LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

R boundaries embed in r cycles

Statement

For every filtered chain complex, $B_{p, q}^{r} \subseteq Z_{p, q}^{r}$ for all $p, q \in Z$ and $r \geq 0$ .

Facts & Assumptions

Given: A filtered chain complex and integers p,q with r≥0.

[F1]

The numerator is $A_{p, n}^{r}$ and the denominator is the displayed sum of lower approximate cycles and differential images (R cycles and r boundaries of an increasingly filtered complex).

Proof

technique · direct

1.1F1algebra

For $r = 0$ , the assertion is $F_{p - 1} C_{n} \subseteq F_{p} C_{n}$ . For $r \geq 1$ , $A_{p - 1, n}^{r - 1} \subseteq F_{p - 1} C_{n} \subseteq F_{p} C_{n}$ and its differential lies in $F_{(p - 1) - (r - 1)} C_{n - 1} = F_{p - r} C_{n - 1}$ . It therefore factors through $A_{p, n}^{r}$ .

2.1F1step 1.1algebra∎

For the other summand, $d (A_{p + r - 1, n + 1}^{r - 1}) \subseteq F_{p} C_{n}$ , and applying $d_{n}$ to it gives zero since $d_{n} d_{n + 1} = 0$ . It too lies in $A_{p, n}^{r}$ . The sum lies there by the least-upper-bound property of the image sum in [F1]. This proves the assertion for every r, including r=1.

Source notes

Weibel, Chapter 5, Construction 5.4.6 and Lemma 5.4.7, pp.133–134; Sharifi, Theorem 4.2.3, pp.91–92. Increasing homological indices are used here.

Depends on

R cycles and r boundaries of an increasingly filtered complex