# Dimensional Chess C2: Full-Chess Gate ## Exact interval minimax, certified disadvantage pruning, and the quantum boundary **Version:** 1.0.0 **Date:** 15 September 2026 **Status:** Formal full-chess target plus exact KQK and quotient/pruning experiments; full initial-position value remains unsolved ## 1. Verdict The project goal is now frozen: determine the game-theoretic value of the normal chess initial position and emit a machine-verifiable strategy certificate. A principal variation, a strong engine, or a large set of solved endgames is not enough. C2 turns the proposed “discard disadvantageous possibilities” idea into an exact theorem. Each state carries a lower and upper value bound. A move may be deleted only when those bounds prove that another move is at least as good for the player choosing. This is mathematically sound branch-and-bound. A heuristic evaluation may order the search, but it may never authorize deletion because a temporary material or positional disadvantage can be the only drawing or winning continuation. C2 also attacks a second proposed shortcut: merge all states with identical future laws. Exact strong-bisimulation refinement was computed on KRK and KQK. It preserves every certified WDL/DTM value, but compresses very little beyond the eight board symmetries. This negative result rejects the naive quotient and points to sound interval abstraction with counterexample-driven refinement. ## 2. What “fully solved chess” means The target ruleset must be frozen before computation. `FCS-2023` uses legal board moves, checkmate, stalemate, dead position, claimable threefold/50-move draws, and automatic fivefold/75-move draws from the FIDE Laws effective 1 January 2023. Clocks, resignation, and negotiated draw offers are excluded from the mathematical game; a correct claim is represented as an optional move to a draw terminal. FIDE Articles 9.2–9.6 define the repetition and move-count rules [FIDE Laws of Chess](https://handbook.fide.com/chapter/e012023). Under those automatic rules, play has finite depth. A recent preprint gives a maximum of 17,697 plies under the 2023 Laws; this sharpens finiteness but does not make the branching graph small [Yim, “The maximum length of a chess game under the 2023 FIDE Laws”](https://arxiv.org/abs/2608.14762). A full solution requires all of the following: 1. the exact initial value $v(s_0)\in\{-1,0,+1\}$ from White's perspective; 2. a strategy DAG covering every adversarial reply, not one selected line; 3. complete rule-state handling and legal-move verification; 4. local witnesses for every minimax or fixed-point assertion; 5. an independently implemented verifier; 6. complete resource accounting for construction and verification. ## 3. The full state is a path-sensitive dimensional lift The visible board is only a projection. A Markov-complete state is \[ S=(B,\tau,C,E,h,\mathcal R), \tag{C2.1} \] where: - $B$ records every current piece type, colour, and square, including promoted material; - $\tau$ is the side to move; - $C$ records castling rights; - $E$ records the en-passant possibility relevant to legal moves and repetition identity; - $h$ records the reversible-move clock needed for claimable and automatic move-count draws; - $\mathcal R$ records repetition-equivalence keys and occurrence counts needed for future threefold/fivefold decisions. FIDE Article 9.2.3 confirms that side to move, possible moves, en passant, and castling rights affect whether two board appearances count as the same position. Therefore deleting these coordinates creates a projection collision: two apparently identical boards can have different legal futures or terminal status. This preserves the project's path/subdimension identity. History is not an informal narrative attached to a board; the rule-relevant part is a coordinate of the mathematical state. ## 4. The zero axis becomes an exact value interval Use White-relative values \[ v(s)\in\{-1,0,+1\} \] for Black win, draw, and White win. Before a state is resolved, store \[ I(s)=[D^-(s),D^+(s)],\qquad D^-(s)\le v(s)\le D^+(s). \tag{C2.2} \] The computational zero axis is not assigned a guessed value. It begins with \[ I_0(s)=[-1,+1], \] which retains all three possible readouts. Terminals and exact tablebases seed point intervals. Once every legal child has sound bounds, ordinary minimax propagation gives \[ \begin{aligned} D^-(s)&=\max_{a\in A(s)}D^-(T(s,a)),& D^+(s)&=\max_{a\in A(s)}D^+(T(s,a)) &&\text{if White moves},\\ D^-(s)&=\min_{a\in A(s)}D^-(T(s,a)),& D^+(s)&=\min_{a\in A(s)}D^+(T(s,a)) &&\text{if Black moves}. \end{aligned} \tag{C2.3} \] The interval can only tighten when a premise is proved. The Boolean-like W/D/L readout occurs only when $D^-=D^+$. This integrates the theory's non-Boolean truth coordinates with standard exact minimax instead of replacing minimax by an unsupported score. ## 5. Exact Disadvantage Pruning Theorem Let