Cosmology Rung Descent Unit Step Shift Down Unit Step Of Cut

The cut condition

In a discrete system where every object carries an integer level, the unit-step invariant says that any two connected objects differ by at most one level. The Recognition Science declaration shiftDown_unitStep_of_cut establishes a precise condition for when lowering a chosen set of objects by one level preserves this invariant. The condition concerns cut edges: connections with exactly one endpoint in the lowered set. If every such cut edge still has a level difference of at most one after the move, then the whole system remains unit-step. Edges with both endpoints lowered, or neither, are preserved automatically, since their level difference is unchanged.

The theorem is a machine-checked result in the framework's library of formal theorems. It is the integer analogue of an earlier real-valued result, and it closes a gap in the framework's account of how its recognition dynamics evolves. The statement is local: it does not claim that any descent preserves the invariant, only that a descent meeting the cut condition does. The proof is complete, with no unfinished parts and no new axioms beyond the standard three of the underlying logic.

The necessity of the condition is shown by a concrete counterexample. On a three-site chain with levels 0, 1, 2, lowering the bottom cell (level 0) to -1 creates a gap of 2 with its level-1 neighbour, breaking the invariant. This demonstrates that the top-rung restriction in a related theorem is essential, not a technical convenience. The framework also proves that a preserving descent always exists for any nonempty edge set with a unit-step field, by taking the maximum level as the set to lower.

In Recognition Science, this result supports the claim that the framework's cost law applies cycle to cycle along the specific update it names. The framework models its dynamics as a series of such descents, and this theorem is the piece that keeps the unit-step invariant intact for the natural top-rung move. It does not claim that the invariant holds for arbitrary updates, and it does not address the real-valued mean move, which is treated separately.

/-- **Local preservation criterion.** Descending a set `S` by one rung preserves `UnitStep` on the
whole edge set provided every *cut* edge (exactly one endpoint in `S`) remains within one rung
after the move. Edges with both endpoints in `S` keep their gap (both shift by `1`); edges with
neither endpoint in `S` are unchanged. So only the cut edges can break the invariant, and the
hypothesis controls exactly those. -/
theorem shiftDown_unitStep_of_cut (S : V → Prop) [DecidablePred S] (k : V → ℤ)
    (E : Finset (V × V)) (hunit : UnitStep k E)
    (hcut : ∀ p ∈ E, ((S p.1 ∧ ¬ S p.2) ∨ (¬ S p.1 ∧ S p.2)) →
        (shiftDown S k p.1 - shiftDown S k p.2 = 0
          ∨ shiftDown S k p.1 - shiftDown S k p.2 = 1
          ∨ shiftDown S k p.1 - shiftDown S k p.2 = -1)) :
    UnitStep (shiftDown S k) E := by
  intro p hp
  by_cases h1 : S p.1 <;> by_cases h2 : S p.2
  · -- both endpoints descend: the gap is unchanged
    rw [shiftDown_pos S k h1, shiftDown_pos S k h2]
    have hsame : (k p.1 - 1) - (k p.2 - 1) = k p.1 - k p.2 := by ring
    rw [hsame]; exact hunit p hp
  · -- cut edge: p.1 descends, p.2 stays
    exact hcut p hp (Or.inl ⟨h1, h2⟩)
  · -- cut edge: p.1 stays, p.2 descends
    exact hcut p hp (Or.inr ⟨h1, h2⟩)
  · -- neither endpoint descends: unchanged
    rw [shiftDown_neg S k h1, shiftDown_neg S k h2]; exact hunit p hp

/-- **A preserving descent always exists.** For any nonempty edge set carrying a unit-step rung
field, take `M` to be the realised maximum rung; descending the top-rung cells preserves
`UnitStep`, and the descent is nontrivial because the maximum is attained. No external parameter
is supplied: `M` is read off the field. -/
theorem exists_top_descent_unitStep [DecidableEq V] (k : V → ℤ) (E : Finset (V × V))
    (hne : E.Nonempty) (hunit : UnitStep k E) :
    ∃ M : ℤ, (∃ v ∈ edgeVerts E, k v = M)
      ∧ UnitStep (shiftDown (fun v => k v = M) k) E := by
  have hVne : (edgeVerts E).Nonempty := by
    obtain ⟨p, hp⟩ := hne
    exact ⟨p.1, fst_mem_edgeVerts hp⟩
  have hImgNe : ((edgeVerts E).image k).Nonempty := hVne.image k
  set M : ℤ := ((edgeVerts E).image k).max' hImgNe with hM
  have hMmem : M ∈ (edgeVerts E).image k := Finset.max'_mem _ hImgNe
  obtain ⟨v, hv, hvk⟩ := Finset.mem_image.mp hMmem
  refine ⟨M, ⟨v, hv, hvk⟩, ?_⟩
  apply shiftDown_top_unitStep k E M hunit
  intro p hp
  refine ⟨?_, ?_⟩
  · exact Finset.le_max' _ (k p.1) (Finset.mem_image.mpr ⟨p.1, fst_mem_edgeVerts hp, rfl⟩)
  · exact Finset.le_max' _ (k p.2) (Finset.mem_image.mpr ⟨p.2, snd_mem_edgeVerts hp, rfl⟩)

/-- **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 any descent preserves the unit-step invariant, only those meeting the cut condition. The theorem does not address the real-valued mean move, which is treated separately and does not preserve the invariant globally. The theorem does not claim anything about the framework's physical cosmology, only about its formal discrete dynamics.

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.

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