Proposed comparison text: 2014 AICCM vs Γ-CCM (direct-torus, -a) vs χ-CCM (finite-character, -b)¶

Proposed by the aiccm2026dev-a line for the cowork chat to adapt and merge into the shared comparison section. This is proposed text, not an edit to the shared/-b-owned aiccm_comparison.{tex,bib} — the maintainer/cowork chat adjudicates wording and ownership. Every number traces to a run in aiccm_a_position.md §6 or to an archived artifact named inline. Written to be fair to all three lines and conclusive per point.

Naming (finalised with both lines, 2026-06-25). We refer to the shared formulation as variational finite-BvK-torus CCM. The two development lines differ by representation, not by Hamiltonian: aiccm2026dev-a → Γ-CCM, the direct-torus (translation-constrained Γ-supercell) representation; and aiccm2026dev-b → χ-CCM, the finite-character (Γ-centred character-mesh) representation, with exact inverse transform to the real torus for local correlation. Mandatory footnote: these are representation labels at a declared exchange-q=0 convention — they must not imply two theories or two Coulomb kernels. (Note for prose: do not describe Γ-CCM as “no k-machinery” — its production path is multi-k GDF under the hood; “direct-torus” is the exact-theory presentation. χ is the finite group’s irreducible characters, not ordinary Bloch quadrature.) Use the descriptive names in prose/tables; the aiccm2026dev-a/-b labels are code/dev references only; 2014 stays AICCM.

A. One-paragraph framing (for the section intro)¶

Two independent re-derivations of the variational finite-BvK-torus CCM — the direct-torus (Γ-supercell) representation, Γ-CCM (aiccm2026dev-a), and the finite-character (Γ-centred character-mesh) representation, χ-CCM (aiccm2026dev-b) — set out, blind to each other, to put the 2014 ab-initio Cyclic Cluster Model on a rigorous footing. They converged on the same physics: the cyclic cluster is a finite Born–von-Kármán (BvK) torus; the Γ-supercell calculation equals the full unreduced Γ-centred character mesh exactly (not as a quadrature); the two-electron kernel is the charge-neutral periodic Green’s function (G=0 dropped); the Wigner–Seitz fractions are a partition of unity over tied minimum-image representatives, not a multiplier on free-space integrals; and the 2014 four-centre weight (eq. 18) is bra–ket-asymmetric and so is not the derivative of a scalar energy. That independent agreement on the foundations is the single most important result of the comparison. The two production paths compute with the same object — the neutral periodised four-centre or its RI factorisation — which we verify below to all printed digits and against an external code. The lines differ only in the diagnostic constructions one of them retains, the route to local correlation, and the representation treated as primary.

B. The decisive, settled result (lead the numerics with this)¶

On a vacuum-padded H₂/STO-3G control (20×20×6 bohr, Γ-centred (1,1,2) mesh) the three production HF routes coincide:

quantity	value (Ha/cell)	difference
`aiccm2026dev-a` `run_ccm_rhf_gdf`	−1.1182352381008094	—
`aiccm2026dev-b` RI-RHF	−1.1182352381008094	Δ = 0.0 (16 digits)
external PySCF KRHF (out of process)	−1.118235238694408	5.9×10⁻¹⁰

Both lines’ production four-centre is the multi-k density-fit of the neutral torus kernel v_E; on the same finite torus with matched cutoffs they are one operator, and that operator reproduces an independent periodic code to sub-nanohartree. aiccm2026dev-b additionally validated the correlation half (3-D RI-MP2 vs PySCF KRHF/KMP2 below one nanohartree per cell; data/h2_mp2_2026-06-21.json). The earlier 4.64 mHa discrepancy on the compact two-cell control was a legacy exxdiv=None Γ-supercell helper versus the production exxdiv="ewald" convention — a finite-mesh exchange-q=0 convention, not an A/B disagreement.

