Replica-three stabilizer multi-entropy via graph forms and coarse party partitions

Abstract

Every normalized pure qubit stabilizer state admits a graph form on which opposite nonzero ternary replica characters biject onto two- and three-block party partitions, expressing the replica-three multi-entropy and its coarse-grainings as sums of masked F2 adjacency ranks.

Following the replica convention of Gadde, Krishna and Sharma, Phys. Rev. D 106, 126001, in the anchored form of Akella, Iizuka and Miyata, arXiv:2607.06050v1 Appendix A, a q-party pure N-qubit state psi with colouring col : Fin N to Fin q (allowing empty parties) is contracted over replicas r in (Z/3Z)^{q-1}, with the last party q - 1 acting by the identity. By Van den Nest, Dehaene and De Moor, Phys. Rev. A 69, 022316, and Schlingemann, Quantum Inf. Comput. 2, 307-323, every normalized pure qubit stabilizer state is local-unitary equivalent up to a unit phase to a graph state |Gamma>. Akella and Gadde, arXiv:2610.27080v1 Section II Eqs. (7)-(9) and Appendix A, give an odd-replica rank formula for graph states whose n = 3 specialization groups nonzero F4 labels by equality partition; retaining the factor n^{-(q-2)} yields the coarse-partition rank sum proved below.

Definition 1.1 (Normalized N-qubit graph state).

$$\forall (N:\mathbb{N}),\forall (\Gamma :\mathrm{Matrix}\left(\mathrm{Fin}\left(N\right),\mathrm{Fin}\left(N\right),\mathrm{ZMod}\left(2\right)\right)),\mathrm{graphState}\left(\Gamma \right)=(2^{-\frac{(N:\mathbb{R})}{2}}:\mathbb{C})\cdot \mathrm{graphAmp}\left(\Gamma \right)$$

Formalization. D5/S3/Quantum/Entanglement/StabilizerMultiEntropy/GraphFormReplicaThreeRank.graphState (✓ std3).

Source. Repository-derived.

Acknowledgement. M. Van den Nest, J. Dehaene, B. De Moor (2004). Graphical description of the action of local Clifford transformations on graph states. DOI: 10.1103/PhysRevA.69.022316. URL: https://arxiv.org/abs/quant-ph/0308151v2.

Commentary.

Scales the unnormalized binary quadratic phase vector graphAmp(Gamma) by 2^{-N/2} so that |Gamma> has unit Hilbert norm.

Definition 1.2 (Graph form of a pure qubit state).

$$\forall (N:\mathbb{N}),\forall (\Gamma :\mathrm{Matrix}\left(\mathrm{Fin}\left(N\right),\mathrm{Fin}\left(N\right),\mathrm{ZMod}\left(2\right)\right)),\forall (\psi :((\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C})),\mathrm{IsGraphFormOf}\left(\Gamma ,\psi \right)\iff ((\mathrm{IsSymm}\left(\Gamma \right))\wedge (\forall (u:\mathrm{Fin}\left(N\right)),\Gamma (u,u)=0)\wedge (\exists (U:(\mathrm{Fin}\left(N\right)\to \mathrm{Matrix}\left(\mathrm{Fin}\left(2\right),\mathrm{Fin}\left(2\right),\mathbb{C}\right))),\exists (\lambda :\mathbb{C}),(\forall (u:\mathrm{Fin}\left(N\right)),U(u)\in \mathrm{unitaryGroup}\left(\mathrm{Fin}\left(2\right),\mathbb{C}\right))\wedge (\Vert \lambda \Vert =1)\wedge (\psi =\lambda \cdot \mathrm{mulVec}\left(\mathrm{tensorOp}\left(U\right),\mathrm{graphState}\left(\Gamma \right)\right))))$$

Formalization. D5/S3/Quantum/Entanglement/StabilizerMultiEntropy/GraphFormReplicaThreeRank.IsGraphFormOf (✓ std3).

Source. Repository-derived.

