hyperpolymath
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+# Echo Types — Composition
+
+**Status:** working note mixing established results and open
+conjectures. Every result backed by the current Agda development is
+marked *Agda-backed*; every conjectural claim is labelled explicitly.
+
+---
+
+## The central question
+
+Given `f : A → B` and `g : B → C`, how does `Echo(g ∘ f) : C → Set`
+relate to `Echo(f) : B → Set` and `Echo(g) : C → Set`?
+
+Three candidate answers to rule between:
+
+1. **Accumulation.** The echoes stack: `Echo(g ∘ f) y` factors as
+   `(Echo(f) b × Echo(g) y)` integrated over intermediates `b`.
+2. **Weakening.** The echo of `g ∘ f` is *strictly less informative*
+   than `Echo(f)` alone — composition can only lose more.
+3. **Cancellation.** When `g` has a section, `Echo(g ∘ f)` is
+   isomorphic to `Echo(f)`.
+
+The current Agda evidence strongly favours **accumulation** as the
+base case and **cancellation** as a corollary. Weakening is true at
+the shadow level but not generally at the intensional core.
+
+---
+
+## Accumulation — Agda-backed (base case)
+
+*Lemma.* For `f : A → B` and `g : B → C`, the type
+`Echo(g ∘ f) y` is canonically isomorphic to
+`Σ B (λ b → Echo(f) b × (g b ≡ y))`.
+
+*Proof sketch.* Σ-associativity plus propositional-equality
+rearrangement. In Agda terms, given
+```
+Echo f b      = Σ A (λ x → f x ≡ b)
+Echo g y      = Σ B (λ b → g b ≡ y)
+Echo (g ∘ f) y = Σ A (λ x → g (f x) ≡ y)
+```
+the iso is witnessed by
+```
+to   : Σ A (λ x → g (f x) ≡ y) → Σ B (λ b → Σ A (λ x → f x ≡ b) × (g b ≡ y))
+to (x , p)       = (f x , (x , refl) , p)
+from : Σ B (λ b → Σ A (λ x → f x ≡ b) × (g b ≡ y)) → Σ A (λ x → g (f x) ≡ y)
+from (b , (x , refl) , p) = (x , p)
+```
+and both round-trips are `refl`. Formalisation is routine and not
+yet in the repo; it is the cleanest candidate for the next
+composition lemma to land.
+
+*Agda adjacency.* `Echo.map-over-comp` proves functoriality of the
+derived action on echoes; this is the morphism side of the same
+composition law. The *object-side* isomorphism above is not yet
+proved but is a straightforward next milestone.
+
+---
+
+## Cancellation — expected corollary
+
+*Conjecture.* If `g : B → C` has a section `s : C → B` with
+`g ∘ s ≡ id`, then `Echo(g ∘ f) y ≃ Echo(f) (s y)` for every `y`.
+
+*Rationale.* Composing with an injection loses no information, so
+the intermediate factor in the accumulation law collapses to a
+single witnessed point.
+
+*Status.* Not proved. Depends on the accumulation isomorphism plus
+a section lemma that the current repo has pieces of (see
+`no-section-weaken` in `EchoLinear` for the negative direction).
+
+---
+
+## Weakening — true at the shadow, false at the core
+
+*Observation (Agda-backed).* At the extensional shadow,
+`Shadow(g ∘ f) ⊆ Shadow(g)`, because `image(g ∘ f) ⊆ image(g)`. So
+at the shadow level, composition weakens: you cannot learn more
+after an additional forgetful step.
+
+*Counter at the core.* At the intensional core, `Echo(g ∘ f) y` can
+carry *more* witness structure than `Echo(g) y` alone — specifically,
+it records which `b ∈ Echo(g) y` came via `f`. This is the content
+of the accumulation iso above: the composed echo is the sum-total,
+not just the outer fiber.
+
+*Summary.* Weakening is a shadow-level phenomenon. At the core,
+composition *accumulates* witnesses rather than losing them.
+
+---
+
+## Open questions
+
+### Q1. 2-categorical structure
+
+*Question.* Is there a 2-category whose objects are types, whose
+1-morphisms are maps, and whose 2-morphisms are echo-preserving
+transformations? `EchoCategorical.agda` hints at this but does not
+commit to a full 2-categorical axiomatisation.
+
+*Why it matters.* If yes, the composition laws are the coherence
+laws of the 2-category. If no, the composition laws are ad-hoc and
+probably a sign the residue structure is subtler than we modelled.
