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75 lines
2.2 KiB
Text
75 lines
2.2 KiB
Text
/-
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MechanicalLogic.lean - Formalization of Merkle Molecular Mechanical Logic
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Implements the "Locks and Balances" primitive system.
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Ref: arXiv:1801.03534 & arXiv:2505.05693
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-/
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import Semantics.FixedPoint
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import Semantics.Bind
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namespace Semantics.MechanicalLogic
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open Q16_16
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/--
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Mechanical State: A link can be Displaced (1) or Neutral (0).
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In fixed-point, we map 1.0 to Displaced.
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-/
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structure LinkState where
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displacement : Q16_16
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deriving Repr, DecidableEq
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/--
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A Mechanical Lock: Restricts displacement based on a control link.
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Formalized as a conditional constraint.
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-/
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def mechanicalLock (control input : LinkState) : LinkState :=
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-- If control is displaced (>= 0.5), input displacement is blocked (forced to 0).
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if control.displacement.val >= 32768 then
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{ displacement := zero }
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else
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input
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/--
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A Mechanical Balance: A reversible gate equivalent to a Fredkin or Toffoli primitive.
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Sums displacements and outputs the residual.
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-/
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def mechanicalBalance (a b c : LinkState) : LinkState :=
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let total := add a.displacement (add b.displacement c.displacement)
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{ displacement := total }
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/--
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Energy Dissipation:
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The Landauer Limit at 300K is ~2.8e-21 Joules.
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Merkle 2025 claims 1e-24 Joules.
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In our Q16_16 model, we use a normalized 'Entropy Cost' where 1.0 = Landauer Limit.
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-/
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def landauerLimit : Q16_16 := one
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def merkleDissipation : Q16_16 := Q16_16.ofRawInt 65 -- ~0.001 * Landauer Limit (approx 10^-24 vs 10^-21)
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/--
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Verification: Is the operation 'Ultra-Efficient' (below Landauer)?
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-/
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def isUltraEfficient (cost : Q16_16) : Bool :=
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cost.val < landauerLimit.val
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/--
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Mechanical Logic Invariant:
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Conservation of mechanical work (simplified).
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-/
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def mechanicalInvariant (links : List LinkState) : String :=
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let sum := links.foldl (fun acc l => add acc l.displacement) zero
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s!"mech_work[{sum.val}]"
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/-- #eval Witnesses -/
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def neutral : LinkState := { displacement := zero }
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def displaced : LinkState := { displacement := one }
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-- Lock test: Blocked
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#eval (mechanicalLock displaced displaced).displacement.val
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-- Lock test: Clear
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#eval (mechanicalLock neutral displaced).displacement.val
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-- Efficiency check
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#eval isUltraEfficient merkleDissipation
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end Semantics.MechanicalLogic
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