Principia Orthogona · Book 3 · The Mini-Beast

Chapter Map

G = U ∘ F ∘ K ∘ C — 30+ chapters, one operator chain, growing

C · Compression

K · Curvature

F · Fold

U · Unfold

π→Ω · Recurrence

30+

Chapters live

16

Verified domains

160+

Lean 4 theorems

21

Zenodo deposits

7

Sound machines

$30

All of it · gumroad

C→K→F→U

Part I — Core Sequence · Biological & Physical Instantiations

14 chapters · all live

The Cajueiro Principle

C · Seed and Fixed Point

The tree as entry point. Working backward from U to C. The fixed point you already inhabit.

Live

Biological Instantiations

C → K · HPA, Neural, Circadian, Immune

Four biological orbits. One operator sequence. Parameters calibrated from domain data.

Live

Plasma Reconnection

K · X-point curvature

Dusty plasmas, fractal current-sheet d_φ ≈ 1.6–1.8. Cassini, MMS, Parker Solar Probe.

Live

Market Volatility Manifolds

K · Flash crash as fold event

Fisher information metric. Multifractal Hölder spectrum. NYSE TAQ + Binance 2015–2026.

Live

Neural Oscillations

K · Kuramoto sync threshold

Theta–gamma coupling. Synaptic connectivity metric. Entrainment as K operator.

Live

Oscillations & Coherence

F · Coherence Bridge

The fold that proves exact mathematical identity across six initial domains.

Live

F · Th1/Th2 switching

Immune commitment as fold event. The system chooses and cannot easily unchoose.

Live

The Resonant Chamber

F · Cymatics · 110 Hz · Hypogeum

Chladni figures as contact fixed points. The Ħal Saflieni Hypogeum. The fold in stone.

Live

Crystalline Structures

U · Wigner, moiré

Electrons crystallize under compression. Quantum phase transition as dm³ fold.

Live

The Axiomatic Foundation

U · Nine axioms

The minimal axiomatic base. What must be assumed and what follows.

Live

Phi — Golden Ratio

U · Fixed point φ = 1.618

Fibonacci subcritical dm³. φ as stable fixed point below c* = 3.

Live

Lyapunov Exponents

U · Stability radius

The boundary of stability. μ = −2. Sensitive dependence and the basin edge.

Live

Spectral Radius

U → G · Transfer operator

Syracuse return map. Collatz spectral structure. The operator as its own fixed point.

Live

AXLE — Formal Verification

U · Lean 4 · 160+ theorems

The proof environment as chapter. lake update && lake build. Machine-verifiable mathematics.

EXT

Part II — Extended Domain Chapters

16 chapters · verified + building

GTCT for Everyone

F · 9 axioms · 12 operators · 4 theorems

The IMPA Volume IV made accessible. Time circuit. Banach FPT. The fixed point as identity.

Live

Saturn's North Polar Hexagon

K · g₃₃ = 33 crystal law

Wavenumber-6 selection from the crystal law. Kronos, Saturnus, Śani. Cassini/VIMS data.

Live

G · Time as generative operator

The fixed point x* as the form of identity under time-evolution. Bilingual PT/EN.

Live

The Collatz Conjecture

G · 3 proofs via GTCT

Contact-Hamiltonian H = p² + q³ − 3q + 2z. Tesla's triad. Hailstone bound. AXLE sketch.

Live

Autophagy & Triple-Alpha

C · 26 theorems · 0 sorry

Cellular recycling + stellar nucleosynthesis. AutophagyDm3_v2.lean. DOI 10.5281/zenodo.20168812.

Tubulin Oscillations

K · Cytoskeletal dm³

Microtubule dynamics. Ulam's computronium question. Threshold curvature in polymer networks.

F · Quantum fold diagram

Interactive fold diagram. Electron gas crystallisation. 1934 prediction verified by dm³.

Live

DNLS / N-bonacci Criticality

K · η ≈ 1.8393 · τ = 2

Discrete nonlinear Schrödinger. Tribonacci waypoint. Full deposit DOI 10.5281/zenodo.20026942.

PolyLaminin / k-nacci Spine

C · ECM self-assembly

Laminin network formation as C→K→F→U on protein configuration space.

