uniform_cycle_degenerate
plain-language theorem explainer
A constant real value assigned across all eight positions of the cycle collapses the effective period to one. Researchers deriving the instability of the zero-defect ground state from the T7 octave would cite this to exclude uniformity. The proof is a direct term-mode contradiction obtained by introducing the non-degeneracy witnesses and applying reflexivity on the constant function.
Claim. For any real number $v$, the constant function on the eight-element cycle with value $v$ everywhere is degenerate.
background
The StillnessGenerative module derives from the T0-T8 chain that the unique zero-defect state x=1 is not passive equilibrium but the source of all structure. Central to this is the eight-tick octave of T7, which requires the cycle to visit eight distinct states; a uniform assignment reduces the period to one and violates the exact period-8 result from Patterns.period_exactly_8. The module's derivation chain lists Law of Existence (T5), Recognition Forcing (T4), and the eight-tick requirement as premises that together force non-trivial content on the phi-ladder.
proof idea
The proof is a term-mode argument. It introduces the assumed witnesses for non-degeneracy of the constant cycle and immediately applies reflexivity to obtain a contradiction, since every position holds the identical value v.
why it matters
This result is invoked directly by eight_tick_breaks_uniformity and stillness_is_creative in the same module. It supplies the T7 non-degeneracy step in the derivation chain: the eight-tick octave cannot be realized by a uniform ledger, thereby forcing departure from x=1 and populating the phi-ladder. The parent theorem stillness_is_creative assembles this with the zero-defect uniqueness and recognition requirements to conclude that stillness is creative.
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