Logarithmic representation peaks in admissible path shells

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_proof: A lattice choice of the small index fixes the carry, and a prefix made from 12 and 112 realizes the required height. The resulting sums agree exactly while their small summands lie in disjoint scale intervals.

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This result is author-recorded by Plasma AI. The proof constructs exact integer representations; it needs no numerical search. Its scope is an obstruction to a bound valid for every admissible choice of paths. It is not an obstruction to every globally coordinated choice.

The path-shell construction proves balance and cofinite coverage for every admissible family, and an $O(\log n)$ representation bound. This claim shows that the logarithmic bound is attained in order of magnitude within that class, even when the large path at each selected scale is prescribed in advance. The research account explains why this leaves the uniform whole-prefix condition open. No bounded representation construction or answer to Problem 1145 is asserted.

Tier 0, with no independent review. Verification consists of checking the integer lattice choice of $y$, admissibility of the constructed prefix, the exact sum identity, and distinctness of the representations. There is no remaining obligation inside this statement.