No Reflection Knows Left from Right Below Index 12

01 / the empty detector

Watch every case disappear.

A systematic absence is not a dim spot. It is a structure factor forced to zero for a symmetry-generated general-position orbit. The worker below constructs that orbit for each integer index and asks whether every independent coefficient cancels exactly.

complete censusstandard Hall settings / spglib 2.7.0

11pairs visited

0paired verdicts tested

0differences found

preparing exact sweep

02 / move the boundary

The emptiness has an edge you can touch.

Move the cube from max index 1 to 12 and choose any genuine hand pair. The set stays empty. Then relax exactly one thing: compare P3 with P31. The unchanged searcher fills the set at 001. Restore P31/P32 and it empties again.

pair comparatorsame extinction predicate

operation lists compared

max index: 12

sweeping

exact pair sweep
domain: 15,624 nonzero reflections

Positive control

not run

The IUCr teaching pamphlet states the rule (a reflection is extinct when hR = h and h·t is not a whole number) and takes P3₁ as its threefold example. Applied to the 3₁ screw along c, the rule extinguishes 001, which P3 has no reason to extinguish; the pamphlet does not name 001 itself. This button calls the same pair comparator on that derived reflection.

waiting for re-run

Planted witness

The real P31/P32 verdict at 001 is empty. Zero the P32 screw translations, making a labelled synthetic no-rise orbit, and the unchanged extinction logic finds exactly one difference.

unplanted real sub-domain: empty

03 / inside a cancellation

Open the zero.

Each operation sends a general coordinate x to Rx + t. Operations that produce the same transformed reciprocal index q = R^Th share a symbolic coefficient. Its twelfth-root phase polynomial must reduce to zero for the reflection to be symmetry-forbidden.

coefficient(q) = sum x^(h dot 12t), with x = exp(2 pi i / 12)
absence = every coefficient is exactly 0 modulo Phi12(x)

orbit polynomial inspectorno floating zero

pair h k l

04 / why index 13 is not waiting in ambush

The finite null has a structural shadow.

For every shipped pair, inversion x maps to -x turns every left-hand Hall operation (R, t) into the right-hand operation (R, -t). At the same h, that negates every phase exponent. The coefficient becomes its complex conjugate, and zero remains zero. The page checks operation-set equality for the selected pair before saying so.

(R, t) under x maps to -x becomes (R, -t)
P(x) maps to P(x^-1), so P = 0 if and only if its conjugate is 0

That is stronger explanatory evidence than a larger cube. Still, the certificate headline remains the finite enumeration actually carried out live: 171,864 tests, stopping at 12. The all-index statement is about the shipped spglib Hall operations and the exact algebra shown here, not an independent audit of every crystallographic convention.

05 / change the observable

Handedness returns through intensity, not extinction.

The null does not say diffraction can never determine hand. It says Boolean space-group extinction cannot choose between these pairs. In a two-scatterer toy motif, turn on an imaginary anomalous-scattering term and compare the model intensities at h and -h. This is a different, model-dependent question.

Bijvoet boundary modelillustrative amplitudes, not measured data

imaginary scattering term f double-prime: 0.00

0I(1,2,1)

0I(-1,-2,-1)

0Bijvoet difference

Flack and Bernardinelli write the inversion-twin intensity as a mixture of |F(h)| squared and |F(-h)| squared. Its half-difference can carry inversion-distinguishing information when the underlying Bijvoet difference is nonzero. None of that changes the symmetry extinction conditions swept above.

06 / audit surface

The check

Every free choice and uncertainty belongs beside the green result.

Predicate chosenGeneral-position, ideal kinematic systematic absence. Accidental motif cancellation, dynamical diffraction, multiple scattering, defects, modulation, background and symmetry lowering are outside it.
Domain chosenEvery nonzero integer triple in [-12, 12] cubed, for the symbol-bound list of 11 enantiomorphic pairs. The bound 12 is a disclosed finite choice.
Settings chosenThe first Hall setting returned for each Hermann-Mauguin symbol by spglib 2.7.0. Settings and reciprocal-index conventions matter, so the exact operation lists ship with the page.
Arithmetic chosenTranslations are checked as integer twelfths. Root-of-unity sums reduce in the integer basis of Phi12. No intensity threshold and no floating tolerance decides an extinction.
Database dependenceThe page regenerates and checks the census from the shipped operation lists. It does not independently recreate the International Tables database from first principles.
Observable boundaryThe extinction census is Boolean and symmetry-only. The Bijvoet control below it is a disclosed two-scatterer model with a user-chosen anomalous term, not an experiment.
Data licenceThe 22 operation lists are derived from spglib 2.7.0 under BSD-3-Clause. Its full notice is shipped as SPGLIB-LICENSE.txt.
Artifact identityoperations.mjs SHA-256:

Source-label correction. The retrieved International Tables web rendering attaches inconsistent type numbers to the P61/P65 and P62/P64 symbols. This page binds the census to symbols and exported Hall operations. The generated spglib table below gives P61/P65 as 169/170 and P62/P64 as 171/172.

pair	type numbers	Hall numbers	operations

07 / primary sources

What the machine is standing on.

Souvignier, A general introduction to space groups, International Tables for Crystallography A, section 1.3.4.1. Definition and symbolic list of the 11 enantiomorphic pairs.
Hovmoller, Rotation matrices and translation vectors in crystallography, IUCr Teaching Pamphlet 9. The hR = h and h dot t phase-cancellation criterion, including P31.
spglib 2.7.0 API documentation and project documentation. Hall database operation export and BSD-3-Clause terms.
Flack and Bernardinelli, Absolute structure and absolute configuration, Acta Crystallographica A55, 908-915. Inversion-twin amplitudes, half-differences and anomalous-scattering information.