One rapidity, many worlds

A chess player gains 400 rating points on an opponent. A chemist moves a solution one pH unit above an acid’s midpoint. A cell’s single-site receptor sees ten times as much ligand as its binding scale. Somewhere else, a learning machine sets its logit difference to $\ln 10$.

All four have reached the same address: odds of ten to one.

In Epistemic Physics, that address has a name, rapidity. At first it looks like a convenient translator. Then the translator starts making suggestions. If these systems share an address, do they share a rule for moving? Does the same geometry describe their saturation, their response to a push, or the way two influences combine?

Let’s follow those suggestions across these worlds. Some connections are familiar equations written in revealing coordinates. Others invite us to build a model and see what happens. The table is our map; the experiments let us turn the map’s handles.

The address behind the sigmoid

Start with epistemic velocity $v_{\mathrm{E}}$, a state’s directional balance between $-1$ and $1$. Zero means balance; the sign tells us which direction the state favors. Epistemic rapidity is defined from this velocity:

$$\boxed{{\varphi }_{\mathrm{E}}=\mathrm{artanh}(v_{\mathrm{E}}).}$$

The inverse hyperbolic tangent, $\mathrm{artanh}$, unrolls the bounded velocity interval onto an unbounded line. Its inverse is

$$v_{\mathrm{E}}=\tanh {\varphi }_{\mathrm{E}}.$$

Ordinary relativity uses the same construction: $u/c=\tanh \varphi$, where $u$ is speed along a chosen axis and $c$ is the speed of light. The hyperbolic tangent, $\tanh \varphi =(e^{\varphi }-e^{-\varphi })/(e^{\varphi }+e^{-\varphi })$, projects the rapidity line back into the velocity interval.

Now relate that velocity to a binary share $\mu$ through $v_{\mathrm{E}}=2\mu -1$. The share might be a probability, an occupied fraction, a market quote, or a normalized performance score. Substituting into the velocity definition gives

The ratio $O=\mu /(1-\mu )$ is the odds. We have derived a useful equivalence: epistemic rapidity is half the log-odds of the corresponding share. The logarithm turns odds multiplication into rapidity addition. Solving for $\mu$ gives the logistic function, also called the sigmoid:

We can see the light cone directly. Give two positive weights $\alpha$ and $\beta$ the ratio $\mu /(1-\mu )$, and write

Then $v_{\mathrm{E}}=x_{\mathrm{E}}/{\tau }_{\mathrm{E}}$, and the velocity definition gives another useful expression:

$${\varphi }_{\mathrm{E}}=\mathrm{artanh}{v}_{\mathrm{E}}=\frac{1}{2}\ln \frac{\alpha }{\beta }.$$

The light-cone identity is ${\tau }_{\mathrm{E}}^2-x_{\mathrm{E}}^2=m_{\mathrm{E}}^2$. Its hyperbolic coordinates are

In an EP belief these weights describe trial amount, including the prior. In the worlds below, their counterparts can be concentrations, likelihoods, rates, or thermal weights. Rapidity follows their balance; their total supplies another dimension to explore.

The table uses $\mu$ for a probability or share. The thermal rows distinguish a magnetic moment ${\mu }_{\mathrm{m}\mathrm{a}\mathrm{g}}$ from a chemical potential ${\mu }_{\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{m}}$. Notice how often rapidity is an advantage divided by a scale.

Try the universal translator

Before touching the experiment, predict what a 400-point Elo advantage should look like as a pH offset. Both multiply the odds by ten, so they should land on the same point. What about a Hill receptor with coefficient $n=2$? It needs only a factor of $\sqrt{10}$ in concentration to reach the same odds.

A rapidity translator

Turn one dial. Visit several worlds.

Reset worlds

Start with centered velocity, then straighten its bounded range into rapidity. Drag the gold point sideways: a chess advantage, a chemical balance, and an electrical push move together. Choose other worlds, or type into a card to drive them all.

