# Dimensional Chess Experiment C1 ## Exact symmetry-quotiented solution of the KRK universe **Version:** 1.0.0 **Date:** 15 September 2026 **Status:** Exact finite computation for the declared king-and-rook versus king abstraction; not a solution of full chess ## 1. Verdict Chess is finite once its complete rules and history-dependent draw state are fixed, so its initial position has a definite game-theoretic value: White win, draw, or Black win. Finiteness proves that an exhaustive solution exists in principle. It does not prove that the solution has a physically feasible computation or a compact polynomial representation. Experiment C1 applies the Theory of Reality's strongest computational rules to a closed, nontrivial chess universe: - ordinary 8×8 board; - White king and rook versus Black king; - legal king and rook movement and check; - checkmate is a win; - stalemate, rook capture, or infinite non-mating play is a draw; - FIDE halfmove and repetition-history clocks are deliberately outside this first universe. Every legal position in that universe has now been solved simultaneously. The result is an exact endgame tablebase with an independently checked proof certificate. ## 2. The complete state is the first dimensional lift For C1 the state is \[ s=(k_W,r_W,k_B,\tau), \] where the three squares and side to move \(\tau\) determine every legal future under the declared rules. The transition relation \(T(s)\) is therefore closed. The visible board alone is not a complete state for full chess. A full-rule solver needs at least \[ S=(B,\tau,C,E,h,R), \] where \(B\) is piece placement, \(C\) castling rights, \(E\) en-passant state, \(h\) the no-pawn/no-capture clock, and \(R\) the repetition-relevant history. Two visually identical boards can have different legal moves or draw status when these hidden coordinates differ. That is exactly the framework's projection-collision criterion: compressing states with different future laws destroys solvability information that downstream computation cannot reconstruct. ## 3. A proved projection instead of a heuristic compression The square board has eight dihedral symmetries: four rotations/reflections and their diagonal counterparts. Let \(G=D_4\) act on positions and define \[ \kappa(s)=\min_{g\in G}\operatorname{encode}(g s). \] This canonicalizer is admissible only if it is transition preserving: \[ \boxed{ \{\kappa(s'):s'\in T(s)\} = \{\kappa(s'):s'\in T(gs)\} \quad\forall g\in D_4. } \tag{C1.1} \] The production solver exhaustively checked (C1.1), together with check status, on every canonical position and every group element: 400,120 comparisons passed. Therefore canonicalization is an exact quotient/lift, not an evaluation heuristic. It reduces 399,112 raw legal states to 50,015 equivalence classes: \[ \frac{399112}{50015}=7.979846046\ldots \] The near-eightfold reduction is the measurable value of recognizing the correct geometry. ## 4. Cycles require a fixed point, not a tree Chess positions can repeat. Treating move sequences as a tree duplicates the same state indefinitely. C1 instead solves the directed state graph. Let \(M\) be the checkmate states with the losing side to move. Beginning with \(L_0=M\), retrograde propagation applies \[ W_{n+1}=W_n\cup \{s:\exists s'\in T(s),\ s'\in L_n\}, \tag{C1.2} \] and \[ L_{n+1}=L_n\cup \{s:T(s)\neq\varnothing, T(s)\subseteq W_{n+1}\}. \tag{C1.3} \] States outside the least fixed point \(W\cup L\) are draws: the defender has a draw-preserving continuation or the state is terminal stalemate. This is the rigorous form of the project's time-cycle rule. A repeated coordinate is not expanded again; it is solved as the same graph state. Distance to mate obeys two different optimal controls: \[ d(s)=1+\min_{s'\in T(s)\cap L}d(s') \quad(s\in W), \tag{C1.4} \] \[ d(s)=1+\max_{s'\in T(s)}d(s') \quad(s\in L), \tag{C1.5} \] because the winning player accelerates mate while the losing player delays it. ## 5. Exact solved result | Quantity | Result | |---|---:| | Raw legal states | 399,112 | | Exact \(D_4\) canonical states | 50,015 | | Canonical transitions, including rook-capture draw | 558,816 | | Symmetry-transition comparisons | 400,120 | | Side-to-move wins | 21,959 | | Side-to-move losses | 25,260 | | Draws | 2,796 | | White-to-move losses | 0 | | Black-to-move wins | 0 | | Maximum forced-mate distance | 32 plies = 16 moves | One maximum-distance canonical state is: > White: king a1, rook b2. Black: king c2. Black to move. With Black delaying and White mating as quickly as possible, the certificate supplies a