Geometry-informed temperature band screening for crop microclimate management in a commercial tomato greenhouse

Commercial greenhouses require temperature strategies that coordinate natural ventilation and screens with supplementary heating while maintaining a suitable crop environment. We evaluated a low input framework that combines a spectral geometry descriptor, one hour autoregressive prediction with exogenous inputs (ARX), temperature-conditioning degree-hour indices, vapor pressure deficit (VPD) exposure, and recorded vent, screen, and heating water data to screen temperature bands for field testing. The case study used 24,132 complete five minute interval starts from a 104 m × 68 m × 6.5 m Venlo tomato compartment operating at 1.25 plants/m², 3.75 productive stems/m², and an annual yield of approximately 45 kg/m². The Rayleigh-Ritz solution gave C_G = 7.188 m², and the maximum single mode error was 4.26% across the numerical test grid. A preliminary fourth root scaling hypothesis mapped the declared reference half widths of 0.8-1.2 °C to candidate half widths of 0.97-1.45 °C; the 1.21 °C midpoint is an illustrative scenario conditional on the declared benchmark and reference half width. Alternative benchmarks produced multipliers of 1.047-1.290, confirming benchmark dependence. Within-site temporal holdout evaluation of ARX yielded RMSE 0.798 °C and R ² = 0.911; this result evaluates temporal prediction separately from geometry scaling. On the same observed temperature series, crop reference band membership increased from 21.02% at ±1.0 °C to 25.22% at ±1.21 °C, while the total degree-hour index decreased by 9.17%. The 9.17% value is a retrospective index difference rather than a measured energy saving. Only 28.38% of daytime records were within the 0.5-1.2 kPa VPD reference interval, showing that the prevailing moisture environment remained a constraint beyond temperature-band flexibility. The framework therefore supplies candidate bands and crop-environment criteria for controlled evaluation of energy use, plant responses, yield, and fruit quality.