Principia Orthogona Volume Two: Contact Realization of Generative Transitions
Year: 2026
Type: preprint
Abstract: Principia Orthogona Volume Two: Contact Realization of Generative Transitions https://totogt.github.io/geometry/book2 MSC 2020 — 37C10, 53D10, 37C75, 58K05, 37G10, 70H05 Volume Two of the Principia Orthogona series constructs the contact-geometric realization of the singularity-theoretic foundations laid in Volume One. Three main results: (A) a correspondence between the geometric fold operator F and the dm³ contact-Hamiltonian dissipation; (B) equivalence of the curvature threshold κ* and the embodiment threshold τ, with τ = 2 and ε₀ = 1/3 in the dm³ toy model; (C) a bifurcation analysis relating the four dm³ bifurcations to the Whitney A₁–A₃ singularity types. Version 5 corrects three things and adds one. Appendix A is now generated from the Lean file rather than maintained beside it. Six of the twelve entries in the V4 table named declarations that have never existed in any version of VolumeTwo.lean — two of them listed as proved. The file was in no build target until 26 August 2026, so nothing had ever elaborated it; its first build reported eight errors, three of them in theorems the table listed as proved. The file now builds and all nineteen declarations are kernel-checked, with no sorryAx and no axiom outside {propext, Classical.choice, Quot.sound}. Two of them depend on no axioms at all. Result (C) is restated as a surjection. Four bifurcations map onto three Whitney types, two-to-one at A₁; that is not a bijection, and §5.2's own table said so while the abstract and Proposition 5.1 said otherwise. The A_k indexing convention is now declared explicitly, since Arnold's convention places the fold at A₂. The Lean file path is corrected. The path cited in V2a–V4 does not resolve. A new §5.3 records the D₆-equivariant DNLS results of Working Paper 77 — where the degenerate eigenspaces of the coupling carry the two-dimensional irreducible representations of D₆, the structure a torus bifurcation requires — and uses them to sharpen rather than discharge the Neimark–Sacker obligation, which is restated as a checkable step (OP-NS). Five declarations were added while preparing this version, all kernel-checked. Two close the rows V4 listed as proved against names that existed nowhere. Two give Theorem A real content beside its True placeholder, which is left in the file with its OPEN comment rather than quietly replaced. One proves that τ = |μ_max| holds exactly when |μ_max| = 2 — a property of the canonical value, not an identity. Passing an axiom check is necessary and not sufficient, and the paper says so. A theorem whose conclusion is True, or whose hypotheses already contain its conclusion, is a true theorem and reports clean axioms. Three such are named explicitly in Appendix A. Theorem B's real direction needs Has'minskii-style stochastic stability; the Gronwall asymmetry needs the DOP853 numerics carried into the kernel; integrability Levels 2d and 2d+t need Lie brackets of vector fields, which a pointwise model cannot express. Those remain open, with reasons. The reproducibility bundle dropped between V3 and V4 is restored. To reproduce the verification: git clone https://github.com/TOTOGT/AXLE && git checkout e44e8d1 && bash tools/verify-vol2/run.sh. The commit is pinned deliberately — a link to a branch resolves to whatever is there today. Deposit contents: principia_v5.pdf, principia_v5.tex (XeLaTeX, run twice), VolumeTwo.lean (the current file, not the V2a/V3 skeleton), verify-vol2.zip, figures.py, certify_rstar.py, dashboard.html, README.md.
Author: Nogueira Grossi, Pablo
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
OpenAlex ID: W7148338159