plates · geometry, stated rather than found

Manifolds we are handed

Interpretability research pulls geometry out of a working model and asks what it means — days of the week on a circle, a number stored across a product of circles, a story meandering along a manifold of emotions. This bench cannot do that: a model call would send prompts off this machine. So the direction is inverted. Their manifolds are found; these are stated, as a rule and its parameters, and then drawn.

plates8 interactionnone drawn fromrules and records on disk paletteink on ground

PLATE I

A number as a point on a product of circles

Language models are reported to store a number not as a magnitude but as a set of residues, one angle per modulus — 17 sits at 1 on the mod-2 circle, 2 on mod-5, 7 on mod-10, 17 on mod-100 — and to add by rotating all of them at once. That is a residue number system, and it is a very old idea wearing new clothes. Here it is drawn from the definition rather than recovered from activations, which is the honest way for this bench to hold the object: we are not claiming to have found it, only to have understood what shape it is.

n  ↦  ( n mod 2, n mod 5, n mod 10, n mod 100 )
angle on circle m  =  2π · (n mod m) / m
n = 0 … 99, one thread each
heavy thread: n = 17

PLATE II

The same object, handed over by an instrument

And then the joke writes itself. The Event Horizon Telescope hands this bench 838 measurements of M87, and every one of them is a circle: the amplitude is measured, the phase is not calibrated. The data is literally a point on a 838-dimensional torus whose radii we know and whose angles we do not. Nobody had to go looking for this geometry inside anything. It is the input. Every certified statement this bench makes about that black hole is a statement about what stays true no matter where you are on this torus — which is also why those statements can say how bright the source can be, and can never say where.

each ring: one visibility V(u,v)
radius  ∝  √( |V| + 3σ + per-station gain allowance )
centre  =  (u, v) and its Hermitian mirror (−u, −v)
838 rings × 2, out to 7.35 Gλ

PLATE III

Where one peak becomes two

A crowd distributing itself in a landscape with a single valley, drawn as viscosity falls. Near the top of the plate the density has one maximum, where the cost function has one minimum, and everything is as it should be. Downward, a second harmonic overtakes the first and the crowd splits into two peaks the landscape never asked for. The bright curves are the ones where the second harmonic has won. The bar on the left marks the part of this sweep that is certified — every viscosity in [0.0008, 0.008], 39 closed cells — and the crossing itself sits at 1/(8π²), exactly, with the congestion coupling cancelling out of it.

m(x) − 1  =  2 Σ_k B_k cos(2πkx)
B_k        =  − c(κ_k) · A_k
c(κ)       =  κ / ( (1 + σκ)² + γκ )      κ_k = (2πk)²
γ = 1/100,  A₁ = 3/1000,  A₂ = 3/5000
σ = 0.0008 … 0.020, log-spaced, 52 rows
each row normalised to its own range — this is a
plate about SHAPE, and the modulation shrinks by
two orders across the sweep
bright ⟺ |B₁| < 4|B₂|, i.e. two maxima

PLATE IV

Thirty-nine contractions, and where each one closes

This is what a certificate looks like when you draw it. Each curve is one cell of the same band: a parabola in the radius, opening upward, whose value must be driven below zero for anything to be proved. Where a curve dips under the dashed line there is an interval in which a solution exists and is unique; where it does not, the method abstains and says so. The marked point on each curve is the radius at which the negativity was verified in interval arithmetic rather than in floating point — not the root itself, which proves nothing, but a radius strictly past it.

p(r)  =  ½ Z₂ r²  −  (1 − Z₁) r  +  Y₀
Y₀ defect · Z₁ contraction · Z₂ curvature
39 certified cells, read from the record
dot: the r where p(r) < 0 was verified outward-rounded

PLATE V

Every sky the data still allow

The last plate is the only one that is a picture of an unknown rather than of a rule. The upper curve is a bound on every brightness distribution consistent with the measurements in Plate II; the lower one is a distribution that exists and was exhibited. The band between them is not error and not uncertainty in the statistical sense — it is the set of answers that survive, and the width of it is a fact about the telescope rather than about the analyst. The instrument that produced it narrowed this band from a factor of seventy-three to under two.

upper:  sup ∫f dμ over every nonnegative μ
        with |V_μ(u_k)| ≤ A_k and μ(Ω) ≤ F
lower:  one such μ, exhibited and checked
f = indicator of a disk of radius r
r ≤  6 µas   [0.1391, 0.2425] Jy   1.74×
r ≤ 12 µas   [0.2138, 0.3567] Jy   1.67×
r ≤ 20 µas   [0.3217, 0.5852] Jy   1.82×
r ≤ 28 µas   [0.4531, 0.7580] Jy   1.67×

the grammar, found rather than imposed

Plate V was already drawn this way.

