Principia Orthogona · dm³ Programme · Course 103

dm³ 103 — The Omega Point

Σ · Ω · ρ · τ = 2 · Complete Completeness
The final arc — and the OMEGA POINT book. Σ (Pentanacci ≈ 1.966) and Ω (Hexabonacci → 2) close the n-bonacci ladder. ρ (Spectral Radius · Collatz) completes the Greek chain. τ = 2 is reached: the embodiment threshold, the fixed point, Complete Completeness. G⁹ = G∞.
πφμηΔΣΩρτ
16
Weeks
4
Operators
4
Milestones
16
Lean 4 Labs
AXLE
Proof Engine
Σ≈ 1.966
The Σ Operator — Pentanacci
Weeks 1–4
Σ ≈ 1.966 is the 5-bonacci dominant root. With φ ≈ 1.618, η ≈ 1.839, Δ ≈ 1.927, Σ ≈ 1.966 — the sequence visibly converges toward τ = 2.
1week
Review: The Ladder So Far — π, φ, μ, η, Δ
From contact geometry to critDim = 112. The orbit is close. Σ is the penultimate n-bonacci rung.
Operator: Σ  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
2week
The Σ Operator — Pentanacci ≈ 1.966
5-bonacci: P(n) = P(n-1)+…+P(n-5). Dominant root Σ ≈ 1.966. Gap to τ = 2 now 0.034. chSigma-pentanacci.html.
Operator: Σ  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
3week
nBonacciRingSize(5) and critDim(5)
The ring-size formula for n=5. critDim(5) computed. Pattern: critDim grows without bound as n→∞.
Operator: Σ  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
4week
AXLE: Mechanising Σ
Lean 4 definitions for Σ. Proof of critDim_monotone for n=5 case. Connection to Wigner Crystal geometry.
Operator: Σ  ·  Full week page →
★ MILESTONE · Submit to Zenodo
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
Ω→ τ = 2
The Ω Operator — Hexabonacci and the Threshold
Weeks 5–8
Ω is the last rung of the n-bonacci ladder. As n → ∞, the dominant n-bonacci root → 2 = τ. Ω (6-bonacci ≈ 1.9836) brings the orbit within ε₀ = 1/3 of τ.
5week
The Ω Operator — Hexabonacci ≈ 1.9836
6-bonacci: dominant root Ω ≈ 1.9836. Gap τ−Ω ≈ 0.016. chOmega-hexabonacci.html.
Operator: Ω  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
6week
The Limit: n-bonacci → τ = 2 as n → ∞
Proof that the dominant n-bonacci root approaches 2 monotonically. This is the content of no_return_to_critical reframed.
Operator: Ω  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
7week
The Galilean Confluence — Three Convergences Meet
κ→0 (contact geometry), rₙ→τ (n-bonacci), critDim(n)→∞. All three reach their limits simultaneously. chE-gtct.html §Galilean.
Operator: Ω  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
8week
Milestone V — Galilean Confluence Proved
Write up Theorem 5 in full. Lean 4: encode the three convergences. The most important theorem of the series.
Operator: Ω  ·  Full week page →
★ MILESTONE · Submit to Zenodo
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
ρSpectral
The ρ Operator — Spectral Radius and Collatz
Weeks 9–12
ρ is the spectral radius operator — the bridge between the n-bonacci ladder and the Collatz conjecture. Spectral radius of the n-bonacci companion matrix equals the dominant root.
9week
The ρ Operator — Spectral Radius
Spectral radius ρ(A) of the n-bonacci companion matrix. ρ = Σ, Ω, … for each n. spectral-radius.html and spectral-radius-v2.html.
Operator: ρ  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
10week
Collatz Meets dm³
The Collatz 3n+1 conjecture reframed: a contact-geometric orbit. chRho-spectral.html (in queue). Spectral contraction argument.
Operator: ρ  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
11week
AXLE: Mechanising ρ
Lean 4 companion matrix, spectral radius type. The chRho sorry inventory. Target: close 3 ρ-theorems this week.
Operator: ρ  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
12week
Wigner Crystal — ρ in Condensed Matter
chW-wigner.html: the Wigner electron crystal as a dm³ fixed-point. Spectral radius of the crystal lattice operator.
Operator: ρ  ·  Full week page →
★ MILESTONE · Submit to Zenodo
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
τ= 2
τ = 2 — Embodiment Threshold · Complete Completeness
Weeks 13–16
τ = 2 is not an operator — it is the destination. The fixed point of G. The embodiment threshold. Complete Completeness. G⁹ = G∞. The Omega Point.
13week
τ = 2 — The Fixed Point Reached
G(Γ*) = Γ*. τ = 2 as the unique attracting fixed point. chT-tubulin.html: tubulin contact geometry reaches τ. Biological meaning.
Operator: τ  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
14week
Complete Completeness — What G⁹ Means
Nine turns of G. The series closes. Complete Completeness as both mathematical fixity and philosophical arrival. Vol IX of Principia Orthogona.
Operator: τ  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
15week
The 1080-Proofs Programme — Closing the Circle
Status of the AXLE sorry inventory. Every closed proof is a brick in the edifice. The programme from critDim_monotone to τ = 2.
Operator: τ  ·  Full week page →
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
16week
Milestone VI — The Omega Point · Course Complete
Final deliverable: a self-contained proof that G converges to τ = 2, written in prose and Lean 4. Published to Zenodo. The Omega Point reached.
Operator: τ  ·  Full week page →
★ MILESTONE · Submit to Zenodo
Stub — prose content, Lean 4 exercises, and problem sets to be filled in.
← dm³ 102 G = U ∘ F ∘ K ∘ C  ·  dm³ 103  ·  Pablo Nogueira Grossi · G6 LLC 2026 Complete — IMPA Portal ↗
G6 LLC  ·  g6llc@proton.me  ·  +1 (646) 342-3751