# Topolei — References

Papers and code referenced for implementation. Organized by subsystem.

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## Cubical Type Theory — Foundational Papers

**CCHM — the primary reference for our cubical core.**
Cohen, Coquand, Huber, Mörtberg (2016)
"Cubical Type Theory: a constructive interpretation of the univalence axiom"
arXiv:1611.02108
https://arxiv.org/abs/1611.02108
> De Morgan interval algebra, face formulas, hcomp, transport, Glue types, univalence.
> This is the spec for Cells/Cubical/*.lean.

**De Morgan Implementation Tutorial — how to actually build it.**
Mörtberg (2022)
"A tutorial on implementing De Morgan cubical type theory"
arXiv:2210.08232
https://arxiv.org/abs/2210.08232
> Type-checking algorithms, cofibration handling, evaluator structure.
> Closest thing to a recipe for our Eval.lean and Transport.lean.

**ABCFHL — Cartesian cubical variant (alternative to de Morgan).**
Angiuli, Brunerie, Coquand, Favonia, Harper, Licata
"Syntax and Models of Cartesian Cubical Type Theory"
https://www.cs.cmu.edu/~rwh/papers/uniform/uniform.pdf
> Read if de Morgan interval causes problems with Lean's kernel; Cartesian
> variant has different composition rules that may embed more cleanly.

**Univalence in cubical sets.**
Bezem, Coquand, Huber (2017)
arXiv:1710.10941
https://arxiv.org/abs/1710.10941

**Axioms for cubical type theory in a topos.**
Orton, Pitts (2017)
arXiv:1712.04864
https://arxiv.org/abs/1712.04864

**Unifying cubical and multimodal type theory.**
Aagaard, Kristensen, Gratzer, Birkedal (2022)
arXiv:2203.13000
https://arxiv.org/abs/2203.13000

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## Cubical Type Theory — Reference Code

**cubicaltt — original Haskell implementation by Mörtberg et al.**
https://github.com/mortberg/cubicaltt
> Reference for hcomp algorithm, face formula solver, evaluator structure.
> Read the source, do not depend on it.

**Agda Cubical Library — target architecture for our Lean embedding.**
https://github.com/agda/cubical
> Key files to read:
>   Cubical/Core/Primitives.agda   — interval, face, transport primitives
>   Cubical/Foundations/Transport.agda — transport lemmas
>   Cubical/Core/Glue.agda         — Glue type and univalence
> We are reimplementing this structure in Lean 4 as a deep embedding.
> Do not import; use as architectural reference only.

**Ground Zero — Lean 4 synthetic HoTT library.**
https://github.com/rzrn/ground_zero
> Shows how to avoid Lean's native equality and build HoTT synthetically in Lean 4.
> Read for: eliminator construction patterns, HIT techniques via quotients.
> cells-spec constraint: do not take it as a dependency.

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## EML — Exp-Minus-Log Binary Primitive

**The EML paper. Primary computational reference for this project.**
Odrzywolek, Andrzej (2026)
"All elementary functions from a single operator"
arXiv:2603.21852
https://arxiv.org/abs/2603.21852
Local copy: exp-log.pdf
> eml(x,y) = exp(x) − ln(y) with constant 1 generates all elementary functions.
> Grammar: S → 1 | eml(S,S). Constructive proof for sin, cos, +, ×, π, etc.
> Foundation for H1. See HYPOTHESES.md.

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## Factorization Homology — FM^fr Notation Layer

**The primer — best entry point.**
Ayala, Francis (2019)
"A factorization homology primer"
arXiv:1903.10961
https://arxiv.org/abs/1903.10961
> E_n-algebras, framed manifolds, ⊗-excision. Read this first for H4.

**Original paper.**
Ayala, Francis (2012)
"Factorization homology of topological manifolds"
arXiv:1206.5522
https://arxiv.org/abs/1206.5522

**Higher categories.**
Ayala, Francis (2015)
"Factorization homology I: higher categories"
arXiv:1504.04007
https://arxiv.org/abs/1504.04007

**Stratified spaces (for mixed text/graph layouts).**
Ayala, Francis, Tanaka (2014)
"Factorization homology of stratified spaces"
arXiv:1409.0848
https://arxiv.org/abs/1409.0848
> Relevant when notation mixes 1D (text) and 2D (diagram) regions.

**Traces in dimension 1 (for linear/sequential syntax).**
arXiv:2105.01143
https://arxiv.org/abs/2105.01143
> Circle-invariant traces; relevant for cyclic/recursive notation structures.

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## Verified Compiler / Shader IR

**Lean4Lean — verified Lean typechecker in Lean. Pattern reference.**
Carneiro (2024)
"Lean4Lean: Verifying a Typechecker for Lean, in Lean"
arXiv:2403.14064
https://arxiv.org/abs/2403.14064
> Reference for: how to structure a verified evaluator/compiler in Lean 4.
> Our EML → ShaderIR compiler should follow similar patterns.

**SPIR-V specification.**
Khronos Group
https://www.khronos.org/spirv/
> Binary format target for native GPU path. Formal grammar maps onto our ShaderIR.

**MLIR SPIR-V dialect — IR structure reference.**
https://mlir.llvm.org/docs/Dialects/SPIR-V/
> Reference for what a typed shader IR looks like before binary encoding.
> Informs the design of Topolei/Shader/IR.lean.

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## Recommended Reading Order

1. CCHM (1611.02108) — understand the cubical core we are embedding
2. De Morgan tutorial (2210.08232) — implementation recipe
3. cubicaltt source — see the evaluator and hcomp in action
4. Agda Cubical library — see the Lean-side target architecture
5. EML paper (2603.21852) — already read; revisit Sect. 3 for derivation tables
6. Ayala-Francis primer (1903.10961) — FM^fr foundation for H4
7. Lean4Lean (2403.14064) — verified compiler patterns before writing ShaderIR