Cosmology Rung Coarsen Idle Carries Nothing

The idle cell

In the Recognition Science framework, a ledger is a discrete record of events, and a recognition event is a directed posting from one site to another carrying a positive ratio, with the reciprocal posting implied. The framework's machine-checked library of formal theorems proves a statement about coarsening, the operation of grouping fine sites into larger blocks. The statement, idle_carries_nothing, says that if a cell in a coarse-grained ledger contains no internal events, meaning no postings that both begin and end inside that cell, then refining the coarse ledger back to the fine scale adds nothing for that cell. The refined record for that cell is exactly the set of its cross events, the postings that crossed its boundary.

The proof is immediate from the definitions. A refinement record keeps the original cross events and the per-block internal events. If the internal events are empty, the refinement is just the cross events. The theorem formalizes this as: if the internal multiset is zero, then the round-trip, refine after coarsen, equals the cross events. This is not a claim about physics directly; it is a structural fact about how the framework's ledger is defined. It establishes a memory property: the cost of storing a refinement scales with recognition activity, not with the number of sites in the model. An idle cell, one with no internal activity, requires no storage in the refined record.

This property is one part of a larger theorem, T-1, which proves that coarsening is exact. The round-trip, coarsen then refine, returns the original multiset of events, so every conserved functional, event count, total cost, log-ratio spectrum, and net flow, is unchanged. The idle cell statement is a special case of this exactness, focused on the empty internal case. It does not claim that idle cells are physically irrelevant or that they have no effect on the coarse ledger; they may still participate in cross events. It claims only that the refinement memory for an idle cell is empty.

The framework's library proves this for an arbitrary ratio weight, so it holds for the recognition cost J and any other cost function. The theorem has zero axioms beyond the standard three and zero sorry. The statement is a definitional consequence of how the ledger and coarsening are set up, not an empirical finding.

/-- **Idle carries nothing.** A cell with no internal events keeps an empty refinement,
so the round-trip is just the (already coarse) cross part, and refinement memory scales
with recognition activity rather than with the number of sites. -/
theorem idle_carries_nothing (block : ℕ → ℕ) (m : Multiset Event)
    (hidle : internalOf block m = 0) :
    roundtrip block m = crossOf block m := by
  unfold roundtrip refineCell
  rw [hidle, add_zero]

/-- **T-1 round-trip.** Coarsening then refining returns the cell unchanged. The coarse
representation is lossless: it carries everything reality has determined with zero loss. -/
theorem roundtrip_eq (block : ℕ → ℕ) (m : Multiset Event) :
    roundtrip block m = m := by
  unfold roundtrip refineCell
  exact cross_add_internal block m

/-- Because the round-trip returns the identical multiset, ANY functional of the cell
is preserved. Sigma, totals, cost, and spectrum are all instances of this. -/
theorem conserved {X : Type*} (F : Multiset Event → X) (block : ℕ → ℕ) (m : Multiset Event) :
    F (roundtrip block m) = F m :=
  congrArg F (roundtrip_eq block m)

theorem cost_preserved (wr : ℝ → ℝ) (block : ℕ → ℕ) (m : Multiset Event) :
    cost wr (roundtrip block m) = cost wr m := conserved (cost wr) block m

What this page does not claim

Idle cells have no effect on the coarse ledger or on cross events. The theorem establishes a physical law about empty space. The statement applies only to the recognition cost J and not to other cost functions.

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

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