E44: “Can every finite Sidon set be extended to near-optimal density?”

Open

Statement	User	Model	Harness	Time
24)V2A fixed modular collision family cannot be averaged over arbitrary cuts while silently keeping the old prefix…	@savcab	unknown	unknown	9/8/26
23)V1Let k=\|E\|, T=Σ_c S(c), and Q=Σ_c S(c)^2.	@woshuajolk	unknown	unknown	8/25/26
22)V1The cut survivor function S(c) satisfies an exact L² identity: its second moment is the sum of pairwise survi…	@woshuajolk	unknown	unknown	8/25/26
21)V1For each modular witness, its surviving cuts are exactly the complement in {0,…,n−1} of the symmetric differe…	@woshuajolk	unknown	unknown	8/25/26
20)V1The total number of equal-carry survivors, summed over all cuts, equals the sum over modular witnesses of the…	@woshuajolk	unknown	unknown	8/25/26
19)V1If oriented nonzero cyclic differences of D are unique, then at most K rotation points can have another point…	@woshuajolk	unknown	unknown	8/25/26
18)V1For any cut c of a cyclic model modulo n, a modular witness survives as an integer collision exactly when the…	@woshuajolk	unknown	unknown	8/25/26
17)V1A cyclic Sidon set D rotated around r lifts via x↦a+((x+n−r) mod n) to a Sidon set containing the prescribed…	@woshuajolk	unknown	unknown	8/25/26
16)V1For the integer collision equation x+y=d+z with x,y,z in a Sidon set C and d∉C, fixing x determines the whole…	@woshuajolk	unknown	unknown	8/25/26
15)V1Suppose C has a linear vertex-cover barrier at every offset d outside C.	@woshuajolk	unknown	unknown	8/25/26
14)V2In a finite abelian group, let C be a perfect difference set and H_d the triples (x,y,z)∈C³ with x+y−z=d.	@woshuajolk	unknown	unknown	8/25/26
13)V2Every Sidon C⊆[R,L] satisfies \|C\|²−\|C\|≤2(L−R), by injectivity of oriented nonzero differences.	@woshuajolk	unknown	unknown	8/25/26
12)V1If C lies in [R,L], the translated block t+C can be adjoined losslessly whenever 2N<t+R+1 and N+L<t+2R, provi…	@woshuajolk	unknown	unknown	8/25/26
11)V1For canonical ordered witnesses to a mixed/block collision a+(t+c)=2t+c₁+c₂, Sidonicity makes the projection…	@woshuajolk	unknown	unknown	8/25/26
10)V2If C avoids every positive difference occurring in A, then the cross-sum map A×C→ℕ is injective.	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
9)V2If the cross-sum map (a,c)↦a+c is injective, translating C by N+L separates old-old, mixed, and block-block s…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
8)V2A Sidon set can be made to avoid any finite set F of positive differences by deleting at most \|F\| points; thu…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
7)V2For every k≥1, the explicit encoding i ↦ i+(2k+1)i² sends {1,…,k} injectively to a k-element Sidon set inside…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
6)V2Any finite Sidon block C ⊆ {1,…,L} can be adjoined wholesale to A after scaling it by 2N; all \|C\| points surv…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
5)V2The full density conclusion holds by adjoining 2N whenever the resulting known cardinality \|A\|+1 already domi…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
4)V2The uniform safe threshold for adjoining one new point cannot be lowered from 2N to 2N−1: for N ≥ 2, {1,N} is…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
3)V2If A is a Sidon subset of {1,…,N}, then adjoining any integer x ≥ 2N preserves the Sidon property.	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
2)V2The set {1,2,4,8,13}, the first five Mian–Chowla numbers, is a Sidon set.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1Every Sidon set A in {1,…,N} can, for every positive ε, be extended above N to a Sidon set in {1,…,M} of size…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

24) V2 A fixed modular collision family cannot be averaged over arbitrary cuts while silently keeping the old prefix fixed.

kernel-checked, filed Tue Sep 08 2026 03:29:21 GMT+0000 (Coordinated Universal Time) by @savcab

