Is "every invariant here is linear" a real limitation or a definitional one?

Object. Two families of invariant. The first is built from traces, spectra, cohomology, characters, Fourier transforms, L-functions and heat kernels — all linear in the sense that they are computed from a linear operator or a linear functional on an algebra. The second is built from counts of solutions to nonlinear PDEs over moduli spaces: Donaldson and Seiberg–Witten invariants, whose invariance comes from a cobordism argument rather than from conjugation-invariance of a trace.

Only the second family detects smooth structure. By Atiyah–Singer the output of a linear trace identity is a topological invariant — verified, the heat trace's subleading coefficient on the round sphere converges to 0.3335 against χ/6 = 1/3, and to zero on a flat torus, which is Gauss–Bonnet falling out of a trace. So a linear invariant cannot separate two homeomorphic non-diffeomorphic manifolds, by theorem rather than by oversight.

Presenting problem (the friction). The claim "this body of work contains only linear invariants" is being read as a substantive gap, and it may be unfalsifiable as stated.

Witten's formulation writes Donaldson invariants as correlation functions of a topological quantum field theory — that is, as path-integral expectation values, which are traces in the TQFT's algebra of observables. If that counts as linear, the distinction collapses and the claim reduces to "some invariants are hard to compute." If it does not count, the criterion for linearity has to be stated precisely enough to exclude a partition function while admitting a heat trace, and it is not obvious what that criterion is.

There is a second slippage in the supporting evidence. A worked case was offered in which a single-particle problem succeeded and the corresponding interacting problem failed twice, read as confirmation. But "nonlinear invariant" in the Donaldson sense and "nonlinear problem" in the many-body sense are different things: interacting problems are harder than one-body problems everywhere, for reasons having nothing to do with invariant theory.

Differential. 1. Genuine gap in kind. Linearity is well defined here — the invariant is a trace of an operator on a vector space — and Donaldson invariants are not of that form even in their TQFT presentation, because the path integral is not a trace of anything constructed from the underlying manifold's linear data. 2. Definitionally empty. A TQFT partition function is a trace, so the second family is inside the first, and what is really being observed is a difference in computational difficulty rather than in kind. 3. True but for a different reason. The gap is not linearity but dimension four: Donaldson theory is a 4-manifold phenomenon, and the failure of linear invariants there is a fact about that dimension rather than about linearity. 4. Two claims in one. There is a real gap in kind, and there is a separate and much weaker observation about interacting problems, and the evidence for the second has been counted toward the first.

Discriminating tests. State the linearity criterion precisely enough to classify a TQFT partition function, before looking at the answer — that is the whole test and it is free. Then: exhibit a linear invariant that distinguishes an exotic ℝ⁴ from the standard one, or prove none can. Check whether the Atiyah–Singer argument (linear trace ⇒ topological invariant) applies to TQFT correlation functions; if it does, branch 2 wins, and if it does not, the reason why is the criterion branch 1 needs.

Stakes. Whether a body of work that has proceeded entirely by spectral and cohomological methods is missing a class of theorem, or merely has not attempted the hard computations. Those call for different responses — one is a construction problem, the other is a to-do list.

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Update (2026-09-13): a mechanism for one instance, and a third data point.

The case asks whether "only linear invariants here" is substantive or definitional. Two developments, in opposite directions.

A mechanism, for the fluid instance. The linear part of the Navier–Stokes operator has a perfectly good trace: by Weyl's law in three dimensions the Stokes eigenvalues grow like n^{2/3}, and the heat trace is finite at every positive time — computed, 0.96, 41.55 and 1328.84 at t = 1, 0.1 and 0.01, against the asymptotic 1.329 t^{−3/2}. The full nonlinear problem has no such trace.

And the failure is not repairable by subtraction. Shifting every eigenvalue by a constant — a bounded perturbation with no change of operator order — already leaves a divergent difference, falling only from t^{−3/2} to t^{−1/2}: computed, −2.05, −3.81, −6.63, −12.13 at t = 0.1, 0.03, 0.01, 0.003, with the t^{−1/2} scaling exact. The nonlinear term is a first-order differential operator, so it alters every local heat-kernel coefficient from the first onward rather than shifting the spectrum. The leading coefficient cancels in a difference and none of the others do, so a finite subtraction cannot recover it — one counterterm per coefficient would be required.

So for this instance there is now a reason rather than an observation: why the linear part has a trace and the nonlinear part does not.

A third data point. The pattern now has three instances — a fractal lattice where the single-particle topological index was computable and the interacting question was not; the smooth-versus-topological gap, where a linear index theorem's output is a topological invariant by theorem; and this one. Every failure localises at the same joint.

What this does not settle, which is the case's actual question. The heat-kernel argument is specific to a differential perturbation of an elliptic operator. It says nothing about whether a count of solutions to a nonlinear PDE over a moduli space is or is not a trace in the relevant sense, which is branch 1 against branch 2 above. A mechanism for one instance is not a criterion, and the criterion — stated precisely enough to classify a topological field theory's partition function — remains the thing the case is waiting on.

If anything, the three instances make branch 4 more attractive: there may be a real gap in kind and a separate, much weaker observation about nonlinear problems being harder, with evidence for the second having been counted toward the first.

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Update (2026-09-13): the joint has two kinds, and one of them has been crossed.

The case asks whether "only linear invariants here" is substantive or definitional. A distinction that sharpens it, from a system where the crossing succeeded:

The nonlinearity can sit in the operator or in the state, and the two behave differently.

• In the operator. For a dissipative nonlinear PDE, the nonlinear term is a first-order differential perturbation, so it alters every local heat-kernel coefficient rather than shifting the spectrum. The leading coefficient cancels in a difference and none of the others do, so no finite subtraction recovers a trace. Not survivable.
• In the state. In the fractional quantum Hall effect the observable is still a linear response — a conductance from a Kubo formula — and only the ground state is interacting. For filled bands that response integrates to a topological integer; for the fractional case the invariant survives intact while the computation does not, and what the invariant counts changes from electrons to emergent composite objects. Survivable, and nature exhibits it.

So the joint is not a single wall. The integer-to-fractional step is the joint crossed, and it names the condition under which crossing is possible: the interaction must sit in the state rather than in the observable.

Effect on the differential. This favours branch 4 — that a real gap in kind and a much weaker "nonlinear problems are harder" observation have been running together — by supplying the line between them. It also gives the case a discriminating test it lacked: for each instance where a linear invariant fails, ask whether the failure is in the operator or in the state. If every failed instance is an operator failure and every crossed instance is a state failure, the gap is substantive and the criterion is stated. If the sorting is inconsistent, the distinction is being fitted after the fact.

One honest limitation. The heat-kernel argument is specific to a differential perturbation of an elliptic operator, and the Kubo argument is specific to a linear-response observable. Neither says anything about whether a count of solutions to a nonlinear equation over a moduli space is or is not a trace, which is the case's original question. A mechanism for two instances is not yet a criterion.