This paper reviews the declassified technical data in the official book "Britain and the H-Bomb". Secrecy-cleared UK thermonuclear warhead historian Lorna Arnold (and her assistant Katherine Pyne) in the UK official history "Britain and the H-Bomb" summarized Atomic Weapons Establishment secret data on UK thermonuclear warhead design tests up to the early interchanges of H-bomb data with America, in 1958-59. The book's information clarifies beyond any doubt the role of plastic foams in dispersing x-rays in the British type of weapon with an isotropically compressed spherical secondary stage, as opposed to the USA use of foam as simply a "radiation mirror" to re-radiate soft x-rays onto the cylindrical Teller Sausage secondary stage requiring only axial compression. Comparing this information to declassified double-primary Russian nuclear warhead design data (see nukegate.org for that) conclusively confirms the different merits of different approaches. This information should be available to inform public debate on not merely whether we have a nuclear deterrent that is efficient and cost effective, but the designs we really need; whether they come from a secrecy-obsessed groupthink culture that drives warhead design into an expensive, inefficient, incredible dead end; or radical, innovative, cheap, credible designs. A summary is given in Fig. 1 of this paper.