E5: make ν operational and add Buchholz example/smoke modules#6
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…#70) ## Summary Extends `proofs/agda/EchoApprox.agda` with the first slice of the axis-2 composition rung from the design note (drafted in `/tmp/echo-types-exploration/axis2-approximate.md`, §7 obligations 2 + 6, plus the canonical-split retract direction). Refs `docs/echo-types/roadmap.md` Axis 2 entry and `docs/echo-types/composition.md` §Q3 — does **not** close either. ## Obligations landed - **`echo-strict→approx`** (§7 #2). General strict ⇒ zero-tolerance approximate. Generalises `echo-approx-intro` from own-fibre points to arbitrary `y` via the codomain equation `p : f x ≡ y`. One extra `subst` along `p`. - **`echo-approx-comp-sound`** (§7 #6). Sound RHS-to-LHS direction of the retract shape from `composition.md` §Q3 / design-note §5. Unpacks the existential and calls `echo-approx-compose`. - **`echo-approx-comp-retract-to`**. Canonical-split LHS → RHS-Σ section of the retract: picks `b := f x`, `ε₁ := zero`, `ε₂ := ε`. Uses `echo-approx-intro` for the inner echo, original bound for the outer. - **`echo-approx-comp-retract-A`**. A-component round-trip `proj₁ ∘ sound ∘ retract-to ≡ proj₁`, proved by `refl`. The retraction direction holds definitionally on the A-component as the design note (§5) predicts. ## Obligations deferred (out of this rung) - §7 #7 separated zero-collapse — needs a separation predicate on the `PseudoMetric` record. - §7 #8 axis-1 shadow agreement — cross-axis classification. - Full retract B-component and tolerance-budget round-trip — needs a `+`-left-identity axiom on `Tolerance` (`zero + ε ≡ ε`) not in the current record. Adding it commits the carrier to a left-unital monoid; deferred as a separate decision. - Lipschitz generalisation (`L_g ≠ 1`) — needs scalar multiplication on `Tol`. ## Docs updated - `docs/echo-types/composition.md` §Q3 promoted from "entirely speculative" to landed-retract-shape with deferred items called out. - `docs/echo-types/roadmap.md` adds a "landed" entry for the composition rung first slice. ## Invariants preserved - `--safe --without-K` - No postulates - No escape pragmas - Funext untouched ## Test plan - [x] `LC_ALL=C.UTF-8 agda -i proofs/agda proofs/agda/All.agda` exits 0 - [x] `LC_ALL=C.UTF-8 agda -i proofs/agda proofs/agda/Smoke.agda` exits 0 - [x] No new `postulate`, no escape pragmas, funext not imported - [x] `Smoke.agda` already pins `module Approx` following project convention (per-lemma pins inside parameterised modules require instantiation, which the project does not do for `EchoApprox`). Co-authored-by: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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