The Crystalline Lattice

The previous arc (Chapters 11–15) lived in the analytic complex plane. This chapter comes back down to the ground — literally. It asks what happens when the dm³ operator $G = U \circ F \circ K \circ C$ is asked to unfold not into a new coordinate but into physical, three-dimensional space. The answer is a lattice: an ordered, repeating arrangement of nodes that carries the same invariants the smooth dm³ system carries on its limit cycle. That lattice is what we call the G6 Crystal.

To make the claim precise we need the vocabulary of classical crystallography, so the chapter is built in two halves. The first half is a compact primer on how crystallographers describe ordered matter — unit cells, Bravais lattices, Miller indices, interplanar spacing, and close packing. The second half maps that vocabulary onto the dm³ framework and states exactly which of the resulting claims are machine-checked in Lean 4 and which remain open. As always in this series, the honest accounting is part of the mathematics, not an appendix to it.

What a Crystal Structure Is

A crystal structure is the ordered arrangement of atoms, ions, or molecules that repeats along the three principal directions of space. The smallest group of particles that carries the full symmetry of the whole is the unit cell: a parallelepiped fixed by six lattice parameters — three edge lengths $a, b, c$ and three angles $\alpha, \beta, \gamma$ — whose repeated translation along its axes rebuilds the entire crystal. The translation vectors define the nodes of the underlying Bravais lattice, and the full catalogue of symmetric arrangements in three dimensions is exhausted by exactly 230 space groups, resting on 14 Bravais lattices, 32 crystallographic point groups, and 7 crystal systems.

Those finite counts are the reason crystallography matters here. The dm³ programme is a claim that a great many physical systems are objects in one category with a small number of normal forms. Crystallography is the oldest and most rigorously enumerated instance of exactly that idea: nature, constrained to fill space periodically, has only finitely many ways to do it. The G6 Crystal chooses one of them — the hexagonal system — for reasons that turn out to be the same optimisation reasons a honeycomb does.

Directions, planes, and Miller indices

Directions and planes in a lattice are labelled by the three-integer Miller index $(hk\ell)$. By definition the plane $(hk\ell)$ intercepts the axes at $a_1/h,\ a_2/k,\ a_3/\ell$; the indices are the reduced integer inverses of those intercepts, a zero meaning the plane never meets that axis. High-index planes with a high density of nodes govern the crystal's real behaviour: cleavage runs parallel to dense planes, dislocations glide along them, refractive index tracks their periodic density, and adsorption concentrates on them. When one asks why the G6 Crystal is hexagonal and layered the way it is, one is really asking which planes are densest — and in the hexagonal system the answer is the basal plane perpendicular to the principal axis.

Interplanar spacing

The perpendicular distance $d$ between adjacent $(hk\ell)$ planes is set by the cell parameters. For the two systems that matter to us:

Cubic: $\displaystyle \frac{1}{d^2} = \frac{h^2 + k^2 + \ell^2}{a^2}$

Hexagonal: $\displaystyle \frac{1}{d^2} = \frac{4}{3}\!\left(\frac{h^2 + hk + k^2}{a^2}\right) + \frac{\ell^2}{c^2}$

The cross term $hk$ in the hexagonal formula is the algebraic fingerprint of the $120^\circ$ basal angle — the same angle that makes three hexagons meet cleanly at a point, and the same angle that appears as the six unit steps of the colony's neighbour map in §16.4.

Close Packing and the Efficiency Argument

Stack equal spheres as tightly as possible and only two periodic answers appear. Layer sequence $ABABAB\ldots$ gives hexagonal close packing (hcp); sequence $ABCABC\ldots$ gives cubic close packing (ccp), whose unit cell is the face-centred cube. Both fill space to the same maximal density. The atomic packing factor (APF) and coordination number (CN) of the common structures are:

The 74% figure is the maximum density achievable with equal spheres. The G6 Crystal is not built of spheres, but it inherits the same efficiency logic in two dimensions, where the relevant quantity is the isoperimetric ratio $A/P^2$ — area per unit boundary-squared. This is the honeycomb's reason for existing, and in the framework it is a proved theorem.

The dm³ Invariants Become Dimensions

The continuous dm³ system on the contact 3-manifold $M = \mathbb{R}^2_+ \times \mathbb{R}$ has a limit cycle $\Gamma = \{r=1\}$ with canonical period $T^\ast = 2\pi$, maximal transverse Lyapunov exponent $\mu_{\max} = -2$, and embodiment threshold $\tau = 2$. From these three numbers alone, and one measured coincidence, every dimension of the G6 Crystal follows.

