What this is

D8 is a geometric model of the Standard Model fields built on the Clifford algebra $Cl(1,7)$, read not as a single eight-dimensional space but as a compact three-complex-dimensional fiber over four-dimensional spacetime, with the two joined by a soldering that identifies part of the fiber with the base.

Previous geometric unification programs derive the Higgs field, and sometimes its potential, from geometry. D8 aims further: the gauge symmetry itself arises as the stabilizer of the fermion vacuum, while its electroweak breaking — together with the three-generation structure and its hierarchy — follows from the solder, the single geometric object the theory already requires in order to produce spacetime.

The three generations are three inequivalent spacetime readings of a single fermionic algebra, rendered physically independent by their orthogonal fiber profiles. A unique, soldering-independent fermion vacuum anchors all three, while the dynamical alignment of the solder selects the physical fiber-to-spacetime identification, from which the Higgs couplings, generation hierarchy, and eventually the sector mixing matrices arise.

Status

Established

Result	Basis
Gauge group as vacuum stabilizer. $U(3)$ on the right is the stabilizer of the sterile fermion corner: $U P_l \tilde{U} = P_l$.	Algebraic
Fermion vacuum and full state spectrum. The vacuum is the right-handed-neutrino spin-up quartet; every fermion state is reached as the two-sided orbit under the four commuting bivectors $S = \gamma^{31}, X = \gamma^{04}, I = \gamma^{57}, D = \gamma^{26}$	Constructed, checked against the 16-block dictionary
Charge assignments. $Q = T_3 + Y$ with $Y = \tfrac{1}{2} Y_L - \tfrac{1}{2} Y_R$; reproduces the SM hypercharge of every Weyl component exactly.	Verified per block
$u(3)$ basis and closure. $\lambda_1 ... \lambda_8, Y_R$; left electroweak $T_a \equiv \lambda_{1,2,3}, Y_L$; Higgs coset $\lambda_{4...7}$ with $Y_H = K_1 = i\gamma_{12}$. All commutators and closure relations hold.	Verified sector-by-sector over $1 \oplus 3 \oplus \bar{3} \oplus 1$
Higgs as coset spin connection. The Higgs is the $U(3) / U(2)$ coset-valued component of the fiber spin connection.	Construction
Single shared hypercharge; no $Z'$. Zeroing the broken $U(1)$ (the $\gamma_{12}$ element) forces the coefficient of $Y_L$ to be the same field $B$ arriving from the right, and forces $Z' = 0$.	Maurer–Cartan computation
Coset annihilation theorem. $\lambda_{4...7}$ annihilate the lepton and antilepton columns from both sides.	Machine-verified
Algebraic mass degeneracy (negative result). For any $H$, the surviving mass operator is a real Clifford vector with $\Omega^2 = (v \cdot v)1$, so it has a single eigenvalue magnitude. Masses cannot be algebraic — they must come from fiber profiles.	Machine-verified to machine precision
Gravity from $\epsilon \omega \langle R^4 \rangle$. Lovelock cascade with computed coefficients: cosmological term, Einstein–Hilbert, ghost-free Gauss–Bonnet, topological Eule$r_8$. AdS sign forced by $\omega^2 = -1$.	Derived
Anomaly structure. Mixed traces vanish except $[SU(2)]^2 \omega$ and $Y^2 \omega$, which are mirror-reinforced and absorbed by the eaten Stückelberg master phase; $tr(\omega Q^2) = 0$, so no anomalous master-$\gamma \gamma$ coupling.	Trace computation
$\theta_{QCD} = 0$ by definition. No scalar can be formed from the right-side curvature and its dual in six fiber dimensions. Strong CP without an axion.	Grade-theoretic
Generation count from flux. Per-sector count is $min(F_L, F_R)$ with $F_i = \Phi_{\omega} + Y_i \Phi_B$. Requiring exactly three for every Weyl species is a nontrivial constraint on the flux fractions, and is satisfiable.	Zero-mode analysis

