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54 lines
2 KiB
Text
54 lines
2 KiB
Text
import Semantics.Bind
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import Semantics.FixedPoint
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namespace Semantics
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/-- Represents the local state of an atom within the 14-axis AMMR manifold. -/
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structure AtomicState where
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manifold : Array Q16_16
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entropy : Q16_16
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phiGate : Q16_16
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deriving Repr, Inhabited
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/-- The Landauer Limit threshold for thermodynamic stability. -/
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def landauerThreshold : Q16_16 := Q16_16.ofInt 10
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/-- The Golden Ratio threshold for phase-gating (approx 0.618 * 65536). -/
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def goldenRatio : Q16_16 := ⟨40501⟩
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/-- Determines if the state is within the physically stable bounds. -/
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def isStable (s : AtomicState) : Bool :=
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Q16_16.le s.entropy landauerThreshold && Q16_16.ge s.phiGate goldenRatio
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/-- The type-level invariant for the atom. Unstable atoms map to a drift state. -/
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def atomicInvariant (s : AtomicState) : String :=
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if isStable s then "crystalline_resonance" else "dissipative_drift"
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/--
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The geometric cost of binding two atoms.
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Uses the Q16.16 scalar cost from the metric.
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-/
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def interatomicCost (_a _b : AtomicState) (g : Metric) : UInt32 :=
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g.cost
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/--
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The primary Interatomic Potential Bind.
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Replaces the ML "soft" equivariance with a hard topological resonance bind.
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-/
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def interatomicBind (a b : AtomicState) (g : Metric) : Bind AtomicState AtomicState :=
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geometricBind a b g interatomicCost atomicInvariant atomicInvariant
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/--
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THEOREM: Hardware-Native Stability.
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Proves that if two atoms are independently stable within the Landauer limit
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and the Golden Ratio phase-gate, their geometric bind is universally lawful.
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This formally verifies that the manifold will not experience "ML drift"
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as long as the SNN hardware enforces the `isStable` bounds.
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-/
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theorem lawful_resonance_of_stable_atoms (a b : AtomicState) (g : Metric)
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(hA : isStable a = true) (hB : isStable b = true) :
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(interatomicBind a b g).lawful = true := by
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dsimp [interatomicBind, geometricBind, informationalBind, thermodynamicBind, physicalBind, controlBind, bind, atomicInvariant]
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simp [hA, hB]
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end Semantics
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