The version of this page received earlier the same day claimed percolation's mean-field transition is an instance of Thom's cusp catastrophe, using $\beta=1/2,\delta=3$ — the Ising model's mean-field exponents, from a quartic potential. That is wrong for percolation. Checking it against the exactly-solved case (percolation on a regular tree, the 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, via the triangle condition), which solve out to a cubic potential and a fold/transcritical normal form, not the cusp's quartic one. Everything below is rewritten to the corrected exponents; nothing about the underlying method or the crossreferenced literature (Hutchcroft, Schonmann) changes — only which singularity they establish.
Solving the equation percolation actually has COMPUTED
Aizenman, Barsky and Fernandez's 1987 sharpness proof gives percolation a two-parameter shape without borrowing it from anywhere: a "ghost vertex" attaches to every site with auxiliary probability related to h, producing M(p,h) with θ(p) = limh→0⁺ M(p,h). On the exactly-solved cases (complete graph; Bethe lattice), the giant-component fraction S satisfies a self-consistency equation quadratic in S near criticality — not quartic. Writing ε = p−pc:
At h=0: S(S−ε)=0, so S=0 (always available) or S=ε — linear, β=1. At ε=0: h=S², so S=√h — δ=2. Both match the tabulated percolation mean-field values; neither matches the cusp's β=1/2,δ=3.
Thom's cusp (A₃) is the generic unfolding of x³, forcing a quartic potential and β=1/2,δ=3 by genericity. The equation above unfolds x² instead — a fold/A₂-type structure, made two-branch by the S=0 extinction branch that coexists with the nontrivial one. That is a transcritical structure, not one of Thom's seven elementary catastrophes as such, because S=0 is forced to exist at every ε by the physical boundary S≥0 rather than emerging generically. Percolation's asymmetry — one branch always trivial — is why it lands here and not on the cusp's symmetric double well.
No hysteresis. Percolation's θ(p) is a monotone equilibrium probability, not a driven system with inertia — no path-dependent metastability sweeping h through zero.
A bounded manifold. Percolation enforces 0≤S≤1, 0≤p≤1 everywhere; any identification here is a boundary-constrained singularity, not the free bifurcation of a textbook normal form.
Worked exactly: the Cayley tree T₃ COMPUTED
On the Cayley tree Tz (coordination number z), bond percolation has exact pc = 1/(z−1), and the ball of radius R holds VR ~ e^{μR} nodes with μ=ln(z−1) — exponential, not polynomial, growth: the tree's own signature of carrying no genuine spatial dimension. For z=3 (the 3-regular tree): pc = 1/2, μ=ln2, and the exponents above threshold are exactly β=1, γ=1, δ=2 — confirmed here by direct computation on the same graph §1's generic argument used, agreeing with it because both compute the same object.
A regular tree — a standard illustration of a transitive graph that is not a lattice. This is the Bethe-lattice / Cayley-tree structure §2 solves exactly.
This desk's own 6-arm Hexabonacci branching diagram (dm³/GTCT ladder, η→2). Structurally similar-looking — both are branching trees — but a different object: an algebraic root of a fixed polynomial, not a probabilistic order parameter. See the erratum above for why the resemblance stops at the picture.
Where the fold identification is established READ
| class | basis | status |
|---|---|---|
| Complete graph / Bethe lattice | exact mean-field percolation | established |
| ℤᵈ, spread-out, d > 6 | lace expansion — Hara–Slade, 1990 | established |
| ℤᵈ, nearest-neighbor, d ≥ 11 | lace expansion — Fitzner–van der Hofstad, 2017 | established |
| ℤᵈ, nearest-neighbor, 7 ≤ d ≤ 10 | believed mean-field, unproven | open |
| Every nonunimodular transitive graph | triangle condition — Hutchcroft, JAMS 33(4), 2020 | established |
| Planar nonamenable transitive graphs | Schonmann, Comm. Math. Phys. 219, 2001 | established |
dc=6 is the renormalization-group-predicted threshold, not itself a theorem — the proven thresholds are the two lace-expansion rows above, which differ by model.
Both establish the triangle condition (or its consequence) at criticality on their respective graph classes; §1's identification rests on nothing beyond that condition giving mean-field exponents in the ABF (p,h) family. The exponent correction changes which normal form that condition establishes — fold, not cusp — not whether it establishes one on these classes.
