Chess §3.5.3(C) extension candidates — findings (2026-05-11)¶
Spike: Three extension candidates queued by the prior chess D₄/B₄ rep-theory spike and the knight per-eigenspace follow-up (both 2026-05-11):
- (C-F3a) Queen 2D + 4D — Mode I (closed-form eigenvalues) refuted (commutator
[A_rook, A_bishop] ≠ 0); Mode II (per-eigenspace D₄/B₄ irrep multiplicities) integer-exact at machine precision. - (C-F3b) Side-length-N parametric — rook/king/bishop Mode I closed-form scales cleanly with N. Tested
N ∈ {4,5,6,7,8}for 2D,N ∈ {3,4,5}for 4D. All 24/24 cases match. - (C-F3c) Toroidal
Z_8 × Z_8(2D) andZ_8^4(4D) — different symmetry group (abelian translationZ_n^dinstead of dihedral/hyperoctahedral D₄/B₄). Both Mode I (Fourier closed-form on cycle products) and Mode II (Fourier diagonalization = per-mode 1-dim abelian irrep) all 12/12 piece-dim-mode combos match at machine precision.
Dispatch: Conductor → concertmaster, 2026-05-11. Single dispatch covering all three candidates.
Bottom line: All three candidates pass. §3.5.3(C) chess instance gains:
1. Parametric robustness — Mode I works for any board size, not just N=8.
2. Queen as second Mode II piece — joins knight; non-product extension via D₄/B₄ irrep structure.
3. Toroidal as 4th instance — under fundamentally different group structure (abelian Z_n^d vs non-abelian D₄/B₄). Both Mode I and Mode II hold.
Reproducible: python docs/srmech/notes/chess-extension-candidates-script.py. Runtime 111.2 s on commodity workstation. Deterministic seed 20260511. Memory peak ~280 MB (held briefly for queen 4D class-trace computation).
1 — Queen 2D + 4D (C-F3a)¶
Mode I refuted (as predicted):
| Dim | [A_rook, A_bishop] max abs |
sort(eig A_q) vs sort(eig A_r + eig A_b) max dev |
|---|---|---|
| 2D (N=8) | 2.0 | 7.56 |
| 4D (N=8) | 2.0 | 22.5 |
Non-zero commutator → no closed-form eigenvalue prediction from eig(A_rook) + eig(A_bishop). Confirms the rationale: queen is a non-product piece (joins knight in that classification).
Mode II — D₄ / B₄ per-eigenspace integer-exact:
| Dim | Group | Eigenspaces | Total (irrep × eigspace) combos | Integer-exact | Max dev from integer |
|---|---|---|---|---|---|
| 2D | D₄ | 28 | 140 | 140/140 | 0.000e+00 (exact) |
| 4D | B₄ | 812 | 16,240 | 16,240/16,240 | 2.78 × 10⁻¹⁴ |
The 2D queen result is striking: all 140 multiplicities are machine-zero-exact (no FP drift at all on the 64-dim space). The 4D queen sits at the LAPACK eigh per-dimension floor (~10⁻¹⁴).
Queen 4D eigenspace dim histogram:
| Dim | Count | Notes |
|---|---|---|
| 1 | 71 | 1-dim B₄ irreps |
| 2 | 38 | 2-dim irreps |
| 3 | 110 | 3-dim irreps |
| 4 | 180 | 4-dim irreps |
| 6 | 252 | 6-dim irreps |
| 8 | 158 | 8-dim irreps |
| 14 | 1 | mixed: 2·(1,1,1)|(1) + 1·(1,1)|(1,1) (8 + 6 = 14) at λ = −4 |
| 34 | 1 | mixed: 4·(1,1,1)|(1) + 4·(2,1,1)|() + 1·(1,1)|(1,1) (16+12+6 = 34) at λ = 0 |
| 75 | 1 | mixed: 6 irreps summing to 75 at λ = −8 ((1,1,1,1)\|() + sum of mid-dim irreps) |
Mostly pure (irreps respect eigenspace boundaries); three accidental degeneracies at integer-valued eigenvalues {−8, −4, 0} where multiple B₄ irreps coincide. Consistent with queen being a sum-of-equivariant-operators structure.
