A counterexample to real spectra with positive couplings

Abstract

A positive-coupling quantum-group spin chain has a nonreal eigenvalue.

Definition 1.1 (Tensor-basis digits).

$$\forall (n:\mathbb{N}),\forall (N:\mathbb{N}),\forall (x:\mathrm{Fin}\left((n+1{)}^N\right)),\forall (k:\mathrm{Fin}\left(N\right)),\mathrm{digit}\left(x,k\right)=\mathrm{fin}\left(\mathrm{mod}\left(\mathrm{div}\left(\mathrm{val}\left(x\right),(n+1{)}^{N-1-\mathrm{val}\left(k\right)}\right),n+1\right),n+1\right)$$

Formalization. D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.digit (✓ std3).

Source. Repository-derived.

Commentary.

The tensor basis is indexed by base-(n+1) words of length N, with the leftmost digit first. Throughout, val(x) is the natural value of a finite index, and fin(t,d) is the element of Fin(d) with value t; its bound is supplied by the displayed domains. div and mod are natural-number division and remainder, and subtraction between natural numbers is truncated. cast(A,B,t) denotes the canonical coercion of t from A to B, including every real-to-complex and natural-to-real coercion in the formulas.

Definition 1.2 (Real q-numbers).

$$\forall (\gamma :\mathbb{R}),\forall (t:\mathbb{R}),\mathrm{qnum}\left(\gamma ,t\right)=\frac{\sin \left(\gamma \cdot t\right)}{\sin \left(\gamma \right)}$$

Formalization. D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.qnum (✓ std3).

Citation. A. G. Bytsko (2011). Non-Hermitian spin chains with inhomogeneous coupling. DOI: 10.1090/S1061-0022-2011-01148-3. URL: https://arxiv.org/abs/0911.4476v3.

Commentary.

The q-number [t] is sin(γt)/sin(γ). Real division is total, so this definition also has a value when sin(γ)=0. The spectral claim below restricts γ to the nonzero admissible range.

Definition 1.3 (The deformation parameter).

$$\forall (\gamma :\mathbb{R}),q\left(\gamma \right)=\mathrm{cast}\left(\mathbb{R},\mathbb{C},\cos \left(\gamma \right)\right)+I\cdot \mathrm{cast}\left(\mathbb{R},\mathbb{C},\sin \left(\gamma \right)\right)$$

Formalization. D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.q (✓ std3).

Citation. A. G. Bytsko (2011). Non-Hermitian spin chains with inhomogeneous coupling. DOI: 10.1090/S1061-0022-2011-01148-3. URL: https://arxiv.org/abs/0911.4476v3.

Commentary.

The parameter q=exp(iγ) lies on the unit circle; I denotes the imaginary unit. The defining algebra relations are [E,F]=(K²−K⁻²)/(q−q⁻¹), KE=qEK and KF=q⁻¹FK, as in Eq. (Uq).

Definition 1.4 (Weights in the finite basis).

$$\forall (n:\mathbb{N}),\forall (i:\mathrm{Fin}\left(n+1\right)),\mathrm{weight}\left(n,i\right)=\mathrm{cast}\left(\mathbb{N},\mathbb{R},\mathrm{val}\left(i\right)\right)-\frac{\mathrm{cast}\left(\mathbb{N},\mathbb{R},n\right)}{2}$$

Formalization. D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.weight (✓ std3).

Source. Repository-derived.

Commentary.

The natural parameter n is twice the spin S. Basis index i has weight val(i)−n/2, so the increasing index order runs from −S to S.

Definition 1.5 (The raising operator).

$$\forall (n:\mathbb{N}),\forall (\gamma :\mathbb{R}),\forall (i:\mathrm{Fin}\left(n+1\right)),\forall (j:\mathrm{Fin}\left(n+1\right)),E\left(n,\gamma \right)\left(i,j\right)=\mathrm{ite}\left((\mathrm{val}\left(i\right)=\mathrm{val}\left(j\right)+1),\mathrm{cast}\left(\mathbb{R},\mathbb{C},\sqrt{\mathrm{qnum}\left(\gamma ,\mathrm{cast}\left(\mathbb{N},\mathbb{R},n-\mathrm{val}\left(j\right)\right)\right)\cdot \mathrm{qnum}\left(\gamma ,\mathrm{cast}\left(\mathbb{N},\mathbb{R},\mathrm{val}\left(j\right)+1\right)\right)}\right),0\right)$$

Formalization. D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.E (✓ std3).

Citation. A. G. Bytsko (2011). Non-Hermitian spin chains with inhomogeneous coupling. DOI: 10.1090/S1061-0022-2011-01148-3. URL: https://arxiv.org/abs/0911.4476v3.

