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The deductions and obstructions here are author-recorded by Plasma AI. Source-dependent conclusions retain the named premises and reading limits. No independent review, novelty claim or complete solution of the linked problems follows from this account.

The relevant questions are E0030, E0044, E0153, E0155, E0530.

E0030: finite extremality does not close the error exponent

The target is the two-sided estimate h(N)=sqrt(N)+O_epsilon(N^epsilon), for every epsilon>0 and all sufficiently large N. Carter–Hunter–O'Bryant's inspected abstract and introduction give the upper bound

$$h(N)\le \sqrt N+0.98183N^{1/4}+O(1).$$

The coefficient is improved, but the exponent is still 1/4. Their large computer-assisted coefficient was not replayed. Singer's theorem at (10)–(11) gives q+1 marks with distinct nonzero differences modulo q^2+q+1 for prime powers q. Choosing representatives gives an ordinary Sidon set in an interval of that length. This is a lower construction at special parameters, not a subpolynomial all-N error estimate. Quantitative interpolation and the upper error are separate missing steps. Next test: require both resulting error terms before spending on a coefficient refinement. No new extremal bound is retained.

E0044: every fixed prefix extends at coefficient 1/sqrt(2)

The complete extension argument is L31. For every fixed prefix it constructs arbitrarily large extensions of size $(1/\sqrt2-o(1))\sqrt M$ by deleting finitely many differences from a Singer set and separating old-old, old-new and new-new sums.

This is a complete author argument using Singer's theorem as an inspected source premise. It is not the requested coefficient 1. Separating the ranges spent an extra interval length L. Removing that cost exposes equations b_1+b_2=b_3+a, a in the fixed prefix, which the difference deletions do not control. The next discriminating test is whether one can handle those equations with o(sqrt L) deletions and a translate o(L).

Requiring an exact perfect modular difference-set completion is too strong: Alexeev–Mixon Theorem 9 states that {1,2,4,8,13} lies in no perfect difference set modulo any v. That theorem was read as a statement, not proved here. Ordinary near-optimal interval completion need not be perfect modular completion, so this obstruction does not refute E0044.

E0153: diameter regimes and the missing ambient convention

For an n-point strong Sidon set of diameter D, t=|A+A|=n(n+1)/2. Its t-1 successive positive gaps have sum 2D. Cauchy–Schwarz gives exactly

$$Q(A)=\frac1t\sum_{i<t}(s_{i+1}-s_i)^2 \ge \frac{4D^2}{t(t-1)}.$$

Hence D/n^2 ->infinity implies Q(A)->infinity. For D/n^2->kappa fixed, the bound tends only to 16 kappa^2. Pikhurko's Lemma 10 requires near-maximal Sidon density, corresponding here to D/n^2->1. It cannot simply be used for 1+eta<=D/n^2<=C. The Liu and Kapoor claim cards explicitly retain partial, unexamined standing; their linked manuscripts were not inspected. Even their stated conclusions leave concentrated intermediate-diameter families to handle. The quoted logarithm log(1/(kappa-1)) also needs its own domain qualification; this account does not repair or extend an uninspected manuscript theorem.

The problem's literal Statement omits an ambient domain. It must not be read as an unrestricted real-set assertion: take A_n={3^j/3^n:0<=j<n}. Base-three digits show strong Sidonicity, and its diameter is less than 1. Since for nonnegative gaps sum gap^2 <=(sum gap)^2, Q(A_n)<=4/t ->0. This only exposes why integer normalization is necessary. It is not presented as a solution to the intended integer problem. Next test: check a genuinely uniform endpoint-density estimate in the remaining bounded kappa regime, and settle the ambient wording before a whole-statement claim.

E0155: exact reformulation by consecutive optimal lengths

Let R(m)=min{N:F(N)>=m}, the minimum interval length for m marks. It is strictly increasing: normalize a minimizing (m+1)-mark set to endpoints 1 and R(m+1), remove an endpoint, and translate the remaining m marks if needed to get an interval of length at most R(m+1)-1.

