diff --git a/tracks/qmc/solutions/zhong-guo-ren-neng-fei/AGENTS.md b/tracks/qmc/solutions/zhong-guo-ren-neng-fei/AGENTS.md
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--- /dev/null
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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.
diff --git a/tracks/qmc/solutions/zhong-guo-ren-neng-fei/README.md b/tracks/qmc/solutions/zhong-guo-ren-neng-fei/README.md
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+# Total positivity and ordered fermion propagation for sign-free determinants
+
+## Team
+
+|  |  |
+|---|---|
+| **Team name** | zhong-guo-ren-neng-fei |
+| **Members** | Zong-yue Liu |
+
+## Challenge
+
+| Row |  |
+|---|---|
+| **Challenge** | Test total nonnegativity and ordered one-dimensional fermion propagation as a determinant-QMC sign-free principle beyond fixed split-metric, Kramers, and Majorana constructions. |
+| **Catalog issue** | `Addresses #121` — “Sign-problem free hunter,” released by Lei Wang, Institute of Physics, Chinese Academy of Sciences. |
+| **Track** | `tracks/qmc/` — selected from the issue’s `Method: Quantum Monte Carlo` field. |
+
+## Status and scope
+
+We prove an arbitrary-depth determinant inequality for exponentials of real
+tridiagonal Metzler matrices, supply a randomized and exact verification
+harness, identify an interacting open-chain auxiliary-field realization, and
+give algebraic obstructions to the fixed structures listed in issue #121.
+
+The underlying theory of totally nonnegative matrices is classical. We do not
+claim to have invented total positivity, nor do we claim worldwide priority
+for its use in determinant QMC. The research claim at this stage is narrower:
+
+1. the determinant theorem and physical auxiliary-field construction below
+   are proved;
+2. the full generator family is not contained in the previous
+   conformal-orthogonal candidate or in the fixed split, Kramers, or 2024
+   Majorana contraction conditions;
+3. the surveyed sign-problem literature did not reveal this ordered
+   total-nonnegativity formulation as a DQMC guiding principle;
+4. an independent expert novelty review remains necessary before making a
+   publication-priority claim.
+
+## 1. Candidate generator family
+
+Define
+
+\[
+\mathcal T_n=
+\left\{
+A\in M_n(\mathbb R):
+A_{ij}=0\ \text{when}\ |i-j|>1,\quad
+A_{i,i+1}\geq0,\quad A_{i+1,i}\geq0
+\right\}.
+\]
+
+The diagonal entries are arbitrary real numbers. Thus \(\mathcal T_n\)
+consists of tridiagonal Metzler matrices.
+
+### Theorem
+
+For every depth \(L\geq1\) and every
+\(A_1,\ldots,A_L\in\mathcal T_n\), let
+
+\[
+P=e^{A_1}e^{A_2}\cdots e^{A_L}.
+\]
+
+Then every minor of \(P\) is nonnegative, and
+
+\[
+\boxed{\det(I+P)\geq1.}
+\]
+
+In particular, the fermion determinant is strictly positive for every
+configuration and every product depth.
+
+## 2. Proof
+
+### 2.1 Additive compounds
+
+For \(1\leq k\leq n\), let \(A^{[k]}\) be the generator induced by \(A\)
+on the exterior-power space \(\bigwedge^k\mathbb R^n\). Let \(C_k(M)\)
+be the \(k\)-th multiplicative compound matrix; its entries are all
+\(k\)-by-\(k\) minors of \(M\), indexed by increasing row and column
+subsets. Exterior-power functoriality gives
+
+\[
+C_k(e^A)=e^{A^{[k]}}.
+\]
+
+### 2.2 Every additive compound is Metzler
+
+An off-diagonal entry of \(A^{[k]}\) replaces one index in an ordered
+\(k\)-subset by another index. For a general dense matrix this replacement
+may carry either sign because the resulting wedge basis must be reordered.
+
+For \(A\in\mathcal T_n\), a nonzero replacement can only move an index
+between adjacent sites \(r\) and \(r+1\). If the replacement does not
+create a repeated index, its position in the ordered subset is unchanged.
+The wedge reordering sign is therefore positive. The corresponding entry
+is \(A_{r,r+1}\) or \(A_{r+1,r}\), both nonnegative.
+
+Hence every \(A^{[k]}\) is Metzler.
+
+### 2.3 Metzler exponentials are entrywise nonnegative
+
+If \(M\) is Metzler, choose \(c\) large enough that \(M+cI\) is
+entrywise nonnegative. Then
+
+\[
+e^M
+=e^{-c}e^{M+cI}
+=e^{-c}\sum_{r=0}^{\infty}\frac{(M+cI)^r}{r!}
+\]
+
+is entrywise nonnegative. Therefore
+
+\[
+C_k(e^A)=e^{A^{[k]}}\geq0
+\]
+
+entrywise for every \(k\). Thus \(e^A\) is totally nonnegative.
+
+### 2.4 Arbitrary products
+
+The Cauchy–Binet formula expresses every minor of a product as a sum of
+products of minors of the factors. Therefore totally nonnegative matrices
+are closed under multiplication, and
+
+\[
+P=e^{A_1}\cdots e^{A_L}
+\]
+
+is totally nonnegative.
