harmony_zero
plain-language theorem explainer
Recognition cost vanishes at the harmony point where the scaling ratio equals 1. Ethics researchers working in the Recognition Science program cite this zero baseline when verifying that symmetric actions leave the agent's ledger unchanged. The proof is a direct one-line application of the upstream unit lemma for the J-cost function.
Claim. At the harmony scaling ratio $r=1$, the recognition cost satisfies $J(1)=0$.
background
The module develops an ethical framework in which an action is ethical precisely when it does not increase the recognition cost J on the agent's ledger and avoids asymmetric costs on others. The J-cost takes the explicit form $J(x)=(x-1)^2/(2x)$, which is zero at x=1 by direct substitution. The upstream lemma establishes this vanishing by simplification of the cost expression.
proof idea
The proof is a one-line wrapper that invokes the upstream lemma establishing J(1)=0 by simplification of the J-cost definition.
why it matters
This supplies the harmony-zero field required by the EthicsCert structure, which also encodes the five ethical frameworks and the golden-rule symmetry J(r)=J(1/r). It anchors the structural claim in the Recognition Science ethics module that the golden rule follows from J-symmetry.
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