Audit nutrient efficiency denominators and missing pathways

Define the useful service

Nitrogen-use metrics are meaningful only when their numerator and denominator are stated. Harvested nitrogen divided by fertilizer input is a partial balance ratio; it does not count all biological fixation, deposition, imported material or soil-stock contributions. A ratio above one can therefore reveal omitted inputs or stock drawdown rather than miraculous nutrient creation.

Construct the comparable ledger

The original packet supplies fertilizer F plus 30 kg N/ha of explicitly declared other inputs. Harvest removes H. Partial ratio is H/F; the total-input ratio is H/(F+30). The residual F+30−H must equal stock change plus other outputs under the complete boundary, subject to measurement error.

Interpret the competing metrics

A positive residual is not automatically pollution. It can include retained soil or biomass nitrogen, gaseous losses, leaching and other exports. A negative residual cannot represent negative physical nitrogen: it requires stock depletion, a missing source or inconsistent observations. The data do not identify those alternatives without additional measurements.

Challenge the boundary and scenario

Crop nutritional quality, yield, soil fertility and water/air effects are different outcomes. Comparing harvest ratios alone can reward depletion of a finite soil stock. A defensible audit retains the full input list, inventories, time interval and uncertainty instead of turning a single number into a fertilizer application recommendation.

Worked original packet

With fertilizer 100, other inputs 30 and harvested N 100 kg/ha, partial ratio is 1, total-input ratio≈0.769 and residual 30 kg/ha. That residual has not been assigned among stock change and environmental losses.

Investigation

Compare fertilizer 50 with harvest 100 and 150. Identify when the supplied total inputs cannot explain harvest without stock drawdown or missing sources, then propose an independent observation.

Changed-case transfer

Fertilizer 75, other inputs 30 and harvest 125: calculate both ratios and the residual.

Feedback

Partial ratio 125/75≈1.667; total-input ratio 125/105≈1.190; residual−20 kg N/ha. A stock decline of at least the missing net amount, another input or measurement correction is needed; no atoms are created.

Sources and evidence boundary

Original prose, numerical packets and exercises. Primary method/mechanism source, checked 30 October 2026. Supplied values are not measured operating data, universal coefficients or recommendations. No physical equipment work, chemical application, organism manipulation or water extraction is requested. Retain first reasoning and critique in the course workspace; public feedback is practice, not independent assessment.