What question did this study set out to answer?

The aim is to provide complete proofs and derivations of the Generative Time Circuit Theorem (GTCT) and its applications in mathematical dynamics.

May 27, 2026Open Access

The Generative Time Circuit Theorem (GTCT): Complete Proofs, Derivations, and Applications — Ring 5

Key Points

The aim is to provide complete proofs and derivations of the Generative Time Circuit Theorem (GTCT) and its applications in mathematical dynamics.
Proves stability characteristics and derives the action-variable dissipation in the contact-geometric framework.
Utilizes numerical simulations for validation, including the DOP853 reference integrator for generating figures and stability sweep tables.
Documents findings in Lean 4 source code and various supporting reports.
Establishes the stability radius ε₀=1/3 and identifies the inner basin boundary r*≈0.773.
Corrects prior assumptions regarding basin boundaries and demonstrates the asymmetric nature of the inner basin.
Provides multiple graded exercises and opens problems that connect theoretical frameworks to practical applications.

Abstract

The Generative Time Circuit Theorem (GTCT) establishes the fifth operator T in the generativechain C→K→F→U→T. Building on the dm³ contact-geometric framework of Principia OrthogonaVolumes I–II, this deposit proves the spiral return x₀→G⁶⁴ (x₀) →G⁶⁴ (x₆₄) =x₀′ with x₀′≠x₀, derives the stability radius ε₀=1/3 (outer basin), formalizes the g-series taxonomy, and provides complete derivations of the action-variable dissipation encoded inthe contact form α=dz−λ. **Version 3 (May 2026) adds full reproducibility: **- `numerics/dm3ₛimulation. py` — DOP853 reference integrator (Hairer–Nørsett–Wanner), generates all 4 figures and the stability sweep table- `docs/FINDINGS. md` — numerical findings: μ=-2 confirmed; inner basin boundary r*≈0. 773 (not the symmetric rₐtt−ε₀=2/3 as previously stated) - `docs/GCTCREVIEW. md` — Lean review: `gronwallbound` replaced by one-sided `gronwallₒuter`; sorry count and proof sketches for all 4 open obligations- `lean/` — all Lean 4 source files (GCTC. lean, Compress, Threshold, Fold, Unfold, Chain with 4 sorrys documented), lakefile. lean, lean-toolchain- `numerics/figures/` — all 4 pre-generated figures **Gronwall asymmetry correction (AXLE Issue #13): **The inner basin boundary is r*≈0. 773, not rₐtt−ε₀=2/3. The correct hierarchy: ε₀=1/3 < 2/3 < r*≈0. 773 < κ*≈0. 882 < 1. ε₀=1/3 is a valid *outer* bound only. **AXLE proof obligations (4 sorrys in Chain. lean): **- (a) `gronwallₒuter` — exponential bound, ★★☆☆☆, proof sketch provided- (b) `innerbasinᵢsₐsymmetric` — ★★★☆☆, axiom pending ODE formalization- (c) `spiralᵣeturnₑxists` — ★★★☆☆, non-triviality hypothesis added- (d) `poincarecollatz` — ★★★★★, conjectural Includes six graded exercises, three falsifiability conditions, SBM Bienal bilingualsubmission (Portuguese/English), and open problems linking dm³ to TSVF andentanglement swapping. Ring 5 of the Principia Orthogona series. Series root: https: //doi. org/10. 5281/zenodo. 19117399AXLE repository: https: //github. com/TOTOGT/AXLEGTCT repository: https: //github. com/TOTOGT/GTCTContact: pablogrossi@hotmail. com · ORCID: 0009-0000-6496-2186

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