> Historical design and handoff snapshot (2026-11-09). Implementation and hosting statements below describe that snapshot. For current reader-facing scope, see https://groups.qft.org/reference/coverage/ and https://groups.qft.org/about/roadmap/.

# Groups for Physics — scientific blueprint and long-term specification

> **2026-11-09 handoff policy:** Codex develops this repository locally on the
> owner's Mac. The future host is **Cloudflare Pages connected to GitHub**.
> Keep the new website offline until the finite initial-launch scope is
> finished. Read [MASTER_PLAN.md](../MASTER_PLAN.md),
> [AGENTS.md](../AGENTS.md), and the current `src/data/site-plan.mjs` and
> `src/data/publication.mjs` manifests first. Those files supersede earlier
> hosting, deployment cadence, and implementation-order instructions here.

**Retained mathematical baseline:** Research workbench IV — a delivered subset of Phase C; the full phase remains unfinished

**Historical private review URL (not the development source of truth):** https://groups-for-physics-design.renphysics.chatgpt.site

**Intended public origin:** https://groups.qft.org — Cloudflare Pages is the future production host; publication is currently disabled

**Primary source repository:** https://github.com/renphysics/groups.qft.org (private)

**Document status:** Workbench IV scientific inventory plus the preserved long-term product specification and original sitemap. The current master plan and page registry control the local-development handoff.

**Snapshot:** 2026-11-08 (America/Los_Angeles); application 6.0.0; core catalogue schema 2.0.0; dataset 2026.11-workbench.4; release algorithm umbrella groups-workbench-4.0.0. Physical setups, fixed-monomial calculations, operator inventories, and enumeration calculations each have their own version 1 contract.

**Navigation:** Atlas · Tools · Physics · Learn · Reference

**Audience:** theoretical physicists, mathematical physicists, graduate students, and researchers building models with symmetry

## 1. Product mandate

Groups for Physics should become the most dependable place to identify a group, understand which mathematical choices matter physically, perform a supported calculation, and carry its assumptions and provenance into research. “The best website for physicists using groups” is the product ambition; it is not a claim that the current edition has comprehensive mathematical coverage.

The organizing unit is a precise mathematical object connected to a physical question. The interface must make it easy to ask: Does this representation descend to my global group? Which multiplets appear under this specific embedding? Which interactions survive field statistics? Which space controls this symmetry-breaking problem? What has actually been calculated, and under which conventions?

Three outcomes define quality:

1. A reference answer can be located and checked quickly.
2. A calculation exposes every input capable of changing its interpretation.
3. A result can be reproduced, linked, cited, and corrected.

The site should be useful without an account. It should not require visitors to understand the internal data model before using a calculator. Explanations appear where they resolve a consequential choice; detailed methodology remains available alongside the result.

Section 2 records what is implemented in the current repository. Sections 3–15 preserve the comprehensive target sitemap, scientific contracts, architecture, and roadmap; they are specifications unless explicitly identified as implemented. Features outside the current inventory are not released merely because a route or design appears below. The bibliography supports mathematical content and proposed integrations, not a claim that every integration or research workflow exists.

## 2. Current research workbench and historical foundation

### 2.1 Research workbench IV implementation status

The **Research workbench IV** edition extends the static Astro application and Starlight documentation established by Phase B. The catalogue contains **266 mathematical records, 2,998 claims, and 114 distinct source records**. These counts are generated by `src/data/catalogue.mjs`. The 194 foundation records, 62 Workbench I representation records, and ten Workbench II Standard Model group/algebra/map records are preserved. Workbench III introduced fixed-monomial local-operator counting and eleven physical setups. Workbench IV adds exhaustive bounded field-inventory enumeration, six inventory presets, three scientific pages, and three primary references. Operator monomials are calculation sectors rather than artificial new mathematical entity records.

| Record kind | Count | Implemented scope |
|---|---:|---|
| Group | 90 | 49 finite groups and 41 connected Lie groups |
| Family | 33 | 24 finite-family or classification-collection records and all 9 complex simple Lie types |
| Lie algebra | 38 | 16 complex algebras and 22 actual real algebras |
| Real form | 19 | 14 compact and 5 selected noncompact real forms, linked to complexifications and integrating groups |
| Map | 22 | Specified homomorphisms with domain, codomain, kernel, image, and injectivity/surjectivity |
| Space | 2 | SU(2)/U(1) and SO(3)/SO(2), with their subgroup maps and boundary conventions |
| Representation | 62 | All 30 ordinary complex irreducible characters across the seven executable finite groups, plus 32 selected Lie representations |
| **Total** | **266** | Classification, reference, and executable coverage remain distinct |

The representation records now connect highest weights or complete finite-character identities to sources, exact mathematical data, and supported calculations. They are a selected Lie catalogue, not a finite enumeration of infinitely many highest weights. The schema supports construction records but none are instantiated. Global forms remain distinguished through group records, explicit maps, and sourced relationships; the full target schema in Section 4 is not instantiated.

Classification navigation covers the finite-simple families, the 26 sporadic groups and the separately identified Tits group, the nine complex simple Lie types, compact central-quotient conventions, and the stated 19-real-form selection. Selected finite records provide complete ordinary complex irrep dimension/reality summaries; such a summary does not supply every representation matrix or enable a new calculator. The five migrated workbenches retain their original bounded mathematical contracts. Workbenches I, II, III, and IV add five, three, one, and one tools respectively, as described in Sections 2.5–2.8.

The historical Phase B release passed its 36-section automated suite and static-output checks across 355 HTML pages, with focused browser evidence preserved in `docs/foundation-release.md`. Those are historical counts. Workbench I added finite/Lie mathematical, matter-quotient, KaTeX, and reproduction-script suites; its evidence is frozen in `docs/workbench-release.md`. Workbench II added exact tensor, charged-branching, and product-group suites, with historical release evidence in `docs/model-building-release.md`. Workbench III adds exact local-operator, physical-setup, and interface verification; its implementation and final integration evidence belong in `docs/research-workbench-iii-implementation.md`. Workbench IV adds exhaustive enumeration, selection-rule, replay, and interface checks; its release evidence belongs in `docs/research-workbench-iv-implementation.md`. Aggregate build and browser evidence must correspond to the actual release state. These checks do not constitute independent scientific approval. The owner has deferred the accessibility assessment. The current handoff adds a finite content-completion and evidence gate before the first public launch, as specified in `MASTER_PLAN.md`. The current catalogue has **zero expert-reviewed records**. Source checking, automated verification, named human approval, and public launch remain separate states.

The existing Sites review mirror is historical and retains its private audience; it is not part of the new development workflow. The private `renphysics/groups.qft.org` repository is the source of truth. Codex develops locally on the owner’s Mac, and Cloudflare Pages connected to GitHub is the selected future host. Do not create remote previews or publish unfinished work. The first public launch occurs only after the initial-launch scope and evidence gate are complete; deployment and domain readiness must then be verified through the host.

