[qmc] zhong-guo-ren-neng-fei: total positivity for sign-free determinants

tracks/qmc/solutions/zhong-guo-ren-neng-fei/AGENTS.md

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+# Verification instructions for AI reviewers
+
+These instructions apply to every file in this directory.
+
+## Purpose
+
+Act as a skeptical, read-only verifier of the proposed sign-free determinant
+construction. Do not treat `results.json`, the prose in `README.md`, or a
+successful randomized run as proof by itself. Do not edit the mathematical
+claim, the verification code, or recorded results merely to make a check pass.
+
+The submission makes three different kinds of statements. Keep them separate:
+
+1. proved statements about products of exponentials of real tridiagonal
+   Metzler matrices;
+2. computational evidence and exact finite-dimensional certificates;
+3. literature-search and novelty statements, which are explicitly not claims
+   of worldwide priority.
+
+## Before running code
+
+State the setup that will be checked:
+
+- real `n`-by-`n` tridiagonal generators;
+- nonnegative entries immediately above and below the diagonal;
+- arbitrary real diagonal entries;
+- arbitrary finite product depth;
+- the grand-canonical weight `det(I + P)`;
+- open-chain geometry for the interacting realization.
+
+Ask the user to confirm or correct this setup before starting computation, as
+required by the repository-level instructions. The default verification is a
+small local CPU run and should take only a few seconds.
+
+## Reproduce the computational checks
+
+From the repository root, run:
+
+```bash
+python -m pip install -r tracks/qmc/solutions/zhong-guo-ren-neng-fei/requirements.txt
+python tracks/qmc/solutions/zhong-guo-ren-neng-fei/verify.py
+```
+
+The second command must exit with status zero. Independently inspect the newly
+written `results.json`; do not rely only on terminal text. At minimum require:
+
+- `tn_semigroup.samples` equals 8000;
+- `tn_semigroup.negative_determinant_count` equals zero;
+- `tn_semigroup.determinant_below_one_count` equals zero;
+- `tn_semigroup.minor_violation_count` equals zero;
+- `tn_semigroup.minimum_checked_minor` is nonnegative up to the declared
+  floating-point tolerance;
+- the Fock-space trace and one-particle determinant agree to numerical
+  precision;
+- `exact_closing_edge_counterexample.exact_integer_determinant` equals `-1`;
+- all Kramers commutant dimensions equal one;
+- all tested Majorana skew-`J2` nullities equal zero.
+
+Re-running the fixed-seed script should not produce an unexplained change in
+the tracked `results.json`. A numerical failure is evidence against the
+implementation or claim and must be reported, not patched around.
+
+## Audit the proof independently
+
+Check each logical implication in `README.md` rather than only its conclusion:
+
+1. For every exterior-power degree `k`, derive the additive compound
+   `A^[k]` and verify that tridiagonality prevents a negative wedge-reordering
+   sign on every allowed off-diagonal transition.
+2. Verify that an exponential of a Metzler matrix is entrywise nonnegative.
+3. Use `C_k(exp(A)) = exp(A^[k])` to conclude that every minor of `exp(A)` is
+   nonnegative.
+4. Use Cauchy-Binet to check closure under arbitrary finite products.
+5. Expand `det(I + P)` as the sum of principal minors, including the empty
+   minor, to obtain the lower bound one.
+6. Multiply the seven integer shear factors in the boundary counterexample
+   directly and recompute `det(I + P) = -1`.
+
+Reject or qualify the theorem if any step silently assumes symmetry, strict
+positivity, invertibility of minors, fixed product depth, or periodic boundary
+conditions. None of those assumptions belongs to the stated theorem.
+
+## Audit the physical construction
+
+Check the auxiliary-field identity on all four occupations of two neighboring
+spinless-fermion sites. Confirm that:
+
+- attractive nearest-neighbor density interaction gives a real Gaussian
+  field;
+- each field value changes only real diagonal one-body terms;
+- open-chain hopping produces nonnegative nearest-neighbor entries in the
+  imaginary-time generator under the sign convention stated in `README.md`;
+- arbitrary chemical potentials do not spoil the tridiagonal Metzler
+  property;
+- the conclusion concerns the determinant weight for each auxiliary-field
+  configuration, not the absence of every possible algorithmic difficulty.
+
+Do not extend the claim to a periodic chain. The exact directed closing-edge
+certificate shows why such an extension requires a separate proof.
+
+## Audit independence from earlier sufficient conditions
+
+Treat this as a family-level question. An individual matrix may satisfy an
+older condition even when the full family does not.
+
+Verify separately:
+
+- the nonscalar diagonal-matrix obstruction to a fixed positive-definite
+  conformal-orthogonal metric;
+- the simultaneous presence of `I` and `-I` in the obstruction to a fixed
+  one-sided split-metric cone;
+- the scalar commutant of the matrix units in the no-common-Kramers argument;
+- the block calculation forcing a proposed fixed Majorana contraction
+  structure to vanish.
+
+If a prior construction is found that covers the complete family after an
+allowed fixed change of basis, report the exact theorem and source. Do not
+infer containment merely from overlap with one-dimensional, bipartite, or
+worldline-sign-free models.
+
+## Literature and novelty discipline
+
+The total-nonnegative matrix theory, Vandermonde determinants, and
+Karlin-McGregor noncrossing kernels are classical. The submission does not
+claim otherwise. A literature search may support the statement that the
+specific determinant-QMC formulation was not found, but absence from a search
+is not proof of novelty.
+
+Any stronger priority assessment must:
+
+- search current primary literature;
+- identify exact theorem statements rather than keyword similarities;
+- provide links or bibliographic identifiers;
+- classify each result as identical, containing, overlapping, or unrelated;
+- state unresolved uncertainty explicitly.
+
+## Verification report
+
+Return a concise report with:
+
+1. `PASS`, `FAIL`, or `INCONCLUSIVE`;
+2. separate verdicts for the theorem, implementation, physical mapping,
+   independence checks, and novelty search;
+3. the first concrete failing equation, file line, or numerical field, if any;
+4. commands run and their exit status;
+5. remaining claims that require expert or literature review.
+
+A `PASS` means the stated bounded claims survived these checks. It must not be
+reported as proof of worldwide publication priority.