# Dimensional Chess C4: Constraint Waves and Human-Style Search **Version:** 1.0.0 **Date:** 15 September 2026 **Status:** Independently verified endgame abstraction; full chess remains unsolved ## Result The proposal that strong chess play is a changing wave of controlled squares, safe mobility, confinement, and opponent restriction is mathematically useful. Across the complete KRK and KQK universes, a constraint-wave signature proves every win/draw/loss value and gives sound mate-distance intervals. The combined signature makes mate distance exact for 44,633 of 50,015 KRK states (89.24%) and 45,276 of 46,137 KQK states (98.13%). An independently written verifier checks all 96,152 state rows against the exact tablebase certificates, recomputes legal-successor dominance, and passes with zero WDL violations, zero distance violations, and zero summary mismatches. This is not yet the full-chess solution. The same signature reduces unique transition profiles by only 1.00010× in KRK and 1.00030× in KQK. It predicts the known endgame structure well by becoming almost as detailed as the original state graph. ## 1. Human search as a constraint field Let $\mathcal A_P(s)$ be the squares attacked by player $P$ in state $s$, and let $\mathcal L_P(s)$ be the legal actions. A move $a$ creates \[ s'=T(s,a). \tag{C4.1} \] Human search does not expand every legal action equally. It estimates how the move changes: - king-safe squares; - defended and hanging pieces; - own and opponent mobility; - checks, captures, and forced threats; - confinement near edges and corners; - future escape routes; - the opponent's strongest legal reply. A non-king move onto an attacked square is not automatically irrational. A sacrifice may force mate or win material. Therefore an action is removable only when a sound bound proves that another action is no worse. With a White-relative value interval $I(s)=[D^-(s),D^+(s)]$, the constraint wave is the repeated operator \[ \begin{aligned} D^-_{k+1}(s)&=\max_a D^-_k(T(s,a)), & D^+_{k+1}(s)&=\max_a D^+_k(T(s,a)) &&\text{at White nodes},\\ D^-_{k+1}(s)&=\min_a D^-_k(T(s,a)), & D^+_{k+1}(s)&=\min_a D^+_k(T(s,a)) &&\text{at Black nodes}. \end{aligned} \tag{C4.2} \] Mate, stalemate, and draw facts are the boundary disturbances. Repeated application propagates them backward through the legal graph. This supplies an exact meaning for the proposed wave. ## 2. Frozen table-hidden experiment The bound phase receives legal transitions, terminal rules, the theorem that a lone black king cannot checkmate, and declared geometric features. It does not receive WDL or DTM labels. All variants and their intervals are frozen before the exact tablebases are opened for evaluation. The base signature includes side to move, terminal kind, black-king edge distance, check status, capture-draw availability, king separation, piece protection, corner distance, current legal-move count, piece/king distances, and piece edge positions. Four declared additions are tested: 1. relational orientation through dot product and absolute cross product; 2. the black king's connected safe-region size; 3. own/opponent mobility and edge-direction profile; 4. their combination with the remaining axis distances. ## 3. Exact results ### KRK | Signature | Exact DTM states | DTM fraction | Certified ignored moves | State compression | Profile compression | |---|---:|---:|---:|---:|---:| | Distance base | 3,911 | 7.82% | 21.21% | 1.7608× | 1.1000× | | + orientation | 13,775 | 27.54% | 74.25% | 1.0700× | 1.0005× | | + safe region | 6,759 | 13.51% | 60.27% | 1.3621× | 1.0811× | | + mobility restriction | 8,791 | 17.58% | 70.56% | 1.4164× | 1.0937× | | Combined wave | 44,633 | 89.24% | 90.65% | 1.0067× | 1.0001× | ### KQK | Signature | Exact DTM states | DTM fraction | Certified ignored moves | State compression | Profile compression | |---|---:|---:|---:|---:|---:| | Distance base | 6,959 | 15.08% | 54.31% | 1.6326× | 1.0800× | | + orientation | 30,554 | 66.22% | 90.02% | 1.0660× | 1.0011× | | + safe region | 22,008 | 47.70% | 87.84% | 1.3205× | 1.0556× | | + mobility restriction | 21,496 | 46.59% | 86.47% | 1.2828× | 1.0745× | | Combined wave | 45,276 | 98.13% | 92.45% | 1.0071× | 1.0003× | Every variant proves 100% of WDL values in both universes. Distance is harder: positions that share a game-theoretic result can require different optimal mate lengths. ## 4. What the evidence supports The human-search proposal survives three tests: - safe-region restriction contains mate-distance information; - mobility restriction contains additional mate-distance information; - relative orientation is especially informative, confirming that independent scalar distances omit a relational phase. The combined coordinates also certify that 506,548 of 558,816 KRK choices (90.65%) and 563,800 of 609,873 KQK choices (92.45%) may be ignored while preserving WDL and the available DTM ordering. These counts are certificate-backed. They do not mean an attacked square is automatically deleted or that a human-looking heuristic is exact. ## 5. What the evidence falsifies The absolute constraint field is not yet the scalable missing variable. The combined wave forms 49,681 KRK blocks from 50,015 states and 45,813 KQK blocks from 46,137 states. Almost every state also retains its own successor profile. The computation still had to enumerate the complete concrete graphs before building those profiles. Thus high predictive coverage and high move elimination do not imply a small proof. A method that stores nearly one signature per position is a compressed tablebase, not a general solution to chess. ## 6. The next lift: constraint deltas Humans usually notice what a move changes rather than memorizing a complete attack map. Define a constraint vector \[ C(s)= \bigl( \text{safe mobility}, \text{opponent mobility}, \text{confinement}, \text{attack dependencies}, \text{forcing rights} \bigr) \tag{C4.3} \] and the move-relative change \[ \Delta C(s,a)=C(T(s,a))-C(s). \tag{C4.4} \] The next candidate quotient should group actions by equivalent constraint deltas: which lines open or close, which pieces become pinned or defended, which king regions split, and which replies disappear. This could reuse one transition law across many absolute boards. The C5 falsification test is strict: 1. construct local attack-dependency graphs without table labels; 2. canonicalize their move-induced deltas; 3. compute sound WDL/DTM intervals on delta classes; 4. require transfer across rook, queen, bishop, knight, pawn, and mixed-material strata; 5. count construction, recognition, refinement, and certificate costs; 6. reject the lift if successor-profile compression again approaches 1×. ## Exact claim C4 validates constraint waves as informative coordinates in two complete endgames and independently verifies every emitted state bound. It also shows that the current absolute encoding is too close to enumeration to scale. It does not determine the normal starting position's value or provide a complete full-chess strategy.