ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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.

A complete spectral-sequence computation record

Example

A complete record for the integer UCT example takes $C_{1} = Z a \oplus Z b$ , $C_{0} = Z c$ , $d a = 2 c, d b = 0$ , and $M = Z / 2$ . Use cohomological $E_{r}^{p, q}$ , $d_{r}$ of degree $(r, 1 - r)$ and a decreasing filtration by projective resolution degree $p$ . The Hom convention is $Hom (C, M)^{n} = Hom (C_{n}, M)$ with $δ f = f d_{C}$ . The computation is in all total degrees, under the AC convention of general integer UCT.

Its complete page data are $E_{2}^{0, 0} = E_{2}^{1, 0} = E_{2}^{0, 1} = Z / 2$ and zero elsewhere. All $d_{r}$ for $r \geq 2$ vanish and $E_{\infty} = E_{2}$ . The target is $H^{0} = M$ , $H^{1} = M^{2}$ , zero otherwise. In degree one the filtration is $M^{2} \supset M \oplus 0 \supset 0$ , with quotient map $(x, y) \mapsto y$ and inclusion $x \mapsto (x, 0)$ . A section is $y \mapsto (0, y)$ ; there is no general natural section in $C$ .

Facts & Assumptions

Given: The displayed complex, coefficients, indexing and full-degree computation range.

[F1]

A complete record resolves its page, differential, convergence, reconstruction and edge obligations (Spectral-sequence computation record).

[F2]

The integer UCT example computes these three page entries, the finite target filtration and evaluation map under AC (UCT as a two-column spectral sequence over the integers).

Verification

1.1

The input homology is $H_{0} C = Z / 2$ , $H_{1} C = Z$ , zero elsewhere. Applying Hom into $M$ to $Z \overset{2}{\to} Z$ gives $M \overset{0}{\to} M$ , so the degree-zero and degree-one Ext entries of $H_{0} C$ are $M$ . The one-term projective resolution of $H_{1} C$ contributes only $M$ at $(0, 1)$ . The AC free-kernel argument in F2 kills every $p > 1$ entry, giving exactly the stated page.

2.1

No later arrow can join columns zero and one: its first-coordinate change is $r \geq 2$ . Every possible incoming arrow also starts in a zero column, unless its target has first coordinate at least two, in which case that target is zero. Thus every later differential vanishes at every bidegree, not just those displayed, and $E_{2} = E_{\infty}$ . F2 supplies first-quadrant finite-filtration convergence to Hom cohomology. Directly this Hom complex is $M \overset{0}{\to} M^{2}$ , confirming the target and vanishing in all other degrees.

F1F2step 1.1

3.1

In degree zero the only quotient has filtration index zero, so $F^{0} H^{0} = M, F^{1} H^{0} = 0$ . In degree one, evaluation on $b$ gives $(x, y) \mapsto y$ , with kernel $M \oplus 0$ ; these are the two graded pieces at $(0, 1)$ and $(1, 0)$ . Thus the exact extension is $0 \to M \overset{x \mapsto (x, 0)}{\to} M^{2} \overset{(x, y) \mapsto y}{\to} M \to 0$ . The lower edge in degree one is the displayed injection; the upper edge is evaluation. In degree zero both edges are the identity under the kernel identification. All edges in degrees at least two have zero target and zero source here. This fixes every endpoint and reconstructs the actual extension.

F1F2step 2.1

4.1

The section $y \mapsto (0, y)$ exists explicitly. Under the chain automorphism $a \mapsto a + b$ , Hom cohomology transforms as $(x, y) \mapsto (x + y, y)$ while the graded endpoints are fixed; no lift of $1$ is fixed. Consequently the section is not natural, and no alternative section restores naturality for all chain maps. AC has been used only through the general UCT free-submodule/projective and replacement conventions of F2; all computations for these specified finite free complexes are explicit. Every obligation of F1 is now resolved in all degrees, with no unknown differential, extension or convergence qualification.

F1F2step 3.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

Weibel, Sections 5.2 and 5.7 (standard reference, not scraped)