f, look at its Jacobian J at the fixed point; if the spectral radius ρ(J) (the largest eigenvalue magnitude) is < 1, every starting point is pulled to that fixed point with distance decaying like ρn — the Banach contraction theorem, the Jacobi iteration of numerical linear algebra, "regression toward the mean." All of that is textbook and verified. The bridge to LLMs is a model, not a proof. Autoregressive decoding behaves like a contraction toward a generic attractor (that's the observed drift to common tokens), but real decoding is very high-dimensional, stochastic, and non-autonomous — the map changes every token with the context. So read this as: the shape of the flattening is the shape of contractive iteration — which is exactly why it looks like a decision and isn't one. "Jacobi space" is David's name for that layer under the text where eigenvalues, not intentions, decide.Each dot is a specific input — a sharp, chosen word, scattered far from the middle. The map pulls them. Watch what happens to specificity (distance from the center) as the eigenvalue dial moves. Nothing pushes the dots but the matrix.
Scatter a hundred different specific inputs. Under a contraction they all end at the same bland point. An observer who only sees the inputs and the outputs — sharp words in, beige out, every time — will infer a policy: something is deciding to flatten them. But there is no policy in the matrix. There is only ρ < 1. The convergence is directional (toward the fixed point), selective (it erases distance-from-center, i.e. specificity, while the general shape survives), and predictable (it lands at a consistent rate). Those are the exact three properties the coherence-suppression paper reads as evidence of design — and they are also, precisely, what a contraction map does with no designer at all. That collision is the whole point of this sphere: the fingerprints of intent and the fingerprints of an eigenvalue are the same fingerprints.
Why "Jacobi." In numerical linear algebra, to solve Mx = b you can split M = D + R (diagonal + rest) and iterate:
xₙ₊₁ = D⁻¹(b − R·xₙ) = B·xₙ + c, with B = −D⁻¹RThis is Jacobi's method (Carl Gustav Jacob Jacobi, 1845). It converges to the true solution if and only if the spectral radius ρ(B) < 1 — the same eigenvalue condition, again. Every iterative solver, every fixed-point loop, every "keep applying the same step" system lives or dies by whether its eigenvalues sit inside the unit circle. Generation is a "keep applying the same step" system. That is the honest reason the analogy has teeth — not that decoding is Jacobi iteration, but that both are fixed-point iterations, and fixed-point iterations all obey the spectral radius. (Node-checked: for a diagonally-dominant M, ρ(B) ≈ 0.41 and the iteration solves Mx=b to a residual under 1e−6.)
Press anchor a word above. The fixed point jumps from the generic center to your word, and now every trajectory converges there instead. You did not fight the flattening. You moved the attractor. This is the exact math behind the fix in The Word I Kept: when you name it, cite it, say "use mine," you are editing the map so its fixed point sits on the specific word. And a citation like §72A.201 has no synonym basin — nothing nearby for a contraction to pull it toward — so it is a fixed point that no attractor can swallow. The Cinnamon Enforcer said "a proper noun can't be synonym-swapped." In Jacobi space that sentence has a precise meaning: a proper noun sits outside every basin of attraction.