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ADR-001: Phase Convention for Unitary Moves

Date: 2026-04-29 Branch: chess-spectral/qm-4d-kinematic Track: B (full move-as-unitary dynamics) Ships in: Phase 4 / qm_4d.applyMoveQm / chess-spectral v1.5.0

1. Status

Proposed. Final acceptance gated on a Phase 3 prototype probe (see §6) that materializes a sample applyMoveQm for the bishop and queen on H_bishop_4 / H_queen_4 and verifies the chosen convention produces visually-distinguishable trajectories under the M14.3 visualization acceptance criteria (consumer side).

2. Context

applyMoveQm(origin, dest) is the §17.1 bridge surface method (chess-spectral v1.5.0) that lifts a chess move to a unitary U_move on the QM module's amplitude space C^45056. The §17.1 contract requires U_move to (a) be unitary on the inner product currently in use, (b) act compatibly with the Z_2 superselection structure (ADR-004), and © optionally be returned to the caller as a sparse complex matrix for animation (M14.3 trajectory replay).

The underlying operation in basis space is a permutation: the indicator on square origin moves to square dest (with the captured piece, if any, removed). Permutation matrices are real orthogonal — therefore unitary over C — so the literal "permute basis vectors" lift gives a real-valued unitary already. The question is whether it should stay real, or pick up a phase factor that distinguishes Track B's QM dynamics from a relabeling exercise.

The Aaronson-escape-valve constraint (Aaronson, Read the Fine Print, 2015; cited in 2D notebook §15.6 / §11): a basis-aligned PVM applied to a basis-aligned state returns classical readouts tautologically. If applyMoveQm produces only real moves and getQmExpectation uses only the real Hermitian piece observables (H_rook_4, etc.), the QM module reduces to "classical chess in different notation." Track B's design must include a path to genuine quantum interference — the phase convention is where that path opens.

Pre-flight 2 noted explicitly: "there's no physics-motivated default phase." This is the design decision left open. Four candidate conventions are on the table.

2.1 Existing infrastructure that constrains the choice

  • Encoder structure (encoder_4d.py). 11 channels, 4096 modes each. Each channel carries a B_4 irrep label (see ADR-003 for the channel-by-channel breakdown). The chosen phase convention must compose cleanly with each channel's existing transformation under a move (linear-in-position channels permute the basis; nonlinear channels require special handling).

  • B_4 unitary representation (b4_unitary_rep_full in qm_4d.py). The module already exposes B_4 group elements as block-diagonal sparse unitaries on C^45056. Any phase convention must commute with the B_4 action — the encoder's symmetry is load-bearing for §15.2's "B_4 equivariance for free" claim.

  • Z_2 superselection (ADR-004). state_to_psi already takes the side-to-move bit and multiplies ψ by ±1. The phase convention for U_move must preserve this: a move flips side-to-move, so U_move must carry the −1 in a way that's consistent with the chosen Z_2 sector (ADR-004 fixes this).

3. Decision

Adopt Option D: per-channel phase derived from the channel's B_4 irrep label.

Concretely, when piece P of type t moves from coords o = (x,y,z,w) to coords d = (x',y',z',w'), the unitary on channel c is

U_move_c = e^(i * theta_c(o, d, t)) * Pi_c

where Pi_c is the channel-c lift of the basis permutation (described in ADR-003) and theta_c(o, d, t) is the channel-specific phase below.

3.1 The channel phase function

The 11 channels group into three irrep classes. Per encoder_4d.py:115-127:

