Key References

Papers and their specific contributions to the program, with the mathematical details that matter for future work.

Todorov — Octonion Internal Space (arXiv:2206.06912)

What it does: Derives $m_H = 2\cos\theta_W m_W$ from the split $\mathbb{O} = \mathbb{C} \oplus \mathbb{C}^3$ and a Quillen superconnection normalisation condition. Claims “within one percent accuracy” of experiment.

Numerical output: \(m_H^\text{pred} = 2\sqrt{5/8} \times m_W = \sqrt{5/2} \times 80.377 = 127.090 \pm 0.019 \text{ GeV}\) The $\pm 0.019$ GeV error comes purely from $\delta m_W = \pm 0.012$ GeV (PDG input). The Weinberg angle is fixed at $\sin^2\theta_W = 3/8$ (no error). Experimental: $m_H = 125.10 \pm 0.14$ GeV. Discrepancy: $1.99$ GeV $= 13.7\sigma$ in units of $\delta m_H^\text{exp}$, but only $1.6\%$ in relative terms. The paper characterises this as “within one percent” (loosely).

Starting postulate: $\mathbb{O} = \mathbb{C} \oplus \mathbb{C}^3$. In the $u$-framework this split is derived: $\mathbb{C} = \mathbb{C}_u$ (span of $1, u$), $\mathbb{C}^3 = u^\perp$.

Connection to $u$-framework: The normalization condition $|\Phi|^2_Q = 3|\Phi|^2_L$ is now formalized in the $u$-framework as a conditional proposition: it follows from $G_2 = \mathrm{Aut}(\mathbb{O})$ transitivity together with orthogonal additivity of the quadratic form and diagonal Higgs embedding across the three colour channels. What remains open is showing that Todorov’s precise superconnection norm is exactly this quadratic form. See higgs-todorov.md.

Useful internal detail: Todorov treats the lepton projector $\ell$ and the colour projectors $q_j$ as mutually orthogonal idempotents in $\mathrm{Cl}_6$. The Higgs normalization is fixed by comparing the leptonic $\Phi^2$ contribution with that of a single coloured quark; in his restricted-projector parametrization this is encoded by $\rho^2 = 1/2$, which then gives $m_H^2/m_W^2 = 5/2$.

Useful structural detail: In the octonion paper the particle projector is written as a singlet-plus-triplet decomposition, $\mathcal{P}=\ell+q$ with $q=q_1+q_2+q_3$. This is the Clifford-algebra counterpart of the octonionic split $\mathbb{O}=\mathbb{C}\oplus\mathbb{C}^3$, and is the natural candidate for the intertwiner between the $u$-framework channels and Todorov’s idempotent basis.

Concrete check to do later: Use Todorov’s colour generators given by the traceless bilinears in the colour oscillators $(b_j,b_k^\dagger)$ and compute their action on the leptonic states built from $\ell$ and the coloured quark states built from $q_m$. If the leptons are colour singlets and the three quark channels mix as a single three-dimensional irreducible colour representation, then the representation part of the intertwiner is established.

Likely simplification: With the projector convention $\ell=p_1p_2p_3$ and $q_j=p_jp_k’p_\ell’$, the leptons sit in the colour vacuum sector $\Lambda^0(\mathbb{C}^3)$ while the quark channels sit in the two-particle sector $\Lambda^2(\mathbb{C}^3)$. So the representation part may already be reducible to Fock-space occupation counting; the harder remaining step is the supertrace metric matching.

Updated read of the gap: The representation half of the normalization problem is now close to settled in the notes. The source formulas sharpen the metric half too: once Todorov’s odd Higgs operator is decomposed channelwise as $\Phi=\Phi_\ell+\sum_j\Phi_{q_j}$, orthogonality of distinct channels and equality of the three quark diagonal weights follow formally from $\ell q_j=0$, $q_jq_k=\delta_{jk}q_j$, and factor separation between the weak oscillators and the colour projectors. So the live issue is no longer the full $4\times4$ matrix from scratch, but the remaining scalar comparison between the leptonic diagonal entry and one single-colour quark entry. Equivalently, one still has to show that the channel form is not merely $a\oplus bI_3$ but actually $cI_4$ on $(\ell,q_1,q_2,q_3)$.

