Start with an observable relation
Normal symmetry is the book’s entry point from the experienced physical world into the wider hierarchy. Readers already encounter symmetry through repeated structures, stable relationships, and transformations that preserve a measurement. The useful question is always which feature remains unchanged.
A rotation can preserve the distances between points while changing their coordinates. A transformation can preserve the equations of a model while moving a particular solution into another solution. The symmetry of the equations and the symmetry of one observed configuration therefore need to be stated separately.
From generation to a completed description
DTT asks how the description available to an observer arises from generative activity. Suppose a map P sends a source state z to a completed record P(z). A source transformation g produces P(gz). It induces a well-defined record transformation only if source states that produce the same record also produce the same transformed record.
That requirement matters when some generative information is hidden. If P(z₁) = P(z₂) but P(gz₁) differs from P(gz₂), then the visible record alone does not determine the transformation’s result. A model may need additional retained information, an approximate regime, or a different class of allowed transformations.
This gives a direct way to connect the book’s language with a research problem: identify which source symmetries remain available after completion, and which distinctions the observer must retain to represent them.
Reading route
Use Transformations and invariants for precise examples, Symmetry breaking for the distinction between laws and selected configurations, and the memory-after-projection example for a calculation showing how hidden source information can affect later observations.
The broader five-level map then places this observed description within the book’s proposed hierarchy.
