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 generative
chain C→K→F→U→T. Building on the dm³ contact-geometric framework of Principia Orthogona
Volumes 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 in
the 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 bilingual
submission (Portuguese/English), and open problems linking dm³ to TSVF and
entanglement swapping.

Ring 5 of the Principia Orthogona series.
Series root: https://doi.org/10.5281/zenodo.19117399
AXLE repository: https://github.com/TOTOGT/AXLE
GTCT repository: https://github.com/TOTOGT/GTCT
Contact: pablogrossi@hotmail.com · ORCID: 0009-0000-6496-2186