  The Parisi formula for the free energy is among the crown jewels in the
theory of spin glasses. We present a simpler proof of the lower bound in the
case of the spherical mean-field model. Our method follows the TAP approach
developed recently in e.g. (Subag, 2018): we obtain an ultrametric tree of pure
states, each with approximately the same free energy as the entire model, which
are hierarchically arranged in accordance with the Parisi ansatz. We construct
this tree ``layer by layer'' given the minimizer to Parisi's variational
problem. On overlap intervals with full RSB, the tree is built by an
optimization algorithm due to Subag. On overlap intervals with finite RSB, the
tree is constructed by a new truncated second moment argument; a similar
argument also characterizes the free energy of the resulting pure states.
Notably we do not use the Aizenman--Sims--Starr scheme, and require
interpolation bounds only up to the 1RSB level. Our methods also yield results
for large deviations of the ground state, including the entire upper tail rate
function for all 1RSB models without external field.