# Channel Irrep type Phase formula
0 A1 trivial / A_1 theta_0 = 0
1 STD4_X std-4D coord residual (x-axis) theta_1 = (pi/4) * (x' - x)
2 STD4_Y std-4D coord residual (y-axis) theta_2 = (pi/4) * (y' - y)
3 STD4_Z std-4D coord residual (z-axis) theta_3 = (pi/4) * (z' - z)
4 STD4_W std-4D coord residual (w-axis) theta_4 = (pi/4) * (w' - w)
5 FIB_SYM_1 rank-3 fiber SVD direction 1 theta_5 = (2pi/3) * d_path
6 FIB_SYM_2 rank-3 fiber SVD direction 2 theta_6 = (4pi/3) * d_path
7 FIB_SYM_3 rank-3 fiber SVD direction 3 theta_7 = (2pi) * d_path
8 FA_PAWN_W pawn antisymmetric W-axis theta_8 = (pi/2) * sgn(w' - w)
9 FA_PAWN_Y pawn antisymmetric Y-axis theta_9 = (pi/2) * sgn(y' - y)
10 FD_DIAG diagonal deviation theta_10 = pi * (1 if diagonal_step else 0)

where d_path is the Chebyshev path-length of the move (number of single-step hops the piece traverses; for non-sliders this is the displacement; for sliders it is the slide length). sgn(w' - w) is +1 for white-direction push, −1 for black-direction.

3.2 Why this satisfies the constraints

  • Unitarity: Each e^(i * theta_c) is a global phase on a 4096-block; it commutes with Pi_c by construction. U_move = ⊕_c e^(i * theta_c) * Pi_c is therefore unitary on C^45056 (each block is unitary; the direct sum is unitary).
  • Aaronson escape: Different channels accumulate different phases under the same move. A bishop e1 → c3 picks up theta_1 = pi/2, theta_3 = pi/2 in STD4 channels but theta_2 = theta_4 = 0 (no y or w displacement). On the FIB channels the phase is (2pi/3) * 2 = 4pi/3 for d_path = 2. A superposition input ψ spanning multiple channels accumulates different phase factors per channel, so |<phi|U_move ψ>|^2 is no longer just the classical permutation overlap — interference arises.
  • B_4 commutativity: Each B_4 element acts within a channel block (per the existing b4_unitary_rep_full = I_11 ⊗ U_4096(g) construction). The per-channel phase e^(i * theta_c) is a scalar on the channel block, so it commutes with B_4 on that block. Therefore U_move commutes with B_4 as long as theta_c does — and theta_c depends only on the move (its channel-irrep type, displacement vector, and path length), not on position-of-pieces details that B_4 would scramble. So B_4-equivariance is preserved.
  • Z_2 superselection (ADR-004): A move flips side-to-move; the −1 from state_to_psi propagates through U_move as a global factor and lands in the opposite Z_2 sector cleanly. The channel-phase choice is independent of the Z_2 grading.

3.3 Why these specific phase values

  • A_1 channel (theta_0 = 0): The trivial irrep transforms identically under all of B_4. Any nonzero phase would single out a direction in B_4 — inconsistent with the irrep label. Zero is the only natural choice.
  • STD4 channels (theta_a = (pi/4) * delta_a): Linear-in-displacement phase along the irrep's dedicated axis. The factor pi/4 is chosen so a single-step move accumulates pi/4 (smallest nonzero phase that survives to displacement = 8 without wrapping).
  • FIB_SYM channels (cube roots of unity scaled by d_path): The three fiber-SVD directions span a 3D rank space; phases 2pi/3, 4pi/3, 2pi encode the three irrep components as 3rd roots of unity. Path-length scaling makes longer slides accumulate more phase, capturing "how far the piece walked" in the wave function.
  • FA_PAWN channels (pi/2 * sgn): Pawn directionality is the defining feature of the antisymmetric pawn channel. A pi/2 phase per push captures the directed-push asymmetry as a quarter-turn — half the phase of a Berry "out and back" loop, since a pawn cannot reverse.
  • FD_DIAG channel (pi if diagonal step): The diagonal-deviation channel measures how far the position is from a pure diagonal lattice. A diagonal step (bishop, queen-diag) lands on a pi phase (sign flip); axis-aligned steps stay on 0. Captures the channel's geometric character.

3.4 What U_move returns to the caller

Per the §17.1 contract, applyMoveQm may optionally return U_move as ComplexMatrix. With the per-channel-phase convention, U_move is a sparse block-diagonal matrix with 11 sparse blocks of shape (4096, 4096), each block being e^(i * theta_c) * Pi_c. Total nnz scales as 4096 × 11 = 45 056 (each permutation has one nonzero per row). The serialization fits comfortably in the existing ComplexArray real+imag-interleaved Float32 format.

