A little reference shelf

Names for the things we’re exploring.

The subscript E means “epistemic,” or related to knowledge. Keep this shelf open when a new symbol joins the story.

Find a term

✧Trial

An amount in the support or opposition channel. Philosophically, accumulated trials describe how resistant an agent, component, or neuron is to changing its mind. Trials can also represent evidence or observed outcomes.

✧Tracelet

An elementary amount of evidence in the broader framework.

✧Lineage

The source history connecting trials, tracelets, and their appearances. Distinct trial identities do not by themselves establish statistical independence.

✧Covariance

A description of how components vary together. Its matrix also records each component’s variance along the diagonal.

$Y,N$Support and opposition

Trials for and trials against a proposition, also read as true/false or belief/disbelief. Together these channels determine the amount and shape of a belief once its prior is specified.

$n_0$Prior concentration

The total weight of the starting Beta prior, split equally here between support and opposition.

$\alpha ,\beta$Beta parameters

Prior-inclusive parameters: $\alpha =Y+n_0/2$ and $\beta =N+n_0/2$. A proper Beta distribution requires both to be positive.

$B$Belief

An uncertain belief about a proposition, modeled here as $B\sim \mathrm{Beta}(\alpha ,\beta )$. Its distribution describes which probabilities are plausible.

✧Mode

The location of a peak in a belief distribution. One philosophical reading is its degree of true-ness or false-ness: 99% is very sure it is true, 1% very sure it is false, and 50% unsure. Trial amount describes resistance to changing that state. For a Beta with , the mode is $(\alpha -1)/(\alpha +\beta -2)$.

✧Malleability

How readily an agent, component, or neuron can be swayed by a new trial. At a given degree of true-ness or false-ness, few trials make a belief easier to change; many trials make it more resistant. Variance and credible intervals describe the distribution’s spread, which is another aspect of its shape.

✧Dirichlet belief

A belief distribution over the proportions of mutually exclusive choices. One interpretation is weighted votes for A, B, C, and so on. The choice weights generalize the two support/opposition channels of a Beta belief.

$\mu$Mean probability

The average of the Beta distribution: $\mu =\alpha /(\alpha +\beta )$.

$t_{\mathrm{E}}$Epistemic time

The accumulated raw evidence $t_{\mathrm{E}}=Y+N$, excluding the prior. Each new trial advances this clock.

${\tau }_{\mathrm{E}}$Prior-inclusive epistemic time

Epistemic time plus the starting prior: ${\tau }_{\mathrm{E}}=t_{\mathrm{E}}+n_0=\alpha +\beta$. It is the vertical coordinate in the evidence cone.

$x_{\mathrm{E}}$Epistemic position

Support minus opposition: $\alpha -\beta =Y-N$. It is the horizontal coordinate in the evidence cone.

$v_{\mathrm{E}}$Epistemic velocity

The centered mean $2\mu -1=x_{\mathrm{E}}/{\tau }_{\mathrm{E}}$. A dimensionless evidence coordinate with values strictly between −1 and 1 for a proper Beta state.

$m_{\mathrm{E}}$Epistemic mass

The positive cone norm of the accumulated belief: $m_{\mathrm{E}}=2\sqrt{\alpha \beta }$ and $m_{\mathrm{E}}^2={\tau }_{\mathrm{E}}^2-x_{\mathrm{E}}^2$. It includes the starting prior and records contributions from both channels.

$\mathrm{\Delta }{\tau }_{\mathrm{E}},\mathrm{\Delta }{x}_{\mathrm{E}}$Evidence update

The change contributed by an incoming packet: $\mathrm{\Delta }{\tau }_{\mathrm{E}}=\mathrm{\Delta }\alpha +\mathrm{\Delta }\beta$ and $\mathrm{\Delta }{x}_{\mathrm{E}}=\mathrm{\Delta }\alpha -\mathrm{\Delta }\beta$. Its local direction is $\mathrm{\Delta }{x}_{\mathrm{E}}/\mathrm{\Delta }{\tau }_{\mathrm{E}}$ when the incoming amount is positive. The accumulated mean $v_{\mathrm{E}}=x_{\mathrm{E}}/{\tau }_{\mathrm{E}}$ describes the whole notebook so far.