+
+### 2.5 The fermion determinant
+
+For every square matrix \(P\),
+
+\[
+\det(I+P)
+=
+\sum_{S\subseteq\{1,\ldots,n\}}\det P[S,S].
+\]
+
+The empty principal minor equals \(1\). All other terms are nonnegative
+because \(P\) is totally nonnegative. Hence
+
+\[
+\det(I+P)\geq1.
+\]
+
+## 3. Exact boundary counterexample
+
+The theorem does not extend to arbitrary directed graphs. Let \(E_{ij}\)
+denote a three-dimensional matrix unit. Since \(E_{ij}^2=0\),
+
+\[
+e^{E_{ij}}=I+E_{ij}.
+\]
+
+Adding the directed closing edge \(E_{31}\) yields the exact seven-slice word
+
+\[
+P=(I+E_{12})^3(I+E_{31})(I+E_{23})^3
+=
+\begin{pmatrix}
+1&3&9\\
+0&1&3\\
+1&0&1
+\end{pmatrix}.
+\]
+
+Consequently,
+
+\[
+I+P=
+\begin{pmatrix}
+2&3&9\\
+0&2&3\\
+1&0&2
+\end{pmatrix},
+\qquad
+\boxed{\det(I+P)=-1}.
+\]
+
+This is an exact integer certificate. It shows that the open ordered-chain
+geometry is essential. The certificate excludes the enlarged class with an
+independent directed closing edge; it does not by itself prove that every
+Hermitian periodic-chain formulation has a sign problem.
+
+## 4. Reduction and novelty checks
+
+The claims below concern the whole family \(\mathcal T_n\). Individual
+matrices or smaller subfamilies may of course satisfy older sufficient
+conditions.
+
+### 4.1 Previous conformal-orthogonal candidate
+
+The previous candidate required a fixed positive-definite matrix \(H\) and
+
+\[
+A=\alpha I+K,\qquad K^{\mathsf T}H+HK=0.
+\]
+
+The new class contains every nonscalar real diagonal matrix \(D\). If such a
+\(D\) admitted the decomposition above, \(K=D-\alpha I\) would have real
+eigenvalues. But an \(H\)-skew matrix is similar to a real skew-symmetric
+matrix and therefore has purely imaginary eigenvalues. Both statements can
+hold only when \(D-\alpha I=0\), contradicting the choice of \(D\).
+
+Thus \(\mathcal T_n\) is not contained in the previous candidate.
+
+### 4.2 Fixed split metric
+
+The split-orthogonal and split-semigroup conditions require a fixed
+nondegenerate indefinite metric \(\eta\) satisfying
+
+\[
+A^{\mathsf T}\eta+\eta A=0
+\]
+
+or a common one-sided semidefinite inequality.
+
+The family \(\mathcal T_n\) contains \(I\) and \(-I\). Applying the same
+one-sided inequality to both forces equality, while \(A=I\) then gives
+\(2\eta=0\), a contradiction. Similarity transformations cannot remove
+this obstruction because \(I\) remains \(I\).
+
+Even after allowing a central scalar to be removed from each generator,
+all traceless diagonal matrices \(D\) and \(-D\) remain. For \(n\geq3\),
+
+\[
+D\eta+\eta D=0
+\]
+
+for every traceless diagonal \(D\) forces every entry of \(\eta\) to vanish.
+
+### 4.3 No common Kramers symmetry
+
+Suppose a common antiunitary symmetry \(\Theta=UK\) fixes every real
+generator. Then \(U\) commutes with every generator. The family contains
+
+\[
+E_{ii},\qquad E_{i,i+1},\qquad E_{i+1,i}.
+\]
+
+These matrix units generate the full real matrix algebra. Its complex
+commutant consists only of scalars, so \(U=\lambda I\) and
+
+\[
+\Theta^2=U\overline U=|\lambda|^2I,
+\]
+
+which cannot equal the Kramers value \(-I\).
+
+### 4.4 No 2024 Majorana contraction structure
+
+The 2024 contraction-semigroup condition requires a fixed real orthogonal
+skew-symmetric matrix \(J_2\) such that
+
+\[
+i\left(J_2V-\overline VJ_2\right)\preceq0
+\]
+
+for every Majorana kernel \(V\), together with a separate \(J_1\) reality
+condition.
+
+For a real diagonal number-conserving generator \(D\), the Majorana kernel,
+up to a positive scalar Fock factor, is
+
+\[
+V(D)=iR(D),\qquad
+R(D)=
+\begin{pmatrix}
+0&D\\
+-D&0
+\end{pmatrix}.
+\]
+
+Because the family contains both \(D\) and \(-D\), the inequality must be an
+equality:
+
+\[
+J_2R(D)+R(D)J_2=0
+\]
+
+for every real diagonal \(D\).
+
+Write
+
+\[
+J_2=
+\begin{pmatrix}
+X&Y\\
+Z&W
+\end{pmatrix}.
+\]
+
+Taking \(D=I\) gives \(Z=Y\) and \(W=-X\). Varying diagonal \(D\)
+forces \(X\) and \(Y\) to be diagonal. Skew-symmetry then forces
+\(X=Y=0\), hence \(J_2=0\), contradicting orthogonality.
