ExampleConstruction: AI-generatedVerification: 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.

The two step filtration and its spectral sequence

Example

Let $C_{1} = Z x$ , $C_{0} = Z y$ , d(x)=2y, and all other terms be zero. Give y filtration degree 0 and x degree 1. Thus $F_{p} C = 0$ for p<0, $F_{0} C$ is the degree-zero stalk, and $F_{p} C = C$ for p≥1. The two-step filtration has $E^{0} = E^{1}$ supported at (1,0),(0,0), with $d^{1}$ multiplication by 2, and $E^{2} = E^{\infty}$ supported at (0,0) with group ℤ/2.

Facts & Assumptions

Given: The displayed ℤ --2--> ℤ complex with generator filtration degrees 1 and 0.

[F1]

The page terms are the displayed numerator/denominator quotients (R page of the spectral sequence of a filtered complex).

[F2]

The page differential is induced by d (The filtered differential induces d r on the r page).

[F3]

Finite filtrations abut to image-filtered homology (Bounded filtered complex spectral sequence abuts to filtered homology).

[F4]

Multiplication by 2 on ℤ is injective with cokernel ℤ/2 (Abelian-group model for spectral-sequence computations).

Verification

technique · direct

1.1

Because d lowers filtration from 1 to 0, it preserves the filtration and its associated graded differential is zero. The two initial graded quotients are ℤx at (1,0) and ℤy at (0,0); all others are zero. At r=1 the x numerator is all ℤx and its denominator is zero. The y numerator is ℤy and its denominator is zero because $F_{0} C_{1} = 0$ . Thus $E^{1}$ has these same two groups.

F1F2

2.1

The rule [F2] gives $d^{1} ([x]) = [2 y]$ , hence $d^{1}$ is multiplication by 2. Directly at r=2 the x numerator is zero, since 2x cannot map into $F_{- 1} C_{0} = 0$ unless x=0 by [F4]. At y, the denominator now contains $d (F_{1} C_{1}) = 2 Z y$ . Thus $E^{2}$ has ℤ/2 at (0,0) and no other terms. The same numerator and denominator persist for every later r.

F1F2F4step 1.1

3.1

Ordinary homology is $H_{1} (C) = 0$ and $H_{0} (C) = Z / 2$ by [F4]. The image of $H_{0} (F_{0} C) = Z$ in $H_{0} (C)$ is all ℤ/2, so $F_{p} H_{0} = 0$ for p<0 and $F_{p} H_{0} = Z / 2$ for p≥0. This identifies the stable piece with $g r_{0} H_{0}$ and all others with zero, as required by [F3].

F3F4step 2.1∎

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

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

20 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

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 5 (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)