Contact-Geometric Theory of Generative Transitions: Mathematical Foundations, Contact Realization, Seven Proofs of the Tribonacci Constant, and Applications to Nuclear Matter

Zenodo Deposit TOGT for NuclearPhysicsB — First deposit DOI reserved: 10.5281/zenodo.20682934Series concept DOI (always latest): 10.5281/zenodo.19117399 Title Contact-Geometric Theory of Generative Transitions: Mathematical Foundations, Contact Realization, Seven Proofs of the Tribonacci Constant, and Applications to Nuclear Matter Authors Name Affiliation ORCID Pablo Nogueira Grossi G6 LLC, Newark, New Jersey 07104, USA 0009-0000-6496-2186 Description We present a self-contained, comprehensive development of the dm³ contact-geometric framework for generative transitions — localized geometric events in which a trajectory undergoes compression, curvature intensification, fold (rank-1 loss of injectivity), and stabilization. Part I establishes the mathematical foundations: the operator sequence C → K → F → U on a Riemannian manifold (X,g), the Whitney A₁–A₃ singularity classification, the symplectic preservation theorem, the variational characterization of the unfolding, and the g-series lock-in threshold g = 33. All foundational claims are stated under minimal assumptions (Assumptions 2.1–2.6) and proved or given explicit proof sketches completable by standard methods. Part II constructs the contact realization on the contact 3-manifold M = ℝ²₊ × ℝ with contact form α = dz − r²dθ. Theorem A proves contact realization of the fold; Theorem B establishes equivalence of the curvature threshold κ* and the embodiment threshold τ = 2; Theorem C proves the singularity–bifurcation correspondence. The canonical invariants (T* = 2π, μ_max = −2, τ = 2) and stability radii (ε₀ = 1/3, r* ≈ 0.77594, κ* = √(7/9)) are computed in closed form. Part III gives seven independent proofs that the characteristic root of the dm³ fold recurrence w(k+3) = w(k+2) + w(k+1) + w(k) is the Tribonacci constant η ≈ 1.8393: operator algebra, distribution theory, Whitney catastrophe cascade, contact geometry, Perron–Frobenius spectral theory, certified Newton computation, and Fisher information geometry. Every proof is algebraically complete and companions the 48+ sorry-free Lean 4 theorems in the AXLE engine. Part IV applies the framework to nuclear matter: confinement, hadronization, and coherent structures in the quark-gluon plasma are modeled as iterated fold events under a conjectural effective contact form α_QCD. Five explicitly falsifiable conditions are stated. The Hill coefficient n_H ≈ 3.64 is derived from μ_max = −2, not fitted. The elliptic flow prediction v₂ ∝ ε_part · exp(−|μ̂_max| τ_hydro) with |μ̂_max| ≈ 0.45 fm⁻¹ is testable at zero free parameters. Formal verification: 48+ theorems proved without sorry in Lean 4 / Mathlib4 (AXLE engine, github.com/TOTOGT/AXLE). 17 honest admits stated, 0 hidden sorries. This paper is the first self-contained joint presentation of Principia Orthogona Volumes I and II, together with nuclear matter applications. Series ISBN: 979-8-9954416-6-3. Keywords (enter each on a separate line in Zenodo) contact geometry contact 3-manifold generative transitions dm³ framework Tribonacci constant Whitney singularities n-bonacci ladder nuclear matter quark-gluon plasma elliptic flow Lean 4 formal verification AXLE Principia Orthogona operator algebra Perron-Frobenius Fisher information geometry fold map symplectic geometry contact Hamiltonian License Creative Commons Attribution 4.0 International (CC BY 4.0) Version / Notes Upload type: Publication → Preprint Version: 1 (first deposit of this paper; part of series at zenodo.19117399) Language: English Series ISBN: 979-8-9954416-6-3 Related Identifiers (add each in Zenodo's "Related identifiers" section) DOI / URL Relation Resource type 10.5281/zenodo.19117399 Is new version of Publication 10.5281/zenodo.20298665 References Publication (Vol I) 10.5281/zenodo.20159456 References Publication (Vol II) 10.5281/zenodo.19379385 References Publication (toy model) 10.5281/zenodo.20026942 References Publication (DNLS) https://github.com/TOTOGT/AXLE Is supplemented by Software Files to upload TOGTnuclearPhysicsB_Draft1.pdf — compiled paper (primary) TOGTnuclearPhysicsB_Draft1.tex — LaTeX source (supplementary) Communities (optional, search in Zenodo) physmath (Physics and Mathematics) mathlib (Mathlib / Lean community, if exists) Grant / Funding (optional) NYFA fiscal sponsorship (New York Foundation for the Arts) — workshop programme.G6 LLC independent research.