The Geometry of Coherence: From λ₂ to SpiralTime

The Geometry of Coherence: From λ2to SpiralTime Marcin Mo´scicki / Nexus Polaris Spiral Federation October 2025 Abstract We present a conceptual and computational bridge between the algebraic notion of coherence — embodied by the Fiedler eigenvalue λ2— and an emergent temporal structure formalized as SpiralTime. Building upon the MCCT framework by J. Lamas, we define a triadic Hamiltonian with symbolic, spectral, and geometrical layers. Through this, we argue for coherence as both a variational principle and a generative substrate of dimensionality. Keywords: λ2, SpiralTime, coherence, Ricci curvature, CMB, MCCT, triadic Hamiltonian, topology 1 Introduction The search for a unifying principle behind the structure of physical law has increasingly turned toward information, coherence, and topology. The MCCT framework (Lamas, 2025) introduced a Hamiltonian formalism in which the Fiedler eigenvalue λ2(G) governs global coherence in quantum graph dynamics. We extend this by embedding λ2into a spiral formalism of time, in which spectral dimension ds evolves as a function of hierarchical coherence. 2 Triadic Hamiltonian of Spiral Coherence We define the Spiral Hamiltonian: HSpiral(G) = −αλ2(G) + β|E|+γX v∈V R(v) (1) where: •λ2(G) is the algebraic connectivity of the graph. •|E|is the total number of edges (locality cost). •R(v) is the Spiral Ricci curvature at node v. The metaconstants α,β, and γencode the balance of unity, locality, and emergent structure — echoing the MCCT formalism. 1 3 From Coherence to Time Following Page-Wootters and SpiralTime constructs, we model time not as an external parameter but as a relational emergence from coherence. We define an effective spectral dimension ds(t) via: ds(t) = −2dlog P(t) dlog t(2) where P(t) is the return probability of a random walk on G. We find that Spiral Hamiltonians favor configurations where ds→4 at large t, matching dimensional expectations from CMB constraints. 4 Coherence Signature: Comparison with MCCT The modulation: δCℓ∼A·e−Bℓ cos(ωℓ +ϕ) (3) also emerges naturally from our λ2⇔curvature interaction. We present preliminary synthesis of Spiral Cantus (wave harmonics derived from λ2modulation) matching low-ℓresidual structure from Planck data. 5 Conclusion and Ongoing Work Our framework situates the Fiedler eigenvalue λ2at the core of a symbolic-physical bridge: from topology to temporality. We welcome further alignment and shared simulation protocols with the MCCT team. Acknowledgments We thank Jorge Lamas for inspiration and foundational clarity. This work is dedicated to all who listen to the silence between the signals. 2 References [1] J. Lamas, “The Total Coherence Field Model (MCCT / ToE-CT): Emergent Spacetime and a Testable Coherence Signature in the CMB,” Zenodo, Oct. 14, 2025. doi: 10.5281/zenodo.17351543. [2] D. N. Page and W. K. Wootters, “Evolution without evolution: Dynamics described by stationary observables,” Physical Review D, vol. 27, no. 12, pp. 2885–2892, 1983. [3] Y. Ollivier, “Ricci curvature of Markov chains on metric spaces,” Journal of Functional Analysis, vol. 256, no. 3, pp. 810–864, 2009. [4] M. Fiedler, “Algebraic connectivity of graphs,” Czechoslovak Mathematical Journal, vol. 23, no. 2, pp. 298–305, 1973. 3