+
+The obstruction also survives a fixed complex orthogonal Majorana change of
+basis. Pulling back its antiunitary operator gives \(CK\), where
+\(CR(D)+R(D)C=0\). On each orbital the general block has the form
+
+\[
+C_i=
+\begin{pmatrix}
+x_i&y_i\\
+y_i&-x_i
+\end{pmatrix}.
+\]
+
+The diagonal entries of \(C_i\overline{C_i}\) are
+\(|x_i|^2+|y_i|^2\), so \(C\overline C=-I\) is impossible.
+
+Since the 2024 paper identifies its symmetric-\(J_1\) branch with Majorana
+reflection positivity and its equality cases with the earlier
+Majorana-symmetry constructions, this excludes a fixed old structure
+covering the full family.
+
+## 5. Interacting determinant-QMC realization
+
+Consider an open chain of one-component spinless fermions:
+
+\[
+H=
+-\sum_{i=1}^{n-1}t_i
+\left(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i\right)
+-\sum_i\mu_i n_i
+-\sum_{i=1}^{n-1}V_i n_i n_{i+1},
+\]
+
+with \(t_i\geq0\), \(V_i\geq0\), and arbitrary chemical potentials
+\(\mu_i\). The final term is an attractive nearest-neighbor density
+interaction.
+
+For \(a=\Delta\tau V_i\) and \(N_i=n_i+n_{i+1}\),
+
+\[
+n_i n_{i+1}=\frac{N_i^2-N_i}{2},
+\]
+
+and the real Gaussian identity gives
+
+\[
+e^{a n_i n_{i+1}}
+=
+e^{-aN_i/2}
+\int_{\mathbb R}\frac{d\phi}{\sqrt{2\pi}}\,
+e^{-\phi^2/2}
+e^{\sqrt a\,\phi N_i}.
+\]
+
+The auxiliary-field measure is positive, and every field value contributes
+only a real diagonal one-body generator. The kinetic imaginary-time
+generator has adjacent off-diagonal entries
+\(\Delta\tau t_i\geq0\). Thus every kinetic, chemical-potential, and
+auxiliary-field factor lies in the semigroup generated by
+\(\mathcal T_n\).
+
+For every auxiliary-field configuration,
+
+\[
+w(\phi)
+=
+f(\phi)
+\det\left(I+\prod_\ell e^{A_\ell(\phi)}\right)
+\geq f(\phi)>0,
+\]
+
+where \(f(\phi)\) is the positive Gaussian factor.
+
+The construction uses one fermion flavor and arbitrary chemical potential,
+so it is not restricted to half filling. One-dimensional versions may also
+admit worldline or Jordan–Wigner sign-free algorithms; the contribution here
+is the determinant-level theorem.
+
+## 6. Vandermonde and the physical meaning of total positivity
+
+The total-positivity mechanism has a direct fermionic interpretation rather
+than being an arbitrary matrix trick.
+
+### 6.1 Slater determinants
+
+For one-particle orbitals \(\phi_j(x)=x^j\), the \(N\)-fermion Slater
+matrix is Vandermonde:
+
+\[
+\Phi_{ij}=x_i^{j-1},\qquad
+\det\Phi=\prod_{i<j}(x_j-x_i).
+\]
+
+For \(N\) noninteracting spinless fermions in a one-dimensional harmonic
+trap, the occupied orbitals are Hermite polynomials times Gaussians. Their
+ground-state Slater determinant reduces to
+
+\[
+\Psi_0(x_1,\ldots,x_N)
+=
+C e^{-\sum_i x_i^2/(2\ell^2)}
+\prod_{i<j}(x_j-x_i).
+\]
+
+Within the ordered chamber \(x_1<\cdots<x_N\), this wavefunction has fixed
+sign. The zeros at collisions are the Pauli principle in coordinate space.
+
+### 6.2 Imaginary-time heat kernels
+
+The free-particle imaginary-time kernel is
+
+\[
+K_\tau(x,y)
+=
+\frac{1}{\sqrt{4\pi\lambda\tau}}
+\exp\left[-\frac{(x-y)^2}{4\lambda\tau}\right].
+\]
+
+It factors as
+
+\[
+K_\tau(x,y)
+=
+C_\tau
+e^{-x^2/(4\lambda\tau)}
+e^{xy/(2\lambda\tau)}
+e^{-y^2/(4\lambda\tau)}.
+\]
+
+The kernel \(e^{xy}\) is strictly totally positive, and positive row and
+column scaling preserves all minor signs. The harmonic-oscillator Mehler
+kernel has the same positive-Gaussian-times-\(e^{cxy}\) structure.
+
+For ordered initial and final positions, the \(N\)-fermion propagator is
+
+\[
+K_\tau^{(N)}(\mathbf x,\mathbf y)
+=
+\det[K_\tau(x_i,y_j)]_{i,j=1}^N\geq0.
+\]
+
+This is the Karlin–McGregor noncollision determinant: in one dimension,
+worldlines cannot exchange their order without crossing. Total
+nonnegativity is the one-particle linear-algebra encoding of this
+noncrossing fermion propagation.
+
+The Vandermonde matrix is a confluent limit of the same kernel:
+
+\[
+\left.
+\frac{\partial^{j-1}}{\partial y^{j-1}}e^{xy}
+\right|_{y=0}
+=x^{j-1}.
