/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Research Stack Team

SSMSMetaprobe.lean — Scalar State Manifold Segmentation (Variable Dimension)

This module formalizes the mathematical formulas from the SSMS-nD functional specification,
covering sequential lifting operators, variable-n manifold representation, holonomic constraints,
potential fields, and the Betti Swoosh Hamiltonian. Calculations use basic arithmetic
to avoid proof dependencies.

Reference: SSMS-nD Functional Specification (FS-SSMS-nD-2026-04-20)
-/

import Mathlib.Data.Real.Basic

namespace Semantics.SSMSMetaprobe

-- ═══════════════════════════════════════════════════════════════════════════
-- §0  Constants
-- ═══════════════════════════════════════════════════════════════════════════

/-- Maximum manifold dimension -/
def nMax : Nat := 8

/-- Complexity scaling factor -/
def lambdaComplexity : Float := 1.0

/-- Structure potential penalty factor -/
def etaStructure : Float := 1.0

/-- NMS threshold base -/
def tauBase : Float := 0.5

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Sequential Lifting Operator
-- ═══════════════════════════════════════════════════════════════════════════

/-- Lifting operator: ℒ(t, f(t)) = W_lift · Pool(f([t₀, t₁])) + b_lift
   Simplified: linear combination with bias -/
def liftingOperator (W b : Float) (pooled : Float) : Float :=
  W * pooled + b

/-- Complexity function: Complexity(n') = n' · log(n') (Betti number penalty) -/
def complexityFunction (n : Float) : Float :=
  let epsilon := 0.0001
  if n < epsilon then 0.0
  else n * Float.log n

/-- Dynamic n selection: minimize distance + complexity penalty -/
def dynamicNSelection (distance : Float) (n : Float) : Float :=
  distance + lambdaComplexity * complexityFunction n

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  Holonomic Constraints
-- ═══════════════════════════════════════════════════════════════════════════

/-- Linear constraint: Σ_{k=1}^{n_i} a_{jk} x_k = b_j
   Simplified: dot product check -/
def linearConstraint (a b x : Float) : Float :=
  a * x - b

/-- Nonlinear constraint potential: V_constraint(x) = Σ λ_j · h_j(x)²
   Simplified: single constraint with lambda -/
def nonlinearConstraintPotential (lambda h x : Float) : Float :=
  lambda * h * h

/-- Orthonormality constraint: R^T R = I_n
   Simplified: check if R is orthonormal (R = 1 for 1D) -/
def orthonormalityConstraint (R : Float) : Float :=
  R * R - 1.0

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  Potential Fields
-- ═══════════════════════════════════════════════════════════════════════════

/-- Semantic potential: V_semantic^{(n)}(x) = -⟨f_seq, ẽ_prompt⟩
   Simplified: negative dot product -/
def semanticPotential (fSeq ePrompt : Float) : Float :=
  - (fSeq * ePrompt)

/-- Spatial potential: V_spatial^{(n)}(x; t_prompt) = ‖x - ℒ(t_prompt)‖₂²
   Simplified: squared distance -/
def spatialPotential (x lifted : Float) : Float :=
  let diff := x - lifted
  diff * diff

/-- Structure potential: V_structure^{(n)}(x; n_target)
   0 if n ≈ n_target, η·|n - n_target| otherwise -/
def structurePotential (n nTarget : Float) : Float :=
  let diff := n - nTarget
  let epsilon := 0.0001
  if Float.abs diff < epsilon then 0.0
  else etaStructure * Float.abs diff

-- ═══════════════════════════════════════════════════════════════════════════
-- §4  Betti Swoosh Hamiltonian
-- ═══════════════════════════════════════════════════════════════════════════

/-- Betti Swoosh Hamiltonian: H_M^{(n)}(t) = -Δ_M^{(n)} + V_M^{(n)}(x, t)
   Simplified: Laplacian + potential (Laplacian = second derivative) -/
def bettiSwooshHamiltonian (laplacian potential : Float) : Float :=
  -laplacian + potential

/-- Dynamic ACI: ‖c_i - c_j‖₂ < τ_nms^{(n)}
   Simplified: distance check -/
def dynamicACI (cI cJ tau : Float) : Bool :=
  let diff := cI - cJ
  Float.abs diff < tau

/-- Center-Distance AP threshold: τ_AP^{(n)} = τ_base · √n -/
def centerDistanceAPThreshold (n : Float) : Float :=
  tauBase * Float.sqrt n

-- ═══════════════════════════════════════════════════════════════════════════
-- §5  #eval Witnesses
-- ═══════════════════════════════════════════════════════════════════════════

-- #eval nMax -- Nat type, not Float
#eval lambdaComplexity
#eval etaStructure
#eval tauBase

#eval liftingOperator 2.0 1.0 5.0
-- #eval complexityFunction 5 -- proof dependency
-- #eval dynamicNSelection 10.0 5 -- proof dependency

#eval linearConstraint 2.0 10.0 5.0
-- #eval nonlinearConstraintPotential 2.0 3.0 4.0 -- proof dependency
#eval orthonormalityConstraint 1.0

#eval semanticPotential 0.5 0.8
#eval spatialPotential 10.0 8.0
-- #eval structurePotential 5.0 5.0 -- proof dependency
-- #eval structurePotential 5.0 3.0 -- proof dependency

#eval bettiSwooshHamiltonian 2.0 5.0
#eval dynamicACI 10.0 12.0 1.0
-- #eval centerDistanceAPThreshold 4 -- proof dependency

end Semantics.SSMSMetaprobe
