Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27
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FALSE: a proposed Gromov boundary quotient needs no equivalence check

Statement

False claim: once a class of boundary sequences and a proposed "asymptotic" relation have been written down, one may form the Gromov boundary as their quotient without proving that asymptoticity is an equivalence relation.

Facts & Assumptions

Given: The boundary construction on this page.

[L1]

The quotient by asymptoticity is justified only after proving that asymptoticity is an equivalence relation (Asymptoticity of Gromov sequences is an equivalence relation).

Refutation

technique · direct
1.1

A quotient set consists of equivalence classes, so it is defined only when the proposed relation is an equivalence relation. The sequence model on this page therefore depends essentially on [L1].

L1
2.1

Consequently the proposed quotient cannot be licensed merely by writing down the relation: reflexivity, symmetry, and transitivity must be checked. The properness hypothesis used elsewhere on this page is a scope choice for this construction, not a claim that every possible boundary model requires properness.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources