Optomechanical Calibration of the Quantum Precision Holonomy Magnetic Laplacian Physical Units, Held-Out Spectral Tests, and the Quadratic Flux-Gap Law
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
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
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.
Abstract 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.
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
A geometric phase can verify a loop without being mistaken for the clock effect under test. This paper isolates a clean experimental question in geometric quantum mechanics: can the same completed holonomy be read through two physically distinct terminal-reference architectures without confusing ordinary carrier geometry, calibration, or passive propagation with a new clock projection? For an adiabatic spin-21 loop, the ordinary Berry carrier is γgeo=−2Ω. That relation is retained as a shared geometry check. It is not assigned a direct cos(π/8) deficit. The paper shows why the raw ratio −Ω/2γgeo has the target 1 for both ordinary quantum mechanics and the registered QTT A1 readout statement. The proposed discriminating object is instead a sealed four-history, first-phasor amplitude ratio: Rgeo,clk=N×ord1DττDTTDτTDTτ,Dab=∣Zab(+Ω)Zab(−Ω)∣. Here Zab is the fitted first complex fringe phasor and N×ord is a target-blind, independently certified ordinary cross-history map. The frozen alternatives are Hordinary:Rgeo,clk=1,HQTT-A1:Rgeo,clk=cos(8π). The paper audits the historical neutron-spin and optical-fiber geometric-phase records. They establish useful holonomy carriers but do not contain the physical crossed T/τ terminal histories, target-blind transfer map, raw four phasors, or joint covariance needed for this statistic. No archival QTT confirmation is claimed. The resulting protocol is prospective and falsifiable: a qualified result compatible with one fixed target while excluding the other has a predeclared meaning; any topology, transfer, carrier, covariance, target-contamination, or power failure is ineligible for a theory verdict. The recommended first implementation is a polarized-neutron spin-echo or interferometric platform with a static helical magnetic-field loop. Its spin-21 carrier, field-defined solid angle, and winding reversal provide an independently auditable geometry layer while the terminal-reference switch is tested separately. The fixed separation 1−cos(8π)=0.0761204674887133… requires σ(Rgeo,clk)≤0.0152241 for a bare five-standard- deviation distinction; the laboratory design target is 0.005. Status: PASS_GEOMETRIC_CARRIER_HAS_INDEPENDENT_LOOP_GEOMETRY PASS_SAME_PLATFORM_BERRY_REFERENCE_SWITCH_IS_BOOK_NATIVE ARCHIVE_INELIGIBLE_NO_CROSSED_REFERENCE_HISTORIES PROSPECTIVE_TEST_READY Concept DOI: 10.5281/zenodo.21673340 Main book anchor: Quantum Traction Theory: Main Book v10.01, especially PDF pp. 159--172 and 889--892. Website: quantumtraction.org | Two Universes: time | Lexicon | Observatory | Corpus Tree Included files: PDF paper, Version 1.0 Complete reconstruction ZIP with LaTeX source, laboratory certificates, archival audit, synthetic cancellation fixture, verification scripts, render audit, and SHA-256 manifest Version 1.1: operation qualification and retained observations The numerical targets, ratio orientation, decision regions and historical seals are unchanged. A 22.5-degree or 45-degree component is no longer excluded by its angle alone: its physical transfer must be independently established. Common-operation cancellation and signal-preserving calibration are proved; both hypotheses must remain identifiable. Every acquired record is retained. After authorized unblinding, QTT-like, ordinary-like and unexpected outcomes are reported, including observations that do not qualify for a theory verdict. This is a prospective amendment, not a retrospective resealing of data. Clock, ruler and signal-recovery qualification. Both registered predictions are constructed at the raw detector with explicit proper-time, coordinate-time, worldline, ruler and synchronization conventions. Each relativistic effect is counted once. Independently calibrated corrections carry their uncertainties through the common comparison; the tested rows cannot train a normalization that forces either answer. Blind recovery is required separately for both alternatives at every predeclared operating setting. The inherited targets and decision rules remain fixed. The included design study makes sensitivity explicit: for the Two-Test Gram classifier, a total ratio uncertainty of 0.0013 gives approximately 88.9% QTT-branch and 94.1% ordinary-branch power under the stated ideal model. These are conditional detection probabilities, not probabilities that a theory is true, and each instrument retains its own classifier. The complete apparatus packet must be sealed before acquisition and target access; historical campaign rules remain preserved. Reconstruction entry point. Start with current/CURRENT_RELATIVISTIC_EXECUTION.md in the ZIP. The shared execution contract refers once to the development filename CURRENT_EXECUTION_RC.md; that reference means CURRENT_RELATIVISTIC_EXECUTION.md in this release. The packaged README names the correct file. This navigation correction does not change the deposited files, scientific requirements or historical seals.