# Dimensional Chess C3: Full-State and Initial-Position Probe

**Version:** 1.0.0  
**Date:** 15 September 2026  
**Status:** Verified computational checkpoint; full chess is not solved

## Verdict

This checkpoint moves the project from isolated endgames to the orthodox initial
position under the frozen FCS-2023 rules. It implements the complete move-state
coordinates needed for legal play, passes six standard perft positions and ten
targeted rule-transition tests, and then performs an exact four-ply interval
probe from the normal starting position.

The root remains

\[
[D^-,D^+]=[-1,+1].
\]

Therefore this computation does not establish whether the initial position is a
win, draw, or loss. That negative result is useful: the legal-state engine is no
longer the immediate wall. The active wall is obtaining strong sound bounds
before the answer is already known.

## 1. Frozen full state

The encoded state is

\[
S=(B,\tau,C,E,h,n,\mathcal R),
\]

where $B$ is the typed board, $\tau$ the side to move, $C$ castling rights, $E$
the en-passant coordinate, $h$ the halfmove clock, $n$ the fullmove number, and
$\mathcal R$ the repetition-key history. The transition engine includes:

- all orthodox piece moves and captures;
- check and self-check rejection;
- both castling sides, including attacked-transit restrictions;
- en passant, including the pinned-en-passant legality case;
- all four promotion choices;
- irreversible castling-right updates after king moves, rook moves, and rook
  captures;
- claimable threefold and 50-move draws;
- automatic fivefold and 75-move draws, with checkmate precedence.

The repetition key retains an en-passant square only when a legal en-passant
capture exists, matching the requirement that repetition depends on possible
moves. The basic material-dead predicate covers the ordinary bare-king and
minor-piece cases used by this checkpoint. A complete predicate for every
composed dead position remains an explicit boundary.

These semantics follow the official
[FIDE Laws of Chess](https://handbook.fide.com/chapter/e012023), especially
Articles 5 and 9.

## 2. Independent move-generation audit

The verifier checks published reference counts from the official
[Stockfish perft suite](https://github.com/official-stockfish/Stockfish/blob/master/tests/perft.sh).

| Position | Depth | Nodes | Result |
|---|---:|---:|---|
| Initial position | 4 | 197,281 | PASS |
| Kiwipete | 3 | 97,862 | PASS |
| Position 3 | 4 | 43,238 | PASS |
| Position 4 | 3 | 9,467 | PASS |
| Position 5 | 3 | 62,379 | PASS |
| Position 6 | 3 | 89,890 | PASS |
| **Total checked** |  | **500,117** | **PASS** |

Ten direct tests also pass: castling execution, castling-right mutation,
castling through check rejection, en-passant execution, pinned en passant,
promotion multiplicity, rook-capture rights mutation, threefold history,
automatic 75-move draw, checkmate precedence, and the basic dead-material
predicate.

Perft agreement is strong evidence for the tested move transitions. It is not a
proof that every reachable state or every FIDE interpretation is correct.

## 3. Exact interval probe from the initial position

The probe assigns every horizon leaf the uninformative interval $[-1,+1]$.
Only exact terminals or rule-authorized draw actions may create point values.
No material score, neural evaluation, opening book, or guessed probability is
used.

| Quantity | Depth-four result |
|---|---:|
| Root legal moves | 20 |
| Tree nodes including root | 206,604 |
| Legal edges | 206,603 |
| Horizon leaves | 197,273 |
| Checkmate leaf visits | 8 |
| Exact node visits | 16 |
| Certified dominated choices below root | 242 |
| Certified ignored-edge fraction below root | 242 / 206,583 = 0.1171% |
| Certified dominated root choices | 0 |
| Root interval | $[-1,+1]$ |

The eight checkmate leaves occur inside branches beginning with f2-f3, f2-f4,
and g2-g4. Their existence does not prove those first moves lose: White or
Black can choose other continuations. All twenty root children retain
$[-1,+1]$, so no initial move is yet legally ignorable.

## 4. Forcing dominance

The proposed idea that the most valuable path ignores the most legal moves can
be made exact, but it must be lexicographic:

\[
\operatorname{FD}(a)=
\bigl(\operatorname{WDL}(a),\operatorname{mate\ efficiency}(a),
\operatorname{certified\ sibling\ eliminations}(a)\bigr).
\]

For the side to move:

1. preserve the best game-theoretic result;
2. among forced wins, minimize distance to mate; among forced losses, maximize
   it;
3. only then prefer a proof path that certifiably eliminates more rival legal
   continuations.

The third coordinate is proof efficiency, not a replacement for chess value.
A legal move counts as ignored only when an interval inequality certifies that
it cannot improve the minimax result. Otherwise a seemingly bad sacrifice,
defence, or zugzwang resource could be deleted incorrectly.

At depth four, 242 such below-root choice certificates cover 0.1171% of the
206,583 legal edges expanded below the root, but zero initial moves are
eliminated. This directly falsifies any claim that shallow
checkmate-counting or local forcing alone has solved the opening position.

## 5. What this checkpoint falsifies

- A finite game is not thereby a computationally feasible game.
- Legal move generation does not determine the initial minimax value.
- Seeing checkmates in a shallow tree does not determine the root.
- Counting forcing moves without sound bounds is not safe pruning.
- A heuristic evaluation is not a proof interval.
- The word dimensional does not itself supply a value-preserving quotient.

## 6. Required next proof object

A full solution still requires all of the following:

1. a complete dead-position treatment or another exact terminal mechanism;
2. sound pre-solution abstractions that tighten $D^-$ or $D^+$;
3. proof-carrying dominance deletions;
4. transposition and repetition-safe strategy-DAG construction;
5. collapse of the initial interval to one point in $\{-1,0,+1\}$;
6. an independent verifier covering every required adversarial reply;
7. explicit time, memory, and certificate-size accounting.

The most promising next experiment is to hide the known KRK and KQK tables,
force an abstract interval engine to rediscover their values, and measure
forcing dominance before exposing oracle labels. Only a method that succeeds
there without circularity should be promoted into larger material strata.

## Exact claim

C3 provides a tested complete-move state engine and an exact depth-four
uncertainty certificate for the standard starting position. It does not provide
the initial position's game-theoretic value, a complete strategy DAG, a
polynomial-time chess algorithm, a quantum advantage, or evidence for physical
extra dimensions.