{
  "schema_version": 1,
  "description": "Finite integer witnesses for the Fuchs extremal problem. Coefficients and moments are checked by verify_pte.py using only Python integers and fractions.",
  "pairs": [
    {"id":"m1","m":1,"A":[0],"B":[1],"expected_degree":0,"expected_coefficients":[1],"main_example":true,"source":"Elementary identity; displayed explicitly."},
    {"id":"m2","m":2,"A":[0,3],"B":[1,2],"expected_degree":1,"expected_coefficients":[1,1],"main_example":true,"source":"Elementary identity; displayed explicitly."},
    {"id":"m3","m":3,"A":[0,4,5],"B":[1,2,6],"expected_degree":3,"expected_coefficients":[1,2,2,1],"main_example":true,"source":"Classical small PTE identity; displayed explicitly."},
    {"id":"m4","m":4,"A":[0,4,7,11],"B":[1,2,9,10],"expected_degree":7,"expected_coefficients":[1,3,5,6,6,5,3,1],"main_example":true,"source":"Classical small PTE identity; displayed explicitly."},
    {"id":"m5","m":5,"A":[0,4,8,16,17],"B":[1,2,10,14,18],"expected_degree":13,"expected_coefficients":[1,4,9,15,21,26,29,29,26,21,15,9,4,1],"main_example":true,"source":"Classical small PTE identity; displayed explicitly."},
    {"id":"m6","m":6,"A":[0,5,6,16,17,22],"B":[1,2,10,12,20,21],"expected_degree":16,"expected_coefficients":[1,5,14,29,49,71,91,105,110,105,91,71,49,29,14,5,1],"main_example":true,"source":"Classical small PTE identity; displayed explicitly."},
    {"id":"m7","m":7,"A":[0,18,27,58,64,89,101],"B":[1,13,38,44,75,84,102],"expected_degree":95,"main_example":true,"source":"Caley, The Prouhet-Tarry-Escott problem, PhD thesis, University of Waterloo, 2012, Section 5.4.2, printed page 48; attributed there to Borwein's 2002 book.","source_url":"http://hdl.handle.net/10012/7205"},
    {"id":"m8","m":8,"A":[0,4,9,23,27,41,46,50],"B":[1,2,11,20,30,39,48,49],"expected_degree":42,"main_example":true,"pure_product_lengths":[1,2,3,5,7,8,11,13],"source":"Cipu, Upper bounds for norms of products of binomials, LMS J. Comput. Math. 7 (2004), Table 1, first n=8 vector. The expansion of that vector is exactly this signed incidence polynomial.","source_url":"https://doi.org/10.1112/S1461157000001030"},
    {"id":"m9","m":9,"A":[0,24,30,83,86,133,157,181,197],"B":[1,17,41,65,112,115,168,174,198],"expected_degree":189,"main_example":true,"source":"First size-nine solution in Coppersmith et al., Ideal solutions in the Prouhet-Tarry-Escott problem, equation (3), after translation by 99; ultimately due to Letac.","source_url":"https://arxiv.org/pdf/2304.11254"},
    {"id":"m10","m":10,"A":[0,12,125,213,214,412,413,501,614,626],"B":[5,6,133,182,242,384,444,493,620,621],"expected_degree":616,"main_example":true,"source":"Borwein, Lisonek and Percival (2003); also the first solution in equation (4) of Coppersmith et al., after translation by 313.","source_url":"https://www.daemonology.net/papers/pte.pdf"},
    {"id":"m12","m":12,"A":[0,11,24,65,90,129,173,212,237,278,291,302],"B":[3,5,30,57,104,116,186,198,245,272,297,299],"expected_degree":290,"main_example":true,"source":"Kuosa, Meyrignac and Shuwen's 1999 size-twelve solution; equation (5) in Coppersmith et al., after translation by 151.","source_url":"https://arxiv.org/pdf/2304.11254"},
    {"id":"m3_nonpalindromic","m":3,"A":[0,9,10],"B":[1,6,12],"expected_degree":9,"expected_coefficients":[1,2,3,4,5,6,6,5,3,1],"require_nonpalindromic":true,"main_example":false,"source":"Explicit elementary counterexample to a universal palindromicity claim; verified directly."},
    {"id":"m7_caley","m":7,"A":[0,14,16,45,54,73,83],"B":[3,5,28,34,65,66,84],"expected_degree":77,"require_nonpalindromic":true,"main_example":false,"source":"Caley, The Prouhet-Tarry-Escott problem, PhD thesis, University of Waterloo, 2012, Theorem 5.4.2, printed page 48.","source_url":"http://hdl.handle.net/10012/7205"},
    {"id":"m8_original_large","m":8,"A":[0,477,758,1273,1879,2394,2675,3152],"B":[7,422,899,1128,2024,2253,2730,3145],"expected_degree":3144,"main_example":false,"source":"Explicit size-eight pair in the supplied manuscript. This record retains that witness; no additional provenance claim is required for its exact verification."}
  ],
  "pure_product_witnesses": [
    {"m":7,"lengths":[1,2,3,4,5,7,11],"expected_l1":16,"expected_l2_squared":16,"source":"Cipu (2004), Table 1, n=7."},
    {"m":8,"lengths":[1,2,3,5,7,8,11,13],"expected_l1":16,"expected_l2_squared":16,"source":"Cipu (2004), Table 1, n=8."},
    {"m":9,"lengths":[1,2,3,4,5,7,9,11,13],"expected_l1":20,"expected_l2_squared":20,"source":"Cipu (2004), Table 1, n=9."},
    {"m":10,"lengths":[1,2,3,4,5,7,9,11,13,17],"expected_l1":24,"expected_l2_squared":24,"source":"Cipu (2004), Table 1, n=10."}
  ],
  "product_exclusion_intervals": [
    {"m":7,"pair_id":"m7","pure_norm_lower":16,"delta":[1,40],"gap_lower":[1,50],"lower_rational":[14243,1000],"upper_rational":[14222,1000]},
    {"m":9,"pair_id":"m9","pure_norm_lower":20,"delta":[1,60],"gap_lower":[1,4],"lower_rational":[18418,1000],"upper_rational":[18146,1000]},
    {"m":10,"pair_id":"m10","pure_norm_lower":24,"delta":[1,35],"gap_lower":[1,10],"lower_rational":[20392,1000],"upper_rational":[20281,1000]}
  ],
  "pure_norm_lower_bound_source": "Maltby (1997), as explicitly recorded by Cipu (2004), page 39: minimum coefficient l1 norms at 7, 9, 10 factors are 16, 20, 24. Integer coefficients imply the same lower bounds for the squared l2 norm. These literature theorems are inputs, not claims re-proved by this script."
}