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83 lines
3 KiB
Text
83 lines
3 KiB
Text
import Semantics.FixedPoint
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import Semantics.ManifoldStructures
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namespace Semantics.VirtualWarpMetric
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open Semantics
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/--
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Virtual Warp Metric (Layer 7)
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dI² = -dτ² + (dH - v_eff * f * Ω * dτ)²
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This metric governs the information-theoretic displacement of entropy
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across the manifold's virtual surface.
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-/
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structure VirtualWarpParameters where
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kappa : Q16_16 -- κ : Steepness / coupling intensity
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opcodeEfficacy : Q16_16 -- Ω : Performance tier of the instruction
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localVelocity : Q16_16 -- v_local : Native hardware throughput
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coherence : Q16_16 -- φ : Phase coherence with past anchors
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properTime : Q16_16 -- dτ : Execution clock cycles
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entropyDisplacement : Q16_16 -- dH : Target entropy change
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deriving Repr, DecidableEq
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/--
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Effective compression velocity: v_eff = v_local / (1 - φ)
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Encapsulates the 'boost' achieved by high coherence.
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-/
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def effectiveVelocity (params : VirtualWarpParameters) : Q16_16 :=
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let one := Q16_16.one
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let denom := Q16_16.sub one params.coherence
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-- Avoid division by zero: clamp denominator to at least epsilon
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let safeDenom := if Q16_16.le denom Q16_16.zero then Q16_16.epsilon else denom
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Q16_16.div params.localVelocity safeDenom
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/--
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Warp coupling constant: f_warp = f(x_i) * Ω
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Typically derived from the SSS constant Φ_sss.
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-/
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def warpCoupling (params : VirtualWarpParameters) (sssConstant : Q16_16) : Q16_16 :=
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-- f = sigmoid(-κ * Φ_sss)
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-- Approximation: sigmoid(x) ≈ 0.5 + 0.5 * x for small x
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let inner := Q16_16.mul params.kappa sssConstant
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let ramp := Q16_16.sub (Q16_16.ofNat 1 / Q16_16.ofNat 2) inner
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Q16_16.mul ramp params.opcodeEfficacy
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/--
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Calculates the Virtual Warp Metric value: dI²
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-/
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def calculateVirtualWarpMetric (params : VirtualWarpParameters) (sssConstant : Q16_16) : Q16_16 :=
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let dt := params.properTime
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let dh := params.entropyDisplacement
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let veff := effectiveVelocity params
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let f_omega := warpCoupling params sssConstant
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-- dI² = (dH - v_eff * f * Ω * dτ)² - dτ²
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let velocityWork := Q16_16.mul veff (Q16_16.mul f_omega dt)
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let spaceTermInner := Q16_16.abs (Q16_16.sub dh velocityWork)
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let spaceTerm := Q16_16.mul spaceTermInner spaceTermInner
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let timeTerm := Q16_16.mul dt dt
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Q16_16.sub spaceTerm timeTerm
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/--
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Invariant: The metric is stable if the SSS constant and opcode efficacy
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yield a positive coupling (sustained bubble).
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-/
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def isVirtualWarpStable (params : VirtualWarpParameters) (sssConstant : Q16_16) : Bool :=
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Q16_16.ge (warpCoupling params sssConstant) Q16_16.zero
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-- =============================================================================
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-- VERIFICATION
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-- =============================================================================
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#eval calculateVirtualWarpMetric
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{ kappa := Q16_16.one
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, opcodeEfficacy := Q16_16.one
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, localVelocity := Q16_16.one
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, coherence := Q16_16.ofNat 0 -- 0 coherence
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, properTime := Q16_16.epsilon
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, entropyDisplacement := Q16_16.one
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} Q16_16.zero
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end Semantics.VirtualWarpMetric
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