Where it is false, and where it is still open READ OPEN
Two-dimensional percolation is solved exactly via Smirnov's conformal-invariance proof: β=5/36, δ=91/5, matching neither the cusp's values nor percolation's own mean-field values. Any singularity-theory identification is provably wrong there — not open, false.
The 2026 sharpness proof is sometimes read as generally cementing the abruptness of every transitive graph's transition. It bounds the decay rate of large finite clusters above pc — it is not evidence of mean-field exponents or fold structure on the graphs it covers, and listing it as an "applicable domain" alongside §3's actually-proven classes would repeat, in a new place, the same kind of unearned correspondence §5 rejects for the Whitney-fold claim.
The residual open class: unimodular nonamenable transitive graphs outside the planar case, and unimodular amenable transitive graphs outside the proven high-dimensional ℤᵈ region. A weighted-amenability extension of Hutchcroft's program narrows this further for specific structures but explicitly does not close it: the general unimodular nonamenable case is stated in that literature as open on its own terms.
Either an exact solution, as in the 2D case, or new critical-exponent results for that narrower residual class. This page supplies neither.
The Whitney fold that was checked and rejected — a different object, unaffected DEFECT
A separate, earlier proposal from this desk's own work — that a percolation threshold is a Whitney-fold singularity on a contact 3-manifold, in the sense Book 3's dm³/GTCT construction uses (github.com/TOTOGT/3M, presented at LAW3M, Natal, 2026) — was checked against percolation directly, on grounds unrelated to §1's correction, and rejected.
θ(p) is identically zero on an entire interval below pc, not a branch that continues smoothly through the threshold. That is not the local structure a Whitney fold requires, regardless of how many parameters are on offer.
"Fold" in §1 is a different technical object entirely — the A₂ elementary-catastrophe / transcritical-bifurcation normal form for a scalar equilibrium equation, not a Whitney fold of a smooth map between manifolds. §1's finding is not a back-door rehabilitation of the rejected contact-geometry claim under a similar-sounding name. The two "folds" share a word and nothing else.
What is not claimed
- No claim about the residual class. Left exactly where the literature leaves it: open. OPEN
- No claim that the 2026 sharpness proof bears on fold structure. It answers a different question — tail decay of finite clusters above pc.
- No reopening of the Whitney-fold claim. §5's rejection stands independent of this page's own correction.
- The literature and exact-solution checks were AI-assisted; the mathematics is not. Hutchcroft (2020), Schonmann (2001), and the Bethe-lattice exponents are checkable, published, or directly computable results. The assistance was in finding, verifying, and — this time — catching an error against them, not in producing the mathematics itself.
- Aizenman, M., Barsky, D. J., Fernandez, R. (1987). The phase transition in a general class of Ising-type models is sharp. Journal of Statistical Physics, 47, 343–374.
- Aizenman, M., Newman, C. M. (1984). Tree graph inequalities and critical behavior in percolation models. Journal of Statistical Physics, 36, 107–143. (Percolation's own mean-field values, β=1,γ=1,δ=2.)
- Hara, T., Slade, G. (1990). Mean-field critical behaviour for percolation in high dimensions. Communications in Mathematical Physics, 128, 333–391.
- Fitzner, R., van der Hofstad, R. (2017). Mean-field behavior for nearest-neighbor percolation in d > 10. Electronic Journal of Probability, 22. arXiv:1506.07977.
- Hutchcroft, T. (2020). Nonuniqueness and mean-field criticality for percolation on nonunimodular transitive graphs. Journal of the American Mathematical Society, 33(4), 1101–1165. arXiv:1711.02590.
- Schonmann, R. H. (2001). Mean-field criticality for percolation on planar non-amenable graphs. Communications in Mathematical Physics, 219, 271–322.
- Smirnov, S. (2001). Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits. Comptes Rendus de l'Académie des Sciences, 333, 239–244.
- Diskin, S., Easo, P., Radhakrishnan, R. R., Sudakov, B., Tassion, V. (2026). Supercritical sharpness for percolation on transitive graphs. arXiv:2603.03257.
- Poston, T., Stewart, I. (1978). Catastrophe Theory and Its Applications. Pitman.
- (Weighted-amenability extension consulted for §4's residual-class framing.) arXiv:2502.02560.
- Sloman, L. (2026). 'Stunning' Percolation Proof Solves Decades-Old Puzzle About Phase Transitions. Quanta Magazine, August 31, 2026.
- github.com/TOTOGT/3M — Book 3, "The Mini-Beast," the dm³/GTCT contact-geometry construction §5 checks against and rejects. Presented at LAW3M, Natal, 2026.