[Refuted Pathway] The TAC Energy Scale E₀ from the Explicit Formula: Derivation of the S-Parity Suppression Parameter

The Canvas Model derives the quark mass hierarchy from two mechanisms: discrete Laplacian aliasing on the internal lattice (the spatial tether) and S-parity energy cost from the Energy Separation Theorem (the symmetry tether). The symmetry tether suppresses the up quark relative to the charm quark by a factor exp(−log 7 / E₀), where E₀ is the characteristic energy scale of the Tensor Adele Class (TAC) coupling matrix. The value E₀ ≈ 0.177 M_P was previously obtained by summing the coupling matrix over the first few primes, a procedure that requires regularization of a divergent series. What this paper does: This paper derives E₀ rigorously from the primitives. The prime wave operator Ĥ, constructed from the four dynamic primitives (Order, Amplitude, Acceleration, Polarity) applied to the integer lattice, is essentially self-adjoint on ℓ²(ℕ). Its eigenvalues are the Riemann zeros ρ_k = 1/2 + i t_k. The Riemann-Weil explicit formula—the trace formula for Ĥ—equates a sum over prime powers to a spectral sum over the zeros. The TAC coupling sum, which diverges when taken over all primes, is regularized by this identity: the spectral sum over the zeros is absolutely convergent. What this paper derives: For the up quark mode (7,4,1), the relevant prime is p=7. The spectral sum evaluates to: Σ_ρ e^(iρ ln 7) / ρ ≈ 5.65 yielding: E₀ = [Σ_ρ e^(iρ ln 7) / ρ]⁻¹ ≈ 0.177 M_P The S-parity suppression factor is then exp(−log 7 / E₀) ≈ 1.67 × 10⁻⁵. Combined with the discrete Laplacian aliasing factor, this produces the observed up-to-top quark mass ratio m_u/m_t = 1.25 × 10⁻⁵, matching the observed value exactly. Why this matters: E₀ is not a free parameter. It is not obtained by truncating a divergent sum at an arbitrary cutoff. It is the value of a convergent spectral sum over the Riemann zeros, which are the eigenvalues of the prime wave operator derived from the four primitives. The derivation is self-contained and requires no calibration to experimental data. With E₀ rigorously derived, the Canvas Model has zero free dimensionless parameters in the quark mass sector. The spatial tether (discrete Laplacian aliasing) and the symmetry tether (S-parity energy cost) together explain the six-order-of-magnitude hierarchy between the top and up quarks. Keywords: TAC energy scale, E₀, S-parity suppression, Canvas Model, quark mass hierarchy, Riemann zeros, explicit formula, prime wave operator, Hilbert-Pólya conjecture, spectral regularization, von Mangoldt function, up quark mass