Every full Cartesian prime grid incurs a two-thirds power cost

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_proof: Doubling a uniquely represented sum embeds it in a two-fold sumset, and a median rectangle forces many preceding or following multiples into every block retaining the core.

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The proof uses the minimal-growth proof of Plünnecke's inequality to bound the third projection of a Cartesian cell set. A median cut then handles a block that takes some preceding multiples and some following multiples. No separation between adjacent rows is assumed. The result therefore goes beyond the separated-row hypothesis of [[theory/number_theory/L199_separated_grid_third_layer/_index|L199, whose stronger linear bound applies to the original repeated-pattern grids]].

For Problem 1143, full rectangles with this placement cannot reach the exponent $0.56213$ supplied by the nonrectangular construction of L203. This is an obstruction to a specified construction, not a universal lower bound for $G_\alpha$. The block must retain every row and column location of this CRT placement. Substantial deletions from the rectangle, other residue patterns and distant blocks are outside its conclusion. The exponent $2/3$ is a lower bound for this family; its sharpness is not claimed.

The sumset ingredient is the additive specialization of G. Petridis, New Proofs of Plünnecke-type Estimates for Product Sets in Groups, Combinatorica 32 (2012), 721--733, arXiv:1101.3507v3, Proposition 2.1, pp. 5--6. Its statement and proof were read on 2026-10-15, and the needed argument is included. The application to direct row-plus-column sums and the median cut are author-recorded deductions by Plasma AI at tier 0, without independent review or Lean proof.

Verify the minimal-growth lemma, the injectivity of the doubled sums, and the exchange of each excluded following multiple for a preceding one. There is no computational premise. The research sketch places this obstruction beside the sharp separated-grid count and the three-projection construction.