Generative Temporal Contact Theory — 9 axiomas · 12 operadores · 0 sorry
A contact-geometric framework for the operator chain G = U∘F∘K∘C with formal Lean 4 verification
Vol IV é uma introdução leve aos números complexos pela geometria de contato. Se você já sabe que $i^2=-1$, este volume mostra de onde esse sinal de menos vem estruturalmente — não como definição, mas como consequência do operador de dobra $F$ e do campo temporal $T$ deixarem de comutar na variedade de contato $(M,\alpha)$.
Vol IV is a light, six-dimensional introduction to complex numbers via contact geometry. If you already know $i^2=-1$, this volume shows where the minus sign comes from structurally — not as a definition but as a consequence of the fold operator $F$ and temporal field $T$ failing to commute on the contact manifold $(M,\alpha)$. No prior differential geometry required; each chapter builds on the previous one.
Volume V (Ring 5 — quateêrnios ℍ) continúa onde este termina, perdendo a comutatividade. Volume V continues where this ends, losing commutativity.
Um operador dimensional $O_i$ é uma função que move o sistema de uma fase $D_i$ para a fase seguinte $D_{i+1}$, como uma engrenagem no relógio — cada dente avança exatamente um passo.
A dimensional operator $O_i$ is a function that moves the system from one phase $D_i$ to the next, $D_{i+1}$ — like a gear tooth advancing a clock by exactly one step. There are 12 operators in total (4 base operations × 3 invariants), and their composition $G = O_{12}\circ\cdots\circ O_1$ is one full cycle.
Mathematically, each $O_i$ acts on the dimensional field $\Delta(x)\in\mathbb{R}^{12}$ via a $12\times12$ matrix:
$A_i = P^i + u_i w_i^T$ (permutation skeleton + rank-1 correction)The permutation $P^i$ is the bare rotation through $i$ steps — the skeleton. The rank-1 term $u_i w_i^T$ is the minimal perturbation that makes each operator distinguishable while keeping $\|A_i\Delta(x)\| = \|\Delta(x)\|$ (norm-preserving). Its size is bounded by the stability radius[Ch 10] $\varepsilon^* = 1/3$: small enough that the whole chain still contracts to a unique fixed point.
The four base operators are: C Compress (Fold inward) · K Constrain (Curvature) · F Fold (the contact singularity) · U Unfold (project back). Each appears three times in the 12-step cycle, once per invariant class. Their composition $G = U\circ F\circ K\circ C$ is the generator of the entire framework.
Pablo Nogueira Grossi · G6 LLC · Principia Orthogona Vol. IV · AXLE v6.1 · Lean 4 + Mathlib4 · 0 axioms extra
We study a 3D model on a contact manifold $(M,\alpha)$ and establish that the unit circle $r=1$ is a globally attracting helix for all $r(0)>1$, with exponential convergence rate $\mu\to-2$. Concrete instantiation of G = U∘F∘K∘C. Submitted to XII Bienal SBM, UFRN Natal, August 2026.
@book{grossi2026gtct,
author = {Grossi, Pablo Nogueira},
title = {Principia Orthogona, Vol. {IV}: {GTCT}},
year = {2026},
publisher = {G6 LLC},
address = {Newark, NJ},
isbn = {979-8-9954416-8-7},
doi = {10.5281/zenodo.19117400},
url = {https://totogt.github.io/GTCT/}
}