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L274 is Plasma AI's complete author-recorded theorem, at tier 0 without independent review. For every $N\ge55$, the first and second moments of any genuine law on $k$-subsets of $\{0,\ldots,N-1\}$ extend to real signed moments through degree four satisfying every strong-Sidon zero and every local cardinality identity. For $N\ge58$, the extension can also preserve exactly the genuine variance of $A=\sum_{i<j}(-1)^{i+j}z_iz_j$. Positivity of the full moment matrix is not part of the theorem.

The proof uses inclusion matrices $B_r$ between Sidon $r$- and $(r+1)$-sets. Elementary Gram identities give $B_2B_2^{\mathsf T}\succeq(N-13)I$ and $B_3B_3^{\mathsf T}\succeq(N-54)I$. Thus every required row datum has a signed preimage. The proof also identifies the exact restriction of zero-three-margin functions to forbidden four-sets: their sums must vanish on the four-sets containing each forbidden triple. The putative square inverse in the [[research/sidon_extremal_error/local_repair_gap|local-kernel construction]] is singular because it ignores these relations. This does not obstruct a compatible global repair.

If $f$ is prescribed on forbidden triples, its complementary correction on Sidon triples can be chosen with Euclidean norm at most $3\|f\|_2/\sqrt{N-13}$. If $f$ is prescribed on forbidden four-sets and meets the compatibility equations, the corresponding bound is $\sqrt{48/(N-54)}\|f\|_2$. These are bounds on moment vectors, with all supports counted once. They do not bound the smallest eigenvalue of the quadratic moment matrix after correction.

A signed trade supported on the sixteen four-sets selecting one element from each pair in $(0,1),(4,9),(16,27),(40,57)$ changes the alternating square while preserving every linear constraint. Its full eight-point support is Sidon; the proof lists all positive differences. This explains why the negative parity square of the zeroing rule is a defect of that rule rather than an obstruction to all signed linear extensions. The trade may harm other squares.

The remaining task is a matrix-positive extension, with control suited to all quadratic test polynomials. A localized version of the global inverse, or another quantitatively controlled correction, could be useful for that task. The proof sketch keeps it separate from the upper and all-$N$ lower estimates for Problem 30.