# H0346 — perforated hydrogel skin: a complete transport-model proof

**Status:** the stated lumped transport result is proved. Its applicability to a real swelling gel remains an experimental question. The calculation is not a new transport law.

## Original claim and the ambiguity to remove

The original claim says sparse perforations reduce swelling delay by more than their fractional open area. Define that precisely as

    D(φ) = 1 − t_half(φ)/t_half(0) > φ,

where φ is the perforated fraction of skin area, 0<φ<1, and t_half is time to half of equilibrium uptake. This comparison is a defined dimensionless criterion, not a general meaning of “more than expected.” It is not a claim to exceed a correctly specified area-weighted permeability model.

## Model assumptions

Normalize total surface area to one. Let k_s>0 be conductance of intact skin per unit area, k_h≥k_s conductance through a hole, and b>0 bulk conductance behind the skin. Skin and hole pathways act in parallel; the resulting surface layer and bulk act in series. Their conductances are linear, constant, and independent of swelling.

Let S>0 be a fixed storage coefficient. The interior is represented by one well-mixed potential, and the bath potential is a fixed step. Perforating the skin leaves equilibrium capacity S unchanged. There are no access/spreading resistances, inter-hole interactions, cracks, changing modulus, nonlinear osmotic effects, or additional exposed edges.

These are explicit model restrictions. Real poroelastic gels need not behave as a single well-mixed storage element.

## Theorem 1: exact half-time and monotonicity

Parallel conductance is

    g(φ)=(1−φ)k_s+φk_h=k_s+φd,       d=k_h−k_s≥0.

Series addition gives

    G(φ)=1/(1/b+1/g(φ)).

If z is normalized uptake, conservation is S dz/dt=G(φ)(1−z), z(0)=0. Separating variables gives

    z(t)=1−exp(−G(φ)t/S),
    t_half(φ)=S ln(2)[1/b+1/g(φ)].

Differentiation yields

    dt_half/dφ=−S ln(2)d/[k_s+φd]²≤0.

For d>0, half-time strictly decreases with perforated area. For d=0 it is unchanged. ∎

## Theorem 2: exact condition for delay reduction exceeding open fraction

Direct algebra gives

    D(φ)= b φd / [(k_s+φd)(b+k_s)].

For φ>0, all denominators are positive, so

    D(φ)>φ
    iff b d > (k_s+φd)(b+k_s)
    iff d[b−φ(b+k_s)] > k_s(b+k_s).

This last inequality is a necessary and sufficient condition. Equivalently, the effect requires

    φ < b/(b+k_s)

and

    d > k_s(b+k_s)/[b−φ(b+k_s)].

If b−φ(b+k_s)≤0, the claimed inequality is impossible. This establishes both the predicted regime and its failure regime without a numerical fit. ∎

## An exact example

Use consistent arbitrary conductance units and set

    b=1, k_s=1/100, k_h=1, φ=1/100.

Then g=199/10000, and

    D=9900/20099≈0.492562,
    t_half(φ)/t_half(0)=10199/20099≈0.507438.

Thus a 1% opening fraction produces about a 49.26% half-time reduction within this model. This is a constructed mathematical example, not a measured gel result or a predicted performance specification.

## Counterexamples and limits

1. If k_h=k_s, perforations provide no transport contrast and D=0.
2. For fixed k_s,k_h,φ, letting b→0 gives D→0. If bulk transport is the bottleneck, opening the skin has arbitrarily little effect.
3. If k_h<k_s, outside the theorem's beneficial-hole assumption, the derivative changes sign and holes can slow uptake in this same algebraic model.
4. As b→∞, D approaches φd/(k_s+φd). The gain remains explained by ordinary parallel conductance; it is not evidence for extra physics.
5. As φ→0, D/φ approaches b d/[k_s(b+k_s)]. A large initial gain is possible when intact skin is a strong resistance and hole conductance is high. For a finite φ, the exact condition above still controls the result.

## What remains unproved

The proof does not establish that a selected gel has a dense transport-limiting skin, that holes remain open and non-damaging, or that its full uptake curve follows a single exponential. It also does not show an effect beyond an area-weighted transport prediction: that prediction is precisely the mechanism used here.

Measure skin and bulk transport independently, keep equilibrium uptake fixed, and test the complete uptake curves on new specimens. If one parameter set cannot predict intact and perforated specimens, reject this lumped explanation before claiming an enhanced physical effect. See experiments/H0346.md.