IndisputableMonolith.Physics.RecognitionCompositionLawCert
Certificate module packaging the Recognition Composition Law (RCL) for the J-cost: normalisation J(1)=0, symmetry, positivity, and the uniqueness axioms that force J(x)=(x+x^{-1})/2-1. Physicists citing the RCL identity or T5 J-uniqueness pull these lemmas. Structure is a thin cert bundle over Cost, with named props and a top-level RCLCert record.
claimThe module certifies the Recognition Composition Law for the cost $J$: $J(1)=0$, symmetry and positivity of $J$, and the uniqueness axioms implying $J(x)=\frac{x+x^{-1}}{2}-1$, packaged as an $RCLCert$ record.
background
Recognition Science derives physics from a single functional equation on a cost $J:(0,\infty)\to\mathbb{R}$. The Recognition Composition Law (RCL) is
$$J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y).$$
Together with normalisation $J(1)=0$, symmetry, and positivity, this forces the unique smooth solution $J(x)=\cosh(\log x)-1=(x+x^{-1})/2-1$ (forcing step T5). The Cost import supplies the raw $J$ and elementary identities; this module only packages the RCL-facing axioms and certificates used by physics layers.
proof idea
Definition and certificate module, not a single deep proof. It exposes named propositions (normalisation $J(1)=0$, symmetry, positivity, and the J-cost uniqueness axiom bundle) and assembles them into an RCLCert structure with a concrete rclCert instance. Arguments are thin wrappers or direct appeals to lemmas already proved in IndisputableMonolith.Cost.
why it matters in Recognition Science
RCL is the algebraic heart of the forcing chain: T5 J-uniqueness, then T6 $\phi$ as self-similar fixed point, T7 eight-tick octave, and T8 $D=3$. Downstream physics (mass ladder, $\alpha$ band, constants in RS units) assumes a certified J obeying RCL. This module is the physics-facing seal on that assumption; no further used_by edges are recorded here, so it sits as a leaf cert for importers that need an explicit RCLCert rather than raw Cost lemmas.
scope and limits
- Does not re-prove uniqueness of J from RCL; that lives in Cost / T5.
- Does not derive numerical constants (c, hbar, G, alpha) or the mass ladder.
- Does not assert existence of physical units or dimensions; only algebraic RCL facts.
- Does not contain sorry-free end-to-end physics theorems beyond the cert record.