Phase Autonomy and Topological Transition: Two Independent Conditions for First Persistence, and Development as a Sequence of Euler-Characteristic Events

A companion manuscript on evolutionary correspondences begins from configurations already capable of self-replication and does not say where they come from. This manuscript supplies that step, and finds that the question this framework can ask is not the question the origin-of-life literature asks. Replication is constitutive here rather than achieved, so the operative question is phase autonomy: when does a region first hold a reference of its own rather than inheriting the ambient one? We give two conditions, show they are independent, and show that neither implies the other — which positions this framework’s answer outside all three standard programmes rather than within any of them. The first condition is exact rather than simulated, and is the Adler entrainment threshold; three simulations failed before it was reached, for a reason we record, since each wrote the condition as a ratio and it is not one. We then show that cell division is not a coherence failure but a topological event, with the Euler characteristic running 2 → 0 → 4 through a torus, and that this explains what a coherence account cannot: why division is all-or-nothing. We identify what drives the transition — a boundary-to-interior ratio falling as 2/(n+1), with pinching the only available topological repair, worth 26% more boundary for the same contents. We then resolve a standing objection to the companion manuscript’s use of cost-minimisation, by showing that the framework’s bounded quantity is information changed rather than sites updated. Finally we extend the same machinery to development, where differentiation is repeated access-change, and check four classic problems. On the fourth, the relevant result was published in 2003 and is more precise than our own formulation, which we record along with the correction it forces.

The foundational architecture, geometric intuitions, and motivating philosophy of this framework originated independently with the author prior to and separate from any AI involvement. Subsequent mathematical derivation, connection to established physics literature, computational verification, and error-checking were conducted through extended technical work with AI systems (Anthropic’s Claude and Google’s Gemini). The author directed this process, evaluated and selected among proposed derivations, and takes full responsibility for the accuracy and originality of the final work.