Owner's question: “When enough agents interact, is equilibrium a point — or a geometry?” This tree has answered it three times already, at three different evidence levels, without naming it.
A thesis over existing artifacts, not a new result and not a new field. Nothing here is a fresh claim; the three instantiations below are already built, already gated, and sit at three different rungs — one proved, one demonstrated, one measured. Saying so is the point. Unlike the five frontier prospectuses, this cannot be killed by an occupancy check, because it does not bet on any ground being empty.
A radii-polynomial or Krawczyk argument returns: a true solution exists within radius r of the computed one, and is unique there. Both halves are local. “Unique within r” is already a geometric hedge — it is silent about what lies at 2r, and deliberately so. The machinery was built to certify a point and it does that well.
When enough agents interact, the object stops being a point, and it stops in two distinguishable ways. If the coupling is monotone but not strictly, the solution set is a positive-dimensional face — every point on it is an equilibrium. If the coupling is not monotone, the set can have several components far apart. These are different geometries and they need different certificates.
Stated at their real rungs. They are not equal, and pretending they were would be the first dishonesty available here.
Two distinct solutions enclosed in disjoint balls at one parameter set. The page's own words: “a proof of non-uniqueness — a statement about the equations, not about a solver.” Disjointness is the whole argument: two enclosures that cannot overlap cannot describe one solution. This is a geometric claim carrying a certificate, in a regime where classical uniqueness theory makes no claim at all.
Totals unique across reseeds to 6.8e-14; the split moves by ~5.9. The cost c = j¹+j² is monotone but not strictly, so the equilibrium is a face and every point on it is an equilibrium. Demonstrated in float, not certified — and the reason it is not certified is the next section.
The enclosure closes for A ∈ [0.3, 5] and refuses at A = 6 (Z₁ = 1.10 ≥ 1, no contraction), with min m falling 0.9736 → 0.5889 across the window. That wall is an object in parameter space, and it was measured rather than argued.
S1 does not merely fail to certify. It fails in a named direction. The Krawczyk contraction breaks because the split direction lies in the null space of the Jacobian — and a null direction of the Jacobian at a solution is tangent to the solution set. So the failure is not an absence of information. It is the instrument reporting where the set extends.
That inverts what a refusal is for. Everywhere else in this tree a refusal is an honesty device: the method could not close, and we say so instead of loosening a tolerance. Here it is diagnostic: the geometry is exactly what the point-certificate cannot see, and the shape of its failure is a measurement of that geometry. A certificate that refuses in one direction and closes in the others has localised a face without ever being designed to.
If that holds up, it is the contribution: not a better certificate, but reading the existing certificate's failures as data.
Three decidable claims, in increasing difficulty. Each is a statement that could be attacked; none is asserted here.
For a degenerate instance: the solution set through the computed point has dimension exactly k, certified by enclosing the Jacobian's rank — a rank statement over intervals, which is decidable when the singular values separate.
The direction in which the contraction fails spans the tangent to the solution set, certified rather than observed. This is the claim that turns a refusal into an instrument, and it is the one worth attacking first.
There is a parameter value between the last closing and first refusing instance at which the contraction constant crosses 1, bracketed rather than sampled — the refusal frontier as a certified curve instead of a scatter of runs.
It is not new mathematics. Bifurcation theory, degenerate-fixed-point theory and the study of solution manifolds are old and deep; certified rank determination and validated continuation both exist. An occupancy check is owed before any of C1–C3 is called ours, and the honest expectation is that C1 is largely occupied by validated continuation and pseudo-arclength work. C2 is the one most likely to survive, because it is a statement about reading a specific certificate's failure mode rather than about the underlying mathematics.
It also does not claim the three instantiations are one result. They are one pattern, at three rungs, and the pattern is a reason to look — not evidence for anything.