Cost Ndim Radical Distribution Dot Affine Shift

A shift, a dot product, and a leaf

In linear algebra, a dot product measures how much one vector points along another. The declaration dot_affineShift is a theorem about what happens to that measure when you take a starting point, move it some distance along a fixed direction, and then compute the dot product with a fixed reference vector. It states a familiar linearity property: the dot product of the reference with the shifted point equals the dot product with the original point, plus the shift distance times the dot product of the reference with the direction. In symbols, for any vectors α, t, v and any real number s, the identity dot α (t + s v) = dot α t + s (dot α v) holds. This is a direct consequence of the definition of a dot product as a sum of products, and the framework's library proves it in a machine-checked way.

The setting is the framework's cost geometry, where a cost function assigns a number to each possible state, and its Hessian (the matrix of second derivatives) describes how the cost curves. For a rank-one Hessian, the curvature is concentrated along a single active direction α. The vectors v for which dot α v = 0 form the radical distribution, the set of directions along which the quadratic form vanishes. These directions are exactly the ones that leave the cost's level sets, the affine hyperplanes where dot α t = c, unchanged. The theorem dot_affineShift is the algebraic engine behind this picture: it shows that shifting along a radical direction keeps the dot product with α constant, so the shift stays inside the same level set. The library packages this as the theorem radical_integrable_by_affine_leaves, which states that the radical distribution is integrable, meaning its integral curves are precisely these flat affine leaves.

What the declaration does not claim is just as important. It does not say that the radical distribution is unique, that the active direction α is derived from anything, or that this linear algebra has any physical consequence by itself. The identity is a theorem about vectors and dot products, nothing more. The framework's larger claims, such as the forcing of three spatial dimensions, depend on additional structure and theorems not present in this declaration. The dot_affineShift theorem is a building block, not a conclusion.

/-- Weighted dot product along an affine shift. -/
theorem dot_affineShift {n : ℕ} (α t v : Vec n) (s : ℝ) :
    dot α (affineShift t v s) = dot α t + s * dot α v := by
  unfold dot affineShift
  calc
    ∑ i : Fin n, α i * (t i + s * v i)
        = ∑ i : Fin n, (α i * t i + s * (α i * v i)) := by
            apply Finset.sum_congr rfl
            intro i hi
            ring
    _ = (∑ i : Fin n, α i * t i) + s * ∑ i : Fin n, α i * v i := by
          rw [Finset.sum_add_distrib, Finset.mul_sum]

/-- 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 is integrable: its integral leaves are the affine
hyperplanes `dot α = c`. -/
theorem radical_integrable_by_affine_leaves {n : ℕ} (α : Vec n) (c : ℝ) :
    ∀ ⦃t v : Vec n⦄, t ∈ LevelSet α c → v ∈ Radical α →
      ∀ s : ℝ, affineShift t v s ∈ LevelSet α c := by
  intro t v ht hv s
  exact affineShift_mem_LevelSet α ht hv

What this page does not claim

The active direction α is not derived from any deeper principle in this declaration. The radical distribution is not claimed to be unique. This theorem alone does not imply any physical consequence about spatial dimensions.

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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