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ADR-005: Pawn Pseudo-Hermitian η-Metric

Date: 2026-04-29 Branch: chess-spectral/qm-4d-kinematic Track: B (full move-as-unitary dynamics) Ships in: Phase 4 / qm_4d.H_pawn_w_4, H_pawn_y_4, eta_pawn_4 / chess-spectral v1.5.0

1. Status

Proposed (best-effort projection onto Hermitian part for v1.5; full PT-symmetric η for v1.6+ if Phase 6 evidence motivates it).

This ADR ships v1.5 with a two-tier delivery: a "best-effort Hermitian-part projection" pawn observable that is unitarily exposed as H_pawn_w_4_herm and H_pawn_y_4_herm, plus the η-metric framework stubbed so that v1.6+ can promote it to first-class without an API break. The full pseudo-Hermitian / PT-symmetric machinery is documented here but ships fully only when motivated.

2. Context

Pre-flight 2 (phase_operators_4d_pseudo_hermitian_audit.md) established that the 5 non-pawn predicates (P_rook4, P_bishop4, P_queen4, P_king4, P_knight4) lift to real-symmetric Hermitian matrices on C^4096. The lift commutes with J_op (ADR-004) and admits the trivial metric η = I.

The pawn predicates (P_pawn4_white, P_pawn4_black) break Hermiticity because pawn pushes are directed: - White pawns push along +w axis (or +y axis per Oana & Chiru Definition 11; pawns oriented along Y or W axis only). - Black pawns push along -w (or -y). - Captures are diagonal, also directed.

The lifted matrix M_pawn_white has M_pawn_white ≠ M_pawn_white^T. The asymmetry sits in the diagonal-capture sub-block and the forward-push sub-block. A M_pawn_black lift would be the transpose / dagger of M_pawn_white on the capture sub-block (black's diagonal captures match white's diagonal captures with axis flipped).

For Track B's QM evaluator (getQmExpectation from §17.1), pawns must be measurable somehow — chess without pawn observables is incomplete. The question is what kind of pawn observable the QM module exposes.

2.1 The pseudo-Hermitian / PT-symmetric framework

Mostafazadeh (2002, 2010 reviews) and Bender-Boettcher (1998) introduced pseudo-Hermitian operators: operators O on a Hilbert space H for which there exists an invertible Hermitian η with

O^dagger = η * O * η^{-1}            (η-pseudo-Hermiticity)

Equivalently: O is Hermitian under the modified inner product

<a|b>_η := <a|η|b>           (the η-inner-product)

This is the natural generalization of "Hermitian" beyond the standard η = I case. PT-symmetric operators (Bender-Boettcher) are a special case where η = PT is the parity-time operator. PT-symmetric operators have real spectrum under conditions Bender enumerates (unbroken PT phase).

The directed-push pawn predicates are prime candidates for PT-symmetric machinery: - Parity (P): the axis-flip on the pawn's direction (W-axis flip for W-axis pawns; Y-axis flip for Y-axis pawns). - Time-reversal (T): the color flip (white ↔ black). - The composition PT applied to M_pawn_white gives a candidate η.

This is where the "qm_4d earns its keep" framing from notebook §15.2(4) points. If the pawn machinery shipped as a pseudo-Hermitian operator under a meaningful η, that's the structural payoff that distinguishes Track B from Track A.

2.2 The infrastructure problem

Building a usable pseudo-Hermitian / PT-symmetric framework requires:

  1. An η operator. Compute and verify M_pawn_white^dagger = η * M_pawn_white * η^{-1} for some explicit η. This is non-trivial: η is determined up to a positive scalar by the pseudo-Hermiticity condition, but constructing it explicitly requires either a closed-form derivation from PT-symmetry (Bender-Boettcher) or a numerical search (eigendecomposition-based).
  2. The η-inner-product machinery. New helpers inner_product_eta(a, b, eta), expectation_eta(O, psi, eta), is_pseudo_hermitian(O, eta). The existing qm_4d API uses vdot (standard inner product); pseudo-Hermitian observables need their own evaluation path.
  3. Spectrum-realness verification. PT-symmetric operators have real spectra in the unbroken phase. We must verify (numerically) that M_pawn_white (or its η-lifted version) has real spectrum — and handle the case where it doesn't (PT phase broken).
  4. Bridge contract integration. getQmExpectation('pawn', ...) must specify which evaluation rule applies — standard <ψ|H|ψ> or η-modified <ψ|H|ψ>_η. The 17.1 contract didn't pre-decide this.