Commentary.

In the basis of weights −S,−S+1,…,S, the matrix entry with row i and column j raises the weight by one. Its coefficient is the square root in Eq. (EFK), with n=2S. The square root is taken in the reals before its complex coercion.

Definition 1.6 (The lowering operator).

$$\forall (n:\mathbb{N}),\forall (\gamma :\mathbb{R}),\forall (i:\mathrm{Fin}\left(n+1\right)),\forall (j:\mathrm{Fin}\left(n+1\right)),F\left(n,\gamma \right)\left(i,j\right)=\mathrm{ite}\left((\mathrm{val}\left(j\right)=\mathrm{val}\left(i\right)+1),\mathrm{cast}\left(\mathbb{R},\mathbb{C},\sqrt{\mathrm{qnum}\left(\gamma ,\mathrm{cast}\left(\mathbb{N},\mathbb{R},\mathrm{val}\left(j\right)\right)\right)\cdot \mathrm{qnum}\left(\gamma ,\mathrm{cast}\left(\mathbb{N},\mathbb{R},n-\mathrm{val}\left(j\right)+1\right)\right)}\right),0\right)$$

Formalization. D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.F (✓ std3).

Citation. A. G. Bytsko (2011). Non-Hermitian spin chains with inhomogeneous coupling. DOI: 10.1090/S1061-0022-2011-01148-3. URL: https://arxiv.org/abs/0911.4476v3.

Commentary.

This is the lowering operator of Eq. (EFK), in the same increasing-weight basis. The subtraction n−val(j)+1 is performed in the naturals before coercion to the reals.

Definition 1.7 (The diagonal weight operator).

$$\forall (n:\mathbb{N}),\forall (\gamma :\mathbb{R}),K\left(n,\gamma \right)=\mathrm{diagonal}\left((\lambda (i:\mathrm{Fin}\left(n+1\right)),q\left(\gamma \cdot \mathrm{weight}\left(n,i\right)\right))\right)$$

Formalization. D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.K (✓ std3).

Citation. A. G. Bytsko (2011). Non-Hermitian spin chains with inhomogeneous coupling. DOI: 10.1090/S1061-0022-2011-01148-3. URL: https://arxiv.org/abs/0911.4476v3.

Commentary.

The diagonal entry at i is exp(iγ(val(i)−n/2)), the action of K in Eq. (EFK). diagonal(f) has entries f(i) on the diagonal and zero off the diagonal.

Definition 1.8 (The inverse weight operator).

$$\forall (n:\mathbb{N}),\forall (\gamma :\mathbb{R}),\mathrm{Kinv}\left(n,\gamma \right)=\mathrm{diagonal}\left((\lambda (i:\mathrm{Fin}\left(n+1\right)),q\left(-(\gamma \cdot \mathrm{weight}\left(n,i\right))\right))\right)$$

Formalization. D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.Kinv (✓ std3).

Citation. A. G. Bytsko (2011). Non-Hermitian spin chains with inhomogeneous coupling. DOI: 10.1090/S1061-0022-2011-01148-3. URL: https://arxiv.org/abs/0911.4476v3.

Commentary.

The diagonal entries are exp(−iγ(val(i)−n/2)), giving the inverse of K.

Definition 1.9 (The tensor Casimir).

$$\forall (n:\mathbb{N}),\forall (\gamma :\mathbb{R}),\mathrm{tensorCasimir}\left(n,\gamma \right)=(\mathrm{let}de=\mathrm{tensor}\left(E\left(n,\gamma \right),\mathrm{Kinv}\left(n,\gamma \right)\right)+\mathrm{tensor}\left(K\left(n,\gamma \right),E\left(n,\gamma \right)\right);(\mathrm{let}df=\mathrm{tensor}\left(F\left(n,\gamma \right),\mathrm{Kinv}\left(n,\gamma \right)\right)+\mathrm{tensor}\left(K\left(n,\gamma \right),F\left(n,\gamma \right)\right);(\mathrm{let}dk=\mathrm{tensor}\left(K\left(n,\gamma \right),K\left(n,\gamma \right)\right);(\mathrm{let}dki=\mathrm{tensor}\left(\mathrm{Kinv}\left(n,\gamma \right),\mathrm{Kinv}\left(n,\gamma \right)\right);\mathrm{smul}\left(\frac{1}{2},de\cdot df+df\cdot de\right)-\mathrm{smul}\left(\mathrm{cast}\left(\mathbb{R},\mathbb{C},\frac{\cos \left(\gamma \right)}{4\cdot (\sin \left(\gamma \right){)}^2}\right),(dk-dki)\cdot (dk-dki)\right)))))$$

Formalization. D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.tensorCasimir (✓ std3).

