The Spacing Bound: Where Planting Lettuce Closer Stops Paying, and the Spacing Rule That Follows

Few pieces of gardening advice are repeated as confidently as the claim that planting closer means harvesting more. This paper tests that claim for headed lettuce using a stated competition model, the reciprocal-yield relation 1/w = a + bN, with a = 3.33 per kg, so that an isolated plant approaches 0.30 kg, and a competition coefficient b of 0.20 under standard water, 0.30 under dry conditions, and 0.15 under irrigation. Under standard water, yield per plant falls from 0.2421 kg at 4 plants per square metre to 0.12 kg at 25 plants per square metre and 0.0429 kg at 100 plants per square metre, while total yield per square metre rises from 0.969 kg to 3.001 kg and 4.286 kg, approaching a ceiling of 5.00 kg per square metre that it reaches only slowly: 80 percent of the ceiling requires 66.6 plants per square metre, 90 percent requires 149.9, and 95 percent requires 316.3. The claim is therefore true for total yield and false for the yield anyone can sell or serve, because at a stated marketable grade of 0.12 kg per head the mean head falls below grade beyond 25 plants per square metre, and marketable yield is at its maximum exactly there, at 3.001 kg per square metre. The spacing rule that follows is 20.0 cm square under standard water, widening to 24.5 cm when dry and narrowing to 17.3 cm when irrigated. Seed sown beyond the useful density is a measurable waste: planting at 100 plants per square metre instead of 25 costs an extra USD 1.67 per square metre in pelleted seed (USD 16.66 on a 10 square metre bed) for total growth whose mean head weighs 0.043 kg, and a third of the seed spend at that density lies beyond even the 80-percent-of-ceiling plateau. The study is simulated, not measured.