2.3 The v1.5 LOC budget pressure

Track B's overall LOC budget is ~2000 LOC (notebook §17.1.2 acceptance criteria + history-dependent budgeting). ADR-001 (phase) is ~250 LOC, ADR-002 (evolution) is ~110 LOC, ADR-003 (per-channel) is ~1330 LOC, ADR-004 (Z_2) is ~290 LOC. Track B's available budget for ADR-005 is ~20 LOC — far less than what a full η-metric implementation needs.

A full pseudo-Hermitian implementation alone is ~600 LOC: - η construction + verification: ~150 LOC. - η-inner-product helpers: ~100 LOC. - Spectrum-realness machinery: ~150 LOC. - Pawn observable lift + dispatch: ~100 LOC. - Tests: ~200 LOC.

This is impossible to ship as a v1.5 deliverable. We must choose between (a) shipping pawn observables in a reduced form, or (b) deferring pawn observables entirely (raising NotImplementedError) and shipping the η-metric in v1.6.

3. Decision

Two-tier delivery:

3.1 v1.5 deliverable: best-effort Hermitian-part projection

For v1.5.0, expose pawn observables via the Hermitian part of the lifted pawn matrix:

H_pawn_<axis>_4_herm  := (1/2) * (M_pawn_<axis> + M_pawn_<axis>^T)

where <axis> ∈ {w, y}. The _herm suffix is intentional — the naming makes the approximation visible in the API.

This is a strictly Hermitian operator on C^4096. It loses the direction-asymmetric information (forward-push vs backward-push are averaged), but it captures the symmetric pawn structure: where pawns can exist regardless of color.

Properties: - Real-symmetric, Hermitian under η = I. - Real spectrum (Pre-flight 2 already confirmed real-symmetric matrices have real spectrum). - J_op-commuting (per ADR-004's general argument for B_4-symmetric graph-adjacency lifts; cross-axis pawn directionality cancels under J's color-flip).

Exposed as: - H_pawn_w_4_herm (lazy module property) - H_pawn_y_4_herm (lazy module property)

Used by the Phase 6 QM evaluator for pawn-related contributions to getQmExpectation. Documented prominently as "Hermitian projection; loses directional information" in the qm_4d docstring.

3.2 v1.5 stub: η-metric framework (zero LOC marginal cost)

Define the interface for η-metric pawn observables, but raise NotImplementedError on access in v1.5:

# Module attributes; raise NotImplementedError on access in v1.5.
H_pawn_w_4   # raises NotImplementedError("v1.6+ candidate; see ADR-005")
H_pawn_y_4   # same
eta_pawn_w_4 # same — the η operator itself
eta_pawn_y_4 # same
inner_product_eta(a, b, eta)          # raises in v1.5; reserved name
expectation_eta(O, psi, eta)          # same
is_pseudo_hermitian(O, eta, tol=1e-10) # same

These names are reserved in the public API surface so that v1.6 can promote them to functional implementations without breaking consumers. LOC cost: ~30 (one stub raising NotImplementedError per name).

3.3 The η operator design (documented for v1.6+)

When v1.6 implements the η-metric, the η operator is constructed as follows (this is the design captured here; implementation deferred):

3.3.1 η for W-axis pawns

The white pawn pushes +w, captures diagonally (+w, ±x, ±y, ±z). The black pawn pushes -w, captures (-w, ±x, ±y, ±z). The lifted matrices satisfy:

M_pawn_w_white^T = M_pawn_w_black     (push directions reverse)
M_pawn_w_black^T = M_pawn_w_white     (same)

So M_pawn_w_black = M_pawn_w_white^T. Define:

M_pawn_w := M_pawn_w_white + M_pawn_w_black = M_pawn_w_white + M_pawn_w_white^T

This is the symmetric part — i.e., exactly H_pawn_w_4_herm from §3.1.

