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ADR-003 Amendment: Strict-unitary tier restricted to same-orbit moves

Date: 2026-04-29 Branch: chess-spectral/adr-003-amendment-strict-unitary-orbit-restriction Triggering PR: #76 — Phase 4 milestone B1 (A_1 channel move-as-unitary). Amends: ADR-003-per-channel-move-transformation.md §3.1. Companion amendment: PHASE_3_5_PROBE_RESULTS.md (FIB → measurement-only).

1. Status

Accepted (2026-04-29; post-PR-#76 / commit 79ba509).

This is the second amendment to ADR-003. The first (Phase 3.5 Probe-2) moved FIB_SYM_½/3 to measurement-only re-encode. This one restricts the strict-unitary tier itself: the projector-sandwich construction U_c = P_c @ swap @ P_c is only sub-unitary on range(P_c) for same-orbit moves; cross-orbit moves drop to the measurement-only path that the Phase 3.5 amendment already established.

The amendment is documentation-level for v1.5 (Option (a) below). A v1.7+ refactor (Option (b)) is sketched but not adopted here.

2. Context

ADR-003 §3.1 sketches Pi_0 (the A_1 channel's per-move unitary) as a bare 4096-square swap matrix and asserts the verification claim

Pi_0 @ encode_4d_A1(pos_pre) = encode_4d_A1(pos_post) exactly, provided sig values for non-o, non-d squares are unchanged.

PR #76 implemented the projector-sandwich form U_a1 = P_A1 @ swap @ P_A1 (which is the construction that lives inside range(P_A1) and matches the encoder's A_1 subspace) and discovered that the verification claim fails for typical chess moves.

The B1 implementation ships as-is with two strict-xfail tests pinning the cross-orbit failure (test_qm_4d_dynamics_b1.py test_b1_subunitarity_cross_orbit_xfail, test_b1_re_encoding_match_cross_orbit_xfail) and a "Findings" block in the module docstring (qm_4d_dynamics.py "Phase 3.5 Probe-Result Amendments" section, lines 24-46) calling for this amendment before B3a launches. This document is that amendment.

3. Finding precision

3.1 Numerics from PR #76

Move geometry ‖U†U − P_A1‖_F ‖U·ψ_pre − ψ_post‖ Test outcome
Same-orbit non-capture 1e-14 1e-14 13/13 strict tests PASS
Cross-orbit non-capture (typical) 1e-2 to 3e-1 1e-2 to 3e-1 2/2 strict-xfail (trips on unexpected PASS)

Same-orbit residuals are at machine precision. Cross-orbit residuals exceed the 1e-10 strict-unitary tolerance by 8 to 12 orders of magnitude — this is a structural failure, not a numerical one.

3.2 Why: projector-sandwich commutativity

P_A1 is the A_1 orbit-mean projector on the B_4 action over Z_8^4. By construction, P_A1 @ e_s = (1/|orbit(s)|) Σ_{s' ∈ orbit(s)} e_{s'} for every basis vector e_s. Two facts follow algebraically:

  1. swap_{from,to} commutes with P_A1 iff from and to lie in the same B_4 orbit. When they share an orbit, swapping them permutes within an orbit-mean and leaves the orbit-mean invariant. Cross-orbit swap mixes two distinct orbit-means and cannot be absorbed into P_A1.

  2. The sandwich P_A1 @ swap @ P_A1 is sub-unitary on range(P_A1) exactly when [swap, P_A1] = 0. Otherwise U†U picks up off-diagonal mass from the non-commuting pair.

Z_8^4 has 35 B_4 orbits, of sizes {16, 64, 96, 192, 384} (sum = 4096). Most ordered pairs of distinct squares belong to different orbits; "same-orbit" is a sparse subset of legal chess moves. The strict-unitary path therefore covers only a small fraction of the move space if interpreted under ADR-003 §3.1's original scope.

3.3 Propagation to the rest of the strict-unitary tier

ADR-003 §3.1's strict-unitary roster is {A_1, STD4_X, STD4_Y, STD4_Z, STD4_W, FA_PAWN_W, FA_PAWN_Y}. Their constructions share the projector-sandwich shape:

  • STD4_X/Y/Z/W (B3a): Pi_a = D_a @ swap @ D_a^{-1} (rescaled swap; ADR-003 §3.1 channels 1-4). The rescaling is diagonal and commutes with the orbit projector iff the swap does — i.e., the same orbit dichotomy applies. Predicted to fail cross-orbit by the same mechanism; B3a's tests will quantify.
  • FA_PAWN_W/Y (B3b): block scatter on 16-mode (sx, sy, sz, *) blocks via W_ANTI_DCT / Y_ANTI_DCT (ADR-003 §3.1 channels 8-9). The relevant projector here is the antisymmetric DCT block selector, not P_A1; B3b's structure is orthogonal sub-channel decomposition (rather than orbit-quotient) and may avoid the dichotomy. To be confirmed by B3b's tests before declaring strict-unitary scope.
  • FD_DIAG (B3d/e, channel 10): rank-1 update on a piece-coefficient diagonal. Orbit issue does not apply (no projector sandwich; the rank-1 + renormalization path validated in Phase 3.5 Probe-2 is unchanged).

