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+# Tropical Correspondence (echo-types ↔ tropical-resource-typing)
+
+Last updated: 2026-05-20
+
+This appendix records the citation-level correspondence between
+`proofs/agda/EchoTropical.agda` in this repo and the adjacent
+`hyperpolymath/tropical-resource-typing` repo (Isabelle + Lean4). The
+alignment is **statement-level only**: there is no FFI surface between
+Agda and Isabelle/Lean, no shared definition module, and no cross-prover
+extraction pipeline. Each side carries its own independent proof of the
+named theorems; this document is a cross-reference that lets a reader
+verify "the same fact is established in all three systems," but does
+not constitute a unified verification artefact. It closes the
+"Adjacent repo not recently audited" blocker noted in
+`cross-repo-bridge-status.md`.
+
+## Source files
+
+- **Agda (this repo).** `proofs/agda/EchoTropical.agda` — the
+  Tropical-Echo bridge (E10): max-plus semiring on `ℕ` plus the
+  echo-retention bridge theorems.
+- **Isabelle.**
+  `tropical-resource-typing/Tropical.thy` — max-plus tropical semiring
+  over the lifted carrier `ℕ ∪ {-∞}`, wired into Isabelle's
+  `comm_semiring_1` typeclass hierarchy, with idempotence proved
+  separately (the structure is a dioid, not a ring).
+- **Lean4.**
+  `tropical-resource-typing/TropicalSessionTypes.lean` — max-plus
+  tropical semiring on the lifted carrier `Nat ∪ {bot}`, used to grade
+  session types so that speculative parallel cost is the bottleneck
+  (`max`) rather than the sequential sum.
+- **Canonical adjacent remote.** `hyperpolymath/tropical-resource-typing`
+  (active; primary language Isabelle; the `.thy` files were last
+  touched in the `Tropical_Semirings` close sweep, with subsequent CI
+  hardening commits on top).
+- **Local clone for this audit.**
+  `/home/hyperpolymath/dev/repos/repos-monorepo/verification-ecosystem/tropical-resource-typing`.
+
+## Name-by-name correspondence
+
+| Agda (`EchoTropical.agda`) | Isabelle (`Tropical.thy`) | Lean4 (`TropicalSessionTypes.lean`) | Notes |
+|---|---|---|---|
+| `_⊕_` (max-plus add on `ℕ`, line 23) | `trop_add` (function, line 41) | `tAdd` (function, line 96) | Same operation (max with identity element absorbed at the left). DIVERGES (carrier): Agda's `_⊕_` operates on raw `ℕ` with `zero` acting as the additive identity by the recursion shape; Isabelle's `trop_add` and Lean's `tAdd` both operate on a *lifted* carrier (`tropical = Fin nat \| NegInf` / `Tropical = .val Nat \| .bot`) where the additive identity is the bottom element `−∞`, not `0`. The `ℕ` quotient on the Agda side is intentional (the bridge only needs scores in `ℕ`) but means Agda's `_⊕_` is the restriction of the Isabelle/Lean operation to the finite sub-semiring; the algebraic laws below match on that restriction. |
+| `⊕-idem` (line 30) | `trop_add_idem [simp]` (lemma, line 73) | — | Match (Agda ↔ Isabelle): Agda `⊕-idem : ∀ m → m ⊕ m ≡ m` ↔ Isabelle `trop_add_idem : trop_add a a = a`. The Isabelle file also restates this at typeclass level as `tropical_add_idem` (theorem, line 266: `(a :: tropical) + a = a`); the bare-function lemma is the closer match. Lean side has NO named idempotence theorem — the file ships 13 CommSemiring laws (commut/assoc/identity/distrib) but not `tAdd a a = a`. The Lean docstring explicitly flags this gap: `tropical_grade_le_sequentialTotal` is offered as the "Lean analogue of Isabelle `tropical_add_idem`" because `max a b ≤ a + b` is what `add_idem` buys in a dioid. So the Lean cell is `—` for the bare law and `tropical_grade_le_sequentialTotal` for the downstream consumer. |
+| `score-⊕-idem` (line 82) | — | — | Unilateral (Agda-only). The Agda side specialises `⊕-idem` to scores of the 3-candidate set; the adjacent repo has no `Candidate` type and no `score` function, so the specialisation has no analog. |
+| `tropical-non-injective` (line 55) | — | — | Unilateral (Agda-only). Headline of the echo-retention bridge: there exist distinct candidates with the same tropical score. The adjacent repo does not type candidates (its tropical semiring is generic), so there is nothing to be non-injective about. |