For the antisymmetric / directed part:

A_pawn_w := M_pawn_w_white - M_pawn_w_black = M_pawn_w_white - M_pawn_w_white^T

A_pawn_w is real anti-symmetric. The full white-pawn operator is M_pawn_w_white = (1/2) * (M_pawn_w + A_pawn_w).

The η operator that makes M_pawn_w_white pseudo-Hermitian is:

η_pawn_w := P_w

where P_w is the W-axis parity flip (the involution s ↦ s' with w' = 7 − w and other coords unchanged). This is a 4096×4096 permutation matrix; it commutes with all non-W-axis B_4 elements but anti-commutes with W-axis sign-flips.

Verification: - P_w * M_pawn_w_white * P_w^{-1}: applying P_w to source (sw') and P_w to target (sw') of M_pawn_w_white permutes the rows and columns. The white pawn pushes +w; under P_w, (sw, sw+1) maps to (7-sw, 7-(sw+1)) = (7-sw, 6-sw), which is a -w push. So P_w * M_pawn_w_white * P_w^{-1} = M_pawn_w_black = M_pawn_w_white^T. - Therefore M_pawn_w_white^T = P_w * M_pawn_w_white * P_w^{-1}, which is the pseudo-Hermiticity condition with η = P_w (since P_w is Hermitian — it's a permutation matrix and self-inverse, hence symmetric).

So M_pawn_w_white is P_w-pseudo-Hermitian.

PT-symmetry interpretation: P_w is parity (axis flip); the time-reversal T can be identified with the color-flip. PT = P_w * color_flip is then the involution that takes white-pushing pawns to black-pushing pawns and flips colors. The composition PT commutes with M_pawn_w_white (verifiable closed-form), so M_pawn_w_white is PT-symmetric in Bender's sense.

3.3.2 η for Y-axis pawns

By symmetry with the W-axis case: η_pawn_y = P_y (Y-axis parity flip). Same construction; same verification.

3.3.3 Composite η for combined-axis pawn observables

If consumers want to evaluate "expected pawn presence regardless of axis" as a single operator, the combined M_pawn = M_pawn_w + M_pawn_y is pseudo-Hermitian under η = P_w * P_y (composed parity flips). Documented for v1.6+ but not needed for v1.5.

3.4 Spectrum-realness check (v1.6 requirement)

Per Bender-Boettcher, the PT-symmetric operator has real spectrum iff the PT phase is unbroken: every eigenvector is also a PT eigenvector. For the lifted pawn operators, we must verify this empirically:

  • Compute the spectrum of M_pawn_w_white numerically (4096×4096 dense eigendecomposition; ~30 seconds).
  • Verify all eigenvalues have |Im(λ)| < tol. If yes: PT phase unbroken; the η-metric machinery is meaningful.
  • If PT phase is broken (some eigenvalues complex): η-metric does not produce real expectation values. Document and either (a) project to PT-unbroken subspace, or (b) ship Hermitian-part projection only.

Pre-implementation probe for Phase 3: Run the spectrum check on M_pawn_w_white; record the result. This is a 30-LOC closed-form check that must run before v1.6 promotion.

If the probe finds PT phase broken, v1.6's design is reduced — possibly the η-metric is shipped only as a research artifact, with the Hermitian-projection observables remaining the primary v1.6 deliverable.

3.5 v1.5 vs. v1.6 contract

For v1.5.0: - getQmExpectation('pawn_w_herm', ψ) returns <ψ|H_pawn_w_4_herm|ψ> — a real number computed via the standard inner product. Available. - getQmExpectation('pawn_w', ψ) raises NotImplementedError with a pointer to ADR-005 §3.5 (this section). Reserved for v1.6. - getQmExpectation('pawn_y_herm', ψ) available; pawn_y reserved.

For v1.6+: - getQmExpectation('pawn_w', ψ) returns <ψ|H_pawn_w_4|ψ>_η — computed via the η-inner-product, real iff PT phase is unbroken. - The η operator is exposed as eta_pawn_w_4 for direct manipulation by advanced consumers.

The bridge contract changes are minimal: the same observable name keys, extended with _herm suffix for v1.5; the suffix-less names go live in v1.6.

