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As you already described in post #5, voltage divider is loaded by summer circuit. Using 10k voltage divider resistors makes the effect worse. The experimental setup should focus on demonstration of summer circuit rather than side effects.

This only works with low impedance and clean DC sources. Transformer + rectifier is neither a good solution. I would either use multiple 1.5 V batteries as test sources or a "black box" with multiple buffered voltage outputs.