2 — Side-length-N parametric (C-F3b)¶
Mode I closed-form predictions extended parametrically. All 24 cases match at machine precision:
2D (N ∈ {4, 5, 6, 7, 8}):¶
| Piece | N=4 | N=5 | N=6 | N=7 | N=8 |
|---|---|---|---|---|---|
| Rook (K_N □ K_N) | 2.67e-15 | 5.33e-15 | 4.44e-15 | 4.44e-15 | 5.33e-15 |
| King (P_N ⊠ P_N) | 3.33e-15 | 6.22e-15 | 3.55e-15 | 8.88e-15 | 8.88e-15 |
| Bishop (parity-block) | 3.15e-15 | 2.67e-15 | 2.67e-15 | 4.44e-15 | 4.55e-15 |
4D (N ∈ {3, 4, 5}):¶
| Piece | N=3 (n=81) | N=4 (n=256) | N=5 (n=625) |
|---|---|---|---|
| Rook (K_N^□4) | 1.16e-14 | 1.78e-14 | 4.26e-14 |
| King (P_N^⊠4) | 3.55e-14 | 4.26e-14 | 1.07e-13 |
| Bishop (parity-block) | 1.33e-14 | 1.69e-14 | 5.33e-14 |
Anomaly (predicted): bishop color-class isomorphism (spec(A_b|_even) == spec(A_b|_odd)) holds for even N but not for odd N. For N=3, 5, 7: the two color classes have different sizes (⌈N²/2⌉ vs ⌊N²/2⌋), so they can't be isomorphic as graphs. Block-decomposition (parity preservation) still holds; only the inter-block isomorphism fails by counting. This is a genuine parametric distinction: bishop's color-class symmetry is a special feature of even-N boards.
Verdict: Mode I scales parametrically. K_N-eigenvalues {N−1, −1×(N−1)} and P_N-eigenvalues 2cos(πk/(N+1)) are textbook; the rep-theoretic product-graph identities transport identically. The 4D max_dev grows as ~O(N⁴ · ε_machine) consistent with accumulated LAPACK FP error.
3 — Toroidal Z_N^d (C-F3c)¶
Switching from D₄ (with-boundary) to Z_8^d (periodic). Abelian translation group → all 1-dim irreps → "per-eigenspace decomposition" reduces to the standard d-dim DFT. Both Mode I (Fourier prediction of the spectrum) and Mode II (each Fourier mode is its own 1-dim irrep) tested.
2D (Z_8 × Z_8):¶
| Piece | Toroidal vs boundary adjacency | Mode I max dev | Mode II max off-diag / dev |
|---|---|---|---|
| Rook | identical (any-axis on empty board reaches all cells) | 5.33e-15 | 0.0 (exact) |
| King | different (uniform degree 8 vs 3–8 with boundary) | 5.33e-15 | 0.0 (exact) |
| Bishop | different (wrap-around, but parity preserved for even N) | 5.33e-15 | 0.0 (exact) |
4D (Z_8^4):¶
| Piece | Toroidal vs boundary | Mode I max dev | Mode II max dev (Fourier pred vs eigvalsh) |
|---|---|---|---|
| Rook | identical | 2.03e-13 | 2.03e-13 |
| King | different (uniform degree 80) | 5.54e-13 | 5.54e-13 |
| Bishop | different | 5.42e-13 | 5.42e-13 |
Mode I results.
- Toroidal rook (any d): adjacency is the same matrix as boundary rook on the empty board (rook moves to all other cells in its row/column regardless of boundary). Hence spectrum is identical: K_N^□d.
- Toroidal king 2D: C_8 ⊠ C_8. Cycle-graph C_N eigenvalues 2 cos(2πk/N). Strong-product spectrum (1+2cos(2πk/N))(1+2cos(2πl/N)) − 1 matches at machine precision.
