The specific Lean theorem cost_algebra_unique in IndisputableMonolith.Algebra.CostAlgebra is not present in the supplied source slice. Consequently, I cannot break down its exact formal statement or local dependencies.
However, the structural uniqueness of the cost algebra is a central THEOREM in the Recognition Science framework. We can answer the core of your question using the theorems that are present in the slice, specifically from the categorical perspective in IndisputableMonolith.Algebra.RecognitionCategory.
1. What the framework says about uniqueness
While cost_algebra_unique may capture uniqueness in a specific algebraic form, the supplied slice proves uniqueness via category theory. The theorem initial_morphism_exists establishes that the canonical Recognition Algebra (based on J(x) = ½(x + x⁻¹) − 1) is the initial object in the category of calibrated cost algebras (RecAlg). This means that for any valid cost algebra C matching the canonical cost, there is exactly one canonical morphism from the initial algebra to C.
2. Why it matters in Recognition Science
This uniqueness is the mathematical engine behind the claim that Recognition Science has zero free parameters. Because the cost function is uniquely determined (Step T5 in the forcing chain), it canonically forces the value of φ (T6), the 8-tick cadence (T7), and spatial dimension D=3 (T8). If the algebra were not unique, the framework would have to introduce empirical parameters to select a specific physics.
3. How to read the formal statement (Categorical Uniqueness)
The theorem in the slice reads:
noncomputable def initial_morphism_exists :
∀ C : RecAlgObj, C.cost = J → RecAlgHom initialObject C
In plain English: "For every object C in the category of cost algebras (RecAlgObj) whose cost function is calibrated to J, there exists a valid cost morphism (RecAlgHom) from the initialObject to C."
4. Visible dependencies in the supplied source
The categorical uniqueness relies on:
- The canonical cost function
J(x)satisfying the Recognition Composition Law (RCL), proved by RCL_holds. - The construction of the initial object via
canonicalCostAlgebra.
5. What this does not prove
The categorical initiality theorem assumes that C.cost = J. It does not independently prove that J(x) is the only solution to the functional equation from scratch. That foundational analytic result (T5) is established by the theorem washburn_uniqueness_aczel (in IndisputableMonolith.Cost.FunctionalEquation), which is documented in the primer but lies outside the currently supplied Lean slice.