Clarify finite precision positivity of log density

experiments/LOG_DENSITY_ASSESSMENT_2026-09-19.md

@@ -17,8 +17,10 @@ eventual EOS to a barotrope. The important choice is what gets exponentiated:
    coordinate change on positive coefficient vectors. The measured negative
    density fields already have positive nodal coefficients.
 2. Evaluating `rho = rho_scale * exp(u_h(x))` pointwise guarantees strictly
-   positive density wherever `u_h` is finite. This is a different approximation
-   family from polynomial density. Every density-dependent operator must use it.
+   positive density in exact arithmetic wherever `u_h` is finite. Floating-point
+   evaluation also needs checks for underflow and overflow. This is a different
+   approximation family from polynomial density. Every density-dependent
+   operator must use it.
 3. A finite polynomial `u_h` cannot represent an exact zero-density vacuum
    surface with that raw exponential. A vanishing reference factor can encode
    the surface, but its asymptotic behavior becomes part of the representation.
@@ -256,11 +258,16 @@ corresponding numbers are `6.65377`, `13.7153`, and `74.9786`.
 
 The first-order identity `delta rho/rho=delta u` makes a log step a relative
 density step. That can be useful across large density contrasts, but large log
-steps create exponential changes. Underflow, overflow and weak response near
-vacuum need explicit finite-value checks and meaningful globalization; silently
-clamping the exponential would change the equations. A vanishing factor removes
-the singular log profile when it matches the asymptotics, but retains the
-density-weighted sensitivity near vacuum.
+steps create exponential changes. Underflow can produce zero from finite log
+coefficients, so finite-value checks alone do not ensure positivity. A raw
+exponential implementation must also detect or reject underflow and nonpositive
+reconstructed density in the material interior, including underflow when
+multiplying by its reference scale or factor. Prescribed zero-density traces
+from a vanishing reference factor remain intentional. Overflow and weak
+response near vacuum also need explicit checks and meaningful globalization;
+silently clamping the exponential would change the equations. A vanishing
+factor removes the singular log profile when it matches the asymptotics, but
+retains the density-weighted sensitivity near vacuum.
 
 With the explicit constant test retained, a converged weak exponential closure
 preserves its target cell mass under the closure quadrature. Exponentiation