Worked representations, quotient tests and anomaly sums

Specify the functor before counting particles

Let E=E₃⊕E₂, with dimensions three and two. The primitive relative circle acts on E₃ with weight −2 and on E₂ with weight +3. The full finite CAR vacuum module is ΛE of dimension 2⁵=32. Retaining all even pair-connected states gives

Every even monomial is a product of pair creators, so this is an algebraic reachability statement. It is not yet a charge-conserving physical time evolution. Pair creators can themselves carry gauge charge and need suitable interaction fields or a reservoir in an actual implementation.

The degree-zero component is an internal representation label. Calling it a left-handed neutral Weyl field uses the separate physical interpretation; it does not identify it with the no-particle vacuum of a spacetime field theory. CAR, even retention, chirality and the untwisted exterior action are the source’s additional assumptions M1–M4.

Source: Sevenfold Gauge, PDF pp. 47–48.

Derive all six multiplets

The identity Λᵏ(E₃⊕E₂)=⊕_p+r=kΛᵖE₃⊗ΛʳE₂ organizes the calculation. Each bidegree has q=−2p+3r and conventional Y=q/6.

For example Λ²E₃≅E₃*⊗det E₃. Its SU(3) type is an antitriplet, but its circle weight is +2−6=−4. Omitting the determinant factor would give the wrong charge. Likewise L=det E₃⊗E₂ has weight −6+3=−3.

The product volume identifies Λ⁴E with E*, yielding the familiar algebraic 1⊕10⊕5̄ packaging. This notation does not introduce off-block SU(5) gauge bosons: the retained splitting still defines the gauged group.

Source: Sevenfold Gauge, PDF pp. 48–49.

Use one character to check dimension and charge traces

The full exterior character is generated by the five defining weights. Taking the even part gives the Laurent polynomial

With D_z=z∂_z, evaluation at z=1 gives χ₊(1)=16, D_zχ₊(1)=0, D_z³χ₊(1)=0 and D_z²χ₊(1)=120. These are respectively the dimension, mixed gravitational–circle sum, cubic circle anomaly sum and quadratic integral-charge trace.

The derivative is z∂_z, not the ordinary derivative alone: it returns powers of the weight q from each monomial z^q. This compact calculation independently checks all multiplicities in the table. It does not replace the simple-group anomaly tests, which also depend on the non-Abelian representation carried by each weight.

Source: Sevenfold Gauge, PDF pp. 48–50. The generating-character calculation is developed here.

Test the center before testing anomalies

The kernel generator of the covering group acts on a multiplet by exp[2πi(q+2t₃+3t₂)/6], where t₃ is triality and t₂ weak center parity. Descent requires this phase to equal one.

For an exterior sector, t₃=p modulo three and t₂=r modulo two. Substituting q=−2p+3r makes the residue equal to 6r. Thus every row descends automatically. Explicitly Q gives 1+2+3=6, uᶜ gives −4+4=0 and L gives −3+3=0.

The tests are logically independent. Reverse every Abelian charge while retaining the same non-Abelian representations. Every local anomaly still cancels because the relevant charge sums reverse sign or are unchanged at zero. But Q now gives −1+2+3=4, and fails quotient descent. An allowed Lie-algebra action is not automatically a representation of the specified global group. The anomaly explorer makes this distinction explicit.

Source: Sevenfold Gauge, PDF pp. 49–51.

Calculate the remaining local sums and their scope

Normalize the color fundamental cubic coefficient to one. Q contributes two fundamentals, while uᶜ and dᶜ contribute one antifundamental each. Hence A₃₃₃=2−1−1=0. For the mixed anomalies, common nonzero quadratic-index factors can be suppressed:

The character already verified Σdim(R)q=0 and Σdim(R)q³=0. A weak cubic local anomaly vanishes for these pseudoreal representations; terms with a single simple-group generator vanish by tracelessness. The usual weak global test counts three colored Q doublets plus one L doublet, giving an even total of four.

That global parity test assumes ordinary spin backgrounds. It is not a complete classification of anomalies for arbitrary quotient-group bundles or non-spin manifolds. Cancellation also does not construct a quantum chiral measure or prove the existence of an interacting continuum theory. It is a consistency check on the supplied left-handed spectrum.

