unity_has_no_phi_structure
plain-language theorem explainer
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.
Claim. Let $N$ 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
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.
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