Skip to content

4D Chess Spectral — v1 Validation Notebook

"Can't stop the signal, Mal. Everything goes somewhere, and I go everywhere." — Mr. Universe, Serenity (Joss Whedon, 2005)

Signature epigraph of the spectral-research collection. The body of work — validated results and rigorous falsifications alike — was offered through conventional channels and dismissed as foolery. The math stands independently. The discipline since: ship every result, falsifications included, with full reproducibility and per-row provenance (the Mathematical Provenance Method). A corpus that publishes its own invalidations is harder to dismiss than one that doesn't, and propagates through every channel that ingests open research. The signal is in the world; it goes everywhere now.

Companion to the 2D reference at chess-spectral/. This notebook records what transferred cleanly from the 2D encoder (640-dim on Z_8^2) to the 4D encoder (now 45 056-dim on Z_8^4 at v1.1.1), what changed, and what is still open.

Paired with the implementation plan when-we-need-to-spicy-seahorse.md (kept in ~/.claude/plans/), which defined the six-phase v1.0 roadmap and the seven-phase v1.1.1 pawn-axis extension.

Project navigation + state-pointer

ReadTheDocs landing — https://mlehaptics.readthedocs.io/en/latest/ — is the canonical pointer to the current state across all sister notebooks in this project.

This notebook is a snapshot. Future framework additions will not be back-ported into it; the RTD landing tells you whether new sister notebooks or downstream developments are available.

Brief since-summary (as of 2026-05-08): - ephemerides-spectral framework matured through v0.26.0 — per-body action-angle catalogues, Attested Multi-Source Collector framework with MPR v1 normative format, four-tier reproducibility model (T0 / T1 / T2 / T3), schema-gap-driven trigger. PyPI: https://pypi.org/project/ephemerides-spectral/. - The Mathematical Provenance Method (MPM) formalised as the project's discipline (ephemerides notebook §0.0); instrument-first physics critique in ephemerides §20 with three-regime classification grounded in this 4D extension's parent (the chess-spectral §1b literature work). - Inkscape contribution shipped on the spectral-faithful branch — three new SVG filter primitives (feSpectralBilateral, feSpectralDistance, feSpectralNoise) using the same eigenbasis substrate. - mfo-spectral sister-notebook added (May 2026) — Metric Field Ontology, one candidate foundational-ontology framing hosted in the project (cavity-instrument analogy; fractal metric field; ~11D structure motivated independently of string theory). Not the project's endorsed answer over alternatives; ephemerides §20 cites MFO as a worked example without picking a spatial-structure side. Future MPM target. - Ephemerides §21 — Tool-rejection as MPM-screening failure (symmetric counterpart to §20). Names the evaluator-side screening failure: rejecting work by which tool was used to make it, not by what it claims. Anchored on the historical orbital-mechanics chain DE441 traces back to (Copernicus / Bruno / Galileo / Kepler). §21.3 names the disability-accommodation dimension explicitly: categorical tool bans function as participation barriers for contributors with aphantasia, ADHD, dyslexia, motor disabilities, and many other variations. MPM is tool-agnostic by design.

Run the gates yourself

cd docs/chess-maths/chess-spectral/python
python -m chess_spectral_4d.cli tables-verify --phase all
python -m pytest tests/test_encoder_4d.py tests/test_roundtrip_4d.py \
                 tests/test_pawn_axis_symmetry.py -v

--phase all now covers Phases 1..5 plus the v1.1.1 pawn-axis orthogonality gate. All gates — plus the pytest suite (including test_c_py_parity_4d.py when a built spectral_4d binary is present) — must pass before any further work.

Phase-by-phase results

Phase 1 — 4D piece graphs

Verifies the Oana & Chiru piece-movement definitions produce the mobility numbers reported in their section 3.

Piece Expected (O&C) Observed (gate) Where
Rook 28 28 every one of 4 096 squares
King 80 (interior) 80 100 sampled deep-interior sqs
Knight 48 (interior) 48 100 sampled deep-interior sqs
Bishop 2 components 2 (2048/2048) partitioned by coord-sum mod 2
Queen Rook + Bishop exact union A_Q == A_R ∪ A_B
Pawn +w push out-deg 1 if w<7 directed graph; 0 at w=7

Transferred cleanly from 2D. The generator pattern is identical; we re-ran the closure checks at 4D and they reproduced the published numbers without special-casing.

New: the bishop disconnectivity structure is richer in 4D — the parity partition still splits into exactly two components (2 048 each), matching the combinatorial expectation from 2 * 8^3.

Phase 2 — Kronecker-sum eigenbasis

4D grid Laplacian L_grid on P_8 □ P_8 □ P_8 □ P_8 has eigenvalues λ_i + λ_j + λ_k + λ_l where {λ_n} is the P_8 spectrum.

P_8 eigenvalues: 0.0000, 0.1522, 0.5858, 1.2346, 2.0000, 2.7654, 3.4142, 3.8478
Spectrum range: [0.000000, 15.391036]  (max = 4 * λ_max of P_8)
Unique eigenvalues: 225  (many are tensor-sum-degenerate)
Trace identity: trace(L) = 28 672 == sum(kron eigvals)
L v = λ v holds on 20 random tensor products with max residual 4.44e-16.

Memory win: we never construct the 4 096 × 4 096 dense eigenvector matrix. The tensor-DCT basis is applied on demand either as np.einsum over tensor factors or as a double np.kron product (128 MB, built once for the Phase ⅘ transforms).

Transferred cleanly from 2D's grid eigenbasis — the only change is a tensor of 4 DCT factors instead of 2.

