Continuity from Compression: Refinement-Stable Semigroups, Generator Tiers, and a Graded Bracket

Near-identity refinement forces a graded generator structure for interaction. This paper derives a continuity-like bookkeeping constraint from representational compression: refinement stability implies a Q>=0-parameterized semigroup for repeated near-identity, yield-preserving updates. Generator content is defined as tier quotients N_e/N_e2 and N_e2/N_e4. To preserve interaction residue under non-commutation, the commutator-induced bracket is forced to be graded, mapping first-tier generator classes into the next tolerance tier rather than closing at the first tier. No calculus, manifolds, or Lie algebra limits are assumed; only tier nesting, refinement stability, and commutator placement in finer tolerance.