Lang Latin · LTR
Principia Orthogona · Book 4 · GTCT · G6 LLC · 2026

PRINCIPIA
ORTHOGONA

Generative Temporal Contact Theory · Topographical Orthogenetics

9Axioms · Lean 4
12Operators
0Sorry in Chain.lean
x*Unique Fixed Point
33Orth. Constraints
G = U ∘ F ∘ K ∘ C  ·  ε* = 1/3  ·  κ ≤ √(7/9) < 1  ·  T* = 2π  ·  doi:10.5281/zenodo.19117399
C∘ K∘ F∘ U = G  ·  0 sorry  ·  Lean 4 verified
The Principia Orthogona Series · 7 Volumes
Vol I GOMC Operator algebra · G-chain · Ring structure ● Live Vol II TOGT Contact geometry · Reeb orbits · dm³ ● Published Vol III Mini-Beast Language + Science · 121 HTML chapters · CEFR A2–C1 ● Live Vol IV GTCT Formal theory · 9 axioms · 12 operators · Lean 4 · IMPA/Bienal ● Here Vol V Dim. Theory Dimensional Theory · v8 final submission · Higher Dimensions Arc ↑ Submitted
Vol VI Explicit Ops Explicit operator realization · sorry-free A_i matrices Planned
Vol VII — In preparation Planned
§ Vol IV · Formal Theory — GTCT

9 axioms · 12 operators · 4 theorems · 0 sorry in Chain.lean · Lean 4 verified · Bilingual EN/PT · doi:10.5281/zenodo.19117400 · ISBN 979-8-9954416-8-7

Ch 0 Prose
Preface — Lemniscata & Analemma Origin story · two parental lessons · 26 years of ESL in Newark · PT primary
Ch 0 · IMPA IMPA Ed.
Preface — IMPA Edition Stars background · EN primary · Cajueiro roots · SBM Bienal 2026
Ch 1 ✓ Lean 4
The Seven Axioms Contact structure · G-chain · Temporal field · I₁I₂I₃ triad · 12-phase · Correspondence · Recursion
Ch 2 ✓ Lean 4
The Dimensional Field Δ : M → ℝ¹² Axioms 8–9 · unit sphere S¹¹ · temporal compatibility T* = 2π · 12 phase subspaces
Ch 3 ✓ Lean 4
Dimensional Operators Aᵢ = Pⁱ + uᵢwᵢᵀ · ε*=1/3 · σ_min ≥ 2/3 · 4×3=12 colour-coded phase table · 7 languages
Ch 4 ✓ Lean 4
The Correspondence Theorem Unique bijection Φ : {D₁…D₁₂} ↔ {O₁…O₁₂} · the cycle is rigid · four-step proof
Ch 5 ✓ Lean 4 · SH
The Orthogonality Theorem ⟨Δ_⊥(Oᵢ), Δ_⊥(Oⱼ)⟩ = 0 for i≠j · Structural Hypothesis SH · v3 proof · g₃₃ = 33
Ch 6 ✓ Lean 4 · SH
The Recursion Theorem Pythagorean contraction κ < 1 · Banach fixed point · Spiral Return T1 · 0 sorry
Ch 7 ✓ Lean 4 · SH
Emergence as Fixed Point E(x*)=x* unique · Eⁿ(x₀)→x* geometrically · identity as dynamic invariant · g₃₃ g₆₄
Ch 8 Interpretive
The Nature of Time in GTCT 9 readings: constraint · generator · memory · resolver · identity · irreversibility · bridge · present · emergence
App A–F Reference
Appendices Notation · Operator tables · 12-phase diagram · Brouwer & Banach · Glossary EN/PT · Correspondence table
§ Higher Dimensions Arc — 1D → 5D+t

1D fermion seed (ψ²=0) → 2D+t phase plane → 3D contact manifold → 4D tesseract → 5D jet space → 5D+t GTCT closes · doi:10.5281/zenodo.19117399

Ch 0 · Student Guide
Student Edition — How to Read Bilingual EN/PT · mathematical reading guide · three-pass structure
Ch 11D
The Fermion Seed — ψ² = 0 1D · six directions of potential · the decision as C · pre-braid
Ch 22D+t
2D+t — Extended Phase Space α = dy − f dx · Darboux · phase plane lifts to helix in 3D
Ch 33D
Contact 3-Manifold + S³ dm³ ODE · Theorem 3.2 · Hopf fibration · §3.9 T*=2π forced
Ch 44D
The Tesseract 4D · S³ as hypersurface in ℝ⁴ · symplectisation · holomorphic curves
Ch 55D
OPERA SATOR — Jet Space J¹(ℝ,ℝ²) 5D · palindrome square as operator map · five coordinates = GTCT operators
Ch 65D+t
666 · The 111 Square 5D+t · 6×6 magic square · time as sixth slot · 666 = T(36)
Ch 7Crystal
The Crystalline Return Wigner crystal · moiré · E = kT·ln(Ω)
Ch 8Lean 4
The Axiomatic Turn Γ ⊢ φ · formal proof as U · Lean 4 and the sorry-free paper
Ch 9φ
φ — The Subcritical Approach Golden ratio · Fibonacci · fold frozen at q* < 1
Ch 10✓ dm³
Helical Attractors on Contact 3-Manifolds dm³ ODE · r*≈0.773 basin · SBM Bienal paper
§ Extended Chapters & Special Files

Additional material, popular editions, and extended applications beyond the main arc.

Ch 11CatGT
CatGT — Catalytic Generative Theory Category-theoretic formulation of GTCT · functorial operator chains
Ch τBilingual
The Cosmic Tortoise / Tartaruga Cósmica τ=2 · embodiment threshold · bilingual EN/PT
FaradayEM
Hidden Geometry of EM Forces Faraday + dm³ · electromagnetic contact geometry · Grossi 2026
Ch EPopular
GTCT for Everyone 9 axioms · 12 operators · 4 theorems · time as 5 properties
Ch E · AltAlt
GTCT for Everyone — Alternate Alternative formatting of the popular edition
GOMCFull
GOMC Opus — CatGT + Plasma + Coherence Bridge Full operator coherence bridge · plasma + biology domains
§ PDFs & Preprints
PDF
GTCT 2026 v3
SBM Bienal paper · doi:10.5281/zenodo.20239928
 PDF
Faraday EM + dm³
Grossi 2026 · electromagnetic geometry
 PDF
k-nacci Spine v4
N-bonacci criticality · DNLS · doi:10.5281/zenodo.20026942
 PDF
Zeolite Selectivity
Coherence Bridge domain · crystal lattice geometry
§ Repositories & Records
GitHub · AXLE GitHub · GTCT GitHub · DM3-lab Zenodo Series Root Vol IV DOI ORCID SSRN · Book 3 Gumroad AXLE Live Book