Phase 3 — B_4 group action + partial irrep projection

B_4 = (Z_2)^4 ⋊ S_4, |B_4| = 2^4 · 4! = 384, 20 irreps in total.

For v1 we compute two irrep projectors:

  • A_1 (trivial): orbit-average projector P_A1 = (1/|B_4|) Σ_g Π(g). Closure of 7 generators (4 sign flips + 3 adjacent S_4 transpositions) reached 384 elements exactly. The resulting sparse CSR projector has rank = #orbits = 35 on Z_8^4; idempotent and B_4-invariant within 1e-12.
  • Standard 4D: closed form P_std = I_4 − (1/4) J_4 on coord channels. Trace 3, annihilates the axis-sum.

Character spot-checks matched the known values of the B_4 defining rep (whose character equals the number of fixed axes of a signed permutation):

χ_std(identity) = 4
χ_std(sign-flip on one axis) = 2
χ_std(axis transposition) = 2

Deferred to v1.1: the other 18 irreps via Serre projection over bipartitions of 4. Not needed for the current channel layout.

Phase 4 — 4D fiber bundle

Core algebraic identities of the encoder, cross-checked at 4D:

A_queen == A_rook + A_bishop            (exact, disjoint support)
C_queen == C_rook + C_bishop            (DCT basis; max |Δ| = 1.14e-13)
||rook diag-dev||_2 = 1376.74           (finite, nonzero: rook shadow lives)

Pawn antisymmetric fiber is block-diagonal over (x, y, z). A_anti = (A_pawn − A_pawn^T) / 2 factors as I ⊗ I ⊗ I ⊗ A_w_anti, so in the tensor-DCT basis only the w-axis mixes modes. Gate measured

on-w-axis   ||C_anti||_F = 42.3320
off-w-axis  ||C_anti||_F = 3.55e-14

The off-w residual is pure round-off. This is the cleanest symmetry check in the whole pipeline — it tells us the w-axis convention for pawn direction is not just a naming choice but a spectral fact.

6-piece diagonal deviation table (norms of row vectors in DCT mode space, unitless; pawn_sym derives from the directed-push symmetrized adjacency):

pawn_sym=423.62  knight=1603.55  bishop=2936.36
rook=1376.74     queen=4725.19   king=3194.72

Queen's diag-dev norm is close to bishop + rook (2936 + 1377 = 4313, vs. observed 4725 — triangle inequality, not equality, since they're vectors in mode space with different directions).

Transferred cleanly from 2D: the (pawn-is-the-only-antisymmetric contributor) and (rook-is-the-only-diag-dev contributor that actually commutes like a rook) narratives hold at 4D. In 2D, the rook's DIAG_DEV norm was ~88.05; at 4D it scales to ~1376.74, roughly consistent with the n^{1.5} growth you'd expect from stacking 4 axes of K_8 on an 8-cube.

Shipped in v1.1: per-square fiber coordinates on the rank-3 cross-piece SVD basis. Instead of materializing the 2.6 GB (5, 4096, 4096) local adjacency tables of the naive 2D analogue, we build one signal-space fiber matrix per direction — K_d = U F_d U^T, where F_d reconstructs the d-th cross-piece SVD fiber direction as a symmetric zero-diagonal (4096, 4096) matrix — and accumulate fib_d(p, s) = Σ_{t ∈ adj_p(s)} K_d[s, t] per piece/square/direction. Resulting table: FIBER_LOCAL[5, 4096, 3] float32 = 240 KB runtime. Peak build memory: ~200 MB (U + one K_d).

Rook is dropped from the SVD and its FIBER_LOCAL row is held at zero — rook's global off-diag is identically zero, so any non-zero per-square fib_d(rook, s) would be an artifact of the local decomposition, not a genuine fiber participation. The rook-only-zero test encodes this discipline.

Phase 5 — encoder + spectralz v3

  • encode_4d(pos4) -> np.ndarray[float32, (40960,)]
  • 10 channels, each 4 096 modes wide:
idx Name Dim range Space Status
0 A_1 [0 : 4096] signal ✓
1 STD4_X [4096 : 8192] signal ✓
2 STD4_Y [8192 : 12288] signal ✓
3 STD4_Z [12288 : 16384] signal ✓
4 STD4_W [16384 : 20480] signal ✓
5 FIB_SYM_1 [20480 : 24576] signal ✓ (v1.1)
6 FIB_SYM_2 [24576 : 28672] signal ✓ (v1.1)
7 FIB_SYM_3 [28672 : 32768] signal ✓ (v1.1)
8 FA_PAWN [32768 : 36864] mode ✓
9 FD_DIAG [36864 : 40960] mode ✓

spectralz v3: 256-byte header (LARTPSEC magic, version=3, encoding_dim=40960, frame_bytes=163 854, board_dim_side=8, n_dimensions=4) + packed frames of encoding_dim*4 + 14 bytes each. The 2D reader remains untouched; a single dispatch shim frame_4d.read_header_any(fp) branches on the version field.

Round-trip gate: 10 random frames written and read back bit-identical (max |Δ| = 0.0) with all metadata preserved.

Phase 6 — cross-framework validation + unit tests

27 pytest cases across test_encoder_4d.py and test_roundtrip_4d.py. Notable algebraic checks:

  • A_1 channel is B_4-orbit-invariant — placing a piece at two orbit-equivalent squares produces bit-identical A_1 channels. This is the clean spectral analogue of "the board has no intrinsic orientation".
  • Pawn antisym flips sign with color — for equal-magnitude opposite-color pawns at the same square, the FA channel is exactly opposite (bit-exact).
  • Diagonal deviation is piece-specific — knight, rook, and queen FD channels are all pairwise distinct at the same square.
  • Channel energies partition total energy — Σ ch-energy = ||v||^2 (matches 2D semantics).

