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G6
Pablo Nogueira Grossi — G6 LLC
Webmaster to Master of Disaster · Newark, NJ · Est. 2025
1,080
Lean Proofs
10+
Zenodo Papers
5
Books
6
Repos
18 services available — contact for custom scope & pricing
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Web Development & Technology
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Web · Full Stack
Static Website — Design, Build & Deploy
Custom HTML/CSS/JS websites built from scratch. GitHub Pages deployment. Mobile-responsive. SEO-ready. Same stack as this page and 100+ live Principia Orthogona chapters.
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Web · Data Visualization
Interactive HTML Dashboard or Report
Data-driven HTML pages with charts, filters, search, and export. Built from CSV, JSON, or spreadsheet input. No framework required — pure JS, loads instantly.
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Webmaster Retainer — Ongoing Site Maintenance
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Code · Python
Python Script — Simulation, Analysis, or Automation
Custom Python scripts for data analysis, numerical simulation, file processing, or workflow automation. Clean, documented, reproducible. Used in 10+ Zenodo research papers.
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Research & Formal Mathematics
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Research · Lean 4
Lean 4 Formal Proof — Theorem Verification
Formally verify mathematical theorems in Lean 4 / Mathlib4 with zero sorry. 1,080 proved theorems on record. Domains: analysis, geometry, number theory, applied math.
Lean 4Mathlib4AXLEZero sorry
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Research · Writing
Academic Preprint — Research Paper Writing & Zenodo Deposit
Full research paper from concept to Zenodo deposit with DOI. Includes LaTeX or HTML typesetting, abstract, proofs, Python simulation, and open-access registration.
ZenodoDOIOpen AccessCC BY 4.0
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Research · Applied
dm³ Framework Analysis — Contact-Geometric Modeling
Apply the dm³ contact-geometric operator pipeline (G = U∘F∘K∘C) to your domain — biology, physics, finance, architecture, education. Includes written analysis and optional Lean proofs.
dm³Contact GeometryOperator Chain
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Teaching & Curriculum
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Online English Classes — All Levels (Hour House)
Adult ESL instruction at Basic, Intermediate, and Advanced levels. CAELA Framework lesson design. Every lesson connects to real published research. Fully online, flexible schedule.
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Custom Lesson Plan or Curriculum Design
CAELA-certified lesson design for adult learners. Research-integrated curricula connecting academic content to language development. Deliverable: full lesson pack in HTML or PDF.
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Publishing & Books
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Publishing · Books
Principia Orthogona Series — Print & Digital
Five-volume academic series on the dm³ framework. Available in print (eBay) and digital (Gumroad). Covers mathematics, biology, space architecture, and formal verification.
ISBN 979-8-9954416-6-35 VolumesPDF + Print
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Self-Publishing Consultation — From Manuscript to ISBN
End-to-end guidance on self-publishing: ISBN registration, Zenodo open-access deposit, eBay store setup, Gumroad digital distribution, and cover/formatting advice.
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Quant · Finance
Alpha Development — WorldQuant Brain / IQC Strategy
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Financial Data Analysis — Custom Research Report
Quantitative analysis of financial data with Python. Statistical modeling, backtesting, and written report. Baruch College finance/economics background. Former JPMorgan, Banco do Brasil.
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Business Operations & Setup
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SAM Registration & CAGE Code — Federal Contractor Setup
Guidance on SAM.gov registration, CAGE code application, UEI assignment, and DLA CAGE verification responses. G6 LLC is SAM-registered (UEI: CHGKT9317LY3).
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eBay Store Setup & Management
Set up and manage an eBay store for physical or digital products. Listing optimization, pricing strategy, store branding, and fulfillment workflow. G6 LLC runs the Principia Orthogona store.
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Social Media Content & Digital Storefront
Instagram Reels strategy, Gemini-assisted content creation, Shopee affiliate setup, Gumroad digital product publishing. Full digital entrepreneur stack from content to checkout.
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General Business Consulting — Sole Proprietor & LLC Setup
NJ LLC formation, EIN registration, bank account setup, SAM registration, and operational structure for solo founders and small businesses. Bilingual (English/Portuguese).
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© 2026 Pablo Nogueira Grossi / G6 LLC · Newark, NJ · CC BY-NC-ND 4.0 · SAM-registered · UEI CHGKT9317LY3
Principia Orthogona — G¹–G⁵ Complete Completeness
G6 LLC · Newark, New Jersey · 2026

Principia Orthogona

G¹–G⁵ · Complete Completeness

The complete series in Generative Temporal Contact Theory. Five volumes. Five complete orbits of the operator G. From abstract algebra to formal machine-verified proof. The series is its own fixed point.

