Part Three introduces the formal account. The mathematics is not decoration — it is the most useful language available for accumulation, weighting, decay, resonance, and the axis on which imagination runs — out of phase with the sensory present, but alongside it rather than cut off from it. It works through the model in four stages — the integral, resonance, imagination, and the large language model as a comparative case.

A few people, from Buddha to Parfit, have come to the same conclusion: that we ought to focus on the moment. Now consider what it would mean if our minds work by accumulating those moments. There is no recording device built into our eyes and ears. Instead, through some iterative evolutionary process, the mind lays down a synaptic memory — something more like a matrix than a recording: the moment transposed across different parts of the brain, distributed, weighted. An integral is the area under the curve, but in our use case an n-dimensional shape, faded by time in parts. Think of an accumulating spider web.

The self is the accumulation of a transformation function operating on two inputs — what you chose and what happened to you — across time. The function f(·) remains partially unspecified, which is honest rather than a limitation. Newton worked with gravity without knowing what it was.

How the matrix holds all this has a shape worth seeing. Many researchers now treat parts of the mind as a Hopfield network — an n-dimensional web of near-binary nodes that stores experience as a minimum-energy landscape. Picture a contour map of far more than three dimensions: the valleys are attractor basins the system settles into, deepened by every repetition; the hills are the boundaries between them. Young, the landscape is steep and easily reshaped; as the parameter set consolidates, it broadens and flattens, and grows harder to move. The Jacobian is the local map of that surface — how susceptible you are to influence in each direction — and it flattens with age. That is the formal account of what it means to find your own voice.

Of the four stages, three are the pillars.

The Integral is your mind accumulating experience not as a recording but as patterns — a synaptic matrix built by everything you have lived through and chosen.

Resonance is the largely subconscious way those patterns communicate with others — the way music moves you before you have decided to be moved, the incoming signal finding purchase in what is already there, with depth ranging from surface activation to permanent rebuilding.

Imagination is the third pillar — the capacity to run content the senses are not currently delivering, built from the same accumulated patterns and running out of phase with the sensory present — alongside it, not in place of it.

Three pillars, three loose ends in the existing literature: parameter-based memory (the substrate that large language models have independently converged on), memes (which Dawkins named without specifying how they work), and the cognitive revolution (which Harari named without explaining). One substrate underneath all three.

Running through all of it is an information axis. What accumulates is not raw experience but experience with meaning — semantic information that, like the information carried in our genes, can persist and be transmitted to those who follow. This is how a personal integral becomes a cultural deposit.

The fourth stage is the large language model — the framework's abductive proof. A trained language model is a frozen integral of the cultural deposit — accumulation without ongoing decay, without resonance, without the oscillatory axis that imagination runs on. It integrates in a recognisable way but does not resonate. That makes it not just an existence proof for the architecture, but the most powerful instrument currently available for seeing what an integration substrate looks like from the inside.