Clausen's Hochschild companion conjecture: HH of the rapid-decay operator ideal over ℚ

In the Joint PU/IAS Arithmetic Geometry Seminar lecture [*What is the K-theory of the Complex
Numbers?*](https://www.ias.edu/video/what-k-theory-complex-numbers) (6 November 2023), Dustin
Clausen states a *companion conjecture* to the modified Hodge conjecture of his Copenhagen notes
[*Three Perspectives on Deligne
Cohomology*](https://www.math.ku.dk/english/calendar/events/masterclass-continuous-k-theory/Clausen1.pdf)
(Main Conjecture 4.5). At 1:05:01–1:06:55 of the
[recording](https://www.youtube.com/watch?v=5QE__xdYtA0) he describes it, remarks that it "unwinds
to something that looks completely unbelievable", and invites attempts: "if you manage to prove
it's acyclic in positive degrees, then you'll be halfway to proving our conjecture."

### Statement

Let `H = ℓ²(ℕ, ℂ)` and let `J∞ ⊆ B(H)` be the ideal of compact operators whose singular values
decay faster than every inverse power of the index. `J∞` is a nonunital ring. Form its ordinary
**algebraic** Hochschild complex over `ℚ`,

```
C_n = J∞ ⊗_ℚ ⋯ ⊗_ℚ J∞ (n + 1 algebraic tensor factors),
```

with the usual alternating boundary — adjacent products, plus the cyclic last face
`a_{n+1} a₀ ⊗ a₁ ⊗ ⋯ ⊗ a_n`. Note these are algebraic tensor products, not completed ones, and
`J∞` is used as-is rather than unitalized: this is what makes the statement surprising, and what
distinguishes it from the existing literature on entire and local cyclic homology of Schatten
ideals, which is topological.

**Conjecture.**

- `HH₀(J∞/ℚ) ≅ ℂ`, the isomorphism being induced by the operator trace; equivalently, the
rank-one corner `z ↦ [z P₀]` is bijective, where `P₀` is a rank-one projection.
- `HH_n(J∞/ℚ) = 0` for every `n > 0`.

In the lecture the degree-zero value is stated for the category whose invariants `J∞` computes,
and positive-degree acyclicity for the Hochschild complex of `J∞` itself; the two differ by the
shift relating `J∞` to the Calkin-type quotient `B(H)/J∞`. The normalization above — the base
field in degree zero, zero above — is the one Clausen states outright for the non-archimedean
analogue, and is the only one available to a complex concentrated in nonnegative degrees.

Clausen notes that the non-archimedean analogue, with `ℚ_p` in degree zero, was proved by his
student Adriano Córdova, by methods that do not proceed by analyzing the Hochschild complex.

### Status of the two clauses

The degree-zero clause is expected to be classical. `HH₀` is `J∞/[J∞, J∞]`, which coincides with
`J∞/[J∞, B(H)]` because every element of `J∞` is a product of two (polar decomposition and a
square root), so `[ab, c] = [a, bc] + [b, ca]` rewrites a commutator with a bounded operator
inside the ideal. That quotient is one dimensional, spanned by the trace: the tail sums of a
rapidly decaying sequence again decay rapidly, so `J∞` is stable under the arithmetic mean at
infinity, and for such ideals the commutator description of Dykema–Figiel–Weiss–Wodzicki
([Adv. Math. 185 (2004) 1–79](https://math.berkeley.edu/~wodzicki/prace/Advances-185.pdf),
Theorem 5.6; see also Kaftal–Weiss, [arXiv:0707.3169](https://arxiv.org/abs/0707.3169),
Theorem 6.6) leaves a unique trace up to scalars. The open content is the positive-degree clause,
which is the half Clausen asks for.

### Scope note

This is the companion conjecture only. The modified Hodge conjecture itself (Conjecture 4.5:
that a specific regulator map `K^nuc → K^Del` is an equivalence) is a separate and larger
formalization target, and is not covered here.

### AMS classification

16 (associative rings and algebras), 19 (K-theory), 46 (functional analysis).

The issue provides a mathematical conjecture but names no repository files, tests, or Lean entry points. First locate how formal conjecture statements are organized in this Lean repository, then determine whether the algebraic Hochschild complex and rapid-decay operator ideal can be represented with the available definitions. Done means adding a reviewed formal statement or proof target for the positive-degree acyclicity clause.

Written by the indexing model from the issue text.