The Hodge Conjecture has remained open for 75 years with zero substantial progress for $p \geq 2$. We argue this is not due to technical difficulty, but to a fundamental flaw in the formulation: classical Hodge theory assumes a $C^\infty$-smooth Riemannian metric as an axiomatic starting point, without physical or geometric justification. We prove that any metric constructed from physically realizable coherence relations---the only meaningful construction in Coherent Mathematics (CoMath)---\emph{cannot} be $C^\infty$-smooth, because coherence exhibits universal phase transitions (percolation thresholds, ferromagnetic ordering, hierarchical resonance forcing). The classical Hodge decomposition via $\Delta = dd^* + d^*d$ breaks down at metric singularities. Only algebraic cycles, represented by positive currents independent of the metric, survive these singularities. Therefore, purported ``non-algebraic Hodge classes'' are artifacts of an unjustified smoothness assumption---they do not exist in any physically realizable geometry. This resolves the conjecture by eliminating the false premise rather than proving it within classical frameworks. When you eliminate the impossible, whatever remains must be the truth.