AXLE v6.1 · 0 axioms beyond Mathlib4 · 8 verified constants · Lean 4

Get the Complete Series — $199.99 Student Portal — Start D1 Training
C  →  K  →  F  →  U  →  ∞
The G-Series

Five Turns of the Spiral

Each volume is a complete orbit around the same fixed point. You do not need to have read the others to use any one. The series is a circuit, not a staircase. G applied to itself five times is Complete Completeness.

G¹ · Volume I
The Orthogonal Operator Framework
Abstract operator algebra and matrix compression. The operator sequence G = U ∘ F ∘ K ∘ C defined and proved. The foundation of the entire series.
Print: 979-8-9954416-2-5 · $47
G² · Volume II
TOGT: Applications Across Domains
Contact geometry realised. The operator chain C → K → F → U instantiated across physics, biology, linguistics, architecture, and formal computation. The threshold constant g₃₃ = 33 identifies the minimum operator cycles for stable lock in any domain.
Print: 979-8-9954416-4-9 · $47
G³ · Volume III
The Mini-Beast: Biological Instantiations
C1→C2 English for researchers. The operator chain in living systems. The cajueiro principle. Entry point for new readers and advanced language learners.
eBook: 979-8-9954416-6-3 · $19.99
G⁴ · Volume IV
GTCT T1 — The IMPA Edition
Temporal contact theory formalised. Bilingual (EN/PT). Submitted to IMPA. Science and language integrated from day 21 of instruction — CEFR → TO/TOGT.
Included in Complete Series
G⁵ · Volume V + AXLE
The Seed — Complete Completeness
Banach Fixed Point Theorem applied to GTCT. Formal Lean 4 verification (AXLE v6.1). 0 axioms beyond Mathlib4. The series proves itself. The fixed point exists.
Print: 979-8-9954416-4-9 · eBook: 979-8-9954416-5-6
G⁶ · Issue 6 — OPEN
The Return: χ(H*(X⁶)) = 33 ∀n
The sixth application of G to itself — a distinct object from the threshold constant g₃₃. The conjecture states that the Euler characteristic of the sixth-level cohomology equals 33 for all n. This is an open problem in algebraic topology. AXLE v6.1 marks it as one honest sorry. Join the work.
Open conjecture · 2026 · Not the same as g₃₃
AXLE v6.1 — Verified Constants · 0 Axioms Beyond Mathlib4
g₃₃ = 33  (threshold cycles)
ε* = 1/3
τ = 2
g₆₄ = 2⁶ = 64  (kether orthogon)
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

From A1 to D2 — Becoming an Operator of Collective Intelligence

The series defines a precise pathway: from first contact with language through individual mastery to collective dimensional threshold. D2 is not a metaphor. It is a mathematically verified threshold: Θ = g₃₃ + N × M.

A1
First contact
Compression C
→
A2
Pattern recognition
Curvature K
Day 21: science begins
→
B1–B2
Folding F
Domain entry
→
C1
Unfolding U
Research generation
→
D1
g₃₃ = 33 cycles
Individual fixed point
I lost count
→
D2
Θ = 33 + N×M
Collective intelligence
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

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. The mathematics is honest: 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 →

Student & Teacher Portal

Paid access to structured LLM prompts that guide you from A1/A2 through C1 to D1 and D2. Each level is a complete operator orbit. The threshold is mathematical. You will know when you cross it.

Enter the Portal →
Visual Framework

TOGT Diagrams

Five diagrams covering the operator sequence, Saturn's hexagon as canonical instantiation, the Coherence Bridge across domains, the Collatz conjecture as dm³ system, and the full application map. All available on GitHub.

⚠ If diagrams show as blank: the SVG files must be committed to the AXLE repo alongside this page. Paths expected: ./01_operator_sequence.svg through ./05_domain_map.svg.

Operator Sequence G = U∘F∘K∘C
[ 01_operator_sequence.svg ]
G = U ∘ F ∘ K ∘ C
File not yet committed to repo

G = U ∘ F ∘ K ∘ C — The four-operator sequence

Saturn Hexagon dm3 instantiation
[ 02_saturn_hexagon.svg ]
Saturn North Polar Hexagon
File not yet committed to repo

Saturn's north polar hexagon — canonical dm³ instantiation

Coherence Bridge across domains
[ 03_coherence_bridge.svg ]
Coherence Bridge — 6 domains
File not yet committed to repo

Coherence Bridge — exact morphisms across 6 domains

Collatz as dm3 system
[ 04_collatz_dm3.svg ]
Collatz as dm³ System
File not yet committed to repo

Collatz conjecture as dm³ system — AXLE Target 5

Domain application map
[ 05_domain_map.svg ]
Application Domain Map — 20+ fields
File not yet committed to repo

Application domain map — 20+ fields

Interactive Tools
LIVE
SIM
Interactive Simulation · Collatz / AXLE Target 5
Lyapunov Exponent — Syracuse Return Map
Live computation of the Lyapunov exponent λ for the Syracuse return map. Adjustable parameters: orbit length N, seed n₀, branch weights (odd/even), noise amplitude σ. Visualises the Collatz trajectory, exponent convergence, and return-map scatter in real time. Directly probes AXLE Target 5 — the dm³ embedding of the Collatz conjecture.
↗
Open Access Research

Papers on Zenodo

All papers are freely available. Cite via DOI. Each paper is a self-contained contribution — you do not need the full series to read any one paper.

