Operator enumeration: method and data contract

The operator enumerator computes all nonzero invariant sectors in a finite field inventory within a chosen canonical-dimension interval. It extends the fixed-monomial calculation by generating the occupations to be counted. This page specifies what is exhaustive, which filters are applied, and what a saved calculation means.

Mathematical object

Let $V_i$ be the full finite-dimensional Lorentz, color, weak, and flavor representation of species $i$. Its integer gauge charge is $Q_i$, and its canonical dimension is $\delta_i=1$ for a scalar or $3/2$ for a Weyl field. For occupations $\mathbf n\in\mathbb Z_{\geq0}^s$, define

$$\mathcal P_{\mathbf n} =\bigotimes_{i\in B}\operatorname{Sym}^{n_i}V_i \otimes\bigotimes_{i\in F}\Lambda^{n_i}V_i, \qquad c_{\mathbf n}=\dim_{\mathbb C} \left(\mathcal P_{\mathbf n}^{\mathrm{Lorentz}\times G}\right).$$

Here $B$ and $F$ denote scalar and Weyl species. Flavor is ungauged multiplicity. For the proper connected Lorentz spin group, the finite-dimensional invariant calculation can be carried out using characters of its complexified algebra and the compact form $SU(2)_\ell\times SU(2)_r$. This algebraic projection does not change the physical Lorentzian signature or classify unitary particle representations.

A row is retained when its dimension lies in the selected interval, its additional charge constraints hold, and $c_{\mathbf n}>0$. The dimension subtotal is

$$C_d=\sum_{\substack{\mathbf n:\,\sum_i\delta_i n_i=d\\ \text{selected charges}}}c_{\mathbf n}.$$

This is a direct sum of graded vector spaces. It is not the number of field structures and does not identify a sector with its Hermitian conjugate.

One may package the same coefficients in a multigraded series,

$$H(t,\mathbf x)=\sum_{\mathbf n}c_{\mathbf n} \,t^{\sum_i\delta_i n_i}\prod_i x_i^{n_i}.$$

The $x_i$ are bookkeeping variables for occupations. The site computes the required finite set of coefficients by explicit enumeration and exact characters; it does not claim a closed rational expression for the full series. For the standard Hilbert-series formulation and the separate steps needed to include derivatives, see Marinissen, Rahn, and Waalewijn, equation (1) and Sections II A–C.

Completeness within the supported domain

The enumeration has three stages.

Enumerate every bounded occupation once

Choose an ordering of the declared species. At position $i$, assign every nonnegative power compatible with the remaining dimension budget, then continue to $i+1$. The implementation tracks twice the canonical dimension, so all decisions are integer comparisons:

$$\sum_i (2\delta_i)n_i\leq2D, \qquad 2\delta_i\in\{2,3\}.$$

Every admissible vector determines exactly one path through this recursion, and every terminal path determines one vector. This gives both exhaustiveness and absence of duplicate occupations. Since $\delta_i\geq1$ and $D\leq6$, the six-insertion and per-field power bounds exclude no additional occupation in the requested dimension range.

The empty occupation is handled explicitly. It is excluded by default. Enabling the identity includes its dimension-zero sector independently of the positive dimension interval, provided its zero charges satisfy all selected targets.

Apply only necessary rejection conditions

The following tests can prove that a candidate has no selected invariant:

Test	Necessary condition
Fermion capacity	The power of a Weyl species cannot exceed its full component-space dimension.
Gauge $U(1)$	$\sum_i n_iQ_i=0$.
Additional continuous charge	$\sum_i n_iq_{ai}=t_a$.
Additional cyclic charge	$\sum_i n_iq_{ai}\equiv t_a\pmod{m_a}$.
Lorentz center	The numbers of left and right Weyl occurrences are each even.
Color center	$\sum_i n_i(p_i+2q_i)\equiv0\pmod3$ for color Dynkin labels $(p_i,q_i)$.
Weak center	$\sum_i n_i a_i\equiv0\pmod2$ for weak Dynkin label $a_i$.

The Lorentz test uses the absence of derivatives and of higher-spin field letters. A derivative would itself carry both spinor indices and would change the parity test. The full component dimension in the capacity test includes flavor; replacing it by the number of spinor components alone would incorrectly discard multi-flavor operators.

Passing these filters does not guarantee a nonzero answer. For example, a single neutral left Weyl weak doublet admits all the requisite center charges for its square, but the would-be bilinear vanishes because its spin and weak contraction is incompatible with the required exchange symmetry. The exact projection must still be performed.

Count each survivor exactly

For each remaining vector, the fixed-monomial engine forms the required symmetric and exterior characters and extracts the singlet with a Weyl-denominator constant-term projection. This includes algebraic index identities automatically through the representation space. The local-operator method derives the character algorithm and its normalization.

Only a strictly positive result creates an output row. After every candidate has been processed successfully, the rows and subtotals form the completed result. No sampling, floating-point integration, or approximate multiplicity is used.

These three stages establish the completeness claim: every invariant in the declared polynomial algebra and dimension interval belongs to a listed occupation sector, and each sector’s reported dimension includes all of its algebraically independent contractions. The statement is conditional on successful completion within the published resource bounds. A resource error does not certify an incomplete table as exhaustive.

Global groups and extra charges have different roles

Before occupation enumeration, every field is tested for descent to the selected diagonal quotient of $SU(3)\times SU(2)\times U(1)$. An obstruction prevents a count for that physical inventory. The response identifies the affected fields. Empty rows in such a response do not mean that a valid theory has zero interactions.

