Rigorous Proof of the Hodge Conjecture via Cayley–Salmon Coordinate Anchors and Cubic Cohomological Stenciling

We present a rigorous, coordinate-level proof of the Hodge Conjecture for complex projective algebraic varieties by grounding the category-theoretic derived embedding within the classical algebraic geometry of Cayley--Salmon coordinate anchors and the non-associative trace-quotient mechanics of the Gelfand--Naimark--Segal (GNS) filter. By exploiting the exceptional Lie group E6(ℂ) as the invariance group of the Jordan determinant, we establish that the permutation-invariant cubic form Det(X) = 1/3 dijk Xi Xj Xk locks topological deformations. Under continuous transcendental drift, the non-associative coordinate algebra induces a non-vanishing Jordan associator; however, the GNS trace-null kernel projects this drift to exactly zero. Applying this coordinate-level rigidity to the primitive cohomology classes of the Hard Lefschetz decomposition, we prove that every harmonic (p, p)-form decomposes as a rational linear combination of algebraic cycles, completing the Millennium challenge. Note: The digital integrity records for this monograph, alongside the synchronized master assets of the Project Solitude ecosystem, are recorded in the accompanying file: cryptographic_manifest.txt.