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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 E0039, E0040, E0041, E0158, E1191.

E0039: the one-set and all-cutoff obligations remain

Fabian's Theorem 1.1 at alpha=0 gives an infinite strong Sidon set with counting exponent sqrt(2)-1, not 1/2. The target asks for one set that works for every epsilon>0. A family of finite near-optimal sets, or a different infinite set for each epsilon, does not establish that target.

The existing compatible-tower discussion for E0158 is relevant to this logical interface, not a solution for strong Sidon sets. Suppose a proposed nested construction reaches size N_j^(1/2-o(1)) at cutoffs N_j. Monotonicity alone only gives A(x)>=A(N_j) for N_j<=x<N_(j+1). The condition log N_(j+1)/log N_j ->1 would let this interpolate to x^(1/2-o(1)); without some comparable scale control it does not. Next test: an extension lemma must state both prefix preservation and scale growth. The coefficient-$1/\sqrt2$ prefix extension does not supply this tower.

E0040: polynomial losses are excluded by a precise specialization

Write G for the density denominator and r for a fixed representation cap. Pliego's Theorem 1.1 states that, for each integer r>=2, one infinite B_2[r] set has

$$A(x)\gg x^{r/(2r+1)} =x^{1/2-1/(4r+2)}.$$

For a fixed delta>0 choose one integer r with 1/(4r+2)<=delta. This set satisfies A(x)>>sqrt(x)/x^delta. Its ordered convolution is at most 2r, because an unordered off-diagonal representation contributes two ordered ones and a diagonal contributes one. It therefore disproves the implication in E0040 for G(x)=x^delta, conditional on the inspected source theorem. More generally it does so for G(x)>=c x^(1/(4r+2)) eventually, c>0 fixed. This is a complete specialization; the construction proof was not audited.

The cap r must stay fixed for the entire infinite set. Letting r grow with x would not produce bounded convolution. At the other end, O'Bryant I excludes the density premise with G=o(sqrt(log x)) within the ordinary Sidon subclass: the premise forces A(x)sqrt(log x)/sqrt(x) to diverge, contradicting its finite liminf bound. This does not settle arbitrary bounded sum multiplicity, by the E0158 counterexample below. Next test: find a theorem whose hypothesis bounds unordered sums, not differences. The subpolynomial classification is unresolved.

E0041: the even-order hypothesis is a genuine obstruction

O'Bryant II Theorem 1 applies to even h and bounds liminf A(x)/(x/log x)^(1/h) by an explicit finite constant. Its Sections 1.1–1.2 explicitly retain the odd-order question. B_3 implies B_2, but sqrt(x/log x) is much larger than x^(1/3); this gives no required thinning at the cubic-root scale.

B_3 does not imply B_4. The ten unordered triple sums of {1,2,5} are 3,4,5,6,7,8,9,11,12,15, all different, whereas 1+1+1+5=2+2+2+2. Thus substituting h=4 requires a false implication. This finite example is an obstruction to specialization, not an infinite counterexample to E0041. Next test: isolate the first four-fold equality in a proposed proof and explain why the B_3 hypothesis controls it.

E0158: bounded sums can coexist with infinitely repeated differences

Set

$$A=\{5^j,5^j+1:j\ge1\}.$$

Every pair sum has form 5^i+5^j+e+f with e,f in {0,1}. In base five, the units digit is e+f in {0,1,2}, without carry. The higher digits determine the unordered pair of indices: two digits 1 at different indices, or one digit 2 at a repeated index. For distinct indices, units digit 1 permits two unordered pairs, whereas units digit 0 or 2 permits one. For equal indices, each units digit permits only one unordered pair, including a repeated element when appropriate. Thus A is B_2[2]. But (5^j+1)-5^j=1 for every j, so difference 1 has infinitely many representations.

This completely disproves the proposed implication to any fixed g-Golomb condition. It does not refute E0158: this particular set has only O(log x) elements up to x. O'Bryant's new difference-thinning theorem therefore does not settle the question by specialization. Existing two-representation constructions and tower accounts were read; their exponents remain below 1/2 and their standing is unchanged. Next test: a thinning inequality using bounded unordered sum counts that remains valid on this example.

E1191: a printed zero attribution requires source reconciliation

At difference cap g=1, O'Bryant I, arXiv:2606.28651v3, Theorem 1 supplies

$$\liminf_{x\to\infty}\frac{A(x)\sqrt{\log x}}{\sqrt x} \le\frac2{\sqrt{\log2}}.$$

A finite positive bound does not imply zero, but is compatible with it; it does not disprove the stronger assertion. O'Bryant II, arXiv:2607.23795v1, gives the same normalization and describes the historical result as a finite constant. Pliego, arXiv:2405.04154v1, in the introduction immediately after (1.4), instead prints exactly the zero-liminf assertion, attributing it to Erdős via Halberstam–Roth, Sequences, Section 2 Theorem 8. The zero agrees with the printed PDF, so it is not a transcription error. It is not a misleading transcription to repair.

The cited book theorem has not been inspected. Its publisher record identifies the 1983 edition, but supplies no inspected theorem here. The actual book theorem and the relation between the cited editions remain uninspected. The zero and finite-constant assertions must be distinguished at that source. The reported-source conflict concerns the strength attributed to that historical result. Until then this is an explicit source_gap, not an accepted proof or a negative theorem. The source discrepancy is not resolved by choosing the newer or older sentence.

For the construction half, an exponent sqrt(2)-1 is strictly below 1/2, and x^(sqrt(2)-1+o(1)) is eventually smaller than sqrt(x)/(log x)^c for every fixed c. No inspected construction reaches the required fixed polylogarithmic loss. Both subquestions remain unresolved by this account.

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/fabian_2019_strong_infinite_sidon_b_h_sets/_index|fabian 2019 strong infinite sidon b h sets]]: Introduction and Theorems 1.1-1.3, especially alpha=0 (statement reading). Theorem 1.1 at alpha=0 (statement reading).
• [[../library/additive_bases/green_2001_number_squares_b_h_g_sets/_index|green 2001 number squares b h g sets]]: Section 3: finite Bh definitions and bounds (statement reading).
• [[../library/additive_bases/obryant_2026_thickness_infinite_generalized_sidon_sets_i/_index|obryant 2026 thickness infinite generalized sidon sets i]]: Bibliographic identity, exact arXiv version and theorem account (account reading). Introduction, Theorem 1 and Corollary 2: g-Golomb definition and liminf bound (statement reading). Theorem 1 and Corollary 2, cap g=1 (statement reading). Theorem 1 and g-Golomb definition (statement reading).
• [[../library/additive_bases/obryant_2026_thickness_infinite_generalized_sidon_sets_ii/_index|obryant 2026 thickness infinite generalized sidon sets ii]]: Bibliographic identity, exact arXiv version and theorem account (account reading). Theorem 1 and Sections 1.1-1.2 historical normalization (statement reading). Theorem 1, Corollary 2, Sections 1.1-1.2 (statement reading).
• [[../library/additive_bases/pliego_2024_erdos_turan_conjecture_growth_b_2/_index|pliego 2024 erdos turan conjecture growth b 2]]: Bibliographic identity, exact arXiv version and theorem account (account reading). Definition of B2[g]; Theorem 1.1 and (1.2) (statement reading). Introduction, display after (1.4), and reference [18]; printed PDF page 2 compared (statement reading).