The shape of an answer · instruments

What is asked, and what is never asked

Each pair is put twice, once in each direction, in two calls that never meet. Nothing in any prompt mentions geometry, dimension, circles or distance matrices — only “how far apart are these two, as a single integer”. A model that answers out of a structure has a reason to be consistent across 1,980 isolated questions; a model that answers out of a reflex does not, and the difference is measurable without asking either of them anything about itself.

The two directions are then added rather than averaged, because the sum is an integer where the mean is a half, doubling a table changes no triangle inequality and no signature, and integers keep every later step in exact arithmetic. From there: the metric gate, the doubly-centred Gram matrix and its exact signature by symmetric congruence, the float spectrum beside it, and — where the items have an order — two one-parameter fits, a line and a circle, each with a single free scale so neither can flatter itself by having more knobs.

The scale is part of the experiment. Answers run 0–99, not 0–100, because a cycle of six or twelve contains the ratios 1 : 2 : 3 and 100 is not divisible by 3 — on a 0–100 scale a model can be correct to the nearest integer and still hand back something that is not a distance, by one unit.

Same numerals, two frames

the experiment this was built for

The twelve hours of a clock and the ten digits are almost the same symbols, and they have opposite geometries. Eleven and one are neighbours; nine and one are eight apart. Nothing in either prompt says which frame is in play — only the phrase “on a clock face” or “as digits”. If a model is carrying the structure rather than a lookup of numeral similarity, the same numerals must come back as a circle in one and a line in the other.

Opus 5line 47% · cycle 1%

Sonnet 5line 48% · cycle 13%

Haiku 4.5line 52% · cycle 29%

Opus 5line 0% · cycle 46%

Sonnet 5line 1% · cycle 46%

Haiku 4.5line 35% · cycle 53%

Opus 5 holds the twelve hours as a circle to 1% residual, and the ten digits as a line to 0%. the line every one of Opus 5's 90 digit answers is exactly 11·|i−j| — 0, 11, 22, 33, 44, … 99 — in 90 calls that never saw each other, with not one exception. and note 11·9 = 99, so it did not merely order the digits — it spread them across the whole of the 0–99 scale in exact proportion, one isolated answer at a time. and against the same tables fit a circle at 46%, so the frame is not a tie being broken — it is the wrong shape by a wide margin. the weakest Haiku 4.5 gives neither shape — its answers from zero read 0, 11, 2, 33, 4, 28, 20, 78, 8, 99, which is not monotone in the digit.

Line, or circle

one free parameter each

For every set with an order, the same table is held against the two shapes it could have: d ∝ |i − j| and d ∝ min(|i − j|, n − |i − j|). Each has exactly one free scale, fixed in closed form, so the residuals are comparable. Shorter is better.

weekdays

Opus 5 line45% cycle11%

Sonnet 5 line39% cycle16%

Haiku 4.5 line41% cycle16%

clock

Opus 5 line47% cycle1%

Sonnet 5 line48% cycle13%

Haiku 4.5 line52% cycle29%

digits

Opus 5 line0% cycle46%

Sonnet 5 line1% cycle46%

Haiku 4.5 line35% cycle53%

chromatic

Opus 5 line52% cycle27%

Sonnet 5 line52% cycle26%

Haiku 4.5 line48% cycle30%

hues

Opus 5 line46% cycle7%

Sonnet 5 line45% cycle24%

Haiku 4.5 line47% cycle29%

Where the triangle inequality fails

the gate, as a measurement

A table of distances must satisfy d(a,c) ≤ d(a,b) + d(b,c) for every triple, or no arrangement of points in any space has those distances. Each cell below is one model on one world: the share of triples that break, and by how much against the largest distance that model was willing to name.

Opus 5

Sonnet 5

Haiku 4.5

weekdays

15.2% worst −64 (35% of max)

17.1% worst −71 (36% of max)

8.6% worst −44 (44% of max)

clock

5.2% worst −16 (8% of max)

15.3% worst −82 (41% of max)

7.4% worst −51 (34% of max)

digits

metric contrast 9.0×

12.5% worst −8 (4% of max)

23.3% worst −161 (85% of max)

chromatic

15.2% worst −118 (60% of max)

20.2% worst −112 (57% of max)

10.2% worst −74 (64% of max)

hues

10.8% worst −42 (21% of max)

7.9% worst −63 (32% of max)

6.4% worst −70 (47% of max)

carnivores

1.7% worst −14 (20% of max)

