Add comprehensive academic proofs and formal verification

docs/academic/README.md

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+# AffineScript Academic Documentation
+
+This directory contains formal academic documentation for the AffineScript programming language, including proofs, specifications, white papers, and mechanized verification.
+
+## Document Index
+
+### Proofs and Metatheory
+
+| Document | Description | Status |
+|----------|-------------|--------|
+| [Type System Soundness](proofs/type-soundness.md) | Progress and preservation theorems | Complete |
+| [Quantitative Type Theory](proofs/quantitative-types.md) | Linearity and quantity proofs | Complete |
+| [Effect Soundness](proofs/effect-soundness.md) | Algebraic effects metatheory | Complete |
+| [Ownership Soundness](proofs/ownership-soundness.md) | Affine/linear type safety | Complete |
+| [Row Polymorphism](proofs/row-polymorphism.md) | Extensible records metatheory | Complete |
+| [Dependent Types](proofs/dependent-types.md) | Indexed types and refinements | Complete |
+
+### White Papers
+
+| Document | Description |
+|----------|-------------|
+| [Language Design](white-papers/language-design.md) | Design rationale and related work |
+| [Type System Design](white-papers/type-system.md) | Bidirectional typing with quantities |
+| [Effect System Design](white-papers/effect-system.md) | Algebraic effects and handlers |
+
+### Formal Verification
+
+| Document | Description | Status |
+|----------|-------------|--------|
+| [Operational Semantics](formal-verification/operational-semantics.md) | Small-step semantics | Complete |
+| [Denotational Semantics](formal-verification/denotational-semantics.md) | Domain-theoretic model | Complete |
+| [Axiomatic Semantics](formal-verification/axiomatic-semantics.md) | Hoare logic for AffineScript | Complete |
+
+### Mathematical Foundations
+
+| Document | Description |
+|----------|-------------|
+| [Categorical Semantics](mathematical-foundations/categorical-semantics.md) | Category theory models |
+| [Logic Foundations](mathematical-foundations/logic-foundations.md) | Curry-Howard and proof theory |
+| [Complexity Analysis](mathematical-foundations/complexity-analysis.md) | Decidability and complexity bounds |
+
+### Mechanized Proofs
+
+| Document | Description | Status |
+|----------|-------------|--------|
+| [Coq Formalization](mechanized/coq/README.md) | Coq proof development | Stub |
+| [Lean Formalization](mechanized/lean/README.md) | Lean 4 proof development | Stub |
+| [Agda Formalization](mechanized/agda/README.md) | Agda proof development | Stub |
+
+## Citation
+
+```bibtex
+@misc{affinescript2024,
+  title = {AffineScript: A Quantitative Dependently-Typed Language with Algebraic Effects},
+  author = {AffineScript Contributors},
+  year = {2024},
+  howpublished = {\url{https://github.com/hyperpolymath/affinescript}}
+}
+```
+
+## Status Legend
+
+- **Complete**: Theoretical content complete, pending implementation verification
+- **Stub**: Placeholder awaiting implementation of corresponding compiler component
+- **TODO**: Section identified but not yet written
+
+## Dependencies on Implementation
+
+Many proofs in this documentation depend on compiler components that are not yet implemented. These are marked with `[IMPL-DEP]` tags throughout the documents, indicating which compiler phase must be completed before the proof can be fully verified against the implementation.
+
+| Proof Area | Required Implementation |
+|------------|------------------------|
+| Type soundness | Type checker |
+| Effect soundness | Effect inference |
+| Ownership soundness | Borrow checker |
+| Termination | Termination checker |
+| Refinement verification | SMT integration |