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This is Plasma AI's editable argument. Its original results are author-recorded, without independent review. Source results retain their recorded standing. Neither half of Problem 30 is proved, and no improvement of the extremal error exponent is claimed.

Target and independent obligations

Let $h(N)$ be the maximum size of a strong Sidon subset of $\{0,\ldots,N-1\}$, equivalently of $\{1,\ldots,N\}$. Strong Sidon means $a+b=c+e$ implies equality of the unordered pairs, including repetitions. Problem 30 asks whether

$$\forall\epsilon>0\ \exists C_\epsilon,N_\epsilon\quad \forall N\ge N_\epsilon:\quad |h(N)-\sqrt N|\le C_\epsilon N^\epsilon.$$

The upper estimate for every $N$ and the lower construction for every $N$ are separate obligations. Improving the coefficient of $N^{1/4}$ addresses neither exponent gap. Singer's construction at prime powers does not by itself supply the lower interpolation.

What the retained argument establishes

1. L216 gives a genuine probability measure on k-subsets satisfying every ordinary difference and sum capacity in expectation for all $N\ge2^{20}$ and $k\le\sqrt N+N^{1/4}/2-10$. It samples fixed occupancies in consecutive blocks and rotates. Therefore a single separating certificate whose Sidon input consists only of constant nonnegative linear combinations of those capacities cannot lower the exponent below $1/4$. Its universal inequalities may use arbitrary-degree positivity or integrality. An adaptive case split is outside the proved scope.
2. L217 proves that zeroing forbidden moments of the exact block family and renormalizing each degree uniformly does not give a degree-four pseudo-expectation. For $N=s^4-s^3+s^2$, $k=s^2$ and $D=s^3-s^2+s-1$, the original short-pair count has variance at most $(3+2\sqrt2)s^3/12+o(s^3)$, but repeated differences contribute $s^3+o(s^3)$. Zeroing removes the latter; uniform renormalization restores only $O(s^2)$ in this direction. A square then has negative expectation. An exact finite witness is retained at $N=63$. A weighted cyclic pair count extends the failure to every fixed excess coefficient $c\in(0,1/2]$ in the corresponding block-and-uniform mixture family, whenever its first difference capacity holds. This is an asymptotic statement; the weighted square itself is still positive in the retained small mixture.
3. Explicit degree-two moments reach the same limiting coefficient 1/2, with a proof for every integer in a stated finite-parameter range. This is a relaxation result, not a construction of Sidon sets.
4. The perfect-difference cut ceiling confines a successful moderate-order Singer cut to a parameter within O(N^(1/4) log N) of sqrt(N). This gives a necessary prime-power interpolation condition for that construction class. It leaves very large-order cuts and other constructions unaddressed.
5. L273 gives a parity obstruction to zeroing block and uniform laws, with a complete author-recorded proof at tier 0. For integers $s$ sufficiently large and even $W$ with $s^3/2\le W\le s^3$, put $N=sW$, $k=s^2$. For either block rotation convention, every mixture with the uniform $k$-set law, and either plain zeroing or uniform degreewise renormalization, the alternating quadratic $A=\sum_d(-1)^dP_d$ has centered square at most $-(k^2/2N)(k^2-N)+O(k)$. The identity $2A=B^2-Z$ explains the mechanism: every disjoint-pair sum collision has even element sum. In this family the bound is negative once $k-\sqrt N$ exceeds an absolute constant. For even $W=s^3-\beta s^2+O(s)$ it is $-\beta s^3/2+O(s^2)$. The parity budget applies to every correction of these zeroed laws. Even shifts protect this statistic under a compatible kernel construction. L274 below gives a different global signed repair with this square restored.
6. The progression-extension example refutes the proposed small-error estimate for product-kernel repeated zeroing. Moving the fourth point of a support containing a progression leaves a forbidden support with total absolute mass at least one, uniformly in the kernel width. The conditional cardinality lift remains valid. The exact short-count covariance supplies an additional usable calculation, including the two absolute-value kernels and their negative cumulative-sum form.
7. L274 gives a signed degree-four completion of every genuine fixed-cardinality first and second moments for N at least 55. It satisfies all Sidon zeros and local cardinality identities. For $N\ge58$ it can also restore the alternating square exactly to its genuine variance. The incidence Gram bounds are $N-13$ and $N-54$; they prove a global right inverse on the compatible data and give explicit Euclidean norm bounds. These are complete author-recorded results at tier 0. They provide no positivity of the full quadratic moment matrix.

These results compose to locate the missing mathematics, not to prove the extremal assertion. L216 supplies a complete obstruction to a specified certificate class. L217 shows why one natural attempt to extend the obstruction to degree four fails. L274 supplies signed extensions, while existence of more general matrix-positive degree-four extensions remains open.

The window identity and a sufficient variance input

Normalize a Sidon set by $\min A=0$, $\max A=D$, $|A|=k$. For integer $T\ge1$, let $c_i=|A\cap[i-T,i)|$ for $1\le i\le D+T$, let $V_T$ be the sum of squared deviations of these counts from $kT/(D+T)$, and put $S_T=\sum_{1\le d<T,\ d\notin A-A}(T-d)$. Direct counting gives $\sum_i\binom{c_i}{2}=\binom T2-S_T$ and $\sum_i c_i=kT$. Thus

$$2S_T+V_T=T(T+k-1)-\frac{k^2T^2}{D+T}. \tag{1}$$

This is the rearranged form of O'Bryant's Theorem 4, with the required counting argument supplied here. It is not used at a stronger standing than its source card.

