Neutrinos in Phase‐Biased Geometry

Neutrinos in Phase-Biased Geometry (PBG)

Purpose. Present a self-contained neutrino chapter: structure from PBG first principles, core predictions, controlled variants, and cross-calibration methods—all in Obsidian-ready Markdown.

1 Neutrino as a Zero-Winding Coherence Mode

PBG treats every particle as a coherence mode of the scalar phase field $Φ (t, x)$ . Neutrinos are the simplest:

Winding number $N = 0$ : no trapped $2 π$ spiral (unlike baryons $N = 3$ , mesons $N = 1$ ). They are plane waves in $Φ$ .
Coherence envelope (static Yukawa tail): $Φ_{ν} (r) \propto \frac{e^{- r / ℓ_{ν}}}{r}, ℓ_{ν} = \sqrt{\frac{α}{β}} \sim 4 \times 10^{34} m,$ so neutrinos are effectively delocalised
.

Why this matters: Zero winding guarantees minimal internal structure; all nontrivial effects come from the tiny anchoring cost $β$ .

2 Structural Mass Derivation

From the PBG action density:

L = \frac{1}{2} γ (\partial_{t} Φ)^{2} - \frac{1}{2} α | \nabla Φ |^{2} - \frac{1}{2} β Φ^{2},

the plane-wave dispersion is

γ ω^{2} = α k^{2} + β ⟹ ω^{2} = c^{2} k^{2} + m_{ν}^{2}, m_{ν}^{2} = \frac{β}{γ} .

Using current anchors ( $β = 5.3 \times 10^{- 54} J / m^{3}$ , $γ = 1.00 \times 10^{- 18} J m^{- 3} s^{2}$ ) one finds

m_{ν} = \sqrt{β / γ} \approx 2.3 \times 10^{- 18} s^{- 1} (ℏ m_{ν} \sim 10^{- 33} eV) .

No see-saw or Higgs mechanism is needed: neutrino mass is purely the residual anchoring penalty.

3 Core Predictions

Probe	Observable	PBG prediction
KATRIN / Project 8	Tritium-β endpoint $m_{β}$	$< 10^{- 30}$ eV (null result)
Neutrinoless double-β (LEGEND)	$T_{1 / 2}^{0 ν β β}$	$\infty$ (no decay)
TOF (OPERA/ICARUS/JUNO)	$	v_\nu-c
Cosmology (Planck/Euclid)	$\sum m_{ν}$	$\sim 10^{- 33}$ eV (negligible)
Oscillations (DUNE/Hyper-K)	$P (ν_{α} \to ν_{β})$	geometric phase drift, $δ_{C P} = 0$

All predictions follow from $N = 0$ and $m_{ν}^{2} = β / γ$ with ±0.3 % anchor errors.

4 Variant A — Micro-Winding Hypothesis

Allow a fractional winding $N = \frac{1}{2}$ in weak-isospin space. Consequences:

Tiny core radius $r_{m i c r o} = α γ / (2 π m_{ν})$ ⇒ small Pauli sheet.
Nonzero 0νββ amplitude $\propto r_{m i c r o}^{2}$ .
Neutrino form factors: charge radius $⟨ r_{ν}^{2} ⟩ \sim r_{m i c r o}^{2}$ , magnetic moment $μ_{ν} \sim e r_{m i c r o} / 2$ .

Key test: any signal in LEGEND-1000 or GEMMA/COHERENT above Variant A levels falsifies minimal ( $N = 0$ ) PBG.

5 Cross-Calibration: Oscillation Rulers & Sirens

Oscillation ruler: nodes at

L_{n} (E) = \frac{2 π n E}{Δ m^{2}},

measure $L_{n}$ for $n = 1, 2, \dots$ from a transient.

Standard siren: neutrino–photon lag

Δ t (ζ) = \int_{0}^{ζ} \frac{d ζ^{'}}{{\dot{ζ}}^{'}} [\frac{1}{v (ζ^{'})} - \frac{1}{c}] .

Both tie to the modal-decay marker $ζ$ for the same source. Matching $L_{n} (ζ)$ and $Δ t (ζ)$ to photon-based $D (ζ)$ empirically builds the PBG distance ladder.

6 Speculative Frontiers

CνB coherence: relic neutrinos as a phase-locked field—deviations probe IR anchoring.
Density tomography: flavor-phase shifts through Earth map $ρ (x)$ in real time.
Sterile modes: excited anchoring resonances as heavy neutrino states, fixed by ${α, β, γ}$ (no new couplings).

Last updated: 2025-05-15