+| `tropical-collapse-visible` (line 121) | — | — | Unilateral (Agda-only). `score a ≡ score b` for the concrete 3-candidate choice; no analog. |
+| `Echo` / `echo-intro` / `TropEcho` / `IsArgmin` (lines 59–73) | — | — | Unilateral (Agda-only). The echo type itself (`Echo f y := Σ A (λ x → f x ≡ y)`) is an echo-types invention; the adjacent repo has no fibre type and no echo bridge. |
+| `echo0-to-tropical` (line 113) | — | — | Unilateral (Agda-only). Bridge map echo → tropical residue; no analog (no echo on the other side). |
+| `distinct-candidates-same-visible-distinct-echo` (line 130) | — | — | Unilateral (Agda-only). The main bridge theorem: collapse on the visible (tropical) side, retention on the echo side. No analog. |
+| `tropical-echo-retention-simple` (line 135) | — | — | Unilateral (Agda-only). Simplified restatement of the headline bridge. |
+| (no Agda analog) | `trop_mul` (function, line 46) | `tMul` (function, line 103) | DIVERGES: the adjacent repo carries a full semiring (add + mul), the Agda side does not. The Agda bridge needs only the additive (max) structure of the dioid; multiplicative tropical structure is out of scope for the echo bridge as it currently stands. |
+| (no Agda analog) | `trop_add_comm` (line 65) | `add_comm_trop` (line 114) | Adjacent-side commutativity. Match between Isabelle and Lean. No Agda analog by design — `_⊕_` on `ℕ` is commutative but the lemma is not stated because the bridge headlines do not consume it. |
+| (no Agda analog) | `trop_add_assoc` (line 69) | `add_assoc_trop` (line 117) | Adjacent-side associativity. Match between Isabelle and Lean. No Agda analog by design (same reason as commutativity). |
+| (no Agda analog) | `trop_distrib_{left,right}` (lines 122/126); semiring instance | `left_distrib_trop` / `right_distrib_trop` (lines 167/173); `CommSemiring Tropical` instance (line 208) | Adjacent-side distributivity + semiring typeclass wiring. Match between Isabelle and Lean. Out of scope for the echo bridge. |
+
+## Alignment caveats
+
+- **No Agda↔Isabelle/Lean import surface.** Agda cannot `import` a
+  `.thy` or a `.lean`; Isabelle and Lean4 cannot `import` an `.agda`.
+  Any "alignment" between these files is therefore citation-level only
+  — a reader verifies the names and statements line up, but each
+  prover runs its own independent proof. None of the three is a
+  trusted oracle for either of the others.
+- **Independent proofs, identical claims.** The first alignable
+  theorem pair is `⊕-idem` (Agda) ↔ `trop_add_idem` (Isabelle). Both
+  are proved (clean, no axioms, no `sorry`); the Agda side under
+  `--safe --without-K`, the Isabelle side as a `simp` lemma against
+  the algebraic kernel of `Tropical.thy`. The Lean side does not name
+  this fact directly but consumes it inside the QTT refinement
+  theorem `tropical_grade_le_sequentialTotal` (`max a b ≤ a + b` is
+  the dioid consequence of additive idempotence).
+- **Carrier mismatch is intentional, not a defect.** Agda's `_⊕_`
+  lives on `ℕ` (the score type); Isabelle/Lean live on the lifted
+  carrier with an explicit `−∞` bottom. The Agda side never needs
+  `−∞` because every candidate has a finite score; the bridge is
+  consciously narrower than the full max-plus semiring.
+- **Echo-side machinery is Agda-exclusive.** The `Echo`/`TropEcho`
+  fibre type, `IsArgmin`, the candidate datatype, and all bridge
+  headlines (`tropical-non-injective`, `echo0-to-tropical`,
+  `distinct-candidates-same-visible-distinct-echo`) have no analog
+  on the Isabelle or Lean sides. The adjacent repo's tropical
+  semiring stands on its own and is consumed by *session-type
+  grading* (`grade : Session → Tropical` in Lean), which is the
+  symmetric Agda-exclusive direction.
+- **Long-game alignment target.** When the echo-types ordinal track
+  reaches Bachmann–Howard (ψ₀(Ω_ω); see `docs/echo-types/roadmap.md`
+  and `buchholz-plan.adoc`), the adjacent repo's
+  `Tropical_Ordinal_Bridge.thy` becomes the natural cross-repo
+  alignment target (Agda Buchholz BT ↔ Isabelle `tropO` carrier).
+  This target is **firewalled** until the ordinal track lands the
+  milestone — do not pull it forward.
+
+## Revision history
+
+- 2026-05-20: created (initial citation-level correspondence).