jcost_ground

In Recognition Science the cost of a positive ratio is $J(x) = (x + x^{-1})/2 - 1$, equivalently $\cosh(\log x) - 1$. It is the unique continuous solution of the Recognition Composition Law forced at T5, and satisfies $J(1) = 0$ with $J(x) > 0$ for $x \neq 1$.

Masses live on the $\varphi$-ladder: a torsion integer $n$ is scored by $J(\varphi^n)$. The module derives the generation torsions ${0, 11, 17}$ as the unique values compatible with the 8-tick Hamiltonian cycle on $Q_3$, projected through RCL additivity of ladder exponents and the CW lower-set constraint on passive couplings.

Variational dynamics already forces the ground-state torsion to zero. The present identity records that this ground rung is costless under $J$, which is the base case of any strict cost ordering along the forced torsion set.

One-line simp proof. Rewrite $\varphi^{(0:\mathbb{Z})}$ by the integer power law $z\mapsto z^0 = 1$, then apply the unit identity $J(1) = 0$. No case splits or arithmetic beyond those two rewrites.

Supplies the zero conjunct of forced_jcost_ordering, which asserts $J(\varphi^0) = 0 < J(\varphi^{11}) < J(\varphi^{17})$. That ordering is the cost half of the torsion-forcing chain: once RCL makes ladder exponents additive, the 8-tick Gray cycle on $Q_3$ partitions passive cells, and CW attachment forces lower-set profiles, the only admissible generation torsions are ${0, 11, 17}$.

Module step (D) states that any nonzero torsion has positive $J$-cost, so the variational ground state is costless. This lemma is exactly that costless base. It sits downstream of T5 ($J$-uniqueness) and T6 ($\varphi$ as self-similar fixed point), and upstream of the mass-ladder yardstick comparisons that rank the three generations.