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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” is a standard term with a decision procedure attached, and the procedure 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 reads the eleven bridge rows that carry both numbers and tests every pair. Linear similarity: 0 of 55. Allowing the clock to be rescaled — comparing only the ratio μ/ω — 0 of 55. The closest pair is Immune adaptation (−2.4444) against Market volatility (−2.3929): near, not equal. What all eleven do share is that every 2D linear spiral sink is topologically conjugate to every other — true of eleven damped oscillators picked at random, so it carries no information about these eleven. The claim is true at the level where it says nothing and false at every level where it would say something. See ch20.
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.