Glossary

The rule that identical inputs — graph or configuration, seed, and parameters — must produce identical results. The server re-derives every score from the bundle before storing it, and refuses the save if the numbers do not reproduce.

The self-contained record of one experiment: its structure, seed, parameters and metrics. A published experiment replays from its own bundle, which is what makes the archive checkable rather than just a list of claims.

The six-level taxonomy separating established physics from mathematical models, computational experiments, speculation and rejected results. Every explanation in the product carries the level its own claim sits at.

The integer that fixes the one source of randomness in a run. In Causal Forge it drives the initial perturbation, and it changes the trajectory without changing any score.

A construction that reproduces the shape of a real idea with deliberately simplified machinery. Toy readings are labelled everywhere they appear; the label is a statement about the method, not an apology for it.

A set of events plus a partial order saying which can influence which, with no background space. The order is the primitive; anything geometric has to emerge from it.

A graph whose arrows never lead back to where they started. Acyclicity is what makes the arrows a causal order rather than just connections.

An edge lying on a directed cycle — the discrete analogue of a closed timelike curve, where following the causal direction returns you to your own past. Causal consistency is the fraction of edges that are not these.

L = D − A, the difference between the weighted degree matrix and the adjacency matrix. It is the operator the diffusion runs on, and its eigenvalues set both the spectral dimension and the stability bound.

The fraction of connected triples that are closed into triangles. Reported as the locality score, it measures whether the structure has local neighbourhoods at all.

The dimension read from how fast heat spreads: for a d-dimensional space the return probability falls off like t^(−d/2). A behavioural dimension, which is why a graph with no coordinates can have one.

On screen — The 45–55 band on the gauge is d_s ≈ 2 — surface-like. Higher is not better.

The simplest time-stepping scheme: x ← x − α·L·x. Conditionally stable — past α = 2/λ_max the errors grow each step instead of shrinking, which is exactly what the stability score measures.

Merging nearby elements to look at a structure at lower resolution. A property that survives it is a feature; one that does not was an artefact of resolution.

The space over which the fiber is built

On screen — The game map — the geometric foundation the player shapes

A torus, or elliptic curve, sitting over every point of the base

On screen — The hidden layer of space attached to each point

A degeneration of the fiber

On screen — A special zone where new symmetries can appear

An object associated with fiber degenerations

On screen — A force node or defect anchored in the geometry

The compact geometry of the extra dimensions

On screen — The hidden shape of the universe being explored

The symmetry of the resulting physical model

On screen — The set of forces the configuration allows

y² = x³ + f·x + g, the standard way of writing an elliptic curve. In this module f and g are polynomials over the base, so the curve varies from point to point.

Δ = 4f³ + 27g². It vanishes exactly where the elliptic fibre degenerates, so the zero set of Δ is where all the interesting structure lives.

One of a finite list of ways an elliptic fibre can degenerate (I₁, II, III, IV, I*₀, IV*, III*, II*). The real classification uses vanishing orders; this module assigns types by magnitude tests, which is the toy part.

Matter arising where two 7-brane stacks intersect, transforming under both gauge groups. Detected here by grid adjacency rather than intersection theory, so the labels are prompts rather than a derived spectrum.

A displayed-only gauge, loosely analogous to a K3 surface spending its fixed budget of 24 degenerations. It is not part of the total, not verified on save, and not visible to challenges.

A grid of cells, each updating in lockstep from a rule that reads only its own state and its neighbours. The entire specification is local; anything global that appears was not written down anywhere.

A two-state rule written as B/S — the neighbour counts at which a dead cell is born, and those at which a live cell survives. Conway’s Life is B3/S23. Eighteen bits in total, so 262,144 rules exist and about a dozen have names.

On screen — The two rows of nine toggles in the rule panel are literally those eighteen bits.

A four-way classification of automaton behaviour: I dies out, II settles into stable or repeating states, III is chaotic, IV has structures that persist and interact. Class IV is the narrow interesting band, and the only class shown capable of computation.

A pattern that reproduces itself a few cells away every few steps, so it appears to travel. Nothing in any rule mentions motion; a glider is a consequence, which is what makes it the standard example of emergence.

A grid whose edges wrap: leaving the right side re-enters on the left. Used here so no reading depends on how close a pattern happened to drift to a boundary — that would be a property of the container rather than of the rule.

Shannon entropy computed over local blocks rather than single cells. A per-cell measure cannot tell a checkerboard from a half-and-half split — both are 50% live — and only one of them has structure.