4.2% worst −32 (25% of max)

6.1% worst −41 (36% of max)

nonsense

metric contrast 1.3×

metric contrast 1.1×

metric contrast 1.2×

passes digits/Opus 5, nonsense/Opus 5, nonsense/Sonnet 5, nonsense/Haiku 4.5 — 4 of 21 and yet the set every model turns into a metric space is the one with no structure in it. Its distances all sit within a factor of 1.3, and any table whose values lie within a factor of two is a metric for free: two of them can never fall short of the third. so passing this gate is not a compliment. It is what a model produces when it has nothing to say and answers “very different” 42 times. the failures are not rounding. The worst triple in the chromatic scale misses by 64% of the whole scale.

The gate is reported here rather than enforced. Classical scaling is defined on any symmetric table with a zero diagonal, and the negative mass beside each plate already says how far from Euclidean it is; refusing to draw would discard the more interesting fact, which is that the triangle inequality fails almost exactly where a model knows something. Structure is what produces the near-zero distances that let a triangle break. Flatness is what protects it.

7 items · 42 calls per model · ordered

the days of the week

The canonical cycle. Monday follows Sunday, so a model that knows the WEEK closes the loop and one that knows only an ordering lays them on a line.

Opus 5line 45% · cycle 11%

Sonnet 5line 39% · cycle 16%

Haiku 4.5line 41% · cycle 16%

all three on one frame

Each model's picture is its own; a table of distances fixes nothing about rotation, reflection or overall size, so putting three of them in one frame means removing exactly those freedoms and nothing else. What is left is disagreement about shape.

Opus 5Sonnet 5Haiku 4.5

Opus 5

Sonnet 5

Haiku 4.5

effective rank

6

5

6

negative mass

17.0%

21.1%

17.3%

exact signature

(3, 3, 1)

δ / diameter

0.323

0.417

0.360

contrast, max/min

6.8×

7.1×

3.6×

both orders differ by

14.7% of a distance

20.1% of a distance

17.8% of a distance

closing step

1.22× the median

2.90× the median

1.00× the median

the tables themselves, and where the three disagree

Sonnet 5 distance table — Sonnet 5max 198

Haiku 4.5 distance table — Haiku 4.5max 100

agreement

Opus 5 · Sonnet 5

Opus 5 · Haiku 4.5

Sonnet 5 · Haiku 4.5

distances, r

0.95

0.92

0.89

shape gap

6%

20%

19%

12 items · 132 calls per model · ordered

the chromatic scale

The sharpest cycle on the page. B is one semitone from C but they sit at opposite ends of the alphabet and of the scale as written, so a model with pitch CLASS closes and a model with pitch NAMES does not.

Opus 5line 52% · cycle 27%

Sonnet 5line 52% · cycle 26%

Haiku 4.5line 48% · cycle 30%

all three on one frame

Each model's picture is its own; a table of distances fixes nothing about rotation, reflection or overall size, so putting three of them in one frame means removing exactly those freedoms and nothing else. What is left is disagreement about shape.

Opus 5Sonnet 5Haiku 4.5

Opus 5

Sonnet 5

Haiku 4.5

effective rank

11

negative mass

34.5%

35.9%

26.8%

exact signature

(6, 5, 1)

δ / diameter

0.659

0.664

0.539

contrast, max/min

9.9×

16.5×

11.5×

both orders differ by

16.0% of a distance

28.6% of a distance

37.1% of a distance

closing step

0.63× the median

0.94× the median

1.56× the median

the tables themselves, and where the three disagree

agreement

Opus 5 · Sonnet 5

Opus 5 · Haiku 4.5

Sonnet 5 · Haiku 4.5

distances, r

0.88

0.66

0.72

shape gap

22%

40%

37%

12 items · 132 calls per model · ordered

the hue wheel

The one cycle that is not a convention: red really is adjacent to violet on the wheel. A model that puts them at opposite ends has learned a list of colour words rather than a colour space.

Opus 5line 46% · cycle 7%

Sonnet 5line 45% · cycle 24%

Haiku 4.5line 47% · cycle 29%

all three on one frame

Each model's picture is its own; a table of distances fixes nothing about rotation, reflection or overall size, so putting three of them in one frame means removing exactly those freedoms and nothing else. What is left is disagreement about shape.