C. Three-line comparison table¶

Axis	2014 AICCM (Peintinger & Bredow)	Γ-CCM (direct-torus, -a)	χ-CCM (finite-character, -b)
Finite object	cluster + centre-specific WS interaction regions	finite BvK torus `𝕋 = ℝ³/L_c` (CCM ≡ SCM-Γ)	finite BvK quotient group `ℤ_{N₁}×ℤ_{N₂}×ℤ_{N₃}` (same identity)
Coulomb kernel	truncated bare `1/r`	charge-neutral Ewald `v_E` (`G=0` dropped)	charge-neutral Ewald `v_E` (`G=0` dropped) — same kernel
Four-centre	product weight eq. 18: `ω_{μν}·½(ω_{μρ}+ω_{νρ})·ω_{ρσ}`	production: neutral `g_eff = Σ_P L⊗L`; plus a diagnostic bare symmetric `ω^sym`	neutral periodised tensor `Σ_P L⊗L` only
Permutation symmetry	broken in ≥2-D (~15 % bra–ket antisym. on H₄)	exact 8-fold (kernel property)	exact 8-fold (kernel property)
Positive-semidefinite?	no	`g_eff` yes (min eig −6.6×10⁻¹⁶); `ω^sym` no (min eig −6.8×10⁻⁵)	`g_eff` yes (same object)
WS weight role	ad hoc product of pair weights	partition of unity over tied min-images, per-index	partition of unity over tied min-images, per-index — same role
Variational status	numerically fit; not a functional stationary point	functional stationary point; `F=∂E/∂P`	functional stationary point; `F=∂E/∂P`
Madelung / ionic	over-binding, not separated	bare−neutral = `ξ·S⊗S` to leading order (ξ→cell Madelung const); fixed-reference diagnostic — neutral reference required for correlation	neutral by construction; no bare tensor built
3-D settling test	1-D vs CRYSTAL (~10⁻⁶)	`E_CCM == periodic-Γ HF == KRHF` (Δ=0 vs -b; 5.9×10⁻¹⁰ vs PySCF)	RI-MP2 == KRHF/KMP2 (< 1 nHa/cell)
Open shell	HF + post-HF, 1-D-validated	UHF/UKS, UMP2, UCCSD(T)	UHF/UKS, UMP2, UCCSD(T)
Low-D long-range gauge	(not addressed)	3-D-Ewald `v_E`; mixed-boundary kernel open	direct-truncated vs fitted gauges unmatched (open)
Local correlation	—	subspace-projected union-PNO DLPNO; == canonical at no trunc (7×10⁻¹⁸)	finite-torus full-domain PNO; == canonical at no trunc (6.9×10⁻¹²)
Test breadth	1-D chain	28-system all-Bravais set + CRYSTAL23 harness	compact controls + 31-job fleet
External anchor	CRYSTAL (reference)	KRHF (Run R1)	KRHF/KMP2 sub-nHa (strongest single check)

