Mixed-boundary (wire/slab) Green’s function for the low-D CCM four-center¶

The aiccm2026dev-a side of joint deliverable item 6 (the open construction of the converged -a/-b theory result, docs/manuscripts/aiccm_a_position.md §0.5). Derived Hamiltonian-first: define the mixed-boundary Poisson problem, then read both the Coulomb kernel and the exchange-q=0 seam off the same Green’s function G – not a minimum-image truncation. Built as -a’s own implementation. A future χ-CCM comparison must bind the separately derived B Hamiltonian; current χ-CCM-B 1-D/2-D absolute energies fail closed.

D90 construction boundary. This document describes an A-line mixed-boundary wire construction/control. It is not currently classified as union-and-weight Γ-CCM because no binding from this Hamiltonian to the Wigner–Seitz integral-weighting map has been derived. A-line namespace ownership and real-Gamma evaluation of the finite cluster do not confer Γ-CCM identity. This wire/slab Green-function work also does not define a χ-CCM kernel, and a same-Green-function representation control would not prove equality of the two constructions.

Status: M1 (this document) → M2a/M3a scalar kernels, M2b/M4/M5 d=1 (wire) DONE (2026-07-04): the full wire four-center + V_ne/E_nn + SCF (run_ccm_rhf_wire) with the neutral-cell Hartree gauge cancellation gated and the M5 4c-vs-RI gap collapsed to machine ε. d=2 (slab) four-center + the wire-Madelung exchange seam remain. Progress tracked in handovers/HANDOVER_AICCM_FOLLOWON.md.

0. Why the 3-D neutral control is wrong for genuine 1-D/2-D¶

The historical neutral-torus control object is the charge-neutral 3-D-torus Ewald Green’s function v_E^{3D} – the unique solution of the torus Poisson equation with a uniform compensating background −1/V_c over the 3-torus 𝕋³ = ℝ³/L_c, with the G=0 mode dropped (python/vibeqc/periodic/ccm/neutral.py; ccm_neutral_cderi is its Γ density fit). It was previously mislabelled as a shared Γ-CCM/χ-CCM production object; D89 withdraws that claim. A vacuum-padded 1-D chain or 2-D slab is modelled with this same 3-D kernel by inflating the open directions with vacuum. That is a supercell model, not the right Hamiltonian: the neutralizing background is spread through the vacuum, the long-range field in the open directions is the wrong functional form, and the resulting four-center carries a system-size-dependent gauge offset that is not an RI fitting error. Historical pre-guard comparisons showed four-center-vs-RI gaps of hundreds of kJ/mol in low-D, including a vacuum-padded B H₄ control with 4c - RI = 22.2 mHa. Those B values are defect evidence, not model or accuracy numbers (Q-lowD). Current χ-CCM-B blocks every lower-dimensional SCF backend before SCF.

The honest fix is a Green’s function that is periodic in the d lattice directions and open (decaying, free-space) in the 3−d transverse directions, with the neutralizing background a uniform charge only over the periodic directions (a sheet for d=2, a line for d=1). This is the mixed-boundary kernel G^{(d)}. The 3-D kernel is recovered as the transverse-periodic sum of G^{(d)} – the hard, dependency-free validation anchor (§5).

0.1 Correction (2026-07-10) - the low-D production kernel was not `v_E^{3D}`¶

The paragraph above understates the defect, and the correction changes how low-D numbers must be read. A vacuum-padded low-D cell was not being modelled with the 3-D torus kernel v_E^{3D}. rsgdf_dense_g_mesh (and rsgdf_g_mesh) size the reciprocal mesh over the periodic axes only (axis < dim) and pin every non-periodic axis at G_⊥ = 0; consumers then apply 4π/|G+q|²/V to that mesh. Dropping the transverse Fourier components replaces each AO-pair density by its transverse average, so the kernel is not 1/r at all but a transverse-uniform sheet term ∝ 1/V, which vanishes as the vacuum grows.

Measured (H₂ chain, dim=1, sto-3g, 6-bohr period): |g_neutral|_max · V = 92.46 is constant across transverse vacuum D = 12/18/24 bohr, while the wire four-center is D-invariant at 1.895512; the two differ by 14 % of the wire scale even after removing the best-fit c·S⊗S. At SCF level the dim=1 route ran -6.917 → -6.568 Ha for D = 12 → 30 bohr, and the production multi-k run_krhf_periodic_gdf ran -3.458 → -3.324 Ha on the same sweep.

