Experimental Platform for Discrete Topological Dynamics

Clifford Introduction and Observables

Algebra set: Clifford Reversi is one game with configurable algebra and opening rules. Standard Pauli keeps the ordinary four-stone Reversi opening. Choose Stabilizer Algebra to use qubit sites, alternate initial rules, measurements, phases, and ancillas.

Lab goal: compare how the same Pauli/Clifford flipping rule changes when topology, seams, lattice, noise, and time settings alter the rays that carry local information.

3D boards: R3 uses a rotatable cubic graph, T2 uses a solid torus, and S2 uses a latitude-ring sphere. Click graph vertices to place stones; drag to rotate and wheel to zoom. The same topology-aware Clifford rules remain active on these surfaces.

Move rule: place a Pauli-labelled stone on an empty vertex only when it brackets at least one opponent chain along a topology-aware ray. Rays use the selected topology, so 2D RBC, torus, Klein bottle, RP2, S2 latitude, and 4D grid boards use their graph normalization instead of flat boundary assumptions.

Local operator choice: Clifford Insertion Field opens a small operator palette at the clicked site, so X/Y/Z are chosen where the insertion happens. The side Pauli selector remains a default for exports and keyboard-style operation.

Worldline mode: Clifford Worldline Operators use X, Z, and Y tokens moving on the graph. Their labels propagate through hops and exchanges so you can compare operator transport on the same topology.

Flip rule: each locally coupled opponent operator is transformed by the selected Clifford generator, and twisted seams apply their transport while the graph ray is resolved.

Pauli math: I=(0,0), X=(1,0), Z=(0,1), Y=(1,1). The symplectic product is x1*z2 + z1*x2 mod 2. Labels anticommute when this value is 1.

H and S: without phase signs, H swaps X and Z and leaves Y as Y; S maps X to Y, Y to X, and Z to Z. With Phase / Sign Display on, H maps Y to -Y and S maps Y to -X, so flipped stones display +X, -Y, etc.

Dynamics choices: Pauli noise changes labels with seeded random rolls. Time evolution can age pieces on the selected clock after moves or full rounds, while noise can be applied separately from the same dynamics panel.

Results: the export records topology, lattice, board labels, move history, position history, probability/noise events, time-evolution state, and final counts. These results let you compare how the same Clifford rule behaves on flat, random-boundary, torus, Klein, Mobius, RP2, S2, R3, and 4D graphs.

Stabilizer Introduction and Observables

Algebra set: this is the Stabilizer Algebra configuration of the same Clifford Reversi game. Switch Algebra Set back to Standard Pauli Reversi to restore the ordinary opening and controls.

Qubit sites: empty vertices are identity I with no active excitation. Occupied vertices store owner, Pauli I/X/Y/Z, sign, phase modulo four, and optional ancilla data. Black is the positive sector and white is the negative sector.

Initial states: Stabilizer Vacuum is empty with positive checks. Paired Defects creates nearby opposite-sign pairs. Domain Wall Seed separates positive and negative domains. Prepared Circuit applies ancilla preparations, H, S, CNOT, CZ, and measurement to a vacuum start. Custom Setup lets you place or clear X/Y/Z/I sites first, then start recovery from that designed board.

Reversi action: placement still requires a normal topology-aware bracket. Each flipped site changes owner and sign, then receives the selected H or S transform plus any seam transport. Pauli phase is tracked modulo four.

Ancillas: prepare Z0, Z1, X+, X-, or magic ancillas on empty sites. Entangle an occupied control and target with CNOT or CZ when at least one endpoint is an ancilla. Ancillas may be measured in X/Z or discarded.

Phase actions: S, S dagger, and Z are Clifford actions. Phase Kick accepts a numeric angle and marks non-quarter-turn use as a non-stabilizer approximation.

Pauli error-correction goal: recovery means using local Pauli-frame actions to reduce measured X/Z syndrome defects and return the board toward the stabilizer vacuum or intended logical sector. It adds graph-neighborhood X/Z syndromes, topology-cycle logical checks, recovery time, seeded measurement errors, and a dedicated QEC export without changing ordinary Reversi.

Measurements: measure a local Pauli, connected-domain parity, line-interval parity, neighborhood stabilizer, or logical-cycle parity. Seeded measurement errors use the configured error rate, and the measured site collapses to the selected X or Z basis.

Observables: every action records Pauli/sign/phase distributions, ancilla count, syndrome weight, local X/Z violations, global parity, commutation conflicts, logical sector, recovery state, measurement errors, and non-Clifford resource use.

