Three model results: reference layouts, surge buffering, and perforated skins

Conditional mathematical results · Physical hypotheses remain unproved.

Three proposals in Study 1 have reached a useful stage: their assumptions are explicit, their mathematical consequences are worked out, and their limits can be checked. They concern where to place reference measurements, what a surge buffer actually guarantees, and how small holes can change transport through a skin. No physical experiments were performed. These results describe specified models, rather than demonstrated material performance.

H0199: put references where they cover the field

Suppose a spatial field has a known bound on how rapidly it can change, and each calibrated point reading has a bounded error. The derived best possible worst-case reconstruction error is L × coverage radius + measurement-error bound. Here, L bounds spatial variation and the coverage radius is the greatest distance to the nearest reference.

On a unit square, the specified 3 × 3 interior grid has coverage radius √2/6; every boundary-only layout leaves the center at least 1/2 away. That establishes an advantage for this worst-case objective when L is positive. It does not establish superiority for every individual field, random-noise metric or physical deposition experiment. References are assumed not to disturb the field. Read the complete derivation and checks.

H0836: lower peaks need the right comparison

A model with constant compliance and linear resistance smooths a matched inlet waveform and reduces nonzero-frequency and rectangular-pulse peak flow. A stronger statement about total modeled release requires an additional assumption: release must be a convex, nonnegative, memoryless function of flow, zero at zero.

The counterexamples matter. A threshold-style release rule can produce more total release despite a lower peak. Matching the inlet waveform does not match hydraulic work; a constructed equal-work comparison even gives a buffered peak above its unbuffered comparator. Volume, waveform and work are different controls. The model excludes leakage, inertia, resonance and changing resistance, and establishes no measured reduction in sand release. See the conditions and counterexamples.

H0346: a small open area can change the bottleneck

In a fixed-storage transport model, skin and hole conductances act in parallel, with bulk conductance in series. The derivation identifies exactly when the reduction in half-time exceeds the hole area fraction. One constructed example gives about 49.26% reduction with 1% open area. That is an arbitrary-unit model example, not a gel specification.

The effect follows ordinary conductance accounting. With no transport contrast there is no gain; when the bulk is the bottleneck, the gain can be arbitrarily small. Hole interactions, cracks, spreading resistance and changing material properties are outside the model. Read the model and limits.

What was checked

An internal consistency review read all three derivations and ran an isolated copy of the supplied code once. It reproduced 333 output rows, including exact parameter cases, ODE comparisons and counterexamples; generated files matched the supplied outputs byte for byte. Finite checks supplement the written arguments. This was not external peer review or formal verification, and establishes no worldwide novelty. The unchanged originals, code, results and provenance are archived together.

 

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