4. Consequences

4.1 What this enables

  • M14.3 visualization differentiability. The consumer's M14.3 milestone requires "visually-distinguishable trajectories" — the phase convention produces channel-specific phase rotations during a slide, which colorize the M14.2 phase-as-color render. Tested at the prototype stage.
  • Aaronson escape pathway opens. Channel-distinct phases break the classical-relabeling reduction. Whether the resulting interference is useful (per §15.6's escape valve) is a Phase 6 / 7 empirical question, but the structural pathway exists.
  • Chess physics intuition lands. The path-length-dependent FIB phases encode "the piece walked through positions" as accumulated geometric phase, matching the user's instrument-and-string framing (notebook §42.1).
  • B_4 equivariance preserved. Phase factors are channel-block scalars; they commute with the existing B_4 action.

4.2 What this forecloses

  • No path-history dependence on the phase. The phase depends only on origin, destination, piece type, and channel — not on how the piece's ψ-amplitude got to origin. This is intentional (keeps U_move Markov on the QM state) but rules out "true" path-integral phase that integrates over all paths the piece might have taken. If Phase 6 / 7 finds that M14.3 visualizations need path-integral richness, the convention can be upgraded in v1.7+ — see §6 below.
  • No piece-type-specific Berry phase beyond the per-channel formulas. Bishop-vs-rook are distinguished by their channel-decomposition profile (bishop populates FIB_SYM, FD_DIAG; rook populates STD4 + A1) but not by per-piece-type phase coefficients. Option C (sign-parity from piece type) is rejected — see §5 below.

4.3 LOC implications

applyMoveQm total: ~250 LOC including doc strings. - _compute_channel_phase(channel_idx, origin, dest, piece_type) -> float: ~80 LOC (one branch per channel, reading from the table in §3.1) - _build_channel_permutation(channel_idx, origin, dest) (decision deferred to ADR-003): ~120 LOC — the load-bearing piece for linear channels; nonlinear channels add their own LOC budget per ADR-003. - applyMoveQm driver assembling the block-diagonal sparse matrix: ~50 LOC.

4.4 Test surface

  • Unitarity: For a sample of 100 random legal moves on random non-trivial positions, assert is_unitary(U_move, tol=1e-10) from existing qm_4d.py validation helpers. Cost: ~30 LOC test.
  • B_4 commutativity: For a sample of 50 moves and 20 B_4 elements, assert ‖U_move @ B - B @ U_move‖_F < 1e-10 where B = b4_unitary_rep_full(g). Cost: ~40 LOC.
  • Z_2 sector flip: For a sample of moves, assert that state_to_psi(state', NOT side) = U_move @ state_to_psi(state, side) (where state' is the post-move state and side flips). Cost: ~30 LOC.
  • Per-channel phase formulas: Direct numeric test of the table in §3.1 against _compute_channel_phase. Cost: ~50 LOC.
  • Aaronson escape demonstration: A single hand-rolled test showing superposition input + nontrivial move produces an output where the channel marginals' phases differ. Cost: ~30 LOC.

Total test LOC: ~180.

5. Alternatives Considered

5.1 Option A: Real (+1) only — moves are real orthogonal matrices

Rejected. This is the simplest convention: U_move = ⊕_c Pi_c, each block real. Implementing this is ~100 LOC less than Option D (no _compute_channel_phase function, no per-channel branches). However:

  • Aaronson trap. Pre-flight 2 and 2D notebook §15.6 explicitly call this out: real-orthogonal U_move + basis-aligned PVM observables = classical chess in QM notation. The whole point of Track B is to escape this trap.
  • No interference under M14.x. The M14.3 trajectory replay would render identically to a classical board-state animation, defeating the visualization milestone's purpose.
  • Wastes the Pre-flight 3 spectral identity. The encoder is the simultaneous eigenbasis of (Δ, B_4 commutant). Real-only moves don't exercise this structure beyond the kinematic view in Track A.

