= Half-size prime cyclic Freiman model
{title2=$|A'|\geq|A|/2,\quad p>4K^4|A|$}
For a nonempty <integer> set with <doubling constant> at most $K$, <Plünnecke-Ruzsa inequality> bounds $|2A-2A|$ by $K^4|A|$. Embed the set modulo a sufficiently large auxiliary prime $q$, randomly dilate it, and select the more populated of two half-intervals of representatives. Differences of two pair sums on that half have absolute value less than $q$, so the lift preserves original relations. Every nonzero original difference is uniformly dilated among nonzero residues; avoiding all residues represented by nonzero multiples of $p$ of absolute value less than $q$ has positive <probability> when $p>4|2A-2A|$. Reduction modulo $p$ then yields a <Freiman 2-isomorphism> on the selected half.
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