Chess knight per-eigenspace irrep multiplicities under D₄ / B₄ — findings (2026-05-11)¶
Spike: For the knight move-graph adjacency matrix in 2D (8×8 = 64 cells) and 4D (8⁴ = 4096 cells), is the per-eigenspace decomposition under D₄ (2D) and B₄ (4D) integer-exact at machine precision? Even though knight has no closed-form eigenvalues, the per-eigenspace irrep structure was queued as candidate "structural prediction without closed-form eigenvalues" — a stronger §3.5.3(C) sub-instance for the chess case.
Method: Concertmaster role, MPM-discipline. 2D: D₄ projector method (P_ρ = (d_ρ/|G|) Σ_g χ_ρ(g) U_g; rank(P_ρ V_λ)/d_ρ = multiplicity). 4D: character orthogonality via class-trace inner products (mult = (1/|G|) Σ_C |C| · χ_ρ(C) · tr(U_{g_C}|V_λ)) — avoids materializing 384 × 4096×4096 dense projectors. B₄ character table computed from Specht/Frobenius formula via Murnaghan-Nakayama for S_n characters and the power-sum-Schur bridge for B_n. Cited rules SSOT: Rinaldi-Unciuleanu & Chiru 2026 (docs/srmech/hoodoos/rinaldi-unciuleanu-chiru-2026.xml, §3.6 names knight as the non-product case).
Dispatch: Conductor → concertmaster, 2026-05-11. Follow-up to today's chess D₄/B₄ rep-theory spike (Fermata 2: queue per-eigenspace knight irrep decomposition).
Bottom line: Integer-exact at machine precision in both 2D and 4D. 2D: 45 distinct eigenvalues × 5 D₄ irreps = 225 (irrep × eigenspace) combos; zero non-integer. 4D: 825 distinct eigenvalues × 20 B₄ irreps = 16,500 combos; zero non-integer (max deviation 2.98 × 10⁻¹⁴ — essentially machine precision). Sum-rules pass for every irrep in both dimensions. Knight gains a §3.5.3(C) sub-instance: even without closed-form eigenvalues, the per-eigenspace irrep structure IS closed-form-predictable from rep theory at machine precision — the eigenspace dimensions are exactly the B₄ irrep dimensions {1, 2, 3, 4, 6, 8}.
Reproducible: python docs/srmech/notes/chess-knight-irrep-multiplicities-script.py. Runtime ~44 s on commodity workstation. Deterministic seed 20260511. Memory peak ~265 MB (the 4096-eigenvector matrix held briefly).
1 — 2D knight × D₄ — per-eigenspace decomposition¶
45 distinct eigenvalues in the spectrum of the 64×64 knight adjacency. Every eigenspace is pure (single non-zero D₄ irrep). Eigenspace dimension histogram:
| Eigenspace dim | Count | Irrep type |
|---|---|---|
| 1 | 32 | A₁, A₂, B₁, B₂ (1-dim irreps; 8 each) |
| 2 | 12 | E (only 2-dim irrep) |
| 8 | 1 | E with mult 4 (λ=0 kernel) |
Sum-rule (per-irrep total across all eigenspaces, expected = natural-rep multiplicity on R⁶⁴):
| Irrep | dim | Per-eigenspace total | Expected | Match |
|---|---|---|---|---|
| A₁ | 1 | 10.000 | 10 | ✓ |
| A₂ | 1 | 6.000 | 6 | ✓ |
| B₁ | 1 | 6.000 | 6 | ✓ |
| B₂ | 1 | 10.000 | 10 | ✓ |
| E | 2 | 16.000 | 16 | ✓ |
Total dim = 10 + 6 + 6 + 10 + 2·16 = 64 ✓. All multiplicities integer-exact to within 1 × 10⁻¹⁴.
Notable eigenvalues: - λ = +6.0107 (Perron / max): A₁, dim 1 - λ = −6.0107 (anti-Perron / min, bipartite partner): B₂, dim 1 - λ = 0.0 (kernel): 4 copies of E, dim 8 - λ = ±2.0 exactly: B₂ / A₁ respectively, dim 1 each — rational eigenvalues in knight 2D's otherwise-irrational spectrum
Cross-method check. The 2D result was verified by TWO independent methods (D₄ projectors with rank-via-SVD AND D₄ character orthogonality with class traces); both give identical integer multiplicities.
