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+# Phase 3 — String-Level Universal Round-Trip Proof Plan
+
+## Goal
+
+Prove the universal theorem
+
+```lean
+∀ v, MetaCTerm.fromLeanSource? (MetaCTerm.toLeanSource v) = some v
+```
+
+(and parallel theorems for `Lean.Name`, `MetaClassifier`, `MetaArtifact`)
+without using `decide`, `native_decide`, or closed-instance hacks. Full
+structural induction over the meta-mirror types.
+
+## Current state
+
+### Proven (`Foundation/MetaParse.lean`)
+
+**Phase 1 — Canonical token forms** (definitions):
+- `nameToTokens : Lean.Name → List Token`
+- `MetaClassifier.toTokens : MetaClassifier → List Token`
+- `MetaCTerm.toTokens : MetaCTerm → List Token`
+- `MetaArtifact.toTokens : MetaArtifact → List Token`
+
+**Phase 2 — Parser correctness on `toTokens`** (universal, structural induction):
+- `parseName?Aux_correct`
+- `parseClassifier?Aux_correct`
+- `parseMetaCTerm?Aux_correct`
+- `parseArtifact?Aux_correct` (modulo `.declAt`)
+
+**Length-vs-depth lemmas**:
+- `nameToTokens_length_bound`, `classifierToTokens_length_bound`,
+  `cTermToTokens_length_bound`
+
+**Token-level universal round-trip** (the headline results):
+```lean
+nameFromTokens?_round_trip       : ∀ n,        fromTokens? (toTokens n)   = some n
+classifierFromTokens?_round_trip : ∀ φ,        fromTokens? φ.toTokens     = some φ
+cTermFromTokens?_round_trip      : ∀ t,        fromTokens? t.toTokens     = some t
+artifactFromTokens?_round_trip   : ∀ a, a.supported → fromTokens? a.toTokens = some a
+```
+
+**Atomic Phase 3 witnesses** via `decide` (kernel-rooted, closed inputs):
+- `tokenize_render_name_anonymous`
+- `tokenize_render_classifier_always` / `_never`
+- `tokenize_render_cterm_empty`
+- `tokenize_render_artifact_empty`
+
+**Foundation tokenize lemmas**:
+- `tokenize_lparen_cons : tokenize ('(' :: rest) = .lparen :: tokenize rest`
+- `tokenize_rparen_cons` (analogous for `)`)
+- `tokenize_space_cons` (whitespace skip)
+
+### Open (the four distribution lemmas)
+
+```lean
+readIdent_app    : -- ident chars + clean boundary → exact accumulation
+readStrLit_app   : -- escapeStrLit body + close → recovers original String
+tokenize_app_clean: -- tokenize distributes over clean concatenation
+tokenize_render_X: -- by induction over each meta-mirror type
+```
+
+These compose to `∀ v, tokenize (toLeanSource v).toList = toTokens v`,
+which combined with Phase 2 yields the full String-level universal.
+
+## The obstacle
+
+Lean 4.30's `String` is **UTF-8 ByteArray-backed**, not a `List Char`
+structure. Consequently:
+
+```lean
+example (s : String) (c : Char) (xs : List Char) :
+    s.push c ++ xs.asString = s ++ (c :: xs).asString := by rfl  -- FAILS
+```
+
+This identity is **not definitionally true**. The kernel sees `String.push`,
+`String.append`, and `List.asString` as primitive (or natively-implemented)
+operations that don't unfold to a common normal form via `rfl`.
+
+This blocks the inductive step of `readIdent_app`: the inductive
+hypothesis returns `(acc.push d ++ ds.asString, rest)` but we want to
+conclude `(acc ++ (d :: ds).asString, rest)`. The two are
+**propositionally equal** but require a String-internals lemma to bridge.
+
+## Two solution paths
+
+### Path A: Refactor to `List Char` accumulators
+
+Replace the `String` accumulator in `readIdent` and `readStrLit` with a
+`List Char`. Convert to `String` only at the emission point in
+`tokenizeAux`.
