Physical setup library

Start with the physical question. A fixed-monomial setup asks for the invariants in one specified product of fields. An inventory asks which products and contractions survive throughout a dimension range. Both retain spacetime, the gauge quotient, representation labels and flavor multiplicities. Neither specifies a complete dynamical theory.

All examples below use four-dimensional Lorentzian spacetime, ordinary complex polynomials in scalar and Weyl fields, and algebraic equivalence. Scalars have canonical dimension one; Weyl fields have dimension $3/2$. Derivatives, gauge-field strengths, equations of motion, integration by parts, automatic Hermiticity and flavor-basis reduction are outside the implemented calculation.

Fix the group and field conventions

The supported gauge groups are

$$G_d=\bigl(SU(3)\times SU(2)\times U(1)\bigr)/\Gamma_d, \qquad d=1,2,3,6,$$

where $\Gamma_d$ is generated by the $(6/d)$th power of $\gamma=(e^{2\pi i/3}I_3,-I_2,e^{2\pi i/6})$. The $U(1)$ angle has period $2\pi$ and every field carries integer charge $Q_Y$. Standard Model fields use $Q_Y=6Y$, with $Q_{\rm em}=T_3+Y$. A quotient order refers to this embedded subgroup, not an arbitrary group of that order. See Tong, §3, p.7 and the global-group guide.

The presets use $d=6$ except the neutral weak-doublet mass and opposite-chirality examples, which explicitly use $d=1$. In particular, a weak doublet of charge zero fails descent to $G_6$ even though a pair may form a singlet. The tools check every declared field before counting a monomial.

A flavor number is the number of copies of a field representation. A daggered field is an explicit additional polynomial variable, with conjugate gauge charges and opposite Weyl chirality. The real coefficient conditions of a Lagrangian are a subsequent step. For the detailed contract, use physical setups and replay.

Choose one field product

Each link opens the complete preset in the fixed-monomial workbench. The count is the complex dimension of its invariant space. A zero is a valid answer, not a failed computation.

Question and exact preset	Monomial	Flavors	Dimension	Count	Input file
Weinberg, one flavor	$L^2H^2$	$n_L=1$	5	1	JSON
Weinberg, three flavors	$L^2H^2$	$n_L=3$	5	6	JSON
Up Yukawa	$Qu^cH$	$n_Q=n_u=3$	4	9	JSON
Higgs quartic	$H^2(H^\dagger)^2$	One Higgs	4	1	JSON
Higgs sextic	$H^3(H^\dagger)^3$	One Higgs	6	1	JSON
Sterile Weyl mass	$N^2$	$n_N=3$	3	6	JSON
Neutral weak doublet, one flavor	$\Psi^2$	$n_\Psi=1$	3	0	JSON
Neutral weak doublet, two flavors	$\Psi^2$	$n_\Psi=2$	3	1	JSON
Neutral weak doublet, three flavors	$\Psi^2$	$n_\Psi=3$	3	3	JSON
Baryon-violating QQQL	$Q^3L$	$n_Q=n_L=3$	6	57	JSON
Opposite-chirality bilinear	$NN^\dagger$	One each	3	0	JSON

The Weinberg derivation explains the symmetric lepton-flavor tensor. The Yukawa/four-fermion guide derives the unrestricted Yukawa matrix and the QQQL permutation constraints. The spin/statistics lesson explains the sterile versus weak-doublet mass contrast and why a left/right bilinear is a Lorentz vector. These links supply the derivations behind the table; the setup index JSON also links each recomputable result and its source metadata.

For example, the neutral-doublet candidate is

$$\mathcal O_{ab}=\epsilon_{\alpha\beta}\epsilon_{ij} \Psi_a^{\alpha i}\Psi_b^{\beta j}.$$

Exchanging the two fermions contributes a minus sign, while exchanging both antisymmetric tensors contributes a plus sign. Thus $\mathcal O_{ba}=-\mathcal O_{ab}$, leaving $n_\Psi(n_\Psi-1)/2$ contractions: zero, one and three for the presets. This is why an abstract doublet tensor square containing a singlet is insufficient. These examples are algebraic representation tests; they do not establish that an arbitrary chiral matter set is anomaly-free.

Enumerate an inventory

The following presets include every nonzero monomial sector within their stated alphabet and range. Here all use $G_6$, dimensions one through six, and exclude the constant. A sector is a field occupation; its invariant multiplicity may exceed one.

Inventory	Nonzero sectors	Total complex dimension	Input file
Standard Model, one generation	46	55	JSON
Standard Model, three generations	46	2874	JSON
Three-generation SM, conserving B and L	36	2316	JSON
One Higgs doublet and its conjugate	3	3	JSON
Three sterile Weyl flavors and conjugates	5	60	JSON
One neutral scalar generator with Z₂ selection	3	3	JSON

The last row keeps the even powers of a single generator $S$: $S^2,S^4,S^6$. It does not impose $S=S^\dagger$ or add a conjugate automatically. In the Higgs inventory the three invariants are instead the first three positive powers of $H^\dagger H$. The field alphabets differ even though their dimension totals agree.

Follow the Standard Model walkthrough for a complete inventory-to-sector calculation. The published sector comparison checks its flavor counts against the corresponding classes in the literature; the enumeration method explains its exact projection and resource bounds. All inventory and result files remain available together.

Save a result you can reconstruct

1. Open the desired preset and inspect all field rows, including zero-power fields and conjugates. Check the quotient, integer charges, flavors and range.
2. Change inputs if needed, then run the calculation. Charge selection rules only filter candidates; the exact invariant projection decides the count.
3. Save the setup/inventory to retain inputs, or save the calculation to retain inputs, results, conventions, sources and algorithm versions together.
4. Load that JSON in the corresponding tool. It validates the input and recomputes; a saved answer is not trusted as a new calculation.

In enumeration results, open a sector in the fixed-monomial tool to inspect its full field list and powers. That handoff preserves the gauge quotient and all declared fields. Selection rules belong to the inventory; a single-monomial file retains the chosen powers but is not a record of those filters. Save the inventory as well when the selection was part of the scientific question.

A resource refusal means the requested computation exceeded an explicit bound. It is not a proof of zero invariants. Counts do not supply a normalized operator basis, coupling constants, anomaly cancellation, a vacuum, or a complete EFT Lagrangian. Use reproducibility when citing a result.