Acknowledgement. M. Van den Nest, J. Dehaene, B. De Moor (2004). Graphical description of the action of local Clifford transformations on graph states. DOI: 10.1103/PhysRevA.69.022316. URL: https://arxiv.org/abs/quant-ph/0308151v2.

Commentary.

Asserts that Gamma is symmetric over ZMod 2 with zero diagonal and that psi equals lambda * (U_0 x … x U_{N-1}) |Gamma> for single-qubit unitaries U(u) and a scalar lambda with ||lambda|| = 1.

Definition 1.3 (Anchored party translation vector).

$$\forall (q:\mathbb{N}),\forall (c:\mathrm{Fin}\left(q\right)),\forall (i:\mathrm{Fin}\left((q-1)\right)),\mathrm{partyDelta}\left(c,i\right)=\mathrm{ite}\left((i:\mathbb{N})=(c:\mathbb{N}),1,0\right)$$

Formalization. D5/S3/Quantum/Entanglement/StabilizerMultiEntropy/GraphFormReplicaThreeRank.partyDelta (✓ std3).

Source. Repository-derived.

Acknowledgement. S. Akella, N. Iizuka, A. Miyata (2026). Genuine Multi-Entropy in the Toric Code. DOI: 10.48550/arXiv.2607.06050. URL: https://arxiv.org/abs/2607.06050v2.

Commentary.

Assigns the c-th standard basis vector of (Z/3Z)^{q-1} to party c < q - 1 and the zero vector to the last party q - 1.

Definition 1.4 (Party evaluation of a ternary character).

$$\forall (q:\mathbb{N}),\forall (t:(\mathrm{Fin}\left((q-1)\right)\to \mathrm{ZMod}\left(3\right))),\forall (c:\mathrm{Fin}\left(q\right)),\mathrm{charEval}\left(t,c\right)=\sum _{(i:\mathrm{Fin}\left((q-1)\right))}t(i)\cdot \mathrm{partyDelta}\left(c,i\right)$$

Formalization. D5/S3/Quantum/Entanglement/StabilizerMultiEntropy/GraphFormReplicaThreeRank.charEval (✓ std3).

Source. Repository-derived.

Acknowledgement. S. Akella, N. Iizuka, A. Miyata (2026). Genuine Multi-Entropy in the Toric Code. DOI: 10.48550/arXiv.2607.06050. URL: https://arxiv.org/abs/2607.06050v2.

Commentary.

Evaluates f_t(c) = t . delta_c in Z/3Z, taking value t_c for c < q - 1 and 0 on the last party q - 1.

Definition 1.5 (Fibre partition of a ternary character).

$$\forall (q:\mathbb{N}),\forall (t:(\mathrm{Fin}\left((q-1)\right)\to \mathrm{ZMod}\left(3\right))),\mathrm{charPartition}\left(t\right)=\mathrm{ofSetoid}\left(\ker \left(\mathrm{charEval}\left(t\right)\right)\right)$$

Formalization. D5/S3/Quantum/Entanglement/StabilizerMultiEntropy/GraphFormReplicaThreeRank.charPartition (✓ std3).

Source. Repository-derived.

Acknowledgement. S. Akella, N. Iizuka, A. Miyata (2026). Genuine Multi-Entropy in the Toric Code. DOI: 10.48550/arXiv.2607.06050. URL: https://arxiv.org/abs/2607.06050v2.

Commentary.

Partitions Fin q into the nonempty fibres of charEval(t).

Definition 1.6 (Coarse-grained replica-three multi-entropy).

$$\forall (N:\mathbb{N}),\forall (q:\mathbb{N}),\forall (\pi :\mathrm{PartyPartition}\left(q\right)),\forall (col:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(q\right))),\forall (\psi :((\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C})),\mathrm{coarseMultiEntropy}\left(\pi ,col,\psi \right)=\mathrm{S}\left(3,\mathrm{card}\left(\mathrm{parts}\left(\pi \right)\right),\mathrm{coarseCol}\left(\pi ,\mathrm{symm}\left(\mathrm{equivFin}\left(\mathrm{parts}\left(\pi \right)\right)\right),col\right),\psi \right)$$

Formalization. D5/S3/Quantum/Entanglement/StabilizerMultiEntropy/GraphFormReplicaThreeRank.coarseMultiEntropy (✓ std3).