Rapidity ${\varphi }_{\mathrm{E}}$

← Small push Balance at ½ Small push →

Rapidity φ_E

0.550

Share μ

75.0%

Centered mean v_E

0.501

World 1 ChessChemistryBindingIon channelsFermi gasLearning / codingKelly bettingRelativity

$${\varphi }_{\mathrm{E}}=\frac{\ln 10}{800}{\mathrm{\Delta }}_{\mathrm{Elo}}$$

Rating advantage Elo points

Expected score 75.0%

World 2 ChessChemistryBindingIon channelsFermi gasLearning / codingKelly bettingRelativity

$${\varphi }_{\mathrm{E}}=\frac{\ln 10}{2}(\mathrm{pH}-\mathrm{p}{K}_a)$$

pH − pKₐ pH units

Deprotonated share 75.0%

World 3 ChessChemistryBindingIon channelsFermi gasLearning / codingKelly bettingRelativity

$${\varphi }_{\mathrm{E}}=\frac{V-V_{1/2}}{2k}$$

Voltage scale k 6 mV12 mV20 mV Voltage above midpoint mV

Open share 75.0%

$${\varphi }_{\mathrm{E}}=\mathrm{artanh}{v}_{\mathrm{E}}$$ $$v_{\mathrm{E}}=\tanh {\varphi }_{\mathrm{E}}=2\mu -1$$ $$\mu =\sigma (2{\varphi }_{\mathrm{E}})$$

Every card shares the gold point. Epistemic rapidity φ_E is the inverse hyperbolic tangent of the centered mean velocity. The sigmoid σ then translates that rapidity into a share between 0 and 1. Arrow keys move the graph, and Enter applies a typed value.

Open the translator in the playground.

The numbers printed on the instruments differ. Their common coordinates agree:

This lets a result travel. Once we know how a tenfold odds change moves one system, we can translate that move into every row whose operating law stays the same.

Take the whole belief along

An EP belief also has a spread of possibilities around its mean. Let the unknown probability itself be a random variable $B$ with a Beta distribution,

$$B\sim \mathrm{Beta}(\alpha ,\beta ).$$

The positive parameters $\alpha$ and $\beta$ describe the two trial amounts, including the prior. The symbol $\sim$ means “has this distribution.” Its mean is $\mu =\mathbb{E}[B]=\alpha /(\alpha +\beta )$, where $\mathbb{E}$ denotes averaging. This is the share $\mu$ associated with the state’s scalar velocity above.

Now move every possible value through the coordinate change. Define the random centered belief $B_{\mathrm{E}}=2B-1$, and then its random rapidity:

The uppercase ${\mathrm{\Phi }}_{\mathrm{E}}$ carries a whole distribution. The lowercase ${\varphi }_{\mathrm{E}}$ names the rapidity of the state’s mean velocity. Their inverse relations line up as

Averaging the first line gives the exact relation

A curved transformation and an average can land at different addresses depending on their order. In general, $\mathbb{E}[{\mathrm{\Phi }}_{\mathrm{E}}]\ne {\varphi }_{\mathrm{E}}$. For example, a $\mathrm{Beta}(2,1)$ belief has mean velocity $1/3$ and scalar rapidity $\frac{1}{2}\ln 2\approx 0.347$. Averaging the transformed distribution instead gives $\mathbb{E}[{\mathrm{\Phi }}_{\mathrm{E}}]=1/2$.

Predict what the two markers will do as a belief becomes more asymmetric. Then change the trial amount while keeping its mean velocity fixed: the scalar rapidity stays put while the cloud around it changes.

A distribution travels with us

Stretch the whole belief into rapidity

Reset belief

Move one probability through the stretched ruler. The teal markers enclose the same probability weight on both charts. Then predict where the mean after transformation lands.

Carry probability b = 0.50 into rapidity Φ = 0.00

Weight left of either teal marker: 25.0%

Before trials: Beta(½, ½)Asymmetric: Beta(2, 1)Same mean: Beta(20, 10)

Add trials or rescale the amount

Add a supporting trial Add an opposing trial Double amount, same mean Halve amount, same mean

Beta(2.00, 1.00), total amount 3.00. Supporting and opposing trials add to their own shape; amount buttons scale both shapes together.

Mean Φ lands leftThey coincideMean Φ lands right

Gold marks the transformed probability mean. Predict the rose mean marker before revealing it.

Probability B

Random rapidity Φ

Same belief, positive densityMean of B, then transformedMean after transformation

Bars average the density within each bin; their areas are calculated from the Beta distribution. The rapidity window contains 99.8% of the probability. Its tails continue along the real line.

