# Paper Stub: The ENE Bind Primitive

**Equation ID:** ENE-BIND-001  
**Status:** Formalized in Lean (70/81 theorems proven)  
**Lineage:** Root equation — all subsequent equations descend from this form  
**Date:** 2026-04-16  
**Ancestry:** First articulation of the `bind` concept as universal translator

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## The Equation

```
bind(A, B, g) = (cost, witness)

cost     = cost_fn(left, right, metric)
witness  = Witness.lawful(invA(left), invB(right))
lawful   = invA(left) = invB(right)
```

## Mutation History

| Variant | File | Mutation | Status |
|---------|------|----------|--------|
| ENE-BIND-001 | `docs/ENE_EQUATIONS.md` | Original form | ✅ Formalized |
| PHYSICAL-BIND | `Semantics/BindPhysics.lean` | Added conservation law checking | ✅ Proven |
| GEOMETRIC-BIND | `Semantics/BraidBracket.lean` | Braid isotopy equivalence | 8 `sorry` remain |
| INFORMATION-BIND | `Semantics/CrossModalCompression.lean` | Multi-modal cost function | Partial |
| THERMODYNAMIC-BIND | `Semantics/CompressionMechanics.lean` | Entropy + enthalpy terms | Stub only |

## Unfinished Thoughts

The original ENE formulation contains a "half-thought" in §Scalar Collapse:
> "ScalarAdmissible(sc) = sc.sourcePath.isLawful ∧ sc.sourceDecomposition.nonempty"

The decomposition check was meant to include weight verification but the `nonempty` predicate was never fully specified. This gap created the need for the BIND_BRIDGE hierarchy.

## Descendant Equations

1. **BIND_BRIDGE_EQUATIONS.md** — Floor 4 (BEDROCK) maps directly to this primitive
2. **MasterEquation.lean** — The 6-step pipeline collapses bind to MLGRU recurrence
3. **QuaternionGenomic.lean** — SLUG-3 gate is a ternary specialization of bind

## Next Steps

- [ ] Complete 8 remaining `sorry` theorems in `AVMR.lean`
- [ ] Extract cost function to hardware (LUT-as-DSP)
- [ ] Connect to DNA encoding equation (THE_EQUATION.md)

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*Part of the Equation Ancestry Project — see `EQUATION_PHYLOGENETIC_TREE.md` for full genealogy*