Citation. A. G. Bytsko (2011). Non-Hermitian spin chains with inhomogeneous coupling. DOI: 10.1090/S1061-0022-2011-01148-3. URL: https://arxiv.org/abs/0911.4476v3.

Commentary.

Here tensor is the Kronecker product with multiplication of complex entries. The four local expressions de, df, dk and dki are the represented coproducts Δ(E), Δ(F), Δ(K) and Δ(K⁻¹). Thus the formula is (π⊗π)Δ(C), using the coproduct of Eq. (DelEFK) and the Casimir C=(EF+FE)/2−cos(γ)(K−K⁻¹)²/(4sin²(γ)) of Eq. (C). It follows the definitional left side of Eq. (dC2). smul denotes scalar multiplication.

Definition 1.10 (Casimir values on irreducible summands).

$$\forall (\gamma :\mathbb{R}),\forall (l:\mathbb{N}),\mathrm{casimirValue}\left(\gamma ,l\right)=\mathrm{qnum}\left(\gamma ,\mathrm{cast}\left(\mathbb{N},\mathbb{R},l\right)\right)\cdot \mathrm{qnum}\left(\gamma ,\mathrm{cast}\left(\mathbb{N},\mathbb{R},l+1\right)\right)$$

Formalization. D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.casimirValue (✓ std3).

Citation. A. G. Bytsko (2011). Non-Hermitian spin chains with inhomogeneous coupling. DOI: 10.1090/S1061-0022-2011-01148-3. URL: https://arxiv.org/abs/0911.4476v3.

Commentary.

On the spin-l irreducible summand the Casimir value is [l][l+1], as stated after Eq. (C). The successor l+1 is a natural number before coercion.

Definition 1.11 (Products in increasing index order).

$$\forall (\alpha :\mathrm{Type}),([\mathrm{Monoid}\left(\alpha \right)])\to (\forall (f:(\mathbb{N})\to (\alpha )),(\mathrm{orderedProduct}\left(f,0\right)=1)\wedge (\forall (k:\mathbb{N}),\mathrm{orderedProduct}\left(f,k+1\right)=\mathrm{orderedProduct}\left(f,k\right)\cdot f\left(k\right)))$$

Formalization. D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.orderedProduct (✓ std3).

Source. Repository-derived.

Commentary.

The empty product is one. Each recursive step multiplies on the right by the next factor, fixing an order even in a noncommutative monoid. Bracketed typeclass assumptions in the displays are anonymous instance arguments.

Definition 1.12 (The Casimir spectral projector).

$$\forall (n:\mathbb{N}),\forall (s:\mathbb{N}),\forall (\gamma :\mathbb{R}),\mathrm{projector}\left(n,s,\gamma \right)=\mathrm{orderedProduct}\left((\lambda (l:\mathbb{N}),\mathrm{ite}\left((l\ne s),\mathrm{smul}\left(\mathrm{inv}\left(\mathrm{cast}\left(\mathbb{R},\mathbb{C},\mathrm{qnum}\left(\gamma ,\mathrm{cast}\left(\mathbb{N},\mathbb{R},s\right)-\mathrm{cast}\left(\mathbb{N},\mathbb{R},l\right)\right)\right)\cdot \mathrm{cast}\left(\mathbb{R},\mathbb{C},\mathrm{qnum}\left(\gamma ,\mathrm{cast}\left(\mathbb{N},\mathbb{R},s+l+1\right)\right)\right)\right),\mathrm{tensorCasimir}\left(n,\gamma \right)-\mathrm{smul}\left(\mathrm{cast}\left(\mathbb{R},\mathbb{C},\mathrm{casimirValue}\left(\gamma ,l\right)\right),1\right)\right),1\right)),n+1\right)$$

Formalization. D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.projector (✓ std3).

Citation. A. G. Bytsko (2011). Non-Hermitian spin chains with inhomogeneous coupling. DOI: 10.1090/S1061-0022-2011-01148-3. URL: https://arxiv.org/abs/0911.4476v3.

Commentary.

With n=2S, this is P^{S,s} from Eq. (PjC). The product runs through l=0,…,n in increasing order, inserting the identity at l=s. Each other factor is (C^{S,S}−[l][l+1]I)/([s−l][s+l+1]). The denominator factors are separately coerced to the complexes before multiplication and inversion. In the spectral claim all these denominators are nonzero.