+
+**Before**:
+```lean
+def readIdent : Nat → List Char → String → String × List Char
+  | n+1, c :: rest, acc =>
+      if isIdentRestChar c then readIdent n rest (acc.push c)
+      else (acc, c :: rest)
+```
+
+**After**:
+```lean
+def readIdent : Nat → List Char → List Char → List Char × List Char
+  | n+1, c :: rest, acc =>
+      if isIdentRestChar c then readIdent n rest (acc ++ [c])
+      else (acc, c :: rest)
+
+-- in tokenizeAux:
+| n+1, c :: rest =>
+    if isIdentStartChar c then
+      let (consumed, rest') := readIdent (n+1) (c :: rest) []
+      Token.ident consumed.asString :: tokenizeAux n rest'
+```
+
+Now `readIdent_app` becomes a pure `List Char` lemma, structurally provable:
+
+```lean
+theorem readIdent_app (chars rest : List Char) (acc : List Char) (fuel : Nat)
+    (hf : fuel > chars.length)
+    (hcs : ∀ c ∈ chars, isIdentRestChar c = true)
+    (hr : rest = [] ∨ ∃ c rest', rest = c :: rest' ∧ isIdentRestChar c = false) :
+    readIdent fuel (chars ++ rest) acc = (acc ++ chars, rest) := by
+  induction chars generalizing acc fuel with
+  | nil => rcases fuel with _ | f
+           · simp at hf
+           · simp [readIdent]
+             cases hr with
+             | inl h => subst h; simp [readIdent]
+             | inr h => obtain ⟨c, r', hcr, hcb⟩ := h
+                        subst hcr; simp [readIdent, hcb]
+  | cons d ds ih =>
+      rcases fuel with _ | f
+      · simp at hf
+      · have hd : isIdentRestChar d = true := hcs d (List.mem_cons_self _ _)
+        have hds : ∀ c ∈ ds, isIdentRestChar c = true :=
+          fun c hc => hcs c (List.mem_cons_of_mem _ hc)
+        have hf' : f > ds.length := by simp [List.length_cons] at hf; omega
+        have ih' := ih (acc ++ [d]) f hf' hds
+        simp [readIdent, hd]
+        rw [ih']
+        simp [List.append_assoc]
+```
+
+The String-internals lemma is replaced by `List.append_assoc` —
+trivially provable.
+
+**Cost**: ~50 lines refactor of `readIdent`/`readStrLit`/`tokenizeAux`,
+~200 lines of proof.
+
+### Path B: Use Mathlib's String API
+
+Mathlib provides the structural String lemmas needed:
+- `String.toList_append : (s ++ t).toList = s.toList ++ t.toList`
+- `String.push_eq : s.push c = ⟨s.toList ++ [c]⟩` (or equivalent)
+- `String.eq_iff_toList_eq` (extensionality)
+
+With these, the original `String`-accumulator `readIdent` can be
+proven correct without refactor. The proof structure is the same as
+Path A but uses the Mathlib lemmas to bridge between `String.push`
+and `List.cons`.
+
+**Cost**: Add `mathlib` as a dependency to `infoductor` (already
+present transitively via `Mathlib` import in some Foundation files,
+but not directly used). Then ~200 lines of proof using Mathlib's
+String API.
+
+## Lemma-by-lemma proof plan
+
+### Lemma 1: `readIdent_app`
+
+**Statement** (Path A version):
+```lean
+theorem readIdent_app : ∀ (chars rest : List Char) (acc : List Char) (fuel : Nat),
+    fuel > chars.length →
+    (∀ c ∈ chars, isIdentRestChar c = true) →
+    (rest = [] ∨ ∃ c rest', rest = c :: rest' ∧ isIdentRestChar c = false) →
+    readIdent fuel (chars ++ rest) acc = (acc ++ chars, rest)
+```
+
+**Proof**: structural induction on `chars`. Base case (`[]`) splits on
+the boundary form. Inductive case unfolds `readIdent` once, applies the
+ident-rest hypothesis, and recurses via IH with `acc ++ [d]`.
+
+**Difficulty**: Low (Path A) / Medium (Path B).
+
+### Lemma 2: `readStrLit_app`
+
+**Statement**:
+```lean
+theorem readStrLit_app (s : String) (rest : List Char) (acc : List Char) (fuel : Nat) :
+    fuel > (escapeStrLitBody s).length + 1 →
+    readStrLit fuel ((escapeStrLitBody s).toList ++ '"' :: rest) acc =
+      some (acc ++ s.toList, rest)
+```
+
+Where `escapeStrLitBody s` is the inner part of `escapeStrLit` (without
+the surrounding `"`).
