4.0.0 — Research workbench II

Application: 4.0.0 · Dataset: 2026.11-workbench.2 · Schema: 2.0.0
Release algorithm umbrella: groups-workbench-2.0.0 · Edition date: 2026-11-09

Research workbench II implements the next bounded model-building milestone within Phase C. It connects representation products, restriction along a specified homomorphism, and matter descent through an embedded quotient. The broader research-workbench and public-launch criteria remain open.

Three new tools

Tool	Released contract
SU(N) tensor products	SU(2)–SU(6), input labels 0–12, at most 16 canonical Young boxes across both inputs; exact Littlewood–Richardson decomposition
Branching with U(1) charges	SU(3)–SU(6), split $N=p+q$, labels 0–12, at most 12 canonical parent Young boxes; the primitive block map and its complete charged restriction
Product gauge groups and matter	One or two SU(N) factors with $2\leq N\leq8$ and one U(1); at most 16 multiplets, labels 0–12, integer charges −24…24; at most 3 quotient generators, phase denominators ≤60, generated subgroup order ≤4096

The tensor engine enumerates Littlewood–Richardson tableaux and retains all nonnegative multiplicities. It checks dimensions, center charges, singlet duality, and an independent Schur-character specialization. Full determinant columns are removed when identifying SU(N) labels and retained in the original polynomial characters. A defensive 2,000,000-state search guard fails explicitly without returning a partial decomposition. See the method reference.

The block map is

$$(A,B,z)\longmapsto \operatorname{diag}(z^{q/g}A,z^{-p/g}B), \qquad g=\gcd(p,q).$$

Its U(1) has period $2\pi$, image $S(U(p)\times U(q))$, and cyclic kernel of order $pq/g$. The Schur coproduct retains both block determinant powers until assigning the integer charge $X=(q|\mu|-p|\nu|)/g$. Every output passes dimension, kernel, and exact first/second charge-moment checks. The block reference supplies the derivation.

The product-group tool evaluates the exact phase of every specified quotient generator on every matter multiplet. It reports finite or continuous common kernels, invariant factors, compatible cyclic subgroup choices where applicable, and the fundamental group of the supplied finite quotient. The abstract order of a quotient is not substituted for its embedding. See product center lattices.

A complete SU(5) convention path

The color-first split uses $X=\operatorname{diag}(2,2,2,-3,-3)$ and $Y=-X/6$, with electric charge $Q_{\rm em}=T_3+Y$. The SU(5) page derives the fundamental, antifundamental, ten, and adjoint branchings and the resulting Standard Model multiplets.

Transferring those multiplets to the product-group tool uses integer $Q_Y=6Y=-X$ and the corresponding diagonal-kernel generator

$$\gamma=(\omega_3\mathbf1_3,-\mathbf1_2,e^{2\pi i/6}).$$

The interface preserves that sign conversion. It also tests the receiving tool’s bounds before offering a transfer.

The ten new mathematical records describe the four Standard Model global groups with quotient orders 1, 2, 3, and 6; their common real Lie algebra; three quotient projections; the SU(5) block homomorphism; and the injective map induced by quotienting its full kernel. All four global forms admit the usual local matter, while this specified SU(5) embedding has the full diagonal $\mathbb Z_6$ kernel. The global-group guide explains the remaining physical choices.

Save, load, and inspect a calculation

The three new tools provide named JSON files, calculation links, LaTeX, worked inputs, and print. A saved file contains the inputs, exact result, convention profile, source references, and application/dataset/algorithm versions. Loading restores the inputs and recomputes with the current engine; an algorithm-version change is reported. The saved answer is not accepted as an unchecked fresh result.

The 4 MiB import limit accommodates the tested large product-group exports. Formatted files as large as approximately 1.49 MB were loaded and recomputed intact, including a 1,536-element common kernel with a 3,840-element selected quotient and a continuous-kernel example.

The worked-file index supplies an SU(5) tensor example, a charged ten-dimensional restriction, and Standard Model matter with the diagonal $\mathbb Z_6$ quotient. Use Load calculation in the tool named by the file. In product-group inputs, the key ranks means SU factor sizes $N_i$, not Lie-algebra ranks $N_i-1$.

These files encode individual mathematical calculations. The complete proposed physical-setup model, including Lorentz content, field statistics, spacetime, interactions, and additional quantum data, is a later milestone.

Edition inventory and mathematical rendering

The edition contains 266 records, 2,998 claims, and 105 source records:

Record kind	Count
Groups	90: 49 finite and 41 Lie groups
Families	33
Lie algebras	38
Real forms	19
Homomorphisms	22
Spaces	2
Representations	62

There are thirteen tools and nine convention profiles. The original ten tools retain their earlier algorithms and domains. The convention manifest and method manifest include the three additions.

Six new scientific pages connect methods to worked physics. KaTeX renders their mathematics and the interactive results, including exact fractions, multiplets, maps, and quotient generators. Dimensions and large exact integers remain strings in JSON. No new external calculation backend is invoked.

Verification and remaining boundaries

The new mathematical suites include independent SU(2)/SU(3) comparisons, Pieri rules, direct tableau character checks, published SU(4)–SU(6) fixtures, an exhaustive census of 11,666 supported SU(6) input pairs, and all 2,009 admitted block-branching shape/split cases. Branching also uses independent Jacobi–Trudi character evaluations. Product-group fixtures distinguish continuous kernels, noncyclic finite groups, and quotients with different fundamental groups despite equal abstract subgroup orders.

The complete release suite passed 139 verification sections plus catalogue validation. The production build passed with 470 HTML pages, all 266 record definitions and 2,998 claims present as static content, and checks of 51,528 internal references, 15,830 fragments, and 11 aliases. An HTML scan found 4,014 KaTeX expressions and no KaTeX errors.

Focused browser checks covered calculation presets, invalid-input recovery, real JSON download and restoration, exact clipboard output, the SU(5) transfer, and obstructed and continuous-kernel cases. At a 375 CSS-pixel viewport, the three new tools, homepage, and three worked guides had no horizontal document overflow; wide mathematics and tables scrolled locally. The checked light/dark views and mobile navigation also worked. These are scoped browser checks, rather than comprehensive accessibility certification.

These automated checks and source verification are separate from named human mathematics and physics approval. Independent expert review remains pending. The prior 3.0.0 release remains a historical record with its own counts and verification boundaries.

The roadmap identifies the next steps: broader compact types and embeddings, several U(1) factors, full physical setups and operator bases, reviewed computational adapters, specialist topology and representation categories, and notebook/collaboration features. Independent scientific review, accessibility assessment, final reuse terms, correction operations, and the groups.qft.org public-domain/audience transition remain release gates. The current deployment retains its private review audience.