Let X1, . . . , XN be a fixed finite family of centered self-adjoint observables in a faithful state ω, and let Γij = ω(XiXj ) be their full Hermitian GNS Gram matrix. We study the normalized inverse Gramian Bij = −Qij/ p QiiQjj , Q = Γ−1 , and show that it is the negative Gram matrix of a canonical normalized dual frame formed by conditional residual directions in the GNS Hilbert space. Consequently, cyclic products of the entries of B are Bargmann invariants of this conditional-residual frame. Combining B with the scale-free conditional-embedding levels pi = ΓiiQii yields a Hermitian commutator generator F = i[B, Hg], where Hg = diag(g(pi)) and g is strictly increasing. The phases of the nonzero entries of F define a magnetic connection on the precision graph. We prove four principal results. First, the induced cycle holonomy factorizes into a discrete ordering skeleton and the continuous Bargmann phase of the inverse GNS Gramian. Second, at a real Gramian C, the commutator-visible antisymmetric perturbation Γ(ε) = C + iεΣ maps with full local rank onto the entire cycle-flux space of the precision graph; hence the flat locus has local codimension equal to the first Betti number. Third, the lowest magnetic-Laplacian eigenvalue opens quadratically from a flat point, with coefficient equal to the weighted Hodge norm of the non-exact edge phase divided by total vertex measure. A complete perturbative proof is supplied. Fourth, every positive Hermitian Gramian admits an explicit faithful finite-dimensional realization by centered self-adjoint observables, so the construction is not merely formal. The relation to covariance selection and complex partial spectral coherence is stated explicitly: Hermitian precision matrices and complex partial-coherence phases are estab- lished objects. The proposed contribution is the GNS provenance, the conditional-residual Bargmann interpretation, the state-to-magnetic-connection map, the full-cycle-rank theo- rem, and the resulting spectral-gap transduction law. A deterministic four-cycle witness and a bitwise reproducible verification package are included. No identification of the resulting gap with a particle mass is claimed.
Let X1, . . . , XN be a fixed finite family of centered self-adjoint observables in a faithful state ω, and let Γij = ω(XiXj ) be their full Hermitian GNS Gram matrix. We study the normalized inverse Gramian Bij = −Qij/ p QiiQjj , Q = Γ−1 , and show that it is the negative Gram matrix of a canonical normalized dual frame formed by conditional residual directions in the GNS Hilbert space. Consequently, cyclic products of the entries of B are Bargmann invariants of this conditional-residual frame. Combining B with the scale-free conditional-embedding levels pi = ΓiiQii yields a Hermitian commutator generator F = i[B, Hg], where Hg = diag(g(pi)) and g is strictly increasing. The phases of the nonzero entries of F define a magnetic connection on the precision graph. We prove four principal results. First, the induced cycle holonomy factorizes into a discrete ordering skeleton and the continuous Bargmann phase of the inverse GNS Gramian. Second, at a real Gramian C, the commutator-visible antisymmetric perturbation Γ(ε) = C + iεΣ maps with full local rank onto the entire cycle-flux space of the precision graph; hence the flat locus has local codimension equal to the first Betti number. Third, the lowest magnetic-Laplacian eigenvalue opens quadratically from a flat point, with coefficient equal to the weighted Hodge norm of the non-exact edge phase divided by total vertex measure. A complete perturbative proof is supplied. Fourth, every positive Hermitian Gramian admits an explicit faithful finite-dimensional realization by centered self-adjoint observables, so the construction is not merely formal. The relation to covariance selection and complex partial spectral coherence is stated explicitly: Hermitian precision matrices and complex partial-coherence phases are estab- lished objects. The proposed contribution is the GNS provenance, the conditional-residual Bargmann interpretation, the state-to-magnetic-connection map, the full-cycle-rank theo- rem, and the resulting spectral-gap transduction law. A deterministic four-cycle witness and a bitwise reproducible verification package are included. No identification of the resulting gap with a particle mass is claimed.