Definition 1.13 (Embedding an adjacent two-site operator).

$$\forall (R:\mathrm{Type}),([\mathrm{Zero}\left(R\right)])\to (\forall (n:\mathbb{N}),\forall (N:\mathbb{N}),\forall (m:\mathrm{Fin}\left(N-1\right)),\forall (P:\mathrm{Matrix}\left((\mathrm{Fin}\left(n+1\right))\times (\mathrm{Fin}\left(n+1\right)),(\mathrm{Fin}\left(n+1\right))\times (\mathrm{Fin}\left(n+1\right)),R\right)),\forall (x:\mathrm{Fin}\left((n+1{)}^N\right)),\forall (y:\mathrm{Fin}\left((n+1{)}^N\right)),\mathrm{siteOperator}\left(m,P\right)\left(x,y\right)=\mathrm{ite}\left((\forall (k:\mathrm{Fin}\left(N\right)),(k\ne \mathrm{fin}\left(\mathrm{val}\left(m\right),N\right))\to ((k\ne \mathrm{fin}\left(\mathrm{val}\left(m\right)+1,N\right))\to (\mathrm{digit}\left(x,k\right)=\mathrm{digit}\left(y,k\right)))),P\left((\mathrm{digit}\left(x,\mathrm{fin}\left(\mathrm{val}\left(m\right),N\right)\right),\mathrm{digit}\left(x,\mathrm{fin}\left(\mathrm{val}\left(m\right)+1,N\right)\right)),(\mathrm{digit}\left(y,\mathrm{fin}\left(\mathrm{val}\left(m\right),N\right)\right),\mathrm{digit}\left(y,\mathrm{fin}\left(\mathrm{val}\left(m\right)+1,N\right)\right))\right),0\right))$$

Formalization. D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.siteOperator (✓ std3).

Citation. A. G. Bytsko (2011). Non-Hermitian spin chains with inhomogeneous coupling. DOI: 10.1090/S1061-0022-2011-01148-3. URL: https://arxiv.org/abs/0911.4476v3.

Commentary.

The operator P acts at the adjacent positions val(m) and val(m)+1, numbered from zero. Its matrix entry vanishes unless all other tensor digits agree. This is the adjacent-site embedding used before Eq. (Hal), expressed in the base-(n+1) tensor basis.

Definition 1.14 (The open-chain Hamiltonian).

$$\forall (n:\mathbb{N}),\forall (s:\mathbb{N}),\forall (N:\mathbb{N}),\forall (\gamma :\mathbb{R}),\forall (a:(\mathrm{Fin}\left(N-1\right))\to (\mathbb{R})),\mathrm{hamiltonian}\left(n,s,N,\gamma ,a\right)=\sum _{m:\mathrm{Fin}\left(N-1\right)}(\mathrm{smul}\left(\mathrm{cast}\left(\mathbb{R},\mathbb{C},a\left(m\right)\right),\mathrm{siteOperator}\left(m,\mathrm{projector}\left(n,s,\gamma \right)\right)\right))$$

Formalization. D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.hamiltonian (✓ std3).

Citation. A. G. Bytsko (2011). Non-Hermitian spin chains with inhomogeneous coupling. DOI: 10.1090/S1061-0022-2011-01148-3. URL: https://arxiv.org/abs/0911.4476v3.

Commentary.

Equation (Hal) sums the adjacent projectors with real couplings. The index m in Fin(N−1) corresponds to the source’s bond m+1. Each coupling is coerced from the reals to the complexes before scalar multiplication.

Definition 1.15 (Positive couplings and the proposed real spectrum).

$$\mathrm{claim}\iff (\forall (n:\mathbb{N}),\forall (s:\mathbb{N}),\forall (N:\mathbb{N}),\forall (\gamma :\mathbb{R}),(1\le n)\to ((s\le n)\to ((2\le N)\to ((0<\left|\gamma \right|)\to ((\mathrm{cast}\left(\mathbb{N},\mathbb{R},n\right)\cdot \left|\gamma \right|<\pi )\to ((\forall (l:\mathbb{N}),(l\le n)\to ((l\ne s)\to (\mathrm{qnum}\left(\gamma ,\mathrm{cast}\left(\mathbb{N},\mathbb{R},s\right)-\mathrm{cast}\left(\mathbb{N},\mathbb{R},l\right)\right)\cdot \mathrm{qnum}\left(\gamma ,\mathrm{cast}\left(\mathbb{N},\mathbb{R},s+l+1\right)\right)\ne 0)))\to (\forall (a:(\mathrm{Fin}\left(N-1\right))\to (\mathbb{R})),(\forall (m:\mathrm{Fin}\left(N-1\right)),0<a\left(m\right))\to (\forall (v:(\mathrm{Fin}\left((n+1{)}^N\right))\to (\mathbb{C})),\forall (z:\mathbb{C}),(v\ne 0)\to ((\mathrm{mulVec}\left(\mathrm{hamiltonian}\left(n,s,N,\gamma ,a\right),v\right)=\mathrm{smul}\left(z,v\right))\to (\mathrm{im}\left(z\right)=0))))))))))$$

Formalization. D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.claim (✓ std3).

