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Algorithmic construction of exact extremal states for fermionic moment polytopes: a certified census through rank 10

James Orlando

Published Jul 11, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.21313736

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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.
A proof assistant's kernel evaluates by reducing terms symbolically—and that is slow. Asking the kernel to sum a stabilizer code's weight enumerator over its 224 elements is hours of reduction; the same sum, compiled, is milliseconds. This gap is not a footnote. It means a machine-checked certificate is not one thing: for an object too large for the kernel to reduce, certification splits into a ladder. On the first rung the kernel proves the method exact—that bucket elimination equals the flat sum, a small general theorem with no compiler trust. On the second the compiler executes the instance—the actual census over a real code's generators, carrying only the compiler-correctness axiom. On the third the consequences are axiom-free: the quantum MacWilliams identity forces the normalizer spectrum, the code distance, and the sub-threshold error-rate prefactor from the executed spectrum, with an empty footprint. We make this ladder the object of study, draw its trust boundary exactly, and certify the rotated surface code [[25, 1, 5]]—distance 5, prefactor 160—past a census the kernel cannot touch. The ladder has an outer wall that is a theorem, not a limitation: computing an arbitrary code's exact spectrum is #P-hard, so the bounded-treewidth codes—the surface and toric family that hardware actually deploys—are the tractable island by complexity theory, and the determinant shortcut is provably blocked on the coupled CSS enumerator by a local free-fermion obstruction. We also state the discipline the ladder demands: a certificate that reads back a constant it was handed certifies nothing; certification must be tied to the actual object, and we show the difference is a theorem that fails when the tie is cut. Published by The Institute for Applied Ontological Mathematics.
This research establishes an exact, matrix-free framework for spectral analysis of Fibonacci-constrained quantum systems, demonstrating how exponentially expanding Hilbert spaces can admit computational descriptions whose complexity grows only linearly with chain length at any fixed spectral moment order. The investigation focuses on the open PXP quantum chain, whose admissible configurations obey a nearest-neighbor exclusion constraint and form a Hilbert space of dimension dim(Hₙ) = Fₙ₊₂. By combining Fibonacci combinatorics, Zeckendorf indexing, temporal-mask transfer operators, and exact integer arithmetic, the study develops a unified mathematical approach to spectral moments, constrained quantum dynamics, and gauge-sensitive observables without explicitly constructing the exponentially large Hamiltonian matrix. The central result is a universal rational hierarchy of even spectral moments, expressed through the generating function Σₙ₌₀∞ Tr(Aₙ²ᵐ) zⁿ = Pₘ(z) / (1 − z − z²)ᵐ⁺¹, with deg Pₘ ≤ 3m + 2. This theorem establishes that every fixed-order even spectral moment possesses a rational generating function governed by a power of the Fibonacci characteristic polynomial. Consequently, infinitely many spectral moments across increasing chain lengths can be reconstructed from a finite collection of exact initial values. Explicit rational expressions are derived for the fourth, sixth, and eighth spectral moments, accompanied by integer recurrence relations and independently reproducible computational certificates. The underlying temporal-mask transfer identity reduces fixed-order moment evaluation to O(n) arithmetic operations, even with spatially inhomogeneous hopping amplitudes and local diagonal fields. Its computational constants depend exponentially on moment order, but not on the exponentially growing Hilbert-space dimension. A complementary result introduces an exact gauge-sensitive fourth-moment certificate for quantum chains carrying occupation-dependent hopping phases. The spectral response obeys the identity Tr(Hₙ,φ⁴) − Tr(Hₙ,₀⁴) = 8(cos φ − 1) Sₙ, where Sₙ is obtained through a Fibonacci-based weighted recurrence requiring only O(n) arithmetic operations. Remarkably, this identity remains valid under arbitrary static real diagonal disorder, provided the diagonal energies and hopping magnitudes remain unchanged between phase settings. It therefore isolates a precisely defined gauge-dependent contribution to the fourth spectral moment without requiring diagonalization or knowledge of the complete spectrum. The certificate reconstructs a dimensionless flux response rather than an absolute quantum-action scale. The broader significance lies in demonstrating that exponential state-space growth does not necessarily imply exponential computational complexity for every exact spectral observable. The results connect constrained quantum many-body physics, Fibonacci graph theory, rational generating functions, and finite-state transfer methods within a common algebraic framework. All reported coefficients are supported by exact-integer verification rather than numerical fitting, and the established complexity bounds concern arithmetic operations rather than total bit complexity. The work provides a reproducible mathematical foundation for further investigations of spectral invariants, gauge responses, quantum constraints, and scalable symbolic computation, while restricting its formal claims to the specified finite-dimensional Hamiltonian models. Keywords: Fibonacci-constrained quantum chains · PXP Hamiltonian · Rational spectral hierarchy · Matrix-free computation · Temporal-mask transfer · Zeckendorf indexing · Gauge invariants · Exact spectral moments · Quantum many-body systems · Integer verification
We prove, computer-assisted, that every elliptic quadric Q−(7,2) of the four-qubit symplectic polar space W(7,2) has contextuality degree 315. Hence d(W(7,2)) = 1575, closing the published bracket 1071 ≤ d ≤ 1575 (Muller et al., J. Phys. A 58 (2025) 215302), and every five-qubit hyperbolic quadric has d(Q+(9,2)) = 6975. These results settle the first open problem of Li, Gupta and Ganesh (arXiv:2610.26465) and answer their question whether d(Q+(9,2)) = 4743 in the negative. The optimal assignments of Q−(7,2) (391,680 gauge classes), W(7,2) (1,175,040) and Q+(9,2) (2,350,080) each form a single orbit of the symmetry group. For the hyperbolic quadric Q+(7,2), the degree 315 and the 240 optimal classes (one orbit) were first proved by Li, Gupta and Ganesh; our independent count agrees. The note also proves a section–square law: for a quadric Q of the n-qubit space and a point d off it, the completion problem of a section carrier on Q ∩ d⊥ ≅ W(2n−3,2) is governed by the anticommutator action of the (n−1)-qubit carrier operator on the exterior square (elliptic Q) or symmetric square (hyperbolic Q), with spectrum {(μi + μj)/2}. At four qubits this explains the two tight carriers of Q−(7,2): the Hoggar carrier (spectrum {21, −3^7}; its optimal completions are exactly the 144 sign vectors of one eigenspace) and a second carrier with characteristic polynomial (t+3)^5 (t^3 − 15t^2 − 117t + 579). The proof is regenerated here with new code in intrinsic three-qubit coordinates, and every input is shipped. The package contains the exhaustive census program (C), 42 exact lower-bound certificates checked by integer Bareiss elimination, a standard-library Python verifier that rebuilds the Pauli algebra from 2×2 matrices (ALL CHECKS PASS in seconds; 13 single-input mutations rejected), the classification programs, every run record, and the report of an independent re-implementation (a separate AI agent writing its own code from the definitions) that reproduced the census, every completion minimum and all 42 certificates. A proof log for the census step (LRAT, checked with the formally verified checker cake_lpr) was produced separately and is not part of this record. Computer-assisted; not formally verified; not peer reviewed. Computations and drafting were carried out with an AI assistant (Claude, Anthropic) under the author's direction.