Across physics, biology, linguistics, architecture, and computation, systems change in ways that appear domain‑specific — yet the underlying transitions obey a single deeper structure. Principia Orthogona presents that structure in full.
A single operator chain C → K → F → U governs how form compresses, constrains, folds, and unfolds across scales. Developed over twenty‑five years, this series establishes GOMC Science as a mathematically rigorous, empirically testable framework for understanding generative transitions in natural and artificial systems.
The series continues: G⁴ (GTCT — Generative Time Circuit Theorem), G⁵ (AXLE Lean 4 formal proofs), and G⁶ (the complete six‑iterate circuit) are in development. This trilogy is the foundation.
All core invariants are formally verified in Lean 4 through the AXLE proof engine, with zero axioms beyond Mathlib. Page numbers refer to this combined three‑volume edition.
| Theorem | Pages | What It Proves |
|---|---|---|
| Thm 0.5.1 — Existence & Well‑Posedness | 258–261 | G = U∘F∘K∘C is well‑defined on any piecewise C² trajectory |
| Thm 0.5.2 — Local Determinism | 261–263 | Operator output is locally unique given initial conditions |
| Thm C / 0.14.3 — Singularity–Bifurcation | 241–246 | Four dm³ bifurcations correspond to Whitney singularity types |
| Thm B / 0.14.2 — Threshold Equivalence | 279–285 | Curvature threshold κ* and embodiment threshold τ are one event |
| Separation Theorem — Tr(M■) ≠ 33 | 14, 22–23 | Stable representations below g₆ = 33 cannot reach the threshold |
| GTCT / T1 — Generative Time Circuit | 35–41 | Time is the circuit operator T; retrocausal enrichment without paradox |
All operator‑level results are encoded as Lean 4 types. Canonical dm³ invariants are proved theorems — 0 axioms beyond Mathlib. 7 documented open problems (Issue 6). Clone and verify:
github.com/TOTOGT/AXLEPages 239–310 of the combined edition present a fully explicit two‑dimensional contact‑Hamiltonian system instantiating every definition, operator, and theorem in the trilogy.
The author's characterization: "a complete verification witness — the entire abstract framework is jointly realizable, computable, and dynamically consistent."
Examine this before evaluating the abstract claims. The toy model does not require trust — it requires computation.
This pre‑print release is part of the early circulation of a new mathematical framework. It is intended for those engaged with the development of generative science — whether from a formal or applied angle.
Volume III (The Mini‑Beast) provides an intentional pedagogical entry at the C1→C2 research‑reading level. A reader new to the framework can begin there and work backward into Volumes I and II as their fluency with the operator chain develops.
The series carries registered ISBNs and Zenodo DOIs. Use the format required by your discipline. Click any tab to switch format, then copy.
ORCID: 0009-0000-6496-2186 · Zenodo: 10.5281/zenodo.19117400 · Publisher: G6 LLC, Newark, NJ · First Edition 2026
Twenty‑five years of mathematical development. Three volumes. One framework. Direct from the author — available today.
Your purchase directly funds editing, typesetting, proof verification, printing, and distribution of the complete hardcover edition. This is the only authorized early release.