GQ128 and a live demo

All concepts in the talk are demonstrated through interactive visualizations generated by AI in real time, without traditional slides.

Full paragraph (translation of the abstract)

The talk concludes with the first public presentation of GQ128 — a complex floating-point format whose mantissa lives in $\mathbb{Z}[\omega]$ rather than $\mathbb{Z}$. This yields isotropic quantization error — a property structurally impossible on the square lattice of the standard IEEE 754 complex representation.

All concepts are demonstrated through interactive visualizations generated by AI in real time, without traditional slides.

What's worth knowing about GQ128

• Base. $Z = 3 + \omega$, $|Z|^2 = 7$. Consistent with the base-7 system on slide 6: same geometry, but in floating-point form.
• Mantissa. 42 slots, 3-bit digits. The number $42 = \varphi(7^2)$ is not a coincidence; it is precisely what ensures self-consistency of the exact 98-point NTT over $\mathbb{Z}/343\mathbb{Z}$ used for lossless multiplication.
• Exponent. Not fixed, but a "floating END-marker" — a spiral coordinate with step $\arg(Z) \approx 19.1°$, irrational with respect to $\pi$.
• Arithmetic. Addition is carry-bounded via a 343-element LUT; multiplication by $\omega$ is a cyclic shift; negation is two's complement in Doppler-EGA encoding.
• Serialization. Values are laid out in memory along the Gosper curve — the same one that bounds the fractal from slide 7. Locality on the curve = proximity in the $\mathbb{C}$ plane.

Why "isotropic error" matters

On the square lattice $\mathbb{Z}[i]$, the quantization error has a different scale along the axes versus along the diagonals: a factor of $\sqrt{2}$ in the corner, a factor of $1$ on the axis. This makes IEEE 754 complex anisotropic: the error depends on the argument. On $\mathbb{Z}[\omega]$, the six nearest neighbors are equidistant — the error is the same in all directions.

GQ128's place in the hierarchy of sphere packings

GQ128 is the 2D member of a broader family "kissing number + 1 = number of digits", which we unfold separately in the discussion:

$d$	Lattice	$\kappa$	$p = \kappa + 1$	Format
1	$\mathbb{Z}$	2	3	Setun (Brusentsov, MSU, 1959)
2	$A_2$ hex	6	7	GQ128 (128 bits)
3	FCC $A_3$	12	13	G626 (626 bits)
8	$E_8$	240	241	— (Viazovska, 2016)
24	Leech $\Lambda_{24}$	196,560	196,561	— (Cohn–Kumar–Miller–Radchenko–Viazovska, 2016)

GQ128 is the only member of the family that fits entirely in a standard 128-bit register. And the only one in which all arithmetic operations are $O(1)$ on a suitable AXR architecture.