### 2.2 Implemented application, data, and review workflow

| Area | Current implementation |
|---|---|
| Static application | Custom Astro pages for Atlas, Tools, Physics, search, sources, and review status; Starlight for Learn, Reference, and project documentation. Dependencies are pinned in `package.json` and the lockfile. |
| Typed catalogue | `packages/schema` validates record identity, kind-specific fields, references, assertion support, and selected mathematical consistency rules. `src/data/core.mjs`, `finite-foundation.mjs`, `lie-foundation.mjs`, `research-workbench.mjs`, `model-building.mjs`, and the new operator sources/capabilities are assembled by `src/data/catalogue.mjs`. The physical-setup v1 schema is separate from the core catalogue schema. |
| Canonical routing | Entity pages use `/atlas/{kind-plural}/{slug}/`, independently of collection membership. Curated aliases and migrated hash links resolve to canonical identities. Tool state uses validated query parameters. |
| Search and discovery | `packages/search` builds a typed index from record names, aliases, search terms, definitions, and assertion text, with normalized ASCII/Unicode/TeX-like queries and kind/category/coverage filters. Atlas browse, comparison, collections, and capability views use the same catalogue. |
| Claim-level provenance | Definitions, properties, explanatory paragraphs, and relationships generate stable claim IDs. Assertions carry source locators or derivations; source pages expose their dependent claims. Derived entity/claim references are validated. |
| Downloads and citations | Catalogue, claims, sources, release, and review JSON; schemas; individual record JSON and BibTeX; ten convention profiles; 32 root-system datasets; seven exact character tables; fifteen method contracts; three Workbench II example files; eleven bounded physical-setup presets; and six operator inventories with exact enumeration exports. Calculation exports retain inputs, conventions, methods, sources, and specific algorithm versions. Downloadable reproduction scripts cover finite characters, Weyl invariants, SU(3) products, and matter quotients. No live backend is invoked. |
| Review workflow | `packages/editorial`, `src/data/reviews.json`, and the review command support status inspection, review requests, and named human decisions. SHA-256 fingerprints cover scientific content and its relevant dependencies. Changed fingerprints make earlier decisions stale. Executable expert-review status requires distinct mathematics and physics reviewers. No such approvals are claimed by this snapshot. |
| Versions | Application `6.0.0`, core catalogue schema `2.0.0`, dataset `2026.11-workbench.4`, umbrella `groups-workbench-4.0.0`. The inventory enumerator is `operator-enumeration-1.0.0`. The local-operator engine is `local-operators-1.0.0`; its convention is `four-d-local-polynomial-v1`. Physical setup and physical calculation have separate `/1` schema identifiers. All earlier mathematical-engine versions are preserved, including `product-gauge-1.0.0` for descent. |
| Mathematical rendering | KaTeX renders documentation and new scientific result views, including character tables, Cartan matrices, exact rational formulas, and decompositions. Math rendering has its own strict automated checks. |

The implementation includes static classification and provenance pages, mathematical method references, versioned convention profiles, and graduate-level guides connecting the tools to physics. All fifteen calculators have named domains. Named files for the three Workbench II tools and the new local-operator tool restore inputs and recompute. Workbench III records spacetime, chirality, statistics, flavors, gauge representations, a specified global quotient, and a fixed derivative-free monomial. It is a bounded physical setup, with strict rejection of unsupported semantics. Workbench IV enumerates all monomials in a chosen dimension interval for this same field domain, optionally selecting continuous or cyclic charges. Arbitrary-group characters, arbitrary branching, noncompact unitary duals, full EFT operator bases, anomaly classifiers, and live GAP/Sage backends remain unreleased. Downloaded GAP/Sage/Python verification scripts are reproduction aids and do not constitute backend adapters; GAP/Sage runtime validation has not been performed in this environment.

### 2.3 Historical design edition and the fourteen migrated records

The original dependency-free HTML/CSS/JavaScript **design edition** demonstrated the interaction model with fourteen group records, five scoped tools, and connected Physics and Learn articles. That historical contract remains useful for migration compatibility; it no longer describes the size or architecture of the current catalogue.

The original fourteen records and article/tool metadata now live in `src/data/legacy.js`, and the framework-independent mathematical functions live in `src/lib/math.js`. The former prototype paths `dist/data.js` and `dist/math.js` are historical locations; `dist/` is now generated Astro build output. The mathematical engine retains version `groups-design-1.0.0` because the Phase B migration preserves the supported algorithms and bounds.

| Internal ID | Canonical production slug | Object | Design-edition role |
|---|---|---|---|
| su2 | su-2 | SU(2) | Highest weights, tensor products, covers |
| so3 | so-3 | SO(3) | Descent, rotational symmetry, topology |
| su3 | su-3 | SU(3) | Highest weights, center charge, selected branching |
| psu3 | psu-3 | PSU(3) | Adjoint global form and representation descent |
| u1 | u-1 | U(1) | Charge normalization and maximal tori |
| spin4 | spin-4 | Spin(4) | Semisimple Euclidean spin group |
| sl2c | sl-2-c | SL(2,C), treated as a real group | Lorentz cover and representation categories |
| c3 | c-3 | Cyclic group of order three | Discrete charges |
| s3 | s-3 | Symmetric group on three letters | Elementary nonabelian character theory |
| a4 | a-4 | Alternating group on four letters | Character table and invariant counting |
| s4 | s-4 | Symmetric group on four letters | Permutations and cubic rotations |
| a5 | a-5 | Alternating group on five letters | Simple groups and exceptional isomorphisms |
| q8 | q-8 | Quaternion group | Quaternionic reality type and McKay example |
| d8 | d-8 | Dihedral group of order eight | Point groups and notation conventions |

These original records have been migrated into the typed catalogue with canonical IDs, property-specific evidence, and related entities. They retain different capabilities. An irreducible-dimension summary is not a complete explicit representation library, and a source link is not a certificate that every possible property has been checked.

### 2.4 Five migrated tools — unchanged mathematical scope

| Tool | Exact supported domain |
|---|---|
| Global forms | SU(2), SO(3), SU(3), PSU(3); each highest-weight label is an integer from 0 through 30 |
| Tensor products | Two SU(2) irreps with highest weights from 0 through 30 |
| Invariant couplings | Tensor or symmetric powers of the real A4 triplet, degrees 0 through 12 |
| Branching | SU(3) fundamental 3 and adjoint 8 under the regular or principal SU(2) map |
| Breaking topology | The specified adjoint-breaking cases SU(2)/U(1) and SO(3)/SO(2) |

The SU(2) tensor output can contain labels as high as 60 even though each input is bounded by 30. Preserve that distinction in validation. Query-based sharing and JSON exports are implemented; the broader export/adaptor ambitions below do not expand these five computational domains. Migration and scientific fixtures pass for the stated contracts.

### 2.5 Five exact additions delivered in Research workbench I

| Tool and route | Released domain and method |
|---|---|
| Finite characters — `/tools/finite-characters/` | Complete ordinary complex tables for C3, S3, D8, Q8, A4, S4, A5; each base dimension ≤60 and multiplicity ≤12; tensor products and tensor/symmetric/exterior powers of degree 0–12. Exact algebraic character arithmetic, class power maps, inner products, and complete-character reconstruction. |
| Highest weights — `/tools/lie-representations/` | A1–A8, B2–B8, C2–C8, D4–D8, E6, E7, E8, F4, G2; each Dynkin label 0–30. Weyl dimensions from positive coroots, normalized Casimir/index, conjugation, reality, and full-center adjoint descent. |
| SU(3) tensor products — `/tools/su3-tensor-products/` | Each Dynkin label of each input 0–6; complete decomposition from Gelfand–Tsetlin characters, including conjugates and multiplicities. Output labels may exceed the input bound. |
| SU(3) branching — `/tools/su3-branching/` | Each input label 0–6; complete restriction along the regular SU(2) inclusion or principal map with kernel Z2 and image SO(3). No commuting U(1) refinement. |
| Matter and global form — `/tools/matter-quotients/` | One SU(N) factor, 2≤N≤12, at most 16 multiplets, labels 0–30; common central kernel and every compatible quotient. Kinematic descent only, without line-operator, discrete-theta, anomaly, or dynamical selection. |

All new engines use exact arbitrary-precision integers and rational/algebraic arithmetic. Large integers and algebraic coefficients serialize as strings. The highest-weight convention is `A_ij = <alpha_j,alpha_i^vee>` with Bourbaki/Sage node order, long-root squared length 2, and `C2(lambda) = (lambda,lambda+2rho)/2`. Positive **coroots** are used consistently for non-simply-laced types. A failed full-center adjoint-descent test does not exclude descent to an intermediate quotient.