Todorov — Superselection of Weak Hypercharge (arXiv:2010.15621)

Published: JHEP 04 (2021) 164. The Clifford algebra version preceding the octonion paper.

What it does: The earlier, more algebraically explicit paper. Derives the same $m_H/m_W$ relation. Contains the cleaner mathematical derivation of the superconnection normalisation.

Key algebra: Built on $Cl_4 \hat\otimes Cl_6$ — the $\mathbb{Z}_2$-graded tensor product of:

• $Cl_4$: Clifford algebra of spacetime (4 real dims), generates Lorentz/Poincaré structure
• $Cl_6$: Clifford algebra of internal space (6 real dims = $u^\perp$ in the $u$-framework)

The idempotent: \(P = \frac{1}{2}(1 - i\omega_6), \qquad \omega_6 = \gamma_1\gamma_2\gamma_3\gamma_4\gamma_5\gamma_6, \quad \omega_6^2 = -1\) where $\gamma_1,\ldots,\gamma_6$ are the $Cl_6$ generators. $P$ satisfies $P^2 = P$ and projects the full 32-dimensional Majorana spinor space of $Cl_{10}$ onto the 16-dimensional particle subspace. This idempotent uses the $Cl_6$ volume form (not a Witt decomposition product like Furey’s $qq^\dagger$) — but both are derived from the same $u^\perp$ decomposition.

Pati-Salam intermediate: The subgroup $G_{PS} = Spin(4) \times Spin(6)/\mathbb{Z}_2$ is left-invariant by $\omega_6$. The $Cl_6$ volume form selects Pati-Salam as the maximal subgroup preserving the complex structure on $u^\perp$.

Sterile neutrino exclusion: The weak hypercharge $Y$ is promoted to a superselection rule, not merely a gauge generator. The zero eigenspace of $Y$ — which contains right-handed neutrinos (sterile neutrinos) — is excluded from the particle subspace by the $P$ projection. Sterile neutrinos are allowed to exist but are structurally outside the observable sector defined by $P$.

The claim: “A relation between the masses of the W boson and the Higgs that fits the experimental values within one percent accuracy” — this is the original statement of the $m_H/m_W$ result.

What to extract next: For the normalization problem, the sharp target is no longer “derive the factor of 3 somehow”. It is:

• show the odd-sector supertrace on the lepton-plus-colour channels has $SU(3)$ block form $a \oplus bI_3$;
• show the octonionic lift to pre-selection $G_2$ data forces $a=b$;
• then read Todorov’s $\rho$ as the basis-dependent encoding of that equality in the restricted projector formalism.

Use for future work: For a proper derivation of $\lambda = g^2\cos^2\theta_W/2$ from first principles (rather than stating it follows from the normalisation condition), this earlier paper has the explicit supertrace calculation. Priority to read if pursuing the EW-threshold correction calculation for the residual Higgs-mass gap.

Todorov — Superconnection in the Spin Factor Approach (arXiv:2003.06591)

Published: 2020. Precursor to both papers above.

The Quillen superconnection structure (from what was retrieved): \(\mathbb{D} = D + L\) where $D$ is the covariant derivative (even, gauge fields) and $L$ is the Higgs field (odd). The curvature: \(\mathbb{D}^2 = \nabla^2 + \chi[D, \Phi] + \Phi^2\) where $\chi$ is a grading operator. The action is $\mathrm{Str}(\mathbb{D}^2 \wedge \ast \mathbb{D}^2)$. The scalar potential term comes from $\mathrm{Str}(\Phi^4) \sim \lambda|\Phi|^4$.

Furey & Hughes — Division Algebraic Symmetry Breaking (arXiv:2210.10126)

Published: Phys. Lett. B 831 (2022) 137186.

What it does: Demonstrates the cascade $Spin(10) \to \text{Pati-Salam} \to \text{Left-Right} \to \text{SM} + B-L$ driven by sequential complex structure selections from $\mathbb{O}$, then $\mathbb{H}$, then $\mathbb{C}$. Derives the Higgs doublet as the scalar component $V$ of the quaternionic triality triple $(\Psi_+, \Psi_-, V)$ of $\mathrm{tri}(\mathbb{H})$.