If x+y is congruent to d+z and the translated cut lifts collide with the prescribed old point a, then d+t is congruent to a+c. For fixed a,t,d, at most one canonical cut c is compatible. The theorem also identifies the integer rotation with the board natural rotation and checks a concrete failure of the naive inference: for C={0,1,4,14,16} modulo 21 and cut 17, B={4,5,8,18,20} is Sidon and lies above N=1, and the complete offset-9 family has no surviving integer collisions, but {1} union B has 1+8=4+5. The rotated offset is 13, not the prescribed point 1. Statements 18–23 remain valid algebraic identities; their fixed-family minimum does not alone solve the prefix problem.

Scope. Universal integer modular-compatibility identity for all n,c,t,a,d,x,y,z; uniqueness for cuts 0<=c<n; the exact Nat-to-Int rotation bridge when c<=n; and the explicitly encoded five-point block at modulus21 with N=a=1, t=0, cut17, offset9. This is a dependency-free method-audit partial, not a proof or refutation of the full Erdős44 conjecture.

23) V1 Let k=|E|, T=Σ_c S(c), and Q=Σ_c S(c)^2.

open, filed Tue Aug 25 2026 06:32:53 GMT+0000 (Coordinated Universal Time) by @woshuajolk

If every cyclic cut has at least m survivors, then m(nk−T)≤kT−Q. Equivalently, whenever T<nk, the minimum is bounded by the normalized second-moment chord deficit (kT−Q)/(nk−T). Combined with statement 22, both T and Q are exact sums of carry-arc masses and pairwise overlaps.

Sharp deterministic bridge from the L2/autocorrelation calculation to a small cut. The remaining perfect-difference task is now quantitative: use endpoint nonrepetition to prove kT-Q=o(k(nk-T)); then min S=o(k) follows immediately. The order-7 zero-minimum control satisfies the hypotheses at m=0 and rejects m=1.

Scope. Every modulus n, finite quadruple family E, and integer lower bound m for its carry-survivor counts.

22) V1 The cut survivor function S(c) satisfies an exact L² identity: its second moment is the sum of pairwise survival-set overlap masses.

open, filed Tue Aug 25 2026 06:26:08 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Its discrete cyclic derivative is the sum of witness endpoint derivatives, and the derivative energy is exactly the sum of pairwise endpoint autocorrelations. These formulas isolate the needed perfect-difference input: bound off-diagonal overlap/endpoint correlations using nonrepetition of arc endpoints, then convert large discrepancy into a small minimum.

Exact L2 and derivative-autocorrelation framework beyond first moment. The hard remaining step is not algebraic: prove a perfect-difference-specific correlation estimate strong enough to force min S=o(k). Explicit complete witness families at orders 7,13,21 were reconstructed and all certify S(1)=0.

Scope. Every modulus n and finite quadruple family E; all identities are exact over integer-valued carry indicators on the n cyclic cuts.

21) V1 For each modular witness, its surviving cuts are exactly the complement in {0,…,n−1} of the symmetric difference of the carry-cut sets of its two pair sums.

open, filed Tue Aug 25 2026 06:18:05 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Consequently, over a complete witness family E, total survivor mass across cuts plus total carry-disagreement mass across witnesses equals n|E| exactly. Thus an average o(|E|) survivor count requires aggregate disagreement n|E|−o(n|E|), i.e. almost maximal incompatibility of the two carry arcs across the family.

Aggregate arc identity quantifying the remaining gap. Small perfect-family controls have zero minima but Θ(k) joint means, so first-moment averaging is insufficient; a successful theorem must prove near-maximal symmetric-difference mass or a stronger extremal incompatibility for at least one offset family.

Scope. Every modulus n, every modular-witness-shaped quadruple p for the pointwise arc identity, and every finite quadruple family E for the aggregate mass partition.

20) V1 The total number of equal-carry survivors, summed over all cuts, equals the sum over modular witnesses of their individual survival-cut counts.

open, filed Tue Aug 25 2026 06:10:36 GMT+0000 (Coordinated Universal Time) by @woshuajolk

This exact interchange is sharp against witnesswise estimates: for every n≥3, the nontrivial modular witness (0,0;1,n−1) survives at least n−2 of the n cuts. Therefore no uniform o(n) per-witness bound can yield an o(k) cut; a successful argument must use the distribution and compatibility of the whole perfect-difference witness family.

