Certification and result states

Fail-closed principle

The workflow distinguishes four questions:

1. Was the projected-AGP reference built consistently?
2. Is the saved QPCCSD state a root of the original equations?
3. Is the fixed-amplitude PAV quadrature numerically stable?
4. For a continued curve, has an independent branch-closure check been done?

A finite energy answers none of these questions by itself. Missing evidence is reported as missing, not silently promoted to success.

Reference gates

The production reference must have finite RDMs and fit diagnostics, the requested CAS electron/orbital counts, exact orbital partition coverage, the correct target particle number, and Bogoliubov canonical defects below 1e-12. The occupation and pair-transfer fit errors are retained as model diagnostics. They are not hard convergence gates unless a campaign explicitly declares additional scientific acceptance criteria.

Certification also requires the exact versioned source-RDM conventions, successful finite projected-AGP fit diagnostics, finite relative/scaled geminals, an explicit global number-setting scale, and two Bogoliubov canonical defects strictly below 1e-12. Missing legacy metadata is a hard failure rather than a serialization default.

QPCCSD root gates

The terminal amplitudes are evaluated once more using the original, unshifted Hamiltonian and residual equations. Certification requires strict inequalities:

max(abs(R_pair), abs(R_quadruple)) < 1e-8
abs(Im(E_QPCCSD))                  < 1e-8 Eh.

Denominator floors, trust radii, line searches, and Newton shifts may be used to propose iterative steps. They do not appear in the final equations or reported energy.

For a potential-energy curve, a stronger branch certificate may additionally require independent forward/reverse continuation closure:

abs(Delta E)            <= 1e-8 Eh
max(abs(Delta amplitude)) <= 1e-6.

That campaign-level closure state is kept distinct from single-point root convergence.

PAV gates

PAV reuses an immutable copy of the terminal unprojected amplitudes. The implementation verifies that projection did not alter them. The base grid is the reported value; the doubled grid is an independent discretization check.

abs(E_2L - E_L) < 1e-8 Eh
abs(Im(E_L))    < 1e-8 Eh.

The projected denominator must also remain finite and above the configured overlap safety threshold. No Richardson-extrapolated energy is reported.

Labels

Use these semantic states in tables and plots:

State	Meaning	Plotting guidance
certified	all required reference, root, and PAV gates pass	filled marker
root_certified	single-point root passes; campaign closure not requested or not yet supplied	record closure separately
diagnostic_only	PAV was evaluated from a finite but uncertified source root, or a required PAV gate failed	open marker; exclude from certified statistics
failed	no finite or contract-valid value is available	omit energy; report failure

If a curve requires independent branch closure, an otherwise valid root may remain an open marker until that closure is available. The reason must be stored explicitly so it is not confused with a residual or quadrature failure.

One-geometry molecular regression

The repository’s remote molecular test executes one frozen-core N2/STO-3G geometry through both QPCCSD and PAV. It compares the fresh values with a checked-in, provenance-stamped regression record and verifies the method identifiers, coordinate counts, residual, direct-energy convention, frozen amplitudes, doubled-grid difference, and imaginary-energy gates.

The certified Medora anchor at R = 1.10 angstrom is:

Quantity	Expected value
direct QPCCSD energy	-107.65269549964435 Eh
fresh unshifted residual maximum	9.660306061257431e-10
fixed-amplitude Ser2 PAV energy	-107.65060143314642 Eh
9/18-grid difference	3.47388253863627e-18 Eh
PAV imaginary-energy magnitude	7.422711739519054e-18 Eh

The machine-readable record is tests/methods/qpccsd/fixtures/n2_sto3g_cas66_r1p10_certified.json; its source result, amplitude, and reference SHA-256 values are part of that fixture. The fresh clean-repository run must agree within 1e-8 Eh while independently passing all numerical gates.

The final clean-repository validation rebuilt CASSCF and the projected-AGP reference from scratch on Medora and started QPCCSD from zero amplitudes. It obtained -107.65269549939507 Eh for direct QPCCSD with a fresh unshifted residual of 9.33506029421327e-10, and -107.65060143315307 Eh for PAV. The differences from the certified anchor were 2.493e-10 Eh and 6.651e-12 Eh, respectively; all 54 tests passed. The complete receipt is tests/methods/qpccsd/validation/n2_sto3g_cas66_r1p10_medora_evidence.json.

Updating that regression reference requires an independent rerun on an approved host and a documented scientific reason. A test failure must not be resolved by simply replacing expected energies.