Settings for a developmental separation proof [ftip-00NE]

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A promising mathematical setting couples related developmental tasks to a fresh target family through reusable structure. The contributor must acquire that structure by a bounded process. The recipient must then learn to use it under the common checker. The central choice is a family whose structure can both explain the developmental benefit and support a lower bound over the admitted closed alternatives. The following possibilities are research directions, not established separations.

One direction is a family of transformation puzzles with public local move rules. Development offers related tasks on which a learner can experiment, notice conserved quantities or discover a compositional decomposition. Fresh tasks ask for a checked move sequence or a certificate of impossibility. The acquired method must apply to new instances, and its certificates must fit the shared checker and deployment cap. This makes the intended assistance concrete without giving the contributor unrevealed answers. The missing argument is why the closed lineage cannot affordably discover any comparably useful invariant, decomposition or direct search method from the same public rules and its admitted tools.

A second direction studies the order and interaction of developmental experience. A learner's partial understanding can determine which question or intervention makes the next relation visible. Compare that process with the available passive traces and with any interactive or simulated substitutes admitted to the closed lineage. A proof must derive the cost of obtaining enough useful experience or processing the available data; simply withholding the interaction and declaring its outputs necessary would not explain conceptual discovery. Giving the lineage the same interaction channel is a stronger comparison when the intended mechanism is the cost of choosing how to use it.

A third direction studies a contributor drawing on cumulative cultural search. Several bounded learners develop, test and teach reusable concepts across earlier tasks. A current contributor's short suggestion may transmit structure produced by that longer process. This naturally motivates a marginal advantage and an amortization calculation. A lifetime advantage additionally requires accounting for that earlier population's work and for analogous shared data, teachers and parallel search available to model lineages. Culture must be generated by the declared process; a collection of perfect hints would assume its success.

For each direction, the proof needs a quantity that can be bounded through every admitted operation and connected to fresh-task performance. Information arguments apply when there is residual uncertainty; query arguments apply to explicitly limited response interfaces; computational arguments must address the available program class and its costs. If a separation is conditional on an independent computational hardness assumption, state that assumption and exhibit the reduction. Merely renaming the desired discovery difficulty as a hardness assumption adds no explanation. Restricted models can yield useful first results, provided their restrictions are not silently transferred to contemporary agents.

The most immediate constructive target is a development process that learns a reusable invariant on one task distribution and transfers it to fresh certified tasks. Alongside that construction, seek a family-specific lower bound covering every equally useful acquired method as in § [ftip-00N7]. Keep the model's own discovery, simulations and learning inside this argument. Neither the construction alone nor a failed search for substitutes proves the prohibitive closed cost.

§ [ftip-00NF] develops the first direction as a representation learner whose developmental experience also teaches it to construct a useful curriculum for another recipient. The concrete objects are the grounded representation, the learned teaching policy and their fresh-task effects.