Integer Emergence from Transcendental Root Constants presents a fully reproducible mathematical and computational framework demonstrating that the positive integers need not be assumed as primitive objects, but can emerge as stable attractors from continuous structure. Starting from the transcendental constants φ (golden ratio), π (circular structure), and e (exponential growth), the paper constructs a perception function that maps continuous geometric and exponential processes into discrete integer outputs. An initial uncoupled model demonstrates partial integer emergence, while an extended coupled model achieves near-perfect integer reconstruction (<10⁻⁶ mean error). Crucially, the five parameters required for perfect integer emergence are not fit ad hoc. Each parameter is independently derived from prior theoretical principles within the ENSO framework: λ = −1 from geometric bound-state (Acrolysis) conditions β = 1/12 from topological closure via pentagonal defect theory α = 2(Δ_res − a) from Resolution Gap structure γ = 1 − Δ_res² from coherence attenuation physics δ_s = 1 + Δ_res²/12 from discretization cost Theoretical values match empirically optimized values with a mean deviation below 1%, demonstrating that the integer structure is constrained by geometry, topology, and information loss rather than numerical coincidence. A complete validation script is provided, including versioning, fixed random seeds, and machine-verifiable outputs, ensuring full reproducibility. Beyond its technical results, the paper explores the physical and ontological implications of integer emergence, proposing that discreteness itself may arise from coherence loss and phase-space inflation when continuous structure becomes observable. These interpretive sections are clearly delineated from the formal derivations. This work contributes to foundational questions in mathematics, physics, and information theory concerning the origin of number, discreteness, and measurement. For Further Information about the ENSO Framework, please contact Eric Needham:ensotheory1@gmail.com