+\]
+
+Thus heat kernels, noncrossing paths, Vandermonde Slater states, and the
+tridiagonal Metzler theorem are different presentations of one ordered
+one-dimensional mechanism.
+
+### 6.3 Relation to the grand-canonical determinant
+
+For a one-particle propagator \(P\),
+
+\[
+\det(I+P)
+=
+\sum_{k=0}^n\operatorname{Tr}_{\wedge^k}P.
+\]
+
+The matrix entries of \(\wedge^kP\) are the \(k\)-particle Slater
+minors. The determinant-QMC weight is therefore the sum of the fixed-particle
+number propagation traces.
+
+An additional direction, not yet claimed as a theorem here, is projector QMC
+with Vandermonde or Hermite boundary Slater matrices:
+
+\[
+\det\left(
+\Phi_{\mathrm L}^{\mathsf T}
+e^{A_1}\cdots e^{A_L}
+\Phi_{\mathrm R}
+\right).
+\]
+
+The Cauchy–Binet formula suggests a natural positivity route when both the
+bulk propagators and boundary orbital matrices lie in compatible
+total-positivity cones.
+
+## 7. Verification
+
+Run:
+
+```bash
+python tracks/qmc/solutions/zhong-guo-ren-neng-fei/verify.py
+```
+
+The fixed-seed default run records:
+
+- dimensions \(1\) through \(8\);
+- depths \(1,2,4,8\);
+- 250 random samples per dimension-depth cell;
+- 8000 candidate products in total;
+- zero negative determinants;
+- zero determinants below \(1\) at tolerance;
+- 64 exhaustive all-minor checks with no negative minor;
+- 96 principal-minor identity checks;
+- a five-orbital direct Fock-space trace cross-check;
+- 600 split-Lie and 600 split-cone positive anchor samples;
+- exact negative and zero controls for the other split-group components;
+- commutant and Majorana-\(J_2\) linear-algebra obstruction checks;
+- the exact integer directed-cycle counterexample.
+
+The generated `results.json` records the seed, protocol, extrema, residuals,
+and exact certificate.
+
+Randomized tests support the implementation but do not replace the proof.
+
+## 8. Literature checked
+
+- L. Wang, Y.-H. Liu, M. Iazzi, M. Troyer, and G. Harcos,
+  [Split orthogonal group: A guiding principle for sign-problem-free
+  fermionic simulations](https://arxiv.org/abs/1506.05349).
+- Z. C. Wei, C. Wu, Y. Li, S. Zhang, and T. Xiang,
+  [Majorana Positivity and the Fermion sign problem of Quantum Monte Carlo
+  Simulations](https://arxiv.org/abs/1601.01994).
+- Z.-X. Li, Y.-F. Jiang, and H. Yao,
+  [Majorana-time-reversal symmetries](https://arxiv.org/abs/1601.05780).
+- Z.-C. Wei,
+  [Semigroup approach to the sign problem in quantum Monte Carlo
+  simulations](https://arxiv.org/abs/1712.09412), v3.
+- M. Margaliot and E. D. Sontag,
+  [Revisiting Totally Positive Differential Systems](https://arxiv.org/abs/1802.09590).
+- M. Katori and H. Tanemura,
+  [Noncolliding Brownian Motion and Determinantal
+  Processes](https://arxiv.org/abs/0705.2460).
+- P. K. Panigrahi and M. Sivakumar,
+  [Laughlin Wave Function and One-Dimensional Free
+  Fermions](https://arxiv.org/abs/cond-mat/9509039).
+- MathOverflow,
+  [How to prove this determinant is positive?](https://mathoverflow.net/questions/204460/how-to-prove-this-determinant-is-positive)
+  and
+  [How to prove this determinant is positive II?](https://mathoverflow.net/questions/229788/how-to-prove-this-determinant-is-positive-ii).
+
+Targeted searches for total nonnegativity, tridiagonal Metzler generators,
+fermion determinants, and auxiliary-field QMC did not reveal a source using
+the theorem above as a determinant-QMC guiding principle. This is a search
+record, not proof of publication priority.
+
+## 9. Reproducible claim
+
+The following statement is supported by the proof and artifacts in this
+directory:
+
+> Arbitrary finite products of exponentials of real tridiagonal Metzler
+> generators are totally nonnegative. Their grand-canonical fermion
+> determinants are at least one. The generator family has an open-chain,
+> off-half-filled interacting auxiliary-field realization and is not covered
+> as a whole by the fixed split, Kramers, or 2024 Majorana contraction
+> conditions tested here.