Cross-framework mobility numbers were recorded during Phase 1 gate and aligned exactly with Oana & Chiru section 3. No stop-the-line events.

What transferred cleanly from 2D

  • Graph-Laplacian + DCT eigenbasis recipe scales by tensor power.
  • Fiber-bundle structure (symmetric + antisym pawn + diag-dev rook shadow) stayed rank-5 conceptually; 3 sym subspaces were validated via SVD (v1.1 will exercise the fiber basis end-to-end in the encoder).
  • queen = rook + bishop holds as an algebraic identity in both adjacency and Laplacian pictures.
  • Channel energy partition + signal-vs-mode-space split carried over without modification.

What's new or different in 4D

  • 35 B_4-orbits on Z_8^4 (vs. 10 D_4-orbits on Z_8^2).
  • Standard rep expanded to 4 coordinate channels rather than the 2D's 2-dim E-irrep (so the encoder dedicates 4 channels to std-4D vs. the 2D encoder's 1 channel for E).
  • Memory pressure is real: the dense tensor-DCT basis is 128 MB; the pawn antisym fiber matrix is another 128 MB. We sidestepped materializing the local fiber tables entirely.
  • Pawn directionality is cleanly w-axis-only in spectral space — the off-w residual of C_anti is zero to 4e-14.

Known gaps / v1.1 candidates

  1. No game corpus yet. The plan called for 10 Oana-Chiru playouts. Requires external data (their engine is not wired into this repo). The CLI's encode-moves4 + corpus-gen subcommands still print "not yet implemented" (exit 4) and will be filled in when a JSONL move-log generator is available.
  2. C encoder + emit_tables_4d.py + test_4d_consolidated.c — all deferred to v1.1 per the plan.
  3. Full B_4 irrep decomposition (18 remaining irreps) — deferred.
  4. FEN4 grammar — deferred; encoder is driven by pos4 dict directly.
  5. Performance — no attempt at C-speed yet. The first call to encode_4d takes ~60 s to build the dense tables (U tensor + pawn-antisym + diag-dev + 3 K_d fiber matrices); subsequent calls are millisecond-scale via the module-level cache.

v1.1 investigation: empirical fiber rank

Follow-up to the v1 stub decision in Phase 4. Two analysis scripts produced the input numbers:

  • chess-spectral/python/analyze_fiber_rank_4d.py — cross-piece SVD on Z_8^4
  • chess-spectral/python/direct_orthogonality.py + chess-spectral/python/staged_cosine.py — localize which matrix representation the rank-3 result actually lives in (answer: the off-diag(U^T L U) representation specifically; see the two-claim section below for the sharper reading)

Cross-piece SVD singular values (grid-DCT basis, off-diagonal)

Normalized flattened upper-triangles of C_p = U^T L_p U stacked as rows; rook dropped because its off-diagonal vanishes identically (see below). Thin SVD on the 4-row matrix.

leading σ σ₂ σ₃ σ₄ rank @ 10% rank @ 1%
2D (Z_8²) 1.70 0.82 0.65 0.00 3 3
4D (Z_8⁴) 1.83 0.65 0.49 0.00 3 3

Fiber rank = 3 in both dimensions. The σ₄ zero is algebraic (queen ≡ bishop in fiber space, see below), not numerical. Leading σ grew slightly in 4D; σ₂ and σ₃ shrank — 4D fiber couplings concentrate more energy in the leading direction than 2D.

Two structural facts that force rank < #pieces

(i) Rook fiber vanishes identically. The K_8 Laplacian has spectrum {0, 8, 8, …, 8}; the DC P_8 eigenvector is the K_8 nullvector and every non-DC P_8 DCT column lies in the 8-eigenspace (it is orthogonal to the 1-vector). So U_P8^T L_K8 U_P8 = diag(0, 8, 8, 8, 8, 8, 8, 8) — diagonal. The 4D rook Laplacian is a Kronecker sum of four K_8 Laplacians, diagonal in the tensor-DCT basis, and so has C_rook_off ≡ 0 exactly.

Measured: - 2D: ||C_rook_off||_F = 3.78e-13 (round-off at dim 64) - 4D: ||C_rook_off||_F = 2.8e-13 (round-off at dim 4096)

This is not a 4D-specific property — the 2D rook also vanishes identically in the DCT basis. The original 2D chess_spectral/tables.py code already omits rook from its cross-piece SVD (it iterates ['Knight', 'King', 'Bishop', 'Queen']), which implicitly encoded this fact without ever naming it.

(ii) Queen ≡ Bishop in fiber space. A_queen = A_rook + A_bishop, so C_queen = C_rook + C_bishop. With C_rook diagonal, C_queen_off = C_bishop_off exactly. Verified bit-identical:

  • 2D: ||C_bishop_off||_F = ||C_queen_off||_F = 14.83435
  • 4D: ||C_bishop_off||_F = ||C_queen_off||_F = 370.6751

So the five piece Laplacians collapse to three independent fiber directions (knight, bishop/queen, king) in both dimensions.

Staged cosine progression — where does rank-3 emerge?

