The Twelve Answers Problem Why a grain boundary has one angle and twelve rotations, and what that costs a defect map

ABSTRACT

A disorientation angle is one number. The rotation that produces it is not.

Reducing a misorientation by the crystal’s point group returns a unique minimum angle and a non-unique rotation: twelve equally-valid rotors for hexagonal 622, twenty-four for cubic 432. None of that is news. The multiplicity of symmetrically equivalent misorientations is textbook, and conventions for selecting a unique representative were published decades ago. Our claim is narrower: the convention in everyday use specifies an angle, not a rotor, and the gap between those two things has a measurable, platform-dependent cost.

The identity is harmless for any quantity built from angles. It is not harmless for any quantity built by composing rotors around a loop, because a loop multiplies rotations and therefore multiplies arbitrary choices.

We measure the cost on a public EBSD map of twinned magnesium. Between two machines running identical code on identical bytes, 453 plaquettes move by more than one degree, 201 change defect classification, and one cell reads 0.0000° on one platform and 146.6018° on the other. Across five valid tie-break rules the boundaryseam
count ranges 951 to 2207 — a factor of 2.3 from a bookkeeping choice. Grain-ring closures lose rank order entirely (Spearman 0.02) while moving a median of 67°.

Everything angle-based reproduces bit-exact across the same platforms.

We report the fix — exact-constant group elements, a quantised selection rule, and a record split into invariant and gauge tiers — verified on a third platform. We report two of our own results that the discipline killed: a striking spike at exactly 60° that was a tie-break artifact, and a grain-network defect census that was spanning-tree dependent. And we note that one established route, Bollmann’s O-lattice construction, is structurally immune because it never composes reduced rotors at all — a reason to prefer it that we have not seen stated.

A number that moves ten percentage points when you reorder a list of twelve matrices was never data.

Keywords: EBSD · electron backscatter diffraction · disorientation · symmetry reduction · crystallographic gauge · reproducibility · cross-platform determinism · disclination · triple junction · Frank vector · O-lattice · Bollmann · geometrically necessary dislocations · kernel average misorientation · Mackenzie distribution · noise floor · point group · quaternion · holonomy · metrology