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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 E0156, E0340, E0864, E0963.

For a maximal strong Sidon set $A\subset[1,N]$ of size $k$, the existing blocker proof gives

$$N\le(k^3+k)/2.$$

Every greedy prefix in $[1,N]$ is maximal there: each omitted integer was already blocked by earlier marks, and those marks remain. Immediately before the next greedy mark $a_{k+1}$, the first $k$ marks are maximal in $[1,a_{k+1}-1]$. Thus

$$a_{k+1}\le(k^3+k)/2+1.$$

This supplies the cubic-root baseline for E0340. An existentially small maximal set does not control this particular greedy sequence. An improvement needs overlap information for its actual blockers.

For E0156 the [[research/erdos_156/foundations/ruzsa_and_carries|existing construction account]] identifies the union-bound logarithm and the integer carry obstruction. No new saturation computation is supplied here.

For E0864, retain the [[research/erdos_864/established_bounds|existing difference and window bound]] $|A|\le\sqrt{2N}+O(N^{1/4})$ and the exact-reflection obstruction. Neither establishes the requested coefficient $2/\sqrt3$, and the exponential diameter of the obstruction leaves dense asymptotic completion separate.

For E0963 the existing dissociation sketch already separates maximal ternary spanning from the desired binary-size bound and proves the reduction to positive sets. Bounds for the largest element of a dissociated subset of $[1,n]$ bound its size above, hence the universal guarantee above; they do not give the required lower bound. No finite dissociation search is part of 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/huber_2026_saturated_sidon_sets_consecutive_intervals_eight_mark_threshold_144/_index|Huber: Saturated Sidon Sets in Consecutive Intervals: The Eight-Mark Threshold Is 144]]: fixed-eight theorem and evidence scope (account reading).
• [[../library/additive_bases/redman_2021_small_maximal_sidon_set_z_2/_index|redman 2021 small maximal sidon set z 2]]: Introduction, Definitions 1.1-1.2 and main bound (statement reading).
• [[../library/additive_bases/ruzsa_1998_small_maximal_sidon_set/_index|ruzsa 1998 small maximal sidon set]]: Theorem and final Remark on minimum maximal cardinality (account reading). Theorem, Lemma and full construction through maximal extension (proof reading).
• [[../library/additive_combinatorics/lev_yuster_2010_size_dissociated_bases/_index|On the size of dissociated bases]]: Introduction, Theorems 1-3 (statement reading).