Lieb-Thirring — two uses, shared inequality, unsharp constant

Instrument audit — Lieb–Thirring

Flagged as an audit target when the same inequality turned up in a second subject with its hypotheses stated in neither. Run 2026-09-13. Same procedure as the earlier instrument audits: state what the tool requires, then check each use against it.

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The two uses

1. Stability of matter. Bulk matter has energy E ~ −N rather than −N^{7/5}; without fermionic statistics the energy per particle at N = 10²³ would be inflated by 1.58×10⁹ and merging two equal lumps would release 0.639·N^{7/5}.

2. The Navier–Stokes attractor dimension. The finite-N trace of the linearised operator is finite for every N, and requiring it negative at large N gives d_f ~ Re^{9/4} — 5.6×10⁶ degrees of freedom at Re = 10³, 3.2×10¹³ at 10⁶.

What the tool requires

Both uses invoke the kinetic-energy form, not the eigenvalue-sum form:

> for an orthonormal family {φⱼ} ⊂ H¹(Ω), Σⱼ ∫|∇φⱼ|² ≥ K_n ∫ ρ^{1+2/n}, ρ = Σⱼ|φⱼ|²

Three preconditions, none stated at either use:

• orthonormality of the family — the inequality is false without it;
• H¹ regularity and a domain on which the embedding holds;
• the constant K_n, which is not sharp. The sharp value is the Lieb–Thirring conjecture and is open.

Findings

Finding 1 — same inequality, not two senses. Unlike duality, spectrum, closure, dimension and trace, this word is not carrying two notions. Both uses are the kinetic form. The audit's usual result does not occur here, which is worth recording: a tool used twice is not automatically a collision, and four prior audits finding one made that easy to assume.

Finding 2 — the orthonormality step is load-bearing and silent. In the attractor application the natural family is the first m eigenfunctions of the linearised Navier–Stokes operator, which is not self-adjoint — it carries the advection term. Its eigenfunctions are therefore not orthonormal, and an orthonormalisation step is required before the inequality applies. That step is standard and correct; it is also never mentioned, and it is exactly the kind of precondition the earlier audits found load-bearing.

Finding 3 — and this is the one that matters: the constant is not sharp, so the two uses inherit it differently.

So the exponent 9/4 is forced and the coefficient in front of it inherits an unsharp constant. Which is Part II's which-coefficient-is-forced test, applied to a tool rather than to a formula — and the first time that test has been pointed at an instrument.

Any numerical attractor-dimension estimate is therefore an upper bound with an unknown multiplicative slack, and quoting "5.6×10⁶ degrees of freedom at Re = 10³" as a figure rather than as a bound overstates what the inequality delivers.

Verdict

Not a collision. One genuine unstated precondition (orthonormality), and one real limit on what the tool can deliver (an unsharp constant, harmless for a scaling and not for a number).

The audit's value here is mostly negative, and that is informative. Four prior instrument audits each found a collision, and a fifth finding none is evidence the pattern was about those tools rather than about tools in general. A hit rate on audits that were triggered measures the trigger, not the base rate — the same selection effect that had to be corrected once already when a four-for-four record was offered as a prior.

Correction, 2026-09-13. "Four prior instrument audits" was a partial recollection. The genesis chat holds ten audits — eight word audits and two instrument audits — now filed as A-001–A-010, and this one is renumbered A-011 to sit after them. The four-for-four record, and the base-rate remark built on it, are withdrawn; see audits/README.md.

What this changes

• The attractor-dimension figures in this project should read as bounds with unknown slack, not estimates.
• R18's writeup should state the orthonormalisation step explicitly, since the recipe invites someone to reproduce the bound.
• The instrument-audit pattern drops from 4/4 to 4/5 — and the corrected way to say it is that precondition failures are a recurring error class, not that audits usually find them.