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EP-0099: Master Expansion Roadmap & Unified Algebraic Architecture

Field Value
EP 0099
Title Master Expansion Roadmap & Unified Algebraic Architecture
Author Eran Rivlis & Antigravity
Status Active
Type Informational
Created 2026-08-01
Updated 2026-09-10

Abstract

This proposal establishes the strategic roadmap and architectural blueprint for expanding algebrax into advanced mathematical domains. It connects core primitives (SparseVector, SparseMatrix, Semiring, AlgebraicTrie) through two foundational engine extensions (EP-0100 and EP-0101) to four specialized domain modules (EP-0110 through EP-0113), an interactive visual studio (EP-0120), a maturity & consistency track (EP-0130 through EP-0134), the Phase 3.5 Council Refinement Track (EP-0140 through EP-0146), formalized micro-benchmarking (EP-0148), universal binomial convolution statistical moment semirings (EP-0149), standardized mathematical docstrings (EP-0150), algebraic PageRank contraction solvers over semirings (EP-0151), Spectral Graph Theory with Graph Laplacians and Fiedler partitioning (EP-0152), and the Phase 4 Algebraic Continents & Next Frontiers (EXP-0001, EP-0160 through EP-0163) extending into Discrete Exterior Calculus and continuous Lie symmetries.

The Master Dependency & Connection Graph

flowchart TD
    CORE["algebrax Core Foundations\n(Sparse Mappings + Semirings + Tries)"]

    CORE --> EP0100["EP-0100\nQuotient Monoid Algebra"]
    CORE --> EP0101["EP-0101\nSparse Chain Complex"]
    CORE --> EP0149["EP-0149\nBinomial Moments & Divided Powers"]
    CORE --> EP0163["EP-0163\nLie Bracket & BCH Dynamics"]

    EP0100 --> EP0111["EP-0111\nClifford Algebra"]
    EP0100 --> EP0112["EP-0112\nGalois Fields"]
    EP0100 --> KNOT["Knot Skein Modules\n(Existing)"]

    EP0101 --> EP0110["EP-0110\nSimplicial Homology"]
    EP0101 --> SHEAF["Sheaf Consensus\n(Existing)"]
    EP0101 --> EP0113["EP-0113\nCategory Kleisli"]
    EP0101 --> EP0160["EP-0160\nDiscrete Exterior Calculus"]

    EP0110 --> P3
    EP0111 --> P3
    EP0111 --> EP0163
    EP0112 --> P3
    EP0113 --> P3
    EP0149 --> P35

    subgraph P3["Phase 3 — Explorer, Maturity & Curiosity"]
        EP0120["EP-0120 Web Explorer"]
        EP0130["EP-0130 API Consistency ✅"]
        EP0131["EP-0131 Law Verification ✅"]
        EP0132["EP-0132 Matrix Decompositions ✅"]
        EP0133["EP-0133 Jupyter & CLI ✅"]
        EP0134["EP-0134 Semiring Namespaces ✅"]
    end

    P3 --> P35

    subgraph P35["Phase 3.5 — Council Refinement Track"]
        EP0140["EP-0140 API Symmetry ✅\n⚖️ Noether"]
        EP0141["EP-0141 Taxonomy Cleanup ✅\n🧩 Russell"]
        EP0142["EP-0142 Performance ✅\n⚡ Shannon"]
        EP0143["EP-0143 Documentation ✅\n💡 Feynman"]
        EP0144["EP-0144 Testing ✅\n🔬 Popper"]
        EP0145["EP-0145 Type Safety ✅\n🛡️ Golem"]
        EP0146["EP-0146 Ergonomics ✅\n🤝 Steward"]
        EP0148["EP-0148 Benchmarking ✅\n⏱️ CodSpeed"]
        EP0149_N["EP-0149 Binomial Moments ✅\n∑ Noether & Russell"]
        EP0150_N["EP-0150 Math Docstrings ✅\n💡 Feynman"]
        EP0151_N["EP-0151 Algebraic PageRank ✅\n⚡ Shannon & ⚖️ Noether"]
        EP0152_N["EP-0152 Spectral Graph Theory ✅\n⚖️ Noether & ⚡ Shannon"]
    end

