A crystal twinned by reticular merohedry of index n - obverse/reverse
rhombohedral domains, or a cubic crystal twinned by a 60 deg rotation about
[111] (the spinel law, index 3) - has its two domains' reflections on a
common lattice of n times the primitive volume of either domain's own cell.
That coincidence cell indexes more frames and more spots than the domain's
cell, so the first pass adopted it (a hexagonal/orthohexagonal or R:R cell
for an F- or I-cubic crystal), and nothing downstream could recover the
true lattice or Laue class.
A new step after the sub-lattice hypothesis (TwinCompositeHypothesis), for
n = 2 and 3, counts the validation spots on the standing cell in the four
classes of the pair of index-n sub-lattices whose union holds the most
spots: on both, on each alone, and on neither. Two domains leave the
(1 - 1/n)^2 of the nodes on neither empty but for chance coincidences; one
crystal, whatever sub-lattice a pseudo-translation favours, populates the
nodes off the stronger sub-lattice alike whether they lie on the second
sub-lattice or on neither. The step adopts the stronger domain's cell when
the two sub-lattice cells are congruent (2 % / 2 deg), the weaker domain's
own class holds at least 20 spots at more than four times the per-node density
of the class on neither, and the domain's cell indexes a majority of the
validation frames. The other domain is left to the leftover-lattice census.
Evaluated on the private, inhouse and a twin/tNCS/rhombohedral/cubic/
long-axis selection of the open arm: it fires on the two known index-3
twins (a spinel-twinned F-cubic crystal, now indexed F-cubic; the in-house
I-cubic insulin set with a 60 deg [111] second domain, R3:R -> I23) with a
weak/neither per-node density ratio of 60 and more (the class on neither
holds 0-8 spots against 250-2850 on the weaker domain), and nowhere else.
On every other set the ratio stays between 1.0 and 1.6, the top of that
range only on runs with a hundred or so validation spots, where the pair
picked as the emptiest of 7 or 13 forms carries the scatter; hence the
factor 4. Both twins now merge in their cubic group, and the sets
re-run against the release baseline otherwise did not change. Cost when it
does not fire: tens of milliseconds per first pass.
Merges two prototypes: the class census and n = 2 coverage of one, the
sub-cell congruence requirement of the other; no environment switch.
Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01SVmAWnzCmRKAXVUCdc4iNi
leonarski_fandClaude Opus 5.5
e25b74dd1c
