A cell that measures in rapidity

Imagine you are a single bacterium swimming through a microscopic droplet of water.

You possess no eyes to scan your surroundings, no compass to orient yourself, and no digital gauge to read out chemical concentrations. Instead, your entire universe consists of molecules colliding with your outer membrane.

Some molecules strike your surface receptors, attach briefly, and trigger a conformational ripple through the cell wall. Other molecules break loose and drift back into the liquid. Your immediate reality is nothing more than a flickering, dynamic pattern of occupied and unoccupied binding sites.

How does a microscopic organism translate that chaotic dance of individual molecular impacts into a steady sense of direction?

One of the most thrilling things I encountered while exploring Epistemic Physics is seeing how nature arrived at these coordinates billions of years before we wrote down the mathematics. The thermodynamic equilibrium of a cell receptor, the buffering curve of an acid in water, and the differential survival of competing genes across generations all operate in the exact same mathematical language: the half-log-odds rapidity coordinate of spacetime.

By tracing this single coordinate across disciplines, we can watch a single cell membrane become a relativistic velocity sensor, an acid-base buffer become a center of epistemic inertia, and natural selection become a steady march along a hyperbolic trajectory.

One receptor, two states: The Michaelis-Menten bridge

Let us begin with the simplest sensory device in biology: a single receptor protein embedded in a lipid bilayer.

The receptor can bind to a specific signaling molecule, known as a ligand (such as a sugar, an amino acid, or a hormone). Let $c$ denote the concentration of free ligand molecules in the surrounding fluid.

In basic physical chemistry, binding and unbinding are opposing reactions. Molecules in solution bind to free receptors at a rate proportional to concentration, $k_{\mathrm{on}}c$, while bound complexes spontaneously dissociate at a constant thermal rate, $k_{\mathrm{off}}$.

At chemical equilibrium, the rate of association matches the rate of dissociation:

$$k_{\mathrm{on}}c(1-p)=k_{\mathrm{off}}p,$$

where $p\in (0,1)$ is the fraction of receptors that are occupied, and $(1-p)$ is the fraction that remain free.

Define the dissociation constant $K_d=k_{\mathrm{off}}/k_{\mathrm{on}}$, which carries the same units of concentration as $c$. Solving for the occupancy probability $p$ yields the famous Langmuir binding isotherm. It shares its shape with the Michaelis-Menten relation, though that enzyme model generally uses $K_M=(k_{\mathrm{off}}+k_{\mathrm{cat}})/k_{\mathrm{on}}$, where $k_{\mathrm{cat}}$ is the rate of converting bound substrate into product:

$$p=\frac{c}{K_d+c}.$$

Look at what $K_d$ represents: when the ligand concentration matches the dissociation constant ($c=K_d$), the occupancy is exactly $p=K_d/(2K_d)=1/2$. The constant $K_d$ is the midpoint of the receptor’s operating range. A smaller $K_d$ corresponds to tighter, stickier binding.

Now let us compute the odds of occupancy:

$$\frac{p}{1-p}=\frac{\frac{c}{K_d+c}}{1-\frac{c}{K_d+c}}=\frac{\frac{c}{K_d+c}}{\frac{K_d}{K_d+c}}=\frac{c}{K_d}.$$

Look at that cancellation!

The fractional odds that a receptor is occupied equal the ratio of the ambient ligand concentration to the dissociation constant.

Now take the natural logarithm of both sides and divide by two to construct the epistemic rapidity coordinate $\rho$ (rho):

$$\rho =\frac{1}{2}\ln \left(\frac{p}{1-p}\right)=\frac{1}{2}\ln \left(\frac{c}{K_d}\right).$$

Define the dimensionless logarithmic concentration $\ell =\ln (c/K_d)$. Then epistemic rapidity is simply half the log concentration:

$$\boxed{\rho =\frac{\ell }{2}.}$$

This is a breathtaking mathematical bridge.

In chemistry, concentrations vary across orders of magnitude: a bacterium might encounter a micromolar signal in one corner of a pond and a millimolar surge near a food source. Environmental shifts are naturally multiplicative: a signal doubles, quadruples, or drops tenfold.

In special relativity, velocity boosts compose additively in rapidity: ${\rho }_{\mathrm{combined}}={\rho }_1+{\rho }_2$.

