rcl_identity
The Recognition Composition Law equates the J-cost of a product and quotient to an expression involving the product and sum of individual J-costs for nonzero reals. Foundation paper F1 researchers cite it to establish geometric-mean minimization and simultaneous versus sequential descent in log-domain geometry. The proof is a direct algebraic reduction: unfold the Jcost definition, clear denominators with field_simp, and verify the polynomial identity by ring.
claimFor all nonzero real numbers $x$ and $y$, $J(xy) + J(x/y) = 2 J(x) J(y) + 2 J(x) + 2 J(y)$, where $J(z) := ½(z + z^{-1}) - 1$.
background
The module defines the canonical reciprocal cost $J(x) = ½(x + x^{-1}) - 1$ in the log-domain geometry of foundation paper F1. This identity is the Recognition Composition Law (RCL) listed as F1.5.1. It extends core J-cost identities from JcostCore and supports results on geometric-mean optimality and log-domain descent. The local setting is the foundation paper F1, which derives physics from this functional equation and cites the law for applications in NS, P vs NP, Yang-Mills, and RH.
proof idea
The proof introduces auxiliary facts that the product and quotient are nonzero, then unfolds the definition of Jcost. It applies field_simp to clear the denominators under the given nonzeroness hypotheses, and concludes with the ring tactic to verify the resulting polynomial equality.
why it matters in Recognition Science
This theorem is the core Recognition Composition Law (RCL) in the F1 foundation paper, directly supporting geometric-mean minimization of total bond cost and the distinction between simultaneous and sequential descent. It aligns with the Recognition Science framework's use of the RCL to derive the phi-ladder and eight-tick octave structure. No open questions are touched here as the identity is fully proved.
scope and limits
- Does not extend the identity to zero or infinite values.
- Does not address complex arguments or higher-dimensional generalizations.
- Does not derive the J-cost definition itself.
formal statement (Lean)
190theorem rcl_identity {x y : ℝ} (hx : x ≠ 0) (hy : y ≠ 0) :
191 Jcost (x * y) + Jcost (x / y) = 2 * Jcost x * Jcost y + 2 * Jcost x + 2 * Jcost y := by
proof body
Term-mode proof.
192 have hxy : x * y ≠ 0 := mul_ne_zero hx hy
193 have hxdy : x / y ≠ 0 := div_ne_zero hx hy
194 simp only [Jcost]
195 field_simp [hx, hy, hxy]
196 ring
197
198/-- **F1.5.2**: The golden ratio -/