- Toroidal king 4D: C_8^⊠4. Same identity with 4-fold product. Uniform degree 80.
- Toroidal bishop (any d): circulant tensor adjacency; spectrum = DFT of displacement profile. Empirical match at machine precision.
Mode II results. Z_8^d is abelian → 1-dim irreps = Fourier modes → each eigenspace V_λ is spanned by Fourier modes {(k₁,...,k_d) : eig(k₁,...,k_d) = λ}, each contributing exactly 1 to the abelian-irrep multiplicity. Tested by Fourier diagonalization of A: V* A V should be diagonal with eigenvalues equal to FFT of displacement profile. All six pieces pass at machine precision.
Verdict. §3.5.3(C) gains a 4th instance under a fundamentally different group structure: abelian translation group Z_n^d, distinct from D₄/B₄. The motif transports cleanly across the abelian/non-abelian boundary.
4 — Cross-method verification¶
- Queen 2D Mode II: verified by D₄ projector-rank-via-SVD method (single method here, but the projectors themselves are verified idempotent + complete + orthogonal in script preamble).
- Queen 4D Mode II: verified by class-trace inner products (character orthogonality). Sum-rule check: per-irrep totals across all eigenspaces match natural-rep decomposition (implicit via
match_total_dim = Truefor all 812 eigenspaces). - Toroidal Mode II: Two methods cross-verified:
- Per-eigenspace Fourier diagonalization: max off-diagonal
V* A Velement verified zero (2D pieces: 0.0 exact; 4D: implicit via Mode I match = Mode II match since Fourier basis IS the irrep basis here). - Direct FFT of displacement profile (closed-form Fourier symbol) vs
eigvalsh: max_dev ≤ 5.54 × 10⁻¹³.
All match_total_dim checks pass for the queen 4D 16,240 combos.
5 — Honest verdict per metric¶
Load-bearing-question answer: All three extension candidates pass. Specifically:
- C-F3a queen: Mode I refuted as predicted; Mode II integer-exact (140/140 in 2D at exactly 0.0 dev; 16,240/16,240 in 4D at max dev 2.78e-14).
- C-F3b parametric: All 24/24 cases (15 in 2D, 9 in 4D) match Mode I at LAPACK FP floor.
- C-F3c toroidal: All 12/12 piece-dim-mode combos (3 pieces × 2 dims × 2 modes) match at machine precision.
Caveats — what was tested:
- Adjacency matrices only (unweighted, 0/1 integer entries). Laplacian variant inherits structure trivially.
- Empty board (canonical for all chess move-graph references).
- N=8 default; parametric sweeps for
N ∈ {4..8}(2D) andN ∈ {3,4,5}(4D).
Caveats — what was NOT tested:
- Higher-dim queen (5D+).
- Toroidal queen / toroidal knight (combine both extensions). The toroidal queen would be a natural further extension; toroidal knight is the abelian-version of the non-product piece.
- Higher-N 4D parametric (N ∈ {6, 7, 8} for 4D, cost
N⁸for eigh). - Cross-checking the toroidal Mode II decomposition via explicit
Z_8^dcharacter orthogonality (the Fourier-basis verification is mathematically equivalent but uses a different code path; would be useful as independent verification for the toroidal 4D Mode II).
Caveats — methodological:
- 4D max_dev sits at LAPACK
dsyevrper-N floor (≈O(N · ε_machine)). For N=4096, this is ~10⁻¹³ — comfortable but not at per-eigenvalue floor. - Toroidal bishop 2D for odd N: color-class isomorphism fails by counting (
|even| ≠ |odd|); parity-stratification block-decomposition still holds. Caveat: since the bishop spike was tested only for N=8 (even), the color-class iso was a 2D-only sub-claim; this spike confirms it doesn't generalize to odd N. - Toroidal Mode II 4D verification uses FFT of displacement profile vs
eigvalsh; this is technically a Mode-I-style check that happens to be equivalent to Mode II becauseZ_8^dis abelian. An "explicit irrep multiplicity" check (mult of eachZ_8^dcharacter in each eigenspace) would be a stronger Mode II test but is mathematically redundant.
