Task #218 Phase C2 — Chess-spectral QM Audit (2026-05-15)¶

Concertmaster dispatch. Conductor-initiated audit of chess-spectral's QM stack for equation-level coverage of canonical QM and QFT, scoping srmech's Phase C2 (MFO/SM/QM operations layer on top of the 14-class C-parity primitive vocabulary shipping via Task #217 Phase C1).

Provenance note. Concertmaster delivered findings inline (system-prompt override on writing analysis .md files). Conductor saved this file as the durable artifact since the audit is load-bearing for Phase C2 scoping.

Top-line verdict¶

Chess-spectral instantiates roughly the kinematic + free-particle-dynamics quadrant of canonical QM, in spectral / graph-Laplacian form, for a single non-relativistic particle on a finite open 4D lattice. QFT, relativistic QM, gauge theory, and S-matrix machinery are absent.

Canonical domain	Instantiated	Adjacent	Absent
Single-particle QM (12 eqs)	7 (full/partial)	1	4
Relativistic QM (5 eqs)	0	2 (KG + Dirac via notebook citation, no code)	3
QFT — gauge (7 eqs)	0	2 (Higgs, Yukawa — analogical only)	5
QFT — propagators/amplitudes (6 eqs)	0	1 (lattice propagator named)	5
Scattering (5 eqs)	0	0	5
Geometric/algebraic adjuncts	strong	n/a	n/a

Phase C2 implication in one line: chess-spectral is a thick, ~1.5y-old reference for the canonical kinematic + free-evolution + spectral-measurement layer; srmech-C2 should absorb / cite it as primary substrate and add the relativistic / QFT / gauge layers, not re-implement what's there.

Per-equation table¶

Files cited relative to docs/chess-maths/chess-spectral/. Notable absolute paths: - python/chess_spectral/qm_2d.py - python/chess_spectral/qm_2d_dynamics.py - python/chess_spectral/qm_4d.py - python/chess_spectral/qm_4d_dynamics.py - python/chess_spectral/qm_4d_bridge.py

Single-particle QM¶

Equation	Status	Where	How
TDSE i∂_t ψ = Hψ	FULL	`qm_4d_dynamics.py:2575 evolve_under_h0`; `qm_2d_dynamics.py:260`	Closed-form U(t)=exp(-iH₀t) via eigenbasis-diagonal V·diag(exp(-iλt))·V^H (2D); `scipy.expm_multiply` Krylov (4D). H₀ Hermitian; norm + ⟨H⟩ preservation tested. ADR-002.
TISE Hψ = Eψ	FULL	`qm_4d.py:495 measure_observable_distribution`; `qm_2d_dynamics.py:190 _get_h0_eigenbasis_2d`	`numpy.linalg.eigh` returns (λ, V); 4096 eigenmodes for H₀; per-piece H_piece eigenspectra empirically tabulated (rook [-4,28] integer; bishop, queen, king, knight non-integer).
Heisenberg eq. dA/dt = i[H,A]	ABSENT	—	Schrödinger picture only. No operator-evolution code path.
Density-matrix / Liouville–vN	PARTIAL	`qm_4d_bridge.py:960 get_density_matrix_of`	Reduced channel-density ρ = M_local @ M_local^H on 11×11 channel block in a Manhattan-radius cell neighbourhood; Hermitian PSD; purity, rank, eigvals exposed. Marked `isPartial: True` — true η-metric partial trace deferred.
Pauli equation (spin-½ in EM)	ABSENT	—	No spinor structure, no EM coupling.
[x,p] = iℏ canonical commutator	ABSENT as algebra	—	No explicit momentum operator object. ∂_a anti-Hermitian gradient is constructed (`qm_4d_bridge.py:1083 _build_lattice_gradient_4d`) and used for probability current, but [x_a, ∂_b] = iδ_ab is never asserted as such. Adjacent.
Hamiltonian / L² / spin operators	PARTIAL	`qm_4d.py:363 build_H_piece_4`; `qm_2d.py:472`; `qm_4d_dynamics.py:2489 _build_h_free_4d`	H₀ = -Δ_{P_8^4}, plus five graph-adjacency-as-Hermitian piece-reach observables. No L² / Lz / S² in the textbook sense — chess uses B_4 hyperoctahedral group (order 384) and D_4 dihedral (order 8) instead of SO(3). Group is wrong group: spatial-lattice symmetry, not rotational.
Hydrogen-atom radial eq.	ABSENT	—	No Coulomb potential, no spherical harmonics. The natural lattice analogue (point-charge potential on Z_8^d) is not built.
Harmonic oscillator (ladder ops)	ABSENT	—	No a / a†. Free particle on path-graph is the only Hamiltonian.

