Foundation Entanglement

Ledger Entanglement

Foundation entanglement is the structural consequence of the Recognition Composition Law (RCL) for pairs of entries in the ledger. The RCL states that for any two positive real costs a and b, the expression J(a·b) + J(a/b) equals 2·J(a)·J(b) + 2·J(a) + 2·J(b). When two entries are created by the same recognition event, their costs must satisfy this identity. That shared constraint, not any physical connection, is what entanglement means in the ledger framework.

The constraint is non-factorizable. The joint cost J(a·b) + J(a/b) cannot be written as f(a) + g(b) for any functions f and g. This is established as the theorem no_local_decomposition. Because the joint cost cannot be decomposed, no local hidden variable model can reproduce the correlations. The non-locality comes from the global nature of the variational update, which selects the minimum-defect configuration at each tick.

Entanglement persists because breaking the constraint would increase defect. The theorem coherence_preserved_by_dynamics proves that a variational step never increases total defect. Spatial separation is irrelevant to the constraint, which is algebraic, not geometric. The theorem separation_irrelevant proves that coherence holds regardless of the lattice distance between the two entries. Entanglement ends only when a recognition event with the environment provides enough budget to absorb the defect increase, a process called decoherence.

The module also proves that the entanglement excess, the difference between joint defect and independent defect, is positive for any pair whose entries are not equal to 1. This excess is exactly the cross term 2·J(a)·J(b). The theorem entangled_not_independent states that joint defect never equals independent defect for non-trivial entries. Finally, measurement_constrains_partner shows that measuring one entry of a pair with known product determines the other entry's value and cost, a formal statement of correlation without signaling.

THEOREM no_local_decomposition · IndisputableMonolith/Foundation/Entanglement.lean

THEOREM coherence_preserved_by_dynamics · IndisputableMonolith/Foundation/Entanglement.lean

THEOREM separation_irrelevant · IndisputableMonolith/Foundation/Entanglement.lean

THEOREM entangled_not_independent · IndisputableMonolith/Foundation/Entanglement.lean

THEOREM measurement_constrains_partner · IndisputableMonolith/Foundation/Entanglement.lean

What this page does not claim

This page does not claim that entanglement is a physical force or a geometric connection. This page does not claim that the Bell inequality violation is experimentally confirmed in the ledger framework. This page does not claim that the decoherence process is formalized in the module.

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expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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