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

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/dm3_simulation.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 r_att−ε₀=2/3 as previously stated)- `docs/GCTC_REVIEW.md` — Lean review: `gronwall_bound` replaced by one-sided `gronwall_outer`; 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 r_att−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) `gronwall_outer` — exponential bound, ★★☆☆☆, proof sketch provided- (b) `inner_basin_is_asymmetric` — ★★★☆☆, axiom pending ODE formalization- (c) `spiral_return_exists` — ★★★☆☆, non-triviality hypothesis added- (d) `poincare_collatz` — ★★★★★, 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