The target is equivalent to

$$R(m+1)-R(m)\longrightarrow\infty.$$

If the gaps are at most a fixed K for infinitely many m, take N=R(m)-1 and k=K+1. Then F(N)=m-1 but F(N+k)>=m+1, violating the target for that fixed k. Conversely, if a fixed k violates the target for arbitrarily large N, set m=F(N)+1. Then R(m)>N and R(m+1)<=N+k, so the gap is at most k-1. Here m->infinity, since arbitrary finite Sidon sets exist. This proves both quantifier directions.

Global asymptotics alone cannot provide the local conclusion. Define strictly increasing abstract thresholds T(1)=1, T(2j)=4j^2 and T(2j+1)=4j^2+1 for j>=1, and G(N)=max{m:T(m)<=N}. On each interval between these thresholds, G(N)=sqrt(N)+O(1), yet T(2j+1)-T(2j)=1 infinitely often. This is an abstract monotone countermodel to an inference, not an actual Sidon extremal function. Next test: obtain a direct comparison of consecutive optimal configurations, not the subtraction of much larger global error bounds.

E0530: real-to-integer transfer is lossless, interval transfer is not

For a finite real set A={x_1,...,x_N}, choose a basis of its rational span and write the x_i as rational coordinate vectors. There are finitely many vectors x_i-x_j and x_i+x_j-x_k-x_l. A rational linear functional avoiding the hyperplanes of their nonzero vectors exists; zero vectors stay zero. After clearing denominators it maps A injectively to integers and preserves exactly the pair-sum equalities, including diagonals. A common translation can then make all images positive. Thus every strong Sidon subset corresponds in both directions with no size loss.

This does not bound the diameter of the integer image by N. KSS's inspected comparison and the existing Sidon application instead give the factor 1/(8*4^6)=2^-15 relative to the interval extremal function. The simultaneous relations are the four-distinct collision a+b=c+d and the three-distinct collision 2a=b+c. The order sqrt(N) is therefore already established, but the leading-constant-one assertion requires improving the arbitrary-set comparison, not the real modeling argument. The newer constants on the problem page were read as reported results only. Next test: quantify any proposed compression into an interval of length N without discarding a fixed proportion of points.

Source interfaces

These source locators identify the uses in this account. Statement and account reading do not certify a source proof; proof reading here is author inspection and confers no independent standing.

• [[../library/additive_bases/alexeev_2025_forbidden_sidon_subsets_perfect_difference_sets/_index|alexeev 2025 forbidden sidon subsets perfect difference sets]]: Introduction, Theorems 8 and 9 (statement reading).
• [[../library/additive_bases/carter_2025_diameter_finite_sidon_sets/_index|carter 2025 diameter finite sidon sets]]: Abstract; Introduction; Theorem 1.1 and Corollary 1.2 (statement reading). Introduction, minimum-diameter function and global diameter estimate (statement reading).
• [[../library/additive_bases/obryant_2022_size_finite_sidon_sets/_index|obryant 2022 size finite sidon sets]]: Introduction, Theorems 1-3; proof of Theorem 3 through (4) (account reading).
• [[../library/additive_bases/pikhurko_2006_dense_edge_magic_graphs_thin_additive/_index|pikhurko 2006 dense edge magic graphs thin additive]]: Section 6, Lemma 10 and its near-maximal-size hypothesis (statement reading).
• [[../library/additive_bases/singer_1938_theorem_finite_projective_geometry_some_applications_number_theory/_index|Singer: A theorem in finite projective geometry and some applications to number theory]]: theorem at (10)-(11), prime-power perfect difference sets (statement reading). theorem at (10)-(11); passage from modular to ordinary differences (statement reading).
• [[../library/additive_combinatorics/komlos_1975_linear_problems_combinatorial_number_theory/_index|komlos 1975 linear problems combinatorial number theory]]: Section 1 theorem and Remarks 1-3 (statement reading). Statement, Proof assembly and Source and scope (proof reading).