Research-Stack/6-Documentation/docs/papers/EQUATION_01_ENE_BIND.md

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