Resolving the Lorentzian Signature Problem in Noncommutative Geometry via an Immersive Jordan–Clifford Bridge

We present a resolution to the Lorentzian signature problem in noncommutative geometry (NCG) by constructing an explicit immersive bridge between the exceptional Jordan algebra \(h_3(\mathbb{O})^\mathbb{C}\) and the associative Clifford algebra on a 14-dimensional complex vector space. The construction yields a finite-dimensional operational subspace \(\mathcal{H} = \mathbb{C}^7_+ \oplus \mathbb{C}^7_- \cong \mathbb{C}^{14}\) on which a Dirac-type operator \(\mathcal{D}\) has finite discrete spectrum:\[\sigma(\mathcal{D}) = \{ +3 \text{ (×1)}, -3 \text{ (×1)}, +\sqrt{2} \text{ (×6)}, -\sqrt{2} \text{ (×6)} \}.\] The real algebra generated by the Clifford blocks \(A = \sigma_z \otimes I_7\) and \(B = \mathcal{D}\) contains a subalgebra isomorphic to the real Clifford algebra \(\mathrm{Cl}(3,1)\) of signature \((+,-,-,-)\). The embedding is algebraically exact and strictly real:\[\gamma^0 = A, \qquad\gamma^i = \frac{1}{\sqrt{2}} A \hat{B}_i \quad (i=1,2,3),\]where \(\hat{B}_i\) are mutually anticommuting spatial blocks of \(B\) corresponding to the eigenvalue \(3\) of \(X^2\), constructed from a spatial frame decomposition of the 12-dimensional eigenspace. These satisfy the strict Clifford relations:\[\{\gamma^\mu, \gamma^\nu\} = 2\eta^{\mu\nu} I_{14}, \qquad \eta = \mathrm{diag}(+1,-1,-1,-1).\] The minimal polynomial of the difference operator \(X = A - B\),\[m_X(x) = x^4 - 13x^2 + 30 = (x^2 - 10)(x^2 - 3),\]gives the algebraic origin of the spectral splitting: \(10\) and \(3\) correspond respectively to the timelike/curvature and the spacelike dimensions. The construction contains no free parameters and is falsifiable via the condition \(q \le 45\) for the continued fraction expansion of \(\alpha^{-1}\), where \(45 = \dim(\text{adj } SO(10))\) is the bond dimension \(d_B\). This work extends the framework initiated by Boyle's identification of the Standard Model gauge group from \(h_3(\mathbb{O})^\mathbb{C}\) and the subsequent Jordan--Clifford immersive bridge, connecting to the unified spectral framework of circular tensor networks on Fano moduli space.