diff --git a/tracks/qmc/solutions/zhong-guo-ren-neng-fei/requirements.txt b/tracks/qmc/solutions/zhong-guo-ren-neng-fei/requirements.txt
new file mode 100644
index 000000000..f861e4049
--- /dev/null
+++ b/tracks/qmc/solutions/zhong-guo-ren-neng-fei/requirements.txt
@@ -0,0 +1,2 @@
+numpy>=1.26
+scipy>=1.11
diff --git a/tracks/qmc/solutions/zhong-guo-ren-neng-fei/results.json b/tracks/qmc/solutions/zhong-guo-ren-neng-fei/results.json
new file mode 100644
index 000000000..9850c7cff
--- /dev/null
+++ b/tracks/qmc/solutions/zhong-guo-ren-neng-fei/results.json
@@ -0,0 +1,138 @@
+{
+  "exact_closing_edge_counterexample": {
+    "I_plus_product": [
+      [
+        2,
+        3,
+        9
+      ],
+      [
+        0,
+        2,
+        3
+      ],
+      [
+        1,
+        0,
+        2
+      ]
+    ],
+    "each_factor_is": "exp(Eij) = I + Eij because Eij^2 = 0",
+    "exact_integer_determinant": -1,
+    "product": [
+      [
+        1,
+        3,
+        9
+      ],
+      [
+        0,
+        1,
+        3
+      ],
+      [
+        1,
+        0,
+        1
+      ]
+    ],
+    "word": [
+      "E12",
+      "E12",
+      "E12",
+      "E31",
+      "E23",
+      "E23",
+      "E23"
+    ]
+  },
+  "fock_trace_check": {
+    "absolute_difference": 7.105427357601002e-15,
+    "determinant": 35.852592547734055,
+    "fock_dimension": 32,
+    "fock_trace_imag": 0.0,
+    "fock_trace_real": 35.85259254773405,
+    "one_particle_dimension": 5
+  },
+  "known_anchors": {
+    "component_controls": {
+      "O_minus_minus_determinant": -0.5,
+      "mixed_component_determinant": 0.0
+    },
+    "split_cone": {
+      "maximum_cone_violation": 0.0,
+      "minimum_determinant": 3.6150254149035583,
+      "negative_count": 0,
+      "samples": 600
+    },
+    "split_lie": {
+      "maximum_metric_residual": 8.87334158954865e-15,
+      "minimum_determinant": 4.000008416107035,
+      "negative_count": 0,
+      "samples": 600
+    },
+    "trials_per_dimension_depth_cell": 50
+  },
+  "novelty_obstruction_checks": {
+    "dimensions": [
+      2,
+      3,
+      4,
+      5,
+      6,
+      7,
+      8
+    ],
+    "dirac_commutant_dimensions": [
+      1,
+      1,
+      1,
+      1,
+      1,
+      1,
+      1
+    ],
+    "expected_for_no_2024_contraction_J2": "all nullities equal 0",
+    "expected_for_no_kramers": "all commutant dimensions equal 1",
+    "majorana_skew_J2_nullities": [
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0
+    ]
+  },
+  "seed": 121,
+  "tn_semigroup": {
+    "depths": [
+      1,
+      2,
+      4,
+      8
+    ],
+    "determinant_below_one_count": 0,
+    "dimensions": [
+      1,
+      2,
+      3,
+      4,
+      5,
+      6,
+      7,
+      8
+    ],
+    "exhaustive_all_minor_checks": 64,
+    "maximum_determinant": 156608.52410555657,
+    "maximum_generator_structure_violation": 0.0,
+    "maximum_principal_identity_error": 1.0186340659856796e-10,
+    "minimum_checked_minor": 5.881419941595498e-29,
+    "minimum_determinant": 1.0895854257783073,
+    "minor_violation_count": 0,
+    "negative_determinant_count": 0,
+    "principal_identity_checks": 96,
+    "samples": 8000,
+    "trials_per_dimension_depth_cell": 250
+  }
+}
\ No newline at end of file
diff --git a/tracks/qmc/solutions/zhong-guo-ren-neng-fei/verify.py b/tracks/qmc/solutions/zhong-guo-ren-neng-fei/verify.py
new file mode 100644
index 000000000..5a947e2a0
--- /dev/null
+++ b/tracks/qmc/solutions/zhong-guo-ren-neng-fei/verify.py
@@ -0,0 +1,500 @@
+#!/usr/bin/env python3
+"""Numerical and exact checks for the issue #121 TN-semigroup construction.
+
+The proof in README.md is the primary result. This program independently
+checks known split-group anchors, total nonnegativity, the principal-minor
+identity, the Fock-trace identity, novelty obstructions, and an exact
+directed-closing-edge counterexample.
