The Lichtenbaum–Quillen conjecture (LQC) relates special values of zeta functions to algebraic K-theory groups. The Ausoni–Rognes red-shift conjectures generalize the LQC to higher chromatic heights in a precise sense. In this paper, we propose an alternate generalization of the LQC to higher chromatic heights and give evidence for it at height two. In particular, if the
n
n
-th Greek letter family is detected by a commutative ring spectrum
R
R
, then we conjecture that the
n
+
1
n+1
-st Greek letter family will be detected by the algebraic K-theory of
R
R
. We prove this in the case
n
=
1
n=1
for
R
=
K
(
F
q
)
R=\mathrm {K}(\mathbb {F}_q)
modulo
(
p
,
v
1
)
(p,v_1)
where
p
≥
5
p\ge 5
and
q
=
ℓ
k
q=\ell ^k
is a prime power generator of the units in
Z
/
p
2
Z
\mathbb {Z}/p^2\mathbb {Z}
. In particular, we prove that the commutative ring spectrum
K
(
K
(
F
q
)
)
\mathrm {K}(\mathrm {K}(\mathbb {F}_q))
detects the part of the
p
p
-primary
β
\beta
-family that survives mod
(
p
,
v
1
)
(p,v_1)
. The method of proof also implies that these
β
\beta
elements are detected in iterated algebraic K-theory of the integers. Consequently, one may relate iterated algebraic K-theory groups of the integers to integral modular forms satisfying certain congruences.