The "predicted same-orbit-only" outcome for STD4 is a load-bearing guess, not yet probed; B3a's test surface must include the same-orbit / cross-orbit dichotomy explicitly.

4. Decision

4.1 v1.5: adopt Option (a) — restrict + measurement-only fallback

ADR-003 §3.1's strict-unitary tier is restricted to same-orbit non-capture moves for channels A_1 and (predicted) STD4_*. Cross-orbit non-capture moves drop to the §3.3 measurement-only re-encode path established by the Phase 3.5 FIB amendment.

Rationale:

  • Ship velocity. Option (a) is configuration-level: which moves take which path. No architectural redesign. B2, B3a, B3b can proceed in parallel with this amendment landing.
  • Phase 3.5 precedent. The FIB amendment established measurement-only re-encode as an honest QM-formalism fallback (a measurement-then- evolution sequence). Re-using the same fallback for cross-orbit strict-unitary moves is conceptually consistent and re-uses the same infrastructure.
  • Test surface preserved. B1's xfail-strict tests already pin the cross-orbit residual. Cross-orbit measurement-only re-encode produces correct ψ_post without claiming strict unitarity, so the re-encoding-match assertion still holds — it just routes through a different code path. No test changes needed in B1; B3a/B3b's tests must mirror the same-orbit / cross-orbit split.
  • Zeno semantics. Cross-orbit transitions naturally fit ADR-002's Zeno-style evolution as projector-mediated jumps, which is exactly what the measurement-only re-encode implements operationally.

4.2 Acceptance criteria for the v1.5 strict-unitary tier

For each channel claiming strict-unitary status in v1.5, the implementation must:

  1. Detect the same-orbit / cross-orbit case at construction time (cheap: precomputed orbit_id_of_sq lookup; 35 orbits over 4096 squares).
  2. Build the projector-sandwich form (or its channel-specific analogue) only when the move is same-orbit; otherwise route to the measurement-only re-encode of that channel.
  3. Document the dichotomy in the module docstring and in the per-move return value (e.g., a flag indicating which path was taken).
  4. Test surface: ‖U†U − P_c‖_F < 1e-10 on a sample of ≥ 30 same-orbit non-capture moves; cross-orbit cases verified to route to measurement-only and recover ψ_post from encode_4d re-encoding within 1e-10.

4.3 Updated strict-unitary tier roster

# Channel v1.5 strict-unitary scope v1.5 fallback path Status
0 A_1 Same-orbit non-capture only Cross-orbit → measurement-only re-encode ✅ Verified by B1 (PR #76)
1 STD4_X Same-orbit non-capture only (predicted) Cross-orbit → measurement-only ⚠️ To verify in B3a
2 STD4_Y Same-orbit non-capture only (predicted) Cross-orbit → measurement-only ⚠️ To verify in B3a
3 STD4_Z Same-orbit non-capture only (predicted) Cross-orbit → measurement-only ⚠️ To verify in B3a
4 STD4_W Same-orbit non-capture only (predicted) Cross-orbit → measurement-only ⚠️ To verify in B3a
8 FA_PAWN_W Strict (orthogonal sub-channel — to confirm) Cross-orbit issue may not apply; capture → partial isometry per §3.1 ⚠️ To verify in B3b
9 FA_PAWN_Y Strict (orthogonal sub-channel — to confirm) As FA_PAWN_W ⚠️ To verify in B3b
10 FD_DIAG Rank-1 + renormalization (orbit issue N/A) Phase 3.5 path unchanged ✅ Probe-2 validated
5-7 FIB_SYM_½/3 — (measurement-only) Phase 3.5 amendment ✅ Probe-2 validated

The aggregate v1.5 unitarity claim weakens correspondingly: for non-capture moves, same-orbit moves are strictly unitary on the strict-tier channels; cross-orbit moves use measurement-only re-encode on those channels (with FA_PAWN possibly retaining strict behavior pending B3b).

5. Implications for downstream milestones

5.1 B3a (STD4_X/Y/Z/W)

  • Test surface MUST include the same-orbit / cross-orbit split. The test module should generate same-orbit and cross-orbit pairs (re-using B1's _gen_same_orbit_pairs / _gen_cross_orbit_pairs helpers or equivalents) and assert strict unitarity only on the same-orbit subset.
  • Implementation pattern follows B1. Routing logic at the u_move_std4_* builder level: detect orbit case, dispatch to projector-sandwich (same-orbit) or measurement-only re-encode (cross-orbit). The dispatch should be a shared helper across A_1 and STD4_*.
  • D_a boundary case (ADR-003 §6 Open Q4). Where coord_resid[a, *] has zero entries (256 central-axis squares per channel), the similarity transform is ill-defined. ADR-003's pad-with-identity fix remains applicable; orthogonality to those rows means the orbit dichotomy plays no role there.