Key result used: The Higgs as the scalar component of triality — not inserted by hand. The cascade mechanism of complex structure selections structurally matches the $u$-selection cascade in this program.

Mapping (in higgs.md): The quaternionic slice $\mathbb{H} \subset u^\perp$ in this framework corresponds exactly to the quaternionic complex structure selected second in Furey & Hughes. The $SU(2)$ residual freedom (which direction in $\mathbb{H}$ is the vev direction) corresponds to the remaining step not fixed by Furey & Hughes.

Historical Context: Ne’eman & Fairlie (1979)

Ne’eman: Nucl. Phys. B 158 (1979) 131. Fairlie: Phys. Lett. B 82 (1979) 97.

How Todorov differs: Todorov’s construction uses Quillen’s superconnection (1985) rather than the original Lie-superalgebra gauge theory. The Quillen approach allows the Higgs self-coupling $\lambda$ to be fixed differently — yielding $m_H = 2\cos\theta_W m_W$ rather than $m_H = m_Z$. The key difference is in how the action is constructed from the curvature (supertrace of $\mathbb{F}^2$ vs. gauge-theory kinetic term).

Community Reception: Physics Forums Discussion

A Physics Forums thread (accessible) discussed the Todorov $m_H/m_W$ claim. Key points from the discussion:

1. The $13.7\sigma$ discrepancy (in experimental error units) is flagged as significant, even though the percentage is only 1.6%.
2. The “within one percent” phrasing is characterised as misleading given that $\delta m_H^\text{exp} = 0.14$ GeV makes 1.9 GeV a large sigma-deviation.
3. The community distinguishes between “close in percentage” and “a genuine prediction” — the latter requires the gap to close under radiative corrections in a way that is calculable and not fine-tuned.

Implication for this framework: The claim must be presented as a tree-level relation with an unresolved correction term. The earlier “GUT-to-EW running closes the gap” framing no longer holds up; the live issue is an explicit electroweak-threshold correction calculation. Without that calculation, the “1%” language invites the same criticism. See the gap analysis in higgs.md.

Quillen — Superconnections (1985)

Quillen, D.: J. reine angew. Math. 346 (1985) 163–196.

What it does: Defines the superconnection formalism. A superconnection on a $\mathbb{Z}_2$-graded vector bundle $E = E^+ \oplus E^-$ is a map $\mathbb{A}: \Omega^(M, E^+) \oplus \Omega^(M, E^-) \to \Omega^*(M, E^- \oplus E^+)$ that is a graded derivation. The curvature $\mathbb{F} = \mathbb{A}^2$ is used to define characteristic classes.

Physics use: Bismut (1986) applied superconnections to the index theorem. Connes (1996) applied the formalism to derive the SM action from NCG. Todorov applies it to get the Higgs-gauge mass relation.

Gogberashvili — Octonionic Geometry and Physics (arXiv:1602.07979)

What it does: Derives the conformal group $SO(2,4)$ from $G_2^{\text{split}}$ acting on a cone of split-octonions. Used in the $G_2 \to SO(2,4)$ arrow of the chain. See chain/g2-so24.md.

Status in this project: This check is now completed. The image of $\tilde{u}$ under the cone map is taken to be spacelike in $\mathbb{R}^{2,4}$, giving the required $SO(2,3)$ stabilizer and hence the $Spin(2,3)$ arrow. See chain/so24-spin23.md.

Baez — Octonions (2002) and the Leech Connection

Baez, J.C.: Bull. AMS 39 (2002) 145–205. arXiv:math/0105155.

Egan, G.: “Symmetries of the Standard Model” (blog, 2022). Explicitly works out the Leech lattice embedding in $J_3(\mathbb{O})$ and the $E_8$ connection.

Key result for this program: $J_3(\mathbb{O})$ contains a 24+3 structure (24 off-diagonal octonion entries + 3 real diagonal) whose 24-dimensional sublattice maps to the Leech lattice. See statics.md.

Status in this project: This is no longer treated as a single open check. The current formulation is: continuous $SU(3)$ equivariance is impossible for a discrete lattice; discrete equivariance is conditional on fixing the triality frame; and the Jordan product $D \circ U$ does not preserve the Leech tier except for $D=\pm I$. See statics.md.