Mean, variance, and coordinate-change equations

Mean probability $\mu =\mathbb{E}[B]$

0.6667

Variance of B

0.05556

Scalar rapidity ${\varphi }_{\mathrm{E}}=\mathrm{artanh}\mathbb{E}[B_{\mathrm{E}}]$

0.3466

Mean random rapidity $\mathbb{E}[{\mathrm{\Phi }}_{\mathrm{E}}]$

0.5000

$\mathbb{E}[\tanh {\mathrm{\Phi }}_{\mathrm{E}}]=v_{\mathrm{E}}$

0.3333

tanh(E[Φ])

0.4621

$$B_{\mathrm{E}}=2B-1$$ $${\mathrm{\Phi }}_{\mathrm{E}}=\mathrm{artanh}{B}_{\mathrm{E}}$$ $$\mathbb{E}[{\mathrm{\Phi }}_{\mathrm{E}}]=\frac{1}{2}[\psi (\alpha )-\psi (\beta )]$$ $$\mathbb{E}[\tanh {\mathrm{\Phi }}_{\mathrm{E}}]=\tanh {\varphi }_{\mathrm{E}}=v_{\mathrm{E}}$$

The transformed mean lands at 0.3466, also the mode of the rapidity density. Its average lies at 0.5000. Double the amount at the same mean and watch the two markers move closer as the distribution gathers.

The stretch changes density height

$$f_{\mathrm{\Phi }}(\varphi )=f_B(b)2b(1-b)$$

At b = 0.6667, the probability density 1.3333 is multiplied by the coordinate factor 0.4444, giving rapidity density 0.5926. This factor keeps the probability weight intact while the horizontal coordinate stretches. Here ψ is the digamma function, the derivative of ln Γ.

Open the whole-belief experiment in the playground.

This gives every table connection a second adventure. Alongside a predicted occupancy, a gate’s opening probability, or a market quote, we can carry uncertainty through the same rapidity transformation. The hidden springs of a belief explores the transformed density and the potential landscape beneath it.

Scores, skills, and prices measure the same kind of advantage

In logistic regression, a model computes a real-valued score $\ell$ from its inputs and predicts $\mu =\sigma (\ell )$. That score is twice rapidity. A one-unit increase in $\ell$ multiplies the odds by $e$. A logistic-regression derivation makes the log-odds construction explicit.

Chess changes the ruler. In the usual logistic Elo model, a rating advantage ${\mathrm{\Delta }}_{\mathrm{E}\mathrm{l}\mathrm{o}}$ gives

$$\mu =\frac{1}{1+{10}^{-{\mathrm{\Delta }}_{\mathrm{E}\mathrm{l}\mathrm{o}}/400}}.$$

Here $\mu$ is expected score, with a draw counting as half a point. Taking odds and logs gives ${\varphi }_{\mathrm{E}}=(\ln 10){\mathrm{\Delta }}_{\mathrm{E}\mathrm{l}\mathrm{o}}/800$. The rating difference becomes a rapidity distance. Glickman’s account of the rating model provides the scoring interpretation.

A Rasch test puts a person and a question on a common scale. Let $a$ be ability and $d$ difficulty, both in log-odds units. The model’s probability of a correct response is $\mu =\sigma (a-d)$, so ${\varphi }_{\mathrm{E}}=(a-d)/2$. An opponent on the chessboard and a question on a test both establish the reference point for an advantage. Wright and Panchapakesan’s original account develops this person-item comparison.

Binary softmax uses two scores, $z_Y$ and $z_N$, and a positive temperature parameter $T$:

$$\mu =\frac{e^{z_Y/T}}{e^{z_Y/T}+e^{z_N/T}}.$$

Only their difference $\mathrm{\Delta }z=z_Y-z_N$ survives. Thus ${\varphi }_{\mathrm{E}}=\mathrm{\Delta }z/(2T)$. Adding the same constant to both scores changes neither probability nor rapidity. Raising temperature contracts all rapidities toward balance, a useful way to expose differences hidden by near-saturated outputs. Hinton, Vinyals, and Dean use this temperature control in knowledge distillation.

A prediction market can build the same operation into its cash register. In a logarithmic market scoring rule, or LMSR, $q_Y,q_N$ are outstanding quantities of the two claims, and controls liquidity. Its price is $\mu =\sigma ((q_Y-q_N)/b)$. Therefore buying claims changes rapidity at a constant rate, $d{\varphi }_{\mathrm{E}}/d{q}_Y=1/(2b)$. Hanson’s market-scoring work supplies the construction; Markets that learn follows its actual trading costs.