5.2 Option B: Berry phase from phi4

Rejected for v1.5; revisit for v1.7+. The proposal: accumulate the geometric phase along the actual lattice path the piece traverses (e.g., sum phi4(square_k) for each square k on the move's path).

  • Path-dependence ambiguity for non-sliders. Knights teleport to the destination — the "path" is ill-defined. Berry phase as a path integral needs a continuous path, which the lattice doesn't have. Defining the phase only for sliders introduces piece-type-specific code branches that ADR-003 already has to handle for nonlinear channels; we'd be doubling the conditional surface.
  • Computation cost. phi4 has modulus MODULUS_4D = 145451; summing over the path produces phases mod 2pi * 145451 / 145451, which doesn't give the simple closed-form e^(i * theta) factors that compose cleanly in the channel-block structure.
  • Less expressive than per-channel. A single Berry phase from a path doesn't distinguish A_1 from STD4 from FIB_SYM channel evolution. The per-channel-irrep convention (Option D) does, which is what the visualization-differentiability test requires.

If Phase 6 / 7 finds that Option D's interference structure is too rigid to capture chess insight, Option B becomes a v1.7+ extension where sliders accumulate Berry phase in addition to the per-channel-irrep base phase from Option D.

5.3 Option C: Sign-parity from piece type

Rejected. The proposal: assign each piece type a fixed phase (e.g., e^(i*pi/3) for knight, e^(i*pi/2) for bishop, etc.) accumulated per move.

  • Arbitrary. No representation-theoretic motivation. The choice of which rational multiples of pi to assign is unprincipled and would introduce a "magic numbers" violation by chess-spectral SSOT discipline.
  • No B_4 equivariance. Piece-type is a global label, not a B_4 irrep component. A pi/3 phase on knight moves doesn't commute with the B_4 action on the channel decomposition — the construction would break the B_4 commutativity test in §4.4.
  • Less informative than per-channel. A bishop e1 → c3 and a bishop e1 → b4 would carry the same phase under Option C, even though they traverse different displacement vectors. The M14.3 visualization acceptance criterion explicitly requires "different moves → distinguishable trajectories," which Option C only weakly satisfies (different piece types, yes; different moves of the same piece type, no).

5.4 Option D variants we considered but didn't pick

  • Real-coefficient per-channel formulas (theta_c real, noi`). Same effect as Option A; no interference. Rejected.
  • Per-channel phase from Krawtchouk polynomial values. The encoder basis decomposes via Krawtchouk; using Krawtchouk values at displacement argument is mathematically natural. Rejected because the formulas produce phases of the form K_n(d, q) / N_norm, which is opaque to the consumer (no clean intuition about phase-as-rotation). The pi/4, 2pi/3, pi/2, pi rationals in §3.1 are more readable; the channel-distinctness property is preserved either way.

6. Open Questions / Future Work

  1. Phase 3 prototype probe. Before implementation, build a 50-LOC prototype that applies one U_move (e.g., bishop e1 → c3) to a superposition state and renders the channel-marginal phases. The prototype validates that the M14.3 visualization differentiates "same-piece different-destination" pairs.

  2. Berry phase v1.7+ extension. If §16 self-play tournament finds that QM-evaluator with Option D underperforms — i.e., chess-relevance is less than spectral evaluator at the representative depths the tournament harness exercises — revisit whether path-integral phase (Option B) on sliders adds discriminative signal. This becomes a Phase 7 question, not a Phase 4 question.

  3. Phase calibration via tournament. Phase 7's learned-weight experiments (§16.8) could fit the rational multipliers in §3.1 against tournament outcomes. This is "tune the phase convention to maximize Elo" — a natural follow-up if Option D has unconstrained parameters.

7. References

8. Phase 4 Implementation Pointer

This is the design we will implement in Phase 4 B[1] (qm_4d phase convention + scalar phase tables). The two helpers _compute_channel_phase and the channel_phase_table constant (encoding §3.1's rational multipliers) ship together; _build_channel_permutation (ADR-003) and the applyMoveQm driver ship in subsequent Phase 4 steps. Acceptance: the §4.4 test surface passes, and a hand-checked bishop-move example produces the §3.1 phases at machine precision.