Breakpoint graphs originate in comparative genomics, where their alternating cycles encode relationships between genomes. We study a constrained class of three-colored breakpoint graphs associated with permutations and develop cycle-refined generating functions for two extremal families. These families have a natural topological interpretation: their canonical surfaces are, respectively, the sphere and the projective plane. The spherical family is characterized by noncrossing configurations, while the projective-plane family admits a different decomposition involving a distinguished family of Möbius ladders.
The resulting generating-function equations retain the full cycle structure but nevertheless admit substantial reductions. This leads to explicit Catalan-weighted evaluations, polynomiality results for refined cycle statistics, and a connection between a natural diagonal specialization and noncrossing trees. The two topological families exhibit markedly different combinatorial mechanisms, providing complementary examples of how local transformations of breakpoint graphs can control refined permutation enumerations.
As a further application, the same Catalan-weighted sums arise in asymptotic unitary-Weingarten expansions for entanglement of random Gaussian states in linear optics. The combinatorial results determine the leading and constant-order moment polynomials entering the Rényi entropy expansion, with the projective-plane contribution giving the finite-size constant correction.
Breakpoint graphs originate in comparative genomics, where their alternating cycles encode relationships between genomes. We study a constrained class of three-colored breakpoint graphs associated with permutations and develop cycle-refined generating functions for two extremal families. These families have a natural topological interpretation: their canonical surfaces are, respectively, the sphere and the projective plane. The spherical family is characterized by noncrossing configurations, while the projective-plane family admits a different decomposition involving a distinguished family of Möbius ladders.
The resulting generating-function equations retain the full cycle structure but nevertheless admit substantial reductions. This leads to explicit Catalan-weighted evaluations, polynomiality results for refined cycle statistics, and a connection between a natural diagonal specialization and noncrossing trees. The two topological families exhibit markedly different combinatorial mechanisms, providing complementary examples of how local transformations of breakpoint graphs can control refined permutation enumerations.
As a further application, the same Catalan-weighted sums arise in asymptotic unitary-Weingarten expansions for entanglement of random Gaussian states in linear optics. The combinatorial results determine the leading and constant-order moment polynomials entering the Rényi entropy expansion, with the projective-plane contribution giving the finite-size constant correction.
We extend the Shannon-blind contextual information residue separation theorem fromcycle graphs to arbitrary finite graph families and introduce a partition-function formalismfor predicting error-regime boundaries. Our contributions are threefold. (1) We generalize theČech cohomological obstruction to parameterized n-cycles (proved for all n ≥ 3), completegraphs (K4 , with dim H 1 = 3), and trees (negative control: dim H 1 = 0, separation collapses).The polynomial degree barrier persists across all tested cycle lengths n ∈ {3, 4, 5, 7, 10, 15, 20}.P(2) We define a finite Čech Hamiltonian and partition function Z(β) = ω exp(−β · H(ω))on the measurement-cover graph, whose heat-capacity peak identifies a critical inversetemperature β ∗ predicting the error regime. (3) We present a falsifiability surface: the regimeclassifier (exact-decoder-required vs. surrogate-sufficient) is correct for all 8 tested graphfamilies, while the quantitative threshold predictions are falsified for 5 of 8 families undertolerance 0.03. All proofs are verified in Lean 4 with zero sorry declarations; all experimentsare exhaustive finite-graph enumerations reproducible from a single command. The separationis binary: any non-trivial cycle topology (H 1 ≥ 1) activates the full polynomial barrier,making exact symbolic computation the only reliable decoder for topologically structuredworkloads.
The generalized Pauli constraints of Altunbulak and Klyachko [Comm. Math. Phys. 282, 287 (2008)] characterize the moment polytopes of fermionic natural occupation numbers. Two vertices of the ∧4H9 polytope—(16,16,16,6,6,6,6,6,6)/21 and (20,14,14,14,14,4,4,4,4)/23—were verified in that work “only numerically,” and the completeness of the published ∧4H10constraint list rests on them. We close both gaps with explicit states whose extremality admits elementary, hand-checkable proofs: a seven-determinant real state for the first vertex, and an eight-determinant state for the second carrying a single nontrivial phase, cos γ = 3/(4√14)—to our knowledge the first genuinely complex extremal state exhibited for these polytopes. Real amplitudes are provably insufficient within its mechanism class; whether any real state attains this vertex remains open. The construction motivates a general notion of weighted-design extremal states (independent-set support, integer weights, exactly diagonal one-body reduced density matrix), for which vertex attainability reduces to integer combinatorics. An exhaustive census shows that all vertices of the ∧3H6 and ∧3H7 polytopes and 22 of the 38 vertices of ∧3H8 are design-attainable, while the second open ∧4H9 vertex is provably not (mixed-integer infeasibility certificate): its extremal states necessarily involve phase interference between one-hop-connected configurations, and an exhaustive enumeration further shows no real-amplitude single-block state exists, so the interference is irreducibly complex. The historically difficult vertices—including the exotic (5,5,5,5,2,2,2,2) family of ∧3H8—all fall in the interference class. We relate the dichotomy to the superselection rule of Liebert et al. [Phys. Rev. Research 7, 023247 (2025)], report the first computations of canonical forms and adjoint degrees (in the sense of Arkani-Hamed–Bai–Lam) for all rigorously solved fermionic moment polytopes, and describe the numerical methods—an alternating-projection solver for prescribed occupation spectra and a sparsifying flow run as a multi-start swarm—by which the explicit state was found before being proved.
