== A. Golod-Shafarevich polynomials P(t) = 1 - d t + r t^delta
PASS p=5: d=3, r=6, delta=5, P(1/2): -5/16
PASS p=3: d=5, r=15, delta=3, P(1/3): -1/9
PASS p=2: d=7, r=12, delta=2, P(7/24): -1/48
PASS relator counts 3+3 = 6 and 5+10 = 15
PASS 1 - 7t + 12t^2 = (1-3t)(1-4t) (coefficients)
PASS 7/24 is the midpoint of (1/4, 1/3)
== B. Kinyon-Urban polynomial f(t) = 1 - 2t + 3t^p
PASS p=2: 2/(3p) = 1/3 < (p/(2(p-1)))^(p-1) = 1: f > 0 on [0,oo)
PASS p=3: 2/(3p) = 2/9 < (p/(2(p-1)))^(p-1) = 9/16: f > 0 on [0,oo)
PASS p=5: 2/(3p) = 2/15 < (p/(2(p-1)))^(p-1) = 625/4096: f > 0 on [0,oo)
PASS p=7: 2/(3p) = 2/21 >= (p/(2(p-1)))^(p-1) = 117649/2985984: f has a negative minimum
PASS p=7: f(2/3) = -115/729
PASS p=2: discriminant 4 - 12 < 0 (f > 0 everywhere)
== C. Lemma 3 in W_p
PASS p=2: |W_p| = p^(p+1) = 8
PASS p=2: inverse formula
PASS p=2: Lemma 3(ii) w^p = 1 <=> s(v)a = 0, every element: 6 elements with w^p = 1 (= p^p + (p-1)p^(p-1) = 6)
PASS p=2: Lemma 3(i) s is invariant under the shifts (all a, w)
PASS p=2: pi is a homomorphism (all 64 pairs)
PASS p=2: W_p = <(e_1,0), t>
PASS p=2: Z(W_p) = {((c,..,c),0)} (order p) and Z(W_p) lies in ker pi
PASS p=2: ker pi = {(v,0): s(v)=0} (order p^(p-1)) is generated by commutators [(e_k,0), t]
PASS p=3: |W_p| = p^(p+1) = 81
PASS p=3: inverse formula
PASS p=3: Lemma 3(ii) w^p = 1 <=> s(v)a = 0, every element: 45 elements with w^p = 1 (= p^p + (p-1)p^(p-1) = 45)
PASS p=3: Lemma 3(i) s is invariant under the shifts (all a, w)
PASS p=3: pi is a homomorphism (all 6561 pairs)
PASS p=3: W_p = <(e_1,0), t>
PASS p=3: Z(W_p) = {((c,..,c),0)} (order p) and Z(W_p) lies in ker pi
PASS p=3: ker pi = {(v,0): s(v)=0} (order p^(p-1)) is generated by commutators [(e_k,0), t]
PASS p=5: |W_p| = p^(p+1) = 15625
PASS p=5: inverse formula
PASS p=5: Lemma 3(ii) w^p = 1 <=> s(v)a = 0, every element: 5625 elements with w^p = 1 (= p^p + (p-1)p^(p-1) = 5625)
PASS p=5: Lemma 3(i) s is invariant under the shifts (all a, w)
PASS p=5: W_p = <(e_1,0), t>
PASS p=5: Z(W_p) = {((c,..,c),0)} (order p) and Z(W_p) lies in ker pi
PASS p=5: ker pi = {(v,0): s(v)=0} (order p^(p-1)) is generated by commutators [(e_k,0), t]
== C'. axis argument in F_p^2 and the pair statement in W_3
PASS p=3: P, Q, P+Q on the axes => P, Q on a common axis (all 25 pairs)
PASS p=5: P, Q, P+Q on the axes => P, Q on a common axis (all 81 pairs)
PASS p=3: pairs u, v with u^3 = v^3 = (uv)^3 = 1 have pi(u), pi(v) on a common axis: 2025 pairs, 1377 with (uv)^3 = 1
== D. Lemma 4 in D = C_2 wr C_2
PASS Z(D) = {1, z}, z = ((1,1),0)
PASS D is dihedral of order 8 (element orders 1,2,2,2,2,2,4,4; non-abelian)
PASS three maximal subgroups, all of order 4; Phi(D) = {1, z}
PASS g^2 = z^(l1(g) l2(g)) for all g
PASS [g,h] = z^(l1(g)l2(h) + l2(g)l1(h)) for all g, h (both commutator conventions)
PASS <g,h> = D iff (l1,l2)(g), (l1,l2)(h) are independent in F_2^2 (all 64 pairs)
== E. certificate C (p = 2, d = 7, twelve relators)
PASS table: 12 relators, 7 diagonal and 21 upper entries each, all 0/1
PASS check A: ordered independent pairs orthogonal for S_1..S_m, m = 0..12: [16002, 7938, 3938, 1954, 1002, 530, 274, 122, 46, 22, 8, 2, 0]
