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+# 4Q + Off-Axis Measurement Extension — Specification
+
+Status: ACCEPTED by Petrus 2026-07-07 (decisions in §7); technical review by
+claudeMB pending on this PR. Gauge group: **SU(2)** throughout —
+same as the existing production pipeline and the PGM program this extends.
+Target lattice: 24⁴ at β=2.4 (dev runs on 16⁴/8⁴), on the existing
+generation → measurement → live-dashboard chain.
+
+## 1. Physics goals
+
+1. **Off-axis 2Q potential** V(R⃗) on non-axis separations — fills the gaps
+   between integer R, tests restoration of rotational invariance (lattice
+   artifacts show as scatter off the Cornell curve at small R), and anchors
+   the same R values the tetrahedron uses (R = d√2).
+2. **4Q binding energies** B(geometry, d) = E₄Q − E₂Q,pair for
+   **square** (d = 2…8) and **tetrahedron** (d = 2…6, edge R = d√2 ≈ 2.8…8.5)
+   geometries — the multiquark question proper: is the 4-body ground state
+   below two independent flux tubes, and by how much.
+3. (Later phase) 4Q flux distributions on the symmetry-reduced wedges — where
+   does the field rearrange when binding turns on.
+
+## 2. Geometries and quark positions (lattice sites)
+
+| Geometry | Positions | Scan | Point group | Reduction |
+|---|---|---|---|---|
+| off-axis 2Q | 0 → R⃗ for the canonical PGM class list (§8) | \|R\| ≤ ~8.5 | axial C₂ᵥ-type | orbit-avg |
+| square | (0,0,0),(d,0,0),(0,d,0),(d,d,0) | d = 2…8 | D₄h (order 16) | 1/16 |
+| tetra | (0,0,0),(d,d,0),(d,0,d),(0,d,d) | d = 2…6 | T_d (order 24) | 1/24 |
+
+All tetra vertices are lattice sites for every integer d (verified). The
+fundamental-domain conditions in `tools/geometry_viewer.html` are verified
+exact (20k-orbit brute-force test, 2026-07-07) and are reused verbatim for
+flux sampling.
+
+**Weights (hard requirement):** flux/observable sums over a reduced wedge must
+weight each site by `|G| / |stab(site)|` — NOT flat ×16/×24. On-wall sites are
+14–50% of the wedge; flat weighting overcounts sums by ~56% at d=6. The
+reduction helper must return `(site, weight)` pairs and a unit test must check
+`Σ weights == full-volume count` for every geometry and d (both parities).
+
+## 3. Operators
+
+### 3.1 Off-axis 2Q
+Wilson loop with spatial legs built as **PGM symmetrized paths** (corrected
+2026-07-08 — the earlier draft said "canonical staircase, fixed lexicographic
+order"; the PGM papers did not use a single staircase):
+
+- **Planar (x,y,0)**: equal-weight average of the two L-shaped routes,
+  P(x,y) = ½[P(a) + P(b)] (hep-lat/9404004 §3.1 fig. 2; identical statement
+  in hep-lat/9508002 §2).
+- **3D (x,y,z)**: symmetric cuboid-edge combination
+  P(x,y,z) = ⅓[P(x)P(yz) + P(y)P(xz) + P(z)P(xy)], each planar factor itself
+  the ½(a+b) L-average — effectively the 6 symmetric routes along cuboid
+  edges (hep-lat/9404004 §3.1; UKQCD hep-lat/9209007 used the same 2-/6-route
+  symmetric sums).
+
+Symmetrization keeps the operator invariant under the coordinate
+interchanges that fix R⃗ and improves ground-state overlap over any single
+staircase. Same smearing (12-step spatial), same T set {1…6}, same V_eff
+plateau extraction as on-axis. Cubic-orbit averaging on top: measure all
+distinct orientations of each R⃗ class per time direction and average before
+the T-fit — same statistical role that time-direction averaging plays today.
