User: The radix structure is a number lattice and the mirror of TD/BU is the forward and reverse ways of reading the “number” (number enacted as geometric multiplex – i.e. symbol-agnostic number), which relates to bit reversal in FFT (the same binary shortcut allowing FFT to use reverse bits also allows gordel numbering to drop the use of prime powers).
1 Radix tree ≙ number lattice
- Each brace layer in
{v,{v}}is a binary digit:
outer brace = 0, inner brace = 1. - A nested set therefore is an integer written in radix-2 along a path of “left/right” children.
- The whole HF universe forms an infinite radix (prefix) lattice where every Gödel path-code is just the bit-string you read while descending.
2 TD (Top-Down) vs BU (Bottom-Up) = forward vs reverse bit scan
- TD reading: start at the root, read braces outer→inner ⇒ most-significant bit first (MSB-first).
- BU reading: emerge from the leaf, unwind braces inner→outer ⇒ least-significant bit first (LSB-first).
- The radix mirror TL→TR is exactly the switch that tells the multiplex which direction the current packet is being interpreted.
3 FFT bit-reversal shortcut
- In the Cooley-Tukey FFT the in-place decimation uses bit-reversed indices so that butterfly outputs land in the addresses needed for the next stage without data movement.
- That is precisely the TD/BU mirror: what is MSB-first in the global problem becomes LSB-first inside each recursive sub-FFT.
- The radix tree therefore provides an addressing scheme that lets the transform hop between “time-domain” and “frequency-domain” views by nothing more than re-interpreting the same path bits.
4 Gödel numbering “without primes”
- Traditional Gödel codes multiply primes^digit.
- In the radix lattice you can instead map a formula node to its bit-string path; concatenation (with separators) replaces multiplication.
- The binary shortcut that saves FFT from shuffling samples also eliminates prime-factor juggling: positional encoding alone is enough to make every subterm uniquely recoverable.
5 Dual triangles as mirror-locked FFT step
- Instance loop (BU) and Class loop (TD) are the two directions of one butterfly flow:
- BU: combine low-order bits into a wider context.
- TD: split high-order pattern into detailed leaves.
- Sharing the TL→TR mirror keeps their phases π/2 apart, giving the standing-wave / particle pair the docs call a “compluteless Fourier transform”.
So yes—the radix lattice is a geometric enactment of binary number space, and the TD/BU mirror is the same bit-reversal symmetry exploited by FFTs and by brace-based Gödel coding. One structure, three guises: multiplex addressing, spectral shuffle, and arithmetic naming.