eight_tick_breaks_uniformity
Any constant assignment of a single real value v across all eight positions of a cycle fails to visit distinct states and is therefore degenerate. Researchers tracing the T7 octave in the Recognition Science forcing chain cite this to establish that the ground state x=1 cannot sustain period-8 dynamics. The proof is a one-line application of the uniform degeneracy lemma.
claimFor every real number $v$, the constant assignment of value $v$ to each of the eight positions in a cycle does not visit two distinct values.
background
The module derives the ground state instability from the T0-T8 chain with no external assumptions. Key definition: a cycle of length 8 is non-degenerate precisely when it visits at least two distinct real values; otherwise it collapses to effective period 1. The 8-tick requirement (T7) demands eight distinct states on the three-dimensional lattice Q3. Uniform cycles violate this by reducing all entries to one value, which the module shows cannot support recognition events under the Law of Existence (T5). Upstream results establish that x=1 is the unique zero-defect state and that any uniform ledger carries zero information content.
proof idea
The proof is a direct term-mode application of the uniform cycle degeneracy theorem. That lemma states that any constant function on Fin 8 yields a degenerate cycle because no pair of positions can differ. The target theorem simply instantiates the lemma for arbitrary v in R.
why it matters in Recognition Science
This result supplies the T7 non-degeneracy step inside the Stillness as Generative Source derivation. It shows that the initial configuration x=1 must depart from uniformity to satisfy the eight-tick octave, thereby forcing non-trivial content on the phi-ladder. The declaration closes the argument that the ground state is maximally creative rather than passive equilibrium, directly supporting the later claims that every non-trivial entry is a power of phi.
scope and limits
- Does not identify which specific values replace the uniform assignment.
- Does not extend to cycles whose length differs from eight.
- Does not compute the J-cost of the required departure from uniformity.
- Does not address continuous-time limits or infinite lattices.
formal statement (Lean)
244theorem eight_tick_breaks_uniformity :
245 ∀ v : ℝ, ¬ cycle_nondegenerate (fun _ : Fin 8 => v) :=
proof body
Term-mode proof.
246 uniform_cycle_degenerate
247
248/-! ## Part VII: Perturbation Cost Bounds -/
249