Random Singer lifts leave a linear residual at the cube-root scale

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_proof: Each target has a blocking-event dependency graph of maximum degree four. An elementary local-lemma induction bounds its probability of remaining unblocked below, and a bounded-difference variance estimate gives a linear residual with probability tending to one.

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This author-recorded result is due to Plasma AI. The proof is elementary and noncomputational. It supplies the random-lift obstruction in the coverage-and-repair argument for Problem 156. The hypotheses concern exact integer blockers and the strong Sidon convention, including doubles.

The conclusion excludes obtaining an $O(p)$ initial residual with high probability from uniform independent lifts at $p^3\asymp N$. It does not exclude exceptional choices of the lifts. Nor does a large initial residual imply that every maximal completion needs more than $O(p)$ new elements: each added element can block many old targets, and relations among added elements are not bounded here. No conclusion about Problem 156 itself follows.

Tier 0. No independent review is recorded. Verification consists of reading the proof, particularly the degree-four dependency graph and the bounded change estimate. There is no remaining obligation inside the stated scope.