    P35 --> P4

    subgraph P4["Phase 4 — Algebraic Continents & Next Frontiers (EXP-0001)"]
        EP0160_N["EP-0160 Discrete Exterior Calculus (DEC) 📝\n⚖️ Noether & ⚡ Shannon"]
        EP0161_N["EP-0161 Cellular Sheaves & Laplacians 💡\n🧩 Russell & 🤝 Steward"]
        EP0162_N["EP-0162 Persistent Homology (TDA) 💡\n🔬 Popper & 🧭 Explorer"]
        EP0163_N["EP-0163 Lie Algebras & BCH Dynamics ✅\n⚖️ Noether & 🛡️ Golem"]
        EP0164_N["EP-0164 Root Systems & Chevalley 📝\n⚖️ Noether & ⚡ Shannon"]
    end

    EP0152_N --> EP0160_N
    EP0110 --> EP0162_N
    EP0163_N --> EP0164_N

    style CORE fill:#4a90d9,color:#fff
    style P3 fill:#2d7d46,color:#fff
    style P35 fill:#8b5cf6,color:#fff
    style P4 fill:#d97706,color:#fff
    style EP0100 fill:#3b82f6,color:#fff
    style EP0101 fill:#3b82f6,color:#fff
    style EP0110 fill:#10b981,color:#fff
    style EP0111 fill:#10b981,color:#fff
    style EP0112 fill:#10b981,color:#fff
    style EP0113 fill:#10b981,color:#fff
    style EP0149 fill:#8b5cf6,color:#fff
    style EP0160_N fill:#f59e0b,color:#fff
    style EP0161_N fill:#f59e0b,color:#fff
    style EP0162_N fill:#f59e0b,color:#fff
    style EP0163_N fill:#f59e0b,color:#fff
    style KNOT fill:#6b7280,color:#fff
    style SHEAF fill:#6b7280,color:#fff

Architectural Principles & Core Connections

The expansion adheres strictly to The Council Framework (PRINCIPLES.md):

  1. Zero Bloat (Shannon Efficiency): Rather than building 4 isolated monolithic modules, the roadmap reduces the expansion to two core foundational engine extensions:

    • EP-0100 (QuotientMonoidAlgebraSemiring): Powers Clifford Algebra, Galois Fields, and Knot Skein Modules through a single quotient reduction callback quotient_fn.
    • EP-0101 (SparseChainComplex): Unifies 1D Graph Laplacians, Sheaf Coboundary Gradients, Simplicial Boundary Operators \(D_k\), and Categorical Morphisms through nilpotency \(D_{k-1} \circ D_k = \mathbf{0}\).

  2. Falsifiable Invariants (Popper): Every extension introduces strict algebraic invariants (\(D_{k-1} D_k = \mathbf{0}\), \(\mathbf{e}_i \mathbf{e}_j + \mathbf{e}_j \mathbf{e}_i = 2 g_{ij}\), \(P^2 = P\)) verified via automated test suites. Phase 3 deepens this with property-based law verification (EP-0131).

  3. Self-Documenting Symmetry (Noether & Feynman): Each domain track includes a standalone Python recipe, an interactive Jupyter notebook, and a dedicated Graphical Laboratory view in recipes/lab.py. Phase 3 completes the symmetry with matrix decompositions (EP-0132).