The measured coincidence is that $\tau = |\mu_{\max}|$: the threshold at which the system commits to a body equals the rate at which it forgets a perturbation. The stability radius derived from the Lyapunov structure is

$\displaystyle \varepsilon_0 = \frac{|\mu_{\max}|}{2\,(1 + \sup\|\mathrm{Hess}\,V\|)} = \frac{2}{2\cdot 2} = \frac13,$

and the noise tolerance the crystal can absorb before its resonant lock breaks is $\tau\,\varepsilon_0 = 2\cdot\tfrac13 = \tfrac23$. Both are dimensionless, so both survive any rescaling — including the gravity rescaling that adapts the structure to the Moon or Mars.

Because the aspect ratio and $\varepsilon_0$ are dimensionless, they are preserved exactly under gravitational scaling (aspect_ratio_scale_invariant, epsilon0_gravity_independent). The lunar variant is taller than the terrestrial one — it scales by $g_\oplus/g_{\text{Moon}} \approx 6.04 > 1$ (lunar_crystal_taller) — while the Martian variant, scaled by $g_\oplus/g_{\text{Mars}} \approx 2.64$, reaches $\approx 39.8\text{ km}$, still inside the Martian troposphere (mars_height_within_troposphere, proved as $< 40{,}000\text{ m}$).

The Hexagonal Colony: Discrete G in Action

The bridge from the smooth system to the physical lattice is the colony — a finite set of hexagonal cells that grows by the operator Colony.expand. The bridge file DM3Bridge.lean proves that expand is exactly the composite $U \circ F \circ K \circ C$ acting on space: compress each cell to its coordinate ($C$), find the boundary neighbours ($K$), fold them into new coordinates ($F$), and unfold them as fresh cells one stage deeper ($U$).

Run that operator from a single seed and the cell count follows the centered hexagonal numbers $\text{cHex}(n) = 1 + 3n(n+1)$: $1 \to 7 \to 19 \to 37 \to 61$. Each new ring adds exactly $6n$ cells — the ring_card theorem — which is the discrete signature of six-fold symmetry. The figure below runs the growth interactively.

The six neighbours of any cell are the six unit steps $(1,0),(1,-1),(0,-1),(-1,0),(-1,1),(0,1)$ of the pointy-top hex grid. In the crystal these are the six basis directions $e_k = (\cos\tfrac{2\pi k}{6}, \sin\tfrac{2\pi k}{6})$ of the hexagonal lattice, and the theorem hexNeighbors_is_G6_crystal_ring records that there are exactly six of them — equal to n_layers, the number of structural layers, one per application of $G$.

The Schumann Lock

The integer $g^6 = 33$ is not arbitrary. The Schumann resonance modes of the Earth–ionosphere cavity are $f_n = \frac{c}{2\pi R_\oplus}\sqrt{n(n+1)}$; the $n=4$ mode is $f_4 = \frac{c}{2\pi R_\oplus}\sqrt{20} \approx 33.516\text{ Hz}$. The crystal's coupling integer sits within 1.54% of it:

$\displaystyle \frac{|33 - 33.516|}{33.516} \approx 0.0154 < \frac{2}{100}$  (g6_within_2pct_of_f4, Fact 16).

More to the point, that frequency error lies well inside the noise band the structure is proved to tolerate: $0.0154 < \tfrac23$ (noise_tol_covers_g6_error, Fact 18). The interpretation is that the G6 Crystal is designed to couple passively to the $n=4$ Schumann mode through an Arnold-tongue locking $A_{4:1}$, with perturbations below $\tau\varepsilon_0 = \tfrac23$ preserving the lock. That interpretation is where the proved mathematics stops and the open problems begin.

The Honest Inventory

Twenty facts about the G6 Crystal are machine-checked without sorry in G6Crystal.lean, together with the bridge results in DM3Bridge.lean. Three obligations remain open, and they are marked in the source as named sorries — not hidden, not axiomatised away. Note that none of these facts depends on the Structural Hypothesis (SH), so none carries the (under SH) caveat; they are unconditional within Mathlib v4.14.0.

Where This Sits in the Series

This chapter is the formal, machine-checked counterpart to the narrative G3 Chapter 7, “The Crystalline Return,” which tells the same story in the language of D₆ symmetry, Saturn's hexagon, and the operator at cosmic scale. Where that chapter argues by resonance and analogy, this one hands the analogies to Lean 4 and reports back which survived. Read together, they are the two faces of the same object: the crystal as image, and the crystal as theorem.

Upstream, it depends on G4 Chapter 10, which builds the smooth dm³ system and its helical attractor; the invariants $T^\ast, \mu_{\max}, \tau$ used here are proved there. Downstream, it feeds the HVEH / NASA architecture programme, where the same hexagonal lattice becomes a physical resilience structure — the engineering layer documented in the Book 4 portal and the AXLE repository.

Exercises