Book 3 · The Mini-Beast · Chapter 41 of 44
About the Author
Author bio, ORCID, contact, and the dedication.
Pablo Nogueira Grossi
OrientationAuthor
Pablo Nogueira Grossi. G6 LLC, Newark, New Jersey.
The origin of this book is a classroom coincidence. Two professors, two rooms: a mathematician drawing the lemniscate — the figure-eight of infinite return — and a geologist drawing the analemma, the figure-eight the sun traces across a year. Neither knew the other was drawing the same curve. The student sitting in both rooms did. Principia Orthogona is the mathematics built to prove it, and Book 3 is that proof applied — not to one system, but to four classes of them.
| ORCID | 0009-0000-6496-2186 |
| g6llc@proton.me | |
| Telephone | +1 (646) 342-3751 |
| Organisation | G6 LLC · Newark, New Jersey, USA |
| Repository | github.com/TOTOGT/DM3-lab |
| Deposits | Zenodo · Principia Orthogona community |
| Licence | CC BY-NC-ND 4.0 |
MethodHow This Work Is Done
Two commitments shape the series, and they are worth stating because they explain its unusual features.
Do the math, then write the claim — in that order
The corpus audits itself in public. Working papers in Volume VI have refuted an algebraic bridge in this author’s own material, traced a five-domain convergence table to a row with no source, and corrected an ESTABLISHED badge on a number that the chapter owning it lists as an open conjecture. Those documents were published rather than quietly fixed.
State what is not claimed
The dm³ system is a toy: an explicit closed form, one attracting limit cycle, and a basin whose inner edge has been located numerically to eight digits and not proved. The Lean development verifies the surrounding structure — the eigenvalue, τ = 2, ε₀ = 1/3, the singularity classification — and states the basin result only as an existence claim over (2/3, 1), which is far weaker than r★ = 0.77594058 suggests. The integrator, the Lean files and the failing cases are public.
Lean StatusWhat Has Been Checked
analemma_not_gerono and bernoulli_not_analemma find a point on one that is not on the other. What they do share is the crossing. A1Node.lean proves that the Gerono and Bernoulli lemniscates and the solar and lunar analemmas each have an A₁ node at the origin: zero gradient, a non-degenerate indefinite quadratic part, and — for the parametrised curves — two branches crossing with independent velocities. So the curve the student saw in both rooms is the same kind of crossing on two different curves, and that is a theorem. The four curve files were first written in May 2026 in TOTOGT/DM3-lab and never built there; they were corrected and kernel-checked in 2026-09 (see Chapter 1 and Week 1). The repository to cite for the Lean is TOTOGT/geometry.BridgesWhere This Connects
- On PublicationThe argument for self-publishing this particular work, and the costs it accepts.
- Pointers · XII Bienal SBM, UFRN, August 2026Seven plenaries, and what would actually be asked at each. A list of questions, not of claims.
- Pointers · ICHEP 2026, NatalWhere the honest answer is mostly ‘not much’ — and the outreach track where the work does belong.
- Chladni Plate · Cimática com MáquinasStanding-wave modes of a square plate, driven by what it hears. The public-facing version of a nodal-set theorem.