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141 lines
6.8 KiB
Text
141 lines
6.8 KiB
Text
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Research Stack Team
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SSMSMetaprobe.lean — Scalar State Manifold Segmentation (Variable Dimension)
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This module formalizes the mathematical formulas from the SSMS-nD functional specification,
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covering sequential lifting operators, variable-n manifold representation, holonomic constraints,
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potential fields, and the Betti Swoosh Hamiltonian. Calculations use basic arithmetic
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to avoid proof dependencies.
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Reference: SSMS-nD Functional Specification (FS-SSMS-nD-2026-04-20)
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-/
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import Mathlib.Data.Real.Basic
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namespace Semantics.SSMSMetaprobe
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 Constants
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Maximum manifold dimension -/
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def nMax : Nat := 8
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/-- Complexity scaling factor -/
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def lambdaComplexity : Float := 1.0
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/-- Structure potential penalty factor -/
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def etaStructure : Float := 1.0
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/-- NMS threshold base -/
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def tauBase : Float := 0.5
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Sequential Lifting Operator
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Lifting operator: ℒ(t, f(t)) = W_lift · Pool(f([t₀, t₁])) + b_lift
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Simplified: linear combination with bias -/
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def liftingOperator (W b : Float) (pooled : Float) : Float :=
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W * pooled + b
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/-- Complexity function: Complexity(n') = n' · log(n') (Betti number penalty) -/
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def complexityFunction (n : Float) : Float :=
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let epsilon := 0.0001
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if n < epsilon then 0.0
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else n * Float.log n
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/-- Dynamic n selection: minimize distance + complexity penalty -/
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def dynamicNSelection (distance : Float) (n : Float) : Float :=
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distance + lambdaComplexity * complexityFunction n
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Holonomic Constraints
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Linear constraint: Σ_{k=1}^{n_i} a_{jk} x_k = b_j
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Simplified: dot product check -/
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def linearConstraint (a b x : Float) : Float :=
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a * x - b
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/-- Nonlinear constraint potential: V_constraint(x) = Σ λ_j · h_j(x)²
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Simplified: single constraint with lambda -/
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def nonlinearConstraintPotential (lambda h x : Float) : Float :=
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lambda * h * h
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/-- Orthonormality constraint: R^T R = I_n
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Simplified: check if R is orthonormal (R = 1 for 1D) -/
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def orthonormalityConstraint (R : Float) : Float :=
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R * R - 1.0
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Potential Fields
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Semantic potential: V_semantic^{(n)}(x) = -⟨f_seq, ẽ_prompt⟩
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Simplified: negative dot product -/
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def semanticPotential (fSeq ePrompt : Float) : Float :=
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- (fSeq * ePrompt)
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/-- Spatial potential: V_spatial^{(n)}(x; t_prompt) = ‖x - ℒ(t_prompt)‖₂²
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Simplified: squared distance -/
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def spatialPotential (x lifted : Float) : Float :=
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let diff := x - lifted
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diff * diff
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/-- Structure potential: V_structure^{(n)}(x; n_target)
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0 if n ≈ n_target, η·|n - n_target| otherwise -/
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def structurePotential (n nTarget : Float) : Float :=
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let diff := n - nTarget
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let epsilon := 0.0001
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if Float.abs diff < epsilon then 0.0
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else etaStructure * Float.abs diff
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Betti Swoosh Hamiltonian
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Betti Swoosh Hamiltonian: H_M^{(n)}(t) = -Δ_M^{(n)} + V_M^{(n)}(x, t)
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Simplified: Laplacian + potential (Laplacian = second derivative) -/
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def bettiSwooshHamiltonian (laplacian potential : Float) : Float :=
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-laplacian + potential
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/-- Dynamic ACI: ‖c_i - c_j‖₂ < τ_nms^{(n)}
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Simplified: distance check -/
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def dynamicACI (cI cJ tau : Float) : Bool :=
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let diff := cI - cJ
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Float.abs diff < tau
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/-- Center-Distance AP threshold: τ_AP^{(n)} = τ_base · √n -/
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def centerDistanceAPThreshold (n : Float) : Float :=
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tauBase * Float.sqrt n
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 #eval Witnesses
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-- ═══════════════════════════════════════════════════════════════════════════
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-- #eval nMax -- Nat type, not Float
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#eval lambdaComplexity
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#eval etaStructure
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#eval tauBase
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#eval liftingOperator 2.0 1.0 5.0
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-- #eval complexityFunction 5 -- proof dependency
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-- #eval dynamicNSelection 10.0 5 -- proof dependency
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#eval linearConstraint 2.0 10.0 5.0
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-- #eval nonlinearConstraintPotential 2.0 3.0 4.0 -- proof dependency
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#eval orthonormalityConstraint 1.0
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#eval semanticPotential 0.5 0.8
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#eval spatialPotential 10.0 8.0
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-- #eval structurePotential 5.0 5.0 -- proof dependency
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-- #eval structurePotential 5.0 3.0 -- proof dependency
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#eval bettiSwooshHamiltonian 2.0 5.0
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#eval dynamicACI 10.0 12.0 1.0
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-- #eval centerDistanceAPThreshold 4 -- proof dependency
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end Semantics.SSMSMetaprobe
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