MATH.9:4.4 - Cover the requested input range

For a finite group acting on a finite input set, enumerate input orbits as in MATH.8. Pick one representative d0 per orbit, compute B(d0), choose one of its members and extend it by :4.3. This supplies a rule on every processed orbit. Empty B(d0) blocks a total rule on a range containing that orbit.

For a larger or infinite range, give the representative construction and a compatible answer for each orbit needed by the claimed rule. These choices can be supplied by formulas or previously established mathematical results. Pointwise nonempty B(d) alone has not provided an evaluation algorithm or the choices across an arbitrary infinite family.

When an existing direct formula can be checked against A(d) and equivariance, use it without building an orbit table. The stabilizer argument can still expose a failed input or explain why its tie-break works.

Before returning the rule, check the other properties actually requested for it. If a geometric or learned model must vary continuously with its data, continuity requires its own argument: the pointwise construction here permits unrelated choices on different input orbits. If the result will be computed repeatedly, compare its evaluation cost with the available direct rule.