A Heuristic Approach to P ≠ NP Based on the Hyperuniformity of Langford Sequences

A Heuristic Approach to P ≠ NP Based on the Hyperuniformity of Langford Sequences Wino Research* Published on 2025-09-02; revised on 2025-09-19 *[email protected] Introduction to Langford’s Problem Arrange 𝑚 sets of numbers 1 to 𝑛 in a sequence, so that any two consecutive occurrences of 𝑘 are separated by exactly 𝑘 numbers [1–5]. Let 𝐿(𝑚,𝑛) denote the number of distinct Langford sequences up to a reversal symmetry. We have 𝐿(2,3)=𝐿(2,4)=1 and 𝐿(3,9)=3: 3 1 2 1 3 2 4 1 3 1 2 4 3 2 1 9 1 6 1 8 2 5 7 2 6 9 2 5 8 4 7 6 3 5 4 9 3 8 7 4 3 1 9 1 2 1 8 2 4 6 2 7 9 4 5 8 6 3 4 7 5 3 9 6 8 3 5 7 1 8 1 9 1 5 2 6 7 2 8 5 2 9 6 4 7 5 3 8 4 6 3 9 7 4 3 Wino Research, “A Heuristic Approach to P ≠ NP Based on the Hyperuniformity of Langford Sequences” 1 / 17 Asymptotic Formulas for Counting Langford Sequences Conjecture 1. The number of Langford sequences 𝐿(𝑚,𝑛) has the following asymptotic formula [6] 𝐿(𝑚,𝑛)∼𝑛!𝑒−ℓ(𝑚)𝑛,where ℓ(𝑚) is an exponential coefficient depending only on 𝑚, i.e. ℓ(𝑚,𝑛)=1𝑛log𝑛!𝐿(𝑚,𝑛)converges to a constant when 𝑛→∞. Wino Research, “A Heuristic Approach to P ≠ NP Based on the Hyperuniformity of Langford Sequences” 2 / 17 Conjecture 2. The exponential coefficient ℓ(2,𝑛) for the number of Langford sequences 𝐿(2,𝑛) converges to Taniguchi’s constant lim𝑛→∞ℓ(2,𝑛)=∏𝑝∈ℙ(1−3𝑝3+2𝑝4+1𝑝5−1𝑝6)=0.678234491⋯,where the product runs over the primes ℙ. More precisely, the specific value can be approximated as ℓ(2,𝑛)≃∏𝑁𝑖=1(1−3𝑝3𝑖+2𝑝4𝑖+1𝑝5𝑖−1𝑝6𝑖).Wino Research, “A Heuristic Approach to P ≠ NP Based on the Hyperuniformity of Langford Sequences” 3 / 17 Table 1. Number of Langford sequences 𝐿(2,𝑛), OEIS A014552. In Ref. [7], the approximate values 𝐿(2,31)≃5.381⋅1024 and 𝐿(2,32)≃8.812⋅1025 are obtained using a parallel tempering algorithm. exact approximate 𝑛error 𝐿(2,𝑛)ℓ(2,𝑛)ℓ(2,𝑛)𝐿(2,𝑛)310.5972531∼0%410.7945131∼0%0.7656257260.75243824−7.7%81500.699246147−2.0%11177920.7014371.777⋅104−0.1%0.701560121081440.6996661.057⋅105−2.2%Wino Research, “A Heuristic Approach to P ≠ NP Based on the Hyperuniformity of Langford Sequences” 4 / 17 15398096400.6933104.367⋅107+9.7%163267218000.6917033.514⋅108+7.6%192568148912800.6878032.600⋅1011+1.3%0.6871482026363378612000.6867602.616⋅1012−0.8%2337994559425154880.6840454.006⋅1015+5.4%24468451580565159360.6832964.862⋅1016+3.8%0.681745271116836110987649032320.6813091.104⋅1020−1.2%2816073832606093823931520.6807451.627⋅1021+1.2%315.701⋅10240.680305329.240⋅1025354.861⋅10290.679426368.871⋅1030Wino Research, “A Heuristic Approach to P ≠ NP Based on the Hyperuniformity of Langford Sequences” 5 / 17 Relations between Permutations and Langford Sequences For any permutation 𝜎(𝑛) of the set {1,2,…,𝑛}, we know that its Lehmer code forms a factoradic number 𝑥, which can be used to index a permutation in the lexicographic ordering. Since every Langford sequence can be represented as a permutation, such as 1 4 1 5 6 7 4 2 3 5 2 6 3 7 can be represented as (1,4,5,6,7,2,3), an intriguing question arises: how many permutations on the index interval [𝑥,𝑥+𝑑) correspond to Langford sequences? Wino Research, “A Heuristic Approach to P ≠ NP Based on the Hyperuniformity of Langford Sequences” 6 / 17 The Hyperuniformity of Langford Sequences For the Langford pairing problem 𝕃(2,𝑛), the pairing ratio 𝑟(𝑛,𝑑) is defined as 𝑟(𝑛,𝑑)=𝑛!