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1263695 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 | #!/usr/bin/env python3
"""Exact scalar zero-mean transformation with three forced terminal sentinels.
All arithmetic is rational or integer. The zero-endpoint orders of an input
instance from zero correspond exactly to those of the output: each old order
is followed by the three labeled sentinels in their designated order.
"""
from fractions import Fraction as F
from math import prod
def balance_scalar_centers(centers, rho=F(1, 2)):
"""Return a polynomial-bit exact-hitting reduction with sum(centers)==0.
The output centers and all reachable states lie strictly in (-1/2,1/2).
The guarantee concerns exact zero endpoints from initial state zero.
With the designated sentinel suffix, the new endpoint equals gain times
the old endpoint, where gain = rho**3 * denominator_scale / R.
"""
t = list(map(F, centers))
rho = F(rho)
if not t:
raise ValueError('at least one input center is required')
if not 0 < rho < 1:
raise ValueError('rho must be strictly between zero and one')
N = len(t)
K = N + 3
r, s = rho.numerator, rho.denominator
weights = [r**(K-p)*s**(p-1) for p in range(1, K+1)]
T = sum(weights)
U = sum(weights[:N])
A, B, C = weights[N:]
v1 = (0, C-B, A-C, B-A)
v2 = (B-A, U-N*B, N*A-U, 0)
V = max(map(abs, v1+v2))
L = 2*T*V + 1
markers = tuple(x+L*y for x,y in zip(v1,v2))
S = sum(t)
data = t + [-B*S/(B-A), A*S/(B-A), F(0)]
P = prod(x.denominator for x in data)
integer_data = [int(P*x) for x in data]
data_bound = T*max(map(abs, integer_data))
Q = 2*data_bound + 1
categories = [0]*N + [1,2,3]
raw = [d+Q*markers[c] for d,c in zip(integer_data,categories)]
R = 1+2*max(map(abs,raw))
new_centers = [F(a,R) for a in raw]
return dict(original_centers=t, rho=rho, N=N, K=K, weights=weights,
T=T, U=U, A=A, B=B, C=C, v1=v1, v2=v2, V=V,
marker_base=L, markers=markers, integer_data=integer_data,
denominator_scale=P, data_bound=data_bound, Q=Q, raw=raw,
R=R, categories=categories, centers=new_centers,
gap=(1-rho)/F(s**(K-1)*R),
gain=rho**3*F(P,R), sentinel_suffix=(N,N+1,N+2))
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