Source: Sevenfold Gauge, PDF p. 50.

Exhibit anomaly-free alternatives algebraically

Forget the inherited action, but keep the six non-Abelian representations and Dirac Yukawa structures with a doublet of charge h. Write q_Q=x. The weak mixed anomaly and Yukawa charge sums allow

The quark contribution to the cubic sum is 6x³+3(−x−h)³+3(−x+h)³=−18xh². The lepton contribution is 2(−3x)³+(3x+h)³+(3x−h)³=18xh². They cancel for every x,h.

Imposing the same compact quotient and a descended Higgs still permits x≡1 and h≡3 modulo six, hence infinitely many integral choices. Requiring q_ν=0 gives h=3x and fixes the relative ratios, with an overall normalization still to be chosen. The exterior rule selects the primitive pair (1,3) directly; anomaly cancellation alone does not do so.

Source: Sevenfold Gauge, PDF p. 51. The cubic expansion and quotient-compatible lattice are worked out here.

Distinguish a charge trace from a coupling prediction

Normalize a fundamental simple-group generator by Tr T²=1/2. For one physical left-handed generation, the quadratic indices are K₃=2(1/2)+1/2+1/2=2 and K₂=3(1/2)+1/2=2. Since Y=q/6, the character result gives K_Y=120/36=10/3.

The defining five-dimensional carrier instead has Tr Q²=3(−2)²+2(3)²=30. Its trace is not the sixteen-dimensional matter trace of 120. Each response calculation must state which module and normalization it uses. Adding charge-conjugate copies doubles all three matter indices; it should not happen silently.

With the source’s common spectral-action moment, these indices give g₃²=g₂²=(5/3)g_Y² at the declared matching scale. Their common magnitude still depends on the action profile. Additional representations or different weights change the calculation. The neutral singlet contributes to gravitational counting but not to these gauge traces.

Source: Interactions Unification, PDF pp. 21–22.

Allowed tensors do not select masses or chirality

Add a scalar H in E₂ with q=3. The charge sums for QuᶜH, QdᶜH*, LeᶜH* and LνᶜH are respectively 1−4+3, 1+2−3, −3+6−3 and −3+0+3, all zero. Color uses the triplet–antitriplet pairing; weak indices use the alternating tensor or dual pairing.

These invariant structures admit independent coefficients under the retained group. Gauge symmetry also allows higher-order structures such as QQQL and uᶜuᶜdᶜeᶜ; additional restrictions would be needed to exclude them. A common exterior notation does not fix interaction strengths.

The finite parity penalty Δ(I−(−1)^N)/2 places sixteen odd internal states above the even sector. It does not supply a Lorentzian Weyl operator or a local mirror-decoupling mechanism. Keeping both parities as independent left-handed fields produces conjugate mirror pairs; choosing only one parity remains a physical assumption separate from its finite internal gap.

Source: Sevenfold Gauge, PDF pp. 50–52.

Test whether three modes are really three families

A three-dimensional multiplicity space F gives three identical generations only when the gauge group acts trivially on F. Reusing the color carrier’s dual V* instead changes every representation. In particular V*⊗V is the space of three-by-three matrices under conjugation, splitting into its scalar identity and traceless parts: 3̄⊗3=1⊕8.

The difference can be checked by characters. For g=diag(e^iφ,e^−iφ,1), three neutral copies of V have character 3(1+2cosφ). The color-dual construction has character (1+2cosφ)². At φ=π these are −3 and 1, despite both spaces having dimension nine.

The source’s three holomorphic sections on the projective space of the same color V transform as V*, so their dimension cannot be reused as a neutral family count. A viable multiplicity construction must specify the group action, operator and stability, followed by flavor interactions. Counting modes alone leaves those tests unresolved.

Source: Sevenfold Gauge, PDF pp. 52–53. The explicit character comparison is developed here.

Continue reading

• The conditional matter reconstruction
• Compare local anomalies and quotient descent
• Identify the globally retained group
• Match representation traces to physical parameters