Source. Repository-derived.

Acknowledgement. S. Akella, N. Iizuka, A. Miyata (2026). Genuine Multi-Entropy in the Toric Code. DOI: 10.48550/arXiv.2607.06050. URL: https://arxiv.org/abs/2607.06050v2.

Commentary.

Evaluates S_3^{(|pi|)} on psi under the coarse-grained colouring induced by the canonical block enumeration of pi.

Theorem 1.7 (Unit replica-one contraction, graph-form existence, and replica-three rank contraction).

$$(\forall (N:\mathbb{N}),\forall (q:\mathbb{N}),\forall (col:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(q\right))),\forall (\psi :((\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C})),\sum _{(x:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right)))}{\Vert \psi (x)\Vert }^2=1\Rightarrow (\mathrm{Z}\left(1,q,col,\psi \right)=1))\wedge (\forall (N:\mathbb{N}),\forall (\psi :((\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C})),((\mathrm{StabilizedBy}\left(pauliSet,\psi \right))\wedge (\sum _{(x:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right)))}{\Vert \psi (x)\Vert }^2=1))\Rightarrow (\exists (\Gamma :\mathrm{Matrix}\left(\mathrm{Fin}\left(N\right),\mathrm{Fin}\left(N\right),\mathrm{ZMod}\left(2\right)\right)),\mathrm{IsGraphFormOf}\left(\Gamma ,\psi \right)))\wedge (\forall (N:\mathbb{N}),\forall (q:\mathbb{N}),\forall (\Gamma :\mathrm{Matrix}\left(\mathrm{Fin}\left(N\right),\mathrm{Fin}\left(N\right),\mathrm{ZMod}\left(2\right)\right)),\forall (\psi :((\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C})),\mathrm{IsGraphFormOf}\left(\Gamma ,\psi \right)\Rightarrow (\forall (col:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(q\right))),\forall (P:\mathrm{Finset}\left((\mathrm{Fin}\left((q-1)\right)\to \mathrm{ZMod}\left(3\right))\right)),((\neg (0\in P))\wedge (\forall (t:(\mathrm{Fin}\left((q-1)\right)\to \mathrm{ZMod}\left(3\right))),t\ne 0\Rightarrow (t\in P\iff \neg (-t\in P))))\Rightarrow ((\sum _{(x:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right)))}{\Vert \psi (x)\Vert }^2=1)\wedge (0<\mathrm{Re}\left(\mathrm{Z}\left(3,q,col,\psi \right)\right))\wedge (\mathrm{Im}\left(\mathrm{Z}\left(3,q,col,\psi \right)\right)=0)\wedge (\mathrm{Z}\left(3,q,col,\psi \right)=(2^{{\sum }_{t\in P}\mathrm{rank}\left(\mathrm{C}\left(\Gamma ,\mathrm{delta}\left(col\right),t\right)\right)}{)}^{-1}))))$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Entanglement/StabilizerMultiEntropy/GraphFormReplicaThreeRank.graph_form_existence_and_replica_rank_formula (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. S. Akella, N. Iizuka, A. Miyata (2026). Genuine Multi-Entropy in the Toric Code. DOI: 10.48550/arXiv.2607.06050. URL: https://arxiv.org/abs/2607.06050v2.

Commentary.

Shows that Z(1, q, col, psi) = 1 whenever sum_x ||psi(x)||^2 = 1; extracts a symmetric zero-diagonal graph form Gamma for every normalized pure qubit stabilizer state by factoring the reciprocal norm 2^{-N/2} out of the stabilizer normal-form scalar; and proves for every graph form Gamma of psi and every opposite-pair representative set P in (Z/3Z)^{q-1} that psi has unit norm and Z(3, q, col, psi) is a positive real number equal to 2^{-sum_{t in P} rank C(Gamma, delta(col), t)}.