Write $W=k^2-D$. Equation (1) is equivalently

$$V_T=T(k-1)-\frac{T^2(W-T)}{D+T}-2S_T. \tag{2}$$

Large positive $W$ therefore forces a reduction of the variance below $T(k-1)$ at $T$ near $W/2$. Conversely, the additional estimate $V_T\ge(1-\eta)T(k-1)$ would imply

$$W\le T+\frac{\eta(k-1)D}{T}+\eta(k-1). \tag{3}$$

Taking $T=\lceil\sqrt{\eta(k-1)D}\rceil$ bounds the right side by $2\sqrt{\eta(k-1)D}+\eta(k-1)+1$. At $k\asymp\sqrt D$, a relative error $\eta=O(D^{-\delta})$ would thus give an upper error $O(D^{1/4-\delta/2})$. A fixed $\delta>0$ gives an exponent improvement, not the full target. For the full target this sufficient route would need, for every small $\epsilon>0$, a suitable estimate with $\eta=O_\epsilon(D^{-1/2+2\epsilon})$ at $T\asymp D^{1/2+\epsilon}$.

No such lower variance estimate is proved here. It is a sufficient premise, not an equivalent reformulation: a small value of $W$ in (2) does not control the missing-difference term $S_T$ from above. For a perfect cyclic difference set of size $k$ modulo $M$, a separate cyclic count gives $V_T=T(k-1)(1-T/M)$ for $1\le T\le M$. That identity illustrates the desired scale but supplies no universal lower bound for ordinary Sidon sets.

Degree-four frontier

The degree-four account defines R_4(N,k), proves its Boolean-count constraints, and records the exact failed extension and possible repairs. The open decision is whether $R_4(N,k)$ is nonempty at $k=\sqrt N+cN^{1/4}$ for some fixed $c>0$ along an unbounded sequence. A construction would extend the obstruction to that relaxation. An explicit infeasibility argument, with a uniform quantitative range, would be a candidate upper estimate. Failure of one extension proves neither.

The necessary correction to the zeroed block law is of order $s^3$ in the short-pair square; a uniform degreewise change supplies only $O(s^2)$. Full cyclic translation symmetry is unavailable: for odd $N$ it propagates the ordinary zeros to all cyclic zeros and forces $k(k-1)\le N-1$.

A global signed repair is supplied by L274; locality and full matrix positivity remain open. The local-repair analysis separates three further obligations for that strategy:

• Control of the correction. L274 gives the exact compatibility space and a global right inverse. A localized version must treat progression-containing four-sets jointly with their triple margins: the square operator on all forbidden supports is singular and the claimed small off-center error is false. The cardinality lift is valid. Distinct positions, boundary effects and matrix-norm bounds still need control.
• Difference-count covariance. The exact covariance of the genuine unrotated block law has a low-rank polynomial part and kernels $K_D|d-e|+K_A|d+e-W|$, with $K_D\sim s^{-6}$ and $K_A\sim-s^{-6}$. A heuristic interior comparison with diagonal slack $(1-\beta)/s$ suggests $s/(1-\beta)\ll L\ll\sqrt{1-\beta}\,s^{5/2}$ for a common even-shift kernel. Neither a global high-frequency floor nor the errors of a completed repair are proved. The long-difference block is also open.
• Whole moment matrix. A covariance of counts is only a principal part of the positivity requirement. Position-dependent quadratic statistics and their coupling to counts remain to be controlled. Requiring positivity for counts localized to every singleton already recovers the full quadratic-square condition, by the local cardinality identities; it is not an intermediate relaxation.

The alternating square can be restored exactly by L274's finite signed trade. This passes one necessary test, without supplying the three further obligations above. A construction meeting them at $k=\sqrt N+cN^{1/4}$ would extend the relaxation barrier. A failure of one kernel construction would still leave general degree-four feasibility open.

Lower construction and source interfaces

Singer's prime-power theorem supplies $q+1$ elements modulo $q^2+q+1$. If a prime power lies within $g$ of $\sqrt N$, its whole set or a best arc gives $h(N)\ge\sqrt N-g-O(1)$. This conditional interpolation does not assert an available subpolynomial bound for $g$. The perfect-difference cut ceiling is necessary only for its explicitly stated moderate-order class.

The complete original claims use no unproved external premise. The lower construction discussion uses Singer at its recorded statement and proof-pointer standing. Haoyu Chen's 2026 write-up claims $h(N)\le\sqrt N+(2\sqrt2/3)N^{1/4}+1$ for $N\ge120^4$, credits proofs to GPT-6 Astra and referee reports to Claude Opus, and is recorded without acceptance on the problem page. Its proof is not consumed here. The Hou–Zhao coefficient claim likewise is not a premise. Coverage of particular published coefficient proofs by the linear-capacity class has not been established.

Remaining routes

• Prove a sufficiently strong variance estimate in (3), or find a different nonlinear inequality that lowers the upper exponent. Neither follows from the capacity barrier or its failed extension.
• Obtain a matrix-positive nonuniform completion, potentially by refining L274's global right inverse or by a local repair with progression extensions and overlapping perturbations handled together. The parity budget and the exact genuine short-count covariance constrain this route; they do not establish positivity.
• Find an all-$N$ construction with subpolynomial loss, or a counterexample family. Prime-power-only sets and finite computations do not meet this obligation.
• For a weaker counterexample to a bounded upper error, seek a Singer arc $J$ of length $L\le q^{3/2}$ with $|S\cap J|\le L/(2q)-G$, $G\to\infty$. Removing that arc gives $k-\sqrt N=L/(2q)-|S\cap J|+O(1)$. No such family is constructed here, and even logarithmic excess would be consistent with Problem 30.