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A+C+E. Exhaustive enumeration, n = 1..26 (patterns with 1 in A; the symmetry A -> -A is checked for n <= 14)
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n= 1: max#missing=1  dist(up to sign)={1: 1}  pairs odd-diff:True sum>=n+1:True; full 2^n enumeration = 2 x reduced: True  [0.0s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[1, 1, 1] min=[0, 0, 0]; floor((m-1)^2/4)=[0, 0, 0]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..1; #patterns violating the count bound 2t+2: 0
n= 2: max#missing=2  dist(up to sign)={1: 1, 2: 1}  pairs odd-diff:True sum>=n+1:True; full 2^n enumeration = 2 x reduced: True  [0.0s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[1, 1, 1] min=[0, 0, 0]; floor((m-1)^2/4)=[0, 0, 0]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..2; #patterns violating the count bound 2t+2: 0
n= 3: max#missing=2  dist(up to sign)={1: 3, 2: 1}  pairs odd-diff:True sum>=n+1:True; full 2^n enumeration = 2 x reduced: True  [0.0s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[1, 1, 2] min=[0, 0, 0]; floor((m-1)^2/4)=[0, 0, 0]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..3; #patterns violating the count bound 2t+2: 0
n= 4: max#missing=2  dist(up to sign)={0: 2, 1: 3, 2: 3}  pairs odd-diff:True sum>=n+1:True; full 2^n enumeration = 2 x reduced: True  [0.0s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[1, 1, 2] min=[0, 0, 0]; floor((m-1)^2/4)=[0, 0, 0]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..3; #patterns violating the count bound 2t+2: 0
n= 5: max#missing=2  dist(up to sign)={0: 6, 1: 7, 2: 3}  pairs odd-diff:True sum>=n+1:True; full 2^n enumeration = 2 x reduced: True  [0.0s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[1, 2, 3] min=[0, 0, 1]; floor((m-1)^2/4)=[0, 0, 1]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..4; #patterns violating the count bound 2t+2: 0
n= 6: max#missing=2  dist(up to sign)={0: 16, 1: 11, 2: 5}  pairs odd-diff:True sum>=n+1:True; full 2^n enumeration = 2 x reduced: True  [0.0s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[1, 2, 3] min=[0, 0, 1]; floor((m-1)^2/4)=[0, 0, 1]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..5; #patterns violating the count bound 2t+2: 0
n= 7: max#missing=2  dist(up to sign)={0: 40, 1: 19, 2: 5}  pairs odd-diff:True sum>=n+1:True; full 2^n enumeration = 2 x reduced: True  [0.0s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[1, 2, 4] min=[0, 0, 2]; floor((m-1)^2/4)=[0, 0, 2]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..5; #patterns violating the count bound 2t+2: 0
n= 8: max#missing=2  dist(up to sign)={0: 92, 1: 27, 2: 9}  pairs odd-diff:True sum>=n+1:True; full 2^n enumeration = 2 x reduced: True  [0.0s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[1, 2, 4] min=[0, 0, 2]; floor((m-1)^2/4)=[0, 0, 2]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..6; #patterns violating the count bound 2t+2: 0
       claimant's signed_q2.log for n=8: (0, 0, 2)  -> agree: True
n= 9: max#missing=2  dist(up to sign)={0: 204, 1: 43, 2: 9}  pairs odd-diff:True sum>=n+1:True; full 2^n enumeration = 2 x reduced: True  [0.0s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[1, 3, 5] min=[0, 1, 4]; floor((m-1)^2/4)=[0, 1, 4]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..7; #patterns violating the count bound 2t+2: 0
