Yeah you are right to basics of the paper and I am extrapolating a bit here:

I think the remarkable result of this paper is that they add a local Lagrange multiplier λ at each layer, which accumulates constraint/prediction error over the inference dynamics.

At equilibrium, in the linear case, those local multipliers converge to exactly the same gradient signal that backpropagation would calculate globally

Now what is Predictive coding: its a network that can minimize prediction errors through local recurrent interactions instead of an explicit global backward pass.

Now I am making the leap to Fristons more philosophical and mathematical work not the paper - so that is me making the allusion

But a light bulb moment for me dawned when I read it

This process (PC-ALM) gives us a concrete example of how globally coherent inference/credit assignment can emerge from purely local dynamical interactions.

PC-ALM lets a recurrent dynamical system relax toward a state in which the backprop gradient is represented locally throughout the network.

That distinction is potentially important for neuroscience.

A brain doesn’t obviously have a central routine saying:

loss.backward()

it certainly has recurrent neural populations whose states continuously influence neighboring populations.

This paper is demonstrating that, at least mathematically, those sorts of local recurrent dynamics can generate the same credit information that backprop obtains through the chain rule. The authors explicitly motivate predictive coding as a biologically plausible local-learning alternative because standard BP requires globally coordinated error variables and update ordering.

Think about it also give plausible evolutionary to chain intelligence through cells coming together and creating nested networks

This has got to be how the neurological intelligence sausage gets made

What it eventually means for ML I’m Not sure but hopeful it opens a door

Backprop Alternative: Augmented Lagrangian Predictive Coding