Cosmology Rung Coarsen Cost Coarse Eq Cross

The coarse ledger's cost

In the Recognition Science framework, a ledger is a discrete record of events, each event a directed posting from a source site to a target site carrying a positive ratio. The framework's library of machine-checked formal theorems proves a family of results about what happens when such a ledger is coarsened: grouped into larger blocks, with events internal to a block absorbed into a summary and events crossing block boundaries promoted to coarse events between the blocks.

The specific theorem cost_coarse_eq_cross establishes that the cost of the coarse ledger, computed by summing a weight function over each coarse event's ratio, is exactly equal to the cost of the original cross-block events before relabeling. In plainer terms: when you zoom out to a coarser grid, the total cost you compute at the coarse level is precisely the cost of the events that actually crossed between blocks. The relabeling step, which changes site names but preserves ratios, does not alter the cost. This holds for any weight function, not just the framework's own recognition cost J, because the proof is purely algebraic: it relies only on the fact that mapping a function over a list and then summing equals summing the function's values on the original list.

This theorem is one component of a larger exactness result. The round-trip identity, refine after coarsen, returns the original ledger exactly, which implies that every conserved functional, including event count, total cost, log-ratio spectrum, and net flow, is preserved. The cost partition theorem then states that the coarse cross-block cost plus the sum of block-internal costs equals the fine cost. Together these say that coarsening loses nothing that reality has determined; the coarse view is a faithful summary, not an approximation.

What the theorem does not claim is also precise. It does not claim that the coarse ledger is identical to the fine ledger, only that certain functionals agree. It does not claim that the weight function has any particular form, such as the recognition cost J; it holds for arbitrary weights. And it does not claim that the physical recognition-to-linking bridge, which would connect this formal ledger structure to actual three-dimensional space, is established; that bridge remains an open target. The theorem is a formal statement about multisets of events and a relabeling map, proved with zero axioms beyond the kernel's standard three.

/-- Relabeling endpoints to coarse sites preserves the ratio, hence the cost. -/
theorem cost_coarse_eq_cross (wr : ℝ → ℝ) (block : ℕ → ℕ) (m : Multiset Event) :
    cost wr (coarseLedger block m) = cost wr (crossOf block m) := by
  unfold cost coarseLedger
  rw [Multiset.map_map]
  rfl

/-- **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

/-- **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]

/-- Relabeling endpoints to coarse sites preserves the ratio, hence the cost. -/
theorem cost_coarse_eq_cross (wr : ℝ → ℝ) (block : ℕ → ℕ) (m : Multiset Event) :
    cost wr (coarseLedger block m) = cost wr (crossOf block m) := by
  unfold cost coarseLedger
  rw [Multiset.map_map]
  rfl

What this page does not claim

The coarse ledger is identical to the fine ledger, only that certain functionals agree. The weight function has any particular form, such as the recognition cost J. The physical recognition-to-linking bridge to three-dimensional space is established by this theorem.

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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