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3699 bytes, as of the version from 2026-10-27 20:42 (current). Recorded here, not run.
"""Independent exact checks for T560; Python rational arithmetic, no generator data.
CMSZ section 3.1: B(u)=(u/2)/sinh(u/2), F(u)=sum (j-1)! B_j u^j,
A(t)=1/Q^{-1}(t), Q(u)=u exp(F(u)). Lagrange inversion gives
[t^n](t A(t)) = [u^n]exp((1-n)F(u))/(1-n), n >= 2.
For the signed spin series use G(u)=-sum B_(2j)u^(2j)/(2j*2^j)
and Q_spin(u)=u exp(-G(u)); see CMSZ section 6.3. The conversion is
2*(-1)^g*2^(2g)/((2g-1)*(2g-1)!) in both cases. This includes the
factor 2 from the EMZ volume convention, also documented in the independent
surface-dynamics implementation. Hyperelliptic spin is floor((g+1)/2) mod 2.
The code uses coefficient recurrences and Lagrange inversion, not Sage's
series inverse or surface-dynamics' executable implementation.
"""
from fractions import Fraction as F
from math import factorial
def inverse_series(a):
b = [1 / a[0]]
for n in range(1, len(a)):
b.append(-sum(a[k] * b[n-k] for k in range(1, n+1)) / a[0])
return b
def exp_series(a):
assert a[0] == 0
b = [F(1)]
for n in range(1, len(a)):
b.append(sum(k * a[k] * b[n-k] for k in range(1, n+1)) / n)
return b
def controls():
assert exp_series([F(0), F(1)] + [F(0)]*7) == [F(1, factorial(n)) for n in range(9)]
assert inverse_series([F(1), F(-1)] + [F(0)]*7) == [F(1)]*9
assert minimal_total(1) == F(1, 3)
assert spin_difference(1) == F(-1, 3)
assert hyperelliptic(2, 1) == F(1, 120)
assert hyperelliptic(2, 2) == F(1, 135)
def minimal_total(g):
n = 2*g
sinh_over = [F(0)]*(n+1)
for j in range(g+1):
sinh_over[2*j] = F(1, 2**(2*j)*factorial(2*j+1))
B = inverse_series(sinh_over)
log_q = [F(0)] + [factorial(j-1)*B[j] for j in range(1,n+1)]
tA_n = exp_series([(1-n)*x for x in log_q])[n] / (1-n)
return F(2 * (-1)**g * 2**n, (n-1)*factorial(n-1)) * tA_n
def spin_difference(g):
n = 2*g
# t/(exp(t)-1): coefficient B_j/j!.
b = inverse_series([F(1, factorial(j+1)) for j in range(n+1)])
log_p = [F(0)]*(n+1)
for j in range(1,g+1):
log_p[2*j] = -b[2*j]*factorial(2*j) / (2*j*2**j)
a_n = exp_series([(n-1)*x for x in log_p])[n] / (1-n)
return F(2 * (-1)**g * 2**n, (n-1)*factorial(n-1)) * a_n
def double_factorial(n):
product = 1
for j in range(n,0,-2):
product *= j
return product
def hyperelliptic(g, zeros):
if zeros == 1:
return F(2*double_factorial(2*g-3), factorial(2*g+1)*double_factorial(2*g-2))
if zeros == 2:
return F(8*double_factorial(2*g-2), factorial(2*g+2)*double_factorial(2*g-1))
raise ValueError('Hyperelliptic components have one or two zeros')
def partitions(n, maximum=None):
if n == 0:
yield ()
else:
for first in range(min(n, n if maximum is None else maximum),0,-1):
for rest in partitions(n-first, first):
yield (first,) + rest
def components(g, mu):
if g == 2:
return ('hyp',)
if g == 3:
return ('hyp','odd') if mu in ((4,),(2,2)) else ('connected',)
if mu == (2*g-2,):
return ('hyp','even','odd')
if mu == (g-1,g-1):
return ('hyp','nonhyp') if g % 2 == 0 else ('hyp','even','odd')
if all(m % 2 == 0 for m in mu):
return ('even','odd')
return ('connected',)
def selection():
return [(g, mu, c) for g in range(2,5)
for mu in partitions(2*g-2) for c in components(g,mu)]
if __name__ == '__main__':
controls()
for g in range(1,5):
print(g, 'total/pi^(2g)', minimal_total(g), 'spin difference/pi^(2g)', spin_difference(g))
print('Profiles', [len(list(partitions(2*g-2))) for g in range(2,5)])
print('Components', len(selection()))