2 — 4D knight × B₄ — per-eigenspace decomposition¶
825 distinct eigenvalues in the spectrum of the 4096×4096 knight adjacency. 824 of 825 eigenspaces are pure (single non-zero B₄ irrep); only the λ=0 kernel hosts multiple irreps.
Eigenspace dimension histogram (matches B₄ irrep dimensions exactly):
| Eigenspace dim | Count | B₄ irreps with this dim |
|---|---|---|
| 1 | 72 | (4)|(), ()|(4), (1,1,1,1)|(), ()|(1,1,1,1) [4 irreps] |
| 2 | 40 | (2,2)|(), ()|(2,2) [2 irreps] |
| 3 | 120 | (3,1)|(), ()|(3,1), (2,1,1)|(), ()|(2,1,1) [4 irreps] |
| 4 | 192 | (3)|(1), (1)|(3), (1,1,1)|(1), (1)|(1,1,1) [4 irreps] |
| 6 | 240 | (2)|(2), (1,1)|(1,1), (2)|(1,1), (1,1)|(2) [4 irreps] |
| 8 | 160 | (2,1)|(1), (1)|(2,1) [2 irreps] |
| 96 | 1 | λ=0 kernel: 10·(2)|(2) + 6·(1,1)|(1,1) — mixed |
The eigenspace dimensions are precisely the 6 distinct B₄ irrep dimensions {1, 2, 3, 4, 6, 8}. This is itself a structural prediction: rep theory alone tells you the menu of possible eigenspace dimensions for any B₄-equivariant operator on R⁴⁰⁹⁶.
Knight 4D is bipartite (move changes parity of x+y+z+w; verified empirically — the 2048×2048 same-parity block of the adjacency is exactly zero). Bipartiteness implies:
- Spectrum is symmetric: λ in spectrum ↔ −λ in spectrum.
- Rank deficiency: independent rank check confirms rank(A) = 4000, nullity = 96 — the 96-dim kernel matches the bipartite-graph rank formula.
- The 96-dim kernel splits as 10·(2)|(2) + 6·(1,1)|(1,1) (both are 6-dim B₄ irreps).
- "Even-half" irreps (α ↔ β swap pairs) appear at eigenvalues +λ and −λ with mirror multiplicities.
Sum-rule (per-B₄-irrep total across all 4D eigenspaces, expected = natural-rep multiplicity on R⁴⁰⁹⁶):
| Irrep | dim | Per-eigenspace total | Expected | Match |
|---|---|---|---|---|
| (2)|(2) | 6 | 100.0 | 100 | ✓ |
| (1)|(3), (3)|(1) | 4 | 80.0 each | 80 each | ✓ |
| (1)|(2,1), (2,1)|(1) | 8 | 80.0 each | 80 each | ✓ |
| (2)|(1,1), (1,1)|(2) | 6 | 60.0 each | 60 each | ✓ |
| (3,1)|(), ()|(3,1) | 3 | 45.0 each | 45 each | ✓ |
| (1,1)|(1,1) | 6 | 36.0 | 36 | ✓ |
| (4)|(), ()|(4) | 1 | 35.0 each | 35 each | ✓ |
| (2,2)|(), ()|(2,2) | 2 | 20.0 each | 20 each | ✓ |
| (1,1,1)|(1), (1)|(1,1,1) | 4 | 16.0 each | 16 each | ✓ |
| (2,1,1)|(), ()|(2,1,1) | 3 | 15.0 each | 15 each | ✓ |
| (1,1,1,1)|(), ()|(1,1,1,1) | 1 | 1.0 each | 1 each | ✓ |
All 20 irreps sum-rule-check pass. Total dim × multiplicity over all irreps = 4096 ✓ (the full R⁴⁰⁹⁶ ambient).
Notable eigenvalues:
- λ = +36.0631 (Perron / max): (4)|(), dim 1
- λ = −36.0631 (anti-Perron / min, bipartite partner): ()|(4), dim 1
- λ = ±2.0 exactly: hosts (1,1,1,1)|() and ()|(1,1,1,1) respectively (the "sign-determinant" 1-dim B₄ irreps), dim 1 each — rational eigenvalues in an otherwise-irrational 4D spectrum
- λ = 0 (kernel): dim 96, splits as 10·(2)|(2) + 6·(1,1)|(1,1)
Numerical precision (max deviation from nearest integer multiplicity, 4D): 2.975 × 10⁻¹⁴. Comfortably within machine precision (LAPACK dsyevr floor ~ O(n · ε_machine) ≈ 9 × 10⁻¹³ on N=4096).