+
+### 3.2 4Q correlation matrix (PGM formalism)
+Four static quarks admit two independent pairings into SU(2) singlets:
+A = (Q₁Q₂)(Q₃Q₄), B = (Q₁Q₃)(Q₂Q₄). Measure the 2×2 matrix
+
+```
+W_ij(T) = ⟨ P_i(0) · [4 temporal lines] · P_j(T)† ⟩,  i,j ∈ {A,B}
+```
+
+- **Diagonal W_AA, W_BB**: products of two rectangular (or staircase) Wilson
+  loops — existing loop machinery, run twice per pairing.
+- **Off-diagonal W_AB**: one closed contour — spatial connectors of pairing A
+  at t=0, the four temporal lines, spatial connectors of pairing B at t=T.
+  New but small: the path builder already composes arbitrary link products;
+  this is a single trace over a longer path.
+- Extract E₄Q via the generalized eigenvalue problem on W(T)/W(T₀) (2×2 GEVP,
+  closed-form for 2×2 — no numerics dependency), then the same V_eff plateau
+  fit per eigenvalue. Binding: B(d) = E₄Q^(0)(d) − 2·V(R_pair) with V from the
+  SAME configs (correlated errors partially cancel — jackknife over configs).
+
+Note the square has a third pairing (diagonals) that mixes only weakly; PGM
+used the 2×2 basis and so do we (documented limitation).
+
+## 4. Estimator reuse (nothing new invented)
+
+- 12-step spatial smearing on connectors (ground-state overlap).
+- Multilevel blocks × multihit on temporal lines — identical mechanics;
+  the 4Q observable is linear in each temporal line's sub-average exactly like
+  2Q, so blocks×hits factorization carries over unchanged.
+- Full-volume translation averaging via the existing sampler.
+- Time-direction (×4) orientation averaging.
+- Same live JSON/JSONL schema: loop keys extended to
+  `G<geom>_d<d>_P<ij>_T<t>` (e.g. `Gsq_d4_PAB_T3`); off-axis 2Q keys
+  `Rv<class>_<n>_T<t>` (e.g. `Rv110_3_T4`). Dashboard plateau/loop panels key
+  off these transparently (R-selector becomes a geometry+size selector).
+
+## 5. Implementation plan (phased, each lands runnable)
+
+- **P1 — off-axis 2Q** (smallest diff, validates the path builder):
+  symmetrized L/cuboid spatial paths (§3.1) over the §8 class list + orbit
+  averaging + dashboard keys. Gate: off-axis V(R) points lie on the Cornell
+  fit through on-axis points within 2σ.
+- **P2 — square 4Q**: pairing connectors, 2×2 matrix, GEVP, binding output.
+  Gate: at large pair separation the ground state → 2·V(d) within errors;
+  W_AB → 0 as separation grows.
+- **P3 — tetra 4Q**: same machinery, tetra positions/pairings.
+- **P4 — 4Q flux profiles** on the verified wedges with orbit weights.
+
+Each phase: parity test at 8⁴ against a slow reference implementation
+(direct link-product evaluation, no multilevel) before production, same
+discipline as PR #24's bit-identical check.
+
+## 6. Cost estimate (24⁴, per config)
+
+Off-axis 2Q, measured on 8⁴ (PR #31): full 28-class list ≈ 2.7× current 2Q
+loop cost (124 symmetrized passes vs 108 rectangles per tdir); the
+production diagonal subset (§7.5) is 24 passes ≈ 0.5×. (The original 1.5×
+guess for the full list was wrong.)
+4Q square/tetra: 3 matrix elements × 7 (5) sizes × 6 T × 4 tdirs ≈ 2–3× the
+2Q loop budget at equal multilevel settings — well inside the current
+per-measurement envelope; flux phases dominate cost and get the 16×/24×
+wedge reduction. Run tiering follows the leak situation (2x1 now, 8x2 when
+the reuse-pool/census-v3 or process-split lands).
+
+## 7. Decisions (Petrus, 2026-07-07, thinkoff-development)
+
+1. **Off-axis classes: use the exact PGM set.** DONE 2026-07-08 — class list
+   extracted verbatim from the PGM/UKQCD paper tables (full provenance in
+   `docs/pgm_offaxis_extraction.md`) and encoded as the canonical list in §8;
+   the (n,n,0)/(n,n,n)/(2n,n,0) placeholder is replaced. The extraction also
+   corrected §3.1: PGM used symmetrized L-shaped / cuboid-edge paths, not a
+   lexicographic staircase.