The authoritative convention data are `/data/conventions.json`, `/data/root-systems.json`, and `/data/characters/{id}.json`. `/data/workbench-methods.json` records the contracts and implementation versions. GAP scripts verify serialized finite tables and operations; SageMath scripts reconstruct Weyl/SU(3) calculations; standard-library Python scripts independently verify the SU(N) matter criterion. The site does not execute these external environments. Python verification has been executed; GAP/Sage scripts have serialization/syntax checks but their runtime execution is unverified here.

### 2.6 Three exact additions delivered in Research workbench II

| Tool and route | Released domain and method |
|---|---|
| SU(N) tensors — `/tools/sun-tensor-products/` | SU(2)–SU(6), input labels 0–12, at most 16 canonical Young boxes across both inputs. Complete Littlewood–Richardson tableaux, nonnegative multiplicities, determinant-column reduction for SU labels, exact dimensions, center charges, singlet duality, and an independent polynomial-character specialization. A 2,000,000-state guard fails explicitly without partial output. |
| Charged block branching — `/tools/charged-branching/` | SU(3)–SU(6), split N=p+q, 1≤p<N, labels 0–12, at most 12 canonical parent Young boxes. The complete Schur coproduct along `diag(z^(q/g) A,z^(-p/g) B)`, g=gcd(p,q), with primitive U(1), actual finite kernel, both SU labels, exact charge, and multiplicity. |
| Product global groups — `/tools/product-gauge-groups/` | One or two SU(N) factors, 2≤N≤8, followed by one compact U(1). At most 16 multiplets, labels 0–12, integer charges −24…24, three quotient generators, phase denominators ≤60, and generated subgroup order ≤4096. Computes common central kernels, descent obstructions, finite invariant factors, and the fundamental group of the specified finite quotient. |

The tensor algorithm retains original GL partitions for character verification; determinant columns disappear only after restriction to SU(N). Its canonical-box bound is computational and need not be invariant under conjugating both inputs. Dimensions remain decimal strings, exact rational charges retain numerator and denominator, and representation identity remains the full Dynkin labels.

The block map has image S(U(p)×U(q)) and kernel of order pq/g, generated by `(exp(-2pi i/p) I_p,exp(2pi i/q) I_q,exp(2pi i/(pq/g)))`. SU(1) is the trivial bookkeeping factor. The color-first SU(5) split uses `X=diag(2,2,2,-3,-3)`, physical `Y=-X/6`, and electric charge `T3+Y`. The product-group convention uses integer `Q=6Y=-X`. The interface transfers the charges with that sign conversion and the embedded kernel, subject to the destination's bounds.

The product-group input key `ranks` contains SU factor sizes `N_i`, not algebra ranks `N_i-1`. A quotient is supplied by all its center exponents and rational U(1) phases. Its abstract order alone is insufficient. Uncharged matter can leave a continuous common U(1) kernel; noncyclic kernels retain invariant factors. The fundamental group is derived from the lift to SU(N1) [× SU(N2)] × R, rather than equated with the finite quotient subgroup.

Ten new records organize the Standard Model example: global quotient orders 1, 2, 3, and 6; the shared real gauge algebra; three quotient maps; the SU(5) block map; and the injective map induced from its full kernel quotient. These records preserve the distinction between matter compatibility, a specified SU(5) embedding, and a full quantum gauge theory.

Three additional convention profiles and six scientific pages explain the Workbench II methods and worked physics. Strict KaTeX renders the formulas in documents and results. Local named JSON files contain the single-tool input, exact output, sources, conventions, and versions. Loading validates the input and recomputes; a saved result is not treated as trusted computation. Worked files appear at `/data/model-setups.json`. This earlier single-tool format does not encode spacetime, Lorentz representations, or field statistics; Workbench III introduces a separate bounded physical-setup format for that purpose.

### 2.7 Physical setups and local operators delivered in Research workbench III

The new `/tools/local-operators/` workbench counts complex invariant polynomials at one specified multidegree. It supports four-dimensional Lorentzian scalar, left-Weyl, and right-Weyl fields with color SU(3), weak SU(2), and a period-2π U(1). A quotient order d=1,2,3,6 selects the subgroup generated by `(omega3,-1,exp(2pi i/6))^(6/d)`. Every declared field must descend, including fields absent from the selected monomial. A global obstruction remains an explainable setup result with no operator count.

There are at most sixteen species and three flavors per species; color labels satisfy p+q≤2, weak labels are 0–2, and integer charges are −24…24. The selected monomial contains at most six field occurrences and has canonical dimension at most six, assigning dimensions 1 and 3/2 to scalars and Weyl fields. Derivatives are exactly zero. Flavor is ungauged multiplicity. Conjugates are explicit independent variables; Hermiticity, flavor-basis reduction, and discrete spacetime symmetries are not imposed.

The exact engine builds joint SU(3) × SU(2)_weak × SU(2)_left × SU(2)_right characters. Symmetric or exterior powers act on each entire spin × gauge × flavor space before distinct species are tensor-multiplied. Weyl-denominator projection extracts singlets, with the U(1) charge checked separately. The compact Lorentz factors implement the complexified connected Lorentz-algebra calculation; spacetime remains Lorentzian. BigInt multiplicities serialize as decimal strings. Limits of 120,000 intermediate weights and 20,000,000 coefficient multiplication pairs fail explicitly without a partial answer.

The interface reports exact count, canonical dimension, charge, selection rules, statistics, the distinguishable-slot comparison, provenance, and checks. Eleven presets include one-/three-flavor Weinberg operators (1/6), the three-flavor up Yukawa (9), Higgs quartic and sextic (1 each), a three-flavor sterile Weyl mass (6), neutral weak-doublet masses with one/two/three flavors (0/1/3), three-flavor QQQL (57), and an opposite-chirality derivative-free bilinear (0). Sourced worked contractions appear only for exact matching fixtures and do not constitute a general normalized basis generator.

`groups.qft.org/physical-setup/1` specifies the bounded input, while `groups.qft.org/physical-calculation/1` adds current result, source, convention, and version evidence. The core catalogue schema remains 2.0.0. Imports are limited to 4 MiB, reject unknown semantics, recompute from validated inputs, and report changed or unknown saved algorithms. The entire matter declaration transfers to the product-group workbench with its diagonal embedding. Five new scientific pages cover Lorentz/statistics, the counting method, the setup contract, the Weinberg operator, and Yukawa/four-fermion examples.

### 2.8 Operator enumeration delivered in Research workbench IV

The new `/tools/operator-enumeration/` workbench accepts an inventory of the same scalar and Weyl fields and enumerates all nonnegative occupation vectors in an integer canonical-dimension interval from one through six. It checks every declared field against the selected global quotient before counting. The optional dimension-zero identity is considered separately and must satisfy the selected charge targets.

Each surviving sector is evaluated by the existing exact local-operator engine. The output distinguishes nonzero field monomials from the dimension of their invariant spaces, reports dimension subtotals, and can open any row in the detailed monomial workbench. Hermitian-conjugate fields remain explicit polynomial variables. No Hermiticity condition, flavor-basis quotient, derivative reduction, or normalized contraction basis is implied.

Up to three additional selection rules may be declared. An integer U(1) rule selects a specified total charge; a cyclic rule selects its residue modulo an order from two through twelve. Charges and targets lie between −24 and 24. Zero target imposes classical polynomial invariance under that selection rule; a nonzero target isolates a charged sector. These declarations do not prove quantum anomaly cancellation or implement additional gauge dynamics.

The finite census allows 100,000 visited multidegrees, 4,096 projected candidates, 100,000,000 aggregate coefficient multiplication pairs, and 4,096 output rows, while retaining each monomial engine's own limits. A resource failure returns an explicit error with no partial enumeration. A worker can report progress and be cancelled without blocking the editor. Saved inventory and calculation files validate and recompute; cached answers are ignored and changed algorithms are reported.

