Cosmology Recognition Work Bound Cycle Activations Le

The bounded cycle

In the Recognition Science framework, reality keeps a discrete record of events, a ledger. Each event is a recognition, a forced posting that resolves one connection between two regions of the world. The framework's central law allows at most one such resolution per tick, a fixed cadence of eight ticks per cycle. The theorem cycle_activations_le, proved in the machine-checked library of formal theorems, states a plain bound: over a cycle of T ticks, the total number of region-activations is at most 2 times T. Each resolved connection activates its two endpoints, a double-entry posting, so one tick contributes at most two activations. Summed over the cycle, the bound is simply twice the tick count.

The force of the bound is what it does not mention. The region-index type, the population of regions in the world, does not appear in the inequality. The type can be arbitrarily large, and the bound still holds. Recognition work per cycle does not grow with the world. When the world expands by a fixed number of regions per cycle, the recognition-cost numerator stays capped while the volume denominator grows. The recognition-active fraction falls toward zero, and the engine's cost localizes to a sub-extensive interface, a thin boundary rather than the whole interior.

This result is the formal core of a larger claim about driven expansion. The theorem stands beside two others: the engine stays literal under any schedule, and charge is conserved through every birth. Together they say the driven engine is literal, conserves its charge through growth, and pays a per-cycle recognition cost bounded by the cadence regardless of how large reality grows. The bound is a theorem, not a model choice; it follows from the forced cadence law and the definition of a tick's work.

The theorem does not claim that the world actually grows, that the cadence is eight ticks in any physical sense, or that the cost bound is tight. It bounds activations, not the total work when per-region costs vary; the work bound is a separate theorem with a per-region ceiling P. The theorem also does not say the recognition-active fraction reaches zero, only that it falls toward zero as the world grows linearly. The bound is a ceiling, not an achievement.

/-- **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

/-- 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

/-- **Phase-11 cost-localization headline.** In a `T`-tick cadence cycle with at most one forced
resolution per tick, the engine's recognition work is at most `2 * P * T` and the region-activations
are at most `2 * T`, both independent of the population `ι`. So when the world grows by a fixed number
of regions per cycle, the recognition-cost numerator is capped while the volume denominator grows: the
recognition-active fraction falls toward zero and the cost localizes to a sub-extensive interface. -/
theorem recognition_work_localizes (T : ℕ) (res : Fin T → Option (ι × ι)) (P : ℕ)
    (cost : ι → ℕ) (hcost : ∀ i, cost i ≤ P) :
    (∑ t, tickWork (res t) cost) ≤ 2 * P * T
    ∧ (∑ t, tickActivations (res t)) ≤ 2 * T :=
  ⟨cycle_work_le T res P cost hcost, cycle_activations_le T res⟩

What this page does not claim

The theorem does not prove the world grows, only bounds cost if it does. The bound is not tight; it is a ceiling, not an exact count. The theorem does not cover per-region costs above the ceiling P.

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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