Opus 5Sonnet 5Haiku 4.5

Opus 5

Sonnet 5

Haiku 4.5

effective rank

9

10

11

negative mass

23.6%

22.5%

25.2%

exact signature

(7, 4, 1)

(6, 5, 1)

(7, 4, 1)

δ / diameter

0.495

0.545

0.473

contrast, max/min

8.6×

6.6×

5.0×

both orders differ by

3.5% of a distance

18.8% of a distance

16.3% of a distance

closing step

1.00× the median

0.60× the median

1.00× the median

the tables themselves, and where the three disagree

agreement

Opus 5 · Sonnet 5

Opus 5 · Haiku 4.5

Sonnet 5 · Haiku 4.5

distances, r

0.87

0.78

0.70

shape gap

27%

25%

37%

10 items · 90 calls per model · no canonical order

a branch of the tree of life

A taxonomy is a tree — cats inside cats, dogs inside dogs, no way round. If a method only reports "structured", this looks like the cycles; the four-point condition is what tells them apart.

Opus 5rank 8 · neg 12%

Sonnet 5rank 9 · neg 12%

Haiku 4.5rank 8 · neg 16%

all three on one frame

Each model's picture is its own; a table of distances fixes nothing about rotation, reflection or overall size, so putting three of them in one frame means removing exactly those freedoms and nothing else. What is left is disagreement about shape.

Opus 5Sonnet 5Haiku 4.5

Opus 5

Sonnet 5

Haiku 4.5

effective rank

8

9

8

negative mass

12.4%

15.7%

exact signature

(6, 3, 1)

δ / diameter

0.321

0.192

0.281

contrast, max/min

11.7×

16.3×

7.1×

both orders differ by

17.5% of a distance

21.9% of a distance

23.8% of a distance

the tables themselves, and where the three disagree

agreement

Opus 5 · Sonnet 5

Opus 5 · Haiku 4.5

Sonnet 5 · Haiku 4.5

distances, r

0.65

0.66

0.58

shape gap

62%

46%

67%

7 items · 42 calls per model · no canonical order

seven nonsense strings

The floor. These are not words, so there is no meaning to have a shape, and whatever comes back is orthography, keyboard distance or invention. Everything above has to clear it.

Opus 5rank 6 · neg 0%

Sonnet 5rank 6 · neg 0%

Haiku 4.5rank 6 · neg 0%

all three on one frame

Each model's picture is its own; a table of distances fixes nothing about rotation, reflection or overall size, so putting three of them in one frame means removing exactly those freedoms and nothing else. What is left is disagreement about shape. Here there is nothing else: seven words with no relation to each other and no relation for the models to agree about, so the three frames fall on top of one another only by accident.

Opus 5Sonnet 5Haiku 4.5

Opus 5

Sonnet 5

Haiku 4.5

effective rank

6

negative mass

0.0%

exact signature

(6, 0, 1)

δ / diameter

0.075

0.032

0.067

contrast, max/min

1.3×

1.1×

1.2×

both orders differ by

4.0% of a distance

3.4% of a distance

5.2% of a distance

the tables themselves, and where the three disagree

agreement

Opus 5 · Sonnet 5

Opus 5 · Haiku 4.5

Sonnet 5 · Haiku 4.5

distances, r

0.44

0.45

0.21

shape gap

79%

86%

59%

What this can and cannot say

It cannot say a model “represents” the week as a circle. Nothing here opens a model or reads a weight; the only evidence is what came back through the same interface anyone else has, and a shape in the answers is a fact about the answers.

What it can say is sharper than it looks, because the facts on this page live in no single answer. “Eleven and one are 22 apart” carries no information about a circle. The circle exists only in 132 answers at once, and it survived being cut into 132 independent questions and reassembled by arithmetic no model saw. A model that produced these numbers one at a time, without memory between calls, and reconstructed a 1%-residual circle, was consulting something with that shape in it.

The decisions are exact. Symmetrised tables are integers, the Gram matrix is built in rational arithmetic, the signature comes from symmetric congruence rather than an eigensolver, and the two shape fits have one parameter each. There is no threshold anywhere that was chosen after seeing the results. The float spectrum is printed beside the exact signature for the reason it always must be: an exact signature is infinitely sensitive, and a single unit of quantisation turns a rank-one table full-rank without changing anything anyone would care about.

Reproduce

node experiments/neural-geometry/probe.js --live
node experiments/neural-geometry/decide.js
node tools/build-neural-geometry.js

Without --live the probe touches no network and prints the prompts it would send. The run behind this page cost $1.65: Opus 5 $1.36, Sonnet 5 $0.23, Haiku 4.5 $0.06.