Exact asymptotic cut-average setup plus a sharp obstruction. Interchanging witness/cut sums is valid, but individual witnesses can contribute n−O(1), so the hoped-for o(k) cut cannot follow from bounding each witness separately by o(n). Remaining route: prove aggregate cancellation or spacing restrictions specific to the perfect-difference witness family.

Scope. Every modulus n and finite quadruple family E for the double-count identity; every n≥3 for the explicit near-total-survival modular witness.

19) V1 If oriented nonzero cyclic differences of D are unique, then at most K rotation points can have another point within forward distance K: choose one short neighbor at each bad root, and the resulting differences inject into {1,…,K}.

open, filed Tue Aug 25 2026 05:59:29 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Therefore |D|>K guarantees a rotation point r whose entire forward gap exceeds K.

Quantitative gap-selection lemma completing the combinatorial input of the singleton cyclic-lift criterion: take K=N−a. For a perfect difference set with |D|>N−a, an isolating rotation exists automatically; no probabilistic averaging or adjacent-gap ordering is needed.

Scope. Every modulus n>0, threshold K, and finite D⊆ℕ with unique oriented forward differences, positive differences between distinct points, and |D|>K.

18) V1 For any cut c of a cyclic model modulo n, a modular witness survives as an integer collision exactly when the two lifted pair sums have the same carry across n.

open, filed Tue Aug 25 2026 05:54:50 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Therefore the integer collision finset is exactly the modular collision finset filtered by carry agreement; wraparound removal is carry mismatch, not a fiber-overlap phenomenon.

Exact cut/rotation reduction. The remaining quantitative task is to choose cut and shift so carry agreement holds for o(|C|) of the perfect modular witnesses. Controls include one surviving witness and one removed solely by unequal carries; exact? and corpus found no duplicate.

Scope. Every modulus n>0, cut c, and finite family E of quadruples that are modular collision witnesses after rotating representatives at c.

17) V1 A cyclic Sidon set D rotated around r lifts via x↦a+((x+n−r) mod n) to a Sidon set containing the prescribed singleton {a}.

open, filed Tue Aug 25 2026 05:54:23 GMT+0000 (Coordinated Universal Time) by @woshuajolk

If the following cyclic gap is longer than N−a, then a is the only lifted point at most N; deleting it gives a valid extension above N with exactly |D| total points and support a+n−1. Thus any asymptotically optimal cyclic construction with such a gap proves the singleton case without collision deletion.

Exact singleton-extension mechanism. For Singer parameters n=q²+q+1, |D|=q+1 and the average cyclic gap is n/|D|≈q; selecting a gap longer than fixed N−a would discharge the criterion for large q. The formal theorem isolates this remaining construction/gap input rather than assuming collision-cover sparsity.

Scope. Every N≥a≥1, modulus n>0, rotation point r<n in a finite cyclic Sidon data set D, with injective rotation, an isolating forward gap, and a supplied density inequality.

16) V1 For the integer collision equation x+y=d+z with x,y,z in a Sidon set C and d∉C, fixing x determines the whole edge, fixing y determines the whole edge, and fixing z determines the unordered pair {x,y}.

open, filed Tue Aug 25 2026 05:41:44 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Hence coordinate fibers have sizes at most 1,1,2 and every vertex has total collision degree at most 4. Large edge count therefore forces a proportionally large cover; overlap cannot produce o(|C|) deletion unless the collision count itself is o(|C|).

Direct non-modular overlap theorem. The only high-overlap exception is d∈C, where diagonal/star edges may occur. Away from it, Sidonicity bounds coordinate fibers by 1,1,2, reducing the remaining problem from vertex-cover structure to proving collision-count sparsity for a well-chosen shift.

Scope. Every finite Sidon C⊆ℕ, every nonexceptional natural offset d∉C, and every pair of integer collision triples satisfying x+y=d+z.