n=10: max#missing=2  dist(up to sign)={0: 432, 1: 67, 2: 13}  pairs odd-diff:True sum>=n+1:True; full 2^n enumeration = 2 x reduced: True  [0.0s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[1, 3, 5] min=[0, 1, 4]; floor((m-1)^2/4)=[0, 1, 4]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..7; #patterns violating the count bound 2t+2: 1, e.g. signs(1..n)=+-----++++ sorted r=[1, 1, 2, 2, 2, 2, 2, 4, 5, 7]
       claimant's signed_q2.log for n=10: (0, 1, 4)  -> agree: True
n=11: max#missing=2  dist(up to sign)={0: 912, 1: 99, 2: 13}  pairs odd-diff:True sum>=n+1:True; full 2^n enumeration = 2 x reduced: True  [0.0s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[2, 3, 6] min=[0, 1, 6]; floor((m-1)^2/4)=[0, 1, 6]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..8; #patterns violating the count bound 2t+2: 1, e.g. signs(1..n)=+------++++ sorted r=[1, 1, 2, 2, 2, 2, 2, 3, 5, 6, 8]
n=12: max#missing=2  dist(up to sign)={0: 1876, 1: 155, 2: 17}  pairs odd-diff:True sum>=n+1:True; full 2^n enumeration = 2 x reduced: True  [0.0s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[2, 3, 6] min=[0, 1, 6]; floor((m-1)^2/4)=[0, 1, 6]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..9; #patterns violating the count bound 2t+2: 11, e.g. signs(1..n)=+-+++++----- sorted r=[2, 2, 2, 2, 2, 3, 3, 3, 3, 6, 7, 8]
       claimant's signed_q2.log for n=12: (0, 1, 6)  -> agree: True
n=13: max#missing=2  dist(up to sign)={0: 3860, 1: 219, 2: 17}  pairs odd-diff:True sum>=n+1:True; full 2^n enumeration = 2 x reduced: True  [0.0s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[2, 4, 7] min=[0, 2, 9]; floor((m-1)^2/4)=[0, 2, 9]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..9; #patterns violating the count bound 2t+2: 9, e.g. signs(1..n)=++-+++++----- sorted r=[2, 2, 2, 3, 3, 3, 3, 3, 3, 5, 6, 7, 9]
n=14: max#missing=2  dist(up to sign)={0: 7834, 1: 335, 2: 23}  pairs odd-diff:True sum>=n+1:True; full 2^n enumeration = 2 x reduced: True  [0.0s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[2, 4, 7] min=[0, 2, 9]; floor((m-1)^2/4)=[0, 2, 9]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..10; #patterns violating the count bound 2t+2: 23, e.g. signs(1..n)=+-+-+++++----- sorted r=[2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 6, 8, 9]
       claimant's signed_q2.log for n=14: (0, 2, 9)  -> agree: True
n=15: max#missing=2  dist(up to sign)={0: 15898, 1: 463, 2: 23}  pairs odd-diff:True sum>=n+1:True  [0.0s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[2, 4, 8] min=[0, 2, 12]; floor((m-1)^2/4)=[0, 2, 12]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..11; #patterns violating the count bound 2t+2: 27, e.g. signs(1..n)=+-+++++++------ sorted r=[2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 6, 9, 10, 11]
n=16: max#missing=2  dist(up to sign)={0: 32034, 1: 703, 2: 31}  pairs odd-diff:True sum>=n+1:True  [0.0s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[2, 4, 8] min=[0, 2, 12]; floor((m-1)^2/4)=[0, 2, 12]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..11; #patterns violating the count bound 2t+2: 77, e.g. signs(1..n)=++-+++++++------ sorted r=[2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 6, 8, 9, 10, 12]
       claimant's signed_q2.log for n=16: (0, 2, 12)  -> agree: True
n=17: max#missing=2  dist(up to sign)={0: 64546, 1: 959, 2: 31}  pairs odd-diff:True sum>=n+1:True  [0.0s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[2, 5, 9] min=[0, 4, 16]; floor((m-1)^2/4)=[0, 4, 16]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..12; #patterns violating the count bound 2t+2: 92, e.g. signs(1..n)=+--++++++-------- sorted r=[3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 8, 10, 10, 13]
