verify-degree-budget-counterexample.py

#!/usr/bin/env python3
"""Exact arithmetic checks for the degree-budget counterexample.

The analytic inputs are compact resolvent and the quantum virial theorem.
This script checks the degree inequalities, the dilution estimate, and the
normalized Hankel escape that corrects the recession-ray claim.
"""

from fractions import Fraction


def outer(v):
    return [[a * b for b in v] for a in v]


def det2(a, b, c, d):
    return a * d - b * c


print("DEGREE BUDGET")
for m in range(3, 9):
    K = m
    energy_row_degree = 2 * K - 2 * m
    conjectured_k = K - 1
    admissible_k = K - m + 1
    assert energy_row_degree == 0
    assert conjectured_k >= 2
    assert admissible_k == 1
    print(
        f"m={m}: deg(V)={2*m}, K={K}, literal k<=K-1 gives {conjectured_k}, "
        f"energy-shell mechanism gives k<=K-m+1={admissible_k}"
    )


print("\nDILUTED HIGH-ENERGY LOWER BOUND")
ground = Fraction(1)
fixed_energy = Fraction(2)
m = 3
virial = Fraction(2 * m, m + 1)
previous = Fraction(0)
for excited in (10, 100, 1000, 10000):
    lam = Fraction(excited)
    alpha = (fixed_energy - ground) / (lam - ground)
    lower = alpha * virial * virial * lam * lam
    assert 0 < alpha < 1
    assert lower > previous
    previous = lower
    print(
        f"lambda={excited:5d}: alpha={alpha}, "
        f"lower bound on mixed <p^4>={lower}"
    )


print("\nHANKEL ESCAPE")
for t in (0, 1, 2, 10):
    v = [1, t, t * t]
    M = outer(v)
    assert M[0][0] == 1
    assert M[0][2] == M[1][1] == t * t
    assert det2(M[0][0], M[0][1], M[1][0], M[1][1]) == 0
    assert det2(M[0][0], M[0][2], M[2][0], M[2][2]) == 0
    assert det2(M[1][1], M[1][2], M[2][1], M[2][2]) == 0
    print(f"t={t:2d}: second moment={M[0][2]:3d}, rank-one PSD Hankel matrix verified")

print("\nAll exact checks passed.")