The Bridge: A Multiplicative Completion of the Weil Quadratic Form

We construct a bridge operator satisfying

M = H + iC = QW_S + iC = (d/ds log K_d)(1/2)

unifying the multiplicative and additive descriptions of L'/L(1/2,χ) across all Dirichlet characters simultaneously. Here K_d(s) is the parametrised primorial kernel, H is the Hermitian part (Connes' Weil quadratic form QW_S restricted to the finite set S of primes dividing d), and C is the anti-Hermitian part encoding zero locations. The operator M is diagonal in the character basis with eigenvalues −L'/L(1/2,χ), verified to relative error < 10⁻¹⁴ at d = 30 and to 12-digit accuracy across all 480 characters at d = 2310.

The matrix K_d(1/2) is real but not symmetric — a consequence of the functional equation giving entrywise conjugation, not transposition. This non-symmetry is the source of the entire imaginary channel: a real non-symmetric matrix has complex eigenvalues in conjugate pairs, and these complex eigenvalues carry the zero-dependent information that Connes' real symmetric Weil form cannot access.

The decomposition M = H + iC separates arithmetic from analytic content. The real part h_χ = −2Re[L'/L(1/2,χ)] depends only on the conductor and parity of χ, not on zeros (proved via the functional equation). The imaginary part c_χ = −2Im[L'/L(1/2,χ)] encodes zero locations through the Hadamard product. The deficiency δ_χ := c_χ^max − c_χ ≥ 0 vanishes if and only if all zeros of L(s,χ) lie on the critical line; GRH is equivalent to δ_χ = 0 for all primitive characters.

The central new result is a computable formula for the total analytic energy: Σ_χ c_χ² = Σ_χ h_χ² − 4Tr(M²), where both terms on the right are computable from K_d alone without knowledge of zero locations. To our knowledge, this identity does not appear in the existing literature. It establishes that a zero-dependent aggregate — the total squared imaginary content of L'/L(1/2,χ) across the character family — is an arithmetic invariant.

This identity refines to the per-prime level. At each primorial transition d_k → d_{k+1}, the Euler factor shift Δ_χ(p) = χ(p)log p/(√p − χ(p)) satisfies per-prime H ⊥ C orthogonality: Σ_χ h_χ · Im[Δ_χ(p)] = 0 at every prime individually, so each prime's imaginary injection is invisible to the arithmetic channel. The total analytic energy obeys a computable recurrence: Σ_χ c_χ²(d_{k+1}) = Σ_χ c_χ²(d_k) + 2Σ_χ c_χ · Im[Δ_χ] + Σ_χ Im[Δ_χ]², where the cross-term is the zero-arithmetic correlation and the injection term scales as (φ(d)/2) · log²(p)/p. The per-prime arithmetic projection vectors span the full complex character subspace when sufficiently many primes are included, meaning zero information is fully recoverable from arithmetic projections.

Additional results include: Tr(C) = 0 unconditionally (quadruplet cancellation); H ⊥ C trace orthogonality Σ_χ h_χ c_χ = 0; equipartition C% peaking uniquely at σ = 1/2 (numerical); a Poisson kernel connection at the critical point; and a derivative hierarchy providing computable necessary conditions for GRH at each order. The full moment hierarchy Im[Tr(M^k)] = 0 for all k is algebraically equivalent to M being real (Newton's identities); its non-trivial content is the computable Tr(C²), the per-prime orthogonality, and the energy recurrence described above.

The paper establishes a precise dictionary between the bridge operator and five classical GRH-equivalent criteria: Connes' self-adjoint operator, Li's positivity criterion, the Weil explicit formula, zero-density estimates, and the Selberg moment hierarchy. Each reduces to a statement about deficiency δ_χ. The open problem is a single inequality: proving δ_χ = 0 for all primitive characters, which is equivalent to GRH.