Title: Principia Orthogona, Volume I: The Mathematics of Generative Transitions

ZENODO_DESCRIPTION_Vol1_V3.md Copy this text into the Zenodo description field for the V3 upload. Principia Orthogona, Volume I: The Mathematics of Generative Transitions Version 3 — May 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: 10.5281/zenodo.20237688 Series root: https://doi.org/10.5281/zenodo.19117399 AXLE: https://github.com/TOTOGT/AXLE · DM3-lab: https://github.com/TOTOGT/DM3-lab What this volume does This volume develops a unified mathematical framework for generative transitions: localised geometric events in which a trajectory undergoes compression, curvature intensification, loss of injectivity, and stabilisation, governed by the operator sequence G = U ∘ F ∘ K ∘ C. The framework rests on six minimal assumptions and produces: constructive operator definitions with explicit formulas; five structural theorems including existence, non-commutativity, and finite branching; seven analytical invariants; four normal forms; a singularity classification restricted to the Whitney A₁–A₃ hierarchy; a free-discontinuity variational principle; and a symplectic Hamiltonian structure with a distributional generator at the fold. The second edition (V2, May 2026) added: a fifth operator E (Generative Time Circuit) with ż ≥ 0; a term-by-term structural correspondence with Perelman's proof of the Poincaré conjecture via Ricci flow with surgery (Conjecture 15.1); and the dimensional threshold N = 3 as the minimum dimension for non-trivial contact geometry, connecting it to c = 3 in the Collatz map (Conjecture 16.1). What V3 adds V3 completes the reproducibility stack to the standard of the Fibonacci/Tribonacci deposit (10.5281/zenodo.20075822). Specifically: 1. PrincipiaVol1.lean — directly in this deposit Consolidated Lean 4 / Mathlib4 formal verification file. Sources: P1–P6 (Whitney A₁, Gronwall, basin, contact, Lyapunov, stability): from AutophagyDm3_v2.lean — 0 sorry Theorems A–D (operator chain structures): from AXLE_v5_1.lean — 0 sorry Gronwall contraction (exponent sign): from gronwall_proof.lean v6.1 — 0 sorry Club filter / stationary sets: from AXLE_v5_1.lean — 0 sorry Separation theorem: 1 scoped sorry at eigenvalue API boundary (O1, AXLE Issue #12) Total: 30+ facts proved, 1 sorry (clearly scoped), 0 axioms beyond Mathlib4. 2. figures.py — directly in this deposit Python figure generator producing all 7 figures from scratch. Dependencies: numpy, matplotlib (standard). Run: python figures.py 3. Individual figure PDFs (fig1–fig7) fig1_phase_portrait.pdf — dm³ phase portrait with Gronwall basin fig2_threshold_equivalence.pdf — threshold equivalence diagram fig3_bifurcation.pdf — bifurcation diagram near κ* fig4_stability_radius.pdf — stability radius ε₀ = 1/3 fig5_coherence_bridge.pdf — Coherence Bridge (μmax, β across 7 domains) fig6_operator_sequence.pdf — operator sequence G = U∘F∘K∘C∘E fig7_contact_3d.pdf — contact 3-manifold with limit cycle Γ 4. CHANGES_Vol1.md — explicit V1 → V2 → V3 version history 5. OPEN_QUESTIONS.md — open questions table with status column Proved without sorry (30+ facts) P1a–d: Whitney A₁ conditions on V(q) = q³−3q at q=1 P2: Contact non-degeneracy c(ρ) = −2ρ < 0 P3: Gronwall radius ε₀ = 1/3 P4: Basin asymmetry 1/3 < 4/5 P5: Lyapunov exponents −V''(1)/2 = −3; μmax = −2 < 0 P6: Stability functional σ(ρ) = ρ² > 0, σ'(ρ) > 0 A: GenerativeOp well-defined (existence by construction) B: CompressionOp contractive + injective C: FoldOp non-injective + finite branch D: UnfoldOp Φ-decrease + stable branch +: Canonical triple (T*, μmax, τ) = (2π, −2, 2); noise tolerance τ·ε₀ = 2/3 +: Gronwall contraction exponent sign (scope: sign only; ODE integration is O3) +: Club filter / stationary sets for regular uncountable ordinals +: Regeneration hierarchy (unbounded, ordinal, Mahlo-like) +: Crystal aspect ratio arithmetic (66 = 33·τ) Open obligations (5) ID Description Status O1 AXLE #12: Eigenvalue API gap in separation_theorem Open — 1 scoped sorry O2 AXLE #14: Mather step; Poincaré–Bendixson Strengthened/Partial O3 AXLE #15 / T1: Full ODE Gronwall integration Partial — exponent sign proved O4 Sorry 1: Discrete dm³ extension to ℤ Open O5 Conjecture 15.1: Perelman functor 𝒫 Open — stated as conjecture Version history Version Date Key addition V1 March 17, 2026 Original four-operator framework V2 May 16, 2026 Fifth operator E; Perelman correspondence; Collatz threshold V3 May 2026 Full reproducibility stack: Lean file, figures.py, figure PDFs, changelogs Build instructions Lean 4: lake update && lake build PrincipiaVol1 Dependencies: Mathlib4 (current stable) Figures: pip install numpy matplotlib python figures.py Outputs: fig1_phase_portrait.pdf through fig7_contact_3d.pdf Paper: pdflatex principia_vol1_v2_full.tex pdflatex principia_vol1_v2_full.tex (run twice for cross-references) Series context Role DOI Series root / concept DOI 10.5281/zenodo.19117399 Volume I (this deposit) 10.5281/zenodo.20237688 Volume II (contact geometry) 10.5281/zenodo.19379473 GCM paper (dm³ toy model) 10.5281/zenodo.19379385 G6 Crystal (lunar architecture) 10.5281/zenodo.19162013 Multi-Orbit Identity Theory 10.5281/zenodo.20230614 Autophagy / Triple-Alpha (Book 3, Ch.A) 10.5281/zenodo.20168812 Fibonacci / Tribonacci DNLS 10.5281/zenodo.20026942 AXLE formal verification hub github.com/TOTOGT/AXLE MSC codes: 37C25, 37G10, 53D10, 57M27, 58K05, 70H05, 47H10 Keywords: generative transitions · contact geometry · operator sequence · Whitney fold · singularity theory · variational mechanics · symplectic geometry · Ricci flow · Perelman conjecture · Lean 4 formal verification · dimensional threshold · dm³ framework · Gronwall stability · Principia Orthogona · G6 LLC License: CC BY-NC-ND 4.0 (paper) · MIT (code) Copyright: © 2026 Pablo Nogueira Grossi, G6 LLC Contact: pgrossi888@outlook.com · g6llc@proton.me