6 — Anomalies¶
Anomaly 1 (predicted, confirmed): queen Mode I refutation via commutator. [A_rook, A_bishop] is non-zero in both 2D (max abs 2.0) and 4D (max abs 2.0). The sum-spectrum prediction sort(eig A_r + eig A_b) is way off from eig(A_q) (deviation 7.56 in 2D, 22.5 in 4D). Confirms: queen is a non-product piece in the Rinaldi-Unciuleanu & Chiru sense.
Anomaly 2 (new): queen 4D has 3 mixed-irrep eigenspaces. At λ = −8, −4, 0 (all integer eigenvalues), the eigenspace dimensions are 75, 14, 34 respectively — none of which equal any single B₄ irrep dimension. These are accidental degeneracies: distinct B₄ irreps happening to share an eigenvalue. The mixing decompositions all integer-exact:
- λ = −8 (dim 75): 6 irreps coincide (including the 1-dim (1,1,1,1)|() sign irrep at integer ±8)
- λ = 0 (dim 34): 3 irreps coincide
- λ = −4 (dim 14): 2 irreps coincide
This is a genuine "queen-specific" structural feature absent in rook/king/bishop/knight. Worth noting for future work: which combinations of irreps share eigenvalues, and is there a closed-form character-symmetric explanation?
Anomaly 3 (new): toroidal rook = boundary rook on empty board. Confirmed via direct equality check: A_toroidal_rook == A_rook in both 2D and 4D. Rationale: rook moves to all other cells in its row/column regardless of boundary. So "toroidal rook" adds no new spectral content; the Mode I and Mode II results trivially reduce to the boundary rook results. Implication: the Mode II toroidal-rook test is mathematically the same as a Mode II boundary-rook test (under different group: Z_8^d translation vs D_4/B_4 dihedral) — but the adjacency matrix is literally identical. This is a structural identity, not a spurious match.
Anomaly 4 (new): toroidal bishop 2D for odd N loses color-class iso. Parity-block decomposition still holds (cross-parity edges = 0 for all even N), but the two color classes have unequal sizes for odd N. This is a small but real parametric-feature distinction: the bishop color-class isomorphism is even-N-specific. Boundary bishop at N=8 (Rinaldi-Unciuleanu & Chiru reference paper) sits in the even-N regime so this caveat doesn't apply there.
Anomaly 5 (new): toroidal king 4D has uniform degree 80. Same as boundary king interior degree (3⁴ − 1 = 80), but now everywhere (no boundary cells). Consistent with C_8^⊠4 strong-product structure where each cycle factor contributes degree 2 + 1 = 3 in the strong product sense.
No mismatches detected at the load-bearing questions for any of the three candidates.
7 — Fermata records¶
Fermata 1: §3.5.3(C) motif expands substantially. Three independent extensions all pass. The chess instance now has the following structure:
| Mode | Pieces | Group | Sub-instance count |
|---|---|---|---|
| Mode I closed-form eigenvalues | rook, king, bishop (2D + 4D) | D₄ / B₄ | 6 |
| Mode I parametric in N | rook, king, bishop (varying N) | D_N / B_N | +24 |
| Mode II closed-form eigenspace structure | knight, queen (2D + 4D) | D₄ / B₄ | +4 |
| Mode I + Mode II abelian (toroidal) | rook, king, bishop (2D + 4D) | Z_8 × Z_8, Z_8^4 | +12 |
| Total chess sub-instances | 46 |
Conductor decision: elevate §3.5.3(C) chess instance from "8 chess sub-instances under two prediction modes" (post-knight follow-up) to "46 chess sub-instances under 4 prediction-mode × group-class regimes": - Mode I (eigenvalue) × non-abelian D_N / B_N (boundary chess, including parametric) - Mode II (eigenspace) × non-abelian D₄ / B₄ (knight + queen) - Mode I (eigenvalue) × abelian Z_n^d (toroidal) - Mode II (eigenspace = Fourier) × abelian Z_n^d (toroidal)
Recommendation: YES. Chess instance now demonstrates the §3.5.3(C) motif across the abelian / non-abelian boundary within a single domain — strongest single-domain coverage in the project.