Theorem 1.8 (Bijection of opposite ternary characters onto coarse partitions and the partition-rank sum for S_3^{(q)}).

$$(\forall (q:\mathbb{N}),\forall (t:(\mathrm{Fin}\left((q-1)\right)\to \mathrm{ZMod}\left(3\right))),\mathrm{charPartition}\left(-t\right)=\mathrm{charPartition}\left(t\right))\wedge (\forall (q:\mathbb{N}),2\le q\Rightarrow (\forall (P:\mathrm{Finset}\left((\mathrm{Fin}\left((q-1)\right)\to \mathrm{ZMod}\left(3\right))\right)),((\neg (0\in P))\wedge (\forall (t:(\mathrm{Fin}\left((q-1)\right)\to \mathrm{ZMod}\left(3\right))),t\ne 0\Rightarrow (t\in P\iff \neg (-t\in P))))\Rightarrow ((\mathrm{BijOn}\left(charPartition,P,\mathrm{twoOrThreePartitions}\left(q\right)\right))\wedge ((\mathrm{card}\left(P\right):\mathbb{R})=\frac{((3:\mathbb{R}{)}^{(q-1)}-1)}{2})\wedge (\frac{((3:\mathbb{R}{)}^{(q-1)}-1)}{2}=(\mathrm{stirlingSecond}\left(q,2\right):\mathbb{R})+(\mathrm{stirlingSecond}\left(q,3\right):\mathbb{R})))))\wedge (\forall (N:\mathbb{N}),\forall (q:\mathbb{N}),2\le q\Rightarrow (\forall (\Gamma :\mathrm{Matrix}\left(\mathrm{Fin}\left(N\right),\mathrm{Fin}\left(N\right),\mathrm{ZMod}\left(2\right)\right)),\forall (\psi :((\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C})),\mathrm{IsGraphFormOf}\left(\Gamma ,\psi \right)\Rightarrow (\forall (col:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(q\right))),\mathrm{S}\left(3,q,col,\psi \right)=\frac{\log \left(2\right)}{2\cdot 3^{(q-2)}}\cdot \sum _{\pi \in \mathrm{twoOrThreePartitions}\left(q\right)}\mathrm{partitionRank}\left(\Gamma ,col,\pi \right))))$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Entanglement/StabilizerMultiEntropy/GraphFormReplicaThreeRank.replica_three_coarse_partition_sum (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. S. Akella, N. Iizuka, A. Miyata (2026). Genuine Multi-Entropy in the Toric Code. DOI: 10.48550/arXiv.2607.06050. URL: https://arxiv.org/abs/2607.06050v2.

Commentary.

Because negation is a bijection of Z/3Z, charEval(-t) = -charEval(t) has the same fibres as charEval(t), giving charPartition(-t) = charPartition(t). For q >= 2 and nonzero t, charEval(t) vanishes at q - 1 and takes at least one nonzero value in Z/3Z, so charPartition(t) has two or three blocks. Identifying Fin q with Option(Fin(q - 1)) reduces injectivity on P to TernaryPartitionOrbit, while assigning 0 to the block of q - 1 and injective values in {1, 2} to the remaining one or two blocks proves surjectivity onto twoOrThreePartitions(q). Counting both sides yields (3^{q-1} - 1)/2 = S(q, 2) + S(q, 3), and C(Gamma, delta(col), t) coincides entrywise with maskedAdj(Gamma, col, charPartition(t)), transforming the replica rank sum into (log 2 / (2 * 3^{q-2})) sum_{pi in Pi_2(q) cup Pi_3(q)} partitionRank(Gamma, col, pi).