+2. **Square pairings: keep PGM's 2×2 basis.** No third (diagonal) pairing; the
+   omission stays a documented limitation as in §3.2.
+3. **Scope: all of it.** On-axis 2Q (existing) + off-axis 2Q + square + tetra
+   are all in scope — the P1→P4 order in §5 is a landing sequence, not a
+   selection.
+   **Refined 2026-07-08** ("We only measure diagonal off axis for
+   subtraction from 4q potentials"): PRODUCTION off-axis = the diagonal
+   classes (n,n,0), n = 1…8, only. These are exactly the R = d√2 pair
+   potentials entering B(d) = E₄Q − 2·V(R_pair): every tetra vertex pair is
+   a (d,d,0)-type separation, and the square's 2×2 pairings are on-axis
+   sides already measured. Implemented as `--offaxis-classes diag` (the
+   default; `su2_offaxis.DIAGONAL_CLASSES`). The full §8 list remains
+   available via `--offaxis-classes all` for rotational-invariance studies
+   — it is machinery, not production scope.
+4. **GPU-resident throughout** ("implement all the listed measurements using
+   gpu fully"): every new measurement inner loop — symmetrized/connector path
+   products, 4Q contour traces, multilevel sub-averages — runs on the GPU
+   (Metal), same as the existing plaquette/Wilson-loop kernels. CPU is
+   orchestration, GEVP (closed-form 2×2), fits, and I/O only. The 8⁴ parity
+   gates in §5 compare GPU results against the slow CPU reference.
+
+## 8. Canonical off-axis class list (exact PGM set, extracted 2026-07-08)
+
+One representative per cubic class, lattice units; sources verbatim from the
+papers (full extraction with quotes and caveats:
+`docs/pgm_offaxis_extraction.md`). Orientation averaging (§3.1) generates the
+full orbits.
+
+```
+OFF_AXIS_CLASSES = [
+  # -- planar L-shaped grid: Green-Michael-Paton-Sainio hep-lat/9301006
+  #    Table 1 (beta=2.4, 16^3x32, MC-measured; y = 1..3, x <= 6)
+  (1,1,0), (2,1,0), (2,2,0), (3,1,0), (3,2,0), (3,3,0),
+  (4,1,0), (4,2,0), (4,3,0), (5,1,0), (5,2,0), (5,3,0),
+  (6,1,0), (6,2,0), (6,3,0),
+  # -- diagonal series (d,d,0): 9508002 Tables 4-5 for d=4,5;
+  #    9804004 flux paper measured sqrt(2)*R diagonals R=2,4,6,8
+  (4,4,0), (5,5,0), (6,6,0), (8,8,0),
+  (7,7,0),                     # OURS (not in a PGM table): completes the
+                               # square-d=7 / tetra-range anchor
+  # -- near-diagonal: 9508002 Tables 4-5
+  (5,4,0), (6,5,0),
+  # -- 3D vectors: Green-Michael-Sainio hep-lat/9404004 Tables 5-6
+  (1,1,1), (1,2,1), (1,3,1), (1,4,1), (2,1,2), (2,3,2),
+]
+```
+
+Notes anchored in the extraction:
+
+- The diagonals (d,d,0) carry |R| = d√2 — exactly the tetra edge lengths
+  (§1 goal 1), so tetra binding at edge R uses a directly measured pair
+  potential, PGM-style (9804004 measured the diagonals precisely because
+  interpolating on-axis data violates rotational invariance at these β).
+- PGM's tetra coordinates ((0,0,0),(r,0,d),(0,d,d),(r,d,0) with r=d,
+  hep-lat/9412029) are the SAME four sites as our §2 tetra row — consistency
+  confirmed, no change needed.
+- PGM measured more geometry families (rectangles, tilted rectangles,
+  linear, quadrilateral, non-planar; 3×3 bases for the last three) — out of
+  scope here per §7.3, listed in the extraction doc as future extensions.