$E_{\mathrm{packet}}$Energy carried by an exchange

The received packet's energy is $E_{\mathrm{packet}}=\epsilon \mathrm{\Delta }{\tau }_{\mathrm{E}}$. Choose as the energy per trial in units with $c=1$. Fully receiving the packet changes the recipient's physical energy by $\mathrm{\Delta }E=E_{\mathrm{packet}}$; the starting belief prior contributes no exchanged energy.

$p_{\mathrm{packet}}$Momentum carried by an exchange

The received packet's signed momentum is $p_{\mathrm{packet}}=\epsilon \mathrm{\Delta }{x}_{\mathrm{E}}$, using the same positive energy-per-trial calibration and $c=1$. It changes the recipient's physical momentum by $\mathrm{\Delta }p=p_{\mathrm{packet}}$. A closed transfer gives the donor the opposite energy and momentum changes.

$E,p$Physical energy and momentum

For a free massive particle, $E=\gamma m{c}^2$ and $p=\gamma mv$. Energy carries amount; momentum also carries direction. A closed collision preserves their totals. With $c=1$, velocity is $v=p/E$.

$M$System mass

The invariant mass of a collection: $M^2=E_{\mathrm{tot}}^2-p_{\mathrm{tot}}^2$ when $c=1$. It includes the energy of relative motion and can exceed the sum of the individual rest masses. Opposing light pulses can have positive system mass although each pulse has zero rest mass.

$K$Kinetic energy

Total energy above rest energy: $K=E-m{c}^2$, or $K=E-m$ when $c=1$. In a closed merge, incoming relative kinetic energy becomes internal energy of the composite, contributing to its rest mass.

$(t,x)$Spacetime event and worldline

An event specifies a time t and position x. A worldline is a path of such events; its local slope $dx/dt$ is velocity. For a massive particle in units with $c=1$, $E=mdt/ds$ and $p=mdx/ds$, where s is proper time.

$s$Proper time

Time measured along a timelike worldline by a traveling clock. In units with $c=1$, $d{s}^2=d{t}^2-d{x}^2$ and $ds/dt=1/\gamma =m/E$. This physical clock has a different definition from the accumulated epistemic time ${\tau }_{\mathrm{E}}$.

${\gamma }_{\mathrm{E}}$Epistemic Lorentz factor

The factor $1/\sqrt{1-v_{\mathrm{E}}^2}$. It describes mean polarization, not the full uncertainty of a belief.

${\varphi }_{\mathrm{E}}$Mean-state rapidity

The coordinate $\mathrm{artanh}(v_{\mathrm{E}})=\frac{1}{2}\log (\alpha /\beta )$. Unlike velocity, it is not bounded.

${\mathrm{\Phi }}_{\mathrm{E}}$Random belief in rapidity

The full transformed random belief ${\mathrm{\Phi }}_{\mathrm{E}}=\mathrm{artanh}(B_{\mathrm{E}})=\frac{1}{2}\log (B/(1-B))$. Its expectation generally differs from the rapidity ${\varphi }_{\mathrm{E}}$ of the centered mean.

$\varpi ,{\varpi }_{\mathrm{E}}$Jeffreys prior densities

The symmetric Beta prior with shapes one half. In probability coordinates, $\varpi (\mu )=1/(\pi \sqrt{\mu (1-\mu )})$; in centered coordinates, ${\varpi }_{\mathrm{E}}(v_{\mathrm{E}})=1/(\pi \sqrt{1-v_{\mathrm{E}}^2})$. The coordinate in a density is its possible argument, rather than a fixed posterior mean.

$1/K$Directional prior per bin

For K equal bins, each gets 1/K so the total directional prior remains one when the representation is refined.