Because a single receptor maps concentration ratios directly into rapidity, multiplicative chemical stimuli become straight, additive strides along a relativistic axis! Every doubling of concentration advances the receptor’s internal rapidity by the exact same distance: $\mathrm{\Delta }\rho =\frac{1}{2}\ln 2\approx 0.3466$.

Inside the light cone: Why a membrane is a velocity sensor

Now let us translate receptor occupancy into the centered coordinates of Epistemic Physics.

In our evidence framework, an expected probability $p$ maps to a centered epistemic velocity $v=2p-1\in (-1,1)$.

Substituting our receptor occupancy $p=c/(K_d+c)$ into this transformation:

$$v=2\left(\frac{c}{K_d+c}\right)-1=\frac{2c-(K_d+c)}{K_d+c}=\frac{c-K_d}{c+K_d}.$$

Notice the remarkable symmetry:

• When the ambient concentration matches the receptor’s tuning ($c=K_d$), the velocity is zero: $v=0$. The receptor is at rest.
• When the ligand is completely absent ($c\to 0$), the velocity reaches the negative speed of light: $v\to -1$.
• When the ligand floods the membrane ($c\to \mathrm{\infty }$), the velocity reaches the positive speed of light: $v\to +1$.

Now recall the hyperbolic connection between velocity and rapidity from special relativity: $v=\tanh \rho$.

Using our formula $\rho =\frac{1}{2}\ln (c/K_d)$:

$$\tanh \rho =\frac{e^{\rho }-e^{-\rho }}{e^{\rho }+e^{-\rho }}=\frac{\sqrt{c/K_d}-\sqrt{K_d/c}}{\sqrt{c/K_d}+\sqrt{K_d/c}}=\frac{c/K_d-1}{c/K_d+1}=\frac{c-K_d}{c+K_d}=v.$$

The correspondence is exact down to the last decimal!

Now calculate the relativistic Lorentz factor $\gamma =1/\sqrt{1-v^2}$ for the receptor:

$$1-v^2=1-{\left(\frac{c-K_d}{c+K_d}\right)}^2=\frac{(c+K_d{)}^2-(c-K_d{)}^2}{(c+K_d{)}^2}=\frac{4c{K}_d}{(c+K_d{)}^2}.$$

Taking the square root and inverting:

$$\gamma =\frac{c+K_d}{2\sqrt{c{K}_d}}=\cosh \rho .$$

The Lorentz factor of a biological receptor is the hyperbolic cosine of half its logarithmic concentration ratio, $\gamma =\cosh [\frac{1}{2}\ln (c/K_d)]$!

When the cell is tuned to its environment ($c=K_d$), the Lorentz factor is at its minimum: $\gamma =1$. As the chemical concentration moves toward either extreme (starvation or saturation), $\gamma$ climbs toward infinity, signaling that the receptor is approaching the boundary of its dynamic range.

Saturated blindness: Why a busy receptor is a poor sensor

Suppose you are designing an artificial nanobot or engineering a synthetic bacterium. You want your sensor to detect tiny proportional changes in an environmental chemical.

Which receptor should you choose:

1. One that is almost always empty?
2. One that is occupied about half the time?
3. One that is almost always full?

At first intuition, an almost-full receptor sounds attractive: it binds molecules constantly, providing continuous chemical engagement.

Yet in information theory, an almost-full receptor is virtually useless.

To understand why, we compute the Fisher information of a single receptor observation.

Suppose the cell takes an instantaneous binary reading of the receptor state: $X=1$ if bound, $X=0$ if free. The probability of binding is $p(\ell )$, where $\ell =\ln (c/K_d)$ is the true log concentration the cell wants to estimate.

The Fisher information with respect to log concentration $\ell$ measures how sensitively the observation distribution responds to shifts in $\ell$:

$$I_{\ell }=\frac{1}{p(1-p)}{\left(\frac{dp}{d\ell }\right)}^2.$$

Let us evaluate the derivative $dp/d\ell$. Since $p=\frac{e^{\ell }}{1+e^{\ell }}$, the quotient rule gives:

$$\frac{dp}{d\ell }=\frac{e^{\ell }(1+e^{\ell })-e^{\ell }(e^{\ell })}{(1+e^{\ell }{)}^2}=\frac{e^{\ell }}{(1+e^{\ell }{)}^2}=\left(\frac{e^{\ell }}{1+e^{\ell }}\right)\left(\frac{1}{1+e^{\ell }}\right)=p(1-p).$$