We develop a coherence-geometric framework in which transport processes on projective state spaces generate spectral determinant structures and geometric state-counting relations. The construction begins from a normalized complex coherence variable and its extension to a three-component coherence manifold whose physical state space is naturally identified with a complex projective geometry. Closed coherence cycles define holonomy operators analogous to Wilson loops in gauge theory. The corresponding spectral determinant generates a partition function whose cycle invariant is determined by the accessible volume of the underlying coherence manifold. This geometric state-counting principle establishes a direct connection between projective geometry, spectral theory, and coherence transport. A fundamental spectral invariant emerges from the equality between spectral and geometric partition structures and subsequently generates a hierarchy of higher-order coherence quantities. The resulting framework exhibits close mathematical connections to geometric quantum mechanics, Kähler geometry, geometric quantization, spectral geometry, and non-Abelian gauge theory. Rather than modifying established gauge theories, the formalism is proposed as a possible deeper coherence-geometric layer whose effective transport structures reproduce familiar holonomy properties while suggesting a common geometric origin for certain spectral hierarchies appearing in theoretical physics.
We investigate the entanglement dynamics of quantum states defined on graphs with non-local Ising interactions governed by the Katz kernel of the underlying network. The interaction pattern is physically motivated by a gapped fermionic mediator propagating on the same graph, whose perturbative elimination yields an effective Katz-weighted Ising Hamiltonian. Using the Entanglement Distance, we derive an exact analytical expression for the entanglement generated from an initially separable state and apply it to representative deterministic graph families.
We then characterize the dynamics in different propagation regimes. In the weak-Katz regime, the dynamics admits a systematic motif expansion with triangles entering at first order order and four-cycles, local degree structure, and overlappin triangles appearing at second order. In the strong-propagation regime, the interaction is instead dominated by the principal adjacency mode and by the localization properties of its eigenvector. For Erdős--Rényi graphs, the weak-propagation expansion can be averaged analytically, revealing a locally tree-like contribution in the sparse regime and saturation of the Entanglement Distance density in the dense regime.
Our results connect entanglement dynamics with both the walk-based and spectral structure of complex networks.
In Part I, we introduced a general framework for constructing logical gates in quantum CSS codes from cohomology invariants, focusing on cup products and their generalizations from topological to qLDPC codes. In particular, we produced a family of Ck−1Z logical gates that we called copy-cup on k copies of the same quantum code implemented by finite-depth circuits. Here, we vastly extend this framework by introducing twists incorporating code automorphisms, higher twisted cohomology operations, and folding operations in relative cohomology. The framework includes operations lifting coefficients from Z/2 to Z/2k, enabling all diagonal logical gates such as single-qubit rotations R(2π/2k). Composing these operations with maps that select cohomology classes yields addressable logical gates. Examples include addressable T gates on three- and four-tori and gates in qLDPC codes beyond topological ones. We introduce cohomology operations, such as the higher twisted Pontryagin powers, that to our knowledge have not previously appeared in the mathematics literature. These operations arise from E∞-structures on cochains, with explicit constructions using cyclic resolutions and the surjection operad. We expect these results to be of independent interest in homotopy theory. Our formalism applies much more broadly to chain complexes carrying suitable operadic structures, and we expect this operadic perspective to uncover further connections between homotopy theory and fault-tolerant quantum gates.
Complex multiplication, coherent addition, and quadratic readout are related ingredients of quantum calculation, but they answer different reconstruction questions. We separate them on a supplied reciprocal carrier. The centralizer of a complex structure on a real two-plane is the complex algebra; its invertible transports are nevertheless not closed under the addition needed for destructive interference at a selected output. A one-outcome dephaser and a reversible phase kick show that certainty of a public outcome does not imply nondisturbance. Within an explicitly assumed positive quadratic response class, calibrated zero-leakage comparisons on a connected tree select the all-ones Gram kernel. We prove quantitative stability estimates and retain the distinction between unconditional record-local operations, selective conditioning, and a controlled inverse interaction. A local normalized-weight ansatz, arbitrary anchored orthogonal refinement, and additive coarse-graining of recorded outcomes force weights proportional to quadratic size. Equal binary refinement alone does not: a positive continuous increasing log-periodic function supplies a counterexample, while binary and ternary refinement at every size, with continuity, recover linearity in size. Real rotations implement the requisite refinements and controlled tagging without full complex control. Fixed-complement preservation and reciprocal reversal admit different generator sectors; independently switchable real mixers on a connected graph plus one addressable local phase generate the unitary Lie algebra. A common scalar phase does not. Supplementary finite action–phase and history examples expose the remaining calibration assumptions. The contribution is an assumption-controlled synthesis of conditional selectors and obstructions, not the introduction of the standard component constructions. The carrier, comparison response, control resources, tensor interpretation, and empirical probability interface are not derived from primitive complementarity.