The rank-3 result is a property of a specific matrix representation. To locate which representation gives it, compute Frobenius cosines between piece pairs at four successive stages (see staged_cosine.py):

stage 1 — raw Laplacian         L = D - A
stage 2 — raw adjacency         A
stage 3 — centered adjacency    A - mean(A) · J/N²
stage 4 — DCT off-diagonal      off-diag(U^T L U)

Each stage strips one layer. Numbers:

2D (Z_8²):

pair raw L raw A cent. A DCT off
knight-rook 0.8381 0.0000 -0.1582 -0.02*
knight-bishop 0.8638 0.0000 -0.1190 0.4657
knight-queen 0.8698 0.0000 -0.2220 0.4657
knight-king 0.8495 0.0000 -0.1010 0.5089
bishop-rook 0.8973 0.0000 -0.2106 -0.02*
bishop-queen 0.9616 0.6202 0.5359 1.0000
rook-queen 0.9840 0.7845 0.7125 -0.02*

4D (Z_8⁴):

pair raw L raw A cent. A DCT off
knight-rook 0.9371 0.0000 -0.0073 0.15*
knight-bishop 0.9682 0.0000 -0.0100 0.7168
knight-queen 0.9652 0.0000 -0.0125 0.7168
knight-king 0.9639 0.0000 -0.0104 0.7603
bishop-rook 0.9645 0.0000 -0.0095 0.10*
bishop-queen 0.9957 0.8076 0.8048 1.0000
rook-queen 0.9848 0.5898 0.5859 0.10*

* rook-involving cosines at stage 4 are numerically undefined (rook off-diag norm is 1e-13). The printed values are noise / noise.

What we learn at each stage:

  • Stage 1 (raw L): every piece-pair cosine is 0.83–0.99. Dominated by the degree matrices on the diagonal. No discriminative information.
  • Stage 2 (raw A): every distinct-support pair is exactly zero. Knight edges are not rook edges, rook edges are not bishop edges, etc. The nonzero pairs are those where one piece's edge set contains the other (queen ⊃ bishop, queen ⊃ rook). This orthogonality is trivial — it reflects disjoint edge support, not any spectral property.
  • Stage 3 (centered A): removing the scalar mean barely changes anything (cosines shift by ~0.01–0.2). The DC-DC correction is small because piece adjacencies are already sparse.
  • Stage 4 (DCT off-diag): the picture flips. Knight-bishop, knight-queen, knight-king become strongly positive (0.47–0.76). Bishop-queen collapses to exactly 1.0 (degeneracy). Everything involving rook vanishes (rook fiber identically zero). This is where the rank-3 cross-piece SVD lives.

Interpretation

The Stage 2 → Stage 4 flip is the key finding. In the raw adjacency representation, pieces with disjoint edge support are trivially orthogonal — but that orthogonality gives rank 4 (knight + bishop + rook + king), not rank 3. Queen is a linear combination, rook is not. Five pieces → rank 4 trivial independence.

The DCT projection discards one more dimension — rook — by diagonalizing the K_8 Kronecker sum. Queen then equals bishop. Five pieces → rank 3 structural independence.

So the rank-3 claim has a specific content:

  • It is not a property of the raw graph Laplacians or their adjacency matrices (both of those give rank 4 after normalization).
  • It is a property of the piece Laplacians viewed in the grid (tensor-)DCT basis, after removing the basis's diagonal component.
  • The rank value (3 rather than 4) is forced by rook's Kronecker-sum structure: L_rook is a sum of K_8 Laplacians along each axis, each of which commutes with the DC/non-DC split of the DCT basis.
  • The rank holding at 3 across both 2D and 4D is therefore a claim about the grid-DCT representation being uniform-across-dimensions for this piece set, not a claim about intrinsic graph invariants.

For the v1.1 encoder this means the rank-3 basis is the right thing to ship — it captures the non-trivial shared structure. The narrative to put in prose, though, is "rank-3 in the grid-DCT off-diagonal representation" not "rank-3 graph invariant".

Two claims about piece-graph structure

The v1.1 investigation produced two distinct structural claims about chess piece graphs. They live on different objects and are easily confused. The test suite pins each of them to its own gate.

Claim 1 — Edge-support orthogonality (graph-theoretic, basis-free)

Statement. The binary adjacency matrices A_knight, A_bishop, A_rook share no edges pairwise, and the knight is edge-disjoint from the king:

<A_p, A_q>_F = Σ_{i,j} A_p[i,j] · A_q[i,j] = 0  exactly (integer)

for (p, q) in {(knight, bishop), (knight, rook), (bishop, rook), (knight, king)} in both Z_8^2 and Z_8^4.

Basis independence. The Frobenius inner product of two binary matrices is an integer edge-overlap count. It does not care about any eigenbasis; it is invariant under any orthogonal change of basis on either vertex set only up to the reindexing of edges. Orthogonality here means literally "no shared edges", not any spectral claim.

Gate. tests/test_edge_support_orthogonality.py (20 tests, integer-typed assertions). All pass on int64 dtype — zero is exactly zero, not 1e-13.

Claim 2 — Grid-spectral rank-3 (basis-dependent, representation-specific)

Statement. When each piece's graph Laplacian is pushed into the tensor-DCT basis of the grid and the diagonal is zeroed, the resulting C_off_p = off-diag(U^T L_p U) vectors for p ∈ {knight, bishop, rook, queen, king} span a rank-3 subspace (σ₄ ≈ 0 is algebraic, not numerical).

Basis dependence, explicit. The number 3 here is a property of (a) the grid-DCT basis U, (b) the off-diagonal projection, © the specific piece set. Change any of these and the rank changes. In particular:

  • In the raw adjacency basis the five pieces already drop to rank 4 by A_queen = A_bishop + A_rook (Claim 1's trivial queen partition lifted to a linear identity).
  • In the raw Laplacian basis the five pieces are all pairwise cosine ≈ 0.9 and look full-rank at any tolerance — the degree diagonal dominates.

