Proposition 4. For every pair of integers m,R≥1 , set d_0=64m4^R, d_j=4^j d_0 (0≤j≤R). There are a compact smooth manifold Y , Hermitian complex vector bundles E_j→Y for all j≥0 , and continuous bundle maps X_i^ (j) :E_j→E_ j+1 , for …

Proposition 4. For every pair of integers m,R≥1 , set d_0=64m4^R, d_j=4^j d_0 (0≤j≤R). There are a compact smooth manifold Y , Hermitian complex vector bundles E_j→Y for all j≥0 , and continuous bundle maps X_i^ (j) :E_j→E_ j+1 , for 1≤i≤24 and j≥0 , such that E_0 is trivial of rank d_0 , the rank of E_j is d_j for 0≤j≤R , and ∑_ i=1 ^ 24 (X_i^ (j) )^*X_i^ (j) =I_ E_j , ∑_ i=1 ^ 24 X_i^ (j) (X_i^ (j) )^* ≤23 I_ E_ j+1 . Moreover, (_ j=0 ^R E_j)^ ⊕m ; contains no trivial subbundle of rank 2md_0.