G6 LLC · Newark, New Jersey · AXLE v6.1 · 0 axioms beyond Mathlib4 · 8 verified constants

Principia Orthogona

G¹–G⁵ · Complete Completeness. Five volumes. Five complete orbits of the operator G. The series is its own fixed point.

G = U ∘ F ∘ K ∘ C · g₃₃ = 33 · τ = 2 · ε* = 1/3 · T* = 2π
Student Portal → IMPA Portal →
C  →  K  →  F  →  U  →  ∞
Principia Orthogona · The Complete G-Series

Five Turns of the Spiral

G6 LLC · Newark NJ · DOI 10.5281/zenodo.19117400

🇺🇸 English English
The G-Series · A Série G

Five Volumes · Cinco Volumes

AXLE v6.1 — Verified Constants · 0 Axioms Beyond Mathlib4
g₃₃ = 33
ε* = 1/3
τ = 2
g₆₄ = 2⁶ = 64
T* = 2π
κ ≤ √(7/9) ≈ 0.882
τ·ε* = 2/3
ε₀ = 1/3

Note: g₃₃ (threshold invariant) and G⁶ (sixth operator application, open conjecture) both involve 33 — they are distinct mathematical objects.

The Student Journey · A Jornada do Estudante

From A1 to D2

The series defines a precise pathway. D2 is not a metaphor — it is a mathematically verified threshold: Θ = g₃₃ + N × M.

A1
First contact
Compression C
A2
Curvature K
Day 21 · Dia 21
B1–B2
Folding F
Domain entry
C1
Unfolding U
Research gen.
D1
g₃₃ = 33 cycles
Individual fixed point
D2
Θ = 33 + N×M
Complete Completeness

"Your education is yours. No one can take it away from you."

— Pablo Nogueira Grossi, Newark NJ · The Seed (Principia Orthogona Vol V)
Formal Verification · Verificação Formal

AXLE — The Series Proves Itself

AXLE (Automated eXtensible Lean Engine) is the formal verification backbone of Principia Orthogona. All 8 structural constants are machine-verified in Lean 4 + Mathlib4, with zero additional axioms. 9 open problems are named precisely as sorrys — each a conjecture with a known missing lemma, not an evasion.

/- Mathematics is a language. These theorems have been proved in every language simultaneously. A matemática é uma linguagem. (Portuguese) Las matemáticas son un idioma. (Spanish) Les mathématiques sont une langue. (French) Mathematik ist eine Sprache. (German) 数学は言語である。 (Japanese) 数学是一种语言。 (Mandarin) الرياضيات لغة. (Arabic) Математика — это язык. (Russian) Hisabati ni lugha. (Swahili) गणित एक भाषा है। (Hindi) -/ -- 0 axioms beyond Mathlib4 -- 8 verified constants · 9 honest sorrys -- g₃₃=33 · ε*=1/3 · τ=2 · g₆₄=64 · T*=2π · κ≤0.882

View AXLE on GitHub ↗

Open Access Research · Pesquisa de Acesso Aberto

Papers on Zenodo

All papers are freely available. Cite via DOI. Each paper is a self-contained contribution.

Zenodo · 10.5281/zenodo.19117400
Principia Orthogona, Volume One
Abstract operator algebra. The operator sequence C → K → F → U defined and proved. Singularity classification (Whitney A1–A3). The foundation.
doi.org/10.5281/zenodo.19117400 ↗
Zenodo · 10.5281/zenodo.19379473
Principia Orthogona, Volume Two
Contact Realization of Generative Transitions. Threshold Equivalence theorem. Four dm³ bifurcations ↔ Whitney A1–A3. Submitted to IMPA.
doi.org/10.5281/zenodo.19379473 ↗
Zenodo · 10.5281/zenodo.19122168
Generative Contact Mechanics
A Geometric Framework for Dissipative Systems. Universal contact normal form (μ_max, ω, β). Stability radius ε₀ = 1/3. Submitted to Journal of Geometric Mechanics.
doi.org/10.5281/zenodo.19122168 ↗
Zenodo · 10.5281/zenodo.19379385
The dm³ Operator — Explicit Toy Model
Global Dynamic Analysis on contact manifold M = ℝ²×ℝ. Canonical invariant triple (T*, μ_max, τ) = (2π, −2, 2). Global attractor Γ₁₂.
doi.org/10.5281/zenodo.19379385 ↗
Zenodo · 10.5281/zenodo.19162013
The G6 Crystal
A dm³-derived architectural form. Hexagonal tower with aspect ratio 66 = 33·τ. Passive Schumann n=4 resonance coupling at 33.516 Hz.
doi.org/10.5281/zenodo.19162013 ↗
Zenodo · 10.5281/zenodo.19208015
Biological Transitions as Multi-Agent Realisations
Neural oscillations, HPA-axis, circadian regulation, immune adaptation, and protein conformational change modelled as realisations of G = U∘F∘K∘C.
doi.org/10.5281/zenodo.19208015 ↗
Zenodo · 10.5281/zenodo.19210137
Fruit-Fly Connectome Toy Model
Drosophila connectome toy model. Neural clusters as coupled agents whose trajectories stabilise under six applications of G. Reproduces neural coherence.
doi.org/10.5281/zenodo.19210137 ↗
Zenodo · 10.5281/zenodo.19208284
The Swarm Simulator
A Dynamical Systems Model of Collective Intelligence using the TO/TOGT Operator Pipeline. Four collective operators: shared-intent stability, coordination efficiency, type-propagation, diffusion.
doi.org/10.5281/zenodo.19208284 ↗
Zenodo · 10.5281/zenodo.19378742
The Collatz Conjecture as a Corollary of Crystal Geometry
The coefficient c = 3 in 3n+1 is the fingerprint of triad stabilisation (3×11 = 33). The conjecture is visible from within crystal geometry before it is provable within it.
doi.org/10.5281/zenodo.19378742 ↗
Zenodo · 10.5281/zenodo.19210058
Mathematical Foundations of Multi-Orbit Identity Theory
Identity orbits defined as operator-generated closed trajectories with invariant structure within the TO/TOGT framework. Categorical invariants.
doi.org/10.5281/zenodo.19210058 ↗