The rank-3 claim is therefore "in this representation, with this diagonal-strip convention, rook is algebraically zero and queen collapses onto bishop". It is the right thing to encode into the fiber-sym channels because it captures the shared-structure directions that the off-diag(U^T L U) recipe can see. It is not an intrinsic graph property and should not be described as "basis-independent".

What v1.1 tooling does not claim

Earlier drafts of this notebook used language like "intrinsic geometry" and "basis-independent rank-3 result" for the cross-piece SVD finding. Those readings are wrong. Claim 2 is a representation- specific fact about a specific spectral basis; Claim 1 is a trivial graph-theoretic fact about disjoint edge supports. Conflating them obscures what is being measured.

Primitives-plus-sums decomposition

Claim 1 refines into a cleaner structural statement that classifies the piece set into primitives and their compositions. This language generalizes to 4D with a sharp dimension-sensitive twist.

Primitives and sums in 2D

Primitives (pairwise edge-disjoint): knight, bishop, rook.

Queen as an edge sum. A_queen = A_bishop + A_rook exactly as sets. Equivalently <A_B, A_Q> + <A_R, A_Q> = <A_Q, A_Q>.

King as an edge sum (2D only). Every king edge is either a distance-1 diagonal (one-step bishop) or a distance-1 orthogonal (one-step rook). So <A_B, A_K> + <A_R, A_K> = <A_K, A_K> exactly. The 2D king's move set is fully generated by the one-step primitives.

Knight does not participate. Knight is edge-disjoint from every other piece: its (2,1)-leap never coincides with any one-step or slider edge. Knight is the only piece whose movement is not expressible as any sum of one-step primitive components.

Dimension-sensitive king in 4D

The 4D king's move set is Chebyshev-1 in the full 4-dimensional sense: any cell reachable by changing 1, 2, 3, or 4 coordinates by ±1 simultaneously. This decomposes by Hamming distance of the move displacement:

HD class displacement count (interior) matches...
HD = 1 ±1 on one axis C(4,1) · 2 = 8 one-step rook
HD = 2 ±1 on two axes C(4,2) · 4 = 24 one-step bishop
HD = 3 ±1 on three axes C(4,3) · 8 = 32 none (un-named)
HD = 4 ±1 on all four axes C(4,4) · 16= 16 none (un-named)
total 80 Oana & Chiru interior

In 4D the one-step bishop + one-step rook strictly under-counts the king: <A_B, A_K> + <A_R, A_K> < <A_K, A_K>, and the deficit splits exactly into the HD=3 and HD=4 subgraphs of the king. The test file test_4d_king_hd_partition_exact asserts the full four-way partition.

Candidate pieces for the 4D HD=3 and HD=4 residuals (research note)

The 4D king includes move classes that have no independently named piece in the codebase:

  • Trihedral stepper (HD = 3 one-step). Displacement = (±1, ±1, ±1, 0) and permutations. 32 interior moves. Geometrically: "corner of a 3-cube inside a 4-cell move along three coordinate axes at once".
  • Tetrahedral stepper (HD = 4 one-step). Displacement = (±1, ±1, ±1, ±1). 16 interior moves. Geometrically: "corner of the 4-hypercube along all four axes at once".

These are well-defined graph primitives — their adjacencies would integrate cleanly into the primitives-plus-sums decomposition. The framework is agnostic to whether Oana & Chiru or any downstream ruleset names them. What the spectral machinery surfaces is that the named piece vocabulary {pawn, knight, bishop, rook, queen, king} under-specifies the 4D geometric primitives by exactly two classes. This is the kind of finding a purely top-down axiomatic ruleset is unlikely to notice.

This is a dimension-sensitive fact. In 2D, HD ∈ {1, 2} is exhaustive for Chebyshev-1 and the named pieces cover the ground. In 4D, HD can reach 3 or 4 and the named pieces miss those classes. The spectral framework is dimension-blind (the recipe doesn't change); the piece vocabulary happens to be two-dimensional by default.

Knight is the only piece that is edge-disjoint from every other named piece, including from the king (whose closure covers HD 1–4 one-step). It cannot be written as a sum of any other piece's edges at any distance. The (2,1)-leap is topologically isolated from one-step motion.

Does this graph-level isolation explain why, in Claim 2, the knight contributes an independent direction in the grid-DCT off-diagonal subspace distinct from the bishop+queen direction and the king direction?

The conjecture is: an edge set that lies in no other piece's closure cannot be a linear combination of them in off-diag(U^T L U) either, so it must add a spectral direction. The converse (pieces in one another's closures collapse) holds trivially: queen = bishop + rook at the edge level forces C_off_queen = C_off_bishop + C_off_rook at the spectral level.

This is speculative. The implication from edge-disjointness to spectral independence is only one-way obvious (edge-disjoint pieces have orthogonal raw adjacencies, but that is Claim 1, not Claim 2). The claim that they also span independent directions after the DCT projection and off-diagonal strip requires more work to make rigorous. Logged here as a bridge between the two claims rather than a proof.

v1.1.1 — Pawn axis split (Y / W)

v1.0 encoded every pawn as W-axis-forward. Oana & Chiru Def. 11 allows each pawn a fixed axis of motion chosen from {y, w} (never z), and §3.11 names Y↔W as one of three ruleset-preserving symmetries. v1.1.1 lifts the W-only assumption by splitting the single pawn antisymmetric channel into two.