PASS alpha^T S_k beta equals the exponent sum of Lemma 4 (sampled alpha, beta, all k)
PASS bit evaluation = evaluation of the words by multiplication (20000 random tuples)
PASS check B: generating 7-tuples = 8^7 - 3*4^7 + 2*2^7 = 2048256
PASS check B: epimorphisms onto D killing rho_1..rho_m equal 2^7 x check A for every m = 0..12: [2048256, 1016064, 504064, 250112, 128256, 67840, 35072, 15616, 5888, 2816, 1024, 256, 0]
PASS certificate C: no epimorphism of the 12-relator group onto C_2 wr C_2
== F. single-bit changes of the table
PASS of the 336 single-bit changes of the table, 114 break the certificate and 222 keep it: 114 break, 222 keep
== G. relation elements f = w(1+X) - 1 in truncated free algebras
PASS p=5, d=3: the 6 relation elements lie in F_5, with non-zero degree-5 parts
PASS p=3, d=5: the 15 relation elements lie in F_3, with non-zero degree-3 parts
PASS p=2, d=7: the 12 relation elements lie in F_2 and their degree-2 parts have rank 12 over F_2
== H. the table: checksum, scripts/d8_data.py, and the table printed in ../main.tex
PASS SHA-256 of data/p2_relator_table.json: ae6e2e599e6af8742255beaa3a75042860c6f17a562b49fb12d828b8679c58e4
PASS data/p2_relator_table.json equals the DATA of scripts/d8_certificate.py (read by scripts/d8_data.py)
PASS the table printed in ../main.tex equals the data: 12 rows found
== I. Lemma 5 tested on D_8 and W_3 (a test, not a proof)
PASS W_2: dim F_p[Q]/I^N <= sum_(n<N) c_n, N = 1..6 (lower p-central ranks [2, 1]): dims [1, 3, 5, 7, 8, 8]; bounds [1, 3, 5, 7, 8, 8]
PASS W_2: Jennings ranks [2, 1] give equality dim F_p[Q]/I^N = sum_(n<N) c_n (Jennings' theorem)
PASS W_3: dim F_p[Q]/I^N <= sum_(n<N) c_n, N = 1..16 (lower p-central ranks [2, 1, 1]): dims [1, 3, 7, 12, 19, 27, 36, 45, 54, 62, 69, 74, 78, 80, 81, 81]; bounds [1, 3, 7, 12, 19, 27, 36, 45, 54, 62, 69, 74, 78, 80, 81, 81]
PASS W_3: Jennings ranks [2, 1, 1] give equality dim F_p[Q]/I^N = sum_(n<N) c_n (Jennings' theorem)
== J. sanity values of B_N printed in the note satisfy Lemma 1
PASS p=5: B_1..B_9 = [1, 4, 13, 40, 121, 358, 1057, 3118, 9193] satisfy B_(N+1) >= 1 + d B_N - r B_(N+1-delta)
PASS p=3: B_1..B_6 = [1, 6, 31, 141, 631, 2781] satisfy B_(N+1) >= 1 + d B_N - r B_(N+1-delta)
PASS p=2: B_1..B_5 = [1, 8, 45, 220, 1001] satisfy B_(N+1) >= 1 + d B_N - r B_(N+1-delta)
PASS p=2: 1, 8, 45, 220, 1001 are the partial sums of the coefficients of 1/((1-3t)(1-4t))
== K. arithmetic of the sketch in Remark 7.5 (not part of the theorem)
   Delta(3,3,3): orbifold Euler characteristic 0; index-9 kernel: chi = 0, genus 1
PASS p=3: genus (p-1)(p-2)/2 = 1
   Delta(5,5,5): orbifold Euler characteristic -2/5; index-25 kernel: chi = -10, genus 6
PASS p=5: genus (p-1)(p-2)/2 = 6
   Delta(7,7,7): orbifold Euler characteristic -4/7; index-49 kernel: chi = -28, genus 15
PASS p=7: genus (p-1)(p-2)/2 = 15
   Delta(11,11,11): orbifold Euler characteristic -8/11; index-121 kernel: chi = -88, genus 45
PASS p=11: genus (p-1)(p-2)/2 = 45
PASS p=5: 1 - 12t + t^2 at t = 1/2 equals -19/4 < 0
PASS F_5[C_5 x C_5]: the augmentation ideal has nilpotency index 9 (I^8 != 0, I^9 = 0): dim F_5[C_5^2]/I^N, N = 1..10: [1, 3, 6, 10, 15, 19, 22, 24, 25, 25]
TOTAL failures: 0
running time: 2.2 s