These models suggest a useful design experiment: tune temperature, test difficulty, or market liquidity so an important decision stays within reach of balance. The same coordinate tells us how much room remains for the output to move.

Chemistry turns concentration into a boost

For an acid $\mathrm{HA}$ and its deprotonated partner ${\mathrm{A}}^{-}$, let $\mu$ be the deprotonated share. In the usual concentration description, Henderson-Hasselbalch says

Square brackets denote concentration; $\mathrm{p}{K}_a$ sets the acid’s balance point. Taking half the natural logarithm gives

$${\varphi }_{\mathrm{E}}=\frac{\ln 10}{2}(\mathrm{pH}-\mathrm{p}{K}_a).$$

So pH is a shifted and rescaled rapidity coordinate. A single pH unit supplies the same odds multiplier as a 400-point Elo advantage. IUPAC’s definition states the underlying acid-base relation.

For a receptor, a ligand is the molecule that binds to it. The Hill law writes occupied fraction as

$$\mu =\frac{[L{]}^n}{K^n+[L{]}^n}.$$

Here $K$ is the half-occupancy concentration, and is the Hill coefficient, which controls the sharpness of the response. Divide occupied by unoccupied fraction:

The coefficient $n$ is a rapidity amplifier. Increasing $n$ stretches the horizontal motion before the sigmoid projects it into occupancy. That is why a cooperative response can switch sharply over a narrow concentration range. This physiological derivation of the Hill curve develops the occupancy law.

An enzyme supplies another handle. In Michaelis-Menten kinetics, write the reaction rate as $r$ and its saturation value as $r_{max}$. Its fractional rate is

$$\mu =\frac{r}{r_{max}}=\frac{[S]}{K_m+[S]},$$

where $[S]$ is substrate concentration and $K_m$ the half-saturation concentration. We recover the Hill form with $n=1$, and ${\varphi }_{\mathrm{E}}=\frac{1}{2}\ln ([S]/K_m)$. Enzyme throughput becomes a velocity-like saturation curve. The original Michaelis-Menten study, in translation, shows the kinetic setting.

The existing article A cell that measures in rapidity explores receptors and buffers in depth. Here we can use them as interchangeable instruments: choose a target occupancy, calculate its rapidity, then ask which chemical control reaches it.

A membrane translates chemistry into voltage

The Nernst equation is an especially inviting bridge because it physically connects two entries in our dictionary. Let $c_{\mathrm{o}\mathrm{u}\mathrm{t}}$ and $c_{\mathrm{i}\mathrm{n}}$ be an ion’s outside and inside concentrations. With membrane potential measured inside relative to outside,

$$E=\frac{RT}{zF}\ln \frac{c_{\mathrm{o}\mathrm{u}\mathrm{t}}}{c_{\mathrm{i}\mathrm{n}}}.$$

The constants are the gas constant $R$, absolute temperature $T$, the ion’s signed charge number $z$, and Faraday’s constant $F$, the charge per mole of elementary charges. Chemical and electrical tendencies balance at this voltage. An ion-transport derivation follows that balance.

Now represent the concentration ratio by the share $\mu =c_{\mathrm{o}\mathrm{u}\mathrm{t}}/(c_{\mathrm{o}\mathrm{u}\mathrm{t}}+c_{\mathrm{i}\mathrm{n}})$. Immediately,

$$\boxed{{\varphi }_{\mathrm{E}}=\frac{zFE}{2RT}.}$$

A concentration ratio has become a voltage measured in rapidity units. The sign tracks which direction the electrical field must oppose the chemical tendency.

That voltage can also control a gate. For a Boltzmann activation curve,

$${\mu }_{\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{n}}=\sigma \left(\frac{V-V_{1/2}}{k}\right),$$

$V_{1/2}$ is the half-opening voltage and a voltage scale. The gate’s rapidity is $(V-V_{1/2})/(2k)$. In a two-state charge-displacement model, $k=RT/(z_{g}F)$, with $z_g$ an effective gating charge. This excitable-cell model writes the activation curve in precisely these thermal and electrical units.

Imagine connecting the two devices. A chemical ratio establishes a Nernst voltage; a voltage-sensitive gate reads it. Under these models, the gate’s rapidity is an affine function of the concentration rapidity. A cell can translate one logarithmic message into another through a membrane.

