Appendix N — Derivation 14: Neutrino Mass from Anchoring Suppression

Overview

In the Standard Model, neutrinos were originally massless. Later, mass was added via the see-saw mechanism, involving heavy sterile states and symmetry breaking.

In modal dynamics, no such mechanism is needed.
Neutrinos acquire mass because they are weakly anchored modes that propagate near-latently—experiencing structural drift relative to fully latent modes like photons.

This appendix derives that drift, and shows how it leads to a nonzero but small effective mass.

1. Neutrino as a Weakly Anchored Mode

A neutrino is represented by a coherence function $ψ_{ν} (x, t)$ with:

Low anchoring density: $ρ_{c} ≪ ρ_{crit}$
Minimal internal phase winding: $w = 0$
Weak interaction with coherence field: low $Γ (ψ_{ν}, B)$
High structural persistence over large distances

Its internal phase evolves slowly, and it travels almost—but not exactly—at the speed of light.

2. Anchoring Cost and Latency Drift

The coherence cost functional (from Appendix A) is:

C [ψ] = \int (γ | \partial_{t} ψ |^{2} + α | \nabla ψ |^{2} + β | ψ |^{2}) d^{3} x

For a fully latent mode (photon), the anchoring term $β | ψ |^{2}$ vanishes, and the motion occurs at $v = c = \sqrt{α / γ}$ .

For the neutrino, $β \neq 0$ —there is a nonzero anchoring cost, which alters the balance between spatial and temporal coherence.

The photon satisfies:

γ ω_{γ}^{2} = α k_{γ}^{2}

But the neutrino satisfies:

γ ω_{ν}^{2} = α k_{ν}^{2} + β

This implies a lower frequency for the same $k$ , or a lower group velocity.

3. Effective Mass from Temporal Lag

Let the neutrino propagate with dispersion relation:

ω_{ν}^{2} = \frac{α}{γ} k^{2} + \frac{β}{γ}

Compare to a massive wave equation:

ω^{2} = c^{2} k^{2} + m^{2}

Then the effective mass of the neutrino is:

m_{ν}^{2} = \frac{β}{γ}

Thus, the neutrino’s mass is set by:

Its anchoring penalty $β$ (mode–field tension)
Its phase rigidity $γ$ (resistance to distortion)

This mass is structural, not a coupling constant or symmetry-breaking effect.

4. Stability and Oscillation

Because anchoring is weak, neutrinos exhibit:

Slow phase drift over long distances
Structural rotation in internal topology
Apparent flavour oscillation as the modal coherence profile reconfigures

There is no flavour mixing matrix. Oscillation is a geometric effect of non-rigid internal phase under bias drift.

5. Neutrino Hierarchy

Different neutrino types (electron, muon, tau) correspond to modes with slightly different:

Anchoring stiffness $β$
Temporal phase rigidity $γ$
Internal topologies ${\vec{κ}}_{i}$

These lead to small but distinct mass values and coherence drift rates.

Conclusion

Neutrinos are not massless.
But their mass is not fundamental—it is the residual effect of propagating with minimal anchoring in a deforming coherence field.

Appendix M | [Index](./Appendix Master) | Appendix O