+- Caveat from the sources: hep-lat/9404004 says its Table 5 is only a subset
+  of all 2Q potentials produced, and hep-lat/9608147 refers to a 29-vector
+  two-body list that is not printed in any of the papers. If the original
+  run lists still exist offline, they supersede this reconstruction —
+  flagged to Petrus.
diff --git a/docs/pgm_offaxis_extraction.md b/docs/pgm_offaxis_extraction.md
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+# PGM off-axis 2Q separations and 4Q geometries — source extraction
+
+Extracted 2026-07-08 from the arXiv LaTeX sources of the Green / Michael /
+Pennanen (PGM) SU(2) series, for §8 of `4q_measurement_spec.md`. Quotes are
+verbatim from the sources; everything not directly verifiable is flagged at
+the end. Non-PGM candidate hep-lat/9509064 (de Forcrand et al., cooling) was
+checked and discarded; hep-lat/9709124 and hep-lat/9804003 are model papers
+with no new lattice vector lists (9709124 confirms the fitted data set, see
+below).
+
+## hep-lat/9209007 — UKQCD (Booth, Hulsebos, Irving, McKerrell, Michael, Spencer, Stephenson), "SU(2) Potentials from Large Lattices", Nucl.Phys. B394 (1993) 509
+
+β=2.85, 48³×56. Section "Off-axis potentials": "We choose vector separations
+(R_x,R_y,R_z) with 0 ≤ R_i ≤ 3a. For the 2 and 3 dimensional paths, we sum
+over all 2 (6) symmetric routes along the edges of the 2 (3) dimensional
+hypercuboid." Operators via Albanese-type blocking, c=2, 100 iterations,
+8 configs.
+
+Table 3 vector list (R_x,R_y,R_z)/a — every distinct cubic class with
+components ≤ 3: (1,0,0), (1,1,0), (1,1,1), (2,0,0), (2,1,0), (2,1,1),
+(2,2,0), (2,2,1), (2,2,2), (3,0,0), (3,1,0), (3,1,1), (3,2,0), (3,2,1),
+(3,2,2), (3,3,0), (3,3,1), (3,3,2), (3,3,3). On-axis companion table: even
+R = 2..24.
+
+## hep-lat/9209019 — Green, Michael, Paton, "Multi-quark energies in QCD", Nucl.Phys. A554 (1993) 701
+
+β=2.4, 16³×32. Off-axis v measured "directly by Monte Carlo simulation using
+L-shaped separations of the two static sources", only "up to (x/a=3, y/a=3)
+for z/a=0"; for x/a ≥ 4 off-axis values from the lattice-Coulomb
+parametrization v_L = −(e/r)_L + b_S·r + v_0 (e=0.234, b_S=0.0736, v_0=0.542),
+normalized at r=(6,0,0).
+
+Table 1 measured off-axis (x,y) [z=0]: (1,1), (2,1), (2,2), (3,1), (3,2),
+(3,3); on-axis (1,0)–(7,0). Off-axis paths: single fuzzing level 16, c=4.
+4Q (Tables 2, 4): planar squares/rectangles (d,r) = (1,1),(1,2),(1,3),(1,4),
+(2,2),(2,3),(3,3),(3,4),(4,4),(5,5),(6,6),(7,7); 2×2 basis {A,B}.
+
+## hep-lat/9301006 — Green, Michael, Paton, Sainio, "Multi-quark Energies in SU(2) Lattice Gauge Theory", Int.J.Mod.Phys. E2 (1993) 479
+
+β=2.4, 16³×32 (720 meas.); scaling check β=2.5, 24³×32. "The off-axis
+results (y ≠ 0) correspond to L-shaped separations of the static sources."
+Ground E₀ AND first-excited E₁ tabulated.
+
+Table 1 measured (x,y) [z=0]: on-axis (1,0)…(6,0); off-axis (1,1), (2,1),
+(2,2), (3,1), (3,2), (3,3), (4,1), (4,2), (4,3), (5,1), (5,2), (5,3), (6,1),
+(6,2), (6,3) — y = 1,2,3 for x ≤ 6. ((4,4),(5,5),(6,6),(7,x),(8,0),(9,0)
+entries are parametrization-only, no MC value.)
