Derive, don't store · z = x·y

The whole picture below is computed from one small definition. There is no array of values anywhere on this page. Move the dials and every pixel is re-derived from the definition. This is the core idea: store the generator, not the data, and read the data out of geometry on demand.

The definition (this is all that is stored)

At the cursor (derived, not looked up)

x, y·
z = a·x·y·
lens output·

Stored vs derived (honest)

definition·
values derived·
if stored as a field·
ratio·
costa few multiplies / value

One surface, many lenses: what each is for

The same field z = a·x·y, read four ways. A lens does not change the surface; it chooses what to read off it.

Colour reads a value as a hue. For: heatmaps, a field you can see at a glance, low-to-high status.
Contours draw the level sets, every place the value equals K. For: thresholds and bands, elevation lines on a map, quantising a smooth field into steps, finding the boundary where a quantity hits a limit.
Logic gate reads the sign, an AND of x and y. For: regions and masks, pass/fail gating, a yes/no decision boundary, "is this point in the allowed quadrant".
Gradient reads the slope, how steep and which way. For: edges, flow direction, sensitivity (how fast it changes here), and which way to move to climb, the basis of optimisation.
Where this holds, and where it does not. This works because the field is derivable: a tiny definition can regenerate every value. A photograph or arbitrary text is irreducible, it cannot be regenerated from a small definition, so it must be stored. Derive-not-store trades disk for compute: it saves hard-drive storage on the machine it runs on, not RAM, and at run time it still processes normal bits and bytes. (Storage is also the squeeze at data-center scale, but one local instance does not touch that; that is a direction, if widely adopted.) It applies to data a definition can generate, not to all data. Stated plainly so it is not oversold.

Why there is genuinely no store here. Open the source. You will not find a data array, a Map, or a saved field. The render function computes each pixel directly from z = a·x·y and the active lens. The pixels are a transient rendering of the definition, thrown away and regenerated whenever you move a dial. The definition is the source of truth; the picture is derived from it.

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Concept and direction: Kenneth W. Bingham. Built with the help of Claude AI under a standing directive to be skeptical, to insist on proof, and to allow no claim that is not demonstrated in tested code. The ideas are the author's; the AI implemented and verified them, it did not originate them.