Principia Orthogona · dm³ Programme · Course 101

dm³ 101 — Foundations

Contact Geometry · The π and φ Operators
The opening arc of the dm³ programme. Sixteen weeks from contact manifolds and the operator chain G = U ∘ F ∘ K ∘ C through the first two rungs of the recurrence ladder: π (period T* = 2π) and φ (Fibonacci ratio ≈ 1.618). Every theorem is mechanised in Lean 4 / AXLE.
πφμηΔΣΩρτ
16
Weeks
2
Operators
4
Milestones
16
Lean 4 Labs
AXLE
Proof Engine
Gchain
The Operator Chain — Contact Geometry Foundations
Weeks 1–4
Contact manifold (M, ξ), the four operators U, F, K, C, and the fixed-point iteration G = U ∘ F ∘ K ∘ C. η weighting introduced.
1week
Contact Manifolds — (M, ξ)
The arena: contact structure ξ on M, Legendrian submanifolds, contact forms. Why contact geometry, not Riemannian.
Operator: G  ·  Full week page →
Full week content published — prose, real theorem citations, and Lean 4 references.
2week
The Four Operators — U, F, K, C
Each operator defined: U (unfolding), F (folding), K (kernel projection), C (contact lift). Composition rules.
Operator: G  ·  Full week page →
Full week content published — prose, real theorem citations, and Lean 4 references.
3week
Fixed-Point Iteration — G = U ∘ F ∘ K ∘ C
G as a single map. First iteration G¹. Convergence criterion. The embodiment threshold τ = 2 as destination.
Operator: G  ·  Full week page →
Full week content published — prose, real theorem citations, and Lean 4 references.
4week
η Weighting and Phase Weights
The η⁻ᵏ weighting scheme — why η, not geometric. Phase weights over contact forms. AXLE: first Lean 4 definitions.
Operator: G  ·  Full week page →
★ MILESTONE · Submit to Zenodo
Full week content published — prose, real theorem citations, and Lean 4 references.
πT* = 2π
π — Period and the Recurrence Ladder
Weeks 5–8
"π Operator" is this phase's working label, kept from the course's original design — but π is a proved period constant (T* = 2π, from Reeb-flow compactification), not a sixth operator alongside U, F, K, C. It sets the fundamental recurrence scale the n-bonacci ladder is read against.
5week
The π Operator — Period T* = 2π
Periodic orbits in contact geometry. T* as minimal recurrence period. Connection to Fourier decomposition over ξ.
Operator: π  ·  Full week page →
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6week
Recurrence Ladders — First Rung
How T* = 2π anchors the ladder π → φ → μ → η → Δ → Σ → Ω → τ. The spacing between rungs as convergence rate.
Operator: π  ·  Full week page →
Full week content published — prose, real theorem citations, and Lean 4 references.
7week
AXLE: Mechanising π
Lean 4 definitions for the π operator. Type-theoretic encoding of T*. First sorry-free theorems.
Operator: π  ·  Full week page →
Full week content published — prose, real theorem citations, and Lean 4 references.
8week
Milestone I — π Operator Complete
Assessment: prove the π recurrence bound by hand. Submit Lean 4 file. Zenodo timestamp.
Operator: π  ·  Full week page →
★ MILESTONE · Submit to Zenodo
Full week content published — prose, real theorem citations, and Lean 4 references.
φ≈ 1.618
φ — Fibonacci and the Subcritical Rung
Weeks 9–12
"φ Operator" is likewise a working label, not a literal operator: φ = (1+√5)/2 ≈ 1.618 is the n=2 rung of the recurrence ladder — proved subcritical (Theorem φ.1), meaning it never completes a dm³ fold on its own. Nested self-similarity, fixed-point structure, and the honest gap in AXLE's Lean coverage (η is formalized, φ isn't yet) are this phase's real content.
9week
The φ Operator — Golden Ratio
φ as dominant eigenvalue of the 2-bonacci recurrence. Self-similar contact structures. Connection to quasicrystals.
Operator: φ  ·  Full week page →
Full week content published — prose, real theorem citations, and Lean 4 references.
10week
Fibonacci Sequences in Contact Geometry
F(n+1)/F(n) → φ. Rate of approach. η weighting for n=2. Chapter ch9-phi.html as primary text.
Operator: φ  ·  Full week page →
Full week content published — prose, real theorem citations, and Lean 4 references.
11week
Nested Infinities — φ and Self-Reference
φ = 1 + 1/φ as fixed-point equation. The same structure as G* = G(G*). Philosophical implications for dm³.
Operator: φ  ·  Full week page →
Full week content published — prose, real theorem citations, and Lean 4 references.
12week
AXLE: Mechanising φ
Lean 4: Fibonacci type, φ as limit, convergence proof. Link to chPI-recurrence theorems.
Operator: φ  ·  Full week page →
★ MILESTONE · Submit to Zenodo
Full week content published — prose, real theorem citations, and Lean 4 references.
G²synthesis
First Synthesis — G² and the Road Ahead
Weeks 13–16
Two operators in, seven to go. G² applies: the iterated chain begins to linearise. Preview of μ, η, Δ, Σ, Ω and the Galilean Confluence.
13week
G² — Second Iteration
What changes after two turns of G. Linearisation near fixed points. Gap τ−φ ≈ 0.382 as the remaining distance.
Operator: G²  ·  Full week page →
Full week content published — prose, real theorem citations, and Lean 4 references.
14week
Applications — Neural Circuits and Cardiac Resonance
ch4-neural.html and ch6-resonance.html: dm³ applied to biological rhythms. φ in heart-rate variability.
Operator: G²  ·  Full week page →
Full week content published — prose, real theorem citations, and Lean 4 references.
15week
Preview: μ, η, Δ — The Middle Operators
Road map to dm³ 102. The n-bonacci family for n = 3, 4. critDim formula. Lyapunov stability.
Operator: G²  ·  Full week page →
Full week content published — prose, real theorem citations, and Lean 4 references.
16week
Milestone II — 101 Complete · Ready for 102
Final assessment: operator chain essay + Lean 4 proof of φ convergence. Portfolio submission.
Operator: G²  ·  Full week page →
★ MILESTONE · Submit to Zenodo
Full week content published — prose, real theorem citations, and Lean 4 references.
G = U ∘ F ∘ K ∘ C  ·  dm³ 101  ·  Pablo Nogueira Grossi · G6 LLC 2026 dm³ 102 →
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