Cost Ndim Radical Distribution Affine Shift Mem Level Set

The affine shift theorem

In the Recognition Science framework, a cost function assigns a number to each possible state of a system, and the framework studies how that number changes as the state moves. The declaration affineShift_mem_LevelSet concerns a special kind of motion called an affine shift, which moves a state t along a straight line in a direction v by an amount s. The theorem states that if the starting state lies on a level set, meaning a set of states where the cost is constant, and if the direction v is in the radical distribution, meaning that moving along v does not change the cost to first order, then every point along the entire affine shift also lies on that same level set. In plainer terms: if a small step in a direction leaves the cost unchanged, then any size step in that same direction leaves the cost unchanged, provided you start on a level set.

The proof is a direct calculation. The framework defines the level set as the set of vectors t satisfying dot α t = c, where α is a fixed vector and c is a constant. The radical distribution is defined as the set of vectors v with dot α v = 0. The theorem dot_affineShift shows that the dot product of α with an affine shift affineShift t v s equals dot α t + s * dot α v. Combining this with the assumptions dot α t = c and dot α v = 0 gives dot α (affineShift t v s) = c, which is exactly the membership condition for the level set. The calculation is a few lines of algebra, and the framework's machine-checked library of formal theorems verifies every step.

The theorem does not claim that the radical distribution is the only way to stay on a level set, nor that every level set is connected or path-connected. It also does not claim anything about the global structure of the cost function beyond the level sets themselves. The theorem is purely local and algebraic: it describes how affine shifts behave with respect to a single linear constraint. It says nothing about whether the radical distribution is integrable in a differential-geometric sense, though a related theorem in the same file, radical_integrable_by_affine_leaves, does establish that the affine leaves foliate the space. The theorem also does not claim that the cost function is constant on the level set, only that the affine shift stays within it.

The practical consequence is that the framework can identify directions of symmetry for a cost function by checking a single linear condition. If a direction is in the radical distribution, then moving along it never changes the cost, regardless of the distance. This is a useful tool for simplifying the analysis of cost functions in high-dimensional spaces, where checking every possible motion would be infeasible. The theorem gives a clean criterion for when a whole family of motions is cost-preserving.

/-- Directions in the radical stay inside the affine leaves `dot α = c`. -/
theorem affineShift_mem_LevelSet {n : ℕ} (α : Vec n) {c s : ℝ} {t v : Vec n}
    (ht : t ∈ LevelSet α c) (hv : v ∈ Radical α) :
    affineShift t v s ∈ LevelSet α c := by
  rw [mem_LevelSet_iff] at ht ⊢
  have hv' : dot α v = 0 := hv
  rw [dot_affineShift, ht, hv']
  ring

/-- The radical distribution of the rank-one Hessian metric. -/
def Radical {n : ℕ} (α : Vec n) : Set (Vec n) :=
  { v | dot α v = 0 }

/-- The affine leaves orthogonal to the active direction. -/
def LevelSet {n : ℕ} (α : Vec n) (c : ℝ) : Set (Vec n) :=
  { t | dot α t = c }

What this page does not claim

The theorem does not claim that the radical distribution is the only way to stay on a level set. The theorem does not claim that every level set is connected or path-connected. The theorem does not claim that the cost function is constant on the level set.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cost/Ndim/RadicalDistribution.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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