Substitute this derivative back into the Fisher information formula:

$$I_{\ell }=\frac{[p(1-p){]}^2}{p(1-p)}=p(1-p).$$

Now express $p(1-p)$ in terms of our relativistic Lorentz factor $\gamma$:

$$\boxed{I_{\ell }=p(1-p)=\frac{1}{4{\gamma }^2}.}$$

Look at what this elegant equation reveals:

Fisher information is maximized when $\gamma =1$, which occurs at the resting center where $c=K_d$ and $p=1/2$. At that sweet spot, each independent binding reading delivers $I_{\ell }=1/4$ of Fisher information.

As the receptor becomes saturated ($c\gg K_d,p\to 1$) or starved ($c\ll K_d,p\to 0$), the Lorentz factor $\gamma$ diverges, and Fisher information plummets toward zero:

$$I_{\ell }=\frac{1}{4{\gamma }^2}\to 0.$$

If a receptor is occupied $99.9\mathrm{\%}$ of the time, observing it in the bound state provides virtually zero information about whether the concentration just doubled. Both “a high concentration” and “a ten-times-higher concentration” yield the exact same response: bound. A saturated receptor has been blinded by its own success.

Maximum sensitivity demands an open, uncommitted channel: a receptor operating near $p=1/2$.

A membrane's view of the world

Full receptors can have very little new to say.

Reset lab

Each equal step on the concentration dial multiplies the chemical supply by the same factor. Find the region where a small proportional change is easiest to detect.

Log concentration ratio, base ten

0.00

Hill response steepness

1.0

Expected occupancy

An illustration of the expected fraction, rounded to 40 icons.

$$c/K_d=1.00$$ $$p=0.500,\rho =0.000$$ $$I_{\ell }=0.250$$

Near its sweet spot: both answers occur often, and occupancy changes readily.

Go to half occupancy

Hill occupancyRapidity projectionBernoulli informationLorentz-factor form

Each dashed curve lies on its solid partner: two expressions for the same quantity. Both plots use the same concentration axis and selected point. The information scale adjusts when steepness changes.

The horizontal axis uses base-ten logs for readable concentration ranges; the information calculation uses natural log concentration ℓ = ln(c/K_d). Rapidity is ρ = hℓ/2, and the Lorentz factor is γ = cosh ρ. These coordinate changes give the matching curves above.

Explore this trade-off in the interactive laboratory above. Drag the concentration slider and watch the green Fisher information curve peak sharply at $c=K_d$, falling away as the receptor approaches saturation or starvation.

Cooperativity and the Hill equation: Sharpening the switch

Living organisms frequently need to sharpen their sensory decisions. When a white blood cell hunts a bacterial pathogen or an enzyme triggers a metabolic cascade, a gradual response across two orders of magnitude may be too slow.

To achieve razor-sharp transitions, cells use cooperative binding, modeled by the Hill equation:

$$p=\frac{(c/K_d{)}^h}{1+(c/K_d{)}^h}=\frac{e^{h\ell }}{1+e^{h\ell }},$$

where is the Hill coefficient.

When multiple binding subunits cooperate (as in hemoglobin, where the binding of one oxygen molecule increases the affinity of neighboring subunits), .

Let us calculate the rapidity and Fisher information for a cooperative Hill receptor:

Taking the odds:

$$\frac{p}{1-p}={\left(\frac{c}{K_d}\right)}^h=e^{h\ell }.$$

Taking half the logarithm:

$$\rho =\frac{1}{2}\ln \left(\frac{p}{1-p}\right)=\frac{h}{2}\ell .$$

The derivative with respect to $\ell$ is now:

$$\frac{dp}{d\ell }=hp(1-p).$$

The resulting Fisher information is:

$$I_{\ell }=\frac{(dp/d\ell {)}^2}{p(1-p)}=h^{2}p(1-p)=\frac{h^2}{4{\gamma }^2}.$$

Raising the Hill coefficient $h$ amplifies the peak Fisher information by a factor of $h^2$!

For $h=4$ (similar to hemoglobin), the maximum information delivered per reading jumps from $1/4$ to $4$: sixteen times higher than an uncooperative receptor.

However, nature exacts a price for this heightened sensitivity. Because the curve rises much more steeply, the width of the sensitive window contracts. A high Hill coefficient creates a sharp, decisive digital switch, but it sacrifices broad dynamic range.

