Cosmology Recognition Work Bound

The bounded work of recognition

Cosmology recognition work bound is a theorem in Recognition Science about the cost of maintaining a discrete record of events as a modeled universe grows. The framework models reality as a ledger, a discrete record of recognition events, where each event is a forced posting. The theorem states that the recognition work per cycle is bounded by the cadence, independent of the population of regions. In plain terms, the cost of updating the ledger in a fixed time step does not grow with the size of the world being tracked.

The result is established by modeling a cycle as a fixed number of ticks, with at most one resolved edge per tick. Each resolved edge activates its two endpoints, a double-entry posting, so a tick contributes at most two region-activations. When each endpoint costs at most P forced postings to expand, a tick contributes at most 2P units of recognition work. Over a cycle of T ticks, the total recognition work is at most 2PT, and the number of region-activations is at most 2T. These bounds mention only the tick count and the per-region cost ceiling, never the number of regions.

The significance is that the recognition cost per cycle does not grow with the world. In an expanding dynamics where the world grows by a fixed number of regions per cycle, the recognition-active fraction falls toward zero. The numerator, recognition work, is capped while the denominator, volume, grows. This is the formal core of the claim that the engine never expands the locked interior; the cost localizes to a sub-extensive interface.

In Recognition Science, this theorem is part of a larger forcing chain. The cadence of eight ticks is itself derived from the golden ratio, and the cost function J(x) = (x + 1/x)/2 - 1 is proved unique from five plain conditions. This work bound is a consequence of that structure, not an assumption. The result sits beside two other results: the engine stays literal under any schedule, and charge is conserved through every birth. Together they say the driven engine is literal, conserves sigma through growth, and pays a per-cycle recognition cost bounded by the cadence regardless of how large reality grows.

/-- **Recognition work per cycle is bounded by the cadence, independent of the population.** Over a
cycle of `T` ticks with at most one resolved edge per tick and per-region expansion cost at most `P`,
the engine's total recognition work in the cycle is at most `2 * P * T`. The bound mentions only the
tick count `T` and the per-region ceiling `P`; the region-index type `ι` (the population) does not
appear, so the per-cycle recognition cost does not grow with the world. -/
theorem cycle_work_le (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 := by
  have h : (∑ t : Fin T, tickWork (res t) cost) ≤ ∑ _t : Fin T, 2 * P :=
    Finset.sum_le_sum (fun t _ => tickWork_le (res t) P cost hcost)
  have hconst : (∑ _t : Fin T, 2 * P) = 2 * P * T := by
    rw [Finset.sum_const, Finset.card_univ, Fintype.card_fin, smul_eq_mul, Nat.mul_comm]
  exact h.trans (le_of_eq hconst)

/-- **Recognition work per cycle is bounded by the cadence, independent of the population.** Over a
cycle of `T` ticks with at most one resolved edge per tick and per-region expansion cost at most `P`,
the engine's total recognition work in the cycle is at most `2 * P * T`. The bound mentions only the
tick count `T` and the per-region ceiling `P`; the region-index type `ι` (the population) does not
appear, so the per-cycle recognition cost does not grow with the world. -/
theorem cycle_work_le (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 := by
  have h : (∑ t : Fin T, tickWork (res t) cost) ≤ ∑ _t : Fin T, 2 * P :=
    Finset.sum_le_sum (fun t _ => tickWork_le (res t) P cost hcost)
  have hconst : (∑ _t : Fin T, 2 * P) = 2 * P * T := by
    rw [Finset.sum_const, Finset.card_univ, Fintype.card_fin, smul_eq_mul, Nat.mul_comm]
  exact h.trans (le_of_eq hconst)

/-- **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

This theorem does not claim the universe itself has a bounded information content. This theorem does not derive the value of the cadence; it takes the eight-tick cadence as given. This theorem does not establish that the physical universe is a ledger; that is a modeling choice within the framework.

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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