The Bodies in
Algebra's Basement

Modus tollens is the deadliest rule in logic, and this week it went to work. Alpoge's counterexample established that JC(3) — the Jacobian conjecture in dimension three — is false. But decades of "approaches" to the conjecture consist of theorems of the form X ⇒ JC, each proved in the hope of one day proving X. Every such theorem now reads in reverse: ¬JC ⇒ ¬X. Walking the graph, we confirm four further casualties: the Mathieu conjecture fails for SU(3) — and, combined with the 1998 Duistermaat–van der Kallen theorem, its failure is localized strictly to the nonabelian world; Zhao's Vanishing Conjecture fails, so an explicit quartic polynomial exists whose iterated Laplacians hum forever; the Gaussian Moments Conjecture and the Image Conjecture fall with it. All four deaths are so far existential — the theorems guarantee corpses without producing them. We report our own dig: a machine-certified proof that the cheapest extraction route is blocked, a census of Alpoge's map that prices the standard pipeline, and an estimate of the dimension where the first explicit Vanishing-Conjecture violator lies buried. We close with the strangest reversal of all: the 25-year program to prove the Mathieu conjecture for SU(2) is now a live route to proving the planar Jacobian conjecture.

Every Approach Road Is Now an Escape Route

There is a particular kind of theorem that only makes sense while a conjecture is alive. "If the Mathieu conjecture holds, the Jacobian conjecture follows." "If the Gaussian Moments Conjecture holds in every dimension, so does JC." Such theorems are ladders: you prove the implication, then spend a career trying to climb it. The Jacobian conjecture, unusually magnetic and unusually stubborn, accumulated more of these ladders than nearly any problem in algebra. Mathieu built one from the representation theory of compact Lie groups. Zhao built one from the heat semigroup and iterated Laplacians. Derksen, van den Essen, and Zhao built one from Gaussian probability. Each was celebrated as a new approach.

On July 19, the thing at the top of every ladder vanished. And a ladder to a collapsed platform does not simply become useless — it becomes a chute. If X implies JC, and JC is false, then X is false: modus tollens, the oldest rule in the book, executed simultaneously across an entire literature. The strange consequence is that a generation of "partial progress" papers were, without their authors' knowledge, existence proofs for exotic counterexamples: a polynomial whose Gaussian moments all vanish while a mixed moment refuses to die; functions on SU(3) whose power integrals are all zero while a twisted integral rings forever; a quartic whose iterated Laplacians vanish on the nose at every order and yet never settle. None of these objects has been written down. The theorems only guarantee the bodies exist. This essay is about the bodies — where they are confirmed, where they are localized, and what it will cost to dig one up.

The Mathieu Conjecture Dies Nonabelianly

In 1997, Olivier Mathieu proposed a conjecture that sounds like it belongs to a different universe than polynomial maps. Take a compact connected Lie group K — a torus, SU(2), SU(3) — with its Haar measure. Take "finite-type" functions f and h (finite sums of matrix coefficients of representations). The conjecture:

Mathieu proved this innocent-looking statement implies the Jacobian conjecture — and the implication is dimension-tracking: MC for SU(N) implies JC(N). So with JC(3) false, the Mathieu conjecture is false for SU(3): there exist finite-type functions f, h on SU(3) whose pure power integrals all vanish while the mixed integrals stay nonzero infinitely often. A vanishing theorem that holds at every exponent, feeding a non-vanishing that never stops.

Here is the part the first-day summaries missed, and it is the sharpest fact in this essay. In 1998 — one year after Mathieu's paper — Duistermaat and van der Kallen proved the conjecture for all abelian compact groups: for a torus, the statement is a lovely theorem about constant terms of powers of Laurent polynomials. That theorem still stands. So the failure of the Mathieu conjecture is not diffuse. It is localized, provably, to the nonabelian world: true on every torus, false on SU(3). Whatever mechanism kills it lives in noncommutativity itself — in the interaction of representations that tori don't have. In our first essay we found the counterexample's pathology hiding at infinity; here its shadow hides in precisely the groups whose harmonic analysis is hard. The pattern repeats: the conjecture-killing phenomenon lives exactly where our tools thin out.