  4. Consistency (Russell): Phase 3 resolves the __init__.py export gap for Phase 2 modules (EP-0130).

Phased Implementation Sequence

Phase 0: Architecture Roadmap (EP-0099)
  │
  ├── Phase 1: Foundational Engine Extensions  ✅ COMPLETE
  │     ├── EP-0100: QuotientMonoidAlgebraSemiring (algebrax.semiring)     [Final]
  │     └── EP-0101: SparseChainComplex & Hodge-Laplacian (algebrax.analysis) [Final]
  │
  ├── Phase 2: Specialized Domain Tracks  ✅ COMPLETE
  │     ├── EP-0110: Simplicial Homology & Betti Numbers (algebrax.homology) [Final]
  │     ├── EP-0111: Clifford Geometric Algebra & Rotors (algebrax.clifford) [Final]
  │     ├── EP-0112: Galois Finite Fields & Cryptographic Matrices (algebrax.galois) [Final]
  │     └── EP-0113: Categorical Morphisms & Kleisli Composition (algebrax.category) [Final]
  │
  ├── Phase 3: Explorer, Maturity & Curiosity  ⏳ IN PROGRESS
  │     ├── EP-0120: Algebraic Web Explorer & Interactive Visual Studio     [Draft]
  │     ├── EP-0130: API Consistency & Public Export Audit (Russell)        [Final]
  │     ├── EP-0131: Algebraic Law Verification Engine (Popper)            [Final]
  │     ├── EP-0132: Matrix Decompositions — LU, QR, SVD (Noether)         [Final]
  │     ├── EP-0133: Jupyter & CLI Integration (Steward)                   [Final]
  │     └── EP-0134: Semiring Namespace Refactoring (Russell)              [Final]
  │
  ├── Phase 3.5: Council Refinement Track  ✅ COMPLETE
  │     ├── EP-0140: API Symmetry Restoration (Noether)                    [Final]
  │     ├── EP-0141: Structural Taxonomy Cleanup (Russell)                 [Final]
  │     ├── EP-0142: Performance & Efficiency Optimizations (Shannon)      [Final]
  │     ├── EP-0143: Documentation Clarity & Freshman Test (Feynman)       [Final]
  │     ├── EP-0144: Testing & Falsifiability Hardening (Popper)           [Final]
  │     ├── EP-0145: Type Safety & Contract Hardening (Golem)              [Final]
  │     ├── EP-0146: Developer Ergonomics & Ecosystem Bridges (Steward)    [Final]
  │     ├── EP-0147: Optional Loop Pragmas & Concurrency (Shannon)         [Deferred]
  │     ├── EP-0148: Formalized Micro-Benchmarking & CodSpeed (Popper)     [Final]
  │     ├── EP-0149: Universal Binomial Moment Semirings (Noether/Russell) [Final]
  │     ├── EP-0150: Standardized Mathematical Docstrings (Feynman/Russell) [Final]
  │     ├── EP-0151: Algebraic PageRank & Semiring Random Walks (Shannon/Noether) [Final]
  │     └── EP-0152: Spectral Graph Theory & Graph Laplacians (Noether/Shannon/Popper) [Final]
  │
  └── Phase 4: Algebraic Continents & Next Frontiers (EXP-0001)  🚀 IN PROGRESS
        ├── EP-0160: Discrete Exterior Calculus & Helmholtz-Hodge (Noether/Shannon/Feynman) [Draft]
        ├── EP-0161: Cellular Sheaves & Network Laplacians (Russell/Steward)               [Queued]
        ├── EP-0162: Persistent Homology & Topological Barcodes (Popper/Explorer)          [Queued]
        ├── EP-0163: Lie Algebras, Root Systems & BCH Dynamics (Noether/Shannon/Golem)     [Final]
        └── EP-0164: Root Systems, Weyl Groups & Chevalley Construction (Noether/Shannon)  [Draft]

