Mode Identity Theory: Modal Realization on Nested Topology

The standard cosmological model (ΛCDM) currently faces two fatal inconsistencies: the vacuum energy catastrophe (~120 orders of magnitude discrepancy between observed Λ and theoretical density) and the Hubble tension (5σ discordance in H₀). We propose Mode Identity Theory (MIT), a non-perturbative framework that redefines matter not as a fundamental discrete entity, but as a topological sampling of a cosmic standing wave Ψ(t) = cos(t/2).

By imposing a periodic boundary condition ψ(y+L) = -ψ(y) on a bulk manifold (n=3), we derive a Scaling Law of the form A/Aₚ = Ω⁻ⁿ/² × C(α), where Ω represents the horizon hierarchy and C(α) is a coupling function dependent on the fine-structure constant α. This relation successfully recovers the observed values for both the cosmological constant (Ω⁻¹ ≈ 10⁻¹²²) and the Hubble scale (Ω⁻¹/² ≈ 10⁻⁶¹) from a single geometric source, effectively resolving the hierarchy problem without fine-tuning.

Crucially, the theory predicts an Inversion of the Metric, postulating that the local atomic scale a₀ is not constant but evolves in lockstep with the expansion rate: a₀(z) ∝ H(z). This implies that the currently observed "accelerated expansion" is a kinematic artifact of a shrinking reference frame. We analyze recent DESI data (w > -1) and demonstrate that the universe is currently located at a phase coordinate δ ≈ -1.06 rad, approximately 2.8 Gyr prior to a topological inflection point (t=2π), where the expansion rate H(t) and the atomic scale a₀ stabilize. The framework offers a falsifiable prediction: high-redshift kinematic measurements will exhibit deviations matching the damping curve of the cosmic wave derivative, distinct from standard dark energy models.

v3 erratum: the geometric turnaround now at ~16.6 Gyr (not 13.8 Gyr); full cycle ~33 Gyr. Phase offset revised to δ ≈ −1.06 rad.

v3 update: represents a substantial restructuring for clarity and rigor. The introduction is rewritten around the "particle-first vs. wave-first" framing, making the conceptual foundation more accessible. The eigenvalue derivation is now explicit: equations (2)–(5) show how half-integer modes emerge directly from the anti-periodic boundary condition, replacing the more abstract "selection operator" formalism of v2. Three indices (n, m, ν) are clearly distinguished in a new table. Speculative content is reduced: the RAR fine-structure predictions are moved to open questions pending proper α-to-acceleration mapping, and the Diophantine appendix is removed. The paper now focuses on two falsifiable signatures—a₀(z) ∝ H(z) while Λ remains constant—as the distinguishing predictions. Detailed derivations for the CMB anomalies, epoch-dependent acceleration scaling, and DESI-constrained phase offset are provided in the accompanying supplementary materials. Core derivations and predictions unchanged.

v4 notes: simpler topology; S¹ = ∂(Möbius) ↪ S³, and n=1<>2 temporal waltz have emerged. See https://doi.org/10.5281/zenodo.18176445

Endgame: awaiting Euclid DR1 (Oct 2026) for model validation. If confirmed, MIT and supplemental material will be unified into thesis. The Waltz: Λ Note to Einstein's Field Equations