tickSucc_index
The successor on discrete time ticks increments the index by one. Researchers establishing the natural-number object property of the tick sequence in Recognition Science cite this when verifying the successor axiom for the forced iteration object. The proof is a direct reflexivity after unfolding the successor definition.
claimFor every tick $t$, the index of the successor tick equals the index of $t$ plus one.
background
Tick is the atomic unit of temporal progression: a structure carrying a natural-number index, with time advancing discretely by index steps rather than on a background manifold. The successor operation advances any tick by incrementing its index by one, making the sequence of ticks a Lawvere natural-number object under this successor map. The module establishes that this Tick coincides with the canonical arithmetic object forced by recognition, via the universal property of natural-number objects proved in the upstream NaturalNumberObject construction.
proof idea
One-line wrapper that applies reflexivity after the successor definition unfolds directly to the index increment.
why it matters in Recognition Science
This declaration supplies the successor axiom needed to identify Tick as the natural-number object of recognition, closing the combinatorial identification of time with the orbit construction. It supports the module claim that recognition steps generate the iteration object without adding time as an external structure. The result sits inside the forcing chain that derives the temporal sequence from the Recognition Composition Law and the initial Peano object.
scope and limits
- Does not derive metric time or continuous evolution.
- Does not address the eight-tick octave or spatial dimension forcing.
- Does not treat the arrow of time or Berry-phase monotonicity.
- Does not supply the universal property proof itself.
formal statement (Lean)
49@[simp] theorem tickSucc_index (t : Tick) : (tickSucc t).index = t.index + 1 := rfl
proof body
Term-mode proof.
50
51/-- The canonical zero tick. -/