Cosmology Rung Coarsen

Coarsening without loss

In the Recognition Science framework, a recognition cell at one scale is a multiset of directed events, each a posting from a source site to a target site carrying a positive ratio. Coarsening means grouping fine sites into blocks. The framework's rung-coarsening module proves that this grouping loses nothing: the round trip from fine to coarse and back returns the original multiset exactly, so every conserved functional of the cell, event count, total cost, log-ratio spectrum, and net flow, is unchanged. The theorem is stated for an arbitrary ratio weight, so it holds for the framework's recognition cost J and any other weighting.

The proof is a partition argument. Each event is either internal, both endpoints in one block, or cross, endpoints in different blocks. Internal events are absorbed into their block's summary; cross events are promoted to coarse events between coarse sites with the same ratio. Refinement keeps exactly the original cross events and per-block internal events, and reassembles them. The round trip is the identity because the partition recombines to the original multiset. Cost partitions exactly: the coarse cross-block cost plus the sum of block-internal cost equals the fine cost. Idle cells carry nothing: a cell with no internal events keeps an empty refinement, so refinement memory scales with recognition activity, not with the number of sites.

The module is theorem-backed with zero sorry and zero new axioms. It discharges the statement itself, mirroring the Python side's coarsen and refine functions. The result is that a coarse ledger is not a lossy summary but a faithful repackaging. What this changes is the meaning of scale in the framework: a coarse description is not an approximation of the fine one, it is the same ledger with some sites merged. The framework can therefore reason at any rung without discarding information determined at a finer rung.

/-- **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]

/-- **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]

What this page does not claim

The module does not prove that any particular coarsening is physically preferred or optimal. The module does not claim that the coarse ledger is a separate physical object; it is the same multiset repackaged. The module does not establish the recognition cost J itself; it holds for any ratio weight.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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