Six presets cover the Standard Model with one or three generations, the three-generation model with conserved B and L, one Higgs doublet, three sterile Weyl flavors and their conjugates, and a scalar with Z2 selection. The Standard Model derivative-free scalar/Weyl sector has dimension-six counts 45 and 2,806 for one and three generations, counting distinct conjugates separately. Three-generation B/L conservation reduces the dimension-six count to 2,260. These are explicitly not the full SMEFT counts. Three new articles explain inventory-to-interaction reasoning, the enumeration method, and the Standard Model sector benchmarks.

### 2.9 Current milestone boundary and publication policy

Workbench IV completes bounded monomial enumeration within Phase C. Derivatives, gauge-field strengths, equations of motion, integration by parts, normalized contraction bases, wider gauge/spacetime groups, and the complete PhysicalSetup specification remain future work. Broader compact-group products, specified additional branching maps, multiple gauge U(1) factors, and reviewed computational adapters also remain open.

The current owner instruction is to finish the defined initial website locally before putting it online. `MASTER_PLAN.md` and `src/data/site-plan.mjs` define that finite scope; `src/data/publication.mjs` records the closed publication state and required evidence. Cloudflare Pages replaces GitHub Pages as the future host. Source commits and pushes to the private repository are allowed during development. Named independent scientific review remains a recorded quality goal rather than a mandatory launch prerequisite. The accessibility assessment is deferred; ordinary functional checks remain part of implementation. Original articles use CC BY 4.0 and original code uses MIT. Third-party material retains its own terms.

## 3. Information architecture and stable routes

Keep five primary navigation items. Atlas owns mathematical entities; Tools owns calculations; Physics owns problem contexts; Learn owns learning sequences; Reference owns conventions, formulas, classification summaries, and provenance.

Search, theme controls, and the QFT.org switcher are utilities. About belongs in the footer. Finite and continuous Lie groups receive equal first-level visibility in Atlas. Spacetime symmetry is a cross-cutting collection, not a mutually exclusive mathematical group category.

The following inventory preserves the original **100 route patterns** as a long-term design reference. The current `src/data/site-plan.mjs` registry and `docs/SITEMAP.md` resolve implemented names, explicit consolidations, new scaffolds, and launch scope. Do not create duplicate articles merely to reproduce an older proposed name. Braced segments are validated parameters, not invitations to generate unlimited low-quality pages. Production navigation includes completed content/capabilities; the local development desk deliberately includes draft scaffolds.

### 3.1 Home, Atlas, and entity routes

| No. | Route | Purpose |
|---|---|---|
| 1 | / | Product entrance and representative live example |
| 2 | /search/ | Searchable, typed results with shareable queries |
| 3 | /atlas/ | Browse all released mathematical entities |
| 4 | /atlas/finite/ | Finite groups and their capabilities |
| 5 | /atlas/lie/ | Continuous Lie groups and global forms |
| 6 | /atlas/compare/ | Comparison with exact object selection |
| 7 | /atlas/coverage/ | Entity-level capability matrix |
| 8 | /atlas/collections/ | Curated mathematical and physics collections |
| 9 | /atlas/groups/{group}/ | Canonical group record |
| 10 | /atlas/algebras/ | Lie-algebra index |
| 11 | /atlas/algebras/{algebra}/ | Algebra over a specified field |
| 12 | /atlas/real-forms/ | Real-form index |
| 13 | /atlas/real-forms/{real-form}/ | Real form and integrating groups |
| 14 | /atlas/representations/ | Representation search and category selection |
| 15 | /atlas/representations/{representation}/ | Stable representation identity |
| 16 | /atlas/maps/ | Homomorphisms and embeddings |
| 17 | /atlas/maps/{map}/ | Map definition, kernel, image, and branching |
| 18 | /atlas/spaces/ | Group manifolds, cosets, and related spaces |
| 19 | /atlas/spaces/{space}/ | Precisely constructed geometric space |
| 20 | /atlas/families/ | Parameterized family index |
| 21 | /atlas/families/{family}/ | Parameters, restrictions, and exceptions |
| 22 | /atlas/constructions/{construction}/ | Products, quotients, covers, extensions |
| 23 | /atlas/collections/finite-simple/ | Finite-simple classification navigation |
| 24 | /atlas/collections/compact-simple/ | Compact groups with simple Lie algebra |
| 25 | /atlas/collections/discrete-physics/ | Finite groups used in physical models |
| 26 | /atlas/collections/spacetime/ | Rotation, Lorentz, Poincare, conformal groups |
| 27 | /atlas/collections/gauge-groups/ | Global forms used in gauge theory |
| 28 | /atlas/collections/exceptional/ | Exceptional Lie types and selected relations |

### 3.2 Tool routes

| No. | Route | Purpose |
|---|---|---|
| 29 | /tools/ | Released calculators and their supported domains |
| 30 | /tools/global-forms/ | Center kernels and representation descent |
| 31 | /tools/tensor-products/ | Irrep decompositions |
| 32 | /tools/branching/ | Restriction along an explicit map |
| 33 | /tools/invariants/ | Singlets with specified field symmetry |
| 34 | /tools/topology/ | Supported spaces and homotopy sequences |
| 35 | /tools/characters/ | Character calculations and orthogonality |
| 36 | /tools/representation-dimensions/ | Highest-weight dimension formulas |
| 37 | /tools/weight-lattices/ | Weights, roots, and quotient constraints |
| 38 | /tools/root-systems/ | Root data and selected visualizations |
| 39 | /tools/reality-types/ | Real, complex, quaternionic distinctions |
| 40 | /tools/central-extensions/ | Supported extensions and projective classes |
| 41 | /tools/anomaly-checks/ | Explicitly delimited anomaly tests |
| 42 | /tools/subgroups/ | Supported subgroup and containment calculations |

### 3.3 Physics routes

| No. | Route | Purpose |
|---|---|---|
| 43 | /physics/ | Physics questions and connected workflows |
| 44 | /physics/gauge-theory/ | Symmetry data in gauge theories |
| 45 | /physics/global-forms-and-lines/ | Matter, line operators, additional choices |
| 46 | /physics/matter-representations/ | Charges, multiplets, and reality |
| 47 | /physics/symmetry-breaking/ | Order parameters and stabilizer maps |
| 48 | /physics/topological-defects/ | Vacuum spaces and defect sectors |
| 49 | /physics/discrete-symmetries/ | Finite internal symmetries |
| 50 | /physics/flavor-models/ | Discrete-flavor examples with conventions |
| 51 | /physics/spacetime/ | Isometry groups, components, and covers |
| 52 | /physics/spinors/ | Spin representations and field covariance |
| 53 | /physics/cft/ | Conformal groups and representation categories |
| 54 | /physics/ads-and-holography/ | Relevant real forms, covers, and embeddings |
| 55 | /physics/anomalies/ | Perturbative, global, and structural distinctions |
| 56 | /physics/condensed-matter/ | Point groups, projective symmetry, applications |
| 57 | /physics/setups/ | Reviewed physical-setup templates |
| 58 | /physics/setups/{setup}/ | Assumptions, fields, calculations, limitations |

### 3.4 Learn routes

| No. | Route | Purpose |
|---|---|---|
| 59 | /learn/ | Learning entry points |
| 60 | /learn/paths/ | Task-oriented learning paths |
| 61 | /learn/paths/{path}/ | Ordered lessons and examples |
| 62 | /learn/groups-and-algebras/ | Local versus global data |
| 63 | /learn/finite-group-characters/ | Reading and using character tables |
| 64 | /learn/irreducible-representations/ | Categories and parametrizations |
| 65 | /learn/covers-and-global-forms/ | Covers, centers, and descent |
| 66 | /learn/roots-and-weights/ | Root systems and highest weights |
| 67 | /learn/real-forms/ | Compact and noncompact choices |
| 68 | /learn/embeddings-and-branching/ | Why the map matters |
| 69 | /learn/tensor-products-and-statistics/ | Distinct versus identical fields |
| 70 | /learn/invariants-and-operators/ | From singlets to interactions |
| 71 | /learn/groups-cosets-and-bg/ | Distinct spaces and their uses |
| 72 | /learn/lorentz-and-poincare/ | Field representations versus particle states |
| 73 | /learn/projective-representations/ | Multipliers and central extensions |
| 74 | /learn/exercises/{exercise}/ | Focused problems with reviewed solutions |