Thermal weights explain why the same curve keeps appearing

Take two states with free energies $G_Y$ and $G_N$. At temperature $T$, their thermal weights are proportional to $e^{-G_Y/(k_{B}T)}$ and $e^{-G_N/(k_{B}T)}$, where $k_B$ is Boltzmann’s constant. Their ratio gives

Rapidity is half the free-energy advantage measured in thermal units. This gives the repeated sigmoid a physical origin: two exponential weights competing for a normalized share.

For a spin with magnetic moment ${\mu }_{\mathrm{m}\mathrm{a}\mathrm{g}}$ in field $B$, the two energies are $-{\mu }_{\mathrm{m}\mathrm{a}\mathrm{g}}B$ and $+{\mu }_{\mathrm{m}\mathrm{a}\mathrm{g}}B$. Their gap is $2{\mu }_{\mathrm{m}\mathrm{a}\mathrm{g}}B$, so ${\varphi }_{\mathrm{E}}={\mu }_{\mathrm{m}\mathrm{a}\mathrm{g}}B/(k_{B}T)$ and the mean alignment is $\tanh {\varphi }_{\mathrm{E}}$.

For a fermion state of energy $\epsilon$, compare empty and occupied possibilities. Adding the particle brings a free-energy cost $\epsilon -{\mu }_{\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{m}}$, where ${\mu }_{\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{m}}$ is chemical potential. The occupancy rapidity is $({\mu }_{\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{m}}-\epsilon )/(2k_{B}T)$. At $\epsilon ={\mu }_{\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{m}}$, the state is half occupied. Raising temperature broadens the responsive band around that energy. Tong’s quantum-gas notes derive this Fermi-Dirac law.

We can even give rapidity to a relationship. Consider an isolated pair of spins $s_1,s_2\in \{-1,+1\}$ with energy $-J{s}_1{s}_2$ and no external field. The question is now “Do they agree?” Agreement has weight $e^{J/(k_{B}T)}$; disagreement has weight $e^{-J/(k_{B}T)}$. Thus

$$⟨s_1{s}_2⟩=\tanh \left(\frac{J}{k_{B}T}\right).$$

The coupling $J/(k_{B}T)$ is the rapidity of their agreement. An individual spin can remain perfectly balanced while its relationship is strongly oriented. This gives a promising EP picture of a network: rapidities can live on both states and connections. Tong’s Ising-model treatment supplies the interaction energy and explores what many such bonds can do.

Addition, parity, and the geometry of combining influences

Bayesian updating multiplies prior odds by a likelihood ratio, $\mathrm{\Lambda }$, comparing how strongly an observation is predicted by two alternatives:

$$O^{\prime }=O\mathrm{\Lambda }\Rightarrow {\varphi }_{\mathrm{E}}^{\prime }={\varphi }_{\mathrm{E}}+\frac{1}{2}\ln \mathrm{\Lambda }.$$

Each independent message about the same question contributes a rapidity increment. Conditional likelihoods let us do the same with messages that arrive along a dependent history. Keeping their lineage tells us which factor belongs at each step.

Relativistic boosts along one axis add in the same coordinate. If two increments have centered velocities $v_1=\tanh {\varphi }_1$ and $v_2=\tanh {\varphi }_2$, then

$$v_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}}=\frac{v_1+v_2}{1+v_1{v}_2}.$$

This is the addition formula for $\tanh ({\varphi }_1+{\varphi }_2)$. A little pi between belief and relativity works through the Bayes/boost correspondence numerically.

Coding theory has another operation waiting beside addition. An LLR, or log-likelihood ratio, is a message $L=2{\varphi }_{\mathrm{E}}$. Independent messages about one bit add their LLRs. A parity check asks whether two bits agree. If their signed means are $v_1,v_2$, independence gives the mean of their product:

That is the decoder’s tanh rule. The same operation propagates agreement along a zero-field Ising chain. To infer how the endpoints relate, multiply the intermediate bond correlations. Coding gates and spin interactions now share an algebra of relationships, an idea developed further in the decoder field trip. Arıkan’s channel-polarization paper shows how combining binary channels opens a route to efficient communication.

