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STEINER DIVISIBILITY SIEVE for S(k-1,k,2k), k = 2..300
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Claim under test:  S(k-1,k,2k) satisfies design divisibility  <=>  k+1 is prime
Mismatches found: NONE  (equivalence holds on 2..300)

Survivors (divisibility passes) up to 300:
  k        : [2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 256, 262, 268, 270, 276, 280, 282, 292]
  k+1      : [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293]
  all k+1 prime? -> True

Worked obstruction for a few composite k+1 (first failing design equation):
  k=  3 (k+1=  4, composite): NO  -- lambda_1 = C(5,1)/2 = 5/2 (not integer)
  k=  5 (k+1=  6, composite): NO  -- lambda_2 = C(8,2)/3 = 28/3 (not integer)
  k=  7 (k+1=  8, composite): NO  -- lambda_1 = C(13,5)/6 = 1287/6 (not integer)
  k=  8 (k+1=  9, composite): NO  -- lambda_2 = C(14,5)/6 = 2002/6 (not integer)
  k=  9 (k+1= 10, composite): NO  -- lambda_3 = C(15,5)/6 = 3003/6 (not integer)
  k= 11 (k+1= 12, composite): NO  -- lambda_2 = C(20,8)/9 = 125970/9 (not integer)
  k= 13 (k+1= 14, composite): NO  -- lambda_6 = C(20,6)/7 = 38760/7 (not integer)
  k= 14 (k+1= 15, composite): NO  -- lambda_2 = C(26,11)/12 = 7726160/12 (not integer)
  k= 15 (k+1= 16, composite): NO  -- lambda_1 = C(29,13)/14 = 67863915/14 (not integer)

Strict-gap check: if A(2k,4,k) <= C_k - 1 then chi >= ceil(binom(2k,k)/(C_k-1)):
  k=   3:  ceil( binom(2k,k) / (C_k - 1) ) = 5   (= k+2 ? True)
  k=   5:  ceil( binom(2k,k) / (C_k - 1) ) = 7   (= k+2 ? True)
  k=   7:  ceil( binom(2k,k) / (C_k - 1) ) = 9   (= k+2 ? True)
  k=   8:  ceil( binom(2k,k) / (C_k - 1) ) = 10   (= k+2 ? True)
  k=   9:  ceil( binom(2k,k) / (C_k - 1) ) = 11   (= k+2 ? True)
  k=  10:  ceil( binom(2k,k) / (C_k - 1) ) = 12   (= k+2 ? True)
  k=  16:  ceil( binom(2k,k) / (C_k - 1) ) = 18   (= k+2 ? True)
  k= 112:  ceil( binom(2k,k) / (C_k - 1) ) = 114   (= k+2 ? True)

Open YES-candidates (sieve cannot kill) with 3<=k<=500  ==  {k : k+1 prime}:
  first several k: [4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42] ...
  smallest: 4  (k+1 = 5 , prime: True )

Wrote sieve_835_table.json