E–M–I: Energy, Matter and Information — A Unified Model of Cosmic Recurrence

We derive one exact statement and use it to organize the levels of nature. Main result. Give any bounded system of rest mass M, areal radius R and information capacity I (bits) the size temperature TR = ħc/(2πkR), which is the Unruh temperature of the proper acceleration c²/R. Then Mc² = E ≥ kTR ln2·I for ordinary matter (a restatement of the Bekenstein bound), and the three quantities (mass, energy, information) become exactly equal on every Schwarzschild horizon and on the Hubble horizon of a flat universe at every epoch. For rotating or charged black holes the information on the horizon equals exactly the irreducible mass, and the extractable energy is precisely what stays outside the equality. The dimensionless gap δ = ln[Mc²/(kTR ln2·I)] ≥ 0 measures how far any system is from being a horizon: about 55 for the Sun, ln√2 for an extremal Kerr hole, 0 for a Schwarzschild hole and for the Hubble horizon. The ingredients are known; our contribution is a compact common form for these cases, the irreducible-mass reading and the gap δ. Framework. We then thread the forms that recur across physics, chemistry, life and mind on one mathematical line of nine steps (counting, sign, ratio, logarithm, first- and second-order change, nonlinear flow, entropy, topological invariant), each forced by the non-closure of the step before and each taking the completed whole of the previous step as its new unit. The levels of nature use these steps in broadly the same order; the ninth step counts a completed level as one integer, the "1" of the next level, and the horizon equality is where the line closes. We test three trends against established data, report one preregistered forward test (passed) and preregister four further falsifiable hypotheses. Throughout we mark what is established and what is a reading.