t6_derived
Any initial configuration of size at least 2 that satisfies T4 recognition non-triviality, contains a ground-state entry of 1, and lies on the geometric phi-ladder must possess phi-structure. Researchers closing the T0-T8 forcing chain cite this result to obtain phi-structure as a theorem rather than an added axiom. The proof is a one-line wrapper that invokes the nontrivial closed configuration lemma.
claimLet $N$ be a natural number with $N$ at least 2, and let $c$ be a configuration of $N$ entries. If $c$ is non-trivial (in the sense that it supports recognition events), contains at least one entry equal to 1, and every entry equals $phi^n$ for some integer $n$, then $c$ has phi-structure.
background
T4_Recognition is the structure asserting that any configuration supporting recognition must be non-trivial: all entries equal to 1 yields zero distinguishable comparisons and therefore no recognition events. The phi-ladder condition states that every entry is a power of the golden ratio phi, as required by the T6 closure theorem on closed geometric scale sequences. The ground-state hypothesis records the existence of an entry equal to 1, the unique zero-defect state from the Law of Existence (T5).
proof idea
The proof is a one-line wrapper that applies nontrivial_closed_has_phi_structure, passing the non-triviality projection from h_T4, the ground-state witness, and the ladder membership hypothesis.
why it matters in Recognition Science
This theorem derives phi-structure directly from the T0-T8 chain by combining T4 non-triviality with T6 closure, removing the need to postulate phi-structure. It supplies the final step for the ground-state instability argument in the same module, which shows that the uniform state x=1 cannot support the eight-tick cycle and must generate non-trivial content. The result closes one link in the derivation that every physical ledger carries phi-structure.
scope and limits
- Does not prove existence of a ground-state entry equal to 1.
- Does not apply to configurations that fail the phi-ladder condition.
- Does not treat infinite or continuous state spaces.
- Does not derive explicit rung values or the mass spectrum.
formal statement (Lean)
161theorem t6_derived {N : ℕ} (hN : 2 ≤ N)
162 (c : InitialCondition.Configuration N)
163 (h_T4 : T4_Recognition c)
164 (h_ground : ∃ j : Fin N, c.entries j = 1)
165 (h_on_ladder : ∀ i : Fin N, ∃ n : ℤ, c.entries i = PhiForcing.φ ^ n) :
166 has_phi_structure c :=
proof body
Term-mode proof.
167 nontrivial_closed_has_phi_structure hN c h_T4.nontrivial h_ground h_on_ladder
168
169/-! ## Part V: Ground State Instability (now fully derived)
170
171The incompatibility between x = 1 (all entries uniform) and T4 (recognition
172forces non-triviality) is now a THEOREM, not an assumption. -/
173
174/-- **Ground State–T4 Incompatibility**: The uniform ground state
175 cannot support recognition. Recognition requires distinguishable
176 entries, but the unity config has all entries = 1. -/