Anacardium occidentale · Pirangi-RN · 8.500 m² · one organism
A healing walking meditation. An acoustic machine. A bilingual exhibition. A community in formation. The cajueiro establishes itself in difficult soil — branches, touches the ground, re-roots, becomes new trunks. One organism. 8,500 m².
Not a hierarchy, not a ladder — four surfaces of the same mathematical object, each of which you can enter first. They are Ler, Simular, Provar, and Ensinar — to read, to simulate, to prove, to teach. They travel in Portuguese through every Level. Meet them once, in the legend below, and afterward they will carry you.
Every Level offers all four faces. Each closes by leaving one open problem in your hands. Not because the book forgot to solve it — because that is how the book knows you are learning. This is Axiom 9, Honest Incompleteness, made pedagogical.
You, the reader, this book will call camará — companion in the circle. When a Level hands you back an open problem and you carry it out of the book, the sound of your work returning to the roda has a name in Portuguese: ressonância.
Não é hierarquia, não é escada — são quatro superfícies do mesmo objeto matemático. Ler, Simular, Provar e Ensinar. Atravessam todos os Níveis sem tradução.
As quatro faces respondem às quatro cores do encadeamento operatorial: C · Comprimir (Ler), K · Curvatura (Simular), F · Filtro (Provar), U · Desdobrar (Ensinar).
O cajueiro é a meditação de caminhada. A raiz aérea toca o solo, se enraíza de novo, se torna novo tronco. Um organismo. Esta é a prática: comprimir até a semente, sentir o solo, cruzar o limiar, desdobrar para a copa. C K F U. Cada passo é um operador.
A caminhada da rua Tiffany Blvd até Forest Hill, é uma passagem completa do operador G = U ∘ F ∘ K ∘ C. Coloque o som. Comece a andar.
Cada Nível se fecha deixando um problema em aberto nas suas mãos — não porque o livro esqueceu de resolvê-lo, mas porque é assim que o livro reconhece que você está aprendendo. Isso é o Axioma 9 (Incompletude Honesta) feito didática.
Você, leitor, este livro chama de camará. Não aluno, não pupilo. Companheiro na roda. Quando um Nível devolve um problema em aberto e você o carrega para fora do livro, o som do seu trabalho voltando à roda se chama ressonância.
The former Delaware, Lackawanna & Western Railroad Boonton Line runs past 90 Tiffany Blvd. The street is named for the Tiffany & Co. manufacturing plant that stood at Forest Hill station — where American silver was made, where the train served the factory, where the station planned a subway connection that was never built. Passenger service abandoned in 2002. Now proposed as the Ice & Iron Greenway. Put the soundworks on. Begin.
The soundworks is the score for this walk. Play it before you come. Play it while you walk. After you donate, you are invited to visit the cajueiro — the physical installation at Uceda School of Elizabeth (November 2026) and, when it opens, the homebase at Forest Hill.
The dm³ dynamical system passes through five regimes before the helical attractor locks. Each regime has a name, a cycle count, a place on the walk, and a place in the learning body. The g-series is the formal taxonomy — proved in Lean 4, walked in Newark, taught in the book.
The g-series is not a metaphor. It is a formally proved regime taxonomy in GCTC.Operators.Chain, built on Mathlib4, zero axioms beyond the library. The poincare_collatz_contracting theorem guarantees entry into the r* basin at n ≥ 33 for any contracting G-chain. The Spiral Return theorem T1 guarantees a non-trivial orbit at g64. github.com/TOTOGT/GTCT →
Not in 2020. That is the whole point, and it is why this project exists.
Minutes a day spent alone: 2003, 2019, 2020. The one-year jump equals what the previous sixteen years produced. Everyone saw this. Every newspaper reported it.
The year companionship — shared leisure, done for no reason but itself — began its measurable fall. Seven years before the pandemic. From 202 minutes a day to 182 by 2019.
Time with household family, 2003 to 2019 — falling along a straight line with no change of slope anywhere in eighteen years. The pandemic did not bend it. In 2020 it went back up.
This is the Cajueiro Principle in a public dataset. The pandemic is the canopy: visible from the air, impossible to miss, and the last thing made. The root — a slow withdrawal from shared leisure, from time spent with people for no purpose at all — had been growing in the dark since long before anyone looked up.
A cajueiro does not spread by being planted again. A branch grows out, touches the ground, and roots where it touched. One organism, 8,500 square metres, from a single seed — by contact, repeatedly, over decades.
Uma roda é isso. A branch touching the ground.
That is what a roda is, and it is why this one meets in person.
V. D. Kannan & P. J. Veazie, US trends in social isolation, social engagement, and companionship — nationally and by age, sex, race/ethnicity, family income, and work hours, 2003–2020, SSM — Population Health 21 (2023) 101331. Open access, CC BY-NC-ND. Joinpoint analysis, University of Rochester.
The authors make no claim about dm³, about contact geometry, or about this project. Their data does not support one and is not offered as if it did. What their paper gives us is a fact and a reading: the decline predates the event everyone blames, and you only see that if you look earlier than the canopy. The soundworks, the walk and the roda are our response to that fact — not evidence for a theory.
The Principia Orthogona series is a living book — updated as the research advances, the proofs close, and the workshops run. Members receive all volumes, full soundworks access, and GitHub collaborator status. Pay what you can. Every amount funds the same work.
