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.

#	Sheet	Focus	Why it matters for matheology
	Structural Identity
1	Reference Sheet 1: Category Theory for Cross-Model Reasoning	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	Reference Sheet 2: Homotopy Type Theory for Identity and Equivalence	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	Reference Sheet 3: Mechanism Design for Axiomatic Economics	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	Reference Sheet 8: Social Choice Theory for Collective Decisions	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	Reference Sheet 5: Dynamical Systems & Bifurcation Theory	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	Reference Sheet 6: Ergodic Theory & Ergodicity Economics	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	Reference Sheet 4: Paraconsistent Logic for Graceful Contradiction	LP, relevant logic, chunk-and-permeate	LP preserves classical theorems in consistent fragments. The variable-sharing property provides formal firewalls between model domains.
7	Reference Sheet 7: Topos Theory for Multi-Logic Unification	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	Reference Sheet 9: Coalgebra & Bisimulation for Infinite Processes	Final coalgebras, bisimulation, coinduction	Connects th11’s infinite game to th8’s attractors via bisimulation. ax21 (Permanent Mediator) becomes a bisimulation class.
10	Reference Sheet 10: Constructive Mathematics & Intuitionistic Logic	BHK, excluded middle, decidability, proof relevance	Constructive audit sketch of all 25 axioms. Essential companion to Sheet 7 for topos-internal reasoning.
11	Reference Sheet 11: Information Theory & Model Complexity	Shannon entropy, Kolmogorov complexity, MDL	Pairwise mutual information for all 300 axiom pairs identifies redundancy candidates. Rate-distortion formalizes audience-depth.
12	Reference Sheet 12: Proof Theory & the Curry-Howard Correspondence	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.

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:

Write a new pre-forge prompt (pre-forge-compiler-refsheet4.rst)
Run it to produce new sheets in wb/
Update the table and dependency map above
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)

Sheets