So the low-D four-center-vs-RI gap is not merely “a gauge offset rather than an accuracy error”. Combined with the bare (conditionally convergent) V_ne/E_nn lattice sums the dim<3 route also used, the total additionally diverged with the nuclear lattice-sum cutoff. Both the 3-D-torus route and the multi-k GDF therefore fail closed for dim < 3 as of 2026-07-10; the gauge-consistent low-D Hamiltonians (run_ccm_rhf_wire, the four-center route) are selected by name, never substituted silently - they are different operators, coinciding only in the non-interacting limit. See tests/test_ccm_lowd_gauge_consistency.py and handovers/HANDOVER_AICCM_FOLLOWON.md finding #6.

1. The mixed-boundary Poisson problem (Hamiltonian-first)¶

Let the cell be periodic in d directions with in-plane/along-axis cell area / length A (d=2) or L (d=1), and open in the transverse coordinate(s), written z (a scalar for d=2; ρ=(x,y) for d=1). Write r = (R∥, z) with R∥ the periodic coordinates. The electrostatic kernel G(r,r') is the periodic solution of Poisson’s equation with the periodicity’s neutralizing background:

−∇² G(r, 0) = 4π [ δ^{(3)}(r) − b_d(z) ] ,      (Gaussian units, e=1)

where b_d is the compensating background that makes the source charge-neutral per periodic cell, smeared uniformly over the periodic directions and localized in the transverse coordinate exactly as the unit point source is:

d=3: b_3 = 1/V_c (uniform over the 3-torus) → the production v_E^{3D} (G=0 dropped). The transverse background is absent (no open direction).
d=2 (slab): b_2(z) = δ(z)/A – a neutralizing sheet in the source plane. The in-plane average density vanishes; the z-profile is a δ at the source.
d=1 (wire): b_1(ρ) = δ^{(2)}(ρ)/L – a neutralizing line along the axis.

Boundary conditions (the part that is declared, not derived – they fix the otherwise-undetermined harmonic piece):

Periodic in R∥ (Born-von-Kármán): G(R∥+T, z) = G(R∥, z) for lattice T.
Open / decaying in the transverse direction(s): the oscillatory (G∥≠0 / G_z≠0) Fourier components decay as e^{−|G∥||z|} (d=2) / K₀(|G_z|ρ) (d=1); the zero transverse-mean (G∥=0/G_z=0) channel is the field of the neutralizing sheet/line and is only conditionally convergent – its linear-in-|z| (d=2) / logarithmic-in-ρ (d=1) growth is the low-D analogue of the 3-D G=0 conditional convergence (§4). A finite constant C (the gauge) is fixed by a reference (z→0 / ρ→ρ₀).
Dipole / asymptotic BC (polar systems). A slab with a net normal dipole M_z ≠ 0 (or a wire with a net transverse dipole) carries a dipole-layer term in the conditional channel – a discontinuity in the asymptotic potential across the slab, the low-D analogue of the surface term in the 3-D conditionally convergent lattice sum. It must be declared and is the physical content of the M_z≠0 correction; see §4 and the known limitation in §6.

2. The 2-D slab kernel `G^{2D}` (periodic in the plane, open in `z`)¶

Solving §1 for d=2 in a plane-wave basis over the in-plane reciprocal lattice {G∥} (area A per cell), each G∥≠0 mode solves (|G∥|² − ∂_z²) ĝ_{G∥}(z) = 4π δ(z), giving the decaying Green’s function ĝ_{G∥}(z) = (2π/|G∥|) e^{−|G∥||z|}. The G∥=0 mode solves −∂_z² ĝ_0(z) = 4π[δ(z) − 1/A]·A per area, the field of a sheet pair, giving ĝ_0(z) = −2π|z|/A (up to the gauge constant). Summing:

G^{2D}(ρ, z) = (2π/A) Σ_{G∥≠0} (e^{iG∥·ρ} e^{−|G∥||z|}) / |G∥|   −  (2π/A)|z|  +  C
                └──────── oscillatory, exponentially screened ────────┘   └ sheet ┘

