Hyperbolic Fixed Points, a Curvature Invariant, and a Conjectured β=1 Universality Class in Unified Constraint Dynamics
- 1Independent Researcher
Abstract
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. Precisely, β′_UCD(κ*_g) = K(κ*_g), which immediately gives β=1 scaling for the UCD flow when K(κ*_g) ≠ 0. Both results follow from exact identities of the exponential family; no approximations are made.
We further observe that the quartic exponential family of Paper I exhibits analogous β=1 scaling, with a curvature invariant K_I as slope. We conjecture that the UCD and Paper I instances reflect a common geometric mechanism: hyperbolicity of a critical point with non-vanishing curvature invariant, realised as an information-geometric gradient flow.
For static horizons in the area-radius representation, we record only the following. The horizon zero of the squared lapse f is simple (β=1) exactly when f′(r_h) ≠ 0, and its linear coefficient f′(r_h) has magnitude 2κ_J. We also compute a horizon shape quantity K_J = f″(r_h)/f′(r_h)² for several standard families. K_J is not part of the β=1 condition and can vanish (Reissner–Nordström with r₋ = r₊/2). A relation to Jacobson's thermodynamic derivation is not constructed, and whether gravitational horizons belong to the conjectured mechanism is left open. The comparison with Jacobson and Verlinde is recorded as Scaling Ansätze.
For the Paper I instance, we give a computer-assisted proof, using exact rational and outward-rounded interval arithmetic, that the even block of the Ricci endomorphism of the three-parameter quartic family is negative definite at the critical point ΔV*. The slope formula F′(ΔV*) = K_I(ΔV*) (Proposition 4) is therefore unconditional. By the same methods we prove, for all ΔV ≥ 0, that the Gaussian curvature K_even of the two-parameter even sub-family is negative and that κ₃ = −∂_t log det G_even > 0, which implies a moment inequality. Large-ΔV asymptotics are proved with explicit remainders. The proof of the conjecture remains open.
Supplementary material: the verification scripts (Python 3.11, SymPy 1.13.1, mpmath 1.3.0), their logs and a proof note.
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BibTeX
@misc{tanaka2026hyperbolic,
title = {{Hyperbolic Fixed Points, a Curvature Invariant, and a Conjectured β=1 Universality Class in Unified Constraint Dynamics}},
author = {Yusuke Tanaka},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23642476},
url = {https://zenodo.org/records/23642476},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hyperbolic Fixed Points, a Curvature Invariant, and a Conjectured β=1 Universality Class in Unified Constraint Dynamics
AU - Yusuke Tanaka
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23642476
UR - https://zenodo.org/records/23642476
AB - 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. Precisely, β′_UCD(κ*_g) = K(κ*_g), which immediately gives β=1 scaling for the UCD flow when K(κ*_g) ≠ 0. Both results follow from exact identities of the exponential family; no approximations are made. We further observe that the quartic exponential family of Paper I exhibits analogous β=1 scaling, with a curvature invariant K_I as slope. We conjecture that the UCD and Paper I instances reflect a common geometric mechanism: hyperbolicity of a critical point with non-vanishing curvature invariant, realised as an information-geometric gradient flow. For static horizons in the area-radius representation, we record only the following. The horizon zero of the squared lapse f is simple (β=1) exactly when f′(r_h) ≠ 0, and its linear coefficient f′(r_h) has magnitude 2κ_J. We also compute a horizon shape quantity K_J = f″(r_h)/f′(r_h)² for several standard families. K_J is not part of the β=1 condition and can vanish (Reissner–Nordström with r₋ = r₊/2). A relation to Jacobson's thermodynamic derivation is not constructed, and whether gravitational horizons belong to the conjectured mechanism is left open. The comparison with Jacobson and Verlinde is recorded as Scaling Ansätze. For the Paper I instance, we give a computer-assisted proof, using exact rational and outward-rounded interval arithmetic, that the even block of the Ricci endomorphism of the three-parameter quartic family is negative definite at the critical point ΔV*. The slope formula F′(ΔV*) = K_I(ΔV*) (Proposition 4) is therefore unconditional. By the same methods we prove, for all ΔV ≥ 0, that the Gaussian curvature K_even of the two-parameter even sub-family is negative and that κ₃ = −∂_t log det G_even > 0, which implies a moment inequality. Large-ΔV asymptotics are proved with explicit remainders. The proof of the conjecture remains open. Supplementary material: the verification scripts (Python 3.11, SymPy 1.13.1, mpmath 1.3.0), their logs and a proof note.
ER -