+
+### Q2. Negative echoes
+
+*Question.* Is there a systematic dual to `Echo(f)` — call it
+`CoEcho(f)` — that records *what has been lost* rather than what
+remains? For a linear map this would correspond to the kernel; for
+a general map, to a typed analogue of the fibre-wise "information
+loss".
+
+*Candidate.* `CoEcho(f)(y) = (something like) "equivalence class of
+preimages of y modulo identity"`. The tropical / metric echo
+(`EchoTropical.agda`) may be the first instance.
+
+*Status.* Speculative. Worth developing alongside approximate
+echoes.
+
+### Q3. Composition of approximate echoes
+
+*Question.* Under the approximate-echo definition (taxonomy,
+axis 2), does composition give a clean tolerance calculus?
+
+*Conjecture.* For metric-tolerance echoes,
+`ε₁-echo(f) ⊙ ε₂-echo(g) ⊑ (ε₁ + L_g · ε₂)-echo(g ∘ f)` where `L_g`
+is a Lipschitz constant of `g`. This is a crude first guess — the
+right form may involve sup-norms, dilation-operators, or
+coarser bounds.
+
+*Status.* Entirely speculative. Requires a formal definition of
+approximate echo first.
+
+### Q4. Associativity
+
+*Question.* Does the accumulation isomorphism above satisfy the
+pentagon coherence for three-fold composition? I.e., for
+`f : A → B`, `g : B → C`, `h : C → D`, does the two ways of
+associating `Echo((h ∘ g) ∘ f) = Echo(h ∘ (g ∘ f))` yield equivalent
+iso's?
+
+*Expectation.* Yes, but the proof requires the iso to land as a
+`map-over` morphism whose `commute` field is itself proved by
+pentagon-style transitivity. `Echo.map-over-comp` already bottles
+the relevant shape. Proving pentagon on top is routine but not
+written down.
+
+### Q5. Interaction with role-indexing, gradings, linearity
+
+*Question.* The existing repo modules `EchoIndexed`, `EchoGraded`,
+`EchoLinear` each decorate the basic echo with extra structure
+(role index, grade label, mode tag). Does composition commute with
+these decorations, or do some decorations require refined
+composition laws?
+
+*Evidence.* `EchoGraded.degrade-comp` is the first hint of a
+graded-composition law. Linear echoes via `EchoLinear.weaken` behave
+by weakening along mode transitions. No systematic cross-check
+between these decorations has been attempted.
+
+### Q6. Composition in the presence of recovery / echo-erasure
+
+*Question.* When a downstream stage "uses" the echo — extracts the
+preimage `x` and re-applies `f` to reconstruct `y` — the echo is
+temporarily made definite. Does the composition law respect this
+extraction?
+
+*Formalisation hint.* Probably expressible as a 2-cell in the
+hypothetical 2-category of Q1. Not attempted.
+
+---
+
+## Composition laws — a compact statement
+
+Collecting the above:
+
+1. **(Agda-backed) Base accumulation iso.**
+   `Echo(g ∘ f) y ≃ Σ B (λ b → Echo(f) b × (g b ≡ y))`.
+
+2. **(Agda-backed) Functorial action.** `map-over` respects
+   composition: `map-over (g' , c₁) ∘ map-over (f' , c₂) ≡ map-over
+   ((g' ∘ f') , coherence)`. Proved in `Echo.map-over-comp`.
+
+3. **(Expected) Cancellation.** `Echo(g ∘ f) y ≃ Echo(f) (s y)` when
+   `g` has a section `s`.
+
+4. **(Open) Pentagon.** Three-fold composition associates.
+
+5. **(Open) Tolerance calculus.** For approximate echoes, tolerances
+   compose with a Lipschitz-like law.
+
+6. **(Open) Decoration commuting.** Role, grade, linearity, and
+   modal decorations commute with composition under conditions to be
+   identified.
+
+---
+
+## What to formalise next
+
+Ranked by unblock-value:
+
+1. **Base accumulation iso.** Single Agda proof, no new modules
+   needed. Would add `Echo-comp-iso` to `Echo.agda`.
+2. **Cancellation corollary.** One paragraph on top of (1).
+3. **Pentagon coherence.** Moderate proof, probably one more lemma.
+4. **Approximate-echo skeleton.** New module
+   `EchoApprox.agda` defining ε-echoes and restating (1) in the
+   approximate setting. This is where axis 2 of the taxonomy gets
+   teeth.
+5. **Decoration commuting.** Per-decoration lemmas in the existing
+   `EchoGraded`, `EchoLinear`, `EchoIndexed` modules.
+
+None of these depend on the blocked Buchholz-WF / shared-binder
+work. All are Sonnet-class proofs; (4) is Opus 4.7 design and
+Sonnet execution.