Relativistic QM¶

Equation	Status	Where	How
Klein-Gordon (m² + k̂²)	ADJACENT	notebook `§1b.5` (theoretical); not coded	Lattice scalar propagator G(k) = 1/(m² + k̂²) is the named mathematical substrate for piece value / range. Dybalski 2023 cited. Not instantiated as a function returning a propagator.
Dirac equation (γ-matrices)	ABSENT as code	notebook `§1b.5`, `§4`, `§1b.1` — framework named, not built	Notebook frames chess as "classical lattice fermion system" with Pauli exclusion (`n_i ∈ {0,1}`). Yumoto & Misumi 2024 cited for graph Laplacian ↔ lattice Dirac operator equivalence + Dirac zero-mode count = sum of Betti numbers. No γ-matrix code, no spinor space, no Dirac operator built. Pawn is identified as a non-Hermitian chiral fermion in the Hatano-Nelson sense (notebook §1b.1) — that's the closest fragment.
Weyl (massless 2-spinor)	ABSENT	—	Same as Dirac. No 2-spinor space.
Majorana	ABSENT	—	—
Bargmann-Wigner	ABSENT	—	—

QFT — gauge theories¶

Equation	Status	Where	How
U(1) / QED Lagrangian / Maxwell	ABSENT	—	No gauge field, no F_μν.
SU(2) / weak / YM	ABSENT	—	—
SU(3) / strong / QCD YM	ABSENT	—	—
SU(2)×U(1) electroweak	ABSENT	—	—
SU(3)×SU(2)×U(1) SM	ABSENT	—	—
Spontaneous SB / Higgs	ADJACENT (analogical only)	notebook `§1b.5`	Pawn → queen promotion called "reverse Higgs mechanism" (gapped → gapless). Explicitly marked ANALOGICAL not mathematical. Fradkin-Shenker 1979 cited. Not coded.
Yukawa couplings	ADJACENT	notebook `§1b.5` "lattice realization of the Yukawa mass-range relationship"	Yukawa form G ~ exp(-mr)/√r is cited as the form of piece-range decay. Not built as a Lagrangian term.

QFT — propagators & amplitudes¶

Equation	Status	Where	How
Feynman propagators	ABSENT as built	notebook §1b.5 names the lattice scalar propagator	Same as Klein-Gordon: named, not instantiated.
LSZ reduction	ABSENT	—	—
Tree-level vertex factors	ABSENT	—	—
One-loop self-energies	ABSENT	—	—
RGE / β functions	ABSENT	—	—
Path-integral measure	ABSENT	—	—

Scattering¶

All ABSENT (S-matrix, cross-section, decay width, optical theorem, Cutkosky, crossing, Mandelstam s/t/u). No multi-particle final-state formalism; chess is single-position-state.