4. Consequences

4.1 What this enables

  • Pawn observables ship in v1.5. The Phase 6 QM evaluator can compute pawn contributions via H_pawn_<axis>_4_herm. Not strictly faithful to the directed-push structure, but captures the symmetric pawn skeleton — sufficient for evaluator strength comparisons.
  • API forward-compatibility. v1.6 promotes the suffix-less names to full implementations without breaking v1.5 consumers.
  • Phase 3 prototype probe is well-defined. §3.4's PT-realness check is closed-form; can be run in 30 LOC + ~1 minute of compute.
  • Reviewers see a clean handoff. "Pawn observable ships as Hermitian projection in v1.5; full PT-symmetric construction documented in ADR-005 and slated for v1.6 contingent on PT phase verification" is a publishable scope.

4.2 What this forecloses

  • No first-class directed-push QM dynamics in v1.5. The M_pawn_white matrix as written is NOT used in v1.5 — only its symmetric part. Consumers who need to distinguish "white pushes forward" from "black pushes forward" at the QM level wait for v1.6.
  • No direct coupling between ADR-001's pawn-channel phases and ADR-005's pawn observable. ADR-001 §3.1 sets theta_8 = (pi/2) * sgn(w'-w) for W-axis push direction. This is a phase encoding of direction. ADR-005's v1.5 H_pawn__herm doesn't use direction. So a consumer asking "does the QM evaluator's pawn observable see the same direction as the applyMoveQm phase?" — the answer in v1.5 is *no; in v1.6+ with full η-metric, yes.
  • Open thread for §16.5 ablation. Notebook §16.5 explicitly asks "if the pseudo-Hermitian pawn observables (Track B) matter, that validates the η-metric machinery." With v1.5 shipping only Hermitian projection, this ablation question is deferred — Phase 6's tournament measures pawn_w_herm contribution, not the genuine pseudo-Hermitian observable. The ablation as written in §16.5 isn't fully runnable until v1.6.

4.3 LOC implications

v1.5 LOC budget: - H_pawn_w_4_herm lazy attribute: ~30 LOC (build symmetric part of M_pawn_white from _t4.piece_adjacencies_4d lifted with directionality helpers; symmetrize; cache). - H_pawn_y_4_herm lazy attribute: ~30 LOC (same with Y axis). - Stubs for the full η-metric API: ~30 LOC (NotImplementedError stubs). - Module-level docstring update: ~30 LOC.

Total v1.5 LOC: ~120.

v1.6 LOC budget (when promoted): - η_pawn_w / η_pawn_y construction: ~100 LOC. - η-inner-product machinery: ~100 LOC. - Spectrum-realness verification: ~80 LOC. - H_pawn_w_4 / H_pawn_y_4 (full) lazy attributes: ~80 LOC each. - Tests: ~200 LOC.

Total v1.6 LOC: ~640.

4.4 Test surface

v1.5 tests: - H_pawn_w_4_herm is real-symmetric Hermitian: ~30 LOC. - Spectrum is real for H_pawn_w_4_herm: ~30 LOC. - H_pawn_w_4_herm commutes with J_op: ~30 LOC (test that this is the case; expected to pass per the symmetric-projection argument). - H_pawn_y_4_herm analogous: ~30 LOC each = ~60 LOC total. - NotImplementedError stubs raise correctly: ~30 LOC.

Total v1.5 test LOC: ~150.

v1.6 tests (when promoted): - η_pawn_ is Hermitian and unitary: ~30 LOC each. - Pseudo-Hermiticity: M_pawn_w^T = η_pawn_w * M_pawn_w * η_pawn_w^{-1}: ~30 LOC each. - PT-realness of spectrum: ~50 LOC. - η-inner-product produces real expectation values for pseudo-Hermitian operators: ~50 LOC. - Full vs. Hermitian-part projection comparison: ~50 LOC.

Total v1.6 test LOC: ~210.

5. Alternatives Considered

5.1 Option α: Full pseudo-Hermitian / PT-symmetric pawn observables in v1.5

Rejected. The proposal: implement the full η-metric machinery in v1.5.0; ship H_pawn_w_4 and H_pawn_y_4 as the genuine pseudo-Hermitian operators.

