Erratum: this page originally claimed percolation's mean-field transition is Thom's cusp catastrophe, with $\beta=1/2,\delta=3$. Those are the Ising model's mean-field exponents. Checking against the exactly-solved case — percolation on a regular tree (Bethe lattice), prompted by comparing a plain transitive-graph illustration against this desk's own branching-ladder diagrams — shows percolation's own mean-field exponents are $\beta=1,\gamma=1,\delta=2$ (Aizenman–Newman), which is a fold/transcritical structure, not a cusp. Everything below is corrected; the crossreferenced literature (Hutchcroft, Schonmann) is unaffected — only which singularity it establishes changes.
Aizenman, Barsky and Fernandez's 1987 "ghost field" gives percolation a two-parameter shape: a second parameter $h$ alongside edge-probability $p$, producing $M(p,h)$ with $\theta(p)=\lim_{h\to0^+}M(p,h)$. On the exactly-solved cases, the giant-component fraction $S$ solves $h=S^2-\epsilon S$ near criticality ($\epsilon=p-p_c$) — quadratic, not quartic. At $h=0$: $S=0$ or $S=\epsilon$, linear ($\beta=1$). At $\epsilon=0$: $S=\sqrt h$ ($\delta=2$). Both match percolation's own tabulated mean-field values; neither matches the cusp's $\beta=1/2,\delta=3$. On the 3-regular Cayley tree specifically ($p_c=1/(z-1)=1/2$ for $z=3$), these exponents are exact, computed directly — not merely asymptotic.
| Where the fold equation actually holds | Basis | Status |
|---|---|---|
| Complete graph / Bethe lattice | Exact mean-field percolation | established |
| $\mathbb{Z}^d$, spread-out, $d>6$ / nearest-neighbor, $d\geq11$ | Hara–Slade 1990 / Fitzner–van der Hofstad 2017 | established |
| Every nonunimodular transitive graph | Triangle condition (Hutchcroft, 2020) | established |
| Planar nonamenable transitive graphs | Schonmann, 2001 | established |
| 2D lattices (triangular, exact) | Smirnov conformal invariance, $\beta=5/36$, $\delta=91/5$ | false |
| Unimodular nonamenable, non-planar | — | open |
| Unimodular amenable outside proven $\mathbb{Z}^d$ region | — | open |
A regular tree — a generic transitive graph, and the Bethe lattice this page solves exactly.
This desk's own Hexabonacci ladder diagram — visually similar, mathematically a different object (see erratum above).
A separate, earlier proposal — that a percolation threshold is a Whitney fold singularity on a contact manifold, in the sense Book 3's dm³/GTCT framework uses (github.com/TOTOGT/3M, LAW3M, Natal, 2026) — was checked against percolation directly and rejected: $\theta(p)$ is identically zero on an entire interval below $p_c$, not a smoothly continuing branch. "Fold" above is the unrelated A₂ catastrophe/transcritical normal form for a scalar equation, not a Whitney fold of a manifold map. The two share a word and nothing else; this page's correction does not touch that rejection.
Hutchcroft's (2020) and Schonmann's (2001) theorems establish the triangle condition on their respective graph classes; this page's identification rides on nothing beyond that condition giving mean-field exponents in the ABF $(p,h)$ family. The exponent correction changes which singularity that condition establishes, not whether it establishes one on these classes.
Where the question still stands open, honestly: unimodular nonamenable transitive graphs outside the planar case, and unimodular amenable transitive graphs outside the proven high-dimensional $\mathbb{Z}^d$ region. That residual class still contains graphs the 2026 sharpness proof covers — a proof about cluster-decay rate above $p_c$, silent on the fine structure of $\theta$ at $p_c$. A weighted-amenability extension of Hutchcroft's program narrows this further but states the general unimodular nonamenable case as open in its own right.
How hard is the $\mathbb{Z}^d$, $7\le d\le10$ piece of that residual class? Mean-field exponents are universally expected there on physical grounds; whether the current lace-expansion/NoBLE proof technique can reach it without new analytic ingredients is separate and unresolved. The obstruction is the triangle sum $\sum_{x,y}G(x)G(x-y)G(y)\sim1/(d-6)$ (three propagators — not the two-propagator bubble $\|G\|_2^2$, whose own threshold is $d=4$, the Ising/self-avoiding-walk value, misattributed to this problem in an earlier pass at scoping it). Fitzner–van der Hofstad's own account of their closely related lattice-trees/animals extension of the same method says further progress "would require substantially new ingredients and insights" — a statement about that adjacent model, not this one, offered here as the closest available evidence rather than as a quotation about percolation specifically. Full breakdown: WP-93, Book 6.
A note on method: both the literature chase (Hutchcroft, Schonmann) and the exponent check that caught today's error used AI-assisted web search and direct recomputation on the exactly-solved tree case. The mathematics is checkable and, in the tree case, exact; the assistance was in finding and verifying it — and this time, in catching where an earlier pass had it wrong.
Pablo Nogueira Grossi · G6 LLC · draft 2, 2 September 2026.