Detailed Proposal Matrix

Proposal Title Pillar Target Module Status Deliverables
EP-0099 Master Expansion Roadmap — Docs Active EP-0099-expansion-roadmap.md
EP-0100 Quotient Monoid Algebras Shannon algebrax.semiring Final QuotientMonoidAlgebraSemiring, tests
EP-0101 Sparse Chain Complexes Shannon algebrax.analysis Final SparseChainComplex, hodge_laplacian, tests
EP-0110 Simplicial Homology Explorer algebrax.homology Final SimplicialComplex, betti_numbers, Lab View 21
EP-0111 Clifford Geometric Algebra Explorer algebrax.clifford Final CliffordSemiring, rotor_rotation, Lab View 22
EP-0112 Galois Finite Fields Explorer algebrax.galois Final GaloisFieldSemiring, gf_matrix_mul, Lab View 23
EP-0113 Categorical Morphisms Explorer algebrax.category Final kleisli_compose, kan_extension, Lab View 24
EP-0120 Algebraic Web Explorer Feynman Web / Visual Draft site/explorer/index.html, HTML5/Canvas studio
EP-0130 API Consistency Audit Russell algebrax.__init__ Final Public re-exports, Semiring catalog
EP-0131 Algebraic Law Verification Popper algebrax.verification Final Property-based axiom tests, CLI auditor python -m algebrax.verify
EP-0132 Matrix Decompositions Noether algebrax.decompose Final Sparse LU, QR, SVD, Cholesky on dict-matrices
EP-0133 Jupyter & CLI Integration Steward algebrax.display Final _repr_html_(), python -m algebrax inspect
EP-0134 Semiring Namespace Refactor Russell algebrax.semiring/ Final Categorical sub-modules, consolidated Clifford/Galois
EP-0140 API Symmetry Restoration Noether matrix, transforms, homology Final Recomposition helpers, inverse transforms, coboundary operator
EP-0141 Taxonomy Cleanup Russell analysis, tensor, __init__ Final Relocate SparseChainComplex, permute_tensor, clean imports
EP-0142 Performance Optimizations Shannon matrix, transforms, tensor Final Local binding, catalog cache, twiddle precompute, backtracking
EP-0143 Documentation Clarity Feynman docs/, docstrings Final Freshman summaries, typo fixes, concepts.md rewrite
EP-0144 Testing Hardening Popper tests/ Final Property-based tests, edge cases, numerical stability
EP-0145 Type Safety Hardening Golem typing, analysis, converters Final Future annotations, semiring normalization, collision fix
EP-0146 Developer Ergonomics Steward __init__, converters, display Final Namespace org, NumPy/SciPy bridges, Jupyter display
EP-0147 Optional Loop Pragmas Shannon benchmarks/, core loops Deferred Non-invasive lucen pragmas, free-threaded GIL-less scaling
EP-0148 Formalized Benchmarking Popper benchmarks/, .github/ Final Standardized pytest-benchmark suite & CodSpeed CI tracking
EP-0149 Universal Binomial Moments Noether & Russell algebrax.semiring Final Divided power quotient ring, MultivariateMomentSemiring, decoders
EP-0150 Standardized Math Docstrings Feynman & Russell algebrax.semiring, display Final Unified AMDS standard, MathJax semiring card, LaTeX repr, tests
EP-0151 Algebraic PageRank Shannon & Noether algebrax.analysis Final Sparse semiring contraction solver, random walk with restart, PPR
EP-0152 Spectral Graph Theory Noether & Shannon algebrax.analysis Final Graph Laplacians (unnormalized, sym, rw), Fiedler vector, cuts, smoothing
EP-0160 Discrete Exterior Calculus Noether & Shannon algebrax.homology / dec Draft \(\Omega^k\) forms, exterior derivative \(d\), Hodge star \(\star\), Helmholtz-Hodge
EP-0161 Cellular Sheaves & Laplacians Russell & Steward algebrax.homology Queued CellularSheaf, coboundary \(\delta\), sheaf Laplacian, consensus dynamics
EP-0162 Persistent Homology (TDA) Popper & Explorer algebrax.homology Queued Simplicial filtrations, boundary reduction over \(\mathbb{Z}_2\), barcodes
EP-0163 Lie Algebras & BCH Dynamics Noether & Golem algebrax.lie Final LieAlgebra, StructureConstants, so3, sl2, se3, BCH solver
EP-0164 Root Systems & Chevalley Noether & Shannon algebrax.lie Final RootSystem, dynkin_diagram, to_euclidean, Chevalley construction

Change Log

Date Author Description
2026-08-01 Eran Rivlis & Antigravity Initial Master Roadmap EP created.
2026-08-02 Eran Rivlis & Antigravity Implemented Phase 1 & Phase 2 proposals; status updated to Final.
2026-08-02 Eran Rivlis & Antigravity Phase 3 track added: EP-0120, EP-0130, EP-0131, EP-0132, EP-0133.
2026-08-02 Eran Rivlis & Antigravity EP-0134 (Semiring Namespace Refactoring) added to Phase 3.
2026-08-02 Eran Rivlis & Antigravity Phase 3.5 Council Refinement Track: EP-0140 through EP-0146.
2026-09-04 Eran Rivlis & Antigravity Phase 3.5 completed; added EP-0148 (CodSpeed CI), EP-0149 (Binomial Moment Semirings), and User Guide architectural overhaul.
2026-09-05 Eran Rivlis & Antigravity Added EP-0150: Standardized Mathematical Docstrings & Algebraic Signature Registry.
2026-09-05 Eran Rivlis & Antigravity Added EP-0151: Algebraic PageRank & Semiring Random Walks with Restart.
2026-09-05 Eran Rivlis & Antigravity Added EP-0152: Spectral Graph Theory, Algebraic Connectivity & Graph Laplacians.
2026-09-08 Eran Rivlis & Antigravity Initiated EXP-0001 (Next Frontiers); drafted EP-0160 (DEC) and EP-0163 (Lie Algebras & BCH Dynamics).
2026-09-10 Eran Rivlis & Antigravity Implemented EP-0163 (Lie Algebras & BCH Dynamics), added user guide, and updated status to Final.
2026-09-10 Eran Rivlis & Antigravity Added EP-0164: Root Systems, Weyl Groups & Chevalley-Serre Construction for Simple Lie Algebras.
2026-09-10 Eran Rivlis & Antigravity Transitioned EP-0150 to Final; added Phase 4 roadmap integration (EP-0160 through EP-0163) and landed sparse commutator.
2026-09-12 Eran Rivlis & Antigravity Implemented EP-0164: added multi-format Dynkin diagrams (ASCII/Mermaid/SVG), Jupyter rich display, benchmarks, and updated status to Final.