+"""
+
+from __future__ import annotations
+
+import argparse
+import json
+from itertools import combinations
+from pathlib import Path
+
+import numpy as np
+from numpy.typing import NDArray
+from scipy.linalg import expm
+
+
+RealArray = NDArray[np.float64]
+ComplexArray = NDArray[np.complex128]
+
+
+def product_exponentials(generators: list[RealArray]) -> RealArray:
+    product = np.eye(generators[0].shape[0])
+    for generator in generators:
+        product = product @ expm(generator)
+    return product
+
+
+def det_i_plus(matrix: RealArray) -> float:
+    return float(np.linalg.det(np.eye(matrix.shape[0]) + matrix))
+
+
+def split_metric(n: int) -> RealArray:
+    return np.diag(np.r_[np.ones(n), -np.ones(n)])
+
+
+def split_lie_generator(
+    rng: np.random.Generator, n: int, scale: float
+) -> RealArray:
+    eta = split_metric(n)
+    raw = rng.normal(size=(2 * n, 2 * n))
+    return scale * 0.5 * (raw - eta @ raw.T @ eta)
+
+
+def split_cone_generator(
+    rng: np.random.Generator, n: int, scale: float
+) -> RealArray:
+    eta = split_metric(n)
+    lie_part = split_lie_generator(rng, n, scale)
+    raw = rng.normal(size=(2 * n, 2 * n))
+    positive = raw.T @ raw
+    positive /= max(1.0, float(np.linalg.norm(positive, ord=2)))
+    return lie_part + 0.5 * scale * eta @ positive
+
+
+def test_known_anchors(
+    rng: np.random.Generator, trials_per_cell: int
+) -> dict[str, object]:
+    lie_values: list[float] = []
+    cone_values: list[float] = []
+    max_lie_residual = 0.0
+    max_cone_violation = 0.0
+
+    for n in (1, 2, 3):
+        eta = split_metric(n)
+        for depth in (1, 2, 4, 8):
+            for _ in range(trials_per_cell):
+                lie_generators = [
+                    split_lie_generator(rng, n, scale=0.22)
+                    for _ in range(depth)
+                ]
+                lie_product = product_exponentials(lie_generators)
+                lie_values.append(det_i_plus(lie_product))
+                residual = np.linalg.norm(
+                    lie_product.T @ eta @ lie_product - eta, ord=2
+                )
+                max_lie_residual = max(max_lie_residual, float(residual))
+
+                cone_generators = [
+                    split_cone_generator(rng, n, scale=0.14)
+                    for _ in range(depth)
+                ]
+                for generator in cone_generators:
+                    cone_matrix = generator.T @ eta + eta @ generator
+                    minimum = float(np.linalg.eigvalsh(cone_matrix).min())
+                    max_cone_violation = max(
+                        max_cone_violation, max(0.0, -minimum)
+                    )
+                cone_values.append(
+                    det_i_plus(product_exponentials(cone_generators))
+                )
+
+    tolerance = 1e-9
+    return {
+        "trials_per_dimension_depth_cell": trials_per_cell,
+        "split_lie": {
+            "samples": len(lie_values),
+            "negative_count": sum(value < -tolerance for value in lie_values),
+            "minimum_determinant": min(lie_values),
+            "maximum_metric_residual": max_lie_residual,
+        },
+        "split_cone": {
+            "samples": len(cone_values),
+            "negative_count": sum(value < -tolerance for value in cone_values),
+            "minimum_determinant": min(cone_values),
+            "maximum_cone_violation": max_cone_violation,
+        },
+        "component_controls": {
+            "O_minus_minus_determinant": det_i_plus(
+                np.diag([-2.0, -0.5])
+            ),
+            "mixed_component_determinant": det_i_plus(
+                np.diag([-1.0, 1.0])
+            ),
+        },
+    }
+
+
+def random_tridiagonal_metzler(
+    rng: np.random.Generator, dimension: int, scale: float
+) -> RealArray:
+    generator = np.diag(rng.normal(scale=scale, size=dimension))
+    if dimension > 1:
+        upper = np.abs(rng.normal(scale=scale, size=dimension - 1))
+        lower = np.abs(rng.normal(scale=scale, size=dimension - 1))
+        generator += np.diag(upper, 1) + np.diag(lower, -1)
+    return generator
+
+
+def minimum_minor(matrix: RealArray) -> float:
+    dimension = matrix.shape[0]
+    answer = float("inf")
+    indices = range(dimension)
+    for order in range(1, dimension + 1):
+        index_sets = list(combinations(indices, order))
+        for rows in index_sets:
+            for columns in index_sets:
+                value = float(
+                    np.linalg.det(matrix[np.ix_(rows, columns)])
+                )
+                answer = min(answer, value)
+    return answer
+
+
+def principal_minor_sum(matrix: RealArray) -> float:
+    dimension = matrix.shape[0]
+    total = 1.0
+    indices = range(dimension)
+    for order in range(1, dimension + 1):
+        for subset in combinations(indices, order):
+            total += float(np.linalg.det(matrix[np.ix_(subset, subset)]))
+    return total
+
+
+def generator_structure_violation(generator: RealArray) -> float:
+    dimension = generator.shape[0]
+    violation = 0.0
+    for row in range(dimension):
+        for column in range(dimension):
+            if abs(row - column) > 1:
+                violation = max(violation, abs(float(generator[row, column])))
+            elif abs(row - column) == 1:
+                violation = max(
+                    violation, max(0.0, -float(generator[row, column]))
+                )
+    return violation
+
+
+def test_tn_semigroup(
+    rng: np.random.Generator,
+    trials_per_cell: int,
+    minor_checks_per_cell: int,
+    identity_checks_per_cell: int,
+) -> dict[str, object]:
+    determinants: list[float] = []
+    minimum_checked_minor = float("inf")
