⚜ PRINCIPIA ORTHOGONA · Vol VI · Roots · WP-103 ← WP-102 · What the Limb Allowed
Same ground, elsewhere in the series Bk4 · Ch 16 · The Crystalline Lattice · Bk4 · Ch 21 · The Closing Field · Σ · Pentanacci · Bk8 · Spectral Hierarchies
#Derivation
Vol VI · Roots · WP-103 · Received 2026-09-06 · Keystone for WP-104, WP-105 · Not novel — standard crystallography, stated because the corpus leans on it

Why Six, and Why Not Five

The crystallographic restriction is asserted in two chapters of this corpus and derived in none of them. It is three lines. Those three lines are also the unanswered question under Chapter 16’s own figure — why the colony grows six-fold — and the reason Chapter 21’s shell needs exactly twelve defects.
Methodone trace argument, one field degree, one companion matrix
reproduced by wp103-verify.py
Inputsnone external. This is nineteenth-century crystallography and standard algebraic number theory
Claim typeno novelty claimed — a derivation of something the corpus already uses
closes a gap, opens nothing
Statussettled
the theorem is classical; what is new here is only that it is now written down in this corpus
This note claims nothing. It exists because two chapters of this corpus rest on a theorem that appears nowhere in it, and because a reader who follows Chapter 16’s colony to 1 → 7 → 19 → 37 → 61 is entitled to ask why the number under all of it is six.
COMPUTED produced by the companion script CHECKED verified against a primary source OPEN not established here

§ 1 What the corpus asserts, and does not derive

Two chapters use the crystallographic restriction. chSigma-pentanacci.html states that five-fold symmetry “is forbidden in crystal lattices by the crystallographic restriction theorem” and that quasicrystals are the exception; book8/ch-q1-spectral-hierarchies.html uses it in the same way. CHECKED Neither derives it. A corpus-wide search returns zero occurrences of totient and no statement of the integrality condition anywhere. COMPUTED

Meanwhile book4/ch16-crystal-lattice.html builds the hexagonal colony explicitly: Colony.expand is proved in DM3Bridge.lean to be the composite U ∘ F ∘ K ∘ C acting on space, the cell count follows the centered hexagonal numbers cHex(n) = 1 + 3n(n+1), and ring_card gives exactly 6n new cells per ring. The chapter proves the six-fold growth and never says why six-fold is one of the orders available to it.

§ 2 The theorem, in the plane

Let L be a lattice in the plane and R a rotation of order n mapping L to itself. Written in any basis of L, R has integer entries, because it carries lattice vectors to lattice vectors. Trace is invariant under change of basis, and in the standard orthonormal basis the trace of a rotation by 2π/n is 2 cos(2π/n). Therefore

2 cos(2π/n) ∈ ℤ

and since |2 cos| ≤ 2 the trace lies in {−2, −1, 0, 1, 2}, giving n ∈ {1, 2, 3, 4, 6} and nothing else. COMPUTED

n  =  1   2   3   4   6
2cos =  2  −2  −1   0   1

Five is absent because 2 cos(2π/5) = (√5 − 1)/2, which is irrational. That is the whole of it.

§ 3 The same statement as a field degree

The trace argument is elementary but it hides the structure. Write ζ = e2πi/n; then 2 cos(2π/n) = ζ + ζ−1, which generates the maximal real subfield ℚ(ζn)+, of degree φ(n)/2 over ℚ. A number of degree k is rational exactly when k = 1. So

2 cos(2π/n) ∈ ℤ  ⇔  φ(n) ≤ 2

and φ(n) ≤ 2 holds for n = 1, 2, 3, 4, 6 alone. COMPUTED The restriction is a statement about Euler’s totient, which is why it generalises cleanly and the trace version does not.

§ 4 Any dimension: the answer is φ(n)

In dimension d, a lattice carrying a rotation of order n exists if and only if φ(n) ≤ d. The construction is explicit: the companion matrix of the n-th cyclotomic polynomial is an integer matrix of size φ(n) whose order is exactly n, and it acts on ℤφ(n). wp103-verify.py builds that matrix for each n below and confirms both properties. COMPUTED

φ(3) = φ(4) = φ(6) = 2  →  the plane
φ(5) = φ(8) = φ(10) = φ(12) = 4  →  four dimensions
φ(7) = φ(9) = 6  ·  φ(11) = 10  ·  φ(13) = 12  ·  φ(30) = 8

So no symmetry is forbidden outright; each one has a price, denominated in dimensions. What the plane forbids, four dimensions permit — and that is the whole subject of WP-104.

§ 5 What this settles in the corpus

Chapter 16. Six-fold is available because φ(6) = 2: it fits the plane, so Colony.expand can run forever without leaving it. The colony grows flat not by construction but by arithmetic, and ring_card’s 6n is the discrete signature of the one non-trivial order the plane allows besides 3 and 4.

Chapter 21. The same fact is why closing that sheet costs twelve defects. A six-fold plane tiling has zero angular defect at every vertex; a sphere needs 4π; the elementary disclination is 2π/6 because six is what the lattice carries; and 4π ÷ (2π/6) = 12. ring_card and the twelve pentagons are two readings of one theorem.

Σ · Pentanacci. Five-fold is forbidden because φ(5) = 4. The chapter’s assertion is correct and now has a derivation to point at.

One date, while we are here

chSigma-pentanacci.html gives Shechtman’s discovery as 1984. That is the publication year — Shechtman, Blech, Gratias and Cahn, Phys. Rev. Lett. 53, 1951 (1984). The observation was made on 8 April 1982. CHECKED

§ 6 What this note does not establish

Nothing here is novel and none of it is claimed as such. The trace argument is Fedorov-era crystallography; the totient form is standard; the companion-matrix construction is textbook algebraic number theory. The note exists so that two chapters of this corpus stop resting on an unstated theorem, and so WP-104 and WP-105 have something to cite.

It says nothing about dm³. That a lattice admits six-fold symmetry is a fact about lattices, not about the operator chain that happens to grow one. OPEN The connection between the crystal arithmetic and the dm³ systems — and what closure costs thermodynamically — is chapter work, not a working paper, and is not attempted here.

§ 7 Sources

[1] Shechtman, Blech, Gratias & Cahn, “Metallic phase with long-range orientational order and no translational symmetry”, Phys. Rev. Lett. 53, 1951 (1984); observation 8 April 1982; Nobel Prize in Chemistry 2011.
[2] The trace argument is classical; see any crystallography text on the restriction theorem.
[3] Washington, Introduction to Cyclotomic Fields, for [ℚ(ζn)+ : ℚ] = φ(n)/2.
[4] This corpus: book4/ch16-crystal-lattice.html (Colony.expand, ring_card, DM3Bridge.lean); book4/ch21-the-closing-field.html; chSigma-pentanacci.html; book8/ch-q1-spectral-hierarchies.html.