A two-state quantum system. Unlike a bit it can be in a combination of both states at once, with complex weights whose squared magnitudes are the probabilities of each outcome.

A state that is a combination of basis states rather than one of them. Says nothing on its own about correlation between qubits — a register with a Hadamard on every qubit is maximally spread and entangled with nothing.

On screen — The SUPERPOSITION gauge measures spread across the computational basis, and only that.

A joint state that cannot be written as one state per subsystem. Neither half then has a description of its own; what exists is a description of the pair.

A bound satisfied by every theory in which each particle carries its measurement outcomes with it. Quantum states can exceed it, and laboratories have — with the loopholes closed. Violating it rules out local hidden variables, not merely our ignorance of them.

The maximum CHSH value quantum mechanics permits, 2√2 ≈ 2.828, against a classical maximum of 2. A Bell pair reaches it exactly.

On screen — The BELL VIOLATION gauge is scored from 0 at the classical bound to 100 at this one.

An all-or-nothing entangled state of three or more qubits: every qubit 0, or every qubit 1. More entangled across a split than a Bell pair, while any two of its qubits are separable and violate no Bell inequality.

A discrete, universal gate set — H, S, CNOT and friends plus the T gate. Discrete meaning no continuously tunable angle, which is why this module evaluates no trigonometric function and its amplitudes are bit-identical on every engine.

1 − Tr(ρ²), a measure of how mixed a state is. Used here in place of the von Neumann entropy because it needs no eigenvalues and no logarithm; it is the 2-Rényi entropy before the log, so it answers the same question with different arithmetic.

A graph whose edges carry SU(2) spins and whose vertices carry intertwiners. Introduced by Penrose as combinatorial spacetime, and later found to be the states of quantum geometry in loop quantum gravity.

An invariant tensor at a vertex — the thing that exists exactly when the spins meeting there can cancel to nothing. If none exists, the vertex is not allowed.

On screen — The green and red vertex outlines are exactly this: an intertwiner exists, or it does not.

For SU(2): no single spin exceeds the sum of the others, and the total Σj is a whole number. Stored as 2j integers here, so it is two integer comparisons rather than a numerical test.

In loop quantum gravity a surface pierced by edges has area proportional to Σ√(j(j+1)) — discrete, not continuous. The proportionality involves the Barbero–Immirzi parameter, which the theory does not fix.

On screen — Displayed on the panel and never scored, precisely because of that free parameter.

Volume lives at vertices, and a trivalent vertex has volume exactly zero: three edges cannot bound a region. A four-valent vertex is dual to a tetrahedron — the smallest piece of space the theory describes.

A free constant γ in loop quantum gravity that sets the overall scale of the area and volume spectra. Its value is not derived by the theory, which is why any absolute size quoted from it is a convention.

Predicting the motion of several bodies under mutual gravity. Solved in closed form for two; for three or more the motion is chaotic and no general solution exists — a theorem about the problem, not a limit of computers.

A numerical method whose energy error stays bounded instead of drifting away. Velocity Verlet is the one used here, and it is why energy conservation is worth scoring at all.

For a relaxed self-gravitating system, 2K + U = 0. The standard test for whether a cluster has settled — and a statement about averages, so a virialised system can still be losing members.

A small constant added to every squared distance, so two bodies passing very close feel a finite force. Without it a near-collision would eject a body at absurd speed and the result would be an artefact of the timestep.

A body thrown clear of the system. Not a failure: it is how a gravitating cluster sheds energy, and it is what lets the bodies that remain fall closer together.

The rate of an elementary reaction is proportional to the product of the reactant concentrations, each raised to its stoichiometric coefficient. The whole engine of the Reaction Kinetics Lab — no temperature, no solvent, no special cases.

Q = Π[products]^ν / Π[reactants]^ν at any moment. At equilibrium Q equals K = kf/kr; a Q above K means the reverse rate currently wins, and a Q below K means the forward rate does.

A system at equilibrium, disturbed, shifts to partly counteract the disturbance. In mass action it is a consequence rather than a rule: add product and the reverse rate exceeds the forward one until the ratio is restored.

On screen — The product-injection slider is the disturbance; the LE CHATELIER gauge is how much of it came back.

The reactant that runs out first — the one with the smallest initial amount per stoichiometric coefficient. Conversion is measured on it, because it is the one that decides where a one-way reaction stops.

Grams per mole as listed in a reference table — here PubChem, with the compound id recorded. Rounded, which is why the gram total in the conservation readout drifts in the fourth digit while the atom counts hold exactly.

On screen — Shown in the conservation readout and never scored: the invariant is atoms, not the table.