--- title: "One rapidity, many worlds" author: "Miraia S. Chiou" creator: "Miraia Chiou" published: "2026-09-13" identifier: "ep_post_c4a89fcf422645149702806b8825efba" source: "https://epistemicphysics.com/articles/one-rapidity-many-worlds/" citation: "https://epistemicphysics.com/cite/?work=ep_post_c4a89fcf422645149702806b8825efba" --- # One rapidity, many worlds A chess advantage, a chemical gradient, a neural score, and a relativistic boost can share one coordinate. Translate 21 connections and explore the predictions they suggest. [Read the illustrated article](https://epistemicphysics.com/articles/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: $$ \boxed{\phi_{\mathrm E}=\operatorname{artanh}(v_{\mathrm E}).} $$ The inverse hyperbolic tangent, $\operatorname{artanh}$, unrolls the bounded velocity interval onto an unbounded line. Its inverse is $$ v_{\mathrm E}=\tanh\phi_{\mathrm E}. $$ Ordinary relativity uses the same construction: $u/c=\tanh\phi$, where $u$ is speed along a chosen axis and $c$ is the speed of light. The hyperbolic tangent, $\tanh\phi=(e^\phi-e^{-\phi})/(e^\phi+e^{-\phi})$, 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 $$ \begin{aligned} \phi_{\mathrm E} &=\frac12\ln\frac{1+v_{\mathrm E}}{1-v_{\mathrm E}}\\ &=\frac12\ln\frac{\mu}{1-\mu}. \end{aligned} $$ 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**: $$ \begin{aligned} \sigma(x)&=\frac1{1+e^{-x}},\\ \mu&=\sigma(2\phi_{\mathrm E}). \end{aligned} $$ We can see the light cone directly. Give two positive weights $\alpha$ and $\beta$ the ratio $\mu/(1-\mu)$, and write $$ \begin{aligned} \tau_{\mathrm E}&=\alpha+\beta,&x_{\mathrm E}&=\alpha-\beta,\\ m_{\mathrm E}&=2\sqrt{\alpha\beta}.&& \end{aligned} $$ Then $v_{\mathrm E}=x_{\mathrm E}/\tau_{\mathrm E}$, and the velocity definition gives another useful expression: $$ \phi_{\mathrm E}=\operatorname{artanh}v_{\mathrm E} =\frac12\ln\frac\alpha\beta. $$ The light-cone identity is $\tau_{\mathrm E}^2-x_{\mathrm E}^2=m_{\mathrm E}^2$. Its hyperbolic coordinates are $$ \begin{aligned} \tau_{\mathrm E}&=m_{\mathrm E}\cosh\phi_{\mathrm E}=\alpha+\beta,\\ x_{\mathrm E}&=m_{\mathrm E}\sinh\phi_{\mathrm E}=\alpha-\beta. \end{aligned} $$ 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. 21 ways to meet $\phi_{\mathrm E}=\operatorname{artanh}v_{\mathrm E}$. Here $\mu$ is a probability or share, and $\sigma(x)=1/(1+e^{-x})$. | Field | Familiar law | Rapidity $\phi_{\mathrm E}$ | | ------------------------- | ------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------- | | Statistics | Logistic regression $\displaystyle \mu=\sigma(\ell)$ | $\displaystyle \ell/2$ | | Coding theory | LLR messages and the tanh rule $\displaystyle \mu=\sigma(L)$ | $\displaystyle L/2$ | | Chess | Elo expected score $\displaystyle \mu=\frac{1}{1+10^{-\Delta_{\rm Elo}/400}}$ | $\displaystyle \frac{\ln 10}{800}\Delta_{\rm Elo}$ | | Psychometrics | Rasch model $\displaystyle \mu=\sigma(a-d)$ | $\displaystyle (a-d)/2$ | | Chemistry | Henderson-Hasselbalch $\displaystyle \frac{\mu}{1-\mu}=10^{\mathrm{pH}-\mathrm{p}K_a}$ | $\displaystyle \frac{\ln 10}{2}(\mathrm{pH}-\mathrm{p}K_a)$ | | Biochemistry | Hill occupancy $\displaystyle \mu=\frac{[L]^n}{K^n+[L]^n}$ | $\displaystyle \frac n2\ln\frac{[L]}K$ | | Enzymology | Michaelis-Menten fractional rate $\displaystyle \mu=\frac{[S]}{K_m+[S]}$ | $\displaystyle \frac12\ln\frac{[S]}{K_m}$ | | Electrophysiology | Boltzmann gating $\displaystyle \mu=\sigma\!\left(\frac{V-V_{1/2}}k\right)$ | $\displaystyle \frac{V-V_{1/2}}{2k}$ | | Membrane biology | Nernst equilibrium $\displaystyle \frac{c_{\rm out}}{c_{\rm in}}=e^{zFE/(RT)}$ | $\displaystyle \frac{zFE}{2RT}$ | | Statistical mechanics | Spin-½ magnetization $\displaystyle \langle s\rangle=\tanh\!