(Parry, Surf. Sci. 49, 433 (1975); de Leeuw-Perram, Mol. Phys. 37, 1313 (1979); Bertaut.) This is exactly the gauge realized in vibe-qc’s already-landed slab Hartree: python/vibeqc/ewald_composed_slab.py builds the oscillatory G∥≠0 channel as the L_z→∞ continuous-G_z 3-D-FT and the G∥=0 channel as the analytic −(2π/A)∫dz∫dz' n(z)|z−z'|n(z') |z|-convolution, with V_ne (periodic_v_ne_slab.py) sharing the same gauge so the bilinear total E = E_nn + Tr[D·V_ne] + ½Tr[D·J] is gauge-consistent (tests/test_j_slab_ewald_2d.py). The 2-D Coulomb/Hartree side of G^{2D} therefore already exists and is validated against the 3-D kernel (§5). What D3 adds for d=2: the four-center / ERI form of the same G^{2D} and the matching exchange-q=0 seam (§3, §6), plus closing the polar-slab dipole BC (§4/§6).

3. The 1-D wire kernel `G^{1D}` (periodic along `z`, open in `ρ`)¶

For d=1 (axis z, length L per cell, transverse ρ=(x,y)), each G_z≠0 mode solves (G_z² − ∇_ρ²) ĝ_{G_z}(ρ) = 4π δ^{(2)}(ρ), the 2-D screened-Poisson (Yukawa-in-2-D) equation, whose decaying solution is the modified Bessel function ĝ_{G_z}(ρ) = 2 K₀(|G_z|ρ). The G_z=0 mode solves the 2-D Poisson equation of a neutralized line, −∇_ρ² ĝ_0 = 4π[δ^{(2)}(ρ) − 1/(L·∞)]·L, giving the logarithmic line potential ĝ_0(ρ) = −2 ln(ρ/ρ₀) (up to gauge). Summing:

G^{1D}(ρ, z) = (2/L) Σ_{G_z≠0} K₀(|G_z|ρ) e^{iG_z z}   −  (2/L) ln(ρ/ρ₀)  +  C
                └──── oscillatory, Bessel-screened ────┘   └─── line ───┘

This kernel is new to vibe-qc (the slab workstream marked NEUTRALIZED_1D explicitly out of scope, handovers/HANDOVER_SLAB_EWALD_2D.md). M2 implements it. K₀ is available via scipy.special.k0; the G_z≠0 sum converges exponentially (K₀(x) ~ √(π/2x) e^{−x}), and the ρ→0 short-range singularity is K₀(|G_z|ρ) → −ln(|G_z|ρ/2) − γ, which combines with the −(2/L)ln(ρ/ρ₀) line term to give the correct bare 1/r Coulomb singularity as ρ,z→0 (the short-range K exchange piece is built molecular-limit/real-space, as in the 3-D and slab paths). ρ₀ (the gauge radius) and C are fixed by the reduction anchor (§5).

4. The conditional (zero-transverse-mean) channel and the dipole BC¶

The G∥=0 (d=2) / G_z=0 (d=1) channel is where all the finite-size physics and the convention freedom live – it is the low-D image of the 3-D G=0 term:

Its growing part (−(2π/A)|z|; −(2/L)ln(ρ/ρ₀)) is forced by the neutralizing sheet/line – the analogue of the forced J/Coulomb part in 3-D.
Its constant C is the gauge, fixed by a reference point; differences of energies (and the MO-basis correlation numerator) are C-independent.
For a polar cell (net normal dipole M_z, slab; net transverse dipole, wire) the asymptotic potential is discontinuous across the system – a dipole-layer term Δφ = 4π M_z/A (slab). This is the genuine low-D analogue of the 3-D conditionally-convergent surface term, and it is a declared boundary condition: the modeller chooses tin-foil (short-circuit, Δφ=0) or open (the dipole layer is kept). -a declares the open BC (the dipole layer is physical for an isolated slab/wire) and records it alongside exchange_q0. The existing slab Hartree carries a known inaccuracy here (the M_z≠0 z-profile first moment is off ~6.7e-3, blocking the slab SCF un-gate, §6) – D3 must get the dipole-layer term right, because it is exactly the term that distinguishes a polar slab’s Hamiltonian.

