Exact-arithmetic bounds, a flat-tail special case, and the remaining sparsity gap

For a sparse nonsingular matrix with at most $k$ singular-value outliers and
tail condition number at most $\kappa$, conjugate gradients on the implicit
normal equations gives a deterministic
$O(N[k+1+\kappa\log(2/\varepsilon)])$ arithmetic bound. We give its
residual-norm proof and identify regimes where it already meets the target
cost. We then prove an input-sparsity bound for explicitly sparse positive
definite matrices whose spectral tail is exactly constant, including an
exact algebraic recovery of the unknown tail value. Sparse sketches and an
inner preconditioned solve avoid a dense ambient-dimensional basis. A
normal-equations corollary covers exactly flat singular tails when row
supports are small. Explicit Gram-fill and sketch examples show why these
arguments do not establish the general target. All computational checks use
rational arithmetic; they check finite identities and examples, not
asymptotic complexity or the full problem.