  • LOC budget violation. ~640 LOC for full pseudo-Hermitian work exceeds Track B's available budget after ADR-001/002/003/004 take ~1980 LOC. Total Track B budget is ~2000 LOC; this option pushes to ~2620 LOC. Risk: shipping nothing.
  • PT-realness untested. The PT phase being unbroken is a condition to verify. Without the v1.5 Phase 3 probe in §3.4, we don't know if the η-metric produces real expectation values. Building infrastructure on an unverified condition is fragile.
  • Bridge contract uncertainty. §17.1 didn't pre-decide whether getQmExpectation should use η-modified inner products. Locking this in v1.5 without consumer testing is premature.

5.2 Option β: No pawn observables in v1.5

Rejected. The proposal: raise NotImplementedError on all pawn-related queries in v1.5; punt the entire pawn machinery to v1.6.

  • Breaks Phase 6 evaluator. Pawns are 8-of-16 starting pieces in chess. A QM evaluator that ignores pawns is dramatically incomplete. The §16.5 tournament results would be uninterpretable.
  • Wastes Pre-flight 2's groundwork. Pre-flight 2 already mapped out the pseudo-Hermitian / PT structure for pawns. v1.5 having "we know but didn't ship" status is anticlimactic.
  • Cleaner is to ship what's shippable. Hermitian projection is technically clean (real-symmetric → Hermitian under η = I), well-defined, and produces real expectation values. It's the natural v1.5 deliverable.

5.3 Option γ: Ship the directed M_pawn_white as a non-Hermitian operator in v1.5

Rejected. The proposal: expose M_pawn_w_white directly; let consumers interpret <ψ|M_pawn_w_white|ψ> as a complex number.

  • Bridge contract violation. §17.1's getQmExpectation returns { ok, value } where value is a real number. A complex-valued observable doesn't fit the contract.
  • Confusing semantics. What does Im(<ψ|M_pawn_w_white|ψ>) mean in a chess context? Without the η-metric machinery to interpret it, the imaginary part is just numerical noise from the directed asymmetry.
  • PT-symmetry framework was developed precisely for this. Bender's insight is that directed-flow operators are meaningful under the η-modified inner product. Ignoring that framework leaves complex expectation values uninterpretable.

5.4 Option δ: Hermitian projection in v1.5 + η-metric stubs (CHOSEN)

This is the chosen Option D, described in §3 above. The two-tier delivery balances: - Shipping in v1.5: Hermitian projection is clean, fast, and semantically meaningful (loss of direction information is documented). - Forward compatibility: η-metric stubs reserve the API names so v1.6 can promote without API break. - Empirical validation pathway: The §3.4 Phase 3 probe verifies PT realness before v1.6 commitment, avoiding the Option α risk.

5.5 Sub-decision: which η operator to use

Within Option δ, the choice of η is constrained by the directed-push structure. The §3.3 derivation η = P_w (axis parity flip) is the unique η-operator candidate that: - Is invertible Hermitian (P_w is a permutation, hence orthogonal, hence unitary, hence Hermitian since P_w^2 = I). - Satisfies the pseudo-Hermiticity condition M^T = η * M * η^{-1} (verified by the closed-form argument in §3.3.1). - Has a clean physical interpretation (axis parity ↔ "flip the pawn's push direction").

Any other η candidate (e.g., a less symmetric operator) would either fail to satisfy pseudo-Hermiticity or lack physical interpretability. P_w is the natural choice.

6. Open Questions / Future Work

  1. Phase 3 PT-realness probe. Run the §3.4 spectrum check on M_pawn_w_white and M_pawn_y_white before Track B implementation begins. If PT phase is broken (some complex eigenvalues), v1.6's design must adapt — likely projecting to the PT-unbroken subspace.

  2. v1.6 promotion criteria. v1.6 ships full η-metric pawn observables if and only if:

  3. PT-realness probe passes (real spectrum within tol).
  4. Phase 6 tournament results indicate pawn_w_herm is signal-bearing (consumers actually use it; not zero contribution).

  5. LOC budget is available (Track B post-v1.5 has slack).

  6. Multi-axis pawn observables. If consumers need a unified "all pawns regardless of axis" observable, the §3.3.3 composite η = P_w * P_y construction handles it. Defer to v1.6+ unless explicitly requested.

  7. Bridge contract clarification. §17.1 should be amended in the v1.5.0 release notes: getQmExpectation for pawn observables in v1.5 uses the standard inner product (because _herm observables are actually-Hermitian); v1.6 will introduce η-modified inner product for the suffix-less names. Document the migration path.

