Erdős 257: add the half-value membership question and its finite exclusion

Planned contribution to `FormalConjectures/ErdosProblems/257.lean`, stacked on #5290.

1. `erdos_257.variants.half_infinite_support` (research open): is there an infinite `A ⊆ ℕ` with `∑ n ∈ A, 1/(2^n - 1) = 1/2`? A positive answer refutes the universal statement in `erdos_257`.
2. `erdos_257.variants.half_not_finite_support` (research solved, `formal_proof` link): no finite `A` has that value, by the odd reduced denominator of every finite subsum.

The linked corpus reduces a positive answer to one inequality holding for infinitely many rows of an explicit greedy remainder sequence and records its observed frequency; that is numerical evidence only, and the question remains open.

Start in FormalConjectures/ErdosProblems/257.lean and compare the surrounding erdos_257 variants with the work in #5290. Add the named half_infinite_support question and half_not_finite_support exclusion, using the linked formal_proof material for the finite result; done means both declarations are present and the file checks.

Written by the indexing model from the issue text.