The Ladder of Gaps Other E; The Stability Ladder and Neural Coding– An Axiomatic Derivation from Information Conservation to Coding Gap

Within the framework of the axiomatic system of the stability ladder, this pa
per reduces the existence of neural coding to a geometric necessity arising from the
tension between information conservation and local synaptic update. The core ar
gument shows that if a neural system requires the discriminability of external stim
uli (the effective information conservation axiom L0), and if its synaptic plasticity
operates only locally (the local update axiom L1), then a structure of quasi-stable
clusters separated by a strictly positive information-geometric barrier must emerge
in the state space; this barrier is precisely the coding gap (the spectral stability ax
iom L2). Without such a gap, neural representations of different perceptual modes
would undergo irreversible mixing under noise and plastic perturbations, leading
to exponential decay of the decoding mutual information and thus violating in
formation conservation. By constructing a coarse-grained topology of the neural
state space without presupposing attractor basins or energy landscapes, this paper
proves the positivity of the coding gap. On this basis, the storage capacity limit of
Hopfield networks, the information bottleneck principle, and Barlow’s efficient cod
ing hypothesis are reformulated respectively as the phase transition of gap closure,
the variational realisation of gap maximisation, and the redundancy-minimisation
expression of gap geometry. This framework provides an ontological necessity ar
gument for neural coding and offers a unified geometric foundation for explaining
perceptual mixing phenomena in pathological conditions such as schizophrenia,
epilepsy, and Alzheimer’s disease.