diff --git a/lake-manifest.json b/lake-manifest.json
index e8ffcb435b3b516c7e7e6f32008d4a78dcc9271c..5ba329fb1fb3ff704cdeabb2e0b58cda469ccdd3 100644
--- a/lake-manifest.json
+++ b/lake-manifest.json
@@ -1,7 +1,17 @@
 {"version": "1.2.0",
  "packagesDir": ".lake/packages",
  "packages":
- [{"url": "https://github.com/lana-agents/heights",
+ [{"url": "https://github.com/lana-agents/belyi.git",
+   "type": "git",
+   "subDir": null,
+   "scope": "",
+   "rev": "9ce4d3f793d3bc831f2fc7214b7dbad90ea012d8",
+   "name": "belyi",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "9ce4d3f793d3bc831f2fc7214b7dbad90ea012d8",
+   "inherited": false,
+   "configFile": "lakefile.toml"},
+  {"url": "https://github.com/lana-agents/heights.git",
    "type": "git",
    "subDir": null,
    "scope": "",
@@ -11,31 +21,31 @@
    "inputRev": "721496ca4c158e511d25d8ccf6bfe8503eda73c1",
    "inherited": false,
    "configFile": "lakefile.toml"},
-  {"url": "https://github.com/leanprover-community/mathlib4",
+  {"url": "https://github.com/leanprover-community/mathlib4.git",
    "type": "git",
    "subDir": null,
-   "scope": "leanprover-community",
-   "rev": "81a5d257c8e410db227a6665ed08f64fea08e997",
+   "scope": "",
+   "rev": "d13f23b723b8a846827a245b89c10fc7d3f11612",
    "name": "mathlib",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "v4.32.0",
+   "inputRev": "d13f23b723b8a846827a245b89c10fc7d3f11612",
    "inherited": false,
    "configFile": "lakefile.lean"},
-  {"url": "https://github.com/lana-agents/belyi.git",
+  {"url": "https://github.com/lana-agents/oka.git",
    "type": "git",
    "subDir": null,
    "scope": "",
-   "rev": "9ce4d3f793d3bc831f2fc7214b7dbad90ea012d8",
-   "name": "belyi",
+   "rev": "75dcdc3faadd10b986afb9283e89902bcdddecd4",
+   "name": "oka",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "9ce4d3f793d3bc831f2fc7214b7dbad90ea012d8",
+   "inputRev": "75dcdc3faadd10b986afb9283e89902bcdddecd4",
    "inherited": true,
    "configFile": "lakefile.toml"},
   {"url": "https://github.com/leanprover-community/plausible",
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "e12c1910fe855cbfc38803cd4e55543906d5fa62",
+   "rev": "118aa17ee84656b8bd727fef7c458ee8c833385c",
    "name": "plausible",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
@@ -45,7 +55,7 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "c5d5b8fe6e5158def25cd28eb94e4141ad97c843",
+   "rev": "ddf04cf3949fa556442341e87d47f9f6e6074707",
    "name": "LeanSearchClient",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
@@ -55,7 +65,7 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "7e9612bf0b9ee66db3cb5b9988a35afc706f5a12",
+   "rev": "e928b72544873815af278d38681b31c0293588e3",
    "name": "importGraph",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
@@ -65,7 +75,7 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "6e311e2a844da9b2cc3971187df2fe0066947b93",
+   "rev": "106ff4fafc74ef4ac99d81dbf3ab399118f497a5",
    "name": "proofwidgets",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
@@ -75,7 +85,7 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "a7dbf0c63b694e47f425f3dcddbc0e178bb432d3",
+   "rev": "355695d523e41d0554926416cba2a2b3544fbbc9",
    "name": "aesop",
    "manifestFile": "lake-manifest.json",
    "inputRev": "master",
@@ -85,7 +95,7 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "38d591e778f100aec9762bb582f9c7f55f50e9dc",
+   "rev": "6a489d9af5d0c47e5b259e2e8bcdfc1811b5a259",
    "name": "Qq",
    "manifestFile": "lake-manifest.json",
    "inputRev": "master",
@@ -95,30 +105,20 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "023ce7d62a0531e22a5331e20b587817a80d49ff",
+   "rev": "f2effa3d803fda822b1f97b806c47cf2adfbcbc2",
    "name": "batteries",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
    "inherited": true,
    "configFile": "lakefile.toml"},
-  {"url": "https://github.com/lana-agents/oka.git",
-   "type": "git",
-   "subDir": null,
-   "scope": "",
-   "rev": "da228a2cf9671aaba08ddc96274d75e098b28f67",
-   "name": "oka",
-   "manifestFile": "lake-manifest.json",
-   "inputRev": "da228a2cf9671aaba08ddc96274d75e098b28f67",
-   "inherited": true,
-   "configFile": "lakefile.toml"},
   {"url": "https://github.com/leanprover/lean4-cli",
    "type": "git",
    "subDir": null,
    "scope": "leanprover",
-   "rev": "88679d088c9720c27ebdf2ba4dafe17341747f94",
+   "rev": "e92c9f15fdfacc8536f31cfb3b7ad26c3c8cd204",
    "name": "Cli",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "v4.32.0",
+   "inputRev": "v4.34.0",
    "inherited": true,
    "configFile": "lakefile.toml"}],
  "name": "genl",
diff --git a/lakefile.toml b/lakefile.toml
index 8be38dbd83af2475de6b65a66e460a7709cb702f..24cede3b9b7f23a9cfc2f4edd6d0e8cf2bb6c458 100644
--- a/lakefile.toml
+++ b/lakefile.toml
@@ -4,6 +4,8 @@ keywords = ["math"]
 defaultTargets = ["Genl"]
 
 [leanOptions]
+# Preserve the type-unification behavior expected by the pinned upstream sources.
+backward.isDefEq.respectTransparency.types = false
 pp.unicode.fun = true # pretty-prints `fun a ↦ b`
 autoImplicit = false
 relaxedAutoImplicit = false
@@ -12,16 +14,22 @@ maxSynthPendingDepth = 3
 
 [[require]]
 name = "mathlib"
-scope = "leanprover-community"
-rev = "v4.32.0"
+git = "https://github.com/leanprover-community/mathlib4.git"
+rev = "d13f23b723b8a846827a245b89c10fc7d3f11612"
 
 # Heights on curves over number fields (the genuine height theory, Genl.Curves), which also
 # brings lana-agents/belyi (curves as function fields).
 [[require]]
 name = "heights"
-git = "https://github.com/lana-agents/heights"
+git = "https://github.com/lana-agents/heights.git"
 rev = "721496ca4c158e511d25d8ccf6bfe8503eda73c1"
 
+# Imported directly by the curve height and covering developments.
+[[require]]
+name = "belyi"
+git = "https://github.com/lana-agents/belyi.git"
+rev = "9ce4d3f793d3bc831f2fc7214b7dbad90ea012d8"
+
 [[lean_lib]]
 name = "Genl"
 
diff --git a/lean-toolchain b/lean-toolchain
index 94b9f495baff80fd9cb44aad8f4762cb3b2066fe..ba8ebf2dbaf6a668cd2a0e086186d6d569b69ff5 100644
--- a/lean-toolchain
+++ b/lean-toolchain
@@ -1 +1 @@
-leanprover/lean4:v4.32.0
+leanprover/lean4:v4.34.1