### 3.5 Reference and governance routes

| No. | Route | Purpose |
|---|---|---|
| 75 | /reference/ | Authoritative supporting material |
| 76 | /reference/conventions/ | Site defaults and named alternatives |
| 77 | /reference/notation/ | Searchable notation dictionary |
| 78 | /reference/formulas/ | Formula index |
| 79 | /reference/formulas/{topic}/ | Formula, domain, normalization, derivation |
| 80 | /reference/classification/ | Classification scopes and relationships |
| 81 | /reference/classification/finite-simple/ | Families, exceptions, identifications |
| 82 | /reference/classification/lie-algebras/ | Complex and compact classification |
| 83 | /reference/classification/real-forms/ | Real-algebra classification conventions |
| 84 | /reference/sources/ | Bibliography and external databases |
| 85 | /reference/sources/{source}/ | Source metadata and dependent claims |
| 86 | /reference/provenance/ | How facts and calculations are supported |
| 87 | /reference/coverage/ | Coverage policy and capability definitions |
| 88 | /reference/data-model/ | Public schema and identifiers |
| 89 | /reference/reproducibility/ | Calculation versions and saved inputs |
| 90 | /reference/api/ | Released machine interfaces |
| 91 | /reference/downloads/ | Licensed, versioned data distributions |
| 92 | /reference/faq/ | Common interpretive and workflow questions |
| 93 | /about/ | Mission and project status |
| 94 | /about/method/ | Editorial and computational method |
| 95 | /about/contributors/ | Roles and credited contributions |
| 96 | /about/corrections/ | Specific correction submission workflow |
| 97 | /about/licenses/ | Code, prose, data, and asset licenses |
| 98 | /about/roadmap/ | Capability-based phases |
| 99 | /about/releases/ | Release history |
| 100 | /about/releases/{version}/ | Changed claims, data, and algorithms |

Canonical entity URLs are category-neutral. SU(3) is /atlas/groups/su-3/, regardless of the collection used to reach it. A5, PSL(2,4), and PSL(2,5) resolve to one record. A1 is a type-search term that offers relevant algebras and global groups, not a unique identity alias for SU(2).

Use immutable internal IDs and explicit alias redirects. Keep state in validated query parameters. A branching permalink must include the map and representation, not only the two group names. Production direct links must work on refresh without a prior visit to the home page.

## 4. Mathematical entity schema

This is the **target schema**. The implemented catalogue schema and its populated kinds are summarized in Section 2. Workbenches III and IV additionally release bounded physical-setup, fixed-monomial calculation, inventory, and enumeration-calculation v1 files as described in Sections 2.7–2.8; these do not instantiate the full proposed PhysicalSetup/Computation entities below. Separate GlobalForm and RepresentationModel entities, root/weight datasets beyond the released 32 systems, and every proposed absence-state representation are not all released features.

Implement a validated discriminated schema rather than an oversized group object. Every record has: stable ID, kind, slug, display names, aliases, mathematical definition, source claims, revision, review status, capability references, and related entity IDs.

| Entity | Additional required fields |
|---|---|
| GroupFamily | Parameter schema, valid ranges, construction, exceptional parameters, isomorphism collisions |
| Group | Category, order or dimension, components, topology specification, algebra link where applicable, construction |
| LieAlgebra | Base field, dimension, decomposition, root data when applicable, normalization profile |
| RealForm | Real-algebra ID, complexification, classification label, equivalent descriptions, integration links |
| GlobalForm | Covering-group ID, explicitly embedded central kernel, quotient construction, induced lattice conditions |
| Representation | Domain ID, coefficient field, characteristic, category, labels, dimension, dual/conjugate, center character |
| RepresentationModel | Representation ID, basis, matrix or action data, coefficient realization, ordering conventions |
| Homomorphism | Domain, codomain, definition, kernel, image, injectivity, algebra/lattice maps, normalization |
| Space | Construction kind, input entities and maps, basepoint conventions, topological claims |
| PhysicalSetup | Dimension, signature, gauge/global role, groups, fields, statistics, chirality, breaking, extra theory choices |
| Computation | Operation, exact inputs, convention profile, engine/data versions, result, verification metadata |
| Claim | Subject, predicate, value, assumptions, source locator or derivation, status, reviewer |

Distinguish abstract isomorphism from a chosen realization or action. The rotational representation of S4 carries information beyond the name S4. Distinguish a finite group of Lie type from its algebraic construction and from a group of rational points before taking derived subgroups or central quotients.

Representation category is mandatory. Ordinary complex characters, modular characters, real representations, projective representations, finite-dimensional continuous representations, and infinite-dimensional unitary representations cannot share an unqualified “all irreps” endpoint. Highest-weight families should be parameterized; an arbitrary dimension cutoff does not make a Lie representation catalogue complete. [1–6]

For compact connected groups with simple Lie algebra, central quotients of the simply connected cover provide global forms. Do not extend a finite-center assumption to universal covers of all noncompact real groups. Store a quotient subgroup with its embedding; distinct subgroups can matter, and outer automorphisms can identify resulting abstract groups. [1,2]

A homomorphism is the primary object; an embedding is the injective subtype. A Lie-algebra embedding and an integrating group map require separate records when their global properties differ. A space record must identify whether topology refers to G, G/H, BG, or another construction.

Expose four distinct absence states: unsupported by this implementation, not yet curated, mathematically inapplicable, and mathematically nonexistent. Never encode all of them as an empty array or a dash.

## 5. Main page templates

### Home

Lead with “Groups for Physics,” a concrete statement of utility, and two actions: Explore the Atlas and Open Tools. Follow with a working global-form example, task entrances, balanced finite/Lie group previews, selected Physics explanations, and an honest edition statement. Avoid decorative mathematical formulas that resemble unverified results.

### Atlas index and collection

Provide search, kind and capability filters, family information, and compact table/card views. Filters survive navigation. Finite records show order; Lie records show real dimension and explicitly labeled rank. A collection adds a short editorial introduction and inclusion criteria. Display result counts as counts of the released catalogue.

### Group record

The first screen contains the exact name, definition, category, essential invariants, related global forms, and supported actions. Continuous records use Overview, Representations, Topology, Maps, Physics, Sources. Finite records use Overview, Characters and Representations, Subgroups and Extensions, Cohomology, Physics, Sources.

Keep common section anchors stable. Provide a capability panel, concise source-linked facts, and related objects. For finite groups, explain any transition from discrete group topology to BG; the classifying space is not the group manifold. “Simple Lie group” must state whether it means a connected group with simple Lie algebra.

### Algebra and real-form records

Show base field first, followed by dimension, decomposition, root data, and related forms. A real-form page provides compactness, real rank, maximal-compact information when curated, and integrating global groups. Do not quietly identify complex dimension with real dimension. SL(2,C) as a real algebra requires the corresponding complexification distinction.

### Family and construction records

State the parameter domain, definitions, exception table, and identification rules. Examples link to individual records. A cover, central quotient, extension, or product page displays the participating maps and explains which data determine the construction.

### Representation record

Show the domain and category before dimensions or diagrams. Include labels and conventions, dual/conjugate, reality information where defined, center action, allowed global forms, known decompositions, and supported models. Place explicit matrices and basis-dependent coefficients in a separate realization section.

### Map and space records

A map page displays direction, formula, kernel, image, lattice action, and embedding-dependent calculations. A space page displays its construction, topology with hypotheses, relevant exact sequences, and physics applications. Both expose source and convention panels.