𝜇(𝑛,𝑑)2𝐿(2,𝑛)𝑑,where 𝜇(𝑛,𝑑) is the sampling mean of the number of Langford sequences on the index interval [𝑥,𝑥+𝑑). From Table 2, we can see that 𝑟(𝑛,𝑑)≃1−1𝑑⇒𝜇(𝑛,𝑑)≃2(𝑑−1)𝑒−ℓ(2,𝑛)𝑛.Wino Research, “A Heuristic Approach to P ≠ NP Based on the Hyperuniformity of Langford Sequences” 7 / 17 Table 2. Pairing ratios 𝑟(𝑛,𝑑) for different interval lengths. 107 samples 108 samples 109 samples 𝑟(𝑛,𝑑)𝑛=11𝑛=12𝑛=15𝑛=16𝑛=19𝑛=20𝑑=20.4979500.5042750.5047120.5052640.5048120.494178𝑑=50.7940280.8021890.7951890.7988170.7998360.811909𝑑=100.8983640.9014060.8949490.8953550.8975540.908530𝑑=200.9489100.9505270.9529010.9518050.9532460.943667𝑑=500.9799040.9827320.9832970.9793220.9761780.980197𝑑=1000.9904960.9915680.9877250.9867860.9882950.989223𝑑=2000.9953670.9949590.9931110.9953480.9957930.992925𝑑=5000.9985190.9989650.9980341.0010880.9982810.997641𝑑=10000.9993760.9994800.9988311.0002530.9992080.999534Wino Research, “A Heuristic Approach to P ≠ NP Based on the Hyperuniformity of Langford Sequences” 8 / 17 Bibliography 1. Langford CD. Problem 228. The Mathematical Gazette. 1958;42. 2. Miller JE. Langford's Problem. Available: https://dialectrix.com/ langford.html 3. Walsh T. CSPLib Problem 024: Langford's Number Problem. Jefferson C, Miguel I, Hnich B, Walsh T, Gent IP, editors. Available: https://www. csplib.org/Problems/prob024 Wino Research, “A Heuristic Approach to P ≠ NP Based on the Hyperuniformity of Langford Sequences” 15 / 17 4. Krajecki M, Loiseau J, Alin F, Jaillet C. Many-Core Approaches to Combinatorial Problems: case of the Langford Problem. Supercomputing Frontiers and Innovations. 2016;3: 21–37. doi:10.14529/jsfi160202 5. Akgün Ö, Miguel I. Modelling Langford's Problem: A Viewpoint for Search. 2018. doi:10.48550/arXiv.1808.09847 6. Pan Z. Conjectures on the Number of Langford Sequences. 2021. doi:10.5281/zenodo.13824675 7. Assarpour A, Barnoy A, Liu O. Counting Skolem Sequences. 2015. doi:10.48550/arXiv.1507.00315 Wino Research, “A Heuristic Approach to P ≠ NP Based on the Hyperuniformity of Langford Sequences” 16 / 17 Appendix: Commands in Wino Studio •oeis.langford.count_pairings(n, start, end) •oeis.langford.find_pairings(n, start, end, count) •oeis.langford.pairing_moduli(n, start, end, m) •oeis.langford.estimate_pairings(n, d, samples) •oeis.langford.pairing_density(n, d, samples) •oeis.langford.pairing_ratio(n, d, samples) Examples Input: oeis.langford.count_pairings(12, 1000000, 2000000) Output: 622 Wino Research, “A Heuristic Approach to P ≠ NP Based on the Hyperuniformity of Langford Sequences” 17 / 17