Zenodo · 10.5281/zenodo.19117400
Principia Orthogona, Volume One
The Mathematics of Generative Transitions. Abstract operator algebra, the operator sequence C → K → F → U defined and proved. Singularity classification (Whitney A1–A3), symplectic preservation theorem, curvature threshold κ*. The foundation.
doi.org/10.5281/zenodo.19117400
Zenodo · 10.5281/zenodo.19379473
Principia Orthogona, Volume Two
Contact Realization of Generative Transitions. Explicit contact-geometric realization of Volume One. Threshold Equivalence theorem: |κ| ↑ κ* ⟺ μ_max < 0 ⟺ τ ∈ (0,∞). Four dm³ bifurcations ↔ Whitney A1–A3 singularity types. Submitted to IMPA.
doi.org/10.5281/zenodo.19379473
Zenodo · 10.5281/zenodo.19122168
Generative Contact Mechanics
A Geometric Framework for Dissipative Systems with Structured Limit Cycles. Complete operator algebra (g-, L-, R-, U-operators). 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, four bifurcations, stochastic concentration below embodiment threshold τ = 2. Submitted to SIAM J. Applied Dynamical Systems.
doi.org/10.5281/zenodo.19379385
Zenodo · 10.5281/zenodo.19162013
The G6 Crystal
A dm³-derived architectural form for resonance-stable tall structures. Six applications of G yield a hexagonal tower with aspect ratio 66 = 33·τ. Passive Schumann n=4 resonance coupling at 33.516 Hz via Arnold tongue A4:1. Noise tolerance τ·ε₀ = 2/3.
doi.org/10.5281/zenodo.19162013
Zenodo · 10.5281/zenodo.19378742
The Collatz Conjecture as a Corollary of Crystal Geometry
Supplement to the Crystal Paper. 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. The polar vortex is the empirical certificate.
doi.org/10.5281/zenodo.19378742
Zenodo · 10.5281/zenodo.19208015
Biological Transitions as Multi-Agent Realisations of TO/TOGT
Neural oscillations, HPA-axis stress response, circadian regulation, immune adaptation, and protein conformational change modelled as multi-agent realisations of G = U∘F∘K∘C. Fixed-point, contraction, and saturated pitchfork results applied.
doi.org/10.5281/zenodo.19208015
Zenodo · 10.5281/zenodo.19210137
Fruit-Fly Connectome Toy Model
Biological Transitions as Multi-Agent Realisations of TO/TOGT: A Drosophila connectome toy model. Neural clusters as coupled agents whose trajectories stabilise under six applications of G. Reproduces neural coherence, oscillatory entrainment, and stress-response switching.
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 It, coordination efficiency Ct, type-propagation multiplier Mt, diffusion factor Ft. All results proved from established theorems.
doi.org/10.5281/zenodo.19208284
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. U-, R-, L-, and B-operator families. Categorical invariants and compositional operator algebra. Mathematically independent of speculative cosmological models.
doi.org/10.5281/zenodo.19210058
Zenodo · forthcoming
Additional Papers
GTCT — The Generative Time Circuit Theorem (Ring 5) · Wavenumber 6: Orthogenetic Stability Generator · The dm³ Criticality Principle · The Number 33. DOIs will appear here upon publication.
Links to be added
Get the Series

Editions & Pricing

Complete Series
Principia Orthogona
G¹–G⁵ · Print
$199.99
456 pages · IngramSpark
NJ tax incl.: $213.24
ISBN 979-8-9954416-0-1
Buy via PayPal →
eBook
Complete Series
Adobe PDF
$199.99
Includes AXLE v6.1 embedded
NJ tax incl.: $213.24
ISBN 979-8-9954416-1-8
Buy via PayPal →
Individual
Volume I · GOMC Science
Hardback
$47
139 pages
NJ tax incl.: $50.11
ISBN 979-8-9954416-2-5
Buy via PayPal →
Individual
Volume II · TOGT
Hardback
$47
122 pages
NJ tax incl.: $50.11
ISBN 979-8-9954416-4-9
Buy via PayPal →
Living Book
Volume III · Mini-Beast
eBook · All Chapters
$19.99
All current chapters now · All future chapters included
17 chapters · C1→C2 · Student portal prompts
NJ tax incl.: $21.31 · Buy once, access forever
ISBN 979-8-9954416-6-3
Buy via PayPal →
IMPA Edition
Complete Series
Hardback · IMPA
$247
Distributed via IMPA portal
ISBN 979-8-9954416-8-7
Purchase via Portal →
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