"""Exact numerical guards for the explicit remainder appendix.

These rational checks support, but do not replace, the all-n proof.
Python standard library only. No sampling or floating point is used.
"""
from fractions import Fraction as Q
import json

n0=10000
beta=Q(25,129); gamma=Q(125,258); mu=Q(125,16641)
c=Q(8242,49923); alpha=Q(8257,49923); gain=mu/25
cap=Q(21119,11094000)
checks={
    'dual_weights_admissible_at_threshold':
        Q(4,5)+Q(4*(n0+1),n0*n0)/beta**2 < Q(9,10),
    'mu_lower_bound_positive':mu*n0*n0-n0-1>0,
    'g_below_m':(Q(1,2)-gamma)*n0>1,
    'g_above_m_over_two':(gamma-Q(1,4))*n0>1,
    'entire_rank_band_below_g':(gamma-Q(9,20))*n0>1,
    'high_D_mass_defect_below_six':1/beta+Q(1,2)<6,
    'high_D_product_error_below_one':12*mu+Q(1,25)<1,
    'dual_objective_error_below_two':
        Q(1,2)+3*mu/4+Q(1,n0)+Q(1,2*n0*n0)<2,
    'small_D_band_error_below_402':1-9*mu+Q(7,20)+400<402,
    'low_D_gain_stronger':cap>gain,
    'small_rho_gain_stronger':Q(2975,399384)>gain,
    'large_rho_gain_stronger':Q(186,79507)>gain,
    'tail_gain_stronger':Q(1822061,103041072)>gain,
    'small_n_covered_by_16':n0*(Q(1,6)-c)<16,
    'coefficient_identity':alpha-gain==c,
    'largest_SC_quadratic_allowance_below_500':409<500,
}
assert all(checks.values()), checks
print(json.dumps({'status':'all exact rational guards passed',
                  'threshold':n0,'coefficient':str(c),'quadratic_constant':500,
                  'checks':checks},indent=2))