5. The transverse-periodic-reduction gate (M3 – the hard, dependency-free check)¶

v_E^{3D} is the transverse-periodic sum of the mixed-boundary kernel: stacking G^{2D} at every transverse image z → z + nL_z (a 3-D lattice with the open direction re-periodized at period L_z) must reproduce the already-validated v_E^{3D} up to the gauge constant:

Σ_{n∈ℤ} G^{2D}(ρ, z + nL_z)  ≟  v_E^{3D}(ρ, z; L_z)  +  c·(gauge)        [d=2 → 3]
Σ_{m,n}  G^{1D}(ρ + m a_x + n a_y, z)  ≟  v_E^{3D}(·; a_x,a_y)  + c       [d=1 → 3]

Realized on the four-center / kernel level, the gate is: the low-D effective four-center g_eff^{(d)} = (μν| G^{(d)} |rs) re-periodized in the transverse directions equals ccm_eri_neutral/ccm_neutral_cderi (v_E^{3D}) up to a rank-1 c·S⊗S gauge shift – the four-center analogue of the already-passing tests/test_j_slab_ewald_2d.py::test_j_matches_3d_vacuum_limit_up_to_gauge (J_2D − J_3D → c·S + resid, residual → 0 as the gap grows). This needs no external dependency: the right-hand side is -a’s own validated 3-D kernel. Passing it for both d=2 and d=1, on the four-center, is M3.

6. The exchange-`q=0` seam from the same `G` (M4) – and the D1 tie-in¶

The exchange contribution K_{μν} = Σ_{rs} D_{rs} (μs| G |rν) carries a q=0 (G∥=0/G_z=0) self-interaction seam – the low-D analogue of the 3-D exxdiv term. It is read off the same G, not chosen independently: the seam is the signed probe-charge limit of the conditional channel of G^{(d)} (the C-gauge value at the probe-charge self-distance), i.e. the low-D analogue of madelung.apply_exxdiv_ewald_to_K / exxdiv_ewald_energy_shift (the 3-D BvK-Madelung exchange). Because the seam shifts the HF occupied/virtual spectrum, it propagates into correlation through the gap – exactly the dependence D1 made a machine-recorded provenance field (exchange_q0, python/vibeqc/periodic/exchange_convention.py). As landed (2026-07-04) the wire SCF (run_ccm_rhf_wire) records exchange_q0 = strict-zero-mode – the well-defined, g_perp_min-independent default (the S⊗S monopole projected out of K; §10). The BvK-ewald low-D seam (the signed-probe-charge limit of the conditional channel of G^{(d)}, whose 3-D sibling is the ξ_N·S D S term of run_ccm_rhf_direct) is the residual freedom – the D2 low-D exchange-q=0 question – to be added, recorded, and asserted matched in the A/B/KMP2 comparison alongside the dipole BC.

