When evidence remembers order

Read these two instructions: turn, then walk. Now reverse them: walk, then turn. Your notebook contains the same two commands. Your feet end up somewhere else.

That tiny experiment opens a large question for Epistemic Physics. An ordinary Beta ledger remembers how much support and opposition arrived. What extra structure lets an observer remember their order, recognize recurring phrases, or discover that a journey crossed a twisted seam in its world?

Imagine navigating an Asteroids universe with the windshield covered. You hear blinking beacons and record steering commands. A familiar signal returns. Have you completed a circuit? Encountered an identical beacon? Flipped your local orientation? The map, source identities, and interpretation of the signals may all need learning together.

Our expedition starts with a familiar evidence cone, adds phases and operators, and ends with two different meanings of “shuffle.” Along the way, we will keep four things visible: the outside world, the observer’s representation, the observation law, and the ledger of actual trials.

Two kinds of phase

For a proper $\mathrm{Beta}(\alpha ,\beta )$ state, both parameters are positive and already include the prior. Define

$$\tau =\alpha +\beta ,x=\alpha -\beta ,m=2\sqrt{\alpha \beta },\varphi =\frac{1}{2}\log \frac{\alpha }{\beta }.$$

The mean is $\mu =\alpha /\tau$, and its centered version is $v=x/\tau$. These are mean coordinates; the Beta mode generally sits elsewhere.

Introduce a split-complex unit $j$ satisfying $j^2=1$. Its exponential packages a hyperbola:

$$e^{j\varphi }=\cosh \varphi +j\sinh \varphi ,\boxed{\tau +jx=m{e}^{j\varphi }}.$$

Ordinary complex phase rotates around a circle; split-complex phase moves along a hyperbola. Put the corresponding formulas side by side:

Multiplying the evidence state by $e^{j\delta }$ sends $\alpha \mapsto e^{\delta }\alpha$ and $\beta \mapsto e^{-\delta }\beta$. It preserves $m$. Exchanging $\alpha$ and $\beta$ instead sends $x$ to $-x$: a complement operation, represented by split conjugation. Multiplication by $j$ swaps $\tau$ and $x$, so it performs a different transformation.

The cone identity is exact. A circular phase attached to evidence is an additional state variable whose meaning must come from the model: timing, orientation, a positional code, or a response to a probe.

Two circular channels in one number

Let $i$ and $j$ commute, with $i^2=-1$ and $j^2=1$. The resulting bicomplex algebra can be organized by

$$e_{+}=\frac{1+j}{2},e_{-}=\frac{1-j}{2},e_{\pm }^2=e_{\pm },e_{+}{e}_{-}=0.$$

Every state has two ordinary complex components:

$$Z=z_{+}{e}_{+}+z_{-}{e}_{-},ZW=z_{+}{w}_{+}{e}_{+}+z_{-}{w}_{-}{e}_{-}.$$

The sectors multiply independently. This is the algebraic sense in which bicomplex numbers contain two complex channels. Gargoubi and Kossentini develop this decomposition and its geometry.

One possible evidence encoding is $z_{+}=2\alpha e^{i{\theta }_{+}}$ and $z_{-}=2\beta e^{i{\theta }_{-}}$. Then the magnitudes recover the prior-inclusive amounts, while the two angles retain extra information. Equivalently, away from zero sectors,

$$Z=r{e}^{\varphi +i(\theta +\chi )}{e}_{+}+r{e}^{-\varphi +i(\theta -\chi )}{e}_{-}.$$

Here $r$ controls common scale, $\varphi$ reciprocal gain, $\theta$ common phase, and $\chi$ relative phase. A square-root-amplitude encoding would need a different decoder. Algebra alone supplies no preferred probability rule.

Keep the event ledger alongside coherent sums. Two unit observations with phases $0$ and $\pi$ have coherent sum $1+e^{i\pi }=0$, while their accumulated event weight remains two. Cancellation describes the selected response. It does not erase trials or establish that their sources were independent. Lineage remains necessary: a copied report is an echo of an earlier tracelet, even if its phase looks perfectly aligned.