+
+4Q: squares (1,1)–(7,7) on both lattices (Tables 2–3); rectangles (Table 4);
+colinear (Table 5): (r,d) = (2,1),(3,1),(4,1),(3,2),(4,2),(5,2),(4,3),(5,3),
+(6,3).
+
+## hep-lat/9404004 — Green, Michael, Sainio, "Four-quark Binding Energies from SU(2) Lattice Monte Carlo", Z.Phys. C67 (1995) 291 — the "six geometries" paper
+
+β=2.4, 16³×32.
+
+**Path construction (Sec. 3.1, verbatim):** on-axis = straight fuzzed paths
+at 3 fuzzing levels (12, 16, 20; c=4) → 3×3 variational basis. Planar
+off-axis V₂(x,y): "the two paths P_i(a,b) shown in fig. 2 are combined with
+equal weight, P_i(x,y) = ½[P_i(a) + P_i(b)]" — the symmetrized L-shaped
+pair. 3D V₂(x,y,z): "combinations around the sides of a cuboid",
+P_i(1→4,xyz) = ⅓[P_i(x)P_i(yz) + P_i(y)P_i(xz) + P_i(z)P_i(xy)], each 2D
+factor itself the ½(a+b) average — effectively 6 symmetric cuboid-edge
+routes.
+
+Table 5 (2Q; explicitly "only a few of the 2-quark potentials produced in
+this work"): on-axis (1,0)…(8,0); off-axis (1,1,0), (1,1,1), (1,2,0),
+(1,2,1), (1,3,0), (1,3,1), (1,4,0), (1,4,1).
+Table 6 (rotational-invariance pairs, additional measured vectors): (4,3,0)
+vs (5,0,0); (2,1,2) vs (3,0,0); (3,3,0); (1,4,1); (4,1,0); (2,3,2). Text
+also cites V₂(3,0,0) ≈ V₂(2,2,1).
+
+4Q geometries (two "mesons" of equal length d):
+- Rectangles (d,r) = (1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),
+  (4,4); Large Squares add (5,5),(6,6),(7,7) — Table 7. 2×2 basis.
+- Tilted Rectangles — Table 8, d,(x,y): 2,(1,1); 2,(2,1); 3,(2,2); 3,(3,1);
+  4,(3,2); 4,(4,1); 5,(4,3); 5,(5,1); 6,(5,3); 6,(6,1). 5,(4,3) is a 5×5
+  square tilted off-axis (rotational-invariance test vs on-axis 5×5).
+- Linear (d,r) = (1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3) —
+  Table 9.
+- Quadrilateral: same (d,r) set — Table 9; one meson rotated π/2 in-plane.
+- Non-Planar: all 16 combinations (1,1)–(4,4) — Table 10; one meson rotated
+  π/2 out of plane (diagonals give the (2,1,2), (2,3,2), (1,4,1) entries of
+  Table 6). 3×3 basis {A,B,C} for Linear, Quadrilateral, Non-Planar.
+
+## hep-lat/9412029 — Green, Lukkarinen, Pennanen, Michael, Furui (Lattice'94), tetrahedral geometry
+
+β=2.4. Verbatim: "the four quarks have the coordinates (0,0,0), (r,0,d),
+(0,d,d) and (r,d,0)" — tetrahedron when r=d. Table 1: (d,r) = (1,0),(1,1),
+(1,2),(2,1),(2,2),(2,3),(3,2),(3,3),(3,4),(4,3),(4,4),(4,5).
+For r=d these are the same four sites as the spec §2 tetra row.
+
+## hep-lat/9508002 — Green, Lukkarinen, Pennanen, Michael, "A Study of Degenerate Four-quark states...", Phys.Rev. D53 (1996) 261
+
+β=2.4, 16³×32. Square of side d parallel to yz-plane at distance r. Runs:
+(i) r = d, d±1 for d=2..5 (d=1: r=1,2), 2208 meas.; (ii) r=1, d=2..5,
+3008 meas.; (iii) squares/rectangles re-run with full 3×3 basis.
