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69 lines
2.1 KiB
Text
69 lines
2.1 KiB
Text
/-
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MasterEquation.lean - Anisotropically Frustrated Torsional Gradient Flow
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This module formalizes the "Minimal Compact System" (Section 7) as the
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authoritative governing evolution for the Sovereign Informatic Manifold.
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Equations (Discrete Time):
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1. Phase Flow: ϕ_{t+1} = ϕ_t + Δt [ ∇_i(M^ij ∇_j μ) - σ (∂I_lock/∂ϕ) ]
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2. Local Potential: μ = δF/δϕ
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3. Embedding Flow: X^A_{t+1} = X^A_t + Δt [ -Λ^AB(X^B - X_0^B) - δF/δX^A + τ T^A ]
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One-line interpretation: The fabric folds back into n-space, snagging on
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anisotropic frustration, storing stress as torsional geometry.
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-/
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import Semantics.DynamicCanal
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import Semantics.BraidBracket
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import Semantics.ManifoldFlow
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namespace Semantics.MasterEquation
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open DynamicCanal
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open Semantics.BraidBracket
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open Semantics.ManifoldFlow
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-- =============================================================================
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-- THE MINIMAL COMPACT SYSTEM (Section 7)
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-- =============================================================================
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/--
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Executes one step of the "n-space foldback-lock" equation.
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Encapsulates the hyperfluid phase evolution and embedding dynamics.
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-/
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def foldbackLockStep
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(p : ManifoldPoint)
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(dt : Fix16)
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(prevX : PhaseVec)
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: ManifoldPoint :=
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let nextPhi := flowPhi p dt
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let nextX := flowEmbedding p dt prevX
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{ p with phi := nextPhi, x_pos := nextX }
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/--
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Master Equation: Recursive Manifold Evolution
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Σ̂_{t+1} = FoldbackLockStep(Σ_t, Δt, Σ_{t-1})
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-/
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def masterEquation
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(curr : ManifoldPoint)
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(prev : ManifoldPoint)
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(dt : Fix16)
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: ManifoldPoint :=
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foldbackLockStep curr dt prev.x_pos
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-- =============================================================================
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-- LEGACY MAPPING (Braid Compatibility)
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-- =============================================================================
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-- Keeping these as shims for the SLUG-3 decoder which operates on segments of
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-- the manifold generated by these flows.
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structure CMYK where
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c : Fix16
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m : Fix16
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y : Fix16
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k : Fix16
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deriving Repr, DecidableEq, BEq
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def CMYK.zero : CMYK := ⟨Fix16.zero, Fix16.zero, Fix16.zero, Fix16.zero⟩
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end Semantics.MasterEquation
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