Feasible pair moments for a minimal Johnson coloring

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_proof: Binomial inversion gives the projection onto the highest-degree Johnson space; its color decomposition supplies the feasible two-point moments and their forced intersection counts.

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The proof is an author-recorded argument by Plasma AI. It constructs formal moments on the entire vertex and color set, so it tests a partition into classes rather than the existence of one design. No independent review or formalization is recorded.

The Steiner-partition research explains its use for Problem 835. The displayed relaxation cannot exclude a $(k+1)$-coloring in the remaining prime regime. Feasible formal moments need not come from a probability distribution on colorings. This claim constructs no coloring and leaves all constraints involving three or more vertices outside its scope.

Verification and remaining obligations

The proof is finite linear algebra and binomial inversion; it consumes no other native claim or external theorem as an established premise. To check it, verify the projection onto the kernel of the downward inclusion matrix, then its decomposition over the color coordinates. There is no computational evidence entry point. No mathematical gap remains in this stated relaxation; obtaining a full partition remains the separate problem in the linked research.