7. Milestones¶

M	Deliverable	Gate	State
M1	Poisson problem + BCs + kernels declared (this doc)	derivation + dependency-free reduction anchor stated	DONE
M2a	Scalar mixed-boundary kernels: `G^{1D}` (new, `K₀`/`ln`) + `G^{2D}` (Parry, `e^{−	G∥
M2b	Four-center routing `(μν\|G^{(d)}\|rs)`	short-range → bare `1/r` pair; contracts as chemists’ ERIs	d=1 (wire) DONE (`periodic/ccm/lowd_four_center.py` `ccm_wire_cderi`/`ccm_eri_wire`, the §9 quadrature; factorises as a real cderi → the whole RI/DLPNO stack rides it. Gate: exact erf-image-sum four-center matched to 2.8e-8 mod one `c·S⊗S`, `test_ccm_lowd_four_center.py` 3/3. NB the 3-D vacuum-ladder is not a clean gate: `c` drifts as `−(2/L)ln D` ✓ physics, but the non-gauge residual is the 3-D route’s own finite-`D` artifact.) d=2 (slab): the wire’s construction does NOT carry over (finding 2026-07-05). The naive full-3-D-FT + open-axis quadrature works for the wire because the open direction is 2-D and its `G∥=0` conditional channel is a mild log that reduces to a `c·S⊗S` monopole gauge. The slab’s open direction is 1-D, so its `G∥=0` channel is a **`
M3a	Scalar reduction: each kernel `== lim` (large transverse period) of `v_E^{3D}` up to a gauge `C`	reference Ewald `v_E^{3D}` self-validated (η-invariant + Madelung); residual ↓ monotonically (wire ~1/D² to <2e-3; slab ~1/L_z)	DONE (`test_lowd_greens_reduction.py` 3/3)
M3b	Four-center reduction: `g_eff^{(d)}` re-periodized `== ccm_eri_neutral` up to `c·S⊗S`	residual → 0 with gap	SUPERSEDED by a stronger, reference-free gate: the wire four-center is matched to the exact `erf`-image-sum four-center (`Σ_n erf(μ d_n)/d_n`) to 2.8e-8 mod `c·S⊗S` (M2b). The 3-D vacuum ladder is explicitly not a clean gate (§0/§7-M2b: its non-gauge residual is the 3-D route’s own finite-`D` artifact), so exact image sums replaced it throughout M2b/M4/M5
M4-foundation	Low-D self-energy ξ = `lim_{r→0}[G − 1/r]` (the Madelung-constant analog)	r² Richardson, gauge-shift exact `(2/L)ln(ρ₀'/ρ₀)`; anchored to the validated 3-D `madelung_constant_for_cell`: `−madelung(D⊥) − ξ_wire − (2/L)ln(D⊥)` is D⊥-independent (the gauge is exactly the line-vs-uniform `(2/L)ln(D⊥)`)	DONE (`wire_self_energy`/`slab_self_energy`, `test_lowd_greens{,_reduction}.py`)
M4	Wire `V_ne`/`E_nn` on the shared gauge + the exchange-`q=0` seam	neutral-cell Hartree total `g_perp_min`-independent (gauge cancels); exchange `q=0` handled by the strict-zero-mode	d=1 DONE (`lowd_four_center.py` `ccm_wire_v_ne` (`Δc'/Δc=−N_nuc`), `ccm_wire_e_nn` (1-D mini-Ewald, β-independent); Hartree total invariant to 1.1e-5 while each term swings ~6 Ha, `test_ccm_lowd_four_center.py` 10/10). The wire-Madelung seam (`exxdiv="ewald"` analogue) is the residual D2 low-D question; the strict-zero-mode (`exxdiv="strict"`, S⊗S projected out of `K`) is the well-defined default
M5	Re-run 1-D H-chain control	4c-vs-RI gap (`-b` H₄ `4c−RI=22.2 mHa`) collapses to the RI fitting error	d=1 DONE (`lowd_scf.py` `run_ccm_rhf_wire`: dense four-center SCF == RI-cderi SCF to 5e-13 – the wire `4c−RI` is machine-zero; and the strict total == the SCF from exact image-sum integrals to 8.4e-7, `test_ccm_lowd_scf.py` 4/4)

For the declared A-line wire Hamiltonian, d=1 now yields accuracy numbers, not model numbers. This statement does not bind the implementation to Γ-CCM. d=2 (slab) still awaits its four-center + reduction gate.

8. Reuse / boundaries¶

Reuse vibe-qc’s own slab Ewald (ewald_composed_slab.py, periodic_v_ne_slab.py) and the 3-D v_E (neutral.py) – vibe-qc code, not an external program (CLAUDE.md §10). The 1-D wire kernel is new.
The slab SCF un-gate (the polar-dipole §7 gate) is the periodic-SCF chat’s workstream (HANDOVER_SLAB_EWALD_2D.md, blocked on M_z≠0); D3 builds the CCM four-center + exchange-seam layer on top and contributes the corrected dipole-layer term back to that gate.
Any future parity check against a separately specified -b mixed-boundary construction must run out-of-process (subprocess), never by importing -b modules; -a keeps its own implementation. Current χ-CCM-B 1-D/2-D absolute energies remain fail-closed, and parity would not assign either implementation to an approach by ancestry.
Heavy runs (the dense g_eff / all-FT GDF) go to vq, never local (the 137 GB hazard); M3/M5 numerics are vq jobs.

