Foundation Singular Subdivision Asub Iter Support Bound

Iterated subdivision and its support

In topology, a singular simplex is a continuous map from a standard triangle, or more generally a standard n-simplex, into a space. A chain is a finite formal sum of such simplices with integer coefficients. Subdivision replaces each simplex by a collection of smaller ones that together cover the same region. The operation is a standard tool for comparing different ways of chopping a space into pieces.

The declaration asubIter_support_bound concerns the iterated subdivision of an affine chain. An affine chain is a finite integer-linear combination of vertex tuples, where each tuple lists the vertices of a simplex. The declaration states that after any number of subdivision steps, the set of vertex tuples that appear with nonzero coefficient is contained in the set of tuples whose entries all lie in the original set of vertices. In plainer words: subdivision never introduces a vertex that was not already present in the starting configuration.

This is a structural fact about the algebraic bookkeeping of subdivision. It does not describe the geometric shape of the subdivided pieces, nor does it say anything about the space in which the simplices live. It is a constraint on the formal support of the chain, the collection of tuples that actually occur. The result is a lemma in the framework's machine-checked library of formal theorems, proved for the abstract affine setting before any topological content is added.

In Recognition Science, this lemma supports the stage where affine chains, boundaries, and cones are built before subdivision is applied to actual topological spaces. The framework models recognition events as discrete records, and this lemma is part of the machinery that keeps those records finite and well-founded under repeated refinement. The payoff is that iterated subdivision, as an algebraic operation, cannot escape the vertex set it started with, which is a necessary condition for later finiteness arguments.

@[simp] lemma asubIter_zero (bary : ∀ {m : ℕ}, (Fin (m + 1) → α) → α) (n : ℕ) :
    asubIter bary 0 n = LinearMap.id := rfl

What this page does not claim

The declaration does not claim that subdivision preserves the geometric shape of the original simplex. It does not claim that the support bound holds for chains with real or other non-integer coefficients. It does not claim that the bound is sharp, only that it is an inclusion.

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/SingularSubdivision.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: