The Identity of Extremes: A Generative Framework From Fixed Points to Cosmology: Mapping the Invariant Across Disciplines

Different disciplines routinely arrive at strikingly similar organizing relationships: a stable center flanked by two limiting extremes, described in vocabularies that share no common history and no obvious reason to agree. This paper asks a question that is logically prior to any specific cross-domain claim: what remains once disciplinary vocabulary is stripped away, and why should its recurrence be taken seriously rather than dismissed as coincidence or projection? The paper develops a single structural invariant, 1 = (0 ↔ ∞), and its minimal mathematical expression as a reciprocal deviation from a fixed point, δ(x) = x + 1/x − 2, which is provably zero only at x = 1. It introduces a tiered framework of Tier I, II, and III for evaluating how much weight a given cross-domain analogy can bear, and demonstrates the method of abstraction directly rather than surveying every domain it touches. The detailed mathematical, physical, and psychological development of this invariant is carried out in companion papers; this paper's task is to establish why that development is legitimate in the first place.