This presents cells with a profound design trade-off: should they deploy a single, ultra-steep cooperative sensor, or a diverse bank of uncooperative receptors with staggered $K_d$ values? Real biological membranes choose the latter, maintaining overlapping receptor lineages to tile the entire concentration landscape.

Acid in a beaker: Buffering as epistemic rest mass

This geometric structure is not confined to biological receptors. It governs fundamental inorganic chemistry.

Consider a weak acid $\mathrm{HA}$ dissolved in water. It exists in equilibrium with its conjugate base ${\mathrm{A}}^{-}$ and hydronium ions ${\mathrm{H}}^{+}$:

$$\mathrm{HA}+{\mathrm{H}}_2\mathrm{O}\rightleftharpoons {\mathrm{A}}^{-}+{\mathrm{H}}_3{\mathrm{O}}^{+}.$$

The acid dissociation constant is:

$$K_a=\frac{[{\mathrm{A}}^{-}][{\mathrm{H}}^{+}]}{[\mathrm{HA}]}.$$

Taking the base-ten logarithm and rearranging yields the celebrated Henderson-Hasselbalch equation:

$$\mathrm{pH}=\mathrm{p}{K}_a+{\log }_{10}\left(\frac{[{\mathrm{A}}^{-}]}{[\mathrm{HA}]}\right),$$

where $\mathrm{pH}=-{\log }_{10}[{\mathrm{H}}^{+}]$ and $\mathrm{p}{K}_a=-{\log }_{10}{K}_a$.

Now let $p=[{\mathrm{A}}^{-}]/([{\mathrm{A}}^{-}]+[\mathrm{HA}])$ be the fraction of acid molecules that have lost their proton (the deprotonated fraction).

The ratio of deprotonated to protonated molecules is simply the odds:

$$\frac{p}{1-p}=\frac{[{\mathrm{A}}^{-}]}{[\mathrm{HA}]}.$$

Substitute this into the Henderson-Hasselbalch formula:

$$\mathrm{pH}-\mathrm{p}{K}_a={\log }_{10}\left(\frac{p}{1-p}\right)=\frac{\ln \left(\frac{p}{1-p}\right)}{\ln 10}.$$

Multiply by $\frac{\ln 10}{2}$:

$$\boxed{\rho =\frac{1}{2}\ln \left(\frac{p}{1-p}\right)=\left(\frac{\ln 10}{2}\right)(\mathrm{pH}-\mathrm{p}{K}_a)\approx 1.1513\times (\mathrm{pH}-\mathrm{p}{K}_a).}$$

What a marvelous identity!

The pH scale of chemistry is literally an affine transformation of epistemic rapidity!

When a solution is at $\mathrm{pH}=\mathrm{p}{K}_a$, the acid and conjugate base are present in exact 1:1 balance ($p=1/2$). The rapidity is zero: $\rho =0$.

In chemistry, this point is called the maximum buffer capacity. A buffered solution resists changes in pH most effectively when it is balanced at $\mathrm{pH}=\mathrm{p}{K}_a$.

Why does it resist change? Because in the language of Epistemic Physics, the clashing presence of equal amounts of acid $[\mathrm{HA}]$ and base $[{\mathrm{A}}^{-}]$ maximizes the epistemic rest mass of the chemical system:

$$m_{\mathrm{E}}=2\sqrt{[\mathrm{HA}][{\mathrm{A}}^{-}]}.$$

A chemical buffer is a high-mass epistemic state. It resists acceleration (shifts in pH) under the addition of acidic or basic tracelets because its opposing components absorb the perturbation, stabilizing the solution.

Acid, base, and resistance to change

Give a buffer a little shove.

Reset balanced buffer

Add the same small dose of acid or base. Does a balanced mixture move as far in pH as one that is already mostly acid or mostly base?

Total buffer amount

20 mmol

Starting pH minus pK_a

0.00

HA · acid 10.00 mmolA⁻ · conjugate base 10.00 mmol

Add 1 mmol acid Add 1 mmol base Undo additions

pH

7.000

Rapidity ρ

0.000

Epistemic mass

20.00 mmol

Buffer capacity

11.51 mmol/pH

One curve, two rulers: the pH values on the left and rapidity values on the right mark the same mixtures. At pH 7, rapidity is zero.

