Planck Bits Foundations Using Hyperbit Ternary Math

Abstract

This paper contains a short history and summary of the mathematical properties developed during my thirty year effort researching hyperdimensional bits (aka hyperbits).  Hyperbits are expressed in Geometric Algebra (GALG) [1] so are compatible with physics [2], and interestingly has a large number of three-ness relationships. In my recent research connecting hyperbits to blackholes [3], I discovered that 6 hyperbits (aka a BekBit) are the math root of the “physical” nature of fermions (particles with mass) [4] and the vacuum [5]. So therefore, for each hyperbit, Landauer’s Principle energy per bit formula [6] applies and along with Einstein’s Energy to mass formula [7], so that at Planck temperature T_Planck, resulting in each bit has an effective mass of M_bit = ln2 / (2π) * M_Planck = 0.11 M_Planck (see details in [3]), thus making it a first class Planck Bit physics entity, as related to other Planck units. This term was suggested by Melvin Vopson at his Aug 2026 ICIP conference [8]. Since hyperbits (using GALG) are not well known, this paper forms a baseline education of the properties and ubiquitous three-ness relationships of hyperbits including the ternary logic of symmetric GF(3) (a trit {-1,0,+1} vs a bit {0,1}), which is then used in complexified behaviors of GF(9) with coefficients {0,±1, ±i }. This complex trit valued (or ternary) logic is symmetric mod 3 arithmetic bound by unit coefficients ±1 so addition’s carry/borrow is inline by inverting the sign (i.e. +1+1=2=-1 and -1-1=-2=+1). This GF(3) consistent GALG convention is denoted as G(p, n, Z3C), a symmetric ternary Galois subset of CL(p, n). Next importance are the squaring classifications of unitaries, quaternions, nilpotents  idempotents, involutions, and followed by the cubing properties of 3nilpotents, trines, anti-trines, tripotents, anti-tripotents, Frobenius identity, Affine invariance and higher order roots of unity. I built a GALG symbolic math tool (in Python) [8] to systematically represent and preform the math formulas consistently in Z3C GALG framework used in this paper and my research. This paper was written manually, supported by all research interactions with my GALG tools and with some math conceptual/proofs support from Claude AI tool.