5.2 B3b (FA_PAWN_W/Y)

  • Verify whether the orbit dichotomy applies. FA_PAWN's projector is the W/Y antisymmetric DCT block selector, not P_A1. Construct a probe analogous to B1's cross-orbit test and measure ‖U_pawn† @ U_pawn − P_pawn‖_F on cross-block pawn moves. If the residual is at machine precision regardless of pawn-block geometry, FA_PAWN_W/Y stays strict-unitary unconditionally for non-captures; if not, this amendment's restriction extends to FA_PAWN with a similar measurement-only fallback.
  • Capture path unchanged. FA_PAWN's capture handling remains the partial-isometry path of ADR-003 §3.1 + ADR-002 renormalization, independent of the orbit issue.

5.3 B5 (capture-move dispatcher)

  • The capture-move path was already a partial-isometry / rank-1-update case under ADR-003. No additional restriction from this amendment — capture handling routes through ADR-002's renormalization regardless of orbit geometry. The amendment narrows the non-capture strict- unitary path; capture handling is orthogonal.
  • The dispatcher in apply_move_qm (per qm_4d_bridge.py) needs two conditions at routing time: capture-vs-non-capture (existing) AND same-orbit-vs-cross-orbit (new). The four-way table is well-defined per channel.

5.4 ADR-002 (time-evolution semantics)

No change required. The Zeno-style evolution and capture renormalization are unchanged. Cross-orbit measurement-only re-encode fits naturally under the Zeno framing as a projector-mediated jump between channel subspaces.

5.5 ADR-004 (Z_2 superselection)

No change required. Phase 3.5 Probe-4's amendment moved Z_2 sector flip to state_to_psi's sign multiplier; the orbit-restriction here is upstream of that and does not interact.

6. Option (b) — orbit-quotient refactor (deferred to v1.7+)

The v1.7+ candidate refactor noted in B1's findings docstring works on the 35-dim orbit quotient rather than the full 4096-dim channel space. range(P_A1) is naturally 35-dimensional (one mean value per B_4 orbit); a move from o (in orbit a, size n_a) to d (in orbit b, size n_b) induces the deterministic quotient-space update

m_a → m_a − sig[o] / n_a
m_b → m_b + sig[o] / n_b

— a non-unitary translation by (−v/n_a, +v/n_b) on the two affected orbit-mean components, identity on the other 33 orbits.

Pros: * Geometrically natural. Uses the full B_4-orbit reduction structure rather than embedding it in a redundant 4096-dim representation. * Eliminates the dichotomy: same-orbit and cross-orbit moves are both expressible in the same closed form (same-orbit reduces to m_a → m_a because the from / to terms cancel). * Could simplify the spectral identity (Pre-flight 3) since the orbit quotient is the natural reduction modulo B_4.

Cons: * Semantic shift in the API. apply_move_qm returns a per-channel ψ in C^4096; switching one channel to live in the 35-dim quotient breaks the uniform per-channel interface. Either every consumer learns about the quotient embedding, or there's a 4096↔35 lift layer that re-introduces the projector machinery. * Reconciling with the channel PVM. Notebook §15.2(3) frames the 11-channel decomposition as a built-in PVM on the encoder. The quotient view changes which Hilbert space the A_1 PVM lives on (35-dim quotient vs. range(P_A1) ⊂ C^4096). Channel-projector observables in Phase 6's QM evaluator need re-derivation. * Substantial refactor of qm_4d.py / qm_4d_dynamics.py. The existing P_A1 / encoder integration would need a parallel quotient-space implementation that interoperates with Track A's kinematic operators. ~600-1000 LOC; defers v1.5 ship.

Recommendation: Treat Option (b) as a v1.7+ research direction rather than a v1.5 production change. Revisit if Phase 6's QM evaluator empirically benefits from the orbit-quotient picture, or if the v1.7+ Stinespring dilation work (already deferred per ADR-002) makes the larger refactor opportune. Until then, the v1.5 strict-tier ships under Option (a) and accepts the same-orbit-only narrowing as documented.

7. References

  • ADR-003 (original): ADR-003-per-channel-move-transformation.md §3.1 — the strict-unitary tier construction this amends.
  • Phase 3.5 probe results: PHASE_3_5_PROBE_RESULTS.md — first ADR-003 amendment (FIB → measurement-only); structural template for this document.
  • PR #76: chess-spectral Phase 4 B1: A_1 channel move-as-unitary + bridge stub — implementation that surfaced the finding; merged at commit 79ba509.
  • B1 implementation: qm_4d_dynamics.py — module docstring "Phase 3.5 Probe-Result Amendments" section (lines 24-46) + u_move_a1 builder.
  • B1 tests: test_qm_4d_dynamics_b1.py — module docstring "Findings" block (lines 33-59); test_b1_subunitarity_cross_orbit_xfail, test_b1_re_encoding_match_cross_orbit_xfail (strict-xfail markers pinning the cross-orbit residual at lines 287-326 and 407-445).
  • ADR-001 / ADR-002 / ADR-004 / ADR-005: unchanged by this amendment. See README.md for the full set.