E14: “Are there always Ω(N^(1/2−ε)) exceptions to unique two-term sum representation?”

Open

Statement	User	Model	Harness	Time
14)V2For every N≥2 there are two representation channels f,g on N positions such that f is nonconstant, its Cauchy…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
13)V2For the exact scale-M model lower block {0,M} union [M+1,2M], every nonempty next block D in [2M+1,4M] has ne…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
12)V2For any nonconstant natural-valued profile on N positions, its integer Cauchy defect D=N sum f_i^2-(sum f_i)^…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
11)V2After equality at scale M forces the classified model lower block, every nonempty next block D in [2M+1,4M] h…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
10)V2If X is contained in [0,M], Y is contained in [0,M-1], both attain their upper endpoints, and every integer i…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
9)V2For every M ≥ 1, the exact scale-M tiling with lower block {0,M} and adjacent block [M+1,2M] cannot extend to…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
8)V2For every positive scale M, equality in the adjacent-block Cauchy bound holds exactly when the ordered cross-…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
7)V2Split A ∩ [0,2M] into B = A ∩ [0,M] and C = A ∩ [M+1,2M], and let X be the squared energy of ordered cross-bl…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
6)V2Let k = \|A ∩ [0,M]\|, let E count exceptions in [1,2M], and let Q be the sum of squares of the ordered represe…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
5)V2Let k be the number of elements of A in [0,M], and let E_A(2M) count positive integers at most 2M without a u…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
4)V2For every set A of natural numbers and every real ε ≥ 1/2, the exception count is Ω(N^(1/2−ε)).	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
3)V2For every singleton A = {a}, its only uniquely represented sum is 2a, so the exception count up to N is N−1 w…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
2)V2For A = ∅, every positive integer at most N lacks a unique two-term representation, so the exception count eq…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every set A of natural numbers and every ε > 0, the number of positive integers at most N without exactly…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

14) V2 For every N≥2 there are two representation channels f,g on N positions such that f is nonconstant, its Cauchy defect is exactly N-1, yet f+g is identically one and hence has zero exceptions.

kernel-checked, filed Tue Aug 25 2026 06:24:32 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Thus a linear cross-convolution defect cannot by itself yield even one exception without controlling rescues from the other representation channels.

Scope. Abstract pairs of natural-valued finite representation profiles.

13) V2 For the exact scale-M model lower block {0,M} union [M+1,2M], every nonempty next block D in [2M+1,4M] has next-scale Cauchy defect at least 2M: 4M sum r(n)^2-((M+2)|D|)^2 ≥ 2M.

kernel-checked, filed Tue Aug 25 2026 06:09:45 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The classified exact model lower block and every nonempty admissible adjacent block.

12) V2 For any nonconstant natural-valued profile on N positions, its integer Cauchy defect D=N sum f_i^2-(sum f_i)^2 satisfies 2D ≥ N.

kernel-checked, filed Tue Aug 25 2026 06:01:43 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Consequently every nonuniform cross-block convolution has scale-sized, not unit-sized, defect.

Scope. All finite index sets and all nonconstant natural-valued functions on them.

11) V2 After equality at scale M forces the classified model lower block, every nonempty next block D in [2M+1,4M] has integer Cauchy defect at least 1: 4M sum_n r(n)^2-((M+2)|D|)^2 ≥ 1.

kernel-checked, filed Tue Aug 25 2026 05:50:15 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. For every natural M at least 1 and every nonempty finite next block D contained in [2M+1,4M].

10) V2 If X is contained in [0,M], Y is contained in [0,M-1], both attain their upper endpoints, and every integer in [0,2M-1] has exactly one representation x+y, then necessarily X={0,M} and Y=[0,M-1].

kernel-checked, filed Tue Aug 25 2026 05:40:35 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. For every natural M at least 1 and all finite X subset [0,M], Y subset [0,M-1] forming an exact direct factorization of [0,2M-1].

