Cosmology Rung Descent Unit Step Ck Levels Descend Min Breaks

The necessity counterexample

In the Recognition Science framework, a ledger (a discrete record of events) assigns each cell an integer rung, and a unit-step invariant demands that adjacent cells differ by at most one rung. The framework's engine relaxes a configuration by descending a region by exactly one rung. A central question is whether this move preserves the unit-step invariant cycle to cycle.

The declaration ckLevels_descend_min_breaks answers that question for a specific case. It considers the simplest possible chain of three cells, with rungs 0, 1, and 2. The theorem proves that descending the bottom cell, the one at rung 0, sends it to -1. The edge between this cell and its neighbour at rung 1 then has a gap of 2, which violates the unit-step rule. The move is unsafe.

This is not a failure of the framework's dynamics. It is a precise, machine-checked necessity result. It shows that the top-rung restriction in the companion theorem shiftDown_top_unitStep is not a cosmetic convenience but a logical requirement. Descending the coarsest, topmost rung is the only parameter-free relaxation that provably keeps the field unit-step. The counterexample pins down exactly where the boundary lies.

The theorem is tagged THEOREM in the framework's machine-checked library of formal theorems, with zero unproved assumptions beyond the standard three axioms of the ambient type theory. It does not claim that all descents fail, nor that the unit-step invariant is always preserved. It establishes a single, sharp fact: on a three-site chain, descending the minimum rung breaks the invariant, and therefore the safe move is the top-rung descent.

/-- **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

/-- **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

The theorem does not claim that all descents of non-top cells fail on every graph. The theorem does not claim that the unit-step invariant is always preserved by the framework's dynamics. The theorem does not claim that the three-site chain is the only configuration where a non-top descent breaks 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)

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Derived articles

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