This paper resolves the Continuum Hypothesis (CH) and Generalized Continuum Hypothesis (GCH) by rejecting ordinal-based enumeration as the foundation for infinite cardinal growth. Instead, it introduces a closure-based framework where infinite sets are treated as structural strata rather than sequences. Key contributions: Introduces the identity (x + 1) = ∞ for any x ∈ ℵ₀, demonstrating that infinite cardinality is not additive Proves that no intermediate cardinal exists between ℵ₀ and the continuum 𝒞 because no intermediate closure stratum exists Shows that CH and GCH appear undecidable only within frameworks that conflate enumeration with structure Reframes the continuum as a topological shell rather than an extended sequence The resolution is structural, not axiomatic: cardinalities grow by closure completion, not interpolation. The continuum is the immediate structural closure of the naturals, with no admissible intermediate shell. Subject: Mathematics / Set Theory / Foundations of Mathematics