The geometry of balance

The invariant interval and epistemic mass

In special relativity, a particle’s invariant rest mass obeys $m^2=E^2-p^2$, where $E$ is energy, $p$ is momentum, and the speed of light is set to 1. The spacetime interval $s^2=t^2-x^2$ has the same algebraic form, but measures squared proper time for timelike separation.

Calculate this difference of squares using our cumulative epistemic coordinates:

The squared difference between total evidence ${\tau }_{\mathrm{E}}$ and net balance $x_{\mathrm{E}}$ simplifies to $4\alpha \beta$!

Its square root, $m_{\mathrm{E}}=2\sqrt{\alpha \beta }$, is the epistemic mass of the accumulated belief. Both channels contribute to this cone norm. To meet energy and momentum, the next chapter follows changes in the ledger: packets exchanged during a collision.

Bernoulli photons and lightlike trials

What happens when you record a single new observation?

A single support trial adds $1$ to support and $0$ to opposition: $\mathrm{\Delta }\alpha =1,\mathrm{\Delta }\beta =0$.

Its interval increment is $\mathrm{\Delta }{s}_{\mathrm{E}}^2=4\mathrm{\Delta }\alpha \mathrm{\Delta }\beta =4(1)(0)=0$.

In physics, any trajectory with $d{s}^2=0$ travels along the light cone at the speed of light: only massless particles, like photons, travel this way. Each individual trial is an epistemic photon!

When positive and negative photons collide and are absorbed into your notebook, they generate rest mass.

What about retracting evidence? A negative trial increment (an un-observation) acts like a past-directed photon, stepping backward along the cone. An update that adds support while subtracting opposition is spacelike, moving faster than belief light.

The formula for belief acceleration

Let $s\in \{+1,-1\}$ denote the direction of a new incoming trial (+1 for support, -1 for opposition).

When this trial arrives, how much does our epistemic velocity shift?

The denominator contains the total accumulated evidence ${\tau }_{\mathrm{E}}+1$.

If ${\tau }_{\mathrm{E}}=2$, a new opposing observation ($s=-1$) produces a massive jolt. If ${\tau }_{\mathrm{E}}=\mathrm{10,000}$, that exact same evidence produces an imperceptible nudge. The response is set by the accumulated amount. The cone identity ${\tau }_{\mathrm{E}}={\gamma }_{\mathrm{E}}{m}_{\mathrm{E}}$ expresses that amount through epistemic mass and the Lorentz factor ${\gamma }_{\mathrm{E}}=1/\sqrt{1-v_{\mathrm{E}}^2}$.

Keep the two directions in view: $v_{\mathrm{E}}=x_{\mathrm{E}}/{\tau }_{\mathrm{E}}$ summarizes the notebook so far, while $\mathrm{\Delta }{x}_{\mathrm{E}}/\mathrm{\Delta }{\tau }_{\mathrm{E}}=s$ describes this incoming trial. The update changes the accumulated mean by blending that new direction into the existing amount.