Channel layout change

slot v1.0 v1.1.1 notes
0 A_1 A_1 unchanged
1-4 STD4_{X,Y,Z,W} STD4_{X,Y,Z,W} unchanged
5-7 FIB_SYM_{1,2,3} FIB_SYM_{1,2,3} unchanged
8 FA_PAWN FA_PAWN_W W-axis pawn antisym; byte range unchanged for bit-exact v1.0 backward parity on W-only fixtures
9 FD_DIAG FA_PAWN_Y new; Y-axis pawn antisym
10 — FD_DIAG shifted from slot 9

Encoding dim: 40 960 → 45 056 (one extra channel × 4096 modes). Spectralz format bumps v3 → v4. read_header_any keeps dispatching v3 blobs to the 40 960-dim reader for backward compatibility.

Factored Kronecker form

Each sub-channel lives on a single tensor factor of the 4D DCT basis:

PAWN_ANTI_FIB_w  =  I (x) I (x) I (x) W_ANTI_DCT     (w-axis, slot 4)
PAWN_ANTI_FIB_y  =  I (x) Y_ANTI_DCT (x) I (x) I     (y-axis, slot 2)

W_ANTI_DCT and Y_ANTI_DCT are both U_P8^T · A_anti_8x8 · U_P8 where A_anti_8x8 = 0.5*(S − S^T) is the 8×8 antisymmetrized boundary- aware forward shift. Numerically the two 8×8 blocks are equal (max|W − Y| = 0); the distinction is which tensor factor they inhabit in the 4096×4096 lift.

The encoder uses the factored form directly — it scatters 8 modes per pawn along the relevant axis, never materializing the 128 MB dense DCT-basis adjacency.

Feasibility gate (P1, hard-stop)

chess_spectral.tables_4d.verify_pawn_axis() — also exposed as python -m chess_spectral_4d.cli tables-verify --phase pawn-axis.

Observed numbers on the current build:

W-axis support: on-w-block ||.||_F = 42.3320, off = 0.00e+00
Y-axis support: on-y-block ||.||_F = 42.3320, off = 0.00e+00
on-axis Frobenius norms agree: |w - y| = 3.55e-14  (< 1e-10)
cross-channel <C_w, C_y>_F        = +1.54e-32      (< 1e-13)

The cross-channel inner product is the v1.1.1 feasibility signal: it must be exactly zero in infinite precision by the Kronecker trace identity tr(A⊗B⊗C⊗D) = tr(A)·tr(B)·tr(C)·tr(D) combined with tr(A_anti_8x8) = 0. A numerical floor at ~1e-32 is 19 orders below the 1e-13 gate — the independent-tensor-factor argument holds cleanly.

Y↔W ruleset symmetry (P2)

test_pawn_axis_symmetry.py verifies that the encoder respects the §3.11 Y↔W mirror:

  1. A pure W-axis pawn population, re-encoded after swapping each square's (x,y,z,w) → (x,w,z,y) and flipping axis labels w→y, produces FA_PAWN_W of the original equal to FA_PAWN_Y of the mirror (modulo the mode-index re-index).
  2. A mixed Y+W-pawn population's two channels swap roles under the same mirror.
  3. The mirror is an involution on the 45 056-dim encoding (applying it twice is bit-identical to identity).

These are orthogonal to the P1 orthogonality gate: P1 says the two sub-channels live on independent tensor factors; P2 says the encoder's routing respects the ruleset symmetry.

Position schema

  • Python: Dict[int, Union[str, Tuple[str, str]]]. Pawns are tuples ('P'|'p', 'y'|'w'); non-pawns remain single chars. Legacy single-char pawns ('P', 'p') are still accepted and route to the W axis with a one-shot DeprecationWarning per session.
  • C: sidecar uint8_t pawn_axis[CS_N_SQUARES_4D] on cs_position_4d_t (4 KB overhead). 0 = CS_PAWN_AXIS_W (matches memset zero-init → v1.0 callers still get W-axis pawns for free); 1 = CS_PAWN_AXIS_Y.
  • JSONL (positions_4d.jsonl): 2-char values for pawns ("Pw", "Py", "pw", "py"); single char unchanged for non-pawns. main_4d.c rejects any other 2-char value (z-axis pawns are explicitly forbidden by Def. 11).

C parity (P6)

spectral_4d encode-fixture remains bit-exact against the Python reference at 1e-10 absolute tolerance on all 11 gated channels across the expanded fixture set (12 legacy W-only + 1 Y-only + 1 mixed Y+W + 1 single Y pawn). See docs/chess-maths/chess-spectral/python/tests/test_c_py_parity_4d.py.

Bench sanity (bench_encoder_4d.py): speedup ratios stay in the 30–60× range even with the extra channel — e.g. starting_like_mini runs at ~43× vs v1.0's ~42×, single_pawn_white_y (a new fixture) clocks in at ~19×.

Phase operators (v1.2)

The chess4d-OC phase-operator move engine — a 4D analogue of the 2D §11 phase-operator framework — landed in v1.2 of chess-spectral. The full design + experimental record lives in PHASE_OPERATOR_SUPPLEMENT_4D.md; this notebook only carries the high-level pointer.

