jcost_positive_of_nonzero

The module derives the generation torsion set ${0,11,17}$ as the unique values compatible with the 8-tick Hamiltonian cycle on the cube $Q_3$, projected onto the $\varphi$-ladder through the Recognition Composition Law. In log-coordinates RCL becomes the d'Alembert equation whose solution is $J=\cosh-1$, so independent couplings add in the $\varphi$-exponent.

The cost functional is $J(x)=(x+x^{-1})/2-1$. Upstream, Jcost_pos_of_ne_one states that $J(x)>0$ whenever $x>0$ and $x\neq 1$. The golden ratio $\varphi>1$ has $\varphi^n=1$ if and only if $n=0$, so every nonzero integer rung is a genuine distinction.

Module part (D) records the variational fact that ground-state torsion is forced to zero; positivity of $J$ on nonzero rungs is the cost side of that claim.

One-line term proof. Apply the upstream lemma that $J(x)>0$ for $x>0$, $x\neq 1$, at $x=\varphi^n$. Positivity of the power follows from $\varphi>0$. The remaining obligation $\varphi^n\neq 1$ is discharged by the characterization $\varphi^n=1\Leftrightarrow n=0$ together with the hypothesis $n\neq 0$.

Feeds forced_jcost_ordering, which asserts $J(\varphi^0)=0$, $0<J(\varphi^{11})$, and $J(\varphi^{11})<J(\varphi^{17})$. That ordering is the cost ranking of the forced generation torsions ${0,11,17}$ produced by the 8-tick cycle $\times$ RCL $\times$ $\varphi$-ladder chain.

In the module narrative this is step (D): any nonzero torsion has positive $J$-cost, so the variational ground state is uniquely costless. It sits under T5 ($J$-uniqueness) and T6 ($\varphi$ as self-similar fixed point), and under the eight-tick octave (T7) that partitions $Q_3$ into the passive subcell counts feeding the torsion values. Without this positivity lemma the strict hierarchy among generations would not be cost-justified.