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Principia Orthogona
G = U ∘ F ∘ K ∘ C · dm³ framework · τ = 2 · ε₀ = 1/3
G = U ∘ F ∘ K ∘ C · ε* = 1/3 · κ < 1 · T* = 2π
The Nine Volumes
Each volume is one step in the Cayley-Dickson algebra ladder. The same operator chain G = U∘F∘K∘C acts in every volume — only the underlying algebra changes. Each extension loses one property: ordering, commutativity, associativity, division.
G-chain · Ring structure · Toy model equations ṙ = r(1−r²)+2(r−1)e⁻ᶻ · The invariant triple (T*,μ_max,τ) = (2π,−2,2) · AXLE Lean 4 companion.
Reeb orbits · dm³ contact manifold M = ℝ²₊ × ℝ · Symmetry s↔1−s · GCM Theorems A–D · RK4 toy model · Stability radius ε₀ = 1/3.
121+ HTML chapters · CEFR A2–C1 · Biology, physics, consciousness, language pedagogy · 17 orders of magnitude · The Cajueiro Principle · g-series g⁰→g⁶⁴ · Pedagogical entry point for the series.
Generative Temporal Contact Theory · 9 axioms · 12 operators Aᵢ = Pⁱ + uᵢwᵢᵀ · 4 theorems · 0 sorry in Chain.lean · Lean 4 + Mathlib4 · [F,T]=iJ complex structure · ε* = 1/3 · κ ≤ √(7/9) · g₃₃ = 33 · Bilingual EN/PT · SBM Bienal 2026.
G(x*) = x* · Banach fixed-point applied to GTCT · 0 axioms beyond Mathlib4 · 8 verified constants · 1,080 theorems, 0 sorry (Project 1080, June 22 2026) · The series proves itself.
Baker's theorem · log p as root length in the Lie algebra · bridge from ℍ/𝕆 to exceptional algebras · division is lost here.
G = U∘F∘K∘C IS the Dynkin diagram of E₈ · J² = −id · 12 bio-domain proofs in preparation · Enceladus · CatGT · Spectral Radius · Banking Butterfly · TEFL · EMMEs · Transport · The Crystal G6 conjecture χ(H*(X⁶)) = 33.
The Scientist Gallery · Ada, Curie, Dirac, Escher, Faraday, Hawking, Lattes, Ramanujan, Thoreau · each scientist as a dm³ instance · César Lattes · pion discovery · Brazilian physics.
Monster group · Monstrous Moonshine · the hidden identity of the series. Galaxy mergers, nebulae, embryogenesis, Bitcoin, VOA, mycelium, nested infinities — one operator, one attractor, one Monster. 25 chapters including the closing threshold.
G = Genesis · L = Logos · R = Resonance · U = Union · τ = 2 = Omega Point. The same operator chain in theological vocabulary. Trinity Triptych · Gallery of Mathematical Mystics (al-Kindi, Baudhayana, Cusanus, Hildegard, Ramanujan, Rumi, Teilhard…) · 10 numbered chapters · Belief · Transmission · Union.
17 Orders of Magnitude
Volume III instantiates the same G = U∘F∘K∘C operator chain at scales from string theory (10⁻³⁵ m) to galactic structure (10⁶ ly). The framework is scale-invariant — τ = 2, ε₀ = 1/3, and κ < 1 hold at every scale. These are the currently instantiated domains:
10⁻³⁵
Planck / String
Vol V / E₈
10⁻¹⁰
Atomic / Molecular
Zeolite (CatGT)
10⁻⁹
Cytoskeletal
Tubulin (T)
10⁻⁷
Viral / Nanoscale
Polylaminin (Λ)
10⁻⁵
Cellular
Immune (Ch 5)
10⁰
Human / Walk
Cajueiro / Neural
10²
Tree / Ecosystem
Cajueiro de Pirangi
10⁶
Plasma / MHD
DustyPlasma (GOMC)
10¹⁰
Stellar
Compact objects
10²²
Galactic
Wigner lattice (W)
Lean 4 Audit Trail
Mathematical content is mechanised in two Lean 4 environments: AXLE (github.com/TOTOGT/AXLE) for Vol I–III applications and DM3-lab, and GTCT (github.com/TOTOGT/GTCT) for Vol IV formal theory. Zero sorries in the main chain. One sorry marked in P vs NP (acknowledged open problem).
| File | Content | Volume | Sorries | Status |
|---|
| Chain.lean | G-chain; κ < 1; Banach fixed point | IV (GTCT) | 0 | ✓ Lean 4 verified |
| Dm3RHToy.lean | Riemann Hypothesis toy reformulation | I / II | 0 | ✓ Lean 4 verified |
| Dm3NSToy.lean | Navier–Stokes toy model | I / II | 0 | ✓ Lean 4 verified |
| Dm3GoldbachToy.lean | Goldbach toy model | I / II | 0 | ✓ Lean 4 verified |
| Dm3Comp.lean | P vs NP (acknowledged open) | I / II | 1 | Open · sorry marked |
| GCTC.Operators.Chain | poincare_collatz_contracting; Spiral Return T1; g33 entry | IV | 0 | ✓ Lean 4 · Mathlib4 |
| AutophagyDm3.lean | Tubulin polymerization dm³ | III | 0 | ✓ Lean 4 verified |
| vitruvian-approximation.pdf | Rhind Papyrus / G-cycle · selection functional 𝒞_rat | III | 1 | Open · sorry marked · Zenodo ↗ |
Drifting Chapters — All Domains
These chapters appear across volumes and sometimes across both Book 3 and Book 4. They are listed here so auditors can find every instantiation. The dm³ framework is domain-free — these chapters show it operating in unexpected places.
Biology · ℝ³ contact manifold (Vol III)
Physics · condensed matter · EM
Number Theory · discrete dm³
Formal Theory · Book 4 (ℂ / Vol IV)
Pedagogy · Econophysics · Language (Book 3)
n-bonacci Recurrence Ladder (Greek operator stubs)