We study the precision geometry generated by three self-adjoint observables in a faithful quantum state. The centered GNS Gramian Γ = C + iΣ, its inverse Q, normalized precision coefficients B, and conditional levels pi = ΓiiQii define a weighted magnetic triangle. The paper separates exact structure, reconstruction, and open prediction. First, a reduction identity gives pi = (1 − b 2 i )/ det(I − B). For the logarithmic embedding this implies an exact global rigidity theorem: if all three induced edge weights are positive, then the logarithmic circulant deviation obeys D > J∗ = 0.394202716596 . . . , with sharp infimum J ∗ approached only in the degenerate equal-b limit. The former numerical global-floor claim is therefore replaced by an analytic theorem, independently cross-checked by an interval branch-and-bound certificate. Second, the Pauli triple has Q = (I −i[n]×)/(1−|n| 2 ) and can realize only Hol = ±1. Real precision geometry gives instead Hol = ±i; the charged- lepton phase lies outside both quantized sectors. Third, the Koide–Foot square-root mass family is exactly the spectrum of an offset uniform magnetic Schr¨odinger operator on C3. Current 2026 PDG masses show that r = √ 2 and δ = 2/9 are each individually compatible after refitting the other parameter, but their simultaneous exactness is not: it fixes mµ/me about 451.7 quoted standard deviations from the measured ratio. Fourth, for the explicit modelling prescription µi = pi , a Krawczyk interval certificate proves an exact realization of the measured square-root mass ratios at the idealized Brannen flux Φ0 = π+2/3 on a fixed transversal slice; further Krawczyk certificates prove realizations at the exact decimal fluxes Φ = 1.105 and Φ = 1.4, and a continuation scan locates the feasibility boundary at Φ∗ ≈ 1.1044604 (record level). Finally, an implicit-function theorem turns a former caveat into a structural result: near the certified witness, the exact mass-ratio solutions form a three-dimensional manifold when (b1, Φ, α) are allowed to vary in the family µ = p α. Hence the two mass ratios cannot locally select the flux, the measure exponent, or the witness coordinate. The framework therefore provides exact rigidity, exclusions, and realizability, while identifying a parameter-free upstream selector as the precise missing ingredient required for prediction.
We study the precision geometry generated by three self-adjoint observables in a faithful quantum state. The centered GNS Gramian Γ = C + iΣ, its inverse Q, normalized precision coefficients B, and conditional levels pi = ΓiiQii define a weighted magnetic triangle. The paper separates exact structure, reconstruction, and open prediction. First, a reduction identity gives pi = (1 − b 2 i )/ det(I − B). For the logarithmic embedding this implies an exact global rigidity theorem: if all three induced edge weights are positive, then the logarithmic circulant deviation obeys D > J∗ = 0.394202716596 . . . , with sharp infimum J ∗ approached only in the degenerate equal-b limit. The former numerical global-floor claim is therefore replaced by an analytic theorem, independently cross-checked by an interval branch-and-bound certificate. Second, the Pauli triple has Q = (I −i[n]×)/(1−|n| 2 ) and can realize only Hol = ±1. Real precision geometry gives instead Hol = ±i; the charged- lepton phase lies outside both quantized sectors. Third, the Koide–Foot square-root mass family is exactly the spectrum of an offset uniform magnetic Schr¨odinger operator on C3. Current 2026 PDG masses show that r = √ 2 and δ = 2/9 are each individually compatible after refitting the other parameter, but their simultaneous exactness is not: it fixes mµ/me about 451.7 quoted standard deviations from the measured ratio. Fourth, for the explicit modelling prescription µi = pi , a Krawczyk interval certificate proves an exact realization of the measured square-root mass ratios at the idealized Brannen flux Φ0 = π+2/3 on a fixed transversal slice; further Krawczyk certificates prove realizations at the exact decimal fluxes Φ = 1.105 and Φ = 1.4, and a continuation scan locates the feasibility boundary at Φ∗ ≈ 1.1044604 (record level). Finally, an implicit-function theorem turns a former caveat into a structural result: near the certified witness, the exact mass-ratio solutions form a three-dimensional manifold when (b1, Φ, α) are allowed to vary in the family µ = p α. Hence the two mass ratios cannot locally select the flux, the measure exponent, or the witness coordinate. The framework therefore provides exact rigidity, exclusions, and realizability, while identifying a parameter-free upstream selector as the precise missing ingredient required for prediction.
