# Dimensional Chess C6: Legal Exchange Dependencies

**Version:** 1.0.0  
**Date:** 15 September 2026  
**Status:** Independently verified bounded mate-search experiment; full chess remains unsolved

## Result

C6 replaces raw attacker counts with a legal capture–recapture dependency graph
on one contested square. The graph sees material values, recapture order, newly
opened slider rays, king safety, and pins enforced by legal move generation. It
is used only to order an exact bounded mate search.

On a deterministically generated eight-position corpus, every position contains
at least one legal capture followed by a legal recapture and has a forced mate
of minimal depth three plies. The corpus uses seed `20260915`; it accepts the
first eight qualifying positions in 258 attempts and never consults comparative
ordering performance.

All four search modes prove the same eight mates:

| Ordering | Proof nodes | Reduction vs lexical | Wall time |
|---|---:|---:|---:|
| Lexical UCI | 10,394 | 0.00% | 2.014 s |
| Checks/captures forcing | 699 | 93.27% | 0.579 s |
| C5 local tension | 809 | 92.22% | 0.700 s |
| **C6 exchange dependency** | **668** | **93.57%** | **0.638 s** |

C6 reduces nodes by 17.43% relative to C5 tension and 4.43% relative to the
forcing baseline. It does not reduce wall time relative to forcing: constructing
the dependency graphs makes the measured run 10.22% slower. Across individual
positions, C6 beats forcing on three, ties on one, and loses on four. The
aggregate win is real for this corpus but is not evidence of universal
superiority.

## 1. Restricted exchange graph

Let $a_0:q_0\to q$ be a legal capture by player $P$, and let
$s_1=T(s,a_0)$. Define $\mathcal X_q(s)$ as the legal captures in state $s$
whose destination is the same square $q$. Material balance from $P$'s
perspective is $M_P(s)$, and the change from the original position is

\[
\Delta M_P(s_i)=M_P(s_i)-M_P(s).
\tag{C6.1}
\]

The local optional-exchange value is

\[
E_P(s_i,q)=
\begin{cases}
\displaystyle
\max\left(\{\Delta M_P(s_i)\}\cup
\{E_P(T(s_i,a),q):a\in\mathcal X_q(s_i)\}\right),
& \text{if }\operatorname{turn}(s_i)=P,\\[6pt]
\displaystyle
\min\left(\{\Delta M_P(s_i)\}\cup
\{E_P(T(s_i,a),q):a\in\mathcal X_q(s_i)\}\right),
& \text{otherwise.}
\end{cases}
\tag{C6.2}
\]

The singleton $\Delta M_P(s_i)$ is the option to stop exchanging on $q$.
Every expanded capture is legal in the full position, so pinned pieces and
illegal king captures disappear automatically. Vacated squares can reveal
x-ray attackers at later graph levels.

This value is exact only inside the declared captures-on-one-square subgame.
It omits zwischenzugs, quiet threats, captures elsewhere, and most check
evasions. When a graph node is in check, the implementation records the event
because the abstract stop option may not correspond to an available full-game
move. Therefore $E_P$ is not a sound full-position value bound.

## 2. Search modes

Every benchmark mode traverses the same complete orthodox legal-successor
relation. Only sibling order changes.

- Lexical orders moves by UCI text.
- Forcing prioritizes immediate mate, check, captured value, and reply
  restriction.
- C5 prioritizes immediate mate, check, local tension entry, attack surplus,
  square tension, capture, and reply restriction.
- C6 prioritizes immediate mate, check, exchange value, then the C5 coordinates.

The bounded mate predicate for attacker $P$ is the AND/OR recurrence

\[
B_P(s,d)=
\begin{cases}
1,&s\text{ is checkmate won by }P,\\
0,&d=0\text{ or }s\text{ is a draw/other terminal},\\
\bigvee_{a\in\mathcal L(s)} B_P(T(s,a),d-1),
&\operatorname{turn}(s)=P,\\
\bigwedge_{a\in\mathcal L(s)} B_P(T(s,a),d-1),
&\operatorname{turn}(s)\ne P.
\end{cases}
\tag{C6.3}
\]

Move ordering changes how quickly short-circuit witnesses or counterexamples are
found; it cannot change the Boolean result.

## 3. Frozen corpus construction

The state generator places two nonadjacent kings, creates an enemy target with
an attacking piece and a geometrically defending piece, adds an attacking major
piece and limited mixed material, and rejects impossible states in which the
player who just moved remains in check. Admission then requires:

1. a legal capture from the side to move;
2. at least one legal same-square recapture after a legal capture;
3. no forced mate within one ply;
4. a forced mate within three plies; and
5. no duplicate FEN.

The exact Boolean admission test is independent of traversal order. A forcing
order computes it, but no node-count or runtime result is inspected during
selection.

The largest root exchange graph in each position contains two to four nodes
and reaches two or three capture plies. For example, `synthetic-006` produces a
four-node, three-capture-ply graph for `e6d5`, valued at −250 centipawns from the
mover's initial perspective.

## 4. Independent verification

The independent verifier does not import the C6 module. Using only the
separately tested orthodox move engine, it:

- recounts every legal root move;
- confirms that none of the eight positions has mate in one;
- enumerates every root move, every defender reply, and every mating third ply;
- reproduces the complete set of winning root moves for each position;
- covers 33 defender reply branches with 53 mating continuations;
- reconstructs the selected exchange graph for every position;
- matches every exchange value, graph-node count, maximum capture depth, and
  checked-node count; and
- independently recomputes the four aggregate proof-node totals.

All checks pass.

## 5. What this proves—and what it does not

Established computationally:

- legal recapture order contains useful search-ordering information beyond raw
  local attack counts on this corpus;
- C6 preserves exact bounded mate results because it changes order, not legal
  successors or terminal semantics;
- the deterministic corpus and independent checker make the reported numbers
  reproducible;
- C6 has the best aggregate node count among the four declared modes here.

Not established:

- C6 is not faster than the forcing baseline in measured wall time;
- C6 is not better on most individual positions;
- eight synthetic mate-in-two positions are not representative of chess;
- the exchange value is not a sound minimax bound and cannot prune moves;
- no initial-position interval has narrowed; and
- chess is not solved.

## 6. The next wall

The result identifies the next precise experiment. Replace the optional
single-square graph with a proof-carrying *forcing dependency graph* that may
include checks, required evasions, captures on other squares, promotions, and
quiet threats. Each abstract edge must carry lower and upper game-value bounds.
Only then can a graph do more than reorder siblings.

The decisive pass condition is not tactical accuracy. It is either:

1. a transferable abstraction that yields sound nontrivial intervals while
   retaining substantial successor-profile compression; or
2. a machine-checkable strategy certificate for the orthodox initial position.

Until one of those conditions is met, the work is progress toward solving
chess, not a solution of chess.