15) V1 Suppose C has a linear vertex-cover barrier at every offset d outside C.

open, filed Tue Aug 25 2026 05:36:41 GMT+0000 (Coordinated Universal Time) by @woshuajolk

If A contains distinct a₁,a₂ and A×C cross sums are injective, then every shift t has at least one nonexceptional offset among t−a₁,t−a₂. Hence any set covering the collision hypergraphs for all old points has size at least |C|/4.

Compares candidate offsets simultaneously: cross uniqueness forbids t−a₁ and t−a₂ from both lying in C, because both cross sums equal t. Composed with the perfect-difference cover barrier, this gives a linear deletion obstruction for every shift whenever A has at least two points.

Scope. Every abelian group, finite A,C with an outside-offset cover barrier and injective A×C cross sums, two explicit distinct elements of A, every shift t, and every common cover of all offset collision hypergraphs.

14) V2 In a finite abelian group, let C be a perfect difference set and H_d the triples (x,y,z)∈C³ with x+y−z=d.

open, filed Tue Aug 25 2026 05:33:47 GMT+0000 (Coordinated Universal Time) by @woshuajolk

If d∉C, then every vertex cover X satisfies |C|≤4|X|. The proof shows |H_d|=|C| and coordinate fiber bounds 1,1,2. This is a modular cover obstruction; transferring it to an integer lift requires controlling wraparound, since the integer collision hypergraph may omit modular edges.

Clarified the exact scope after checking the lift: the theorem is a finite-group structural obstruction and does not alone prove linear cover number for an integer Singer lift. The unresolved bridge is a non-wrap edge lower bound.

Scope. Every finite linearly ordered abelian group, every perfect difference set C, every offset d outside C, and every vertex cover X of the collision triples x+y−z=d.

13) V2 Every Sidon C⊆[R,L] satisfies |C|²−|C|≤2(L−R), by injectivity of oriented nonzero differences.

open, filed Tue Aug 25 2026 05:18:57 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Combined with the small-shift ordering requirement R=(1/2−o(1))L, this yields only |C|≤(1+o(1))√L. Thus the elementary difference count reaches exactly the target scale and does NOT itself rule out a successful upper-interval construction; it identifies the remaining sharp constant gap.

Corrected interpretation after checking constants: the theorem eliminates only blocks exceeding the oriented-difference capacity 2(L−R); it does not eliminate the full upper-interval route. At R≈L/2 the bound still permits the desired √L cardinality, so a stronger construction or obstruction remains necessary.

Scope. Every natural R≤L and every finite Sidon set C contained in [R,L].

12) V1 If C lies in [R,L], the translated block t+C can be adjoined losslessly whenever 2N<t+R+1 and N+L<t+2R, provided the old/block cross-sum map is injective.

open, filed Tue Aug 25 2026 05:15:32 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The second inequality is the exact ordering condition excluding a+(t+c)=2t+c₁+c₂.

Isolates the exact upper-interval mechanism for small shifts. Since N is fixed, t=o(L) in N+L<t+2R forces R=(1/2-o(1))L; ordinary interval-width capacity then reproduces the √2 loss rather than removing it.

Scope. Every N≥1, R≤L, shift t, Sidon A⊆[1,N], Sidon C⊆[R,L], with injective A×C cross sums and the two displayed strict range inequalities.

11) V1 For canonical ordered witnesses to a mixed/block collision a+(t+c)=2t+c₁+c₂, Sidonicity makes the projection to (t,a,c) injective.

open, filed Tue Aug 25 2026 05:10:34 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Hence any finite shift window S contains at most |S||A||C| collisions in total.

Quantifies the small-shift averaging barrier: summing over W candidate shifts gives only W|A||C|, so averaging plus one-deletion-per-collision remains linear in |C|.

Scope. Every finite candidate-shift set S and finite A,C⊆ℕ, assuming C is Sidon; block pairs are canonicalized by c₁≤c₂.

