# H0836 — compliant buffering: complete conditional proofs and counterexamples

**Status:** the circuit-model results below are proved. Reduced physical sand breakthrough is not established. No new buffering principle is claimed.

## Model and assumptions

Consider a fluid-storage node with constant positive compliance C and a linear downstream hydraulic resistance R>0. Pressure p is measured relative to the downstream outlet; inflow is u(t), outflow is y(t)=p(t)/R. Conservation gives

    C dp/dt = u−p/R,
    τ dy/dt + y = u,             τ=RC>0.

The baseline is subtracted and initially equilibrated. Assume no inertia, resonance, cavitation, leakage, hysteresis, or deformation-dependent resistance. Comparing the rigid limit uses the same R and specified inlet waveform, with C=0 giving y=u. These assumptions are not yet a model of particle transport.

## Theorem 1: exact attenuation

Multiplying the differential equation by exp(t/τ) and integrating gives the zero-state solution

    y(t)=∫₀∞ k(s)u(t−s)ds,
    k(s)=exp(−s/τ)/τ for s≥0.

For a nonzero-frequency sinusoid of amplitude A, inserting a complex exponential gives transfer function

    H(iω)=1/(1+iωτ),
    output amplitude = A/√(1+(ωτ)²) < A.

The DC gain is 1; this model does not remove the mean flow.

For a positive rectangular pulse with A>0, u=A on [0,T] and zero otherwise,

    y=A(1−exp(−t/τ))                     for 0≤t≤T,
    y=A(1−exp(−T/τ))exp(−(t−T)/τ)       for t≥T.

The first branch increases and the second decreases, so

    peak(y)=A(1−exp(−T/τ))<A.

Integrating the differential equation over the whole pulse and recovery gives ∫y dt=∫u dt=AT. A lower peak is accompanied by a longer tail, not the destruction of fluid volume. ∎

## Theorem 2: a sufficient particle-release closure

For nonnegative bounded inflow pulses about a zero-flow baseline, suppose, additionally, that release mass per unit time is Φ(flow), where Φ is convex, nonnegative, memoryless, Φ(0)=0, and both full-event integrals exist. This is an assumption requiring measurement, not a proven law for sand.

Because k is nonnegative and integrates to 1, Jensen's inequality gives

    Φ(y(t)) = Φ(∫k(s)u(t−s)ds)
            ≤ ∫k(s)Φ(u(t−s))ds.

Integrating over all real time and changing order of the nonnegative integrals yields

    ∫Φ(y(t))dt ≤ ∫Φ(u(t))dt.

Thus, under this closure and a matched inlet waveform, buffering cannot increase total modeled release. An example is Φ(q)=a max(q−qcrit,0), a≥0. Strict reduction requires further conditions; Φ(q)=aq produces exactly equal total release because volume is conserved. ∎

## Counterexample 1: a lower peak can release more material

The original claim does not specify convexity. Let release be constant whenever flow exceeds a threshold:

    Φ(q)=1 if q>0.1, otherwise 0.

Take A=T=τ=1. Rigid flow exceeds the threshold for exactly one time unit. Buffered flow crosses it during the rise at

    t↑=−ln(0.9),

and during decay at

    t↓=1+ln((1−exp(−1))/0.1).

Its modeled release duration is t↓−t↑≈2.73855>1, although its peak is only 1−exp(−1)≈0.6321. This explicitly refutes the inference that a lower peak alone guarantees less integrated particle release. The counterexample closure is deliberately nonconvex and is not asserted to describe real sand.

## Counterexample 2: equal inlet waveform is not equal hydraulic work

For this energy comparison use zero initial and final flow and pressure, not an unspecified nonzero operating baseline. Pump work is W=∫p u dt. Multiplying conservation by p and integrating over a complete return to equilibrium gives

    W = R∫y²dt + (C/2)[p²]initial to final = R∫y²dt.

For the rectangular pulse,

    Wbuffer=R A²[T−τ(1−exp(−T/τ))],
    Wrigid=R A²T.

Consequently the same inlet pulse does not supply the same work. With R=T=τ=1, compare a rigid amplitude 1 with a buffered amplitude exp(1/2). Both have work 1, but the buffered peak is

    exp(1/2)−exp(−1/2)=2sinh(1/2)>1.

The strict inequality follows from sinh(x)>x for x>0. This equal-work example does not also match total injected volume; it demonstrates a limitation of energy matching alone, not a refutation of every jointly controlled physical experiment.

## Exact conclusion

Linear compliant storage attenuates matched-waveform fluctuations. A specific convex release closure then bounds integrated modeled release. Neither attenuation nor that closure establishes the original sand-breakthrough claim under all the original matching conditions. Experiments must establish actual compliance, resistance, flow and pressure histories, energy, volume, and release kinetics.