n=18: max#missing=2  dist(up to sign)={0: 129576, 1: 1459, 2: 37}  pairs odd-diff:True sum>=n+1:True  [0.0s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[2, 5, 9] min=[0, 4, 16]; floor((m-1)^2/4)=[0, 4, 16]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..13; #patterns violating the count bound 2t+2: 199, e.g. signs(1..n)=+--+++++++-------- sorted r=[3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 9, 11, 11, 14]
       claimant's signed_q2.log for n=18: (0, 4, 16)  -> agree: True
n=19: max#missing=2  dist(up to sign)={0: 260136, 1: 1971, 2: 37}  pairs odd-diff:True sum>=n+1:True  [0.1s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[2, 5, 10] min=[0, 4, 20]; floor((m-1)^2/4)=[0, 4, 20]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..13; #patterns violating the count bound 2t+2: 320, e.g. signs(1..n)=+-++-++++++-------- sorted r=[4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 7, 8, 8, 12, 13, 13]
n=20: max#missing=2  dist(up to sign)={0: 521264, 1: 2979, 2: 45}  pairs odd-diff:True sum>=n+1:True  [0.2s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[2, 5, 10] min=[0, 4, 20]; floor((m-1)^2/4)=[0, 4, 20]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..14; #patterns violating the count bound 2t+2: 656, e.g. signs(1..n)=+-+-+++++++--------- sorted r=[4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 10, 10, 12, 14, 15]
       claimant's signed_q2.log for n=20: (0, 4, 20)  -> agree: True
n=21: max#missing=2  dist(up to sign)={0: 1044528, 1: 4003, 2: 45}  pairs odd-diff:True sum>=n+1:True  [0.3s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[3, 6, 11] min=[1, 6, 25]; floor((m-1)^2/4)=[1, 6, 25]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..15; #patterns violating the count bound 2t+2: 979, e.g. signs(1..n)=+-++-+++++++--------- sorted r=[3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 7, 9, 10, 10, 14, 15, 15]
n=22: max#missing=2  dist(up to sign)={0: 2091066, 1: 6031, 2: 55}  pairs odd-diff:True sum>=n+1:True  [0.7s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[3, 6, 11] min=[1, 6, 25]; floor((m-1)^2/4)=[1, 6, 25]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..15; #patterns violating the count bound 2t+2: 2217, e.g. signs(1..n)=+---++++++++---------- sorted r=[4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 10, 12, 13, 15, 18]
       claimant's signed_q2.log for n=22: (1, 6, 25)  -> agree: True
n=23: max#missing=2  dist(up to sign)={0: 4186170, 1: 8079, 2: 55}  pairs odd-diff:True sum>=n+1:True  [1.3s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[3, 6, 12] min=[1, 6, 30]; floor((m-1)^2/4)=[1, 6, 30]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..16; #patterns violating the count bound 2t+2: 3383, e.g. signs(1..n)=+--+-+++++++----------- sorted r=[5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 10, 10, 12, 16, 16, 17]
n=24: max#missing=2  dist(up to sign)={0: 8376386, 1: 12159, 2: 63}  pairs odd-diff:True sum>=n+1:True  [2.7s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[3, 6, 12] min=[1, 6, 30]; floor((m-1)^2/4)=[1, 6, 30]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..17; #patterns violating the count bound 2t+2: 7271, e.g. signs(1..n)=+-+--++++++++----------- sorted r=[5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 10, 12, 13, 16, 17, 18]
       claimant's signed_q2.log for n=24: (1, 6, 30)  -> agree: True