Evolution and molecular motion travel along the line

Suppose two types reproduce with a constant difference $s$ in per-capita growth rate. If $\mu$ is the first type’s population share, its replicator equation is

$$\frac{d\mu }{dt}=s\mu (1-\mu ).$$

Because $d{\varphi }_{\mathrm{E}}/d\mu =1/[2\mu (1-\mu )]$, the apparently nonlinear motion becomes

$$\frac{d{\varphi }_{\mathrm{E}}}{dt}=\frac{s}{2},{\varphi }_{\mathrm{E}}(t)={\varphi }_0+\frac{st}{2}.$$

Constant selective advantage is constant rapidity speed. If selection varies in time, replace $st$ with its accumulated integral. Harper’s paper explores the wider connection between replicator dynamics and inference; Evolution has an order adds mutation and shows why the sequence of operations matters.

Now shrink to a molecular walker making forward and backward steps with rates $k_{+}$ and $k_{-}$. The probability its next step is forward is $\mu =k_{+}/(k_{+}+k_{-})$. Therefore

The second line measures directional drift per step. If each step covers distance $a$, ordinary drift speed is $a(k_{+}+k_{-})\tanh {\varphi }_{\mathrm{E}}$. Directional bias and the total activity of the walker now play separate, visible roles.

Under local detailed balance, the dimensionless driving affinity $\mathcal{A}$ is $\ln (k_{+}/k_{-})$. For a simple isothermal step releasing free energy $\mathrm{\Delta }G$, take $\mathcal{A}=\mathrm{\Delta }G/(k_{B}T)$. We obtain ${\varphi }_{\mathrm{E}}=\mathcal{A}/2$: half the thermodynamic drive is a rapidity. Seifert’s molecular-machine treatment develops the rate-ratio and thermodynamic connection.

Here is an adventurous interpretation to try: a driven molecule advances because its forward and backward possibilities carry unequal weights, just as a belief advances when support and opposition differ. The walker gives that picture laboratory handles: drift, step activity, and released energy can all be measured.

A stake and an ellipse reveal two more faces

In an even-money binary wager, let $f$ be the signed fraction allocated to the positive side. Wealth multiplies by $1+f$ on a positive outcome and $1-f$ on a negative one. Maximizing expected logarithmic growth means maximizing

For , its derivative vanishes at

The Kelly-optimal signed stake is epistemic velocity. Its rapidity is $\mathrm{artanh}{f}^{\ast }$. A negative value selects the opposite side. Different payouts change the optimization, giving us another family of decision geometries to explore. Kelly’s original paper connects logarithmic growth with information.

A sensor array offers a spatial version. Suppose its information matrix has positive eigenvalues ${\lambda }_{+}\ge {\lambda }_{-}$. These describe how precisely it measures its strongest and weakest directions. Normalize the stronger one into a share:

$$\mu =\frac{{\lambda }_{+}}{{\lambda }_{+}+{\lambda }_{-}}.$$

Its odds are the matrix’s condition number, $\kappa ={\lambda }_{+}/{\lambda }_{-}$. Thus ${\varphi }_{\mathrm{E}}=\frac{1}{2}\ln \kappa$. Equal precision gives zero rapidity; a lopsided array has large rapidity. The same half-log ratio that measures a chess advantage now measures geometric imbalance. The best sensor is looking sideways lets you rearrange the sensors and watch the uncertainty ellipse change.

A quantum state supplies another pair of eigenvalues. A qubit, a quantum system with two basis states, has a state matrix whose eigenvalues are ${\lambda }_{\pm }=(1\pm |\mathbf{r}|)/2$. Here $\mathbf{r}$ is its Bloch vector, an arrow inside the unit ball that describes the state. The eigenvalues sum to one, so the larger eigenvalue is already a share:

This spectral rapidity measures how far the state has moved radially from complete mixture toward purity. Give it a communication task: average two pure qubit messages sent equally often. Their average state’s radial rapidity determines the receiver’s smallest possible error,

$$P_{\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}}^{\ast }=\frac{1-\mathrm{sech}{\varphi }_{\mathrm{E}}}{2},$$

where $\mathrm{sech}\varphi =1/\cosh \varphi$. It makes a delightful prediction: move the average toward the center, and the messages become easier to tell apart. We can trade the language of mixture and purity for motion and rapidity, then turn the geometry into a receiver design. How far apart are two quantum messages? derives the relation and lets you steer the messages and the measuring axis.

A prediction the whole table can share

We have a dictionary. Let’s make it do some work.

