totalCost_card

This module weights the Phase-54 edge-count ledger by the recognition cost $J$. Every adjacency of the polarized birth field is either a monochromatic carried edge or a bichromatic interface edge. The charge is $\mathrm{sign}(x)\in{+1,0,-1}$, and an ordered edge costs $J(\varphi^{\mathrm{charge},p-\mathrm{charge},q})$.

Carried edges have equal charges, so they span zero phi-rungs and cost $J(1)=0$. Interface edges join the spine $x=0$ to a cell at $x=\pm 1$, span one rung, and cost $J(\varphi^{\pm 1})=J(\varphi)$ by reciprocal symmetry. Thus the whole field cost is the interface count times $J(\varphi)$.

Upstream, the local identity that total cost equals interface cost (carried bulk contributes nothing) and the exact interface formula $(8t-4),J(\varphi)$ for $t\ge 1$ are already proved. The 2D diamond closed form is the target here; the 3D octahedron analogue is stated separately in the module doc.

Term-mode proof by two rewrites. First apply the local fact that total cost equals interface cost (carried sum vanishes). Then apply the already-proved interface closed form, which multiplies the interface edge count $8t-4$ by $J(\varphi)$. No further algebra.

This is the exact cost-unit statement of the Phase-55 2D headline: carry the bulk free, pay only for the codimension-1 interface. It is the middle conjunct of the bundled ledger theorem (carried cost zero, total equals $(8t-4),J(\varphi)$, total equals interface, and $J(\varphi)>0$).

Downstream it feeds the real-multiplication form $(8t-4)\cdot J(\varphi)$, and the graded-rung recovery that Phase 55 is the binary special case of the general graded cost law. The module frames this as the compute-watch law: cost scales with recognition activity on the interface, not bulk volume. $J$ is the T5-unique cost; $\varphi$ is the T6 fixed point. The linear-in-$t$ growth underpins the run-total $\Theta(T)$ cost of growing the world.