Capacity / maximum capacity(Mass / total amount)²

The two curves coincide: both equal 4p(1 − p), where p is the fraction in the base form. Here each is 1.000 of its maximum. Capacity reaches 11.51 mmol/pH and mass reaches 20 mmol at balance.

Start with equal amounts, then try a mixture away from balance. At a fixed total amount, both buffer capacity and epistemic mass reach their maximum at a 50:50 mixture.

Here a and b are the acid and base amounts. Added strong base converts HA to A⁻; added strong acid does the reverse. The model holds volume fixed, uses pK_a = 7, and neglects water's contribution. Capacity means the local amount of added base per unit pH: ln(10) ab/(a + b). The shared curve compares their shapes after dividing out their different units and maxima. Additions stop before either component falls below 1%.

Natural selection as a relativistic velocity boost

Now let us zoom out from molecules and beakers to an entire population of self-replicating organisms.

Consider two competing genetic variants, type $A$ and type $B$, reproducing in discrete generations in a large population.

Let $p_t$ be the frequency of type $A$ in generation $t$, so that $(1-p_t)$ is the frequency of type $B$.

Let $w_A$ and $w_B$ be the absolute fitness values (expected surviving offspring per individual) of types $A$ and $B$.

In the next generation, the number of $A$ individuals is multiplied by $w_A$, and the number of $B$ individuals is multiplied by $w_B$. After normalizing by the total population, the new frequency is:

$$p_{t+1}=\frac{p_t{w}_A}{p_t{w}_A+(1-p_t)w_B}.$$

This is the standard replicator dynamic of population genetics.

Now calculate the odds of variant $A$ in the next generation:

$$\frac{p_{t+1}}{1-p_{t+1}}=\frac{\frac{p_t{w}_A}{p_t{w}_A+(1-p_t)w_B}}{\frac{(1-p_t)w_B}{p_t{w}_A+(1-p_t)w_B}}=\left(\frac{p_t}{1-p_t}\right)\left(\frac{w_A}{w_B}\right).$$

Take half the natural logarithm of both sides:

$$\frac{1}{2}\ln \left(\frac{p_{t+1}}{1-p_{t+1}}\right)=\frac{1}{2}\ln \left(\frac{p_t}{1-p_t}\right)+\frac{1}{2}\ln \left(\frac{w_A}{w_B}\right).$$

In terms of epistemic rapidity ${\rho }_t=\frac{1}{2}\ln \left(\frac{p_t}{1-p_t}\right)$:

$$\boxed{{\rho }_{t+1}={\rho }_t+\frac{1}{2}\ln \left(\frac{w_A}{w_B}\right).}$$

Look at that result!

Natural selection is collinear relativistic velocity addition!

Each generation of selection applies an identical rapidity boost:

$$\mathrm{\Delta }\rho =\frac{1}{2}\ln \left(\frac{w_A}{w_B}\right).$$

If type $A$ has a constant selective advantage (say, $w_A/w_B=1.10$, a $10\mathrm{\%}$ fitness advantage), the population frequency $p_t$ traces a classic S-shaped sigmoidal trajectory. In raw frequencies, change seems painfully slow when the allele is rare ($p\approx 0.01$), accelerates to its fastest pace near the middle ($p\approx 0.50$), and then slows down to a crawl as it approaches fixation ($p\approx 0.99$).

Standard evolutionary biology often describes this sigmoidal trajectory as a complex nonlinear curve.

In the rapidity coordinate of Epistemic Physics, there is no nonlinearity whatsoever. The population advances in a perfectly straight, uniform line, adding an identical increment $\mathrm{\Delta }\rho$ every single generation. The apparent slowdown near $0$ and $1$ is merely the foreshortening of the Lorentz projection near the speed of light!

Build the cell that lives on this axis

Let us give our imagined cell a membrane containing many receptor types.

Each receptor type has its own dissociation constant $K_d$, meaning each is informative in a different concentration range. Together, an array of staggered receptors can cover an environment that varies across six orders of magnitude, far beyond the narrow window of any single protein hook.

Occupancy events feed local evidence notebooks inside the cytoplasm. Those notebooks record both a current directional balance and the amount of independently acquired experience $\tau$.

