costIncrement
plain-language theorem explainer
Advancing a polarized 2D diamond by one cadence cycle raises total recognition cost by exactly $8 J(\varphi)$, for every radius $t \ge 1$. Cosmologists tracking the compute-watch law cite this differential form: each shell posts a constant interface cost, independent of world size. The proof rewrites both totals via the closed form $(8t-4)J(\varphi)$ and cancels over $\mathbb{R}$.
Claim. For every natural number $t \ge 1$, the total recognition cost of the polarized diamond of radius $t+1$ minus that of radius $t$ equals $8 \cdot J(\varphi)$, where $J(x)=(x+x^{-1})/2-1$ is the recognition cost and $\varphi$ is the golden-ratio fixed point.
background
This module weights the Phase-54 edge ledger by actual recognition cost. Every adjacency of the polarized birth field is either monochromatic (carried) or bichromatic (interface). Cost of an ordered edge is $J(\varphi^{\mathrm{charge},p-\mathrm{charge},q})$ with $J(x)=(x+x^{-1})/2-1$, unique by the recognition functional equation.
Carried edges have equal charges, span zero phi-rungs, and cost $J(1)=0$. Interface edges join the spine charge $0$ to $\pm 1$, span one rung, and cost $J(\varphi^{\pm 1})=J(\varphi)$ by reciprocal symmetry. Thus all recognition cost lives on the interface: in 2D, $\mathrm{totalCost},t=(8t-4),J(\varphi)$ and carried cost vanishes.
The module states the compute-watch law in cost units: cost scales with the codimension-1 recognition-active interface, not bulk volume. $J(\varphi)=(\sqrt{5}-2)/2>0$ is a genuine positive unit cost.
proof idea
One algebraic cancellation from the closed form. Rewrite $\mathrm{totalCost}(t+1)$ and $\mathrm{totalCost},t$ by the multiplicative formula that expresses each as $(8r-4)\cdot J(\varphi)$ (valid for $r\ge 1$). Cast the two natural-number interface counts $8(t+1)-4$ and $8t-4$ to $\mathbb{R}$, discharging the subtraction side-conditions by omega. The difference simplifies by ring to $8\cdot J(\varphi)$.
why it matters
This is the differential form of the 2D compute-watch law (Phase 55 headline): carry the bulk free, pay only for the interface, and the per-cycle bill is constant $O(1)$ rather than growing with volume. It converts the static identity $\mathrm{totalCost},t=(8t-4)J(\varphi)$ into a cadence increment, matching the geometric fact that the 1D interface curve gains eight ordered edges per shell.
In the broader forcing chain it sits on T5 ($J$-uniqueness) and T6 ($\varphi$ as self-similar fixed point): the unit cost is forced, not fitted. No downstream consumers are wired yet; the natural parents are any global energy or coarsening budgets that integrate shell costs over cosmic time, and the 3D octahedron analogue whose quadratic total yields a linear increment.
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