sourceCost1

The module builds a convergent least-cost flow on hinge-link strain so the C2 bridge can adopt a sourced stationary carrier. The sourced action on a link splits by channel as $\Phi_c(t) = \sum_i(\cosh t_i - 1) - (c/n)\sum_i t_i$. Each summand is the scalar cost $\psi_a(s) = \cosh s - 1 - a, s$ with $a = c/n$.

The term $\cosh s - 1$ is the additive form of the Recognition J-cost: $J(x) = \cosh(\log x) - 1$ when $s = \log x$. The linear source $-a s$ tilts the well so the unique critical point sits at $s^* = \mathrm{arsinh}, a$. The residual (gradient) is $\sinh s - a$; the rest of the module runs a self-tuned gradient step on that residual and proves Lyapunov decrease and global convergence.

Pure definition: the map is introduced by the closed-form expression $\mathrm{fun}, s \mapsto \cosh s - 1 - a\cdot s$. No lemmas, tactics, or reduction steps.

This scalar cost is the Lyapunov function for the entire strain-descent construction and the predicate in the universality theorems that discharge C2 stationarity. Downstream, cost_decreasing_dynamics_converges and its minimal form state that any continuous dynamics that strictly lowers $\psi_a$ off $\mathrm{arsinh}, a$ has every orbit converging to that point; the residue theorem packages continuity, rest, and never-gaining-cost as the exact remaining adoption. Related results identify the derivative with the residual, prove weighted descent fields never raise cost, and show continuity is load-bearing. In the RS chain this supplies the missing dynamical half of hinge stationarity after the eight-tick update wall rules out non-constant descent on the canonical tick image.