+
+**Proof**: induction on `s.toList`. Each char either escapes (`"` → `\"`,
+`\` → `\\`) or passes through. The `readStrLit` decoder reverses the
+escape exactly.
+
+**Difficulty**: Medium. The escape/unescape pairing is straightforward
+but tedious: 3 sub-cases per char (escape `"`, escape `\`, normal).
+
+### Lemma 3: `tokenize_app_clean`
+
+**Statement**:
+```lean
+theorem tokenize_app_clean (xs ys : List Char) :
+    isCleanlyTerminated xs →
+    tokenize (xs ++ ys) = tokenize xs ++ tokenize ys
+```
+
+Where `isCleanlyTerminated xs` says `xs` ends in a single-char-token
+(paren), whitespace, or string-literal close — *not* in the middle of
+an ident, num, or open string.
+
+**Proof**: induction on `xs`. Use the foundation tokenize lemmas
+(`tokenize_lparen_cons` etc.) for the constructor cases. The
+"cleanly terminated" predicate ensures no token spans the boundary.
+
+**Difficulty**: Medium. Defining `isCleanlyTerminated` precisely and
+proving each clean-prefix case takes care.
+
+### Lemma 4: `tokenize_render_name`
+
+**Statement**:
+```lean
+theorem tokenize_render_name : ∀ (n : Lean.Name),
+    tokenize (nameToLeanSource n).toList = nameToTokens n
+```
+
+**Proof**: induction on `n`.
+
+- **`.anonymous`**: `nameToLeanSource .anonymous = "Lean.Name.anonymous"`.
+  Tokenize this via `readIdent_app` (Lemma 1) with `rest = []`, gives
+  `[.ident "Lean.Name.anonymous"]`. Equals `nameToTokens .anonymous`.
+
+- **`.str p s`**: `nameToLeanSource (.str p s) = "(Lean.Name.str " ++ nameToLeanSource p ++ " " ++ escapeStrLit s ++ ")"`.
+
+  Use `String.toList_append` (Mathlib) or the chars-version of the
+  renderer to break `(...)`.toList into:
+  `['('] ++ "Lean.Name.str ".toList ++ (nameToLeanSource p).toList ++ [' '] ++ (escapeStrLit s).toList ++ [')']`
+
+  Then apply `tokenize_app_clean` (Lemma 3) repeatedly:
+  - `'('` → `lparen :: tokenize ...`
+  - `"Lean.Name.str ".toList` → `ident "Lean.Name.str" :: tokenize ...`
+    (via `readIdent_app` with rest starting at `' '`)
+  - `(nameToLeanSource p).toList` → `nameToTokens p ++ tokenize ...`
+    (via IH)
+  - `[' ']` → `tokenize ...` (whitespace skip)
+  - `(escapeStrLit s).toList` → `[strLit s] ++ tokenize ...`
+    (via `readStrLit_app`)
+  - `[')']` → `[rparen]`
+
+  Combine: `[lparen, ident "Lean.Name.str"] ++ nameToTokens p ++ [strLit s, rparen]`
+  = `nameToTokens (.str p s)`. ✓
+
+- **`.num p k`**: analogous to `.str`, using `numLit` instead of `strLit`.
+
+**Difficulty**: Medium-High. Each step requires careful invocation of
+Lemmas 1–3.
+
+### Lemma 5: `tokenize_render_classifier`
+
+**Statement**:
+```lean
+theorem tokenize_render_classifier : ∀ (φ : MetaClassifier),
+    tokenize (MetaClassifier.toLeanSource φ).toList = MetaClassifier.toTokens φ
+```
+
+**Proof**: 9 cases mirroring the 9 classifier arms. The atomic arms
+(`.always`, `.never`) are direct. The `.atDecl`/`.inFile`/`.underAttribute`/`.dependencyOf`/`.inNamespace` cases use Lemmas 1+3+4. The `.meet`/`.join`
+cases use Lemmas 1+3+IH.
+
+**Difficulty**: Medium. Mostly mechanical after Lemmas 1–4.
