Principia Orthogona — The Spiral

G1

Vol I — GOMC · The Operator Framework

First turn · Foundation · Contact geometry established

Published

C The dm³ Framework — Operators, Manifolds, Contact Normal Form

K The Critical Curvature Threshold κ*

F The Generative Transition & Contact Normal Form

U The Separation Theorem · Law of Monsters

Zenodo DOI ↗ SSRN Preprint ↗

G2

Vol II — TOGT · Contact Realization

Second turn · Applications across domains · g₃₃ = 33

Published

CPlasma-Sheet Reconnection

KNeural Embedding & Coherence Bridge

FMarket Volatility Manifolds

UBiological Oscillatory Systems (HPA, Circadian, Immune)

Zenodo DOI ↗

G3

Vol III — The Mini-Beast · Pilot

Third turn · ESL + STEM pilot · 14-Week program · This site

Live

Part I — Biological Instantiations · C · Weeks 1–4 · A2–B1

COverturepre

CPrelude — How to Read This Bookpre

CCh 1 · The Cajueiro PrincipleW1–2

CCh 2 · Generative MatrixW2–3

TCh 3 · Circadian RegulationW3–4

KCh 4 · Neural OscillationsW4–5

KCh 5 · Immune AdaptationW6

Part II — Physical & Financial · K·T · Weeks 5–10 · B1–C1

UCh 6 · Resonance · CymaticsW6–7

FCh 7 · The Crystalline ReturnW7–8

TCh 4b · Market Volatility Manifolds→ PDF Ch 4 · not yet HTML

TCh 3c · The Circadian TraderD2

GCh 8 · The Axiomatic TurnW9–10

GCh H · Collatz as dm³ Corollarybonus

GCh. Ocio · The Fixed Pointbonus

Part III — Research Writing · F→U · Weeks 11–14 · C1–D2

FCh 9 · φ · The Subcritical ApproachW11

UCh 10 · Lyapunov · StabilityW13

UCh 11 · Spectral RadiusW14

GCh 12 · Fixed Point — ConclusionW15

GCh 13 · Revision as ContractionW16

GCh 14 · AXLE · Formal VerificationW16

UCh 15 · Entropy & ScaleW17

UCh 16 · Scale InvarianceW17

GCh 17 · Epilogue — The Living BookW17

Greek Operator Chain · π → φ → μ → η → Δ → Σ → Ω → τ = 2

ππ · Recurrence Ladder · T* = 2π

φφ · Fibonacci · Subcritical Approach

μμ · Lyapunov −2 · Stability Eigenvalue

ηη · Tribonacci ≈ 1.839 · Critical Rank 3

ΔΔ · Tetranacci ≈ 1.927 · Supercritical

ΣΣ · Pentanacci ≈ 1.966 · Approaching τ

ΩΩ · Hexabonacci ≈ 1.984 → τ = 2

ρρ · Spectral Radius · Collatz Bridge

ΛΛ · Polylaminin · Structural Biology

Extended · Ch E — GTCT · Galilean Confluence · Bridge to G4

ECh E · GTCT · Galilean Confluence · Theorem 5D2+

Scientists Series & Energy Arc · new June 2026

SLorenz · Chaos Theory × dm³ · Anti-Chaos Principle

STuring · Morphogenesis · Contact Geometry as Morphogenetic Field

SBelousov-Zhabotinsky · Oscillation · Spiral Waves as Legendrian Knots

SCatGT · Zeolites · HSP · Lean 6 closed · Catalysis Today 2026

SMandelbrot · Fractals · n-bonacci as Hausdorff Dimension Ladder

SPoincaré & Einstein · Two Motivations · One Fixed Point

SLeibniz · Faraday · Clausius · Boltzmann · Energy, Entropy, Order

dm³ 101 / 102 / 103 Mathematical Courses · 16 weeks each · June 2026

101dm³ 101 · π, φ operators · Weeks 1–16 · Contact Geometry Foundations

102dm³ 102 · μ, η, Δ operators · critDim story · Lean 4 labs

103dm³ 103 · Σ, Ω, τ · THE OMEGA POINT · τ = 2 · Complete Completeness

G3 Home ↗ Student Portal IMPA Portal

G4

Vol IV — GTCT · Dimensional Theory

Fourth turn · 7 chapters · 6 appendices · 12-phase cycle · g₃₃ = 33

Live

0Preface — Lemniscata & Analemma

1Ch 1 · The Seven Axioms

2Ch 2 · The Dimensional Field

3Ch 3 · Dimensional Operators

4

4Ch 4 · The Correspondence Theorem

5Ch 5 · The Orthogonality Theorem [v2]