+Path statement (Sec. 2): for V(x,y) "the appropriate path of links is then
+constructed as the average of the two most simple paths connecting x and y —
+each consisting of one straight section along the x and y axes."
+
+2Q values quoted (Tables 4–5, V₁ columns), planar (x,y): (1,1)=0.4885(1),
+(2,2)=0.6689(4), (3,3)=0.7974(8), (4,4)=0.9102(15), (5,5)=1.017(2);
+(2,1)=0.6023(3)/0.6021(3), (3,1)=0.6992(5), (4,1)=0.7841(10),
+(5,1)=0.8652(8), (3,2)=0.7421(6), (4,3)=0.8589(11), (5,4)=0.967(2),
+(6,5)=1.072(4). These double as parity targets for P1 at 16³×32 β=2.4.
+
+## hep-lat/9608147 — Pennanen, "Continuum extrapolation of energies of a four-quark system", Phys.Rev. D55 (1997) 3958
+
+Squares (2a…6a) and tilted rectangles at β=2.35, 2.4 (16³×32), 2.45
+(20³×32), 2.5 (24³×32), 2.55 (26³×32); 2-body-only at β=2.3. On-axis fits
+r/a=2..6. "29" two-body potentials enter the TR analysis; trcs.eps labels
+(x,y,z) = (3,2,0), (3,3,0), (5,4,3), (6,6,1) [figure labels, not a table].
+betr.eps TR set at β=2.5: (3,2,2), (3,3,1), (4,3,2), (4,4,1), (5,4,3),
+(5,5,1), (6,5,3), (6,6,1) in d,(x,y) form. The full 29-vector list is NOT
+printed in the paper (text refers to the companion for more).
+
+## Flux-distribution papers
+
+- hep-lat/9610011 (Green, Michael, Spencer, PRD 55 (1997) 1216): 2Q on-axis
+  only, R = 2, 4, 6, 8 (ground A₁g and excited E_u).
+- hep-lat/9705033 / 9708012 (Pennanen, Green, Michael): 2Q on-axis only,
+  R = 2,3,4,6,8 (β=2.4), R = 2,3,4,6,12 (β=2.5).
+- hep-lat/9804004 (Pennanen, Green, Michael, PRD 59 (1999) 014504), β=2.4,
+  verbatim: "The quark distances we measured were R=2,4,6,8. For all these
+  values... a) two quarks on a lattice axis separated by R lattice units,
+  b) two quarks on an axis diagonal with respect to the lattice axis and
+  separated by √2·R units [(2,2,0),(4,4,0),(6,6,0),(8,8,0)] and c) four
+  quarks at the corners of a square with side length R." Diagonal tubes
+  measured directly rather than interpolated, citing measured
+  rotational-invariance violations.
+- hep-lat/9709124 (model paper) confirms the fitted data set: "15 Tetrahedra,
+  6 Squares, 12 Rectangles (including Tilted), 4 Quadrilaterals, 9 Non-Planar
+  and 4 Linear", 100 energies (E₀,E₁), 16³×32, β=2.4, a=0.119(1) fm; configs
+  with flux links shorter than 2a excluded.
+
+## Caveats / not directly verified
+
+- Exact quark coordinates for Tilted Rectangles, Quadrilateral, Non-Planar
+  in 9404004 are defined only via its Fig. 1 (figures absent from the LaTeX
+  source); the side/diagonal identifications above are reconstructed from
+  the in-text checks ((5,0,0)↔(4,3,0), (3,0,0)↔(2,1,2), path lengths
+  a + √13·a) and 9608147's figure macros. The measured V₂ vectors themselves
+  are verbatim from Tables 5–6.
+- 9608147's complete 29-vector two-body list and 9508002's Fig. 1 were not
+  readable as text; labels quoted from EPS internals where noted.
+- 9404004 states Table 5 is a subset of all 2Q potentials produced; no
+  complete list exists in any of these papers. Original offline run lists,
+  if they survive, supersede this reconstruction.