Pablo Nogueira Grossi · G6 LLC · Newark, NJ · 2026 · ORCID 0009-0000-6496-2186
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).
|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.
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.
“Identity” has a standard test attached, and it had never been run. Two matrices are similar exactly when they represent one linear map in different bases; near Γ each domain is a 2×2 system with eigenvalues μ ± iω. tools/coherence_similarity.py parses the table and runs it on the eleven rows carrying both. Linear similarity: 0 matching pairs out of 55. Up to rescaling the clock — the ratio μ/ω — 0 out of 55. The closest pair is immune adaptation against market volatility, −2.4444 to −2.3929: near, not equal, and nothing else is within 0.11. What the rows do share is being spiral sinks, and every 2D linear spiral sink is topologically conjugate to every other — eleven of eleven qualify, and so would eleven damped oscillators picked at random. The honest statement is the one Ch 20 already gives: the same normal form with different invariants, which is real and checkable, and not a categorical equivalence.
All sorry statements are open proof obligations.
None are hidden. Green badges = closed in Lean 4. Red = open.
λ(0) = 0 — neutral stability at embodiment threshold.
λ(z) < 0 for z > 0 — attracting post-embodiment.
τ = √(4/1) = 2 in closed form.
ε₀ = 2 / (2·(1+2)) = 1/3.
A2, A3 have unique preimages in the singularity correspondence.
λ(z) → μ_max as z → ∞.
H_diss → S(γ) as β→∞ in the distributional sense. The declaration's conclusion is True, not a sorry — so it passes every axiom check, which is why it is marked here rather than left to a badge.
The declaration proves μ_max < 0 ⟺ τ > 0, but from assumptions on both sides: each branch discards its hypothesis, and μ_max < 0 is a field of the DM3System structure. Each half is independently true and the arrow carries nothing.
Every τ-stable dm³ system arises from a fold globally on X.
k:m correspondence between higher Ak and higher resonances.
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→∞.
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.
sorryAx. A clean axiom report is not a reading of the statement: per R20, a theorem can assume its conclusion and still report clean. Follow the link before citing one as evidence.