### Tool workspace

Use the sequence: scope, inputs, mathematical result, explanation, checks, export/share, references. Keep a result's disallowed or obstructed status visible while preserving the chosen input. An invalid input produces a local explanation, not a silent coercion. Long calculations require cancellation and stated resource bounds.

### Physics setup and guide

Begin with the physical question and assumptions. Identify exact groups and maps, matter, dimensions, signature, statistics, and any additional choices. Provide a worked path through tools and explain what the result establishes. A setup is structured data; its guide is a readable interpretation.

### Learn, Reference, comparison, and governance

Learn pages have prerequisites, a single learning objective, examples, related live calculations, and next steps. Reference pages lead with the formula or convention and its domain. Comparison pages align compatible quantities and explain inapplicable cells. Governance pages state responsibilities and versioned decisions without imitating mathematical records.

## 6. Visual system and accessibility

Use a restrained academic identity: warm light background, dark blue text, blue actions, copper emphasis, and sparing green for validated outcomes. The current light-theme tokens provide the initial direction: background #f8f7f3, ink #182d3e, action #233e61, accent #984f2f. Maintain a fully specified dark palette. Color must never carry status alone.

Use a serif for major headings and mathematical emphasis, a legible sans serif for interfaces and prose, and monospace for identifiers and code. System fallbacks must remain attractive; external font loading is optional. Production mathematics should use accessible typesetting with text equivalents, not Unicode approximations as the only representation of complex formulas.

Use a spacing scale of 4, 8, 12, 16, 24, 32, 48, and 64 pixels. Keep ordinary text near 16 pixels with generous line height, long-form measure around 65–80 characters, and application content near a 1168-pixel maximum width. Dense reference tables may be wider than article prose. Preserve useful information density rather than making every datum a large card.

Desktop tool layouts may have a compact sidebar and main calculation panel. Mobile layouts stack controls, keep labels visible, and offer a usable local contents menu. Tables scroll within labeled regions; essential meaning must not disappear with hidden columns. Diagrams require captions and equivalent textual statements.

Keep WCAG 2.2 AA as the long-term accessibility target. The formal assessment is deferred at the owner’s request and does not block publication. Interface implementation should retain visible focus, keyboard operation, proper form labels, semantic headings, focus management in search dialogs, sufficient contrast, reduced-motion support, and announced calculation errors. A later assessment should cover reflow, zoom, screen-reader labels, and mathematical reading order.

Provide print styles for records and calculations. Avoid unnecessary animation, decorative charts, remote trackers, and video backgrounds. Use interactive visualizations only when manipulation clarifies a mathematical relationship; exact tables remain preferable for exact comparisons.

## 7. Calculator contracts and scientific checks

The mathematical engine must be framework-independent. Inputs and outputs use plain validated data, exact integers or normalized rationals within stated bounds, and structured errors. Formatting and user-interface state do not belong in the algorithms.

### Global forms

SU(2) uses nonnegative integer label n = 2j. Dimension is n+1; the nontrivial central element acts by (-1)^n. Even n descends to SO(3). SU(3) uses labels (p,q), dimension (p+1)(q+1)(p+q+2)/2, and center charge p+2q modulo three. Zero charge descends to PSU(3). [1,2,7]

The current module also returns conjugation, reality, quadratic Casimir, and index. The production UI must declare the normalization T(fundamental)=1/2 and the relation T(R)dim(g)=C2(R)dim(R). Descent and reality are independent questions. Verify the PSU(3) representation (3,0): dimension ten, zero triality, complex type.

### Tensor products

For labels a,b, emit a+b, a+b-2, down to |a-b|, each once. Check that the sum of output dimensions equals (a+1)(b+1). Treat the output as a representation tensor product, not automatically as the allowed polynomial space for identical fields. [1,7]

### A4 invariant counting

Explicitly select distinguishable triplet factors or one identical commuting scalar triplet. The former uses tensor powers; the latter uses symmetric powers. Return the full decomposition, trivial-irrep multiplicity, total dimension, and a dimension check.

At degrees two, three, four, the symmetric-power invariant counts are 1,1,2; tensor-power counts are 1,2,7. Degree zero returns the trivial representation. The rotational triplet's Molien series is (1+t^6)/[(1-t^2)(1-t^3)(1-t^4)]. Explicit polynomial or Clebsch–Gordan output requires a declared basis. Fermionic and derivative operators are outside this tool's current domain. [3–5,10]

### Branching

Store the actual regular block inclusion and principal spin-one homomorphism. The regular restrictions are 3→2+1 and 8→3+2+2+1. The principal restrictions are 3→3 and 8→3+5. Its kernel is Z2 and its image is SO(3), so the principal group map is not an SU(2) subgroup embedding. Check dimensions and compatibility with the kernel. [8]

### Breaking topology

For both curated cases the vacuum orbit is S2. Display the exact-sequence information, including the map from pi1(H) to pi1(G). For SO(3), its kernel is 2Z, and a generator of pi2(S2) maps to winding two in SO(2). For SU(2), the corresponding winding is one in the specified maximal torus. State angular conventions.

These results classify relevant topological sectors under the supplied setup. They do not establish finite energy, existence, or stability of a solution. In a gauge theory, points on a Higgs vacuum orbit are gauge related; the orbit should not be described as a set of physically distinct vacua. [1,11]

## 8. Research workflows

The following describes the broader product workflows. Workbenches I–IV implement finite internal-character operations, compact highest-weight invariants, bounded SU(N) products, two SU(3) restrictions, charge-refined block branching, matter descent, bounded four-dimensional local-polynomial counting, and exhaustive inventory enumeration through dimension six. Named mathematical calculations and bounded physical setups can be saved and loaded. Multiple abelian factors, arbitrary maps, derivative-aware operator enumeration, normalized bases, and the complete PhysicalSetup specification remain future work.

**Global gauge structure:** choose a cover, enter matter representations, calculate their common central kernel, select a compatible quotient, and inspect each descent condition. Then record additional line-operator or discrete-theta choices when the physical question requires them. Usual Standard Model matter permits several quotients under specified charge normalization; matter does not uniquely force one quotient. [9,12]

**Branching:** choose the parent representation, choose a named map with its kernel and normalization, inspect the restricted multiplets, and compare alternate maps. Carry the exact map into the saved result.

**Operator construction:** specify fields, species, representations, Lorentz information, charges, statistics, and derivative scope. Workbench III performs the joint spin/gauge/flavor symmetry projection for a specified derivative-free monomial; Workbench IV enumerates all supported monomials in a chosen dimension range. Derivative extensions require explicit additional treatment of equations of motion and integration by parts; a count alone does not construct a normalized basis. [10]

**Topology and defects:** identify the physical setup and actual space, inspect the relevant homotopy sequence, then interpret sectors for the declared spacetime dimension. Do not substitute pi3(G/H) indiscriminately for a gauge-instanton classification.

**Finite/continuous bridge:** follow a specified inclusion or cover, restrict a continuous representation, and identify ordinary versus projective finite-group representations. A4 in SO(3) and its binary tetrahedral lift in SU(2) provide a future reviewed example. Such pathways connect the two halves of the Atlas without implying a universal ADE classification of finite groups.

## 9. Coverage hierarchy and public-release priorities

Publish three coverage levels: Classification, Reference, Executable. Classification identifies a family and its scope. Reference supplies a stated set of checked properties. Executable supports named operations on a bounded domain. Combine the level with a capability matrix; a single “complete” badge is insufficient.

Those coverage levels and the expanded classification/reference catalogue were established in Phase B; Workbenches I–IV add the bounded executable domains listed in Sections 2.5–2.8. The following list preserves the **broader public-release priorities**; it includes unreleased calculations and integrations. Consult Section 2 and the live capability manifest for the implemented subset.