The companion theory paper Quantum Precision Holonomy constructs a magnetic graph Laplacian LU from inverse GNS-Gramian data and proves a quadratic opening law for its lowest eigenvalue. The present paper addresses a narrower physical question: what does the dimensionless operator mean, and how is it assigned physical units in a system where the same operator class is an actual Hamiltonian? We use the programmable nano-optomechanical network of Slim, del Pino, and Verhagen. In the opposed-flux diamond at the special ratio Jc = √ 2J, one antisymmetric apex mode decouples and the remaining three modes are exactly those of an equal-coupling Aharonov– Bohm loop. For the unit-weight magnetic triangle, LU = 2I − AU , and the experimentally realized coupled-mode Hamiltonian satisfies Hphys h = νrefI + jeff(2I − LU ). Thus spectral splittings are linear in differences of eigenvalues of LU ; no particle-mass map is introduced. The independently reported perimeter coupling is J/(2π) = 5.000 kHz, implying jeff = 7.071 067 812 kHz. A one-parameter calibration on ten digitized gaps gives jeff = 6.877 941 342 kHz and hence J/(2π) = 4.863 438 964 kHz, a difference of −2.731%. On nine frozen held-out gaps, MAE = 0.344 229 kHz and RMSE = 0.423 010 kHz. The raster scale is approximately 0.303 kHz per vertical pixel; reported mechanical linewidths are approximately 1 kHz to 7 kHz. The small-flux QPH law becomes ∆νQPH = jeff 9 Φ 2 + O(Φ4 ), with independently calibrated coefficient 0.785 674 201 kHz rad−2 . This is a figure-level held- out consistency test of the downstream map (G, w, U) → LU → physical spectrum. It does not test the upstream derivation of (w, U) from a measured complex GNS Gramian. Keywords: magnetic graph Laplacian; optomechanics; Aharonov–Bohm phase; spectral calibra- tion; held-out test; Quantum Precision Holonomy. Scope. This paper is a self-contained empirical companion to the mathematical QPH paper. It defines every object needed for the calibration, derives the C3 spectral map, freezes one fit parameter and a prospective train/test split, and reports physical units and held-out errors. It does not repeat the full inverse-GNS-Gramian proofs and does not claim that the
This study develops a rigorous symplectic spectral framework for analyzing the relationship between the covariance matrix and an invertible antisymmetric commutator form in finite-dimensional quantum systems. The earlier positive-eigenvalue interpretation based directly on is replaced by the Hermitian operator H(\Sigma,\Omega) = i\Sigma^{1/2}\Omega^{-1}\Sigma^{1/2}. It is shown that the spectrum of this operator is real, symmetric about zero, and composed of pairs , where the positive quantities are the symplectic eigenvalues of relative to . The study defines the Özbil symplectic gap as g_{\mathrm{Ö}}(\Sigma,\Omega) = \min_j \nu_j, and proves that under the strict Robertson–Schrödinger condition, \Sigma+\frac{i}{2}\Omega>0 \quad\Longrightarrow\quad g_{\mathrm{Ö}}(\Sigma,\Omega)>\frac12. The paper also establishes the invariance of this quantity under simultaneous coordinate transformations, determinant and trace identities, continuity in the operator norm, and local Lipschitz stability on uniformly positive classes of covariance matrices. The scope of the result is stated explicitly. The derived quantity is not a Hamiltonian energy gap, a construction of a continuum quantum field theory, or a proof of the Yang–Mills mass gap. Instead, the work introduces a rigorous finite-dimensional geometric invariant for covariance–commutator pairs and outlines a research programme for extending the framework to finite-volume lattice systems, constructive renormalization-group analysis, and infinite-dimensional operator settings.
