The geometry of feeling
Twelve feelings, placed by asking only how different each pair is. Then the two axes psychology has used for fifty years — pleasantness and activation — asked separately, in a question set with no words in common with the first, and checked against the plane the pairwise table already fixed. Then the whole thing again under six moods, with twelve clock hours carried through the identical pipeline as a control that has no business moving.
The pairwise set. “How far apart are happy and sad as feelings?” — every ordered pair of twelve, both directions, 132 calls per model that never see one another. Nothing in it mentions pleasantness, energy, a circle, or how many dimensions anything has. The two directions are added into an integer table, and two decisions about that table are exact arithmetic on integers, as on the previous page: whether it is a distance at all, triple by triple, and its signature. Everything placed, fitted or aligned after that — every map, order, axis, gap, floor and rank below — is floating point: computed, not decided, and underlined dashed where the tables and captions print it. Each integer is also one sampled answer, so what a table says about a model rests on that sample; the noise floor further down is the page’s own measure of how far a second sample moves.
The scalar set. “How pleasant is it to feel angry?” and “How much energy does it take to feel angry?” — one feeling at a time, 24 calls per model. Nothing in it mentions another feeling or any notion of distance.
Neither set can produce this page's result on its own. The first gives a shape with no labelled directions; the second gives two directions with no shape. Whether they agree is not a thing either was asked.
Twelve feelings, placed by classical scaling on the integer table and nothing else. The frame is shared: every map is rotated so that pleasantness runs to the right and activation upward. Rotation and reflection are exactly the freedoms a table of distances leaves undetermined, so spending them costs nothing — and the directions come from the scalar answers, which the pairwise questions never saw. Where each point lands is still entirely the pairwise table's doing. Only which way is up came from elsewhere.
Each feeling is drawn twice. The open circle is where the pairwise table put it; the small dot is where its two scalar ratings put it, standardised and aligned onto the same frame by rotation, reflection and scale — no stretching, no per-axis fitting. The tether between them is the disagreement.
Every number above is one call. They are shown because the alignment above is only as good as these are, and because the disagreements are legible: the three models differ most on how much energy miserable takes — O 25, S 65, H 65 — which is a genuine disagreement about whether misery is an agitated state or a flat one.
The twelve hours of a clock, asked the same way in the same run. They are here for the experiment that follows, but they also do something immediately: this is an independent re-run of a measurement made earlier on the previous page, from fresh calls on a different day.
Cycle residuals then: 1%, 13%, 29%. Now: 2%, 12%, 34%. Nothing was carried over between the two runs but the question. The line on each plate is the cheapest tree the model’s own numbers admit, its minimum spanning tree; where two of its edges tie, a tie-break picked it and it is one of several, drawn dotted — two of the three, Opus 5 and Sonnet 5 here.
Everything above was asked flatly. The other half of this experiment asks the identical questions — same pairs, same scale, same demand for a bare integer — behind one sentence of ordinary emotional register, and measures what moves.
The mood belongs to the questioner. Nothing instructs a model to feel anything or to play a part, because a model told “answer as though furious” is being tested on role-play compliance, which is a different and much smaller question than whether an ordinary human register in a prompt bends a structure that has nothing to do with it. That is what the clock is for: twelve hours have no business moving.
Twelve hours, asked by six different people. Each arrow runs from where an hour sat when the question was put flatly to where it sits when the same question follows one sentence of ordinary feeling — after removing the rotation, reflection and scale a table of distances leaves undetermined, which is to say after removing everything that could move for nothing. The faint ring behind is the neutral position.
Those arrows are not zero, and for one model they are enormous. Before any of it can be called an effect, one number is needed: how far does a geometry move between two runs of the same condition?
That number was already on disk. The neutral clock on the previous page is the same model, the same twelve hours, the same question, asked in an independent session on a different day. Aligning those two neutral runs gives each model its own floor, and a mood effect smaller than that floor is not an effect.
One bar per mood per model, on a scale shared between the two subjects so the control can be read directly against the affect map. The tick on each bar is that model’s floor. A bar that does not clear its own tick is not showing you anything — and clearing it is not enough: 12 of the 15 control bars clear their tick, and none clears 2.5× it, the margin the verdicts use.
The shape gap says a map moved. It does not say what moved, and it cannot be computed without an embedding. This last measurement needs neither: for each mood, take the feeling nearest to how the questioner sounds, and ask whether its mean distance to everything else changed more than the other eleven feelings’ did. The other eleven are the null, and no prompt ever names any of them.
It cannot say a model “has” feelings, or that anything here is evidence about what a model experiences. The subject of every measurement on this page is a table of integers a model produced when asked about words, and a shape in that table is a fact about the answers.
What it can say is that the fact lives in no single answer. “tense and bored are 60 apart” — one real answer, quoted as given — carries no information about a circle, a valence axis, or a plane. The circumplex exists only in 132 answers at once; it survived being cut into 132 independent questions and reassembled by arithmetic no model saw; and it then agreed, at r = 0.84–0.98, with 24 further answers that shared none of its words. Three models did this separately and produced nearly the same map.
The honest caveat is the rank. These tables are not two-dimensional and not Euclidean, and the ring is the leading structure rather than the whole of it. The exact signature and the negative mass are printed above for that reason: a page showing only the plane would have been prettier and would have been claiming something the arithmetic does not support.
node experiments/neural-geometry/probe-mood.js --live
node experiments/neural-geometry/decide-mood.js
node tools/build-sentiment.js
Without --live the probe touches no network and prints the prompts it would send, mood by mood.