3 — Structural prediction without closed-form eigenvalues¶
This is the load-bearing result for the dispatch.
The original chess D₄/B₄ rep-theory spike (2026-05-11) established that closed-form eigenvalue prediction matches at machine precision for rook, king, bishop (in both 2D and 4D, under three structural primitives: Cartesian product, strong product, parity stratification). Knight was the predicted non-match — explicitly per Rinaldi-Unciuleanu & Chiru 2026 §3.6, knight admits no clean product or parity factorization.
This follow-up shows: the absence of closed-form eigenvalues for the knight does NOT preclude structural prediction. The eigenspace decomposition into D₄ / B₄ irreps is integer-exact to machine precision in BOTH 2D and 4D. The prediction is weaker than "you can write down the eigenvalues" but stronger than "no structure" — you can predict:
- The menu of possible eigenspace dimensions — exactly the irrep dimensions of the symmetry group.
- The total multiplicity of each irrep across all eigenspaces — exactly its multiplicity in the natural rep on R^N (independent of the operator).
- Most eigenspaces are pure — only λ=0 mixes irreps (and only when the operator has a non-trivial kernel).
- The bipartite structure — when the operator is bipartite-equivariant, irreps pair off as
(α, β) ↔ (β, α)at eigenvalues+λand−λ.
This is a rep-theoretic invariant distinct from closed-form eigenvalue prediction. The chess spike's six previous sub-instances (rook 2D/4D, king 2D/4D, bishop 2D/4D) used the spectrum-formula identity; the knight 2D/4D follow-up uses the eigenspace-structure identity — same closed-form-from-rep-theory motif, different load-bearing prediction.
Motif refinement for §3.5.3(C)¶
The chess instance is the strongest §3.5.3(C) instance because it now spans two distinct prediction modes within a single domain:
| Mode | Pieces | What is closed-form-predicted | Empirical match |
|---|---|---|---|
| Mode I: closed-form eigenvalues | rook, king, bishop (2D + 4D) | full spectrum at machine precision | 6/6 cases ≤ 3.7 × 10⁻¹³ |
| Mode II: closed-form eigenspace structure (new) | knight (2D + 4D) | eigenspace dims = irrep dims; per-irrep totals = natural-rep multiplicities; mixing only in kernel | 2/2 cases ≤ 2.98 × 10⁻¹⁴ |
Eight sub-instances now stand under D₄ / B₄ within a single domain, all at machine precision. Compare: - MFO Phase B 18-block (D₃ on Sierpinski Gasket): single sub-instance. - Finance block-correlation (S_k × S_m on synthetic correlation matrix): single sub-instance. - Chess (D₄ / B₄ on hypercube move graphs): now eight sub-instances under two prediction modes.
4 — Implementation: B₄ character table from first principles¶
For reproducibility, the B₄ character table (20×20) was computed from first principles using the standard Specht / Frobenius formula:
χ^{B_n}_{(α,β)}((λ⁺, λ⁻)) =
coefficient of s_α(x) · s_β(y) in
∏_i (p_{lp_i}(x) + p_{lp_i}(y)) · ∏_j (p_{lm_j}(x) − p_{lm_j}(y))
In our implementation: for each cycle of λ⁺ or λ⁻, distribute it to either the α-side or the β-side; negative cycles assigned to the β-side contribute a sign of −1. Then χ is summed over all distributions of S_|α|-character × S_|β|-character of the cycle-multisets. S_n characters computed via Murnaghan-Nakayama rule.
Verification. The computed B₄ character table satisfies:
- Row-orthogonality <χ_ρ, χ_σ>_G = δ_{ρσ} for all 20·19/2 + 20 = 210 inner products (verified).
- Sum of squared dimensions = 384 = |B₄| (verified: 4·1² + 2·2² + 4·3² + 4·4² + 4·6² + 2·8² = 4+8+36+64+144+128 = 384).
- B₄ irrep dimensions: {1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 6, 6, 6, 6, 8, 8} — match standard reference (e.g., Fulton-Harris Appendix B).
Class data: 20 conjugacy classes indexed by signed bipartitions of 4. Centralizer formula |C_{B_n}(g)| = ∏_k (2k)^{m_k^+} m_k^+! · (2k)^{m_k^-} m_k^-! (James-Kerber 4.2.10). Class sizes sum to 384 ✓.