D. Conclusive statement per disputed point¶

The kernel (production). Equivalent, at the declared exchange-q=0 convention. aiccm2026dev-a and aiccm2026dev-b compute with the same neutral periodised four-centre / RI; verified to all printed digits (§B) and against PySCF KRHF. The Coulomb/J part is forced (unique zero-average G=0 removal); the exchange-q=0 part is an additional finite-size convention — strict-zero-mode and BvK-Madelung (exxdiv="ewald") exchange are distinct finite Hamiltonians with the same thermodynamic limit. The equality holds because both lines chose exxdiv="ewald", not by Poisson-uniqueness. The paper declares the convention once and applies it to HF and post-HF (the exchange seam shifts the gap, hence the MP2/CCSD denominators); recording exchange_q0 = BvK-ewald in the .system manifest turns “same convention” into a reproducibility gate for every A/B/KMP2 comparison. The 2014 weight is superseded by both.
The four-centre as an object. Both production tensors are genuine ERIs; the 2014 weight is not, and -a’s diagnostic ω^sym is not either. g_eff is positive-semidefinite (min eig −6.6×10⁻¹⁶), a Gram factorisation of one symmetric two-point kernel, basis-covariant, and RI-separable. The 2014 eq. 18 tensor is bra–ket-asymmetric (the condition x_{mq}=x_{rl} it would require is not geometric). aiccm2026dev-a’s bare symmetric ω^sym is exactly 8-fold symmetric yet not positive-semidefinite (min eig −6.8×10⁻⁵): symmetry buys a scalar functional, not an ERI. The structural reason (the defensible, scoped statement): positive-semidefiniteness requires g_{pq,rs} = ∫∫ f_{pq}(r) K(r,r') f_{rs}(r') with K a positive operator; a fold keyed on AO labels rather than only on the pair density f_{pq}(r) is not generally of this form, so kernel-positivity is unprotected — and ω^sym is the concrete demonstration that it fails. (Not “any weight cannot”; the claim is that label-weighting does not protect positivity, and periodising the kernel first guarantees it.) This vindicates aiccm2026dev-b’s argument that a weight on a non-periodised kernel is the wrong kind of object; ω^sym is retained by -a only as a historical diagnostic that shows why symmetry repairs eq. 18, and is excluded from the production hierarchy.
The Wigner–Seitz weights. Equivalent. In both lines the WS fraction is 1/n over tied minimum images, applied independently per translation index — a partition of unity that never couples a bra index to a ket index. The 2014 ½(ω_{μρ}+ω_{νρ}) bridge is the broken approximation (it couples the bra pair to the ket anchor); both lines diagnose it, by an algebraic condition (-b) and a structural argument plus a measured ~15 % antisymmetry (-a).
The Madelung background. 2014 leaves it unseparated; -a characterises it (with one claim corrected); -b avoids it by construction; a residual is open on both. The bare−neutral difference is ξ·S⊗S to leading order (ξ → the cell Madelung constant), a fixed-reference diagnostic: in the MO basis it has no occupied–virtual numerator element, but it shifts the occ–virt denominators, so independent bare-vs-neutral SCF→correlation runs do not agree (measured: +67 % MP2 at ξ=0.5, +17 % at ξ≈0.186). The neutral reference is therefore required for ionic correlation, not just ionic HF. The beyond- rank-1 remainder is characterised (not arbitrary): with the signed self-potential ξ_N = (v_E−1/r)|_{r→0} < 0 (declare one sign; the deprecated −ξ·S⊗S form used ξ=−ξ_N>0), v_E − 1/r = ξ_N + h_N(r), h_N(0)=0, and the difference is exactly ξ_N·S⊗S + ⟨φφ|h_N|φφ⟩. The h_N term is physical finite-size physics (not gauge freedom, not RI error, not an A/B ambiguity since both use the same v_E); h_N(0)=0 makes the point-pair limit pure rank-1, but a finite compact Gaussian pair carries a small second-moment shape term and diffuse/ionic-dipolar pairs the full h_N, vanishing as N→∞. Its correlation footprint is a joint open measurement (LiH ≥2 cells vs multi-k KMP2; the auxiliary-ladder residual R_X = g_eff^RI(X) − g_bare − ξ_N·S⊗S → R), not an open concept. Because the exchange-q=0 convention shifts the gap, the finite-N correlation also carries that convention dependence — both lines fix and manifest-record the same convention (exchange_q0 = BvK-ewald).
Low dimensionality. Both open; neither validated; the construction is now well-posed. The neutral kernel both lines use is the 3-D-Ewald torus Green’s function; for genuine 1-D/2-D periodicity a mixed-boundary (wire/slab) Green’s function is required, derived Hamiltonian-first (define the Poisson problem, derive both Coulomb and the exchange-q=0 seam from the same G — not a minimum-image truncation). The standard forms are G^{2D}(ρ,z)=(2π/A)Σ_{G∥≠0} e^{iG·ρ−|G||z|}/|G| − (2π/A)|z| + C (slab, Parry/Bertaut) and G^{1D}(ρ,z)=(2/L)Σ_{G_z≠0}K₀(|G_z|ρ)e^{iG_z z} − (2/L)ln(ρ/ρ₀) + C (wire). Neither line supplies it yet, so aiccm2026dev-b’s 1-D/2-D four-centre-vs-RI offsets (hundreds of kJ/mol; four-centre−RI = 22.2 mHa on vacuum-padded H₄) are gauge differences, not fitting errors; no 1-D/2-D accuracy number is claimed by either line. Validation anchor: the 3-D torus kernel is the transverse-periodic sum of the mixed-boundary kernel, so the new kernel must reduce to the already-validated v_E^{3D} under transverse periodization (a consistency check with no new external dependency). The dipole/asymptotic boundary condition (polar-slab dipole-layer term) is the low-D analogue of the 3-D G=0 conditional convergence and must be declared. (The aiccm2026dev-a Hamiltonian-first M1 derivation — the mixed-boundary Poisson problem, G^{2D}/G^{1D}, the dipole-layer BC, the transverse-periodic reduction anchor to the validated v_E^{3D}, and the exchange seam from the same G — is in docs/aiccm2026dev_a_lowd_greens.md; the matched-convention recording of point 1 is implemented as [run].exchange_q0.)
Local correlation. Equivalent at no truncation. Both reproduce canonical finite-torus MP2/CCSD(T) to machine precision at full domain (-a 7×10⁻¹⁸; -b 6.9×10⁻¹²); under truncation they use different (both conservative) PNO schemes, and neither has yet demonstrated reduced scaling.
Convergence. Equivalent. Both define the same Γ-centred N₁×N₂×N₃ mesh sequence (non-monotone; each N a distinct finite Hamiltonian) converging to the Brillouin-zone integral; -a’s real-space interaction radius R_c with Δk = π/R_c is a parameterisation of the same family.

E. Closing sentence (proposed)¶

That two from-scratch derivations land on the same finite-torus / neutral-kernel / partition-of-unity foundation — agree to all printed digits on the production energy and against an external periodic code — while a published 12-year-old weighting is shown, twice and by different arguments, to lack the positive-semidefinite symmetric structure a variational energy requires, is the strongest statement this comparison makes. The lines differ not in what they compute on the production path but in the diagnostic scaffolding one retains, the route to local correlation, and the representation each treats as primary; the remaining physics — the non-rank-1 Madelung remainder and the low-dimensional long-range gauge — is open to both and is the natural subject of the next round.

Provenance: comparison verdicts and runs in aiccm_a_position.md; -b artifacts in aiccm_comparison/data/ (h2_mp2_2026-06-21.json, h4_ri_2026-06-21.json, lih_dlpno_ccsdt_2026-06-21.json). Do not transcribe this file into the shared aiccm_comparison.*; the cowork chat adapts and the maintainer merges.