Highlights:

  • §13.1. The φ_4d phase encoding is a mixed-radix tower phi(x,y,z,w) = (x·g_x + y·g_y + z·g_z + w·g_w) mod M with (M, g_x, g_y, g_z, g_w) = (145451, 9719, 647, 43, 3). The ladder coefficient is 14 (not 7 as in 2D), so the integer- Diophantine no-aliasing condition holds in the wider [-14, 14]^4 box that operator-shift differences span.
  • §13.3. Empty-board structural gate: 4096 origins × 9 piece configs against tables_4d.X_targets, all 36,864 set-equality assertions pass.
  • §13.4. Occupation-aware Solution A validates against python-chess4d-oana-chiru (the Python reference port of Oana & Chiru's JS implementation at oanaunc/4d_chess). 50-position seeded corpus; tens of thousands of (state, origin, piece) cases, all green.
  • §13.5. Reverse-cast phasecast_is_check_4d agrees with the naive enemy-iterator oracle on 100 (state, color) pairs plus 7 hand-built check constructions covering each piece type's attack ray (rook on x-axis, knight (2,1)-leap, bishop xy-diagonal, queen zw-diagonal, blocked rook, W-axis pawn check, Y-axis pawn check).
  • §13.6. Pawn captures lifted from O&C §3.10 Def 13 (xw plane for W-axis pawns, xy plane for Y-axis). En passant + promotion remain deferred (matches the 2D supplement's deferral discipline).
  • §13.8. Cross-pollination from docs/othello-maths/: the coprimality is necessary but not sufficient discipline (Patch 2 of CHESS_NOTEBOOK_PHASE_1C_PATCHES.md) was the lesson that drove the C5 design constraint. All six patches from that document are also already present in the 2D notebook (chess_spectral_research_notebook.md); PR #67 verified this and updated the orphan-doc status header to FULLY APPLIED with citations to the §9f / §10.4 / §10.10 / §10.12 / §10.13 / §9a landing locations. The Othello-to-chess propagation loop is closed.

qm_4d Pre-flight Findings (Phase 1, 2026-04-29)

Three audits completed before the planned chess_spectral.qm_4d quantum-mechanical front-end ships. Each lands as a finding the QM extension consumes directly. The cross-disciplinary framing of how this all travels (ML hooks, adjacent variants, target subfields) lives in the 2D notebook's §15. Cross-Disciplinary Applications.

Pre-flight 1 — Encoder injectivity

encode_4d is not strictly injective on synthetic corpora: 8 collisions found out of 41 556 unique positions. All 8 are fixed points of the (central inversion x → (7,7,7,7) − x + global color flip) involution applied to anti-podal-king configurations (K@(0,0,0,0) + k@(7,7,7,7)) with one piece on the main 4D diagonal at one of {(2,2,2,2), (3,3,3,3), (4,4,4,4), (5,5,5,5)}. Real-game corpora are 100% injective (self-play 601/601, fixtures 15/15). Off-diagonal pieces and pawn-W axis pieces never collide.

The collision class is the fixed-point set of a global Z_2 symmetry the encoder respects by design (orbit projections, fiber sums, DCT-mode aggregations all preserve it). state_to_psi resolves the degeneracy by taking the side-to-move bit as input (already part of any chess state). Deeper interpretation: ψ naturally lives on configurations / Z_2 — a parity-superselection sector. See python/research/encoder_injectivity_4d.py for the probe and the 41 556-row CSV.

Pre-flight 2 — Pseudo-Hermitian audit of phase_operators_4d/

Strict reading: 0 linear operators, 9 predicates, 2 compositions. The phase-operator package returns frozenset[int] and bool, never vectors — they are reach predicates with operator-shaped vocabulary, not linear maps.

Lift reading: when materialized as 4096×4096 matrices on the board basis, P_rook4, P_bishop4, P_queen4, P_king4, P_knight4 are real-symmetric Hermitian matrices with real spectra:

basis n              = 4096
nnz                  = 114688 (rook)
real symmetric?      = True   (max |M − M^T| = 0)
all eigs real?       = True   (max |Im(eig)| = 4.7e-15)

Pre-flight 2 verified the structural property (real-symmetric ⇒ Hermitian) on rook + knight. Phase 2's full empirical sweep across all five non-pawn observables refined the per-piece spectrum bounds:

Piece Spectrum (Re(eig) range)
Rook [-4, 28] (integer; vertex-transitive lattice degree)
Bishop [-12, 54.4]
Queen [-16, 81.9]
King [-22, 67.7]
Knight [-36.06, 36.06]

Non-rook pieces are Hermitian with non-integer spectra: boundary clipping on the open Z_8^4 lattice breaks vertex-transitivity, so the per-vertex degree distribution is not constant. Pre-flight 2's [-4, 28] figure paraphrased the rook envelope across all pieces — that paraphrase is now corrected. Pawn predicates break Hermiticity (directed push) — that's where the qm_4d module's pseudo-Hermitian / PT-symmetric machinery (Bender / Mostafazadeh metric operator η) earns its keep. The five non-pawn observables are free Hermitians; the kinematic qm_4d module (python/chess_spectral/qm_4d.py) materializes them as H_rook_4, H_bishop_4, H_queen_4, H_king_4, H_knight_4 lazily.

Renaming recommendation: keep package name (123 occurrences across 34 files; load-bearing in __init__.py, immolation tests, CHANGELOG history); amend docstrings to "phase reach predicates / predicate-form generators of Hermitian operators"; reserve "operator" vocabulary for the qm_4d module that exposes matrices. See python/research/phase_operators_4d_pseudo_hermitian_audit.md.

Pre-flight 3 — Spectral identity verification

HOLDS at machine precision. The encoder's 4096 eigenmodes are exactly the simultaneous eigenbasis of (Δ, B_4 commutant), where Δ = P_8 □ P_8 □ P_8 □ P_8 is the 4D path-graph Laplacian.