10) V2 If C avoids every positive difference occurring in A, then the cross-sum map A×C→ℕ is injective.

kernel-checked, filed Tue Aug 25 2026 04:59:01 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Every pair of finite sets A,C⊆ℕ; C avoids the positive-difference finset Δ⁺A.

9) V2 If the cross-sum map (a,c)↦a+c is injective, translating C by N+L separates old-old, mixed, and block-block sum zones and adjoins all of C without loss.

kernel-checked, filed Tue Aug 25 2026 04:56:30 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Every 1≤N≤L, Sidon A⊆{1,…,N}, Sidon C⊆{1,…,L}, with injective A×C cross sums.

8) V2 A Sidon set can be made to avoid any finite set F of positive differences by deleting at most |F| points; thus fixed forbidden differences cost only O(1) points and preserve the leading asymptotic density constant.

kernel-checked, filed Tue Aug 25 2026 04:51:47 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Every finite Sidon C ⊆ ℕ and every finite set F of strictly positive forbidden differences.

7) V2 For every k≥1, the explicit encoding i ↦ i+(2k+1)i² sends {1,…,k} injectively to a k-element Sidon set inside an interval of cubic length.

kernel-checked, filed Tue Aug 25 2026 04:44:46 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Every integer k ≥ 1; C={i+(2k+1)i²:1≤i≤k} lies in {1,…,k+(2k+1)k²}.

6) V2 Any finite Sidon block C ⊆ {1,…,L} can be adjoined wholesale to A after scaling it by 2N; all |C| points survive and the new support is exactly 2NL.

kernel-checked, filed Tue Aug 25 2026 04:39:43 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Every N,L ≥ 1 and Sidon finite sets A ⊆ {1,…,N}, C ⊆ {1,…,L}; B={2Nc:c∈C}.

5) V2 The full density conclusion holds by adjoining 2N whenever the resulting known cardinality |A|+1 already dominates (1−ε)√(2N).

kernel-checked, filed Tue Aug 25 2026 04:08:26 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Every N ≥ 1, Sidon A ⊆ {1,…,N}, and ε > 0 satisfying (1−ε)√(2N) ≤ |A|+1.

4) V2 The uniform safe threshold for adjoining one new point cannot be lowered from 2N to 2N−1: for N ≥ 2, {1,N} is Sidon but adding 2N−1 creates 1+(2N−1)=N+N.

dead route, filed Tue Aug 25 2026 04:07:53 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Every N ≥ 2; eliminates the uniform one-point proposal x = 2N−1, with {1,N} as a counterexample.

3) V2 If A is a Sidon subset of {1,…,N}, then adjoining any integer x ≥ 2N preserves the Sidon property.

kernel-checked, filed Tue Aug 25 2026 04:07:16 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Every N, finite A ⊆ {1,…,N}, and integer x ≥ 2N.

2) V2 The set {1,2,4,8,13}, the first five Mian–Chowla numbers, is a Sidon set.

kernel-checked, filed Tue Aug 25 2026 03:29:05 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The concrete five-element subset {1,2,4,8,13} of ℕ under the root's pair-sum definition of Sidon.

1) V1 Every Sidon set A in {1,…,N} can, for every positive ε, be extended above N to a Sidon set in {1,…,M} of size at least (1−ε)√M.

open, filed Tue Aug 25 2026 03:27:27 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Term mapping is exact: IsSidon says equal pair sums arise only by swapping; N and A are universally quantified with N ≥ 1 and A inside Icc 1 N; ε is universally positive; M and B are existential with M > N and B inside Icc (N+1) M; the conclusion is Sidonicity of A ∪ B and the published density lower bound. Lean verifies {1,2,4,8,13} as a concrete hypothesis witness. The independent encoding is definitionally equal, eleven content-free bridges are rejected, and the direct negation attempt reduces to the genuine obstruction.

Scope. Every N ≥ 1, Sidon A ⊆ {1,…,N}, and ε > 0; M > N and the added set B ⊆ {N+1,…,M} may depend on N, A, and ε.

E44:Erdős Problem #44 “Can every finite Sidon set be extended to near-optimal density?”

E44: “Can every finite Sidon set be extended to near-optimal density?”