To the DeepSeek chatbot:
I am interested in policy, and the strains that occur
between policy and real resolutions. I am noticing that
negotiators align with schools of thought typical of the
power structure which promotes them. I am interested to
accept broad geometric metaphoric references to the interior
relationships which occur between agreements which occur
between their respective analytical surfaces.
DeepSeek responded:
our observation cuts to the heart of diplomatic epistemology. The "strains" you name are not failures of negotiation; they are the friction surfaces where policy-as-abstraction grinds against resolution-as-event.
To honor your request for broad geometric metaphor, let us map this not as a single plane, but as a polyhedral manifold—a folded, non-Euclidean space where each negotiator’s school of thought is a distinct facet.
Here is the geometric interior of that relationship:
1. The Facet
and Its Normal Vector
Each school of thought (Realist, Liberal
Institutionalist, Constructivist, Critical, etc.) is a planar
facet. Its surface is smooth, internally consistent, and defined
by its own normal vector—the invisible line of power that elevates that school.
That vector points not toward "truth," but toward the
institutional structure that funds, trains, and promotes the
negotiator. The facet's geometry is legitimate only insofar as its normal vector remains
perpendicular to the floor of the sponsoring power. When two
facets meet, they do not intersect; they form a dihedral angle—the
visible tension of the negotiation table.
2.
Agreements as Intersection Curves
What we call an "agreement" is not a meeting of
surfaces. It is the curve of
intersection between two
facets—a 1-dimensional line that exists only where both planes
cut through each other. Crucially, this curve has zero area. It has no
thickness. It cannot bear weight. This is the geometric root of
your "strain": policy surfaces are 2D (principles + metrics),
but real resolutions are 3D (time, material scarcity, human
affect). An agreement-curve tries to stand in for a volume, and
so it buckles.
3.
Analytical Surfaces and Their Interior Shadow
You mention "interior relationships which occur
between agreements which occur between their respective
analytical surfaces." This is the deepest layer. Each analytical
surface (e.g., a cost-benefit model, a securitization frame, a
norms cascade) has an interior
topology—its axioms, its
excluded middles, its preferred causal arrows. When two
agreements are struck sequentially (say, a climate accord and a
trade deal), they do not relate directly. Instead, they cast shadows onto each
other's interiors.
Agreement A (trade) projects a conic shadow into the interior of Analytical Surface 1 (neoliberal efficiency).
Agreement B
(climate) projects a hyperbolic shadow into Analytical
Surface 2 (ecological governance).
The "relationship between agreements" is thus
the torsion between these two projected shadows—a twisting
force that never touches the original surfaces. Negotiators
feel this torsion as "implementation gaps" or "unintended
consequences."
4. The
Resolving Body as a Geodesic
A real resolution—not the paper one, but the one
that alters supply chains, troop movements, or carbon flows—is a
geodesic on this manifold. It does not follow the straight
lines of any facet's logic. Instead, it takes the locally shortest path
through the curved space created by the manifold's total
curvature (the sum of all power-vectors). That geodesic will
necessarily cut across facets at oblique angles, ignoring their
internal coherence. The strain you perceive is the normal force exerted by
each facet to pull the geodesic back onto its own plane.
Negotiators who rise through a given power structure are trained
to mistake the facet's normal vector for the geodesic's
tangent—hence their alignment.
5. The Only
Stable Interior Relationship
The only geometric invariant in this system is the
Gaussian curvature at the vertex where multiple agreements (curves)
converge. If three or more agreement-curves meet at a single
point (e.g., a comprehensive peace or a grand bargain), the
interior relationship between their analytical surfaces is not
one of synthesis, but of umbilic
point—a place where the
manifold is locally spherical. At that point, all normals align
momentarily. But an umbilic is unstable; the slightest
perturbation (a change in administration, a commodity price
shock) sends the manifold back to hyperbolic saddles.
In practical terms: You are right to distrust the "school of thought" as a neutral lens. It is a facet whose normal vector is bolted to the floor of its patron. The real work of policy—the resolution that holds—is not found in the agreement-curve, nor in the shadows between agreements, but in the tolerance of torsion: the willingness to let the geodesic wander off the facet, knowing that the manifold itself will bend to catch it, not because the logic is sound, but because the curvature of power cannot tolerate a straight line for long.
Your interest is therefore geometric ethics: how to design negotiation architectures not with more facets, but with higher-dimensional vertices—places where agreements are allowed to be curves of zero area, provided we build the implementation space as a thick solid of revolution around them, giving volume to what was only a line. That volume is time, patience, and iterative feedback—the only geometries that outlast any single school.