We solve cyclic finite-dimensional quantum histories for arbitrary time-dependent unitary steps, without assuming that one step has finite order. The propagation Hamiltonian is a unitary connection Laplacian on a cycle; its complete gauge invariant is the monodromy M=UL−1⋯U0. Its spectrum is λa,k=1−cos((2πk−θa)/L), where eiθa∈spec(M). Thus the exact history sector is isomorphic to Fix(M), while frustration, the gap above a nonempty zero-energy sector, the determinant, and the finite-temperature trace are obtained in closed form. Ordinary spectral data recover the multiset of monodromy phase cosines but not phase orientation; low energy certifies proximity to an exact relational history. We then define the predictive quotient of a sharp finite clock relative to an accessible operator system as the unique coarsest event alphabet preserving all conditional statistics on a history sector. A finite-error theorem shows that threshold clustering recovers this quotient when the minimum diamond separation of inequivalent event channels exceeds four times the estimation error, and proves an optimal record-count bound. With full matrix access and homogeneous step U, the minimal number of clock events is the projective order of U. We distinguish the normalizer of the clock algebra from transformations preserving the coherent history code and classify oriented exact sharp clock changes by U(r)×ZL on a rank-r history sector; without orientation the cyclic factor becomes dihedral. Reversible changes of full-information clock fibers are necessarily unitary, so irreversible coarse-graining is not exact clock covariance. Minimal realizations of a complete history Gram kernel are uniquely unitarily equivalent, with a finite-data Procrustes bound. Independent finite-matrix code verifies the main results.
Holonomy-first quantum frameworks treat closed-loop transformations — holonomies — as the primitive operations from which native gates are mined, rather than as derived quantities. We develop this viewpoint in two parts. First, we construct a gauge-invariant commutator-strength matrix S(ε) from trace data of short group-commutator words and use it to drive an explicit native-operation promotion protocol. The bracket-norm interpretation of S(ε) is asymptotic with a general O(ε) remainder; for inversion-symmetric primitive families all odd-order corrections vanish, giving O(ε2) accuracy, and a counterexample shows the symmetry cannot be dropped. The promotion criteria are operationalized as declared statistical tests — sampling-cost laws for interferometric and readout-floor regimes, a bias-corrected scale-equivalence rule, and multiplicity control — and a deterministic simulation campaign reports decision replication, false-promotion control, inverse-defect detection, and execution cost under finite-shot readout, calibration error, imperfect inverses, and visibility loss. Second, for two-particle preparations of Schmidt rank two, the common-basepoint return-amplitude residue factorizes exactly as ΔPθ = C(θ)HE(χ, γ), with C(θ) the concurrence and HE = sin2(2γ) sin4(χ/2) a calibrated loop factor, determining negativity, entanglement entropy, product fidelity, and geometric entanglement within the family. A two-boson construction realizes the encoded geometric gate −σx on span{|20⟩, |02⟩}; under the same two-mode, same-dimension projected-scalar condition, no corresponding one-particle holonomic cycle exists unless the one-body Hamiltonian is scalar. The mathematical ingredients are established; the contribution is the invariant-readout protocol, its quantified cost and uncertainty analysis, and precise statements of what can and cannot be inferred without full operator reconstruction. This record contains the abstract only. It is not the full text of the article. The complete peer-reviewed article is published open access in Discover Quantum Science and is freely available from the publisher at https://doi.org/10.1007/s44464-026-00037-5 — including all figures, tables, appendices and the supplementary code and data. Cite as: Kazadi J. Holonomy-first native operations geometric quantum framework and entanglement-sensitive multi-particle holonomic gates. Discover Quantum Science. 2026;2:35. https://doi.org/10.1007/s44464-026-00037-5