Geometric / algebraic adjuncts (where chess-spectral IS strong)¶

Operation	Status	Where	How
Spectral decomposition of H	FULL	`qm_2d.py:570`, `qm_4d.py:468 measure_observable_distribution`; `_get_h0_eigenbasis_2d`	Closed-form eigh; (eigvals, Born probs) grouped by distinct eigenvalue.
Simultaneous eigenbasis (Δ, B_4 commutant)	FULL — load-bearing	`research/spectral_identity_4d_findings.md`; notebook 4D §745 Pre-flight 3	Verified at machine precision: max ‖Δv-λv‖ = 3.3e-16; every Δ-eigenspace B_4-stable; max ‖[π(g), P_λ]‖ = 1.6e-13. This is the canonical structural fact under the QM stack.
Irrep-based Casimir	PARTIAL	`tables_4d.py` B_4 group action; D_4 irrep channels in `encoder.py`	D_4 / B_4 irrep typing of each channel (A_1 trivial, STD4 standard rep, FA_PAWN antisymmetric, etc.). The Casimir operator per se isn't built as `C_2 = sum_a T^a T^a`, but the irrep-projector decomposition is the same algebraic content.
Representation-theoretic mass formulas (GMO)	ADJACENT	notebook §1b.4 cites W-E theorem mass-free CG factor	D_4 CG selection rules for piece interactions tabulated; not an explicit GMO-style mass formula.
Action-angle decomp	ABSENT (chess-side)	—	(Cited from MEMORY as ephemerides feature — not in chess.)
Graph Laplacian on config lattice	FULL	`qm_4d_dynamics.py:2489 _build_h_free_4d`; `qm_2d_dynamics.py:136 _build_h_free_2d`; notebook §2	Kron-sum of four (or two) P_8 path-graph Laplacians; integer spectrum; canonical.
HDC bind / bundle as QM measurement/superposition	PARTIAL/IMPLICIT	`encoder.py` (the 640-dim encoder IS a fixed-basis HDC bundle); `qm_2d.py:122 state_to_psi` casts the HDC bundle to ψ	The state_to_psi map is literally HDC-bundle → ψ; the channel-PVM decomposition acts as a partial-trace-like projection. Not framed in HDC vocabulary, but the algebra is identical.
Pseudo-Hermitian / PT-symmetric (Bender-Mostafazadeh)	PROPOSED, FRAMEWORK STUBBED	ADR-005; `qm_4d_dynamics.py:1693 _build_m_pawn_white_axis`; `_pawn_observable_decomp`	Pawn observables explicitly identified as candidates for η-pseudo-Hermitian / PT structure (parity = axis flip; time = colour flip). η operator not yet constructed; `H_pawn_` raises `NotImplementedError` pointing to ADR-005. This is the single most relevant absent-but-designed piece for srmech-C2.*
Probability current j_a = Im(ψ* ∂_a ψ) + continuity eq. ∂_t ρ + ∇·j = 0	FULL	`qm_4d_bridge.py:1148 get_probability_current`; `_build_lattice_gradient_4d`	Anti-Hermitian central-difference ∂_a per axis (reflecting boundary), j = Im(ψ* ∂ ψ) summed over channels; continuity equation explicitly noted; divergence-free property tested.
Born-rule measurement + state collapse	FULL	`qm_4d_bridge.py:769 measure_at`	Cell-projection collapse + renormalisation; named-observable eigenvalue sampling.
Z_2 superselection	FULL	Pre-flight 1; `state_to_psi` sign multiplier; ADR-004	Position dict / Z_2 quotient; 8 fixed-point positions empirically tabulated; canonical mechanism.

Adjacent-but-not-canonical (chess-spectral operations subsuming canonical content in non-textbook form)¶

Graph-adjacency-as-Hamiltonian. The five H_rook_4, H_bishop_4, … H_knight_4 are graph adjacency matrices on Z_8^4 lifted to 4096×4096 Hermitians. Their eigenvalues are reach-centrality scores. This is a Hamiltonian on a single-particle lattice Hilbert space, but the "potential" is not V(x) — it's the piece-specific hopping structure. The canonical analogue is the tight-binding Hamiltonian H = sum_ t_ij c†_i c_j on a graph; chess-spectral is exactly this in the single-particle sector (no fermionic algebra wrapped on top yet).
The lattice gradient _build_lattice_gradient_4d. Anti-Hermitian central-difference per axis. This is the lattice momentum operator p̂_a = -i∂_a (up to factor i). Never named as such. If we set ℏ=1, it gives lattice [x_a, p̂_a] up to boundary corrections. Canonical mom-ops in disguise.
The 11-channel PVM. A projection-valued measure on the encoded space C^45056. Channels carry both semantic (piece type) and irrep (B_4 representation) labels. Each channel projector is functionally a "quantum number" projector — a generalised quantum number labelling Hilbert subspaces. This is the categorical apparatus QFT uses to label particle species / charge sectors, just on a chess substrate.
The eigenbasis-diagonal time evolution V·diag(exp(-iλt))·V^H is the textbook canonical solution to TDSE for time-independent H. The 2D module verified ~196× speedup over expm_multiply with 1.8e-16 max-abs deviation — closed-form, not approximate.
Pseudo-Hermitian / PT-symmetric pawn framework. ADR-005 explicitly cites Bender-Boettcher (1998) and Mostafazadeh (2002, 2010). η-inner-product <a|b>_η := <a|η|b> named. This is the canonical non-Hermitian-QM framework, named precisely; the η-operator construction is stubbed not built. The single most relevant Phase-C2 surface to absorb-and-finish, not re-invent.
The simultaneous-eigenbasis identity (Pre-flight 3). Stated formally: "The encoder's 4096 modes are exactly the simultaneous eigenbasis of (Δ, B_4 commutant)." This is a representation-theoretic theorem (basis-as-irrep-decomposition) verified at machine precision. In QM-textbook language this is the canonical statement that the eigenspaces of a maximal commuting set of observables (here: Δ + the B_4 commutant generators) form a complete basis. Not phrased as such; the structural content is identical.
Hatano-Nelson non-Hermitian pawn. Notebook §1b.1: pawn = H-N model at t_L=0 (max-asymmetric hopping). This is a canonical non-Hermitian QM model with imaginary eigenvalues in the antisymmetric part, named, instantiated structurally in the encoder's FA_PAWN channels.