n=25: max#missing=2  dist(up to sign)={0: 16760898, 1: 16255, 2: 63}  pairs odd-diff:True sum>=n+1:True  [5.6s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[3, 7, 13] min=[1, 9, 36]; floor((m-1)^2/4)=[1, 9, 36]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..17; #patterns violating the count bound 2t+2: 11095, e.g. signs(1..n)=+-+-++-+++++++----------- sorted r=[5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 9, 10, 10, 13, 15, 17, 17, 18]
n=26: max#missing=2  dist(up to sign)={0: 33529934, 1: 24423, 2: 75}  pairs odd-diff:True sum>=n+1:True  [11.7s]
       Q2: min_A S_m for m=ceil(eps n), eps=(0.1, 0.25, 0.5): m=[3, 7, 13] min=[1, 9, 36]; floor((m-1)^2/4)=[1, 9, 36]; min_(A,i)[r_(i)-ceil((i-2)/8)]=0, [r_(i)-ceil((i-2)/4)]=0, per-parity [r_(i)-floor(i/2)]=0 (all >=0: proved bounds)
       all m=1..n: min_A S_m >= floor((m-1)^2/4): True; equality for m=1..18; #patterns violating the count bound 2t+2: 23722, e.g. signs(1..n)=++-+-++++-++++------------ sorted r=[4, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 13, 14, 14, 15, 17, 18, 18]
       claimant's signed_q2.log for n=26: (1, 9, 36)  -> agree: True
[section A/E exhaustive part done, 22.6s]
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A'. Sharpness: A = [1,m] u -[m+1,n] misses exactly {m, m+1} for every m with n/2 <= m <= n-1
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all n in [2,40], all m in [ceil(n/2), n-1]: True  (the claimant's example is m = ceil(n/2))
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B. (L1) is an exact characterisation: d missing  <=>  s_{j+d} != s_j (1<=j<=n-d)  and  s_j = s_{d-j} (1<=j<d)
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n <= 16, every d, every pattern: equivalence holds: True
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C. Extremal sets: which pairs {d<d'} can be the two missing values?  Rule R: d+d' odd, d+d' >= n+1, gcd(d,d')=1
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n=6: exhaustive pairs [(1, 6), (2, 5), (3, 4), (4, 5), (5, 6)]; each realised by exactly one A up to sign: True; equals rule R: True
n=7: exhaustive pairs [(2, 7), (4, 5), (4, 7), (5, 6), (6, 7)]; each realised by exactly one A up to sign: True; equals rule R: True
n=10: exhaustive pairs [(1, 10), (2, 9), (3, 8), (3, 10), (4, 7), (4, 9), (5, 6), (5, 8), (6, 7), (7, 8), (7, 10), (8, 9), (9, 10)]; each realised by exactly one A up to sign: True; equals rule R: True
n=11: exhaustive pairs [(2, 11), (3, 10), (4, 9), (4, 11), (5, 8), (6, 7), (6, 11), (7, 8), (7, 10), (8, 9), (8, 11), (9, 10), (10, 11)]; each realised by exactly one A up to sign: True; equals rule R: True
n = 2..26: exhaustive extremal pairs = rule R, each realised by exactly one A up to sign: True
GF(2) solver, n = 2..64: feasible pairs = rule R, each with a unique solution up to sign: True
number of extremal A (exactly n-2 differences hit) up to A -> -A, n = 2..64:
  [1, 1, 3, 3, 5, 5, 9, 9, 13, 13, 17, 17, 23, 23, 31, 31, 37, 37, 45, 45, 55, 55, 63, 63, 75, 75, 87, 87, 95, 95, 111, 111, 127, 127, 139, 139, 157, 157, 173, 173, 185, 185, 205, 205, 227, 227, 243, 243, 263, 263, 287, 287, 305, 305, 329, 329, 357, 357, 373, 373, 403, 403, 435]
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D. Source's remark: variant 'exactly one of k, 1-k' (A in [1-n, n]); max #missing d in [1,n], exhaustive n <= 18
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n= 2: max #missing = 1; the even-integers choice misses 1 = ceil(n/2) = 1
