Bipartite Spectral Action and the Magnitude of 1/G in Entropy-Response Gravity: A Candidate Closure 1/G = M_Pl^2 at the a_2 level from a Bipartite Wightman Cutoff and Pell-Lucas Arithmetic

In Entropy-Response Gravity (ERG), the MOND interpolating function and acceleration scale a₀ arise from a Rindler–de Sitter product vacuum with modular Hamiltonians K_R, K_dS. A companion paper (ERG Paper XII v2, DOI 10.5281/zenodo.20527216) treats the matter spectral triple, in which the silver mean ε = 1+√2 appears as the wedge coupling constant of a three-level bipartite construction with Pell–Lucas eigenvalues. This work treats the complementary gravity spectral triple, on which the magnitude of Newton's constant G is determined.

We derive an explicit cutoff function f(t) = (6/π)/(1+t²)² from the bipartite Wightman two-point function on the Bunch–Davies vacuum, via the Cayley transform of the modular two-point function W_b(τ) ∝ 1/sin⁴(τ/2). The numerator 6 in the prefactor is identified with the second Pell–Lucas number A₂ = ε² + ε̄² = 6 from the matter-side three-level construction. Combined with a modular-period cutoff scale Λ = 2π M_Pl, the Chamseddine–Connes spectral action gives the exact closure 1/G = M_Pl² at the a₂ level in non-reduced Planck convention.

We compute the spectral dimension of the matter triple explicitly (d_s = 0 with logarithmic corrections), prove a general identity Tr(D_b²) = ε^(4k-2) · Σ A_{4j-2} for N-level truncations, and document open problems for the full infinite-dimensional extension. The higher-order Seeley–DeWitt coefficients diverge with the bipartite Wightman cutoff, consistent with ERG's framework-level resolution of the cosmological constant problem via H₀ from the Pythagorean identity (4π²-1) a₀² = (cH₀)².