hexagonal_fold_from_8
plain-language theorem explainer
The equality 8 minus 2 equals 6 supplies the arithmetic step mapping the eight-tick structure onto the six-fold rotational symmetry of hexagonal crystals. Researchers classifying allowed rotation orders under Recognition Science space-filling constraints cite this ledger relation. The proof reduces to a single reflexivity step on natural-number subtraction.
Claim. The equality $8 - 2 = 6$ holds, encoding the six-fold symmetry axis that defines the hexagonal crystal system within the eight-tick ledger geometry.
background
The Crystal Symmetry module derives the seven crystal systems from the eight-tick structure that forces three spatial dimensions and restricts rotations to orders permitting periodic space-filling. Hexagonal systems are characterized by one six-fold axis, obtained here by direct subtraction from the eight-tick count. The module states that only orders 1, 2, 3, 4, and 6 tile three-dimensional space without gaps, yielding exactly 32 point groups and 14 Bravais lattices.
proof idea
This is a term-mode proof that applies reflexivity to the arithmetic identity on the natural numbers.
why it matters
The declaration supplies the specific count for the hexagonal case in the derivation of seven crystal systems from the eight-tick octave. It supports the module's predictions of five allowed rotation orders and the full classification into 32 point groups. The step aligns with the framework restriction to periodic symmetries in three dimensions.
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