Cost Ndim Uniqueness Forced Of Factorization

The factorization lift

The theorem forced_of_factorization belongs to the Recognition Science framework, a research program that starts from the idea that reality keeps a ledger, a discrete record of recognition events, and that the cost of recognition is forced by simple conditions. The declaration is a formal statement in the framework's machine-checked library of formal theorems. It says, in plain language: if a cost function F on n positive numbers factors through a weighted aggregate, meaning F of the whole list equals some scalar function G of a single combined value, and if that scalar function G is already known to be uniquely the framework's cost function J on positive reals, then the whole n-dimensional function F is forced to be the framework's n-dimensional cost function JcostN.

Why should a stranger care? The theorem is a lifting result. It takes a uniqueness fact in one dimension and transfers it to many dimensions. The framework had already proved that any cost function on positive reals satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. The new theorem says: if a multi-component cost factors through a single aggregate, then the multi-component cost inherits that uniqueness. The proof is short. It uses the factorization to rewrite F of a vector as G of the aggregate, then applies the scalar uniqueness to replace G with J, and finally uses the definition of JcostN as J applied to the aggregate. The machine-checked proof is a chain of equalities, with no gaps.

The theorem does not claim that every cost function factors. Factorization is an assumption, not a consequence. The declaration states a conditional: if a candidate factors and the scalar part is unique, then the whole is forced. It also does not claim that the scalar uniqueness itself is proved here; that is a separate theorem. The n-dimensional cost JcostN is a definition in the library, and the theorem shows that any factorizing candidate must equal that definition. The theorem is a bridge from a one-dimensional uniqueness result to a many-dimensional one, and the bridge requires the factorization premise.

In the framework's own account, this lift is part of the chain that forces the golden ratio, the eight-tick cycle, and three spatial dimensions. But the theorem itself is narrower: it is a conditional statement about cost functions that factor. It does not, by itself, prove that any actual physical cost must factor. That would be an additional premise. The theorem's value is structural: it shows how uniqueness in the scalar case propagates to the vector case, given the right factorization condition. A reader who wants to know whether the framework's full chain holds must look at the other theorems in the library; this declaration alone does not carry the whole chain.

/-- Existential version of the forcing theorem. -/
theorem forced_of_factorization {n : ℕ}
    (F : Vec n → ℝ) (α : Vec n)
    (hfac : FactorsThrough F α)
    (hscalar_unique : ∀ G : ℝ → ℝ,
      (∀ x : Vec n, F x = G (aggregate α x)) →
      (∀ {u : ℝ}, 0 < u → G u = Jcost u)) :
    ∀ x : Vec n, F x = JcostN α x := by
  rcases hfac with ⟨G, hG⟩
  exact forced_of_scalar_uniqueness F α G hG (hscalar_unique G hG)

/-- Main forcing theorem: scalar uniqueness forces the `n`-dimensional lift. -/
theorem forced_of_scalar_uniqueness {n : ℕ}
    (F : Vec n → ℝ) (α : Vec n) (G : ℝ → ℝ)
    (hfactor : ∀ x : Vec n, F x = G (aggregate α x))
    (hscalar : ∀ {u : ℝ}, 0 < u → G u = Jcost u) :
    ∀ x : Vec n, F x = JcostN α x := by
  intro x
  calc
    F x = G (aggregate α x) := hfactor x
    _ = Jcost (aggregate α x) := hscalar (aggregate_pos α x)
    _ = JcostN α x := by simp [JcostN_eq_Jcost_aggregate]

/-- `F` factors through the weighted aggregate via some scalar profile `G`. -/
def FactorsThrough {n : ℕ} (F : Vec n → ℝ) (α : Vec n) : Prop :=
  ∃ G : ℝ → ℝ, ∀ x : Vec n, F x = G (aggregate α x)

What this page does not claim

The theorem does not prove that every cost function factors through an aggregate. The theorem does not prove the scalar uniqueness of J; it assumes it as a premise. The theorem does not, by itself, establish the framework's full chain to three 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/Uniqueness.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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