\left(\frac{\mu_{\rm mag}B}{k_B T}\right)$ | $\displaystyle \frac{\mu_{\rm mag}B}{k_B T}$ | | Quantum statistics | Fermi-Dirac occupancy $\displaystyle \mu=\sigma\!\left(\frac{\mu_{\rm chem}-\varepsilon}{k_B T}\right)$ | $\displaystyle \frac{\mu_{\rm chem}-\varepsilon}{2k_B T}$ | | Prediction markets | LMSR price $\displaystyle \mu=\sigma\!\left(\frac{q_Y-q_N}b\right)$ | $\displaystyle \frac{q_Y-q_N}{2b}$ | | Machine learning | Binary softmax, temperature T $\displaystyle \mu=\sigma(\Delta z/T)$ | $\displaystyle \Delta z/(2T)$ | | Evolution | Replicator solution $\displaystyle \mu(t)=\sigma(st+2\phi_0)$ | $\displaystyle st/2+\phi_0$ | | Decision theory | Kelly, even-money stake $\displaystyle f^*=2\mu-1$ | $\displaystyle \operatorname{artanh}f^*$ | | Special relativity | Velocity and boosts $\displaystyle u/c=\tanh\phi_{\mathrm E}$ | $\displaystyle \operatorname{artanh}(u/c)$ | | Bayesian inference | Likelihood-ratio update $\displaystyle O'=O\Lambda$ | $\displaystyle \phi'_{\mathrm E}=\phi_{\mathrm E}+\tfrac12\ln\Lambda$ | | Interacting spins | A zero-field Ising bond $\displaystyle \langle s_1s_2\rangle=\tanh\!\left(\frac J{k_B T}\right)$ | $\displaystyle \frac J{k_B T}$ | | Stochastic thermodynamics | Forward and backward hopping $\displaystyle \mu=\frac{k_+}{k_++k_-}$ | $\displaystyle \frac12\ln\frac{k_+}{k_-}=\frac{\mathcal A}{2}$ | | Sensor geometry | Share of the stronger eigenvalue $\displaystyle \mu=\frac{\lambda_+}{\lambda_++\lambda_-}$ | $\displaystyle \frac12\ln\kappa$ | | Quantum state spectra | A qubit’s larger eigenvalue $\displaystyle \lambda_+=\frac{1+|\mathbf r|}{2}$ | $\displaystyle \frac12\ln\frac{\lambda_+}{\lambda_-}$ | The table uses $\mu$ for a probability or share. The thermal rows distinguish a magnetic moment $\mu_{\rm mag}$ from a chemical potential $\mu_{\rm chem}$. 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. [Turn one dial. Visit several worlds.](https://epistemicphysics.com/articles/one-rapidity-many-worlds/#rapidity-translator-lab) [Open the translator in the playground](https://epistemicphysics.com/lab/?demo=rapidity-translator#experiment). The numbers printed on the instruments differ. Their common coordinates agree: $$ \begin{aligned} O&=10, &\mu&=\frac{10}{11},\\ \phi_{\mathrm E}&=\frac{\ln10}{2}, &v_{\mathrm E}&=\frac9{11}. \end{aligned} $$ 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**, $$ B\sim\operatorname{Beta}(\alpha,\beta). $$ 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: $$ \begin{aligned} \Phi_{\mathrm E} &=\operatorname{artanh}(B_{\mathrm E})\\ &=\frac12\ln\frac{1+B_{\mathrm E}}{1-B_{\mathrm E}}\\ &=\frac12\ln\frac B{1-B}. \end{aligned} $$ The uppercase $\Phi_{\mathrm E}$ carries a whole distribution. The lowercase $\phi_{\mathrm E}$ names the rapidity of the state’s mean velocity. Their inverse relations line up as $$ \begin{aligned} B_{\mathrm E}&=\tanh\Phi_{\mathrm E},\\ B&=\frac{1+\tanh\Phi_{\mathrm E}}2,\\ v_{\mathrm E}&=\tanh\phi_{\mathrm E}. \end{aligned} $$ Averaging the first line gives the exact relation $$ \begin{aligned} \mathbb E[B_{\mathrm E}] &=\mathbb E[\tanh\Phi_{\mathrm E}]\\ &=v_{\mathrm E}=\tanh\phi_{\mathrm E}. \end{aligned} $$ A curved transformation and an average can land at different addresses depending on their order. In general, $\mathbb E[\Phi_{\mathrm E}]\ne\phi_{\mathrm E}$. For example, a $\operatorname{Beta}(2,1)$ belief has mean velocity $1/3$ and scalar rapidity $\tfrac12\ln2\approx0.347$. Averaging the transformed distribution instead gives $\mathbb E[\Phi_{\mathrm E}]=1/2$. 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. [Stretch the whole belief into rapidity](https://epistemicphysics.com/articles/one-rapidity-many-worlds/#belief-rapidity-lab) [Open the whole-belief experiment in the playground](https://epistemicphysics.com/lab/?demo=belief-rapidity#experiment). 