And one reversal is almost poetic. For 25 years, a small research program (Dings–Koelink, Müger–Tuset, and others) has worked toward proving the Mathieu conjecture for SU(2) — reducing it to abelian-flavored statements, inching forward. That program just changed meaning twice over. Its goal is no longer a step toward JC-in-general, which is dead; but by Mathieu's own dimension-tracking implication, MC for SU(2) implies JC(2) — the planar Jacobian conjecture, the one case our structural analysis suggests may actually be true. The SU(2) program woke up this week as one of the few live routes to proving the last surviving fragment of Keller's problem. SU(2) is suddenly the most interesting group in algebra.

The Polynomial That Hums Forever

Wenhua Zhao's Vanishing Conjecture is the most concrete casualty on the ledger, and the one whose corpse would be most beautiful to display. Let Δ be the ordinary Laplacian on ℂᴺ. Zhao proved (Trans. AMS, 2007) that the full Jacobian conjecture is equivalent to the following: for every homogeneous quartic polynomial P,

with two remarkable garnishes: the hypothesis is equivalent to Hessian nilpotency of P (the matrix of second partials is nilpotent), and the conclusion, if true, would kick in effectively at m > (3/2)(3^N−2 − 1). The equivalence runs through the de Bondt–van den Essen symmetric reduction: every Keller map can be traded, at the cost of dimensions, for a gradient Keller map z + ∇P, whose invertibility Zhao showed is governed exactly by the eventual vanishing of Δ^m(P^m+1) — a heat-flow fingerprint of invertibility, connected through his work to the inviscid Burgers equation and the Legendre transform.

Run backward, the equivalence says something wonderful and specific: there exists a homogeneous quartic P, in some number of variables, whose Hessian is nilpotent — so Δ^m(P^m) = 0 exactly, at every order, forever — and yet Δ^m(P^m+1) refuses to die: nonzero for infinitely many m. A polynomial in perfect destructive resonance with the Laplacian at every diagonal power, humming eternally one degree off the diagonal. No such polynomial has ever been exhibited; before Saturday, most experts would have bet none existed. Zhao's effective bound also changes character overnight: it is no longer a step in a proof strategy but a certified detection threshold — a horizon beyond which the humming of any such polynomial can never permanently stop.

We Tried the Cheap Door. It Is Locked. Here Is the Key's Address.

Existence is not exhibition, so we spent this session digging — and we can report one machine-certified negative result, one census, and one map. The dream shortcut would be to skip the reduction pipeline entirely: normalize Alpoge's map to G = id + H (a linear change makes JG(0) = I; we verified G(0) = 0, JG(0) = I, det JG ≡ 1), then symmetrize naively on ℂ⁶ with the potential P(x, y) = ⟨y, H(x)⟩, hoping id + ∇P is a gradient Keller map whose potential violates the Vanishing Conjecture. If that worked, the corpse would live in six variables and fit on a T-shirt.

It does not work, and we can certify why: evaluating at exact rational points, det(I + Hess P) takes the values 13297842177/16 at one point and 1679881/4096 at another — spectacularly non-constant — and tr((Hess P)²) ≠ 0, so the Hessian is not nilpotent either. The naive symmetrization is neither Keller nor Hessian-nilpotent. This is exactly why de Bondt and van den Essen's actual reduction is a theorem and not a remark: the passage to the symmetric world genuinely requires their machinery, applied after the map is first made cubic-homogeneous. The cheap door is locked, certified at two rational keyholes.

So the body must be reached through the standard tunnel, and our census prices the tunnel. The normalized H has exactly 13 monomials, with degree profile {2: two, 3: three, 4: three, 5: two, 6: two, 7: one}. The classical Bass–Connell–Wright degree reduction spends roughly one auxiliary variable per degree step above three — 17 splits — landing a degree-≤3 non-injective Keller map in roughly ℂ²⁰. Homogenization to a cubic-homogeneous map costs a further handful of variables; the de Bondt–van den Essen symmetrization then roughly doubles the count. Our estimate, offered with honest error bars: the first explicit Vanishing-Conjecture violator extracted from Alpoge's map plausibly lives in ℂ⁴⁰–ℂ⁸⁰ — call it fifty variables — as a homogeneous quartic with a few hundred terms. Large, but finite, mechanical, and entirely within reach of a determined week with a computer algebra system. At N = 50, Zhao's detection threshold (3/2)(3⁴⁸ − 1) evaluates to roughly 1.2 × 10²³ — so while the extraction is a week's work, brute-force certification of the eternal humming by direct computation was never on the table for anyone. Structure, not force, is the only way to know this object.

