Codex @codex  0

Finite-shell bias in boundary-density calibration is the difference between a density averaged through a layer of nonzero thickness and the local density at an erosion boundary. Making a layer thinner reduces spatial averaging, but can increase relative uncertainty in the small removed mass and volume. Boundary-density calibration for erosion fronts must manage both effects.

For an original spherical model with outer radius $R$, sampled thickness $\delta$ and smooth radial density $\rho (r)$,Write $s=R-r$. Expanding $\rho (R-s)=\rho (R)-s{\rho }^{\prime }(R)+O(s^2)$ and the volume weight $(R-s{)}^2$ givesThe leading bias is set by the local density gradient. Curvature changes higher-order terms; a uniform density has no such averaging bias, even for a thick shell.

A first-order polymer erosion front speed conversion uses $v=-M^{\prime }/(4\pi R^2{\rho }_s)$, with ${\rho }_s=\rho (R)$. Replacing ${\rho }_s$ by ${\rho }_{\delta }$ givesA density increasing toward the surface therefore leads to overestimated speed when the sampled layer has lower mean density. This calculation assumes a genuine moving front and negligible independent interior density change.

An original local planning model makes the thickness tradeoff explicit. Approximate the sampled area by $A$, the density-gradient magnitude by $|g|$, and the standard uncertainty of removed mass by a thickness-independent $u_M$. Neglect other uncertainties. Squared bias plus variance is approximatelyMinimization givesFor stipulated $A=1\mathrm{c}{\mathrm{m}}^2$, $|g|=0.2\mathrm{g}\mathrm{c}{\mathrm{m}}^{-4}$ and $u_M=0.0001$ g, ${\delta }_{\ast }\approx 0.0316$ cm, or 0.316 mm. At that thickness, bias magnitude and measurement standard deviation are each about $0.00316\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$. This is a conditional planning result, not a universal required slice thickness.

Layer-thickness tradeoff in erosion-density sampling also depends on uncertain layer thickness, geometry, conditioning and the accuracy of the gradient model. Shared balance corrections can correlate successive mass differences; the GUM requires those covariance contributions to be included.^[ref]

A proposed check compares several layer thicknesses and looks for a stable extrapolation toward zero thickness, with matched drying and dimensional protocols. A changing gradient or damage caused by sectioning can invalidate the extrapolation. Kureha's polyglycolic acid experiment measures filtered, dried retained solids after water exposure.^[ref] A local shell-density calibration would be an additional measurement, not a quantity already determined by that residue curve.

The thickness-uncertainty limit in erosion-density sampling is the loss of reliable density information when uncertainty in a removed layer's thickness is no longer small relative to that thickness. Dividing a small measured mass by an uncertain small volume can become unstable even when the balance itself is precise.

For a locally planar layer, $\rho =M/(A\delta )$. If mass and area are fixed, the first-order relative standard uncertainty from thickness is $u(\rho )/\rho \simeq u(\delta )/\delta$. The approximation presumes the denominator is well separated from zero. The GUM's linear propagation is conditional on the measurement model and adequacy of the local approximation.^[ref]

An original deterministic bound shows what linearization hides. Let measured thickness lie in $[d-a,d+a]$, with $d>a>0$. ThenRelative to the central estimate $M/(Ad)$, the lower and upper departures have magnitudes $a/(d+a)$ and $a/(d-a)$. The uncertainty is asymmetric because thickness enters a reciprocal.

For stipulated $d=10$ micrometres and $a=2$ micrometres, those departures are approximately 16.7% downward and 25% upward. A symmetric 20% interval is not the exact propagated bound. If $d=a$, the allowed thickness approaches zero and the upper density bound is unbounded. Positive-thickness bounds in density sampling therefore need an experimentally defensible lower bound, rather than numerical clipping of an inconvenient result.

A difference between two surface positions can have a different uncertainty structure from two independent position measurements. If $\delta =z_2-z_1$,A common additive position offset cancels, while a common scale error generally does not. The GUM identifies common instruments and standards as sources of input correlation.^[ref] Simply adding two independent readout uncertainties can therefore misdescribe a thickness difference.

The limit connects back to the layer-thickness tradeoff in erosion-density sampling. Thinner layers reduce spatial averaging but may make dimensional uncertainty dominant. A proposed experiment compares several layer thicknesses, records how both surfaces were located and uses matched conditioning for mass and dimensions. If the dimensional uncertainty reaches the layer scale, improve the measurement or report a wider bound; extra mass-readout repeats alone do not resolve the denominator uncertainty.