docs(wiki): replace stale betlang placeholder with an accurate page

The betlang language page was a thin placeholder describing a generic
sampling/inference PPL (bernoulli/sample/observe/infer). It omitted the
actual language: the ternary (bet A B C) lazy core, the probabilistic-CAS
framing, Echo structured-loss types (functor/comonad surface), the 14-system
uncertainty number tower, and the Lean 4 axiom-free metatheory. Rewritten
accurately in the umbrella house style, cross-linking betlang's in-repo wiki
and the upstream echo-types spine.

https://claude.ai/code/session_01QGi8GND5yNWgDyfReVEPYs

Author: claude

wiki/languages/betlang.md

@@ -4,60 +4,71 @@ Copyright (c) Jonathan D.A. Jewell <j.d.a.jewell@open.ac.uk>
 -->
 # betlang
 
-> Probabilistic programming language
-
-**Status:** 📝 Placeholder - contributions welcome
+**Tagline:** A symbolic probabilistic metalanguage — a *probabilistic computer algebra system*, not a betting language.
 
 ## Overview
 
-betlang is a probabilistic programming language built on Racket, enabling explicit modeling of uncertainty through sampling and inference.
-
-## Core Invariant
-
-**All uncertainty must be explicitly modeled—no hidden randomness.**
+BetLang is built on one idea: **computation is structured choice under uncertainty.**
+Its core primitive is the ternary form `(bet A B C)` — a probabilistic, **lazy** choice
+between three branches (only the selected branch is evaluated). This makes BetLang closer
+to a *probabilistic CAS* than a simulation DSL: it is symbolic-first, compositional, and
+treats uncertainty as a first-class **typed** object rather than as noise to average away.
 
-## Key Features
+## Philosophy
 
-- First-class probability distributions
-- Bayesian inference primitives
-- Conditioning and observation
-- Sampling and expectation
+- **Ternary over binary** — real-world decisions are rarely yes/no; the musical ternary
+  form (A–B–A) gives compositional structure, and Dutch-book coherence needs ≥3 outcomes.
+- **Uncertainty as structure** — not noise, but a first-class object the type system tracks.
+- **Symbolic first** — computation stays inspectable and composable.
 
-## Example
+## Core Primitive
 
 ```racket
 #lang betlang
 
-; Define a biased coin
-(define coin (bernoulli 0.7))
+(bet 'win 'draw 'lose)                       ; uniform ternary choice (lazy)
+(bet/weighted '(common 7) '(uncommon 2) '(rare 1))  ; non-uniform
+(bet/conditional pred A B C)                 ; predicate-driven
+(bet (expensive) (cheap-approx) (fallback))  ; only one branch is evaluated
+```
 
-; Model with observation
-(define (model)
-  (let ([heads (sample coin)])
-    (observe (= heads #t))  ; condition on heads
-    heads))
+## Type System
 
-; Run inference
-(define posterior (infer model 1000))
-```
+BetLang's checker (`bet-check`, Rust) is Hindley–Milner with two domain-specific pillars:
 
-## Use Cases
+- **Echo types** — *structured loss* as a typed object. When a `bet` collapses three
+  branches to one (or a `sample` marginalises a draw), `Echo T` retains a proof-relevant
+  residue of what was lost; `Echo T` is deliberately **distinct from `T`** (no implicit
+  forgetting). The operations form a functor + comonad surface (`echo`, `echo_output`,
+  `echo_map`, `echo_duplicate`, `echo_to_residue`, `sample_echo`), the ungraded ghost
+  shadow of the graded comonad proved in [`echo-types`](https://github.com/hyperpolymath/echo-types) (Agda).
+- **The uncertainty number tower** — 14 uncertainty-aware number systems (Gaussian,
+  interval/affine, fuzzy, Bayesian, VaR/CVaR, surreal, p-adic, imprecise, Dempster–Shafer,
+  …). These *are* the type system, not add-ons.
 
-- Bayesian modeling
-- Machine learning prototyping
-- Statistical inference
-- Uncertainty quantification
-- A/B testing analysis
+## Formal Verification
 
-## Getting Started
+The core calculus is mechanised in **Lean 4** (`proofs/BetLang.lean`, CI-checked):
+**Progress**, **Preservation**, and the **distribution monad laws** — with an
+**axiom-free core** (the `substTop_preserves_typing` axiom was discharged) and **zero
+`sorry`**. This is BetLang's differentiator versus untyped numeric/CAS tools (R, Octave,
+Scilab, Mathematica, Maple): *typed, proof-anchored uncertainty*.
 
-```bash
-cd betlang
-racket main.rkt
-```
+## Architecture
+
+Four layers with separated responsibilities: **Racket** (authoritative semantics) ·
+**Julia** (compute kernel / number tower) · **Lean 4** (proofs) · **Rust** (compiler
+tooling — `bet-core`, `bet-check`, `bet-parse`).
+
+## Status
+
+Active. Racket frontend authoritative; Lean proofs machine-checked (axiom-free core);
+Rust checker handles Echo formers **and** the typed echo operations; Julia backend in
+active development.
 
 ## See Also
 
-- [[Duet Language]] - AI-assisted verification
-- [[Ensemble Language]] - AI integration
-- [[NEUROSYM.scm]] - Neuro-symbolic patterns
+- BetLang repository wiki (in-repo `wiki/`): Home · Type-System · Echo-Types · Formal-Verification · Number-Tower
+- [[Ephapax]] — linear/affine once-only semantics (sibling language)
+- [[AffineScript]] — affine-typed primary application language
+- Upstream: [`echo-types`](https://github.com/hyperpolymath/echo-types) (Agda, source of truth) · [`EchoTypes.jl`](https://github.com/hyperpolymath/EchoTypes.jl) (executable shadow)