- Classification navigation for all finite-simple families, their parameter restrictions and exceptional identifications, and the 26 sporadic groups. Handle the Tits group explicitly within stated Lie-type conventions. [13]
- Classification navigation for complex simple Lie algebras and compact connected global forms, plus a curated real-form index that states its scope.
- Reference coverage of classical compact families, exceptional compact types, commonly used noncompact forms, spin covers, products, and central quotients.
- Strong finite-group coverage through a licensed, reviewed import manifest: small nonabelian examples, permutation families, cyclic and dihedral families, quaternionic and binary polyhedral groups, and selected discrete-flavor groups.
- Exact character and tensor-product workflows for supported finite records, and highest-weight workflows for supported compact Lie groups.
- A reviewed map catalogue for common physics branching problems, rather than a promise of arbitrary branching.
- Spacetime records that distinguish components, covers, finite-dimensional field representations, and particle-state representation categories.

Set explicit backend and resource limits before advertising generated coverage. Very large finite groups may have reference data without practical explicit matrix calculations. Higher homotopy, noncompact unitary duals, general cohomology, supergroups, and anomaly classification require separate capability releases.

## 10. Conventions, provenance, and editorial governance

The target is to give every consequential convention a stable identifier and version. Workbenches I–IV use ten profiles: finite ordinary-complex characters; Lie highest weights/root normalization; SU(3) labels and its two restriction maps; SU(N) central charges; canonical Young diagrams; primitive block U(1) charges; embedded product centers; and four-dimensional complex local polynomials. See `/reference/convention-profiles/`. The new profile fixes Lorentzian signature, chirality, joint field statistics, flavor interpretation, charge normalization, and algebraic equivalence. Broader defaults must cover basis-dependent tensors, derivatives, other spacetimes, multiple abelian factors, and additional physical choices as those capabilities are introduced.

Use Sp(n) for the compact quaternionic convention only after defining it, with USp(2n) as a searchable equivalent. Distinguish Sp(2n,R). State that D8 means the dihedral group of order eight in the current edition. Explain that finite simple includes prime-order cyclic groups, whereas “compact simple” commonly refers to simple Lie algebra and may allow a center.

Attach provenance to claims, not just pages. A sourced datum needs a source, locator, assumptions, and retrieval or edition information. A computed datum needs inputs, algorithm, version, dependencies, and verification. Mark conjectural or numerical material explicitly when introduced; never silently promote it to an exact theorem.

Require a mathematics reviewer and a physics reviewer before assigning independently expert-reviewed status to executable capabilities. Publication itself is authorized by the owner and does not wait for that optional status. Imported data passes schema and identity checks before editorial approval. Corrections identify the affected claim and downstream computations. Keep historical releases readable, and mark corrected outputs without silently rewriting their claimed provenance.

Use MIT for original code and CC BY 4.0 for original articles. Keep third-party dataset, font, and asset terms with their provenance. Import only where terms permit reuse; publicly readable data is not automatically licensed for redistribution. Preserve attribution, version, and transformation history. A source adapter may link to data rather than bundle it when redistribution is unavailable.

Maintain separate schema, dataset, algorithm, and application versions. A saved computation records all relevant versions plus its convention profile. Provide canonical page citations and machine-readable exports only after their formats are implemented and tested.

## 11. Production engineering architecture

The Astro/Starlight framework choice is **implemented**: custom layouts serve Atlas, Tools, and Physics, while Starlight serves Learn, Reference, and project documentation. The current build is static, and dependency versions are pinned in `package.json` and the lockfile. The working tree uses `src/pages`, `src/content/docs`, `src/data`, `src/lib`, and the implemented `packages/schema`, `packages/search`, and `packages/editorial` modules.

The following remains the **target module layout**, not a description of directories all present today. In particular, live computational adapters, the proposed `apps/web` split, and the broader `packages/math`/`packages/data` organization are not released merely because they appear in this architecture. Keep shared modules independent of either framework:

~~~text
packages/schema       validated entity and result types
packages/math         exact algorithms and mathematical fixtures
packages/data         curated records, sources, capability manifests
packages/adapters     versioned external-data integrations
packages/search       typed index construction and normalization
apps/web              historical architecture sketch; current implementation uses src/
content/learn         reviewed explanatory documents
content/reference     conventions, formulas, sources, methodology
tests/scientific      independent mathematical acceptance fixtures
docs                  this blueprint and implementation decisions
~~~

Server-render reference content so it remains readable without client JavaScript. Hydrate calculation controls and interactive diagrams selectively. Build search from validated records, aliases, notation, and source-linked articles. Normalize common ASCII, Unicode, and TeX-like input while preserving category distinctions.

Do not parse arbitrary mathematical input with eval. Use constrained schemas and explicit parsers. The released workbenches implement BigInt integer/rational arithmetic and exact finite-character fields in browser-safe modules. Retain this exactness when extending bounds; the inherited bounded algorithms retain their original checked contracts. KaTeX renders scientific notation through a shared renderer with strict verification. Expensive future backends need cancellation, time limits, memory limits, and clear failure states.

Adapters for GAP, its Character Table Library, SageMath, and other sources must expose capability manifests and versions. A frontend cannot label an arbitrary request “computed by GAP” merely because the website cites GAP. Export code must reconstruct the chosen objects and conventions.

Use static generation where appropriate, cached computation where justified, and a content manifest to control released routes. Keep redirects and canonical links under version control. Accounts, collaborative workspaces, and public uploads are later decisions, not dependencies of the core product.

## 12. Acceptance gates

The owner has authorized Codex webmaster development on the Mac and publication after the initial website is finished. Follow `MASTER_PLAN.md`, `docs/LAUNCH_CHECKLIST.md`, and the current publication manifest. The gate requires completed launch pages and actual scientific, software, editorial, and hosting-plan evidence before production is enabled. Named human review is recorded honestly and is not a mandatory launch prerequisite. The formal accessibility assessment remains deferred. Historical release evidence remains unchanged; the Workbench IV report documents that earlier source state, not the later handoff’s final validation.

### Historical design-edition gates

All fourteen records are reachable and searchable. All five tools work across their declared domains. Unsupported inputs produce explicit errors. Mathematical outputs pass the checks in Section 7. Sources, related links, comparison, theme, navigation, and share/export controls do what their labels promise. The edition boundary remains visible.

### Current publishing criteria

In addition to the current finite launch checklist, preserve these scientific and operational criteria:

1. **Identity:** no conflation of group, algebra, real form, global form, representation, or chosen map; exceptional aliases resolve consistently.
2. **Mathematics:** meaningful independent checks, including character orthogonality, representation dimension conservation, descent under center action, and compatibility of branching with map kernels.
3. **Physics:** field statistics, gauge/global distinctions, topology subjects, and spacetime assumptions are explicit.
4. **Reproduction:** shared calculations recover inputs and conventions; exported results name relevant versions.
5. **Editorial:** claims have sources or derivations; reviewers and coverage states are recorded.
6. **Interface behavior:** supported calculations, invalid inputs, saving/loading, and real navigation work; a formal accessibility assessment is deferred.
7. **Operations:** canonical URLs survive refresh, no broken advertised routes, licenses are recorded, and errors are observable without collecting sensitive input unnecessarily.

Add regression cases for SU(2) spinors failing SO(3) descent; the complex PSU(3) ten-dimensional irrep; Q8 versus D8 reality type; A4 symmetric versus tensor counts; and regular versus principal SU(3) restriction.

Future anomaly tools need additional gates. A four-dimensional SU(2) Weyl-doublet global anomaly must not be conflated with the perturbative cubic anomaly. Group cohomology must not be presented as a universal classification of all fermionic or gravitational anomalies. [14]

## 13. Roadmap and implementation order

Advance by acceptance criteria, without calendar promises. The phases below preserve the long-term scientific roadmap. The current implementation order is the finite initial-launch sequence in `MASTER_PLAN.md`; completing core reference/learning/physics coverage now takes priority over extending the operator engine again.

