Jcost_phiLadder_symm
plain-language theorem explainer
Physicists resolving the Yang-Mills mass gap in Recognition Science cite this result to establish that J-cost is identical for phi-ladder excitations at rung n and -n. The symmetry follows from the inversion invariance built into the recognition cost functional. The proof is a one-line wrapper that substitutes the ladder definition and applies the Jcost symmetry lemma to a positive power of phi.
Claim. $J(x) = J(x^{-1})$ for every $x = phi^n$ with $n$ an integer, where $J(y) = 1/2 (y + y^{-1}) - 1$ and $phi$ denotes the golden ratio.
background
The module derives the Yang-Mills mass gap from the J-cost functional on the golden-ratio lattice. J-cost is defined by $J(x) = 1/2 (x + x^{-1}) - 1$ and satisfies the Recognition Composition Law. PhiLadder n is the element $phi^n$ for integer n, forming the discrete substrate forced by the T5-T6 chain. The upstream Jcost_symm lemma states that J(x) equals J(x^{-1}) whenever x > 0, proved by direct expansion and field simplification.
proof idea
The proof rewrites both sides via the PhiLadder definition and the zpow_neg lemma, reducing the claim to J(phi^n) = J(phi^{-n}). It then applies the Jcost_symm lemma directly to the positive quantity phi^n, which is supplied by zpow_pos on Constants.phi_pos.
why it matters
This symmetry is used by the downstream spectral_gap theorem to treat positive and negative rungs uniformly when proving that every nonzero excitation costs at least Delta = J(phi). It fills the symmetry step required by the module's central claim that the gap is topological and exact on the phi-lattice. The result closes one link in the T5-forced uniqueness of J to the spectral gap without additional hypotheses.
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