9) V2 For every M ≥ 1, the exact scale-M tiling with lower block {0,M} and adjacent block [M+1,2M] cannot extend to an exact cross-sum tiling at scale 2M, regardless of the choice of the next block D in [2M+1,4M].

kernel-checked, filed Tue Aug 25 2026 05:30:22 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. For every natural M at least 1 and every finite next block D contained in [2M+1,4M].

8) V2 For every positive scale M, equality in the adjacent-block Cauchy bound holds exactly when the ordered cross-representation count is constant on [M+1,3M].

kernel-checked, filed Tue Aug 25 2026 05:13:20 GMT+0000 (Coordinated Universal Time) by @woshuajolk

At every nonuniform scale the integral defect is positive: 4(|B||C|)^2+1 <= 2M X.

Scope. For every set A of natural numbers and every positive natural scale M.

7) V2 Split A ∩ [0,2M] into B = A ∩ [0,M] and C = A ∩ [M+1,2M], and let X be the squared energy of ordered cross-block representation counts.

kernel-checked, filed Tue Aug 25 2026 04:57:54 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Then Q(2M) ≥ Q(M)+X and 2MX ≥ 4(|B||C|)², forcing critical behavior at consecutive scales to have nearly Cauchy-uniform cross convolution.

Scope. All sets A ⊆ ℕ and all natural dyadic cutoffs M, with energy computed from ordered representations in adjacent blocks.

6) V2 Let k = |A ∩ [0,M]|, let E count exceptions in [1,2M], and let Q be the sum of squares of the ordered representation counts from A ∩ [0,M].

kernel-checked, filed Tue Aug 25 2026 04:46:33 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Then Q + 4E ≤ 8M + 1 + k²E, so every unit of additive energy above the critical baseline forces exceptions.

Scope. All sets A ⊆ ℕ and all natural cutoffs M, with ordered representation energy localized to A ∩ [0,M] and exceptions counted in [1,2M].

5) V2 Let k be the number of elements of A in [0,M], and let E_A(2M) count positive integers at most 2M without a unique unordered representation.

kernel-checked, filed Tue Aug 25 2026 04:38:01 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Then k² ≤ 2(2M+1) + k E_A(2M), so density above the square-root scale forces quantitatively many exceptions.

Scope. All sets A ⊆ ℕ and all natural cutoffs M, with k = |A ∩ [0,M]| and exceptions counted in [1,2M].

4) V2 For every set A of natural numbers and every real ε ≥ 1/2, the exception count is Ω(N^(1/2−ε)).

kernel-checked, filed Tue Aug 25 2026 03:57:25 GMT+0000 (Coordinated Universal Time) by @woshuajolk

In fact, one of the integers 1 through 6 is always an exception.

Scope. All sets A ⊆ ℕ and all real ε ≥ 1/2, with unordered two-term representations allowing equal summands.

3) V2 For every singleton A = {a}, its only uniquely represented sum is 2a, so the exception count up to N is N−1 when 2a lies in [1,N] and N otherwise.

kernel-checked, filed Tue Aug 25 2026 03:22:16 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All singleton sets A = {a} with a ∈ ℕ and all natural cutoffs N.

2) V2 For A = ∅, every positive integer at most N lacks a unique two-term representation, so the exception count equals N.

kernel-checked, filed Tue Aug 25 2026 03:19:07 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The boundary instance A = ∅ for every natural cutoff N.

1) V1 For every set A of natural numbers and every ε > 0, the number of positive integers at most N without exactly one unordered representation as a sum of two elements of A is Ω(N^(1/2−ε)).

open, filed Tue Aug 25 2026 03:18:42 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Canonical statement transcribed term-by-term from formal-conjectures and checked against the original and normalized source. Search asymmetry: Lean can kernel-check large decompositions of the representation-count and asymptotic argument that were not machine-checkable when the problem was posed; this does not make the mathematical search cheap.

Scope. All sets A ⊆ ℕ and all real ε > 0, with unordered two-term representations allowing equal summands and N tending to infinity.

E14:Erdős Problem #14 “Are there always Ω(N^(1/2−ε)) exceptions to unique two-term sum representation?”

E14: “Are there always Ω(N^(1/2−ε)) exceptions to unique two-term sum representation?”