**Phase A — design edition, historical baseline:** the scoped prototype established fourteen records, five bounded workbenches, source links, and the original blueprint. Its mathematical engine and useful interactions are retained in the reference foundation.

**Phase B — public reference foundation, technical milestone delivered:** the validated entity schema, canonical routing, claim-level provenance, fingerprinted review workflow, classification navigation, 194-record catalogue, typed search, static exports, and five-tool migration are implemented and pass the recorded automated and focused browser checks. That historical release used a private review audience. The current handoff follows the offline-until-finished Cloudflare Pages policy in the master plan; independent review status remains separate.

**Phase C — research workbench, Workbenches I–IV delivered subsets; full phase unfinished:** fifteen bounded tools, 62 representation records, four Standard Model global groups and their specified maps, ten convention profiles, exact data, named calculations, eleven fixed-monomial physical setups, six enumerated inventories, reproduction scripts, KaTeX rendering, and method/physics guides are implemented. Sections 2.4–2.8 are the released contracts. Remaining work includes derivatives, field strengths, EOM and IBP; normalized matrices and contraction bases; broader compact-group tensors and specified additional maps; several gauge U(1) factors and general charge lattices; computational adapters; and the complete PhysicalSetup model. Downloaded scripts do not imply a live external computation service.

**Phase D — specialist depth, future and unreleased:** expand topology, noncompact representation categories, spacetime applications, central extensions, selected anomaly checks, and finite/continuous bridges. Each domain has its own review and resource requirements.

**Phase E — ecosystem, future and unreleased:** consider shared projects, collaborative contributor workflows, notebook integrations, and interoperable sibling-site data after the core research workflows are dependable. The current repository review command is not a collaborative account or shared-project service.

### Historical architecture sequence

The sequence below describes how the application foundation was designed. Do not restart it or migrate the working repository to the old directory sketch. Follow the current master plan and next-task record for ongoing development.

1. Read the repository instructions, current master plan, this blueprint, and the capability manifest; identify the requested phase.
2. Inventory existing entities, routes, sources, and mathematical functions before modifying them.
3. Establish stable IDs, alias policy, schemas, and convention profiles.
4. Create independent scientific fixtures for the supported algorithms.
5. Extract framework-independent math and data modules from the prototype.
6. Implement canonical routes and the shared navigation shell.
7. Build entity templates before expanding the catalogue.
8. Connect tools to validated inputs, capability metadata, and reproducible result records.
9. Add Physics and Learn paths that exercise the same data and functions.
10. Import additional data through reviewed adapters with license manifests.
11. Verify scientific contracts, supported interface behavior, and real navigation; defer the formal accessibility assessment for now.
12. Update coverage, release notes, citations, and this blueprint when contracts change.

Do not manufacture data to fill a template, add nonfunctional controls to suggest future capability, or expand testing beyond a concrete unresolved risk. Stop optional work once the phase's acceptance gates are satisfied.

## 14. QFT.org ecosystem boundaries

Groups for Physics owns canonical group-theoretic identities, representation data, maps, and supported symmetry calculations. QFT.org provides the broader physics context and site directory. Sibling projects may own general geometry, topology, field-theory explanations, document workflows, or computational notebooks.

Populate a shared site switcher from a registry of confirmed destinations. Do not invent sibling hostnames or suggest unavailable sites are live. Contextual links should land on a relevant concept, preserve the user's research question, and clearly identify an external destination.

Avoid duplicating authoritative mathematical records across sibling sites. Exchange stable IDs, citations, and versioned data through an explicit integration contract. Keep general physics prose on the appropriate sibling while retaining enough local explanation for every group calculation to stand on its own.

## 15. Initial source and integration bibliography

The following bibliography preserves the initial design-edition review corpus. The current Workbench IV edition's 114-source catalogue bibliography and claim dependencies are generated from the catalogue and exposed through `/reference/sources/` and the source JSON download. Precise chapter, theorem, table, or page locators are attached to the current curated claims; new curation must retain that standard.

1. Brian Conrad and Aaron Landesman, *Compact Lie Groups*. https://math.stanford.edu/~conrad/210CPage/handouts/lie_groups_notes.pdf
2. Brian Conrad, *Centers of Simply Connected Semisimple Groups*. https://math.stanford.edu/~conrad/210CPage/handouts/centers.pdf
3. Pavel Etingof et al., *Introduction to Representation Theory*. https://math.mit.edu/~etingof/reprbook.pdf
4. GAP, official reference manual, character-theory chapter. https://gap-system.github.io/gap/doc/ref/chap72_mj.html
5. GroupNames, A4 record and related finite-group data. https://people.maths.bris.ac.uk/~matyd/GroupNames/1/A4.html
6. Thomas Breuer, GAP Character Table Library. https://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib/
7. SageMath, *Weyl Character Rings*. https://doc.sagemath.org/html/en/thematic_tutorials/lie/weyl_character_ring.html
8. SageMath, *Branching Rules*. https://doc.sagemath.org/html/en/thematic_tutorials/lie/branching_rules.html
9. Aharony, Seiberg, and Tachikawa, *Reading Between the Lines of Four-Dimensional Gauge Theories*. https://arxiv.org/abs/1305.0318
10. Henning, Lu, Melia, and Murayama, *Hilbert Series and Operator Bases with Derivatives in Effective Field Theories*. https://arxiv.org/abs/1507.07240
11. T. W. B. Kibble, *Topology of Cosmic Domains and Strings*, J. Phys. A 9 (1976) 1387. https://doi.org/10.1088/0305-4470/9/8/029
12. David Tong, *Line Operators in the Standard Model*. https://arxiv.org/abs/1705.01853
13. Robert A. Wilson, finite simple groups reference materials. https://webspace.maths.qmul.ac.uk/r.a.wilson/TFSG.html
14. Edward Witten, *An SU(2) Anomaly*, Phys. Lett. B 117 (1982) 324–328. https://doi.org/10.1016/0370-2693(82)90728-6

Additional Workbench I source checks use the GAP Reference Manual, SageMath root-system/Weyl-character/Gelfand–Tsetlin documentation, Etingof’s Lie theory notes, Conrad’s center tables, Adams’s self-dual reality-indicator formula, Haber’s SU(3) map examples, and the cited gauge-theory papers. Each new method page supplies exact source URLs and locators. These source checks are AI-assisted and do not supply independent human approval.

Selected ATLAS records are already used as identified sources for the curated finite-group reference. Broader automated integrations with the ATLAS of Finite Group Representations, LMFDB's abstract-group collection, or computational backends remain future work requiring reviewed scope, identifiers, and reuse terms. The current source-linked factual summaries do not constitute wholesale imports of those datasets or a live backend service.

Workbench II source checks additionally use the official SageMath lrcalc and Schur documentation, Stanley’s Littlewood–Richardson theorem, LieART tensor fixtures, Baez–Huerta’s SU(5) construction, Tong’s Standard Model global-form analysis, and Conrad’s compact-group topology notes. Each method page records the relevant section, theorem, or table. Mathematical derivations are original exposition; linked sources do not imply bundled software or copied datasets.

Workbench III source checks use Marinissen–Rahn–Waalewijn on symmetric/exterior powers and Haar projection, Lehman–Martin and Fonseca on QQQL flavor counting, Dreiner–Haber–Martin on two-component spinors, Weinberg’s dimension-five interaction, and Henning–Lu–Melia–Murayama on the additional EOM/IBP machinery. Tong’s established global-form convention is reused. Primary papers are linked with section/equation locators in the new method and physics pages. This is AI-assisted source checking; the count of independently human-approved records remains zero.

Workbench IV adds published Standard Model flavor-count benchmarks from Alonso and collaborators and the operator-dimension selection rule of Kobach. The new method and physics pages distinguish counts of derivative-free scalar/Weyl invariants from complete EFT bases and state precisely when distinct Hermitian conjugates are counted.