Cosmology Rung Descent Unit Step Ck Levels Unit Step

The unit-step chain

In the Recognition Science framework, a ledger (a discrete record of events) can carry an integer-valued field, a rung number for each cell. A field is called unit-step when every edge in a given set connects cells whose rungs differ by at most one. The declaration ckLevels_unitStep establishes the simplest possible case: the three-site chain with rungs 0, 1, 2, and edges between 0 and 1, and between 1 and 2, is unit-step. This is a machine-checked theorem in the framework's library of formal theorems.

The result matters because it anchors a larger claim about how the framework's dynamics can evolve. A separate theorem, shiftDown_top_unitStep, proves that descending the top rung of any unit-step field preserves the invariant. The chain 0, 1, 2 is the concrete witness that this top-rung restriction is necessary. If one descends the bottom cell (rung 0) instead, it becomes -1, and the edge to its neighbor at rung 1 now has a gap of 2, breaking unit-step. The theorem ckLevels_descend_min_breaks records exactly this failure.

What the declaration does not claim is broader. It does not prove that any descent preserves unit-step; in fact, the bottom-rung counterexample shows the opposite. It does not assert anything about real-valued moves or continuous dynamics, which are handled separately and are known not to preserve the invariant globally. It also does not claim that the chain 0, 1, 2 represents any physical cosmology; it is purely a finite combinatorial example used to test a formal property.

/-- The chain `0, 1, 2` is unit-step before any move. -/
theorem ckLevels_unitStep : UnitStep ckLevels ckEdges := by
  unfold UnitStep; decide

/-- **Necessity counterexample.** Descending the *bottom* cell (rung `0`) of the chain `0, 1, 2`
sends it to `-1`, so the edge to the rung-`1` neighbour has gap `2` and `UnitStep` fails. Only
descending the top rung is safe; the top-rung hypothesis of `shiftDown_top_unitStep` is necessary,
not cosmetic. -/
theorem ckLevels_descend_min_breaks :
    ¬ UnitStep (shiftDown (fun v => v = (0 : Fin 3)) ckLevels) ckEdges := by
  unfold UnitStep; decide

What this page does not claim

That any descent preserves unit-step. That the chain 0, 1, 2 represents a physical cosmology. That real-valued moves preserve the invariant.

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/RungDescentUnitStep.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.

Derived articles

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