n= 3: max #missing = 2; the even-integers choice misses 2 = ceil(n/2) = 2
n= 4: max #missing = 2; the even-integers choice misses 2 = ceil(n/2) = 2
n= 5: max #missing = 3; the even-integers choice misses 3 = ceil(n/2) = 3
n= 6: max #missing = 3; the even-integers choice misses 3 = ceil(n/2) = 3
n= 7: max #missing = 4; the even-integers choice misses 4 = ceil(n/2) = 4
n= 8: max #missing = 4; the even-integers choice misses 4 = ceil(n/2) = 4
n= 9: max #missing = 5; the even-integers choice misses 5 = ceil(n/2) = 5
n=10: max #missing = 5; the even-integers choice misses 5 = ceil(n/2) = 5
n=11: max #missing = 6; the even-integers choice misses 6 = ceil(n/2) = 6
n=12: max #missing = 6; the even-integers choice misses 6 = ceil(n/2) = 6
n=13: max #missing = 7; the even-integers choice misses 7 = ceil(n/2) = 7
n=14: max #missing = 7; the even-integers choice misses 7 = ceil(n/2) = 7
n=15: max #missing = 8; the even-integers choice misses 8 = ceil(n/2) = 8
n=16: max #missing = 8; the even-integers choice misses 8 = ceil(n/2) = 8
n=17: max #missing = 9; the even-integers choice misses 9 = ceil(n/2) = 9
n=18: max #missing = 9; the even-integers choice misses 9 = ceil(n/2) = 9
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F. Q2 at larger n
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pair lemma, |Def(d)| = 2r(d), mirror symmetry of Def(d), 2 distinct charged defects per pair, and the per-parity bound p <= 2t+1: True on 69 sets (n = 30, 61, 120)
simple families at n = 1200: S_m / n^2 for m = eps*n
   [1,n]                       : eps=0.5: 0.1248 (eps^2/4=0.0625)  eps=0.25: 0.0311 (eps^2/4=0.0156)  eps=0.1: 0.0050 (eps^2/4=0.0025)
   odd positives u -(evens)    : eps=0.5: 0.0831 (eps^2/4=0.0625)  eps=0.25: 0.0207 (eps^2/4=0.0156)  eps=0.1: 0.0033 (eps^2/4=0.0025)
   half-and-half m=n/2         : eps=0.5: 0.0831 (eps^2/4=0.0625)  eps=0.25: 0.0207 (eps^2/4=0.0156)  eps=0.1: 0.0033 (eps^2/4=0.0025)
   two-block m=0.625n          : eps=0.5: 0.0623 (eps^2/4=0.0625)  eps=0.25: 0.0155 (eps^2/4=0.0156)  eps=0.1: 0.0025 (eps^2/4=0.0025)
simulated annealing from random starts (2 runs x 30000 flips) vs best two-block set and floor((m-1)^2/4):
   n= 40 eps= 0.1: m=  4  annealing best=    2  two-block best=    2  floor((m-1)^2/4)=    2  proved lower bound ceil((m-1)(m-2)/8)=1
   n= 40 eps=0.25: m= 10  annealing best=   20  two-block best=   20  floor((m-1)^2/4)=   20  proved lower bound ceil((m-1)(m-2)/8)=9
   n= 40 eps= 0.5: m= 20  annealing best=   90  two-block best=   90  floor((m-1)^2/4)=   90  proved lower bound ceil((m-1)(m-2)/8)=43
   n= 80 eps= 0.1: m=  8  annealing best=   12  two-block best=   12  floor((m-1)^2/4)=   12  proved lower bound ceil((m-1)(m-2)/8)=6
   n= 80 eps=0.25: m= 20  annealing best=   90  two-block best=   90  floor((m-1)^2/4)=   90  proved lower bound ceil((m-1)(m-2)/8)=43
   n= 80 eps= 0.5: m= 40  annealing best=  380  two-block best=  380  floor((m-1)^2/4)=  380  proved lower bound ceil((m-1)(m-2)/8)=186
   n=160 eps= 0.1: m= 16  annealing best=   56  two-block best=   56  floor((m-1)^2/4)=   56  proved lower bound ceil((m-1)(m-2)/8)=27
   n=160 eps=0.25: m= 40  annealing best=  380  two-block best=  380  floor((m-1)^2/4)=  380  proved lower bound ceil((m-1)(m-2)/8)=186
   n=160 eps= 0.5: m= 80  annealing best= 1560  two-block best= 1560  floor((m-1)^2/4)= 1560  proved lower bound ceil((m-1)(m-2)/8)=771
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ALL CHECKS PASSED: True    [total 35.5s]