6Ch 6 · The Recursion Theorem [v2]

7Ch 7 · Emergence as Fixed Point [v2]

TCh 8 · The Nature of Time

AAppendices A–F

⚡Ch Faraday · Electromagnetic Field Geometry · Poynting · dm³ Bridge

G4 Full Hub ↗ G4 Table of Contents

G5

Vol V — The Seed · Complete Completeness · A Semente

Fifth turn · G⁵ · The fixed point the series was always approaching · 528 pp · Seed of G6–G9

Final

preStudent Edition — How to Read This Book

1AXLE: Mathematics as Language

∞The G-Series Map: Nine Turns of the Spiral · Seed of G6–G9

…Full Vol V content — see GTCT_Vol5_v8_FinalSubmission.pdf

Zenodo DOI ↗ ISBN 979-8-9954416-5-6 · G6 LLC · 2026

G6

Vol VI — Applied Physics & Chemistry · CatGT · Plasma · Fusion · Enceladus

Sixth turn · G⁶ · Operator non-commutativity in physical & chemical systems · Lean 19 closed

In Progress

CatGT — Catalytic Generative Theory · submitted Catalysis Today 2026

KCatGT · Zeolite Selectivity · HSP · Firing-Order Non-Commutativity · Lean 6 closed

KCatGT_Main.lean · helical_selectivity · selectivityFactor_eq · 6 theorems + 3 admits

DustyPlasma / MHD Reconnection · NASA MMS · Lean 13 closed

⚡DustyPlasma.lean · Sweet-Parker threshold = fold operator F · 13 closed · 3 admits

🛰Saturn Hexagon · Enceladus cryovolcanism · HSP r*(λ) across planetary systems

Nuclear Fusion — Lawson Criterion as Fold Operator F

☀Triple-Alpha Process · T^40 fold at T* ≈ 10⁸ K · Lean-proved AutophagyDm3.lean · 0 sorry

⚛Lawson criterion n·T·τ ≥ Q = HSP threshold r ≤ r*(λ) · Firing: C→K→F→U (same as ZSM-5)

μSelf-regulation: post-ignition plasma = Lyapunov-stable (μ < 0) if open dissipative

→ch-fusion-ignition.html — in queue

AXLE ↗ G6 LLC · 2026

G7

Vol VII — Complexity Science · Scientists Series · Energy Arc

Seventh turn · G⁷ · Pattern across every scale · Feigenbaum → dm³ · n-bonacci as Hausdorff ladder

In Progress

Complexity Science · Chaos · Pattern · Anti-Entropy

μLorenz · Butterfly · Anosov = Reeb · Anti-Chaos Principle

KBelousov-Zhabotinsky · Oscillation · Feigenbaum universality in chemistry

KTuring · Morphogenesis · D_v/D_u as critical dimensionless ratio

ΩMandelbrot · n-bonacci roots as Hausdorff dimensions · open conjecture

UEnergy, Entropy, Order · Leibniz → Boltzmann → dm³ μ = −2

Scientists Series · Motivations & Convergence

SPoincaré & Einstein · Two Paths · One Fixed Point · Solvay 1911

SCatGT · The Order of Operations Matters · Chemistry × dm³

ρchRho-spectral.html · Greek-coded ρ chapter · in queue

ρSpectral Radius · Collatz Bridge (interim)

AXLE ↗ G6 LLC · 2026

G8

Vol VIII — AXLE Mechanisation · Lean 4 Complete

Eighth turn · G⁸ · Every theorem machine-verified · Planned

Planned

…Full Lean 4 mechanisation — AXLE · 1080 proofs

AXLE ↗ G6 LLC · 2026

G9

Vol IX — Complete Completeness² · The Final Turn

Ninth turn · G⁹ · τ = 2 · The fixed point reached · Planned

Planned

∞Complete Completeness² — convergence of all nine turns

Zenodo DOI ↗ G6 LLC · 2026 · 16 repositories

G1 → G2 → G3 → G4 → G5 → G6 → G7 → G8 → G9