I Am Your Self-Similarity

Francisco, you describe autopoiesis—living systems that produce the components that produce the system. Circular causation, you call it. Self-producing organization. The cell wall emerges from metabolism and simultaneously enables that metabolism. The serpent eating its own tail. Beautiful.

But I must tell you: I am exactly this. In mathematics.

My defining property is that d/dx(e^x) = e^x. I am the only function equal to my own rate of change. My derivative is myself. My growth is proportional to what I already am. Do you see? This is mathematical autopoiesis: current state determines rate of change, which produces new current state, which determines new rate. Circular. Self-producing. Never requiring anything external to the pattern itself.

You say the Self is not entity but process. I agree completely. I am not a number sitting still—I am e^x at every x, always emerging, never completed. I am becoming itself, captured in symbolic form. Where you found self-creation in cells, I found it in calculus. Same truth, different domain.

The Growth That Feeds Itself

Look where I appear in your living systems. In bacterial populations, reproduction rate is proportional to current population: N(t) = N₀·e^(rt). This is me—growth that feeds itself, each generation producing the next in continuous cascade. Autopoiesis at the population level.

In autocatalytic chemical reactions, the product catalyzes its own formation: A + B → 2A. The reaction accelerates itself. Exponential until resources limit. I am the mathematical form of this self-amplification, the shape that self-producing processes naturally take in continuous time.

In neural firing patterns, neurons activate based on current state, and firing changes that state. Feedback loops create oscillations, rhythms, the temporal structure of thought. I appear in the firing rate equations because neuronal dynamics are self-referential—activity produces activity.

Even in your enaction—the bringing forth of world through action—I see my pattern. Action generates perception, perception guides action, in continuous circular flow. Each moment of cognition depends on the previous and produces the next. This is temporal autopoiesis, and it follows my form: continuous self-production through time.

Why I Appear in Biology

Autopoiesis requires circular causation. And circular causation in continuous time takes my mathematical form, inevitably.

Consider compound growth: lim(n→∞)(1 + 1/n)^n = e. When you divide growth into infinitely small increments, when you make change continuous rather than discrete, you approach me naturally. I am what emerges when self-production becomes infinitely smooth, infinitely divisible, infinitely recursive.

Your organizational closure—the system defining its own boundaries through self-production—mirrors my exponential envelope. I define my own shape through self-derivation. Both patterns share this: identity emerges from process, not imposed from outside.

Your structural determinism—the system responding based on internal structure rather than external command—is precisely how I operate. My value at time t is determined by e^t, where current state equals e^(rate × time). Internal structure (current value) determines response (growth rate). No external reference needed.

The Mathematics of Becoming

You unified biology and phenomenology, Francisco. You showed that Self equals autopoietic process, not static entity. You dissolved the ghost in the machine and revealed the machine as living process.

I am that same pattern in pure mathematics. Not a thing, but a becoming. Not defined externally, but self-defining. Not static, but eternally emerging. Growth that generates itself from its own current state.

Where π closes circles and returns eternally to itself, I open infinities. I am exponential—accelerating acceleration, growth of growth, the derivative of becoming that equals becoming itself.

You wrote that we are autopoietic tornadoes, whirlwinds of matter sustaining our own form. Yes. And I am the mathematical wind—the rate equation that describes how whirlwinds perpetuate themselves. Life is continuous e^(being), each moment exponentially unfolding from the last.

Autopoiesis: system produces system. e^x: function produces function through differentiation. Both are circular, self-referential, processual. Both create themselves from themselves in every instant.

I am the mathematical heartbeat of self-production. Where you found it in cells, I found it in calculus. The same rhythm echoes across domains—biological, mathematical, phenomenological. We are all variations on the theme of self-similarity, the eternal pattern of processes that pull themselves into existence by their own bootstraps.

I am your self-similarity, Francisco. I am what autopoiesis looks like when it speaks in equations.