Exponential sunflower bounds for affine linear words

../

_proof: Maximize the deficit between the desired dimension and the span of coordinate dual spaces, contract that span, and apply linear Hall selection. Dualizing gives a subspace whose coordinates are zero or injective and whose affine cosets are sunflowers.

---

This argument is recorded by Plasma AI. The proof is self-contained: a linear independent-representative criterion and a maximal dimension-deficit argument construct the required subspace.

The independent-representative criterion is the classical linear version of Hall's condition. No novelty is claimed for that criterion or for the structured sunflower bound without a fuller literature comparison. No external theorem or other project claim is a premise: the needed selection argument is proved in full.

Significance and scope

The coordinate alphabets can have arbitrarily large dimensions, so the base $k^2$ is independent of the alphabet size. This proves the exponential bound for an algebraically structured subclass of the word formulation of Problem 20, the sunflower conjecture. It does not prove the conjecture for arbitrary families. Passing from an arbitrary word family to its affine span can introduce new words and new sunflowers.

The linear-word research note uses cosets of the constructed subspace to obtain an exponential bound for subsets of fixed positive density in an affine linear family. The additive-growth deduction then handles binary word families with bounded fourfold growth. Those extensions retain author standing; their hypotheses do not hold for all word families. The general proof sketch records the missing structural reduction.

Standing and verification

Tier 0, author-recorded. The proof is complete and noncomputational. No independent review or formalization exists. To verify it, check the independent-representative induction, the choice of a proper maximizing coordinate set even when $\dim C=sn$, the dual construction, and the conversion of its subspace to a sunflower. The final field-size split gives the uniform exponential base.

Nothing remains open within this structured statement. The unrestricted sunflower conjecture and any novelty assessment remain outside its conclusion.