#!/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))