Results: Stabilizer Pauli correction/recovery exports the initial and final observables, full physics history, measurement log, circuit history, recovery time, logical-error flag, final sector, Pauli/phase distributions, and a compact final answer for whether recovery stayed in the intended stabilizer sector.

Physical Cluster Field Introduction and Observables

Physical meaning: black is species A, phase A, or spin sector A. White is species B, phase B, or spin sector B. Empty vertices are local growth, oxygen, or resource sites on the selected topology graph.

Physical problem / goal: test whether competing local growth rules create survival, extinction, percolation, or topology-wrapping clusters on different spaces.

Resource contacts: neighboring empty graph vertices feed growth. A connected cluster with zero open contacts undergoes local extinction, annihilation, or confinement.

Possible models: use the model selector to interpret the same rule set as percolation clusters, reaction-diffusion domains, two-species competition, exciton/hole recombination, or spin-domain growth.

Actions: place species on an empty site, grow a connected cluster into a neighboring resource site, automatically remove zero-contact clusters, pass, or run an optional diffusion/noise step.

Initial states: sparse seeds, random density, two-cluster competition, interface seed, and thermal cluster sample all create playable nonempty states with immediate growth/capture opportunities.

Observables: every move records cluster size distribution, largest cluster, percolation probability, removal/extinction events, interface length, correlation length estimate, survival probability, and topology-wrapping cluster count.

Topology: all neighbors come from topology.neighbors(vertex), so torus, Klein, Mobius, RP2, sphere, random boundary, R3, and 4D graphs do not assume rectangular edges.

Answer: export summarizes which species percolated, survival/extinction time, cluster-size exponent estimate when enough cluster statistics exist, and topology-wrapping probability.

Spin & Phase Domain Introduction and Observables

Physical meaning: black is spin up s=+1 and white is spin down s=-1. Empty vertices are unoccupied or undecided graph sites when the selected initial state leaves gaps.

Physical problem / goal: build, move, and stabilize domain walls while checking how topology and lattice geometry change energy, magnetization, and coarsening.

Actions: place or flip one spin, flip a connected domain, rewrite a line interval, pass, or enable Metropolis acceptance for temperature-driven trials.

Observables: every move records energy, magnetization, domain-wall length, domain-wall density, up/down domain counts, accepted or rejected updates, and the final phase.

Topology: neighbors, domains, and walls use the selected graph, so torus, Klein, Mobius, RP2, S2, R3, 4D, square, honeycomb, and triangular boards can produce different wall loops.

Answer: export summarizes whether the position ordered, stayed mixed, formed stable walls, or kept topology-winding interfaces.

Two-Phase Competition Introduction and Observables

Physical meaning: black is phase A, white is phase B, and empty vertices are metastable substrate that can nucleate or be converted.

Physical problem / goal: compare how two competing phases invade, coarsen, pin interfaces, or wrap through the selected topology.

Actions: nucleate a phase on empty substrate, grow from a neighboring same-phase domain, flip an interface when the energy allows it, pass, or add optional droplet noise.

Observables: every move records energy, interface length, area fractions A and B, domain count, nucleation count, coarsening events, and topology-wrapping interfaces.

Topology: the same phase rule uses graph neighbors, so boundary seams and lattice geometry can change which phase connects or gets trapped.

Answer: export reports the winning phase, whether an interface is stable, and whether either phase forms a noncontractible connected region.

Particle Hopping / Reaction Introduction and Observables

Physical meaning: black and white are two particle species or charge signs. Empty vertices are available sites on the selected graph.

Physical problem / goal: study local hopping, exchange scattering, recombination, and path parity on different topologies and lattices.

Actions: hop to an adjacent empty vertex, exchange across an occupied interaction particle, follow a multi-step scattering path, recombine adjacent opposite charges, or measure path parity.

Observables: every move records particle count, recombination count, exchange events, braid-like parity, average path length, energy recovered, and the final reaction state.

Topology: hop, exchange, and scattering paths use graph adjacency, so twisted seams and higher-dimensional boards change which reactions are locally reachable.

Answer: export reports whether charges recombined, remained separated, formed a persistent scattering path, or changed the parity sector.

Spin Ice Vertex Introduction and Observables

Physical meaning: variables live on graph edges, not vertices. Black arrows follow the chosen edge orientation and white arrows point opposite that orientation.

Physical problem / goal: create, move, and annihilate monopole defects while testing how string and loop excitations depend on topology.

Actions: flip one arrow, flip a connected string, flip a closed loop, move a monopole along an edge path, annihilate a monopole pair, or pass.

Observables: every move records energy, ice-rule violations, monopole count, string length, closed-loop count, loop winding, and defect density.

