Functional Symmetry, Zero Orbits, and Quantitative Constraints for Off-Axis Zeros of the Riemann Zeta Function

We establish a self-dual entire-function framework for the nontrivial zeros of the Riemann zeta function via the completed xi-function. Three symmetries—the functional equation, complex conjugation, and their composition—generate a four-element group whose orbits are either conjugate pairs on the critical line or off-axis quadruples. Combining this orbit classification with explicitly stated zero-free half-plane inputs and critical-line proportion inputs, we derive position bands, counting bounds, multiplier modulus windows, derivative modulus ratios, residue modulus ratios, density bounds, orbit height pairings, and explicit-formula contributions for any off-axis zero. The system is stated with a strict separation between exact structural identities, fixed external inputs, and conditionally reported inputs. The half-plane input ZF(7/8) is a fixed external hypothesis; every statement using it is marked as relative to ZF(7/8). The stronger input ZF(0.6) is conditional. The unconditional critical- line simple-zero proportion input is CL(2/3), established by Alpöge–Furman with a Lean 4 formalization; the Montgomery–Taylor refinement p = 0.6725 is recorded as conditional. The paper proves a precise and verifiable system of constraints that any off-axis zero must satisfy, and shows that the system is a monotone operator on a partially ordered state space. The zeros are characterized as the characteristic exponents of an infinite-order differential operator, as the poles of a Laplace transform, and as the convergence-abscissa spectrum of a particular solution. A Weil–Bombieri quadratic form is constructed on the orbit space; its orbit decomposition is proved to be a sum of nonnegative contributions, and the zero-side form is proved to be automatically nonnegative, so that the non-trivial positivity content is isolated in the arithmetic side. The Suzuki isomorphism between the Weil completion and a de Branges space is recorded as a conditional equivalence, and the Connes–Consani–Moscovici spectral criterion is stated with its convergence hypothesis isolated. Galois cohomological compatibility and Picard–Vessiot compatibility are proved: neither the four-element group, its orbit structure, nor its first cohomology with trivial coefficients excludes off-axis quadruples; and the Picard–Vessiot group of any finite truncation preserves the generalized eigenspace decomposition and does not constrain the real parts of the exponents. A one-parameter translation group, a scaling group, and a KMS criterion are analyzed; the critical-line zero set is proved to admit no nonempty translation-invariant subset. Differential V -invariants are constructed in the differential closure DCF0 of the field generated by ζ, and they are shown to distinguish critical-line types from off-axis types. Exact critical-point identities of all orders are proved, including the vanishing of all odd-order sums and the absolute convergence of all even-order sums; the reciprocal square identity is combined with the position band and the counting bound to give two-sided bounds for the off-axis and critical-line weighted densities. The constraint system is reduced to five independent components.