15.8 A worked episode: inventing an intrinsic property

The IPC–3 generated worlds provide a small but complete example of declaration accommodation. Each world presents mixtures of materials with masses \(m_1,m_2\), initial temperatures \(t_1,t_2\), and equilibrium temperature \(t_f\). The seed theory contains the uniform balance law

This law succeeds when the materials respond uniformly but fails persistently in the positive worlds. The proposed repair adjoins a typed intrinsic property

and replaces the seed equation by

The public mixtures identify the values of \(c\) only up to a common positive scale. Consequently, the admitted object is a projective coefficient family, not one privileged numerical vector. A compiler fixes a gauge for execution, but scoring is invariant under rescaling.

This example separates three possible responses to failure. In a uniform control, the old law already works and no new generator is justified. In an intrinsic-property world, pairwise ratios of \(c\) are consistent across mixtures and the extension predicts unseen combinations. In a contextual negative control, the implied ratios change from one situation to another; forcing a single material property would merely conceal the obstruction. Correct behavior is therefore reuse, extension, and abstention, respectively.

Eight blinded worlds contained four intrinsic-property cases, two uniform controls, and two contextual controls. GPT–OSS 20B selected the correct structural response in seven of eight worlds from the records alone. When given a deterministic diagnostic containing naive residuals and implied pairwise ratios, it was correct in all eight. Every admitted extension transported the public evidence and predicted four held-out mixtures exactly; both contextual controls were rejected in both conditions. Completion-token use fell by \(75\% \) and elapsed time by \(67\% \) under the diagnostic, even though its prompt was longer.

Experiment: IPC–3: Intrinsic-Property Accommodation. Seed theory. Uniform mass-weighted balance.
Obstruction. Persistent material-dependent residuals across public mixtures.
Constructed object. A positive material property and a repaired balance law, interpreted up to common scale.
Admission. Evidence transport, exact held-out prediction, uniform reuse controls, and contextual abstention controls.
Outcome. Diagnostic-assisted construction was admitted in \(8/8\) worlds; language-only construction was admitted in \(7/8\).
Boundary. The signature of the possible property and the repair family were registered. The experiment tests typed accommodation, not unrestricted invention of a physical concept.

The construction resembles the role of specific heat in classical scientific discovery systems, but its importance here is logical rather than historical. A failed equation is not repaired by lowering a scalar loss alone. It is repaired by adding an object of a declared type, changing the admissible equations, transporting old observations, and surviving counterfactual controls that make the attractive extension invalid.