Contested

Item	Why
Shift symmetry — action depends only on $\nabla H$	The coset mixing result puts rotor and fiber-spin-connection content in the same $\lambda_{4...7}$ directions, so the pure-Higgs $F$ term is not zero
Masses $\varpropto h'(y)$	Same cause
Fiber vacuum geometry: $\mathbb{CP}^2$ vs doubly-warped $AdS_5 \times S^3$	Which rigidity statement survives the from-scratch rebuild is unsettled
Solder boson	Predicted from the fifth broken coset generator $(\gamma_{12})$; needs reconciling with the step that sets the $\gamma_{12}$ element to zero
Master $U(1)$ mass	Bare $m^2 = -28/l^2$ is tachyonic; the tree Proca mechanism is orphaned. Meissner mass from the charge-2 solder condensate is the candidate, requiring $m^2_{Meissner} \gt 28 \varphi^2$

Open

• The solder potential — the active front; every other open item terminates here
• Fiber anisotropy and the isospin / up–down splitting channel
• Hierarchy magnitude and ordering — needs roughly 4–6 e-folds per step, up-sector steepest, super-geometric
• CKM — identically trivial until transverse structure enters
• Mirror-sector mass exclusion
• Scale setting: ⟨𝜑⟩ by dimensional transmutation under global scale invariance; massless dilaton/radion accepted as the cost
• Right-side kinetic term: loop-induced alone gives $\alpha_s$ far too strong, so a bare term is present; its coefficient is unfixed
• Whether the mirror-CP $\mathbb{Z}_2$ is exact or spontaneously broken by the condensate
• Quantization of the quadratic-gravity sector

Contents

The papers below are ordered by dependency: each assumes only those above it. Conventions and generator definitions live in Paper 0 and are not restated elsewhere.

#	Paper	Covers	Status
0	Conventions & Algebra	$Cl(1,7)$ as a three-complex-dimensional fiber over 4D spacetime; oscillator basis $\alpha_m$ and the $\gamma$ dictionary; the sixteen-block state table; $u(3)$ generators $\lambda_1 ... \lambda_8, Y_R$; left $T_a, Y_L$; coset $\lambda_{4...7}, Y_H$; charge operators and reversion conventions.	Draft
1	The Fermion Vacuum and the Gauge Group	The sterile corner and its four complex functions; the vacuum as the RH-neutrino spin-up quartet; $U(3)$ derived as the stabilizer $U P_l \tilde{U} = P_l$; the state spectrum as the two-sided orbit under $S, X, I, D$; the conjugation lock and its Majorana reading.	In preparation
2	The Solder	Local variation of the fiber-to-spacetime identification; the rotor and its Maurer–Cartan payment; simultaneous breaking to spacetime, $U(2)$, and the Higgs coset; coset mixing and the single shared hypercharge; the mirror sector; the master $U(1)$.	In preparation
3	The Action	$L = k \epsilon \langle \omega R^4\rangle_0$; the Lovelock cascade and its computed coefficients; AdS vacuum with the sign forced by $\omega^2 = -1$; the tiling classification; left/right asymmetry of the action; anomaly traces.	In preparation
4	Fiber Profiles and Generations	Why masses cannot be algebraic; the internal zero-mode equation; flux, Landau and Aharonov–Casher profiles; generations as node states along the solder axis; the generation-universality constraint on flux fractions.	In preparation
5	The Solder Potential and the Mass Spectrum	The potential of the solder field itself; the stretch profile it must produce; hierarchy magnitude and ordering; isospin and sector splitting; CKM.	Active work

Predictions & falsifications

Predictions

Prediction	Status
A mirror sector. A full doubling of the fermion content carrying the same Standard Model charges, with the $W$ coupling to mirror right-handed states instead of left-handed ones, and master $U(1)$ charge reversed. Gauge bosons are shared between the sectors, but there is no mirror-changing vertex — no process converts a Standard Model state into a mirror state.	Derived
A master $U(1)$ along the pseudoscalar direction $\omega = \gamma_{01234567}$, coupling mirror-chirally: $+1$ on Standard Model states, $-1$ on mirror states. No tree-level mixing with the $Z$. Its mass mechanism is unsettled.	Derived; mass open
A solder boson — a massive neutral vector from the fifth broken coset generator, whose $U(1)$ couples to spin as its charge. A singlet under $U(2)$.	Contested
No $Z'$. Follows from the coset calculation, not from assumption.	Derived
No axion, and $\theta_{QCD} = 0$. In six fiber dimensions no scalar can be built from the right-side curvature and its dual. Strong CP is solved by dimension.	Derived
Exactly three generations, for every Weyl species. Requiring universality constrains the flux fractions to $3 \lt \Phi_{\omega} - \Phi_B$ and $\Phi_{\omega} + \tfrac{2}{3}\Phi_B \lt 4$.	Derived
Right-handed neutrinos exist and are sterile — and are the fermion vacuum, not an addition to it.	Derived
No GUT-scale colored mediators and no proton decay from unification structure — the gauge group is never embedded in a larger simple group.	Structural
A computable Higgs quartic. $\lambda / g^2$ is a group-theory ratio rather than a free parameter, making $m_H$ predicted relative to $m_W$.	Conjectured — depends on the contested fiber-$F^2$ result