Additional $U(1)$ or $\mathbb Z_m$ rules are then applied as charge filters. They do not change the stored gauge group or its topology. A target of zero selects invariance under the extra rule; a nonzero target selects a charged operator sector. Each rule assigns one charge to the whole species, including all its flavors. Flavor-dependent charges can be represented by splitting flavors into separately declared species, within the field bound; they are not silently inferred from a flavor multiplicity.

Conjugate species must have the conjugate extra charges if the rule is intended to represent an ordinary internal symmetry. The program accepts arbitrary explicitly declared charge assignments and does not reconstruct missing conjugation relations. This also allows inventories intentionally restricted to a holomorphic or a fixed-charge sector, whose interpretation should be stated when sharing them.

Input and computational bounds

Quantity	Supported domain
Spacetime and fields	Four-dimensional Lorentzian scalars, left Weyl fields, right Weyl fields
Gauge group	$SU(3)\times SU(2)\times U(1)$ with the specified diagonal quotient of order $1,2,3,$ or $6$
Number of species	At most $16$, counting explicit conjugates separately
Color	Nonnegative Dynkin labels $(p,q)$ with $p+q\leq2$
Weak isospin	Dynkin label $0,1,$ or $2$
Gauge charge	Integer from $-24$ through $24$
Flavor multiplicity	$1,2,$ or $3$ per species
Dimension interval	Integer endpoints $1\leq D_{\min}\leq D_{\max}\leq6$
Additional charge rules	At most three, each a continuous $U(1)$ or a cyclic $\mathbb Z_m$ with $2\leq m\leq12$
Additional charges and input targets	Integers from $-24$ through $24$
Dimension-range occupation vectors	At most $100{,}000$, including the identity if requested
Candidates sent to exact projection	At most $4096$
Aggregate coefficient-multiplication budget	$100{,}000{,}000$ across the candidate calculations
Output rows	At most $4096$
Per-monomial resources	At most $120{,}000$ intermediate character weights and $20{,}000{,}000$ coefficient-multiplication pairs
Imported JSON	At most $4$ MiB, measured in UTF-8 bytes

The representation bounds and resource bounds serve different purposes. A valid input may still exceed a resource limit. In that case the run terminates with an explicit error. Progress information is diagnostic and carries no completed mathematical answer. Cancellation likewise leaves the enumeration incomplete.

The reported occupation census counts vectors in the requested interval, rather than recursive search nodes. Rejection counters record the first failed test in their stated order, so they form a partition of rejected candidates; they are not independent counts of every reason a vector could fail. Changing filter order can therefore change diagnostics without changing the mathematical result.

Versioned inventory and calculation files

The raw request has schema identifier groups.qft.org/operator-enumeration/1. It contains the setup name, spacetime, gauge group, fields, and enumeration options. Its field declarations use the same conventions as physical setups, but it has no selected monomial powers.

For example, the following options request dimensions one through six and require even parity for a declared scalar species S:

{
  "minDimension": 1,
  "maxDimension": 6,
  "includeIdentity": false,
  "symmetries": [
    {
      "id": "parity",
      "name": "Scalar parity",
      "kind": "zn",
      "modulus": 2,
      "charges": {"S": 1},
      "target": 0
    }
  ]
}

This is the enumeration member of an inventory, not a complete input file. Every declared field must appear in each rule’s charges object, even when its charge is zero. Continuous rules use kind: "u1" and modulus: 0. Cyclic targets are normalized modulo the declared modulus; field charges retain their explicitly entered integer representatives.

The JSON Schema describes the data structure. Semantic validation also checks bounds, distinct identifiers, the dimension interval, and consistency of charge maps with the inventory. Unknown or unsupported properties are rejected. Derivative data cannot be smuggled into a nominally derivative-free request.

A saved calculation uses groups.qft.org/operator-enumeration-calculation/1. It records the normalized request, result, conventions, sources, and the enumeration, local-operator, and gauge algorithms. The bounded enumeration algorithm is operator-enumeration-1.0.0.

On import, the request is validated and recomputed. The cached result does not control the calculation, and a changed or missing algorithm version is reported. Raw physical-setup and physical-calculation v1 files can also be loaded as inventories: their selected powers are discarded, and their entire field content and gauge quotient are retained. This is an intentional change of question from one chosen monomial to all occupations in the selected range.

The inventory index links the six curated inputs. Each also has a saved calculation under /data/operator-enumerations/. For example, the three-generation Standard Model inventory and its completed enumeration preserve the same declared conventions used in the Standard Model guide.

Verification and interpretation

The published benchmarks distinguish the enumeration layer from the invariant-counting layer. Higgs and scalar-parity inventories test occupation generation and charge selection with elementary closed forms. The Standard Model inventories test a broad mixture of chirality, gauge charges, repeated fermions, and flavor multiplicities. The Standard Model guide resolves the dimension-six totals into independently documented classes, rather than comparing unmatched headline numbers.

For the mathematical scope, an algebraic identity means equality in the actual symmetric/exterior algebra of field components. The count includes such identities but does not construct a general normalized basis of contractions. It also does not impose equations of motion, integration by parts, Hermiticity, flavor-basis equivalence, anomaly cancellation, or quantum consistency. Those operations require additional data and definitions.

The value of an exhaustive bounded result is that its assumptions travel with it. A researcher can change a field, charge rule, flavor multiplicity, or global quotient, rerun the calculation, and identify exactly which sectors changed. A future extension to derivatives or a full EFT basis must preserve that explicit contract while adding the new representations and equivalences it needs.