The Lehmer Totient Chamber Lift: Product Totient Gates via Korselt Supports

This note gives an expository chamber-coordinate presentation of Lehmer’s totient problem.
Lehmer asked whether there exists a composite integer n such that
φ(n) | n − 1,
where φ is Euler’s totient function. For primes p, the equality φ(p) = p−1 makes this divisibility
automatic. A composite solution would therefore imitate a prime through a totient-divisibility
gate.
The classical structure is severe. Any composite solution must be odd, square-free, and
Carmichael. In standard terms, the Carmichael condition is the least-common-multiple gate
λ(n) = lcmp|n(p − 1) | n − 1,
where λ is Carmichael’s function. For square-free n =
Q
i
pi
, the Lehmer condition is the stronger
product gate
φ(n) = Y
i
(pi − 1) | n − 1.
Thus the central obstruction is the gap between the lambda gate and the full phi product gate,
measured by the integer ratio
φ(n)
λ(n)
.
The quotient
M =
n − 1
φ(n)
is the classical Lehmer index when it is integral.
The chamber vocabulary introduced here is only an atlas-style organization of known objects:
prime support, Korselt support, the lambda gate, the phi product gate, and the Lehmer index.
The mathematics assembled here is classical and closely aligned with work of Korselt, Carmichael,
Lehmer, Cohen–Hagis, Hagis, Pomerance, Luca–Pomerance, Pinch, Renze, and others. No proof
of Lehmer’s totient problem is claimed, and no new analytic reduction is claimed.