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122 lines
4.5 KiB
Text
122 lines
4.5 KiB
Text
/-
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Semantics/ManifoldFlow.lean - Anisotropically Frustrated Torsional Gradient Flow
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This module formalizes the "n-space foldback-lock" equation as the authoritative
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governing physics for the Sovereign Informatic Manifold.
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Governing Equation:
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∂_t ϕ = ∇_i(M^ij ∇_j δF/δϕ) - σ ∂ϕ/∂I_lock
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∂_t X^A = -Γ^A_BC ∂_i X^B ∂_i X^C - Λ^AB(X^B - X_0^B) - δF/δX^A + τ T^A
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Lean is the source of truth.
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-/
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import Semantics.DynamicCanal
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import Semantics.BraidBracket
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namespace Semantics.ManifoldFlow
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open DynamicCanal
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open Semantics.BraidBracket
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-- =============================================================================
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-- 1. TENSOR FIELDS (Q16.16)
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-- =============================================================================
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/-- Anisotropic Tensor A^ij -/
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structure AnisotropyTensor where
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xx : Fix16
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xy : Fix16
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yy : Fix16
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deriving Repr, DecidableEq, BEq
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/-- Metric Tensor g_ij -/
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structure MetricTensor where
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xx : Fix16
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xy : Fix16
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yy : Fix16
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deriving Repr, DecidableEq, BEq
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/-- Torsion Tensor T^k_ij (for 2D manifold, k=1,2) -/
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structure TorsionTensor where
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t1_12 : Fix16 -- T^1_{12}
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t2_12 : Fix16 -- T^2_{12}
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deriving Repr, DecidableEq, BEq
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-- =============================================================================
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-- 2. MANIFOLD STATE (Fabric + Hyperfluid)
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-- =============================================================================
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/-- Manifold State at coordinate x -/
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structure ManifoldPoint where
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phi : Fix16 -- Hyperfluid Phase Field
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x_pos : PhaseVec -- Embedding X^A in ambient 2-space
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x0_pos : PhaseVec -- Preferred "fold-back" location X_0^A
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g : MetricTensor
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t : TorsionTensor
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a : AnisotropyTensor
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deriving Repr, DecidableEq, BEq
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-- =============================================================================
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-- 3. ENERGY FUNCTIONAL (F)
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-- =============================================================================
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/-- Locking potential W(z; A) = w * (1 - cos(k * z)) approximation -/
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def lockingPotential (z : Fix16) (weight : Fix16) : Fix16 :=
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-- Periodic frustration: Using a simplified multiwell
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-- Fix16 approximation of (1 - cos(z))
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let z_mod := Fix16.mk (z.raw % 0x00010000) -- mod 1.0
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Fix16.mul weight (Fix16.mul z_mod (Fix16.sub Fix16.one z_mod))
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/-- Interlocking energy I_lock for recursive deposition -/
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def interlockingEnergy (x x_prev : PhaseVec) (a : AnisotropyTensor) : Fix16 :=
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let dx := Fix16.sub x.x x_prev.x
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let dy := Fix16.sub x.y x_prev.y
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-- Frustration modulated by anisotropy
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let frustration := Fix16.add (Fix16.mul a.xx dx) (Fix16.mul a.yy dy)
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lockingPotential frustration (Fix16.mk 0x00008000) -- weight 0.5
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/-- Torsional Stress Σ^ij(T) contribution -/
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def torsionalStress (t : TorsionTensor) : Fix16 :=
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-- χ * T^i_ab T^jab ... simplified to magnitude squared
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Fix16.add (Fix16.mul t.t1_12 t.t1_12) (Fix16.mul t.t2_12 t.t2_12)
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-- =============================================================================
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-- 4. EVOLUTION DYNAMICS (OISC Target)
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-- =============================================================================
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/-- Compute the next Phase Field state (ϕ_{t+1}) via gradient descent -/
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def stableDt (dt : Fix16) : Fix16 :=
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if dt.isNeg then Fix16.zero
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else if dt.raw > Fix16.one.raw then Fix16.one
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else dt
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/-- CFL-style stability guard for the evolution step size. -/
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def cflSatisfied (dt : Fix16) : Bool :=
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(stableDt dt).raw = dt.raw
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/-- Compute the next Phase Field state (ϕ_{t+1}) via gradient descent -/
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def flowPhi (p : ManifoldPoint) (dt : Fix16) : Fix16 :=
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let dt' := stableDt dt
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let gradient := Fix16.sub p.phi (Fix16.mk 0x00008000) -- simplified δF/δϕ
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-- ϕ' = ϕ - dt * (Mobility * gradient)
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Fix16.sub p.phi (Fix16.mul dt' gradient)
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/-- Compute the next Embedding state (X_{t+1}) via fold-back dynamics -/
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def flowEmbedding (p : ManifoldPoint) (dt : Fix16) (prevX : PhaseVec) : PhaseVec :=
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let dt' := stableDt dt
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-- Tendency to return to X0: Pull = -Λ(X - X0)
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let pullX := Fix16.mul (Fix16.mk 0x00004000) (Fix16.sub p.x_pos.x p.x0_pos.x)
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let pullY := Fix16.mul (Fix16.mk 0x00004000) (Fix16.sub p.x_pos.y p.x0_pos.y)
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-- Frustration from locking: snagging on previous pattern
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let snag := interlockingEnergy p.x_pos prevX p.a
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-- Torsional forcing: τ * T
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let forceX := Fix16.mul (Fix16.mk 0x00002000) p.t.t1_12
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let forceY := Fix16.mul (Fix16.mk 0x00002000) p.t.t2_12
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{ x := Fix16.sub p.x_pos.x (Fix16.mul dt' (Fix16.add (Fix16.add pullX snag) forceX))
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, y := Fix16.sub p.x_pos.y (Fix16.mul dt' (Fix16.add (Fix16.add pullY snag) forceY)) : PhaseVec }
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end Semantics.ManifoldFlow
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