move $a$ lead to interval $[L_a,U_a]$. At a White/max state, move $a$ can be discarded if another move $b$ satisfies \[ L_b\ge U_a. \tag{C2.4} \] Indeed, $v_b\ge L_b\ge U_a\ge v_a$, so $a$ cannot improve White's value. At a Black/min state, move $a$ can be discarded if another move $b$ satisfies \[ U_b\le L_a. \tag{C2.5} \] Then $v_b\le U_b\le L_a\le v_a$, so $a$ cannot improve Black's value. Equality is sufficient when the target is only the game-theoretic value; it may remove an alternative equally optimal strategy. Strict inequality identifies a strictly inferior move. These equations subsume the safe idea behind alpha-beta pruning and give the Dimensional Mathematics version a precise proof gate: > A move is not removable because it looks disadvantageous. It is removable > only when a retained bound witness dominates its complete remaining value > interval. ## 6. C2.1: exact KQK material compiler test C1's state/transition/canonicalization/fixed-point pipeline was independently reimplemented for White king and queen versus Black king on the standard board. The declared abstraction treats checkmate as a win and stalemate, queen capture, or infinite non-mating play as a draw; history-dependent FIDE draws are outside this small material universe. | Quantity | Exact KQK result | |---|---:| | Raw legal states | 368,452 | | $D_4$-canonical states | 46,137 | | Canonical edges, including queen-capture draw | 609,873 | | Symmetry-transition comparisons | 369,096 | | Side-to-move wins | 18,081 | | Side-to-move losses | 25,160 | | Draws | 2,896 | | Maximum forced mate | 20 plies = 10 moves | One maximum-distance canonical state is White king a1, White queen b2, Black king e4, Black to move. The stored optimal line has 20 legal plies and ends in checkmate. An implementation sharing no production functions independently re-enumerated all positions, transitions, symmetries, Bellman equations, distances, policies, and concrete moves. It passed. KQK therefore validates a second movement law and the typed-material direction. It still does not test interacting friendly pieces, capture transitions among material strata, pawns, promotion, or full history. ## 7. C2.2: the future-law quotient attack Let states initially be labelled by controller and terminal kind. Repeatedly split every block by the set of current successor blocks, retaining previous block identity so refinement is monotone. At the fixed point, two states remain together only if they are strongly bisimilar under these labels. This is an exact transition-law quotient. | Universe | $D_4$ states | Strong-bisimulation blocks | Extra compression | Rounds | |---|---:|---:|---:|---:| | KRK | 50,015 | 49,885 | 1.002606× | 13 | | KQK | 46,137 | 45,249 | 1.019625× | 9 | Every final block was checked to contain one WDL/DTM value. A separately ordered implementation reproduced both partitions and counts. The result is exact but negative: strict identity of complete future laws is far too fine. It also requires the complete graph before refinement, so it cannot solve a graph too large to construct. After solving, the same positions collapse to only 34 KRK or 22 KQK classes when grouped by side/WDL/DTM—compression factors of 1,471× and 2,097× from the $D_4$ graphs. That partition is circular because it uses the answer we seek. It is nevertheless a target: discover cheap pre-solution invariants whose sound intervals converge toward these semantic classes. ## 8. C2.3: how much does perfect disadvantage pruning buy? Using the completed tables as an oracle, C2 measured the maximum pruning opportunity. These measurements do not constitute a pre-solution classifier. | Universe | Original branching | Strictly worse WDL edges removed | WDL+DTM-optimal branching | |---|---:|---:|---:| | KRK | 11.1810 | 40,266 / 558,816 = 7.21% | 1.8880 | | KQK | 13.2633 | 40,323 / 609,873 = 6.61% | 2.3062 | Pure WDL disadvantage pruning is real but modest because losing positions often have many equally losing moves and winning positions often have many winning moves. When fastest-mate/longest-defense distance is also optimized, the ideal branching factor falls sharply. But those DTM values currently come from the solution, so treating them as a free pruning oracle would hide the entire hard part inside the evaluator. ## 9. The new scalable candidate: interval abstraction plus refinement The next engine should not demand identical concrete future laws. It should: 1. group states by a polynomial-cost relational signature; 2. compute sound block intervals $[D^-,D^+]$ using may/must transitions; 3. apply the dominance rules (C2.4)–(C2.5); 4. retain a witness for every tightened bound and every discarded move; 5. when a block remains ambiguous, extract the concrete counterexample that caused the conflict; 6. add only the missing invariant—material interaction, king safety, pawn reachability, zugzwang parity, repetition phase, or