Mode Identity Theory: Modal Realization from Nested Topology

Mode Identity Theory (MIT) proposes that matter is wave, sampled. From one topological postulate, S¹ = ∂(Möbius) ↪ S³, ∂S³ = ∅, a single scaling law emerges: A/Aₚ ≈ C(Θ) · (√Ω)⁻ⁿ It derives Λ, H₀, a₀, and α across 122 orders of magnitude. Each constant is a location on the cosmic standing wave Ψ, diluted by where the observation takes place. Five cosmological anomalies map one-to-one to five foundations: the a₀ ≈ cH₀ coincidence, the cosmological constant, CMB large-angle departures, the cosmic coincidence, and null dark detection. The framework produces the 3/2 Gauss-Codazzi gravitational embedding interface. At the particle scale, the same topology confines gauge fields; the Yang-Mills mass gap on S³/2I is a consequence. The fine structure constant α is the first step of Λ. Three particle generations follow from three isolated vacua. Two predictions separate MIT from alternatives: a₀(z) tracks H(z) while Λ remains constant. Falsification criteria are pre-registered to Euclid Data Release 1 (October 2026). Every component of the scaling law traces to the topology postulate: the 120-domain (|2I| = 120 native to S³), Fibonacci wells (Hurwitz stability), observer position (IR↔UV fixed point at √Ω ≈ 10⁶¹), manifold assignments (epoch-dependence selects Ω), the phase operator C(Θ) = 2sin²(πΘ) (anti-periodic boundary condition), and the Bosonic Filter (|ψ|² projects 2I → I onto the 60R-grid). The phase field Θ_f = 2/120 is the minimum bosonic step, yielding the 8.4% Hubble tension as environment-dependent sampling of the H₀ well. The cosmological constant Λ_obs = (3/2)Λ_top follows from Gauss-Codazzi embedding of the Möbius surface eigenvalue into S³ spatial curvature. The chronon π/30 is the smallest phase advance the domain can register; the minimum action ΔS = ℏπ/30 holds in every frame. 🔗 github.com/mode-identity-theory/framework to follow the working version