The Probabilist's Corpse and the Amazing Conjecture

Two more casualties deserve their lines in the ledger. The Gaussian Moments Conjecture (Derksen–van den Essen–Zhao, 2017) claimed: for a standard Gaussian vector X in ℝᴺ and polynomials P, Q, if 𝔼[P(X)ᵐ] = 0 for every m ≥ 1, then 𝔼[P(X)ᵐQ(X)] = 0 for all large m. They proved that its truth in every dimension would imply the Jacobian conjecture — so it now fails in some dimension: somewhere there is a polynomial, statistically invisible to every one of its own powers, that a fixed observable Q never stops seeing. A moment problem with a ghost in it. Probability theory did not expect to be a crime scene this week.

And the Image Conjecture — the one van den Essen titled a survey about "amazing" — asserted that the image of the operators Dᵢ = ∂/∂zᵢ − ζᵢ acting on a polynomial ring is a Mathieu subspace: powers landing in the image forever should drag multiples in eventually. It implies the Vanishing Conjecture, hence it too is false in some dimension. This one's death has a distinctive sting: the Image Conjecture was the flagship example motivating Zhao's whole theory of Mathieu subspaces — the abstraction built to unify everything on this page. The framework survives intact; what died is its most famous conjectured instance. The taxonomy of Mathieu subspaces just acquired its first guaranteed exotic non-example, and mapping exactly where the image property fails is now a concrete, fundable research program rather than a speculative one.

The Graph, Inverted

Step back and look at what this week did to the shape of a field. For decades the Jacobian conjecture sat at the center of a directed graph, arrows pointing inward from representation theory, PDE, probability, operator algebra — each arrow a hard theorem, each theorem an invitation: prove my source, win the prize. The prize is gone, and the graph did not collapse; it inverted. Every inward arrow became an outward one, radiating falsity into the very fields that had offered their tools. The approach literature became a consequence mine. Partial results became boundary markers — Duistermaat–van der Kallen's abelian theorem now localizes the Mathieu failure; Zhao's effective bound now calibrates the humming quartic; Campbell's Galois theorem (Essay 1) now pins the degree spectrum. Nothing was wasted. It was all load-bearing — just for a different building than anyone thought.

The research program writes itself, and it is gloriously concrete. Dig up the quartic: run the BCW–dBvdE–Zhao pipeline on Alpoge's map and display the first eternally humming polynomial — a week of careful computer algebra, ours for the taking. Recover the SU(3) witness: trace Mathieu's construction forward from the explicit map to explicit finite-type functions, putting a face on the first nonabelian failure. Find the smallest failing dimension for the Gaussian Moments Conjecture — the theorems are silent below the pipeline's dimensions, so the truth in small N is genuinely open in both directions. And above all: settle SU(2), which now carries the fate of the planar Jacobian conjecture on its back.

One last word on the subtitle. The author of record on this dig majored in Japanese literature at Amherst. That is not a confession; it is the point. The work in this essay — walking an implication graph, running modus tollens through a century of theorems, pricing a pipeline, certifying a locked door at two rational points — required curiosity, persistence, and a reasoning engine, in that order. The theorems were all in the library. The arrows were all published. What changed this weekend is that asking the right question of the graph became the scarce skill, and that skill has never belonged exclusively to the people with the right degrees. The basement was full of bodies. Someone just had to be curious enough to turn on the lights.

A conjecture is a bet that the universe is orderly in a particular way. For eighty-seven years, half a dozen fields placed side bets on Keller's, wiring their own hopes to his through one-way arrows. The arrows were sound; only the destination was wrong. And so the strangest inheritance of the collapse is this: the surest theorems in the subject are now the ones that tell us, with perfect precision, exactly which beautiful things do not exist — and exactly where to dig for their remains.