The Averaged Null Energy Budget of a Traversable Throat Against the Quantum Inequality

A traversable throat must violate the averaged null energy condition along every ray that passes through it. Quantum field theory permits such violations within limits set by the Ford-Roman inequality, which caps the product of a negative density and the fourth power of its sampling duration. For a one-metre throat crossed at one tenth of light speed over a ten-kilometre internal path, the crossing time is 3.34e-04 seconds, the required averaged deficit is 1.79e+22 in integrated mass-density units, and the inequality allows 1.00e-51: an overrun factor of 1.78e+73. Slower crossings make the overrun worse, not better, because the deficit accumulates while the allowance falls as the inverse cube of the duration. The verdict is plain: within free-field quantum theory, a macroscopic throat open long enough to cross is excluded by the averaged budget, independently of how its exotic matter is sourced. Computed, not measured.