Skip to content

Spectral Identity Verification (Pre-Flight #3 for chess_spectral.qm_4d)

Date: 2026-04-29 Run: python research/spectral_identity_4d_verification.py (FULL_4D and FULL_4D_ALL) Verdict: HOLDS at machine precision, with one important wording correction.

Wording correction (Z_8 vs P_8)

The original statement of the claim mentions the "Z_8^4 graph Laplacian." The encoder's actual graph is P_8 [box] P_8 [box] P_8 [box] P_8 (Cartesian product of four open 8-vertex paths), not the 8-cycle Cayley graph of Z_8. This is an important factual point because:

  • The 8-cycle Z_8 has 4 distinct eigenvalues with high multiplicity from the abelian Pontryagin dual; the encoder eigenbasis would be Fourier modes (DFT-II), and B_4 acts only by axis permutations + cyclic shift sign-flips.
  • The path P_8 has 8 distinct eigenvalues, eigenbasis is DCT-II, and B_4 acts as signed permutations through the symmetric reflection s -> 7-s on flipped axes (this is exactly what tables_4d.b4_permutation_matrix does).

So the strictly correct claim for the encoder is:

The 4096 eigenmodes used by chess_spectral.encoder_4d are the simultaneous eigenbasis of (a) the P_8^4 Cartesian-product graph Laplacian Delta and (b) the centralizer of the B_4 (signed-permutation) representation.

The script tests this exact statement.

Numerical evidence

2D pilot (P_8^2 = 64-dim, B_2 group order 8)

  • 33 distinct eigenvalues, multiplicities {1: 9, 2: 23, 7: 1}.
  • Every Delta-eigenspace is B_2-stable: max ||[pi(g), P_lambda]||_inf = 3.5e-15.
  • Encoder columns are exact L-eigenvectors: max ||L v - lambda v|| = 1.1e-15.

Full 4D (P_8^4 = 4096-dim, B_4 group order 384)

  • Delta build (sparse Kronecker sum): 0.81s.
  • Diagonalize (np.linalg.eigh, dense): 48.1s.
  • 225 distinct eigenvalues, total dim = 4096 (matches by trace).
  • Maximum multiplicity = 127 at lambda = 8.0 (the spectral midpoint, as expected from convolution-of-eight P_8 spectra).
  • P_8 1D spectrum: [0, 0.15224, 0.58579, 1.23463, 2.0, 2.76537, 3.41421, 3.84776] (the standard 2 - 2 cos((k+0.5) pi / 8) is not what P_8 gives; this is the open-boundary 2 sin^2(k pi / 16) spectrum that tables_4d.eig_p8 returns).
  • Multiplicity histogram (via kron_sum4_eigvals cross-check, identical to the diagonalization output):
  mult     1:    7 eigenvalues   ( 7  vectors, fully axis-symmetric corners)
  mult     4:   43 eigenvalues   (172 vectors)
  mult     6:   18 eigenvalues   (108 vectors)
  mult    12:   96 eigenvalues   (1152 vectors  - dominant interior class)
  mult    24:   28 eigenvalues   (672 vectors)
  mult    34:    6 eigenvalues   (204 vectors)
  mult    42:    1 eigenvalue    ( 42 vectors)
  mult    60:   12 eigenvalues   (720 vectors)
  mult    64:    6 eigenvalues   (384 vectors)
  mult    72:    6 eigenvalues   (432 vectors)
  mult    76:    1 eigenvalue    ( 76 vectors)
  mult   127:    1 eigenvalue    (127 vectors  - lambda = 8.0 spectral midpoint)
                                  -----
                                  4096

Bottom-line metrics

Quantity Value
eigenvalue_match_4d (encoder lambdas == Delta lambdas, with mults) True
encoder_basis_residual_4d (max
all_stable_4d (all 225 eigenspaces B_4-stable) True
max_commutator_err_4d (max
encoder_alignment_residual_4d (max

All four diagnostics are at floating-point noise. The claim holds at 1e-13 tolerance everywhere, ~3 orders of magnitude better than the 1e-10 target the project uses for C-Python parity.

