TITLE: A Priori proof that Covering Radius strictly less than $\sqrt 2$ implies class number one
QUESTION [18 upvotes]: 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: 
Intuition for the last step in Serre's proof of the three-squares theorem
Is the square of the covering radius of an integral lattice/quadratic form always rational?
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. Quoted:

But there is a good way to
             show that some genus of lattices L has class number 1 if its
  covering
             radius is small. What you do is look at the sum of L and a
  2-dimensional
             Lorentzian lattice. Then other lattices in the genus correspond
  to some
             norm 0 vectors in the Lorentzian lattice. On the other hand,
  if the
             covering radius is small enough one can use this to show that a
             fundamental domain of the reflection group of the Lorentzian
  lattice has
             only one cusp, so all primitive norm 0 vectors are conjugate
  and therefore
             there is only 1 lattice in the genus of L. This works nicely when
  L is the
             E8 lattice for example.
             There are some variations of this. When the covering radius is
  exactly
             sqrt 2 the other lattices in the genus correspond to deep holes,
  as in the
             Leech lattice (Conway's theorem). The covering radius sqrt 2
  condition
             corresponds to norm 2 reflections, and more generally one
  can consider
             reflections corresponding to vectors of other norms; see
             http://dx.doi.org/10.1016/0021-8693(90)90110-A
             for details

REPLY [5 votes]: Alright, make this a separate answer... there are tables of positive forms available up to dimension 5. So I revised some C++ programs and found the "odd" lattices that satisfy Allcock's rule, the ambient vector space is covered when one places an open ball of radius 1 around each lattice point of odd norm, then an open ball of radius $\sqrt 2$ around each lattice point of even norm (such as the origin). A lattice is called "borderline" if those open balls do not suffice but closed balls of the same radii do the job. 
So I am posting the table, so far (dimension 1,2,3,4,5), at this LINK. One can see that these are integral lattices as the matrix entries listed (one matrix per line) gives inner products always integral, meaning all f_ij are even when i,j are distinct, but the lattices are "odd" in that at least one diagonal entry f_ii is odd. 
The computer that hosts my websites is down for the next month, so here is the table of "odd" lattice successes up to five variables. It is probably complete up to dimension 4, in dimension 5 I was getting near the upper bound of Nipp's tables so I am less certain about completeness. There are surely some of these in dimensions 6,7,8, probably not 9,10. 
EDIT 24 November 2011: It is my educated guess that the sum of eight squares is one of the "odd" borderline lattices. I'm trying to figure out how to check this.
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
          f11 
           1  
           3  
           5  
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= 
          f11   f12   f22
           1     0     1
           1     0     2
           1     0     3
           1     0     5
           2     0     3
           2     2     3  
           3     0     3
-------------------------
           1     0     4  borderline 
           2     2     5  borderline 
           3     2     3  borderline 
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= 
        disc:  f11 f22 f33 f23 f13 f12 
succeed    4:   1   1   1   0   0   0  
succeed    8:   1   1   2   0   0   0  
succeed   12:   1   1   3   0   0   0  
succeed   12:   1   2   2   2   0   0  
succeed   20:   1   1   5   0   0   0  
succeed   20:   1   2   3   2   0   0  
succeed   24:   1   2   3   0   0   0  
succeed   28:   2   2   3   2   2   2  
succeed   36:   1   2   5   2   0   0  
succeed   36:   1   3   3   0   0   0  
succeed   36:   2   2   3   0   0   2  
succeed   60:   2   3   3   0   0   2  
-----------------------------------------------------------
succeed   16:   1   1   4   0   0   0  borderline  
succeed   16:   1   2   2   0   0   0  borderline
succeed   32:   1   3   3   2   0   0  borderline  
succeed   32:   2   2   3   2   2   0  borderline
succeed   48:   2   3   3   2   2   2  borderline
succeed   64:   3   3   3  -2   2   2  borderline  
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
        disc:  f11 f22 f33 f44 f12 f13 f23 f14 f24 f34  
succeed   16:   1   1   1   1   0   0   0   0   0   0  
succeed   32:   1   1   1   2   0   0   0   0   0   0  
succeed   48:   1   1   1   3   0   0   0   0   0   0  
succeed   48:   1   1   2   2   0   0   0   0   0   2  
succeed   80:   1   1   2   3   0   0   0   0   0   2  
succeed   96:   1   1   2   3   0   0   0   0   0   0  
succeed  112:   1   2   2   3   0   0   2   0   2   0  
succeed  144:   1   1   2   5   0   0   0   0   0   2  
succeed  144:   1   2   2   3   0   0   2   0   0   0  
succeed  240:   2   2   3   3   2   2   0   2   0   0  
succeed  240:   1   2   3   3   0   0   0   0   2   0  
-----------------------------------------------------------
succeed   64:   1   1   1   4   0   0   0   0   0   0  borderline  
succeed   64:   1   1   2   2   0   0   0   0   0   0  borderline  
succeed   64:   1   2   2   2   0   0   2   0   2   0  borderline  
succeed   80:   1   1   1   5   0   0   0   0   0   0  borderline  
succeed  128:   1   1   3   3   0   0   0   0   0   2  borderline  
succeed  128:   1   2   2   3   0   0   0   0   2   2  borderline  
succeed  128:   2   2   2   3   2   2   0   2   0   0  borderline  
succeed  144:   1   1   3   3   0   0   0   0   0   0  borderline  
succeed  192:   1   2   3   3   0   0   2   0   2   0  borderline  
succeed  192:   2   2   2   3   0   0   0   2   2   2  borderline  
succeed  256:   1   3   3   3   0   0   2   0   2  -2  borderline  
succeed  256:   2   2   3   3   0   2   2   2   2   2  borderline  
succeed  256:   2   2   3   3   2   2   0   0   2   0  borderline  
succeed  304:   2   2   3   3   0   2   2   0   2   0  borderline 
succeed  512:   3   3   3   3   2   2  -2  -2  -2   2  borderline  
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=