legal 32-ply line ending in: > White: king b6, rook c8. Black: king a8. Black to move and checkmated. Published KRK verification work independently reports that the optimal strategy mates within 16 moves, matching C1's computed maximum [Stålmarck et al., “Proving Correctness of a KRK Chess Endgame Strategy by SAT Based Constraint Solving”](https://journals.sagepub.com/doi/pdf/10.3233/ICG-2013-36203?download=true). ## 6. Independent proof audit `verify_dimensional_chess.py` imports no production function. It independently reconstructs: 1. all 399,112 raw legal positions; 2. all 50,015 symmetry orbits; 3. every legal transition and checkmate/stalemate condition; 4. all 400,120 symmetry-closure comparisons; 5. every win/loss/draw Bellman equation; 6. every shortest-mate and longest-defense distance; 7. every stored optimal policy edge; 8. all 32 concrete moves in the longest principal variation; 9. the certificate and principal-variation digests. The independent audit passes. A stronger semantic-tamper test altered a WDL value and recomputed the file digest, so a hash-only verifier would accept the change. The independent verifier still rejected state 133 because its draw fixed-point equation became false. This separates data integrity from mathematical correctness. ## 7. Did this solve chess? It solved a complete chess universe, not the initial position of full chess. Claude Shannon's original analysis already explains the central wall: a typical full-game tree is on the order of \(10^{120}\), so straightforward enumeration is infeasible even though the rules are finite [Shannon, “Programming a Computer for Playing Chess” (1950)](https://vision.unipv.it/IA1/ProgrammingaComputerforPlayingChess.pdf). Modern tablebases solve restricted endgames exactly; current Syzygy infrastructure exposes complete seven-piece tables and partial eight-piece work [Lichess tablebase implementation](https://github.com/lichess-org/lila-tablebase). Our eightfold symmetry quotient is real but nowhere near enough to defeat that scale: \[ 10^{120}/8\approx1.25\times10^{119}. \] The next frontier is therefore not “add more brute force.” It is to discover a uniform canonical relation \(\kappa\) satisfying the same closure law as (C1.1) while collapsing astronomically more states. Any proposed lift must include polynomial-cost recognition, transformation, solution, and certificate verification. If two merged positions have different minimax values or successor quotients, the lift fails immediately. ## 8. What our mathematics contributed Retrograde endgame analysis is established mathematics; C1 does not claim its invention. The contribution of this experiment is a clean executable demonstration that the Theory of Reality's disciplines are useful and falsifiable: - **Hidden-state completion:** retain every coordinate needed to determine future legal laws. - **Projection-collision audit:** merge only states whose transition quotients are identical. - **Dimensional lift:** recognize the \(D_4\) orbit and replace eight geometric views by one canonical state. - **Cycle handling:** solve a fixed point on the state graph rather than unfold an infinite tree. - **Zero axis:** the unsolved state initially represents all three game-theoretic potentials \(\{W,D,L\}\); retrograde constraints remove impossible values. - **Proof before rhetoric:** every result carries a machine-checkable Bellman witness. - **Finite-extension wall:** solving KRK does not determine KQK, KBNK, seven-piece chess, or the initial position without a proved extension rule. The experiment succeeds as a demonstration of the method. It does not yet supply the massive new quotient theorem required to solve full chess. ## 9. Next dimensional ladder The disciplined progression is: 1. solve and independently certify KQK, KBBK, and KBNK using the same state algebra; 2. replace piece-specific code with a typed material compiler; 3. add capture transitions between material strata; 4. add pawns, promotion, castling, en passant, halfmove time, and repetition history as explicit coordinates; 5. search for larger exact quotients using material symmetries, pawn-file symmetries, reversible-region components, and independently verified decompositions; 6. measure state reduction against recognition and certificate cost; 7. refuse a “full chess solved” claim until the initial position has a complete verifiable strategy certificate. That is the wall we exposed: chess is finite, but the missing variable is a sufficiently powerful **value-preserving canonicalizer**, not merely more clock cycles.