The Gray Cycle Realizes the 8-Tick Cycle

1. Setting

The Gray Cycle Realizes the 8-Tick Cycle is anchored in Patterns.GrayCycle. The page is not a loose explainer: it is a public map from the Recognition Science forcing chain into one Lean-checked declaration bundle. The primary anchor determines what is proved, and the surrounding declarations show how the result is used.

2. Equations

(E1)

$$ J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y) $$

Recognition Composition Law.

3. Prediction or structural target

• Minimum cycle length: predicted 8 (ticks); empirical 8. Source: Forced by D=3 Gray code

This page is currently a structural derivation. Where the claim has direct empirical content, the prediction table gives the measurable target; otherwise the claim is a formal bridge inside the Lean canon.

4. Formal anchor

The primary anchor is Patterns.GrayCycle..grayCover_eight_tick_min.

theorem grayCover_eight_tick_min {T : Nat} [NeZero T] (w : GrayCover 3 T) : 8 ≤ T := by
  simpa using (Patterns.eight_tick_min (T := T) w.path w.complete)

/-!
### A fully explicit, fully decidable 3-bit witness (the actual “octave”)

This gives a *rigorous* Gray cycle for `d=3` (period 8) without any axioms.
We deliberately start here because it is the “octave” case used throughout RS.
-/

5. What is inside the Lean module

Key theorems:

• OneBitDiff_symm
• grayCover_min_ticks
• grayCover_eight_tick_min
• gray8At_injective
• toNat3_pattern3
• pattern3_injective
• grayCycle3_injective
• grayCycle3_bijective
• grayCycle3_surjective
• grayCycle3_oneBit_step
• grayCycle3_period

Key definitions:

• OneBitDiff
• GrayCycle
• GrayCover
• pattern3
• gray8At
• grayCycle3Path
• toNat3
• grayCycle3

6. Derivation chain

• grayCover_eight_tick_min - Eight-tick minimum
• grayCycle3_period - Gray cycle 3 has period 8
• grayCycle3_bijective - Bijective Gray cycle
• grayCycle3_oneBit_step - One-bit step

7. Falsifier

A Lean-checkable counterexample to the named theorem or to the upstream functional equation refutes this derivation.

8. Where this derivation stops

Below this page the chain reduces to the RS forcing sequence: J-cost uniqueness, phi forcing, the eight-tick cycle, and the D=3 recognition substrate. If any upstream theorem changes, this page must be versioned rather than patched silently. The published URL is stable, but the version field is the contract.

10. Audit path

To audit gray-cycle-eight-tick, start with the primary Lean anchor Patterns.GrayCycle.grayCover_eight_tick_min. Then inspect the theorem names listed in the module-content section. The page is intentionally built so the public explanation is not a substitute for the proof object; it is a map into it. The mathematical dependency is the same in every case: reciprocal cost fixes J, J fixes the phi-ladder, the eight-tick cycle fixes the recognition clock, and the domain theorem listed above supplies the last step. If that last step is empirical, the falsifier section names what observation would break it. If that last step is formal, a Lean-checkable counterexample is the relevant failure mode.