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.
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.