A Complete Proof of the Riemann Hypothesis within the Constraint Network Framework: Densification-Sparseification Contest and Five-Point Full Coverage

The Riemann Hypothesis asserts that all non-trivial zeros of the Riemann zeta function lie on the critical line σ = 1/2 in the complex plane, and that all trivial zeros are strictly the negative even integers. This conjecture, proposed over 160 years ago, is one of the most important unsolved problems in the history of mathematics. This paper presents a complete proof of the Riemann Hypothesis within the axiom system of Constraint Network Dynamics. The Constraint Network Model is defined by five axioms: the ontology of the universe is energy, energy always moves at the speed of light, encounters result in collision, the constraint network limits the free path of energy, and the total energy of the entire system is strictly conserved. Through the rigorous proof of five sub-propositions, this paper fully covers the entire content of the Riemann Hypothesis: Sub-proposition 1 provides an explicit construction function from the constraint network density to the complex variable of the zeta function, rigorously proving a one-to-one correspondence between the non-trivial zeros of the zeta function and the eigenvalues of the Hilbert-Pólya Hermitian operator H; Sub-proposition 2 proves that all non-trivial zeros satisfy Re(s) = 1/2; Sub-proposition 3, starting from the densification-sparseification contest equation, rigorously proves that the density set is continuously complete and fully covers the critical strip, ruling out any stray zeros; Sub-proposition 4 proves that all trivial zeros of the zeta function are strictly the negative even integers; Sub-proposition 5 proves that the zeta function is self-consistently analytically continued to the entire complex plane and that its functional equation holds globally. Together with the physical symmetry proof in [1] and the rigorous operator construction in [2], this paper forms a trilogy for the proof of the Riemann Hypothesis within the Constraint Network framework.