R↻
0
START
Solved cube
Only the next easy step is shown here.
C_0 = I
D_0 = 0
state_source = canonical(C_0)
phi((C_0, D_0, route_0)) = C_0
Structural Cube lets you mix and solve a 3 x 3 x 3 cube directly in your browser.
The cube creates a declared seed, uses the seeded core generator, and shows the seed so the same starting position can be reproduced.
You mix the cube. Drag a row or column of three squares. Tap a face to turn the whole face.
Strategy and Plan appear before every move. The move then carries out that checked purpose. Pause freezes the whole sequence, including a move already in progress.
Structural Cube shows one move. You make that move yourself.
No hint is shown. You can solve the cube in your own way. Hints can be turned on later.
Drag across one coloured square. The row or column of three squares will turn.
Tap a face. Use the Turn button to choose clockwise, anti-clockwise, or halfway around.
Drag the dark edge around the coloured squares. Press Reset View to return to the normal view.
The centre square tells you the final colour of that side. Try to bring matching colours near their centre.
In hint mode, Structural Cube can find a new route from the cube you now have. You do not need to start again.
The Move tab always uses simple English. The Technical tab keeps the formulas separate.
Structural Cube resolves and teaches a standard 3 x 3 x 3 cube through deterministic structural analysis, exact legal transformations, and replayable evidence.
Auto Resolve and With Hint use the graph-governed economy resolver as the primary interactive authority. The resolver reads the cube as it exists now and does not require the earlier mixing history.
interactive_authority = GRAPH_GOVERNED_ECONOMY_PRIMARY_AUTHORITY
history_dependency = NONE
Each guided action is presented in three parts. Strategy states the larger structural goal. Plan explains the next checked step. Move carries out that step.
A specific learning claim is displayed only when the current cube state supports it. When the evidence is not specific enough, the application uses general guidance rather than inventing a purpose.
The resolver builds the current observed obligation graph, ranks certified pure-kernel targets, compares the four strongest admissible choices through a four-action lookahead, and selects the lowest declared compiled face-turn cost.
canonical state -> full graph -> certified targets -> depth 4 lookahead -> exact effect -> strict W descent -> rebuild
reference_route_access = NONE
current_state_reference_distance_access = NONE
fallback_activation = NONE
The current legal cube state is represented as C = (P_e, O_e, P_c, O_c). The structural rank records permutation parity, transposition distance, two-cycle residue, edge flips, and corner twists.
Phi(C) = (rho, D_e + D_c, tau_e + tau_c, F, T)
W(C) = 24*rho + 4*(D_e + D_c) + tau_e + tau_c + F + T
0 <= W(C) <= 126
W(C) = 0 iff C = SOLVED
Every admitted action must reproduce its declared pure effect exactly and strictly reduce W at the completed action boundary.
The resolver uses five declared pure-effect kernel families: EF, CT, E3, C3, and PB. Their face-turn realizations are constructed from isolated pure effects and then verified against the current cube.
compiler_search_input = ISOLATED_PURE_KERNEL_EFFECT_ONLY
observed_effect = predicted_effect
W(C_after) < W(C_before)
Cube states, observed graphs, rank records, action records, route records, and manifests use canonical representations with SHA-256 commitments. Random Start uses a declared xorshift32 seed so the same starting state can be reproduced.
same canonical state + same versioned rules -> same deterministic route
canonical bytes -> SHA-256
wall_clock_authority = NONE
The graph-governed economy route can be exported as a portable SCCERT-1 certificate. The complete 100-seed certificate corpus binds every starting state, graph-aware target decision, compiled move sequence, strict structural-rank descent, final solved state, and certificate chain into one aggregate manifest and bundle root.
economy route -> SCCERT-1 -> independent Python verification
100 certificates + 100 independent passes -> certificate corpus PASS
The versioned P100 corpus completed 100 of 100 routes with final solved states, strict W descent, zero fallback, deterministic repeatability, and no route regression against the declared baseline.
median moves = 172
p95 moves = 207
maximum moves = 223
Structural Cube supports the standard 3 x 3 x 3 cube. It does not claim shortest-route optimality, competition-speed superiority, or proof over every legal cube state.
The observed graph is a declared structural model for this resolver. It is not presented as the only possible mathematical dependency graph for cube solving.
Shunyaya Framework