PhaseOrigin
Independent inverse-problem and experimental-design paper

Which measurements actually identify a three-mode ring?

Published title Spectral–Current Nonidentifiability and Globally Identifying Interventions in Flux Trimers: Exact Invariant Compression, Calibrated Edge Attenuation, and a Public-Data Closure Audit

Complete flux spectroscopy and independently measured persistent currents do not determine the microscopic Hamiltonian of a general Hermitian trimer. This paper proves the obstruction exactly, identifies the remaining two-dimensional ambiguity, and gives the minimal controlled intervention that removes it globally.

Read the 21-page paper Zenodo record Experimental protocol
DOI
10.5281/zenodo.22643651
Publication
2026-09-07 · version 1.0
Passive inverse
Generic two-dimensional fiber
Global identification
Two distinct edge attenuations
Empirical scope
Reproducible figure-level audit
Exact passive obstruction

Spectrum and current share the same information bottleneck.

Complete flux spectra and nondegenerate persistent currents do not identify the individual site offsets and hopping magnitudes of a general Hermitian three-mode ring. They determine only three centered polynomial invariants and generically leave a two-dimensional family of indistinguishable Hamiltonians. Two calibrated attenuations of distinct labelled edges remove the ambiguity and recover the positive-hopping Hamiltonian globally by closed-form equations.

Characteristic polynomial
det⁡(λI−H)=λ3−Qλ−D−2Tcos⁡θ\det(\lambda I-H)=\lambda^3-Q\lambda-D-2T\cos\theta

After the trace is removed, every flux-dependent eigenvalue is fixed by three centered invariants.

Invariant coordinates
Q=12∑iδi2+∑iti2,D=δ1δ2δ3−δ1t22−δ2t32−δ3t12,T=t1t2t3.\begin{aligned} Q&=\tfrac12\sum_i\delta_i^2+\sum_i t_i^2,\\ D&=\delta_1\delta_2\delta_3-\delta_1t_2^2-\delta_2t_3^2\\ &\quad-\delta_3t_1^2,\\ T&=t_1t_2t_3. \end{aligned}

Hellmann–Feynman current is the phase derivative of the same polynomial data; it does not add a fourth microscopic coordinate.

Generic ambiguity
dim⁡FQ,D,T=5−rank⁡ ⁣[∂(Q,D,T)∂(x,y,t1,t2,t3)]=2\dim\mathcal F_{Q,D,T}=5-\operatorname{rank}\!\left[\frac{\partial(Q,D,T)}{\partial(x,y,t_1,t_2,t_3)}\right]=2

Five trace-free microscopic coordinates compress to three invariants, leaving a generic two-dimensional isospectral-current family.

This is structural nonidentifiability, not insufficient sampling. Arbitrarily dense and precise passive flux data remain unable to separate Hamiltonians that share Q, D, and T.

The obstruction has a constructive experimental remedy.

Local identifiability, global uniqueness, and conditioning are treated as separate requirements.

Passive obstruction

Every phase-resolved eigenvalue and conjugate current factors through the same three centered invariants. More flux points and smaller measurement error cannot remove the resulting structural ambiguity.

One-intervention boundary

One labelled edge attenuation is generically locally identifying, but the global inverse is a monic sextic. An exact witness contains four admissible positive-hopping reconstructions.

Global inverse

Spectra after attenuation of two distinct labelled edges recover all labelled site offsets and positive hopping magnitudes by closed-form formulas.

Minimal design at symmetry

One attenuation has first-order rank at most two. Two distinct attenuations are necessary and sufficient, with determinant −4(1−ρ₁²)²(1−ρ₃²)².

Optimal attenuation

Under independent equal invariant noise, complete cuts are simultaneously D-, A-, and E-optimal. The attenuation-dependent variance penalty is explicit.

Model-isolation check

Reconstruction residuals test whether a nominal edge attenuation also shifted site frequencies or untouched couplings, separating nonidentifiability from intervention failure.