+    minor_violations = 0
+    minor_checks = 0
+    identity_checks = 0
+    max_identity_error = 0.0
+    max_structure_violation = 0.0
+    tolerance = 2e-8
+
+    for dimension in range(1, 9):
+        for depth in (1, 2, 4, 8):
+            for trial in range(trials_per_cell):
+                generators = [
+                    random_tridiagonal_metzler(
+                        rng, dimension=dimension, scale=0.24
+                    )
+                    for _ in range(depth)
+                ]
+                max_structure_violation = max(
+                    max_structure_violation,
+                    *(generator_structure_violation(item) for item in generators),
+                )
+                product = product_exponentials(generators)
+                determinant = det_i_plus(product)
+                determinants.append(determinant)
+
+                if trial < minor_checks_per_cell:
+                    checked_minimum = minimum_minor(product)
+                    minimum_checked_minor = min(
+                        minimum_checked_minor, checked_minimum
+                    )
+                    minor_violations += int(checked_minimum < -tolerance)
+                    minor_checks += 1
+
+                if trial < identity_checks_per_cell:
+                    expansion = principal_minor_sum(product)
+                    max_identity_error = max(
+                        max_identity_error, abs(expansion - determinant)
+                    )
+                    identity_checks += 1
+
+    return {
+        "dimensions": list(range(1, 9)),
+        "depths": [1, 2, 4, 8],
+        "trials_per_dimension_depth_cell": trials_per_cell,
+        "samples": len(determinants),
+        "determinant_below_one_count": sum(
+            value < 1.0 - tolerance for value in determinants
+        ),
+        "negative_determinant_count": sum(
+            value < -tolerance for value in determinants
+        ),
+        "minimum_determinant": min(determinants),
+        "maximum_determinant": max(determinants),
+        "exhaustive_all_minor_checks": minor_checks,
+        "minor_violation_count": minor_violations,
+        "minimum_checked_minor": minimum_checked_minor,
+        "principal_identity_checks": identity_checks,
+        "maximum_principal_identity_error": max_identity_error,
+        "maximum_generator_structure_violation": max_structure_violation,
+    }
+
+
+def annihilation_operator(dimension: int, orbital: int) -> ComplexArray:
+    fock_dimension = 1 << dimension
+    operator = np.zeros((fock_dimension, fock_dimension), dtype=np.complex128)
+    lower_mask = (1 << orbital) - 1
+    for state in range(fock_dimension):
+        if state & (1 << orbital):
+            target = state ^ (1 << orbital)
+            sign = -1 if (state & lower_mask).bit_count() % 2 else 1
+            operator[target, state] = sign
+    return operator
+
+
+def quadratic_lift(matrix: ComplexArray) -> ComplexArray:
+    dimension = matrix.shape[0]
+    annihilators = [
+        annihilation_operator(dimension, orbital) for orbital in range(dimension)
+    ]
+    result = np.zeros((1 << dimension, 1 << dimension), dtype=np.complex128)
+    for row in range(dimension):
+        creator = annihilators[row].conj().T
+        for column in range(dimension):
+            result += matrix[row, column] * creator @ annihilators[column]
+    return result
+
+
+def fock_trace_check(rng: np.random.Generator) -> dict[str, float | int]:
+    dimension = 5
+    sequence = [
+        random_tridiagonal_metzler(rng, dimension, scale=0.2)
+        for _ in range(5)
+    ]
+    one_particle_product = product_exponentials(sequence)
+    determinant = det_i_plus(one_particle_product)
+
+    fock_product = np.eye(1 << dimension, dtype=np.complex128)
+    for generator in sequence:
+        lift = quadratic_lift(generator.astype(np.complex128))
+        fock_product = fock_product @ expm(lift)
+    fock_trace = np.trace(fock_product)
+
+    return {
+        "one_particle_dimension": dimension,
+        "fock_dimension": 1 << dimension,
+        "determinant": determinant,
+        "fock_trace_real": float(fock_trace.real),
+        "fock_trace_imag": float(fock_trace.imag),
+        "absolute_difference": float(abs(fock_trace - determinant)),
+    }
+
+
+def matrix_unit(dimension: int, row: int, column: int) -> RealArray:
+    result = np.zeros((dimension, dimension))
+    result[row, column] = 1.0
+    return result
+
+
+def numerical_rank(matrix: RealArray, tolerance: float = 1e-10) -> int:
+    singular_values = np.linalg.svd(matrix, compute_uv=False)
+    if singular_values.size == 0:
+        return 0
+    threshold = tolerance * max(matrix.shape) * singular_values[0]
+    return int(np.sum(singular_values > threshold))
+
+
+def kramers_commutant_dimension(dimension: int) -> int:
+    identity = np.eye(dimension)
+    generators = [
+        matrix_unit(dimension, index, index)
+        for index in range(dimension)
+    ]
+    for index in range(dimension - 1):
+        generators.append(matrix_unit(dimension, index, index + 1))
+        generators.append(matrix_unit(dimension, index + 1, index))
+
+    constraints = [
+        np.kron(generator.T, identity)
+        - np.kron(identity, generator)
+        for generator in generators
+    ]
+    stacked = np.vstack(constraints)
+    return dimension * dimension - numerical_rank(stacked)
+
+
+def majorana_rotation(diagonal: RealArray) -> RealArray:
+    zero = np.zeros_like(diagonal)
+    return np.block([[zero, diagonal], [-diagonal, zero]])
+
+
+def majorana_j2_nullity(dimension: int) -> int:
+    """Dimension of skew J satisfying {J, R(D)} = 0 for all diagonal D."""
