Optimality of Kl{\o}ve Arrays within the Symmetric Kl{\o}ve-Mossige Class

1 Problem and precise statement

• The optimization question arises because increasing the reflection shift can increase aperture while changing overlap and therefore sensor count.At fixed sensor count, the paper compares arrays only after controlling that overlap.Source: §1 · p. 1

1.1 Relation to additive bases and prior constructions

• Earlier work established the classical KA construction and search within its parameters; this proof eliminates non-KA optimizers in the larger S-KMA class.Kohonen’s searches instead address the larger class of restricted additive bases, which need not be symmetric.Source: §1.1 · p. 2

1.2 Definitions and main theorem

• For every N ≥2, every maximum-aperture feasible S-KMA sensor set belongs to the Kløve-array class, so L∗(N) = ℓ(N).The theorem identifies the optimizing sets, not only the optimal aperture value.Source: §1 · p. 1, §1.2 · p. 2

1.3 Proof strategy and computational scope

• The proof computes generator difference information and the first missing sum, then uses reflection intersections to determine feasible shifts and exact sensor counts.It classifies overlaps and bounds hypothetical non-KA maximizers; remaining cases are KA sets, strictly dominated, or satisfy one of two upper bounds.Source: §1.3 · p. 3
• For 2 ≤N ≤329, bounded integer enumeration checks feasible shifts and equality sets, with every equality case belonging to K as a sensor set.The finite certificate supplies the remaining sizes and matches equality cases to Kløve arrays.Source: §1.3 · p. 3
• For N ≥330, a classical Kløve construction exceeds the exclusion bounds for hypothetical non-KA optimizers, ruling out such optimizers.The exclusion bounds apply after domination alternatives are removed; they are not bounds on every non-KA array.Source: §1.3 · pp. 3–4

Generator identities

• The generator blocks are disjoint and symmetric, with A + B = [2M + 1, m + M] and B − A = [M + 1, m].For z = 0, n = a and the B − A identity still holds.Source: §2 · p. 4

First hole

• For z = 0, the first missing sum is H = 2M + 1; for z ≥ 1, its value depends on the interval and exceptional parameter cases.The proof tests the next candidate sums against B + B after covering [0, m + M] with A + A and A + B.Source: §2 · pp. 4–5

Feasible shifts

• A shift λ is feasible exactly when 0 ≤ λ ≤ H, where H is the first missing sum of the generator’s self-sumset.Below λ, only G + G contributes to the sum co-array.Source: §2 · pp. 4–5

Overlap and sensor count

• For z ≥ 1, the sensor count is N = 2n − ρ(λ), where ρ(λ) counts overlap between G and its shifted reflection.For λ > M, the cross-overlap r(λ) is zero.Source: §2 · p. 5

Kløve-array shifts

• At λ = 2M + 1 + kP, the reflected union is KA(x, y, z + k).Each such KA is feasible.Source: §2 · p. 5

Zero-overlap shifts

• A feasible shift λ < H with zero overlap is not optimal at its sensor count, because shifting to H preserves 2n sensors and increases aperture.At H, ρ(H) = 0.Source: §2 · p. 6

Fixed-x relaxation

• The fixed-x aperture relaxation is U_x(V) = V^2 + 2Vx + 6V − 23x^2 + 6x − 7.It is obtained by eliminating t from the sensor-count and aperture formulas.Source: §3 · p. 6

Maximizing the relaxation

• The real-x maximum of the relaxation is attained at x = (N + 6)/23.The fixed-x relaxations are identified as those of [5, Theorem 4].Source: §3 · p. 6

4.1 Positive overlap above the CNA aperture

• For x, y, z ≥ 1 and M < λ ≤ H with positive overlap, the array is a KA or satisfies L ≤ Φ(x, N).The proof also handles the exceptional endpoint H = m + M + 2 when y = 1 and x ≥ 2.Source: §4.1 · p. 7

4.2 Overlap at or below the CNA aperture

• For x, y, z ≥ 1 and 0 ≤ λ ≤ M, each array is a KA, is KA-dominated, or has L ≤ Φ(x, N).The comparison controls the sensor-count difference s against aperture difference τ relative to KA(x, y, t).Source: §4.2 · p. 7

4.3 Degenerate generators

• For degenerate generators, the lemmas likewise yield a KA, KA-domination, or—when z = 0 or y = 0—the bound L ≤ (N + 3)^2/8 − 2.The x = 0 case is covered by KA membership or domination.Source: §4.3 · pp. 8–9

4 Upper bounds for a non-Kløve optimizer

• For an optimal non-Kløve S-KMA with N ≥ 6, Proposition 4.6 reduces the possibilities to two aperture bounds.The proof removes zero-overlap shifts below H, handles positive-overlap shifts with the preceding lemmas, and excludes Kløve-array membership and domination alternatives.Source: §4.3 · p. 9