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](https://epistemicphysics.com/fields/statistics/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](https://stat.cmu.edu/~cshalizi/dm/20/lectures/07/lecture-07.html) makes the log-odds construction explicit. Chess changes the ruler. In the usual logistic Elo model, a rating advantage $\Delta_{\rm Elo}$ gives $$ \mu=\frac1{1+10^{-\Delta_{\rm Elo}/400}}. $$ Here $\mu$ is expected score, with a draw counting as half a point. Taking odds and logs gives $\phi_{\mathrm E}=(\ln10)\Delta_{\rm Elo}/800$. The rating difference becomes a rapidity distance. [Glickman’s account of the rating model](https://glicko.net/research/chance.pdf) 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 $\phi_{\mathrm E}=(a-d)/2$. 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](https://www.rasch.org/memo46.htm) 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 $\Delta z=z_Y-z_N$ survives. Thus $\phi_{\mathrm E}=\Delta z/(2T)$. 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](https://www.cs.toronto.edu/~hinton/absps/distillation.pdf). 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 $b>0$ controls liquidity. Its price is $\mu=\sigma((q_Y-q_N)/b)$. Therefore buying claims changes rapidity at a constant rate, $d\phi_{\mathrm E}/dq_Y=1/(2b)$. [Hanson’s market-scoring work](https://hanson.gmu.edu/mktscore.pdf) supplies the construction; [Markets that learn](https://epistemicphysics.com/fields/economics/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 $$ \begin{aligned} \frac{\mu}{1-\mu}&=\frac{[\mathrm A^-]}{[\mathrm{HA}]}\\ &=10^{\mathrm{pH}-\mathrm{p}K_a}. \end{aligned} $$ Square brackets denote concentration; $\mathrm pK_a$ sets the acid’s balance point. Taking half the natural logarithm gives $$ \phi_{\mathrm E}=\frac{\ln10}{2}(\mathrm{pH}-\mathrm{p}K_a). $$ 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](https://old.goldbook.iupac.org/html/H/H02781.html) 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 $n>0$ is the Hill coefficient, which controls the sharpness of the response. Divide occupied by unoccupied fraction: $$ \begin{aligned} \frac \mu{1-\mu}&=\left(\frac{[L]}K\right)^n,\\ \phi_{\mathrm E}&=\frac n2\ln\frac{[L]}K. \end{aligned} $$ 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](https://journals.physiology.org/doi/10.1152/advan.00012.2007) 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 $\phi_{\mathrm E}=\tfrac12\ln([S]/K_m)$. Enzyme throughput becomes a velocity-like saturation curve. [The original Michaelis-Menten study, in translation](https://www.chem.uwec.edu/Chem352_F18/pages/readings/media/Michaelis_%26_Menton_1913.pdf), shows the kinetic setting. The existing article [A cell that measures in rapidity](https://epistemicphysics.com/articles/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 $c_{\rm out}$ and $c_{\rm in}$ be an ion’s outside and inside concentrations. With membrane potential measured inside relative to outside, $$ E=\frac{RT}{zF}\ln\frac{c_{\rm out}}{c_{\rm in}}. $$ 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](https://www.imagwiki.nibib.nih.gov/sites/default/files/07elect_1.pdf) follows that balance. Now represent the concentration ratio by the share $\mu=c_{\rm out}/(c_{\rm out}+c_{\rm in})$. Immediately, $$ \boxed{\phi_{\mathrm E}=\frac{zFE}{2RT}.} $$ 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, $$ \mu_{\rm open}=\sigma\!