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structure definition def or abbrev high

Tick

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Ticks formalize discrete time steps in the ledger as natural-number indices. Researchers deriving the arrow of time or eight-tick quantization cite this when building temporal structure from recognition updates. The definition equips the index with decidable linear order relations lifted from the naturals.

claimA tick is an element of the structure consisting of an index $n$ in the natural numbers, equipped with the total order $t_1 ≤ t_2$ if and only if the index of $t_1$ is at most the index of $t_2$, together with decidable equality and the induced strict order.

background

In the TimeEmergence module time arises solely from the ledger update cycle with no background manifold. The fundamental time quantum is the tick, set to 1 in RS-native units, and the minimal complete period is the 8-tick octave corresponding to D=3 spatial dimensions. Upstream the constant tick is defined as the RS time quantum τ₀ = 1 and one octave equals 8 ticks.

proof idea

The declaration introduces the structure with a single field for the natural-number index and derives decidable equality. It then defines the less-than-or-equal and less-than relations by direct comparison of indices and supplies decidable relations using the corresponding Nat predicates.

why it matters in Recognition Science

This definition supplies the temporal primitive for downstream results such as the equivalence of nucleosynthesis mass-luminosity ratios to powers of phi and the eight-tick quantization in stellar assembly. It realizes the F-004 registry item on the nature of time and supports the T7 eight-tick octave in the forcing chain. No open scaffolding remains here; it closes the discrete-time foundation for the arrow-of-time theorems.

scope and limits

formal statement (Lean)

  36structure Tick where
  37  index : ℕ
  38  deriving DecidableEq
  39
  40/-- Ticks are ordered by their index. This IS time — no background manifold. -/
  41instance : LE Tick := ⟨fun a b => a.index ≤ b.index⟩

proof body

Definition body.

  42instance : LT Tick := ⟨fun a b => a.index < b.index⟩
  43instance : DecidableRel (· ≤ · : Tick → Tick → Prop) := fun a b => Nat.decLe a.index b.index
  44instance : DecidableRel (· < · : Tick → Tick → Prop) := fun a b => Nat.decLt a.index b.index
  45
  46/-- A ledger epoch: one complete 8-tick cycle. -/

used by (40)

From the project-wide theorem graph. These declarations reference this one in their body.

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depends on (14)

Lean names referenced from this declaration's body.