= Solution
<Theoretical reduction> requires a relation between the quantities and domains of two theories, not just an assertion that their objects are made of the same material. A microscopic model may imply a macroscopic law after identifying temperature, magnetization or another collective variable, fixing admissible states and controlling an approximation. The bridge between descriptions can involve averaging or coarse-graining, and its adequacy must be established for the <observables> of interest. A reduction of a particular class of predictions is weaker than deriving every explanatory claim of an entire theory.
<Emergence> can mean that a collective description displays behaviour novel relative to individual constituents, robust under changes in microscopic details and naturally explained with new macroscopic concepts. In this sense it does not contradict <theoretical reduction>: a derivation can explain why a new pattern appears. A stronger assertion of irreducible causal powers is a different thesis. The existence of a singular mathematical limit does not on its own prove that stronger thesis.
A crucial distinction is between $T_\infty$ and any finite $T_N$. A limit of analytic functions need not be analytic, and a limiting state representation need not retain the structures of each finite representation. Therefore a claim about $T_\infty$ need not be literally true of $T_N$ even when the approximations are excellent for accessible measurements. Deduction of the limiting theory uses the whole family of finite theories together with a limit theorem; it does not deduce a false exact singularity in one finite system. Approximate <theoretical reduction> then explains the observed rounded or scale-limited counterpart.
The role of explanatory autonomy is also important. <Universality> at a <phase transition> can make a macroscopic description insensitive to many microscopic parameters. That robustness supports using the emergent description without tracking every particle. It does not establish that the microscopic laws have ceased to apply. The <renormalization group> helps explain such insensitivity by distinguishing perturbations which persist at large scales from those which become negligible.
\b[The defensible conclusion is that novelty and derivability can coexist; the limiting idealization and its finite-system accuracy must be stated separately.] Whether one calls the resulting behaviour emergent depends partly on which lower-level description and explanatory expectations form the comparison, rather than solely on the size of $N$.
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