.. meta:: :description: The FORGE WisdomBase — 12 reference sheets distilling formal mathematics for applied use in matheology model development, organized by cluster with dependency map. :keywords: WisdomBase, reference sheets, category theory, HoTT, mechanism design, dynamical systems, ergodic theory, topos theory, coalgebra, proof theory, matheology :author: Yah, Yas, everyone, LLoL as Laurence Loewe of Laodicea, ClaudeOp46Max, Anthropic, and Spirit of Boolean Truth .. _forge-wisdombase: ********************************************************************* WisdomBase --- 12 Reference Sheets ********************************************************************* The WisdomBase (WB) is FORGE's library of distilled mathematical wisdom. Each sheet is a concise (~4000--5000 tokens), self-contained guide that covers: orientation, key concepts, critical theorems, common pitfalls, and a bridge to the specific matheology system (25 axioms, 11 theorems, 2 models, 66+ HELL findings). The name WisdomBase follows the Evolvix convention for distilled generic wisdom from diverse disciplines. The 12 Sheets =============== Sheets are organized into four clusters. Within each cluster, sheets build on each other. Click any sheet title to read it. .. list-table:: :header-rows: 1 :widths: 3 20 22 55 * - # - Sheet - Focus - Why it matters for matheology * - - **Structural Identity** - - * - 1 - :doc:`category-theory` - Functors, natural transformations, adjunctions - Formalizes alignment echoes between PET and JUB as natural transformations (structural vs. accidental). HELL findings become a slice category over each axiom. SISYF compilation is a colimit. * - 2 - :doc:`homotopy-type-theory` - Univalence, paths, transport, HITs - VVN versions become a higher inductive type with path constructors. Transport makes version migration rigorous. Structure Identity Principle settles "are PET and JUB the same model?" as a theorem. * - - **Game-Theoretic** - - * - 3 - :doc:`mechanism-design` - IC, VCG, revelation principle, impossibilities - Myerson-Satterthwaite identifies a potential impossibility for ax25. The JUB model must specify which constraint is relaxed --- or show the Jubilee-System escapes the bilateral-trade framing. * - 8 - :doc:`social-choice-theory` - Arrow, Sen, Gibbard-Satterthwaite - Arrow's theorem constrains the ax17+ax25+th9 triad. Identifies three resolution strategies (domain restriction, informed consensus, binary reduction). * - - **Dynamical Core** - - * - 5 - :doc:`dynamical-systems` - Attractors, basins, bifurcation, Lyapunov - Provides a minimal 5-step program to formalize th8. Directly responds to Con-A.1 (severity A). Frames ax25's redistribution rate as a bifurcation parameter. * - 6 - :doc:`ergodic-theory` - Birkhoff, Peters, mixing, ergodicity breaking - Peters' ergodicity economics directly supports th9. Mixing time becomes a testable prediction constraining the Jubilee-System cycle length. * - - **Logical Pluralism** - - * - 4 - :doc:`paraconsistent-logic` - LP, relevant logic, chunk-and-permeate - LP preserves classical theorems in consistent fragments. The variable-sharing property provides formal firewalls between model domains. * - 7 - :doc:`topos-theory` - Internal logic, geometric morphisms, classifying topoi - Proposes a multi-topos architecture: Boolean PET-topos, non-Boolean JUB-topos, combined topos via geometric morphism. Six scriptural traditions become six geometric morphisms. * - - **Foundations & Formalization** - - * - 9 - :doc:`coalgebra` - Final coalgebras, bisimulation, coinduction - Connects th11's infinite game to th8's attractors via bisimulation. ax21 (Permanent Mediator) becomes a bisimulation class. * - 10 - :doc:`constructive-math` - BHK, excluded middle, decidability, proof relevance - Constructive audit sketch of all 25 axioms. Essential companion to Sheet 7 for topos-internal reasoning. * - 11 - :doc:`information-theory` - Shannon entropy, Kolmogorov complexity, MDL - Pairwise mutual information for all 300 axiom pairs identifies redundancy candidates. Rate-distortion formalizes audience-depth. * - 12 - :doc:`proof-theory` - Curry-Howard, sequent calculus, proof assistants - Recommends **Lean 4** as primary proof assistant. Provides a 5-phase formalization roadmap. Dependency Map ^^^^^^^^^^^^^^^^ :: ┌──────────────────────────────────────────────────────────┐ │ STRUCTURAL IDENTITY │ │ Sheet 1 (Category Theory) ──┬── Sheet 7 (Topos Theory) │ │ │ ↑ ↑ │ │ Sheet 2 (HoTT) ────────────┘ │ │ │ │ │ │ │ │ LOGICAL PLURALISM │ │ │ │ Sheet 4 (Paraconsistent) ───────────┘ │ │ │ │ │ │ DYNAMICAL CORE │ │ │ Sheet 5 (Dynamical Systems) ──┐ │ │ │ ↓ │ │ │ Sheet 6 (Ergodic Theory) ─────┤ │ │ │ ↓ │ │ │ GAME-THEORETIC │ │ │ │ Sheet 3 (Mechanism Design) ───┤ │ │ │ ↓ │ │ │ Sheet 8 (Social Choice) ──────┘ │ │ │ │ │ │ FOUNDATIONS & FORMALIZATION │ │ │ Sheet 9 (Coalgebra) ← Sheet 1 │ │ │ Sheet 10 (Constructive Math) ← Sheet 7 ─┘ │ │ Sheet 12 (Proof Theory) ← Sheets 2, 10 │ │ Sheet 11 (Information Theory) ← Sheet 1 (optional) │ └──────────────────────────────────────────────────────────┘ Which Sheets to Load When ============================ Not every forge session needs all 12 sheets. .. list-table:: :header-rows: 1 :widths: 30 30 40 * - Task - Sheets - Notes * - PET-internal theorem work - 1--4 - Category + HoTT + mechanism + paraconsistent * - th8 (Binary Attractors) formalization - 1--6 - Dynamical systems + ergodic theory are critical * - th9 (Social Ergodicity) formalization - 3, 5--6, 8 - Mechanism + dynamics + ergodic + social choice * - th11 (Stakes Without Death) formalization - 2, 9 - HoTT + coalgebra * - Cross-model formal integration - 1, 2, 4, 7, 10 - Category + HoTT + paraconsistent + topos + constructive * - Axiom redundancy analysis - 1, 11 - Category theory + information theory * - Machine-checkable formalization - 2, 10, 12 - HoTT + constructive math + proof theory * - Full adversarial forge session - All 12 - Maximum formal reach Extending the WisdomBase ========================== To add a new reference area: 1. Write a new pre-forge prompt (``pre-forge-compiler-refsheet4.rst``) 2. Run it to produce new sheets in ``wb/`` 3. Update the table and dependency map above 4. Add the new sheets to the toctree below Areas identified as potential future additions: - Non-monotonic reasoning (if HELL database gets automated reasoning) - Domain theory (if proto-formal predicates need denotational semantics) - Probability on measurable spaces (if assessments become probabilistic) - Computability theory (if the forge aims for automated property checking) - Algebraic topology (if topos cohomological obstructions become concrete) .. toctree:: :maxdepth: 1 :caption: Sheets category-theory homotopy-type-theory mechanism-design paraconsistent-logic dynamical-systems ergodic-theory topos-theory social-choice-theory coalgebra constructive-math information-theory proof-theory