# The heart and unbounded t-structures

**Verdict:** Thick(heart) is exactly the t-bounded part; an unbounded zero-differential complex refutes unconditional generation.

21 October 2026; 2 agents (solver, fresh auditor); 14.0 minutes from brief preparation through synthesis, including queue time. Validation: in-run candidate audit only.

## Problem and setting

Let \(\mathcal C\) be a triangulated category with a t-structure \((\mathcal C^{\le0},\mathcal C^{\ge0})\), using the cohomological convention \(\mathcal C^{\le n}=\mathcal C^{\le0}[-n]\) and \(\mathcal C^{\ge n}=\mathcal C^{\ge0}[-n]\). Put \(\mathcal H=\mathcal C^{\le0}\cap\mathcal C^{\ge0}\) and \(\mathcal C_t^b=\bigcup_{a\le b}(\mathcal C^{\ge a}\cap\mathcal C^{\le b})\). Thick means closure under isomorphisms, finite sums, shifts, cones, and retracts which exist in \(\mathcal C\). Decide the universally quantified assertion: every category with a t-structure satisfies \(\operatorname{Thick}_{\mathcal C}(\mathcal H)=\mathcal C\). A complete answer proves or refutes that assertion, certifies a familiar counterexample, identifies exactly what boundedness supplies, and explains the effect of allowing infinite constructions.

## Completion criterion

Give a proof or certified counterexample at the stated level, specify any additional assumptions, and keep stronger claims with exact unmet obligations. Finite numerical checks or analogies do not complete the question.

## Checklist

- Functor domains, codomains, shifts, and ordinary/perverse meanings.
- Finite thick operations versus arbitrary coproducts.
- Hypotheses of every source and preservation on the intended category.
- Geometric versus rational labels; examples at their actual level.

## Portfolio

Table 1. Independent roles.

| agent | round | role | return |
|---|---:|---|---|
| A1 | 1 | Candidate proof | [A1.json](returns/A1.json) |
| A2 | 2 | Fresh adversarial check | [A2.json](returns/A2.json) |

## Registry

The direct candidate and its independent audit support the stated answer. No mathematical gap in the stated conditional answer. Applicability to the original geometry retains the explicit hypotheses; this is an in-run candidate audit, not a separate audit of the root synthesis.

## References

[Verified literature](references.md), with corrections in [audit-summary.md](audit-summary.md). The original source assertions and locators remain in the immutable returns.

## Files

[Verdict](verdict.md) · [attempt tree](tree.md) · [tree data](tree.json) · [candidate proof](candidate-proof.md) · [presentation plan](presentation.md) · briefs/ · returns/ · scripts/.