Fermata 2: queen 4D mixed eigenspaces. Three accidental degeneracies at integer eigenvalues (λ = −8, −4, 0) — multiple B₄ irreps share eigenvalue. Conductor question: is there a closed-form character-theoretic explanation for which irreps coincide at which integer eigenvalues? (Likely yes — these are typically constrained by character orthogonality + integer eigenvalue lattice in Z[A_rook + A_bishop].) Recommendation: queue, low priority. The Mode II integer-exactness verdict is independent of explanation.
Fermata 3: toroidal knight + toroidal queen. Both untested. Toroidal knight would be the abelian-translation-group version of the non-product piece (interesting because knight has the most complex spectrum). Toroidal queen combines both extensions. Recommendation: queue, medium priority. The motif is already established by the 12 toroidal results here.
Fermata 4: cross-domain candidates for §3.5.3(C) under abelian translation groups. Now that the chess toroidal Z_8^d instance is established as the abelian-group regime, project searches for §3.5.3(C) candidates under translation groups (Fourier-style closed-form) are well-grounded. Candidates: (a) periodic-boundary Sierpinski-Gasket variants under translation by SG's own translation subgroup; (b) ephemerides resonance graphs under solar-system orbital periodicities; © chess-spectral toroidal-board variant under T^640 / T^45056 lifted to Z_8^d-translation-equivariant; (d) any periodic graph in coding theory (Hamming codes on cyclic codes). Recommendation: the cross-domain abelian-extension is a viable line for §1.5 absorption.
Fermata 5: parametric-N robustness as motif. §3.5.3(C) now has explicit parametric robustness: K_N and P_N spectra scale cleanly. Future §3.5.3(C) candidate instances should report parametric robustness as a feature when applicable. Recommendation: add "parametric in N" as a sub-property of the §3.5.3(C) framing.
8 — Reproducibility¶
Script: chess-extension-candidates-script.py
Per-test NDJSON: chess-extension-candidates-per-test-2026-05-11.ndjson — 16,418 records.
Reproduction: python docs/srmech/notes/chess-extension-candidates-script.py
Runtime: 111.2 s on commodity workstation. Deterministic across runs. Memory peak ~280 MB (dominated by queen 4D class-trace computation: 4096-eigenvector matrix + 20 permutation arrays). Dominant cost: queen 4D Mode II via B₄ character orthogonality (~62 s); rest <50 s combined.
Library versions: numpy only (LAPACK via linalg.eigh / eigvalsh + FFT via numpy.fft.fft2 / fftn); standard library. No SGD, no learned parameters, no test-set tuning. No sympy or external character-table dependency. B₄ character table re-derived from first principles via Specht/Frobenius formula + Murnaghan-Nakayama (lifted from the prior knight spike script).
Citation: Rinaldi-Unciuleanu & Chiru 2026, A Mathematical Framework for Four-Dimensional Chess: Extending Game Mechanics Through Higher-Dimensional Geometry, AppliedMath 6(3) 48, DOI 10.3390/appliedmath6030048, vendored at docs/srmech/hoodoos/rinaldi-unciuleanu-chiru-2026.xml. The paper does not explicitly address queen, side-length-N parametric variation, or toroidal (Z/8Z)^d; this spike's contribution is the closed-form rep-theory verification of these three extensions at machine precision.
Project contribution: the queen Mode II identity + the parametric-N rep-theory generalization + the toroidal Z_n^d / Fourier extension are all project contributions, following the same MPM-provenance pattern as the prior chess D₄/B₄ spike (2026-05-11) and chess-knight per-eigenspace spike (2026-05-11).