# Follow one conserved constituent

Turning feedstock into an accepted product involves several chances to retain or lose the constituent you want. This is a transparent accounting example, not a prediction of chemistry, material quality or manufacturing performance. All fractions are hypothetical.

Start with 100 kg of feed containing 20% of the tracked constituent: 20 kg is available. Recovering 80% sends 16 kg to fabrication and leaves 4 kg in the separation residue. A 90% fabrication yield places 14.4 kg of that constituent in accepted product and 1.6 kg in fabrication rejects. The ledger balances: 20 = 4 + 1.6 + 14.4 kg. The other 80 kg of feed is identified but not followed through the plant.

For feed mass F, constituent mass fraction c, separation recovery r and fabrication yield y:

- Target entering: F*c.
- Target in separation residue: F*c*(1-r).
- Target in fabrication rejects: F*c*r*(1-y).
- Target in accepted product: F*c*r*y.

The last three sum to F*c. Fractions must lie between zero and one; feed mass must be finite and nonnegative. Perfect recovery cannot compensate for a low fabrication yield, and vice versa. Zero at either stage gives no target in accepted product. Residue and rejects may be recoverable later; they are not automatically permanent waste. A recycling loop would need a separate explicit model.

Every stream must use the same conserved basis. If processing changes chemical species, track an element or an explicitly equivalent constituent basis; do not balance the mass of unlike compounds as though it were identical. Total product mass may include other constituents, added binder or reagents. Those are outside the ledger. This is not a total plant mass balance and gives no energy, emissions, purity, strength or cost advantage.

## Data, figure and verification

Run `python reproduce.py` with matplotlib. It produces ledger-grid.csv (441 combinations of recovery and yield, each from 0 to 1 in 0.05 steps), metadata.json, checks.json and constituent-ledger.png/svg/pdf. The plotted lines use the same exact formula at finer spacing; the stacked example uses 80% recovery and 90% yield. Figure is an original analytical illustration with no measured points or statistical error bars.

Checks verify constituent conservation and nonnegative stream masses in every grid row, all eight fraction corners, zero feed and nine invalid inputs. Adding the untracked other-feed mass to the three constituent destinations recovers the feed mass as an arithmetic identity, not as proof of a complete plant balance. PNG labels and layout were visually inspected.

The method is derived directly from definitions of mass fraction, recovery and yield. No external source supplies the numerical process inputs because none are measurements. Chapter15-coverage-brief.md supplies the separately verified manufacturing and dimensional-inspection references; this ledger does not borrow their performance claims.