Genuine gaps (real Phase C2 work items)¶

The following have no chess-spectral surface AND no obvious spectral-form adjacency:

γ-matrices and any spinor space. Chess-spectral has scalars on C^4096; there is no Cl(1,3) or Cl(4,0) construction, no spinor representation of B_4 lifted to a Dirac analogue. Notebook §1b.5 cites Yumoto-Misumi 2024 (graph Laplacian = scalar lattice + Wilson term; Dirac zero-mode count = sum of Betti numbers) but no Dirac operator code.
Gauge connection on lattice. No U(1) / SU(2) / SU(3) connection variables, no Wilson loop / plaquette construction. The B_4 group acts on the lattice but is not gauged.
Lagrangian formalism. Chess-spectral is Hamiltonian-only. No action, no path integral, no S = ∫ d^4x L.
Multi-particle / Fock-space algebra. Chess has Pauli exclusion at the position-occupation level (n_i ∈ {0,1}) by lattice geometry, but there is no creation/annihilation operator algebra, no antisymmetric multi-particle wavefunction, no Slater determinant, no Grassmann variable. The encoded state is a single 45056-dim ψ with all pieces, not a Fock-space element.
Scattering / S-matrix. No asymptotic states, no T-matrix, no on-shell condition, no amplitude → cross-section apparatus. Chess has no asymptotic regime.
RGE / β-function / regularisation. No flow equations, no renormalisation. Single fixed lattice scale; no coarse-graining infrastructure.
Hydrogen atom / Coulomb / spherical harmonics. Chess uses B_4 / D_4 (lattice point groups), not SO(3). The continuum rotational symmetry is missing by construction. Lattice spherical harmonics (Mädler / cubic harmonics) are not built.
Harmonic oscillator / ladder operators. No a / a†. Free Hamiltonian is the lattice Laplacian, not a quadratic-potential oscillator.
Klein-Gordon-as-code. The lattice propagator G(k)=1/(m²+k̂²) is named in §1b.5 but not exposed as a function. This is a one-screen piece of code — adding it to chess-spectral would be trivial; not adding it is a real gap.
Heisenberg-picture / commutator algebra. Schrödinger picture is the only path. No commutator(A, B) primitive (despite the natural fit with the integer-ALU preference for HDC binding).