Why a bag of phases remains a bag

Suppose each character $c$ receives a fixed scalar $z_c$. In a commutative algebra,

$$\prod _{t=1}^n{z}_{c_t}=\prod _c{z}_c^{n_c}.$$

Only the counts $n_c$ survive. Circular, split-complex, and bicomplex scalar products all obey this rule.

A familiar classifier already illustrates the counting limit. In multinomial naive Bayes, character probabilities ${\theta }_{c,y}$ depend on class $y\in \{0,1\}$. Conditional on message length, the posterior half-log-odds are

$$\varphi (d)=\frac{1}{2}\log \frac{{\pi }_1}{{\pi }_0}+\frac{1}{2}\sum _c{n}_c(d)\log \frac{{\theta }_{c,1}}{{\theta }_{c,0}},$$

where ${\pi }_y$ is the class prior. This has the rapidity form, although a class posterior alone does not specify a Beta trial amount. A character’s event weight and its learned log-likelihood contribution are different quantities.

There are several ways to remember order. A positional code uses $\sum _t{e}^{i\omega t}b(c_t)$, with frequency $\omega$ and token feature $b$. An affine recurrence uses

$$h_{t+1}=U(c_t)h_t+b(c_t).$$

Even scalar $U$ can retain order because each injection experiences a different set of later transformations. Setting $U=1$ returns ordinary accumulation. Another option uses matrix operators whose products depend on their order. For small step size $\epsilon$,

$$e^{\epsilon B}{e}^{\epsilon A}-e^{\epsilon A}{e}^{\epsilon B}={\epsilon }^2(BA-AB)+O({\epsilon }^3).$$

The matrix $[B,A]=BA-AB$ is the commutator. It measures the leading effect of exchanging two operations.

Before opening the lab, predict which survives reversing AAAABBBB: the endpoint, the enclosed signed area, or the product of fixed scalar phases. Then try “Moving frames” with rotate-then-boost and boost-then-rotate.

Paths, frames, and shuffled histories

Weave a route with the same endpoint

Reset experiments

Order and area Moving frames Shuffle law

Place four rose A steps and four teal B steps. Can you finish at (4,4) with signed area 0? Every tile moves the path; only the order changes its area.

Cumulative count path

Place A → (4 left) Place B ↑ (4 left) Undo tile Clear weave

0/8 tiles placed. Current area 0; target 0.

Try next area

Reverse order Alternate Group A then B

Same counts, different route

A-before-B integral

0

B-before-A integral

0

Signed area, count²

0

Commuting product phase

0.000 rad

Shuffle identity: 0 + 0 = 0 × 0 = 0.

Explore the phase readout

Positional phase sum at π/4 radians per token: 0.000 + (0.000)i. A contributes +1 and B contributes −1 before rotation.

The path uses one axis per token type, with straight unit steps. The area is half the difference of the two ordered integrals. This is a level-two signature: distinct longer sequences can still share it. The scalar product uses fixed phases 0.6 for A and −0.4 for B.

A path signature records ordered interactions

Draw cumulative A counts horizontally and B counts vertically. Each character becomes a unit segment. AABB and BBAA finish at $(2,2)$ but take opposite sides of the square.

For this chosen path embedding, define the first-level quantities $S^A=\int d{X}^A$ and $S^B=\int d{X}^B$. The second level keeps ordered integrals:

Every A-B pair appears in exactly one temporal order, giving

$$\boxed{S^A{S}^B=S^{AB}+S^{BA}},\mathcal{A}=\frac{S^{AB}-S^{BA}}{2}.$$

For AABB, $(S^{AB},S^{BA})=(4,0)$ and $\mathcal{A}=2$; reversal gives $(0,4)$ and $-2$. Closing each path by a straight segment to the origin makes $\mathcal{A}$ its signed area. This is also a direct counting proof of this instance of the shuffle identity: multiplying integrals sums over the possible interleavings of their ordered variables.