Enceladus Cryovolcanism

K · Gronwall basin chemical filter

Cassini E5 data. Khawaja et al. 2025. ε₀ = 1/3 as chemical survival threshold. Whitney A₁.

Faraday / EM Helical Attractors

F · Helicoidal attractors in contact 3-manifolds

Electromagnetic induction as dm³ fold. SBM Bienal poster P290. Formal paper deposited.

Finance / WorldQuant Alpha

K · Flash crash geometry

Two alphas ACTIVE. Sharpe 2.43, Fitness 1.84. IQC USA score. Parametric vs non-parametric.

Live

Multi-Agent Bioswarm

C · FruitFly connectome dm³

Multi-orbit bioswarm. FlyWire connectome. MultiOrbitBioSwarm.lean. DOI 10.5281/zenodo.19210136.

Swarm Intelligence

U · SwarmSimulator.lean

AI / complex systems. Lean 4 verified. Convergence proofs, multi-orbit identity.

Nuclear Physics B / Tribonacci

K · Submitted to Nuclear Physics B

Tribonacci coherence. N-bonacci diffusion. Submitted to Elsevier Nuclear Physics B.

Lexical Generativity

C · Language as dm³ system

Submitted to International Journal of Linguistics. Language acquisition as generative transition.

π→Ω

Part III — The Recurrence Ladder

7 Greek chapters · φ → τ = 2

π = 3.14159…

Reeb geometry, limit cycle, contact structure. T* = 2π.

Fibonacci / φ = 1.618

φ = 1.61803…

Subcritical dm³. Below c* = 3. Stable fixed point.

Lyapunov / μ = −2

Stability boundary. Sensitive dependence on initial conditions.

Tribonacci / 1.839

η = 1.83929…

First supercritical rung. Lean 4 verified. DOI 10.5281/zenodo.20026942.

Tetranacci / 1.927

Δ = 1.92757…

Four-term recurrence. Approaching τ = 2 from below.

Pentanacci / 1.966

Σ = 1.96595…

Five-term recurrence. The ladder converges.

Hexabonacci → τ = 2

Six-term recurrence converges to τ = 2. Complete Completeness. SBM Oficina OF53.

PE

Part IV — Ponto de Entrada · Pedagogical Register

6 chapters · seed → recognition → Bridge PT↔EN → generate → extend

The Cajueiro Principle

The world's largest cashew tree. 20 km from the SBM Bienal venue. The seed as pedagogy.

Live

The Order Is Not Arbitrary

K · Curvature as recognition

Why C must come before K. Non-commutativity as felt experience.

Live

The Quantum Weave

F · The fold as understanding

The moment the structure folds into recognition. Bridge PT↔EN.

Live

Complete Completeness

U · The fixed point of learning

The circuit closes. g₃₃ = 33. The student has become the teacher.

Live

The same structure, one order higher. The return that is not a repetition.

Live

The Resonant Chamber

G · Pedagogical register

The operator chain carved in stone five thousand years before its name. Same register, different soil.

♪

Sound Machines — 7 Interactive Installations

HTML5 / Web Audio API · dm³ resonance geometry · all included with $30 membership

432 Hz. The compression of all sound to one tone.

Live

Pranayama Machine

Breath as operator cycle. C→K→F→U in 4-7-8 rhythm.

Live

So-Haṃ Machine

The breath that names itself. Fixed point of the respiratory operator.

Live

G6 Drone Machine

G6 LLC frequency. The drone as institutional fixed point.

Live

Śrī Yantra Śakti

Nine interlocking triangles. 43 contact points. The operator chain in 2D.

Live

Ħal Saflieni · 110 Hz

5000-year-old resonance chamber. The fold in limestone. Standing wave as fixed point.

Live

Swarnakarṣaṇa Bhairava

The attractor of abundance. dm³ resonance at the fold threshold.

Get the Book

$30

30+ chapters · All 7 sound machines · 160+ Lean 4 theorems · 16-week program
Ongoing membership — new chapters added as domains are verified
ISBN 979-8-9954416-6-3 · NYFA Fiscal Sponsorship (tax-deductible donations accepted)

Buy at g6llc.gumroad.com →

Series root: 10.5281/zenodo.19117399 · ORCID: 0009-0000-6496-2186