What does your
pressure space miss?
Compare a prescribed log-volume field, its pressure-space projection, and the component that the space cannot represent.
Quarter steps, −1 to 1. Editing a value prescribes a new case; it does not deform a body.
Weights, field values, and energies are normalized. They are not measured pressures, reference volumes, or joules.
The field, its projection, and the remainder
Discrete samples · one shared vertical scale · no material geometry
Prescribed field
gAuthored log-volume samples
Represented part
PgOne weighted mean, repeated
Unresolved part
rr = g − Pg
| Sample | wᵢ | gᵢ | (Pg)ᵢ | rᵢ | p*ᵢ = K(Pg)ᵢ |
|---|---|---|---|---|---|
| 1 | 1 | −0.75 | 0.125 | −0.875 | 1 |
| 2 | 3 | 0.25 | 0.125 | 0.125 | 1 |
| 3 | 2 | −0.25 | 0.125 | −0.375 | 1 |
| 4 | 2 | 0.75 | 0.125 | 0.625 | 1 |
p* is the optimal mixed variable in the chosen sample space, not a measured anatomical pressure.
Condensed: 4 × 0.125 = 0.5
Residual: 4 × 1.875 = 7.5
Readouts round to five decimal places; the identity diagnostic uses unrounded values.
What refinement recovers
Every row uses the same current g, weights, and K.
| Pressure values | Econd | Gap |
|---|---|---|
| 1 shared | 0.5 | 7.5 |
| 2 groups | 1 | 7 |
| 4 samples | 8 | 0 |
Two groups recover the weighted pair means. Four independent sample values recover the entire prescribed field.
Energy ordering is at a fixed state. It does not establish stability or compare solutions re-equilibrated in different spaces.
Gᵢ is the selected group containing sample i. Q consists exactly of vectors constant on each group. Positive weights make each group denominator positive.
The unresolved part has zero weighted sum in each group. It is therefore weighted-orthogonal to every group-constant vector in Q. Ordinary unweighted averages would give a different projector.