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, μₘax = −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 αQCD. Five explicitly falsifiable conditions are stated. The Hill coefficient nH ≈ 3. 64 is derived from μₘax = −2, not fitted. The elliptic flow prediction v₂ ∝ εₚart · exp (−|μ̂ₘax| τₕydro) with |μ̂ₘax| ≈ 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 TOGTnuclearPhysicsBDraft1. pdf — compiled paper (primary) TOGTnuclearPhysicsBDraft1. 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.