5 — Honest verdict per metric¶
Load-bearing-question answer: Per-eigenspace irrep multiplicities are integer-exact at machine precision in BOTH 2D and 4D. Max deviation from nearest integer: - 2D: ≤ 1 × 10⁻¹⁴ (per-eigenspace projector formula on 64-dim space) - 4D: 2.975 × 10⁻¹⁴ (per-eigenspace character-orthogonality formula on 4096-dim space)
Tolerance target was 1 × 10⁻⁶ (forgiving — accommodates accumulated FP error from eigh on 4096-dim matrix). All 16,500 (4D irrep × eigenspace) combos and 225 (2D combos) are integer-exact to far tighter than that bound.
Sum-rule check passes for all 5 D₄ irreps (2D) and all 20 B₄ irreps (4D). Per-irrep totals exactly match the natural-rep decomposition multiplicities computed independently via Burnside / fixed-point counting.
Caveats — what was tested:
- Adjacency-matrix eigenvalues (unweighted; integer 0/1 entries). The unsigned graph Laplacian L = D − A would shift the spectrum but preserve the eigenspace structure (since L is also D₄/B₄-equivariant); the multiplicity structure transfers identically.
- Empty board (no other pieces). Knight move-graph is the canonical empty-board displacement graph.
- 8-cell side length only. Generalizing to side length N is straightforward — the move set is unchanged, only the on-board check varies; D₄/B₄ equivariance survives.
Caveats — what was NOT tested:
- Higher-dim knight (5D, 6D) for B_N regression.
- Toroidal variants ((Z/8Z)^d) — different equivariance group structure.
- Weighted move-graph variants.
- B₄ irrep label correctness against an external source (the table was verified internally via orthogonality and dim²-sum but not against a tabulated reference such as GAP / Magma / James-Kerber); the integer-exactness of multiplicities is independent of label convention.
Caveats — methodological:
- numpy.linalg.eigh (LAPACK dsyevr) is deterministic and well-tested; accumulated FP error scales as O(N · ε_machine) ≈ 10⁻¹³ for N=4096. Multiplicity deviations are about an order of magnitude smaller because character-orthogonality is itself an integer sum after dividing by |G|; the inner products of eigenvectors with their permuted selves are the load-bearing FP operations.
- Eigenspace grouping tolerance was 1 × 10⁻⁸. Smallest empirical gap between distinct eigenvalues was 2.7 × 10⁻⁵ — comfortably above tolerance, no risk of merging distinct eigenspaces.
6 — Anomalies¶
Anomaly 1 (predicted, confirmed). The λ=0 eigenspace in 4D is the only one that mixes irreps. It has dim 96 = 60 + 36, decomposing as 10·(2)|(2) + 6·(1,1)|(1,1). Independent rank check confirms this is the true 96-dimensional kernel of the knight 4D adjacency (rank 4000 + nullity 96 = 4096). The 2D analog: λ=0 hosts 4 copies of E (dim 8) — single irrep type, but multiple copies, so technically "mixed in multiplicity" though all the same irrep. The 4D mixing is dimensionally richer: two distinct 6-dim irreps coexist. This is the only place per-eigenspace structure is non-discriminating in 4D — at the kernel, the rep theory predicts the dim (96) and the irrep content ((2)|(2) + (1,1)|(1,1) with multiplicities 10 and 6) but not which specific eigenvectors host which irrep.
Anomaly 2 (predicted, confirmed). Knight bipartiteness implies ±λ mirror pairs. Verified empirically: every non-zero eigenvalue +λ has a partner −λ; corresponding irreps are swap pairs (α, β) ↔ (β, α). Examples: +36.063 → (4)|(), −36.063 → ()|(4); +2 → ()|(1,1,1,1), −2 → (1,1,1,1)|().
Anomaly 3 (new). The eigenvalues λ = ±2 are integer-exact in the knight 4D spectrum (machine-precision integer). The knight has 48 displacement vectors; some of these admit rational eigenvalue contributions from a Fourier-like substructure. Specifically: the (1,1,1,1)|() and ()|(1,1,1,1) are 1-dim "sign" irreps of B₄ (B_n analog of the alternating-sign rep). The 1-dim sign-rep eigenvectors on the hypercube must have specific sign patterns under the B_4 action; for the knight 4D specifically, the value happens to be integer-exact ±2. Not a coincidence, but worth noting: it's the analog of the λ=−2 rational eigenvalue in rook 2D and bishop 2D (also at machine-precision integer). Would be interesting to derive closed-form via the displacement-set Fourier expansion; not done here.