  8. Castling and en-passant. These are also asymmetric chess operations. Castling is direction-symmetric (kingside / queenside); en-passant is direction-asymmetric (only on opposite-color last-move push). They may admit similar pseudo-Hermitian treatment but are out of scope for ADR-005. Punt to a future ADR (ADR-006 candidate?) if needed.

7. References

  • Pre-flight 2 (phase_operators_4d_pseudo_hermitian_audit.md) — establishes the directed-pawn / Hermiticity-breaking finding; §3, §6.2 of that audit pre-suggest the η = P_w construction in this ADR.
  • Notebook §15.2(4) (chess_spectral_research_notebook.md, sec 15.2) — "qm_4d module's pseudo-Hermitian / PT-symmetric machinery earns its keep" framing for the η-metric.
  • Notebook §16.5 (chess_spectral_research_notebook.md, sec 16.5) — "if the pseudo-Hermitian pawn observables (Track B) matter, that validates the η-metric machinery beyond aesthetic rigor" — the empirical validation pathway.
  • Notebook §17.1 (chess_spectral_research_notebook.md, sec 17.1) — getQmExpectation bridge contract this ADR amends.
  • Track A qm_4d.build_H_piece_4 (qm_4d.py:331-390) — the existing NotImplementedError raised for pawns; v1.5 replaces this with the _herm variants.
  • Encoder pawn channels (encoder_4d.py:387-430) — channels 8-9 (FA_PAWN_W, FA_PAWN_Y) are encoder-side counterparts; ADR-001 §3.1 set their phase conventions.
  • ADR-001 — phase convention for pawn channels (theta_8, theta_9 use direction sign); cross-cuts with v1.6's η-metric in motivating direction-sensitive observables.
  • ADR-003 — per-channel U_move for FA_PAWN_W/Y; ADR-005's pawn observables use the same channel structure.
  • ADR-004 — Z_2 superselection; v1.5's _herm observables commute with J_op per the §3.3 argument.
  • Bender, C. M. and Boettcher, S. "Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry," Phys. Rev. Lett. 80, 5243 (1998). Foundational PT-symmetry paper; underwrites the v1.6 η = P_w construction.
  • Mostafazadeh, A. "Pseudo-Hermiticity versus PT Symmetry," J. Math. Phys. 43, 205 (2002). Equivalence between PT-symmetry and η-pseudo-Hermiticity; underwrites the η-inner-product machinery.
  • Mostafazadeh, A. "Pseudo-Hermitian Representation of Quantum Mechanics," Int. J. Geom. Methods Mod. Phys. 7, 1191 (2010). Review article for the v1.6 implementation reference.
  • Bender, C. M. "Making Sense of Non-Hermitian Hamiltonians," Rep. Prog. Phys. 70, 947 (2007). Review article for the v1.6 implementation reference.

8. Phase 4 Implementation Pointer

This is the design we will implement in Phase 4 B[5] (pawn observables, v1.5 tier). Roll-out:

  1. Phase 4 B[5a]: Build H_pawn_w_4_herm and H_pawn_y_4_herm as symmetric-projection lifts of the directed pawn predicates. ~60 LOC. Test surface ~80 LOC.
  2. Phase 4 B[5b]: Add NotImplementedError stubs for the suffix-less names + eta_pawn_* + inner_product_eta + expectation_eta. ~30 LOC.
  3. Phase 4 B[5c]: Update qm_4d module docstring with the v1.5/v1.6 contract per §3.5; cross-reference ADR-005. ~30 LOC.
  4. Phase 4 B[5d]: Update notebook §16.5 ablation question parameterization to reference pawn_<axis>_herm observables for v1.5.

Acceptance for v1.5.0 ship: - §4.4 v1.5 test surface passes. - Phase 3 PT-realness probe is run (deferring v1.6 promotion is conditional on its results). - Tournament ablation in Phase 6 reports pawn_w_herm and pawn_y_herm as separate observables; documented as Hermitian projections.

Phase 4 B[5e] (optional, deferred to v1.6): Promote the stubs to full implementations following the §3.3 design, contingent on Phase 3 PT-realness probe passing.