another proved feature; 7. split the block and repeat; 8. stop when the initial state's interval collapses to one value. This is a counterexample-guided abstraction-refinement loop. It preserves the full D-/D+ framework: $D^-$ is what has been forced from below, $D^+$ is what remains possible from above, and the path coordinate is retained whenever it changes the future rule. A new “dimension” is admitted only when a concrete counterexample proves that the old coordinates were insufficient. The cost gate remains mandatory. The signature language, recognition, refinement, transition abstraction, witness construction, and verification all count. A hidden exponential dictionary is not a dimensional shortcut. ## 10. Why this does not yet prove quantum computing An adversarial game tree can be represented as alternating AND/OR logic, and quantum algorithms can accelerate particular oracle models. Farhi, Goldstone, and Gutmann evaluate a balanced NAND tree with $N$ leaves in $\Theta(\sqrt N)$ Hamiltonian-oracle time [Farhi et al., “A Quantum Algorithm for the Hamiltonian NAND Tree”](https://theoryofcomputing.org/articles/v004a008/v004a008.pdf). This does not turn chess automatically into a $\sqrt N$-time solved game. Chess has transpositions, irregular branching, cycles/history, and a costly legal-transition oracle. Supplying a precomputed game tree as a free oracle would again hide the problem in the input mechanism. A valid quantum claim therefore requires: 1. a reversible encoding of the complete state (C2.1); 2. reversible legal-move and terminal predicates; 3. an explicitly constructed oracle, with its gate cost counted; 4. a quantum algorithm operating on that oracle; 5. total logical qubits, gates, depth, error assumptions, and success probability; 6. comparison with the best classical algorithm on the same instances; 7. independent replay or hardware evidence. Solving chess would validate the computational lift. A proved lower resource bound for its reversible quantum implementation would establish a quantum algorithmic result. Neither result alone proves that physical spacetime has eight dimensions. The mathematical lift, a quantum speedup, and a physical ontology remain three separate claims. Generalized $n\times n$ chess is EXPTIME-complete, so any claimed uniform polynomial solver would be a major complexity result and must confront that theorem [Fraenkel and Lichtenstein](https://doi.org/10.1016/0097-3165(81)90016-9). This does not prohibit solving the one fixed 8×8 initial position; it does prohibit inferring a general polynomial method from small endgames. ## 11. Full-Chess Proof Gate The project may state “full chess solved” only when all gates pass: | Gate | Required evidence | |---|---| | Rules | Frozen complete state and independently tested move generator | | Closure | Every generated successor remains inside the declared state law | | Bounds | Every $D^-/D^+$ update has a local soundness witness | | Pruning | Every deleted move satisfies a recorded dominance inequality | | Abstraction | Every merge has proved may/must transition semantics | | Root | $D^-(s_0)=D^+(s_0)$ for the normal initial position | | Strategy | A DAG covers every opponent response required by minimax | | Verification | Independent code reconstructs the rules and checks the certificate | | Cost | Construction and verification resources are reported honestly | | Quantum | Reversible/oracle costs and advantage are proved separately | ## 12. Next executable ladder 1. Generalize the material compiler to KBBK and KBNK. 2. Add captures between material strata so a move can transition into an already certified lower-piece universe. 3. Implement the D-/D+ abstract solver and dominance witnesses on the solved KRK/KQK graphs, hiding their table values during search. 4. Benchmark which relational signatures collapse the initial interval with the least refinement. 5. Add pawns and promotion, followed by castling, en passant, halfmove phase, and repetition phase. 6. Search from the normal initial position for a proof DAG, sharing transpositions rather than attempting to materialize every legal state. 7. Independently verify the root value and complete strategy certificate. 8. Only then compile the same transition/bound machinery reversibly and test a quantum resource claim. ## 13. Exact status after C2 We have not fully solved chess. We have strengthened the theory in three ways: - KQK is now exactly solved and independently verified; - exact future-law quotienting has been tested and shown insufficient by itself; - the user's disadvantage-pruning idea is now a sound interval-dominance theorem, with its achievable ceiling measured on two complete universes. The missing variable is narrower than before: a cheap relational signature and refinement rule that produces strong enough $D^-/D^+$ bounds to certify most move deletions before the underlying states are solved.