einsum: ToT×ToT→T denest via phantom-unit dot (avoid C0 blowup)

[evaleev]

Problem

A denested ToT × ToT contraction (inner indices fully contracted, plain-T result) is evaluated by einsum<DeNest::True> as expand-then-reduce: it forms the full uncontracted product C0 (external × contracted-outer × inner) before reducing it. When a contracted outer index is large (e.g. a DF/RI index), C0 is enormous.

Concretely, in a CSV-CC PNO-CCSD term for C8H18 — I(i₃,i₁,i₂,Κ;a₄) * I(i₂,i₁,i₄,Κ;a₄) → I(i₄,i₃,i₂,i₁) — C0 reaches ~20 GB (256 s, 26.6 GB process peak) to produce a 412 KB result.

Fix

Reformulate as a single contraction whose inner product is a Frobenius dot. The inner reduction is expressed with a phantom unit-extent result mode (a reserved label prefix, ⊗₁; detail::is_phantom_unit_label) so the result inner cell is a genuine order-≥1 tensor (TA has no order-0), and the dot reads the operand inner cells flat:

• no operand carries the phantom mode (no reshape, arena-safe),
• no inner-GEMM rank match, no order-0 tensor,
• C0 is never built,
• correct even when an inner extent depends on a contracted-outer index (each cell dots its own range).

Changes

• util/annotation.h — phantom-unit label prefix + is_phantom_unit_label.
• einsum/tiledarray.h — DeNest::True builds C(c;⊗₁) = A(…) * B(…;…,⊗₁) then unwraps the unit-extent inner cells to scalars.
• tensor/arena_einsum.h — RegimeAInnerKind::phantom_dot (regime-A !e path), ArenaInnerShapeKind::unit_range, and arena_hadamard_phantom_dot (view-cell outer-Hadamard).
• expressions/cont_engine.h — inner phantom-dot op in both the owning and view-cell (ArenaTensor) inner-op paths, for both outer-Contraction and outer-Hadamard regimes.
• tests/einsum.cpp — adds the external-index (e-present) denest case alongside the existing pure-Hadamard and contracted-outer cases.

Validation

• einsum_manual (incl. the new external-index denest) passes; full ta_test (np=1) clean.
• MPQC c4h10 PNO-CCSD converges as before.
• MPQC C8H18 PNO-CCSD: the motivating term drops 256 s / 26.6 GB → 0.25 s / 2.9 GB, and the run (previously incomplete) now finishes all iterations.

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