P_uniqueness
Any two real-valued functions P and Q that both satisfy the multiplicative consistency relation J(xy) + J(x/y) = P(J(x), J(y)) for all positive x and y must coincide on the closed first quadrant. Researchers deriving the Recognition Composition Law from first principles would cite this result to eliminate alternative functional forms. The proof applies the unconditional determination theorem to both P and Q, obtains the explicit bilinear expression for each, and equates the results by rewriting.
claimLet $J$ denote the J-cost function. Suppose $P, Q : [0,∞) × [0,∞) → ℝ$ satisfy $J(xy) + J(x/y) = P(J(x), J(y))$ and $J(xy) + J(x/y) = Q(J(x), J(y))$ for all $x,y > 0$. Then $P(u,v) = Q(u,v)$ for all $u,v ≥ 0$.
background
The J-cost function is defined as $J(x) = (x + x^{-1})/2 - 1$ for $x > 0$, extended by continuity to the non-negative reals. The module establishes unconditional inevitability of the Recognition Composition Law: once a function F satisfies symmetry, normalization at 1, calibration of its second derivative at 0, and smoothness, the functional equation forces F to equal J and simultaneously forces the associated P to equal the bilinear form 2uv + 2u + 2v on the first quadrant. The upstream theorem rcl_unconditional states that any P satisfying the consistency equation with J must equal this bilinear expression everywhere on [0,∞)².
proof idea
The term proof introduces the non-negative arguments u and v, applies rcl_unconditional to P under hypothesis hP to obtain the explicit bilinear value, applies the same theorem to Q under hQ, and rewrites both sides to conclude equality.
why it matters in Recognition Science
This declaration closes the uniqueness half of the unconditional RCL theorem, confirming that the bilinear expression is the only possible P compatible with J. It directly supports the module's claim that no assumption on P is required and that the composition law is forced rather than chosen. The result sits inside the forcing chain after T5 (J-uniqueness) and supplies the P side of the Recognition Composition Law used in later mass and dimension derivations.
scope and limits
- Does not address arguments outside the closed first quadrant.
- Does not assume or prove any regularity properties of P beyond the given functional equation.
- Does not extend the uniqueness statement to signed or complex domains.
- Does not supply an independent construction of P; it only equates any two solutions.
formal statement (Lean)
215theorem P_uniqueness (P Q : ℝ → ℝ → ℝ)
216 (hP : ∀ x y : ℝ, 0 < x → 0 < y →
217 Cost.Jcost (x * y) + Cost.Jcost (x / y) = P (Cost.Jcost x) (Cost.Jcost y))
218 (hQ : ∀ x y : ℝ, 0 < x → 0 < y →
219 Cost.Jcost (x * y) + Cost.Jcost (x / y) = Q (Cost.Jcost x) (Cost.Jcost y)) :
220 ∀ u v : ℝ, 0 ≤ u → 0 ≤ v → P u v = Q u v := by
proof body
Term-mode proof.
221 intro u v hu hv
222 have hP' := rcl_unconditional P hP u v hu hv
223 have hQ' := rcl_unconditional Q hQ u v hu hv
224 rw [hP', hQ']
225
226end Unconditional
227end DAlembert
228end Foundation
229end IndisputableMonolith