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

Contact-Geometric Theory of Generative Transitions: Mathematical Foundations, Contact Realization, Seven Proofs of the Tribonacci Constant, and Applications to Nuclear Matter Version 2 — 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 V1 DOI: 10. 5281/zenodo. 20682934 V2 DOI: https: //doi. org/10. 5281/zenodo. 21124192 AXLE: https: //github. com/TOTOGT/AXLE What this paper does 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. 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), the outer stability radius ε₀ = 1/3, and the curvature threshold κ* = √ (7/9) are closed form; the inner stability boundary r* is now rigorously certified numerically (see "What V2 fixes"). 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. 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) is testable at zero free parameters once τₕydro is fixed. What V2 fixes V2 (July 2026) closes AXLE Issue #21 (r* certification) and corrects a citation error introduced when the AXLE issue tracker was renumbered after V1's deposit. Change Detail §10. 2, Theorem 10. 3 (v) r* rigorously certified via a Lohner-style interval integrator (mpmath high-precision center trajectory + Jacobian-linearized error transport + interval-Hessian Lagrange remainder bound) to r* ∈ 0. 775940575501953125, 0. 77594057550234375 (width 3. 9×10⁻¹³, 13 significant figures), superseding the V1 plain-bisection estimate (8 d. p. , no rigorous error bound). New script certifyᵣstarᵣigorous. py accompanies the deposit. §23, §25. 2 AXLE Issue references corrected from #13 to #21. V1 mis-cited the r*-precision obligation as "AXLE Issue #13" three times; #13 is actually an unrelated AutophagyDm3 Mather-stability/Poincaré–Bendixson obligation. Issue #21 was filed specifically for r* and is now closed at the numerical level (Lean 4 formalisation of the certificate remains open, tracked separately). Figures Four figures added: the operator sequence G = U∘F∘K∘C (§1), the Whitney A₁ fold potential and verified critical-point conditions (§5. 3), the dm³ phase portrait (§10. 2), and the Gronwall contraction exponent / stability functional (§10. 2). All generated from the companion figures. py and placed adjacent to the text they illustrate. Typography Large-print edition: 14pt base (up from implicit 12pt/article default), DejaVu Serif/Sans/Mono/Math throughout (via XeLaTeX + unicode-math, replacing Computer Modern), 1. 18 line spacing, large bold captions. Date Version date corrected to July 2026 (V1 was dated June 2026). Certified without sorry (unchanged from V1 — 48+ facts) Operator sequence: GenerativeOp (Theorem A), UnfoldOp. stablebranch (Theorem D) Whitney A₁ fold: Vcriticalₐtₒne, Vfactored, Vₛecondderivₐtₒne Canonical invariants: contactCoeffₙeg, dm3basincompact, gronwallᵣadius, Vₛecondderivₐtₒne CatGT: iprbetweenᵦeroₐndₒne, helicalₛelectivity, criticalRadiusₚos, criticalRadiusₐntitone, selectivityFactorₑq, reebₒrbitᵢsᵢntegral g-series: nextLevelₗayercountgt Tribonacci proofs 1, 2, 5, 6, 7: algebraically complete, immediately Lean-ready AXLE total: 48+ proved · 17 admits · 0 hidden sorries r* certification status (new in V2) Numerically closed: r* certified to 13 significant figures with a machine-tracked rigorous error radius at every bisection step (not floating-point estimate). Three paths were evaluated before settling on this method: Closed form: ruled out. Checked symbolic solve, Hamiltonian/exactness structure, and substitution reductions — none yield a closed form. r* is confirmed to be a genuine transcendental threshold of a non-integrable coupled planar system. Existence/uniqueness without exact value: partially established (single clean basin-boundary transition confirmed numerically across a wide scan; a full 2D saddle/stable-manifold proof remains a companion open item). Rigorous numerical certificate: this is what closes Issue #21. A naive interval-arithmetic integrator was tried first and shown to fail via the "wrapping effect" (~5×10⁵× artificial error inflation by t=7, benchmarked directly against true trajectory-cloud spread) ; replaced with linear (Jacobian) error transport plus a genuine interval-Hessian Lagrange remainder bound. Open obligations (Lean 4 formalisation — updated from V1) ID Description Status O1 / AXLE #12 Eigenvalue API gap in separationₜheorem Open — 1 scoped sorry O2 / AXLE #14 Mather stability step; Poincaré–Bendixson Open O3 Full ODE Gronwall integration (mainᵥ7. lean) Open — scalar case proved — whitneyFoldconditional: exact coefficients c₁ = c₂ = 1 (Proof 3) Open — algebraically complete, admit in Lean — Contact cohomology dim H¹ (CA1) = 3 (Proof 4, CatGT) Open — algebraically complete, admit in Lean AXLE #21 r* Lean 4 formalisation of the rigorous certificate Open — numerical certificate closed; Lean port is new scope (no Lohner/Taylor-model integrator currently in Mathlib4) Conjecture 15. 1 analog (§25. 1) Perelman-functor-style structural analogies Stated as conjecture, not proved Version history Version Date Key change V1 June 2026 Original deposit: four-part development (foundations, contact realization, seven Tribonacci proofs, nuclear matter applications), 48+ sorry-free Lean 4 theorems V2 July 2026 r* rigorously certified (13 s. f. ), AXLE Issue #13→#21 citation correction, four figures added, large-print typesetting (DejaVu, 14pt), date corrected Deposit contents TOGTnuclearPhysicsBᵥ2. pdf — 42-page paper (this file), large print with 4 figures TOGTnuclearPhysicsBᵥ2. tex — LaTeX source (XeLaTeX required for DejaVu/unicode-math) certifyᵣstarᵣigorous. py — rigorous r* certification script (new in V2) METHODOLOGY. md — full derivation, benchmarks, and negative results for the three r*-proof paths evaluated (new in V2) figures. py — figure generator (fig1, fig2, fig4, fig6 used in this paper) Build instructions LaTeX (requires XeLaTeX for DejaVu fonts): xelatex TOGTnuclearPhysicsBᵥ2. tex (run three times for cross-references) r* certification: pip install mpmath python certifyᵣstarᵣigorous. py Figures: pip install numpy matplotlib python figures. py Series context Role DOI Series root / concept DOI 10. 5281/zenodo. 19117399 This deposit (V2, latest) 10. 5281/zenodo. 21124192 · V1: 10. 5281/zenodo. 20682934 Principia Orthogona Vol. I 10. 5281/zenodo. 20298665 Principia Orthogona Vol. II 10. 5281/zenodo. 20159456 dm³ toy model 10. 5281/zenodo. 19379385 DNLS (Tribonacci/Tetrabonacci chain) 10. 5281/zenodo. 20026942 AXLE formal verification hub github. com/TOTOGT/AXLE MSC codes: 37C25, 37G10, 53D10, 57M27, 58K05, 70H05, 47H10 Keywords: 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 · Lohner method · interval arithmetic License: Creative Commons Attribution 4. 0 International (CC BY 4. 0) Copyright: © 2026 Pablo Nogueira Grossi, G6 LLC Contact: pablogrossi@hotmail. com