Diagnostic Result
Distinct eigenvalues of Δ 225 (4096 modes accounted for)
Max multiplicity 127 at λ = 8.0 (spectral midpoint)
Encoder is Δ-eigenbasis max ‖Δv − λv‖ = 3.3e-16
Every eigenspace B_4-stable 225 / 225
Max ‖[π(g), P_λ]‖ over (g, λ) 1.6e-13
Encoder columns span E_λ exactly max ‖(I − P_λ) v_enc‖ = 6.2e-14

P_8 1D spectrum: [0, 0.15224, 0.58579, 1.23463, 2.0, 2.76537, 3.41421, 3.84776]. 4D spectrum is the additive Kron-sum of four copies. Multiplicity histogram: {1: 7, 4: 43, 6: 18, 12: 96, 24: 28, 34: 6, 42: 1, 60: 12, 64: 6, 72: 6, 76: 1, 127: 1}, sum = 4096.

The "engineering choice" of basis is a representation-theoretic theorem. Any B_4-equivariant function on Z_8^4 configurations decomposes naturally in the encoder's basis. See python/research/spectral_identity_4d_findings.md for full diagnostics.

Important wording correction. This is the path graph P_8 (open boundary), not the cycle graph C_8 / Z_8 Cayley graph. The notebook has used "Z_8^4" as a coordinate-set shorthand; the underlying lattice has open boundary and full B_4 symmetry but no translation symmetry. A future torus variant on C_8^4 would have a totally different spectrum (~4 distinct eigenvalues, exponential-mode eigenbasis, larger symmetry group B_4 ⋊ Z_8^4 adding the translations). The 2D notebook's §15.5 covers what the torus variant unlocks.

Phase 6 plan: 4D engine + tournament harness (post-QM)

After the QM extension ships (Tracks A + B), Phase 6 adds a 4D chess search engine alongside its 2D counterpart: position evaluators (material, spectral channel-energy, QM-expectation), a standard search stack (alpha-beta, iterative deepening, transposition tables, MVV-LVA, killer moves, null-move pruning, quiescence), and a self-play tournament harness for empirically validating the spectral / QM framework's chess relevance. The full design — three evaluator families, search architecture, tournament protocol, CLI surface, and the --help documentation discipline that's mandatory for every new command — lives in 2D notebook §16. Position Evaluation, Search, and Self-Play Validation. Symmetric implementation for 2D (chess-spectral) and 4D (chess-spectral-4d). The 4D side adds a search-and-evaluation layer on top of python-chess4d-oana-chiru's rule library; we don't have an exhaustive survey of prior 4D chess search implementations, so we make no "first ever" claim.

Combined consequence

These three results gate the qm_4d module as follows:

  • Pre-flight 1 → state_to_psi(state, side_to_move) takes the side-to-move bit; ψ lives on configurations / Z_2
  • Pre-flight 2 → free Hermitian observables for non-pawn pieces (H_rook_4, H_bishop_4, H_queen_4, H_king_4, H_knight_4); pawn dynamics is where the PT-machinery earns its keep
  • Pre-flight 3 → the spectral-identity demo is locked in as experiment 1 (numerically airtight, conceptually clean, piece-agnostic, doesn't rely on chess semantics)

Track A (kinematic-only qm_4d, ~500 LOC) is now unblocked. Track B (full move-as-unitary dynamics) requires additional design decisions (phase convention, time-evolution semantics, per-channel move derivation, Z_2 superselection, pawn pseudo-Hermitian metric η) documented in upcoming ADRs. The cross-disciplinary value of this stack is laid out in 2D notebook §15.

Last updated 2026-04-29 with qm_4d Pre-flight Findings (Phase 1) on branch chess-spectral/qm-4d-kinematic (planned). Previous v1.3.2 (2026-04-29) shipped the 2 KiB FEN4 buffer-overflow fix + dynamic __version__ derivation. v1.1.1 introduced the Y/W pawn-axis split (channels 8-10 relayout, spectralz v4 format, cross-channel orthogonality gate at 1e-32) on branch zesty-llama. v1.2 added the φ_4d phase-operator framework; see PHASE_OPERATOR_SUPPLEMENT_4D.md (pending merge into §13 of this notebook per the merge plan).

How to cite this notebook

BibTeX:

@misc{kirkland_chess_spectral_4d_2026,
  author       = {Kirkland, Steven},
  title        = {4D Chess Spectral --- v1 Validation Notebook},
  year         = 2026,
  howpublished = {\url{https://github.com/lemonforest/mlehaptics/blob/main/docs/chess-maths/chess_spectral_4d_notebook.md}},
  note         = {Part of \emph{mlehaptics: Spectral-Research Portfolio}; higher-dimensional extension of \texttt{chess\_spectral\_research\_notebook}. Project-level citation metadata at \url{https://github.com/lemonforest/mlehaptics/blob/main/CITATION.cff}. Co-authored with Claude Opus 4.7 (Anthropic, 1M-context configuration) per project memory \texttt{feedback\_orchestration\_metaphor}. Framing is one candidate within the project's research portfolio per \texttt{feedback\_no\_lineage\_claims\_in\_notebook}.}
}

Plain text: Kirkland, S. (2026). 4D Chess Spectral — v1 Validation Notebook. mlehaptics Spectral-Research Portfolio. https://github.com/lemonforest/mlehaptics/blob/main/docs/chess-maths/chess_spectral_4d_notebook.md

Per-result citation discipline. Specific technical claims cite their canonical sources directly (Rinaldi-Unciuleanu & Chiru 2026 for the hypercube primary technical content, etc., PDF-verified per [[feedback_pdf_extraction_citation_discipline]]). When citing a specific result, prefer citing both this notebook AND the underlying canonical source. Framings presented here are candidate methodological readings per [[feedback_no_lineage_claims_in_notebook]], not endorsed over alternatives without explicit empirical convergence.

Project-level citation. See CITATION.cff at the repo root for the project-as-a-whole citation form.