# Dimensional Chess C5: Tension Fields and Tactical Entry Ordering

**Version:** 1.0.0  
**Date:** 15 September 2026  
**Status:** Verified computation-allocation heuristic; not a full-chess solution or a pruning proof

## Result

The proposed attacker/defender heat map is a useful way to allocate search.
It identifies where opposing geometric dependencies overlap, distinguishes raw
interaction density from control imbalance, and orders tension-breaking entries
before quiet alternatives. It does not establish that the hottest move is best.

On the frozen Scholar's-mate fixture

```text
r1bqkb1r/pppp1ppp/2n2n2/4p2Q/2B1P3/8/PPPP1PPP/RNB1K1NR w KQkq - 4 4
```

the `f7` square is attacked by the bishop on `c4` and queen on `h5`, and defended
by the king on `e8`. Thus its pre-move entry signature is two attackers against
one defender. The C5 ordering places `Qxf7#` first among all 43 legal moves. An
independent verifier reconstructs all 128 colour-square attack-origin lists,
checks all 64 square formulas, recounts 43 legal moves, proves that `Qxf7` is
checkmate, and confirms that `Bxf7+` instead leaves one legal reply.

The same fixture provides a necessary counterexample: global tension increases
from 16 before the move to 28 after checkmate. Therefore global heat is neither
an evaluation function nor a quantity to maximize or minimize. Heat selects
where to calculate; adversarial legal replies determine what the move means.

## 1. Square field

For a state $s$, colour $c\in\{W,B\}$, and square $q$, define the geometric
attack multiplicity

\[
A_c(q;s)=\sum_{p\in P_c(s)} \mathbf 1\{p\leadsto q\}.
\tag{C5.1}
\]

The relation $p\leadsto q$ follows the piece's attack geometry and stops a
sliding ray at its first occupied square. It is intentionally geometric: a
pinned piece still contributes pressure. Legal move generation remains a
separate layer.

Three views are then available:

\[
\tau(q;s)=A_W(q;s)A_B(q;s),\qquad
\nu(q;s)=A_W(q;s)-A_B(q;s),\qquad
\lambda(q;s)=A_W(q;s)+A_B(q;s).
\tag{C5.2}
\]

- $\tau$ is local tension: it is positive exactly when both colours attack the
  square, and grows with the number of cross-colour dependencies.
- $\nu$ is signed control imbalance.
- $\lambda$ is total attack load, including one-sided pressure.

For a piece occupying $q$, define attackers from the opposing colour and
defenders from its own colour. Its raw geometric fragility is

\[
F(q;s)=\max\{0,\operatorname{attackers}(q;s)
                 -\operatorname{defenders}(q;s)\}.
\tag{C5.3}
\]

This is not a claim that the piece is lost. Pins, checks, mating attacks,
intermediate moves, and exchange sequences can reverse the local reading.

## 2. Tension-breaking entries

Let $a:q_0\to q_1$ be a legal action and $s'=T(s,a)$. From the mover's
perspective, record the pre-move destination counts

\[
E_A(s,a)=A_{\rm mover}(q_1;s),\qquad
E_D(s,a)=A_{\rm opponent}(q_1;s)
\tag{C5.4}
\]

and the local release

\[
R_\tau(s,a)=\max\{0,\tau(q_1;s)-\tau(q_1;s')\}.
\tag{C5.5}
\]

The present implementation labels an action a *tension-breaking entry* when it
captures an enemy occupant on an already contested destination, has
$E_A>E_D$, and releases positive local tension. For `Qxf7#`,

\[
(E_A,E_D,E_A-E_D,R_\tau)=(2,1,1,1).
\tag{C5.6}
\]

This captures the proposed idea precisely: focus computation where an entry
converts a contested dependency into a concrete imbalance.

## 3. Search allocation

For every legal action, C5 evaluates the opponent's legal replies one ply later.
Each reply receives a conventional static vector

\[
V=(\text{terminal},\text{material},\text{pressure},
   \text{control},\text{mobility})
\tag{C5.7}
\]

from the original mover's perspective. The action is assigned the minimum reply
vector, because the opponent chooses adversarially. Search is then ordered
lexicographically by:

1. worst-reply terminal, material, pressure, control, and mobility coordinates;
2. immediate mate and captured material;
3. tension-breaking entry, local tension released, and entry surplus;
4. enemy-piece control flips, other resolved tensions, replies eliminated, and
   total control flips.

These coordinates allocate computation. They do not license deleting a legal
action. Deletion still requires the interval-dominance certificate from C2/C3:
one action's proved lower bound must dominate another action's proved upper
bound, with the inequality reversed at minimizing nodes.

## 4. Frozen fixture results

| Fact | Independently checked value |
|---|---:|
| Legal moves | 43 |
| Contested squares before `Qxf7#` | 10 |
| Total tension before `Qxf7#` | 16 |
| White / Black geometric mobility before | 43 / 28 |
| `f7` White attackers | `c4`, `h5` |
| `f7` Black defenders | `e8` |
| `f7` attack/defence counts | 2 / 1 |
| `Qxf7#` opponent replies | 0 |
| `Bxf7+` opponent replies | 1 |
| Total tension after checkmate | 28 |
| Independent verification | PASS |

The move-ordering result is meaningful but intentionally modest. The chosen
fixture contains an immediate mate, so it tests bookkeeping and tactical focus,
not broad playing strength. On the initial position the field has zero contested
squares and the heuristic's top quiet move is not evidence of optimal play.

## 5. What survived and what failed

Survived:

- attacker and defender multiplicity defines an exact, visual local field;
- occupied-square surplus identifies geometrically fragile targets;
- local tension release and reply elimination distinguish forcing entries;
- the field is suitable for move ordering and parallel search allocation;
- the implementation and fixture admit an independent audit.

Failed or not established:

- larger total tension does not imply a better or worse position;
- an attacked or undefended piece is not automatically lost;
- local majority does not prove an exchange sequence is favorable;
- ordering success on one tactic does not demonstrate general strength;
- no legal full-chess move has been soundly pruned by C5;
- full chess remains unsolved.

## 6. Next falsifiable experiment

Freeze a tactically diverse corpus before scoring it: mates, quiet defensive
moves, sacrifices, overloaded defenders, pins, discovered attacks, promotion
races, zugzwangs, and deliberately misleading hot squares. Compare at equal
node budgets:

1. unordered alpha-beta;
2. checks/captures-first ordering;
3. C5 tension ordering;
4. a standard engine ordering baseline.

Measure solved positions, nodes to exact result, ordering regret, false-hotspot
rate, and certificate-backed pruning. C5 advances only if it reduces exact proof
cost on held-out positions without altering the proof semantics. A useful next
lift would represent exchange chains as a dependency graph, because raw counts
do not encode piece value, pins, move tempo, or the order in which defenders can
actually recapture.