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 c*=3 criticality bridge is derived in full, and a real, honestly-open Collatz descent proof (CollatzDescent.lean) becomes this course’s Lean centerpiece. 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 →
Full week content published — prose, real theorem citations, and Lean 4 references.
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 →
Full week content published — prose, real theorem citations, and Lean 4 references.
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 →
Full week content published — prose, real theorem citations, and Lean 4 references.
4week
AXLE: Mechanising μ
Lean 4 encoding of Lyapunov stability. Distinguishing the abstract contraction proof from the certified r* value.
Operator: μ  ·  Full week page →
★ MILESTONE · Submit to Zenodo
Full week content published — prose, real theorem citations, and Lean 4 references.
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 ≈ 1.839, derived via Cardano's formula. 3 Lean files, 0 sorry.
Operator: η  ·  Full week page →
Full week content published — prose, real theorem citations, and Lean 4 references.
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 →
Full week content published — prose, real theorem citations, and Lean 4 references.
7week
The Criticality Bridge — Why c = 3, in Depth
The Whitney A₁ fold at q=1 worked in full: V'(q)=0, V(1)=-2, V''(1)≠0, factoring V(q)+2=(q-1)²(q+2). Why this forces the Collatz map.
Operator: η  ·  Full week page →
Full week content published — prose, real theorem citations, and Lean 4 references.
8week
Milestone III — Deriving the Fold by Hand
Reproduce Theorem C.1 from scratch: the full c*=3 derivation, no reference material. Connect precisely to Collatz. Zenodo timestamp.
Operator: η  ·  Full week page →
★ MILESTONE · Submit to Zenodo
Full week content published — prose, real theorem citations, and Lean 4 references.
Δ≈ 1.927
The Δ Operator — Tetranacci
Weeks 9–12
Δ ≈ 1.927 is the 4-bonacci dominant root: rank-supercritical (n=4>3) while remaining potential-subcritical (c=Δ<3). Phase closes with a real, open Lean proof on Collatz descent.
9week
The Δ Operator — Tetranacci ≈ 1.927
4-bonacci recurrence. Dominant root Δ ≈ 1.928. Depth-supercritical (n=4) but potential-subcritical (c=Δ<3) — two distinct senses of "supercritical."
Operator: Δ  ·  Full week page →
Full week content published — prose, real theorem citations, and Lean 4 references.
10week
Collatz Descent — The Real, Open Lean File
CollatzDescent.lean: step, orbit, and the descent theorem. What's fully proved (even case, native_decide small cases) and what genuinely isn't (2 honestly-documented sorries).
Operator: Δ  ·  Full week page →
Full week content published — prose, real theorem citations, and Lean 4 references.
11week
The Terras/Everett Argument
What closing the two CollatzDescent.lean gaps actually takes — orbit unrolling (tractable) and the Nat.log 2 induction strategy (genuinely hard), read from the file's own honest self-correction.
Operator: Δ  ·  Full week page →
Full week content published — prose, real theorem citations, and Lean 4 references.
12week
no_return_to_critical — Second Closed Proof
Attempt orbit_halving — the tractable open gap in CollatzDescent.lean. A real candidate pull request against AXLE, not a classroom exercise.
Operator: Δ  ·  Full week page →
★ MILESTONE · Submit to Zenodo
Full week content published — prose, real theorem citations, and Lean 4 references.
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 →
Full week content published — prose, real theorem citations, and Lean 4 references.
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 →
Full week content published — prose, real theorem citations, and Lean 4 references.
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 →
Full week content published — prose, real theorem citations, and Lean 4 references.
16week
Milestone IV — 102 Complete · Ready for 103
Final assessment: 2-page proof summary of Theorem C.1 and the Collatz descent gaps, plus your orbit_halving attempt. Lean 4 portfolio.
Operator: G⁵  ·  Full week page →
★ MILESTONE · Submit to Zenodo
Full week content published — prose, real theorem citations, and Lean 4 references.
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