The Forced Core of Physical Reality: A Parameter-Free Foundation for a Theory of Everything

What structure must physical comparison, composition, and retention preserve? We characterize a parameter-free core through explicit operations and exclusion proofs. Finite records support canonical arithmetic. Conservation and register symmetries force quadratic signal energy, and an internal phase reference forces the normalized reciprocal cost J(x) = (x + x^−1)/2 − 1. Independently, a composition law and unit local curvature characterize the same function; continuity and reciprocity follow. For a hierarchy preserving all positive signed comparisons, binary generation of the first new level and closure of the seed sum force the golden ratio. Reversibility on two independent integer registers supplies another derivation of the unequal join, so its positive comparison cost need not be assumed. For spatial retention, globally defined continuous clock-compatible transport, preserving the source reference norm and enacting a uniformly repeated cycle along a closed probe, produces a nonzero additive path reading. The resulting homology class selects dimension three in a sphere, or in Euclidean space when the completed circle leaves its complement path connected. A concrete realization satisfies the transport requirements. The proofs establish uniqueness across realizations of the stated operations and locate the physical identifications needed to apply that uniqueness to observed space and dynamics.