Topology: edge paths, closed loops, and winding sectors come from the selected graph, so noncontractible loops can exist on wrapped spaces but not on ordinary open boards.

Answer: export reports the monopole sector, string/loop sector, energy trend, and whether the board returned to an ice-rule vacuum.

Z2 Gauge Loop Introduction and Observables

Physical meaning: variables live on graph edges with Ue=+1 or Ue=-1. Star checks multiply adjacent edge values, and plaquette checks multiply values around local faces or cycles.

Physical problem / goal: create, measure, repair, and compare Z2 charge, flux, Wilson-loop, and logical-memory sectors on topology-aware graphs.

Actions: flip one edge, flip an open path, flip a closed loop, measure a star check, measure a plaquette check, enable noisy edge flips, or run the simple decoder.

Observables: every move records syndrome weight, star violations, plaquette flux violations, logical sector, Wilson-loop values, decoder actions, and memory status.

Topology: open strings create charge endpoints, closed loops preserve local constraints, and noncontractible loops can change the logical sector when the board supports cycles.

Answer: export reports whether the gauge state stayed in the vacuum sector, accumulated a logical loop error, or was repaired by decoder/recovery moves.

CFT Local OPE Operators Introduction and Observables

Physical meaning: black is the + source/domain sign and white is the - source/domain sign. A stone is a primary field or spin/domain insertion. Nearby occupied graph rays form a discrete OPE interaction interval; local OPE propagation updates channels and phase transport without requiring an enclosing boardgame bracket.

Physical problem / goal: insert primary fields and let local OPE kernels reorganize source signs, domain walls, and channel data across the selected graph.

Default CFT: Ising CFT uses c=1/2 with sigma, epsilon, and identity fields. OPE rules are sigma x sigma -> identity + epsilon, sigma x epsilon -> sigma, and epsilon x epsilon -> identity.

Initial states: choose domain wall seed, four sigma block, boundary condition change, thermal CFT sample, or two-phase interval seed. Vacuum is not a selectable local-OPE CFT initial option.

Actions: insert a primary/domain field, propagate the local OPE kernel along nearby graph rays, update channel labels, measure interval parity, measure OPE channel, measure region entropy, or apply an L_n deformation.

Virasoro actions: N=1 enables L_-1, L_0, L_1. N=2 also enables L_-2 splitting and L_2 concentration. Overlap of N=2 actions can create central-charge anomaly markers.

Observables: every turn records domain-wall length, primary counts, OPE channel transitions, conformalBlockWeights, dominant block, interval entropy estimate, stress proxy, central-charge anomaly events, and twisted/topological sector when supported by the topology.

Estimator note: CFT correlations, conformal blocks, entropy, and stress are discrete graph estimators for gameplay and are not exact continuum CFT values.

Answer: export summarizes final dominant OPE channel, final domain-wall length, whether a stable topological/twisted sector appears, entropy growth, anomaly count, and final CFT sector.

Anyon Introduction and Observables

Physical problem / goal: create mobile topological charges, braid or unbraid their worldlines, then test whether fusion and logical memory return to the intended vacuum or sector.

3D boards: R3, T2, and S2 use rotatable graph embeddings. Legal hops, jumps, braid paths, fusion sites, labels, and braid-state shadows are drawn directly on the 3D board.

Move rule: a token may hop to an adjacent empty vertex or exchange/scatter across one occupied neighboring token into the next empty vertex in the same topology-aware direction. The move stores the visited path as a worldline segment.

2D RBC: when selected, each boundary exit is linked to one fixed random boundary square for the current game. Anyon hops and exchanges use those links through the same graph-neighbor system as the other topologies.

Symbols: toric code labels are 1, e, m, and ψ. Ising labels are 1, σ, and ψ. Fibonacci labels are 1 and τ. Board tokens use these Greek symbols directly.

Vacuum 1: 1 is the trivial topological charge, meaning no nontrivial anyon remains. It acts like the identity in fusion: 1 x a = a.

Braiding: an exchange around a mutually nontrivial target can append a braid generator σ_i or σ_i^-1. Abelian parity toggles a Z2 braid state for e around m or m around e. Word Exact stores the full noncommutative braid word. Non-Abelian Channel also updates a hidden symbolic fusion channel.

Unbraiding: select a braided token. The valid target is marked with a green UNBRAID badge. Clicking that target applies the next inverse generator. Full unbraiding requires the reverse inverse sequence; a wrong target or wrong sign appends a new braid instead of clearing the word. A successful inverse also reduces the visible braid count.

