Cosmology Rung Coarsen Sigma Preserved

Flow survives coarsening

In numerical simulation, coarsening means grouping fine grid points into larger blocks to save memory and compute. The question is always what gets lost. A theorem in the framework's machine-checked library of formal theorems shows that, for a recognition ledger, one quantity survives perfectly: the net flow at every site. The declaration sigma_preserved states that after coarsening a cell and then refining it back, the net flow at each site is identical to what it was before. Net flow here means the count of events leaving a site minus the count arriving at it, a plain integer balance.

The proof is short because the round trip is the identity. Coarsening splits events into two classes: internal events with both endpoints in one block, and cross events whose endpoints land in different blocks. Refinement simply reassembles the two classes. Since the reassembled multiset equals the original, any function of that multiset is unchanged. The theorem conserved captures this generally: for any function F of the cell, F after the round trip equals F before it. Net flow is one such function, so sigma_preserved follows as a one-line instance.

This matters for scale-adaptive cosmology simulations, where the engine changes resolution dynamically. The result guarantees that flow balances at every site are not an approximation that drifts with resolution changes. It also composes with the other preservation theorems: event count, total cost, and the log-ratio spectrum all survive the same round trip. The cost partition theorem goes further, showing the coarse cross-block cost plus the internal block costs exactly equals the fine cost.

What sigma_preserved does not claim is equally important. It does not say that the coarse ledger itself has the same net flow as the fine one at the original site labels; the coarse ledger relabels sites into blocks, so site-level comparison only makes sense after refinement. It does not say that coarsening preserves net flow for an arbitrary grouping of events into cells; the theorem is about the specific round trip defined in the module. It also does not assert that net flow is zero, only that it is unchanged. The structural statement that double-entry ledgers have zero net flow at every site is a separate theorem proved elsewhere in the framework.

/-- **Sigma is preserved by the round-trip.** The coarsen-then-refine cell has the same
net flow at every site as the original; coarsening returns identical sigma. -/
theorem sigma_preserved (block : ℕ → ℕ) (m : Multiset Event) (a : ℕ) :
    netFlow (roundtrip block m) a = netFlow m a :=
  conserved (fun s => netFlow s a) 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]

What this page does not claim

The coarse ledger at block level has the same net flow as the fine ledger at site level. Coarsening preserves net flow for arbitrary groupings of events into cells. Net flow is zero at any site, only that it is unchanged by the round trip.

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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