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 \(h_3(O)^C\) and the associative Clifford algebra on a 14-dimensional complex vector space. The construction yields a finite-dimensional operational subspace \(H = C^7_+ ⊕ C^7_- C¹⁴\) on which a Dirac-type operator \(D\) has finite discrete spectrum:\[σ(D) = \{ +3 (×1), -3 (×1), +√2 (×6), -√2 (×6) \}.\] The real algebra generated by the Clifford blocks \(A = σ_z ⊗ I_7\) and \(B = D\) contains a subalgebra isomorphic to the real Clifford algebra \(Cl(3,1)\) of signature \((+,-,-,-)\). The embedding is algebraically exact and strictly real:\[γ^0 = A, γ^i = {1}{√2} A B̂_i (i=1,2,3),\]where \(B̂_i\) 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:\[\{γ^μ, γ^ν\} = 2ημν I₁₄, η = diag(+1,-1,-1,-1).\] 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 \(q ≤ 45\) for the continued fraction expansion of \(α⁻¹\), where \(45 = (adj SO(10))\) is the bond dimension \(d_B\). This work extends the framework initiated by Boyle's identification of the Standard Model gauge group from \(h_3(O)^C\) and the subsequent Jordan--Clifford immersive bridge, connecting to the unified spectral framework of circular tensor networks on Fano moduli space.