The Dirac spectral determinant on the Poincaré homology sphere: a topological bridge from the scalar IKKT correction

We compute the ρ2-projected Dirac spectral determinant on the Poincaré homologysphere P3 = S3/2I∗ and establish a topological relationship between the Dirac and scalarIKKT corrections via the eigenvalue relation |D|2 = Λ + 1 connecting the Dirac and scalarspectra on P3. The main results are:(i) The eigenvalue relation (k+1)2 = k(k+2) + 1 on P3 implies that the IKKT Hessian correction(factor 2/3 on scalar eigenvalues) transfers to the Dirac sector with an asymptoticallyexact per-mode halving: δD(k)/δΛ(k) → 1/2 as k → ∞. This topological ratio explains thebare spectral gap relation ΔDirac/Δscalar = −0.491 ≈ −1/2 without dynamical input fromthe fermionic sector.(ii) The structural parallel δΛ = log(2/3)/|C2| for the scalar correction extends to δD =log(6/7)/|A5| for the Dirac correction, where 6/7 = j0/(j0+1) is the ClebschGordan ratioat the rst icosahedrally invariant spin j0 = 6 and |A5| = 60. This candidate closes 98.0%of the Dirac gap.(iii) The topological constant c = 1 connects the atomic numbers Z = 83 (bismuth) andZcr = 173 (Unsepttrium, the relativistic Dirac limit) to the spectral geometry of P3 via83 = 2C2(j0)−c and 173 = λ1+5, where λ1 = 168 is the rst P3-eigenvalue and 5 = | Irr(A5)|.(iv) The fth spectral determinant det′ρ5, left open in prior work, admits two candidateidentications: det′ρ5 = π/2 (yielding det′ρ5/ det′ρ4 = 1/2) and det′ρ5 = exp(18ζ(3)/(5π2))(bare match to 0.011%).(v) Five new testable predictions are derived, including a fermionic dark-matter multiplet at∼0.23 GeV from the nρ2(1) = 1 vector mode.(vi) The character table of A5 splits the ve spectral determinants into a pentagonal sector(ρ2, ρ3; targets involving φ) and a spherical sector (ρ4, ρ5; targets involving π). Since theestablished α-coecients c−1 through c3 contain only φ and e, the spherical sector must enterat order ε4, providing a structural selection for the open coecient c4: it must contain π.This undercuts the transcendence economy argument for the pre-registered choice c4 = e2of 1.All computations were performed in Python using mpmath at 50-digit precision.