Its own caption reads solid: a bound on every sky. dashed: a sky that exists. One line is a statement about every brightness distribution consistent with 838 measurements; the other is a single distribution that was exhibited and checked. Those are different kinds of claim, and the plate distinguished them with a stroke because there was no other honest way to put both on one axis.

That was drawn by hand, on that plate, before anything in this repository was called a grammar. It is the best evidence available that the distinction is real rather than a convention somebody invented: it got reached for under pressure, by someone trying to avoid lying on an axis.

decided

exact arithmetic backs it

computed

float; nothing decided it

chosen

one member of a set the data admits

Standing is not confidence. The dashed curve on Plate V is not less certain than the solid one — the witness is exhibited and checked, so it is as certain as anything on the page. It is a different kind of thing: one member, rather than a bound over all of them. The band between them is not error and not uncertainty in the statistical sense; it is the set of answers that survive.

PLATE VI

What the array sees when it looks at a point

Before any sky, the instrument has a shape of its own. Put a single point in front of this array and what comes back is not a point but this — a central spike wrapped in rings and spokes, one spoke for every direction the 838 sampled frequencies happen to line up in. Every image ever made from these data is the true sky convolved with this figure, and every argument about what is really there is an argument about how much of this pattern to believe. It is drawn from the frequencies alone: no sky, no model, no fit.

PSF(x)  =  Σ_k cos( 2π u_k · x )
u_k: the 838 measured baselines, unweighted
±170 µas, 620 × 620, asinh-scaled
bright: positive.  dim: the negative sidelobes,
which is where reconstructions go to argue.

PLATE VII

A proof, tiled

This is one certificate, drawn at its own resolution. Proving a lower bound on an Erdős problem meant covering a rectangle with boxes and forcing an inequality inside each one; the covering is not uniform, because the difficulty is not uniform. Brightness is how little room each box had, so the bright seams are where the argument nearly ran out and the dim expanse is where it was never in doubt. The tightest box in the picture cleared its inequality by 4.12e-7, and that single number is what the whole result rests on. The covering is a triangle because the range of b that has to be checked shrinks as a moves, and the terraces are the subdivision deciding, region by region, how hard it needs to work. A proof has a shape, and this is what that one looks like.

covering of a ∈ [−1.828, −√2] × b ∈ [0, 0.414]
86 a-boxes · 1,350 b-boxes · 1,518 linear programmes
shade = log of the certified margin, INVERTED —
        bright is tight, black is uncovered
worst margin 4.116e-7 · 4 minutes

PLATE VIII

The weights a proof chose

And this is what is inside those boxes. Forcing the inequality means picking nonnegative weights on a comb of translated potentials, and a linear programme picks them — one vector per box, most entries zero. Averaged over each row of the covering and stacked, the choices stop looking like arithmetic and start looking like a material: bands where the same few teeth carry everything, a front where the programme changes its mind and reaches for others. Nothing here was designed. It is what optimisation under a constraint happens to look like when you stack 1,350 of its answers.

w  =  argmax over nonnegative weights of the
       forced minimum, one LP per b-box
rows: 86 a-boxes, in order
cols: up to 61 teeth
brightness = mean weight, normalised per row

On the difference

The geometry found inside a model is evidence: it was not designed, it emerged from the data, and the interesting question is what it means. The geometry drawn here is the opposite kind of object — it is a definition, and the interesting question is what can be proven about it. Plate I is the hinge between them, the same shape read both ways.

What the two directions share is that in both cases the geometry is what survives. For a trained model it is the structure that survived the training data. For an instrument it is the structure that survives every hypothesis you cannot pin down — which, on Plate II, is every angle on a 838-dimensional torus at once.