Research · Report
Clocks and calendars, read from the inside.
Seven findings with their controls, every angle in degrees, and the tests that failed.
1 · The faces · registered
Each day, month and hour has an angle.
For each family we average the final-position state over the prompts that state a value (“Today is Monday. Today, it is”) and take the first Fourier plane of those averages. The plane has no preferred direction, so we turn the reading until the measured positions are, on average, at their ideal ones, and orient it so the days run clockwise. The positions are then a measurement. In the 8B model at layer 32, Monday sits at 8.2°, Tuesday at 62.4° and Wednesday at 103.0°, and no weekday is more than 10.9° from its ideal position. No month is more than 8.8° off. This plane holds 0.56 of the variance of the seven weekday means (an even spread over the three planes would give 0.33). In an untrained copy of the 1.7B architecture the same plane holds at most 0.31 for days and 0.19 for months, which is what an even spread gives. The hour face is looser: two hours sit up to 30.8° from their ideal positions.
Each clock is drawn from the activations: the labels sit at the measured angle of each stated value, the heavy ticks mark the ideal positions, and the hand points at the value you pick (the dashed hand at its ideal position). Choose a face, a model and a layer; the table lists every angle. Early layers show the input words, and the clock tightens in the last third of the stack.
2 · Turn the hand · registered
A shift turns the state by close to the ideal angle.
For every shifted prompt we measure how far its state has turned from the state of the same start day stated plainly. In the 8B model at layer 32, “+1 day” turns 47.0° (ideal 51.4°), “+2” turns 110.6° (ideal 102.9°) and “+3” turns 158.4° (ideal 154.3°). On the year face “+1 month” turns 23.8° (ideal 30.0°) and “+6” turns 190.2° (ideal 180°). For a shift of one day, 1.00 of prompts land nearest the right weekday, with a spread of 9.2°. Synonyms (“tomorrow”, “one day from now”) are pooled; the dots in the figure show each prompt separately.
The short hand points at the start value where the model puts it; the long hand shows the mean measured turn after the shift, with the arc between them; the dashed hand is the ideal turn; each dot is one prompt. “Two steps” shows prompts like “two days from now, and then three days after that”, measured against the ideal turn for the sum.
3 · Through depth · exploratory, except the two-step landing
The small model turns in one layer; the large one turns over six.
Up to layer 21 of the 1.7B model, the final position still holds the start day: the turn reads 0.09 of the ideal angle. At layer 22 it reads 0.38 and at layer 23 1.16, past the target, and it stays there. In the 8B model the angle grows over about six layers: 0.18 of the ideal at layer 24, 0.40 at 25, 0.62 at 26, 0.78 at 27 and 0.94 at 30. Each prompt moves partway around the circle at each step, rather than some prompts jumping while others wait. Two-step shifts follow the same schedule toward the sum of the two shifts. At layer 32, 0.69 of two-step month prompts land on the summed month, and 0.33 of two-step day prompts land on the summed day. The 8B model's answers to the same two-step questions are correct 0.66 (months) and 0.47 (days) of the time; the 1.7B model's are correct 0.17 and 0.18.
Left: the median fraction of the ideal turn reached at each layer, against relative depth. Right: one small clock per layer, its hand at the mean measured turn for the chosen shift (dashed: ideal). Pick the shift.
4 · What the model uses · swaps registered, circle-only tests exploratory
The whole state decides the answer; at 8B the circle alone does not.
Activation patching tests what the model reads. Replacing the final-position state, inside the span of the day means, with the mean of another day makes the 1.7B model name that day 0.85 of the time at layer 25 (random subspace of the same size: 0.04) and the 8B model 1.00 of the time at layer 32. Rotating only the circle by k steps, at one layer, moves the 1.7B model's answer by k in 0.26 to 0.31 of cases for days (random plane: 0.04). In the 8B model, holding the circle at its rotated position at every layer from 25 to 36 moves the answer in at most 0.04 of cases. The 8B clock is drawn accurately, but the model reads its answer from the whole state; the circle is the clearest part of it, not the part that decides.
Bars: how often the model's answer follows the intervention. Grey ticks: the same intervention in a random subspace or plane.
5 · Half steps · registered
Hours land between the marks; days snap to them.
We asked the 8B model about half steps in matched wording: “90 minutes from now” beside 60 and 120, and “36 hours from now” beside 24 and 48. For each half step we measure where it lands between its two whole-step neighbours on the circle (0 at the lower, 1 at the upper). Hours land near the middle: median 0.56, with 0.50 of half steps between a quarter and three quarters of the way. Days do not: median 0.87, with 0.23 in the middle. “36 hours from now” sits on the position of the day after tomorrow. On the hour face at layer 32, 90 minutes turns the state 55.8°, between the 60-minute turn (23.6°) and the 120-minute turn (82.3°). Time of day is kept continuous; the day is kept as a whole unit.
Histograms of where half steps land between their neighbours, for hours and for days, at a chosen layer. Each bar counts prompts; the shaded band is a quarter to three quarters.
6 · Two languages · registered
Chinese weekdays sit on the English circle.
We ran the same questions in Chinese (“今天是星期一。两天后是星期”), centred the Chinese states on their own mean, and placed them on the English circle of the same model, with nothing fitted between the languages. In the last layers, a stated Chinese day lands on the English position of the same day 0.81 of the time in the 1.7B model and 0.96 in the 8B model. A computed Chinese day (“two days from now”) lands on the English position of its result 0.67 and 0.92 of the time. At layer 32 of the 8B model no Chinese day is more than 25.1° from its English counterpart.
One clock, two languages: English day names on the outer ring and Chinese on the inner ring, each at its measured angle on the English clock. Pick a day to point both hands at it; the table gives both angles and the difference.
7 · What did not hold
Failures, reported with the passes.
The 1.7B model does not compose two shifts in context (two-step answers correct 0.18 and 0.17); it more often applies only the second shift. An early hint that its internal circle composes better than its answers did not replicate in Chinese.
On the 1.7B model, the plain hour prompt (“Right now, it is”) is rarely answered with the current hour, so the stated hour positions were not usable as anchors there and every hour test on that model failed. The 8B hour face uses prompts whose examples fix this.
We predicted that pushing the 8B circle in mid-turn would carry the push through to the answer. It did not, at any single layer, before, during or after the turn; holding it at every layer did not either. That is what section 4 reports.
We predicted that both hours and days would land between the marks, and that the size of one whole step would match 360°/n within a quarter. Only hours land between, and the step size was off by more than a quarter for every face in that run.
Method
How the angles were read.
- Prompts give a start value and a shift in a few-shot format, in two formats per family. The state is the residual stream at the final prompt position after every layer (fp32 for 1.7B, bf16 for 8B, single deterministic runs).
- For each prompt template we subtract the template's mean over start values, which removes the wording and keeps what changes with the day.
- The stated positions are the means of the plain prompts (“Today, it is”) for each value. The clock plane is the first Fourier plane of those means: Σj e−2πij/n·(meanj − overall mean).
- Angles are read in that plane, oriented so the values run clockwise, and turned so the stated positions match their ideal positions on average. A shift's turn is the angle between a prompt's state and the plain prompt with the same start and format; it does not depend on how the plane is turned.
- Patching replaces the final-position state's coordinates in a chosen subspace (the span of the value means, or the clock plane) with those of a target value, at one or more layers, and records the model's answer. Each test is repeated in a random subspace of the same size.
- Each test's outcome was written down before it was run. The report quotes the registered result, and labels the rest as exploration.
Receipt