Cosmology Recognition Work Bound Recognition Work Localizes

The bounded work of recognition

In the Recognition Science framework, reality is modeled as maintaining a discrete record of events, a ledger. The framework's central law forces a specific cost for each act of recognition, the price of updating that record. A new theorem in the framework's machine-checked library of formal theorems, called recognition_work_localizes, establishes a striking bound on that cost: over a fixed cycle of eight ticks, the total recognition work is capped by a constant, no matter how large the population of regions being tracked becomes.

The theorem works from a simple model. A cycle consists of a fixed number of ticks, and at most one edge between two regions is resolved per tick. Resolving an edge activates its two endpoints, and each endpoint costs at most a fixed amount of work to expand. The formal result proves that the total work in a cycle is at most 2 * P * T, where T is the number of ticks and P is the per-region cost ceiling. The key fact is that this bound mentions only T and P, never the number of regions. The region-index type can be arbitrarily large and does not enter the bound.

The consequence is a form of cost localization. If the world grows by a fixed number of regions per cycle, the recognition-work numerator stays capped while the volume denominator grows. The recognition-active fraction falls toward zero, and the cost concentrates on a sub-extensive interface rather than spreading through the whole. This is the formal core behind the framework's claim that the engine never expands its locked interior as the universe grows.

The theorem is a pure bound on a combinatorial model. It does not claim that any physical process realizes this model, nor does it derive the rate at which the world grows. It proves an inequality about a formal sum, not a statement about actual cosmology. The physical bridge from this formal ledger to the expanding universe remains an open target within the framework.

/-- **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)

/-- **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⟩

/-- **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 claim that any physical process realizes this combinatorial model. The theorem does not derive the growth rate of the world, only bounds the work per cycle. The physical bridge from the formal ledger to the expanding universe is not established by this result.

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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