Cosmology Rung Descent Unit Step Fst Mem Edge Verts

The edge vertices lemma

In the Recognition Science framework, a cosmology is built from discrete objects and the connections between them. A ledger, a discrete record of events, assigns an integer rung to each object, and an edge records that two objects are directly connected. The declaration fst_mem_edgeVerts is a formal lemma about this setup: it proves that if a pair of objects is an edge, then the first object of that pair belongs to the set of vertices that actually appear in the edge set. The proof is immediate from the definition of that set, which collects both the first and second components of every edge.

The lemma is a building block, not a standalone discovery. Its role is to support the larger theorem that a specific kind of move, descending the top rung, preserves a key invariant called unit-step, meaning every adjacency changes the rung by at most one. The framework's machine-checked library of formal theorems uses fst_mem_edgeVerts to reason about which vertices are affected by such a descent. Without it, the proof that the top-rung descent keeps the field unit-step would lack a formal handle on the objects involved.

What the lemma does not claim is just as important. It does not say that the first component of an edge is special, that edges are ordered in a meaningful way, or that the set of edge vertices has any particular size or structure. It merely states a containment fact: the first component of any edge is in the set of vertices that the edge set touches. It also does not assert anything about the rung values themselves, only about membership in a set derived from the edges.

In the context of the framework's cosmology, this lemma is a small but necessary piece of the machinery that lets the cost meter be theorem-backed along the top-rung descent. It helps ensure that when the framework proves a descent preserves the unit-step invariant, it is reasoning about the right objects. The lemma itself is a triviality, but it is a triviality that the machine-checked proof needs, and its presence is part of what makes the larger theorem trustworthy.

lemma fst_mem_edgeVerts [DecidableEq V] {E : Finset (V × V)} {p : V × V} (hp : p ∈ E) :
    p.1 ∈ edgeVerts E :=
  Finset.mem_union.mpr (Or.inl (Finset.mem_image.mpr ⟨p, hp, rfl⟩))

What this page does not claim

The lemma does not claim that the first component of an edge is special or that edges are ordered meaningfully. It does not assert any property of the rung values themselves, only membership in a set derived from edges. It does not establish that the edge vertex set has any particular size or structure.

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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