+
+### Lemma 6: `tokenize_render_cterm`
+
+**Statement**:
+```lean
+theorem tokenize_render_cterm : ∀ (t : MetaCTerm),
+    tokenize (MetaCTerm.toLeanSource t).toList = MetaCTerm.toTokens t
+```
+
+**Proof**: 8 cases. Atomic: `.empty`. Compositional: `.ident` (uses
+Lemma 4), `.sym` (uses Lemma 2), `.app` (uses Lemma 6 IH), `.lam`/`.plam`
+(uses Lemma 2 + IH), `.comp`/`.transp` (uses Lemmas 2 + 5 + IH).
+
+**Difficulty**: Medium-High. Most complex case is `.comp` with 5
+sub-fields chained.
+
+### Lemma 7: `tokenize_render_artifact`
+
+**Statement**:
+```lean
+theorem tokenize_render_artifact : ∀ (a : MetaArtifact),
+    a.supported = true →
+    tokenize (MetaArtifact.toLeanSource a).toList = MetaArtifact.toTokens a
+```
+
+**Proof**: 5 cases. `.empty`/`.source`: direct. `.cterm`: uses Lemma 6.
+`.refTo`: uses Lemma 4. `.declAt`: excluded by hypothesis.
+
+**Difficulty**: Low after Lemmas 4–6.
+
+### Composing: the universal String-level round-trip
+
+```lean
+theorem MetaCTerm.toLeanSource_round_trip : ∀ (t : MetaCTerm),
+    MetaCTerm.fromLeanSource? (MetaCTerm.toLeanSource t) = some t := by
+  intro t
+  unfold MetaCTerm.fromLeanSource? tokenizeStr
+  rw [tokenize_render_cterm t]                   -- Phase 3: Lemma 6
+  exact cTermFromTokens?_round_trip t            -- Phase 2 (already proven)
+```
+
+Two-line proof composing Phase 2 with Lemma 6. Same shape for the
+other three meta-mirror types.
+
+## Estimated effort
+
+| Lemma | Path A (refactor) | Path B (Mathlib) |
+|-------|-------------------|------------------|
+| 1. `readIdent_app` | 30 lines proof + 20 lines refactor | 50 lines proof |
+| 2. `readStrLit_app` | 50 lines | 70 lines |
+| 3. `tokenize_app_clean` | 80 lines (incl. predicate def) | 80 lines |
+| 4. `tokenize_render_name` | 40 lines | 40 lines |
+| 5. `tokenize_render_classifier` | 90 lines | 90 lines |
+| 6. `tokenize_render_cterm` | 120 lines | 120 lines |
+| 7. `tokenize_render_artifact` | 30 lines | 30 lines |
+| Composition | 16 lines | 16 lines |
+| **Total** | **~470 lines** | **~500 lines** |
+
+Approximately **2–3 focused days** of Lean proof work for either path.
+
+## Recommendation
+
+**Path A (refactor)** is preferred:
+
+- Stays within Lean core (no new dependencies).
+- The refactor is small (~20 lines) and independently sound.
+- Decouples *all* Phase 3 proofs from String internals — no future
+  proof has to wrestle with `String.push`/`String.append` again.
+- The `List Char` accumulator is arguably more Eulerian: tokens are
+  conceptually sequences of characters, and operating on lists makes
+  the equational reasoning transparent.
+
+The refactor is itself the small Eulerian move; the rest of the
+proofs follow naturally.
+
+## What this unlocks
+
+With Phase 3 complete, the bridge has:
+
+```
+∀ t : MetaCTerm,     fromLeanSource? (toLeanSource t) = some t
+∀ φ : MetaClassifier, fromLeanSource? (toLeanSource φ) = some φ
+∀ n : Lean.Name,     nameFromLeanSource? (nameToLeanSource n) = some n
+∀ a : MetaArtifact,  a.supported → fromLeanSource? (toLeanSource a) = some a
+```
+
+The full String-level universal round-trip — provably, by structural
+induction, with no closed-instance restrictions. The `Edit.apply`
+output (Lean source files written to disk) is then provably
+round-trippable: any consumer reading the file back gets the original
+`MetaCTerm`.
+
+This is the operational soundness of the entire bridge at the source
+level — what topolei or any external consumer needs to trust the
+edit-render-parse loop.