Isotypic spot-check (B_4 trivial-irrep multiplicity)

Burnside-style character projection on the lowest-10 eigenspaces:

lam=-0.00000  dim=  1   trivial-irrep mult ~ 1
lam= 0.15224  dim=  4   trivial-irrep mult ~ 0
lam= 0.30448  dim=  6   trivial-irrep mult ~ 0
lam= 0.45672  dim=  4   trivial-irrep mult ~ 0
lam= 0.58579  dim=  4   trivial-irrep mult ~ 1
lam= 0.60896  dim=  1   trivial-irrep mult ~ 0
lam= 0.73803  dim= 12   trivial-irrep mult ~ 0
lam= 0.89027  dim= 12   trivial-irrep mult ~ 0
lam= 1.04251  dim=  4   trivial-irrep mult ~ 0
lam= 1.17157  dim=  6   trivial-irrep mult ~ 1

lambda = 0 (constant), lambda = 0.58579 (totally symmetric tensor of the first non-zero P_8 mode on each of 4 axes) and lambda = 1.17157 are the first three eigenspaces with a B_4-invariant component. The dim-1 eigenspace at lambda = 0.60896 (the "antisymmetric" eigenvalue lambda_3 + lambda_3 + lambda_3 + lambda_3 minus exchange) carries no trivial irrep — consistent with the encoder's A_1 channel being non-zero only when the signal projects onto these B_4-invariant subspaces.

Interpretation

The encoder's tensor-DCT basis e_i (x) e_j (x) e_k (x) e_l is canonical: it diagonalizes Delta (Kronecker-sum eigenvalues) and every P_lambda commutes with every pi(g) (B_4-action). The choice of basis within each multidimensional eigenspace is therefore not arbitrary engineering — it is determined by the simultaneous spectral decomposition of the operator algebra < Delta, pi(B_4) >. Whatever further refinement we make on a specific eigenspace (e.g., breaking it into B_4-irrep components for the QM extension's "energy basis" / "angular-momentum basis" analogy) is a sub-decomposition within a B_4-stable subspace, not a re-choice of basis.

The encoder basis is therefore representation-theoretically clean:

  • Delta-eigenvalue = "energy quantum number" (kinetic level).
  • B_4-irrep label within each E_lambda = "angular-momentum-like quantum number" (point-group sector).
  • (i, j, k, l) tuple = a specific simultaneous-eigenvector of Delta and of the four commuting axis-position operators (which together form a maximal abelian subalgebra in the B_4 commutant once we restrict to the abelian (Z_2)^4 subgroup of B_4).

This is exactly the "complete set of commuting observables" picture that QM uses for any single-particle problem with a discrete symmetry group — the encoder happens to be exactly that, in disguise.

Recommendation

Ship "experiment 1" of the QM module as the spectral-identity statement. The result is:

  • Numerically airtight (4 independent diagnostics all at 1e-13 or better).
  • Conceptually clean (representation theory + graph Laplacian; no chess semantics needed for the proof).
  • Already proven via existing tables_4d primitives — the QM module just needs to expose the eigendecomposition, the per-eigenspace projector P_lambda, and the irrep-projection convenience function (which can be added incrementally and is independently testable).

Do not fall back to the bishop-wavefunction experiment as the lead result. That experiment is good and complementary, but it relies on a specific piece-Laplacian commutation property which the rook violates (see diag_dev_4d discussion in tables_4d.py:894). The spectral identity is piece-agnostic and therefore the cleaner foundation.

What "experiment 1" should contain

  1. The 225-eigenvalue spectrum table (one figure / one CSV).
  2. Per-eigenspace B_4-stability ledger (already produced by this script).
  3. A worked example of "complete set of commuting observables": pick lambda = 0.58579 (dim 4), show how the four basis vectors are exactly the four 1D eigenvectors permuted by S_4 on the axis index, so they are simultaneous eigenvectors of Delta and the four axis-direction projectors.
  4. The trivial-irrep multiplicity table for at least the bottom 50 eigenspaces (should sum to the total number of B_4-orbits on Z_8^4 = trace(P_A1), which tables_4d.b4_a1_orbit_projector already computes).
  5. A short remark that this generalizes the "lattice momentum + crystal point group" decomposition standard in solid-state physics (Bouckaert-Smoluchowski-Wigner 1936; the usual reference for the diamond-cubic group but the construction is identical here).

Single caveat to flag in the QM module docs

The graph is P_8, not C_8. A reader expecting a translation-invariant plane-wave basis (C_8 / Pontryagin / Bloch) will be surprised. P_8 has sine-cosine Dirichlet/Neumann modes, not exponentials, and the symmetry group is the dihedral D_2 (not Z_8). In 4D this means B_4 acts by signed permutations of the centered coords, and reflections through the box center, not by translations.

If a future "infinite-board" or "torus chess" extension changes to C_8, the entire spectral picture changes (eigenvalues degenerate to 4 distinct values, B_4 grows to W(B_4) ⋊ (Z_8)^4, etc.). The spectral identity in its present form is specific to the open-boundary 4D chess board.

Files added

  • python/research/spectral_identity_4d_verification.py (verification script)
  • python/research/spectral_identity_4d_findings.md (this note)

No shipped source files were modified.