Cosmology Rung Coarsen Coarsening Exact

Rung coarsening

Coarsening is the act of grouping fine-grained data into coarser blocks, a step any large simulation takes to stay tractable. The theorem coarseningExact concerns a specific kind of coarsening: one that groups recognition events, which are directed postings from a source site to a target site carrying a positive ratio. The declaration proves that if you coarsen a multiset of such events and then refine it back, you recover the original multiset exactly. The round trip is the identity, not an approximation.

The proof rests on a simple partition. Each event either has both endpoints in the same block, in which case it is absorbed into that block's internal summary, or its endpoints land in different blocks, in which case it is promoted to a coarse event between the coarse sites with the same ratio. Refinement simply reassembles the internal and cross events. Because the partition recombines to the original multiset, the round trip returns the identical object. The theorem then states that any functional of the cell is preserved: event count, total cost for any ratio weight, the log-ratio spectrum, and net flow at every site. Cost also partitions exactly: the coarse cross-block cost plus the sum of block-internal cost equals the fine cost.

In Recognition Science, a ledger is a discrete record of events, and this theorem earns the word "literal" for coarsening. It says coarsening loses nothing reality has determined, because the refinement record keeps exactly the original cross events and the per-block internal events. A separate consequence, idle_carries_nothing, states that a cell with no internal events keeps an empty refinement, so refinement memory scales with recognition activity, not with the number of sites. The theorem is stated for an arbitrary ratio weight, so it holds for the RS recognition cost J and any other cost function.

The declaration is a theorem in the machine-checked library of formal theorems, with zero sorry and zero new axioms. It does not claim that coarsening is lossless in any physical sense beyond the formal round trip; it does not claim that the block map itself is derived or optimal; and it does not claim that any particular cost function is the RS cost. The theorem is a statement about the formal model, not about which coarsening scheme a simulation should use.

/-- **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)

/-- **Cost partition.** Coarse cross-block cost plus block-internal cost equals the fine
cost. Cost is split exactly across the rung change, with no leakage. -/
theorem cost_partition (wr : ℝ → ℝ) (block : ℕ → ℕ) (m : Multiset Event) :
    cost wr (coarseLedger block m) + cost wr (internalOf block m) = cost wr m := by
  rw [cost_coarse_eq_cross, ← cost_add, cross_add_internal]

/-- **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]

/-- **The coarsening-exactness model holds for every cell, block map, and weight.** -/
theorem coarseningExact (block : ℕ → ℕ) (m : Multiset Event) (wr : ℝ → ℝ) :
    CoarseningExact block m wr where
  roundtrip_identity := roundtrip_eq block m
  count_exact := count_preserved block m
  cost_exact := cost_preserved wr block m
  spectrum_exact := spectrum_preserved block m
  cost_partitions := cost_partition wr block m
  sigma_exact := sigma_preserved block m

What this page does not claim

Not a claim that coarsening is lossless in any physical sense beyond the formal round trip. Not a claim that the block map itself is derived or optimal. Not a claim that any particular cost function is the RS cost.

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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