Cost Ndim Radical Distribution Add Mem Radical

The radical subspace

In the Recognition Science framework, the cost function measures the price of recognizing one state from another. Near any point, the cost's curvature, captured by its Hessian, can be flat along certain directions. The radical distribution, a set of directions where the Hessian vanishes, is the collection of all such flat directions. The theorem add_mem_Radical states that if two directions are both flat, then their sum is also flat. This is the defining property of a subspace: it is closed under addition.

The proof is a direct calculation. The radical is defined as the set of vectors v where the dot product of v with a fixed vector α is zero. If two vectors v and w each satisfy this condition, then their sum does too, because the dot product distributes over addition. The theorem also holds for scaling and subtraction, making the radical a full linear subspace of the vector space. This is a standard result in linear algebra, but here it is formalized within the framework's machine-checked library of formal theorems.

The geometric picture is simple. The radical is the hyperplane perpendicular to α. The theorem says that this hyperplane is a subspace, which is always true. The framework's contribution is not the fact itself but its formalization: the theorem is proved in the Lean proof assistant, meaning the reasoning is verified by a computer. This is part of a larger effort to build a foundation for physics from a few axioms, where even elementary facts are checked mechanically.

What the theorem does not claim is more interesting than what it does. It does not say that the radical distribution is the only flat direction set, nor does it connect this subspace to the golden ratio or the forcing chain that derives physical constants. The theorem is a local, algebraic fact about a single cost function. It does not, by itself, determine the cost function's global behavior or its physical consequences. The radical being a subspace is a necessary condition for the cost framework to be consistent, but it is far from sufficient.

In the broader context of the framework, this theorem is a building block. It shows that the flat directions form a well-behaved structure, which is a prerequisite for the integrability results that follow. The affine leaves, the level sets of the dot product, are preserved by moving along the radical. This is a step toward understanding how the cost function's geometry constrains the physics it describes. The theorem is a small but necessary piece of the framework's logical foundation.

For a reader outside the framework, the takeaway is that even a simple fact like the closure of a subspace can be formalized and checked. The framework aims to build physics from the ground up, and this theorem is a tiny part of that foundation. It is not a claim about the physical world but about the internal consistency of a mathematical structure. The theorem's power lies not in its content but in its provenance: it is a proved theorem in a machine-checked library, not a conjecture or a heuristic.

theorem add_mem_Radical {n : ℕ} (α : Vec n) {v w : Vec n}
    (hv : v ∈ Radical α) (hw : w ∈ Radical α) :
    v + w ∈ Radical α := by
  unfold Radical dot at hv hw ⊢
  have hv0 : ∑ i : Fin n, α i * v i = 0 := hv
  have hw0 : ∑ i : Fin n, α i * w i = 0 := hw
  calc
    ∑ i : Fin n, α i * (v i + w i)
        = (∑ i : Fin n, α i * v i) + ∑ i : Fin n, α i * w i := by
            simp [mul_add, Finset.sum_add_distrib]
    _ = 0 := by rw [hv0, hw0]; 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 }

What this page does not claim

The theorem does not claim that the radical distribution is unique or that it determines the cost function globally. The theorem does not connect the radical subspace to the golden ratio or to the derivation of any physical constant. The theorem does not assert that the cost framework is physically correct; it only establishes a local algebraic property.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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