Cosmology Recognition Work Bound Tick Activations Le Two

A tick's ceiling

In the Recognition Science framework, reality keeps a ledger, a discrete record of events, and each event is a recognition. The framework's engine processes the world in ticks, and each tick can resolve at most one edge between two regions. The theorem tickActivations_le_two states a simple consequence: a single tick contributes at most two region-activations. If a tick resolves nothing, it activates zero regions; if it resolves an edge, it activates exactly the two endpoints of that edge. The bound is two, never more.

This is a pure counting statement about the framework's own model. It does not say how much work those activations cost, nor does it depend on how many regions exist. The type of region indices can be arbitrarily large, even infinite, and the bound still holds. The theorem is proved in the framework's machine-checked library of formal theorems, and the proof is a direct consequence of the definition of a tick's work with unit cost per region.

The importance of this bound appears when it is combined with the cycle-level theorem. Over a cycle of eight ticks, the total number of region-activations is at most sixteen, and this number is independent of the total number of regions in the world. This is the formal core of a cost-localization result: if the world grows by a fixed number of regions per cycle, the recognition work per cycle is capped, while the volume of the world grows without bound. The fraction of the world that is actively being recognized falls toward zero, and the cost of running the engine stays confined to a thin boundary layer.

This result is a theorem about the framework's internal model, not a statement about any physical universe we observe. It does not claim that the actual universe has a boundary, nor that the cost of computation in any real system is bounded. It establishes a property of a formal model, and its value is in showing that the model can describe a growing world without requiring an ever-increasing amount of recognition work.

/-- A tick contributes at most `2` region-activations. -/
theorem tickActivations_le_two (e : Option (ι × ι)) : tickActivations e ≤ 2 := by
  simpa using tickWork_le e 1 (fun _ => 1) (fun _ => le_rfl)

/-- **Region-activations per cycle are bounded by twice the cadence, independent of the population.**
A specialization of `cycle_work_le` with unit cost: at most `2 * T` region-activations occur in a
`T`-tick cycle, regardless of the number of regions. -/
theorem cycle_activations_le (T : ℕ) (res : Fin T → Option (ι × ι)) :
    (∑ t, tickActivations (res t)) ≤ 2 * T := by
  have := cycle_work_le T res 1 (fun _ => 1) (fun _ => le_rfl)
  simpa [tickActivations] using this

What this page does not claim

The theorem does not claim that the actual universe has a finite boundary or a bounded amount of physical computation. The theorem does not claim that the framework's model of recognition is the correct description of physical reality.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cosmology/RecognitionWorkBound.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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