MATH.9:4 - Solution

State the permitted answers and their transformation → find the input stabilizer → choose a permitted fixed answer → extend it along the orbit → construct the needed scope → return the rule or revise the obstructed requirement.

MATH.9:4.1 - State the input, permitted answers and transformation law

Let D be the inputs and Y the possible outputs. Write A(d) for the subset of outputs permitted for input d. An admissible root, minimizing allocation or selected vertex can define this subset.

Give a group G acting on both D and Y. Its action satisfies e*d=d and (g*h)*d=g*(h*d), with analogous rules for outputs; every transformation has an inverse. Different actions on input and output can use the same group elements.

Establish that the permitted answers transform with the input:

A(g*d) = g*A(d),

where g*A(d) means the set of g*y for y in A(d). MATH.8 supplies the solution-preservation argument when it has to be constructed.

The requested deterministic rule f must satisfy both f(d) in A(d) and

f(g*d)=g*f(d).

This relation is equivariance. If the requested output stays unchanged under the transformations, use the trivial output action g*y=y; the relation then expresses invariance.

Keep attributes that the rule is allowed to use inside d. A distinguished label, orientation or mark can change the stabilizer and the existence of a choice. A coordinate label that merely changes under relabeling cannot silently serve as a fixed priority.

MATH.9:4.2 - Find the permitted answers fixed by the input’s stabilizer

The stabilizer of d is the subgroup

K_d = {g in G | g*d=d}.

For every g in K_d, equivariance requires f(d)=g*f(d). Thus the permitted answers for an equivariant rule at d are

B(d) = {y in A(d) | g*y=y for every g in K_d}.

To construct B(d), first obtain the transformations fixing the supplied data, then solve their fixed-point conditions together with membership in A(d). For a finite supplied group and a finite decidable answer set, enumerate those elements and test them. When either is infinite, use an algebraic description or another available construction; a failed finite search leaves the unsearched range open.

If B(d) is empty, no deterministic equivariant rule can answer this input under these requirements. Return the input symmetry and the fixed-output condition that permitted answers fail. This result concerns the supplied data and output requirement; it identifies what a repair must change.

If the original problem has a unique permitted answer y, its preservation under K_d forces every g*y to equal y. The fixed-point equations can therefore restrict or find that answer. Establish existence and uniqueness on the range used by this deduction. With several permitted answers, use the same fixed-point calculation to find a compatible choice; it need not recover every permitted answer.

MATH.9:4.3 - Extend a selected answer over one input orbit

Choose representative data d0 and an answer y0 in B(d0). For input d in the orbit of d0, obtain g with d=g*d0 and define

f(d)=g*y0.

The answer is permitted because the relation in :4.1 carries A(d0) to A(d). It is independent of the transformation used. Indeed, if g1*d0=g2*d0, then k=g2^-1*g1 fixes d0. Since k fixes y0,

g1*y0=g2*(k*y0)=g2*y0.

The rule is also equivariant: for another transformation q,

f(q*(g*d0))=(q*g)*y0=q*f(g*d0).

This constructs the entire rule on that orbit. To evaluate it, retain a way to find g, or an equivalent formula for f(d). An existence argument for g alone may leave the computation unresolved.

If a normalization procedure instead supplies h with h*d=d0, return h^-1*y0. Keeping the direction of this transformation prevents returning a coordinate answer in the normalized frame. The position-selection case in :5.4 carries out both inverse returns.

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.

MATH.9:4.5 - Repair an obstructed choice

Choose the change that answers the receiving problem.

A supplied distinguished attribute can reduce the stabilizer. Include that attribute and its transformation in the new input, then repeat the fixed-output test. The marked-cycle case below constructs such a rule.

A different output may retain the alternatives the input leaves open. Returning the whole set A(d) respects its induced transformation from :4.1. When A(d) is finite and nonempty, a uniform probability distribution on it does too, if a distribution is the requested result. Sampling a single value is a further calculation with its own randomness and comparison requirement.

An average is usable only when it belongs to the permitted output set and has the required transformation law. The mean of two opposite unit directions is zero, which fails a unit-direction requirement.

The receiving task may instead allow coordinate-dependent choice or a narrower input range. State that changed requirement and construct the corresponding rule. Adding distinctions or weakening a requirement is useful only when the resulting answer serves that task; C.11.DUA helps decide whether obtaining further input is worth its cost when that question is unresolved.

MATH.9:4.6 - Return the result and reopen its changed premise

Return the rule with its input range, permitted output and transformation law, or the obstruction that changes the choice problem. A known unique solution can return a smaller fixed-point calculation instead.

When a criterion, attribute or output changes, recalculate the affected stabilizer and fixed-output condition. Retain orbit calculations and proofs whose premises are unchanged. Small perturbations can remove a tie and reverse a selected answer, so use the changed criterion when the choice depends on it.

Use FPF C.29 to establish what this mathematical result means for another subject. The obstruction for an abstract arrangement leaves available attributes and interventions in a real arrangement to that subject comparison.