The dm³ Operator: Explicit Toy Model and Global Dynamical Analysis
Description
webpage: The dm3 Operator: Explicit Toy Model and Global Dynamical Analysis
The dm³ Operator: Explicit Toy Model and Global Dynamical Analysis
Version 3 — July 2026
Pablo Nogueira Grossi · G6 LLC, Newark NJ · ORCID: 0009-0000-6496-2186
Zenodo concept DOI (resolves to latest): https://doi.org/10.5281/zenodo.19117399 V3 DOI: https://doi.org/10.5281/zenodo.21147306 (supersedes V1 and V2) V2 DOI: [prior deposit, see version history below] Series root: https://doi.org/10.5281/zenodo.19117399
Description (Abstract)
We construct and analyze a complete explicit instantiation of the Generative Contact Mechanics framework on the two-dimensional system:
ṙ = r(1−r²) + 2(r−1)e^(−z), θ̇ = 1, ż = r² − 2(r−1)²e^(−z),
on the contact manifold M = R²₍>0₎ × R. Every definition, operator, and boundary from the framework is instantiated explicitly and verified by direct computation. Four main results are established. Theorem A: the global attractor of the full system is the resonant orbit Γ₁₂. Theorem B: the invariant torus conjecture holds for the 1:2 resonant case, with a normally hyperbolic invariant circle and transverse Lyapunov exponent −3. Theorem C: the system undergoes four bifurcations — contact Hopf, saddle-node of limit cycles, Neimark–Sacker, and slow-fast crossover — as parameters vary. Theorem D: the stationary SDE measure concentrates on Γ for noise amplitude below the embodiment threshold τ = 2 and spreads for amplitude above τ. The canonical invariant triple is (T*, μ_max, τ) = (2π, −2, 2) and the stability radius is ε₀ = 1/3.
What V3 fixes
V3 (July 2026) is a hand-verified correction pass following an independent line-by-line audit. Three arithmetic errors were found and fixed, each verified symbolically and/or numerically before correction; two broken internal cross-references were also resolved. No theorem statement, canonical invariant, or main qualitative result changed — all four main theorems (A, B, C, D) hold in substance exactly as in V2.
| Change | Detail |
|---|---|
| §2.2, Proposition (Explicit flow formula) | Corrected the exponent in the closed-form flow solution from e^(−4t) to e^(−2t). The V2 proof's Bernoulli-substitution step asserted u̇ = −4u+4; direct substitution gives u̇ = 2−2u. Verified by direct numerical integration: at r₀=0.5, t=1, the RK-integrated value is r(1)=0.843347, matching the corrected formula exactly; the V2 formula gave 0.973609, a 15% relative error. Does not affect μ_max = −2, which is obtained independently by linearization at r=1. |
| §6.1, Proposition (Contact Hopf bifurcation) | Corrected a spurious factor of 2 in the linearized transverse eigenvalue: λ(γ,z₀) = γe^(−z₀) − 2 (was −2(1−γe^(−z₀)) = 2γe^(−z₀)−2), giving the corrected bifurcation point γ* = 2e^(z₀) (was e^(z₀)). Confirmed by an internal consistency check: the paper's own baseline parameter γ=2, used throughout the paper as the canonical case, already matches the corrected γ* = 2e^(z₀) at z₀=0, not the originally stated value. This correction propagates to Theorem C(i) and the numerical-accessibility discussion (§10.2), both updated accordingly. |
| §7.2, Proposition (Stationary density) | Corrected the stationary Fokker–Planck density from exp(−2(r−1)²/σ₀²) to exp(−4(r−1)²/σ₀²), the same species of factor-of-2 error as above. Verified by direct substitution into the stated Fokker–Planck equation, giving an identically-zero residual only for the corrected form. Corrected variance is σ₀²/8 (was implied σ₀²/4). Propagates to Theorem D's proof: the concentration formula becomes erf(2δ/σ₀) (was erf(δ/(σ₀√2))); the qualitative conclusion (concentration below τ=2, spreading above) is unchanged. |
| §1.4 | Resolved a broken cross-reference ("Section ?? proves Theorem B") to Section 4. |
| §6.3 (saddle-node proof) | Resolved a broken cross-reference ("Definition ??") to §3.2.3, where the anti-collapse potential Q is defined. The numerical value η* ≈ 0.15 itself was not independently re-derived in this audit and is carried forward unchanged. |
Independently re-checked and confirmed unaffected
All other propositions and theorems were read in full during this audit and found correct: Proposition 2.3 (canonical invariants), Proposition 2.4 (all eight axioms), Proposition 2.5 (contact structure), Propositions 2.6–2.7 (behavior on Γ, embodiment threshold), all of §3 (operator instantiation: g-, L-, R-, U-, and boundary operators), Proposition 4.1 (dm³ orbit stability), Proposition 4.2 (resonant orbit stability, δ̇ = −3, independently verified), Proposition 4.3 (collapse boundary), the proof of Theorem A (§5.2, four-step convergence argument), Theorem 6.4 (Neimark–Sacker), Proposition 7.1 (SRB measure), Proposition 8.1 and Corollary 8.2 (contact normal form).
Version history
| Version | Date | Key change |
|---|---|---|
| V1 | March 2026 | Original deposit (part of the four-paper Principia Orthogona / GCM bundle, zenodo.19117400) |
| V2 | May 16, 2026 | Standalone deposit of this paper |
| V3 | July 2026 | Hand-verified correction pass: three arithmetic errors fixed (flow formula exponent, Hopf bifurcation factor, stationary density factor), two broken cross-references resolved. No theorem statements or canonical invariants changed. |
Series context
| Role | DOI |
|---|---|
| Series root / concept DOI | 10.5281/zenodo.19117399 |
| This deposit (V3, latest) | 10.5281/zenodo.21147306 |
| Principia Orthogona, Volume One | 10.5281/zenodo.21146416 (V6, latest) |
| Generative Contact Mechanics (companion paper) | see companion deposit |
| AXLE formal verification hub | github.com/TOTOGT/AXLE |
MSC codes: 37C10, 37C27, 37G15, 37H10, 53D10, 60H10
Keywords: contact geometry · limit cycles · bifurcation theory · invariant measures · structural stability · resonance · normal form · stochastic stability · global attractor · dm³ operator · toy model · embodiment threshold
License: Creative Commons Attribution Non Commercial No Derivatives 4.0 International (CC BY-NC-ND 4.0) Journal title: SIAM Journal on Applied Dynamical Systems. Status: Draft — submission TBD. Copyright: © 2026 Pablo Nogueira Grossi, G6 LLC Contact: pgrossi888@outlook.com · g6llc@proton.me
Notes: Companion paper: Generative Contact Mechanics (GCM). Part of the Principia Orthogona / GCM research program, G6 LLC, Newark NJ, 2026. Series root: https://doi.org/10.5281/zenodo.19117399.
Deposit contents
dm3_toy_model_v3.pdf — 13-page paper (this file), full text with all V3 corrections integrated dm3_toy_model_v3.tex — LaTeX source
Build instructions
pdflatex dm3_toy_model_v3.tex pdflatex dm3_toy_model_v3.tex pdflatex dm3_toy_model_v3.tex (Run three times for cross-references)
Notes
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Additional details
Related works
- Is part of
- Other: 10.5281/zenodo.19117399 (DOI)
- Is supplemented by
- Software: https://github.com/TOTOGT/AXLE (URL)
- Software: https://github.com/TOTOGT/DM3-lab (URL)
- Is version of
- Other: 10.5281/zenodo.19379384 (DOI)