References: Parry 1975; de Leeuw-Perram 1979; Bertaut; Rozzi et al., Phys. Rev. B 73, 205119 (2006) (mixed-boundary Coulomb cutoffs); Sun-Berkelbach-McClain-Chan 2017 (the 3-D periodic GDF this reduces to); Peintinger & Bredow 2014 (CCM). Joint spec: aiccm_a_position.md §0.5 item 6, aiccm_a_proposed_comparison_text.md point 5.

9. M2b – concrete four-center construction (design, 2026-06-28)¶

The scalar kernels (M2a) are translation-invariant (1-D/2-D periodic in free transverse/normal space), so wire_greens(|Δρ⃗|, Δz) / slab_greens(Δρ⃗, Δz) are already the two-point source→field kernels G^{(d)}(r−r'). The four-center (μν|G^{(d)}|rs) = ∫∫ χ_μν(r) G^{(d)}(r−r') χ_rs(r') dr dr' is therefore a mixed reciprocal(periodic)/real(open) object – not a scalar reciprocal contraction:

Wire (d=1, periodic z, open transverse ρ⃗). With g_m = 2πm/L,

(μν|G^{1D}|rs) = (4/L) Σ_{m≥1} ∫dρ⃗ dρ⃗' ρ̃_μν(g_m; ρ⃗) K₀(g_m|ρ⃗−ρ⃗'|) ρ̃_rs(−g_m; ρ⃗')
                 − (2/L) [ln-line / neutralising term],
   ρ̃_μν(g_m; ρ⃗) = ∫dz χ_μ(ρ⃗,z) χ_ν(ρ⃗,z) e^{i g_m z}   (1-D FT in z, RETAINING ρ⃗).

Per longitudinal mode m it is a 2-D transverse convolution of the bra/ket AO-pair profiles with the mode’s K₀(g_m·) (the transverse screened-Poisson Green’s function).

Slab (d=2, periodic in-plane ρ⃗, open normal z). With G∥ the in-plane reciprocal vectors,

(μν|G^{2D}|rs) = (2π/A) Σ_{G∥≠0} ∫dz dz' ρ̃_μν(G∥; z) e^{−|G∥||z−z'|}/|G∥| ρ̃_rs(−G∥; z')
                 − (2π/A) [sheet |z−z'| / neutralising term],
   ρ̃_μν(G∥; z) = ∫dρ⃗ χ_μ(ρ⃗,z) χ_ν(ρ⃗,z) e^{i G∥·ρ⃗}   (in-plane FT, RETAINING z).

Per in-plane mode G∥ it is a 1-D z-convolution with e^{−|G∥||z−z'|}.

Why the wire quadrature must NOT be reused here (finding 2026-07-05). It is tempting to build the slab four-center the way the wire one is built: a full 3-D AO-pair FT ρ̃(G∥,G_z) on a {G∥}×G_z grid with 4π/(|G∥|²+G_z²) folded into the weight, cut at g_z_min. That is wrong for the slab. For the wire the open direction is 2-D and its G∥=0-analogue conditional channel is a log, so the cut leaves a pure c·S⊗S monopole gauge. For the slab the open direction is 1-D: its G∥=0 channel is the −(2π/A)|z| sheet, whose small- G_z weight 4π/G_z² is a 1/g_z_min power divergence coupling to AO-pair dipoles, not a monopole. Empirically (H₂/sto-3g (2,2,1) non-polar slab): the naive slab cderi’s J misses compute_j_slab_ewald_2d_gamma by 24% (after c·S), and its RI-MP2 swings −276.8→−0.13 Ha as g_z_min goes 1e-4→1e-2. The G∥=0 channel therefore must be built as the analytic |z|-convolution below (mirroring the slab-J builder’s J_g0), and only the G∥≠0 channels ride a reciprocal/z-grid factorisation; the partial-FT primitive is not optional.

