Directed Univalence Beyond the Discrete: Extending GWB 2024 to (∞,1)-Categorical Types

The directed univalence theorem of Gratzer, Weinberger and Buchholtz (GWB, 2024) shows that in the universe $$ of discrete types of triangulated type theory (TTT), the canonical comparison map \[ : (A,B) \;\; (A B) \] is an equivalence. The natural strengthening --- directed univalence for the universe of all $(,1)$-categorical (Segal) types --- is the principal open problem in directed type theory and the bottleneck for synthetic representation theory and any HoTT-native formulation of analytic Langlands. This paper does not prove directed univalence for non-discrete types. We position the re…

Lean 4 Proof Verdict

# Lean proof verdict — oq4-directed-univalence-non-discrete

_Generated by codex gpt-5.5 at xhigh reasoning effort, with workspace-write._

## Pre-verification `lake build` output

```
info: oq4-directed-univalence-non-discrete: no previous manifest, creating one from scratch
trace: ././.lake/packages/mathlib> git fetch --tags --force origin
error: ././.lake/packages/mathlib/lakefile.lean:35:40: error: unexpected token ']'; expected term
error: ././.lake/packages/mathlib/lakefile.lean:45:38: error: unexpected token ']'; expected term
error: ././.lake/packages/mathlib/lakefile.lean:64:17: error: unknown identifier 'mathlibLeanOptions'
error: ././.lake/packages/mathlib/lakefile.lean:75:17: error: unknown identifier 'mathlibLeanOptions'
error: ././.lake/packages/mathlib/lakefile.lean:78:17: error: unknown identifier 'mathlibLeanOptions'
error: ././.lake/packages/mathlib/lakefile.lean:143:5: error: invalid field 'baseName', the environment does not contain 'Lake.Package.baseName'
  rootPkg
has type
  Package
error: ././.lake/packages/mathlib/lakefile.lean:143:24: error: invalid field 'baseName', the environment does not contain 'Lake.NPackage.baseName'
  pkg
has type
  NPackage _package.name
error: ././.lake/packages/mathlib/lakefile.lean:149:34: error: invalid field 'trimAscii', the environment does not contain 'String.trimAscii'
  toolchainContent
has type
  String
error: ././.lake/packages/mathlib/lakefile.lean:170:16: error: invalid field 'baseName', the environment does not contain 'Lake.NPackage.baseName'
  pkg
has type
  NPackage _package.name
error: ././.lake/packages/mathlib/lakefile.lean:151:13: error: invalid field 'trimAscii', the environment does not contain 'String.trimAscii'
  toolchainContent
has type
  String
error: ././.lake/packages/mathlib/lakefile.lean:153:29: error: invalid field notation, type is not of the form (C ...) where C is a constant
  toolchainVersion
has type
  ?m.3835 pkg rootPkg y✝ __do_lift✝ toolchainContent
error: ././.lake/packages/

… (truncated)