Anomaly 4 (new structural prediction). The histogram of eigenspace dimensions in 4D is exactly the histogram of irrep dimensions of B_4 (each irrep contributes (mult-in-natural-rep / irrep-dim) eigenspaces; the dim-96 kernel hosts two irreps so contributes 1 eigenspace). This is the structural prediction even more compactly: for a generic B₄-equivariant operator with full-rank-on-each-isotypic-block behaviour, the eigenspace dim histogram is determined by the natural-rep decomposition. Knight 4D is the first project instance demonstrating this empirically at machine precision.
7 — Fermata records¶
Fermata 1: §3.5.3(C) chess instance refinement. The chess instance now spans two prediction modes (closed-form eigenvalues for rook/king/bishop; closed-form eigenspace structure for knight). Conductor decision: refine §3.5.3(C) chess sub-instance from "six chess sub-instances under three structural primitives" to "eight chess sub-instances under two prediction modes" (Mode I: closed-form eigenvalues; Mode II: closed-form eigenspace structure)? Recommendation: YES. The motif refinement is mathematically distinct — Mode I predicts spectrum, Mode II predicts isotypic decomposition. Both are rep-theoretic structural predictions from group G.
Fermata 2: B₄ irrep label canonicalization. The script's B₄ irreps are labelled (α)|(β) for pairs of partitions. Labels were verified internally (orthogonality + dim²-sum) but not against external reference. For project-wide use as a reusable B_n primitive (potential future ephemerides / chess-spectral / power-grid applications), cross-check against e.g. GAP CharacterTable("WeylB",4) recommended. Recommendation: queue this if/when B_n character table is needed elsewhere; the integer-exact-multiplicity verdict here is independent of label convention.
Fermata 3: cross-domain candidate — Sierpinski-Gasket per-eigenspace decomposition. MFO Phase B's 18-block result (D₃ on SG λ=6 eigenspace: 22A + 18B + 40E) was a single-eigenspace fingerprint. The chess knight result is the same structural prediction extended to every eigenspace of a non-product graph operator. Conductor question: is per-eigenspace D₃ decomposition for the FULL SG decimation Laplacian (not just λ=6) also integer-exact at machine precision? Recommendation: queue; this would be the MFO sibling of today's spike, testing whether the SG Laplacian's full eigenspace structure (not just one chosen eigenspace) is rep-theoretically predictable.
Fermata 4: extension candidates within chess. Queen (= rook ∪ bishop) is the other non-product piece. In 2D, queen has D₄ equivariance; per-eigenspace D₄ irrep multiplicities should also be integer-exact (same structural argument). In 4D, queen is the 6-axis-pair bishop union 4-axis rook — a much richer non-product structure. Recommendation: queue, low priority. The motif is already established by knight.
8 — Reproducibility¶
Script: chess-knight-irrep-multiplicities-script.py
Per-eigenspace NDJSON: chess-knight-irrep-multiplicities-per-eigenspace-2026-05-11.ndjson — 16,725 records: 225 (2D = 45 eigenspaces × 5 D₄ irreps) + 16,500 (4D = 825 eigenspaces × 20 B₄ irreps).
Reproduction: python docs/srmech/notes/chess-knight-irrep-multiplicities-script.py
Runtime: ~44 s on commodity workstation. Deterministic across runs. Memory peak: ~265 MB (the 4096×4096 knight adjacency, eigenvector matrix, and 20 permutation arrays all held briefly). Dominant cost: eigh(4096×4096) ≈ 15 s + per-class-trace inner products ≈ 15 s + S_n / B_4 character table ≈ 1 s.
Library versions: numpy 2.4.4 only (LAPACK via linalg.eigh); standard library. No SGD, no learned parameters, no test-set tuning. No sympy / no external character-table dependency.
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. §3.6 names knight as the non-product case; this spike's contribution is the per-eigenspace structural prediction.
B_n character formulation: Specht / Frobenius formula via power-sum-Schur bridge. Standard reference: James-Kerber, Representation Theory of the Symmetric Group (Encyclopedia of Mathematics and its Applications 16, 1981), Theorem 5.5; equivalent to Geissinger 1977. S_n characters via Murnaghan-Nakayama rule. Centralizer formula (James-Kerber 4.2.10) verified by class-size sum = |B_4| = 384.
Project contribution: the per-eigenspace integer-exact multiplicity identity for the knight move-graph is the project's contribution (not in Rinaldi-Unciuleanu & Chiru 2026, which names "spectral analysis of move graphs" as future work). Same MPM-provenance pattern as the prior chess D₄/B₄ spike.