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Case · Scientific and numerical software · Internal record
A proposed analytic gradient, validated blind against finite differences
PySCF, an open-source quantum chemistry library, and pull request #3426 adding analytic nuclear gradients for density-fitted CCSD and CCSD(T).
- The system said
- Gradient implemented; one test molecule agreed with finite differences.
- The evidence showed
- Agreement on 11 molecules and two methods to the conventional floor; the released path wrong in all 22 cells; one of the referee's own thresholds caught sitting below the library's floor.
§ 1
The claim tested
The new gradient is the derivative of the density-fitted energy the calculation minimised, and the released code path is not.
§ 2
What Omega did
- Pre-registered three invariants before any run: agreement with finite differences, translational invariance and rotational invariance, with thresholds fixed in advance.
- Built two environments byte-identical except for the pull request's six files: the 2.14.0 release wheel, and the wheel plus the diff.
- Ran 77 jobs over 11 molecules for both methods: the pull request, the release path, the pull request with auxiliary-basis response switched off, and the conventional-integral path as a floor.
- Recomputed every finite-difference gradient from raw energies in a referee with explicit loops and no library import, and checked the second-order step ratio in every cell.
§ 3
What was found
- The pull request's gradient agreed with finite differences to at most 2.5e-7 Ha/Bohr in 22 of 22 cells, indistinguishable from the conventional floor (largest ratio 1.2).
- The release path was wrong in 22 of 22 cells, by 8.9e-6 to 1.3e-4 Ha/Bohr, reproducing the reported defect on the author's molecule (3.13e-5 against the issue's 3.1e-5) and extending it to ten more.
- Switching off auxiliary-basis response broke translational invariance by 5e-5 to 1.1e-3, so that option is a measurable approximation, not a shortcut.
- The rotational threshold of 1e-8 fired on both the pull request and the established path, to within 1% of each other. Five one-at-a-time hypotheses on the largest cell traced part of it to SCF convergence and part to frozen-core handling. The threshold was Omega's own instrument defect. It was reported, not repaired.
§ 4
Verdict
- SUPPORTED
- The new gradient is the derivative of the density-fitted energy. On these 11 systems and two methods. Eleven systems establish eleven systems.
- CONTRADICTED
- The released path returns a correct gradient for density-fitted CCSD. 22 of 22 cells, matching the defect the issue reported.
- INCONCLUSIVE
- The gradient is rotationally invariant to 1e-8. The pre-registered threshold sat below the floor of the library's own established path. Instrument defect, recorded as such.
§ 5
What this does not establish
Not tested: open-shell references, periodic systems, symmetry-adapted molecules, systems larger than six atoms, auxiliary bases beyond two, or the geometry-optimiser path. The referee differentiates the same energy code the pull request differentiates, so it establishes consistency with that code, not agreement with an external truth. The origin of the shared 1e-7 rotational residual is not identified.
§ 6
The records
Every field of each finding this case rests on, as it appears on the ledger.
MEASURED pyscf-df-ccsd-gradient-matches-finite-differencesverify · hidden trusted base A proposed analytic gradient for a quantum-chemistry method matched independently computed finite differences in 22 of 22 cells; the released code path failed in 22 of 22. One of the referee's own thresholds fired on both arms alike and was reported as an instrument defect.
- Claim
- PySCF pull request #3426 (density-fitted CCSD and CCSD(T) analytic nuclear gradients) agrees with central finite differences of PySCF's own density-fitted energy to at most 2.5e-7 Ha/Bohr on 11 molecules for both methods, indistinguishable from the conventional-integral floor; the release code path differs by 8.9e-6 to 1.3e-4 Ha/Bohr on every cell; disabling auxiliary-basis response breaks translational invariance by 5e-5 to 1.1e-3. A pre-registered rotational-invariance threshold of 1e-8 fired on both the pull request and the established path to within 1%, and was recorded as an instrument defect, not a defect in the pull request.
- Status
- MEASURED Empirical, with a stated sample.
- Question
- Passed is not correct
- Subject
- PySCF, an open-source quantum chemistry library, and pull request #3426, merged on 14 September 2026 (third-party system); the finite-difference referee is Omega's and its evidence was not posted upstream. third-party subject
- Frame
- 11 molecules, CCSD and CCSD(T), 77 jobs; three finite-difference steps Richardson-combined; the pull-request environment is the 2.14.0 wheel plus the pull request's diff.
- Method
- Three invariants pre-registered with thresholds (finite-difference agreement, translational invariance, rotational invariance); four arms (pull request, release path, pull request without auxiliary response, conventional floor); the referee recomputes every gradient from raw energies with explicit loops and no library import.
- Oracle
- Central finite differences of the density-fitted energy, with the second-order step ratio checked in every cell.
- Negative control
- Present The release path had to fail agreement (22 of 22 did); the no-auxiliary-response arm had to break the sum rules (22 of 22 did); a 5e-6 perturbation had to be detected (11 of 11 was); the two environments had to give identical reference energies (11 of 11 did).
- Denominator
- 22 of 22 cells agree to 2.5e-7 Ha/Bohr or better; 22 of 22 release-path cells fail; 11 molecules; 77 jobs.
- Preregistration
- Design, cells, thresholds and controls frozen before any trial; one driver defect before the first trial and the post-falsifier exploratory runs logged as deviations and kept out of every table.
- Limitation
- Agreement on 11 molecules establishes agreement on those 11 molecules. Not tested: open-shell references, periodic systems, symmetry-adapted molecules, systems larger than six atoms, auxiliary bases beyond the two used, or the geometry-optimiser path. The finite-difference referee differentiates the same energy code the pull request differentiates, so it establishes consistency with that code, not agreement with an external truth. The origin of the shared 1e-7 rotational residual is not identified. The evidence has not been posted to the pull request; this is an internal Omega record.
- Source
- artifact not public
- Reproduce
- not reproducible from a public clone
- Independent reproduction
- None known.