Hyperbolic Fixed Points, a Curvature Invariant, and a Conjectured β=1 Universality Class in Unified Constraint Dynamics
Description
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.
Notes (English)
Changes from v12:
- Proposition 4 (slope formula F′(ΔV*) = K_I(ΔV*)) is now unconditional. Its hypothesis concerns the even block of the Ricci endomorphism of the three-parameter family, not the Gaussian curvature of the two-parameter sub-family, and v12 had identified the two. This hypothesis is proved by a computer-assisted argument (Theorem 11): interval Newton enclosure of ΔV* followed by interval arithmetic for the trace and determinant. ΔV* is defined as the unique zero in an explicit interval; global uniqueness is not claimed. K_I(ΔV*) = 0.0370153; the earlier value 0.037016 is corrected.
- New computer-assisted Theorems 12 and 13: K_even < 0 and κ₃ > 0 for all ΔV ≥ 0. The method is a barrier argument for a Riccati equation, using exact resultants, subresultants and Sturm counts, and a rational certificate for Γ(1/4) and Γ(3/4). The new Appendix A summarises the proofs, and the supplementary zip contains the scripts.
- New Lemma 8: large-ΔV asymptotics with rigorous remainders. Remark 4 (K_even → −1/2) and the Laplace-expansion arguments of Propositions 6 and 7 are now complete; in Proposition 6 the ratio is 1 + O(s⁻⁴).
- Corrections in the even sector:
- eq. (12): the claimed minimum 0.092 is replaced by Theorem 13;
- h(ΔV): ΔV³·det G_even, with limit ∞;
- Open Problem 6: rewritten as resolved; the statement "equivalently" is removed, since K_even < 0, κ₃ > 0 and the moment inequality are not equivalent;
- the Stein identity sign, now (2n−1)μ_{2n−2} = 4μ_{2n+2} − 2sμ_{2n};
- the normalisation and score formula in the Remark on K_I;
- Table 2 (0.829 → 0.830) and ΔV* = 3.1041183476.
- UCD side:
- sign of the flow equation, now dκ/dℓ = −μ₂⁻¹dT_c/dℓ;
- α is stated as a definition in Theorem 1;
- κ*_g = 1.563199, with constraint source c₀ = 0.4 and λ(κ*_g) = 0.32581;
- conditions "c non-constant" and "K(κ*_g) > 0 for a maximum" added.
- Gravity side:
- The sign convention is unified: g_tt = f with signature (+,−,−,−), and K_J = f″/f′² is a horizon shape quantity defined in this paper.
- K_J values are corrected; extremal Reissner–Nordström gives +∞ (not −∞).
- Claims are limited to the order of the horizon zero of f in the area-radius representation, with linear coefficient f′(r_h).
- The common mechanism with Jacobson's thermodynamic derivation and Verlinde's entropic gravity is not asserted; it is stated as a conjecture or unconstructed open problem.
- The gradient-flow isomorphism and the numerical quantum-Fisher-information claim are removed.
- verify_gravity.py checks the gravity-side values symbolically.