Module 7 · Page 2 of 3 — The Constraint Manifold

II. The leaf as an optimisation surface

Water-use efficiency \(\mathrm{WUE} = A/E\) is not a number the leaf computes — it is a geometric property of the state space. At any atmospheric state \((c_a, \mathrm{VPD}, P)\), the leaf exists on a constraint surface in \(\{g_s, A, E\}\) space, constrained by Fick's law and the biochemical demand function.

Eq. 5 — WUE as a manifold invariant
\[ \mathrm{WUE}(c_a,\mathrm{VPD},P) = \frac{P(c_a-c_i)}{1.6\,\mathrm{VPD}} \]

g_s cancels — WUE depends only on the atmospheric state. The leaf cannot improve WUE by changing aperture alone.

Eq. 8 — Constraint surface in {A, E, g_s} space
\[ \mathcal{M}:\quad A = g_s(c_a-c_i), \quad E = 1.6\,g_s\,\mathrm{VPD}/P \implies \frac{A}{E} = \mathrm{WUE} = \text{const} \]

As g_s varies, the point (A, E) moves along a ray through the origin with slope WUE — it cannot leave it. The leaf is geometrically constrained.

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Why the manifold matters for Page 3

The constraint manifold establishes that the leaf's operating point is geometrically determined by the atmosphere, not by stomatal control. What stomata can control is the rate of transit along the WUE ray — how quickly assimilation and transpiration scale together. This is where the proton pump timescale enters: if H⁺-ATPase throughput is enhanced by quantum tunnelling (Eq. 7, Page 3), the leaf can track the optimal point faster under rapidly changing conditions.

The leaf does not choose its WUE. The atmosphere inscribes it. The guard cell decides only how fast to arrive.