        disc:  f11 f22 f33 f44 f55 f12 f13 f23 f14 f24 f34 f15 f25 f35 f45 
succeed   16:   1   1   1   1   1   0   0   0   0   0   0   0   0   0   0  
succeed   32:   1   1   1   1   2   0   0   0   0   0   0   0   0   0   0  
succeed   48:   1   1   1   1   3   0   0   0   0   0   0   0   0   0   0  
succeed   48:   1   1   1   2   2   0   0   0   0   0   0   0   0   0   2  
succeed   80:   1   1   1   2   3   0   0   0   0   0   0   0   0   0   2  
succeed  112:   1   1   2   2   3   0   0   0   0   0   2   0   0   2   0  
succeed  144:   1   1   2   2   3   0   0   0   0   0   2   0   0   0   0  
-----------------------------------------------------------
succeed   64:   1   1   1   1   4   0   0   0   0   0   0   0   0   0   0  borderline  
succeed   64:   1   1   1   2   2   0   0   0   0   0   0   0   0   0   0  borderline  
succeed   64:   1   1   2   2   2   0   0   0   0   0   2   0   0   0   2  borderline  
succeed   64:   1   2   2   2   2   0   0   0   0   0   0   0   2   2   2  borderline  
succeed   96:   1   1   1   2   3   0   0   0   0   0   0   0   0   0   0  borderline  
succeed  112:   1   1   1   2   4   0   0   0   0   0   0   0   0   0   2  borderline  
succeed  128:   1   1   1   3   3   0   0   0   0   0   0   0   0   0   2  borderline  
succeed  128:   1   1   2   2   3   0   0   0   0   0   0   0   0   2   2  borderline  
succeed  128:   1   2   2   2   3   0   0   2   0   2   0   0   2   0   0  borderline  
succeed  144:   1   2   2   2   3   0   0   2   0   2   0   0   0   0   2  borderline  
succeed  144:   1   1   1   2   5   0   0   0   0   0   0   0   0   0   2  borderline  
succeed  160:   1   1   2   2   3   0   0   0   0   0   0   0   0   0   2  borderline  
succeed  192:   1   1   2   3   3   0   0   0   0   0   2   0   0   2   0  borderline  
succeed  192:   1   2   2   2   3   0   0   0   0   0   0   0   2   2   2  borderline  
succeed  208:   1   1   2   3   3   0   0   0   0   0   0   0   0   2   2  borderline  
succeed  240:   1   1   2   3   3   0   0   0   0   0   2   0   0   0   0  borderline  
succeed  256:   1   1   3   3   3   0   0   0   0   0   2   0   0   2  -2  borderline  
succeed  256:   1   2   2   3   3   0   0   0   0   2   2   0   2   2   2  borderline  
succeed  256:   1   2   2   3   3   0   0   2   0   2   0   0   0   2   0  borderline  
succeed  256:   2   2   2   2   3   0   0   0   0   0   0   2   2   2   2  borderline  
succeed  256:   2   2   2   3   3   2   2   0   2   0   0   2   0   0   2  borderline  
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

In the "borderline" cases, one may insert extra points at precisely located points to create an overlattice, still integral but possibly "even," that does better than the original as far as covering radius smallness. That is the real end of the project, finding all the borderline guys and describing these overlattices.