Principia Orthogona Vol. II — Interactive Dashboard

dm³ Toy Model — Contact Manifold M = ℝ²₊ × ℝ

σ (noise) 0.0

Initial r 0.4

Initial z 0.01

Time steps 3000

Equations (4.1)–(4.3): ṙ = r(1−r²)+2(r−1)e⁻ᶻ, θ̇=1, ż = r²−2(r−1)²e⁻ᶻ. Limit cycle Γ₁₂ = {r=1} shown as dashed gold line. λ(z)=−2(1−e⁻ᶻ): neutral at z=0, attracting for z>0 (Prop. 4.2).

Theorem B — κ* ⟺ μ_max < 0 ⟺ τ ∈ (0,∞)

c (contraction rate) 4.0

κ_noise 1.0

Theorem B (§3.3)

|κ| ↑ κ* ⟺ μ_max < 0 ⟺ τ = √(c/κ_noise) ∈ (0,∞).
κ* is the geometric precursor of τ: curvature accumulation creates the conditions under which stochastic stability becomes meaningful.

Proof: Forward → Thm 3.2 (Itô correction). Backward → Thm 3.4 (contradiction). Middle → Lemma 3.3.

Theorem 5.4 — Coherence Bridge: Six Domains, One Contact Normal Form

Coherence Bridge Theorem (§5.4)

The six dm³ systems are objects in the same category dm³ and are related by explicit contact morphisms f_ij : X_i → X_j satisfying f_ij(Γ_i) = Γ_j. The systems are not analogies — they are exact mathematical identities in the category dm³.

Contact normal form: ρ̇ = μ_max(1−e^{−βz})ρ + O(ρ²), θ̇ = ω + O(ρ), ż = ω−|μ_max|ρ²e^{−βz}+O(ρ³)

Lean 4 Proof Status — Honest Sorry Tracker

All sorry statements are open proof obligations. None are hidden. Green badges = closed in Lean 4. Red = open.

eigenvalue_at_zero PROVED

λ(0) = 0 — neutral stability at embodiment threshold.

simp [transverseEigenvalue] — closes immediately.

eigenvalue_neg_pos_z PROVED

λ(z) < 0 for z > 0 — attracting post-embodiment.

mul_neg_of_neg_of_pos + exp_lt_one_of_neg.

toyModel_tau PROVED

τ = √(4/1) = 2 in closed form.

norm_num + Real.sqrt_eq_iff_sq_eq.

toyModel_epsilon0 PROVED

ε₀ = 2 / (2·(1+2)) = 1/3.

norm_num. Closes immediately.

thm_C_singularity_bijection PROVED

A2, A3 have unique preimages in the singularity correspondence.

cases + simp on finite inductive type.

eigenvalue_limit OPEN

λ(z) → μ_max as z → ∞.

★★☆ — Needs: Real.tendsto_exp_atBot + filter algebra. Estimated 1–2 days.

thm_A_contact_realization OPEN

H_diss → S(γ) as β→∞ in distributional sense.

★★★★☆ — Requires distribution theory framework in Mathlib. Major effort.

thm_B full chain OPEN

|κ|↑κ* ↔ μ_max < 0: requires Floquet theory formalization.

★★★★★ — Floquet + SDE in Lean 4 is frontier Mathlib work. Collaboration target.

OP1: Global Equivalence OPEN

Every τ-stable dm³ system arises from a fold globally on X.

Open problem (§6.3). Requires global contact topology.

OP2: Higher Resonances OPEN

k:m correspondence between higher Ak and higher resonances.

Open problem (§6.3). Requires Morse theory beyond A3.

-- Proved in Lean 4 (no sorry): theorem eigenvalue_at_zero (sys : DM3System) : transverseEigenvalue sys 0 = 0 := by simp [transverseEigenvalue] ✓ theorem toyModel_epsilon0 : stabilityRadius 2 2 (by norm_num) (by norm_num) = 1 / 3 := by unfold stabilityRadius; norm_num ✓ -- Open obligation (sorry documented): theorem eigenvalue_limit (sys : DM3System) : Filter.Tendsto (transverseEigenvalue sys) Filter.atTop (nhds sys.mu_max) := by sorry OPEN ★★☆ -- Strategy: Real.tendsto_exp_atBot, then ring. Est. 1-2 days.

Operator Sequence G = U ∘ F ∘ K ∘ C and Contact Extension

Volume I → Volume II Bridge

Volume I: C→K→F→U is a piecewise-smooth symplectic map on T*X. The fold F preserves ω = dγ∧dp (Theorem 11.1 [Vol I]).

Volume II: passes to contact extension M = X×ℝ, α = dz−λ, dλ = ω. The fold becomes H_diss = −γVe^{−βz} in the regularized limit β→∞.

Why Contact Geometry?

Liouville's theorem forbids attractors in symplectic systems on compact manifolds. Contact geometry provides: limit cycle attractors, stochastic stability, and variational structure simultaneously. The contact variable z records accumulated dissipation — the orbit earns its stability by accumulating action.