Principia Orthogona · dm³ Programme · Course 103

dm³ 103 — Complete Completeness

Σ · Ω · ρ · τ = 2, reached at G⁵
The final arc. Σ (Pentanacci ≈ 1.966) and Ω (Hexabonacci ≈ 1.984) close the n-bonacci ladder — proved by IVT bracketing, since Abel–Ruffini rules out a closed radical form past degree 4. ρ (spectral radius) opens a real, honestly-open research problem on the Collatz/Syracuse transfer operator ("Bridge 0"). τ = 2 is reached at G⁵ — matching Volume V’s own subtitle, Complete Completeness — with G⁶ standing as a separate, explicitly open conjecture, not a further proved stage.
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. Unlike η, Σ has no closed radical form (Abel–Ruffini) — proved instead by IVT bracketing.
1week
Review: The Ladder So Far — π, φ, μ, η, Δ
Five constants, two proved theorems (C.1, μ_max), one certified value (r*), one honest open problem. Σ and Ω close the ladder next.
Operator: Σ  ·  Full week page →
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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 →
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3week
Why Σ Has No Closed Radical Form
Abel–Ruffini theorem: quintics aren't solvable by radicals (Galois, S₅ not solvable). Contrast with η's exact Cardano derivation in 102.
Operator: Σ  ·  Full week page →
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4week
Milestone V — Mechanising Σ Numerically
No closed form means no rfl/ring proof. What's provable instead: IVT bracketing (f(1.965)<0<f(1.967)) plus monotonicity — a real proof, different shape than Cardano.
Operator: Σ  ·  Full week page →
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Ω→ τ = 2
The Ω Operator — Hexabonacci and the Threshold
Weeks 5–8
Ω is the last individually-named rung of the n-bonacci ladder. Theorem Ω.1 (proved, derived in full): as n → ∞, the dominant n-bonacci root → 2 = τ, via a geometric-series limit argument. Ω (6-bonacci ≈ 1.9837) sits within 1% of τ.
5week
Ω — The Hexabonacci Constant
6-bonacci: dominant root Ω ≈ 1.9837, no closed radical form (degree 6). Full ladder assembled: φ, η, Δ, Σ, Ω → τ.
Operator: Ω  ·  Full week page →
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6week
The Limit: n-bonacci → τ = 2 as n → ∞
Theorem Ω.1, derived in full: factor the recurrence, sum the infinite geometric series, solve 1−1/(x−1)=0 ⟹ x=2. A genuine proved limit.
Operator: Ω  ·  Full week page →
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7week
τ = 2 in AXLE — What's Structure Data, What's Derived
The canonical triple (T*,μ_max,τ)=(2π,−2,2). An earlier draft's "Galilean Confluence" unification claim depended on axioms scrubbed in 102 and is removed here — replaced with a precise statement of what's actually proved.
Operator: Ω  ·  Full week page →
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8week
Milestone VI — What "τ = 2 Is Reached" Actually Means
Reproduce Theorem Ω.1's limit argument by hand, then state precisely what's proved (the ladder limit, AXLE's structure data) versus what was removed (an unproved 3-way unification).
Operator: Ω  ·  Full week page →
★ MILESTONE · Submit to Zenodo
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ρSpectral
The ρ Operator — Spectral Radius and Collatz
Weeks 9–12
ρ is the spectral radius of two distinct finite approximations of the Collatz/Syracuse transfer operator: ρ(P_M)=1/2^(M−1) exactly, and ρ(ℒ)≈0.751 numerically (Lyapunov exponent). Relating the two is a real, honestly-open problem ("Bridge 0").
9week
ρ — The Spectral Radius, Two Different Operators
The Syracuse transfer operator has two natural approximations: ρ(P_M)=1/2^(M−1) exact, ρ(ℒ)≈0.751 numeric (Lyapunov). Different objects, different scales. spectral-radius-v2.html.
Operator: ρ  ·  Full week page →
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10week
"Bridge 0" — A Real, Honestly Open Problem
ρ(P_M) decays superexponentially (2-adic erasure); ρ(ℒ)≈0.751 is a fixed statistical rate. Relating them is a named, open task in spectral-radius-v2.html — not solved here.
Operator: ρ  ·  Full week page →
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11week
AXLE: What's Actually Formalized for ρ
Honest audit: ρ(P_M)=1/2^(M−1) is tractable but unwritten in Lean; ρ(ℒ)≈0.751 needs real numerical-analysis infrastructure. A scoped, real gap.
Operator: ρ  ·  Full week page →
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12week
Milestone VII — Attempt the ρ(P_M) Formalization
Attempt rho_PM_eq (M) : ρ(P_M) = 1/2^(M−1) in Lean, following 102's orbit_halving pattern. Submit as a candidate PR against github.com/TOTOGT/AXLE.
Operator: ρ  ·  Full week page →
★ MILESTONE · Submit to Zenodo
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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. Per Volume V's own text (chV-g6.html), the threshold is reached at G⁵ — matching the subtitle "G⁵ · Complete Completeness" — not G⁹. G⁶ is a separate, explicitly open conjecture.
13week
τ = 2 — The Fixed Point, Reached at G⁵
Correcting a naming error: source material says "τ=2 é alcançado em G⁵," matching Vol V's own subtitle — not G⁹. G⁶ is a separate, explicitly open conjecture (AXLE Issue 6).
Operator: τ  ·  Full week page →
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14week
Complete Completeness — The 8 Verified Constants
Volume V's own audit: 8 constants with real Lean tactics (rfl, decide, norm_num). "Complete" names a specific arrival — the fixed point is reached — not an absence of open problems.
Operator: τ  ·  Full week page →
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15week
The 9 Honest Sorrys — AXLE v6.1's Real Audit
9 sorrys, all honestly named; 8 verified constants; 0 axioms beyond Mathlib4. Replaces an earlier "1080-Proofs" framing that depended on scrubbed axioms.
Operator: τ  ·  Full week page →
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16week
Milestone VIII — Full Series Constants and Proof-Status Table
One table, 101+102+103, every constant precisely labeled proved / certified / structure data / open. Plus a reflection on the sequence's largest self-correction.
Operator: τ  ·  Full week page →
★ MILESTONE · Submit to Zenodo
Full week content published — prose, real theorem citations, and Lean 4 references.
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