Foundation Singular Mayer Vietoris Is Iso Homology Map Chain Succ

The small-chains bridge

In algebraic topology, the Mayer-Vietoris sequence is a standard tool: it computes the homology of a space from the homology of two open subsets and their intersection. The Recognition Science framework's machine-checked library of formal theorems contains a proof that one of the maps in this setup is an isomorphism. The declaration isIso_homologyMap_chain_succ states that a certain induced map on homology groups is invertible, meaning the homology of the whole space can be recovered exactly from the homology of the two pieces and their overlap.

The theorem concerns small chains: singular simplices whose image lies entirely inside one of the two open sets. A singular n-simplex is a continuous map from the standard n-simplex into the space; it is small relative to a pair of subsets when its entire image sits in the first subset or in the second. The framework constructs a subcomplex from these small chains and proves that the natural inclusion of this subcomplex into the full singular chain complex induces an isomorphism on homology. This is the content of the declaration: the map from the homology of the small-chain subcomplex to the homology of the whole space is both injective and surjective, hence an isomorphism.

The proof rests on a short exact sequence of chain complexes, shown in the library as mvSESdeg_exact and mvSESdeg_epi, together with a lemma that the span of the unit generators spans the whole module. The framework's library verifies each step with a proof assistant, so the theorem is checked by machine rather than by hand. This is a formal result about the topology of spaces, not a claim about the physical world; it belongs to the mathematical infrastructure that the framework uses elsewhere.

What the declaration does not claim is broader. It does not assert that the Mayer-Vietoris sequence holds for arbitrary covers; the hypotheses require the two open sets to cover the space and the simplices to be small relative to them. It does not claim that the isomorphism is natural in any category-theoretic sense beyond what the proof states. And it does not say anything about the Recognition Science ledger, cost functions, or the golden ratio; this is a standalone topological theorem, not a step in the forcing chain that derives physical constants.

/-- A singular `n`-simplex of `X` is *small* (relative to the pair of subsets
`U, V`) when its range lies in `U` or in `V`. -/
def Small (U V : Set X) {n : ℕ} (s : Idx X n) : Prop :=
  Set.range ⇑(simplexEquiv X n s) ⊆ U ∨ Set.range ⇑(simplexEquiv X n s) ⊆ V

/-- A singular `n`-simplex of `X` is *small* (relative to the pair of subsets
`U, V`) when its range lies in `U` or in `V`. -/
def Small (U V : Set X) {n : ℕ} (s : Idx X n) : Prop :=
  Set.range ⇑(simplexEquiv X n s) ⊆ U ∨ Set.range ⇑(simplexEquiv X n s) ⊆ V

/-- A singular `n`-simplex of `X` is *small* (relative to the pair of subsets
`U, V`) when its range lies in `U` or in `V`. -/
def Small (U V : Set X) {n : ℕ} (s : Idx X n) : Prop :=
  Set.range ⇑(simplexEquiv X n s) ⊆ U ∨ Set.range ⇑(simplexEquiv X n s) ⊆ V

What this page does not claim

The theorem does not apply to arbitrary covers, only to pairs of open sets that cover the space. The declaration does not assert naturality of the isomorphism beyond what the proof states. This is a standalone topological result, not a step in the forcing chain that derives physical constants.

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/Foundation/SingularMayerVietoris.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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