posting_one
plain-language theorem explainer
The posting potential Π(x), defined as the shifted J-cost J(x) + 1 with J(x) = ½(x + x^{-1}) - 1, satisfies Π(1) = 1. This normalization anchors the multiplicative structure for scale composition derived from the Recognition Composition Law. Researchers deriving additive hierarchies from the forced RCL would cite the result as the base case. The proof is a direct unfolding of the definition followed by algebraic simplification at unity.
Claim. $Π(1) = 1$, where $Π(x) := J(x) + 1$ and $J(x) = ½(x + x^{-1}) - 1$.
background
The module derives additive scale composition from the Recognition Composition Law without assuming linearity a priori. The RCL states J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y) with J(x) = ½(x + x^{-1}) - 1. The posting potential is defined as Π(x) := J(x) + 1 = ½(x + x^{-1}), the shifted cost satisfying the d'Alembert equation for composing events at scales a and b.
proof idea
The proof is a one-line wrapper that unfolds the definition of PostingPotential as Jcost(x) + 1 and simplifies using the fact that the multiplicative inverse of 1 is 1.
why it matters
This normalization anchors the posting potential used to obtain multiplicative posting from the RCL and then additive scale composition via closure, closing Proposition 4.3 of the phi paper. It supports the chain from J-uniqueness through the eight-tick octave in the Recognition Science framework. The result enables later module theorems on integer coefficients from discreteness.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.