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52 lines
2 KiB
Markdown
52 lines
2 KiB
Markdown
# Paper Stub: The ENE Bind Primitive
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**Equation ID:** ENE-BIND-001
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**Status:** Formalized in Lean (70/81 theorems proven)
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**Lineage:** Root equation — all subsequent equations descend from this form
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**Date:** 2026-04-16
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**Ancestry:** First articulation of the `bind` concept as universal translator
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---
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## The Equation
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```
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bind(A, B, g) = (cost, witness)
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cost = cost_fn(left, right, metric)
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witness = Witness.lawful(invA(left), invB(right))
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lawful = invA(left) = invB(right)
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```
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## Mutation History
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| Variant | File | Mutation | Status |
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|---------|------|----------|--------|
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| ENE-BIND-001 | `docs/ENE_EQUATIONS.md` | Original form | ✅ Formalized |
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| PHYSICAL-BIND | `Semantics/BindPhysics.lean` | Added conservation law checking | ✅ Proven |
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| GEOMETRIC-BIND | `Semantics/BraidBracket.lean` | Braid isotopy equivalence | 8 `sorry` remain |
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| INFORMATION-BIND | `Semantics/CrossModalCompression.lean` | Multi-modal cost function | Partial |
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| THERMODYNAMIC-BIND | `Semantics/CompressionMechanics.lean` | Entropy + enthalpy terms | Stub only |
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## Unfinished Thoughts
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The original ENE formulation contains a "half-thought" in §Scalar Collapse:
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> "ScalarAdmissible(sc) = sc.sourcePath.isLawful ∧ sc.sourceDecomposition.nonempty"
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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.
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## Descendant Equations
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1. **BIND_BRIDGE_EQUATIONS.md** — Floor 4 (BEDROCK) maps directly to this primitive
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2. **MasterEquation.lean** — The 6-step pipeline collapses bind to MLGRU recurrence
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3. **QuaternionGenomic.lean** — SLUG-3 gate is a ternary specialization of bind
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## Next Steps
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- [ ] Complete 8 remaining `sorry` theorems in `AVMR.lean`
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- [ ] Extract cost function to hardware (LUT-as-DSP)
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- [ ] Connect to DNA encoding equation (THE_EQUATION.md)
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---
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*Part of the Equation Ancestry Project — see `EQUATION_PHYLOGENETIC_TREE.md` for full genealogy*
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