The key primitive – a partial AO-pair FT. Both need the AO-pair density Fourier-transformed only in the periodic direction(s), keeping the open direction real. python/vibeqc/_aopair_ft.py (general-L McMurchie-Davidson, Bloch variants) provides the full/Bloch AO-pair FT; M2b needs its partial form (periodic dims transformed, open dim retained). Two routes:

extract the partial FT from the Bloch primitive (fix the open-dim reciprocal component to a real-space slice), or
do the open-direction integral analytically – Gaussian × K₀ (wire) and Gaussian × e^{−|G∥||z|} (slab) are closed-form in the MD/Boys machinery _aopair_ft already uses – avoiding a real-space grid.

Screening. The transverse/normal kernels decay (K₀ exponentially in g_m ρ, e^{−|G∥||z|}), so the mode sums truncate at the same n_recip/n_shell as the scalar kernels; only the leading modes carry weight for compact AO pairs.

M3b gate (unchanged). Re-periodise the open direction(s) – make the cell fully 3-D-periodic – and g_eff^{(d)} must reduce to ccm_eri_neutral (the validated 3-D neutral four-center) up to a c·S⊗S gauge, mirroring the passing slab-J reduction test. Contracts as chemists’ ERIs into the existing CCM correlation stack.

Status: design only. Implementation is a dedicated build (the partial-FT primitive + the mode convolution + the M3b reduction gate); dense at first (small/1-D/2-D), then the scalable weighted route. Heavy runs → vq (§8).

10. M4 seam – the conditional gauge in the SCF energy (algebra, pinned 2026-06-29)¶

The wire four-center’s conditional constant enters the closed-shell SCF energy through two exactly known channels (verified against the measured tensor gauge shift Δc between two IR cutoffs; tests/test_ccm_lowd_four_center.py):

E_J[c]  = ½ c (Tr PS)²   = ½ c N_e²      (Hartree;  measured to 1e-6)
E_K[c]  = ¼ c Tr[(PS)²]  ≈ ½ c N_e       (exchange; measured to 9e-5)

The Hartree gauge term cancels for a neutral cell against the matching nuclear-side gauges (½c(N_e² − 2N_eZ + Z²) = ½c(N_e−Z)² = 0) – provided V_ne and E_nn are built with the same wire kernel and regularization (the M4 build item).
The exchange gauge term −½cN_e does not cancel: it is the exchange-q=0 seam. Fixing the exchange-q0 convention ⇔ choosing c (strict-zero vs ewald-like are specific c choices) – connecting directly to D1/D2 (exchange_q0, the 0.24 Ha/atom 3-D finding).
Correlation is seam-free to O(mismatch): the gauge ~vanishes in (ov|ov) MO integrals (occ-virt S-orthogonality; residual only via S^CCM-vs-plain-S, measured 1.1e-5 on H₂ (2,1,1)).

Landed (2026-07-04), the full wire SCF (M4/M5). The wire V_ne/E_nn builders (same quadrature, two-center/zero-center forms) and run_ccm_rhf_wire close this:

ccm_wire_v_ne rides the same real cderi B, so its gauge is −N_nuc·c·S with the four-center’s c (Δc'/Δc = −N_nuc verified to 6 digits).
ccm_wire_e_nn is a 1-D mini-Ewald (bare-1/r erfc real + self, Gaussian reciprocal on the shared quadrature); β drops out; gauge ½c·N_nuc².
Hartree cancellation realised numerically: E_nn + Tr[D·V_ne] + ½Tr[D·J] is g_perp_min-invariant to 1.1e-5 while each term swings ~6 Ha across a 100× IR sweep (½c(N_e−N_nuc)²=0).
S/T convention decided: plain cluster-supercell overlap/kinetic (short-ranged, gauge-free) – the periodicity lives entirely in the Coulomb kernel, consistent with the wire four-center / V_ne / E_nn all built on the plain supercell AO pairs. (The alternative, WSSC-folded S^CCM/T^CCM, would double-count periodicity against the kernel-periodic Coulomb.)
Exchange has no nuclear partner, so the exchange-q=0 ½c·N_e is the one surviving convention. run_ccm_rhf_wire(exxdiv="strict") projects the S⊗S monopole out of K (K → K − c*·S D S) → a g_perp_min-independent HF total (the low-D strict-zero-mode). The wire-Madelung seam (exxdiv="ewald" analogue) is the residual D2 low-D question.

See python/vibeqc/periodic/ccm/lowd_scf.py and tests/test_ccm_lowd_scf.py.