The Jordan-Clifford Immersion and the Hamilton-Jacobi Flow: A Possible Resolution of the Frozen Time Problem in Loop Quantum Gravity

Abstract

We present a self-contained, algebraic derivation of Hamilton--Jacobi dynamics from the Jordan--Clifford immersion. Working on the finite-dimensional operational subspace \(\mathcal{H} = \mathbb{C}^7_+ \oplus \mathbb{C}^7_- \cong \mathbb{C}^{14}\) associated to the Fano incidence geometry, we construct a refinement flow on the space of graphon kernels \(W\) that couples a Hamiltonian component with a dissipative (entropic) component. The central result is the coupled flow equation:\[\frac{\partial W}{\partial k} = -\{W, S\} - \mathcal{M} \frac{\delta S_{\mathrm{ent}}}{\delta W},\]where the reversible part is generated by a Poisson structure induced by the Clifford grading operator \(A\), and the irreversible part is generated by the gradient of an entropy functional. The dissipative metric is fixed algebraically by the curvature of the Mexican-hat potential at its minimum, determined by the root \(\Delta S = 2\sqrt{2}-\sqrt{7}\) of the characteristic polynomial \(\chi_X(x)=x^4-30x^2+1\). The global scale of dissipation is fixed by the normalized spectral gap \(\lambda = \Lambda_1/\Lambda_0 \approx 1.99765\). The cut-norm convergence of the refinement sequence is accelerated by the structural contraction factor \(\mu = 7/3\), derived from the incidence ratio \(v/k\) of the Fano plane. The construction yields an intrinsic algebraic time parameter \(k\), proving:\[\frac{dS_{\mathrm{ent}}}{dk} \ge 0,\]and replaces the canonical Hamiltonian constraint \(\hat H\Psi=0\) by a coupled reversible-irreversible refinement flow in the emergent continuum limit. All numerical results are reproducible via the provided Python script (DOI: 10.5281/zenodo.21681342).