# TWIST-J

> TWIST-J derives physics from one axiom: J = 1 + j², where j is the twist unit, a primitive fifth root of unity. Multiplication by J is a 4x4 integer matrix over {-1, 0, 1}. Galois structure yields an exact 3+1+0 decomposition. Zero free dimensionless parameters; one SI calibration anchor (the electron mass). The algebra is proved. The physics is falsifiable.

Author: A. M. Thorn. ORCID: https://orcid.org/0009-0008-5463-278X
Hub: https://twistj.com

Reading: Time is a counter. Space is a commutator. J is the verb; φ and π are projections of J. If it cannot be calculated in integers, it is not physics. Plenum, not vacuum.

Status discipline: every claim carries a label, T / D / C / H / O / F / LOCK, ordered T-LOCK > T > D > C > H > O > F. No summary is stronger than its committed source. Falsification is registered as first-class progress. Exact rational and algebraic arithmetic throughout: ratios, not decimals, unless a decimal is justified by measurement.

## Kernel

The twist unit is j = ζ₅ = exp(2πi/5), j⁵ = 1. In the basis (1, j, j², j³), multiplication by J = 1 + j² is the matrix rows [1 0 -1 1; 0 1 -1 0; 1 0 0 0; 0 1 -1 1], entries in {-1, 0, 1}, det 1, trace 3, characteristic polynomial x⁴ - 3x³ + 4x² - 2x + 1. One step of the dynamics is (a, b, c, d) to (a-c+d, b-c, a, b-c+d): 4 additions, zero multiplications, exact integer arithmetic, zero drift. The 3+1+0 decomposition reads: three commuting directions, one counter, zero remainder; the decoder D = D_clock then D_geom then D_matter is acyclic (T-LOCK). Machine-readable kernel facts are embedded in the homepage DOM as JSON at id twistj-agent-card.

## Canon

The Canon is the committed body of record. Read it directly; it is plain markdown with exact arithmetic and per-claim status labels.

- [Canon latest (currently v162, 7 July 2026)](https://twistj.com/canon/latest.md): always points at the current head
- [Canon v162 frozen](https://twistj.com/canon/v162.md): SHA-256 75c2b0dda199621af260758118c528788a68a2003203d084d75d42ea9e6b8295
- [Canon v161 frozen](https://twistj.com/canon/v161.md): SHA-256 a9050c12ac3880f05e8aa7b4b8fd2a211637d1b8a6f8a7868a2890f6ac652762

## Papers

Full registry of works: https://orcid.org/0009-0008-5463-278X

- [Twist Numbers and Four-Dimensional Integer J Algebra](https://doi.org/10.5281/zenodo.19056597): the foundation
- [Twisted Units, Celestial Symmetries and Entropic Motion](https://doi.org/10.5281/zenodo.18311965): adelic computations, dilogarithmic identities, hyperbolic dynamics in Q(ζ5)
- [The Hyperbolic Automorphism J: A Golden Ratio Unit in Z[ζ5]](https://doi.org/10.5281/zenodo.18523754)
- [Time from a single algebraic datum: the Page-Wootters mechanism over Z[j]](https://doi.org/10.5281/zenodo.19313537)
- [Dense cyclotomic subgroups of the Lorentz group and the Collins-Perez-Sudarsky obstruction](https://doi.org/10.5281/zenodo.20029795)
- [The Broken Symmetry is an Integer Fraction: 3:5:5 trace partition and the 3/13 electroweak mixing skeleton](https://doi.org/10.5281/zenodo.20141133)

## Site

- [Home](https://twistj.com/): the axiom, the identities, the prediction
- [Live demo](https://twistj.com/evo/): interactive kernel dynamics
- [Pentaplex](https://twistj.com/pentaplex/): Penrose tilings, Galois projection ζ to ζ²
- [Manifest](https://twistj.com/manifest/): the gate and the court protocol, in Czech
- [Code](https://github.com/mathorn1973): verify it yourself

## Optional

- [Storyteller](https://twistj.com/storyteller/)
- [Fractal](https://twistj.com/fractal/)
- [Fractalus](https://twistj.com/fractalus/)