Symmetric-point rank certificate
det⁡M31(ρ1,ρ3)=−4(1−ρ12)2(1−ρ32)2\det M_{31}(\rho_1,\rho_3)=-4(1-\rho_1^2)^2(1-\rho_3^2)^2

Any two distinct nontrivial attenuations have full first-order rank; one attenuation cannot.

Equal-strength attenuation penalty
Var⁡ρ(ε^)Var⁡0(ε^)=(1−ρ2)−2\frac{\operatorname{Var}_{\rho}(\widehat\varepsilon)}{\operatorname{Var}_{0}(\widehat\varepsilon)}=(1-\rho^2)^{-2}

Partial cuts remain exact but become statistically expensive as the retained coupling approaches its original strength.

Numerical verification of the closed-form inverse. 2,000 positive-hopping Hamiltonians give a 1.27 × 10⁻¹⁴ maximum absolute error and a 5.15 × 10⁻¹⁴ maximum residual. The covariance check gives a 0.0118 relative Frobenius discrepancy over 50,000 Gaussian draws. These calculations validate the exact reconstruction equations and their small-noise linearization. They do not substitute for experimental coverage or raw-data uncertainty.
Public-data closure audit

The available figures support a bounded compatibility result, not a formal acceptance claim.

Roushan et al., Nature Physics 13, 146–151 (2017): single-photon spectrum from Supplementary Fig. S4(d) and independently acquired tomographic chiral current from Fig. 4(b).

Registered domain
20 adjacent intervals passing the fixed half-maximum ground-state-gap gate
Closure RMS
0.02103 g₀
Maximum residual
0.04534 g₀
Ideal-model quadrature error
below 8.69 × 10⁻⁵
Secondary amplitude estimate
0.7591 of the ideal adiabatic convention
Evidence boundary. Every conservative figure-resolution enclosure contains zero. The published figures therefore do not support a formal rejection; the amplitude is a figure-derived planning estimate, not an experimental confidence interval.
Source article DOI 10.1038/nphys3930 Source preprint arXiv:1606.00077

A four-stage protocol turns the theorem into an executable experiment.

The same logic applies to superconducting, photonic, microwave, cold-atom, acoustic, and mechanical three-mode networks.

  1. A

    Passive closure

    Acquire spectra and currents independently across the declared flux window. Test work closure and estimate the three passive invariants without interpreting a numerical spectrum derivative as an independent current measurement.

  2. B

    First labelled attenuation

    Attenuate one coupling and report every admissible root of the sextic inverse. A locally stable optimizer is not evidence of global uniqueness.

  3. C

    Second distinct attenuation

    Attenuate a second labelled edge and apply the closed-form global inverse. Calibrate each attenuation directly or infer it from the flux-dependent cubic coefficient.

  4. D

    Isolation and uncertainty

    Propagate the full invariant covariance and test intervention-isolation residuals. Recalculate identifiability if site shifts or cross-talk exceed their calibrated bounds.

The result stands independently of the CHC interpretation.

Independent finite-network inverse-problem paper; it uses the three-mode case identified in the distinguishability paper but neither assumes nor tests a cosmological interpretation of the laboratory phase.

What is established

  • An exact nonidentifiability theorem for the general positive-hopping Hermitian trimer.
  • A finite one-intervention candidate set and an explicit four-solution witness.
  • A closed-form global inverse from two distinct calibrated edge attenuations.
  • A reproducible cross-channel audit of two independently acquired public figure products.

What is not established

  • No preference for CHC over standard magnetic-graph theory follows from the finite-network result.
  • No cosmological meaning is assigned to the laboratory control phase.
  • The figure-level audit supplies neither raw-shot covariance nor an experimental confidence interval.
  • The isolated-attenuation inverse does not cover unbounded cross-talk or unmodelled modes.

Read, reproduce, and reuse the measurement design.

The fixed record contains the paper, exact symbolic certificates, numerical validation, public-data tables, vector figures, source provenance, and prospective protocol.