Principia Attractum

Recursive Operators and Operations of Persistence

Principia Attractum is a nonlinear, dynamic, mathematical framework that extends nonlinear dynamics to the problem of recursive persistence; the conditions under which recursive structure continues to exist, and the cost it pays to continue.

What it extends

Classical nonlinear dynamics describes how systems settle into attractors, occupy basins, cross stability margins, and follow trajectories. Principia Attractum adds a persistence layer to that vocabulary: it asks under what invariant conditions a recursive structure can maintain itself, what continuation costs it must meet, and what happens when that cost is not met. Autocatalysis is the pattern that meets the gate's condition: the structure passes because its action is autocatalytic; the self-sustaining loop is what admissibility is testing for. Autocatalysis does not open the gate, it is the property the gate checks.

How to read it

Non-linear dynamics is cognitively challenging. This particular flavor of it is doubly so, because it also removes time, and time is the crutch most readers lean on to keep their place. There is one source of relief: the framework's own bootstrapping interval metrics are built partly to hold you in the recursive frame, giving the mind something to grip where the time axis used to be. Expect the double difficulty, and let those metrics do some of the work. Principia Attractum resists linear reading because it is not linear. There is no clean chain of the form A causes B, which causes C; a recursive structure is one whose present state is a function of its own prior states, feeding back on itself…tik, tik, tik…so that cause and effect close into a loop rather than run in a line. This framework is not based on time, but on sequence, and you will never see time used within an attractor's logic calculations (t will never appear in an equation). A reader who processes it as a straight sequence will not be able to connect with the material. Linear reading is the wrong tool. The move that unlocks the text is to stop asking what does this do to that and start asking what conditions must hold together, at once, for this to persist. Read it as a system folding back on itself, not as a story told front to back, and the material opens.

For this reason, a set of operators are introduced, the bootstrapping interval metrics, to force the reasoning into the recursive zone: quantities such as recurcline, the stored recursive compression a persistent structure carries. These quantities exist to measure its chances of returning on the next tik. The term “tik” is spelled that way to separate it from time. Removing time from the framework is also required, and these measurements do not exist on a fixed external ruler the way length or temperature do. They exist only while the loop is running.

The metrics introduced in this text are all defined only across the bootstrapping interval, with the exception of one or two, such as the α-trace, which leaves a compressed history of an attractor after it ‘dies’. These metrics come into existence at ignition of an autocatalytic event, the moment the recursion locks and the structure enters active self-maintenance. They end at loss, the moment that maintenance fails. Before ignition there is nothing to measure. After loss there is nothing left to measure.

An example of a bootstrapping interval metric is recurcline. To ask for the recurcline of a structure that has not yet ignited, or one that has already failed, is similar to asking what tempo musicians are playing a song, when the music hasn’t started yet. The metrics are born with the attractor and vanish with it.

This is the shape of the entire subject in miniature. To reason through this material, you must accept that even the ruler is recursive. But these special metrics will aid in keeping your thoughts in a recursive position. The quantities the theory tracks come into being with the thing they track, and they are undefined on either side of its life.

This is why the framework is closer to Stuart Kauffman’s biosphere than to Newtonian mechanics. In a Newtonian system the space of possibilities is fixed in advance, and one computes the trajectory against it. In an evolving autocatalytic system there is no prestated space to compute against; the relevant quantities, and the frame that would measure them, come into being through the very process one is trying to describe. That insight is Kauffman’s. His work on autocatalytic sets, on order arising without external design, and on a biosphere whose next possibilities cannot be stated in advance, opened the ground this framework stands on. Principia Attractum notes these observations and builds upon them.

Deployment

The framework's concepts are exercised in an applied ecosystem simulation project, the Bio-Kernel Series, which deploys the primitives of Principia Attractum in ecosystem-scale settings.

References

The framework's lineage runs through five bodies of work. Complexity and the generated space of possibilities (Kauffman); the survival-under-constraint reframing of viability theory (Aubin); stability and feedback from control theory; the demotion of fundamental time in the foundations of physics (Barbour; causal-set theory); and the parent language of nonlinear dynamics it extends.

• Generated possibility and self-organization. Kauffman, S. A. (1993). The Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press.
• Kauffman, S. A. (1995). At Home in the Universe: The Search for the Laws of Self-Organization and Complexity. Oxford University Press.
• Kauffman, S. A. (2019). A World Beyond Physics: The Emergence and Evolution of Life. Oxford University Press.
• Survival under constraint (viability theory). Aubin, J.-P. (1991). Viability Theory. Birkhäuser. [verify edition/publisher]
• Aubin, J.-P., Bayen, A. M., & Saint-Pierre, P. (2011). Viability Theory: New Directions (2nd ed.). Springer. [verify]
• Stability and feedback (control theory). Khalil, H. K. Nonlinear Systems. Prentice Hall. [verify year/edition — standard Lyapunov / input-to-state stability reference]
• Time as derived, not fundamental. Barbour, J. (1999). The End of Time: The Next Revolution in Physics. Oxford University Press. [verify]
• Sorkin, R. D. Causal sets: overview and status. [verify full citation — representative causal-set-theory reference to be selected]
• Parent field (nonlinear dynamics). Strogatz, S. H. Nonlinear Dynamics and Chaos. [verify year/edition/publisher]
• Guckenheimer, J., & Holmes, P. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer. [verify]

Entries marked [verify] require confirmation of edition, year, and publisher against the primary source before publication.

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