Local excitation choice: in Excitation Energy mode, click an empty site to open only the affordable particle palette. Click an owned anyon to select it, then click an empty site to move or another token to braid/unbraid naturally; use Recombine / Recover for the selected anyon.

General Z_n phase: the selectable non-vacuum charges are α₁ through α_n-1. Fusion adds grades modulo n, so α_i x α_j = α_{(i+j) mod n}, with grade 0 equal to vacuum 1. A positive braid adds +1/n, an inverse unbraid subtracts 1/n, and the token shows the signed net phase as a flat shadow under the piece.

Toric code fusion: e x e = 1, m x m = 1, ψ x ψ = 1, e x m = ψ, e x ψ = m, and m x ψ = e. Fusion to 1 is vacuum fusion and removes or neutralizes the fused pair according to config.

Toric Memory Unbraid: the physical-problem option initializes local e and m pairs with vacuum total charge, records the logical winding sector, and shows QEC observables after every move. Crossing a noncontractible cycle can create a logical error; exact inverse braids reduce braid memory; complete vacuum fusion is recorded as vacuum recovery.

QEC export: JSON includes vertex-based initial state, move history, braid and fusion histories, logical-sector history, memory lifetime, logical-error rate, unbraid success rate, and the final physical answer.

Excitation Energy: both players start with 12 energy. The selector offers every non-vacuum type in the chosen model: e, m, ψ for Toric Code; σ, ψ for Ising; τ for Fibonacci; and every α_k grade for Z_n. Exciting one on an empty vertex consumes one turn and its model cost. Recombining/dropping an owned anyon by button, action, or double-click returns its gap energy after the configured loss rate; completing a vacuum fusion also returns energy.

Non-Abelian fusion: Ising uses σ x σ = 1 + ψ. The plus sign lists alternative channels, not simultaneous products: resolving to 1 removes both incoming σ anyons as vacuum, while resolving to ψ replaces the pair with one ψ excitation. Also, σ x ψ = σ and ψ x ψ = 1. Fibonacci similarly uses τ x τ = 1 + τ. Measurements can reveal hidden channels when measurement rules are enabled.

Entanglement distance: Infinite keeps fusion-channel memory regardless of graph separation. Finite Distance measures shortest legal graph steps and decoheres that hidden pair channel when the anyons separate beyond the selected distance. Braid words remain recorded independently.

Dynamics choices: Anyon noise can create pairs or flip e/m on configured noisy places. Time evolution ages particles on the selected clock; entanglement range, recombination, and measurement settings control the excitation dynamics.

Observables and results: the QEC panel reports total fusion charge, logical sector, memory alive/lost, vacuum recovery, average and maximum braid word length, and successful or failed unbraids. Research export keeps worldlines, braid tokens, fusion outcomes, logical-sector history, energy history, memory lifetime, logical-error rate, unbraid success rate, and final physical answer.

CFT Field Insertion Graph Introduction and Observables

Physical system: the board is a discretized Riemann surface / graph manifold. Empty vertices are identity operators, stones are primary-field insertions, black and white are source sign or player control, and primaryType stores the physical field.

Physical problem / goal: insert primary fields and use local graph/OPE contact updates, then measure which conformal block, correlation pattern, entropy growth, or anomaly response dominates.

Default CFT: Ising CFT is the default with c=1/2, identity h=0, sigma h=1/16, and epsilon h=1/2. Free Boson remains available as an alternate estimator.

Graph contact rule: legal insertion, local graph contacts, topology adjacency, passing, and optional counting use the graph substrate. Local fusion updates replace boardgame capture language with OPE/contact changes on primary-field clusters.

Initial states: choose two-point insertions, four-point block, boundary CFT insertions, thermal sparse insertions, or an identity background seeded with sigma/epsilon defects. The empty vacuum is avoided as the default so the observable game starts with physics to measure.

Measurements: measure OPE channel, two-point correlator, four-point correlator / block, dominant block, region entropy, or stress tensor proxy T(v). Hidden channels collapse when measured and seeded measurement errors are logged.

Virasoro actions: N=1 enables L_-1, L_0, and L_1. N=2 adds L_-2 and L_2 and records central-charge anomaly events when overlapping N=2 stress updates occur.

Observables: every move records primary counts, OPE channel distribution, two-point estimates, four-point cross-ratio, conformalBlockWeights, dominant conformal block, stress tensor proxy T(v), entanglement entropy estimate, mutual information estimate, and anomaly events.

Answer: export summarizes final dominant block, identity/vacuum block dominance, entropy growth, strongest correlations, final OPE sector, and anomaly count.