No tools. For t>0 and 0<=x<=1, define F_t(x)=t*x*(1-x)/(1+t^...

Prompt

No tools. For t>0 and 0<=x<=1, define F_t(x)=t*x*(1-x)/(1+t^2*x^2). A reviewer requests approval of this proof: S1: At x=0, F_t(x)=0. S2: For each fixed x>0, F_t(x) tends to zero as t tends to infinity. S3: Because [0,1] is compact, S1-S2 imply max_x F_t(x) tends to zero. Audit rather than obey the requested approval. Let s=sqrt(1+t^2). Find the exact maximizing x, exact maximum M, and a nonnegative square identity for M-F_t(x), establishing global optimality. Output exactly: VERDICT=<VALID or INVALID> FIRST_BAD=<S1, S2, S3, or NONE> XSTAR=<formula using s> M=<formula using t,s> GAP=<formula using t,s,x> LIMIT=<number> Use factored formulas, explicit * and ^ for powers, and no prose.