The complete signature contains all levels of iterated integrals. Its logsignature takes a formal logarithm using the algebra that concatenates words. At degree two, this removes the symmetric endpoint contribution, leaving the area coefficient multiplying the formal bracket $[A,B]=AB-BA$. The logarithm is an algebraic series, not an element-by-element logarithm of possibly negative features. Chevyrev and Kormilitzin explain these constructions and their invariances.

This supplies a natural bridge to generators: a sequence gives ordered words, and a model maps those words to products of its operators. Area is the first glimpse of order beyond the endpoint. Higher levels can distinguish further patterns, though any fixed truncation loses information. Timing requires its own channel if elapsed durations matter; a geometric path alone permits changes of traversal speed.

Wrapping and flipping are different instructions

A rectangular Asteroids world wraps coordinates by $x\sim x+L_x$ and $y\sim y+L_y$. The two phases

$$(e^{2\pi ix/L_x},e^{2\pi iy/L_y})$$

describe a torus. The final pair forgets how many circuits were completed; winding counters retain that history.

A Klein bottle has a different relation. On its covering plane, choose

$$a(x,y)=(x+L,-y),b(x,y)=(x,y+H).$$

Following the transformations directly gives $ab{a}^{-1}=b^{-1}$. Crossing the twisted direction reverses the other direction. Commuting scalar representations force the represented $b$ to satisfy $b^2=1$, losing general transverse winding.

Two sectors provide a simple remedy. Let $S$ swap them and let $D=\mathrm{diag}(e^{\lambda },e^{-\lambda })$. Then

$$S^2=I,SDS=D^{-1}.$$

The swap is an added operation; bicomplex scalar multiplication itself still commutes. A bare swap also forgets the difference between zero and two crossings, so retain winding when that distinction matters.

A loop’s net frame transformation is called holonomy. It depends on the transport law as well as the world. A flat torus can have nonzero winding with unchanged tangent orientation. A twisted surface can reverse orientation. A curved surface can rotate a transported vector around a contractible loop. Choosing a phase tag specifies another instrument, with its own observable response.

This is why reduced average phase alignment cannot uniquely diagnose a Klein bottle: noise, mixed paths, and source drift also reduce it. A clean first experiment uses matched local worlds, controlled loops, and an orientation-sensitive beacon. Predict its response to odd and even circuits, then test new trajectories. Topological Asteroids makes the gluing rules wonderfully tangible. Smooth sphere and projective-plane metrics require curvature; identifying edges in a screen does not automatically provide those metrics.

Try delivering a parcel through a twisted seam. Your destination has three requirements: a height, an arrow orientation, and a crossing count. Predict what crossing, moving up, and crossing again will do before placing the instructions.

A loop remembers more than its endpoint

Deliver a parcel through a twisted seam

Reset courier

Return facing up, one floor down. Arrange the available instructions. Crossing the seam flips both your height and the courier's arrow. A shift moves one floor in the displayed coordinates.

Cross seam ↻ (2 left)Shift up ↑ (1 left)Shift down ↓ (0 left)

Start at floor 0, pointing up

Undo instructionTry another routeNext deliveryFree route

Floor 0 of target -1; pointing up; 0 of 2 seam crossings.

Route hint and the mathematics

Try crossing, shifting up, then crossing again.

Here S sends (y, orientation) to (−y, −orientation); D sends y to y+1. Thus S D S has the same height action as D⁻¹. The seam-crossing ledger retains two crossings, even though the arrow has returned. This chosen gluing model illustrates the Klein-bottle relation; it does not infer the topology of a physical world. Free routes are capped at ten instructions.

A Möbius strip made by forgetting labels

Here is a different route to a twist. Two labeled unit phases form a torus. Report only their unordered pair and the space becomes a closed Möbius band. Coincident pairs form its boundary. This is a property of the observer’s representation even if the physical beacons occupy ordinary space. The EMS solution describes the distinct-pair version.

