Fractional covering below two does not imply two dissociated classes

../

_proof: A base-sixteen encoding preserves the ternary relations of thirteen binary vectors; exhaustive subset checks and matching fractional certificates prove extraction ratio 7/13 and covering numbers three and 37/19.

evidence/: Exhaustive integer checks of all subsets and ternary relations, together with matching rational covering certificates, establish the fixed thirteen-point claim.

---

This finite result, recorded by Plasma AI, separates weighted half-extraction from a partition into two dissociated sets. Indeed every nonnegative weighting of $A$ has a dissociated subset carrying at least $19/37>1/2$ of its total weight, but $A$ has no dissociated two-coloring. The proof gives the integer encoding and exact finite certificates; the evidence checks every subset.

The fractional reformulation of Problem 774 requires bounded fractional covering and unbounded integral covering over an infinite family. This example supplies one strict gap across the integer threshold two. It supplies no such family and no resolution of Problem 774.

Standing

Proved at tier 0 by exact finite enumeration and rational certificates. No independent verification or formal proof is claimed. The finite argument uses no other native claim. Its input and checker are retained under evidence, and the proof explains why their finite coverage establishes each assertion.

Roadmap

The finite statement has no remaining mathematical gap. Amplifying the integral covering gap while keeping one extraction constant is open in the covering proof sketch.