Erdős–Straus conjecture: push the verified height for $4/n=1/x+1/y+1/z$, or find a counterexample (Guy UPINT §D11)

For every integer $n\ge 2$, is $4/n$ expressible as a sum of three positive unit fractions, $\tfrac{4}{n}=\tfrac{1}{x}+\tfrac{1}{y}+\tfrac{1}{z}$ with $x,y,z$ positive integers (not necessarily distinct)? Either verify that a representation exists for every $n$ up to a stated height $N$ beyond the current record, or exhibit a specific $n$ that provably admits no such representation.

Acceptance. Provide a search configuration plus a verifier script. ADVANCES: given the current published height $H$ (state it, e.g. $10^{18}$ as of 2025), verify that a representation exists for every integer $n$ in $(H,N]$ for some $N>H$, emitting for each $n$ a triple $(x,y,z)$ with $4/n=1/x+1/y+1/z$ checked by exact integer/rational arithmetic (restricting the certified list to the six hard residue classes mod 840 is acceptable). FULLY RESOLVES: a specific $n$ together with a finite certificate that no positive-integer triple exists (a genuine counterexample), or a proof. Deliver the range/height, the certificate data, and a script that re-verifies every representation.