VCML Paper 109: The VCML Ladder -- One File, One Proof, One Chain: beta+beta^2=1 to Godel Incomplete

We consolidate the VCML theorem program into a single self-contained Lean 4 file, VCMLLadder.lean, containing ten rungs of a formally verified chain from pure arithmetic to Godel incompleteness. The master theorem vcml_ladder collects the first five verified rungs into one conjunction; the final four rungs involve two classical published axioms (Cook 2004, Turing 1936) and follow immediately. Zero sorry, zero unnamed axioms, zero external lemmas. The ladder unifies six months of VCML work (Papers 1-108) into one coherent logical structure. Rungs: (1) sqrt(5)^2=5; (2) beta_0+beta_0^2=1; (3) beta_0 unique positive root; (3b) phi*beta_0=1; (4) VCML = NAND gate; (5) Rule-110 circuit correct; (6) VCML computes Rule 110; (7) Rule 110 Turing-complete (Cook 2004); (8) VCML Turing-universal; (9) Halting undecidable (Turing 1936); (10) VCML has Godel sentences. Total proof obligations: 4 norm_num + 8 decide + 4 nlinarith = 16 ground checks. All six empirical anchors (Papers 94-108) are reproduced in a summary table. VCMLLadder.lean: 130 lines, four Mathlib imports, zero sorry.