environment = Python 3.13.5; SymPy 1.14.0
arithmetic = exact rationals / symbolic polynomials / algebraic radicals
route_A_polynomial = (r^4-3*r^2+1)*(r^4+5*r^2+1)
route_A_positive_gram_condition = r^2 + r^(-2) = 3
A_0_through_6 = [2, 3, 7, 18, 47, 123, 322]
admissible_gaps_with_sum_at_most_3 = [(0, 0), (1, 1), (0, 3), (3, 0)]
F_for_those_gaps = [9, 16, 41, 41]
tail_lower_bound_for_gap_sum_at_least_4 = 54
finite_control_gap_bound = 18
finite_control_admissible_cases = 121
finite_control_F16_exponents = [(1, 0, -1)]
finite_control_other_minimum = 41
next_gap_value = 64
rejected_spread_monotonicity_values = (15376, 11561)
norm_only_test_object = S(X)=(N_Albert(X)-1)^2 at the Albert identity
norm_only_hessian_rank = 1
norm_only_hessian_nullity = 26
frame_orbit_count_and_stronger_variational_pointer = NOT VERIFIED: not computed by this script
result = PASS: analytic-gap control and actual Albert norm-only Hessian