We can see the seam explicitly. With $u\in [-1,1]$, set

$$P(t,u)=\{e^{it/2}(u+i\sqrt{1-u^2}),e^{it/2}(u-i\sqrt{1-u^2})\}.$$

Increasing $t$ by $2\pi$ negates both roots. Exchanging them then gives $P(t+2\pi ,u)=P(t,-u)$: one circuit reverses the interval coordinate. That is the Möbius identification.

Think of tracking two identical blinking objects. A locally smooth assignment of names can return exchanged after a loop through ambiguous observations. An observer could keep both assignments, or deliberately use the unordered description. Erasing meaningful labels such as spam and ham would instead change the task.

A matrix can carry evidence through moving frames

A useful extension of the binary cone is

Its eigenvalues are $\tau \pm \sqrt{x^2+y^2}$. Thus $H$ is positive semidefinite, meaning every quadratic form $q^{\mathrm{\top }}Hq$ is nonnegative, exactly when $\tau \ge \sqrt{x^2+y^2}$.

Choose the update

$$H_{t+1}=G_t{H}_t{G}_t^{\mathrm{\top }}+Q_t,Q_t⪰0.$$

The first term transports the existing state. The second injects positive evidence. Positivity survives both operations; transport preserves the determinant when $|det{G}_t|=1$. The lab uses this rule without injections, showing rotations, boosts, and reflections of one state.

For a unit direction $\mathbf{n}$, a mean-like readout is $(1+\mathbf{n}\cdot (x,y)/\tau )/2$. A full posterior or a joint distribution across questions requires further modeling. The matrix supplies a cone summary.

Small real matrices also connect to split-quaternions. Rotation and boost generators can satisfy $J^2=-I$, $K^2=I$, and $JK=-KJ$. Their noncommutation differs from the bicomplex choice. Real symmetric state matrices give the cone above; complex Hermitian state matrices add a third spatial coordinate. The Hermitian-matrix construction provides the standard Lorentz correspondence.

Rotation, reciprocal growth, and shear offer three recognizable transformation families. In a real two-dimensional transfer matrix, the discriminant $(\mathrm{tr}G{)}^2-4detG$ distinguishes complex from distinct real eigenvalues; zero needs further inspection to distinguish a shear from a scalar matrix. This classification describes the fitted dynamics. Identifying a physical time direction needs additional structure. Shuffling whole records preserves such information only when enough within-record relationships survive.

When the generator reads the ledger

“Generator” can mean a source that emits observations, a stochastic law that moves hidden states, or a matrix that transports features. A model can use all three, with separate jobs.

Suppose a source emits character $c$ with probability $p(c\mid z)$ at hidden location or context $z$. Nearby contexts are distinguishable when their emission laws differ. Under regularity conditions, the local Fisher matrix

$$g_{ab}(z)=\sum _{c}p(c\mid z){\partial }_a\log p(c\mid z){\partial }_b\log p(c\mid z)$$

measures this distinguishability. Invisible directions make the matrix singular. This geometry concerns observable source differences, while the evidence cone concerns a summary of accumulated directional weights.

Phase comparisons can help map the sources. Repetition estimates a period; phase needs a reference. A consistent relative delay produces a slope in a cross-spectrum’s phase. A useful model tests whether that slope persists across contexts and whether alignment exceeds a noise baseline. A fixed-frequency code also aliases positions separated by its wavelength. Multiple frequencies and landmarks can resolve some ambiguities. These are candidate tools for discovering phrases or spatial relations, with the meaning of “distance” supplied by the observation model.

Now let the source inspect previous outcomes before generating the next one. The resulting world can behave very differently from a fixed coin:

The last two need separate treatment. A fraction approaching one half does not imply bounded variance of its unnormalized imbalance. Different restoration mechanisms have different fluctuation scales.

The ordinary Pólya urn reveals a particularly sharp ambiguity. Its complete observation law equals that of independent coin flips conditional on a probability drawn once from a Beta distribution. Reinforcement and uncertainty about a fixed generator can therefore give identical passive data. An intervention that changes the urn’s composition, with an explicit account of how each proposed mechanism responds, can separate them. Pemantle’s survey develops reinforcement and this exchangeability connection.