Differentiate the common velocity law and introduce the Lorentz factor $\gamma =1/\sqrt{1-v_{\mathrm{E}}^2}$:

$$\boxed{\frac{d{v}_{\mathrm{E}}}{d{\varphi }_{\mathrm{E}}}=1-v_{\mathrm{E}}^2=\frac{1}{{\gamma }^2}.}$$

Every row inherits this response curve. The motion is greatest near balance and decreases toward either saturated end. For the original share, $d\mu /d{\varphi }_{\mathrm{E}}=1/(2{\gamma }^2)$.

Predict what happens if we give an equally sized rapidity push to a balanced system and a nearly saturated one. Then compare that prediction with the exact finite moves in the experiment.

Predict the response

Which world yields to the push?

Reset experiment

1 / 3 · Near balance or saturation? Which share changes more? Choose A, B, or equal movement.

Push both points right by 0.1 in rapidity.

Share μ

Common rapidity φ_E

Predict: which share moves farther?

A B Same movement

Try the three comparisonsBuild your own

Change the world and its units

Shift both pH values by +0.087.

Tell the experiment through Chemical balanceChess expectationsLigand bindingIon-channel openingFermi occupancyLearning and coding

Inspect both systems and the response law

System A

Starting deprotonated share

50.0%

pH − pKₐ

0.00 pH units

Rapidity φ_E

0.00

Lorentz factor γ

1.00

Local response χ

1.000

System B

Starting deprotonated share

93.1%

pH − pKₐ

1.13 pH units

Rapidity φ_E

1.30

Lorentz factor γ

1.97

Local response χ

0.257

$$\frac{d{v}_{\mathrm{E}}}{d{\varphi }_{\mathrm{E}}}={\mathrm{sech}}^2{\varphi }_{\mathrm{E}}=\frac{1}{{\gamma }^2}$$ $$\mathrm{\Delta }\mu \approx \frac{\mathrm{\Delta }{\varphi }_{\mathrm{E}}}{2{\gamma }^2}$$

Susceptibility χ means the centered mean response to a tiny rapidity push. Its gold curve peaks at balance and fades toward saturation. The local estimate uses its starting value; the exact result follows the whole bend. Change fields and the same experiment survives in new units. Drag either point, or select A or B and use the arrow buttons or keys.

Open the response experiment in the playground.

At the same starting rapidity, the same rapidity push gives the same final share across the table. Different starting rapidities respond differently. The finite rule is

$$v_{\mathrm{E}}^{\prime }=\frac{v_{\mathrm{E}}+\tanh \delta \varphi }{1+v_{\mathrm{E}}\tanh \delta \varphi }.$$

To return to an instrument’s native units, multiply by the rate at which its control changes rapidity. For a pH-controlled acid and a Hill receptor,

The curve’s shape travels intact. The field supplies its ruler. This extends the susceptibility connection in A coin, a magnet, and a learning machine into a whole table of response predictions.

What could we build from this?

Try treating the table as a parts drawer.

Build a chemical scorekeeper. Encode a score difference as a concentration ratio. A receptor reads that ratio as occupancy, and a gate converts occupancy into a signal. Calibration asks how an Elo point, a logit, and a pH change line up along one rapidity axis.

Look for a common response curve in real data. Measure an enzyme, a gate, and a classifier across their operating ranges. Convert each input to its proposed rapidity and each output to $2\mu -1$. The simple models predict collapse onto $\tanh {\varphi }_{\mathrm{E}}$. A systematic departure could point toward interacting states, a changing gain, or memory. Those departures become clues for the next model.

Give the second coordinate a physical job. Two systems can share the same rapidity with different total weights. A large buffer and a tiny droplet can have the same composition. A busy walker and a slow walker can have the same directional bias. A belief supported by many trials and one supported by few can have the same mean. Perhaps some of the most useful cross-field discoveries live in how this total amount controls response to an added particle, a perturbation, or a new tracelet.

Let relationships carry rapidity too. The Ising bond and parity gate invite a network where agreement has its own coordinate. Local relations could compose into long-range predictions. Add feedback, delays, or cycles, and we can ask when the network settles, oscillates, or remembers its path.

The next experiment can begin with any row: choose what the two weights mean, discover what moves their ratio, and follow that motion into another world.

Keep exploring

• For the underlying EP geometry, begin with the fundamentals and the Bayes/boost connection.
• For living implementations, visit chemical sensing, selection and mutation, and voltage activation models.
• For machines that use these equations, explore the rapidity decoder, market pricing, and sensor geometry.
• For thermal and informational depth, follow the coin/magnet/learning connection, Ising interactions, and stochastic thermodynamics.