After payment, send your GitHub username to g6llc@proton.me — you will be added as a collaborator within 24 hours.
All memberships are pay-what-you-can starting at each tier. If $25 is too much right now, reach out — no one is turned away. The roda is open.
g6llc.gumroad.com/l/soundworksThe seed needs two things to take root: soil and witness. We ask you for both — in that order.
Free bilingual public art and wellness project serving immigrant adult learners in Newark/Elizabeth NJ. Five workshops and a public installation use visual art and dialogue to address social isolation among immigrant professionals. Rooted in independent research published on Zenodo and presented at the Sociedade Brasileira de Matemática, 2026.
Fund the soundworks, the workshops, the homebase. Every contribution goes directly to the Newark Wellness Center vision and five free public workshops at Uceda School of Elizabeth. As a donor you are invited to visit the cajueiro in person.
No ticket, no date, no registration. Put the soundworks on and walk. Listen. Read if you feel like reading. Think about whatever strikes you — it does not have to be mathematics, and it does not have to be ours. The invitation is the walk. What it produces is not specified in advance, because it cannot be: the root is made before anyone sees the tree.
If one person comes back from an afternoon in the park and decides to pick up something difficult — nuclear physics, say, from a walk in a park in Newark — the project has done what it was built to do.
XII Bienal SBM, UFRN, Natal — 3–7 August 2026. Five accepted sessions, all bilingual: EXP13 Cimática com Máquinas · Figuras de Chladni (exhibition, daily) · MC48 helical attractors on contact manifolds, dm³ (minicurso, 4 Aug) · OF53 A Escada de Recorrência: do k-nacci ao limiar τ=2 (oficina, 7 Aug) · CO144 O Princípio do Cajueiro (comunicação, 5 Aug) · P290 helical attractors on 3D contact manifolds (pôster, 6 Aug). The Cajueiro de Pirangi is 20 km from that campus. All five, with papers, posters and Lean 4 →
LAW3M — Natal, Rio Grande do Norte, 19–23 October 2026. A contact-geometric model of rotating energy systems on (ℝ³, dz − r²dθ). Papers, posters, theorems and the Lean 4 archive are open now. The LAW3M package →
The moon cajueiro is the digital expansion: soundworks, Lean 4 formalization, Web Audio engine, bilingual education platform. The homebase is the physical one: Forest Hill. We need contributors for both. All code MIT licensed. All educational materials CC BY 4.0.
Portuguese/English facilitators for the Cajueiro Workshops. Newark and Elizabeth NJ. October 2026 — January 2027. Educator, social worker, artist — if you work with immigrant adults or youth in Newark, we want to talk.
Close the kappa_lipschitz sorry in AXLE (Lean 4 + Mathlib4). Extend the dm³ Web Audio engine. Build the interactive Chladni simulator for the Bienal exposition. github.com/TOTOGT/AXLE
Test the TOGT pedagogical framework in your domain — biology, finance, architecture, nursing, music. The Mini-Beast (Vol. III) has 33 task cycles ready to deploy. Partner institutions in Brazil, India, Portugal, and the US invited.
Forest Hill, Newark. The vision: a wellness center, sound studio, workshop space, and community garden along the Boonton Line greenway corridor. Architects, urban planners, community builders — open call.
Tier 1: Portuguese, Russian, Hindi, Mandarin. Tier 2: Spanish, Arabic, Bengali, Tamil. The series has a 40-language roadmap. Translators, educators, and ministry contacts invited. All translations CC BY 4.0.
Newark Arts ArtStart applied. NYFA fiscal sponsorship pending. NJSCA fellowship June 2026. Dodge Foundation LOI July 2026. NSF DRL Q3 2026. Reach out: g6llc@proton.me
The seed tends to a forest with a home. The cajueiro does not plan the canopy — it becomes one. 90 Tiffany Blvd is where the root is. Forest Hill is where it reaches.
The caju forest in Forest Hill, then on the moon.
We dream big.
Tiffany Blvd is named for the Tiffany & Co. manufacturing plant that stood at Forest Hill station on the Boonton Line — where American silver was made from 1877, where the rail served the factory, where early 20th-century planners designed a Newark City Subway connection that was never built. The homebase is planned for Forest Hill. When the Ice & Iron Greenway opens, the walk from this building to will be a public meditation trail. That walk is the practice. That walk is the operator. G = U ∘ F ∘ K ∘ C.
Current funding: Newark Arts ArtStart 2026–27 (applied) · NYFA fiscal sponsorship (pending) · NJSCA Individual Artist Fellowship (applying June 2026) · Dodge Foundation LOI (July 2026) · Personal funds confirmed · Uceda School venue confirmed
G6 LLC · EIN 33-2880433 · Fiscal sponsor: New York Foundation for the Arts (NYFA) · g6llc@proton.me
The former Delaware, Lackawanna & Western Railroad Boonton Line. Passenger service abandoned in 2002. Proposed as the Ice & Iron Greenway — Montclair to Jersey City waterfront, through Glen Ridge, Bloomfield, Belleville, Newark North Ward, Kearny. The Tiffany & Co. factory stood at Forest Hill station. The homebase is planned for Forest Hill. The walk between them is the operator.
One day we hope to walk the line.