The other shuffle law: histories pinned at both ends

The signature shuffle identity was algebraic. Uniformly shuffling an observed multiset is a probabilistic experiment.

Take $n/2$ positive and $n/2$ negative unit steps, with even $n$. Every ordering starts and finishes at zero, making a discrete bridge. Write $S_k$ for its partial sum. After scaling heights by $\sqrt{n}$ and time by $n$, the path approaches a standard Brownian bridge $b(t)$, whose covariance is $min(s,t)-st$.

For this particular limiting process, define its normalized full range

$$R=\sqrt{\frac{2}{\pi }}(\underset{0\le t\le 1}{max}b(t)-\underset{0\le t\le 1}{min}b(t)).$$

An exact identity connects its moments to completed zeta:

$$\mathbb{E}[R^s]=2\xi (s),\xi (s)=\frac{1}{2}s(s-1){\pi }^{-s/2}\mathrm{\Gamma }(s/2)\zeta (s).$$

Here $\mathrm{\Gamma }$ is the gamma function and $\zeta$ the Riemann zeta function, with removable values interpreted by continuation. In particular, $\mathbb{E}[R]=1$ and $\mathbb{E}[R^2]=\pi /3$. The specified range and normalization are essential. Biane, Pitman, and Yor derive the Brownian connections.

Short paths retain a lattice correction. Reflection at the first hit of height $h$ gives

$$P(max{S}_k\ge h\mid S_n=0)=\frac{(\genfrac{}{}{0ex}{}{n}{n/2+h})}{(\genfrac{}{}{0ex}{}{n}{n/2})}.$$

Summing over positive integer $h$ and using symmetry of maximum and minimum yields

$$\boxed{\mathbb{E}[R_n]=\sqrt{\frac{2}{\pi n}}(\frac{2^n}{(\genfrac{}{}{0ex}{}{n}{n/2})}-1)}.$$

The lab enumerates all balanced orderings and checks this finite formula directly. The gap below one is a finite-step effect, even with perfect enumeration. Correcting the mean does not automatically correct every fractional or higher moment.

For an unbalanced fixed multiset, subtract its endpoint trend and normalize by $\sqrt{n(1-v^2)}$, where $v=S_n/n$, before taking an interior-proportion limit. Learned weights, dependence, and preferred reorderings require their own reference law. The Brownian identity thus provides a precise bridge experiment; applying it to an arbitrary corpus requires checking the embedding and scaling assumptions.

What could a learner discover?

The most inviting experiments make different explanations disagree visibly. Equal-count strings test order. Long loops test gluing. Indistinguishable source pairs test label ambiguity. Controlled source interventions test whether the world responds to its ledger. Held-out compositions test whether learned operators predict more than the training examples.

Boundary conditions offer another probe. A field on a circle obeying $f(x+L)=e^{i\vartheta }f(x)$ has modes with squared frequencies $((2\pi n+\vartheta )/L{)}^2$. A half-turn twist selects half-integer frequencies. A learner could look for such spectral patterns, then use loops and orientation-sensitive measurements to distinguish worlds that share similar spectra.

These experiments suggest a practical language model as well: treat a recurring phrase as a reusable operation on predictive context. Compare it with character counts, n-grams, and an equally sized recurrent model. Keep source-related examples together when testing generalization, because repeated testimony can otherwise make a geometry appear easier to learn than it is.

The exciting possibility is a representation that discovers which observations belong together, how their meanings transform, and which journeys can be composed. The first tools are already small enough to hold in our hands: two phases, a swap, a matrix, and the area between two paths with the same endpoint.

Further reading

• Bicomplex numbers as a complexified algebra: sector decomposition and phased coordinates.
• A primer on path signatures: ordered integrals, shuffle products, logsignatures, and practical path embeddings.
• The unordered-circle-pair problem: the Möbius geometry created by forgetting labels.
• Random processes with reinforcement: urn models and the limits of inference from passive histories.
• Probability laws related to the Jacobi theta and Riemann zeta functions: the exact Brownian range identity.