# Reproduction — Soft (tanh) actuator — smooth saturation, band-guarded

Source `examples/soft_actuator.eml` (sha256 `66d899cb3171f438`). Regenerate with `make demo`.

> A transcendental control kernel. tanh gives smooth saturation (no derivative kick at the edge); the clamp is the hard guard. The @verify contract PROVES the output stays in [-1,1]; the forward-error certificate BOUNDS how far the compiled float64 tanh drifts from exact (Certcom.pipeline_nested_std).

| stage | artifact | tier |
|---|---|---|
| emit — software (14) | `c`, `cpp`, `csharp`, `gdscript`, `go`, `java`, `javascript`, `kotlin`, `luau`, `matlab`, `python`, `rust`, `swift`, `wasm` | LOCAL |
| emit — gpu shader (5) | `glsl`, `glsl-es`, `hlsl`, `metal`, `wgsl` | LOCAL |
| emit — compiler IR (1) | `llvm` | LOCAL |
| emit — proof (3) | `coq`, `isabelle`, `lean` | LOCAL |
| emit — safety-critical (4) | `aadl`, `ada/spark`, `autosar`, `ros2` | LOCAL |
| emit — blockchain (2) | `solidity`, `zkproof` | LOCAL |
| **emit total** | **29 targets from one source** | LOCAL |
| proof | `soft_actuator_within_band` — ✓ clean (`proof/soft_actuator_within_band.axioms.txt`) | REPLAY (re-derive: TOOLCHAIN — Lean) |
| simulate | `sim/trace.csv`, `sim/soft_actuator.png` — soft-actuator output samples outside the band [-1,1] = 0 = 0 | LOCAL |
| certificate | fwd-error ≤ `1.11e-13` abs (`1001` ulp of result), all inputs in range — `certificate.json` | LOCAL |
| hardware | — | N/A |

**Forward-error certificate.** For every input in the declared ranges (`|error| ≤ 100`, `|gain| ≤ 10`), the COMPILED kernel's float64 result is within **`1.111e-13`** (absolute, ≈ `1001` ulp of the result) of the exact real value, assuming IEEE-754 binary64 round-to-nearest, unit roundoff u = 2^-53 ≈ 1.110e-16. This is not asserted — it is the `absErr` fold computed over the kernel's AST, and that fold is proved sound (`sorryAx`-free) by machlib's `Certcom.pipeline_nested_std`. Regime: absolute. The soundness is machine-checked (TOOLCHAIN); the per-kernel bound is computed here (LOCAL).

The harness also emits the **instantiated proof** `certificate/soft_actuator_forward_error.lean` — the kernel's arithmetic core as an `EML` term, its generated `IsArith` proof, and the `pipeline_arith` instantiation = a machine-checked per-kernel theorem `soft_actuator_forward_error : AbsEnc (absErr …) (emitted-C value) (exact)`, `sorryAx`-free, holding for any `FPBridge`. Re-check: `lake env lean` it against machlib (TOOLCHAIN).

**The same claim, two ways.** The Lean theorem `soft_actuator_within_band` proves the soft-actuator output is always within the band [-1, 1], for any input; the simulation shows `soft-actuator output samples outside the band [-1,1] = 0 = 0` (across a ±5 error sweep the tanh output saturates smoothly and stays in [-1.0,1.0] (peak |1.0|) — the band the theorem proves, the transcendental the forward-error certificate bounds); Proved, simulated.