\left(\frac{V-V_{1/2}}k\right), $$ $V_{1/2}$ is the half-opening voltage and $k>0$ a voltage scale. The gate’s rapidity is $(V-V_{1/2})/(2k)$. In a two-state charge-displacement model, $k=RT/(z_gF)$, with $z_g$ an effective gating charge. [This excitable-cell model](https://journals.physiology.org/doi/full/10.1152/advan.00039.2014) 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 $e^{-G_Y/(k_BT)}$ and $e^{-G_N/(k_BT)}$, where $k_B$ is Boltzmann’s constant. Their ratio gives $$ \begin{aligned} \frac \mu{1-\mu}&=e^{(G_N-G_Y)/(k_BT)},\\ \phi_{\mathrm E}&=\frac{G_N-G_Y}{2k_BT}. \end{aligned} $$ 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 $\mu_{\rm mag}$ in field $B$, the two energies are $-\mu_{\rm mag} B$ and $+\mu_{\rm mag} B$. Their gap is $2\mu_{\rm mag} B$, so $\phi_{\mathrm E}=\mu_{\rm mag} B/(k_BT)$ and the mean alignment is $\tanh\phi_{\mathrm E}$. For a fermion state of energy $\varepsilon$, compare empty and occupied possibilities. Adding the particle brings a free-energy cost $\varepsilon-\mu_{\rm chem}$, where $\mu_{\rm chem}$ is chemical potential. The occupancy rapidity is $(\mu_{\rm chem}-\varepsilon)/(2k_BT)$. At $\varepsilon=\mu_{\rm chem}$, the state is half occupied. Raising temperature broadens the responsive band around that energy. [Tong’s quantum-gas notes](https://www.damtp.cam.ac.uk/user/tong/statphys/statmechhtml/S3.html) 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 $-Js_1s_2$ and no external field. The question is now “Do they agree?” Agreement has weight $e^{J/(k_BT)}$; disagreement has weight $e^{-J/(k_BT)}$. Thus $$ \langle s_1s_2\rangle=\tanh\!\left(\frac J{k_BT}\right). $$ The coupling $J/(k_BT)$ 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](https://www.damtp.cam.ac.uk/user/tong/sft/sfthtml/S1.html) 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**, $\Lambda$, comparing how strongly an observation is predicted by two alternatives: $$ O'=O\Lambda \quad\Rightarrow\quad \phi'_{\mathrm E}=\phi_{\mathrm E}+\frac12\ln\Lambda. $$ 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 $v_1=\tanh\phi_1$ and $v_2=\tanh\phi_2$, then $$ v_{\rm combined}=\frac{v_1+v_2}{1+v_1v_2}. $$ This is the addition formula for $\tanh(\phi_1+\phi_2)$. [A little pi between belief and relativity](https://epistemicphysics.com/articles/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 $L=2\phi_{\mathrm E}$. 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: $$ \begin{aligned} v_{\rm parity}&=v_1v_2,\\ \phi_{\rm parity} &=\operatorname{artanh}(\tanh\phi_1\tanh\phi_2). \end{aligned} $$ 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](https://epistemicphysics.com/fields/information-theory/a-decoder-that-computes-in-rapidity/). [Arıkan’s channel-polarization paper](https://arxiv.org/abs/0807.3917) 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 $d\phi_{\mathrm E}/d\mu=1/[2\mu(1-\mu)]$, the apparently nonlinear motion becomes $$ \frac{d\phi_{\mathrm E}}{dt}=\frac s2, \qquad \phi_{\mathrm E}(t)=\phi_0+\frac{st}{2}. $$ Constant selective advantage is constant rapidity speed. If selection varies in time, replace $st$ with its accumulated integral. [Harper’s paper](https://arxiv.org/abs/0911.1763) explores the wider connection between replicator dynamics and inference; [Evolution has an order](https://epistemicphysics.com/fields/biology/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 $$ \begin{aligned} \phi_{\mathrm E}&=\frac12\ln\frac{k_+}{k_-},\\ \frac{k_+-k_-}{k_++k_-}&=\tanh\phi_{\mathrm E}. \end{aligned} $$ The second line measures directional drift per step. If each step covers distance $a$, ordinary drift speed is $a(k_++k_-)\tanh\phi_{\mathrm E}$. 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 $\Delta G$, take $\mathcal A=\Delta G/(k_BT)$. We obtain $\phi_{\mathrm E}=\mathcal A/2$: half the thermodynamic drive is a rapidity. [Seifert’s molecular-machine treatment](https://arxiv.org/abs/1012.3119) 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 $$ \begin{aligned} g(f)&=\mu\ln(1+f)\\ &\quad +(1-\mu)\ln(1-f). \end{aligned} $$ For $-1