Φ(D_i) = O_i (unique / único)
(1) Existence. Axiom 6 asserts the existence of a bijection $\Phi$ with $\Phi(D_i) = O_i$. This is the existence clause — Axiom 6 does not prove uniqueness.
(2) Injectivity. Suppose $\Phi(D_i) = \Phi(D_j) = O_k$ for $i \neq j$. Then $O_k$ would need to advance two source phases simultaneously — $D_i$ and $D_j$ — both into $D_{k+1 \bmod 12}$. But Definition 3.2 requires $O_k(D_k) \subseteq D_{k+1 \bmod 12}$ with $k$ uniquely determined. Contradiction.
(3) Surjectivity. $|\{D_i\}| = |\{O_i\}| = 12$. Injectivity on a finite set of equal cardinality implies surjectivity.
(4) Uniqueness. The cyclic ordering forces $\Phi(D_i) = O_i$ by induction from base case $\Phi(D_1) = O_1$: if $O_1$ advances $D_1$ to $D_2$, then $\Phi(D_2)$ must be the operator that advances $D_2$ — which is $O_2$ by Definition 3.2. Continuing cyclically fixes all twelve assignments uniquely. $\square$
∎Theorem 4.1 says the cycle is rigid: given the contact structure and the 12 phases, the operator assignments are forced. You cannot freely choose which operator acts at which phase — the geometry determines it. This is why the GTCT chain is non-commutative in the precise sense: swapping two operators would violate the phase-advance requirement of Definition 3.2.
In terms of the G-chain $G = U \circ F \circ K \circ C$: the four base operators and three invariants per operator produce exactly 12 operators in exactly this order. There is no other valid assignment.
| Phase | Operator | Base | Invariant | Status |
|---|---|---|---|---|
| D₁ | O₁ | C | I₁ (Orthogonality) | ✓ Lean 4 |
| D₂ | O₂ | C | I₂ (Nilpotency) | ✓ Lean 4 |
| D₃ | O₃ | C | I₃ (Spectral) | ✓ Lean 4 |
| D₄–D₆ | O₄–O₆ | K | I₁, I₂, I₃ | ✓ Lean 4 |
| D₇–D₉ | O₇–O₉ | F | I₁, I₂, I₃ | ✓ Lean 4 |
| D₁₀–D₁₂ | O₁₀–O₁₂ | U | I₁, I₂, I₃ | ✓ Lean 4 |