The Ladder of Emergence O: Life as a Phase Transition in the Cost Landscape — Gradient Dynamics from Self-Replicating Molecules to Cognitive Agents

This paper provides a formal argument for the continuous ladder of life, evolu
tion, and cognition within a unified information-thermodynamic framework. The
core proposition is that “being alive” is equivalent to maintaining a net-negative
local minimum on a cost landscape, satisfying the stability conditions that first
order variation vanishes and second-order variation is strictly positive under small
perturbations. Life is defined as an autocatalytic system whose total cost is jointly
given by maintenance energy dissipation, information loss due to replication error,
and catalytic gain. When the information fidelity of a random chemical reaction
network crosses the Eigen error threshold, the system undergoes a phase transi
tion from a non-accumulating information phase to a self-replicating locked phase,
and the first historically continuous negative-cost minimum appears on the cost
landscape—this is the self-replication phase transition. Evolution is explained as
an unceasing constrained gradient descent flow of this minimum on a non-stationary
cost landscape: mutation provides directional sampling in the tangent space, selec
tion performs step acceptance filtering, and drift acts as thermal excitation to help
populations escape shallow minima. Each leap in complexity—from self-replicating
molecules to prokaryotic cells, from metabolic economy to information-processing
economy—corresponds to the opening of a new cost dimension, in which the sys
tem substitutes the metabolic cost of physical trial-and-error with information gain.
The paper further argues that the cost landscape becomes a non-autonomous dy
namical system due to niche construction by organisms, and minima undergo satu
ration and bifurcation under population occupation, continuously generating new
sequences of minima; the principle of algebraic closure failure provides the math
ematical foundation for the unboundedness of minimum splitting, ensuring the
perpetual diversity of evolution. Consequently, from the memoryless minimization
of physical laws to the virtual minimization of cognitive subjects, the gradient flow
on the cost landscape forms a continuous curve, and life is the segment on this curve
that transforms memorylessness into memory, isolated equilibrium into cumulative
search.