Citation. A. G. Bytsko (2011). Non-Hermitian spin chains with inhomogeneous coupling. DOI: 10.1090/S1061-0022-2011-01148-3. URL: https://arxiv.org/abs/0911.4476v3.

Commentary.

Conjecture 3 asserts a real spectrum for positive couplings when |γ|<π/(2S). Here n=2S≥1, 0≤s≤n and N≥2. The nonzero angle and explicit nonvanishing projector denominators make the formulas applicable. Every nonzero vector v satisfying H v=z v is asserted to have Im(z)=0.

Theorem 1.16 (A four-site spin-one counterexample).

$$\neg \mathrm{claim}$$

Proof. Machine-checked in Lean as D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.result (✓ std3). ∎

Resolves. Problems/bytsko-2009-positive-coupling-real-spectrum-refutation (refuted) by D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.result.

Source. Repository-derived.

Acknowledgement. A. G. Bytsko (2011). Non-Hermitian spin chains with inhomogeneous coupling. DOI: 10.1090/S1061-0022-2011-01148-3. URL: https://arxiv.org/abs/0911.4476v3.

Commentary.

Take S=s=1, hence n=2, N=4, γ=γ₀=arccos(4/5), sin(γ₀)=3/5, q=(4+3i)/5 and a=(1,1,1). These satisfy 0<2|γ|<π, and the two required denominator products, for l=0 and l=2, are 8/5 and −112/125.

Order the two-site basis lexicographically by weights −1,0,1. The matrix 14·P^{1,1} is [[0,0,0,0,0,0,0,0,0], [0,7+24i,0,−25,0,0,0,0,0], [0,0,25,0,30i,0,−25,0,0], [0,−25,0,7−24i,0,0,0,0,0], [0,0,30i,0,−36,0,−30i,0,0], [0,0,0,0,0,7+24i,0,−25,0], [0,0,−25,0,−30i,0,25,0,0], [0,0,0,0,0,−25,0,7−24i,0], [0,0,0,0,0,0,0,0,0]].

Index the four-site basis by x=27i₁+9i₂+3i₃+i₄, with iⱼ∈{0,1,2}. Define the 81-by-2 matrix W by its nonzero rows, whose entries are ordered as columns 0 and 1: 26:(0,4900); 44:(−5500−4125i,3080+8190i); 50:(5500−4125i,−7000+5250i); 62:(8800−6600i,−9828+3696i); 68:(7920+10560i,−10080−13440i); 70:(−5500+4125i,7000−5250i); 74:(−8800+6600i,6300−8400i); 76:(−2420−6435i,7000+5250i); 78:(0,−1372+4704i). All other rows vanish.

Let A=[[-11/14,1],[-275/196,25/14]] and B=[[1,0],[0,0]]. For the adjacent embeddings P₁₂,P₂₃,P₃₄ of P^{1,1}, direct matrix multiplication gives (P₁₂+P₃₄)W=WA and P₂₃W=WB. Therefore HW=W(A+B). Put z=1+i√(11/14)=1+i√(22/7)/2 and y=(1,z−1+11/14). The identity z²−2z+25/14=0 gives (A+B)y=zy. Consequently H(Wy)=z(Wy). The component at x=26 is 4900(z−1+11/14), whose imaginary part is positive; hence Wy is nonzero. Since Im(z)=√(11/14)>0, this positive-coupling chain refutes the universal real-spectrum assertion. The construction uses the algebra and projectors of Bytsko’s paper; it does not settle the restriction s=0.

References

• Truth anchor: D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.E
• Truth anchor: D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.F
• Truth anchor: D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.K
• Truth anchor: D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.Kinv
• Truth anchor: D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.casimirValue
• Truth anchor: D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.claim
• Truth anchor: D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.digit
• Truth anchor: D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.hamiltonian
• Truth anchor: D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.orderedProduct
• Truth anchor: D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.projector
• Truth anchor: D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.q
• Truth anchor: D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.qnum
• Truth anchor: D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.result
• Truth anchor: D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.siteOperator
• Truth anchor: D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.tensorCasimir
• Truth anchor: D5/S3/Quantum/SpinChains/QuantumGroupChainRealSpectrumRefutation.weight