It turns out that each of Pete L. Clark's "euclidean" quadratic forms, as long as it has coefficients in the rational integers $\mathbb  Z$ and is positive, is in a genus containing only one equivalence class of forms. In the language of (positive) integral lattices, the condition is that the covering radius be strictly smaller than $\sqrt 2.$ 
 Please see these for background: 

http://mathoverflow.net/questions/3269/intuition-for-the-last-step-in-serres-proof-of-the-three-squares-theorem

http://mathoverflow.net/questions/69428/is-the-square-of-the-covering-radius-of-an-integral-lattice-quadratic-form-always

http://mathoverflow.net/questions/39510/must-a-ring-which-admits-a-euclidean-quadratic-form-be-euclidean

http://www.math.rwth-aachen.de/~nebe/pl.html 

http://www.math.rwth-aachen.de/~nebe/papers/CR.pdf   

I have been trying, for some months, to find an *a priori* proof that Euclidean implies class number one. I suspect, without much ability to check, that any such Euclidean form has a stronger property, if it represents any integral form (of the same dimension or lower) over the rationals $\mathbb Q$ then it also represents it over $\mathbb  Z.$ This is the natural extension of Pete's ADC property to full dimension. Note that a form does rationally represent any form in its genus, with Siegel's additional restriction of "no essential denominator." If the ADC property holds in the same dimension, lots of complicated genus theory becomes irrelevant. 

EDIT: Pete suggests people look at  §4.4 of http://math.uga.edu/~pete/ADCFormsI.pdf . 

EDIT 2: It is necessary to require Pete's strict inequality, otherwise the Leech lattice appears.

So that is my question, can anyone prove *a priori* that a positive Euclidean form over $\mathbb  Z$ has class number one?  

EDIT 3: I wrote to R. Borcherds who gave me a rough idea, based on taking the sum of a given lattice with a 2-dimensional Lorentzian lattice. From page 378 in SPLAG first edition, two lattices are in the same genus if and only if their sums with the same 2-dimensional Lorentzian lattice are integrally equivalent.  I hope someone posts a fuller answer, otherwise I'll be spending the next six months trying to complete the sketch myself. The references: Lattices like the Leech lattice, J.Alg. Vol 130, No. 1, April 1990, p.219-234, then earlier The Leech lattice, Proc. Royal Soc. London A398 (1985) 365-376.