Cosmology Rung Descent Unit Step Exists Top Descent Unit Step

The top-rung descent

In the Recognition Science framework, a ledger is a discrete record of events, and the objects it tracks carry integer rungs, which are numbered levels of distinction. The framework's engine updates these rungs one at a time, and a central invariant, UnitStep, requires that any two objects connected by an edge differ by at most one rung. The declaration exists_top_descent_unitStep proves a fact about this setup: for any nonempty collection of edges with a unit-step field, there is always a way to lower the rung of the top-level cells by exactly one, and this move preserves the unit-step invariant. The theorem constructs the move by taking the maximum rung that actually appears in the field, so no external parameter is needed.

The proof rests on a local criterion. Descending an arbitrary set of cells preserves unit-step exactly when every edge that crosses the boundary of the set, with one endpoint inside and one outside, still has a gap of at most one after the move. For a top-rung cell, this condition is forced: its neighbor cannot be above it, and the invariant forbids it from being two below, so the neighbor must sit exactly one rung down. After the descent, the two endpoints meet at the same level, and the gap is zero. The theorem also includes a counterexample showing the restriction is necessary: on a three-site chain with rungs 0, 1, 2, descending the bottom cell breaks the invariant, so the top-rung choice is not cosmetic.

The declaration is a theorem in the framework's machine-checked library of formal theorems, with no unproved assumptions beyond the standard logical axioms. It does not claim that the real-valued mean move preserves unit-step; in fact, the framework's Phase 58 proved that a global claim of that kind is false. It also does not claim that the dynamics of the actual universe follows this rule, only that within the model, the forced top-rung descent is the safe relaxation step. The practical consequence is that the cost law from an earlier phase applies to the next cycle after this specific update, giving the engine a theorem-backed way to continue its descent.

/-- **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⟩)

/-- **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

/-- **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 apply to real-valued rung updates, which can break the invariant. The theorem does not assert that the actual universe's dynamics obey this rule. The theorem does not claim that any descent, only the top-rung one, is safe.

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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