Open Problem 6(a) (sign of K_even at finite DV*) remains. Corrected: family is on R (not finite state space) per Paper I. v9.0: Adds numerical confirmation that kappa_3 = -d/dt log det G_even > 0 for all DV in [0, 20] (dense 200-point scan, Stein-identity verified to machine precision; minimum approx 0.092 at DV = 20). Identifies the analytic core of Open Problem 6 as the moment inequality G_33^2 + G_22 * C_48 > 2 * G_23 * C_46, and shows via Stein substitution that its s^2-coefficient is unconditionally positive, establishing the result for large DV. Minor expository improvements to the Conclusion. v10.0: Extends Open Problem 2 (Jacobson mechanism) with partial progress and an explicit obstruction. Defines a candidate Fisher metric G_J(r) := exp(-integral from r_h to r of g_tt dr') satisfying a Stein-identity analogue dG_J/dr = -G_J * g_tt and attaining a strict maximum at r_h (Lemma (L1)-(L3)). Shows numerically for a 1+1-dimensional free scalar field (lattice N=150) that the quantum Fisher information of the interval-reduced Minkowski vacuum state is monotonically increasing in r - r_h, contrary to G_J, thereby ruling out this natural candidate and sharpening Open Problem 2 to the identification of the correct quantum state family. v11.0: revises Section 4 (Thermodynamic Correspondence) of v10:- Resolves a sign conflict between the Clausius paragraph and the entropic dual paragraph by unifying S_UCD := log μ_2 as the sole entropy quantity (the auxiliary S_dual has been removed).- The Verlinde correspondence is now expressed via the extremum structure of S_UCD (local maximum at κ*_g) rather than through a gradient-form correspondence.- Table 1 has been updated for sign consistency, and a temperature footnote has been added clarifying the two distinct temperature scales available in UCD: T_standard = 1/κ (first-order structure of log Z) and T_UCD = 1/μ_2 (second-order, information-geometric structure).- All theorems, definitions, and conjectures are unchanged.Paper I is cited as manuscript ID RIMA-D-26-00537, currently under review at Results in Mathematics. No DOI is available for Paper I at the time of this preprint version.