Falsifications

• The solder potential. The mass spectrum is not algebraic — this is proved, not assumed. Everything therefore rests on fiber profiles, and those rest on the stretch profile the solder potential produces. That profile must supply roughly 4–6 e-folds per generation step, with the up sector steepest and the steepening super-geometric in the orbital index. A solder potential that cannot produce this shape breaks the mass sector.
• The flux pinch. Generation universality is a genuine constraint, not a fit. If no flux configuration satisfies it while also reproducing the observed hierarchy, the generation mechanism fails. Current bounded fits saturate the constraint at $\Phi_B \approx 0.52$, which may itself be a prediction rather than a coincidence.
• Mirror-sector masses. The mirror states must be excluded by a mechanism the theory already contains. A bare exclusion term is writable but would cost the universal $U(1)$.
• The master $U(1)$ mass. The bare $m^2 = -28/l^2$ is tachyonic. The Meissner mass from the charge-2 solder condensate must satisfy $m^2_{Meissner} \gt 28 \varphi^2$. If it does not, the vacuum is unstable.
• The strong coupling. A loop-induced right-side kinetic term alone gives $\alpha_s$ roughly sixty times too strong. A bare term is therefore present, and its coefficient is currently unfixed rather than derived.

Open problems / where help is wanted

Listed roughly by how much else depends on them. Discussions are open on each; if you want to work on one, say so there first so effort isn't duplicated.

• The solder potential. The active front, and the bottleneck for everything below it. The solder field is $(\varphi, \hat{n})$ and the question is what potential governs it. Two routes are on the table: an effective potential obtained from $\epsilon \omega F^4$ evaluated on the solder configuration, or a Landau functional with coefficients matched to it. The matching calculation is partly done. Useful here: anyone comfortable with Landau theory of anisotropic order parameters, or with effective potentials from higher-curvature actions.

• Fiber anisotropy and the isospin channel. The current mass insertion is sector-blind, which is why fits fake the up/down spread by pushing the hypercharge flux outside its allowed range. The anisotropic block $h_{22} \neq h_{33}$ was deliberately deferred and now needs to come back. Expected to be worth a factor of a few in the down sector.

• Torsion. The geometry as currently set up is too symmetric to generate the hierarchy or the observed scales, and torsion is the most likely missing ingredient. The question is where it comes from. Color monopoles are ruled out — colour must stay exact.

• Scale setting. Global scale invariance means $\langle \varphi \rangle$ has to arrive by dimensional transmutation. What fixes it, and whether the resulting scale is consistent with the Planck normalization on one side and the fiber scale on the other, is unresolved.

• CKM. Identically trivial until transverse structure enters the fiber. The same deformation that produces the hierarchy should switch it on, which makes it a consistency check rather than a free fit — but the calculation hasn't been done.

• Quantization of the quadratic-gravity sector. The Gauss–Bonnet term is ghost-free at quadratic order, but the full quantization question stands. PT-symmetric and fakeon approaches both look relevant. Useful here: anyone who works on higher-derivative gravity quantization.

• Mirror-CP. Whether the $\mathbb{Z}_2$ relating the sectors is exact or spontaneously broken by the condensate — an exact mirror world and an asymmetric one are different physical scenarios with different cosmology.

Reviewing rather than solving

Independent checks are as valuable as new results, and several claims here are single-source. The algebraic verifications in Papers 0 and 1 are mechanical enough to be redone from scratch by anyone with a Clifford algebra package, and the anomaly traces and Lovelock coefficients are straightforward to recompute. If a check fails, open a Discussion — that is worth more to this project than a new derivation.