IndisputableMonolith.Physics.RecognitionCompositionLawCert
The module certifies algebraic properties of the Recognition Composition Law for the J-cost function, anchored by the normalization J(1) = 0. Researchers deriving the phi-ladder or mass formulas in Recognition Science cite these certificates to confirm the functional equation holds under the standard definition of J. The module organizes its content as a collection of sibling lemmas that verify normalization, symmetry, and positivity via direct algebraic checks imported from the Cost module.
claimThe normalization condition states that $J(1) = 0$, where the J-cost satisfies the Recognition Composition Law $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$ for $x, y > 0$.
background
Recognition Science builds all physics from the single functional equation satisfied by the J-cost, defined in the upstream Cost module as $J(x) = (x + x^{-1})/2 - 1$. The Recognition Composition Law (RCL) encodes the composition rule for this cost under multiplication and division. The present module supplies certificates for the basic properties required before the forcing chain proceeds to J-uniqueness and the self-similar fixed point phi.
proof idea
This is a definition module, no proofs. It collects sibling declarations (rcl_normalised, rcl_symmetric, rcl_positive, jcost_uniqueness_axioms, RCLCert, rclCert) that each record one algebraic consequence of the RCL under the normalization J(1) = 0.
why it matters in Recognition Science
The certificates supply the normalization step that precedes T5 (J-uniqueness) and T6 (phi as fixed point) in the UnifiedForcingChain. Downstream results that construct the eight-tick octave and D = 3 rely on these properties to guarantee the RCL remains consistent when the yardstick and phi-ladder are introduced.
scope and limits
- Does not derive the RCL equation from more primitive axioms.
- Does not compute numerical values for physical constants such as alpha or G.
- Does not address the Berry creation threshold or Z_cf values.
- Does not extend the RCL to non-positive arguments.