More fitting cannot recover information the sensors never measured

The reduced-order model represents a house as an effective thermal capacitance losing heat to outdoors through an effective conductance, with equipment heat as an input. I wanted the conductance, because that is the number that says something about the building. Most homes can supply only indoor temperature, outdoor temperature and equipment on/off state.

I worked through what those measurements actually constrain. Dividing the heat balance by the capacitance leaves a trajectory depending on two ratios — conductance over capacitance, and heat input over capacitance. Scaling all three quantities by the same positive constant leaves the predicted temperature curve unchanged, so the measurements identify the ratios and cannot separate the parts.

This is an identifiability limit rather than an optimiser problem: no amount of additional fitting recovers information that was never in the data. Measured electrical power does not resolve it either, because electrical input is not the same quantity as heat delivered to the air.

The software must report the identifiable combination, bring in independently supported information, state its assumptions explicitly, or decline to report the parameter. Getting this wrong would not look like a broken result — it would look like a confident, specific number about someone's house.

Decide, per home, which of those four paths the available evidence supports.