The cell can compare new observations with older ones. If occupied events become unexpectedly common, it shifts its estimate upward along the log-concentration axis. If the environment stays concentrated for a long time, it adapts by altering gene expression: it down-regulates saturated receptors and produces new receptor types with higher $K_d$, dynamically centering its maximum sensitivity window on the conditions it actually encounters.

Evolution then works on the architecture itself. Organisms whose receptor banks and updating rules lead them toward food leave more descendants. Within a lifetime, molecular events change a notebook. Across lifetimes, reproductive weighting changes the prevalence of the machinery that builds those notebooks.

We can start with a controlled sensing experiment. Give two simulated cells the same total budget of independent receptor readings. Let the first cell invest its entire budget in identical receptors with a single $K_d$. Let the second cell spread its readings across several staggered $K_d$ values.

Sweep both designs through a concentration landscape with sudden, tenfold chemical surges. Compare their total Fisher information about log concentration. The single tuning concentrates its sensitivity near one sweet spot; the staggered bank spreads its sensitivity across a wider range. Try collapsing all five tunings to the same value: with the same reading budget, the two designs become identical.

Two cells, the same sensor budget

Build a membrane with a wider view.

Reset receptor bank

Give both cells 60 independent receptor readings. One cell uses a single tuning; the other splits its readings equally across five tunings. Which notices a proportional chemical change more easily?

Concentration / reference concentration

1×

Spacing between neighboring tunings

10×

Tenfold drop ÷10 Tenfold surge ×10 Return to the middle

One tuning · 60 readings

Expected occupancy

50.0%

Total information

15.000

Every receptor has the reference K_d.

Five tunings · 12 readings each

Tuning range / reference

0.01 to 100

Total information

5.219

Dashed rose: one tuning. Solid teal: five tunings. Both curves use the same reading budget and vertical scale.

At this concentration, the single tuning has 2.87× as much information. Try a tenfold surge, then another. Spreading the tunings trades a tall local peak for wider coverage.

Look inside the five-tuning bank

K_d / reference = 0.01

99.0% occupied

K_d / reference = 0.1

90.9% occupied

K_d / reference = 1

50.0% occupied

K_d / reference = 10

9.1% occupied

K_d / reference = 100

1.0% occupied

$$p_i=\frac{c}{K_{d,i}+c},I_{\ln c}=\sum _i{n}_i{p}_i(1-p_i)$$

Here n_i is the number of independent binary readings at tuning i. Information adds across independent readings and describes sensitivity to a small change in natural log concentration. This equilibrium model compares sensing; movement and adaptation would be further experiments.

A next experiment could put these sensors into swimming cells, then measure how quickly they notice a change and how reliably they steer toward a nutrient source. The sensing comparison gives us a concrete starting point for choosing the movement and adaptation rules.

The cell’s task is concrete: a molecule arrives; a hook becomes occupied; an internal ledger gains an event; an odds ratio shifts; a flagellar motor changes direction. We can follow the entire chain using one coordinate system, from the beaker to the membrane to the population.

The lineage of experience: From tracelet to organism

We have traced a single mathematical idea through four levels of the natural world:

• At the single-molecule scale, a membrane receptor converts multiplicative ligand concentrations into an additive rapidity coordinate $\rho =\frac{1}{2}\ln (c/K_d)$.
• At the sensor level, Fisher information $I_{\ell }=1/(4{\gamma }^2)$ reveals why saturation blinds a detector and why optimal sensing demands balance.
• In physical chemistry, the Henderson-Hasselbalch equation reveals that pH is an affine rapidity scale, and chemical buffering capacity is the physical manifestation of epistemic rest mass.
• Across evolutionary time, natural selection acts as a constant relativistic boost $\mathrm{\Delta }\rho =\frac{1}{2}\ln (w_A/w_B)$ pushing an entire species along an invariant trajectory.

Every single binding event on a bacterial membrane is a discrete tracelet of environmental information. As those tracelets accumulate inside the cell’s metabolic ledger, they update the organism’s internal velocity, steering its flagella toward nutrients.

And when those bacteria reproduce, natural selection weights their lineages by survival, accumulating tracelets across geological epochs.

The universe does not need separate mathematical systems for physics, chemistry, biology, and learning. Underneath them all lies the same geometry of evidence, ticking forward in proper time and marching steadily in rapidity.

In our next excursion, we will explore how this geometry unifies magnetic spin systems, physical coins, and artificial neural networks in A coin, a magnet, and a learning machine.