J_eq_zero_iff
plain-language theorem explainer
J(x) vanishes exactly when x equals unity for positive reals. Researchers deriving the Law of Existence from the cost axioms cite this biconditional to identify the unique existent configuration. The proof is a direct algebraic reduction that rewrites J(x)=0 into (x-1)^2=0 via field simplification and linear arithmetic.
Claim. Let $J(x) := (x + x^{-1})/2 - 1$. For every real $x > 0$, $J(x) = 0$ if and only if $x = 1$.
background
The CostAxioms module encodes the three primitive axioms of Recognition Science: normalization F(1)=0, the Recognition Composition Law F(xy)+F(x/y)=2F(x)F(y)+2F(x)+2F(y), and calibration of the second logarithmic derivative at zero. J is introduced as the canonical cost functional J(x)=(x + x^{-1})/2 - 1 that satisfies these axioms uniquely. The local setting treats J as the measure of deviation cost from unity, with J(1)=0 corresponding to perfect balance and J approaching infinity as x approaches zero.
proof idea
The tactic proof opens the biconditional with constructor. The forward direction simplifies the hypothesis J x = 0, applies linarith to reach x + x^{-1} = 2, then uses field_simp and ring_nf to obtain (x-1)^2 = 0 before concluding x=1 via nlinarith. The reverse direction substitutes x=1 directly into the definition of J.
why it matters
This theorem supplies the existence criterion used by the downstream Law of Existence result, which states that x exists precisely when x=1. It occupies Level 2 in the axiomatic hierarchy and supports the meta-principle that nothing cannot recognize itself. The equivalence anchors the economic reading in which low-cost configurations (J near zero) correspond to logical coherence.
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