Four-point difference conditions and Sidon certificates

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evidence/: Exhaust all four-subsets and nine-subsets and verify an eight-point witness in Ma–Tang's fixed integer set.

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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 E0757.

E0757: exact finite certificate and the first possible improvement size

For a four-point real set B, |B-B|=1+2d(B), where d(B) is the number of positive absolute differences. Thus Bloom's catalog condition is exactly d(B)>=5, the (4,5)-property of Ma–Tang. Every (4,5)-set is weak Sidon, but the inclusion is strict: {1,2,3,5} has six distinct off-diagonal sums and only four positive distances. Also {1,2,3,6} is (4,5) but has 1+3=2+2. Therefore their exact answer for arbitrary weak Sidon sets does not answer E0757.

The inspected Lemma 5.1 supplies the integer set

$$\{0,136,200,243,246,249,272,286,298,323,400,528,596,1056\}.$$

The retained exact checker enumerates all 1001 four-subsets and all 2002 nine-subsets, and checks the eight-point witness

$$\{0,136,200,243,246,298,323,528\}.$$

The four-subset histogram is 132 with five positive distances and 869 with six. No nine-subset was strong Sidon. The witness has all 36 unordered pair sums distinct. Heredity then excludes every larger Sidon subset. The maximum is exactly eight, proving the already-reported upper ratio 8/14=4/7 at this exact finite scope. The checker also covers empty/singleton and diagonal-collision boundaries. This is not an independent global proof audit.

For the next test, retain a constant term that the source discards. In the AP hypergraph H of a weak Sidon set, edges are three-term progressions and strong Sidon subsets are independent sets. Distinct edges have different midpoints, because otherwise two different off-diagonal pairs have equal sums. Neither endpoint of the ordered set is a midpoint. Thus m_H<=n-2. For a (4,5)-set, Ma–Tang also proves linearity and cites exclusion of the five-edge configuration

$$F_7=\{012,034,056,135,246\}$$

on vertices 0,...,6. This is not the seven-edge Fano plane. Its quoted Henning–Yeo theorem gives 17 tau(H)<=5n+3m_H. Therefore

$$h(A)=n-\tau(H)\ge(9n+6)/17, \qquad h(A)\ge\lceil(9n+6)/17\rceil \quad(n\ge2).$$

This specialization is checked; the external transversal theorem and the cited F_7 exclusion were not independently audited. For n<=8 the unrounded bound already exceeds 4n/7. For n=9,...,15 the rounded bounds are respectively 6,6,7,7,8,8,9, none giving a ratio below 4/7. The n=1 case is immediate. Hence no certificate with at most 15 points can improve 4/7 under these source premises. At n=16 the bound is 9, and 9/16<4/7. A 16-point (4,5)-set with h=9 is the first possible improving witness, not an assertion that one exists.

At n=16, a witness must have tau=7. The same inequality forces 119<=80+3m, so m is 13 or 14. A finite investigation could test 16-vertex linear hypergraphs with 13 or 14 edges, the specified F_7 excluded, distinct ordered midpoints and independence number nine, then seek real AP realizations satisfying the whole (4,5)-condition. A rational solution to a fixed consistent pattern of equalities and strict inequalities can be scaled to integers; this does not justify a universal coordinate-box bound. A successful witness needs checks of all 1820 four-subsets and all 8008 ten-subsets. No such search was run. Failure in one coordinate box would settle nothing about all reals.

The source's Lemma 3.6 and Proposition 3.7 also explain why finite ratios are relevant asymptotically. For finite (4,5)-sets A,B, choose a scale for B avoiding all equal nonzero within-set differences of A and the scaled B. Translate B far enough that mixed distances exceed both internal diameters. All mixed differences are distinct. A four-set split 3+1 has three distinct mixed and at least two internal distances; a 2+2 split has four mixed and at least one internal distance. Thus the union remains (4,5), while any Sidon subset has at most h(A)+h(B) points. This proves subadditivity of the minimum h(n), and the usual block division proves lim h(n)/n=inf h(n)/n. The argument does not make the upper ratio optimal.

The exact input and deterministic checker live in [[research/four_point_sidon_certificates/evidence/_index|the certificate evidence]]. The 16-point search described above has not been run. Failure in a bounded coordinate box would not exclude real realizations outside that box.

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/ma_2026_largest_sidon_subsets_weak_sidon_sets/_index|Ma and Tang: Largest Sidon subsets in weak Sidon sets]]: Introduction Theorems 1.2-1.6 and the explicit five-edge F7 definition; Section 5.2 quoted Lemma 5.2 and Theorem 5.3 (statement reading). Propositions 2.1-2.2; Lemmas 2.3-2.5; Lemmas 3.5-3.6 and Proposition 3.7; Lemma 5.1; Section 5.2 lower-bound assembly (external theorem and F7 exclusion only quoted) (proof reading).