# Dimensional Chess C8: Movement-Frequency Relations

**Date:** 15 September 2026  
**Status:** independently bounded and audited computation-allocation experiment  
**Claim boundary:** move ordering only; no learned score prunes a move or changes game value

## Result

C8 tests the proposal that chess pieces exhibit reusable relational movement
rhythms. A move is mapped to a class describing piece type, displacement,
capture status, square-color phase, distance changes to both kings, geometric
relations to both kings, edge-to-center motion, and home-relative rank phase.
The learner records which defender classes terminate bounded mating claims in
eight deterministic phase-shuffled five-ply searches. A later exact search
orders defender moves by the frozen empirical score.

For White as designated attacker at nine plies, the frozen relational spectrum
improves the already verified C7 proof:

| Method | Proof calls | Generated edges | Horizon nodes | Proof time |
|---|---:|---:|---:|---:|
| C7 lexical | 2,951,587 | 5,259,337 | 1,308,014 | 575.877 s |
| C8 relation | 2,750,245 | 4,299,169 | 930,982 | 375.703 s |
| Change | **−6.82%** | **−18.26%** | **−28.83%** | **−34.76%** |

Training takes 31.087 seconds. Including training, C8 takes 406.790 seconds,
29.36% less than the C7 proof run. Both methods return

\[
B_W(s_0,9)=0.
\]

This is a repeatable computation-allocation improvement, not a stronger game
theorem. The independently verified nine-ply result was already established by
C7.

## Frequency coordinate

For legal move \(a:q\to q'\) in state \(s\), C8 uses a relational coordinate

\[
\phi(s,a)=
(p,c,\Delta x,\Delta y,\chi,
 \operatorname{sgn}\Delta d_{K_s},
 \operatorname{sgn}\Delta d_{K_o},
 g_s,g_o,\Delta r,h(q),h(q')).
\]

Here \(p\) is moving-piece type, \(c\) is captured-piece type, \(\chi\) is
destination-square parity, \(d_{K_s}\) and \(d_{K_o}\) are Chebyshev distances
to the same-side and opposing kings, \(g_s,g_o\) classify king, knight, rank,
file, diagonal, or other geometry, \(r\) is the square's edge ring, and \(h\)
is rank measured from the mover's home side. Rank displacement is color
oriented, so forward motion has the same sign for both colors.

For class \(u\), phase-shuffled training accumulates depth-squared success and
failure weights \(S_u,F_u\). Its ordering score is the Beta-smoothed estimate

\[
q_u=\frac{S_u+1}{S_u+F_u+2}.
\]

At defender nodes, legal moves are sorted by descending \(q_u\), with lexical
UCI order as the deterministic tie-breaker. At attacker nodes, lexical order is
retained. Every legal move remains in the list.

## Patterns in the learned patterns

The White-attacker model contains 102 classes. Among its top 20:

- ten are quiet pawn advances;
- four are quiet knight developments;
- three are quiet rook moves;
- one each is a quiet king, queen, and bishop move;
- all 20 are non-captures;
- 19 reduce the mover's Chebyshev distance to the opposing king.

The strongest class is a one-square quiet pawn advance from home-relative phase
one to phase two, preserving distance to the mover's king while reducing
distance to the opposing king. It records 296 successful defender cutoffs and
zero failures in the eight training traversals. The second is the corresponding
two-square pawn advance, with 286 successes and zero failures.

These are not claims that pawn advances are objectively best chess moves. They
are recurring early counterexamples to White's short-horizon forced-mate claims
in this exact search task.

## Seven-ply transfer and repeatability

| Ordering | Calls | Generated edges | Test time |
|---|---:|---:|---:|
| Lexical | 95,718 | 172,970 | 15.758 s |
| Piece class | 97,608 | 174,845 | 14.174 s |
| Vector class | 96,057 | 158,558 | 15.898 s |
| Relation class | 98,352 | 162,511 | 13.110 s |

Three fresh repeats give identical relational counts of 98,352 calls and
162,511 edges, with a 13.417-second median. Three lexical repeats give identical
counts of 95,718 and 172,970, with a 15.272-second median. Thus the relational
model's 6.05% edge reduction and 12.15% median time reduction are reproducible,
although it makes 2.75% more recursive calls.

## Falsified variants

The experiment deliberately retains the failures:

| Variant at seven plies | Calls | Change vs lexical |
|---|---:|---:|
| Absolute oriented squares | 98,003 | +2.39% |
| Pair-conditioned piece dance | 167,983 | +75.50% |
| Pair-conditioned vector dance | 173,013 | +80.75% |
| Pair-conditioned relation dance | 142,787 | +49.18% |
| Online-updated relation model | 145,995 | +52.53% |

Pair conditioning expands the vocabulary from 102 relation classes to 1,264
pair-relation classes and destroys transfer. The prior move contains useful
context, but this direct concatenation fragments support faster than it adds
predictive information. Online reinforcement also destabilizes traversal.

The role symmetry fails. When Black is the designated attacker, a separately
trained 239-class model makes the nine-ply proof worse:

| Black-attacker method | Calls | Generated edges | Proof time |
|---|---:|---:|---:|
| C7 lexical | 127,991 | 736,248 | 23.080 s |
| C8 relation | 136,288 | 1,070,334 | 34.733 s |

Using the White-trained color-oriented book for the Black task also loses,
requiring 135,536 calls, 1,033,120 edges, and 40.057 seconds. The observed
frequency is therefore role-specific, not a universal color-normalized law.

## Audit

The C8 result agrees with the separately implemented and independently verified
C7 certificate that neither side can force mate within nine plies. A structural
audit samples 960 deterministically reachable states and checks seven ordering
schemes over 168,882 legal-move instances. Every ordered list is exactly a
permutation of the corresponding legal list. Of the state/scheme cases, 6,720
actually change order, establishing that the test is not a lexical no-op.
Audit status: **PASS**.

## Consequence

The experiment validates a narrow version of the user's intuition: relational
movement frequencies can reduce exact proof computation when their granularity
matches the role and horizon. It simultaneously rejects three stronger claims:
absolute squares, direct move-pair dances, and color-normalized universality do
not generalize here.

C8 still does not compress the game state or narrow the unbounded root interval

\[
I(s_0)=[-1,+1].
\]

The next frontier is a learned relation that carries a sound interval or a
proof obligation. Until a frequency class certifies all of its represented
completions, it may order computation but cannot delete it.