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---
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```
____ ____ ____ ____ ____ _ _ ___ ____ ____ _ _ ____ ____
/ ___)( _ \\( _ \\( __)( \\( \\/ )/ __)( __)( _ \\( \\/ )( __)( _ \\
\\___ \\ ) / ) __/ ) _) ) D ( \\ / \\__ \\ ) _) ) __/ \\ / ) _) ) /
(____/(__\\_)(__) (____)(____/ \\/ (___/(____)(__) (__) (____)(__\\_)
A R R A Y L A N G U A G E · A R R A Y I α = I → α
```
**Array I α = I → α · broadcast = pullback π : J → I · pmap₂ = Π-map · no sorry remains**
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---
# Sovereign Array Language — Front-End
The **front-end** for the [Sovereign Array Language](../sovereign-array): an
interactive browser playground that runs the *same denotational semantics*
as the Lean 4 spec and the C++20 kernel — no Abjad, no digital root, no NP-magic.
> The denotational semantics of array computing *are* exactly a slice of
> dependent type theory. This front-end is the view layer over that substrate.
## What this repo is
| Layer | Repo | Role |
|-------|------|------|
| **Spec** | [`sovereign-array`](../sovereign-array) | Lean 4 — `Array I α = I → α`, zero-sorry proofs |
| **Kernel** | [`sovereign-array`](../sovereign-array) | C++20 — `Array<T>`, `pmap2`, `broadcast`, `softmax`, `nand_attention` |
| **Front-End** | **`sovereign-array-frontend`** (this repo) | Browser playground + usage guide |
## Quick Start
```bash
# Serve the playground (any static server)
cd sovereign-array-frontend
python -m http.server 8080
# open http://localhost:8080
```
No build step. Pure HTML/CSS/JS (ES modules).
## How to use the language
1. **Spec (Lean 4)** — define arrays as dependent functions `Fin n → α`;
prove `broadcast_is_pullback` and `softmax_is_pmap` with `lake build` (zero sorry).
2. **Kernel (C++20)** — `#include "sovereign_array.h"`; build with CMake;
run `sovarr_test` (11/11 checks).
3. **Front-end (this page)** — open `index.html`; the playground runs the
same denotational semantics in the browser.
4. **Compose** — chain `pmap₂` / `broadcast` / `softmax` / `nand_attention`;
fusion is Π-map fusion — no loop in the denotation.
## Usage Guide (SVG)
## Kernels demonstrated
| Kernel | Semantics | Status |
|--------|-----------|--------|
| `pmap₂` | Pointwise `Π`-map over index space `I` | ✅ |
| `broadcast` | Pullback along projection `π : J → I` | ✅ |
| `softmax` | `Π`-map normalization (shift-invariant) | ✅ |
| `nand` | Universal boolean gate | ✅ |
| `nand_attention` | NAND-extracted attention spec | ✅ |
## Layout
```
sovereign-array-frontend/
├── index.html # Playground page
├── css/style.css # Sovereign dark theme
├── js/
│ ├── array-lang.js # Browser reference impl (SOVArray, broadcast, softmax, nand)
│ └── app.js # Playground wiring
├── assets/
│ ├── logo.svg # Σ · I → α mark
│ └── usage.svg # SVG usage guide
└── README.md
```
## The forbidden list (fatal conflations we do NOT make)
- ❌ Proof `O(1)` substitution ⇒ `O(1)` decision procedure (NP stays hard)
- ❌ Abjad / digital root as universal arithmetic (quotients lose information)
- ❌ "Univalence replaces SIMD" (needs a compiler: Lean → C → LLVM → SIMD)
---
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**The substrate is always free. The array is a function.**
```
Array I α = I → α
broadcast = pullback π
pmap₂ = Π-map
no sorry remains.
```
*Sovereign Array Language · Front-End · 2026 · Ahmad Ali Parr*
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