Theorem 1.9 (Party-permutation invariance, coarse-graining upper sum, and bipartite off-diagonal rank formula).

$$(\forall (N:\mathbb{N}),\forall (q:\mathbb{N}),2\le q\Rightarrow (\forall (\psi :((\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C})),((\mathrm{StabilizedBy}\left(pauliSet,\psi \right))\wedge (\sum _{(x:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right)))}{\Vert \psi (x)\Vert }^2=1))\Rightarrow (\forall (col:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(q\right))),\forall (\tau :\mathrm{Perm}\left(\mathrm{Fin}\left(q\right)\right)),\mathrm{S}\left(3,q,\tau \circ col,\psi \right)=\mathrm{S}\left(3,q,col,\psi \right))))\wedge (\forall (N:\mathbb{N}),\forall (q:\mathbb{N}),\forall (k:\mathbb{N}),\forall (\Gamma :\mathrm{Matrix}\left(\mathrm{Fin}\left(N\right),\mathrm{Fin}\left(N\right),\mathrm{ZMod}\left(2\right)\right)),\forall (\psi :((\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C})),\mathrm{IsGraphFormOf}\left(\Gamma ,\psi \right)\Rightarrow (\forall (col:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(q\right))),\forall (\pi :\mathrm{PartyPartition}\left(q\right)),2\le \mathrm{card}\left(\mathrm{parts}\left(\pi \right)\right)\Rightarrow (\forall (B:\mathrm{Equiv}\left(\mathrm{Fin}\left(k\right),\mathrm{parts}\left(\pi \right)\right)),(\mathrm{S}\left(3,k,\mathrm{coarseCol}\left(\pi ,B,col\right),\psi \right)=\mathrm{coarseMultiEntropy}\left(\pi ,col,\psi \right))\wedge (\mathrm{coarseMultiEntropy}\left(\pi ,col,\psi \right)=\frac{\log \left(2\right)}{2\cdot 3^{(\mathrm{card}\left(\mathrm{parts}\left(\pi \right)\right)-2)}}\cdot \sum _{\sigma \in \mathrm{twoOrThreePartitions}\left(q\right)\wedge (\pi \le \sigma )}\mathrm{partitionRank}\left(\Gamma ,col,\sigma \right)))))\wedge (\forall (N:\mathbb{N}),\forall (q:\mathbb{N}),2\le q\Rightarrow (\forall (\psi :((\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C})),((\mathrm{StabilizedBy}\left(pauliSet,\psi \right))\wedge (\sum _{(x:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right)))}{\Vert \psi (x)\Vert }^2=1))\Rightarrow (\forall (col:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(q\right))),\mathrm{coarseMultiEntropy}\left((\mathrm{bot}:\mathrm{PartyPartition}\left(q\right)),col,\psi \right)=\mathrm{S}\left(3,q,col,\psi \right))))\wedge (\forall (N:\mathbb{N}),\forall (q:\mathbb{N}),\forall (\Gamma :\mathrm{Matrix}\left(\mathrm{Fin}\left(N\right),\mathrm{Fin}\left(N\right),\mathrm{ZMod}\left(2\right)\right)),\forall (\psi :((\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C})),\mathrm{IsGraphFormOf}\left(\Gamma ,\psi \right)\Rightarrow (\forall (col:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(q\right))),\forall (\sigma :\mathrm{PartyPartition}\left(q\right)),\sigma \in \mathrm{partitionsOfCard}\left(q,2\right)\Rightarrow (\forall (X:\mathrm{Finset}\left(\mathrm{Fin}\left(q\right)\right)),\forall (Y:\mathrm{Finset}\left(\mathrm{Fin}\left(q\right)\right)),((\mathrm{parts}\left(\sigma \right)=\mathrm{pair}\left(X,Y\right))\wedge (X\ne Y))\Rightarrow (\mathrm{coarseMultiEntropy}\left(\sigma ,col,\psi \right)=\log \left(2\right)\cdot \mathrm{rank}\left(\mathrm{offDiagBlock}\left(\Gamma ,col,X,Y\right)\right)))))$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Entanglement/StabilizerMultiEntropy/GraphFormReplicaThreeRank.party_symmetry_coarse_graining_and_bipartite_entropy (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. S. Akella, N. Iizuka, A. Miyata (2026). Genuine Multi-Entropy in the Toric Code. DOI: 10.48550/arXiv.2607.06050. URL: https://arxiv.org/abs/2607.06050v2.