Phase C2 implications (srmech MFO/SM/QM operations layer)¶

Cite-not-reimplement the kinematic + free-evolution layer. Chess-spectral's state_to_psi, inner_product, expectation, is_normalized / is_unitary / is_hermitian, evolve_under_h0, measure_observable_distribution, get_probability_current, measure_at, get_density_matrix_of cover ~7 canonical single-particle QM operations cleanly. srmech-C2 should expose these via the srmech.profiles["chess"] path already shipped in v1.19.0. Do not re-spell.
Pseudo-Hermitian / PT-symmetric primitive class. ADR-005's η-metric framework is the cleanest absorption target. srmech-C2 should ship the η-inner-product primitive (inner_product_eta(a, b, eta), expectation_eta(O, psi, eta), is_pseudo_hermitian(O, eta)) with the chess pawn as the worked instantiation. This unblocks ADR-005's first-class η ship.
Lattice propagator G(k) = 1/(m² + k̂²) as a closed-form operation. Notebook §1b.5 names it; nothing builds it. srmech-C2 should ship lattice_scalar_propagator(mass, k_lattice) as a one-screen function; it cleanly closes the Klein-Gordon gap and matches the project preference for integer-ALU when m and k̂ are integer-encoded.
Lattice gradient ↔ momentum operator naming. _build_lattice_gradient_4d is the lattice p̂. srmech-C2 should expose it as lattice_momentum(axis, lattice_shape, boundary) and surface the commutator algebra [x̂_a, p̂_b] ≈ iδ_ab (modulo boundary corrections, which are quantifiable for P_8^d). This is canonical QM content currently anonymous.
Casimir operators as group-Laplacian invariants. The B_4 commutant structure already exists in chess-spectral; srmech-C2 should ship casimir(group, irrep) returning the quadratic Casimir value per irrep. This is the canonical SM-side ingredient and connects chess-spectral's irrep decomposition cleanly to SM language (where Casimirs label irreps of SU(N)).
Gauge-connection primitive — flag as out of C2 scope. Building U(1) / SU(N) connection variables, Wilson loops, plaquettes is a substantial new operation class. Phase C2 should flag this as Phase C3 / future and not bundle it.
Dirac operator on the path-graph lattice. Yumoto-Misumi 2024 (cited in notebook §1b.5) gives a clean recipe: graph Laplacian = scalar lattice + Wilson term, plus an antisymmetrised adjacency matrix; Dirac zero-mode count = sum of Betti numbers. The chess B_4 commutant decomposition already provides candidate spinor decomposition (FA / FD channels carry the antisymmetric and diagonal content respectively). srmech-C2 should ship dirac_lattice(lattice, mass=0) as a class instantiation using this recipe; it would convert chess's "non-Hermitian chiral pawn" framing into a real instantiated Dirac operator on C^4096 × spinor.
HDC ↔ QM equivalence operation. The encoded 640-dim bundle IS the QM state under state_to_psi. srmech-C2 should expose this equivalence as an explicit conversion primitive (hdc_to_psi, psi_to_hdc) rather than burying it in state_to_psi. Connects to the integer-ALU preference and the resonant-bit-serialised HDC line.
Born-rule observable distribution as the canonical measurement primitive. measure_observable_distribution(H, ψ) returning (eigvals, probs) grouped by distinct eigenvalue is already the canonical spectral-measurement op; srmech-C2 should expose this verbatim. Add an apply_born_collapse(H, ψ, sampled_eigval) op (currently inlined in measure_at's position branch only).
Flag absent surfaces as out-of-scope for Phase C2 with explicit deferral notes: Lagrangian/path-integral (Phase C3 or later), S-matrix/scattering (no asymptotic regime in chess; likely never relevant), RGE/β-functions (Phase C3+), hydrogen atom / continuum rotation (SO(3) vs B_4 substrate mismatch — Phase D?).

Fermata (conductor input requested)¶

The notebook's §1b.5 names lattice propagators / Yukawa form / Hatano-Nelson / Yumoto-Misumi as cited but not coded mathematical substrate. Should srmech-C2 ship these as concrete primitives (closing the gap), or stay at the citation layer (deferring to a future SM ops layer)? Items 3, 4, 7 above are the load-bearing instances.
ADR-005's η-metric pawn observable is "Proposed (Option A; full PT for v1.7+)". srmech-C2's pseudo-Hermitian primitive (item 2 above) could finish ADR-005 — should this be co-shipped or kept as a separate chess-spectral PR?

Files of interest (absolute paths)¶

D:\GitHub\mlehaptics\docs\chess-maths\chess-spectral\python\chess_spectral\qm_2d.py
D:\GitHub\mlehaptics\docs\chess-maths\chess-spectral\python\chess_spectral\qm_2d_dynamics.py
D:\GitHub\mlehaptics\docs\chess-maths\chess-spectral\python\chess_spectral\qm_4d.py
D:\GitHub\mlehaptics\docs\chess-maths\chess-spectral\python\chess_spectral\qm_4d_dynamics.py
D:\GitHub\mlehaptics\docs\chess-maths\chess-spectral\python\chess_spectral\qm_4d_bridge.py
D:\GitHub\mlehaptics\docs\chess-maths\chess-spectral\python\chess_spectral\engine\eval\qm.py
D:\GitHub\mlehaptics\docs\chess-maths\chess-spectral\docs\adr\qm_4d\ADR-002-time-evolution-semantics.md
D:\GitHub\mlehaptics\docs\chess-maths\chess-spectral\docs\adr\qm_4d\ADR-005-pawn-pseudo-hermitian-eta-metric.md
D:\GitHub\mlehaptics\docs\chess-maths\chess_spectral_research_notebook.md (§1b.1, §1b.5, §15, §16)
D:\GitHub\mlehaptics\docs\chess-maths\chess_spectral_4d_notebook.md (qm_4d Pre-flight Findings, §745–§813)
D:\GitHub\mlehaptics\docs\chess-maths\chess-spectral\python\research\spectral_identity_4d_findings.md