== (1) small gadgets
small gadgets recomputed: 42 ; split-tau small gadgets and conjugator parity (0=even): {('2.2', '3.1'): 0, ('3.3', '5.1'): 0, ('3.3', '5.3'): 0, ('4.2', '5.1'): 0, ('4.2', '5.3'): 1}
== (2) long gadgets, L = r..45, and Lemma 4 locality
  mu=3 c=(1 2 3) r=3: L-cycle for all L in [3,45]: True; conjugator parities (odd L): {1}; claimed even=False; minL(RESULT)=3
  mu=3 c=(1 3 5) r=5: L-cycle for all L in [5,45]: True; conjugator parities (odd L): {0}; claimed even=True; minL(RESULT)=5
  mu=2.2 c=(1 3)(2 4) r=4: L-cycle for all L in [4,45]: True; conjugator parities (odd L): {1}; claimed even=False; minL(RESULT)=4
  mu=2.2 c=(1 4)(2 6) r=6: L-cycle for all L in [6,45]: True; conjugator parities (odd L): {0}; claimed even=True; minL(RESULT)=7
  mu=5 c=(1 2 3 4 5) r=5: L-cycle for all L in [5,45]: True; conjugator parities (odd L): {1}; claimed even=False; minL(RESULT)=5
  mu=5 c=(1 2 3 5 6) r=6: L-cycle for all L in [6,45]: True; conjugator parities (odd L): {0}; claimed even=True; minL(RESULT)=7
  mu=3.3 c=(1 2 3)(4 5 6) r=6: L-cycle for all L in [6,45]: True; conjugator parities (odd L): {0}; claimed even=True; minL(RESULT)=6
  mu=4.2 c=(1 2 3 5)(4 6) r=6: L-cycle for all L in [6,45]: True; conjugator parities (odd L): {0}; claimed even=True; minL(RESULT)=6
== (3) partner witnesses (local products)
  mu=3 x=(1 2)(3 4) c=(1 2 3) -> xc=(2 4 3) type (3, 1)
  mu=5 x=(1 2)(3 4)(5 6) c=(1 2 3 5 4) -> xc=(2 4)(3 6 5) type (3, 2, 1)
  mu=4.2 x=(1 2)(3 4)(5 6) c=(1 2 3 5)(4 6) -> xc=(2 4 5)(3 6) type (3, 2, 1)
  mu=4.2 x=(1 2)(3 4)(5 6) c=(1 2 3 4)(5 7) -> xc=(2 4)(5 7 6) type (3, 2, 1, 1)
  mu=4.2 x=(1 2 3)(4 5 6) c=(1 2 3 4)(5 6) -> xc=(1 3 5 4 2) type (5, 1)
  mu=4.2 x=(7 8 9) c=(1 2 3 4)(5 6) -> xc=(1 2 3 4)(5 6)(7 8 9) type (4, 3, 2)
== (4) Lemma 7.2: z=(a1..an), t=(a1 a2 a3): zt and tz are n-cycles with odd conjugator from z
  Lemma 7.2 checked for n = 7, 9, 11, 13 (random labellings)
== (5) Theorem B step 1 and step 2 by enumeration of all even types, 24 <= n <= 50 (+ 60, 61, 64, 65)
  even types examined (summed over n): 3567251 ; all five mu: uncovered == E(mu, n) and partners valid: True
== (6) Lemma 7.1 injection, odd n, 7 <= n <= 81
  Lemma 7.1 checked (injective, bounds) for odd n <= 81
== (7) S_n parity identity: #even - #odd partitions = #partitions into distinct odd parts, n <= 60
  identity holds for n <= 60; difference >= 1 for all n >= 3
TOTAL FAILS: 0