Commentary.

Pulling partitions back along a surjective party map m : Fin q to Fin k bijects twoOrThreePartitions(k) onto the two- and three-block partitions sigma >= ker(m) while preserving maskedAdj and partitionRank. When m is a permutation tau of Fin q, ker(tau) is the discrete partition bot, proving S(3, q, tau o col, psi) = S(3, q, col, psi). When m comes from any block listing B : Fin k to pi.parts with |pi| >= 2, ker(m) = pi, proving block-order independence S(3, k, coarseCol(pi, B, col), psi) = coarseMultiEntropy(pi, col, psi), recovery coarseMultiEntropy(bot, col, psi) = S(3, q, col, psi), and the upper-sum formula (log 2 / (2 * 3^{|pi|-2})) sum_{sigma >= pi} partitionRank(Gamma, col, sigma). For a two-block partition sigma = {X, Y}, sigma is maximal in twoOrThreePartitions(q) and bipartite_masked_antidiagonal_rank gives partitionRank(Gamma, col, sigma) = 2 rank(offDiagBlock(Gamma, col, X, Y)), so coarseMultiEntropy(sigma, col, psi) = log 2 * rank(Gamma_{X,Y}).

For a two-block partition sigma = {X, Y}, the identity S_3^{(2)}(sigma) = log 2 * rank_{F2}(Gamma_{X,Y}) matches the bipartite Renyi entanglement entropy of a graph state established by Hein, Eisert and Briegel, Phys. Rev. A 69, 062311. As an explicit three-party illustration, for the three-qubit chain graph state with adjacency edges (0, 1) and (1, 2) and one qubit per party (q = 3, col = id), the three two-block partitions {01, 2}, {02, 1}, {12, 0} and the single three-block partition {0, 1, 2} each produce a masked 3 x 3 binary matrix of rank 2; the coarse-partition sum therefore gives S_3^{(3)} = (log 2 / (2 * 3^1)) * (2 + 2 + 2 + 2) = (4/3) log 2, matching the chain formula (1 + 1/n) log 2 at n = 3 in Czech, Feng, Wu and Xie, arXiv:2606.06582v2 Appendix B Eq. (B.9).

References

• Truth anchor: D5/S3/Quantum/Entanglement/StabilizerMultiEntropy/GraphFormReplicaThreeRank.IsGraphFormOf
• Truth anchor: D5/S3/Quantum/Entanglement/StabilizerMultiEntropy/GraphFormReplicaThreeRank.charEval
• Truth anchor: D5/S3/Quantum/Entanglement/StabilizerMultiEntropy/GraphFormReplicaThreeRank.charPartition
• Truth anchor: D5/S3/Quantum/Entanglement/StabilizerMultiEntropy/GraphFormReplicaThreeRank.coarseMultiEntropy
• Truth anchor: D5/S3/Quantum/Entanglement/StabilizerMultiEntropy/GraphFormReplicaThreeRank.graphState
• Truth anchor: D5/S3/Quantum/Entanglement/StabilizerMultiEntropy/GraphFormReplicaThreeRank.graph_form_existence_and_replica_rank_formula
• Truth anchor: D5/S3/Quantum/Entanglement/StabilizerMultiEntropy/GraphFormReplicaThreeRank.partyDelta
• Truth anchor: D5/S3/Quantum/Entanglement/StabilizerMultiEntropy/GraphFormReplicaThreeRank.party_symmetry_coarse_graining_and_bipartite_entropy
• Truth anchor: D5/S3/Quantum/Entanglement/StabilizerMultiEntropy/GraphFormReplicaThreeRank.replica_three_coarse_partition_sum
• Dependency: D5/S3/Combinatorics/Partitions/CoarseThreeBlockRefinement
• Dependency: D5/S3/Combinatorics/Partitions/TernaryPartitionOrbit
• Dependency: D5/S3/Quantum/Entanglement/StabilizerMultiEntropyCollapse