unity_has_no_phi_structure
unity_has_no_phi_structure shows that the constant-1 unity configuration of any positive size N admits no phi-ladder structure. Derivations of the T4 recognition-forcing step cite it to conclude that a uniform ground state carries zero distinguishing information and cannot host recognition events. The term proof assumes the structure witnesses, simplifies the unity definition to obtain phi^n = 1, and invokes phi_zpow_ne_one for the contradiction.
claimLet $N$ be a positive natural number. The constant function of value 1 on an $N$-element domain (the unity configuration) does not admit a phi-structure: there do not exist indices $i,j$ and nonzero integer $n$ such that the ratio of any two entries equals a power of the golden ratio.
background
The StillnessGenerative module derives from the T0-T8 chain that the unique zero-defect state x=1 is unstable and must generate non-trivial content. unity_config N hN is the constant-1 function on N elements; has_phi_structure asserts the existence of indices i,j and integer n with the ratio of configuration values equal to phi^n (see sibling phi_ladder and phi_zpow_ne_one). The module imports LawOfExistence (T5 J-uniqueness) and PhiForcing to supply the cost function J and the self-similar fixed point phi. Upstream, phi_zpow_ne_one states that no nonzero integer power of phi equals 1, while the module doc records the T4 step: a uniform ledger is informationally equivalent to nothing and cannot support recognition.
proof idea
Term-mode proof. Introduce the assumed phi-structure triple (i,j,n,hn,hratio). Rewrite hratio by the definition of unity_config to obtain 1/1 = phi^n. Simplify to 1 = phi^n. Apply phi_zpow_ne_one to hn and the symmetric equality to reach a contradiction.
why it matters in Recognition Science
The result supplies the key negative step in the T4 recognition-forcing argument of the foundation chain: the initial uniform configuration (from LawOfExistence) cannot carry phi-structure, forcing departure to non-trivial content. It directly supports the module claim that x=1 is the generative source rather than passive equilibrium, and feeds the sibling is_nontrivial and T4_Recognition declarations. No open scaffolding remains; the theorem is fully proved from the T0-T8 premises.
scope and limits
- Does not prove existence of any concrete non-trivial configuration.
- Does not bound the J-cost or probability of departure from unity.
- Does not extend to N=0 or infinite domains.
- Does not address continuous or non-discrete state spaces.
formal statement (Lean)
80theorem unity_has_no_phi_structure {N : ℕ} (hN : 0 < N) :
81 ¬ has_phi_structure (InitialCondition.unity_config N hN) := by
proof body
Term-mode proof.
82 intro ⟨i, j, n, hn, hratio⟩
83 simp only [InitialCondition.unity_config] at hratio
84 have h_div : (1 : ℝ) / 1 = PhiForcing.φ ^ n := hratio
85 simp at h_div
86 exact phi_zpow_ne_one hn h_div.symm
87
88/-! ## Part IV: DERIVED — Recognition Forces Non-Trivial Content (Gap A)
89
90T4 (Recognition Forcing): Recognition requires distinguishing states.
91A configuration where all entries are identical contains zero information
92and cannot support any recognition event. This is the RS formalization
93of "nothing cannot recognize itself" (MP/T1) applied to configurations:
94a uniform ledger is informationally equivalent to nothing.
95
96The derivation:
97 T4 (recognition requires substrate) + T1 (nothing has infinite cost)
98 → at least one entry must differ from the ground state
99 → the configuration is non-trivial
100
101This replaces the old T6_Requirement structure with a THEOREM. -/
102
103/-- A configuration is **non-trivial** if at least one entry differs from 1.
104 Equivalently: the configuration is not the uniform ground state. -/