Compact quartic critical-value parameters

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_proof: Critical-value differences bound the roots, circumcircles parametrize the generic family, and explicit collision calculations complete the coordinate reduction.

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The proof is author-recorded Plasma AI work, with no independent review or formal proof attached. It gives a compact parameter domain, a complete treatment of its collision strata and the comparison for the single-critical-point family. It consumes no other native claim.

The quartic research sketch uses these coordinates for Problem 114. That application separately invokes Eremenko--Hayman's existence and critical-value deformation results. They are not premises of the elementary coordinate theorem here. The separate author-recorded comparison L181 covers both nonunicritical collision families: reflection puts the double-critical family in the form $s^4F_D(z/s)+c$ with $s=2h^{1/4}$, and the repeated-value family has the form $s^4F_E(z/s)+c$ with $s=h^{1/4}$. The unit-critical-value condition makes the circle with center $-c/s^4$ and radius $s^{-4}$ pass through the two critical values of the corresponding normalized polynomial. Thus L181 gives a length deficit greater than $s$ on these families. The generic comparison and practical neighborhoods of the collision pencils remain open. The small-curvature result L180 covers an explicit range near the unicritical limit. These applications are separate from the coordinate theorem and do not change its standing.

Verification. Read the integral identity for critical-value differences, the maximal-root bound, the circumcenter equations, both collision calculations and the one-variable length comparison. No computation is needed for this claim, and no gap remains in its stated scope at author standing. A proof for every monic quartic is outside that scope.