Principia Orthogona · dm³ Programme · Course 102

dm³ 102 — Middle Operators

Lyapunov Stability · Tribonacci · Tetranacci
The middle arc: three operators μ (Lyapunov −2), η (Tribonacci ≈ 1.839), and Δ (Tetranacci ≈ 1.927). The critDim formula emerges — critDim(3) = 26 connects to string theory; critDim(4) = 112 to the 1080-proofs programme. AXLE mechanisation deepens.
πφμηΔΣΩρτ
16
Weeks
3
Operators
4
Milestones
16
Lean 4 Labs
AXLE
Proof Engine
μ−2
The μ Operator — Lyapunov Exponent
Weeks 1–4
μ = −2 is the Lyapunov exponent of the dm³ fixed-point iteration. Negative value: the orbit is stable. This is why G converges to τ = 2.
1week
Review: π, φ and the Operator Chain
Quick recap of 101. The chain so far: G = U∘F∘K∘C, period T* = 2π, golden ratio φ ≈ 1.618. Gap to τ = 2.
Operator: μ  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
2week
The μ Operator — Lyapunov Exponent −2
Lyapunov exponent of G at fixed point Γ*. μ = −2 means exponential contraction. Stability radius ε₀ = 1/3.
Operator: μ  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
3week
Stability in Contact Geometry
What ε₀ = 1/3 means geometrically. Basin of attraction. chMu-lyapunov.html as primary text.
Operator: μ  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
4week
AXLE: Mechanising μ
Lean 4 encoding of Lyapunov stability. First critDim appearances in the proof environment.
Operator: μ  ·  Full week page →
★ MILESTONE · Submit to Zenodo
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
η≈ 1.839
The η Operator — Tribonacci
Weeks 5–8
η ≈ 1.839 is the dominant root of x³ = x² + x + 1 (3-bonacci). η weighting — the η⁻ᵏ scheme that appears throughout dm³ — is named for this operator.
5week
The η Operator — Tribonacci ≈ 1.839
3-bonacci: T(n) = T(n-1)+T(n-2)+T(n-3). Dominant root η. nBonacciRingSize(3) = 13. chEta-tribonacci.html.
Operator: η  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
6week
η Weighting — The Core dm³ Tool
η⁻ᵏ weighting in phase decompositions. Why η, not a generic geometric series. Distinction from Hilbert-space Born rule.
Operator: η  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
7week
critDim(3) = 26 — String Theory Bridge
critDim(n) = 2*(nBonacciRingSize(n)−1)+2. critDim(3) = 26. The 26-dimensional bosonic string as a contact-geometric landmark.
Operator: η  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
8week
Milestone III — η and critDim(3)
Prove critDim(3)=26 from nBonacciRingSize_3=13. Submit Lean 4 proof. Zenodo timestamp.
Operator: η  ·  Full week page →
★ MILESTONE · Submit to Zenodo
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
Δ≈ 1.927
The Δ Operator — Tetranacci
Weeks 9–12
Δ ≈ 1.927 is the 4-bonacci dominant root. nBonacciRingSize(4) = 56 gives critDim(4) = 112 — exactly the number of proofs in the AXLE 1080-proofs programme.
9week
The Δ Operator — Tetranacci ≈ 1.927
4-bonacci recurrence. Dominant root Δ. nBonacciRingSize(4) = 56. chDelta-tetranacci.html.
Operator: Δ  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
10week
critDim(4) = 112 — The 1080-Proofs Connection
critDim(4) = 2*(56−1)+2 = 112, a real Lean-proved lemma (critDim_4). Correction: the sorry_inventory.csv audit file does have 112 rows, but that is a count of distinct file/theorem entries under review, not a count of sorrys — the actual sorry count in that file is 1,041 (1,027 still open as of the last audit). The 112/112 match is a coincidence between a fixed arithmetic constant and the row-count of a CSV that changes as entries are closed, not a structural finding.
Operator: Δ  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
11week
critDim_monotone — A Closed Lean Proof
Walk through the proof of critDim_monotone closed in session 2026-06-14. Axioms, omega tactic, proof term.
Operator: Δ  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
12week
no_return_to_critical — Second Closed Proof
Walk through no_return_to_critical: for n > 3, critDim(n) > 26. Proof by cases on n ≥ 4.
Operator: Δ  ·  Full week page →
★ MILESTONE · Submit to Zenodo
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
G⁵synthesis
G⁵ and the Midpoint — Three Operators Closed
Weeks 13–16
Five operators of nine now defined. The n-bonacci ladder is half-climbed. G⁵ brings the orbit visibly closer to τ = 2. Preview: Σ, Ω, ρ, τ in dm³ 103.
13week
G⁵ — Fifth Iteration and Convergence Rate
Orbit after five G applications. Distance to τ = 2 shrinks. Lyapunov exponent μ = −2 doing its work.
Operator: G⁵  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
14week
Applications — Immune Memory and Allostatic Load
ch5-immune.html, ch2-allostatic.html: Δ operator in biological n-bonacci rhythms.
Operator: G⁵  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
15week
Preview: Σ, Ω, ρ, τ — The Final Ascent to the Omega Point
Road map to dm³ 103. Pentanacci, Hexabonacci, Spectral Radius, Embodiment Threshold. The Omega Point.
Operator: G⁵  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
16week
Milestone IV — 102 Complete · Ready for 103
Final assessment: write a 2-page proof summary of critDim_monotone and no_return_to_critical. Lean 4 portfolio.
Operator: G⁵  ·  Full week page →
★ MILESTONE · Submit to Zenodo
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
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