Learning to discover and teach a representation [ftip-00NH]
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    The contributor begins with the disclosed state and bounded development of § [ftip-00NA]. On related rewrite tasks it can propose predicates, execute public rules, test conjectures, build examples and revise its representation. Neither a completed invariant nor a successful curriculum is placed in its initial state unless that preparation is explicitly counted as an inherited endowment. A constructive success argument must describe how the development process produces useful structure with a stated probability, including failed attempts.

    Teaching is a further learned action. Let \(Z_t\) be the contributor's retained state, \(h_t\) the observed interaction history and \(q_t\) an intermediate task or demonstration. A teaching policy chooses \(q_t\) from \(Z_t\) and \(h_t\); the recipient's response and checked learning progress supply feedback for later choices. The policy may first teach a distinction on small expressions, then ask for a general invariant, and finally require composition on larger expressions. These stages describe a possible mechanism. Their usefulness must be established for the specified learner, rather than assumed from the apparent pedagogical order.

    During development, the contributor can practice with bounded copies of a declared recipient population. It learns which examples correct particular failures and when a new abstraction is worth introducing. The student updates used to evaluate a teaching proposal are real work: cloning, training, progress evaluation, rejected curricula and teacher updates all count. A progress set drawn from the development distribution can provide a verified reward. Final evaluation instances remain unavailable for curriculum selection. Any guarantee must connect development progress to fresh-task success, since a curriculum may overfit either the practiced recipient or its progress measure.

    At assistance time, a fresh recipient starts from the declared model endowment. The contributor supplies an explanation, programs, examples or an adaptive sequence of tasks, within communication and interaction caps. The recipient must interpret the contribution, check the relevant claims and learn to use the representation. Evaluation occurs after the contributor is removed. A retained contributed program is allowed when it fits the common deployment interface and cap; merely consulting an uncharged external solver during evaluation is a different experiment.

    Useful teaching need not require the teacher to solve the final tasks itself. Conversely, a source that can solve them may still fail to teach the recipient. Measuring the contributor's discovery, its teaching skill and the recipient's later capability separately makes these possibilities visible. A short explanation or a short successful curriculum measures transmission after discovery; its length does not establish that finding it was cheap or expensive.