Finite Sidon extremal error

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degree_four_extension: The degree-four Sidon relaxation, a rigorous obstruction to zeroing and uniformly renormalizing the block law, the need to break cyclic symmetry, and the remaining nonuniform-correction problem.

evidence/: Exact finite checks of degree-two moments, lattice capacities, Singer arc discrepancy, and the genuine block law's short-count covariance, with separately scoped exploratory computations.

linear_capacity_barrier: A genuine fixed-size block-sampling law satisfies every Sidon capacity in expectation up to sqrt(N)+N^(1/4)/2-O(1), obstructing single linear-capacity certificates below exponent 1/4; adaptive and nonlinear arguments remain outside this conclusion.

local_repair_gap: A progression family defeats the proposed small off-centre error estimate, and forbidden triples force singularity of the square repair matrix; a compatible global linear completion exists, while local and matrix-positive repairs remain open.

parity_obstruction: The parity identity obstructs zeroing block/uniform moments and sets a necessary budget for repairs; a global signed extension can restore this square, while full matrix positivity remains open.

perfect_difference_cuts: Every arc count of a perfect difference set modulo q^2+q+1 lies within sqrt(q)(1+ln(M/2)) of its mean, so cuts of moderate order reach sqrt(N)-O(N^eps) only from an order within O(N^(1/4) log N) of sqrt(N).

proof_sketch: Editable argument for Problem 30: separate upper and lower obligations, an explicit probability-measure obstruction to linear capacities, an exact failed quartic extension, sufficient variance inputs, and the remaining nonuniform moment and construction gaps.

short_count_covariance: A direct edge-intersection calculation gives the short-count covariance as a low-rank polynomial plus two absolute-value kernels, exposing a negative cumulative-sum form relevant to local moment repairs.

signed_linear_completion: The complete author-recorded linear extension theorem separates Sidon zero and cardinality consistency from matrix positivity; global right inverses and a parity trade leave local and spectral control open.

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This folder holds the sustained attack on Problem 30, whether the largest Sidon set in an interval of length $N$ has size $\sqrt N+O_\epsilon(N^\epsilon)$. The proof sketch is the entry point and records the two obligations, the held links and the gaps. Its results not drawn from a cited source are Plasma AI's, author-recorded and not independently reviewed; they settle neither half of the problem. The earlier scope comparison is in the finite extremality account, which separates the error exponent from coefficient improvements.

The precise original results are L216, the probability-measure barrier, L217, the failure of zeroing followed by uniform moment renormalization, L273, the parity obstruction for even block lengths and arbitrary block/uniform mixtures, and L274, signed linear completion with a parity-square refinement. Each has a complete author-recorded proof at tier 0. L217 has an exact finite negative-square witness; L273 has exhaustive rational controls for its identity and moment calculation. The parity account gives a necessary budget for further repairs. The local-repair analysis identifies the exact compatibility constraints and failures of the proposed local kernel construction. L274 gives a global signed solution of the linear constraints; full matrix positivity remains open. The exact short-count covariance provides a separate calculation for that spectral question.