5.0.0 — Research workbench III

Application: 5.0.0 · Dataset: 2026.11-workbench.3 · Core schema: 2.0.0
Release algorithm umbrella: groups-workbench-3.0.0
Edition date: 2026-11-08, America/Los_Angeles

Research workbench III implements the next bounded Phase C milestone: physical field content becomes an exact local-operator question. A setup now records spacetime, Weyl chirality, gauge representations, flavor multiplicity, statistics, the selected global quotient, and a monomial. The new tool counts independent complex invariants with those assumptions.

From fields to local interactions

The local-operator workbench supports four-dimensional Lorentzian scalars and left- or right-handed Weyl fields with gauge group

$$G_d=\frac{SU(3)\times SU(2)\times U(1)}{\langle\gamma^{6/d}\rangle}, \qquad \gamma=(\omega_3\mathbf1_3,-\mathbf1_2,e^{2\pi i/6}), \qquad d\in\{1,2,3,6\}.$$

The U(1) parameter has period $2\pi$ and integer charge $Q$; Standard Model presets use $Q=6Y$. The established product-group method checks every declared field against the selected quotient, including fields absent from the chosen monomial. Incompatible field declarations produce an obstruction report and no operator count.

Input	Supported range
Field species	At most 16
SU(3) labels	$[p,q]$, with $p,q\geq0$ and $p+q\leq2$
Weak SU(2) label	$n=0,1,2$
Integer U(1) charge	$-24\leq Q\leq24$
Flavor multiplicity	1–3 per species, ungauged
Selected monomial	At most six field occurrences and canonical dimension at most six
Derivative order	Zero
Equivalence	Algebraic identities over $\mathbb C$

The result reports the invariant count, canonical dimension, total charge, selection rules, field-space dimensions, exact checks, and source evidence. A distinguishable-slot comparison makes the effect of identical-field statistics inspectable. A zero count is a successful calculation when a selection rule or algebraic identity forbids the proposed scalar.

Statistics acts on the complete field

For a scalar species the engine takes a symmetric power of its full representation; for a Weyl species it takes an exterior power. Spin, gauge, and flavor components are included together before projecting to a singlet. Applying separate exchange projections to each factor would change the physical counting problem.

Exact joint characters and Weyl-denominator projection implement this calculation with arbitrary-precision integer coefficients. Compact $SU(2)_{\mathrm{left}}\times SU(2)_{\mathrm{right}}$ characters compute invariants of the complexified connected Lorentz algebra while the physical spacetime remains Lorentzian. The method reference derives the procedure and distinguishes the spin factors from weak $SU(2)$.

The resource guards allow at most 120,000 intermediate weights and 20,000,000 coefficient multiplication pairs. Exceeding a guard produces an explicit error without a partial count. Exact values remain decimal strings in JSON. The site invokes no external calculation backend.

Eleven worked setups

The examples expose distinct physical mechanisms:

Sector	Flavor choice	Complex invariant count
Weinberg $L^2H^2$	One / three lepton flavors; one Higgs	1 / 6
Up-type Yukawa $Qu^cH$	Three quark flavors of each species	9
Higgs quartic / sextic	One Higgs and an explicit conjugate	1 / 1
Sterile Weyl mass $N^2$	Three flavors	6
Neutral weak-doublet Weyl mass	One / two / three flavors	0 / 1 / 3
$Q^3L$	Three quark and three lepton flavors	57
Opposite-chirality bilinear	One left and one right gauge singlet	0

The neutral weak-doublet examples use the unquotiented gauge product, since their charge-zero doublets do not descend to the full diagonal $\mathbb Z_6$ quotient. Their counts describe algebraic interactions and do not assert anomaly cancellation for a complete field theory.

The three-flavor QQQL benchmark agrees with Lehman and Martin, §4.1, Eq. (4.2). Weyl bilinears and their flavor symmetries follow the two-component conventions reviewed by Dreiner, Haber, and Martin, §§2–3. Worked contractions are displayed only when the current setup exactly matches the corresponding example. They illustrate selected sectors; the engine does not construct a normalized contraction basis.

Physical setup and calculation files

The new groups.qft.org/physical-setup/1 format records a bounded physical input. The separate groups.qft.org/physical-calculation/1 envelope adds the exact result, sources, convention, and version evidence. These formats leave the existing core catalogue schema at 2.0.0.

Save a setup even when its global form is obstructed. Save a calculation to retain the obstruction report or exact invariant count with its provenance. Both files can be loaded locally, up to a 4 MiB UTF-8 limit. Loading validates the semantics and recomputes from the input; cached answers and source text never enter the computation. Changed or unknown saved algorithm versions are reported. The new engine is local-operators-1.0.0, with convention four-d-local-polynomial-v1; quotient descent continues to use product-gauge-1.0.0.

Conjugate copies are explicit independent variables. The helper reverses color labels and U(1) charge, exchanges Weyl chirality, and preserves flavor. It does not impose Hermiticity or equivalence under flavor rotations. The entire declared matter content can transfer to the product-group tool with its selected diagonal embedding. The physical-setup reference documents the schema, strict validation, examples, and replay behavior.

Edition inventory and scientific pages

The edition contains 266 mathematical records, 2,998 claims, and 111 source records, with fourteen tools and ten convention profiles. The entity inventory is unchanged from Workbench II: 90 groups, 33 families, 38 Lie algebras, 19 real forms, 22 maps, two spaces, and 62 representations. The new physical presets are calculation inputs, rather than new group records.

Five scientific pages connect the method to graduate-level physics:

• Fields, Lorentz indices, and statistics.
• Exact local-operator counting.
• Physical setups and reproducible files.
• The Weinberg operator.
• Yukawa and four-fermion interactions.

KaTeX renders their mathematics and the interactive results. Primary-source checks include graded character methods, two-component spinors, the dimension-five interaction, QQQL flavor counting, and the additional machinery required for derivatives and EFT equivalences.

Verification and milestone boundary

The local-operator suite checks published flavor counts, independent low-degree character and contraction identities, conjugation, and exact selection rules. The physical-setup suite covers all worked inputs, each standard global quotient, unoccupied-field obstructions, strict semantic validation, conjugate transfer, KaTeX formulas, import replay, forged cached answers, and the UTF-8 file-size boundary. Interface verification connects these contracts to share, save, load, edit, and result states. The complete repository gate passes 182 verification sections. The production build generates 483 HTML pages and verifies canonical records, headings, aliases, internal links, and fragment targets. Focused browser checks cover edited and obstructed setups, file and URL replay, keyboard submission, light/dark views, and mobile reflow. These checks support the release without constituting a complete accessibility assessment.

This milestone computes a fixed derivative-free monomial. Full monomial enumeration, gauge-field strengths, derivatives, equations of motion, integration by parts, normalized bases, general gauge groups, and other spacetime representations remain Phase C work. The roadmap also retains broader tensor and branching algorithms, several U(1) factors, and reviewed computational adapters before the later specialist and collaboration phases.

AI-assisted source checking and automated verification are distinct from named human mathematics and physics approval. There are currently zero independently expert-reviewed records. Accessibility assessment, final reuse terms, staffed correction operations, and the groups.qft.org public-domain/audience transition remain separate release gates. The current deployment retains its private review audience. Historical release notes, including the 4.0.0 edition and its UTC date, preserve their original evidence.