Sector 5 — Neutrinos

The \(d=5\) sector has geometry \(S^5\) — the five-sphere — with isometry group SO(6). Like \(S^3\) at \(d=3\), \(S^5\) is a real manifold without a Kähler structure. The Hopf fibration of \(S^5\) takes the form \(S^1 \to S^5 \to \mathbb{CP}^2\), meaning the base space \(\mathbb{CP}^2\) (the colour sector) sits inside \(S^5\) as a quotient. This embedding is what makes the neutrino sector contain the colour directions geometrically while still projecting to zero colour charge.

The critical property of \(d=5\) is the Clifford algebra: \(d \bmod 8 = 5\). At this value of \(d \bmod 8\), the Clifford algebra periodicity theorem (Bott periodicity) makes it impossible to impose a Majorana condition on the spinor. This is a structural fact about the geometry of five-dimensional spaces — it does not depend on the energy scale, the coupling strength, or any dynamical property of the neutrino. The Majorana condition simply cannot be written down for a spinor on \(S^5\).

The neutrino mass scale is not a free parameter. It is derived from the sector coupling constants:

No neutrino mass data enters this derivation. The scale follows from the cross-sector consistency of the coupling constants, which are themselves derived from \(n_s = 4\). The three neutrino masses are then determined by their mode indices alone.

The mass ratio \(\nu_2/\nu_1 = S(15,5)/S(10,5) = 11628/2002 \approx 5.808\), giving \(\Delta m^2_{21} = 7.242 \times 10^{-5}\text{ eV}^2\) (PDG 2024: \(7.53 \times 10^{-5}\text{ eV}^2\), −3.8%). The \(\nu_3\) mass includes a cross-sector correction \(\delta_{\nu_3} = \varepsilon \times g_{33} = 1/35\) from the \(\ell=2\) kernel scale \(\varepsilon\) and the \(d=3\) coupling, derived exactly in Part 2 §9d.

The total neutrino mass sum \(\Sigma m_\nu = 60.39\) meV is within reach of CMB-S4. A measurement outside the 55–65 meV window falsifies IDWT.

Because \(d \bmod 8 = 5\) forbids the Majorana condition, the consequences are total and permanent:

• Neutrinos are strictly Dirac fermions. Not approximately Dirac — the Majorana condition cannot be imposed at any energy scale.
• All Majorana mass terms are forbidden. Every term in a Lagrangian that would give a Majorana mass to a neutrino requires the Majorana spinor type. That type does not exist in the \(d=5\) sector.
• All lepton-number-violating interactions are forbidden. Every such interaction requires a Majorana mass term at some order. Without the spinor type that supports Majorana masses, no lepton-number-violating vertex can be formulated.
• The Majorana mass term is absent at all orders. Any induced effective Majorana operator would require a charge conjugation matrix C satisfying the Majorana condition on the \(S^5\) spinor bundle — but \(d \bmod 8 = 5\) is the one Clifford class where no such C exists. Cross-sector coordinate couplings can correct the Dirac mass but cannot produce \(\psi^T C\psi\) at any order. The 0νββ vertex cannot be written down at any level of the theory.
• The see-saw mechanism is eliminated. The see-saw requires a heavy Majorana neutrino. The \(d=5\) geometry cannot support one.

The 0νββ rate is zero — not as a suppressed coupling but as a consequence of operator non-existence. The \(d=5\) Clifford prohibition eliminates induced Majorana operators at every order, not just at leading order: the charge conjugation matrix C required to write \(\psi^T C\psi\) does not exist on the \(S^5\) bundle at any loop level. KamLAND-Zen, nEXO, LEGEND, and every other 0νββ experiment is searching for a signal that IDWT says cannot be formulated. A positive detection falsifies IDWT's d=5 spinor structure directly.

Neutrinos contain the \(d=4\) colour sector in their coordinates — the nesting gives \(d=4 \subset d=5\). Yet neutrinos have zero colour charge. This is not a coincidence; it is the \(S^5\) Hopf fibration at work.

The Hopf fibration \(S^1 \to S^5 \to \mathbb{CP}^2\) has \(S^1\) (the U(1) fiber) wrapping around \(S^5\) with \(\mathbb{CP}^2\) as the base. The \(S^1\) fiber averages over the \(\mathbb{CP}^2\) colour representation as the field \(\Psi\) winds around it. That averaging selects only the colour-singlet component. Neutrinos have coordinate support in the colour sector but the \(S^5\) geometry projects that support onto exactly zero colour charge. They are not colour-neutral because they lack the \(d=4\) directions — they are colour-neutral because the \(d=5\) geometry has an averaging mechanism that cancels the colour contribution to zero.

What neutrinos get positively from the \(\mathrm{SO}(6) \cong \mathrm{SU}(4)\) Pati-Salam structure of \(S^5\) is baryon-minus-lepton number (B−L charge). This is their genuine quantum number from the \(d=5\) sector geometry.

All three PMNS mixing angles are determined by the sector coupling constant \(g_{55}\) and the four mode indices of the charged leptons and neutrinos:

Predictions: \(\sin^2\theta_{23} = 0.55897\) (PDG 2024: 0.553, +1.07%), \(\sin^2\theta_{12} = 0.30856\) (PDG: 0.307, +0.51%), \(\sin^2\theta_{13} = 0.02211\) (PDG: 0.0220, +0.51%). The CP-violating phase \(\delta_{CP} = \pi + 2\theta_{13} = 197.11°\) follows from spectral flow on the lepton sector manifolds (PDG: ~197°, +0.05%).

← Sector 4 — Up-type Quarks  ·  All Sectors  ·  Next: Sector 6 — Charged Leptons →