+    doubled = 2 * dimension
+    skew_basis: list[RealArray] = []
+    for row in range(doubled):
+        for column in range(row + 1, doubled):
+            basis = np.zeros((doubled, doubled))
+            basis[row, column] = 1.0
+            basis[column, row] = -1.0
+            skew_basis.append(basis)
+
+    rotations = [
+        majorana_rotation(
+            np.diag(
+                [
+                    1.0 if index == selected else 0.0
+                    for index in range(dimension)
+                ]
+            )
+        )
+        for selected in range(dimension)
+    ]
+    columns = []
+    for basis in skew_basis:
+        image = np.concatenate(
+            [
+                (basis @ rotation + rotation @ basis).reshape(-1)
+                for rotation in rotations
+            ]
+        )
+        columns.append(image)
+    constraint = np.column_stack(columns)
+    return len(skew_basis) - numerical_rank(constraint)
+
+
+def novelty_obstruction_checks() -> dict[str, object]:
+    dimensions = list(range(2, 9))
+    return {
+        "dimensions": dimensions,
+        "dirac_commutant_dimensions": [
+            kramers_commutant_dimension(item) for item in dimensions
+        ],
+        "majorana_skew_J2_nullities": [
+            majorana_j2_nullity(item) for item in dimensions
+        ],
+        "expected_for_no_kramers": "all commutant dimensions equal 1",
+        "expected_for_no_2024_contraction_J2": "all nullities equal 0",
+    }
+
+
+def det3_integer(matrix: NDArray[np.object_]) -> int:
+    a, b, c = matrix[0]
+    d, e, f = matrix[1]
+    g, h, i = matrix[2]
+    return int(
+        a * (e * i - f * h)
+        - b * (d * i - f * g)
+        + c * (d * h - e * g)
+    )
+
+
+def exact_closing_edge_counterexample() -> dict[str, object]:
+    identity = np.eye(3, dtype=object)
+
+    def integer_unit(row: int, column: int) -> NDArray[np.object_]:
+        result = np.zeros((3, 3), dtype=object)
+        result[row, column] = 1
+        return result
+
+    e12 = integer_unit(0, 1)
+    e31 = integer_unit(2, 0)
+    e23 = integer_unit(1, 2)
+    elementary_factors = [
+        identity + e12,
+        identity + e12,
+        identity + e12,
+        identity + e31,
+        identity + e23,
+        identity + e23,
+        identity + e23,
+    ]
+    product = identity.copy()
+    for factor in elementary_factors:
+        product = product @ factor
+    determinant = det3_integer(identity + product)
+
+    return {
+        "word": ["E12", "E12", "E12", "E31", "E23", "E23", "E23"],
+        "each_factor_is": "exp(Eij) = I + Eij because Eij^2 = 0",
+        "product": [[int(value) for value in row] for row in product],
+        "I_plus_product": [
+            [int(value) for value in row] for row in identity + product
+        ],
+        "exact_integer_determinant": determinant,
+    }
+
+
+def main() -> None:
+    parser = argparse.ArgumentParser()
+    parser.add_argument(
+        "--trials",
+        type=int,
+        default=250,
+        help="TN trials per dimension/depth cell (default: 250)",
+    )
+    parser.add_argument(
+        "--anchor-trials",
+        type=int,
+        default=50,
+        help="known-class anchor trials per cell (default: 50)",
+    )
+    parser.add_argument("--seed", type=int, default=121)
+    parser.add_argument(
+        "--output",
+        type=Path,
+        default=Path(__file__).with_name("results.json"),
+    )
+    args = parser.parse_args()
+
+    rng = np.random.default_rng(args.seed)
+    results = {
+        "seed": args.seed,
+        "known_anchors": test_known_anchors(rng, args.anchor_trials),
+        "tn_semigroup": test_tn_semigroup(
+            rng,
+            trials_per_cell=args.trials,
+            minor_checks_per_cell=2,
+            identity_checks_per_cell=3,
+        ),
+        "fock_trace_check": fock_trace_check(rng),
+        "novelty_obstruction_checks": novelty_obstruction_checks(),
+        "exact_closing_edge_counterexample": (
+            exact_closing_edge_counterexample()
+        ),
+    }
+    args.output.write_text(
+        json.dumps(results, indent=2, sort_keys=True), encoding="utf-8"
+    )
+    print(json.dumps(results, indent=2, sort_keys=True), flush=True)
+
+    known = results["known_anchors"]
+    tn = results["tn_semigroup"]
+    fock = results["fock_trace_check"]
+    obstruction = results["novelty_obstruction_checks"]
+    counterexample = results["exact_closing_edge_counterexample"]
+
+    failed = (
+        known["split_lie"]["negative_count"] != 0
+        or known["split_cone"]["negative_count"] != 0
+        or known["component_controls"]["O_minus_minus_determinant"] >= 0
+        or abs(known["component_controls"]["mixed_component_determinant"]) > 1e-12
+        or tn["determinant_below_one_count"] != 0
+        or tn["minor_violation_count"] != 0
+        or tn["maximum_principal_identity_error"] > 2e-7
+        or fock["absolute_difference"] > 2e-8
+        or any(
+            value != 1
+            for value in obstruction["dirac_commutant_dimensions"]
+        )
+        or any(
+            value != 0
+            for value in obstruction["majorana_skew_J2_nullities"]
+        )
+        or counterexample["exact_integer_determinant"] != -1
+    )
+    if failed:
+        raise SystemExit(1)
+
+
+if __name__ == "__main__":
+    main()