Kløve’s construction

• For every integer N ≥ 43, Kløve’s construction gives ℓ(N) ≥ U(N) − 72.The reparameterization uses z = 9 and chooses x with parity matching N + 1 near (N − 20)/23.Source: §5 · p. 10

Infinite range

• For N ≥ 330, the bounds for a hypothetical non-KA optimizer contradict max D < U(N) − 72 ≤ ℓ(N) ≤ L*(N) = max D.The finite certificate covers the remaining sensor counts.Source: §5 · p. 10

6 The finite range

• For every 2 ≤ N ≤ 329, the certified aperture bound holds for every feasible S-KMA, and equality implies the set belongs to K.Together with the result for N ≥ 330, this completes the theorem for all N ≥ 2.Source: §6 · p. 10, §6.3 · p. 12

6.1 A complete finite search space

• The finite search covers feasible generators and shifts and retains all maximum-aperture KA sets at each sensor count.The search space includes every feasible shift and every relevant KA comparison set.Source: §6 · p. 10, §6.1 · p. 11

6.2 Checks and their correctness

• The checks compute sensor count from overlap and compare each equality case as a complete coordinate set with the maximizing KA sets.Matching optimum apertures alone would not establish KA membership for every equality case.Source: §6.2 · p. 11

6.3 Execution and independent checks

• 246,028 generators and 1,783 equality occurrences were checked in the direct integer run over N = 2, . . . , 329.A separate Python witness check validated 1,783 witnesses representing 401 distinct sensor sets, but not enumeration completeness.Source: §6.3 · pp. 11–12
• The Python bitset verifier independently checked potentially optimal co-arrays only through N = 44; the direct and formula-based C verifiers share components.The finite computation handles the bounded range; the analytic argument is required for larger sensor counts.Source: §6.3 · p. 12

7 Consequence for exact optimization

• For any N ≥ 2, an optimal contiguous-sum S-KMA and its aperture can be found with O(N log N) exact integer objective evaluations.Keeping all maximizing parameter triples represents every optimal sensor set, possibly more than once.Source: §7 · p. 12
• The O(N log N) bound counts arithmetic evaluations, not bit operations or the cost of materializing and deduplicating all optimal sets.A maximizing triple takes O(N) further operations to construct its set.Source: §7 · p. 12
• The search is the known KA grid search of [5, Proposition 3 and Algorithm 1], now guaranteed to cover the full feasible S-KMA class.The theorem makes the existing search sufficient; no new search algorithm is claimed.Source: §7 · p. 12

8 Concluding remarks

• Classifying overlap yields KA membership, strict domination at the same sensor count, or an exclusion bound for a hypothetical optimizer.Comparing the resulting bounds with a classical construction reduces the proof to finite verification.Source: §8 · p. 13
• The theorem does not count distinct maximizing sets, and the number of such sets across sensor counts remains an open question.The threshold 330 is sufficient for the comparison, rather than a claimed structural transition.Source: §8 · p. 13

A Alignment with the reference definitions

• The middle-block period is P = M + x2 + 1, whereas equation (19) displays the shift increment as M + x2.Using the definitions gives λ = 15 and aperture 25 for (x, y, z) = (1, 2, 1); the displayed increment gives aperture 24 and a different set.Source: A Alignment with the reference… · p. 13
• The observations identify the definitions used in the proof; they do not assert a publisher-issued corrigendum or priority for these observations.The paper says its definitions follow the arXiv v2 version of.Source: A Alignment with the reference… · p. 13
• For (x, y, z) = (1, 1, 1), the first hole in G + G is 9, so the stated exception requires x ≥ 2.The z = 0 case is handled separately.Source: A Alignment with the reference… · p. 13

• Table 1 records the full direct run for Proposition 6.1 through N = 329, counting parameter occurrences and shifts rather than distinct arrays.Source: B Computational record and reproduction · p. 13, Table 1 · p. 13
• A formula-based run visits 28,230,844 candidates and reproduces the aperture and equality-count vectors; independent witness and symbolic checks add corroboration.Source: B Computational record and reproduction · p. 14
• The independent witness checker runs with ordinary Python 3 from the verification directory using python audits/check_witnesses.py.Source: B Computational record and reproduction · p. 14
• 1,783 occurrences and 401 distinct sets are reported with PASS and agreement of all per-N counts; full checks agree with Table 1 through N = 329.Source: B Computational record and reproduction · p. 14
• Full Python checks use integer convolution and NumPy, SymPy is only for auxiliary symbolic checks, and the supplement supplies pinned dependencies and run commands.Source: B Computational record and reproduction · p. 14
• Full Python mode checks certificates through N = 329, while quick mode stops at 44 as an installation check; sanitized direct C output is identical.Source: B Computational record and reproduction · p. 14

• Claude and GPT assisted with code and proof development, logical checks, literature searches, and drafting portions of the text.Source: Declarations · p. 14
• The authors assume responsibility for all content.Source: Declarations · p. 14